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Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science Publishers,

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

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BOSE EINSTEIN CONDENSATES: THEORY, CHARACTERISTICS AND CURRENT RESEARCH

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BOSE EINSTEIN CONDENSATES: THEORY, CHARACTERISTICS AND CURRENT RESEARCH

PAIGE E. MATTHEWS Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

EDITOR

Nova Science Publishers, Inc. New York

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Copyright © 2010 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works.

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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Bose-Einstein condensates : theory, characteristics, and current research / editor, Paige E. Matthews. p. cm. Includes index.

ISBN:  (eBook)

1. Bose-Einstein condensation. I. Matthews, Paige E. QC175.47.B65B665 2009 530.4'2--dc22 2010025399

Published by Nova Science Publishers, Inc. New York Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

CONTENTS

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Preface

vii

Chapter 1

New Approach to Spinor Bose-Einstein Condensates Hiroshi Kuratsuji and Robert Botet

Chapter 2

Quantum Interference in the Time-of-Flight Distribution for Atomic Bose-Einstein Condensates Md. Manirul Ali and Hsi-Sheng Goan

35

Chapter 3

On the Dynamics of Nonconservative Bose-Einsten Condensates in Trapped Dilute Gases Victo S. Filho

63

Chapter 4

Elliptic Vortices in Self-attractive Bose-Einstein Condensates Fangwei Ye, Boris A. Malomed, Dumitru Mihalache, Liangwei Dong and Bambi Hu

115

Chapter 5

Classical Electrodynamics Analogy of Two-Dimensional Bose-Einstein Condensates H.M. Cataldo

125

Chapter 6

Matter Wave Dark Solitons in Optical Superlattices Aranya B. Bhattacherjee and Monika Pietzyk

143

Chapter 7

Bright Matter Waves and Stable Atomic Mode-Locking in Bose-Einstein Condensates J. Nathan Kutz

153

Chapter 8

On the Mathematical Description of the Effective Behaviour of One-Dimensional Bose-Einstein Condensates with Defects Riccardo Adami, Diego Noja and Andrea Sacchetti

169

Chapter 9

Experimental Observation of Some Phenomena in Bose-Einstein Condensates Shuyu Zhou, Zhen Xu, Qiuzhi Qu, Jun Qian, Xiaolin Li, Min Ke, Bo Yan and Yuzhu Wang

199

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1

vi Chapter 10

Contents Bose-Fermi Mixtures of Ultracold Atoms Hiroyuki Yabu and Takahiko Miyakawa

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Index

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233 261

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PREFACE A Bose–Einstein condensate (BEC) is a state of matter of a dilute gas of weakly interacting bosons confined in an external potential and cooled to temperatures very near to absolute zero (0 K or −273.15 °C). Under such conditions, a large fraction of the bosons occupy the lowest quantum state of the external potential, at which point quantum effects become apparent on a macroscopic scale. This new book gathers and presents research in this field including a new approach to Spinor Bose-Einstein condensates, elliptic vortices in selfattractive Bose-Einstein condensates and matter wave dark solitions in optical superlatices, as well as the mathematical description of the effective behavior of one-dimensional BoseEinstein condensates with defects. Chapter 1 presents a review about recent works dealing with the spin dynamics of spinor Bose-Einstein condensates (BEC). Two main topics are considered; the first one concerns typical features of the two-component spinor BEC, namely: the occurrence of ordered states and the effect of randomness. The second part relates to the multicomponent BEC which is formulated in terms of spin-coherent states. The authors address then the problem of quantum tunneling for the collective spin of the Bose-Einstein condensates. In Chapter 2 the authors investigate the matter-wave interference in time domain, specifically in the time-of-flight (TOF) signal or arrival time distribution for an atomic BoseEinstein condensate (BEC). This is in contrast to interference in space at a fixed time observed in the reported BEC experiments. The authors predict and quantify the interference in the centre-of-mass motion by calculating the time distribution of matter-wave arrival probability at some fixed spatial point. Specifically, the authors consider the free fall of an atomic BEC in a non-classical Schr¨odinger-cat state (prepared by coherent splitting) which is the linear superposition of two mesoscopically distinguishable Gaussian wave packets peaked around different heights, namely, z = 0 and z = − d, along the vertical free fall axis. During the free fall, the distinct superposed wave packets of the Schrödinger cat overlap or interfere (spatially as well as temporally) which leads to an interference in the quantum TOF distribution. The interference in the TOF distribution (or signal) can then be observed by taking a note or record of the particle counts over various tiny time windows at a fixed detector location. Under this purely quantum scenario, the classical or semi-classical analyses for calculating TOF distribution are not adequate and a quantum analysis for the TOF distribution is necessary. The authors have used the probability current density approach to calculate the quantum TOF distribution which is logically consistent and also physically motivating. Furthermore, the authors’ proposal of measuring matter-wave interference in TOF

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viii

Paige E. Matthews

distribution has the potential to empirically resolve ambiguities inherent in the various theoretical formulations of the quantum TOF distribution. In Chapter 3, the theory of Bose-Einstein condensation of dilute gases in traps is reviewed, considering mainly the dynamics of realistic condensates with nonconservative processes, in the mean-field approach. It is discussed the main properties of the stationary and dynamical solutions in the cases of attractive and repulsive interactions. Nonlinear phenomena that occur in Bose-Einstein condensates with nonconservative processes as dissipative solitons, spatiotemporal chaos, liquid-gas phase-transition and others are discussed, as well as the limitations of the mean-field formalism in describing the experimental data. The review also includes an analysis of different types of trap potential, a description of the variational and numerical methods used for solving the Gross-Pitaevskii equation and an important description of the stability of Bose- Einstein condensates with nonconservative processes, mainly in the case in which the collapse phenomenum can occur, when the number of particles for negative scattering length is higher than the critical one. Chapter 4 reports solutions for vortex solitons in an anisotropic harmonic trapping potential in a “pancake-shaped” Bose-Einstein condensate with attractive inter-atomic interactions. Elliptic vortices featuring anisotropic patterns bifurcate from dipole states that exist in the linear limit. The elliptic vortices with topological charge S = 1, featuring strongly asymmetric shapes, may be stable in a wide region in their existence domain. The dependence of the stability region on the eccentricity of the elliptic trap is reported. All higher-order vortices, with S > 1, are unstable. In Chapter 5, the previously explored analogy between a two-dimensional homogeneous superfluid and a (2+1)-dimensional electrodynamic system is extended to the case of a confined Bose-Einstein condensate. More precisely, a whole mapping between the hydrodynamics of a two-dimensional Bose-Einstein condensate and the nonrelativistic classical electrodynamics of a charged material medium is developed and analyzed. In such an analogy, macroscopic charges and electromagnetic radiation play the role of vortices and sound radiation, respectively. The mapping is shown to provide a very useful frame to discuss several features of vortex dynamics and induced velocity fields. Particularly, two important local conservation theorems of energy and angular momentum are easily derived. The controversial question of the vortex inertia is also studied from the viewpoint of the electromagnetic analogy, finding a qualitative agreement between the dissipative inertial effects in a uniform superfluid system and the numerical simulation results of vortex motion in a Bose-Einstein condensate. In Chapter 6, the authors study the behaviour of matter-wave band gap spectrum and eigenstates as the periodicity of the optical superlattice is increased. The authors show that the band gap (between the two lowest bands) which opens up in a doubly periodic superlattice decreases as the periodicity increases further. This is interpreted as a decrease in the PerielsNabarro barrier which the dark soliton experiences as it goes from one well to the next. For higher periodicity the mobility of the dark soliton is restored. In Chapter 7 a new method is proposed for the spontaneous generation of mode-locked, atomic bright matter waves. The nonlinear mode-coupling dynamics induced by a three-well, cigar shaped potential generates the intensity discrimination (saturable absorption) necessary to begin the pulse shaping necessary for mode-locking. When combined with proven BEC gain technologies and output coupling, the three critical physical components are shown to allow for the generation of a soliton-like bright matter wave from an incoherent (white-noise)

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Preface

ix

initial state. The mode-locked bright matter wave can be generated in both attractive and repulsive condensates, i.e. a bright matter wave is sustained in both cases. Further, when a Feshbach resonance is used to periodically and rapidly alternate the condensate between attractive and repulsive states, parabolic similarity solutions can be created in a bright matter wave state. Bose-Einstein condensation and the related topic of Gross-Pitaevskiĭ equation have become an important source of models and problems in mathematical physics and analysis. In particular, in the last decade, the interest in low-dimensional systems that evolve through the nonlinear Schrödinger equation has undergone an impressive growth. The reason is twofold: on the one hand, effectively one-dimensional Bose- Einstein condensates are currently realized, and the investigation on their dynamics is nowadays a well-developed field for experimentalists. On the other hand, in contrast to its higher-dimensional analogous, the one-dimensional nonlinear Schrödinger equation allows explicit solutions that simplify remarkably the analysis. The recent literature reveals an increasing interest for the dynamics of nonlinear systems in the presence of so-called defects, namely microscopic scatterers, which model the presence of impurities. In Chapter 8 the authors review some recent achievements on such systems, with particular attention to the cases of the Dirac’s “delta” and “delta prime” defects. The authors give rigorous definitions, recall and comment on known results for the delta case, and introduce new results for the delta prime case. The latter system turns out to be richer and interesting since it produces a bifurcation with symmetry breaking in the ground state. The authors’ purpose lies mainly on collecting and conveying results, so proofs are not included. In Chapter 9 the authors review their experimental research on cold atoms and BoseEinstein condensates, including the properties of the condensate in a tight confinement and across an optical dipole potential, the parametric excitation of cold atoms in a quadrupoleIoffe-configuration trap, and the precise manipulation of atoms on a microchip. The behavior of BEC in a trap without free-falling is described in detail, and the sudden shrink of the long axis of the atomic cloud in a cigar-shape trap was observed directly when the condensation occurs. The experimental exploration supports the use of the reduction in the effective sample size as evidence for the onset of the condensate in the case that the trap cannot be switched off completely, just like permanent-magnet traps, or if the time-of-flightmethod is not convenient in a restricted geometry of the traps. The dynamical evolution of a cold atomic cloud and a condensate passing through a far red-detuned Gaussian beam is studied both numerically and experimentally. Several exotic phenomena, such as the focusing and advancement of atomic clouds, were exhibited. It shows that the BEC can be operate in a controllable manner, and it can be used to simulate a wide variety of quantum phenomena which benefit from the wave-like characteristic of the condensate and flexible manipulation of optical potentials. Furthermore, the authors discuss the parametric excitation of 87Rb atoms in a quadrupole-Ioffe-configuration trap, and measure the dependence of the temperature and the number of the atomic cloud on modulation frequency of the parametric excitation field. It is found that the contribution of atomic collisions to the energy distributions results in a lower temperature of the atomic cloud than the theoretical prediction. Finally, the authors review their efforts in realizing the first BEC on an atom chip The bose-fermi mixture of ultracold atomic gas is a quantum gas composed of two kinds of atomic species with different quantum statistical properties, boson and fermion. In ultralow temperature, the bosons and fermions take the states of the Bose- Einstein condensation

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Paige E. Matthews

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and the Fermi degeneracy as quantum gas. The existence of the inter-atomic interactions, especially the boson-fermion interaction, causes a lot of new phases and phenomena. In Chapter 10, the authors discuss basic properties of bose-fermi mixture of ultracold atoms, ground-state properties, excited states, and its stability, and review peculiar phenomena of the mixture like phase separation, boson-fermion molecular formations.

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In: Bose Einstein Condensates Editor: Paige E. Matthews, pp. 1-34

ISBN 978-1-61728-114-3 c 2010 Nova Science Publishers, Inc.

Chapter 1

N EW A PPROACH TO S PINOR B OSE -E INSTEIN C ONDENSATES Hiroshi Kuratsuji and Robert Botet∗ Department of Physics, Ritsumeikan University-BKC Noji-Hill, Kusatsu City, 525-8577, Japan Laboratoire de Physique des Solides Bˆat.510, CNRS UMR8502 / Universit´e Paris-Sud, Centre d’Orsay, F-91405 Orsay, France

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Abstract We present a review about recent works dealing with the spin dynamics of spinor Bose-Einstein condensates (BEC). Two main topics are considered; the first one concerns typical features of the two-component spinor BEC, namely: the occurrence of ordered states and the effect of randomness. The second part relates to the multicomponent BEC which is formulated in terms of spin-coherent states. We address then the problem of quantum tunneling for the collective spin of the Bose-Einstein condensates.

PACS 03.75.Mn, 03.75.Lm, 03.75.Kk. Keywords: Bose-Einstein spinor, spin tunneling. Key Words: functional integral, Langevin equation, spinors. AMS Subject Classification: 82C, 81T, 60G.

1.

Introduction

In this article we present a novel aspect of the dynamics of the spinor Bose-Einstein Condensates. It is essentially based on two previous advancements recently developed by the present authors [1, 2]. ∗

E-mail address: [email protected]

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H. Kuratsuji and R. Botet

The spin degree of freedom has been a major concept since the advent of quantum mechanics. It is very sensitive to electromagnetic interactions, such as the mutual effect between the spin exchange interaction and external magnetic fields. In such a framework, the Ising model [3] was able to describe in full details the phase transition between the ferromagnetic and the paramagnetic states, which is a paradigm of the quantum many-body problem. Besides the basic cooperative effect for spin, it is known that the magnetic resonance, induced by the oscillating magnetic field, provides a powerful tool to explore efficiently the fine spin structure inside materials [4]. This approach results in very important technological applications, such as MRI [5]. The Rabi oscillations [6] are the atomic counterpart of the magnetic resonance.

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The present work addresses the general question of the nonlinear magnetic oscillations which are expected to occur in spinor Bose-Einstein condensates (BEC). For general references about spinor BEC, one should read for example [7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. This still incomplete approach is partly inspired by previous propositions that occurrences of spontaneous spin polarization in atomic vapors might be caused by cooperative effects involving external magnetic fields and the spin exchange interaction [17, 18]. The spinor Bose-Einstein condensates are very good candidates to study such an effect, since they provide experimentally very well controlled state of atomic vapor. In the first part, we develop two aspects. We first examine the possible occurrence of the ferromagnetic state as it may result from the nonlinear dynamics caused by the cooperative effect between the spin exchange interaction and external magnetic fields (static as well as modulated). In a second part, we consider the effect of random magnetic field for the condensed system. It may arise from magnetic impurities included in the spinor condensate. A stochastic equation is derived for the Bose-Einstein collective spin, and the exact FokkerPlanck equation [19] is derived. The second topics relates to the case of the phenomenon of tunneling in the multicomponent spinor BEC. The problem of tunneling is very typical of quantum mechanics [20, 21]. It deals generally with the finite probability to find the system in a state which cannot be reached in a classical mechanical sense. Quantum condensates may realize tunneling, as for the Josephson effect in which a macroscopic wave function extends through a classically forbidden region of space. In condensed matter physics, the tunneling phenomena have been investigated in various contexts, for example the Josephson effect [22]. or the magnetic spin tunneling [23, 24]. Treatment of tunneling phenomenon in BEC was described previously (e.g. see [25, 26, 27, 28, 29, 30, 31, 32, 33]). Here we propose a novel approach for macroscopic tunneling, which is inspired from the spinor multi-component Bose-Einstein condensate [8, 9, 11, 12, 13, 14, 29]. Here, the essential ingredient of the attempt is based on the collectivespin degree of freedom which is extracted from the order parameter expressed in terms of the Bloch – or spin-coherent – states. The collective degree of freedom is derived within a spinor BEC formalism, following an idea borrowed from the many-body theory. We demonstrate the calculation of the tunneling rate in a few generic cases, giving clear picture of the underlying physics at work behind the tunneling of the collective spin.

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New Approach to Spinor Bose-Einstein Condensates

2.

3

General Formalism for the Spin Dynamics

Both of the first and second topics are essentially based on the general notion of spin, and use techniques of the functional integral. So, our starting point will be the general framework of the spin dynamics.

2.1.

Overview

Let us consider a classical 3-dimensional magnetic moment, M, of constant norm M2 = M02 , submitted to the magnetic field B. The classical equation of motion for M depends generally on the Hamiltonian H = −M · B, and writes [34]: ∂H dM = −γM × , dt ∂M

(1)

where γ is the gyromagnetic factor. The gradient of the Hamiltonian in the space of the magnetic moments, is: ∂H/∂M ≡ (∂H/∂M1 , ∂H/∂M2 , ∂H/∂M3 ) . It is the opposite of the magnetic field, −B, in the present case. We shall see in the following the same equation appearing in various situations related to Bose-Einstein Condensates, either in the context of quantum collective spin systems, or of pseudo-spin dynamics.

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2.2.

Spin Representations of a Quantum 2-States System

Let us consider a general two-states quantum system. The system state can be represented in a number of ways. We recall below three different equivalent representations, which can be useful according to the context. • The 2-components quantum state is the straightforward way to characterize such a quantum system, that is the quantum vector: Ψ ≡ |ψ1 , ψ2 i , with the normalization: |ψ1 |2 + |ψ2 |2 = 1. The positive quantity |ψ1 |2 (resp. |ψ2 |2 ) represents the probability to find the system in the state 1 (resp. 2). • Alternatively, one can define the four Stokes parameters (Sj )j=0,··· ,3 as the averaged Pauli operators: ˆj Ψ , (2) Sj = Ψ† σ in which the {ˆ σj }j=0,··· ,3 can be represented respectively by the four Pauli matrices :         1 0 0 1 0 −i 1 0 , , , . 0 1 1 0 i 0 0 −1

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.

4

H. Kuratsuji and R. Botet Then, the Stokes parameters can be written: S0 = |ψ1 |2 + |ψ2 |2

S1 = ψ2 ψ1⋆ + ψ1 ψ2⋆

S2 = i (ψ1 ψ2⋆ − ψ2 ψ1⋆ ) S3 = |ψ1 |2 − |ψ2 |2 .

(3)

These four parameters define completely the state of the system since Ψ has two complex components. Moreover, the identity: S12 + S22 + S32 = S02 = 1 results from the normalization of the quantum state Ψ. • It is sometimes convenient to introduce the spherical angle variables, θ and φ, of the 3-dimensional Stokes vector S ≡ (S1 , S2 , S3 ) of length S2 = 1, namely: S = (sin θ cos φ, sin θ sin φ, cos θ). The two angles θ and φ are equivalently used in the spinor representation of the state Ψ, namely: Ψ = | cos(θ/2) , eiφ sin(θ/2) i .

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2.3.

The Canonical Variables

The temporal evolution of the system state Ψ is governed by the action, I. Indeed, the Feynman’s path integral for the transition amplitude between two system states Ψ and Ψ′ , at the respective times t0 and t1 , is [35, 36]: Z ′† ˆ Ψ exp[−iHt/~]Ψ = eiI/~D(Ψ′† , Ψ) , where the action functional I, associated to the path in the configuration space, is generally written as:   Z t1 ∂ ˆ Ψdt , I= Ψ† i~ − H ∂t t0

ˆ is the Hamiltonian operator for the system. and H The Schr¨odinger equation results then from the Dirac action principle, δI = 0: i~

∂Ψ ˆ , = HΨ ∂t

which gives the evolution of the state vector Ψ. Within the spinor representation, the Lagrangian, L, of the quantum field Ψ is :   ∂ † ˆ L ≡ Ψ i~ − H Ψ ∂t dφ = −~ sin2 (θ/2) −H, dt

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New Approach to Spinor Bose-Einstein Condensates

5

ˆ that is: H = where the effective Hamiltonian, H, is the average value of the operator H, † ˆ Ψ HΨ. The effective Hamiltonian is expected to depend only on the angular variables θ, φ (or equivalently on the Stokes parameters S1 , S2 , S3 ), and to be independent of the time derivatives of these variables. From the variational principle δI = 0, the evolution of the Stokes vector S, is given by the Lagrange equations : d ∂L d ∂L ∂L ∂L − − =0, =0, ∂θ dt ∂ θ˙ ∂φ dt ∂ φ˙ (with θ˙ = dθ/dt, φ˙ = dφ/dt) or, more specifically, under the form of the canonical equations of motion (using S3 = cos θ): ~ dφ 2 dt ~ dS3 2 dt

∂H ∂S3 ∂H = − . ∂φ

=

The two variables φ and S3 are then conjugated canonical variables for the system.

2.4.

A Consequence of the Ehrenfest Theorem

Applying the Ehrenfest theorem to the constant hermitian Pauli operators σ ˆj , we write:

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i~

 d  † ˆ , Ψσ ˆj Ψ = Ψ† [ˆ σj , H]Ψ dt

which gives the evolution equations for the Stokes parameters, after using the relations (2). ˆ is generally hermitian, then depends on four real parameThe Hamiltonian operator H ters. It can be expanded on the basis of the four Pauli operators, according to: ˆ= H

3 X

αj σ ˆj ,

j=0

with the real parameters {αj }j=0,··· ,3 . Using now the commutation relation [ˆ σ1 , σ ˆ2 ] = 2iˆ σ3 and all cyclic permutations, one obtains the temporal evolution of the 3-dimensional Stokes vector S, as: dS 2 ∂H =− S× , (4) dt ~ ∂S which is of the form (1), with the ‘gyromagnetic’ factor γ = 2/~, and H = α0 + α1 S1 + α2 S2 + α3 S3 . The equation (4) is formally identical to the classical equation (1), though it corresponds here to the temporal evolution of the state of the quantum 2-states system, and is a full alternative to the Schr¨odinger equation. We shall apply now this result to the 2-states BoseEinstein condensate.

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H. Kuratsuji and R. Botet

3.

Spin Dynamics of the 2-states Bose-Einstein Condensate

3.1.

Basic Equations

We consider a system of N identical bosons of mass m, interacting via the s-wave scattering, and condensed in a trapping potential. The system is supposed to appear under two states, and more specifically, the boson spins are supposed to be restricted to two values only. It can for example be the case when two condensates are created in pure orthogonal spin states. The two-components Bose-Einstein Condensate state order parameter, Ψ , can then be written as: Ψ = |ψ1 (x, t), ψ2 (x, t)i , at the location x and time t. The integral of the squared norm of Ψ over the entire space, corresponds to the total number of bosons: Z Ψ† Ψdx = N . Moreover, we define the total spin of the BE condensate through the spin operator: ˆ ≡ (S ˆx , S ˆy , S ˆz ). S

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The evolution of the system state is governed by the Lagrangian [37, 38, 39]:   i~ ∂Ψ† † ∂Ψ L= Ψ − Ψ − H(Ψ′† , Ψ) , 2 ∂t ∂t

(5)

where H(Ψ′† , Ψ) is the Hamiltonian for the N -bosons system [13], namely: H(Ψ′† , Ψ) =

~2 ∇Ψ† · ∇Ψ + U0 + UB + UI . 2m

(6)

• the first term in the Hamiltonian (6) is the kinetic energy. • the quadratic term, U0 ≡ Ψ† µΨ, is the trapping potential, with strength µ. • the mixing term, UB , arises from the interaction between the two modes ψ1 and ψ2 . If the origin of this term is the interaction between two atomic spin states, the Zeeman term writes: µB † ˆ UB = − Ψ (J · B) Ψ , (7) N ˆ ≡ S/~ ˆ of with µB the Bohr magneton coupling the collective angular momentum J the condensate to the applied magnetic field B. We shall essentially take this form below. However, the term UB can more generally be written as: UB = ~Ω Ψ† (ˆ σ · u) Ψ ,

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(8)

New Approach to Spinor Bose-Einstein Condensates

7

with Ω a characteristic frequency, σ ˆ = (ˆ σ1 , σ ˆ2 , σ ˆ3 ) the formal vector of the Pauli matrices, and u a real vectorial field. For example, in the case of the Bose-Einstein Condensate in a double-well potential, (8) holds with Ω the tunneling frequency between the two localized states [25]. • the last term, UI , is the nonlinear two-body interaction term. It is generally a quadratic function in the quantities Ψ† σ ˆj Ψ. In the present section, we shall focus on the anisotropic case: UI =

2 g0 † 2 g3  † ΨΨ − Ψσ ˆ3 Ψ , N N

(9)

with the real parameters g0 and g3 directly related to the s-wave scattering lengths. The interaction term (9) is an extension of the form used in [13]. Moreover, g3 is essentially positive as it generally forces the system to be in either the pure state 1 or 2.

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In the context of the Bose-Einstein Condensate collective spin, the second term in (9) tends to orientate the spin along the direction of easy axis of magnetization, the third axis being then a principal anisotropy axis. In the full Hamiltonian (6), the spin-dependent terms (8)-(9) do not depend on ∇Ψ, and result essentially in the dynamical rotation of the order parameter. If we assume that the interactions act instantaneously everywhere in the condensate, the profile function is simultaneously rotated identically in all the space points. Therefore, in order to take into account the effect of the spin-dependent terms, we look for a simple solution decoupling the time and the space, namely: Ψ = Φ(x)Ψξ (t) ,

(10)

where the angular term Ψξ is a two-dimensional vector whichR depends only on the time. In order to be consistent with the bosons number conservation Ψ† Ψdx = N , we consider the normalizations: Z Φ2 dx = N , Ψ†ξ Ψξ = 1 .

Moreover, we define m4 ≡

R

|Φ|4 dx/N 2 .

The ansatz (10) corresponds to the extraction of the collective degrees of freedom [40]. It is also the main transformation used to find the exact eigenstates of the spin-1 and spin-2 Bose-Einstein condensates in a static uniform magnetic field at zero temperature [41]. As previously, we define the real parameters Sj as the components of the Stokes vector: S = Ψ†ξ σ ˆ Ψξ . The straightest component for the discussion below is the third component,

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H. Kuratsuji and R. Botet

S3 , which is the reduced difference between the population in the pure state 1 and the population in the pure state 2 (see (3)). After using (10) in the Lagrangian (5), we obtain: L′ = iN ~Ψ†ξ

∂Ψξ − H(Ψ†ξ , Ψξ ) , ∂t

˜B + U ˜I ), and the following potentials, with the effective Hamiltonian H = N (µ + g0 m4 + U derived from (7),(9): ˜B = −µB Ψ† (ˆ ˆ2 By + σ ˆ3 Bz ) Ψξ , U ξ σ1 Bx + σ  2 ˜I = −g3 m4 Ψ† σ U . ξ ˆ 3 Ψξ

The equation governing the evolution of the system state Ψξ is then given by the variR ation condition of the action: δ L′ dt = 0, that is, after the same derivation as for (4) [1]: ∂H ′ dS = −γS × , (11) dt ∂S where we have introduced the shifted (removing the irrelevant constant µ + g0 m4 ) Hamiltonian H ′ per particle: H ′ ≡ H/N − µ − g0 m4

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= −µB S1 Bx − µB S2 By − µB S3 Bz − g3 m4 S32 ,

(12)

and γ = 2/~. An alternative derivation of equation (11), based on the Gross-Pitaevskii equations is given in Appendix A.

3.2.

The Zero-Temperature Case: Deterministic Behavior

In this section, we discuss the time-evolution of the 2-states Bose-Einstein Condensate spinor in the magnetic field, B. We consider three specific cases, namely the constant, the oscillating or the modulated transverse magnetic fields. 3.2.1. Constant Magnetic Field along an Axis Perpendicular to the z-Direction We define the x-direction as the direction of the constant magnetic field, B, namely: B = (B⊥ , 0, 0) , with B⊥ > 0. The equation of motion is:   GS2 S3 dS 2µB  B⊥ S3 − GS1 S3  , = dt ~ −B⊥ S2

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with G = 2g3 m4 /µB . From the equation above, we get the two constants of motion: S12 + S22 + S32 = 1 ,

H ′ = −µB B⊥ (S1 + κS32 ) = E ,

(13)

with E a constant energy depending on the initial conditions, and κ = g3 m4 /µB B⊥ , is a positive parameter. Note that the energy per particle, E, is bounded, namely: −µB B⊥ (1 + κ) ≤ E ≤ µB B⊥ . The system is integrable since the three-dimensional system has two constants of motion. The fixed points The two special points (S1 , S2 , S3 ) = (±1, 0, 0) – which are the degenerate trajectories corresponding to the energies E = ∓µB B⊥ ) – always realize equilibrium since: dS/dt = 0 in these cases. Moreover, when p κ > 1/2, two other fixed points exist, namely: (S1 , S2 , S3 ) = (1/2κ, 0, ± 1 − 1/4κ2 ). They both belong to the same energy surface E = −µB B⊥ (κ + 1/4κ). We shall see later that these points play a special role in the case of the thermal equilibrium.

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The initial pure state 1 We focus now on the pure state, say state 1 as the initial condition, that is: S3 (0) = 1. This case corresponds to the special energy E = −g3 m4 in (13). The differential equation for S3 is found, after eliminating S1 and S2 from (49), to be:  

dS3 2 = 1 − S32 1 − κ2 + κ2 S32 , dτ

with the dimensionless scaled time variable τ = 2µB B⊥ t/~. The complete solution for the reduced population difference S3 can be written in terms of Jacobi elliptic functions [42]. Five cases are to be considered: • in the limit case κ → 0 (the linear case), we have: S3 (t) → cos(2µB B⊥ t/~) . • when κ2 < 1, the solution is given by: S3 (t) = cn(2µB B⊥ t/~; κ) . • when κ2 = 1: • when κ2 > 1:

(14)

S3 (t) = sech(2µB B⊥ t/~) . S3 (τ ) = dn(2g3 m4 t/~; 1/κ) .

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10

H. Kuratsuji and R. Botet • in the limit case κ → ∞ (the vanishing magnetic field), we have: S3 (t) → 1 .

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Figure 1. Graph of elliptic functions. The thin curves represent cn functions (14). The dashed curve represents the dn function (15). The bold curve corresponds to the critical value, κ = 1. Therefore, the behavior of the parameters {Sj } exhibits a structural change according to the proper values of κ = m4 g3 /µB B⊥ . The case κ < 1 – which corresponds essentially to the large magnetic fields (B⊥ ) and the small nonlinear coupling (g3 ) –, is a kind of ‘paramagnetic state’, as S3 crosses any value in between −1 and 1 when the ptime runs. On the contrary, when κ > 1, the value of S3 is constrained to stay in the range ( 1 − 1/κ2 , 1), and this behavior can be considered as a kind of ‘ferromagnetic state’. This can also be pictured more precisely using the following lines: • When κ is small enough (κ < 1), the system spin exhibits large oscillations, and the average value of S vanishes. The period, T , of the oscillations (namely: T = 2π2 F1 (1/2, 1/2, 1, κ2 ), with 2 F1 the hypergeometric function [43]) increases with the value of the parameter κ. It is equal to 2π for κ = 0, and diverges at κ = 1. The large oscillations amplitudes are then suppressed by the magnitude of the nonlinear term (g3 ≥ µB B⊥ /m4 ). • When κ is large (κ > 1, which corresponds to the large nonlinearity and/or the small magnetic field), the collective spin is blocked around one of the directions S3 = 1 or S3 = −1. The fig.2 shows example of such behavior, in which is plotted: Z 1 t S3 (t)dt , hS3 i ≡ lim t→∞ t 0 !−1 Z 1 dx p , = 1 − cos2 (πx/2)/κ2 0

(16) (17)

where the S3 function is given by (15). The result (17) is derived from the first term of the Lambert series expansion of the Jacobi elliptic dn-function [42], written for the general Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

New Approach to Spinor Bose-Einstein Condensates

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parameter k < 1: ∞

dn(u, k) =

π 2π X q n cos(πnu/K(k)) , + 2K(k) K(k) 1 + q 2n n=1

√ with the nome q = exp[−πK( 1 − k 2 )/K(k)] – which depends only on the value of k –, and: Z 1 dx p K(k) = , 2 (1 − x )(1 − k 2 x2 ) 0

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the complete elliptic integral of the first kind. The constant term in the Lambert expansion is the only one to give a non-vanishing term, after integration over the time (16). p Moreover, one has hS3 i ≃ π(κ − 1) when κ → 1+ , and hS3 i = 0 when κ ≤ 1.

Figure 2. Plot of the time-averaged difference between the number of bosons in the state 1 and number of bosons in the state 2, hS3 i, for a Bose-Einstein condensate. The quantity , hS3 i, as defined in (16), is plotted vs the applied perpendicular magnetic field B⊥ . The plot is drawn for the particular value κ = 1/B⊥ , and the initial condition S3 (0) = 1. 3.2.2. The Rotating Transverse Magnetic Field The second case we consider is the modulated magnetic field: B = (B⊥ cos ωt, −B⊥ sin ωt, Bz ) , with constant values of the two magnetic fields B⊥ , Bz , and of the frequency ω, turning anti-clockwise around the z-axis when ω > 0. Considering the frame rotating with velocity ω around the z-axis, we define S′ ≡ (S1′ , S2′ , S3′ ), as: S1′ = S1 cos ωt − S2 sin ωt , S2′ = S1 sin ωt + S2 cos ωt , S3′ = S3 .

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H. Kuratsuji and R. Botet

It is now convenient to introduce the shifted scaled frequency: Ω ≡ (~ω−2µB Bz )/2µB B⊥ , which is essentially a combination of the three quantities B⊥ , Bz and ω. The evolution equations for the pseudo-spin S′ are:  −(Ω − 2κS3′ )S2′ = (Ω − 2κS3′ )S1′ + S3′  , dτ −S2′

dS′



(18)

with τ = 2µB B⊥ t/~ and κ = g3 m4 /µB B⊥ defined the same way as in the previous section. From the equations above, we obtain the two constants of motion: ′

′

′

S12 + S22 + S32 = 1 , ′

S1′ + κS32 − ΩS3′ = −E ′ /µB B⊥ , with constant E ′ . The system is then integrable. It has the same form as in the previous section except the term −ΩS3′ . To simplify a little the equations, let us consider the particular initial state S3 (0) = 1, that is the initial system is in the state 1. After eliminating S1′ and S2′ from the system (18), one finds that the quantity S3 is the solution of the differential equation:

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dS3 dτ

2

  = (1 − S3 ) 1 + S3 − (1 − S3 )(κS3 + κ − Ω)2 .

(19)

Appearance of the resonance

In the general case, the value κ = 1 + Ω plays a particular role, as the range of oscillations of S3 jumps suddenly from a localized state (for κ > 1 + Ω) to a wide range (for κ < 1 + Ω). This behavior can be seen noticing that the term in the brackets [· · · ] in equ.(19) is negative (hence, no solution for S3 can exist) when: 1 + S3 < (κS3 + κ − Ω)2 . 1 − S3

(20)

For the system be allowed to cross the S3 = 0 value, one has to realize κ − Ω > 1. This argument can be made more precise looking at the two functions f1 (S3 ) = (1 + S3 )/(1 − S3 ) and f2 (S3 ) = (κS3 + κ − Ω)2 appearing in (20): • when κ − Ω < 1, there are two real solutions to the equation f1 (S3 ) = f2 (S3 ) in the interval [−1, 1], and the system spin is localized above the largest solution. • when κ − Ω > 1, there is just one real solution to the equation f1 (S3 ) = f2 (S3 ) in the same range, and the system spin can take much smaller values.

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This condition for localization, κ > 1 + Ω, is the analogue to the threshold κ > 1 when Ω = 0. At last, note that the condition (20) is always fulfilled when S3 = −1. This means that the pure state S3 = −1 can never be reached when the system – starting from S3 (0) = 1 – is such that Ω > 0. In other words, full oscillations between the states S3 = 1 and S3 = −1 require both Ω = 0 and κ < 1. For this reason, we call Ω = 0 the resonance condition. System at the magnetic resonance We consider now the magnetic resonance, that is: Ω = 0. It is formally identical to the case we studied in the previous subsection and the solution for S3 (t) is the same. Let us comment the results in the present context. The relevant parameter is κ = g3 m4 /µB B⊥ . In this definition, m4 is a constant shape parameter for the Bose-Einstein condensate. The proper value of κ is then essentially given by the ratio g3 /B⊥ between the nonlinear coefficient g3 and the external magnetic field B⊥ . The value κ = 1 separates two domains, namely: a domain (κ > 1) where the population oscillates between the pure state 1 and the pure state 2, and the domain (κ < 1) in which the Stokes vector is localized around the initial value. The transition between √ the two regimes is a critical behavior, as it is clear from fig.2 and the behavior hS3 i ∝ κ − 1 when κ → 1+ .

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System out of the magnetic resonance One can discuss two particular cases: • The linear case (i.e. κ = 0, or, equivalently, B⊥ → ∞) which is generic of the largeoscillations behavior. The solution is then a shifted cosine function, and the average value of S3 is given by: Ω2 hS3 i = , 1 + Ω2 which is indeed 0 at the resonance (Ω ≡ 0), but deviates from 0 when the system is off resonance. Actually, the value of S3 cannot reach the pure state 2 when Ω 6= 0, as: −1 < −1 + 2Ω/(1 + Ω2 ) ≤ S3 ≤ 1. • The zero perpendicular field (i.e. B⊥ = 0, or, equivalently, κ → ∞), for which the solution is still S3 = 1 whatever the value of Ω. 3.2.3. The Modulated Transverse Magnetic Field An important case is related to Bose-Einstein Condensates localized in a double-well potential, when the strength of the trapping potential is regularly modulated [15]. More precisely, when the potential µ (introduced in the term U0 = Ψ† µΨ of (6)), is periodically modulated following: µ(t) = µ0 (1 + ǫ cos ωD t) , Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

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with the driving frequency ωD , and 0 ≤ ǫ < 1 is the modulation strength, the Hamiltonian for the Bose-Einstein Condensate order parameter is still given by (6), with the modulated external transverse magnetic field: µB B =

√ ~Ω0 (µ(t)e−α( µ(t)/µ0 −1) , 0, 0) , µ0

with Ω0 the unmodulated tunneling frequency, and the function µ(t) as in (21). In a sense, the system is formally at the resonance, since Ω = (~ω−2µB Bz )/2µB B⊥ = 0, using the notations of Section 3.2.2.. However, the modulated exponential term makes the discussion more complicated. Solving numerically the full Schr¨odinger equation for such a Bose-Einstein Condensate with small values of the parameter ǫ, Salmond et al [16] have shown that the quantum dynamics can then exhibit dynamical tunneling between regions of regular motion, centered on the fixed points of resonance of the semiclassical dynamics. We will see other forms of dynamical tunneling in the next Section.

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3.3.

The Finite-Temperature Case: Stochastic Behavior

The discussion of the section above concerned the deterministic time-evolution equation of the pseudo-spin S at the 0-temperature. Generally, S is submitted to a number of interactions. Some of them are fully controlled and appears explicitly in the scaled Hamiltonian H ′ (S), as written in (12). The other interactions are random (temperature) or uncontrolled (fluctuations of the magnetic field). They can be phenomenologically introduced in the equations as an effective field, say R(t). The stochastic aspect is then crucial to account for this effective field. In the following, we examine the consequences arising from the stochastic component. General framework The stochastic aspect results in the Langevin-like form of the Landau-Lifshitz-Gilbert’s equation [44] for the movement of the pseudo-spin, namely:   ∂H ′ dS dS = −S × +η + R(t) , (22) dt ∂S dt instead of (11). The parameter η is the dissipation parameter and is assumed to be a constant. We should emphasize here that the Langevin equation (22) still conserves the magnitude of the pseudo-spin, S2 = 1. This approach comes in two different contexts: 1) the random magnetic field, for which R is the actual random component of the magnetic field; 2) the finite temperature BoseEinstein Condensate dynamics. Both can be treated the same way, though interpretation is different.

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The random magnetic field In this context, the random magnetic field, R, is expected to be such that: hRi (t)i = 0 , r2 hRi (t)Rj (t + u)i = o δi,j δ(u) . 3 In the function above, δi,j is the Kronecker symbol, and δ(u) the Dirac distribution. The latter is introduced here to express that the random magnetic field is correlated on time-scales much smaller than the characteristic response time of the spin system. The definition of the random magnetic field leads to hR · Ri = ro2 . The finite temperature When the temperature, T , is finite, it is introduced through the Gaussian white noise random field R: 2D hRi (t)Rj (t + u)i = δi,j δ(u) , (23) 3 where the strength of the thermal fluctuations, D, is a positive constant proportional to T . In particular, (23) leads to hR2 i = 2D. For convenience, we also define the positive parameter m, as: m ≡ 1/hR2 i =

1 . 2D

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3.3.1. The Fokker-Planck Equation In this section we derive and discuss the Fokker-Planck equation related to the phenomenological equation (22). The complete calculation requires the Brownian-motion theory. A useful and straightforward alternative way is to use the functional integral approach, but this way requires a slightly different definition of the random noise R. We shall detail the later approach. 3.3.2. Derivation of the Fokker-Planck Equation Using the Functional Integral Approach The Landau-Lifshitz-Gilbert equation (22), can readily be solved in dS/dt (considering the original equation and its product by ×S), following:   dS ∂H ′ ∂H ′ = −S × + ηS × S × + R′ (t) , (24) dτ ∂S ∂S where we introduced the scaled time τ = t/(1 + η 2 ), and the random force R′ (t) = −S × R + ηS × (S × R). It is convenient to give a name to the deterministic nonlinear vector operator appearing in the right-hand member of (24):   ∂H ′ ∂H ′ F(S) ≡ −S × + ηS × S × . ∂S ∂S

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Temperature, or presence of a random magnetic field, can then be introduced phenomenologically as a Langevin random uncorrelated force R′ : dS = F(S) + R′ (τ ) , (25) dτ with R′ a vectorial Gaussian white noise. However, we keep implicitly the constraint R′ · S = 0, to insure conservation of the magnitude of S. We then take the Gaussian form for the probability distribution of the function R′ , on the time range [0, τ ], namely:   Z m τ ′2 ′ ′ ′ P[R (τ )] ∝ exp − R (τ )dτ . 2 0

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Using this distribution, the propagator K between two definite pseudo-spin states at the two different times 0 and τ , is given by the functional integral [45]:  Z Y  dS ′ − F(S) − R (τ ) K[S; τ |S0 ; 0] = δ dτ τ   Z m τ ′2 ′ ′ × exp − R (τ )dτ D(S)D(R′ (τ )) , 2 0

with δ the Dirac distribution. The integration measure D(S) includes two constraints: 1) the pseudo-spin is equal to S0 at time 0 and to S at time τ ; 2) the magnitude of S is a constant, and is here equal to 1. Performing the integration over all the possible functions R′ (τ ), one finds: # " 2 Z  Z m τ dS − F(S) dτ ′ D(S) . (26) K[S; τ |S0 ; 0] = exp − 2 0 dτ Therefore, the probability distribution for the pseudo-spin function S taking the value S0 at time 0, and the value S at time τ > 0, is calculated through the relation: Z P [S; τ ] = K[S; τ |S0 ; 0] P [S0 ; 0] D(S0 ) . (27) It is convenient to write the differential equation derived from the integral equations (27). We consider the small increment of time τ and the small variation y ≡ S − S0 . The kernel K[S; τ |S0 ; 0] can be approximated following: Z 2 K[S; τ |S0 ; 0] ∼ emy /2τ × 1 + m(y · F) +


m2 ∂ m (y · F)2 + m y · (y · F) − τ F2 dy . 2 ∂S 2 Then, we expand the functions P [S0 ; 0] around the value P ≡ P [S; τ ] and perform the integration over the entire space. We find: +

∂2P ∂(F(S)P ) − , ∂S2 ∂S   ∂H ′ ∂H ′ F(S) = −S × + ηS × S × , ∂S ∂S ∂P ∂τ

= D

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(28) (29)

New Approach to Spinor Bose-Einstein Condensates

17

which is the Fokker-Planck equation for the pseudo-spin distribution. One can easily check that the equation (28) is consistent with the conservation of S2 , namely: dhS2 i/dτ = 0, where the average value is defined generally as: Z hg(S; τ )i ≡ g(S)P [S; τ ]dS , (30) for any function g of the pseudo-spin and of the time. Example of a complete solution using the functional integral approach We give now the time-dependent solution of the equation (27) in a simple but non-trivial case. Let us consider the linear system (i.e. g3 = 0) submitted to a magnetic field in the z-direction, namely:  0 B=0 , Bz 

(31)

where Bz is the magnitude of the longitudinal magnetic field. The initial state at time t = 0 is supposed to be: S3 = 0. Then, one can always choose the x-axis such that the initial value of S is: (S1 , S2 , S3 ) = (1, 0, 0).

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Because of the general relation (24), the deterministic evolution of the third component of the pseudo-vector S is governed by the equation: dS3 = ηµB Bz (1 − S32 ) , dτ

(32)

S3 = tanh (ηµB Bz τ ) .

(33)

whose solution is: Without dissipation (i.e. η = 0), the pseudo-spin would stay stable at S3 = 0, just rotating around the z-axis. Any dissipation process forces the pseudo-spin to reach the pole S3 = 1. This occurs with a characteristic time: (1 + η 2 )/ηµB Bz . Note that the characteristic time is infinite in the both cases η = 0 and η = ∞. The former because dissipation is not effective to drive the system away from the stable orbit S3 = 0, the latter because of the infinitely slow dynamics. When we take the stochastic behavior into account, one has to consider the expression (29) for the vector F, namely:  S2 − ηS1 S3 F = µB Bz −S1 − ηS2 S3  . η(1 − S32 ) 

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The action I:

m I= 2

Z

τ 0



2 dS − F(S) dτ ′ dτ

is then the key quantity to obtain the complete expression for the transition probability (26). Replacing F by (34) in the latter expression, one finds the exact expression: 2   2 Z  dφ m τ  dθ I = + ηµB Bz sin θ + + µB Bz sin2 θ dτ ′ . 2 0 dτ dτ • the first term dθ/dτ + ηµB Bz sin θ vanishes identically because of (32);

• the maximum probability path is given by δI = 0, and leads here to d2 φ/dτ 2 = 0. Then, the first derivative of the angular variable φ is a constant, and can be defined as: dφ/dτ = (φ + 2nπ)/τ , with φ the final value of the azimuthal angle for the time τ , and the winding number [46] n is the number of times the path goes through the angular value φ. n is an integer number ranging from −∞ to ∞; • moreover, using (32) and the initial condition S3 (0) = 0, one has: Z τ Z τ 2 ′ (1 − S32 )dτ ′ , sin θdτ = 0

0

=

S3 , ηµB Bz

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with S3 given by (33). All together, the propagator K, in (26), can be written explicitly under the form: " r   # ∞ ma(τ ) X 2nπ 2 ma(τ )τ φ , + µB Bz + exp − K[φ; τ |0; 0] = 2πτ n=−∞ 2 τ τ where we introduced the time function: a(τ ) =

tanh (ηµB Bz τ ) . ηµB Bz τ

This function decreases from 1 to 0, as the time τ goes from 0 to ∞. The numerical coefficientRbefore the sum in the expression of K[φ; τ |0; 0], comes from the normalization 2π condition 0 K[φ; τ |0; 0]dφ = 1. The propagator K can be expressed in a compact form using the Jacobi ϑ3 -function [47]. Let us recall here the definition of the ϑ3 -function: ϑ3 (z, u) =

∞ X

exp(iπun2 + 2niz) ,

n=−∞

defined for Im(u) > 0. ϑ3 is an analytic function of z in the complex plane. The propagator writes: r ma(τ ) K[φ; τ |0; 0] = ϑ3 (iz, 2iπma(τ )/τ ) e−Λτ (35) 2πτ

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with the real variable: z = πma(τ )





,

2

.

φ + µB Bz τ

and the constant positive coefficient: ma(τ ) Λ= 2



φ + µB Bz τ

Using then the reciprocal relation: √

−iu ϑ3 (y, u) = ey

2 /iπu

ϑ3 (y/u, −1/u) ,

the result (35) writes simply as: P [φ; τ ] = =

 φ + µB Bz τ iDτ , 2 πa(τ ) ∞ 1 1 X −n2 Dτ /a(τ ) + e cos (n(φ + µB Bz τ )) . 2π π 1 ϑ3 2π



n=1

The temperature T comes only in the damping factors exp(−n2 Dτ /a(τ )), through the diffusion coefficient D. In particular, we recover P [φ; τ ] ≃ 1/2π when τ → ∞, which shows (with (33)) that the spin-probability distribution is characteristic of pseudo-spin spiraling and converging to the pole S3 = 1. Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

Angular-variables form The Fokker-Planck equation (28) for the probability distribution density P of the values of S on the sphere S2 = 1, can also be obtained by the general Stratanovich calculus, and has been alternatively derived [48] by the standard Brownian-motion theory applied to the full evolution equation (22), with the random Gaussian white noise R. Considering the angular variables θ and φ (such that: S1 = sin θ cos φ, S2 = sin θ sin φ, S3 = cos θ), the scaled Hamiltonian H ′ , in (12), writes: H ′ = −µB Bx sin θ cos φ − µB By sin θ sin φ − µB Bz cos θ − g3 m4 cos2 θ . After laborious calculations [48], the Fokker-Planck equation reads, in terms of the angular variables:    ∂ ∂H ′ ∂H ′ ∂P ∂P = − η sin θ P + D sin θ + sin θ ∂τ ∂θ ∂θ ∂φ ∂θ    ∂ η ∂H ′ ∂H ′ D ∂P + + P+ , (36) ∂φ ∂θ sin θ ∂φ sin θ ∂φ with the time variable τ = t/(1 + η 2 ). Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

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The integral of the density P over the whole (θ, φ)-space results from (36). In particular, the normalization of the probability density, Z 1 h1i ≡ P (θ, φ) sin θdθdφ = 1 , 4π holds in the rest of the paper. Spin-variables form To check the equivalence between the two expressions (28) and (36), it is useful to consider the three spherical-coordinates relations:  ∂P = ∂S   ∂P ∂H ′ · S× = ∂S ∂S   ∂ ∂H ′ P = ∂S ∂S ∂ ∂S



+

  1 ∂ ∂P 1 ∂2P , sin θ + sin θ ∂θ ∂θ sin2 θ ∂φ2      ∂ ∂H ′ 1 ∂ ∂H ′ P − P , sin θ ∂φ ∂θ ∂θ ∂φ   1 ∂ ∂H ′ sin θ P + sin θ ∂θ ∂θ   1 ∂ ∂H ′ P , sin2 θ ∂φ ∂φ

and the identity: ∂ ∂S

  ∂H ′ S× ≡0. ∂S

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Then, one obtains readily from (36) the compact form:     ∂H ′ ∂ ∂H ′ ∂P ∂P = − ×S−η D P , ∂τ ∂S ∂S ∂S ∂S

(37)

exhibiting naturally the density current: ∂P J(S) = D − ∂S



∂H ′ ∂H ′ ×S−η ∂S ∂S



P.

The equation (37) is identical to (28-29) because of the identity S · ∂H ′ /∂S = 0, as readily checked using the spherical coordinates on the pseudo-spin sphere. Fluctuation-dissipation relation In the case where the field R corresponds to the finite temperature Langevin component, one can derive from the latter expressions, the Einstein relation relating the coefficients η, D and the temperature T [49]. Indeed, we state the thermal equilibrium condition: ∂P =0 ; ∂τ

P ∝ exp(−βH ′ ) ,

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New Approach to Spinor Bose-Einstein Condensates

21

with β = 1/kB T , to obtain the relation: "     # ∂H ′ 2 ∂ ∂H ′ =0, (η − βD) −β ∂S ∂S ∂S which must be realized for all the possible values of S and temperature T . We deduce the relation: η = βD , relating the dissipation coefficient with the diffusion coefficient. More generally, the fluctuation-dissipation relation giving the actual value of the dissipation coefficient, writes here: η = βhR2 i/2, where hR2 i is the time-average value of the squared random field. Case of a constant magnetic field perpendicular to the z-direction We come back to the case of the constant magnetic field introduced in the Section 3.2.1., namely: (Bx , By , Bz ) = (B⊥ , 0, 0). The full Fokker-Planck equation writes:      ∂P ∂ ∂P ∂ ν ∂P sin θ ′ = sin θ a1 (θ, φ)P + ν + a2 (θ, φ)P + , ∂τ ∂θ ∂θ ∂φ sin θ ∂φ with: a1 (θ, φ) = η(2κ sin θ − cos φ) cos θ − sin φ ,

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a2 (θ, φ) = (2κ sin θ − cos φ) cos θ + η sin φ , and: ν = D/µB B⊥ , τ ′ = µB B⊥ t/(1 + η 2 ). We discuss here the asymptotic case t → ∞. Under the assumption that ∂P/∂τ ′ = 0 at the equilibrium, the probability distribution is given by: ′

P ∝ e−βH , with H ′ = −µB B⊥ S1 − g3 m4 S32 , according to (12). Replacing S1 = sin θ cos φ, S3 = cos θ, and taking into account the elementary volume sin θ dθ dφ, one can define the free energy F of the Bose-Einstein system at the thermal equilibrium by: F = µB B⊥ sin θ (κ sin θ − cos φ) − kB T ln sin θ .

(38)

In the right-hand member of (38), the first term corresponds to the system energy, and the second term is the product of the temperature and of the entropy. Of course, since the thermodynamical function defines the equilibrium state of the system, it does not depend on the dissipation parameter η which modifies the dynamics of the system but not its final state. Interestingly, the minima of the free energy change completely for a definite value of κ. More precisely:

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H. Kuratsuji and R. Botet

• when kB T ≥ µB B⊥ (2κ − 1), the minimum of the free energy is realized for φ = 0, θ = π/2, independently of the actual value of the parameters. This represents the ‘paramagnetic’ state: S3 = 0 as the most probable value of the pseudo-spin. • when kB T < µB B⊥ (2κ − 1), the minimum of p the free energy is realized for the two other solutions: φ = 0 or φ = π, and sin θ = (1 + 1 + 8κkB T /µB B⊥ )/4κ. It means that the most probable value of the Bose-Einstein Condensate magnetization at the thermal equilibrium, changes drastically when κ > 1/2 and the value of the temperature T is below the critical temperature Tc such that: kB Tc = 2g3 m4 − µB B⊥ . It is the ‘ferromagnetic’ state v !2 u p u µB B⊥ + (µB B⊥ )2 + 8kB T g3 m4 t . S3 = ± 1 − 4g3 m4 Close to the critical temperature Tc , the two symmetric most probable values for the pseudo-spin S3 can be approximated as: r

S3 ≃ ±

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4.

2κ − 1 4κ − 1



Tc − T Tc

1/2

.

Spin Dynamics of a Multi-component Bose-Einstein Condensate: Tunneling Phenomena

We consider in this section the Bose-Einstein Condensate for which the boson spins can be in any state. The order parameter of the multi-component Bose-Einstein Condensate is called Ψ. We will address the question of collective spin quantum tunneling driven by the magnetic fields.

4.1.

Basic Equations

As in the previous case, the evolution of Ψ is governed by the time-dependent Lagrangian [37, 38, 39], given by L=

i~ 2



Ψ†

∂Ψ ∂Ψ† − Ψ ∂t ∂t



− H(Ψ† , Ψ) ,

(39)

ˆ in the and the Hamiltonian H is still given by the relation (6), with the collective spin S, magnetic field B. We use the same argument as in the section 3.1., to consider here again the case where time- and space-variations can be decoupled, namely: if the spin-dependent terms of the

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New Approach to Spinor Bose-Einstein Condensates

23

Hamiltonian H (namely (7) and the last term of (9)), are neglected, the solution is essentially the spatial function Φ(x), solution of the static Gross-Pitaevskii equation: −

~2 2 2g0 2 ∇ Φ + µΦ + |Φ| Φ = 0 . 2m N

Since the introduction of the spin-dependent terms in the Hamiltonian H results essentially in rotating the profile function Φ(x) simultaneously at all space points, we write: Ψ(x, t) = Φ(x)Ψξ (t) , where the angular term Ψξ depends only on the time t. A natural description of the collective spin state Ψξ (t) is realized through the spin ˆi /~, (i = 1, 2, 3) be the colcoherent states – known as Bloch states [35, 51] –. Let ˆJi ≡ S 2 2 2 2 lective angular momentum operators. Its magnitude is: ˆJ ≡ ˆJ1 + ˆJ2 + ˆJ3 , with eigenvalues J(J +1), and J an integer number. Also let ˆJ± = ˆJ1 ±iˆJ2 be the raising (lowering) operator. General expansion of the multicomponent order parameter Ψξ , – corresponding to the colˆ –, writes as linear combination of the states |J, M i, where M = −J, · · · , +J lective spin S are the eigenvalues of ˆJ3 . The normalized Bloch states are obtained by the rotation of angles (θ, φ), of a particular state, say Ψ0 = |J, −J i, in the angular momentum space. Using the stereographic coordinate ξ = tan(θ/2) exp(−iφ ), a concise expression for Ψξ is [50, 51]: Ψξ = (1 + |ξ|2 )−J exp[ξˆJ+ ]Ψ0 ,

(40)

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which can be expanded in terms of the eigenvectors |J, M i as: Ψξ =



sin θ e−iφ 2

 J X J M =−J

2J J +M

1/2

ξ M |J, M i ,

for the general collective angular momentum J. Therefore, we can rewrite the Lagrangian (39) using the Bloch states. More precisely, we use (40) to write: Ψ†ξ

∂Ψ†ξ ∂Ψ†ξ ∂Ψξ ˙ † ∂Ψξ − ξ˙⋆ − Ψξ = ξΨ Ψξ , ξ ∂ξ ∂t ∂t ∂ξ ⋆

where the dotted quantities correspond to time-derivatives. Then, the normalization condition Ψ†ξ Ψξ = 1, allows to exchange the differentiations: Ψ†ξ

∂Ψ†ξ ∂Ψ†ξ ∂Ψξ ∂Ψξ − Ψξ = ξ˙⋆ Ψ†ξ ⋆ − ξ˙ Ψξ . ∂t ∂t ∂ξ ∂ξ

The last step is to write down the identity: (1 + |ξ|2 )Ψ†ξ ∂Ψξ /∂ξ ⋆ = −Jξ , Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

24

H. Kuratsuji and R. Botet

which is straightforwardly derived from (40), to obtain the Lagrangian: L=

i~J ξ ⋆ ξ˙ − ξ˙⋆ ξ − H(Ψ† , Ψ) . 2 1 + ξ⋆ξ

(41)

Conveniently, the first term in (41) – the canonical term Lc ≡ L + H –, can be ˆ ξ = (S1 , S2 , S3 ), written in terms of the semi-classical collective spin S ≡ Ψ†ξ SΨ or in terms of the angular coordinates (θ, φ), since the collective spin writes: S = ~J (sin θ cos φ, sin θ sin φ, cos θ). Elementary derivation leads to: Lc = =

1 S1 S˙ 2 − S˙ 1 S2 , 2 S + S3 S (1 − cos θ) φ˙ , 2

(42)

where S ≡ ~J. One can remark here that the canonical Lagrangian is such that Lc /~J, depends on the angular variables and their time derivatives only, and not of the value on the collective ˙ angular momentum J. We can write: Lc = SF (θ, φ). We will discuss below the general case where the Hamiltonian H is a quadratic function of the collective spin S, namely: X gj S2j , H = −k S · B −

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j

with k and the gj four real coefficients, and B is a vectorial field. This is a straightforward generalization of the form (12).

4.2.

Tunneling Rate

From the Lagrangian for the collective spin, one can derive a formula for the tunneling in terms of the transition amplitude, Kif (t), between the initial state Si at time 0 and the final state Sf at time t. It can be written in terms of the functional integral [35]: ! Z Z i Kif (t) = exp Ldt′ D[S(t)] , (43) ~ C(Si ,Sf ) where L is the Lagrangian, as defined in (39), and C(Si , Sf ) a path connecting the initial state Si to the final state Sf . When the magnitude of the collective spin J = S/~ is large enough, the most probable path gives the dominant term in the integral (43) over the path measure D[S(t)] (stationaryphase approximation (SPA)). Thus, as a result of the SPA, the path that determines the transition probability for spin is given by the Hamilton equations of motion for the collective spin: 1 ∂H 1 ∂H θ˙ = , φ˙ = − . (44) S sin θ ∂φ S sin θ ∂θ

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New Approach to Spinor Bose-Einstein Condensates

25

Alternatively, it turns out to be similar to the spin-dynamics equations of motion: ∂H dS = −S × . dt ∂S

(45)

Considering now the energy conservative system, it is convenient to consider the Fourier transform of the transition amplitude: Kif (E) =

Z

Kif (t) exp (iEt/~) dt .

The corresponding transition probability is then given by: Pif = |Kif (E)|2 . Hence, we replace the Lagrangian L in (43) by Lc − E, and the SPA leads to:

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Pif

≃ exp

1 ~

! 2 Lc d(it) , C(Si ,Sf )

Z

(46)

along a path on the constant energy surface, H = E. Consequently, Pif = 1 for the energy conservative system, as long as the most probable path exists. When it is no more the case – that is: when a range of states is classically forbidden –, the stereographic coordinates ξ = tan(θ/2) exp(−iφ ) is not defined in terms of real values of (θ, φ). Therefore, one has to consider the Lagrangian Lc in (46) with imaginary values along trajectory in the spin-space, and the value of Pif can be smaller than 1. Because of the remark made after equ.(42), we can also write the tunneling probability Pif as: ! 2S/~ Z ˙ d(it) Pif ≃ exp F (θ, φ) , C(Si ,Sf )

with the function F independent on S. This means that, whenever F takes imaginary values, the value of the tunneling probability will tend to 0 either in the classical limit (~ → 0) or when the magnitude of the collective spin goes to infinity (i.e. when the number of particles becomes infinite).

4.3.

Specific Examples of Tunneling

We will consider from now particular examples. We examine here the cooperative effects between the external magnetic field and the contribution from the two-body interaction. Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

26

H. Kuratsuji and R. Botet

4.3.1. Model(I) The first model arises from the constant magnetic field, B⊥ , along the x-direction and the anisotropic form of the two body interaction. The latter is chosen such that it deviates from the isotropic form along the z-direction. Namely, set: X gi S2i = gS21 + gS22 + g3 S23 . i

Using the conservation of S2 , the Hamiltonian H is, up to to an irrelevant constant value: H = −aS1 − bS23

= −aS sin θ cos φ − bS2 cos2 θ ,

with a = kB⊥ , and b = g3 − g. The equation (45) of motion for the spin S reads  2bS2 S3 dS  = aS3 − 2bS1 S3  . dt −aS2 

Tunneling between the ground states

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Let us consider the system in its ground state, and the coefficients a and b positive (ferromagnetic interaction). Two cases have to be considered. • when 2bS ≤ a, then (θ = π/2, φ = 0) is the single ground state. It corresponds to the energy E = −aS. The system is stable and additional energy is required to reach any other state. Since a is proportional to the magnitude of the magnetic field B⊥ , this case corresponds to experiments in which magnetic trap is so large that the spin degree of freedom is frozen along the direction of the magnetic field. Note that the isotropic case (b = 0) corresponds to this situation, and no tunneling occurs. • when 2bS > a, then the ground state is doubly degenerated, namely: (θ = θo , φ = 0) (state S− ), or: (θ = π − θo , φ = 0) (state S+ ), with the angle θ0 such that: sin θo = a/2bS, θo ∈ [ 0, π/2 ]. Both states have the same energy E = −bS2 − a2 /4b < −aS. Let us then suppose the system to be in the state (θ = θo , φ = 0). Noting the energy conservation, the integral in (46) is written only in terms of S3 : Z

C(S− ,S+ )

Lc d(it) =

Z

S cos θo −S cos θo

L(S ) p 3 dS3 , f (v3 )

L(S3 ) = −bS3 (S − S3 ) +

a2 S3 . 4b S + S3

Here f (S3 ) is given by the quartic function f (S3 ) = −


2 dS3
2 = b2 S23 − S2 cos2 θo . dt

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New Approach to Spinor Bose-Einstein Condensates

27

Elementary integration leads to: = e−4Jγg (θo ) ,   1 + cos θo 1 − cos θo . ln γg (θo ) = 2 1 − cos θo Pif

The values of the auxiliary function γg (θo ) are positive for any θo ∈ [ 0, π/2 [, which insures that Pif < 1 in all cases where θo is defined. One can note that the classical limit ~ → 0 leads to Pif = 0 as it should be expected. In agreement with the final remark of Section 4.2., the less trivial case of the infinite number of particles (i.e. S → ∞) leads to the same result, as no tunneling happens asymptotically (Pif ≃ exp(−4S ln S/~)), recovering the classical result. Tunneling between the poles The system in the state θ = 0, is another case of interest. Generally, it does not correspond to the ground state, and its energy is E = −bS2 . Unlike the previous case which consisted just of two discrete states, the system undergoes now a semi-classical orbit, which is given by the equation: a sin θ = cos φ , bS with θ ∈ [0, π]. Two cases have to be considered.

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• when bS ≤ a, then the orbit passes through both poles (θ = 0, φ = π/2) and (θ = π, φ = 3π/2). The isotropic case (i.e. b = 0) is in this category.

Figure 3. Semi-classical orbit of the point representing the direction of the spin S, when a/bS = 1.5. The magnetic field is large enough to allow complete inversion of the spin by precession.

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28

H. Kuratsuji and R. Botet • when bS > a, then the orbit consists in two disconnected parts, the upper part which includes the pole θ = 0, and the lower part with the pole θ = π. In this case, the system cannot reach the latter through the semi-classical orbit.

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Figure 4. Semi-classical orbit of the point representing the direction of the spin S, when a/bS = 0.5. Only the upper part, around the north pole θ = 0, is shown. The lower part is symmetric around the south pole θ = π. For this parameter, the system in the state θ = 0 cannot reach the lower part by classical trajectory.

This result was previously derived in [25]. The argument above can be used to derive the tunneling probability from the upper part (S+ ) to the lower part (S− ) of the orbit. Indeed, the integral appearing in (46) writes in this case: Z

C(S− ,S+ )

ǫS

L(S ) p 3 dS3 , f (S3 ) −ǫS L(S3 ) = bS3 (S − S3 ) .

Lc d(it) =

Z

Here f (S3 ) is: f (S3 ) = −

dS3
2 = b2 (S2 − S23 )(ǫ2 S2 − S23 ) , dt

with the real coefficient ǫ, (0 ≤ ǫ ≤ 1)), such that: ǫ2 = 1 − a2 /b2 S2 . Elementary integration leads to: Pif

= e−4Jγp (ǫ) ,

γp (ǫ) = K(ǫ) − E(ǫ) , Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

New Approach to Spinor Bose-Einstein Condensates

29

where E and K are the complete elliptic integrals [43]: Z ǫ dx p , K(ǫ) = (1 − x2 )(ǫ2 − x2 ) 0 Z ǫr 2 ǫ − x2 E(ǫ) = dx . 1 − x2 0 The values of γp are positive for any value of the real parameter ǫ. 4.3.2. Model(II) Let now H be the Hamiltonian consisting only of the spin two-body interactions (no magnetic fields). In this case, we consider interaction such that all the coupling constants are different. Without loss of generality, one can set g1 = 0, and g2 , g3 such that: H = g2 S22 + g3 S23 . Moreover, we will discuss the case where both coefficients g2 , g3 are positive, what is the case when the interaction is essentially repulsive. The ground state of the system is E = 0, which is reached for the two states : (θ = π/2, φ = 0) and (θ = π/2, φ = π), that is S1 = ±S. Moreover, the virtual trajectory (the “tunneling path” connecting the two ground states), is given by the equation E = 0, which writes in terms of the variables (θ, φ) as:

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cos2 θ = −

sin2 φ , 1 − α sin2 φ

with α = g2 /g3 > 0. This leads to imaginary values for θ whenever 1 − α sin2 φ > 0. On the other hand, from (44), the equation of motion for φ writes: φ˙ = 2S(g3 − g2 sin2 φ) cos θ , such that the canonical Lagrangian is: Lc = ~J

√

α sin φ

−1 + i p 1 − α sin2 φ

!

φ˙ .

The relation (46) leads to the tunneling probability: Pif

√ = exp −2J α

Z

sin φ

p dφ 1 − α sin2 φ

!

,

in which the range of integration of φ is sin2 φ < 1/α. Let us discuss the two cases : • α ≤ 1. Then the range of integration is [−π, π], and the tunneling probability is given by:  √ 2J 1− α √ . Pif = 1+ α

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30

H. Kuratsuji and R. Botet • α > 1 : the range of integration is [−φo , φo ], with sin2 φo = 1/α, and integration leads to: 4J √ α−1 √ . Pif = 1+ α

In particular, Pif = 0 for α = 1 (y-z isotropy), is a consequence of divergence of the Lagrangian at θ = π/2. It corresponds formally to an infinite potential barrier to cross over. As expected, the tunneling probability decreases for the Bose-Einstein Condensate as the spin S becomes larger. In that way, we see that the relative anisotropy between g2 and g3 plays a key role for the occurrence of the tunneling.

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5.

Conclusion

In this article, we investigated two situations where the dynamics of the spinor BoseEinstein condensate is governed by a collective variable which behaves as a non-linear classical magnetic moment in a magnetic field, though the initial problem relates to quantum mechanics. In the case of the 2-component Bose-Einstein condensate, we showed how a static transverse magnetic field can suppress the macroscopic order. When the magnetic field is oscillating, the dynamical resonance is studied in the same way as the Rabi resonance in a spin population. When randomness is considered (either the finite temperature or the fluctuating magnetic field), the Fokker-Planck equation for the difference of population in the two states, is exactly derived. A few typical examples have explicitly been written down. The second case we discussed above is about the multi-component Bose-Einstein condensate. In this case, the collective-spin can experience quantum tunneling through classically-forbidden regions in the space of the states. A technique based on the functional integral approach was developed, which allows complete calculation of the tunneling rates. Here too, a few typical examples have been discussed.

Appendix A The nonlinear equations of motion of the order parameter Ψ(x, t) are: ∂Ψ ∂H = . ∂t ∂Ψ† In the case of the Hamiltonian (6) with (7) and (9), the equations can be written in the Gross-Pitaevskii form [7] as: i~

~2 2 ∂ψ1 = − ∇ ψ1 + µψ1 + 2g- |ψ1 |2 ψ1 + 2g+ |ψ2 |2 ψ1 ∂t 2m − µB Bz ψ1 − µB (Bx − iBy )ψ2 , ∂ψ2 ~2 2 i~ = − ∇ ψ2 + µψ2 + 2g- |ψ2 |2 ψ2 + 2g+ |ψ1 |2 ψ2 ∂t 2m − µB (Bx + iBy )ψ1 + µB Bz ψ2 , i~

Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

(47)

(48)

New Approach to Spinor Bose-Einstein Condensates

31

where R g- ≡ (g0 −2 g3 )/N and g2+ ≡ (g0 + g3 )/N . One can check from the equations above that (|ψ1 (x, t)| + |ψ2 (x, t)| )dx is well a conserved quantity. If the spin-dependent terms could be neglected, the static order parameter, Φ, would satisfy the scalar Gross-Pitaevskii equation: −

2g0 2 ~2 2 ∇ Φ + µΦ + |Φ| Φ = 0 . 2m N

As the simplest case for the function Φ, the Gaussian shape is generally assumed, though any static wave function can be considered in the following. Writing now the state function Ψ as:   χ1 (t) Ψ = Φ(x) , χ2 (t) the relations (47)-(48) give the evolution equations for the components χ1 , χ2 (written in the compact form as χ1,2 ), namely:

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i~Φ(x)

dχ1,2 dt


2g3 |Φ(x)|2 Φ(x) |χ1 |2 − |χ2 |2 χ1,2 − N − µB Φ(x) (±Bz χ1,2 + (Bx ∓ iBy )χ2,1 ) ,

= ∓

where the upper (resp. lower) sign is used for the leftmost (resp. rightmost) index. To insure finite values of the equation above after integration over x, one has to multiply them by Φ† (x). Then, integration over the entire space gives readily the differential equation for the time evolution of the Stokes vector S ≡ (χ2 χ⋆1 +χ1 χ⋆2 , i(χ1 χ⋆2 −χ2 χ⋆1 ), |χ1 |2 − |χ2 |2 ):   Bz S2 − By S3 + GS2 S3 2µB  dS Bx S3 − Bz S1 − GS1 S3  , (49) = dt ~ By S1 − Bx S2

and G = 2g3 m4 /µB . This is an alternative form for Equ.(11).

References [1] H. Kuratsuji and R. Botet, Eur. Phys. J. D 49 (2008) 111. [2] H. Kuratsuji and R. Botet, Eur. Phys. J. B, 69 (2009) 445. [3] E. M. Lifschitz and L. P. Pitaevskii, Statistical Physics, part 2; Theory of the Condensed State, Pergamon Press, Oxford (1980). [4] C. P. Slichter, Principles of Magnetic Resonance, Springer, New York (1996). [5] Z.-P. Liang and P. C. Lauterbur, Principles of Magnetic Resonance Imaging: A Signal Processing Perspective, Wiley-IEEE Press, Piscataway (1999). [6] N. Bloembergen, Nonlinear Optics, World Scientific, Singapore (1996).

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[7] E. P. Gross, Nuovo Cimento, 20 (1961) 454 ; L. P. Pitaevskii, Sov. Phys. JETP, 13 (1961) 451. [8] C. J. Myatt et al, Phys. Rev. Lett. 78 (1997) 587. [9] D. S. Hall et al, Phys. Rev. Lett. 81 (1998) 1539. ¨ [10] P. Ohberg and S. Stenholm, Phys. Rev. A 57 (1998) 1272. [11] J. Stenger et al, Nature(London) 396 (1998) 345. [12] T. L. Ho and V. B. Shenoy, Phys. Rev. Lett. 77 (1996) 2595. [13] T. L. Ho, Phys. Rev. Lett. 81 (1998) 742. [14] A. J. Leggett, Rev. Mod. Phys. 73 (2001) 307. [15] E. A. Ostrovskaya, Y. S. Kivshar, M. Lisak, B. Hall, F. Catanni and D. Anderson, Phys. Rev. A 61 (2000) 031601 ; P. V. Elyutin and A. N. Rogovenko, Phys. Rev. E 63 (2001) 026610. [16] G. L. Salmond, C. A. Holmes and G. J. Milburn, Phys. Rev. A 65 (2002) 033623. [17] N. Fortson and B. Heckal, Phys. Rev. Lett. 59 (1987) 1281. [18] W. Harper, Rev. Mod. Phys. 44 (1972) 169.

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[19] M. Kac, Probabilities and Related Topics in the Physical Sciences, Interscience, New York (1959). [20] W. Pauli General Principles of Quantum Mechanics, Springer-Verlag, Berlin, 1980. [21] L. I. Schiff , Quantum mechanics, 3rd edition, McGraw-Hill Companies(New York), 1968. [22] P. G. de Gennes, Superconductivity of Metals and Alloys, (Perseus book Publishing), 1999. [23] E. M. Chudonovsky and L. Gunther, Phys. Rev. Lett. 60 (1988) 661. [24] J. L. Van Hemmen and A. S¨ut¨o, Europhys. Lett. 1 (1986) 481. [25] G. J. Milburn, J. Corney, E. M. Wright and D. F. Walls, Phys. Rev. A 55 (1997) 4318. [26] M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch, and I. Bloch, Nature 415, 39 (2002). [27] F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, Science 293 (2001) 843. [28] B. P. Anderson and M. A. Kasevich, Science 282 (1998) 1686. Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

New Approach to Spinor Bose-Einstein Condensates

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[29] D. M. Stamper-Kurn, H.-J. Miesner, A. P. Chikkatur, S. Inouye, J. Stenger, and W. Ketterle, Phys. Rev. Lett. 83 (1999) 661. [30] J. R. Anglin and W. Ketterle, Nature 416 (2002) 211. [31] C. Lee et al, Phys. Rev. A 68 (2003) 053614. [32] J. Williams, et al., Phys. Rev. A 59 (1999) R31. [33] F. Kh. Abdullaev and R. A. Kraenkel, Phys. Rev. A 62 (2000) 023613. [34] A. Sommerfeld, Mechanics : Lectures on Theoretical Physics, vol.I (Academic Press, New York), 1952. [35] A. Inomata, H. Kuratsuji and C. Gerry, Path Integrals and Coherent States of SU(2) and SU(1,1), World Scientific, Singapore, 1992. [36] R. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals , (MacGrawHill, New York), 1965. [37] R. Feynman, Statistical Mechanics: a Set of Lectures (Advanced Book Classics), (Perseus Books, New York), 1998. [38] H. Kuratsuji, Phys. Rev. Lett. 68 (1992) 1746. [39] H. Kuratsuji, Physica B284-288 (2000) 15. [40] T. Skyrme, Proc. Roy. Soc.A260 (1961) 127.

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[41] M. Koashi and M. Ueda, Phys. Rev. Lett. 84, (2000) 1066. [42] D. F. Lawden, Elliptic Functions and Applications, vol. 80 of Applied Mathematical Sciences, Springer-Verlag, New York, 1989. [43] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York (1972). [44] L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part 2, 3rd edition, Pergamon, Oxford, 1980 ; T. L. Gilbert, Phys. Rev. 100 1243 (1955). [45] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th edition, Oxford University Press, Oxford (2003). [46] L. S. Schulman, Techniques and Applications of Path Integration, John Wiley and sons, New York, 1981. [47] R. E. Bellman, A Brief Introduction to Theta Functions, Holt, Rinehart and Winston, New York, 1961. [48] W. Fuller Brown Jr, Phys. Rev. 130 (1963) 1677. [49] R. Kubo and N. Hashitsume, Prog. Theor. Phys. Suppl. 46 (1970) 210. Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

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H. Kuratsuji and R. Botet

[50] H. Kuratsuji and T. Suzuki, J. Math. Phys. 21 (1980) 472.

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[51] F. T. Arechi, E. Courtens, R. Gilmore and H. Thomas, Phys. Rev. A6 (1972) 2211.

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In: Bose Einstein Condensates Editor: Paige E. Matthews, pp. 35-61

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Chapter 2

Q UANTUM I NTERFERENCE IN THE T IME - OF -F LIGHT D ISTRIBUTION FOR ATOMIC B OSE -E INSTEIN C ONDENSATES Md. Manirul Ali1 and Hsi-Sheng Goan 2,3,∗ Research Center for Applied Sciences, Academia Sinica Taipei 11529, Taiwan 2 Department of Physics and Center for Theoretical Sciences National Taiwan University, Taipei 10617, Taiwan 3 Center for Quantum Science and Engineering, National Taiwan University Taipei 10617, Taiwan 1

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Abstract We investigate the matter-wave interference in time domain, specifically in the time-of-flight (TOF) signal or arrival time distribution for an atomic Bose-Einstein condensate (BEC). This is in contrast to interference in space at a fixed time observed in the reported BEC experiments. We predict and quantify the interference in the centre-of-mass motion by calculating the time distribution of matter-wave arrival probability at some fixed spatial point. Specifically, we consider the free fall of an atomic BEC in a non-classical Schr¨odinger-cat state (prepared by coherent splitting) which is the linear superposition of two mesoscopically distinguishable Gaussian wave packets peaked around different heights, namely, z = 0 and z = −d, along the vertical free fall axis. During the free fall, the distinct superposed wave packets of the Schr¨odinger cat overlap or interfere (spatially as well as temporally) which leads to an interference in the quantum TOF distribution. The interference in the TOF distribution (or signal) can then be observed by taking a note or record of the particle counts over various tiny time windows at a fixed detector location. Under this purely quantum scenario, the classical or semi-classical analyses for calculating TOF distribution are not adequate and a quantum analysis for the TOF distribution is necessary. We have used the probability current density approach to calculate the quantum TOF distribution which is logically consistent and also physically motivating. Furthermore, our proposal of measuring matter-wave interference in TOF distribution has the potential ∗

E-mail address: [email protected] (Corresponding author)

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Md. Manirul Ali and Hsi-Sheng Goan to empirically resolve ambiguities inherent in the various theoretical formulations of the quantum TOF distribution.

PACS 03.65.Xp, 03.75.-b, 03.65.Ta. Keywords: matter waves, time of flight, interference, Bose-Einstein condensation.

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1.

Introduction

After the pioneering electron diffraction experiments of Davisson and Germer [1] which demonstrated the wave properties of matter, interferences of matter-waves have been among the most successful confirmation of the wave-particle duality [2]. The first evidence of spatial electron interference is given in a Young’s double slit experiment [3]. The matterwave interferometry with electron and neutron has nowadays reached a very high degree of sophistication [4, 5]. There have been now great progress in the field of atom and molecule interferometry [6, 7]. Matter-wave quantum technology has seen great progress during the last decade. The exotic quantum phenomenon of Bose-Einstein condensation introduces the concept of atom laser which sparks a revolution in the field of matter-wave-optics. Interference between two freely expanding Bose-Einstein condensates (BEC) in space has been observed in a remarkable experiment [18]. Coherent splitting of BEC atoms with optically induced Bragg diffraction have been done experimentally [19, 20]. The spatial coherence of a BEC is measured using interference technique by creating and recombining two spatially displaced, coherently diffracted copies of an original BEC [20]. Most of these experiments mentioned above (particularly when matter-waves are associated with centreof-mass motion or external motion of massive quantum particles) demonstrate matter-wave interference by showing the intensity variation ( time-independent) at an extended region of detection space. In the present paper, in contrast, here we predict and quantify the matter-wave interference in the centre-of-mass motion by calculating the time distribution of matter-wave arrival probability at some fixed spatial point. More specifically, we discuss here the matter-wave interference in the time-of-flight (TOF) distribution for quantum objects freely falling under gravity. For this, we consider here the free fall of quantum objects prepared in non-classical Schr¨odinger-cat states. The interference in the TOF distribution (or signal) can then be observed by taking a note or record of the particle counts over various tiny time-windows at a fixed detector location. There are very few articles [8, 9] which discuss matter-wave diffraction in time domain. Double-slit interference experiments in the time domain were presented recently where the double slit was realized not in position-momentum but in timeenergy domain [8]. The temporal diffraction effect for de Broglie atomic waves is discussed [9] recently by usual interferometer techniques. But, a full quantum analysis of the TOF method is absent in these literatures. TOF measurements have become a very important tool in recent times in the field of laser cooling and trapping of atoms [10]. The temperature of the cold atomic sample is one of its most important characteristics and several methods have been proposed and used for its determination. A well-known technique of measuring this temperature is the time-offlight (TOF) method [11]. It is significant to mention that the first evidence for Bose Einstein condensate (BEC) was emerged from TOF measurements [12]. Most of the samples of cold

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Quantum Interference in the Time-of-Flight Distribution for Atomic BECs

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atoms are initially prepared in magneto-optical traps and the atomic cloud is allowed for a thermal expansion after its release from the trap. These so-called time-of-flight (TOF) measurements are performed either by acquiring the absorption signal of the probe laser beam through the falling and expanding atomic cloud, or by measuring the fluorescence of the atoms excited by the resonant probe light. Most of the theoretical analyses of TOF measurements are as follows. To find the shape of the absorption TOF signal, one assumes to start with the initial Gaussian position and velocity distributions of atoms in the trapped sample. The initial probability distribution of finding an atom in the phase space volume element with coordinates (z0 , v0 ) is given by     1 z02 v02 1 D(z0 , v0)dz0dv0 = exp − 2 exp − 2 dz0 dv0 (1) 2σv 2σ0 (2πσv2)1/2 (2πσ02)1/2 Here for simplicity, we consider the one-dimensional case. The Gaussian width σv of the velocity distribution is associated with the temperature T of the cloud by the relation σv2 = kT /m, where m stands for the atomic mass and k is the Boltzmann constant. Using the Newton’s equations for ballistic motion of a particle accelerated by the earth’s gravitational field (in the vertical z-direction), the velocity is obtained in terms of the time of flight as 1 v0 = (z − z0 + gt2)/t , 2

(z0 + 12 gt2 − z) ∂v0 = . ∂t t2

(2)

Substituting the above expression for v0 from Eq.(2) in Eq.(1), and then finally integrating over z0 , one can obtain the TOF distribution at an arbitrary distance z = H, given by !
1 2 2 2 2 2 2 (H + 12 gt2 )2 1 2 gt (2σ0 + σv t ) − Hσv t dt. (3) exp − D(t)dt =
3/2 2(σ02 + σv2 t2 ) (2π t2 )1/2 σ 2 + σ 2 t2 Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

0

v

The temperature of the atomic cloud is determined by fitting the experimental result to the theoretically predicted TOF signal (3) of the cloud. This kind of purely classical analyses are adopted in most of the discussions on TOF measurements where arrival time of atomic or sub-atomic particles is treated as an elementary well-defined, unique, and classical quantity. Also, the theoretical treatments of the TOF distribution that can be obtained using, for instance, the Green’s function method [13] or any semiclassical method [14], however, are equivalent to the TOF distribution obtained by using Newton’s equations for ballistic motion of particles [11]. The interpretations or theoretical analyses of the results of the various TOF experiments [15, 16, 17] with molecular, atomic or sub-atomic particles where classical trajectories are inferred from Newtonian mechanics remain debatable, especially in the domain of small atomic masses and low temperatures where quantum mechanical effects should be significant and quantum TOF distribution can not be reproduced with classical or semi-classical analyses. Here we provide an example in the context of BEC matter-wave interference, where a quantum analysis for TOF is necessary. We propose a scheme for measuring the TOF distribution of a freely falling atomic BEC prepared in non-classical Schr¨odinger-cat states. The interference in the TOF distribution (or signal) can then be observed by taking a note or record of the particle counts over various tiny time-windows at a fixed detector location. This is different from the interference between two freely expanding BEC’s observed [18]

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Md. Manirul Ali and Hsi-Sheng Goan

in space after a definite time of free fall of the condensates. Coherent splitting of BEC atoms with optically induced Bragg diffraction have been done experimentally [19, 20]. The spatial coherence of a BEC is measured using interference technique by creating and recombining two spatially displaced, coherently diffracted copies of an original BEC [20]. As mentioned above, most of the experiments (particularly when matter-waves are associated with centre-of-mass motion or external motion of massive quantum particles) demonstrate matter-wave interference by showing the intensity variation at an extended region of detection space at a fixed time. In the present paper, in contrast, we predict and quantify the matter- wave interference in the center-of-mass motion by calculating the time distribution of matter-wave arrival probability at some fixed spatial point. More specifically, we discuss here the BEC matter-wave interference in the TOF distribution or arrival time distribution since BEC as a source of coherent matter waves is already routinely demonstrated and thus may be an ideal candidate to show interference signal in the time domain (arrival-time distribution). We use here a particular quantum approach to calculate the TOF distribution and in our analysis we do not use at any point classical or semiclassical ingredients. We consider the free fall of matter-wave associated to quantum particles represented by an initial Schr¨odinger-cat state which is the linear superposition of two mesoscopically distinguishable Gaussian wave packets peaked around different heights viz., z = 0 and z = −d along the vertical z-axis. Then after a certain height of free fall (evolution under the potential V = mgz) of the Schr¨odinger cat, we calculate the quantum TOF distribution at a given detector location z = H. During the free fall, the distinct superposed wave packets of the Schr¨odinger cat overlap or interfere in space, so it is natural to expect that they will also interfere in the time of fall showing an interference pattern in the quantum TOF distribution. We take this particular example of matter-wave interference in the discussion of quantum TOF distribution to pinpoint the necessity of a quantum analysis. So, the need for a quantum analysis of TOF distribution is not merely a conceptual but a practical issue, asking how to predict the TOF distribution using only classical and semi-classical ingredients in a purely quantum scenario like this (interference in the TOF distribution for quantum particles). Now, in spite of the emphasis of quantum theory on the observable concept, there is no commonly accepted recipe to incorporate time observables and their probability distributions in the quantum formalism, and there is considerable difficulty and debate over the issue of defining time (for example, tunneling time, decay time, arrival time) as an observable [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. Even for the simplest case of arrival time problem there is no unique way to calculate the probability distribution in the quantum formalism [26]. Despite this, many researchers have evidently not been discouraged from seeking an expression for the arrival time distribution (or the quantum TOF distribution) within a consistent theoretical framework. Several logically consistent schemes for the treatment of the arrival time distribution have been formulated, such as those based on axiomatic approaches [23], operator constructions [24], measurement based approaches [25, 28], trajectory models [30] and probability current density approach [26, 27, 28, 29, 30, 31, 32, 33, 34]. We will use here probability current density approach to calculate the quantum TOF distribution which is logically consistent and also physically motivating. The main purpose of our paper is two-fold. First, our proposal to experimentally observe or quantitatively predict the matter-wave interference in the time domain, specifi-

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cally in the TOF distribution is itself quite significant which has not been explored in the current literature to the best of our knowledge. BEC as a source of coherent matter waves is already routinely demonstrated in spatial interference experiments, so BEC will also be an ideal candidate to show the interference in the TOF distribution. Second, we have just mentioned that there is an inherent nonuniqueness within the formalism of quantum mechanics for calculating the TOF or arrival time distribution. It remains an open question as to what extent these different quantum mechanical approaches [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34] for calculating the time distributions can be tested or empirically discriminated. Our proposal of measuring matter-wave interference in TOF distribution has the potential to empirically resolve ambiguities inherent in the theoretical formulations of the quantum TOF distribution. In this respect, it would be interesting if the prediction of BEC matter wave interference in the TOF distribution (calculated from different quantum approaches) be verified in actual experiments.

2.

Interference in the Quantum Time-of-flight Distribution for Bose-Einstein Condensate

We begin our analysis with the standard description of the flow of physical probability in quantum mechanics, which is governed by the continuity equation derived from the Schr¨odinger equation given by ∂ |Ψ(x, y, z, t)|2 + ∇.J(x, y, z, t) = 0, ∂t

(4)

i~ [Ψ(x, y, z, t)∇Ψ∗(x, y, z, t) − Ψ∗ (x, y, z, t)∇Ψ(x, y, z, t)] 2m b + Jy (x, y, z, t) y b + Jz (x, y, z, t) b = [Jx (x, y, z, t) x z] (5)

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J(x, y, z, t) =

The quantity J(x, y, z, t) defined as the probability current density corresponds to this flow of probability. In one dimension, the current density J(x, t) tells us the rate at which probability is flowing past the point x. So, interpreting the three dimensional continuity equation in terms of the flow of physical probability, the Born interpretation for the squared modulus of the wave function and its time derivative suggests that one can define the quantum TOF distribution [26, 27, 28, 29, 30, 31, 32, 33, 34] for the atoms crossing a surface element dS as |J.dS|. Hence quantum TOF distribution for the atoms reaching a detector at a finite surface plane S in three-dimension is given by Z Z Z Z J.dS | = | J.b n dS | (6) Π(t) = | S

S

b is the unit vector normal to the surface. It should also be noted that J(x, t) can be where n negative, hence one needs to take the modulus sign in order to use the above definition. It is important to mention here that the quantum flux density |J.dS| have been identified with the “time distribution” of particles crossing the surface element dS by Daumer et al. [35], who were applying Bohm model to the scattering problem for a quantum particle in three dimension. Our aim here is to derive an expression for the TOF distribution through the quantum

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Md. Manirul Ali and Hsi-Sheng Goan

probability current density for the atomic BEC representing the mesoscopic Schr¨odinger cat and showing the interference in TOF signal. Probability current density approach to the TOF distribution is also justified by the Bohmian model of quantum mechanics in terms of the causal trajectories of individual particles [30]. Although the Schr¨odinger probability current density is formally nonunique up to a total divergence term [29], the current can be uniquely fixed if one calculate the current in the non-relativistic limit of a proper relativistic wave equation which provide appropriate spin-dependent corrections to it [32, 36]. We ignore this small spin-dependent contribution here in our present discussion, as the estimated magnitude of the spin-dependent current is roughly 105 to 106 times smaller than the Schr¨odinger current. It was emphasized that the probability current density approach not only provides an unambiguous definition of arrival time at the quantum mechanical level [26, 27, 28, 29, 30, 31, 32, 33, 34], but also adresses the issue of obtaining the proper classical limit of the TOF of massive quantum particles [33, 34]. A magnetically trapped BEC as a source of coherent matter wave or atom laser, where a macroscopic number of atoms occupy the same ground state is now routinely available. After being released from the trapped Bose-Einstein condensates, matter waves fall freely due to the gravity. If the atomic beam is well collimated, we can use the Gross-Pitaevskii equation [12, 37] for the evolution of condensate wavefunction Ψ(x, y, z, t) with the gravitational potential,

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i~

~2 2 ∂ Ψ(x, y, z, t) = − ∇ Ψ(x, y, z, t) + mgzΨ(x, y, z, t) ∂t 2m + U0 |Ψ(x, y, z, t)|2 Ψ(x, y, z, t),

(7)

where |Ψ(x, y, z, t)|2 provides the density profile of the BEC, m denotes the atomic mass, g the gravity acceleration, and U0 the inter-atomic interaction strength. In our present discussion we consider condensate of non-interacting bosons and we neglect [37, 38] the effects of inter-atomic interaction U0 on the freely falling condensate. In the BEC, the whole complex is described by one single wave function Ψ(x, y, z, t) (a macroscopic wave function of the condensate) exactly as in a single atom, and we can speak of coherent matter in the same way as of coherent light in the case of a laser. To show interference in the quantum TOF signal for the freely falling BEC, we consider the initial state of the BEC be prepared in a Schr¨odinger-cat state which is the coherent superposition of two mesoscopically distinguishable states in the configuration space. We will consider two different experimental setup depending upon the initial preparation of the Schr¨odinger-cat state.

2.1.

Vertical Setup

We first consider the initial superposed state as Ψ(x, y, z, 0) = N [ c1 ψ1(x, y, z, 0) + c2 ψ2(x, y, z, 0)],

(8)

      1 x2 y2 z2 exp − exp − exp − ψ1 (x, y, z, 0) = 4σ0 2 4σ0 2 4σ0 2 (2πσ02)3/4

(9)

where

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Quantum Interference in the Time-of-Flight Distribution for Atomic BECs

41

Figure 1. Setup-1 with particles be detected at a surface plane (XY-plane) at z = H. We consider coherently splitted BEC freely falling under gravity along the downward −b z direction. Initially the two wavepackets are separated along the vertical Z-axis. High intensity interference in the quantum TOF distribution Π1 (t) can be observed in this case.

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      1 x2 y2 (z + d)2 exp − exp − exp − ψ2(x, y, z, 0) = 4σ0 2 4σ0 2 4σ0 2 (2πσ02)3/4

(10)

The wave packets ψ1(x, y, z, 0) and ψ2(x, y, z, 0) are respectively peaked around the points (0, 0, 0) and (0, 0, −d) and they are separated along the verticle Z-axis with an initial position spread σ0. A description of the initial 1D wavefunction for two separated BEC’s using the Gaussian form (8) has been made, for example, by the authors of [39], where they consider all the non-interacting bosons are prepared to be condensed in the ground state of √ the harmonic trap [12]. For simplicity, we take c1 = c2 = 1/ 2 which implies that after coherent splitting of the original BEC, each component has equal number of atoms. Then the value of the normalization constant N is p (11) N = 1/ 1 + exp(−d2 /8σ02 ). As we have mentioned, the Schr¨odinger-cat state of matter was generated for a BEC represented by superposition of spatially separated states and the superposition was verified [20] by detecting the quantum mechanical interference ( in space) between the localized wave packets separated by a mesoscopic distance. Under the experimental situtations (where the spatial coherence of the BEC was measured using interference technique by creating and recombining two spatially displaced, coherently diffracted copies of an original BEC) discussed by the authors of [19, 20], the BEC wave function can be written as a linear superposition of spatially separated wave packets [39] which may be inferred as a true macroscopic Schr¨odinger cat. We then consider the free fall of the coherently splitted BEC under gravity along the vertical −b z direction. We calculate the Schr¨odinger time evolution of initial wave function Ψ(x, y, z, 0) under the Hamiltonian H =

p2x 2m

p2

2

pz y + 2m + 2m + mgz. Considering the

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Md. Manirul Ali and Hsi-Sheng Goan

free fall of the coherently splitted BEC under gravity, we calculate the time evolution of the Schr¨odinger-cat state (8) according to equation (7) with U0 = 0, we then obtain N Ψ(x, y, z, t) = √ [ψ1(x, y, z, t) + ψ2(x, y, z, t)] 2

(12)

where "
2 #     − z + 12 gt2 1 x2 y2 exp − exp − exp ψ1 (x, y, z, t) = 4st σ0 4st σ0 4st σ0 (2πst2 )3/4    m 1 (13) × exp −i( ) gtz + g 2t3 ~ 6 "
2 #     − z + d + 12 gt2 1 x2 y2 ψ2(x, y, z, t) = exp − exp − exp 4st σ0 4st σ0 4st σ0 (2πst2 )3/4    1 m (14) × exp −i( ) gtz + g 2t3 ~ 6 with
st = σ0 1 + i~t/2mσ02 .

(15)

The expression for the three dimensional Schr¨odinger probability current density (5) corresponding to the time evolved state Ψ(x, y, z, t) (12) is given by

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b + Jy (x, y, z, t) y b + Jz (x, y, z, t) b z J(x, y, z, t) = Jx (x, y, z, t) x

(16)

where the x-component of the current is given by Jx (x, y, z, t) =

N2 ∗ [J1x (x, y, z, t) + J2x (x, y, z, t) + J3x(x, y, z, t) + J3x (x, y, z, t)](17) 2

with J1x (x, y, z, t) = J2x (x, y, z, t) = J3x (x, y, z, t) = ∗ (x, y, z, t) = J3x

 ~2t |ψ1(x, y, z, t)| x 4m2σ02σ 2   ~2t 2 |ψ2(x, y, z, t)| x 4m2σ02σ 2   ~2t P12 (x, y, z, t) x exp(iδ1) 4m2σ02 σ 2   ~2t P12 (x, y, z, t) x exp(−iδ1) 4m2σ02 σ 2 2



(18) (19) (20) (21)

the y-component of the current is given by Jy (x, y, z, t) =

 N2  ∗ J1y (x, y, z, t) + J2y (x, y, z, t) + J3y (x, y, z, t) + J3y (x, y, z, t) (22) 2

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with 

 ~2 t y 4m2σ02 σ 2   ~2 t 2 J2y (x, y, z, t) = |ψ2(x, y, z, t)| y 4m2σ02 σ 2   ~2 t y exp(iδ1) J3y (x, y, z, t) = P12(x, y, z, t) 4m2 σ02σ 2   ~2 t ∗ J3y (x, y, z, t) = P12(x, y, z, t) y exp(−iδ1 ) 4m2 σ02σ 2 J1y (x, y, z, t) = |ψ1(x, y, z, t)|2

(23) (24) (25) (26)

the z-component of the current is given by Jz (x, y, z, t) =

N2 ∗ [J1z (x, y, z, t) + J2z (x, y, z, t) + J3z (x, y, z, t) + J3z (x, y, z, t)] (27) 2

with    ~2 t 1 2 z + gt − gt 2 4m2 σ02σ 2     2 ~ t 1 2 2 J2z (x, y, z, t) = |ψ2(x, y, z, t)| z + gt − gt 2 4m2 σ02σ 2

J1z (x, y, z, t) = |ψ1(x, y, z, t)|2



∗ (x, y, z, t) = 2 P12(x, y, z, t) (η1 cos δ1 − λ1 sin δ1 ) J3z (x, y, z, t) + J3z

(28) (29)

(30)

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where P12 (x, y, z, t) = |ψ1(x, y, z, t)| |ψ2(x, y, z, t)| is the spatial overlap between the two wave packets and the value of the other parameters are λ1 =


~d ~2t , η = 2z + d + gt2 − gt 1 2 2 2 2 4mσ 8m σ0 σ

(31)

the oscillatory factor δ1 responsible for the interference effect is
~t 2zd + dgt2 + d2 ~t 2 2   δ1 = 2zd + dgt + d = 2 2 8mσ02 σ 2 8m σ04 + ~4mt2




(32)

and the time-dependent position spread is given by
σ 2 = st s∗t = σ02 1 + ~2 t2 /4m2σ04 .

(33)

The quantum TOF distribution can then be calculated using this three dimensional quantum current (16) for the spatially separated BEC Schr¨odinger cat falling freely under gravity. The quantum TOF distribution [26, 27, 28, 29, 30, 31, 32, 33, 34] for the atoms reaching at a finite detector surface plane S in three-dimension can be calculated using Eq.(6). We calculate the quantum TOF distributions in two different situations for this vertical setup depending on the position of the detection plane. First we consider the detection of the atoms at the XY-plane at a distance z = H. This particular experimental configuration

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Md. Manirul Ali and Hsi-Sheng Goan

is shown in Fig.1 which we call Setup-1. Now, according to Eq.(6), the Quantum TOF distribution for the BEC atoms reaching the XY-plane (b n = −b z) after a certain height (z = H) of free fall is then given by Z Z Z Z Π1 (t) = | J.b n dS| = | Jz (x, y, z, t) dxdy| (34) S

x

y

where Jz (x, y, z = H, t) is the z-component of the three dimensional current density (27) at a fixed height z = H. The quantum TOF distribution Π1 (t) will be a function of z = H after Jz (x, y, z = H, t) is integrated out over x and y. Hence, evaluating the integral (34) using Eq.(27), we finally get Π1 (t) =

N2 | [J1 (H, t) + J2 (H, t) + J3 (H, t) + J3 ∗ (H, t)] | 2

(35)

where J1 (H, t) =



 ~2 t 1 2 (H + gt ) − gt |ψ1(H, t)|2 4m2 σ02 σ 2 2



(36)

 ~2t 1 2 (H + d + gt ) − gt |ψ2(H, t)|2 J2 (H, t) = 4m2σ0 2σ 2 2

(37)

J3 (H, t) + J3 ∗ (H, t) = 2 P12(H, t) (η1 cos δ1 − λ1 sin δ1 )

(38)

P12 (H, t) = P12 (z = H, t),

(39)

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with

P12 (z, t) = |ψ1(z, t)| |ψ2(z, t)|,

(40)

and (z + 12 gt2)2 1 exp − |ψ1(z, t)| = 4σ 2 (2πσ 2)1/4

!

(z + d + 12 gt2)2 1 exp − |ψ2(z, t)| = 4σ 2 (2πσ 2)1/4

, !

(41)

.

(42)

We see clear signature of interference in the quantum TOF distribution arising due to the cross terms J3 (H, t) and J3 ∗ (H, t) (38) in the quantum TOF (35). We note here that by evaluating the integral over x and y, the quantum TOF distribution Π1(t) (35) is exactly the same as the quantum TOF distribution Π(t) obtained for one-dimensional analysis discussed in Ref. [40]. Later we will discuss the significance and advantage of this experimental configuration to observe the interference in the quantum time-of-flight distribution. We stress that using this setup (where the BEC wavepackets are initially separated along

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Quantum Interference in the Time-of-Flight Distribution for Atomic BECs

45

the vertical z-axis), one can observe high intensity interference in the quantum TOF distribution if one detect the atoms at a horizontal XY-plane after a certain height ( z = H) of free fall. The interference pattern in the quantum TOF distribution can be detected by using a probe laser, focused in the form of a sheet underneath the falling BEC atoms in the XYplane at z = H. When the trapping forces are turned off, the BEC atoms will fall through the laser probe under the influence of gravity. It is then possible to detect the fluorescence from the atoms excited by the resonant probe light as they reach the detection sheet. The fluorescence can be measured as a function of time to determine the TOF distribution. To understand more clearly the origin of this interference in TOF (35), let us consider the propagation (evolution) of individual wave packets ψ1 and ψ2 under the gravitational potential. Then one will have two different TOF distributions associated to the currents J1 (H, t) and J2 (H, t), having two distinct mean arrival times. This is because the peaks of the component wave packets ψ1 and ψ2 take different times to reach the detector at z = H, since they are spatially separated along the vertical z-axis, and the interference in TOF arises due to the superposition of these two wave packets. In this setup, the cross terms (interfering terms J3(H, t) and J3 ∗ (H, t)) in the quantum TOF distribution arises from the relative phase of the component wave packets ( ψ1 and ψ2) along z-direction, as only z-componets of the component wave packets differ in the time evolution and continue to develope the relative phase, and this relative phase is not canceled out when we perform the integration over XY-plane. Furthermore, in this setup gravity plays an important role which helps to pull down the condensate towards the detection plane. One can also be tempted to calculate a time distribution |Ψ(H, t)|2 directly from the squared modulus of the wave function (12) by integrating |Ψ(x, y, z, t)|2 over XY-plane at a distance z = H. If one defines Z Z 2 |Ψ(x, y, z, t)|2 dxdy |Ψ(z, t)| = Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

x

=

N2 2

y

  |ψ1(z, t)|2 + |ψ2(z, t)|2 + 2P12 (z, t) cos δ1 ,

(43)

one can naively argue that |Ψ(H, t)|2 = |Ψ(z = H, t)|2 may provide the Born probability distribution for quantum TOF for the BEC condensate to arrive at XY-plane at z = H. However, the function |Ψ(H, t)|2 gives a different mathematical distribution compared to the TOF distribution Π1 (t) which is calculated through the quantum probability current density. We will show, in Fig.7, that the characteristic behaviour and magnitude of the two distribution functions |Ψ(H, t)|2 and Π1(t) are not the same. Also, |Ψ(H, t)|2 can not be interpreted as a probability distribution for TOF as |Ψ(H, t)|2dt does not provides us a dimensionless probability. On the other hand, Π1(t) has the proper dimension (time −1 ) for the time distribution since Π1(t)dt gives us the probability for the BEC atoms to have the TOF between t and t + dt. For this vertical configuration (when the superposed wave packets are separated along the vertical z-axis), one can also consider a situation where the detection is made in the YZplane (b n = −b x) at a fixed x = X. This experimental configuration is shown in Fig.2 which we call as Setup-2 in our discussion. In that case, quantum TOF distribution (say, Π2 (t)) can be obtained from the x-component of the three dimensional current ( Jx (x = X, y, z, t)) integrated over YZ-plane using the Eq.(6). By evaluating that integral one can see that there

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46

Md. Manirul Ali and Hsi-Sheng Goan

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Figure 2. Setup-2 with particles be detected at a surface plane (YZ-plane) at x = X. We consider coherently splitted BEC freely falling under gravity along the downward −b z direction. Initially the two wavepackets are separated along the vertical Z-axis. No interference can be observed in the quantum TOF distribution Π2 (t) for this case. will be no interference in the quantum TOF distribution Π2(t) under this situation. This is because the interference term in the quantum TOF distribution is wiped out when we perform the integration over the YZ-plane even though Jx (x, y, z, t) in Eq.(17) contains ∗ (x, y, z, t). It is clear from Eq.(6) that the oscillatory components J3x (x, y, z, t) and J3x the Quantum TOF distribution for the BEC atoms reaching the YZ-plane (b n = −b x) at a distance x = X is given by Z Z Z Z J.b n dS| = | Jx (x, y, z, t) dydz| (44) Π2 (t) = | S

y

z

The quantum TOF distribution Π2(t) becomes a function of x = X after Jx (x = X, y, z, t) is integrated out over y and z. Hence, evaluating the integral (44) using Eq.(17), we finally get   ~2 tX 2 (45) Π2(t) = |ψ1(X, t)| 4m2σ0 2σ 2 where   1 X2 exp − 2 |ψ1(X, t)| = 4σ (2πσ 2)1/4

(46)

Hence we do not see any interference in the TOF distribution Π2 (t) in this case when we detect the atoms in the YZ-plane at x = X. Particles will arrive at this plane only due to Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

Quantum Interference in the Time-of-Flight Distribution for Atomic BECs

47

Figure 3. Setup-3 with particles be detected at a surface plane (XY-plane) at z = H. We consider coherently splitted BEC freely falling under gravity along the downward −b z direction. Initially the two wavepackets were separated along the horizontal X-axis. No interference is seen in the quantum TOF distribution Π3(t) for this case. the free particle evolution along the X and Y directions. It is also clear that less number of atoms will reach to this plane since the gravitational pull is in the perpendicular direction to this plane.

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2.2.

Horizontal Setup

Next, we consider another setup in three dimension where the superposed wave packets φ1 (x, y, z, 0) and φ2 (x, y, z, 0) are separated along the horizontal X-axis, having peaks around the points (0, 0, 0) and (−d, 0, 0), respectively (see Fig. 3). In this case we consider the initial superposed state of the coherently splitted BEC as N Φ(x, y, z, 0) = √ [φ1 (x, y, z, 0) + φ2 (x, y, z, 0)] , 2

(47)

      1 x2 y2 z2 exp − φ1 (x, y, z, 0) = exp − exp − 4σ0 2 4σ0 2 4σ0 2 (2πσ02)3/4

(48)

      1 (x + d)2 y2 z2 exp − exp − exp − φ2 (x, y, z, 0) = 4σ0 2 4σ0 2 4σ0 2 (2πσ02)3/4

(49)

where

The value of the normalization constant N remains the same as that given by Eq.(11). We then consider the free fall of the coherently splitted BEC under gravity along the vertical −b z direction. We again calculate the time evolution of initial wave function Φ(x, y, z, 0) according to equation (7) with U0 = 0. The time evolved state of the coherently splitted BEC under gravitational free fall is given by N Φ(x, y, z, t) = √ [φ1 (x, y, z, t) + φ2 (x, y, z, t)] 2

Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

(50)

48

Md. Manirul Ali and Hsi-Sheng Goan

where "
2 #     − z + 12 gt2 y2 1 x2 exp − exp φ1 (x, y, z, t) = exp − 4st σ0 4st σ0 4stσ0 (2πst2)3/4    m 1 × exp −i( ) gtz + g 2t3 (51) ~ 6 "
2 #     − z + 12 gt2 1 (x + d)2 y2 exp − exp − exp φ2 (x, y, z, t) = 4st σ0 4st σ0 4st σ0 (2πst2)3/4    m 1 × exp −i( ) gtz + g 2t3 (52) ~ 6 The expression for the three dimensional Schr¨odinger probability current density (5) corresponding to the time evolved state Φ(x, y, z, t) (50) is given by b + Jy0 (x, y, z, t) y b + Jz0 (x, y, z, t) b z J0(x, y, z, t) = Jx0 (x, y, z, t) x

(53)

where the x-component of the current is given by Jx0 (x, y, z, t) =

 N2  0 0 0 0∗ J1x (x, y, z, t) + J2x (x, y, z, t) + J3x (x, y, z, t) + J3x (x, y, z, t) (54) 2

with  ~2t = |φ1 (x, y, z, t)| x 4m2σ02 σ 2   ~2t 0 2 (x + d) J2x(x, y, z, t) = |φ2 (x, y, z, t)| 4m2σ02 σ 2

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0 (x, y, z, t) J1x

2



0 0∗ 0 (x, y, z, t) + J3x (x, y, z, t) = 2 P12 (x, y, z, t) (η2 cos δ2 − λ2 sin δ2 ) J3x

(55) (56)

(57)

the y-component of the current is given by Jy0 (x, y, z, t) =

 N2  0 0 0 0∗ J1y (x, y, z, t) + J2y (x, y, z, t) + J3y (x, y, z, t) + J3y (x, y, z, t) (58) 2

where 0 (x, y, z, t) J1y

=

0 (x, y, z, t) = J2y 0 (x, y, z, t) = J3y 0∗ (x, y, z, t) = J3y

 ~2t |φ1(x, y, z, t)| y 4m2σ02σ 2   ~2t 2 |φ2(x, y, z, t)| y 4m2σ02σ 2   ~2 t 0 P12(x, y, z, t) y exp(iδ2) 4m2 σ02σ 2   ~2 t 0 P12(x, y, z, t) y exp(−iδ2 ) 4m2 σ02σ 2 2



Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

(59) (60) (61) (62)

Quantum Interference in the Time-of-Flight Distribution for Atomic BECs

49

the z-component of the current is given by Jz0 (x, y, z, t) =

 N2  0 0 0 0∗ J1z (x, y, z, t) + J2z (x, y, z, t) + J3z (x, y, z, t) + J3z (x, y, z, t) (63) 2

with 0 (x, y, z, t) J1z

=

0 J2z (x, y, z, t) = 0 J3z (x, y, z, t) = 0∗ J3z (x, y, z, t) =

   ~2t 1 2 |φ1(x, y, z, t)| z + gt − gt 2 4m2σ02σ 2     2 1 2 ~ t 2 z + gt − gt |φ2(x, y, z, t)| 2 4m2σ02σ 2     2 ~ t 1 2 0 P12 (x, y, z, t) z + gt − gt exp(iδ2) 2 4m2σ02σ 2     2 ~ t 1 2 0 P12 (x, y, z, t) z + gt − gt exp(−iδ2 ) 2 4m2σ02σ 2 2



(64) (65) (66) (67)

0 (x, y, z, t) = |φ (x, y, z, t)| |φ (x, y, z, t)| is the spatial overlap between the two where P12 1 2 wave packets and the value of the other parameters are

λ2 =

~d ~2t , η = (2x + d) 2 4mσ 2 8m2 σ0 2σ 2

(68)

the oscillatory factor δ2 is given by

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~t 2xd + d2 ~t 2   2xd + d = δ2 = 2 2 8mσ02 σ 2 8m σ04 + ~4mt2

(69)

The quantum TOF distribution can then be calculated using this three dimensional quantum current (53) for the spatially separated BEC Schr¨odinger cat falling freely under gravity. For this setup, we again consider the detection of the particles at a surface plane (XY-plane with b = −b n z) at z = H. This particular experimental configuration is shown in Fig.3 which we call as Setup-3. In this case, quantum TOF distribution (say, Π3 (t)) can be obtained from the z-component Jz0 (x, y, z, t) of the three dimensional current (63) integrated over XY-plane using the Eq.(6). Hence, according to Eq.(6), the Quantum TOF distribution for the BEC atoms reaching the XY-plane (b n = −b z) after a certain height (z = H) of free fall is then given by Z Z Z Z J0 .b n dS| = | Jz0 (x, y, z, t) dxdy| (70) Π3(t) = | S

x

y

The quantum TOF distribution Π3 (t) will be a function of z = H after Jz0 (x, y, z = H, t) is integrated out over x and y. Hence, evaluating the integral (70) using Eq.(63), we finally get Π3(t) = |φ1(H, t)|2



~2 t 4m2σ0 2 σ 2



  1 H + gt2 − gt 2

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(71)

50

Md. Manirul Ali and Hsi-Sheng Goan

with

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(H + 12 gt2)2 1 |φ1 (H, t)| = exp − 4σ 2 (2πσ 2)1/4

!

(72)

Hence, for this setup (Fig.3), we do not see any interference at all in the quantum TOF distribution Π3 (t), even though the z-component of the three dimensional current Jz0 (x, y, z, t) 0 (x, y, z, t) and J 0∗ (x, y, z, t). This is exactly the expercontains the interfering terms J3z 3z imental setup that was used in the Ketterle experiment [18] where the BEC wave packets are initially separated along the horizontal direction and spatial interference pattern over a volume of space was observed at some fixed time. One can calculate the spatial probability by integrating |Φ(x, y, z, t)|2 over a finite volume space. The reason for the quantum TOF interference pattern being wiped out for this setup (Fig.3) is as follows. For this setup (when we detect the atoms in the XY-plane at a height z = H), we obtain the quantum TOF distribution (71) by integrating the z-component of the three dimensional probability current (63) over the XY-plane. Interestingly, in this case, the complex interfering terms in the quantum TOF distribution are wiped out when we perform that integration over XY-plane. Actually, for this setup, the individual wave packets φ1 and φ2 are not separated along the vertical z-axis, so they will have the same TOF distribution with the same mean arrival time to reach the detector at z = H. Hence, in this case we do not expect any interference in the TOF when we consider the horizontal superposition of the wave packets, even if one observes the interference in space at a fixed time. For this horizontal configuration (when the superposed wave packets are separated along the horizontal x-axis), even if one try to observe the interference in the quantum b = −b TOF distribution by detecting the atoms in the YZ-plane ( n x), the intensity of that interference pattern will be too faint to be observed, as only a small fraction of the condensate atoms will arrive at the YZ-plane at a distance, say, x = X due to free expansion (free particle motion) of the wave packets. The experimental configuration for this case is shown in Fig.4 which we call as Setup-4. It is clear from Eq.(6) that the Quantum TOF distribution b = −b for the BEC atoms reaching the YZ-plane ( n x) at a distance x = X is given by Z Z Z Z J0 .b n dS| = | Jx0 (x, y, z, t) dydz| (73) Π4(t) = | S

y

z

The quantum TOF distribution Π4(t) becomes a function of x = X after Jx0 (x = X, y, z, t) is integrated out over y and z. Hence, evaluating the integral (73) using Eq.(54), we finally get Π4(t) =

 N2  0 | J1 (X, t) + J20 (X, t) + J30 (X, t) + J30∗ (X, t) | 2

where J10 (X, t)

2

= |φ1 (X, t)|

J20 (X, t) = |φ2 (X, t)|2





~2 tX 4m2σ0 2σ 2



~2t (X + d) 4m2σ0 2 σ 2



Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

(74)

Quantum Interference in the Time-of-Flight Distribution for Atomic BECs

51

Figure 4. Setup-4 with particles be detected at a surface plane (YZ-plane) at x = X. We consider coherently splitted BEC freely falling under gravity along the downward −b z direction. Initially the two wavepackets are separated along the horizontal X-axis. The interference in the quantum TOF distribution Π4(t) will be too faint to be observed in this case. J30 (X, t) + J30∗ (X, t) = 2 |φ1(X, t)| |φ2(X, t)| (η2 cos δ2 − λ2 sin δ2 )

(75)

  1 X2 exp − 2 |φ1 (X, t)| = 4σ (2πσ 2)1/4

(76)

  1 (X + d)2 exp − |φ2(X, t)| = 4σ 2 (2πσ 2)1/4

(77)

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with

For this Setup-4, although we see the presence of interfering terms J30 (X, t) and in the expression (74) of quantum TOF distribution Π4(t), the intensity to observe this interference will be very low, as only a small fraction of the condensate atoms will arrive at the detection YZ-plane at x = X due to free expansion (free particle motion) of the wave packets φ1 and φ2 . One can also check that the quantum TOF distribution Π4 (t) will have exactly the same expression as Π1 (t) of (35) with g = 0 (no gravity) and with z = H replaced by x = X. So, this situation is exactly equivalent to the situation of Setup-1 in the absence of gravity. Gravity plays an important role in the experimental setup-1, which is quantitatively discussed at the end of next section. Hence the only two situations (in our above discussion) where we see the presence of TOF interference in three dimension are the quantum TOF distribution Π1(t) obtained for the experimental Setup-1 and Π4 (t) for Setup-4. Now, since the temporal distribution Π4 (t) is exactly equal to Π1(t) when g = 0, the whole characteristic of the interference pattern in quantum TOF distribution hinges upon the form of Π1 (t). In the next section we study numerically the parameter dependence of the quantum TOF distribution Π1(t) and the physical interplay between these parameters. We emphasize that gravity plays an J30∗ (X, t)

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52

Md. Manirul Ali and Hsi-Sheng Goan 150

150 d=40 µ m

P ( z , tc )

120 90

90

60

60

30

30

0 -1.03 -1.02 -1.01

-1

-0.99 -0.98 -0.97

150

P (z , t c )

0 -1.03 -1.02 -1.01

-1

-0.99 -0.98 -0.97

150 d=20 µ m

120

90

60

60

30

30 -1

-0.99 -0.98 -0.97

d=10 µ m

120

90

0 -1.03 -1.02 -1.01

0 -1.03 -1.02 -1.01

Position ( z )

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d=30 µ m

120

-1

-0.99 -0.98 -0.97

Position ( z )

2 Figure p5. The position profiles P (z, tc ) = |Ψ(z, tc )| in the z-direction at an arbitrary time tc = 2|H|/g for the coherently split BEC of sodium atoms (representing the macroscopic Schr¨odinger-cat) falling freely under gravity are plotted for varying wave packet separation d. In each curve, position z is plotted (in cm) along the horizontal direction and the coherent position probability density (in the vertical z-direction) |Ψ(z, tc )|2 of BEC Schr¨odinger-cat is plotted (in cm−1 ) along vertical axis with σ0 = 1 µm and H = −1cm.

important role in our Setup-1 to observe the interference in the quantum TOF distribution. In the absence of gravity, condensate atoms will spread in various directions resulting to a small fraction of the condensate reaching the detection plane.

3.

Numerical Results and Discussions

Before going to the TOF signal, let us first focus on the spatial interference pattern in the z-direction. We see that the oscillatory factor δ1 (32) of the cross-term in |Ψ(z, t)|2 (43) describing the interference effect is time-dependent from which it is clear that the local wavelength variations in the interference pattern are also time-dependent. The position profile (in the vertical z-direction) P (z, tc ) = |Ψ(z, tc )|2 of the freely falling BECpSchr¨odinger cat (showing the interference pattern over a spatial region) at time t = tc = 2|H|/g sec with different values of wave packet separations d is plotted in Fig.5. The quantum TOF distribution Π1(t) at a detector location z = H = −1 cm is plotted in Fig.6 with different values of wave packet separation d. We see clear signature of interference in the quantum TOF distribution arising due to the terms J3 (H, t) and J3 ∗ (H, t) of (38) in the expression

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Quantum Interference in the Time-of-Flight Distribution for Atomic BECs

53

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for quantum probability current density (35). Actually, coherence in position distribution (Fig.5) is also reflected in the quantum TOF distribution (Fig.6) (see also the discussion in the next paragraph and Fig.7). During free fall, the spatially separated wave packets of the BEC Schr¨odinger cat overlap or interfere in space and hence they also interfere in the time of fall showing an interference pattern in the quantum TOF distribution. The quantum TOF distribution Π1(t) may be visualized as a coherent pulse of BEC atoms. The interference pattern in the quantum TOF signal (Fig.6a) is very sharp for a typical set of parameter values, for example, H = −1cm, d = 50 µm, σ0 = 1 µm, and the pattern disappears (Fig.6f) when the separation between the BEC superposed wave packets is decreased to d = 1µm for the above mentioned parameter values. For the particular choice of parameter values in Fig.6, the value of λ1 is much smaller than the value of η1 in (31). As a result, the dominant interference cross-term (38) in quantum TOF distribution becomes mainly proportional to cos δ1 which is similar to the cross-term in (43). We can see from the oscillatory factor δ1 (32) in (38) that the number of oscillations and hence the number of fringe increases in both the spatial distribution (in the vertical z-direction) P (z, tc ) = |Ψ(z, tc )|2 (43) and the TOF distribution Π1(t) (35) as one increase the separation d for the range of parameters used even though the distribution P (z, tc ) = |Ψ(z, tc )|2 is evaluated at a fixed temporal point t = tc and the distribution Π1 (t) is evaluated at a fixed spatial point z = H. The interference effect arises mainly because of two factors: one is the temporal overlap P12 (z = H, t) (40) and the other is the oscillatory factor δ1 . When d is very small, the overlap P12 (z = H, t) is very high, but the oscillatory factor δ1 becomes small. As a consequence, the oscillation frequency is too slow or the oscillation period is too large, and we do not see any oscillatory effect in the temporal overlap region of the wave packets. Number of oscillations increases as one increases d, but again after a certain value (d > 400µm) of separation there will be no interference as the overlap P12(z = H, t) becomes very small in that case. For a wide range of parameter values, we see the interference in the quantum TOF signal Π1 (t) as well as in the position distribution (in the vertical z-direction) P (z, tc ) = |Ψ(z, t = tc )|2. We now come back to the question, “whether squared modulus of the wave function |Ψ(z = H, t)|2 can provide us the quantum TOF distribution at the spatial point z = H”. We have seen that the oscillatory behaviour of the function |Ψ(z, t)|2 is similar to that of Π1 (t) for the particular choices of parameters considered in Figs.5 and 6), although their values and units are drastically different. As mentioned, this is because λ1  η1 for this particular choice of parameter values and the dominant interference cross-term in quantum TOF distribution (38) becomes mainly proportional to cos δ which is similar to the cross-term in |Ψ(z, t)|2 (43). However, the functions |Ψ(z = H, t)|2 and Π1 (t) do not always behave in the same fashion that can be shown by reducing the initial width σ0 and the atomic mass m. It should be stressed that |Ψ(z = H, t)|2 is not the same as the quantum TOF signal Π1(t) obtained from quantum probability current density which is depicted in Fig.7. The time distribution calculated from the squared modulus of wave function significantly differ from the quantum TOF distribution Π1 (t) calculated from the probability current density. Hence |Ψ(z = H, t)|2 can not be interpreted as the probability distribution for TOF. We emphasize again that |Ψ(z = H, t)|2dt does not provides us a dimensionless probability. On the other hand, Π1(t) has the proper dimension (time −1 ) for the time distribution since Π1(t)dt gives us the probability for the BEC atoms to have the

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Md. Manirul Ali and Hsi-Sheng Goan

Π( t )

6000

Π(t)

d= 50 µm

4000

4000

2000

2000

0 6000

0.0447

0.045

0.0453

(c)

d= 30 µm

0 6000 4000

2000

2000 0.0447

0.045

0.0453

(e)

0.0456

0 6000

d= 10 µm

4000

4000

2000

2000

0

0.0447

0.045

0.0453

Time (t) Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

0.0456

4000

0 6000 Π( t )

6000

(a)

0.0456

0

(b)

0.0447

d= 40

0.045

0.0453

(d)

0.0447

0.0456

d= 20 µm

0.045

0.0453

(f)

0.0447

µm

0.0456 d= 1 µm

0.045

0.0453

0.0456

Time (t)

Figure 6. Quantum TOF distributions Π1 (t) for the coherently splitted BEC of sodium atoms (representing the macroscopic Schr¨odinger-cat) falling freely under gravity are plotted for varying wave packet separation d. In each curve, time (in sec) is plotted along horizontal direction and the coherent TOF distribution Π1 (t) of BEC Schr¨odinger-cat is plotted (in sec−1 ) along vertical axis. The detector is located at a distance z = H = −1 cm and σ0 = 1 µm.

TOF between t and t + dt. From Fig.8, we see that the interference pattern in quantum TOF signal Π1 (t) gradually disappears as one increase the mass m of the atoms. Fig.9 shows the quantum TOF distribution for different values of wave packet width from σ0 = 1 µm to σ0 = 6 µm. It is clear from Fig.9 that the number of fringe and the contrast of interference pattern in quantum TOF distribution decreases as one increase the value of the initial widths ( σ0 ) of the wave packets. Nevertheless, it is possible to see the interference for a larger value of σ0 . For example, if one chooses σ0 = 10 µm, then to observe good interference pattern (with good contrast and having considerable number of fringe) in Π1 (t), the separation d needs to be considered in the range of 50 µm to 250 µm, with the detector placed at a longer distance (H = −100 cm) for a fixed mass of sodium atoms. The interference in Π1 (t) can

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Quantum Interference in the Time-of-Flight Distribution for Atomic BECs

28 21 14 7 0 0.05 28 21 14 7 0 0.05 28 21 14 7 0 0.05 28 21 14 7 0 0.05

TOF signal Π1 ( t )

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Figure 7. The time distribution |Ψ(H, t)|2 calculated through squared modulus of the wave function at a detector location z = H is shown to be significantly differed from the quantum TOF distribution Π1(t) for the coherently splitted BEC of atomic Hydrogen (representing the macroscopic Schr¨odinger-cat) falling freely under gravity. The time distributions are plotted for four different values of d with the detector located at z = H = −10 cm and σ0 = 0.1 µm.

be observed even with much higher value of the initial width ( σ0 = 20 µm), but then one needs to consider the separation d in the range of 100 µm to 700 µm with the detector be placed at H = −200 cm, and one has to consider BEC of lighter mass atoms like Lithium. The interference in Π1 (t) is sensitive to the parameters σ0 and the atomic mass m, the detector location H and the separation d. We repeat here that the interference in Π1(t) arises mainly because of the temporal overlap P12(H, t) (39) and the oscillatory factor δ1 (32). To increase the temporal overlap P12(H, t), one has to find the condition under which the spreading of the wave packet increases: small σ0, lighter mass atoms, distant detector location (large H) will be helpful in this regard to enhance this effect. The oscillatory factor δ1 can be increased either by reducing the value of σ0 , or by increasing the parameters d and H. Actually, when one considers higher values of the parameter σ0 , then the temporal

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Md. Manirul Ali and Hsi-Sheng Goan 8000

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Figure 8. Quantum TOF distributions Π1 (t) for the coherently splitted BEC of sodium atoms (representing the macroscopic Schr¨odinger-cat) falling freely under gravity are plotted for varying atomic masses. In each curve time (in sec) is plotted along horizontal direction and the TOF signal Π1 (t) is plotted (in sec−1 ) along vertical axis. The detector is located at z = H = −1 cm with d = 50 µm and σ0 = 1 µm.

overlap P12(H, t) and the oscillatory factor δ1 both decrease. This is because for larger values of the parameters σ0 (or mass m), the spreading effect (33) and hence the temporal overlap P12 (H, t) becomes small. As a result, the wave packets try to localize (in time as well as in space) more strongly causing the interference effect to be small. Also, for higher values of σ0 , the oscillatory factor δ1 will be too small due to the presence of σ04 in the denominator of δ1 (32). Then one has to allow the BEC to travel a longer distance (by increasing H) to develope some temporal overlap of the wave packets, and also increasing H helps us to increase δ1 (32). For higher values of σ0 , the parameter δ1 should also be increased by increasing the value of the separation d, keeping in mind that there is a considerable temporal overlap P12 (H, t). The temporal overlap gets reduced if one increases the separation d too much. So, even if there is a delicate choice of the parameters, one can observe the interference in the quantum TOF signal Π1 (t) for a wide range of parameter values. It is significant to mention here that gravity plays an important role in our Setup-1 (Fig.1) to observe the interference in the TOF distribution. In Fig.10 we plot the time distribution Π1 (t) for g = 0 (no gravity) and compare it with Fig.(6d) where we plot Π1(t) in the presence of gravity with the parameter values same as that of Fig.10. We see that

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Π( t )

Quantum Interference in the Time-of-Flight Distribution for Atomic BECs

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Figure 9. Quantum TOF distributions Π1 (t) for the coherently splitted BEC of sodium atoms (representing the macroscopic Schr¨odinger-cat) falling freely under gravity are plotted for varying wave packet width σ0. In each curve, time (in sec) is plotted along horizontal direction and the coherent TOF distribution Π1(t) of BEC Schr¨odinger-cat is plotted (in sec−1 ) along vertical axis. The detector is located at a distance z = H = −1 cm and d = 50 µm.

the magnitude of Π1 (t) (in the absence of gravity) is roughly 105 times smaller than that obtained for gravitational free fall case for z = H = −1 cm. The reason for this is that the magnitudes of J1 (H, t) (36) and J2 (H, t) (37) become very small (roughly 105 times) in the absence of gravity. This magnitude becomes 106 times smaller if we consider the detector location at z = −10 cm. Actually, in the absence of gravity, there will be free particle motion and expansion of the wave packets in every direction. So, if one tries to observe the interference in the quantum TOF distribution in the absence of gravity, the intensity of that interference pattern will be too faint to be observed as only a small fraction of the condensate atoms will arrive at the detector. Hence, in our setup, gravity plays an important role which helps to pull down the condensate towards the detection plane.

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Figure 10. Quantum TOF distribution Π1(t) for the coherently splitted BEC of sodium atoms (representing the macroscopic Schr¨odinger-cat) is plotted in the absence of gravity (g = 0). Time (in sec) is plotted along horizontal direction and the coherent TOF distribution Π1 (t) of BEC Schr¨odinger-cat is plotted (in sec−1 ) along vertical axis. The detector is located at a distance z = H = −1 cm with σ0 = 1 µm and d = 20 µm.

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4.

Summary and Conclusion

To summarize, in this work we propose a scheme to experimentally observe matter-wave interference in the time domain, specifically in the TOF (arrival-time) distribution using atomic BEC. This experimentally testable scheme has the potential to empirically resolve ambiguities inherent in the theoretical formulations of the quantum arrival time distribution. Here we use the probability current density approach to calculate the quantum TOF distributions for atomic BEC Schr¨odinger cat represented by superposition of macroscopically separated wave packets in space. Our definition of the quantum TOF distribution in terms of the modulus of the probability current density is particularly motivated from the equation of continuity, and other physical considerations discussed in the literature [26, 27, 28, 29, 30, 31, 32, 33, 34]. This approach also provides a proper classical limit, as the interference and hence the coherence in the quantum TOF signal disappears in the large-mass limit. We repeat that there is no classical analogue of this TOF distribution Π1 (t) and this is purely a quantum distribution where we quantify the matter-wave interference in the quantum TOF signal. Hence, it will be interesting to see if our prediction of interference in time domain (TOF distribution) can be verified in actual experiments using modern interferometry techniques and sophisticated TOF methods.

Acknowledgments We would like to acknowledge support from the National Science Council, Taiwan, under Grants No. 97-2112-M-002-012-MY3, support from the Excellent Research Projects of the National Taiwan University under Grants No. 97R0066-65 and No. 97R0066-67, and support from the focus group program of the National Center for Theoretical Sciences, Tai-

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wan. H.S.G. is grateful to the National Center for High-performance Computing, Taiwan, for computer time and facilities.

References [1] C. Davisson and L. Germer, Phys. Rev. 30, 705 (1927). [2] L. de Broglie, Ann. Phys. (Paris) 3, 22 (1925). [3] L. Marton, Phys. Rev. 85, 1057 (1952). [4] A. Tonomura, Electron Holography (Springer, New York, 1993). [5] H. Rauch and S. Werner, Neutron Interferometry: Lessons in Experimental Quantum Mechanics (Oxford University Press, Oxford, 2000). [6] J. Baudon, R. Mathevet and J. Robert, J. Phys. B: At. Mol. Opt. Phys. 32, R173-R195 (1999). [7] P. R. Berman (ed), Atom Interferometry (Academic Press, New York, 1997). [8] F. Linder et al., Phys. Rev. Lett. 95, 040401 (2005). [9] P. Szriftgiser et al., Phys. Rev. Lett. 77, 4 (1996); S. Gupta et al., Phys. Rev. Lett. 89, 140401 (2002).

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[10] H. J. Metcalf and P. V. D. Straten, Laser Cooling and Trapping (Springer, New York, 1999). [11] I. Yavin et al., Am. J. Phys. 70, 149 (2002); T. M. Brzozowski et al, J. Opt. B: Quantum Semiclass. Opt. 4, 62 (2002) [12] L. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Oxford University Press, Oxford, 2004); F. Dalfovo et al., Rev. Mod. Phys. 71, 463 (1999). [13] D. S. Weiss et al., J. Opt. Soc. Am. B 6, 2072 (1989); P. D. Lett et al., J. Opt. Soc. Am. B 6, 2084 (1989). [14] J. V. Gomes et al., Phys. Rev. A 74, 053607 (2006). [15] J. M. Butler et al., Anal. Chem. 68, 3283 (1996); T. J. Griffin et al, Nature Biotechnology 15, 1368 (1997); E. Moskovets et al, Rapid Commun. Mass Spectrom. 13, 2244 (1999). [16] W. D. Philips et al., Scientific American March 1987, pp.50-56; C. Salomon et al., Europhys. Lett. 12, 683 (1990). [17] D. Bassi et al. in: “Atomic and Molecular Beam Methods edited by G. Scoles (Oxford University Press, Oxford, 1998); C. Grupen Particle Detectors (Cambridge University Press, Cambridge, 1996); J. H. Gross Mass Spectrometry: A Text Book (Springer, Berlin, 2002).

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Md. Manirul Ali and Hsi-Sheng Goan

[18] M. R. Andrews et al., Science 275, 637 (1997). [19] M. Kozuma et al., Phys. Rev. Lett. 82, 871 (1999). [20] E. W. Hagley et al., Phys. Rev. Lett. 83, 3112 (1999); J. E. Simsarian et al. Phys. Rev. Lett. 85, 2040 (2000). [21] J. G. Muga, R. S. Mayato and I. L. Egusquiza (ed), Time in Quantum Mechanics (Springer, Berlin, 2002). [22] E. H. Hauge and J. A. Stovneng, Rev. Mod. Phys. 61, 917 (1989); R. Landauer and T. Martin, Rev. Mod. Phys. 66, 217 (1994); V. S. Olkhovsky and E. Recami, Phys. Rep. 214, 339 (1992). [23] J. Kijowski, Rept. Math. Phys. 6, 351 (1974). [24] N. Grot et al., Phys. Rev. A 54, 4676 (1996); V. Delgado et al., Phys. Rev. A 56, 3425 (1997). [25] Y. Aharanov et al., Phys. Rev. A 57, 4130 (1998); J. A. Damborenea et al., Phys. Rev. A 66, 052104 (2002). [26] J. G. Muga and C. R. Leavens, Phys. Rep. 338, 353 (2000). [27] R. S. Dumont et al., Phys. Rev. A 47, 85 (1993); C. R. Leavens, Phys. Lett. A 178, 27 (1993); J. G. Muga et al., Ann. Phys. 240, 351 (1995); A. Challinor et al., Phys. Lett. A 227, 143 (1997); V. Delgado, Phys. Rev. A 59, 1010 (1999). Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

[28] A. K. Pan et al., Phys. Lett. A 352, 296 (2006). [29] J. Finkelstein, Phys. Rev. A 59, 3218 (1998). [30] W. R. McKinnon et al., Phys. Rev. A 51, 2748 (1995); C. R. Leavens, Phys. Rev. A 58, 840 (1998); S. V. Mousavi et al., J. Phys. A: Math. Theor. 41, 375304 (2008). [31] Md. M. Ali et al., Phys. Rev. A 75, 042110 (2007). [32] Md. M. Ali et al., Phys. Rev. A 68, 042105 (2003). [33] Md. M. Ali et al., Found. Phys. Lett. 19, 723 (2006). [34] Md. M. Ali et al., Class. Quantum Grav. 23, 6493 (2006). [35] C. R. Leavens in: [21]; M. Daumer in: Bohmian Mechanics and Quantum Theory: An Appraisal, edited by J. T. Cushing, A. Fine, and S. Goldstein (Kluwere, Dordrecht, 1996) pp. 87-98. [36] P. Holland, Phys. Rev. A 60, 4326 (1999); P. Holland, Ann. Phys. (Leipzig) 12, 446 (2003); P. Holland, Phys. Rev. A 67, 062105 (2003); W. Struyve et al., Phys. Lett. A 322, 84 (2004). Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

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[37] T. Tsurumi and M. Wadati, J. Phys. Soc. Jpn. 70, 60 (2001); T. Tsurumi and M. Wadati, J. Phys. Soc. Jpn. 71, 1044 (2002). [38] F. Gerbier et al., Phys. Rev. Lett. 86, 4729 (2001); A. Sinner et al., Phys. Rev. A 74, 023608 (2006). [39] R. W. Robinett, Phys. Scr. 73, 681 (2006); H. Wallis et al., Phys. Rev. A 55, 2109 (1997).

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[40] Md. M. Ali and H.-S. Goan, J. Phys. A: Math. Theor. 42, 385303 (2009).

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Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

In: Bose Einstein Condensates Editor: Paige E. Matthews, pp. 63-114

ISBN 978-1-61728-114-3 c 2010 Nova Science Publishers, Inc.

Chapter 3

O N THE DYNAMICS OF N ONCONSERVATIVE B OSE -E INSTEIN C ONDENSATES IN T RAPPED D ILUTE G ASES Victo S. Filho Laborat´orio de F´ısica Te´orica e Computacional (LFTC), Universidade Cruzeiro do Sul, Rua Galv˜ao Bueno, 868, Liberdade, 01506-000, S˜ao Paulo, Brazil

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Abstract The theory of Bose-Einstein condensation of dilute gases in traps is reviewed, considering mainly the dynamics of realistic condensates with nonconservative processes, in the mean-field approach. It is discussed the main properties of the stationary and dynamical solutions in the cases of attractive and repulsive interactions. Nonlinear phenomena that occur in Bose-Einstein condensates with nonconservative processes as dissipative solitons, spatiotemporal chaos, liquid-gas phase-transition and others are discussed, as well as the limitations of the mean-field formalism in describing the experimental data. The review also includes an analysis of different types of trap potential, a description of the variational and numerical methods used for solving the Gross-Pitaevskii equation and an important description of the stability of BoseEinstein condensates with nonconservative processes, mainly in the case in which the collapse phenomenum can occur, when the number of particles for negative scattering length is higher than the critical one.

1.

Nonconservative Bose-Einstein Condensates

As know from a lot of recent papers [1, 2, 3, 4, 5, 6], the condensation phenomenum can be experimentally verified by means of dilute alcali metal gases trapped by harmonic potential in very low temperatures. Below a critical temperature Tc , one can obtain a coherent state called Bose-Einstein condensate (BEC), in which all the bosonic particles occupy the same state, so-called ground state, and oscillate with same frequency, referred as collective frequency. Bose-Einstein condensates can be experimentally realized in two physically different types, according to the interaction between species: one with positive s-wave scattering

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length (as > 0), that corresponds to the repulsive two-body interactions between atoms, and the negative one (as > 0), that corresponds to the attractive atom-atom interactions. The former is stable and can be more easily realized in laboratory, achieving up to 10 7 or more atoms in the condensate state. The experiments have every year improved, but the lifetime of BEC state is very short yet, with reports of about 75 seconds [7]. The latter is very unstable and when a critical number of atoms is achieved, the system exhibits an implosion, called wave function collapse. The signature of the collapse phenomenum corresponds to the very quickly decreasing of the mean-square root of the condensate state, up to values near to zero, what causes substantial atomic losses from the ground state and the subsequent destruction of BEC state. The first experimental evidences of Bose-Einstein condensation (BEC) in magnetically trapped weakly interacting atoms were report in Refs. [8, 9, 10] and brought a considerable support to the theoretical research on bosonic condensation. As known, the nature of the effective atom-atom interaction determines the stability of the condensed state: the twobody pseudopotential is repulsive for a positive s−wave atom-atom scattering length and it is attractive for a negative scattering length [11]. Ultra-cold trapped atoms with repulsive two-body interaction undergo the Bose-Einstein phase-transition to a stable condensed state for a lot of atomic species as, for instance, in the cases of the two first experimental realizations reported, that is, 87 Rb [8] and 23Na [9]. In the other hand, they undergo a Bose-Einstein phase-transition to a unstable condensed state for other species, as the first realization of this case, that is, 7Li [10]. As earlier explained, the condensed state of atoms with negative s−wave atom-atom scattering length is unstable for a large number of atoms [12, 13]. It was indeed observed in 7 Li gas [10] (atom with s−wave scattering length ˚ that the number of allowed atoms in the condensed state was limited as ∼ = (−14.5 ± 0.4) A) to a maximum value between 650 and 1300, consistently with mean-field predictions [12]. There are a lot of more atoms that can presently be used to obtain Bose-Einstein condensate states, as 1H [14], metastable 4He [15], 85 Rb [16], 41K [17], 133Cs [18] and, more recently, 84 Sr [19]. The theoretical description of BEC is very successful and most of experimental data can be explained by a mean-field approach known as Gross-Pitaevskii formalism [1]. In this formalism, the system is considered conservative in the sense that all the particles come into the ground state and there are no atomic losses. Although, in a realistic scenario, the condensate state is formed inside a dilute gas at very low temperature background, called in the literature as external thermal cloud. If almost all atoms are in BEC, the thermal cloud surrounding the Bose-Einstein condensate can be very thin, but it is always present in realistic scenarios. Another point that favors the nonconservative approach is the real presence of inelastic collisions in BEC, as the dipolar relaxation (two-body inelastic collisions) and three-body recombination (three-body inelastic collisions). In the first inelastic collision type, two atoms in BEC interact and, as an effect, their spins flip, a process that does not conserve the total spin and leads to atom losses from the magnetic trap. In the second one, a bound two-atom state collides with a third atom and one of them is ejected from the bound state and the third one is linked to the remaining one in a new bound state [20]. For describing this realistic case, one can use two models, that describe the time evolution of BEC by means of (1) Boltzmann transport equation [21] or (2) modified (non-

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conservative) Gross-Pitaevskii equation [22]. Both models are not enough to explain all of BEC features, mainly in the attractive two-body interaction BECs. In the attractive BEC, one of the main problems is to describe exactly the remaining number of particles in the condensate after a collapse. It remains a big challenge to describe physically the wave function collapse precisely, so that the final number of particles in the condensate is determined correctly. The main scope of this chapter is to sum up many of the papers that have been written analyzing realistic Bose-Einstein condensates. The review is written as follows: in the section 2, it is explained the mean-field formalism to describe BEC dynamics and the extension to the nonconservative mean-field formalism. In the section 3, there is a brief description of the variational and the numerical methods used for solving the nonlinear equation that describes the evolution of realistic BECs. In the section 4, it is shown the main results and phenomena that appear in realistic BEC. First, it is analyzed the liquid-gas phase transition; after, the presence of collapse phenomenum and spatiotemporal chaos in BEC with attractive interaction. In the following, it is also analyzed the formation of autosolitons in attractive Bose-Einstein condensates. After all, it is described the coherent formation of dimers in BEC with time-dependent high scattering length. Finally, one makes a brief summary in the last section.

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2.

The Mean-field Approach

In the study of trapped many-bosons quantum systems, one considers the Hamiltonian   Z 2 ~ † 2 ˆ r) ˆ = ˆ (~r) − ∇ + Vext(~r) Ψ(~ H d~r Ψ 2m Z 1 ˆ † (~r)Ψ ˆ † (r~0)V (~r − r~0)Ψ(~ ˆ r)Ψ( ˆ r~0 ), d~rdr~0 Ψ + (1) 2 ˆ r) and Ψ ˆ † (~r) are bosonic field operators that annihilate and create one particle in where Ψ(~ the position ~r, respectively, and V (~r − r~0 ) is the two-body interatomic potential. The basic idea for describing such a system in the mean-field approach consists in the separation of the contribution from the condensate and the contribution of other components for the wave function operator. In other words, one can write: ˆ r, t) = Ψ(~r, t) + Ψ0 (~r, t), Ψ(~

(2)

where the ground state function Ψ(~r, t) is defined as the expected value of the field operator: ˆ r, t)i, Ψ(~r, t) = hΨ(~

(3)

that has the meaning of an order parameter and represents the condensate wave function. By supposing very small contribution from the thermal cloud that encloses the condensed state ˆ0 ∼ (Ψ = 0), one can deduce the equation for the evolution of the condensate in the mean-field approach. By using the Hamiltonian (1) and the Heisenberg equation, one has: i~

ˆ r, t) ∂ Ψ(~ ˆ r, t), H]. ˆ = [Ψ(~ ∂t

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(4)

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So, one obtains:  Z ˆ r, t)  ~2 ∂ Ψ(~ 2 † 0 0 0 0 ˆ r~ , t) Ψ(~ ˆ (r~ , t)V (~r − r~ )Ψ( ˆ r, t), = − ∇ + Vext(~r) + dr~ Ψ i~ ∂t 2m

(5)

At last, as the gas is very diluted, one supposes that one parameter, the s-wave scattering length is enough to characterize the atom-atom interaction term, because only low-energy binary collisions are relevant for describing dilute and very cold gases. So, this hypothesis allows the substitution: V (~r − r~0) = λ2δ(~r − r~0),

(6)

in which the coupling constant can be verified as being: 4π~2as , (7) m in which m is the particle mass and as (or simply a) is the s-wave scattering length. By using this effective potential in the Eq. (5), one finds one type of nonlinear Schr¨odinger equation (NLSE), known as Gross-Pitaevskii equation (GPE), that reads:  ˆ r, t)  ~2 ∂ Ψ(~ 4π~2as ˆ 2 i~ = − ∇ + Vext (~r) + Ψ(~r, t), (8) ∂t 2m m λ2 =

which is normalized to the number of atoms in the condensed state. The description of the experimental data concerning the evolution of BEC by using the Gross-Pitaevskii (GP) formalism is very sucessfull in a lot of cases [1, 2, 3, 4, 5, 6].

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3.

Techniques for the Resolution of Nonlinear Schr¨odinger Equations

One can apply either variational or numerical techniques for obtaining the solutions of nonlinear Schr¨odinger equations (NLSE) as, for example, for Gross-Pitaevskii equation (GPE). One mainly has the independent variational method and the dependent variational one, for solving variationally GPE, and Crank-Nicolson and Runge-Kutta methods for solving GPE in the time dependent and time independent form, respectively.

3.1.

Time Independent Variational Approach

One considers here a extended GPE with a term g3|ψ|4 added, for real g3. For obtaining the variational solutions in this stationary case, one can use as trial wave function the Gaussian ansatz ψ(~r), as done in Ref. [24], obtaining:  3   1 mω 4 r2  mω  exp − 2 , (9) ψvar (~r) = πα2 ~ 2α ~ where α is a dimensionless variational parameter. The corresponding mean-square radius is proportional to the variational parameter α, given by:

2 3~ 2 α , r = 2mω

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(10)

On the Dynamics of Nonconservative Bose-Einstein Condensates...

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while the central density is given by ρc (α) = α−3

 mω 3/2 π~

.

The expression for the total variational energy is given by     1 2n2 g3 3 n 2 α + 2 − √ 3+ √ E(α) = ~ωN , 4 α 4 πα 9 3πα6 and, analogously, one can obtain the corresponding chemical potential:     1 n 2n2 g3 3 2 α + 2 − √ 3+ √ . µ(α) = ~ω 4 α 2 πα 3 3πα6

3.2.

(11)

(12)

(13)

Time Dependent Variational Approach

The dynamics of the condensate wavefunction in trapped Bose-Einstein condensates can be obtained in the framework of the mean field approximation, by means of the nonconservative Gross-Pitaevskii equation. By formally adding an averaged nonconservative Lagrangian LR to the averaged Lagrangian L, one uses the property that δLR /δu∗ = −R(u, u∗), where u = u(r, t) is the wave function and R is a function with all of the contributions of inelastic processes, given by
(14) R(u, u∗) = i γ − µ|u|2 − ξ|u|4 u,

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and by applying the Euler-Lagrange equations to L0 ≡ L + LR , with respect to u∗ , one obtains d ∂L0 ∂L d ∂L ∂L0 − = − − R(u, u∗) = 0, ∗ ∗ ∗ ∂u dt ∂ut ∂u dt ∂u∗t

(15)

∂u + ∆u − (ω 2r2 )u + λ2|u|2u + λ3|u|4u = R(u, u∗), ∂t

(16)

that leads to Eq.: i

where ω 2 r2 is the trap harmonic potential; λ2 and λ3 are, respectively, the two- and threebody interaction parameters, with λ2 proportional to the s- wave atomic scattering length. Further, γ µ, and ξ are positive defined coefficients related, respectively, to feeding, dipolar relaxation and three-body inelastic recombination parameters. In the present variational approach, for u ≡ u(r, t), it is used the Gaussian trial function [25]   b(t)r2 r2 + iφ(t) , (17) u = A(t) exp − 2 + i 2a (t) 2 where A, a, b, and φ are, respectively, the amplitude, width, chirp and linear phase. It was not included the center-of-mass coordinate into the ansatz, because the dissipative and amplifying terms have no influence on it.

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Victo S. Filho The variational approach is applied to the averaged Lagrangian of the conservative sys-

tem  i i ∗ λ2 4 λ3 6 ∗ 2 2 2 2 ut u − ut u − |∇u| + |u| + |u| − ω r |u| d3r. L = L(r, t)d r = 2 2 2 3 (18) Substituting the trial function (17) into Eq. (18), one finds the averaged Lagrangian in terms of the condensate wavefunction parameters   √ π π 2 3 6 λ2 4λ3 L=− A a 3a2 bt + 4φt + 2 (1 + a4b2) − √ A2 − √ A4 + 6ω 2 a2 . (19) 4 a 2 9 3 Z

3

Z 

The corresponding variational principle is given by δ

Z

t 0

L dt = δ 0

Z

t

(L + LR)dt = 0,

(20)

0

R where, as in Eq.(18), LR = d3rLR . Taking into account that for a small shift δη of some variational parameter η, one has f (η + δη) = f (η) + δη

∂f , ∂η

(21)

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where f ≡ f (u, u∗) = L or f = LR , one obtains a system of equations for the variational parameters ηi [25, 26]:   Z d ∂L ∂u∗ ∂L 3 ∗ ∂u − = d r R +R , (22) ∂ηi dt ∂ηit ∂ηi ∂ηi where Eq. (15) and its conjugate were used. The substitution of Eqs.(17) and (19) into Eq.(22) yields the following system of ordinary differential equations (ODEs) : µ 2 d(A2a3) = 2γA2a3 − √ A4 a3 − √ ξA6a3 , dt 2 3 3 µ 2 d(A2a5) = 4A2a5 b + 2γA2a5 − √ A4 a5 − √ ξA6a5 , dt 2 2 9 3 2 4 2 λ2 A 4λ3A db = 4 − 2b2 − 2Ω2 − √ − √ , 2 dt a 2 2a 9 3a2 3 7 2 dφ = − 2 + √ λ2A2 + √ λ3A4 . dt a 8 2 3 3

(23)

The Eq. (23a) can also be obtained from the modified form of the conservation law for the number of atoms N , where N is given by Z (24) N = |u|2d3r .

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The other equations of the system (23) can be obtained by using higher moments (see the appendix in Ref. [23]). It is simpler to rewrite the system using the notation x = a2 , y = A2 : µ 4 xt = 4xb + √ xy + √ ξy 2 x, 2 2 9 3 7µ 2 4 yt = −6yb + 2γy − √ y − √ ξy 3 , 4 2 3 3 2 λ2 y 4λ3y 2 2 2 √ √ . − 2b − 2ω − bt = − x2 2 2x 9 3x

(25)

This ODE system can be used for obtaining the solutions of the nonconservative GPE.

3.3.

Numerical Methods

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The methods used for solving GPE were fourth-order degree Runge-Kutta method and Crank-Nicolson method. The fourth-order degree Runge-Kutta method has been verified as a reliable and efficient method for obtaining the stationary solutions of GPE. The CrankNicolson one is a difference finite method, that is very reliable because it is a unconditionally stable method. In the case of Runge-Kutta algorithm, one writes the stationary GPE in a form of a set of first order differential equations, obtaining a system of ODEs: dx = f (x, y, z, t) dt dy = g(x, y, z, t) dt dz = h(x, y, z, t). dt

(26)

For just one equation, du/dt = f (u, t), one defines the parameters: K1 = ∆t f (un , tn )  K2 = ∆t f un +  K3 = ∆t f un +

 1 ∆t K1, tn + 2 2  1 ∆t K2, tn + 2 2 K4 = ∆t f (un + K3 , tn + ∆t),

(27)

so that the function is obtained from: 1 un+1 = un + (K1 + K2 + K3 + K4 ). 6

(28)

Analogously, one defines more two sets of parameters of the system (26), that is, M1 to M4 Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

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and L1 to L4, beside K1 to K4, and the solution of the system becomes: 1 xn+1 = xn + (K1 + K2 + K3 + K4) 6 1 yn+1 = yn + (L1 + L2 + L3 + L4) 6 1 zn+1 = zn + (M1 + M2 + M3 + M4 ). 6

(29)

In the case of Crank-Nicolson algorithm, one writes the parabolic partial equation urr = Ψ(t, r, u, ut, ur )

(30)

in a discretized form, by using the discretization operators: δr2 ui,j+1 = ui+1,j+1 − 2ui,j+1 + ui−1,j+1

(31)

δr ui,j+1 = ui+1/2,j+1 − ui−1/2,j+1
1 ui+1/2,j+1 + ui−1/2,j+1 , µ ui,j+1 = 2

(32) (33)

so that one obtains the discrete equation:

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 1 1 2 δ (ui,j+1 + ui,j ) = Ψ ih, (j + 1/2)k, (ui,j+1 + ui,j ), 2h2 r 2  1 1 µδr (ui,j+1 + ui,j ), (ui,j+1 − ui,j ) . 4h k

(34)

So, one solves both for very small steps in time, for Eq. (26), and in time and space, for Eq. (34), and one obtains the dynamical solutions of the problem. The advantage of Crank-Nicolson method consists in the unconditional convergence of the differences equation solutions to the solutions of the correspondent partial derivatives equation, so that any small value of the stability ratio R = k/h2 , in which k = ∆t and h = ∆r, makes that the iterative solution converges to the exact solution, as demonstrated in Ref. [27]. Any variation of GP equation can, at least in principle, be solved by the numerical techniques early described for nonlinear differential equations, employing the Runge-Kutta (RK) and shooting methods [28], and the early described variational procedure [29], using a trial Gaussian wave-function. The procedure here presented is in the discussion of the physical results obtained, in the next sections.

4.

Unharmonic Trapping Potentials

The main potential used in BEC experiments for trapping atoms is the harmonic one: V (r) =

1 mω 2r2 , 2

(35)

where ω is the trapping frequency. However, it is interesting to consider other types of potential for trapping atoms, as distortions of the harmonic one, as done in Ref. [30] and Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

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possibles power-law potentials, as the cubic or linear one. Such different potentials are important for analyzing the possibility of obtaining BEC with higher number of atoms in the attractive atom-atom interaction case. In Ref. [31], one was studied the dependence of the number of atoms in the condensate against the power of the unharmonic potential. In spherical symmetry, in the corresponding mean-field equation, for attractive twobody interactions, a quantum many-body problem for a very dilute gas can be described by the Gross-Pitaevskii equation [31].   ∂Ψ(~r, t) ~2 ~ 2 4π~2|as | 2 i~ = − ∇ + Vtrap(r) − | Ψ(~r, t) | Ψ(~r, t) , (36) ∂t 2m m

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where Ψ(~r, t) is the wave function and Vtrap the trapping potential, as is the two-body s−wave scattering length and m the mass of the atomic species. In this form, the wave function Ψ ≡ Ψ(~r, t) is normalized to the number N of particles. In the specific case of power-law potentials, in spherical symmetry, one can write the trapping potential as √ !α ~ω 2r , (37) Vtrap(r) = 4 l0 p where l0 ≡ ~/(mω) defines a unit of length. For α = 2 one has the usual 3D harmonic oscillator potential with ω being a geometrical average of the frequencies in the three spatial directions (ω 3 ≡ ω1 ω2 ω3 ). The stationary solutions of Eq. (36) are given by the chemical potential µ, by setting Ψ(~r, t) = e−iµt/~ ψ(~r), such that   ~2 ~ 2 4π~2|a| 2 − ∇ + Vtrap(r) − | ψ(~r) | ψ(~r) = µψ(~r) . (38) 2m m The trap is spherically symmetric and one is interested in the ground-state solutions. So, in Eq. (38), one assumes the system is in the s−wave, and redefine the wave function, the variable r and the chemical potential µ as √ p 2 µ x≡ r, β= . (39) Φ(x) ≡ 8π|a|rψ(~r), l0 ~ω Using the above, one has |Φ(x)|2 d2 Φ(x) xα + Φ(x) − Φ(x) . (40) dx2 4 x2 As ψ(~r) is normalized to N , from (39) one obtains the corresponding normalization of Φ(x) as Z ∞ √ N |a| dx | Φ(x) |2= 2 2 . (41) l0 0 The boundary conditions of (40) are such that Φ(0) = 0 and Φ(x → ∞) = 0. The total energy of the system is given by   l0 √ E, with Etot = ~ω 2 2|a| ) Z ∞ ( 4 dΦ(x) 2 xα |Φ(x)| 2 + |Φ(x)| − dx . (42) E ≡ dx 4 2x2 0 βΦ(x) = −

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Figure 1. The dimensionless chemical potential β = µ/(~ω), versus the number of particles N , scaled by a factor |a|/l0. The numerical results are shown for different confining powerlaw potentials, Eq. (37), with α = 1.0 to α = 2.8, as indicated.

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The size can be obtained from the corresponding mean-square radius hr2i, that can be written in terms of a dimensionless observable hx2i, as R∞ l02 0 dx x2 |Φ(x)|2 l02 2 2 R∞ hr i = ≡ hx i. (43) 2 0 dx |Φ(x)|2 2 By considering variations of the power α, in such equations, one can analyze the behavior of the observables of the system and compare with results obtained for the case of harmonic potential [24]. One important observable is the magnitude of the maximum critical number of particles Nc in the condensate, as one varies the power α. Here, in the present case of stationary solutions, is considered the full numerical approach to obtain the results for the spherically symmetrical equations early described, that is, one has applied the fourth-order Runge-Kutta method, which is combined with the shooting method in order to satisfy the boundary conditions [28]. The results of the calculations are presented for the chemical potential (Fig. 1) and for the mean-square radius (Fig. 2). In Fig. 1, one presents results for the chemical potential µ, given by the corresponding dimensionless observable β, as a function of the number of atoms N , scaled by the factor |a|/l0. The curves point out a stable branch (upper part) and an unstable branch (lower part), that are joined at the critical points and their behavior is qualitatively similar for all cases. However, if is conserved the number of particles, one notes that the values of β are smaller for smaller values of the power α only when the scaled number of particles, N |a|/l0, is not too large. It can happen an inversion of this behavior near the critical limits of potentials with higher powers. This happens because the critical numbers Nc are smaller for larger values of α. To extend the region of stability to higher number of particles, one must decrease significantly the power of the trapping potential. In Fig. 2, one presents the numerical results for the mean-square radius hr2i, given by

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Figure 2. The mean-square radius hr2i, given by the corresponding dimensionless observable hx2i, as defined in Eq. (43), versus the number of particles N , scaled by a factor |a|/l0. The numerical results are shown for different confining power-law potentials, Eq. (37), with α = 1.0 to α = 2.8, as indicated. the corresponding dimensionless observable hx2 i. For a given atomic system with negative two-body scattering length, as one increases the number of particles N the stable condensate becomes smaller, shrinking till its minimum size at the critical point (extreme right-handside of each curve with fixed α). From the critical points, the curves also show the respective sizes of the unstable solutions (lower branches), corresponding to maxima for the energies. By comparing results from the harmonic oscillator confining potential ( α = 2) with the linear one (α = 1), one observes a considerable increase in the mean-square radius at the critical point: hx2iα=1 ∼ = 2hx2iα=2 . In Fig. 3, one shows the numerical results for the maximum critical number of particles for stable solutions, scaled by the factor |a|/l0, as a function of the power α of the confining potential. With α decreasing, the scaled critical number of condensed atoms will increase. For the harmonic trap, in spherical symmetry, one reproduces the well known critical number Nc |a|/l0 = 0.575. In case of linear trap (α = 1), one obtains Nc |a|/l0 ∼ = 0.795.

5. 5.1.

Nonlinear Phenomena in Bose-Einstein Condensates Liquid-Gas Phase Transition in Nonconservative Bose-Einstein Condensates

One possible study in a system of trapped ultra-cold atoms refers to the effects of a repulsive three-body interaction in Bose-Einstein condensed states. The stationary solutions of the corresponding s−wave non-linear Schr¨odinger equation indicate the existence of a firstorder liquid-gas phase transition, observed for the condensed state for a range of values of the strength of the effective three-body force. The time evolution of the condensate with feeding process and three-body recombination losses also shows a longer decay time of

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Figure 3. The critical number of atoms Nc , scaled by the factor |a|/l0, as a function of the power of the trapping potential. The solid line is interpolating the numerical results, represented by circles. the dense (liquid) phase in relation to one expected, due to strong oscillations of the meansquare-radius. The addition of a repulsive three-body interaction can extend considerably the region of stability for a condensate, even for a very weak three-body force, as seen in Ref. [32]. From Refs. [33], one can observe both signs for the three-body interaction. However, here is only considered a repulsive three-body elastic interaction together with an attractive two body interaction. Due to the repulsive three-body force, new physical aspects appear in the time evolution of the condensate. In Ref. [34], in respect to the stationary situation, it was suggested that, for a large number of bosons, the three-body repulsion can overcome the two-body attraction and a stable condensate will appear in the trap. In Ref. [35], the authors have also observed that above a critical value, the only local minimum is a dense gas state, where fails to neglect three-body collisions. For describing this system, one of the types of nonlinear Schr¨odinger equations (NLSE) [36] or GPE is used: GPE is extended to include the effective potential coming from the three-body interaction and then solved numerically. The dimensionless parameters are related to the two-body scattering length, the strength of the three-body interaction and the number of atoms in the condensed state. As one can see in Ref. [37], to describe all two-body scattering processes in that many-particle system, the two-body potential should be replaced by the many-body T −matrix. At very low energies, this is approximated by the two-body scattering matrix, which is directly proportional to the scattering length a [13]. To obtain the desired equation without the relaxation dipolar effect and in a dimensional form, one first considers the effective Lagrangian density, which describes the condensed wave-function in the Hartree approximation, implying in the Gross-Pitaevskii energy functional [36] for the trial wave function Ψ:   ∂Ψ† ~2 † 2 m 2π~2as 2λ3 i~ 6 † ∂Ψ Ψ − Ψ + Ψ ∇ Ψ − ω 2r2 |Ψ|2 − |Ψ|4 − |Ψ| (44) . L = 2 ∂t ∂t 2m 2 m 3! In this formalism, it is considered spherical symmetry, so that the atomic trap is given by a harmonic potential with angular frequency ω and only radial dependence; the last two terms represent the effective atom interactions up to three particles, that is, the effective Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

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interaction Lagrangian density for ultra-low temperature bosonic atoms, including two [13] and three-body effective interaction at zero energy, in which λ3 is the strength of the threebody effective interaction and as the scattering length. The NLSE, which describes the condensed wave-function in the mean-field approximation, is obtained from the effective Lagrangian given in Eq. (44). The stationary solutions are determined as usual, that is, one supposes µ

Ψ(~r, t) = e−i ~ t ψ(~r),

(45)

where µ is the chemical potential and ψ(~r) is normalized to 1 and by rescaling the NLSE for the s−wave solution, one obtains   d2 1 2 |Φ(x)|2 |Φ(x)|4 − 2+ x − + g3 Φ(x) = βΦ(x) (46) dx 4 x2 x4 p p for as < 0, where x ≡ 2mω/~ r and Φ(x) ≡ 8π|as|N rψ(~r). The dimensionless parameters, related to the chemical potential and the three-body strength are respectively given by β ≡ µ/~ω (47) and

λ3~ωm2 . (4π~2a)2

(48)

dx|Φ(x)|2 = n,

(49)

g3 ≡ The normalization for Φ(x) reads Z

∞

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0

where the reduced number n is related to the number of atoms N by p n ≡ 2N |a| 2mω/~.

(50)

The boundary conditions [12] in Eq. (46) are given by Φ(0) = 0 and



   x2 1 ln x lim Φ(x) = C exp − + β − x→∞ 4 2

(51)

From now on, by using the variational and numerical methods described before, one presents the main results and conclusions in this case. As shown from Ref. [24], in Fig. 4 one illustrates the variational procedure considering an arbitrarily small three-body interaction, chosen as g3 = 0.005. In the upper part of the figure, one shows four small plots for the total variational energy E, in terms of the variational width α. Each one of the small plots corresponds to particular values of n. For each number n, one reports the energy of the variational extrema in the lower part of figure 4 [24]. In region (I), where the number of atoms is still small, the attractive two-body force dominates over the repulsive three-body force and just one minima of the energy as a function of the variational parameter α is

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Figure 4. In the lower part, it has been done a comparison between variational (solid curve) and exact (dashed curve) numerical calculations of the condensate energy as a function of the reduced number of atoms n for g3 = 0.005, as shown in figure from Ref. [24]. In the upper part, one shows four plots of the variational energy as a function of the variational parameter α for four ranges of values for n, shown also in the lower frame. (I) (resp IV) corresponds to a small (large) n region, where only one stable solution is encountered; (II) (resp III) to a small (large) n region, where one observes three extrema for the energy. At left of lower part, it is also shown a fifth case, in which the symbol (A) corresponds to a particular n, where one obtains two stable solutions with the same energy E1 = E2 . The energy E is given in units of (N ~ω)/n.

found. If the number of atoms is increased (region (II)), two minima and an unstable maximum appear in the energy E (α) . The lower energy minimum is stable while the solution corresponding to the smaller α is metastable. This system has a higher density and, consequently, it is metastable due to the repulsive three-body force acting at higher densities. The minimum number n for the appearance of the metastable state is characterized by an inflection point in the energy as a function of α. The value of n at the inflection point corresponds to the beak in the plot of extremum energy versus n because for larger n three variational solutions are found, as depicted in the main lower part of figure 4. The attractive two-body and trap potentials dominate the condensed state in the low-density stable phase up to the crossing point (A) [24]. At point A, the denser metastable solution becomes degenerate in energy with the lower-density stable solution, so a first order phase-transition occurs. Since that both differ by their density, this transition looks like a gas-liquid phase transition for which the density difference between the liquid phase and the gas phase is the order parameter. In the variational calculation, this occurs for n ≈ 1.3, while one numerically obtains n ≈ 1.2. In region (III), one observes two local minima with different energies, a

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higher-density stable point and a lower-density metastable point. The metastable solution disappears in the beak at the boundary between region (III) and (IV). In regions (III) and (IV), the three-body repulsion stabilizes a dense solution against the collapse induced by the two-body attraction. The qualitative features of the variational solution is clearly verified by the numerical solution of the NLSE, as shown by the dashed curve. In Fig. 5, considering several values of g3 , using exact numerical calculations, one presents the evolution of two relevant physical quantities, E and ρc , as functions of the reduced number of atoms n. In Fig. 6, one presents for the same values of g3 other two relevant physical quantities, µ and r2 , using exact numerical calculations, representing the behavior of both as functions of the reduced number of atoms n. For g3 = 0, the calculation reproduces the result presented in Ref. [12, 37], with the maximum number of atoms limited by nmax ≈ 1.62. In the plot for the energy as a function of n, it is shown that for values of g3 > 0.0183 there is no phase-transition. At g3 ≈ 0.0183 and n ≈ 1.8, the stable, metastable and unstable solutions degenerate and a critical point associated with a second order phase-transition takes place. As also shown in the figures 5 and 6, for 0 < g3 < 0.0183, the density ρc , the chemical potential µ and the root-mean-squared radius r2 present back bendings typical of a first order phase-transition. In the transition point given by the crossing point in the plot of E versus n, an equilibrated condensate should undergo a phase-transition from the branch extending to small n to the branch extending to large n. From these figures, it is clear that the first branch is associated with large radii, small densities and positive chemical potentials while the second branch presents a more compact configuration with a smaller radius, a larger density and a negative chemical potential. This is the reason for the analogy, with the term gas for the first one and liquid for the second one, adopted in Ref. [24]. With g3 = 0.012, the gas phase happens for n < 1.64 and the liquid phase for n > 1.64. For g3 > 0.0183, a single fluid phase is observed. It is also verified that calculations with the variational expression of hr2i, ρc and µ are in good agreement with the ones ploted in Figs. 5 and 6. Finally, in the lower frame of Fig. 7, it is shown the phase boundary separating the two phases in the plane defined by n and g3 and the critical point at n ≈ 1.8 and g3 ≈ 0.0183. In the upper frame, one sees the boundary of the forbidden region in the central density versus g3 diagram. In summary, the calculation present, at the mean-field level, the consequences of a repulsive three-body effective interaction for the Bose condensed wave-function, together with an attractive two-body interaction. A first order liquid-gas phase-transition is observed for the condensed state, when a small repulsive effective three-body force is introduced. In dimensionless units, the critical point is obtained when g3 ≈ 0.0183 and n ≈ 1.8. The characterization of the two-phases through their energies, chemical potentials, central densities and radius were also given for several values of the three-body parameter g3, as also shown in Ref. [24]. The results here presented can be relevant to determine a possible clear signature of the presence of repulsive three-body interactions in Bose condensed atoms. It points to a new type of phase-transition between two Bose fluids. It is also possible, using the mean-field approximation, to develop the scenario of collapse in this system, which includes two aspects of three-body interaction, that is, threebody recombination and repulsive mean-field interaction, as done in Ref. [38]. By investi-

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Figure 5. Central density ρc and total energy E as functions of the reduced number of atoms n. The three-body strength g3 are given in the upper frame. The corresponding units are: (mω/~)/(4π|a|) for ρc and (N ~ω)/n for E.

 Figure 6. Chemical potential µ and average square radius r2 as functions of the reduced number of atoms n. The three-body strength g 3 are  given in the lower frame. The corre2 sponding units are: ~ω for µ and ~/(2mω) for r .

gating the competition between the leading term of an attractive two-body interaction, originated from a negative two-atom s−wave scattering length, and a repulsive three-body interaction, which can happen in the Efimov limit [39], when |a| → ∞ (see Refs. [40]), one first considers the stationary solutions of the corresponding extension of the Gross-Pitaevskii (GP) or the nonlinear Schr¨odinger equation (NLSE), for fixed number of particles, without dissipative terms, extending an analysis previously reported in Refs. [32]. From the stationary solutions early obtained, the time evolution of the feeding process of the condensate by an external source is introduced and the time-dependent NLSE with repulsive three-body

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Figure 7. Phase-diagram of the Bose-Einstein condensate with repulsive three-body interaction. In the upper frame, the central density ρc is given in units of (mω/~)/(4π|a|).

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interaction (given by g3 > 0) and dissipation due to three-body recombination processes is solved. The dramatic collapse and the consequent atom loss that happens at the critical number of atoms (when g3 = 0) [21] is softened by the addition of the three-body repulsive force [38]. The decay time of the liquid-phase is also unexpectedly long, when compared with the decay time that occurs for g3 = 0, a possible indication of three-body interaction effects. The results pointed out that the mean-square-radius is an important observable to be analyzed experimentally to study the dynamics of the growth and collapse of the condensate [21]. In the present study, in order to enphasize the real part of the three-body interaction, one chooses g3 significantly larger than the magnitude of the dissipative term. The main physical characteristic of the repulsive three-body force is to prevent the collapse of the condensate for the particle number above the critical number found with only two-body attractive interaction. The three-body repulsive potential tends to overcome the attraction of the two-body potential at short distances, as described by Eq. (3), as the repulsive interaction is proportional to x−4 , while the two-body potential to x−2 . Thus, the implosive force that shrinks the condensate at the critical number is compensated by the repulsive three-body force. The time evolution of the growth and collapse of the condensate with attractive interactions [21] should be qualitatively modified by the presence of the repulsive three-body force. The three-body recombination effect [22], which “burns” partially the condensed state should be taken into account to describe quantitatively the dynamics of the condensate. In the case of only two-body attractive interaction, as observed by Kagan et al. [22], by considering the feeding of the condensate from the nonequilibrium thermal cloud, the time evolution is dominated by a sequence of growth and collapse of the condensate. The collapse occurs when the number of atoms in the condensate exceeds the critical number Nc ; and it is followed by an expansion after the atoms in the high density region of the wave-function are lost due to three-body recombination processes and consequently the average attractive potential from the two-body force is weakened. In order to quantitatively study the above features with repulsive three-body interaction, one here considers the time-dependent non-linear Schr¨odinger equation corresponding to Eq. (46), including three-body recombination effects (with an intensity parameter 2ξ) and an imaginary linear term corresponding to the feeding of the condensate (with intensity

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parameter γ): ∂Φ i = ∂τ



 d2 x2 |Φ|2 |Φ|4 iγ − 2+ − 2 + (g3 − 2iξ) 4 + Φ, dx 4 x x 2

(52)

where Φ ≡ Φ(x, τ ) and τ ≡ ωt. For the parameters ξ and γ, one also uses the same notation as given in Ref. [22]. In Fig. 8, it is shown the time evolution of the number of

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Figure 8. Number of condensed atoms and the corresponding mean square radius hr2i (in units of ~/(2mω)) as a function of the dimensionless time τ = ωt, for ξ =0.001, γ =0.1. Nc is the maximum number of atoms for g3 = 0. condensed atoms, starting with N/Nc = 0.75, found by the numerical solution of Eq. (52) with ξ =0.001 and γ =0.1, with and without repulsive three-body potential. The results of a three-body potential with g3 = 0.016 are compared to the case considered in Ref. [22], with g3 =0. In both, Nc is the critical number for g3 =0. The first striking feature with repulsive three-body force is the smoothness of the compression mode in comparison with the results of g3 = 0. This is a result of the explosive force from the repulsion, which oppose to the sudden density increase and damps the loss of atoms due to three-body recombination effects. Even for g3 lower than 0.016, and much closer to g3 = 0, the collapses can no longer “burn” the same number of atoms as in the case of g3 = 0. By extending the calculation presented in Fig. 8 for all cases with g3 > 0.01 and for dimensionless times τ = ωt beyond τ = 50, one has checked that the number of atoms will increase without limit while the condensate is oscillating with frequency about 2ω. In particular, the present approach indicates that the experimental recent observation of the maximum number of 7 Li atoms is compatible with g3 much smaller than 0.01. The mean square radius for g3 = 0, after each strong collapse (when N > Nc ) begins to oscillate at an increased average radius. The collapse “burns” the atoms in the states with higher densities and explain the sudden increase of the square radius after each compression, remaining the atoms in dilute states. The inclusion of the repulsive three-body force, still maintains the oscillatory mode, but the compression is not as dramatic as in the former case, and consequently atoms in

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higher density states are not so efficiently burned. The increase of the mean square radius (averaged with time) is smaller than the one found with only attractive two-body force. This is a remarkable feature of the stabilizing effect of the repulsive three-body force allowing the presence of states with higher densities, as one found in the stationary study. Finally,

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Figure 9. Number of condensed atoms and mean square radius as function of τ = ωt, for γ = 0 and ξ = 0.001. The dimensionless chemical potential β of the initial state and the strengths g3 are given in the plot. one has to consider that, in the situation when the 3-body repulsion dominates over the 2body attraction, the condensate can be in a denser phase where it is expected to be strongly unstable due to recombination losses. The decay time of the condensate in a denser phase is expected to be much smaller than the decay time of the condensate in the less dense phase. However, one observes that the dynamics of the condensate is modulated by an oscillatory mode with a frequency of the order of 2ω, which was already identified by [22] to be ∼ ω even when g3 = 0. In case of g3 > 0, such oscillatory mode dominates the time evolution of the condensate. As the oscillations allow changes in the density, the condensate does not “burn” as fast as expected. In order to study the condensate decay, one considers the original NLSE with the dissipative term and allow different possibilities for the three-body interaction g3 . By using ξ = 0.001, one shows in Fig. 9 the result of this study for g3 = 0 and g3 = 0.016. The initial number of atoms N can be obtained from n, given in the figure. For g3 = 0, one takes n = 1.625, which is close to the critical limit. For g3 = 0.016, one considers three cases: two of them starting with the same number of atoms, n = 1.756, but in different phases (the corresponding chemical potentials are β = −1.2, in a denser phase, and β = 0.3); and another in an even denser phase, with n = 1.965 and β = −2.3. Based on the results obtained in these four different cases, one can verify that there is a relevant role of the oscillatory mode when g3 > 0, related to the frequency of the trap potential, which dominates the dynamics of the condensate. One estimates that the mean-life for the condensate, which

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is initially in a denser phase, is not as small as expected when comparing with g3 = 0. In summary, the variational solutions of Evar (α) are given as a function of n and g3 (where a < 0 and g3 > 0), by finding the extrema of Eq. (12) with respect to α, and are consistent with the numerical calculations. Besides, the present results can be relevant to determine a possible clear signature of the presence of a repulsive three-body interaction in Bose-Einstein condensed states. It points out to a new type of phase-transition between two Bose fluids. Because of the condensation of the atoms in a single wave-function, this transition may present very peculiar fluctuations and correlation properties. The characterization of the two-phases through their energies, chemical potentials, central densities and radius were also given for several values of the three-body parameter g3. One develops a scenario of collapse which includes both three-body recombination and three-body repulsive interaction. From the time-dependent analysis, one shows that the decay time of the condensate which begins in a denser phase is long enough to allow observation. However, the observed strongly oscillating states are quite different from the analysed stationary states. In accordance with the observed strong oscillations of the mean-squaredradius, the condensate density also strongly oscillates and the observed states cannot be characterized as “dense” or “dilute”, justifying the long decay time. Nevertheless, through the amplitude of the oscillations one can differentiate if the system starts in a denser phase.

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5.2.

Spatiotemporal Chaos in Bose-Einstein Condensates

One interesting investigation in the dynamics of grow and collapse of Bose-Einstein condensates (BEC) in a system of trapped ultra-cold atoms with negative scattering lengths is the regime of high values for the atomic feeding ratio. As earlier seen, the condensed state can be described by the solution of the time-dependent nonlinear Schr¨odinger equation (NLSE), which includes the atomic feeding and three-body dissipation. The solutions of NLSE show high complexity and a criterium proposed by Deissler and Kaneko [41] to diagnose spatiotemporal chaos has been applied to this equation. By changing the feeding parameter, one can prove that a chaotic behavior in trapped BEC systems can occur. The results here presented can be seen in detail in Ref. [42]. It has been shown by experiments [10] and numerically [12] that Bose-Einstein condensation (BEC) can occur in atomic traps with attractive two-body interactions, as in 7 Li. It was shown the occurence of a critical maximum number of atoms ( Nc ) in the ground state level; above such limit, the condensate collapses under two-body attraction. Numerical simulations of this process were considered in Ref. [22], by studying the time evolution of the condensed wave-function of atoms of 7 Li [43]. As noticed in [43], there is a qualitative similar behavior of the theoretical simulation in [22] and their experimental measurements, but there is also relevant quantitative difference between the predictions of the remaining number of atoms in the condensate. It is also reported in [43] that their observations could indicate a complex dynamics accompanying BEC in a gas with attractive interactions. This suggests to study the time evolution of BEC atoms for long periods, through the numerical solution of the corresponding time-dependent non-linear Schr¨odinger equation (NLSE), as given in [22], which includes two non-conservative (imaginary) terms: one, linear, related to the feeding of the condensate from the nonequilibrium thermal cloud; another, non-linear and dissipative, corresponding to three-body recombination.

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It is also well known that systems with complex dynamics can present chaotic behaviors for some appropriate range of parameters. In particular, one should note that the transition from a complex dynamics to chaos was previously considered in the time-dependent NLSE by other authors [41, 44]. Deissler and Kaneko, in Ref. [41], have proposed a useful criterium to diagnose spatiotemporal chaos in NLSE, which relies on the determination of the time evolution of a function defined by the integral of the square modulus of the difference between wave-functions with nearby initial conditions. The average slope of this function, when plotted as a function of time, gives the largest Lyapunov exponent. In this subsection of the present review work, one uses the criteria considered in Ref. [41] in order to verify numerically the onset of chaotic behavior of the solution of the time-dependent NLSE, which was considered in Ref. [22] for a trapped gas with attractive two-body interaction. In Ref. [38], the complex dynamics accompanying BEC of 7 Li atoms was observed in the time evolution of the number of atoms in the condensate. It was verified the high sensibility of the numerical accuracy with the change of parameters, such that when a repulsive three-body interaction was considered, the numerical results were more stable for the condensate [38]. Later on, one verifies that the numerical precision decreases very fast by increasing the modulus of the strength of an attractive three-body interaction. This preliminary result lead us to the suspicion of a possible chaotic behavior of the time-dependent NLSE with trapped atoms. One starts the dynamical study by considering the NLSE corresponding to the one given in Ref. [22] for the trapped atoms with attractive two-body interaction. Two nonconservative terms were added to take into account, respectively, the decrease of the density due to three-body recombination (parametrized by ξ), and the feeding of the condensate from the nonequilibrium thermal cloud (parametrized by γ), as in Eq. (52), but with g3 = 0. In dimensionless units, as given in Eq.(2) of [38], the s−wave radial NLSE can be writen as   1 2 |Φ|2 |Φ|4 γ d2 ∂Φ = − 2 + x − 2 − 2iξ 4 + i Φ, (53) i ∂τ dx 4 x x 2 p 2mω/~|~r|, τ ≡ ωt is the diwhere x is related to the physical radius ~r by x ≡ mensionless time variable, with ω the frequency of the harmonic trap interaction. The function Φ ≡ Φ(x, τ ) is related to the physical wave-function Ψ(~r, t) by Φ(x, τ ) ≡ p 8πN (t)|a||~r|Ψ(~r, t), where N (t) is the number of atoms and a is the two-body scattering length (here, assumed to be negative). Using these definitions, Ψ(~r, t) is normalized to one p and Φ(x, τ ) is normalized to the reduced number of atoms n(τ ) ≡ 2N (t)|a| 2mω/~: Z ∞ dx|Φ(x, τ )|2 = n(τ ). (54) 0

In order to obtain numerical solutions of Eq. (53), one applies the semi-implicit CrankNicolson algorithm (CN) as described, for instance, in Ref. [45] for nonlinear problems. By switching off the dissipation( ξ) and the feeding(γ) terms, initially one obtains a stationary stable solution by means of a time independent algorithm [28]. The CN algorithm was used in order to propagate in time the wave function for γ = 0 and ξ = 0, so that the maximum value for the dimensionless space radius x to obtain reasonable and accurate results was about xmax ' 7. For that, a grid with the space and time steps were, respectively, ∆x =

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0.01 and ∆τ = 0.01. By turning on the ξ and γ parameters, in order to obtain accurate and convergent results, one can clearly notice a remarkable increase in the maximum radius and also in the extra care needed for the discretization, particularly when evolving the equation for longer period. This sensibility could already be expected from the results shown in Ref. [22] for the corresponding wave-functions, because they oscillate up to very large distances. One also observes that the results are easily modified by the variation of the γ parameter, for instance. As also presented in Refs. [22, 38], for parameters as small as γ = 0.1 and ξ = 0.001, and for ωt < 50, one observe three strong collapses in the number of atoms when the critical number Nc is exceeded. The three-body recombination process burns approximately half of the number of particles in the course of the collapse, and are followed by dynamical oscillations, with a period that approximately depends on the frequency ω. In order to calibrate the code, and to reach accurate and convergent results up to ωt = 45, one starts by using the same parameters considered in [22]. The numerical discretization in this case corresponds very close to the one used in Ref. [22], but one should note that the radius variable differs from the one given in this reference by a constant factor. So, good accuracy of the results was obtained for ∆x = 0.004, xmax = 40, and ∆τ = 0.001, for ωt < 50. As one aims to access longer times, one observes that large numerical errors appear in this process, even after considering a refined grid. For each small increase in the time interval, for ωt > 60, the numerical effort to obtain accurate and convergent results appears to be not reasonable. In the following, the results are obtained when one considers the same parameters given in Ref. [22] and use the same initial condition they used for the number of atoms in the condensate (N (t)/Nc = n(τ )/nc = 0.75). In Fig. 10, one shows the evolution of the number of atoms for ωt ≤ 1000 (about 20 times larger than the maximum time considered in Fig. 10 of Ref. [22]). As already explained in Ref. [22], some dynamical collapses occur with frequency ∼ ω. Nevertheless a more careful analysis of such dynamical collapses show that the number of small peaks begins to double after each strong collapse, in a kind of fractal pattern. These results support a conjecture that the system, by doubling the peaks indefinitely, can excite a whole spectrum of frequencies. The reconstructed phase-space for the total energy E(τ ), shown in Fig. 11 for ωt < 44, illustrates the fractal pattern previously observed in case of the number of atoms, in Fig. 10. Such results indicate the possibility of a chaotic behavior for the system described by Eq. (53), with the chosen parameters. In order to further analyse the dynamical behavior of Eq. (53), one particular interesting observable isp the mean-square-radius. One defines this observable in dimensionless units d X(τ ) as a function of X(τ ), for a set of by X(τ ) ≡ hx2(τ )i. In Fig. 12, one plots dτ values of the parameter γ (γ =0.01, 0.02, 0.05, 0.1). In all cases, the wave-functions were evoluted up to ωt = 1000 and the strength of the three-body dissipative interaction is kept fixed at ξ = 0.001. In Fig. 12, one observes that a complex dynamical structure starts to appear as the value of the parameter γ increases. For γ = 0.01, the radius decreases from about 1.52 (for τ = 0) to a center near 1.34, then it starts to oscillate with larger radius, but keeping the center fixed. A similar behavior is found for γ < 0.01. For larger values of γ, the center of the oscillation in X grows up to the point it reaches an attractor at very large radius. In case of γ=0.1, the plot clearly resembles a strange attractor and indicates

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Figure 10. Number of condensed atoms as a function of the dimensionless time, for τ = ωt ≤ 1000. A doubling pattern is observed for ωt < 50 (see Fig. 8 for more detailed analysis). The parameters are γ = 0.1 and ξ = 0.001.

Figure 11. Reconstructed phase-space with delayed time ∆τ =0.2, for the total energy Etot, which is given in units of (N/n)~ω, with ωt < 44.

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p Figure 12. Phase-space plots for the mean-square-radius X(τ ) ≡ hx2 (τ )i, in dimensionless units, for a set of values of the feeding parameter γ. In all the cases, the wave-functions were evolved until ωt = 1000 and ξ = 0.001. the presence of a chaotic behavior. One should note that one kept fixed the dissipative term of Eq. (53); however, the indication of transition to chaos can also be found by changing ξ with γ fixed. In other words, by keeping fixed γ to a certain non-zero value, a similar behavior is reached as one decreases the value of the parameter ξ. The aim in the next is to determine the existence of spatiotemporal chaos in the time evolution of trapped atoms. In Ref. [41], the authors have studied the complex quintic Ginzburg-Landau equation and showed that, for an appropriate choice of the parameters the system could present a chaotic behavior. In order to characterize a chaotic behavior, for a spatiotemporal equation, it is necessary to show that the largest Lyapunov exponent related with the solutions of the equation is positive. Here one follows this criterion used by Deissler and Kaneko [41] to characterize spatiotemporal chaos, which prescribes that the largest Lyapunov exponent for the system, in an arbitrary time interval, is obtained by plotting the logarithm of a function ζ, which is defined by ζ(τ ) ≡

Z

∞

1/2 |δΦ(x, τ )| dx , 2

(55)

0

in which δΦ(x, τ ) will give us the separation between two nearby trajectories. It is obtained first by numerically evolving in time an initial Φ0 (x), obtaining Φ(x, τ ). Independently, one evolves Φ0 (x) + (x), and get Φ0 (x, τ ), where (x) is a very small random perturbation. So, δΦ(x, τ ) is given by Φ0(x, τ ) − Φ(x, τ ) and the chaotic behavior is characterized by a positive slope of ln ζ(τ ), which gives the largest Lyapunov exponent [41]. The average slope of this function plotted as a function of time, gives the largest Lyapunov exponent [41]. The chaotic behavior is characterized by a positive slope. The calculation of δΦ as described in [41] can also be related to the excitation of collective modes by an infinitesimal perturbation of the wave-function.

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Figure 13. Logarithmic plot of the separation between two nearby states, as given by Eq. (55), for several values of the parameter γ, shown in the figure. ξ is maintained fixed to 0.001.

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The similarity between complex quintic Ginzburg-Landau equation and Eq. (53) led us to consider the criterium used in Ref.[41], applying Eq. (55) to the wave-functions obtained from Eq. (53). One is mainly interested in studying the time evolution of the condensed wave-function for a negative two-atom scattering lenght, through the investigation of the numerical accuracy of the results for certain parameters used. In order to calculate Eq. (55), one use the same initial wave function Φ(x, τ ) as in Ref. [22], i.e., with initial number of atoms equal to 0.75 Nc. This initial wave function was also evoluted with an added small random perturbation δΦ(x, 0) ∼ 10−14 . The difference between the wave-function and the perturbed wave-function gives the separation of the trajectories δΦ(x, τ ), which is used to obtain the results for Eq. (55) shown in Fig. 13. As one can observe, there is an approximately exponential increase in ζ as the time grows for all cases represented with γ >0.01, such that one can draw a conclusion about the chaotic behavior of Eq. (53). This is better characterized for γ = 0.1. This confirms the suspicion raised when analysing the results obtained in Figs. 10-12, as one can clearly obtain from such results the values of the feeding parameter for the system to become chaotic. In conclusion, the equation used for the description of the dynamics of the Bose condensed wave-function in atomic traps with attractive interactions [22], for certain class of parameters (as, for example, the parameters considered in [22], ξ =0.001 and γ=0.1), is chaotic. Such a chaotic behavior starts to disappear as one decreases γ or increases ξ. This is an important result that should be taken into account in the interpretation of the experimental observations of BEC, as well as in the formalism that describes the Bose-Einstein condensed states, as could be interpreted as the increase of the magnitude of the collective excitations, or alternatively, as the creation of many quasiparticle states or as the population of excited states [42]. Experimentally, by controlling the feeding parameter, it would

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be possible to observe a sudden transition from a chaotic situation to a nonchaotic one, a hypothesis whose validity is based on the mean-field description.

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5.3.

Autosolitons in Bose-Einstein Condensates

In one early work [23], one considered the conditions for the existence of autosolitons in trapped Bose-Einstein condensates with attractive atomic interaction. The variational approach was employed to estimate the stationary solutions for a three-dimensional modified Gross-Pitaevskii equation and compared with exact numerical calculations. One showed that the variational approach gives very reliable analytical results. The exact solution for autosolitons, with arbitrary growth and damping strengths, in the perturbed nonlinear Schr¨odinger equation (NLSE) in one dimension, for homogeneous nonlinear medium with dissipation and amplification, was reported by Pereira and Stenflo [46]. Autosolitons were also discovered in nonlinear fiber optics, namely in fibers with amplifiers and distributed filters [47, 48] and also for waves on the surface of deep water [49]. Correspondingly, in a two-dimensional (2D) homogeneous medium with amplification and nonlinear damping, the possibility of existence of a 2D analog of the Pereira-Stenflo solitons was shown by a variational approach [50]. Autosolitons in a weakly dispersive nonlinear media, described by the Korteweg-deVries equation, were studied in Refs. [51, 52]. The autosolitons can be distinguished from ordinary solitons, that exist in conservative media and are originated from the balance between the nonlinear and dispersive effects of the wave propagation. The properties of ordinary solitons are defined by the initial conditions (their number, parameters like amplitudes, widths, etc.) [51], with the solutions characterized by their corresponding properties. As for to the autosolitons, they can be generated in nonconservative media when effects of amplification and dissipation are present. For the existence of autosolitons, one should add to the equilibrium condition between nonlinearity and dispersion the requirement of a balance between amplification, frequencydependent damping and nonlinear dissipation. One important difference for the autosolitons is that their properties are fixed by the coefficients of the perturbed NLSE and by any initial perturbation that is attracted to this point (attractor in the space of coefficients). The purpose of this subsection is to show that the analog of autosoliton is possible in a trapped Bose-Einstein condensate (BEC), as done previously in Ref. [23]. The existence of bright and dark solitons in BEC has been demonstrated in literature(see theory in [53, 54]), with dark solitons observed in BEC with repulsive interaction between atoms [55] and bright solitons existing for attractive interactions in 1D BEC. It is well known that twoand three-dimensional condensates with attractive interaction between atoms and trapping potential are unstable if the number of atoms exceeds the critical value ( Nc ). Below this value a stable ground state can exist, corresponding to solitary solutions [56, 57]. When the number of atoms exceeds Nc the collapse occurs. At large densities the inelastic scattering processes involving two and three atoms come into play, leading to the effective nonlinear damping of the condensate. The process of feeding atoms from the thermal cloud can be modeled as a linear amplification described in Ref. [22], where the relevance of the threebody inelastic processes was discussed. The statistical data obtained from experiments with 7 Li supports the growing and collapse picture [21, 43]. Numerical simulations of the 3D Gross-Pitaevskii (GP) equation are performed in

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Refs. [22, 23]. The present work shows that periodic oscillations occur in the condensate and, for particular values of atomic feeding and three-body dissipation parameters, stable states of the cloud can exist. Thus, one can expect the occurrence of analog autosolitons in 2D and 3D BECs. The problem is described by the complex Ginzburg-Landau equation with trapping potential in the NLSE limit with small nonconservative perturbations. This equation is nonintegrable and, therefore, analytical solutions can only be obtained by considering approximate methods like the variational approach [58]. In the following, one first use the time-dependent variational approach early described to obtain the solutions in some important cases, by analyzing the fixed points of the system (25). As showed previously, the time-dependent variational approach is quite effective to study the dynamics of 3D BEC in trapping potential with conservative perturbations [59]. These autosoliton solutions can be considered as the nonlinear modes of such system like solitons for the integrable NLSE [60]. Exact numerical simulations are also performed in the present work, so that one can compare with the variational approach, showing that it is a convenient and reliable approximation. One needs to solve the equation (16), so that to include some results with dissipation by dipolar relaxation as well. The autosoliton solution corresponds to the fixed points of the system. The dissipative solitons refer are defined by the balance between amplification and nonlinear dissipative terms. As showed in the analysis of the one dimensional case (the Pereira-Stenflo soliton), the solution is fixed by the parameters related to the amplification and dissipation with chirped phase. It does not depend on the initial conditions. This analysis has been restricted to λ3 = 0. From the system given by Eq. (25), one can obtain the fixed points. In the next, one distinguishes three cases:

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1. Case µ = 0, ξ 6= 0. In this case, ys1

s √ 3 3γ 1 = , bs1 = − γ, ξ 3

and the width is xs1 where

p1 =− + 2

r

λ2 ys1 , p1 = √ 4 2(ω 2 + b2s1 )

p21 + k1, 4 k1 =

ω2

(56)

(57) 1 . + b2s1

(58)

Here one considers that p1 >> k1 . Then the solution is √ k1 4 2ω 4 ≈ . xs1 ≈ p1 λ2ys1 2. Case ξ = 0, µ 6= 0. The fixed points are ys2

√ 2 2γ γ = , bs2 = − . µ 4

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and the width is defined by the same expression as before, given by Eq. (57), with ys2 , bs2. 3. Case µ 6= 0, ξ 6= 0. The fixed points are: ys3 bs3 xs3 where

s √ √ 3 3µ 27µ2 3 3γ = − √ + + , 32ξ 2 ξ 4 2ξ µ γ √ ys3 − , = 3 24 2 r p23 p3 = − + + k3, 2 2

λ2ys3 , p3 = √ 4 2(ω 2 + b2s3)

k3 =

ω2

1 . + b2s3

(60)

(61)

When µ = 0 (ξ 6= 0), one recovers Eqs.(56) and (57). Alternatively, when ξ → 0 (µ 6= 0), one recovers Eq.(59). One now investigates the stability of the fixed points, in cases 1 and 2, by using the linear stability analysis: 1. Case µ = 0, ξ 6= 0. The solutions of the system are x = xs1 +x1 , y = ys1 +y1 , b = bs1 +b1. The linearized system for corrections is 8 √ ξys1 xs1 y1 + 4xs1 b1 ≡ −c2 y1 + c3 b1, 9 3 = −8γy1 − 6ys1 b1 ≡ −d2y1 − d3b1 ,   1 4 λ2ys1 λ2 √ − = x1 − √ y1 − 4bs1b1 ≡ a1 x1 − a2 y1 − a3 b1. (62) xs x2s1 8 2 2 2xs1

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x1t = y1t b1t

With the solutions x1 , y1 and b1 having the same exponential behavior in time, given by ∼ eqt , one obtains the characteristic equation q 3 + α1 q 2 − α2 q − α3 = 0,

(63)

where 20 γ, 3 ≡ (a1c3 + a2 d3 − d2a3 )   16 λ2 ys1 xs1 3λ2ys1 32 2 √ γ + 2 −1 + √ , = 3 xs1 8 2 2xs1   64γ λ2ys1 xs1 √ −1 . ≡ a1(d3c2 + d2 c3) = 2 xs1 8 2

α1 ≡ (d2 + a3 ) = α2

α3

The roots with Re(q) > 0 correspond to the unstable solutions. Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

(64)

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Figure 14. Gaussian variational analysis of stability of the fixed points for the GrossPitaevskii equation including feeding ( γ) and three-body losses (ξ).

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Without loss of generality, in the dimensionless NLSE one can scale the parameters as λ2 = 1 and ω = 1 [22]. The diagram of stability, according to the solutions of Eq. (63), is presented in Fig. 14. The diagram clearly shows that, when γ > 1.84ξ, the system is unstable. If γ >> ξ, the system enters the collapsing process that has been shown in Ref. [42] to be chaotic. If γ is decreased (or ξ increased) the system will eventually achieve a stable region where the formation of autosoliton is possible.

2. Case ξ = 0, µ 6= 0. Analogously, as in case 1, one presents the solutions of the system as x = xs2 + x2 , y = ys2 + y2 , and b = bs2 + b2 . The linearized system for corrections is given by µxs2 √ y2 + 4xs2 b2, 2 2 √ 7 12 2γ = − γy2 − b2, 2 µ   4 λ2 λ2γ xs2 − 1 x2 − √ = y2 − 4bs2b2. 3 xs2 4µ 2 2xs2

x2t = y2t b2t

(65)

The system has the same form as in Eq. (62). Correspondingly, the characteristic equation is given by q 3 + β1 q 2 − β2q − β3 = 0,

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Figure 15. Gaussian variational analysis of stability of the fixed points for the GrossPitaevskii equation including feeding ( γ) and two-body losses(µ). where β1 =

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β2 = β3 =

5 γ, 2   7 2 16 λ2γxs2 6λ2γ γ + 2 −1 + , 2 4µ xs2µ xs2   32γ λ2γxs2 −1 . 4µ x2s2

(67)

With the same scaling used in case 1 (λ2 = 1 and ω = 1), and with the above solutions of Eq.(66), one obtains the diagram of stability, shown in Fig. 15. The diagram clearly shows that, when γ > 0.53µ, the system is unstable. In analogy with the previous case, if γ is decreased (or µ increased), the system will eventually achieve a stable region where the formation of autosoliton is possible. One did a series of time-dependent simulations of the system within the Gaussian variational approach, using Eq.(25), and also by performing exact numerical calculations with Eq.(16). In the numerical calculations, one has used the finite-difference Crank-Nicolson algorithm. The exact initial wave functions were used following the prescription given in [28]. In the next, one presents simulations in a range of parameters that lead to long-time stable autosolitons. One obtains results with autosolitonic solutions for a class of parameters that are near the realistic ones, as indicated by 7 Li experiments in BEC. In Fig. 16, for γ = 10−3 and ξ = 10−3, one shows the time evolution of the number of atoms, in terms of the maximum critical number for stability, Nc . The formation of the autosoliton is demonstrated either by Gaussian variational approach or by exact numerical calculations. There is a remarkable agreement between both approaches. Note that the number of atoms does not depend on the initial conditions, but is related to the equilibration

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Figure 16. Evolution of the number of atoms N in the Gross-Pitaevskii equation with feeding parameter γ = 10−3 and three-body dissipation parameter ξ = 10−3 (µ = 0). The results are represented in dashed lines for the variational approach, and in solid lines for the exact numerical calculations. Cases I and II correspond, respectively, to the initial conditions N (t = 0)/Nc =0.1 and 0.75, where Nc is the maximum critical number for stability and t is given in units of the inverse of the trap frequency ω. of the feeding and dissipation. The variational approach results give a little higher number of atoms than the exact calculations. Corresponding to the results in Fig. 16, one shows in Fig. 17 the results for the time evolution of the mean square radius, where Z 2 (68) hr i = r2 |u|2d3r. In this case, the variational approach results are a bit lower than the ones obtained by exact calculations. In analogy with the case that µ = 0, represented in Figs. 16 and 17, one also presents results for the case that the three-body dissipation parameter ( ξ) is zero. The results obtained for the time evolution of the number of atoms and the mean-square-radius are, respectively, shown in Figs. 16 and 17, for γ = 5×10−5 and µ = 10−4. The formation of the autosoliton is also demonstrated either by Gaussian variational approach or by exact numerical calculations. The remarkable agreement between both approaches, already observed in case that µ = 0 (Figs. 16 and 17), also apply to this case that ξ = 0, with the number of atoms not depending on the initial conditions. In 7 Li experiment, the feeding parameter can be indirectly inferred from measurements done by the Rice Group and will correspond to an average rate of about 600 atoms/s [23]. The two- and three-body losses were also measured [61] and estimated [62], giving atom loss rates of about 2 × 10−14cm3 /s and ∼ 10−28cm6 /s. These rates were measured for non-condensed atoms. For condensed atoms they must be divided by factors of 2! and

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Figure 17. Evolution of the mean square radius, Eq. (68), in the Gross Pitaevskii equation. The same parameters and conventions given in Fig. 16 were used. 3!, respectively [63]. With a scaled equation, such that λ2 = 1 and ω = 1, one have for condensed atoms γ ∼ 10−3 , µ ∼ 10−4 and ξ ∼ 10−6 . Considering this magnitudes, the autosoliton can possibly be observed experimentally in 7 Li, either by results with decreasing γ due to losses, or by increasing the dissipation rate due to other mechanism as the Feshbach resonances [64]. In case of diminishing γ, the autosoliton formation is more likely determined by dipolar relaxation rather than three-body recombination losses. So, in this subsection was reviewed the possibility of existence of autosolitons in trapped 3D BEC in the presence of two- and three-body inelastic processes, that is, dipolar relaxation and three-body recombination. Using the time-dependent variational approach for nonconservative 3D Gross-Pitaevskii equation, one derived expressions for the parameters of autosoliton and checked their stability. The results obtained by using the present time-dependent Gaussian variational approach, in the NLSE with atomic feeding and nonlinear dissipative terms, showed a remarkable agreement with exact numerical calculations, when the parameters were such that stabilization was achieved. One has shown that the transition from unstable (collapsing) to stable point (autosoliton) solely depends on the magnitude of the parameters. This analysis includes non negligible two-body dissipative effects that model the dipolar relaxation losses, and that can be associated with values measured in ultracold 7 Li [61]. In case of decreasing γ due to collapsing and losses, the autosoliton is more likely to be formed first due to dipolar relaxation rather than by three-body recombination processes. These results can be relevant in experiments with attractive scattering length and possibly display a new phenomenon: Pereira-Stenflo soliton formation in BoseEinstein condensates. One believes that such a phenomenon is already occurring in the long time behavior in the actual experiments with 7 Li [43] (ω ∼ 2π× 140Hz), since for longer times (∼60s) the maximum number of atoms ( Nc ∼ 1300 atoms) is considerably reduced, as expected in the simulations. One hopes that experiments with direct observation of the evolution of the condensate can clarify this matter.

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Figure 18. Evolution of the number of atoms in the Gross-Pitaevskii equation with feeding γ = 5×10−5 and two-body dissipation parameter µ = 10−4 (ξ = 0). The initial conditions and conventions are the same as in Fig. 16.

Figure 19. Evolution of the mean square radius in the Gross Pitaevskii equation. The parameters and conventions are the same as in Fig. 18.

5.4.

Coherent Dimer Formation Near Feshbach Resonances in Bose-Einstein Condensates

In this subsection, the results of an experiment with 85Rb Bose-Einstein condensates are analyzed within the mean-field approximation including dissipation due to three-body recombination. The intensity of the dissipative term is chosen from the three-body theory for large positive scattering lengths. The remaining number of condensed atoms in the ex-

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periment, obtained with applied magnetic field pulses, were used to adjust the intensity of the dissipative term. One found that the three-body recombination parameter depends on a characteristic time of the experiment when the field is changed, known as the pulse rise time. For longer rise times, the values found become consistent with the three-body theory, while for shorter pulses this coefficient is found to be much larger. One interprets this finding as an indication of a coherent formation of dimers, as explained for the first time in Ref. [65]. In Ref. [66], it was theoretically predicted that Feshbach resonances could vary the scattering length of atoms in systems of dilute alkali gases over a wide range of values. A Feshbach resonance occurs when the quasimolecular bound state energy is tuned to the energy of two colliding atoms, applying an external magnetic field to the system. The phenomenon was first realized in a Bose-Einstein condensate in Ref. [67]. This opened the possibility of exploring new regimes of BEC, allowing changes of the two-body scattering length from negative to positive and also from zero to infinity. This technique was used for the condensation and collapse control of 85 Rb atoms in the hyperfine state (F = 2, mF = -2) [64, 68, 69]. It was also demonstrated in Ref. [70] that strongly enhanced inelastic three-body collisions occurs near Feshbach resonances. In a more recent experiment, in Ref. [71], it was explored the region of very large scattering lengths (up to ∼ 4000 Bohr radius). The scattering length a has been observed to vary as a function of the magnetic field B, according to the theoretical prediction [72], as   ∆ , (69) a = ab × 1 − B − Br where a is the scattering length, ab is the background scattering length, Br is the resonance magnetic field of 85 Rb, and ∆ is the resonance width. In the case of 85Rb, one has resonance width ∆ ∼ = 11.0 G, resonance field Br ∼ = 154.9 ∼ G and background scattering length ab = −450a0 [71], where a0 is the Bohr radius. So, given the experimental functional dependence B = B(t), one can determine a = a(t) and, consequently, the dynamics of the system in such physical conditions. Further, mainly in strong interaction regime, a Bose-Einstein condensate shows inelastic loss processes that cause its depletion. The dominant process of losses has been verified to be the three-body recombination [20, 71], with a time dependence concerning a simple constant rate equation. All earlier observations in BEC experiments were consistent with a description of mean field including such a loss process. But, in experiments with 85Rb realized in the strong interaction regime [71], this picture indicates model breakdown. Bose-Einstein condensates initially stable were submitted to magnetic field pulses carefully controled in the vicinity of 85 Rb Feshbach resonance, aiming to test the strongly interacting regime for diluteness parameter χ = na3 varying from χ = 0.01 to χ = 0.5. The loss of atoms from BEC occurred in impressively short time scales (up to two hundreds of µs) and disagrees with previous theoretical predictions [64]. Such experiments reveal higher loss of atoms in shorter magnetic field pulses applied on BEC and, previously, one knew that as longer is the time spent near a Feshbach resonance as higher is the loss of atoms from BEC [64] (consistent with a mean field approach where the inelastic loss term has a constant dissipative rate). According to Ref. [71], the results could indicate the existence of a new physics, that cannot

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be described by the Gross-Pitaevskii (GP) formalism. Motivated by this observed discrepancy, one investigates the dynamics of 85Rb Bose-Einstein condensates submitted to such external conditions and time scales, when one varies the s-wave two-body scattering length in the region of strong interaction. So, one considers a generalized mean field approach that includes the time dependence for both mean field coupling and three-body recombination parameter. Previous results considered a mean field approach with constant mean rate for the three-body recombination and constant value for the mean field coupling [22, 73]. The first task was to reproduce the experimental data with time dependent parameters or at least verify possible limitations of the time dependent mean field approach. So, one considered the magnitude of the recombination rate as described in literature [74, 75, 76, 77, 78, 79, 80], but with time functional dependence. For describing a BEC in the framework of the mean field approximation, one reminds that such an approach is valid for very dilute systems when the average inter particle distances d are much larger than |a| and the particle wavelengths are much larger than d [1, 4]. Besides, it is important to pay attention to the time scales present in the system: the physical conditions must not change fast enough in order to allow the replacement of a true interatomic potential by the contact interaction. It is possible in principle for the rate of change to be larger than ~/ma2 for extremely very short changes in the interactions [6], but it is reasonable to assume valid this time dependent approach at least for longer pulses. One begins the description from an effective Lagrangian of the nonconservative system as in Ref. [23], in which one describes the dynamics of a trapped Bose-Einstein condensate in spherical symmetry with such a GPE generalization, in which one also considered losses from BEC by three-body recombination. This Lagrangian leads to the following equation of the system:   ~2 2 mω 2 r2 K3 4 ∂Ψ 2 = − ∇ + + U0|Ψ| − i~ |Ψ| Ψ, i~ ∂t 2m 2 4

(70)

where ω is the geometric mean trapping frequency, U0 ≡ U0 (t) = 4π~2a(t)/m and K3 ≡ K3(t) is the recombination loss parameter. The wave-function Ψ = Ψ(~r, t) is normalized to the number of atoms N . The three-body recombination rate K3 is here introduced for describing atomic losses from the condensate when three atoms scatter to form a molecular bound state (dimer) and a third atom; so, the kinetic energy of the final state particles allows them to escape from the trap. Other nonconservative processes as amplification from thermal cloud and dipolar relaxation are neglected, since the latter has a much smaller effect than three-body recombination [20, 71] and in JILA experiments [71] the thermal cloud is negligible (only 1000 atoms in a sample of ≈ 2x104). A theoretical prediction of K3 is a hard task since it is sensitive to the detailed behavior of the interaction potential [80]. However, such a calculation becomes simpler if one considers that a is the only important length scale (reasonable in the weakly bound s-wave state limit) and this has been considered in many works [74, 75, 76, 77, 78, 79, 80]. Following Ref. [74], the recombination rate is written as ~ K3(t) ∼ = κ [a(t)]4 , m

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(71)

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Figure 20. Triangular (insertion) and trapezoidal shapes of magnetic field pulse (Gauss) applied to 85Rb BEC as function of pulse time (µs). The rise time tr of the pulse amounts to 12.5 µs and the hold field is 156.7 G (scattering length a ∼ = 2300a0). The hold time of trapezoidal pulse is 120 µs.

where κ should correspond to the universal value κ = 3.9. But in Ref. [75] and in Ref. [76], it was found 0 ≤ κ ≤ 65 and in Ref. [77], 0 ≤ κ ≤ 67.9, that is, κ is not universal (in general K3 depends on a three-atom scale [78]). In Fig. 20, one schematically gives two characteristic pulses employed in JILA, in which B0 , Bh , Bm , tr and th , correspond to the initial field, the hold field, the final field, the rise time and the hold time of the pulse, respectively. For describing the dynamics of condensates subjected to (69), one considers in the calculations the same experimental parameters and conditions used in [71]: as a is known to be a function of the magnetic field B by means of (69), one only uses time dependent shapes of experimental magnetic field pulses employed in JILA (triangular and trapezoidal pulses). The hold field is Bh = 156.7 G, and the end field in which one measures the remaining number of particles Nr of the system is Bm = 164.5 G at t = 700 µs. Further, one puts initial field B0 ∼ = 166 G, corresponding to a harmonic oscillator state of the system ( a ∼ 0), applied to an initial sample of N0 =16500 condensed atoms of 85 Rb. Further, in the approach one has used spherical symmetry with mean geometric frequency ω = (ωr2ωz )1/3, for simulating the cylindrical geometry of JILA (radial: ωr = 2π × 17.5 Hz and axial: ωz = 2π × 6.8 Hz). The time dependent calculations started with a Gaussian shape wave-function which one numerically evolves by means of Eq. (70), using the Crank-Nicolson algorithm, as in Refs. [42, 81]. One analyzed the loss of condensed atoms like in [71], by considering hold times from thold = 0 (triangular) or units of µs (shorter trapezoidal pulses) to nearly hundreds of µs (longer trapezoidal pulses). The behavior of the scattering length, as a function of total time of the pulse, in the region of strong interaction atom-atom, follows similarly the behavior of the employed field pulses given in Fig. 20. In triangular pulses, there is a sharp peak in the resonance region; and a plateau with maximum value for a and ξ (minimum value of the field), when B = Bh is kept constant during the time th . In Fig. 21, one used symmetric rise and fall times ( tr = 12.5µs, tr ∼ = 25µs and

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Figure 21. Remaining fraction of atoms in 85Rb condensate versus hold time for some rise times (indicated inside frames). For this behavior, κ ∼ = 100 = 1800 and it decreases up to κ ∼ for higher rise times, up to tr = 252.6µs (not shown). The initial number of particles is 16500. Nr is calculated as if it was measured at τ = 700 µs (Bm =164.5 G).

tr ∼ = 75µs), to determine the remaining fraction of atoms Nr /N0 in the 85 Rb BEC as function of the hold time, by adjusting the curve with the first point of Fig. 21 in Ref. [71]. For tr = 12.5µs, one found κ ∼ = 1800, very far away of the values described earlier in the literature [74, 75, 76, 77, 78, 79, 80]. The results show the same exponential decay and good concordance with experimental data, mainly for short th (circles in Fig. 21). However, as one can realize, for longer hold times, experimental data point out a higher dissipation when compared with the simulations. For other short rise times in this frame or longer rise times (not shown), one has similar behavior, but one has to decrease κ for better adjusting to experimental data. So, the comparison with experiments show that κ depends significantly on values of tr and th . As one knows [82], the mean field approach should make more adequately if one was in a slower process. So, one also tried to verify if the results would give a lower value of κ (inside interval described in literature) if one calibrated the calculation with the last point of the longest pulse of JILA [71] ( tr ∼ = 252.6 µs). Really, one found κ∼ = 100 for this case (closer to values described in literature [74, 75, 76, 77, 78, 79, 80]) and the results reproduce the experimental data for longer rise time but they do not make very well for shorter rise time. The very large value of κ induces us to conclude that there is a coherent formation of dimers, that occurs up to nearly tr ∼ 100µs. It is physically sensible that, for shorter pulses, the presence of coherence in the formation of dimers would be more plausible than for longer pulses. So, the present calculation included the variation of κ with the time parameters of the pulse and the amplification of the three-body recombination rate can be associated with the coherent formation of dimers in the inelastic collisions. Together with the significant loss from the coherent formation of dimers, one should also observe a burst of atoms carrying the excess of energy, which for 85 Rb with the maximum value of a = 4000a0, would be above 70nK. Indeed, in the JILA experiment, it was seen a significant number loss from the condensate for pulses lasting only few tens of microseconds, which were accompanied by a burst of few thousand relatively hot ( ∼ 150nK) atoms

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Table 1. Numerical values of three-body recombination coefficient κ as function of the rise time tr of the magnetic field pulses applied to the 85Rb BEC. At left, one has shorter pulses and longer pulses are shown at right. tr (µs) 12.5 25.3 75.8

κ 1800 1700 1600

tr (µs) 151.6 202.1 252.6

κ 500 200 100

that remained in the trap [71]. In the description, for each hot atom one dimer is also formed. Therefore, the burst of atoms should be accompanied by a burst of weakly bound dimers. Using the observed temperature and momentum conservation of the recombination process, one predicts that the dimers are also relatively hot ( ∼ 75nK). One notes that, if one considers a constant value of κ, in all cases (for any choices of th ) one obtains a decreasing behavior of Nr , as one increases the rise time tr . This is in contrast with the experimental results. So, one considers to adjust the values of κ that approximately better describe the experimental data for each given rise time tr ; i.e., κ is taken as a function of the rise time (κ = κ(tr )). The results are shown in Table 1. There is an obvious uncertainty in the given numbers of Table 1, that are related to the approximate theoretical fitting and experimental data fluctuations. However, based on such results, the behavior of κ(tr ) can be approximately described by a exponential decreasing function. Here one considers the following simple functional time-rise dependence of κ:

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κ(tr ) = 2300 exp (−0.01 × ωtr ).

(72)

With such decreasing functional time-rise dependence of κ, one has verified, as shown in Fig. 22, that one has the qualitative behavior observed in the experimental results given in Ref. [71] of the remaining number of atoms Nr versus tr . One has considered several values of the hold time th . For very small hold time, one can also verify the same experimental results that presents a minimum of Nr as a function of tr . With a linear functional tr dependence of κ, the same qualitative behavior can also be reproduced; which differs quantitatively from the exponential behavior that one shows. So, one concludes that the time dependent mean field approach can describe all the experimental data, if the threebody recombination coefficient depends on the rise time in this short time scale. Such a dependence on the rise time can be explored, once the uncertainties in the experimental results are reduced and by considering a better fitting of data (improving the values given in Table II). In the interpretation, the higher values of κ for smaller values of tr are indicating the coherent formation of another species (dimers) in the condensate. In summary, one reports in this chapter indications based on the calculations that a experiment realized in JILA [71] is evidencing the coherent formation of dimers from inelastic collisions in 85 Rb Bose-Einstein condensates (BEC). One has solved numerically the nonconservative Gross-Pitaevskii equation in spherical symmetry, for condensed systems with very large repulsive two-body interaction, varying in time, due to application of magnetic field pulses, according to Eq. (69). According to the theoretical predictions of three-body

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Figure 22. Remaining number of atoms in 85Rb BEC versus the scaled rise time (factor 1/4) of the applied magnetic pulse, for some hold times, with hold field 156.7 G (2300 a0) and initial number N0=16500. One considers κ given by the exponential dependence of Eq. (72). recombination rates, one used a dissipation parameter proportional to the quartic power of the scattering length a(t) and the observed experimental pulse shapes to calculate the time evolution of the remaining number of atoms in the condensate. One studied this observable as a function of the hold and rise times. The experimental results of Nr versus th can be described in the mean-field approach only with very large κ, when the rise time is small. This indicates a coherent formation of dimers in BEC. For longer pulses, when the coherent dimer formation tends to disappear, one found that κ approaches the maximum value given by the theoretical predictions for large scattering lengths, κ ∼ 70 [75, 77]. One parameterize this surprising behavior of the three-body recombination rate considering κ = κ(tr ) and so it was possible to describe the property of a lower dissipation for longer pulses. Finally, it is natural to see a burst of relatively hot atoms and dimers (carrying the excess of dimer binding energy) accompanying a significant loss from the condensate for short pulses when the coherent dimer formation occurs.

6.

Stability of Nonconservative Bose-Einstein Condensates

In this section, one analyzes the stability of nonconservative Bose-Einstein condensates, by considering the early cited inelastic collisions and external atomic feeding. The dynamics of the nonconservative Gross-Pitaevskii equation for trapped atomic systems with attractive two-body interaction is numerically investigated, considering wide variations of the nonconservative parameters, related to atomic feeding and dissipation. One studies the possible limitations of the mean field description for an atomic condensate with attractive two-body interaction, by defining the parameter regions where stable or unstable formation can be found. The present study is useful and timely considering the possibility of large variations of attractive two-body scattering lengths, which may be feasible in recent experiments.

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A condensed state of atoms with negative s−wave atom-atom scattering length (as in case of 7 Li [10]) is unstable, unless the number of atoms N is small enough such that the stabilizing force provided by the zero-point motion and the harmonic trap overcomes the attractive interaction, as found on theoretical grounds [12, 13]. Particularly, in the case of 7 Li gas [10], it was experimentally observed that 650 < Nc < 1300, a result consistent with the mean-field prediction [12], where the term proportional to the two-body scattering length (negative) dominates the nonlinear part of the interaction. The maximum critical number of atoms for Bose-Einstein condensates with two-body attractive interactions have been deeply investigated by the JILA group, considering experiments with 85Rb [69]. They have considered a wide tunning of the scattering length, a, from negative to positive, by means of Feshbach resonance [83, 84], and observed that the system collapses for a number of atoms smaller than the theoretically predicted number. Their experimental results, when compared with theoretical predictions for spherical traps, show a deviation of up to 20% in the critical number. Such a deviation can also be an indication of higher order nonlinear effects that one should take into account into the meanfield description. In Ref. [38], it was considered the possibility of a real and positive quintic term, due to three-body effects, in the Gross-Pitaevskii formalism. A negative quintic term would favor the collapse of the system for a smaller critical number of atoms, as verified in the JILA’s experiments. However, the real significance of a quintic term in the formalism is still an open question. The main motivation in the present work is to analyze the dynamics represented by an extension of the mean-field or Gross-Pitaevskii approximation, with non-conservative imaginary terms that are added to the real part of the effective interaction, the two-body nonlinear term with a spherically symmetric harmonic trap. For the imaginary part, the interaction is a combination of a linear term, related to atomic feeding, and a quintic term, due to three-body recombination, that is responsible for the atomic dissipation. This is an approximation that is commonly used to study the properties of Bose-Einstein condensed systems. One considers a wide variation of the nonconservative parameters, in particular motivated by the actual realistic scenario, that already exists, of altering experimentally the two-body scattering length [84]. As it will be clear in the following, this possibility will lead effectively to a modification of the dissipation parameter. By changing the absolute value of the scattering length, from zero to very large absolute values, one can change in an essential way the behavior of the mean-field description. As it will be shown from the present numerical approach, the results for the dynamical observables of the system can be very stable (solitonic-type) or very unstable (chaotic-type); the characteristic of the results will depend essentially on the ratio between the nonconservative parameters related to the atomic feeding and dissipation. In the case of positive scattering length, one has a very good agreement of the mean-field calculations with experimental data, as the thermal cloud is practically absent (removed by cooling evaporation) and almost all the particles are in the condensed state. In this chapter, one has mainly concentrated the study in the interesting dynamics that occurs when the scattering length is negative ( a = as = −|as |). In this case, it is well known that the system is unstable without the harmonic trap; and the trapped system has a critical limit Nc in the number of condensed atoms. The extended mean-field approximation has also shown to be reliable in determining the critical number of particles and even

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collapse cycles in the condensate [10, 12, 22]. Actually, systems with attractive two-body interaction are being intensively investigated experimentally [69], by using the so-called Feshbach resonance [83, 84]. The scattering length can be tuned over a large range by adjusting an external magnetic field (for more details, see Ref. [5]). Here, one is interested in the dynamics of a realistic system, where one adds two non-conservative terms: one (linear) related to the atomic feeding from the non-equilibrium thermal cloud; and another, dissipative due to three-body recombination processes (quintic). It is true that other dissipative terms can also be relevant for an arbitrary trapped atomic system, as a cubic one, that can be related with dipolar relaxation or with an imaginary part of the two-body scattering length. However, in order to simplify the study and better analyze the results, one restricts the considerations to the case that one has just one parameter related with the feeding and another related with dissipation. One has considered only the three-body recombination parameter for dissipation also motivated by the observation that, for higher densities, this term dominates the two-body loss [73]. So, for the generalization of Eq. (8), one adds the imaginary terms in the interaction, such that

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i~

∂Ψ ~2 ~ 2 1 4π~2as =− ∇ Ψ + mω 2 r2Ψ + | Ψ |2 Ψ + iGγ Ψ − iGξ | Ψ |4 Ψ , ∂t 2m 2 m

(73)

where Gξ is the dissipation parameter, due to three-body collisions, and Gγ is as parameter related to the feeding of the condensate from the thermal cloud. The Eq. (73) was first suggested in Ref. [22] to simulate the condensation of 7 Li. In order to recognize easily the physical scales in Eq. (73), it is convenient to work with dimensionless units. By making the transformations earlier described: r ~ τ ~x, t≡ , ~r ≡ 2mω ω   γ 4π|as|~ 2 Gξ ≡ 2ξ ~ω and Gγ ≡ ~ω, 2 mω p Φ ≡ Φ(x, τ ) ≡ 8π|as ||~r|Ψ(~r, t), (74) one obtains the radial dimensionless s−wave equation:   ∂2 x2 |Φ|2 |Φ|4 γ ∂Φ = − 2+ − 2 − 2iξ 4 + i Φ . i ∂τ ∂x 4 x x 2

(75)

As Ψ(~r, t) is normalized to the number of atoms N (t) in Eq. (73), the corresponding timedependent normalization of Φ(x, τ ) is given by the reduced number n(τ ): r Z ∞ 2mω 2 dx|Φ(x, τ )| = n(τ ) ≡ 2N (t)|a| . (76) ~ 0 The nonconservative GPE (75) is valid in the mean-field approximation of the quantum many-body problem of a dilute gas, when the average inter-particle distances are much larger than the absolute value of the scattering length; and also when the wave-lengths are much larger than the average inter-particle distance. The nonconservative terms are important when the condensate oscillates, fed by the thermal cloud, while losing atoms due to three-body inelastic collisions, which happen mainly in the high density regions.

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In order to verify the stability and the time evolution of the condensate, as observed in Refs. [42, 81, 85, 86], two possible relevant observables are the number of particles normalized by the critical number of atoms of the static case ( N (t)/Nc) and the mean square radius (msr),   Z ∞ ~ 1 hr2(t)i = dx x2 |Φ(x, τ )|2 , (77) 2mω n(τ ) 0 that can written as:

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hr2(t)i ≡



~ 2mω



h x2 (τ ) i,

(78)

p h x2(τ ) i as the dimensionless root mean square radius and so one can define X ≡ (rmsr). In the analysis of stability, one calculates the time evolution of these quantities. One explores several combinations of the dimensionless nonconservative parameters ξ and γ. One first considers the case in which the atomic feeding is absent or when its parameter is smaller than the atomic dissipation parameter. Next, one explores variations of both parameters of about five orders of magnitude, from 10−5 to 10−1. This wide spectrum includes the parameters considered by Kagan et al. [22], as well as other combinations that can be considered more realistic due to experimental results [61]. Actually, the relevance of a wider relative variation of the nonconservative parameters γ and ξ, presented in Eq. (40), can be better appreciated in face of the experimental possibilities that exist to alter the two-body scattering length [84]. As one should note from Eq. (74), any variation of the scattering length will also affect the effective dissipation parameter ξ and, consequently, its relation with the feeding parameter γ. This implies that, by changing the value of the scattering length, from positive to negative, and from zero to very large absolute values, one can change in an essential way the behavior of the mean-field description. In the present work, one is concerned with negative two-body scattering length, where the collapsing behavior of the Eq. (40) shows a very interesting dynamical structure. Even considering the possible limitations on the validity of the mean-field approach after the first collapse (in cases of parametrization where it can occur), it is worthwhile to verify experimentally the behavior of a system in such a situation, by varying |a|. At least, one can verify how far the theoretical description can be qualitatively acceptable. As already verified for systems with attractive interaction, as the 7Li, it has been possible, via the mean-field approach, to describe properties like the critical number of atoms in the condensate and growth and collapse cycles [10, 22, 12]; besides, in the long time evolution, for certain sets of parameters, the calculations have also shown the presence of strong instabilities of the condensate, with signals of spatiotemporal chaotic behavior. In the next, one presents the most significant results that characterize the time evolution of the normalized number of particles ( N (t)/Nc ), the dimensionless mean square radius hx2i, and, in order to characterize the stability of the system, the function related to the largest Lyapunov exponent. Further, one presents a representative case of the phase-space for the root mean square radius. One has studied a wide region of parameters γ and ξ, covering about five orders of magnitude, from 10−5 to 10−1 , including the case with no feeding (γ = 0).

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In order to have a clear and useful map of the regions where one should expect stable results, as well as regions with instabilities or chaos, one summarizes the present numerical results in a diagrammatic picture that relates these two nonconservative parameters. In general, it is expected that the system is more stable when the parameter related to the feeding of atoms (γ) from the thermal cloud is significantly smaller than the parameter related to the dissipation (ξ). However, it is interesting to find out the region of parameters where this transition (from stable to unstable results) occurs. Analysis of experimental results can provide a test to the present mean-field description in case of negative twobody interaction. As previously observed, one is considering dimensionless observables and parameters. For any realistic comparison with experimental parameters, one should convert γ and ξ to the parameters Gγ and Gξ , as given in Eq. (74). The numerical solutions of Eq. (40) were obtained by applying the semi-implicit CrankNicolson algorithm for nonlinear problems, as implemented in Refs. [23, 42, 81, 85, 86]. This method is stable and, therefore, very convenient and reliable to treat time-dependent nonlinear partial differential equations. The initial condition for the number of atoms N in the condensate was such that N (0)/Nc = n(0)/nc = 0.75. The evolution of the observables have been extended up to τ = ωt = 500. In general, as expected, the smaller is the dissipation parameter, the longer is the life of the condensate. The mean square radius presents an oscillatory behavior while one increases ξ. One observes that, in the regime of small feeding ( γ ≤ 10−4 ), the extended Lyapunov presents no positive slope. For larger values of γ, from ∼ 10−3 and 10−2 , one has studied a few cases where the interplay between the nonconservative behaviors are significant. In Fig. 23, one shows the dynamical behavior of the number of atoms for γ = 10−2 and several values of ξ; and, in the Fig. 24, the corresponding time evolution of hx(τ )2i. One realizes an interesting behavior, that occurs when the dissipation is larger than the feeding process: there are solutions of stability or dynamical equilibrium between both nonconservative processes. This phenomenon was already discussed in Ref. [23], for a few values of the dissipation and feeding parameters, using the time dependent variational approach and also the Crank-Nicolson method. In the present work, one observes a wide region of parameters where it is possible the formation of autosolitons [23]. However, when the feeding process is much larger than the dissipation, of about one or more orders of magnitude, one can also observe chaotic behaviors. See, for example, the case with ξ = 10−3. The time evolution of the number of particles, represented in Fig. 23, shows a collapse for τ ≈30, followed by several other collapses, with the number of particles going above the critical limit Nc . So, after a sequence of collapses, the critical limit for the number of particles is no more followed, as already shown in Ref. [42, 81, 85, 86]. The corresponding time evolution of hx(τ )2i is shown in Fig. 24. One observes that, following each collapse, after the shrinking of the system, the radius is multiplied by a large factor, with indication of being populated by radial excited states. One can also observe the corresponding transition from the stable region (where the system finds the equilibrium at a fixed value of the radius, corresponding to autosoliton formation) to the unstable region in some cases. As shown, the instability starts to occur when ξ = 2 × 10−3, and it can turn into a spatiotemporal chaos.

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Figure 23. Time evolution of the number of condensed atoms N , relative to the critical number Nc , for a set of values of the dissipative parameter ξ shown inside the frame, with feeding parameter γ = 10−2 . All the quantities are in dimensionless units, as given in Eqs. (74).

Figure 24. Time evolution of the dimensionless mean-square radius, hx(τ )2i, for the feeding parameter γ = 10−2 . The results are given for a set of values of the dissipative parameter ξ, shown inside. All the quantities are in dimensionless units, as given in Eqs. (74). In Fig. 25, one illustrates the application of the Deissler and Kaneko criterion to the system given by Eq. (75), for a fixed value of the feeding parameter γ = 0.01, and a set of values of the dissipation parameter ξ. It was plotted the time evolution of the function ln(ζ), where ζ is given by Eq. (55), following the prescription given in Ref. [41] to obtain the largest Lyapunov exponents for the system. Within this prescription, the system becomes chaotic when ln(ζ) has a positive slope. As shown in Fig. 25, this clearly occurs, for example, when ξ = 10−4. In case of ξ = 10−5 one notes a much faster increasing in ln(ζ), with an observed saturation that happens due to the fact that such function has reached the maximum separation between the trajectories. The saturation properties is also verified when studying chaotic behaviors in ordinary differential equations [87]. The plot

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Figure 25. Time evolution of lnζ, related to the separation between two nearby trajectories [See Eq. (55)], for γ = 10−2 and a set of values of ξ indicated inside the figure. All the quantities are in dimensionless units, as given in Eqs. (74). of ln(ζ) corresponds to the same value of γ (= 10−2) used in Figs. 23 and 24. As shown, a clear characterization of chaotic behaviors starts to occur only for values of the dissipation parameter ξ much smaller than γ. In the cases presented in Fig. 25, for ξ ≤ 10−3 . In Fig. 26, one presents another significative illustration of chaotic behavior, through the phase-space behavior of the mean-square-radius, considering one case that was characterized as chaotic by using the Deissler-Kaneko criterion. One has plotted in this figure the mean-square-radius phase space for the case with γ = 0.15 and ξ = 10−3 . The irregular behavior of the trajectories, observed in Fig. 26, with the classical strange attractors being observed, clearly resembles chaos. This behavior is similar to the chaotic behavior observed in ordinary cases [87]. As a general remark that one can make from the presented results, one should note that, in order to observe unstable chaotic behaviors, the dissipation must be much smaller than the feeding parameter. In a diagrammatic picture, given in Fig. 27, one resumes the results. One shows the relation between the two nonconservative parameters, ξ and γ, in order to characterize the parametric regions where one should expect stability or instability in the solutions for the Eq. (75). The stable results of the Eq. (75) are represented by bullets; the nonstable results that clearly present positive slope for ln ζ(τ ) (chaotic behavior) are represented by empty squares; with ×, one shows other intermediate nonstable results, in which the characterization of chaotic behavior was not so clear, through the Deissler-Kaneko criterion. In this figure, in order to observe the approximate consistency of the numerical results, one also includes the variational analysis presented in Fig. 1 of Ref. [23], represented by the dashedline. It is separating the stable region (upper part) from the unstable one (lower part). One should note that, in section V of Ref. [88], it was also considered the dynamics of growth and collapse, with non-conservative terms related to feeding ( γ0) and dissipation (γ1

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Figure 26. Phase-space for the root mean-square-radius, in dimensionless units [ dX(τ )/dt versus X(τ )], considering a collapsing case that leads to chaos. The dimensionless nonconservative parameters are ξ = 10−3 and γ = 0.15, and the time evolution was taken up to t = τ /ω = 500/ω.

Figure 27. Diagram for stability, according to the criterion of Ref. [41] given by Eq. (55), with results for the equation (75), considering the dimensionless non-conservative parameters γ and ξ. Between the unstable results, represented with × and squares, the chaotic ones are identified with squares. The stable results are represented by bullets. Two dotted guidelines are splitting the regions. The dashed line split the graph in two regions according to a variational approach (see Ref. [23]); in the upper part the results are stable, in the lower, unstable.

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and γ2) in a specific example. For the dissipation they have also considered a term related to dipolar relaxation, given by γ2 . Here, in the systematic study of the regions of instability, one took into account previous experimental [69] observations that the dominant process for the dissipation is the three-body recombination. By comparing the parameters of Ref. [88] with the parameters that one has used, and observing that the parameter ξ should be related to both dissipation parameters used in Ref. [88] ( γ = γ0 = 2.6 × 10−3 , ξ ∼ 10−5 ) one can verify from the results given in Fig. 27 that the model of Ref. [88] is inside the intermediate region, where the system is unstable, without a clear signature of chaos.

7.

Final Remarks

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In summary, one has studied the dynamics associated with the extended nonconservative Gross-Pitaevskii equation for a wide region of the dimensionless nonconservative parameters, µ, ξ and γ, that, respectively, are related to dipolar relaxation, three-body recombination and atomic feeding in a trapped atomic condensed system. In such studies, one considered systems with attractive and repulsive two-body interaction in spherically symmetric harmonic traps. Stationary solutions of such an equation were used as initial conditions for the time evolution of the wave function representing BEC dynamics. By obtaining the dynamical solutions of the correspondent extended nonconservative GPE, one found a lot of interesting phenomena, as liquid-gas phase-transition in BEC, formation of autosolitons in repulsive or attractive two-body interaction BEC, spatiotemporal chaos for determined values of the parameters ξ and γ and coherent formation of dimers near Feshbach resonances in nonconservative Bose-Einstein. In Fig. 27, one resumes some results concerning the stability of these systems (for µ = 0), by mapping the space of γ versus ξ, showing the regions of equilibrium and the regions of instability, as well as the regions where one is able to characterize chaotic behaviors, using a criterion given in Ref. [41]. It was also confirmed that chaotic behaviors occur mainly when γ is big enough and γ/ξ is large (at least, when γ is one or two orders of magnitude larger than ξ). A wide variation of the nonconservative parameters was analyzed, in particular motivated by the actual realistic scenario, that already exists, of altering experimentally the two-body scattering length [84]. By changing the absolute value of the scattering length, a real experimental possibility nowadays, one can change in an essential way the behavior of the mean-field description. The model based on GPE extended for including nonconservative effects, as reviewed in this review, is very successful in a lot of features of BEC dynamics. In particular, it can determine precise values of observables as the number of atoms and the root mean-squareradius of BECs, beside describing a very difficult effect as the wave function collapse. However, the model is not enough yet for describing the remaining number of atoms in the condensate after collapses and for explaining in all aspects some experiments as the JILA one, in which strong scattering lengths in condensates vary in time. Such points deserve more investigation for improving the model.

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References [1] Dalfovo, F.; Giorgini, S.; Pitaevskii, L. P.; Stringari S. Rev. Mod. Phys. 1999, 71, 463-512. [2] Parkins, A. S.; Walls, D. F. Phys. Rep. 1998, 303, 1-80. [3] Weiner, J.; Bagnato, V. S.; Zilio, S.; Julienne, P. S. Rev. Mod. Phys. 1999 71, 1-85. [4] Courteille, P. W.; Bagnato, V. S.; Yukalov, V. L. Laser Physics 2001 11, 659-800. [5] Timmermans, E.; Tommasini, P.; Hussein, M.; Kerman, A. Phys. Rep. 1999 315, 199-230. [6] Legget, A. J. Rev. Mod. Phys. 2001 73, 307-356. [7] Tychkov, A. S.; Jeltes, T.; McNamara, J. M.; Tol, P. J. J.; Herschbach, N.; Hogervorst, W.; Vassen, W. Phys. Rev. A 2006 73, 031603(R), 1-4. [8] Andrews, M. R. ; Mewes, M.-O.; van Druten, N. J.; Durfee, Kurn, D. M.; Ketterle, W. Science 1996 273, 84-87; Anderson, M. H.; Ensher, J. R.; Matthews, M. R.; Wieman, C. E.; Cornell, E. A. Science 1995 269, 198-201. [9] Mewes, M.-O.; Andrews, M. R.; van Druten, N. J.; Kurn, D. M.; Durfee, D. S.; Ketterle, W. Phys. Rev. Lett. 1996 77, 416-419.

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[10] Bradley, C. C.; Sackett C. A.; Hulet, R. G. Phys. Rev. Lett. 1997 78, 985-989; Bradley, C. C.; Sackett, C. A.; Tollet J. J.; Hulet, R. G. Phys. Rev. Lett. 1995 75, 1687. [11] Huang, K. Statistical Mechanics; 2nd. Ed.; John Wiley and Sons: New York, 1987. [12] Edwards, M.; Burnett, K. Phys. Rev. A 1995 51, 1382-1385; Ruprecht, P. A.; Holland, M. J.; Burnett, K.; Edwards, M. Phys. Rev. A 1995 51, 4704-4711. [13] Baym G.; Pethick, C. J. Phys. Rev. Lett. 1996 76, 6-9. [14] Fried, D. G.; Killian, T. C.; Willmann, L.; Landhuis, D.; Moss, S. C.; Kleppner, D.; Greytak, T. J. Phys. Rev. Lett. 1998 81 3811. [15] dos Santos, F. P.; Lonard, J.; Wang, J.; Barrelet, C. J.; Perales, F.; Rasel, E.; Unnikrishnan, C. S.; Leduc, M.; Cohen-Tannoudji, C. Phys. Rev. Lett. 2001 86, 3459-3462. [16] Cornish, S. L.; Claussen, N. R.; Roberts, J. L.; Cornell, E. A.; Wieman, C. E. Phys. Rev. Lett. 2000 85, 1795-1798. [17] Modugno, G.; Ferrari, G.; Roati, G.; Brecha, R. J.; Simoni, A.; Inguscio, M. Science 2001 294, 1320-1322. [18] Weber, T.; Herbig, J.; Mark, M.; Ngerl, H.-C.; Grimm, R. Science 2003 299 232-235. Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

On the Dynamics of Nonconservative Bose-Einstein Condensates...

111

[19] Stellmer, S.; Tey, M. K.; Huang, B.; Grimm, R.; Schreck, F. Phys. Rev. Lett. 2009 103, 200401, 1-4; de Escobar, Y. N. M.; Mickelson, P. G.; Yan, M.; DeSalvo, B. J.; Nagel, S. B.; Killian, T. C. Phys. Rev. Lett. 2009 103, 200402, 1-4. [20] Burt, E. A.; Ghrist, R. W.; Myatt, C. J.; Holland, M. J.; Cornell, E. A.; Wieman, C. E. Phys. Rev. Lett. 1997 79, 337-340. [21] Sackett, C. A.; Stoof, H. T. C.; Hulet, R. G. Phys. Rev. Lett. 1995 80, 2031-2034. [22] Kagan, Y.; Muryshev, A. E.; Shlyapnikov, G. V. Phys. Rev. Lett. 1998 81, 933-937. [23] Filho, V. S.; Abdullaev, F. K.; Gammal, A.; Tomio, L. Phys. Rev. A 2001 63, 053603, 1-7. [24] Gammal, A.; Frederico, T.; Tomio, L.; Chomaz, Ph. J. Phys. B 2000 33, 4053-4067. [25] Bondeson, A.; Lisak, M.; Anderson, D. Phys. Scripta 1979 20, 479-485. [26] Ma˘ımistov, A. I. Zh. Eksp. Teor. Fiz. 1993 104, 3620 [JETP 1993 77, 727]. [27] Lees, M. SIAM J. 1959 7, 167; Lees, M. Duke Math. J. 1960 27, 297. [28] Gammal, A.; Frederico, T.; Tomio, L.Phys. Rev. E 1999 60, 2421-2424. [29] Fetter, A. L. Phys. Rev. A 1996 53, 4245-4249. [30] Filho, V. S.; Gammal, A.; Tomio, L. Phys. Rev. A 2002 66, 043605, 1-5.

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[31] Holz, S. M.; Filho, V. S.; Tomio, L.; Gammal, A. Phys. Rev. A 2006 74, 013406, 1-6. [32] Akhmediev, N.; Das, M. P.; Vagov, A. V. “Condensed Matter Theories”, vol. 12, p. 17-25. edited by Clark, J. W.; Panat, P. V. Nova Science Publ., New York, 1997; A. Gammal, T. Frederico, and L. Tomio, “International Workshop on Collective Excitations in Fermi and Bose Systems”, ed. by Bertulani, C.; Canto, L. F.; Hussein, M. (World Scientific, Singapore, 1999). [33] Moerdijk, A. J.; Verhaar, B. J. Phys. Rev. A 1996 53, R19-R22; Zhen, Z.; Macek, J. Phys. Rev. A 1988 38, 1193-1201. [34] Josserand, C.; Rica, S. Phys. Rev. Lett. 1997 78, 1215-1218. [35] Singh, K. G.; Rokhsar, D. S. Phys. Rev. Lett 1996 77, 1667-1670. [36] Ginzburg, V. L.; Pitaevskii, L. P. Zh. Eksp. Teor. Fiz. 1958 34, 1240 [Sov. Phys. JETP 1958 7, 858]; Pitaevskii, L. P. Sov. Phys. JETP 1961 13, 451; Gross, E. P. J. Math. Phys. 1963 4, 195. [37] Houbiers, M.; Stoof, H. T. C. Phys. Rev. A 1996 54, 5055-5066. [38] Gammal, A.; Frederico, T.; Tomio, L. Phys. Rev. A 2000 61, 051602R, 1-4. [39] Efimov, V.Phys. Lett. B 1970 33, 563; Comm. Nucl. Part. Phys. 1990 19, 271. Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

112

Victo S. Filho

[40] Frederico, T.; Tomio, L.; Delfino, A.; Amorim, A. E. A. Phys. Rev. A 1999 60, R9-R12; Amorim, A. E. A.; Frederico, T.; Tomio, L. Phys. Rev. C 1997 56, R2378R2381. [41] Deissler, R. J.; Kaneko, K. Phys. Lett. A 1987 119, 397; Deissler, K.; J. Stat. Phys. 1989 54, 1459-1488. [42] Filho, V. S.; Gammal, A.; Frederico; T.; Tomio, L. Phys. Rev. A 2000 62, 033605; L. Tomio et al., Nucl. Phys. A 2001 684, 681-683. [43] Sackett, C. A.; Gerton, J. M.; Welling, M.; Hulet, R. G. Phys. Rev. Lett. 1999 82, 876-879. [44] Ablowitz, M. J.; Schober, C. M.; Herbst, B. M. Phys. Rev. Lett. 1993 71, 2683; Ablowitz, M. J.; Herbst, B. M.; Schober, C. M. Physica A 1996 228, 212. [45] Ames, W. F. Numerical Methods for Partial Differential Equations , 3rd Ed., Academic Press, New York, 1992, pp. 111-115. [46] Pereira, N. R.; Stenflo, L. Phys. Fluids 1977 20, 1733-1734. [47] Agrawal, G. Nonlinear Fiber Optics, Academic Press, New York, 1994. [48] Abdullaev, F. Kh.; Darmanyan S. A.; Khabibullaev, P. K. Optical Solitons, SpringerVerlag, Heidelberg, 1993. [49] Fabrikant, M. V. Wave Motion 2, 355 1980.

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[50] Anderson, D.; Cattani, F.; Lisak, M. Phys. Scripta 1999 T82, 32-35. [51] Abdullaev, F. Kh. Theory of Solitons in Inhomogeneous Media , Wiley, Chichester, 1994. [52] Abdullaev, F. Kh.; Darmanyan, S. A.; Djumaev, M. R. Izv.AN UzSSR 1986 6, 34; Phys. Lett. A 1989 141, 423-426. [53] Dum, R.; Cirac, J. I.; Lewenstein, M.; Zoller, P. Phys. Rev. Lett. 1998 80, 2972-2975. [54] Michinel, H.; Perez-Garcia, V. M.; de la Fuente, R. Phys. Rev. A 1999 60, 1513-1518. [55] Denschlag, J.; Simsarian, J. E.; Feder, D. L.; Clark, C. W.; Collins, L. A.; Cubizolles, J.; Deng, L.; Hagley, E. W.; Helmerson, K.; Reinhardt, W. P.; Rolston, S. L.; Schneider, B. I.; Phillips, W. D. Science 2000 287, 97-101. [56] Gammal, A.; Frederico, T.; Tomio, L.; Abdullaev, F. Kh. Phys. Lett. A 2000 267, 305-311. [57] Berg´e, L.; Alexander, T. J.; Kivshar, Y. Phys. Rev. A 2000 62, 023607, 1-6. [58] Kaup, D. J.; Malomed, B. A. Physica D 1995 87, 155-159. [59] Ripoll, J. J. G.; Perez-Garcia, V. M. Phys. Rev. A 1999 59, 2220-2231. Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

On the Dynamics of Nonconservative Bose-Einstein Condensates...

113

[60] Akhmediev, N.; Ankiewicz, A. Solitons of the Complex Ginzburg-Landau Equation, Chapter in book: Spatial Solitons 1, Ed. by S. Trillo and W. E. Toruellas, Springer, Berlin, 2000. [61] Gerton, J. M.; Sackett, C. A.; Frew, B. J.; Hulet, R. G. Phys. Rev. A 1999 59, 15141516. [62] Moerdijk, A. J.; Boesten, H. M. J .M.; Verhaar, B. J. Phys. Rev. A 1996 53, 916-920; Moerdijk, A. J.; Verhaar, B. Phys. Rev. A 1996 53, R19-R22. [63] Kagan, Yu.; Svistunov, B. V.; Shlyapnikov, G. V. Pis’ma Zh. Eksp. Teor. Fiz. 1985 42, 169 [JETP Lett. 1985 42, 209]. [64] Cornish, S. L.; Claussen, N. R.; Roberts, J. L.; Cornell, E. A.; C. E. Wieman, Phys. Rev. Lett. 2000 85, 1795-1798. [65] Filho, V. S.; Tomio, L.; Gammal, A.; Frederico, T. Phys. Lett. A 2004 325, 420-425. [66] Tiesinga, E.; Verhaar, B. J.; Stoof, H. T. C. Phys. Rev. A 1993 47, 4114-4122; Tiesinga, E.; Moerdijk, A. J.; Verhaar, B. J.; Stoof, H. T. C. Phys. Rev. A 1992 46, R1167-R1170. [67] Inouye, S. et al., Nature 1998 392, 151-. [68] Roberts, J. L. et al., Phys. Rev. Lett. 2001 86, 4211.

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[69] Donley, E. A.; Claussen, N. R.; Cornish, S. L.; Roberts, J. L.; Cornell, E. A.; Wieman, C. E. Nature 2001 412, 295-299. [70] Stenger, J.; Inouye, S.; Andrews, M. R.; Miesner, H.-J.; Stamper-Kurn, D. M.; Ketterle, W. Phys. Rev. Lett. 1999 82, 2422-2425. [71] Claussen, N. R.; Donley, E. A.; Thompson, S. T.; Wieman, C. E. Phys. Rev. Lett. 2002 89, 010401, 1-4. [72] Moerdijk, A. J.; Verhaar, B. J.; Axelsson, A. Phys. Rev. A 1995 51, 4852-4861. [73] Saito, H.; Ueda, M. Phys. Rev. A 2002 65, 033624, 1-6; Saito, H.; Ueda, M. Phys. Rev. A 2001 63, 043601, 1-8; Saito, H.; Ueda, M. Phys. Rev. Lett. 2001 86, 1406-1409. [74] Fedichev, P. O.; Reynolds, M. W.; Shlyapnikov, G. V. Phys. Rev. Lett. 1996 77, 29212924; [75] Nielsen, E. et al., Phys. Rev. Lett. 1999 83, 1566. [76] Esry, B. D. et al., Phys. Rev. Lett. 1999 83, 1751. [77] Bedaque, P. F. et al., Phys. Rev. Lett. 2000 85, 908. [78] Yamashita, M. T. et al., Phys. Rev. A 2002 66, 052702. [79] Nielsen, E. et al., Phys. Rep. 2001 347, 373. Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

114

Victo S. Filho

[80] Stoof, H. T. C. et al.,Phys. Rev. B 1988 38, 11221. [81] Filho, V. S.; Frederico, T.; Gammal, A.; Tomio, L.Phys. Rev. E 2002 66, 036225, 1-6. [82] Duaine, R. A.; Stoof H. T. C. (2002). Quantum Evaporation of a Bose-Einstein Condensate, arXiv:cond-mat/0204529. [83] Stwalley, W. C. Phys. Rev. Lett. 1976 37, 1628-1631. [84] Courteille, P.; Freeland, R. S.; Heinzen, D. J.; van Abeelen, F. A.; Verhaar, B. J. Phys. Rev. Lett 1998 81, 69-72. [85] Tomio, L. et al., Laser Phys. 2003 V. 13 N. 4, 582. [86] Holz, S. M.; Filho, V. S.; Tomio, L. Phys. Lett. A 2008 Vol. 372, 6778-6783 (2008). [87] Strogatz, S. H. Nonlinear Dynamics and Chaos, Addison-Wesley Publishing Company, 1994.

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[88] Eleftheriou A.; Huang, K. Phys. Rev. A 2000 61, 043601.

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In: Bose Einstein Condensates Editor: Paige E. Matthews, pp. 115-123

ISBN 978-1-61728-114-3 c 2010 Nova Science Publishers, Inc.

Chapter 4

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E LLIPTIC VORTICES IN S ELF - ATTRACTIVE B OSE -E INSTEIN C ONDENSATES Fangwei Ye1 , Boris A. Malomed2 , Dumitru Mihalache3 , Liangwei Dong4 and Bambi Hu1,5 1 Department of Physics, Centre for Nonlinear Studies, and The Beijing-Hong Kong Singapore Joint Centre for Nonlinear and Complex Systems (Hong Kong), Hong Kong Baptist University, Kowloon Tong, China 2 Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel 3 Horia Hulubei National Institute for Physics and Nuclear Engineering (IFIN-HH), Department of Theoretical Physics, 407 Atomistilor, Magurele-Bucharest, 077125, Romania 4 Institute of Information Optics of Zhejiang Normal University, Jinhua, 321004, China 5 Department of Physics, University of Houston, Houston, Texas 77204-5005, USA Abstract We report solutions for vortex solitons in an anisotropic harmonic trapping potential in a “pancake-shaped” Bose-Einstein condensate with attractive inter-atomic interactions. Elliptic vortices featuring anisotropic patterns bifurcate from dipole states that exist in the linear limit. The elliptic vortices with topological charge S = 1, featuring strongly asymmetric shapes, may be stable in a wide region in their existence domain. The dependence of the stability region on the eccentricity of the elliptic trap is reported. All higher-order vortices, with S > 1, are unstable.

PACS number: 42.65.Tg, 42.65.Jx, 42.65.Wi

1.

Introduction

The investigation of the nonlinear dynamics of Bose-Einstein condensates (BECs) is fundamentally important to understanding of the complex physics of degenerate quantum gases. The nonlinearity is at the core of the formation of various self-sustained structures [1], Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

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manifesting itself in such phenomena as four-wave mixing [2], wave amplification [3, 4], super-radiant Rayleigh scattering [5], Faraday patterns [6], etc. Solitons are objects of primary interests in this context. Experimentally, both dark [7, 8] and bright [9, 10] solitons have been created in effectively one-dimensional condensates. In the two-dimensional (2D) geometry, vortices were created by stirring the condensate with a laser beam [11], via imprinting a topological phase pattern onto condensates trapped in Joffe-Pritchard magnetic traps [12], by rotating the BEC cloud [13, 14, 15], and through the decay of solitons of other types [16, 17]. In condensates with attractive inter-atomic interactions, 2D vortices suffer from the collapse instability. However, it was found that the 2D harmonic trap may suppress the collapse, provided that the number of atoms in condensate does not exceeds a certain value [18, 19]. Most of the theoretical works dealing with the vortex generation and their stability were done for isotropic trapping potentials; however, asymmetric (non-circular) traps offer more degrees of freedom. In this respect, in rotating BEC systems with repulsive inter-atomic interactions, effects of asymmetries of the confining potential on vortices were first theoretically considered in Ref. [20] and further developed in Ref. [21]. The role of ellipticity on the stability of vortices in nonrotating traps was investigated in Ref. [22], where stable asymmetric vortices were reported to form for strong enough nonlinearities, a fact that was later recognized to be related to the bifurcation of vortices from dipole states. Such bifurcation occurs when the norm (number of atoms), or the integral power in the context of optics, exceeds a certain value [23]. Dipole solitons, vortex-antivortex pairs (“vortex dipoles”), and vortex clusters in asymmetric nonrotating confining potentials were also investigated [24, 25, 26, 27, 28] in condensates with repulsive interactions. In particular, vortex dipoles exist as stationary soliton solutions of the 2D Gross-Pitaevskii equation for both symmetric and asymmetric traps. However, it should be mentioned that, in circularly symmetric traps, vortex dipoles cannot exist in the linear limit, as they bifurcate from soliton dipoles [25] or dark- soliton modes [29], provided that the corresponding norm (i.e., the strength of the nonlinearity) is large enough. In contrast, in asymmetric traps, vortex dipoles exist even in the linear limit [25]. In spite of the aforementioned works performed for vortices in asymmetric traps, effects of the ellipticity of the trapping potential on the existence and stability of ring-vortex modes were systematically investigated only very recently [23], leading to a conclusion that elliptic vortices emerge via a symmetry-breaking bifurcation from dipole modes, provided that the norm (i.e., the number of atoms in the condensate) exceeds a threshold value; higher-order vortices may feature several spatially separated pivotal points (amplitude zeros), which greatly impacts their stability. These findings were reported in Ref. [23] for a defocusing model (with repulsive inter-atomic interaction in the context of BECs). Thus, a remaining issue to be explored is how the anisotropy of the external trap affects vortices in the case of a focusing nonlinearity (i.e., in the BEC with attractive inter-atomic interactions). In this chapter, we investigate elliptically shaped vortex solitons in BECs with the intrinsic self-attraction. We demonstrate that, similar to the case of the self-repulsive BEC, although the elliptic vortices do not exist in the zero-interaction limit, they emerge when the norm of the condensate is large enough. Featuring highly asymmetric profiles, elliptic vortices can still be stable in a large parameter region. Adjacent to the stability region, there

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exists a small region where the periodic splitting of the vortex into two rotating bright spots and their subsequent recombination are observed. In another instability region, vortices split into two spiraling spots and then converge into a single fundamental soliton. Higher-order elliptic vortices, with topological charge S > 1, also exist in the case of the focusing nonlinearity, with spatially separated pivotal points; however, in contrast with their counterparts in the case of the defocusing nonlinearity, all the higher-order vortices are unstable. The rest of the chapter is structured as follows. In Section 2, we formulate the model and report basic results of the analysis for the fundamental vortex solitons, with S = 1 and S = 2. The shape of these solitons and their stability region are identified (in particular, it is demonstrated that the multi-charged vortices have no stability region). The chapter is concluded by Section 3.

2.

The Model and Numerical Analysis for the Vortex Solitons

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We start with a 2D BEC configuration confined to the (x, y) plane, assuming that the dynamics in the z-direction may be eliminated, as usual, due to the tight confinement. In the framework of the mean-field approximation, the evolution of the BEC obeys the timedependent Gross-Pitaevskii equation for the normalized complex wave functionψ(x, y, τ ):   ∂ψ(x, y, z) 1 ∂2ψ ∂2ψ 1 i =− + + (wx2 x2 + wy2 y 2 )ψ − |ψ|2 ψ (1) 2 2 ∂τ 2 ∂x ∂y 2 where coordinates x, y are measured in units of the trapping length, while time τ is measured in units of the trapping period. In the case of the circular trap (wx = wy ), the vortices are of circular shapes and were studied in detail in Ref. [19]. The general case, wx 6= wy , corresponds to an anisotropic trap in the x − y plane. The anisotropy of the 2D trap can be characterized by the eccentricity, e = (wy2 − wx2 )1/2 /wy , whose value falls in the range of [0, 1]. The circular trap corresponds to e → 0, while a strongly elliptic trap corresponds to e → 1 . Note that Eq. (1) conserves the norm (normalized number of atoms) U and the respective Hamiltonian H, but the conservation of the z−projection of angular momentum Lz is broken by e 6= 0. These three quantities are, respectively, given by: Z Z U = |ψ|2 dxdy, (2) Z Z 1 H = [|∇⊥ ψ|2 − |ψ|4 + (wx2 + wy2 )|ψ|2 ]dxdy, 2 Z Z 1 [r × (ψ ∗ ∇ψ − ψ∇ψ ∗ )]dxdy. Lz = 2i where ∇⊥ = ex ∂/∂x + ey ∂/∂y, r = ex x + ey y, while ex , ey are unit vectors in directions of x and y axes. We looked for vortex solutions to Eq. (1) in the form of ψ(x, y, τ ) = (φr + iφi ) exp(iµz). Here φr,i (x, y)are real functions independent of τ , and −µ stands for the corresponding chemical potential. The topological charge of the vortex solitons, S, is obtained as the circulation of the phase gradient, arctan(φi /φr ), around the phase singularity, divided by 2π. Typical vortex solitons with S = 1, trapped in the elliptic potential, are

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shown in Fig. 1(a)-(c). Similar to the elliptic vortices in the repulsive BECs, elliptic vortices in the attractive condensate also feature an azimuthally modulated intensity profile, whose maximum and minimum are achieved, respectively, at the major and minor semiaxes of the ellipse, see Fig. 1(a). The phase gradient is not uniform either, with the phase varying fastest near the minor axis and slowest at the major one. With the increase of the potential’s eccentricity, the anisotropy becomes more pronounced.

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Figure 1. Amplitude (top panel) and phase (bottom panel) profiles for single-charged vortices, with S = 1 (a)-(c), and double-charged vortices, with S = 2 (d). In panel (a) µ = −1.5, and the vortex is stable; in (b) µ = −2, and the vortex is stable; in panels (c) and (d), µ = 1, and the vortices are unstable. In all plots, the eccentricity is e = 0.5. Figure 2(a) presents the dependence of the norm, U , on µ for different values of the potential’s eccentricities. As said above, in the circular trap vortex solitons exist even in the linear limit (i.e., for U → 0) . However, at e 6= 0, vortex solitons exhibit a threshold Uco (or respectively, µco ), below which they cease to exist. By comparing U (µ) of the vortex with that of the dipoles, for the same eccentricity, we have found that the vortex solitons branch out from dipole solitons at the bifurcation point, Uco (µco ). The value of the norm at the bifurcation increases with the eccentricity, see Fig. 3(a). Figure 1(b) shows the vortex profiles near the bifurcation point for eccentricity e = 0.5. In this case, the vortex features two weakly coupled crescent-like spots, whose phase difference is almost exactly π. Thus, at the bifurcation point, the elliptic vortex indeed resembles a dipole, as concerns its amplitude and phase alike. On the other hand, far from the bifurcation, the vortex becomes strongly localized, with a weak dependence on the external potential, therefore the vortex features nearly isotropic patterns [see Fig. 1(c)], as in the circular trap. Accordingly, regardless of the value of the eccentricity, all U (µ) curves asymptotically converge to the same value, U (µ → ∞) ≈ 24.1. To explore the dynamical stability of the elliptic vortices, we have performed systematic simulations of Eq. (1) with input |ψ|r=0 = (φr +iφi )(1+ρ), where ρ is a random or regular small perturbation [ρ(x, y) ≪ φi,j (x, y)]. The simulations have revealed that, despite the strong asymmetry, the elliptic vortices are stable in a large part of their existence domain. The stability regions are plotted in Fig. 2(b,c), in the parameter planes of (µ, e) and (U, e). These diagrams show that, for a fixed eccentricity, the elliptic vortices are stable in the range

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e e Figure 2. (a) The norm U vs µ for the single-charged vortices (with S = 1) at different eccentricities. (b) and (c): Stability regions are shaded for the vortices with with S = 1 in the (µ, e) and (U, e) planes, respectively. of µ ∈ [µco , µcr ], or, in terms of the norm, U ∈ [Uco , Ucr ] . Naturally, the stability domain gradually shrinks with the increase of the eccentricity. It closes down and disappears at e ≈ 0.89, all the vortices being unstable at still larger values of e. For elliptic vortices belonging to the stability region, they retain their initial shapes over indefinitely long time intervals, even in the presence of considerable input random perturbations, as is clearly seen in Fig. 3(a). In contrast to that, unstable vortices first split into two fragments, which are followed by the disappearance of one of them, see Fig. 3(b). It is interesting to mention that, however, within a very small region adjacent to the stability region, the vortex quasiperiodically splits into a pair of solitons, which subsequently recombine back into the vortex [Fig. 3(c)]. A similar dynamical regime was also observed in circular traps [19]. Higher-order vortices have been also found in the elliptic trap. It has been recently demonstrated that, unlike the conventional multiple-charged vortices with the single pivotal point, elliptic vortices exhibit separated unit-charged phase singularities (multiple pivots) [23]. Here we have found a similar scenario in the case of the self-attraction. Figure 1(d) displays a typical example of a double-charged vortex soliton, with two separated phase singularities, i.e., zero-amplitudes points. However, unlike the higher-order elliptic vortices in the repulsive condensate, we have concluded that all vortex solitons with S ≥ 2 are completely unstable in the attractive condensate, spontaneously transforming into fundamental solitons [Fig. 3(d)]. Finally,we study the excitation of the elliptic vortices with S = 1 from an initial circular

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Figure 3. The evolution of perturbed vortex solitons, with topological charge S = 1, at (a) µ = −1.5, (b) µ = −1, (c) µ = −1.35, and with topological charge S = 2 at µ = 1 (d). The absolute value of the wave function is shown at different times τ . In all cases, e = 0.5. Gaussian matter wave with a embedded phase singularity. We find that the wave function quickly reshapes itself into a double-peaked structure while it well conserves its intrinsic phase singularity during the evolution [Fig. 4]. We have checked that such two-peaked pattern resembles that of the stationary vortex solution at the same value of norm and in the trapping potential with the same ellipticity. Interestingly enough, upon the evolution, the whole structure experiences persistent rotation, and we observe a very slow relaxation of the spontaneously rotating patterns to the corresponding stationary vortex state. The average value of the z−projection of the angular momentum, Lz , is also found to be close to that of of the corresponding stationary elliptic vortex.

3.

Conclusion

We have explored elliptic vortex solitons in the anisotropic harmonic trapping potential in the model of the 2D BECs with attractive inter-atomic interactions. We have found that elliptic vortices feature anisotropic patterns, bifurcating from dipoles, similarly to what was previously found in the model with repulsive inter-atomic interactions. The elliptic vortices with topological charge 1 may be stable in a wide region of their existence domain, despite

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Figure 4. The propagation of a circular Gaussian beam with an embedded phase singularity, corresponding to S = 1. The input amplitude and the followed evolutionary patterns are shown in the top row, at τ = 0, 1000, 1010, and 1020, respectively. (Bottom row) The z−component of angular momentum, Lz , versus propagation distance. e = 0.5. their strongly anisotropic shapes. We have demonstrated the dependence of the shape of the stability region on the eccentricity of the trapping potential. Higher-order vortices feature the splitting of the single pivotal point into a set of separated ones, each corresponding to a charge-1 vortex. However, all higher-order vortices are unstable. The results of this work may also have potential applications in radial lattice, for example, in radial bandgap fibers [30]. To guarantee the single-mode operation of the fiber and preclude polarization mode dispersion (PMD), it is necessary to isolate a single polarization, which usually can be done by making the fiber elliptically deformed, thus arriving at the present model. L. Dong acknowledges the support of the National Natural Science Foundation of China (Grant No. 10704067).

References [1] L. Pitaevskii and S. Stringari, Bose-Einstein Condensate (Oxford University Press, Cambridge, UK, 2003). [2] L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Four-wave mixing with matter waves,” Nature(London) 398, 218 (1999).

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[3] M. Kozuma, Y. Suzuki, Y. Torii, T. Sugiura, T. Kuga, E. W. Hagley, and L. Deng, “Phase-coherent amplification of matter waves,” Science 286, 2309 (1999). [4] S. Inouye, T. Pfau, S. Gupta, A. P. Chikkatur, A. Gorlitz, D. E. Pritchard, and W. Ketterle, “Phase-coherent amplification of atomic matter waves,” Nature(London) 285, 641 (1999). [5] S. Inouye, T. Pfau, S. Gupta, A. P. Chikkatur, A. Gorlitz, D. E. Pritchard, and W. Ketterle, “Superradiant Rayleigth scattering from a Bose-Einstein condensate,” Science 285, 571 (1999). [6] K. Staliunas, S. Longhi, and G. J. de Valc´arcel, “Faraday Patterns in Bose-Einstein Condensates,” Phys. Rev. Lett. 89, 210 406 (2002). [7] S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G. V. Shlyapnikov, and M. Lewenstein, “Dark Solitons in Bose-Einstein Condensates,” Phys. Rev. Lett. p. 5198 (1999). [8] J. Denschlag, J. E. Simsarian, D. L. Feder, C. W. Clark, L. A. Collins, J. Cubizolles, L. Deng, E. W. Hagley, K. Helmerson, W. P. Reinhardt, S. L. Rolston, B. I. Schneider, and W. D. Phillips, “Generating solitons by phase engineering of a Bose-Einstein condensate,” Science 287, 97 (2000). [9] L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a matter-wave bright soliton,” Science 296, 1290 (2002).

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[10] K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, “Formation and propagation of matter-wave soliton trains,” Nature (London) 417, 150 (2002). [11] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, “Vortices in a Bose-Einstein Condensate,” Phys. Rev. Lett. 83, 2498 (1999). [12] A. E. Leanhardt, A. G¨orlitz, A. P. Chikkatur, D. Kielpinski, Y. Shin, D. E. Pritchard, and W. Ketterle, “Imprinting Vortices in a Bose-Einstein Condensate using Topological Phases,” Phys. Rev. Lett. 89, 190 403 (2002). [13] K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, “Vortex Formation in a Stirred Bose-Einstein Condensate,” Phys. Rev. Lett. 84, 806 (2000). [14] J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, “Observation of vortex lattices in Bose-Einstein condensates,” Science 292, 476 (2001). [15] P. C. Haljan, I. Coddington, P. Engels, and E. A. Cornell, “Driving Bose-EinsteinCondensate Vorticity with a Rotating Normal Cloud,” Phys. Rev. Lett. 87, 210 403 (2001). [16] Z. Dutton, M. Budde, C. Slowe, and L. V. Hau, “Observation of quantum shock waves created with ultra-compressed slow light pulsed in a Bose-Einstein condensate,” Science 293, 663 (2001). Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

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[17] B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching Dark Solitons Decay into Vortex Rings in a Bose-Einstein Condensate,” Phys. Rev. Lett. 86, 2926. [18] S. K. Johansen, O. Bang, and M. P. Sørensen, “Escape angles in bulk χ(2) soliton interactions,” Phys. Rev. E 65, 026 601 (2002). [19] D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Vortex stability in nearlytwo-dimensional Bose-Einstein condensates with attraction,” Phys. Rev. A 73, 043 615 (2006). [20] A. A. Svidzinsky and A. L. Fetter, “Stability of a Vortex in a Trapped Bose-Einstein Condensate,” Phys. Rev. Lett. 84, 5919 (2000). [21] J. J. Garc´ıa-Ripoll and V. M. P´erez-Garc´ıa, “Anomalous rotational properties of BoseEinstein condensates in asymmetric traps,” Phys. Rev. A 64, 013 602 (2001). [22] J. J. Garc´ıa-Ripoll, G. Molina-Terriza, V. M. P´erez-Garc´ıa, and L. Torner, “Structural Instability of Vortices in Bose-Einstein Condensates,” Phys. Rev. Lett. 87, 140 403 (2001). [23] F. Ye, D. Mihalache, and B. Hu, “Elliptic vortices in composite Mathieu lattices,” Phys. Rev. A 79, 053 852 (2009).

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[24] L.-C. Crasovan, G. Molina-Terriza, J. P. Torres, L. Torner, V. M. P´erez-Garc´ıa, and D. Mihalache, “Globally linked vortex clusters in trapped wave fields,” Phys. Rev. E 66, 036 612 (2002). [25] L.-C. Crasovan, V. Vekslerchik, V. M. P´erez-Garc´ıa, J. P. Torres, D. Mihalache, and L. Torner, “Stable vortex dipoles in nonrotating Bose-Einstein condensates,” Phys. Rev. A 68, 063 609 (2003). [26] S. Theodorakis, “Piecewise analytic approximation for Bose-Einstein condensates in isotropic harmonic traps,” Phys. Rev. A 70, 063 619 (2004). [27] M. M¨ott¨onen, S. M. M. Virtanen, T. Isoshima, and M. M. Salomaa, “Stationary vortex clusters in nonrotating Bose-Einstein condensates,” Phys. Rev. A 71, 033 626 (2005). [28] V. Pietila, M. Mottonen, T. Isoshima, J. A. M. Huhtamaki, and S. M. M. Virtanen, “Stability and dynamics of vortex clusters in nonrotated Bose-Einstein condensates,” Phys. Rev. A 74, 023 603 (2006). [29] W. Li, M. Haque, and S. Komineas, “Vortex dipole in a trapped two-dimensional Bose-Einstein condensate,” Phys. Rev. A 77, 053 610 (2008). [30] E. Lidorikis, M.Soljacic, M. Ibanescu, Y. Fink, and J. D. Joannopoulos, “Cutoff solitons in axially uniform systems,” Opt. Lett.29,851(2004).

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In: Bose Einstein Condensates Editor: Paige E. Matthews, pp. 125-141

ISBN 978-1-61728-114-3 c 2010 Nova Science Publishers, Inc.

Chapter 5

C LASSICAL E LECTRODYNAMICS A NALOGY OF T WO - DIMENSIONAL B OSE -E INSTEIN C ONDENSATES H.M. Cataldo Departamento de F´ısica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and Instituto de F´ısica de Buenos Aires, Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas, Pabell´on 1, Ciudad Universitaria, 1428 Buenos Aires, Argentina

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Abstract The previously explored analogy between a two-dimensional homogeneous superfluid and a (2+1)-dimensional electrodynamic system is extended to the case of a confined Bose-Einstein condensate. More precisely, a whole mapping between the hydrodynamics of a two-dimensional Bose-Einstein condensate and the nonrelativistic classical electrodynamics of a charged material medium is developed and analyzed. In such an analogy, macroscopic charges and electromagnetic radiation play the role of vortices and sound radiation, respectively. The mapping is shown to provide a very useful frame to discuss several features of vortex dynamics and induced velocity fields. Particularly, two important local conservation theorems of energy and angular momentum are easily derived. The controversial question of the vortex inertia is also studied from the viewpoint of the electromagnetic analogy, finding a qualitative agreement between the dissipative inertial effects in a uniform superfluid system and the numerical simulation results of vortex motion in a Bose-Einstein condensate.

1.

Introduction

There is a well-known correspondence between two-dimensional (2D) homogeneous superfluids and (2+1)D electrodynamics. Popov [1] was the first to explain how the hydrodynamics of vortices in 2D superfluids can be mapped onto (2+1)D relativistic electrodynamics, with vortices playing the role of charged particles and phonons the role of photons. The dynamics of superfluid films was later investigated by exploiting the analogy with the

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dynamics of a 2D plasma [2]. The mapping onto (2+1)D electrodynamics was utilized in Ref. [3] to explore the theory of dynamical vortices in superfluid films, deriving a frequency dependent vortex mass. Such a mapping was also employed to study the phonon radiation by a vortex, which exercises circular motion under the influence of an external force in a 2D homogeneous superfluid [4]. More recently, Fedichev et al. [5], using the analogy to electron-positron pair creation in quantum electrodynamics, investigated the process of vortex-antivortex pair creation in a supersonically expanding and contracting quasi-2D Bose-Einstein condensate (BEC) at zero temperature. However, in spite of the current interest and widespread importance of the phenomenon of Bose-Einstein condensation of dilute gases in traps, the analogy of 2D BECs with electrodynamics remains almost unexplored. This lack led us to attempt amending such a situation, and with this aim, we are presenting in this chapter a thorough mapping of the dynamics of a 2D BEC onto the electrodynamics of a 2D material medium. In fact, after a review of the main features of a 2D BEC in Sec. 2, we are developing such a mapping in Sec. 3. Next, this is applied in Secs. 4 and 5 to deriving local conservation theorems of energy and angular momentum, respectively. In Sec. 6, we discuss the velocity field induced by an off-centered vortex from the viewpoint of the equivalent electromagnetic problem. Our treatment of the vortex motion in Sec. 7. deals with two intriguing effects on the vortex velocity. First, we discuss in 7.1. the recently found ‘core effect’ on such a velocity [6, 7], along with its possible interpretation within the electromagnetic picture. Second, starting again from the electrodynamic analogy, we discuss in 7.2. the highly controversial question of the vortex inertia. We begin by reviewing in 7.2.1. the theoretical treatment of the vortex mass in a homogeneous superfluid, analyzing inertial effects on vortex dynamics, namely cyclotron motion and retardation effect. Next, in the absence of a full theory for the vortex mass in a bounded superfluid, we discuss in 7.2.2. several, sometimes contradicting, theoretical estimates, and perform in 7.2.3. an illuminating comparison of dissipative inertial effects occurring in an infinite superfluid and in a numerically simulated 2D BEC. Finally, the conclusion of this chapter is given in Sec. 8.

2.

Two-dimensional BEC

Our starting point is the effective 2D Gross-Pitaevskii equation [8]  2 2  ∂ ~ ∇ 2 i~ Φ(r, t) = − + Vext (r) + g|Φ(r, t)| Φ(r, t), ∂t 2m

(1)

where Vext (r) denotes the trapping potential and g corresponds to the effective 2D coupling constant between the atoms of mass m. The complex order parameter Φ may be written in terms of a modulus and a phase, as follows: Φ(r, t) =

p n(r, t) exp[iS(r, t)],

(2)

where n(r, t) denotes the particle density and the gradient of the phase S yields the velocity field, v(r, t) =

~ ∇S(r, t). m

Bose-Einstein Condensates: Theory, Characteristics, and Current Research : Theory, Characteristics, and Current Research, Nova Science

(3)

Classical Electrodynamics Analogy...

127

Such a field is irrotational except at points rj , where the phase presents a singularity corresponding to a quantized vortex, i.e. X ∇×v = κj δ(r − rj ) ˆ z, (4) j

κj being the circulation of the velocity field along a path encircling a single vortex located at rj . The Gross-Pitaevskii equation (1) is equivalent to the following two coupled equations for the density and the velocity field [9]: ∂n = −∇ · (vn) ∂t ∂v = −∇ ∂t



Vext gn v2 ~2 2√ √ ∇ n + + − m m 2 2m2 n

(5) 

.

(6)

Assuming that the repulsive interaction among atoms is strong enough, the density profiles are smooth and one can safely neglect the kinetic-pressure term, proportional to ~2 , in the last equation, which then takes the form   Vext gn v 2 ∂v = −∇ + + . (7) ∂t m m 2 This result corresponds to the equation of potential flow for a fluid whose pressure and density are related by the equation of state

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p = g n2 /2.

(8)

The approximation (7), however, overlooks the vortex dynamics represented by the time derivative of eq. (4), which may be explicitly taken into account by adding a Magnus force term [10]:   X ∂v Vext gn v 2 κj δ(r − rj ) r˙ j × ˆ z. (9) = −∇ + + + ∂t m m 2 j

The equation of continuity (5) and the momentum equation (9) represent a system of hydrodynamic-type equations, which after linearization, yields sound waves propagating at the local sound velocity r r 1 ∂p gn cs = = . (10) m ∂n m Having explicitly introduced the vortex coordinates rj into the condensate equations of motion, one must consider additional equations given by vanishing Magnus forces acting upon each vortex: n′ (rj ) m κj ˆ z × [˙rj − v′ (rj )] = 0, (11) where the prime on particle density and velocity field denotes that the self-contribution of the corresponding vortex must be ignored. Note that the above equation simply states that the vortex core will move with the background superfluid velocity. Later we shall comment on various effects causing possible departures from this behavior.

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128

3.

H.M. Cataldo

Mapping onto Electrodynamics

The above two-dimensional dynamics can be mapped onto the electrodynamics of a 2D material medium, with electromagnetic waves representing sound waves and macroscopic (free) point charges representing vortices. We shall utilize in the following the HeavisideLorentz system of units [11]. To begin with, we assume a transverse magnetic induction proportional to the condensate density B=

√

g nˆ z

(12)

and an electric field, whose expression arises from representing vortices as point charges of negligible mass, leading to a vanishing Lorentz force

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E+

v × B = 0, c

(13)

p where c = g nmax /m denotes the maximum speed of sound in the condensate. The above expression, rewritten as E/B = v/c, embodies an important analogy to quantum electrodynamics. In fact, the Schwinger vacuum breakdown [12] is a phenomenon occuring whenever the electric field exceeds the magnetic field, which gives rise to electron-positron pair creation. On the other hand, it is well known that a supersonic superflow becomes inestable with respect to the spontaneous creation of vortex-antivortex pairs [5]. We shall restrict ourselves in this chapter to slow motion superflows v/c