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Table of contents :
THEORETICAL PHYSICS RESEARCH DEVELOPMENTS
THEORETICAL PHYSICS RESEARCH DEVELOPMENTS
CONTENTS
PREFACE
COMMENTARY ON THE PRIOR ERRORS FOR CONSERVATION OF MOMENTUM
1. Foreword and Problem Statement
2. Comprehensive Derivation of Conservation Equation for Momentum on A Fluid Element
Symbols and Notation
DARK ENERGY AS THE SOURCE OF THE TIME DEPENDENT EINSTEIN COSMOLOGICAL CONSTANT
Abstract
Introduction
1. Universal Relaxation Processes
2. Time Dependent Cosmological Constant Λ
References
NON-LINEAR REFRACTIVE INDEX THEORY: I-GENERALIZED OPTICAL EXTINCTION THEORY AND DISPERSION RELATIONS
Abstract
1.Introduction
2.ModelEquations
3.GeneralizedOpticalExtinctionTheorem
4.OpticalBistabilityintheSmallDensityLimit
Summary
AppendixA
References
QUANTUM COHERENCE AND TUNABLE TRANSIENT BEHAVIOR OF A DOUBLE-CONTROL FOUR-LEVEL ATOMIC VAPOR
Abstract
1.Introduction
2.Double-ControlAtomicSystemandItsSteadyOpticalBehavior
3.QuantumInterferencesintheFour-LevelSystem
4.TheTunableTransientEvolutionalBehavior:ConstructiveandDestructiveInterferences
5.ApplicationstothePhotonicLogicGatesandOpticalSwitches
6.ConcludingRemarks
Acknowledgment
References
APPLICATION OF GEANT4 CODE IN GAMMA IRRADIATION PROCESSING
Abstract
1.Introduction
2.DosimetryQualityControlinGammaIrradiationProcessing
2.1.60CoFacilityDesign
2.2.StructureandOperationoftheGEANT4CodeSystem
3.ExperimentalBenchmarks
3.1.SpecialPointswithintheCell
3.2.TransversalDoseRateDistribution
3.3.LongitudinalDoseRateDistribution
3.4.3DDoseRateMappingofaPhantomProduct
4.ApplicationsofGEANT4
4.1.OptimizationofSomeParametersoftheTreatmentProcess
4.2.ImprovementofDoseUniformity
5.Conclusion
References
STUDY OF ATOMIC ENTANGLEMENT IN MULTIMODE CAVITY OPTICS
Abstract
I. Introduction
II. Atom-Field Interaction Hamiltonian
III. Population Dynamics and Atomic Entanglement for a Single Mode Field
IV. Entanglement of Two Atoms through Multimode (1>m) Dynamics
V. Conclusion
Acknowledgment
References
ANALYTICAL EXPRESSION FOR P-3HE TOTAL REACTION CROSS SECTION
Abstract
1. Introduction
2. Mathematical Formulation
3. Results and Conclusion
Acknowledgments
References
RADIATIVE TRANSITIONS OF MESONS IN AN INDEPENDENT-QUARK POTENTIAL MODEL
Abstract
1.Introduction
2.BasicFormalism
3.EnergyCorrectionstotheGround-StateMesonMasses
3.1.Center-of-massCorrection
3.2.One-GluonExchangeCorrection
3.3.ThePhysicalMassoftheGround-StateMeson
4.EffectiveMomentumDistributionFunction
5.RadiativeDecayWidthsbeyondStaticApproximation
6.ResultsandDiscussion
A.HyperfineSplittingsofNon-self-conjugateMesons
B.BeyondStaticCalculationofRadiativeTransitions
Acknowledgment
References
LASER PROTOTYPING OF POLYMER-BASED NANOPLASMONIC COMPONENTS
Abstract
Introduction
Laser Nanomanufacturing of SPP Optical Elements
Fabrication of Optical Waveguiding Dielectric Components
Leakage Radiation Imaging
Nanoplasmonic Components
Acknowledgments
References
AN APPROXIMATION METHOD FOR DETERMINATION OF THE COUPLING CONSTANT AND CHARGED PION FORM FACTOR AT 22Q=0.6-1.6GeVAND 22Q=2.45 GeV
Abstract
1. Introduction
2. The Model
3. Calculations and Results
References
HUBBARD MODEL AND FERROELECTRIC PHASE TRANSITION IN GRAPHENE
Abstract
Introduction
Basic Equations
3. Calculation Results
Conclusion
References
STUDY OF PRESSURE DEPENDENCE SUPERCONDUCTING STATE PARAMETERS OF BINARY METALLIC GLASSES BY PSEUDOPOTENTIAL THEORY
Abstract
Introduction
2. Computational Technique
3. Results and Discussion
4. Conclusion
References
CONTROL OVER GROUP VELOCITY IN A THREE LEVEL CLOSED LADDER SYSTEM WITH SPONTANEOUSLY GENERATED COHERENCE
Abstract
1.Introduction
2.Theory
2.1.TheSystemandDensityMatrixEquations
2.2.TheAnalyticalSolutions
3.ResultsandDiscussions
3.1.EffectofRelativePhaseonSusceptibilityandGroupIndexofProbeField
3.2.EffectofIncoherentPumponSusceptibilityandGroupIndexofProbeField
3.3.RealizationofSGCinLadderSystem
4.Conclusion
References
MODELLING OF MICROCRACKED BODIES USING THE CONCEPT OF CRACK OPENING MODE
1.AimandFrameworkoftheStudy
2.Micro-MacroTransitionfortheUnilateralEffectofDamageandtheDisplacementJumps
3.EquivalentMaterialSymmetryGroup,InducedAnisotropyandConsequences
4.DefinitionoftheCrackOpeningModeofaMicrocrackedBodyintheFrameworkofDamageMechanics
4.1.ExtensionoftheCrackOpeningModeofFractureMechanics
4.2.ExpressionoftheModeandRadiusVariablesUsingMacroscopicDis-placementJumps
4.3.ExpressionoftheModeandRadiusVariablesUsingMacroscopicStrainorStress
4.4.IllustrationoftheCrackOpeningModeforDamageMechanicsandRemarks
5.Extension:TakingintoAccounttheIrreversibleStrainPhenomenon
6.Application:AGeneralHyperelasticBehaviorLawforaMicrocrackedBody
6.1.FrameworkoftheModel
6.2.VectorialDescriptionoftheDamagePhenomenon
6.3.StateFunction
6.4.StateLawsandJustificationoftheHyperelasticCharacteroftheBehaviorLaw
6.5.DamageEvolutionLaw
6.6.ContinuityandConvexityConditions
6.7.RemarksontheInducedAnisotropyandtheDeactivationofDamage
6.8.IdentificationoftheModel
6.9.Application
6.9.1.DescriptionoftheMaterialandDamagePhenomenon
6.9.2.IdentificationoftheBehaviorLaw
6.9.3.IdentificationoftheDamageEvolutionLaw
6.9.4.CharacteristicFunctioningoftheModel
6.9.5.ComparisonbetweenExperienceandSimulation
7.Conclusion
8.Appendix:BuildingoftheObjectiveStateFunction[35][36]
References
INDEX

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PHYSICS RESEARCH AND TECHNOLOGY

THEORETICAL PHYSICS RESEARCH DEVELOPMENTS

No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, 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 herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

PHYSICS RESEARCH AND TECHNOLOGY Additional books in this series can be found on Nova’s website under the Series tab.

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PHYSICS RESEARCH AND TECHNOLOGY

THEORETICAL PHYSICS RESEARCH DEVELOPMENTS

JOHN P. SULLIVAN AND

ANDREW L. MONTEY EDITORS

Nova Science Publishers, Inc. New York

Copyright © 2011 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. 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 Theoretical physics research developments / editors, John P. Sullivan and Andrew L. Montey. p. cm. Includes index. ISBN 978-1-62257-116-1 (E-Book) 1. Physics. I. Sullivan, John P. II. Montey, Andrew L. III. Title. QC28.T44 2011 530--dc22 2011003828

Published by Nova Science Publishers, Inc. † New York

CONTENTS   Preface

vii 

Chapter 1

Commentary on the Prior Errors for Conservation of Momentum Süleyman Mazlum and Nazire Mazlum 

Chapter 2

Dark Energy as the Source of the Time Dependent Einstein Cosmological Constant Miroslaw Kozlowski and Janina Marciak-Kozlowska 



Non-Linear Refractive Index Theory: I-Generalized Optical Extinction Theory and Dispersion Relations S. S. Hassan, R. K. Bullough and R. Saunders 

25 

Quantum Coherence and Tunable Transient Behavior of a Double-Control Four-Level Atomic Vapor Arash Gharibi, Jing Gu and Jian Qi Shen 

41 

Chapter 3

Chapter 4



Chapter 5

Application of GEANT4 Code in Gamma Irradiation Processing O. Kadri, F. Gharbi, K. Farah and A. Trabelsi 

59 

Chapter 6

Study of Atomic Entanglement in Multimode Cavity Optics Papri Saha, N. Nayak and A. S. Majumdar 

75 

Chapter 7

Analytical Expression for P-3He Total Reaction Cross Section M. A. Alvi 

89 

Chapter 8

Radiative Transitions of Mesons in an Independant-Quark Potential Model S. N. Jena, H. H. Muni, P. K. Mohapatra and P. Panda 

Chapter 9

Laser Prototyping of Polymer-Based Nanoplasmonic Components Andrey L. Stepanov, Roman Kiyan, Carsten Reinhardt and Boris N. Chichkov 

97  119 

vi Chapter 10

Contents An Approximation Method for Determination of the Coupling Constant and Charged Pion Form Factor at Q2 =0.6-1.6GeV2 and Q2 =2.45 GeV B. Rezaei 

2

133 

Chapter 11

Hubbard Model and Ferroelectric Phase Transition in Graphene M. B. Belonenko, N. G. Lebedev and N. N. Yanyushkina 

Chapter 12

Study of Pressure Dependence Superconducting State Parameters of Binary Metallic Glasses by Pseudopotential Theory M. Aditya Vora 

151 

Control Over Group Velocity in a Three Level Closed Ladder System with Spontaneously Generated Coherence Sulagna Dutta and Krishna Rai Dastidar 

167 

Modelling of Microcracked Bodies Using the Concept of Crack Opening Mode A. Thionnet 

183 

Chapter 13

Chapter 14

Index

141 

223 

PREFACE This new book presents and discusses current research developments in the study of theoretical physics. Topics discussed include dark energy as the source of the time-dependent Einstein cosmological constant; non-linear refractive index theory; quantum coherence and tunable transient behavior of a double-control four-level atomic vapor; laser prototyping of polymer-based nanoplasmonic components and radiative transitions of mesons in an independant-quark potential model. Chapter 1 - Momentum is the force of a fluid element because of its motion. For incompressible fluids, momentum of fluids at motion, in a cross-section, is expressed as M = ρ . Q . v . At fluid flows having nonlinear and nonparallel streamlines momentum is subject to variations along the flow direction. Conservation of momentum, for dynamic conditions, is commonly derived from Newton’s second law of motion. Chapter 2 - In this paper the authors calculate the Einstein cosmological constant. Following our results presented in the monograph: “From quark to bulk matter”, they obtain a new formula for the time dependent cosmological constant, Λ. Following the general formula for the time dependent Λ, the authors can describe the history of the cosmological constant. Chapter 3 - Starting from the Bloch-Maxwell equations for two-level atoms forming an extended system, taken as a parallele-sided slab for a Fabry-Perot cavity configuration, the authors generalize the optical extinction theorem to the non-linear regime. Generalized form of the Lorentz-Lorenz dispersion relation for the refractive index (m) is derived. Within the context of optical multistability phenomenon, the input-output field relationship is derived in terms of (m). Chapter 4 - The tunable transient evolutional behavior of a double-control four-level atomic vapor is studied and its application to photonic logic gates is suggested. It is shown that the quantum interferences between two external control fields can lead to controllable absorption and transparency characteristics of a multilevel atomic vapor. One of the most remarkable properties involved is that of the absorption (or transmittance) of the probe field in the atomic vapor being dependent upon the ratio of the intensities of the two control fields, allowing the tunable optical features (transparency or opaqueness to the probe field) to be achieved by tuning the quantum interferences between the control transitions (driven by the control fields). The mechanisms in question are applicable to the design of a variety of new photonic and quantum optical devices such as logic and functional devices, optical switches as well as other photonic microcircuit components, which can be employed for controlling the

viii

John P. Sullivan and Andrew L. Montey

wave propagation of a probe field. Two typical photonic logic gates (OR and EXOR gates) based on the tunable four-level optical responses (quantum constructive and destructive interferences) are presented as illustrative examples. Chapter 5 - The present work presents an overview of application of the Monte Carlo code, GEANT4, in the gamma irradiation processing field. In order to check the validity of such code, a successful calculation of expected dose rate and photon flux in the Tunisian gamma irradiation facility was carried out. In the same course of study, an ample set of comparison tests were done using the PMMA dosimeters and the GEANT4 version 8.2 code, for measurement and calculation purposes. Thus, the excellent agreement seen between data and calculations allow us to apply the GEANT4-based tool in order to optimize some process parameters, specific to the studied Co facility, and to systematically improve the dose uniformity within irradiated targets having different densities and volumes. Therefore, three irradiation processing procedures were studied let us to conclude that for a given carrier dimensions, more the product density is higher than a determined value, more a specific procedure will be performed. It is shown that Monte Carlo simulation improves the gamma irradiation process understanding. Chapter 6 - In this work the resonant interaction between two two-level atoms and melectromagnetic modes in a cavity is considered. The authors have considered an ideal cavity that is cavity losses are assumed to be zero. Such a system can be achieved in a microwave cavity with pairs of Rydberg atoms having long lifetime of the order of 0.2 seconds. Entanglement dynamics between two atoms is also examined. In particular, they compare dynamical variations for different cavity modes as well as for different cavity photon numbers. The collapse and revival of entanglement is exhibited by varying the atom-photon interaction times. Chapter 7 - Applying Coulomb correction factor to the Glauber multiple scattering model an analytical expression has been derived for microscopic study of proton-3He total reaction cross section. Using independent single particle model for the density of 3He, reasonably good account of the data has been achieved. Chapter 8 - Taking an independent-quark potential model with its parameters determined from a fit to the mass splittings of ground state meson masses in the strange, charm and bottom flavour sector, the authors investigate the decay widths of the radiative transitions among the vector and pseudoscalar mesons. The authors go beyond static approximation to perform a more realistic calculation of decay widths incorporating some momentum dependence due to recoil of the daughter meson. The results obtained in case of light mesons are in reasonable agreement with the experimental values. The predictions in heavy meson decays are comparable to those of other model predictions. Chapter 9 - Growing interest in the field of surface plasmon polaritons comes from a rapid advance of nanostructuring technologies. The application of two-photon polymerization by pulse laser technique for the fabrication of dielectric and metallic SPP-structures, which can be used for localization, guiding, and manipulation of plasmonic waves on a subwavelength scale, are studied. This fast technology is based on nonlinear absorption of nearinfrared femtosecond laser pulses. Excitation, propagation, and interaction of SPP waves with nanostructures are controlled and studied by leakage radiation imaging. It is demonstrated that created nanostructures are very efficient for the excitation and focusing of plasmonic waves on the metal film. Examples of various plasmonic components are presented and discussed.

Preface

ix

Chapter 10 – The authors represent an approximation method for the charged pion form factor, Fπ , and pion nucleon coupling constant, g πNN (t) , by the longitudinal cross section data analysis for the reaction H(e,e′π + )n at values of the transferred four momentum,

Q 2 =2.45 and Q2 =1.6 GeV2 with invariant mass of the photon- nucleon system of W=2.22 GeV and also at Q2 =0.6-1.6 GeV2 with invariant mass of the photon- nucleon system of W=1.95 GeV . They show an approximation polynomial solution for the coupling constant gives us good solutions for the pion form factor within uncertainties. Chapter 11 - The normal transverse electric field which appears in graphene with Hubbard interaction spontaneously in the presence of a high applied electric field was calculated. The given effect can be associated with non-equilibrium of electron subsystem in graphene. The characteristics of spontaneous field on the parameters of the problem were investigated. Chapter 12 - Theoretical computation of the pressure dependence superconducting state parameters of binary Ca60Al40 and Ca70Zn30 metallic glasses are reported using model potential formalism. Explicit expressions have been derived for the volume dependence of the electron–phonon coupling strength and the Coulomb pseudopotential considering the variation of Fermi momentum and Debye temperature with volume. Well known Ashcroft’s empty core (EMC) model pseudopotential and five different types of the local field correction functions viz. Hartree (H), Taylor (T), Ichimaru-Utsumi (IU), Farid et al. (F) and Sarkar et al. (S) have been used for obtaining pressure dependence of transition temperature and the logarithmic volume derivative of the effective interaction strength for metallic glass superconductor. It has been observed that decreases rapidly with increase of pressure. Chapter 13 – The authors investigate the group velocity of the probe pulse in a closed three level Ladder system with spontaneously generated coherence (SGC). For certain values of the system parameters, high refractive index without absorption has been obtained. The authors have shown that both the subluminal and superluminal propagation can be accompanied with absorption as well as gain without inversion (GWI) of the probe pulse. They have proposed a scheme for experimental realization of SGC in a Ladder system. Chapter 14 - Damage present in a material in the form of plane cracks induces the decrease of some of its mechanical properties. For example, in the case of a tensile loading, and according to the plane in which the cracks are located, a decrease of the axial modulus can be observed. The same observation can be made for the shear modulus in the case of a shear loading. If the sign of the loading changes, the axial modulus is restored. It is not the case for the shear modulus. This effect is usually called the unilateral effect of damage or damage activation/deactivation. Versions of these chapters were also published in International Journal of Theoretical Physics, Group Theory, and Nonlinear Optics, Volume 14, Numbers 1-4, edited by Renat R. Letfullin, Guoping Zhang, and Thomas F. George, published by Nova Science Publishers, Inc. They were submitted for appropriate modifications in an effort to encourage wider dissemination of research.

In: Theoretical Physics Research Developments Editors: J. P. Sullivan and A. L. Montey

ISBN 978-1-61209-446-5 © 2011 Nova Science Publishers, Inc.

Chapter 1

COMMENTARY ON THE PRIOR ERRORS FOR CONSERVATION OF MOMENTUM Süleyman Mazlum* and Nazire Mazlum Süleyman Demirel University, Engineering and Architecture Faculty, Environmental Engineering Dept., Isparta, Turkey

1. Foreword and Problem Statement Momentum is the force of a fluid element because of its motion. For incompressible fluids, momentum of fluids at motion, in a cross-section, is expressed as M = ρ .Q .v

(1)

At fluid flows having nonlinear and nonparallel streamlines momentum is subject to variations along the flow direction. Conservation of momentum, for dynamic conditions, is commonly derived from Newton’s second law of motion as below F = m .a

(2)

da dF dm .m .a + = dt dt dt

(3)

da =0 dt

(4)

Time-derivative of Eq.[2]

If the acceleration doesn’t change

Thus Eq.[3] becomes *

Email address: [email protected]

2

Süleyman Mazlum and Nazire Mazlum dF dm = .a dt dt

Or replacing acceleration

" a"

by time-derivative of velocity

(5) "

dv " dt gives

dF dm dv = . dt dt dt

(6)

And then expressing the mass of the fluid as the product of density and volume (m = ρ .V ) , Eq.[6] becomes dF dV dv . = ρ. dt dt dt

(7)

dV =Q dt

(8)

dF dv = ρ .Q. dt dt

(9)

dF = ρ .Q .dv

(10)

Rate of flow is expressed as

Replacing Eq.[8] into [7] gives

Simplifying Eq.[9]

Integrating Eq.[10] within the boundaries F

∫ 0

v2



dF = ρ .Q . dv v1

(11)

Integration yields the following analytical equation F = ρ .Q .( v2 − v1 )

(12)

F = ρ . Q . v 2 − ρ .Q.v1

(13)

Rearranging gives

Substituting momentum symbol " M " for the pertaining terms in Eq.[13] gives F = M2 − M1

Or

(14)

Commentary on the Prior Errors for Conservation of Momentum

3

F + M1 − M2 = 0

(15)

Eq.[14] suggests that a change in force results in a change in momentum (that is to say “in velocity”), or a change in momentum results in a change in force, such that the total forces acting on a fluid element are always zero and thus the system always remains balanced. However the final equation, (Eq.[14] or [15]), that is just only derived from Newton’s second law, as published in the common literature, doesn’t tell about the details on the distribution “on the components” of the whole forces (F ) acting on the fluid element: more precisely, it doesn’t even inform how the directions of acting on the control volume should be selected (i.e.,

FP ,1

FP ,1

and

FP ,2

hydrostatic forces

should be positive and

be negative in Eq.[15]). If one claims that we should have sense to feel that F

FP ,2

FP ,1

should

and

FP ,2

F

should be in opposite directions together with the claim that P ,1 is positive and P ,2 is negative (whether it is true or not!), some others may claim that “why shouldn’t we have also the same sense to feel that M 1 and M 2 should be in opposite directions too without making above all strenuous mathematical derivations? The equations on the conservation of momentum based on the derivation on Newton’s second law, as demonstrated above, have always been erroneous because such a derivation is not able to demonstrate the all other forces in question during the flow and also how all those forces acting on the fluid element should be introduced into the Eq.[15] (that is, which force terms should be positive and which term should be negative). Because of those ambiguities, it F is commonly but mistakenly considered and reported that P ,1 and M 1 should act from the FP ,2

and M 2 should act from the opposite direction to the fluid element. In a given direction, considering the total resistant force exerted by the pipe (Fr ) , and ignoring the weight of the fluid control element (W x ) , the proposed momentum equation in simplified form is presented in Eq.[16], [17] and [18]. same direction and

P1 . A1 + ρ . Q . v1 − ρ . Q . v 2 . − P2 . A2 − Fr = 0

(16)

FP ,1 + M 1 − FP ,2 − M 2 − Fr = 0

(17)

FP ,1 + M 1 = FP ,2 + M 2 + Fr

(18)

Or

Or

4

Süleyman Mazlum and Nazire Mazlum

2. Comprehensive Derivation of Conservation Equation for Momentum on A Fluid Element The equation on the conservation of momentum should start with applying Newton’s 2.law on a fluid element at motion just as done in the conservation of energy. However, for momentum, it is not needed to deal with the components of the forces (dP and dA) but the forces themselves (dF ) to get the solution for momentum equation. The approach used to derive momentum equation applying Newton’s 2.law on fluid element at motion is depicted in Figure.1

Figure 1. Forces acting on the fluid element flowing upward with a tilted slope.

The total forces acting on the fluid element on the Newton’s second law can be depicted as,

∑ dF

x

= dm .

dv dt

(19)

Or FP − ( FP − dFP ) − dW x − dFr = dm.

dv dt

(20)

Considering that fluid elements flow linearly and have parallel streamlines to each other gets rid of dFl term and reduces dFr to only dFl

Commentary on the Prior Errors for Conservation of Momentum

5

dv dt

(21)

dV .dv dt

(22)

dF P − dW . cos θ − dFl = ρ .dV .

Or dFP − dW . cos θ − dFl = ρ .

Since dV =Q dt

(23)

dFP − dW . cos θ − dFl = ρ .Q.dv

(24)

Then Eq.[22] take the following form

Integration of Eq.[24] within boundaries F p ,2



W



dFP − cos θ . dW − 0

F p ,1

Fl



v2



dFl = ρ .Q . dv

0

v1

(25)

Integration yields the following FP ,2 − FP ,1 − W . cos θ − Fl = ρ .Q.v 2 − ρ .Q.v1

(26)

Ignoring the weight force of the fluid volume ( W ) and the local impulsive forces (Fl ) , Eq.[26) reduces to FP ,2 − FP ,1 = ρ .Q.v 2 − ρ .Q.v1 (27) Or FP ,2 − FP ,1 − ρ .Q .v2 + ρ .Q.v1 = 0

(28)

FP ,1 − M 1 − FP ,2 + M 2 = 0

(29)

Or

F As can be seen, P ,1 and M 1 act on the opposite directions; not in the same direction. F Similarly, P ,2 and M 2 apply in opposite directions.

If the fluid element is moving downwards as in Figure.2, then weight force of the fluid element is opposite to that of slanted upward flow conditions (Eq.[20]) as demonstrated below:

6

Süleyman Mazlum and Nazire Mazlum

Figure 2. Forces acting on the fluid element flowing downward with a slope.

According to laws of thermodynamics, the energy is always less in the flow direction that makes the motion possible. The total forces acting on the fluid element in Figure.2 becomes FP − ( FP − dF P ) + dW x − dFl = dm .

dv dt

(30)

Following simplification and then integration, the analytical equation becomes FP ,1 − FP ,2 + M 2 − M 1 − W . cos θ − Fl = 0

(

(31)

)

F and F f are taken into consideration, If both local and frictional impulsive forces l Fr = Fl + F f and total resistant force is introduced into Eq.[31] to substitute for Fl , it becomes

(

)

(

)

FP ,1 − FP ,2 + M 2 − M 1 − W . cos θ − F f + Fl = 0

(32)

Or Eq.[32] can be rewritten as FP ,1 − FP ,2 + M 2 − M 1 − W . cos θ − Fr = 0

(33)

Commentary on the Prior Errors for Conservation of Momentum

Symbols and Notation F

FP Fr F

f

Force acting on the fluid element, N Hydrostatic force acting on the fluid element, N Total resistant force acting during the flow of fluid element, N Frictional force acting during the flow of the fluid element, N

P

Local (turbulent or minor) forces acting during the flow of fluid element , N Weight force of the fluid element , N Momentum, N Pressure on the fluid element, Pa

m

Mass of the fluid element, kg

V

3 Volume of the fluid element, m

A

Area to where the pressure is applied, m −2 Acceleration of the fluid element, m.s

Fl W

M

a

ρ g t

v x

z h

H L

Q θ

D ΔH

2

−3 Density of the fluid, kg .m

Acceleration due to gravity, m.s Time, s −1 Velocity of fluid element, m.s Travel distance, m

−2

Distance of the fluid element from datum, m Pressure height on the fluid element, m Total depth of the water in reservoir or depth of a tank , m The length that the matter, fluid element, travels , m 3 −1 Flowrate, m .s Angle of the fluid element, Diameter, m Head loss, m

o

7

In: Theoretical Physics Research Developments Editors: J. P. Sullivan and A. L. Montey

ISBN 978-1-61209-446-5 © 2011 Nova Science Publishers, Inc.

Chapter 2

DARK ENERGY AS THE SOURCE OF THE TIME DEPENDENT EINSTEIN COSMOLOGICAL CONSTANT Miroslaw Kozlowski 1,* and Janina Marciak-Kozlowska 2 1

Physics Department, Warsaw University, 00-132 Warsaw, Grzybowska 5/804, Poland 2 Institute of Electron Technology, 02-668 Warsaw, Al. Lotnikow 32/46, Poland

Abstract In this paper we calculate the Einstein cosmological constant. Following our results presented in the monograph: “From quark to bulk matter”, we obtain a new formula for the time dependent cosmological constant, Λ. Following the general formula for the time dependent Λ, we can describe the history of the cosmological constant.

Introduction In this paper following the results obtained in our monograph: “From quark to bulk matter”[1], we develop a model for the time dependent Einstein cosmological constant Λ. We argue that the dark energy can be approximated as the result the dark force with the coupling constant –G, with G Newton’s gravitational constant. As the result we obtain the new formula for the Λ

Λ = 3 (4π N + 3π ) L p ct

(1)

In formula (1) Lp is the Planck length, c is the vacuum light velocity, t is time and N is the natural number. The calculated values: R – Universe radius, υ – velocity of the Universe expansion and t -time are presented in Table1:

*

E-mail address: [email protected]

10

Miroslaw Kozlowski and Janina Marciak-Kozlowska Table 1. Main results T [τPlanck] 5 1020 1060

R [m] 10-35 10-15 1025

υ [c] 0.8 0.8 0.8

Λ [m-2] 1068 > 1 the υ is relatively constant υ ∼ 0.88 c. From formulae (38) and (42) the Hubble parameter H, and the age of our Universe can be calculated

υ = HR,

H=

1 = 5 ⋅10−18 s −1 , 2Mτ p

T = 2 M τ p = 2 ⋅1017 s ~ 1010 years,

(46)

Dark Energy as the Source of the Time Dependent Einstein Cosmological…

19

which is in quite good agreement with recent measurement [13, 14, 15]. As is well known in the de Sitter universe the cosmological constant Λ is a function of R, the radius of the Universe,

Λ=

3 . R2

(47)

Substituting formula (38) to formula (47) we obtain

Λ=

3 , N = 0,1, 2.... π N 2 L2p

(48)

Figure 1 a, b, c. The calulated values of the Spacetime radius (a) , Acceleration (b) and Cosmological constant (c) for times 0 to 105 Planck times.

20

Miroslaw Kozlowski and Janina Marciak-Kozlowska

The result of the calculation of the radius of the Universe, R, the acceleration of the spacetime, a, and the cosmological constant, Λ are presented in Figures. 1, 2, 3, 4 for different values of number N. As can be easily seen the values of a and R are in very good agreement with observational data for present Epoch. As far as it is concerned cosmological constant Λ for the firs time we obtain, the history of cosmological constant from the Beginning to the present Epoch.

Figure 2 a, b, c. The calulated values of the Spacetime radius (a), Acceleration (b) and Cosmological constant (c) for times 0 to 1020 Planck times.

Dark Energy as the Source of the Time Dependent Einstein Cosmological…

21

Figure 3 a, b, c. The calulated values of the Spacetime radius (a), Acceleration (b) and Cosmological constant (c) for times 0 to 1060 Planck times.

22

Miroslaw Kozlowski and Janina Marciak-Kozlowska

Figure 4 a, b, c. The calulated values of the Spacetime radius (a), Acceleration (b) and Cosmological constant (c) for times 0 to 10100 Planck times.

Dark Energy as the Source of the Time Dependent Einstein Cosmological…

23

References [1] Kozlowski M.; Marciak – Kozlowska J. Thermal Processes Using Attosecond Laser Pulses; Optical Science 121; Springer: New York, NY, 2006. [2] Adam G. Riess, Alexei V. Filippenko, Peter Challis, Alejandro Clocchiatti, Alan Diercks, Peter M. Garnavich, Ron L. Gilliland, Craig J. Hogan, Saurabh Jha, Robert P. Kirshner, B. Leibundgut, M. M. Phillips, David Reiss, Brian P. Schmidt, Robert A. Schommer, R. Chris Smith, J. Spyromilio, Christopher Stubbs, Nicholas B. Suntzeff, and John Tonry. Astron. J. 1998, 116, 1009. [3] Glenn Starkman, Mark Trodden, and Tanmay Vachaspati. Phys. Rev. Lett. 1999, 83, 1510. [4] Marciak-Kozłowska, J. Kozlowski M Found. Phys. Lett. 1997, 10, 295. [5] Kozlowski, M. ; Marciak – Kozlowska, J. Found. Phys. Lett. 1997, 10, 599. [6] Dirac, P. A. M. Nature (London) 1937, 139, 323. [7] Damour, T.; Gibbons, G. W.; Gundach, C. Phys. Rev. Lett. 1990, 64, 123. [8] Damour, T.; Esposito Farèse G. Esposito, Classical Quantum Gravity 1992, 9, 2093. [9] La, D.; Steinhard, P. J. Phys. Rev. Lett. 1989, 62, 376. [10] Barton, G. In Elements of Green's functions and propagation, Oxford Science Publications; Clarendon Press: Oxford, 1995; p. 222. [11] Descartes, R. Meditions on the first philosophy. In A discourse on method etc.; Deut: London, 1912; pp 107-108. [12] Whitrow, G. J. In The natural philosophy of time, 2nd Ed.; Oxford Science Publications; Oxford, 1990; p. 204. [13] Cayrel, R. Hill, V. Beers, T. C. Barbuy, B. Spite, M. Spite, F. Plez, Andersen, J. Bonifacio, P. François, P. Molaro, P. Nordström B. and F. Primas Nature, 2001, 409, 691. [14] Spergel D N, Bolte M. W Freedman, PNAS, 1997, 94, 6579. [15] John D. Anderson, Philip A. Laing, Eunice L. Lau, Anthony S. Liu, Michael Martin Nieto, and Slava G. Turyshev Phys. Rev. Lett., 1998, 81, 2858.

In: Theoretical Physics Research Developments Editors: J. P. Sullivan and A. L. Montey

ISBN: 978-1-61209-446-5 c 2011 Nova Science Publishers, Inc.

Chapter 3

N ON -L INEAR R EFRACTIVE I NDEX T HEORY: I - G ENERALIZED O PTICAL E XTINCTION T HEORY AND D ISPERSION R ELATIONS S.S. Hassan∗, R.K. Bullough† and R. Saunders 1 University of Bahrain, College of Science, Department of Mathematics, Kingdom of Bahrain 2 Manchester University, Mathematics Department, Manchester, United Kingdom 3 Manchester Metropolitan University, Faculty of Science and Engineering, Department of Computing and Mathematics, Manchester, United Kingdom

Abstract Starting from the Bloch-Maxwell equations for two-level atoms forming an extended system, taken as a parallele-sided slab for a Fabry-Perot cavity configuration, we generalize the optical extinction theorem to the non-linear regime. Generalized form of the Lorentz-Lorenz dispersion relation for the refractive index (m) is derived. Within the context of optical multistability phenomenon, the input-output field relationship is derived in terms of (m).

1.

Introduction

In this two part series of papers we present a comprehensive report of the fundamental theoretical study of optical multistability (OM) in a non-linear Fabry-Perot (F-P) cavity in the normal vacuum state of the electromagnetic field. The starting point is the coupled Bloch-Maxwell equations for a system of 2-level atoms coupled to the electromagnetic field in an extended medium. The cavity is defined by the region (a parallel sided slab) occupied by the atoms and there are no additional mirrors. A self-consistent scheme is developed ∗ E-mail † E-mail

address: [email protected] (Corresponding author) address: Ceased on 30th August (2008).

26

S.S. Hassan, R.K. Bullough and R. Saunders

which allows subsequent numerical analysis. Very brief reports of some of the analysis presented here have been made in the past [1]-[4] but a complete report of our theory with all the necessary details has never been made. In our theory the F-P cavity is defined by the parallel-sided slab occupied by atoms. No macroscopic Maxwell boundary conditions are imposed at the surface of this slab. Instead, the more fundamental boundary conditions of outgoing free fields at infinity are imposed. The theory yields a non-linear refractive index (m), together with the Fresnel reflection and transmission coefficients expressed in terms of (m). In the many-body sense, the theory is therefore a microscopic theory rather than a macroscopic theory. A comparable theory of the F-P cavity but in the linear optical regime has been reported at some length [5][8] in an all order many-body theory involving all orders of the correlation between the atoms. In this extension, to incident fields intense enough to create non-linear optical multibistability, we restrict the many-body side of the analysis to a one-body theory, except for the occasional remarks connecting the results [5]-[8] of the linear theory. Thus the starting point is the Bloch-Maxwell system defined smoothly at points x inside the slab of the FP cavity [9]. The presented non-linear theory of this paper is therefore a coarse-grained theory where no inter-atomic correlations (2-body, three body etc.) appear. Within the onebody approximations the non-linear theory developed in the paper is essentially exact for two–level atoms. The analytical results of the theory, presented in part (I) of this series, and confirmed by the subsequence numerical work in part (II), show that the cavity detuning depends on the input intensity. The non-linear behavior of the F-P cavity arises from the intensity dependency of the non-linear refractive index, (m), that emerges from the theory through an application of the fundamental “optical extinction theorem” originally due to Ewald-Oseen (cf. references [10]-[14] and references [5]-[8] for the equivalent linear theory). The nonlinear action of the F-B system at the boundaries of the slab-like cavity are derived from the fundamental Bloch-Maxwell system of equations with the outgoing boundary conditions. Also, we show how the derived non-linear refractive index (m) consistently describes the FP cavity action in a non-linear way, leading to multistability, and we see certain corrections to this explicitly in solving atomic inversions. This non-linear action of the F-P cavity contrasts with the non-linear action of a ring cavity where the non-linearity is provided by optical feedback. Part (I) of this series presents the theoretical generalization of the optical extinction theory to the non-linear regime in the normal vacuum state of the electromagnetic field, while part (II) concentrates on a self –consistent numerical scheme to investigate multistable behavior, utilizing the analytical formulas derived in part (I). The organization of this paper, part (I), is as follows. In section 2 we write down the Bloch-Maxwell system of equations in the one-body form explained above. These are coupled partial differential equations in which Maxwell’s equations are linear and can be conveniently put into integral form. In section 3 we make the necessary generalization of the optical extinction theory to the non-linear regime, together with the generalized form of the Lorentz-Lorenz dispersion relation for the refractive index (m). In section 4 we consider a low atomic density approximation to the analytical expressions derived in section 3. This way we draw a corresponding between the present analysis and previous work , ([15], [16]) on OM for a Fabry-Perot cavity which involve spatial mean-field ([17]- [19]) or

Non-linear Refractive Index Theory

27

other approximations: see ([20]). A summary is given in section 5. Appendix (A) provides the derivation for the atomic polarization field, equation (3.3).

2.

Model Equations

Our starting point is the coupled Bloch-Maxwell system of equations for an extended system of many 2-level atoms coupled to the electromagnetic field and taken in the ‘one-body’ approximation in which elements r± (x,t), r3 (x,t) of the atomic Bloch vectors (defined below) are densities with a smooth dependence on the position x of the atoms. In the first instance these are taken without any slowly varying envelope approximation. Later we find precisely what envelope description should be used-one which is essentially exact. In a frame rotating at an angular frequency ω (namely the frequency of a harmonic incident laser field) the c-number Bloch equations for the atoms, in the rotating wave approximation and in the electric dipole approximation are [9] (also see later derivations in the general case of squeezed vacuum state of the electromagnetic field, [21],[22]). h i ∂r+ (x,t) + γ (1 + iδ) r+ (x,t) = 2i E∗ (x,t) − Ωe−ik0 z r3 (x,t) ∂t h i ∂r3 (x,t) + γ (1 + 2r3 (x,t)) = −i E∗ (x,t) − Ωe−ik0 z r− (x,t) + c.c. ∂t

(2.1a) (2.1b)

with r− = (r+)∗ (ω−ω ) The symbols are: γ is the A-coefficient, δ = γ 0 is the normalized atomic detuning, ω0 is the two-level atomic frequency, Ω ≡ p~−1Eo is the amplitude of the incident laser field expressed as a Rabi frequency, p is the magnitude of the atomic dipole matrix element.  In the laboratory frame, the atomic dipole density is n0 pu r+ eiωt + r− e−iωt where uˆ is a unit vector along p, n0 is the atomic number density: r3 (x,t) is the density of the atomic inversion. In this laboratory frame, Maxwell’s equation is curl curl E (x,t) +

∂2 1 ∂2 E (x,t) = −4πn P (x,t). 0 c2 ∂t 2 ∂t 2

(2.2)

The Maxwell equation (2.2) is linear and has a well known Green’s function ([5]-[9],[13], [14]). As long as we are concerned throughout with time dependence e±iωt we can work with the Green’s functions F which is the second rank tensor  eik0 r   , F x, x ; ω = ∇ ∇ + k02 U r 0

(2.3)

in which r = |x−x0 |, U is the unit tensor, and k0 = ωc−1 . The form of the harmonic laser   field assumed is evidently E0 ei(ωt−k0 z) + e−i(ωt−k0 z) with E0 a real amplitude. Thus the total field driving the Bloch equations  (2.1), expressed as a Rabi frequency, has negative frequency part E∗ (x, ω) − Ωe−ik0 z in which [9], E (x, ω) = n0 p ~ ∗

2 −1

Z V −v

 F ∗ x, x0 ; ω : u u r+ (x0, ω) dx0 .

(2.4)

28

S.S. Hassan, R.K. Bullough and R. Saunders

For simplicity we assume that all fields have the unique polarization direction u of the atomic dipole matrix elements. In principle it is possible to average over the assumed isotropic distribution of atomic polarization directions but this complication, in the onebody theory, is easily introduced later in the expressions we shall reach in section 3 for the non-linear refractive index (m). Notice that in the expression (2.4) for E ∗ , the complex conjugate of the atomic polarizing field, a small sphere ν is extracted from the region of integration V, the parallel-sided slab, at the point x where an atom is supposed to be. This sphere acts like the famous Lorentz sphere introduced by Lorentz in linear dielectric theory and is the source of the Lorentz polarizing field (different from the Maxwell field inside the medium [5]-[8]) appearing in the present theory. However, it is mathematically necessary to extract ν from V in Eq.(2.4) since the integral is only conditionally convergent: [5]-[8]: F behaves like r−3 as r → 0. Thus ν is a sphere of vanishingly small radius. Although this situation surrounding the polarization field appears to be non-physical this is not the case. The small sphere acts in the theory precisely as does the Lorentz’s physically based sphere and there is a correction involving the local correlation between the atoms (as in the linear theory [5]-[8]). This correction which is believed small even in the non-linear theory is beyond the one-body theory and is neglected in the present non-linear analysis. All the many-body theoretical details, involving correlation that make physical sense of the sphere ν, in the linear case are carried out in references [5]-[8]. In the following section 3 we outline the steps of the solution of the Bloch equations (2.1) driven by the polarizing field of Eq.(2.4). Since Eq.(2.4) depends under the integral sign on r+ (x, ω) this system of equations is a non-linear integro-differential systems. As we carry out these steps in section III we make further references to the comparable linear theory ([5]-[8]) whenever this seems to be helpful for the understanding of the non-linear theory. Notice that there are radiative damping terms already contained in the Bloch equations (2.1) These describe optical scattering from non-linear medium. The simplest of the manybody theoretical correlation corrections to (2.4) adds a further integral like that in (2.4) in which however the pair correlation function for pairs of atoms g (x, x0 )appears in the form g2 (x, x0) − 1 under the integral sign (compare the linear theory in [5]-[8] and [9] ). This has the effect of changing the damping of the Bloch system to include pair-wise optical scattering from pairs of atoms in the dielectric. In the non-linear context corrections like this become still more complicated. Thus we neglect this at the one-body level in the nonlinear theory. We are now almost in a position to develop the non-linear theory at one-body level analytically, that is we formally solve the system of equations (2.1) with Eq.(2.4). We move the differential operator under the integral sign in (2.4) outside the integral so that [5]-[8],[12] Z  E∗ (x,ω) = F ∗ x, x0; ω : u u r+ (x0 , ω) dx0 2 −1 n0 p ~ V −v   Z eik0 r 4π 2 r r (x0 , ω) dx0. = (x, ω)+u u : ∇ ∇ + k0 U (2.5) 3 + r + V

The first term is entirely due to the small sphere and is the source of the Lorentz-Lorenz form of the expression we derive for the non-linear refractive index (m). The resulting

Non-linear Refractive Index Theory

29

integral is convergent and no small sphere need be removed from the region of integration, the cavity V. We now develop the optical extinction theorem and derive the expression for a nonlinear refractive index (m).

3.

Generalized Optical Extinction Theorem

In the present non-linear theory the atomic inversion density r3 (x) > − 12 , its ground state value in the steady state, and depends on x, the position of the cavity. It is this complication which makes the theory non-linear, for as a consequence of the radiation damping in the Bloch equations (2.1), all the field intensities must damp inside the cavity and since r3 (x,t) depends on these intensities r3 (x,t) necessarily depends on x. Consistency then requires that the polarization fields r± (x,t) are no longer simply of harmonic form in space. In the linear theory r± (x,t) depend on waves exp (±imk0 z) where (m) proves to be the linear refractive index. In the present non-linear theory we find we need to take as ansatz r+ (x,t) = Pf (z) e−imk0 z + ΛPb (z) eimk0 z ,

(3.1)

in response to the incident laser field e±ik0 z . The axis z is the axis of the F-P cavity, taken to be the slab bounded by the two surfaces z = 0 and z = L > 0, and the laser field is, for simplicity, normally incident upon the slab. This slab is the region of integration V (see Eq.(2.4)) and is infinite in value. However, one should really use a parallel-sided cylinder, for example, as the cavity with a large radius R orthogonal to z. Corrections due to the additional diffraction effects from the curved surface of the cylinder are not considered in our theory within the context of OM. In the linear theory [5],[6] these corrections are very complicated but are of small order macroscopically. Notice now that Eq.(3.1) means that a non-linear refractive index (m) has entered the theory. It is the purpose of the analysis to follow to find an expression for (m) and so find the input-output relation for the cavity expressed as far as possible in terms of this refractive index (m). Notice also that Eq.(3.1) has envelope functions Pf (z) (forward) and Pb (z) (backward) depending on z. Pf (z) and Pb (z) are envelopes assumed to be ’slowly varying’ on the scale of a wavelength λ = 2πk0−1 . In practise we approximate the analysis within a ’slowly varying envelop’ approximation by truncating the second and higher derivatives in z of both Pf (z) and Pb (z) (see below). The quantity Λ in Eq.(3.1) is a reflection coefficient which is to be determined. We now turn to the essence of the extinction theorem. The essential physical content of this theorem is that light in the absence of matter travels as a free field [5]-[8]. If it is a harmonic wave along z it has z–dependences exp (±ik0 z). If such waves enter a region occupied by matter, such as the region V of the F-P cavity, a new wave number mk0 6= k0 arises in both linear and non-linear theory inside V. Then scattering from the surface of V exactly cancels the free fields which in the absence of the matter would be inside V and replaces these free fields with fields with new wave numbers. In mathematical terms this exact cancellation of free fields provides the connection between the amplitude of the original free field and the amplitudes of the field induced, with a new wave number, inside

30

S.S. Hassan, R.K. Bullough and R. Saunders

the cavity V. At the same time the ‘refractive index’ (m) has to satisfy a second condition which means in practise that there is a formula, of Lorentz-Lorenz type, which fixes the values of (m). The mathematical analysis implementing this physics is carried out in various ways in the linear case in [5]-[8] and their references. How is it that the theory can be extended to the non-linear case? The point is that the Maxwell equations are linear and the field E ∗ of (2.5) depends on x through the Green’s function f (r) = exp (−ik0 r)/r under the integral (taken over x0 : r = |x − x0 |). By using the result that # " Z Z ±imk0 z0 0 0 e dx0 f (r)e±imk0 z dx0 = k2 (m12 −1) e±imk0 z ∇2x0 f (r) − f (r)∇2x0 0 r V

V

and applying Green’s Theorem one gets Z

0

f (r) e±imk0 z dx0 =

4π e±imk0 z k02 (m2 −1)

V

+ k2 (m12 −1) 0

where

R Σ

Z h

 i 0 0 e±imk0 z ∇x0 ( f (r)) .ds − f (r)ds.∇x0 e±imk0 z ,

(3.2)

Σ

is a surface integral, taken over the surface of V, which depends on the field point

x through the Green’s function f (r). This surface integral satisfies the free-field wave R equation ∇2 + k02 = 0. On the other hand, with m 6= 1 the remaining term in (3.2) plainly Σ  satisfies a differential wave equation ∇2 + m2 k02 e±imk0 z = 0. It follows that the terms in the theory split into two parts : one satisfying the free field wave equation and the other satisfying the different wave equation with m 6= 1. In the linear theory, the first free field condition implements the connection between scattering from dipole sources over the surface of V and the incident free field. The second condition fixes the parameter (m). All of this analysis goes through in the non-linear case also except that we need to take careful account of the envelope functions Pf (z) and Pb (z) modulating the waves exp (±imk0 z). In the appendix A we show how by inserting the ansatz of Eq.(3.1) into the expression (2.4) for the inter-atomic ‘polarizing field’ E ∗ (x,ω) we find the following expression (up to first derivatives): i h 2  0 E∗ (x, ω) 4π m +2 2im = (z) − P (z) e−imk0 z P f 2 3 k0 (m2 −1) f n0 p2 ~−1 (m −1)  0 i h 2im + (m21−1) I (Pf (z)) − k0 (m I Pf (z) 2 −1) + Λ (same terms with P f → Pb and m → −m) , where

0

0

Pf (z) = and I (h(z)) =

Z h Σ

dPf (z) dz

(3.3)

0

,

dP (z) Pb (z) = b , dz 0

i   0 0 h z0 e−imk0 z (∇z0 ( f (r)) .ds) − f (r)(ds.∇z0 ) h z0 e−imk0 z .

(3.4)

Non-linear Refractive Index Theory

31

Note that in Eq.(3.3) all of the surface integrals, as defined from  Eq.(3.4), depend only on exp (±ik0 z) and satisfy the free field wave equation ∇2 + k02 e±ik0 z = 0, with wave number k0 as we show below. On the other hand the remaining terms are envelope functions multiplying exp (±imk0 z) and these cannot satisfy a wave equation with m 6= 1. When Eq.(3.3) is inserted in the first of the steady state Bloch equations in Eq.(2.1a) with ˙r±,3 = 0, we have to equate separately the terms involving e±imk0 z and those involving e±ik0 z . In consequence we get, for the terms involving e±imk0 z , the first order differential equations, i h 2  0 8πin0 p2 m +2 2im (z) − P (z) r3 (z), (3.5a) P γ (1 + iδ)Pf (z)= ~(m f 2 −1) 3 k0 (m2 −1) f i h  0 8πin0 p2 m2 +2 2im (3.5b) Pb (z) + k0 (m γ (1 + iδ)Pb (z)= ~(m 2 −1) 2 −1) Pb (z) r3 (z). 3 In the linear case where P0f (z) = Pb0 (z) = 0 and r3 (z) ' − 12 , Eq.(3.5) then yields exactly the usual Lorentz-Lorenz dispersion relation 4πn0 p2 m2 − 1 = . m2 + 1 3~γ (i − δ)

(3.6)

But in the present non-linear case, Eqs.(3.5) couple to the spatially dependent r3 (z), and m is no longer determined. Since m is to be simply a number in the ansatz Eq.(3.1) and cannot depend on z, we expect that m can nevertheless be related to some z-independent inversion R3 by a relation like Eq.(3.6). The idea here is to introduce some ‘average’ inversion which best describes r3 (z) at all points z inside the slab. To do this we write Eq.(3.5a) , for example, in the form h 2  8πin0 p2 m +2 Pf (z) (R3 − R3 + r3 (z)) γ (1 + iδ) Pf (z) = ~(m 2 −1) 3 −k

0 2im 2 −1) Pf

0 (m

(z)r3 (z) ,

(3.7)

and chose m to satisfy a generalized form of Eq.(3.6), namely, 4πn0 p2 m2 − 1 = (−2R3 ). m2 + 1 3~γ (i − δ)

(3.8)

P0f (z) = fo (z)Pf (z) .

(3.9a)

Then, A similar argument for Eq.(3.5b) gives

where f o (z) is given by

Pb0 (z) = − fo (z) Pb (z) ,

(3.9b)

   k0 m2 − 1 m2 + 2 R3 1 − . fo (z) = 6mi r3 (z)

(3.10)

Equations (3.9) are formally integrated Pf (z)= Pf (0)exp

Z

z 0

 Z Pb (z)= Pb (0) exp −

0



fo z dz z 0

0



(3.11a) 

 fo z0 dz0 .

(3.11b)

32

S.S. Hassan, R.K. Bullough and R. Saunders

To proceed further we treat m as an adjustable parameter, so we choose it so that the F-P action survives in the non-linear case. It is sufficient that we take Pf ,b (0) = Pf ,b (L) = p0 (ω) .

(3.12)

This way we find that the parameter R3 is exactly given in the form R3 = R L

L

−1 0 0 0 r3 (z ) dz

,

(3.13)

that is 1/R 3 is an average of the inverse of r3 (z), but it is not the mean field average [18] (see also [9] and references therein). Note that in the linear theory r3 (z) = − 12 = R3 . Also, in the non-linear theory, Eq.(3.8) with Eq.(3.13) is still an exact result as we are not approximating R3 as an approximate average. Now from Eq.(2.1b) and within the same argument applied to Eq.(2.1a) and the use of Eqs.(3.9), and (3.10) we get h i 2  ∗ ∗ 2r3 (z) r3 (z) + 12 = − Pf (z) e−i(m−m )k0 z + |Λ|2 |Pb (z)|2 ei(m−m )k0 z +terms in e ±i(m+m

∗ )k

0z

(3.14)



The cross terms in e±i(m+m )k0 z appear because r3 (z) depends on the intensity in the cavity 2 which is proportional to Pf (z)e−imk0 z + ΛPb (z)eimk0 z , and are indications of the standing wave effects. Since this causes considerable complications we simply ignore the standing ∗ waves in this paper. The other terms in Eq.(3.14), e±i(m−m )k0 z , are damped and growing (damped backward going) wave profiles. From Eq.(3.8) the scale on which the damping takes place is roughly −i (m − m∗ ) ' α0

4R3 ; 1 + δ2

α0 =

4πn0 p2 ~γ

(3.15)

which is negative since R3 is negative. For Na vapour with n0 ∼ 1012cc−1 , αo ∼ 10−4, so −i (m − m∗ ) leads to damping by e−2π in a length L ∼ 104 cm and wave length ∼ 0.5 cm. Of course, the envelopes Pf ,b are to describe this damping, but we expect that their precise form do not change this picture too much. Thus, within the neglect of the standing waves, Eq.(3.14) with the use of Eq.(3.11) can be put in the form, h i  (3.16a) 2r3 (z) r3 (z) + 12 = − |Po |2 eh1 (z) + |Λ|2 e−h1 (z) , where  2 # 2 m + 2 m − 1 k0 z h1 (z)= −i (m − m∗ ) k0 z + 2Re 6mi # " 2 Z z  m2 − 1 (1 + iδ) γk0 r3−1 z0 dz0 . +2Re 2 −1 m 16πn0 p ~ 0 "

(3.16b)

Non-linear Refractive Index Theory

33

Now we make the second step in the extinction theorem and equate the terms in exp (±ik0 z) in Eq.(2.1a). First we recall that the surface integrals in Eq.(3.3) are evaluated for the parallel-sided slab geometry ((0 ≤ z ≤ L). The results are already available [5]. At the surface z0 = 0,  I Pf ,b (0) = −2πPf ,b (0) (1 ± m) e−ik0 z ,

(3.17a)

 I P0f ,b (0) = −2πP0f ,b (0) (1 ± m) e−ik0 z .

(3.17b)

 I Pf ,b (L) = −2πPf ,b (L) (1 ∓ m) e∓imk0 z e−ik0 (L−z) ,  I P0f ,b (L) = −2πP0 f ,b (L) (1 ∓ m) e∓imk0 z e−ik0 (L−z) .

(3.18a)

and similarly At the surface z0 = L,

(3.18b)

After substituting Eq.(3.17), Eq.(3.18) into Eq.(3.3) and then inserting the result in Eq.(2.1) and using Eqs.(3.9), (3.12) and hence comparing the coefficients of e±ik0 z we find respectively that   m − 1 −2imk0 L e , (3.19) Λ= m+1  2  p0 = h

1−



−Ω 2πn0 p2 ~−1

m2 +2 3



m −1 m2 +1

3 1 − r3R(0)

i 

1−

m−1 m+1

 . Λ

(3.20)

Equation (3.19) represents the Fresnel reflection coefficient, times the phase shift, while Eq.(3.20) means that the usual F-P cavity action takes place. For normal incidence on the F-P cavity (0 ≤ z ≤ L) we use the results of Eq.(3.12), Eq.(3.13) and Eq.(3.20), into Eq.(3.3) to obtain the input-output relation for z > L,  2  i  h 4π −i(m−1)k0 L R3 m +2   e 1 − 1 − 3 r3 (L) (1+m)2 i h    i , (3.21) Eout = Ein h 2 R3  −2imk0 L  1 − m2 +2 e 1 − m−1 1 − m+1 3 r3 (0) 2

where Ein = n~Ωp2 (so Eout is scaled by n0~p ). This is, together with a free field contribution 0 from the surface at z0 = 0, is just sufficient to annihilate the incident field everywhere. The factors in the inner square brackets that contain r3 (0) and r3 (L) reflects the nonlinear nature of the relations (3.21): indeed the non-linearity is in m since it depends on the 2 −2imk L 0 ]. number R3 . The F-P action is represented by the term in [1 − m−1 m+1 e The analytic formulae derived in this section, namely, Eq.(3.8), Eq.(3.11), Eq.(3.13), Eq.(3.16) and Eq.(3.19)-(3.21) are the main results of this part(I) paper. In the following section we show how the conventional optical bistability (OB) behavior lies within the obtained formulae of the present theory.

34

4.

S.S. Hassan, R.K. Bullough and R. Saunders

Optical Bistability in the Small Density Limit

We now derive some transparent form for the output-input relationship of Eq.(3.21), when the atomic density is small. First, we may write the expression 2      2 m − 1 2 −2imk0 L 2 −B 2 −B 2 θ−A , (4.1) e + 4 |R| e sin = 1 − |R| e 1 − m+1 2  iλ where R ≡ m−1 m+1 = |R|e , θ = 2 (λ + νπ − k0 L). With ν an integer and we have put 2 (m − 1) k0 L = A − iB.

(4.2)

For small atomic density n0 , such as in metal vapour, where m ' 1,we assume that m2 − 1 ' 2 (m − 1) and m2 + 2 ' 3. Hence from the generalized dispersion relation (3.8) one finds that    δ+i 4πn0 p2 (4.3) R3 . m−1 ' ~γ 1 + δ2 3 3 and B = −CR From Eq.(4.2) and Eq.(4.3) we then have A = CδR 2 > 0 where the ‘coop1+δ2 1+δ  eration’ number [17], [18] C = 2/ (τR γ) and τR ≡ ~/ 4πn0 p2 k0 L is the well-known cooperative super-fluorescence time [23]. For the second term in Eq.(4.1), θ is a scaled cavity detuning and A depends on both the atomic detuning δ and R3 and hence, via Eq.(3.13), Eq.(3.16), Eq.(3.20) and Eq.(3.21), on the input field intensity |Ein |2 . Consequently this second term is the source of dispersive OB. Note that C can be small for small n0 or small k0 L or both. Further, from Eq.(3.20) we approximately have

p0 '

−~Ω (m − 1)    2πn0 p2 1 − m−1 m+1 Λ

(4.4a)

if we consider the envelopes Pf ,b (z) and hence r3 (z) to be interpreted as mean average values [18], so, in this case, R3 ' r3 . Using Eq.(3.21) and Eq.(4.3) into Eq.(4.4a) we get Eout 2 r32 eB 2 . (4.4b) |p0 | ' δ 2im 2 2 2 (1 + δ ) (m+1)

With Eq.(4.4b) substituted into Eq.(3.14) we get for small enough n0 that, r3 = − 12

1 + δ2

, (4.5) 1 + δ2 + |X|2 2 −1  E 2  2 2 2im out is a scaled output intensity from the cavity. 2 where |X| ' δ 1 + |Λ| (m+1)2  ' (θ − A)2 and e−B ' 1 (small absorption) Further, for small n0 we put 4 sin2 θ−A 2 and the use of Eq.(4.5) and insertion of Eq.(4.1) into Eq.(3.21) yields then the usual cubic expression for the dispersive OB ([19], [24], [25])     !2    Cδ |R|2   (4.6) |Y |2 = |m|2 |X|2 1 +  , 2 θ + 2 2  1 + δ + |X|    1 − |R|2

Non-linear Refractive Index Theory 35    2  2 2 where |Y |2 = Eδin 1 + |Λ|2 / 2(1−|R| ) is the scaled output intensity. If the sine term is retained in Eq.(4.1) then the relation Eq.(4.6) becomes a multiply-branched function of |X|2 with the possibility of optical multistability. Also, if exp (−B) is retained as such in Eq.(4.1), Eq.(4.6) is then corrected by additional absorptive bistability.

Summary The central theme of the present work is to present formal analytical results for a non-linear refractive index theory of optical multistability. (OM). Within the formulation of the theory, the optical extinction theorem has been generalized to the non-linear regime. Apart from our preliminary results [1]-[4] and an earlier work by Bloembergen and Pershan [26] (but not within the context of OM), we know of no other application of this extinction theorem to non-linear optics within the context of optical multistability. Specifically, our derived formulae are: 1. Eq.(3.8) for the generalized Lorentz-Lorenz dispersion relation of the non-linear refractive index m which depends on the intensity of radiation entering the F-P cavity. This dependence arises through the parameter R3 where R−1 3 is an average of the inverse of the atomic inversion r3 (z) arising in the steady state at different points inside the cavity when a sufficient intense field enters the cavity. The actual atomic inversion depends on the cavity action and hence on the non-linear refractive index m. Thus, a self –consistent scheme can be created which converges to essentially exact results for the input-output intensity relation of the F-P cavity, 2. Eq.(3.11) for the spatially dependent forward and backward polarization envelops Pf ,b (z), 3. Eq.(3.13)for the parameter R3 , 4. Eq.(3.16) determines the spatially dependent atomic inversion density, r3 (z) , 5. Eq.(3.19) is the usual form for the Fresnel reflection coefficient, 6. Eq.(3.20) for the dipole envelopes at the boundaries, and, 7. Eq.(3.21) for the input-output field relation outside the F-P cavity (z > L). In the low atomic density limit our analytical results yield the conventional bistable behavior. The self-consistent analytical results derived in this part I are investigated numerically in part II of this series of papers. Finally, we mention that the self-consistent analytical results presented here have been extended to the case of optical rotation via magnetic dipole interactions [27],[28]-but these require a more tedious computational work than that presented in part II.

36

S.S. Hassan, R.K. Bullough and R. Saunders

Appendix A Here we derive the analytical expression for the atomic polarization field E ∗ (x, ω) in Eq.(3.3). From Eq.(2.4) E∗ (x, ω) = n0 p2 ~−1

Z

 F ∗ x, x0; ω : u u r+ (x0, ω) dx0 ,

(A.1)

V −v

where

 eikr r   ; r = x − x0 . F x, x0; ω = ∇ ∇ + k02U r If we consider, in the first instance, a one wave ansatz, that is we put

(A.2)

r+ (x, ω) = Pf (z) e−imk0 z ,

(A.3)

then Eq.(A.1) becomes E (x, ω) = n0 p ~ ∗

2 −1

Z

V −v

  0 Pf z0 e−imk0 z F ∗ x, x0; ω : u u dx0 

= n0 p2 ~−1 

Z

 0 Pf z0 e−imk0 z u u : ∇ ∇ ( f (r)) dx0

V −v

+k02

Z

V −v

 0

z0



Pf z e−imk0 u u : U f (r) dx0  , (A.4)

where f (r) = e−iko r /r. Using Gauss’ theorem to the first term in the right hand side of Eq.(A.4) we get Z

Z   0 0 Pf z0 e−imk0 z u u : ∇ ∇ ( f (r)) dx0 = Pf z0 e−imk0 z u.ds ∇x0 f (r) Σ

V −v



Z

 0 Pf z0 e−imk0 z u.ds ∇x0 f (r) (A.5)

σ

where ∑ is the surface of the volume V in which the medium is contained and σ represents a small sphere at x0 inside ∑. The second term in Eq.(A.5) is the Lorentz local field [4], [5] and [12], Z  4π 0 Pf (z) e−imk0 z . (A.6) − Pf z0 e−imk0 z u.ds ∇x0 f (r) = 3 σ

For the condition u.ds = 0, the first term in Eq.(A.5) then vanishes [5] and hence Eq.(A.5) gives Z  4π 0 Pf z0 e−imk0 z u u : ∇ ∇ ( f (r)) dx0 = Pf (z) e−imk0 z . (A.7) 3 V −v

Non-linear Refractive Index Theory

37

Now for the second term in Eq.(A.4) we have Z

Z   0 0 Pf z0 e−imk0 z u u : ∇2x0 ( f (r)) dx0 = −k02 Pf z0 e−imk0 z ( f (r)) dx0 .

V −v

(A.8)

V −v

Also Z

f (r)∇2x0



0





−imk0 z0

Pf z e

0

  ∂2  0 −imk0 z0 z e P dx0 f ∂z02 V −v Z      0 = f (r) m2 k02 Pf z0 + 2imk0 P0f z0 − P00f z0 e−imk0 z dx0 .

dx =

V −v

Z

f (r)

V −v

(A.9) From Eq.(A.8) and Eq.(A.9) we have Z      0 0 Pf z0 e−imk0 z ∇2x0 ( f (r)) − f (r)∇2x0 Pf z0 e−imk0 z dx0 V −v 2

= m −1 +2imk0



Z

k02

 0 Pf z0 e−imk0 z f (r)dx0

V −v

Z

Z   0 0 P00f z0 e−imk0 z f (r) dx0 . z0 e−imk0 z f (r) dx0 −

P0f

V −v

V −v

So k02

Z

Z 

 0 Pf z0 e−imk0 z f (r) dx0 =

V −v



f (r)∇2x0

+ (m21−1)



−imk0 z0

Pf (z)e

Z

 0 Pf z0 e−imk0 z ∇2x0 ( f (r))

1 (m2 −1) V −v



0

dx −

Z

2imk0 (m2 −1) V −v

 0 P0f z0 e−imk0 z f (r)dx0

 0 P00f z0 e−imk0 z f (r)dx0.

(A.10)

V −v

Applying Green’s theorem for the first integral on the right side of Eq.(A.10) gives, k02

Z

 0 Pf z0 e−imk0 z f (r) dx0=

4π P (m2 −1) f

V −v

+ (m21−1)

(z) e−imk0 z

Z 

Pf (z)e−imk0 z ∇x0 ( f (r)).ds

Σ

   0 − f (r)ds.∇x0 Pf z0 e−imk0 z Z  0 2imk0 P0f z0 e−imk0 z f (r)dx0 − (m2 −1) V −v



Z

1 (m2 −1) V −v

 0 P00f z0 e−imk0 z f (r)dx0.

(A.11)

38

S.S. Hassan, R.K. Bullough and R. Saunders

Hence the use of Eq.(A.7) and Eq.(A.11) into Eq.(A.4) gives E∗ (x, ω) 4π  m2 +2  = 3 m2 −1 Pf (z) e−imk0 z n0 p2 ~−1 Z      0 0 1 + (m2 −1) Pf z0 e−imk0 z ∇x0 ( f (r)).ds − f (r)ds.∇x0 Pf z0 e−imk0 z Σ

Z

2imk0 − (m 2 −1) V −v

Z   0 0 P0f z0 e−imk0 z f (r)dx0 − (m21−1) P00f z0 e−imk0 z f (r)dx0. (A.12) V −v

In a similar way we get for the second integral on the right–hand side of Eq.(A.12), Z  0 0 2 −imk0 z P0f z0 e−imk0 z f (r) dx0 = (m4π k0 2 −1) Pf (z) e V −v

+ (m21−1)

Z 

    0 0 P0f z0 e−imk0 z ∇x0 ( f (r)).ds − f (r)ds.∇x0 P0f z0 e−imk0 z

Σ

Z

2imk0 − (m 2 −1) V −v

P00f

0



−imk0 z0

z e

0

f (r)dx −

Z

1 (m2 −1) V −v

 0 −imk0 z0 P000 f (r)dx0. f z e

(A.13)

The use of Eq.(A.13) into Eq.(A.12) gives, up to ignoring the terms in P00f (z) and higher derivatives, i h 2  0 E∗ (x, ω) 4π m +2 2im = (z) − P (z) e−imk0 z P f 2 2 f 3 (m −1) k0 (m −1) n0 p2 ~−1 h i 0 2im + (m21−1) I (Pf (z)) − k (m I P (z) , (A.14) 2 −1) f 0

with I (h(z)) given by Eq.(3.4). Clearly, the ansatz (3.1), r+ (x,t) = Pf (z)e−imk0 z + ΛPb (z) eimk0 z , will give, along the same lines that led to Eq.(A.14), the inter-atomic field of Eq.(3.3). Hence we have the required result.

References [1] R. K. Bullough and S. S. Hassan, Optical Extinction Theorem in the Non-Linear Theory of Optical Multistability, in The Max Born Centenary Conference Proceedings (September, 1982). eds. M. J. Colles and D. W. Swift, vol. 369, (SPIE, Billingham, W.A., 1982), pp. 257-363. [2] R. K. Bullough, S. S. Hassan and S. P. Tewari, Refractive Index Theory of Optical Bistability, in Quantum Electronics and Electro Optics , ed. P. L. Knight, (Wiley, N.Y., 1982), pp. 229-332. [3] R. K. Bullough, S. S. Hassan, G. P. Hildred and R. R. Puri, Mirrored and Mirrorless Optical Bistability: Exact C-Number Theory of Atoms forming a Fabry-Perot Cavity, in Optical Bistability II, eds C.M. Bowden, H. M. Gibbs and S. L. McCall, (Plenum,1984), pp. 445-62.

Non-linear Refractive Index Theory

39

[4] R. K. Bullough and F. Hynne, in P. P. Ewald and his Dynamical theory of X-Ray Diffraction, eds. D.Q.J.Cruickshank, M.J. Juretschke and N Kato, (OUP, Oxford, 1992), pp. 98-100 [5] R. K. Bullough, Phil. Trans. Roy. Soc. (London), Series A, 254 (1962) 397 [6] F. Hynne and R. K. Bullough, Phil. Trans. Roy. Soc. (London), Series A, 321 (1987) 251 [7] F. Hynne and R. K. Bullough, Phil. Trans. Roy. Soc. (London), Series A, 321 (1987) 305 [8] F. Hynne and R. K. Bullough, Phil. Trans. Roy. Soc. (London), Series A, 330 (1990) 253 [9] S S Hassan and R K Bullough, in Optical Bistability, eds. C M Bowden, M Ciftan and H Robl, (Plenum, N.Y., 1981), pp. 367-404 [10] O.W.Oseen, Ann. Phys. (Leipzig), 48, 1,(1915). [11] P. P. Ewald, Ann. Phys. (Leipzig), 49,1,117 (1916). [12] C. G. Darwin, Trans. Camb. Phil. Soc. 23 (1924) 137. [13] L. Rosenfeld, Theory of the Electrons (North Holland, Amsterdam, 1951); Also reprinted with new prefaces by Dover, N.Y., 1965. [14] M. Born and E.Wolf, Principles of Optics, 6th Edition (Permagon, Oxford,1980). [15] J. H. Marburger and F. S. Felber, Phys. Rev. A 17 (1978) 335. [16] D. A. B. Miller, IEEE Journal of Quantum Electronics QE- 17, (1981), 306. [17] R. Bonifacio and L.A. Lugiato, Opt. Commun. 19 (1976) 172. [18] R. Bonifacio and L. A. Lugiato, Phys. Rev. A 18 (1978) 1179. [19] S. S. Hassan, P. D. Drummond and D. F. Walls, Optics Commun. 27 (1978). 480. [20] M. Orenstein, S. Speiser and J. Katriel, Opt. Commun. 48, (1984) 367-373 and M. Orenstein, S. Speiser and J. Katriel, IEEE J. Quant. Electron. 21 (1985). 1513-1522 [21] S. S. Hassan , H. A. Batarfi, R. Saunders and R. K. Bullough , Eur. Phys. J. D 8 (2000) 417 [22] S. S. Hassan , R. K. Bullough and H. A. Batarfi in Studies in Classical and Quantum Nonlinear Optics, Ed. O. Keller (Nova Sci., NY, 1995) pp 609-623. [23] R. Saunders , R. K. Bullough and C. Feuillade C, The Theory of Far Infrared Superfluorescence in Coherence and Quantum Optics IV, ed. L. Mandel and E. Wolf (Plenum, New York, 1978), pp. 263-286. R. Saunders R and R. K. Bullough, Theory of FIR Superfluorescence in Cooperative effects in Matter and radiation, ed. C. M. Bowden, D. Howgate and H. .P Roble (Plenum New York, 1977), pp. 209-256 and references therein.E. Abraham,.S. S.Hassan and R. K. Bullough,.Optics Commun. 33 (1980) 93.

40

S.S. Hassan, R.K. Bullough and R. Saunders

[24] R. Bonifacio and L.A. Lugiato, Lett. Nuovo. Cim. 21 (1978) 517 [25] E. Abraham,.S. S.Hassan and R. K. Bullough,.Optics Commun. 33 (1980) [26] N. Bloembergen and P. S. Pershan, Phys. Rev. 128 (1962) 606. [27] A-S F Obada, S. S. Hassan and M. H. Mahran, Physica A 142 (1987) 601. [28] S. S. Hassan,.M. H. Mahran and A-S F Obada, . Physica A 142 (1987) 619

In: Theoretical Physics Research Developments Editors: J. P. Sullivan and A. L. Montey

ISBN: 978-1-61209-446-5 c 2011 Nova Science Publishers, Inc.

Chapter 4

Q UANTUM C OHERENCE AND T UNABLE T RANSIENT B EHAVIOR OF A D OUBLE -C ONTROL F OUR -L EVEL ATOMIC VAPOR Arash Gharibi1,2, Jing Gu1,2 , Jian Qi Shen1,2∗ and M.H. Majles Ara3 1 Centre for Optical and Electromagnetic Research, State Key Laboratory of Modern Optical Instrumentations, East Building No. 5, Zijingang Campus, Zhejiang University, Hangzhou 310058, The People’s Republic of China 2 Joint Research Centre of Photonics of the Royal Institute of Technology (Sweden) and Zhejiang University, Zijingang Campus, Zhejiang University, Hangzhou 310058, The People’s Republic of China 3 Photonics Laboratory, Department of Physics, Tarbiat Moallem University, Mofateh 49, Tehran, Iran

Abstract The tunable transient evolutional behavior of a double-control four-level atomic vapor is studied and its application to photonic logic gates is suggested. It is shown that the quantum interferences between two external control fields can lead to controllable absorption and transparency characteristics of a multilevel atomic vapor. One of the most remarkable properties involved is that of the absorption (or transmittance) of the probe field in the atomic vapor being dependent upon the ratio of the intensities of the two control fields, allowing the tunable optical features (transparency or opaqueness to the probe field) to be achieved by tuning the quantum interferences between the control transitions (driven by the control fields). The mechanisms in question are applicable to the design of a variety of new photonic and quantum optical devices such as logic and functional devices, optical switches as well as other photonic microcircuit components, which can be employed for controlling the wave propagation of a probe field. Two typical photonic logic gates (OR and EXOR gates) based on the tunable four-level optical responses (quantum constructive and destructive interferences) are presented as illustrative examples. ∗ E-mail

address: [email protected]

42

Arash Gharibi, Jing Gu, Jian Qi Shen et al.

PACS: 42.50.Gy, 42.50.Nn, 42.25.Bs, 42.50.Md Keywords: Transient evolution, multilevel atomic vapor, quantum interference, photonic logic gates

1.

Introduction

Over the past 50 years, the design and fabrication of artificial materials has attracted considerable attention in a variety of scientific and technological areas [1]. In the 1950s and 1960s, for example artificial dielectrics were exploited for the manufacture of lightweight microwave antenna lenses, and in the 1980s and 1990s the use of artificial chiral materials for microwave radar absorber applications was explored [2]. In recent decades, a type of materials termed photonic crystals, which are patterned so as to show a periodicity in the dielectric constant, allowing them to create a range of forbidden frequencies constituting what is termed the photonic band gap, has captured the attention of many researchers [3]. Recently, artificial composite metamaterials (so-called “left-handed media” or “negative refractive index media”) in which the electric permittivity and the magnetic permeability are negative simultaneously within the microwave frequency band received the attention of researchers in a variety of fields, such as those of material science, condensed-matter physics, optics and classical applied electromagnetism [4, 5, 6]. These are a new kind of materials that possess the ability to mold the flow of light inside media. With the development of photonics and quantum optics, considerable attention was directed by scientists in many different areas at new techniques to manipulate light wave propagations by use of artificial electromagnetic materials [7, 8, 9, 10, 11]. A particularly flexible and promising approach to manipulating light propagation has been that of quantum coherence [10, 11, 12, 13]. Over the past two decades, quantum coherence effects (atomic phase coherence and quantum interferences between atomic transitions) have been demonstrated in the form of many interesting phenomena in physical terms, such as electromagnetically induced transparency (EIT) [10], light amplification without inversion [11], spontaneous emission cancellation [12], multi-photon population trapping [13], coherent phase control [14, 15] as well as sensitive optical responses in EIT waveguides [16]. Recently, there has been increasing interest in quantum coherence in such photonic applications as EIT-based coherent information storage [17, 18] and quantum logic gates among physicists in both fundamental and applied areas [7, 8, 9]. Since a transient evolution accompanies these physical processes (i.e., those of information storage and logic-gate operations) when the control fields are switched on (and off), the time-dependent behavior of the atomic media deserves close consideration. In the literature, Li et al. have investigated the resonant transient properties induced in a three-level EIT system by a quantum interference effect [19]; Greentree et al. have studied both the resonant and the off-resonant transient behavior of a three-level EIT medium [20]. More recently, Yao et al. have been concerned with the problem of transient optical responses [21] that occur under certain approximation conditions in a four-level Nconfiguration system [22]. However, there is another interesting four-level system alongside the N-configuration system. Such a system has a tripod configuration, which can exhibit nontrivial double-control quantum interferences of both destructive and constructive character [23, 24, 25, 26, 27]. Since a four-level EIT vapor exhibits marked dispersion in both

Quantum Coherence and Tunable Transient Behavior...

43

the real and the imaginary parts of the optical ‘constants’ involved, the optical properties are more sensitive to the probe frequency than in a three-level EIT vapor. The quantum interference found in a four-level system is thus potentially applicable to many new techniques, such as those of sensitive optical switches, optical magnetometers and wavelength sensors (optical magnetometers, for example, can be used as highly sensitive detectors of magnetic fields, and multilevel EIT-based wavelength sensors can be used to measure the wavelength of a probe by utilizing the strong dispersion caused by atomic phase coherence). Such devices (e.g., wavelength sensors) can find practical application in color matching and sorting, for example, where precise measurements of light wavelengths or frequencies are required. The transient evolutional behavior of atomic vapors is an important physical phenomenon, which would affect future light-storage technology (and thus quantum coherent information storage) at atomic levels [17, 18]. Obviously, the impact would be enormous if the transient evolution of both atomic vapors and probe fields could be readily tuned by utilizing the quantum interferences found between external control fields. In this paper, we shall describe the tunable steady optical behavior (both transparent and opaque to the probe laser beam) of a tripod-configurated atomic vapor, and then present numerical solutions to the equation of motion of the atomic probability amplitudes, and consider the transient electric permittivity and medium absorption induced by the effects of the quantum interferences and atomic phase coherence involved. We shall analyze the transient evolutional behavior of the atomic electric polarizability, the permittivity and the refractive index (in the case of switching on the external control fields), and show how rapidly the absorption of the atomic medium and the optical responses related to this react to switching on of the external optical fields. It is of interest to note that the quantum interferences between the control fields can lead to the controllable absorption and transparency characteristics of the atomic vapor, so that tunable optical behavior (transparency or opaqueness to the probe laser beam) can be achieved by tuning the ratio of the intensities (or strength) of the external control fields [see, for example, Eq. (3.11)]. Potential applications, such as in photonic logic gates, optical switches and other photonic microcircuit components based on the steady and transient evolutional behavior of the multilevel atomic system will be suggested. This paper is organized as follows: first we study the four-level transitions driven by the three optical fields (the probe and control fields) in an atom. After getting the probability amplitudes of the energy levels, we calculate the microscopic electric polarizability βe of one atom. Then we obtain the electric susceptibility (and hence the electric permittivity) of the four-level atomic vapor by using the Clausius-Mossotti relation. The photonic logic gates (e.g., OR and EXOR gates) are designed based on the double-control quantum constructive and destructive interferences between the atomic transition paths driven by the external optical fields. In the literature, the operations of various optical and optoelectronic logic gates have been demonstrated in bistable semiconductor lasers in combination with optoelectronic switches or photodetectors [28, 29]. One of the aims of this paper is attempt to realize these logic operations by using alternative ways, i.e., the multilevel photonic devices, where the quantum constructive and destructive interferences are involved.

44

2.

Arash Gharibi, Jing Gu, Jian Qi Shen et al.

Double-Control Atomic System and Its Steady Optical Behavior

Now we shall present the steady solutions to the equation of motion of the four-level system, and show its steady optical behavior. Such steady optical properties depend on the intensities of the control fields (via atomic phase coherence). In other words, the optical properties of the present atomic vapor are close related to the quantum interferences between the external two control fields. Consider a four-level atomic system with three lower levels |1i, |2i, |20 i and one upper level |3i (see Fig. 1). Such an atomic system interacts with three optical fields, i.e., the two control laser beams and one probe laser beam, which couple the level pairs |2i-|3i, |20 i-|3i and |1i-|3i, respectively. The three frequency detunings ∆c , ∆c0 and ∆p are defined as follows: ∆c = ω32 − ωc , ∆c0 = ω320 − ωc0 , and ∆p = ω31 − ωp , where ω32 , ω320 and ω31 denote the atomic transition frequencies, and ωc, ωc0 , ωp represent the mode frequencies of the control and probe beams, respectively. Let us now consider the equation of motion of the atomic system. In general, the density matrix formulation should be employed here, since, in accordance with the fundamental principles of quantum mechanics, the observables of a system of ‘mixed state’ (statistic mixture of pure states) are the bilinear combinations of the wave functions. In contrast, the probability amplitude formulation can only be applied to the system of ‘pure state’ because the equations for the probability amplitudes are derived directly from the wave function equations. But under certain conditions (e.g., the density matrix elements ρ23 , ρ20 3 , ρ33 are small, the steady ρ11 is a constant number, and hence the time derivatives of ρ13 , ρ12 , ρ120 , ρ220 can form a complete set of equations), the probability amplitude formulation (as a treatment that leads to approximate results) could also be used for the mixed-state system [21, 22]. For the present atomic system, the equation of motion of the probability amplitudes based on the Schr¨odinger equation is given by i a˙1 = Ω∗p a3 , 2 hγ i i a˙2 = − 2 + i ∆p − ∆c a2 + Ω∗c a3 , 2  20   i γ2 + i ∆p − ∆c0 a20 + Ω∗c0 a3 , a˙20 = − 2 2    Γ3 i + i∆p a3 + Ωp a1 + Ωc a2 + Ωc0 a20 , a˙3 = − 2 2

(2.1)

where the atomic level decay terms have been added phenomenologically [21, 22, 30] (in Refs. [21, 22], the authors treated an N-type atomic system by using the probability amplitude method). The Rabi frequencies of the probe beam and the two control beams are defined through Ωp = ℘31 Ep /~, Ωc = ℘32 Ec /~, and Ωc0 = ℘320 Ec0 /~, respectively. Here Ep , Ec , and Ec0 stand for the field envelopes (slowly-varying amplitudes) of the probe and control fields, respectively. The γ2 and γ02 are collisional dephasing (nonradiative decay rates), and Γ3 is a spontaneous emission decay rate. In general, such a double-control tripod-configuration system can be found in alkali metallic atoms. For example, according to the selection rule (∆L = ±1, ∆J = 0, ±1, ∆mJ = 0, ±1) for the electric-dipole allowed transition, the system {|1i, |2i, |20i, |3i} can be

Quantum Coherence and Tunable Transient Behavior... ∆ c’ Ω



∆ p

45

|3〉 c

c’



|2’〉 Ω

c

p

|2〉 |1〉

Figure 1. The schematic diagram of a double-control four-level system. The two control laser beams, Ωc and Ωc0 , drive the |2i-|3i and |20i-|3i transitions, respectively. The probe transition |1i-|3i can be controllably manipulated via the destructive and constructive quantum interferences between the |2i-|3i and |20 i-|3i transitions.

chosen as {52 S 1 , 42 D 3 , 62 S 1 , 62 P 1 } of neutral Rubidium atom. If the energy level of 2 2 2 2 the ground state 5 2 S 1 is assumed to be zero, the energies of the other three atomic 2 levels 42 D 3 , 62 S 1 , 62 P 1 are 19355.649, 20132.510,23715.081 cm −1 , respectively [31]. 2 2 2 Such a tripod-configuration atomic system can also be found in neutral Lithium atom {22 S 1 , 32 S 1 , 32 D 3 , 42 P 1 } with energy levels {0.000, 27206.066, 31283.018, 36469.714} 2 2 2 2 cm−1 [32], and in neutral Sodium atom {32 S 1 , 42 S 1 , 32 D 3 , 42 P 1 } with energy levels 2 2 2 2 {0.000, 25739.991, 29172.889, 30266.99} cm−1 [33]. In general, the intensity of the probe field is sufficiently weak and its Rabi frequency is small compared with the Rabi frequencies of external control fields as well as some spontaneous emission decay rates. According to Eq. (2.1), the right-handed side of equation a˙1 = (i/2)Ω∗pa3 is negligibly small since both Ωp and a3 are small compared with Ωc and a1 , respectively. In what follows (for the steady case), we assume that a1 is a constant number (this assumption can be verified using the numerical results for transient evolutions in the next section. See, for example, in Fig. 2), which would simplify the problem under consideration. The equations for the probability amplitudes a2 , a20 and a3 in (2.1) can be rewritten in the following matrix form   γ  i ∗   0 Ω − 22 + i(∆p − ∆c ) c 2 h 0 i a  γ ∂ 2   i ∗ 0 − 22 + i(∆p − ∆c0 ) Ωc 0  a2 0 = 2     ∂t Γ i i a3 0 − 23 + i∆p 2 Ωc 2 Ωc     0 a2  . (2.2)  a2 0  +  0 i a3 2 Ωp a 1

46

Arash Gharibi, Jing Gu, Jian Qi Shen et al. The steady solution to Eq. (2.1) is given by   0  a1 ∗ γ2 + i ∆p − ∆c 0 , Ωp Ωc a2 = − 4D 2 hγ i a1 Ωp Ω∗c0 2 + i ∆p − ∆c , a2 0 = − 4D 2   hγ i γ02  ia1 2 Ωp + i ∆p − ∆c + i ∆p − ∆c 0 , a3 = 2D 2 2 (2.3)

where the denominator



  h i γ02  Γ3 γ2 + i∆p + i ∆p − ∆c + i ∆p − ∆c 0 D = 2 2 2 hγ i 1 + Ω∗c0 Ωc0 2 + i ∆p − ∆c 4   02  γ2 1 ∗ + i ∆p − ∆c 0 . + Ωc Ωc 4 2

(2.4)

Note that the atomic electric polarizability of the probe transition |1i-|3i is β(∆p ) = 2℘13 a∗1 a3 /(ε0 Ep ). Substituting the above results (2.3) and (2.4) into β(∆p ), one can obtain the explicit expression for the microscopic electric polarizability of the present doublecontrol atomic vapor:   i γ02  |℘13 |2 a∗1 a1 i h γ2 0 + i ∆p − ∆c + i ∆p − ∆c , (2.5) βe (∆p ) = ε0 ~ D 2 2 where the relation Ep = ~Ωp /℘31 has been inserted. Let us now consider the macroscopic quantity (electric permittivity) that indicates the optical property of the double-control atomic vapor. As is well known, one must make a distinction between the applied fields (macroscopic fields) and the microscopic local fields acting upon the atoms in the vapor when discussing how the properties of the atomic transitions between the levels are related to the electric susceptibility. In a dilute vapor, there is little difference between the macroscopic fields and the local fields that act on any atoms (molecules or group of molecules) [34]. But in a dense medium with closely packed atoms (molecules), the polarization of neighboring atoms (molecules) would give rise to an internal field at any given atom in addition to the external macroscopic field, so that the total fields at the atom are very different from the external macroscopic fields [34]. This difference may lead to a local field correction to the macroscopic quantities (such as the electric permittivity) of the medium. The Clausius-Mossotti relation can reveal the connection between the macroscopic permittivity and the microscopic polarizability. The expression of (LFC) (LFC) − 1)/(εr + 2) [34], where N, βe, εr denote this relation is of the form βe = (3/N)(εr the atomic concentration, the atomic polarizability and the relative permittivity, respectively, of the atomic vapor medium. Thus after the local field correction (LFC) the relative electric permittivity is given by (LFC)

εr

=

1 + 23 Nβe 1 − 13 Nβe

.

(2.6)

Quantum Coherence and Tunable Transient Behavior...

47

In addition to the linear term, there are the nonlinear terms N n in the Taylor expansion (LFC) contains the contribution of atomic dipole-dipole interacseries of Eq. (2.6), because εr (LFC) − 1 whose transient behavior in both tions. The relative electric susceptibility is χ = εr quantum constructive and destructive interferences will be presented in this paper.

3.

Quantum Interferences in the Four-Level System

The double-control four-level system can exhibit a novel quantum coherent effect (constructive and destructive interferences), which can be used to manipulate the probe propagation. It follows from the solution (2.3) that the driving contribution of the two control fields Ωc , Ωc0 from the ground level |1i to the upper level |3i is given by Ωc a 2 + Ωc 0 a 2 0     0 hγ  i a1 γ2 2 ∗ ∗ + i ∆ p − ∆ c 0 + Ωc 0 Ωc 0 + i ∆p − ∆c = − Ωp Ωc Ωc . 4D 2 2

(3.7)

If the driving contribution of the two control fields vanishes, i.e., Ωca2 + Ωc0 a20 = 0,

(3.8)

then we can say that the quantum destructive interference arises between the two control transitions |2i-|3i and |20i-|3i. It can be seen from Eqs. (2.4) and (2.5) that the atomic vapor can exhibit a behavior of two-level resonant absorption, i.e., the atomic electric polarizability (2.5) will be reduced to a simple form: ! 1 |℘13 |2 a∗1 a1 . (3.9) β(∆p ) = i Γ3 ε0 ~ 2 + i∆p In the meanwhile, the quantum destructive interference (3.8) requires that the intensities fields Ωc , Ωc0 agree  0 of the two control    with the following relation: ∗ ∗ Ωc Ωc γ2 /2 + i ∆p − ∆c0 +Ωc0 Ωc0 γ2 /2 + i ∆p − ∆c = 0. In general, the dephasing rate in a vapor is negligibly small compared with the spontaneous decay rate and the detuning frequencies (the detuning frequency range that is of interest has the same order of magnitude as that of the spontaneous decay rate). Thus, from the condition of destructive interference (3.7) and (3.8) one can obtain an expression for a certain probe detuning frequency ∆p =

Ω∗c Ωc ∆c0 + Ω∗c0 Ωc0 ∆c , Ω∗c Ωc + Ω∗c0 Ωc0

(3.10)

where the destructive quantum interference between the two control transitions |2i-|3i and |20 i-|3i would take place. In this case, the double-control atomic vapor is opaque to the probe laser beam, since the two control fields Ωc , Ωc0 seem to be absent because of the quantum destructive interference between the two control transitions, and the present fourlevel system {|1i, |2i, |20i, |3i} is reduced to a two-level system {|1i, |3i}. The requirement (3.10) for the destructive interference can be rewritten as ∆p =

Ω∗c0 Ωc0 Ω∗c Ωc ∆c . Ω∗ Ω 0 1 + Ωc∗0 Ωcc c

∆c 0 +

(3.11)

48

Arash Gharibi, Jing Gu, Jian Qi Shen et al.

This, therefore, means that the destructive interference in the double-control four-level system can occur under proper conditions (related to the intensity ratio of the external optical fields). The probe field at the frequency detuning expressed by Eq. (3.10) will be absorbed by the vapor. However, once we switch off one (e.g., the field Ωc0 ) of the control fields, the atomic vapor is then transparent to the probe field, or the stored probe field would be released. Since the atomic microscopic electric polarizability of the four-level system is more complicated than that of the three-level electric polarizability, there are unavoidably more peaks and valleys in the dispersive curve of the four-level permittivity than in that of the conventional three-level permittivity, and the dispersion in the double-control four-level vapor is therefore more sensitive to the probe frequency than in the three-level atomic vapor. Thus, more intriguing optical properties, e.g., the effects of slow light and negative-groupvelocity propagations, would arise in the four-level atomic vapor. As the atomic polarizability and the vapor permittivity can be manipulated by changing the external fields, the transient evolutional behavior of this multilevel atomic vapor deserves consideration. In the section that follows, the transient optical characteristics of the double-control atomic vapor as well as the constructive and destructive interferences between the two control transitions |2i-|3i and |20 i-|3i will be addressed.

4.

The Tunable Transient Evolutional Behavior: Constructive and Destructive Interferences

Here we present the transient case of optical responses and show how the four-level atomic probability amplitudes and the permittivity (and hence the refractive index and the absorption coefficient) evolve from the initial values of one state to another steady state when the external control fields are simultaneously switched on. Analysis of such turn-on dynamics is necessary for the future technique of quantum coherent light storage . The typical atomic and optical parameters of the present atomic vapor are chosen as follows: the decay rates γ2 = 2.0 × 105 s−1 , γ02 = 2.0 × 105 s−1 , Γ3 = 5.0 × 107 s−1 , and the electric dipole moment ℘31 = 1.0 × 10−29 C·m. The frequency detunings of the three optical fields are ∆p = 1.0Γ3 , ∆c = 0.6Γ3 and ∆c0 = 1.0Γ3 , and the Rabi frequencies are Ωc = 1.0 × 108 s−1 , Ωc0 = 2.0 × 108 s−1 and Ωp = 0.01Ωc. The atomic concentration (atomic number per vol20 3 values of the probability amplitudes are chosen ume) is N = 3.0 × √10 At/m√. The initial√ √ as follows: a1 = 6/4, a2 = 2/4, a20 = 2/4 and a3 = 6/4, which satisfy the normalization relation a∗1 a1 + a∗2 a2 + a∗20 a20 + a∗3 a3 = 1. The typical transient evolutional behavior in the case of constructive interference is shown in Figs. 2-5. The real and imaginary parts for the atomic probability amplitudes a1 , a2 , a20 , and a3 are shown in Fig. 2. The atomic probability amplitude of each level evolves from its respective initial value and oscillatorily increases or decreases within the relaxation time (depending on the decay rates), and approach its own steady value when time t → +∞. It is clearly seen that both the real and imaginary parts of a2 , a20 and a3 change drastically, while a1 does not change much (and its real and imaginary parts can be considered to be constant numbers) during the transient process. This is consistent with the analysis presented in the preceding sections (the discussion of the steady case).

Quantum Coherence and Tunable Transient Behavior...

0.8

49

0.5

0.6

2

a

a

1

0.4 0

0.2 0 −0.2 0

1

2

3

4

5

−0.5 0

1

2

t(µs)

4

5

3

4

5

3

0.5

0

a

a2’

0.5

−0.5 0

3

t(µs)

1

2

3

4

5

0

−0.5 0

1

2

t(µs)

t(µs)

Figure 2. (Color Online) The transient behavior of the real and imaginary parts for the atomic probability amplitudes a1 , a2 , a20 and a3 in the case of quantum constructive interference. The blue solid and red dashed curves represent the real and imaginary parts, respectively. The oscillatory probability amplitudes finally approach their respective steady values.

6 Im(χ) Re(χ) 4

2

χ

0

−2

−4

−6

−8 0

1

2

3

4

5

t(µs)

Figure 3. (Color Online) The transient behavior of the real and imaginary parts of the electric susceptibility of the atomic vapor in the case of quantum constructive interference. The electric susceptibility tends to the steady value after the relaxation process.

50

Arash Gharibi, Jing Gu, Jian Qi Shen et al. 2 Im( n r) Re( n r)

1.5

nr

1

0.5

0

−0.5

−1 0

2

4

6

8

10

t(µs)

Figure 4. (Color Online) The transient behavior of the real and imaginary parts of the relative refractive index of the atomic vapor in the case of quantum constructive interference. The refractive index tends to the steady value after the relaxation process. In Figs. 3 and 4, we plot the transient behavior of the relative electric susceptibility χ and the relative refractive index nr of the four-level atomic vapor. When the two control fields are turned on, the imaginary part of the refractive index nr of the vapor is oscillatorily damped (exponentially) to its steady value (Im (nr ) → 0), and the real part oscillatorily evolves, and finally tends to its own steady value (Re (nr ) → 1). This corresponds to the phenomenon of double-control EIT. The dimensionless absorption coefficient α is shown in Fig. 5 as a function of time t. Here, the dimensionless α is defined as 2πn 00r /n0r, where n0r and n00r denote the real and imaginary parts of nr , respectively (i.e., nr = n0r + in00r ). For a time-harmonic plane wave, the oscillatory electric field amplitude exp (ikx) = exp(in0r ωx/c) exp(−n00r ωx/c), where the wave number k = (n0r + in00r )ω/c. In an absorptive medium, the light wavelength is λ = 2πc/n0r ω. Thus, the oscillatory field amplitude can be rewritten as exp(ikx) = exp(i2πx/λ) exp(−αx/λ). It follows from Fig. 5 that the absorption coefficient, α, can evolve from one initial value to the other steady value (almost zero) once the control fields are switched on. The relaxation time is about 5 microseconds. This can be viewed as the response time of some new photonic devices, e.g., logic gates and switches, which are designed based on the present tunable four-level optical property. The tunable optical property of the present four-level system is that the atomic vapor can be transparent or opaque to the probe field under certain conditions with proper intensities of the external control fields. In the above, we have considered the transient case of the quantum constructive interference between the two control transitions |2i-|3i and |20i|3i. Now we turn to the transient case of quantum destructive interference. We present the destructive-interference transient evolutional behavior in Fig. 6 to Fig. 9, where all the optical and atomic parameters (except for Ωc0 ) are chosen exactly the same as those used before in the constructive-interference transient evolutional behavior. Here, instead, the Rabi frequency of the new control field is taken to be Ωc0 = 1.0 × 108 s−1 . In the case of destructive interference between the two control transitions |2i-|3i and |20i-|3i, the absorption coeffi-

Quantum Coherence and Tunable Transient Behavior...

51

5 4 3 2

α

1 0 −1 −2 −3 −4 −5 0

2

4

6

8

10

t(µs)

Figure 5. (Color Online) The transient behavior of the absorption coefficient, α, of the atomic vapor in the case of quantum constructive interference. It can be seen that α approaches a zero value after the relaxation process. cient α oscillatorily evolves, and approaches a nonzero value ( i.e., α → 0.3). This means that the probe field can propagate through a distance of only several wavelengths, and then it will be dissipated in the atomic medium. The property, in which the absorption and transparency of the atomic medium to the probe beam is induced by the quantum interferences between the two control fields, can be used to realize light storage, optical switches, and photonic logic gates, where the transient evolution (including the dispersion and dissipation of the stored light during the storage and readout processes) deserves consideration when we attempt to achieve such photonic devices with high sensitivity and efficiency. In the literature, some authors treated the problem of light storage with atoms [18, 19], but did not consider the transient evolution of atomic polarizability and vapor permittivity. Since the atomic system has to experience a transient evolution once the control fields are switched on or off, the transient evolution is a very important physical process when one considers the mechanism of storage and readout of pulses for the potential technology of quantum coherent information storage . In the section that follows, we shall give some illustrative examples to demonstrate the application of the double-control four-level vapor to the design of photonic logic gates.

5.

Applications to the Photonic Logic Gates and Optical Switches

Recently, ideas of realizing logic gates by using new optoelectronic materials have attracted attention of some researchers [7, 8, 9]. We expect that the tunable multilevel quantum interference effect could also be utilized to realize some logic and functional operations. Here we design and suggest the working mechanisms of some photonic logic gates based on the double-control optical properties. In Figs. 10 and 11, the logic gates (OR and EXOR gates)

52

Arash Gharibi, Jing Gu, Jian Qi Shen et al.

0.3

0.8

0.2 0.6

a2

a

1

0.1 0.4

0

−0.1 0.2 −0.2 0 0

1

2

3

0

1

t(µs)

0.2

0.2

0.1

0.1

a3

0.3

a2’

0.3

0

−0.1

−0.2

−0.2 1

2

t(µs)

3

2

3

0

−0.1

0

2

t(µs)

3

0

1

t(µs)

Figure 6. (Color Online) The transient behavior of the real and imaginary parts for the atomic probability amplitudes a1 , a2 , a20 and a3 in the case of quantum destructive interference. The blue solid and red dashed curves represent the real and imaginary parts, respectively. The oscillatory probability amplitudes finally approach their respective steady values. are designed using the tunable four-level optical responses. In these schemes, the incident probe field is always present. Note that the incident control fields and the transmitted probe field are viewed as the INPUT signals and the OUTPUT signal, respectively. We propose that the INPUT operation is IN = 1 when one of the control fields is switched on, and IN = 0 when one of the control fields is switched off. If the probe field can propagate through the atomic vapor (i.e., the absorption coefficient α is zero or negligibly small), then this implies that the OUTPUT operation result is OUT = 1; whereas, we would have OUT = 0 if α is large and the probe field is absorbed by the atoms. According to the discussions in the preceding sections, the constructive interference exhibited in the present tripod-configuration system can be employed to realize a two-input OR gate. The operation of the OR gate can produce a high output (OUT = 1) if one or more of its inputs are high. The truth table of the two-input OR gate is given by (Table 1). We show how the constructive quantum interference works for realizing the OR operation: the present four-level system can exhibit a double-control EIT, i.e., OUT = 1, if both of the two control fields are switched on, i.e., INA = INB = 1; and it can also exhibit a single-control EIT (OUT = 1) if one of the control fields is switched off (IN A = 1, INB = 0; or INA = 0, INB = 1). But the atomic medium will be opaque (due to the |1i-|3i resonant absorption), i.e., OUT = 0, when both of the two control fields are absent, i.e., INA = INB = 0.

Quantum Coherence and Tunable Transient Behavior...

53

15 Im(χ) Re(χ) 10

5

χ

0

−5

−10

−15

−20 0

0.5

1

1.5

2

2.5

3

t(µs)

Figure 7. (Color Online) The transient behavior of the real and imaginary parts of the electric susceptibility of the atomic vapor in the case of quantum destructive interference. The electric susceptibility tends to the steady value after the relaxation process. Table 1. The truth table for the OR gate. A 0 0 1 1

B 0 1 0 1

A+B 0 1 1 1

The working mechanism of the two-input OR gate is shown in Fig. 10. Let us now discuss the EXOR gate (‘Exclusive-OR’ gate). The destructive interference exhibited in the present double-control system is required for realizing the two-input EXOR gate: this EXOR gate is a circuit which will give a high output if either, but not both, of its two inputs are high. The truth table of the two-input EXOR gate is given by (Table 2). The EXOR operation process based on the destructive interference is given as follows: if both of the two control fields are absent (IN A = INB = 0), the probe transitions |1i-|3i will certainly lead to the absorption of the probe beam. This, therefore, means that the output signal OUT = 0. As the Rabi frequencies Ωc , Ωc0 have already satisfied the destructive interference condition (3.8) [and hence (3.9)], the probe field in this case can unavoidably be absorbed by the vapor when the two control fields are switched on (IN A = INB = 1), and this gives rise to the output signal OUT = 0. Then, on the other hand, if one of the control fields is switched off, the atomic medium will be transparent to the probe beam (OUT = 1) since it exhibits an effect of single-control EIT. As we have pointed out, the response time (relaxation time) of such photonic logic

54

Arash Gharibi, Jing Gu, Jian Qi Shen et al. 3 Im( n r)

2.5

Re( n ) r

2 1.5

n

r

1 0.5 0 −0.5 −1 −1.5 −2 0

1

2

3

4

5

t(µs)

Figure 8. (Color Online) The transient behavior of the real and imaginary parts of the relative refractive index of the atomic vapor in the case of quantum destructive interference. The refractive index tends to the steady value after the relaxation process.

2 1.5 1

α

0.5 0 −0.5 −1 −1.5 −2 0

1

2

3

4

5

t(µs)

Figure 9. (Color Online) The transient behavior of the absorption coefficient, α, of the atomic vapor in the case of quantum destructive interference. It can be seen that α approaches a nonzero value after the relaxation process.

Quantum Coherence and Tunable Transient Behavior... IN

A

55

(Control field Ω ) c’

Double−control constructive−interference atomic vapor Incident probe field Ωp

OUT (Probe field Ωp)

(always present)

INB (Control field Ωc)

Figure 10. (Color Online) The schematic diagram of a two-input OR logic gate based on the double-control constructive interference. The atomic medium is transparent to the probe beam (OUT = 1) if one or two of the control fields are switched on. Whereas, it would be opaque to the probe field (OUT = 0) if both of the two control fields are switched off. Table 2. The truth table for the EXOR gate. A 0 0 1 1

B 0 1 0 1

A⊕B 0 1 1 0

gates is microseconds, this effect could be applied to some techniques in optical communications, where high-speed signal response is needed. Apart from the logic gates, the optical switches can also be designed by using the tunable optical responses of the present atomic vapor. Such switches might be a useful technique for some future photonic devices, e.g., the photonic microcircuits on silicon, in which light replaces electrons. At present, the all-optical switch on silicon where one controls light with light on chips has been increasingly developed, and we hope that the present photonic devices designed based on multilevel quantum interference effects would have potential applications in this field and other related areas, e.g., the integrated optical circuits.

6.

Concluding Remarks

The optical responses can be controlled by using the tunable quantum interference induced by the external control fields. The four-level system considered here shows more flexible optical responses than the conventional three-level EIT systems. In examining the steady and the transient optical behavior of a double-control four-level atomic vapor, absorption of atomic vapor is found to increase (i.e., the atomic vapor becomes opaque to the probe field) when the ratio of the intensities of the two control fields agrees with the condition of destructive quantum interference (the optical properties of the present four-level system is

56

Arash Gharibi, Jing Gu, Jian Qi Shen et al. INA (Control field Ωc’)

Double−control destructive−interference atomic vapor Incident probe field Ωp

OUT (Probe field Ω ) p

(always present)

INB (Control field Ωc)

Figure 11. (Color Online) The schematic diagram of a two-input EXOR logic gate based on the double-control destructive interference. The atomic medium is transparent to the probe field (OUT = 1) if either, but not both, of its two control fields are switched on. If both of the two control fields are switched on, it would be opaque to the probe field (OUT = 0) because of the double-control destructive interference between the two control transitions |2i-|3i and |20 i-|3i. in fact reduced to those of two-level resonant absorption because of the destructive quantum interference between the two transitions driven by the two control fields). Under this condition, once one of the fields is switched off, the atomic vapor is transparent to the probe field. Thus, the optical properties of the atomic vapor can be manipulated by controlling the quantum interferences between the external control fields. The tunable optical properties induced by quantum interference have several useful applications, such as in the design of new photonic devices—photonic logic gates and optical switches, for example. Some examples of photonic devices in which the two control fields and the probe field act as the input and output signals, respectively, have been presented. The strong dispersion in the susceptibility found here can lead to dramatic modification in the speed of light that propagates in a medium. The ultraslow light and the superluminal propagation (negative group velocity) that can occur in three-level atomic media have received considerable attention in the literature [35, 36]. Since the dispersion in both the real and the imaginary parts of the optical ‘constants’ in the four-level atomic medium is stronger than that in a three-level EIT medium, the tunable ultraslow and superluminal propagation of light induced by the quantum interference between the control fields are highly relevant to the four-level atomic vapor studied here. This means that the topics related to the tunable ultraslow and superluminal propagation in the double-control four-level atomic medium deserve further consideration both theoretically and experimentally.

Quantum Coherence and Tunable Transient Behavior...

57

Acknowledgment This work is supported by the National Natural Science Foundations (NNSF, China) under Project No. 10604046 and by the National Basic Research Program of China under Grant No. 2004CB719800.

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[18] P. Arve, P. J¨anes and L. Thyl´en, Phys. Rev. A 69, 063809 (2004). [19] Y. Q. Li and M. Xiao, Opt. Lett. 20, 1489 (1995). [20] A. D. Greentree, T. B. Smith, S. R. de Echaniz et al., Phys. Rev. A 65, 053802 (2002). [21] J. Q. Yao, H. B. Wu and H. Wang, Acta Sin. Quantum Opt. (in Chinese) 9, 121 (2003). [22] S. E. Harris and Y. Yamamoto, Phys. Rev. Lett. 81, 3611 (1998). [23] E. Paspalakis and P. L. Knight, J. Mod. Opt. 49, 87 (2002). [24] E. Paspalakis and P. L. Knight, J. Opt. B: Quantum Semiclass. Opt. 4, S372 (2002). [25] E. Paspalakis and P. L. Knight, Phys. Rev. A 66, 015802 (2002). [26] J. Q. Shen and P. Zhang, Opt. Express 15, 6484 (2007). [27] J. Q. Shen, New J. Phys. 15, 374 (2007). [28] J. M. Liu and Y. C. Chen, IEEE J. Quantum Electron. 21, 298 (1985). [29] K. Okumura, Y. Ogawa, H. Ito and H. Inaba, IEEE J. Quantum Electron. 21, 377 (1985). [30] M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge Univ. Press, Cambridge, England, Chapt. 5 (1997). [31] I. Johansson, Ark. Fys. 20, 135 (1961). [32] L. J. Radziemski, R. Jr. Engleman and J. W. Brault, Phys. Rev. A 52, 4462 (1995). [33] W. C. Martin and R. Zalubas, J. Phys. Chem. Ref. Data 10, 153 (1981). [34] J. D. Jackson, Classical Electrodynamics (3rd Edn), John Wiley & Sons, New York, Chap. 4, 159-162 (2001). [35] L. V. Hau, S. E. Harris, Z. Dutton and C. H. Behroozi, Nature 397, 594 (1999) and references therein. [36] L. J. Wang, A. Kuzmich and A. Dogariu, Nature 406, 277 (2000) and references therein.

In: Theoretical Physics Research Developments Editors: J. P. Sullivan and A. L. Montey

ISBN: 978-1-61209-446-5 c 2011 Nova Science Publishers, Inc.

Chapter 5

A PPLICATION OF GEANT4 C ODE IN G AMMA I RRADIATION P ROCESSING O. Kadri∗, F. Gharbi, K. Farah and A. Trabelsi National Center for Nuclear Sciences and Technologies, 2020 Tunis, Tunisia

Abstract The present work presents an overview of application of the Monte Carlo code, GEANT4, in the gamma irradiation processing field. In order to check the validity of such code, a successful calculation of expected dose rate and photon flux in the Tunisian gamma irradiation facility was carried out. In the same course of study, an ample set of comparison tests were done using the PMMA dosimeters and the GEANT4 version 8.2 code, for measurement and calculation purposes. Thus, the excellent agreement seen between data and calculations allow us to apply the GEANT4based tool in order to optimize some process parameters, specific to the studied 60 Co facility, and to systematically improve the dose uniformity within irradiated targets having different densities and volumes. Therefore, three irradiation processing procedures were studied let us to conclude that for a given carrier dimensions, more the product density is higher than a determined value, more a specific procedure will be performed. It is shown that Monte Carlo simulation improves the gamma irradiation process understanding.

1.

Introduction

Radiation treatment using 60 Co sources is widely used in research, industry and agriculture. Gamma irradiation plants are constructed for radiation processing of various materials, shapes and dimensions. For the optimum design and use of these facilities, their description and evaluation by means of dose rate distributions is very important. To determine these quantities as well as the minimum and the maximum dose zones, dose mapping of the products in the loading pattern should be carried out [1, 2]. Dosimeters are distributed ∗ E-mail

address: [email protected] (Corresponding address)

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throughout selected product loads to measure minimum and maximum doses. Theses actions, in accordance with the ISO(1995) [3] often consume a large amount of dosimeters, irradiation time and manpower. Furthermore, such facility needs careful characterization to establish plant parameters, such as efficiency, overdose ratio and capacity, since the knowledge of these main parameters with respect to product density is of basic importance for manufacturers and users [4]. These difficulties can be avoided by computer calculation of dose rate distributions. One of the most accurate methods of calculation are Monte Carlo techniques, which have been widely used for existing facilities and for design work and simulation purposes in the last decade. These calculations are generally based on different codes such as MCNP [5], EGS4 [6] and ITS [7]. In this chapter, we aim, firstly, to check the validaty of the GEANT4 code [8] in such calculations at the Tunisian gamma irradiation facility. Currently, Monte Carlo GEANT4 code is used for simulation production in the BaBar experiment [9]; it is widely tested for use as a main simulation engine by large HEP experiments ATLAS [10], CMS [11], and others. Since the first public release of GEANT4 in 1999, several studies focused on medical applications were carried out [12]. For a validity purposes, seven special points were chosen: in air, inside the irradiation cell, and several locations close to the walls. Due to their different localizations relative to the source rack, these points are of extreme cases. Photon flux and dose rates are expected to be different from one point to another. A successful calculation of expected dose rate and flux at these points would be an important check of the validation of GEANT4 for the present simulation of the cell with all its components (source geometry and structure, walls and physics). Transversal and longitudinal dose rate variations relative to the cylindrical source direction were checked in regularly spaced points placed in concentric circles belonging to the middle normal plane to the source rack direction and on one axis parallel to the same direction. To achieve this study, dose uniformity ratios and average dose rates were calculated and measured in a ”dummy” product. Minimum and maximum dose rate positions within the carrier were also established. The second aim of the present chapter consists on the application of the GEANT4 code in the Gamma Irradiation Processing research field. To improve dose uniformity, sawdust carrier was studied, systematically, for three different procedures. Then, to study the effect of package density and volume on the uniformity improvement technique, other different density products and carrier dimensions were taken into account. The next section describes the experimental and the simulation setups followed by a description of an ample set of comparison test of GEANT4 within the Tunisian Gamma Irradiation Facility. After such check of the validity, some applications of GEANT4 is presented in the third section.

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Dosimetry Quality Control in Gamma Irradiation Processing

As mentioned in literature the success of radiation processing largely depends on how accurate one is able to deliver a prescribed dose to an irradiated product. For a state-of-the-art radiation technique such as gamma irradiation this is even more important. In conventional gamma irradiation processing the procedures that confirm that the equipment works, and continues to work, according to its specifications are described as Quality Assurance . An important feature of such a program is the independent testing and verification against existing standards (such as ISO 1995), which is described by the term Quality Control. Since the most important parameter was the dose delivery, we will focus on the dosimetric aspect of the process including the experimental and the theoretical calculations. The next two subsections describe the Tunisian 60 Co gamma irradiation facility, firstly, and the GEANT4 as a Monte Carlo code used to predict dose and to more understanding the process, secondly.

2.1.

60 Co

Facility Design

Mainly designed for preservation of foodstuff and sterilization of medical devices [13], the Tunisian Gamma Irradiation Facility is a box chamber (height=9.4 m, width=length=12.6 m) with high-density concrete walls (1.7 m thick). Source rack is a panoramic dry source storage for continuous operations. Eight 60 Co sources (C-188 MDS Nordion), doubly encapsulated in welded stainless steel, were loaded into the source rack so that two tubes vertically positioned at 1.39 and 1.81 m of height, contain four 60 Co sources with 0.41 m of height each. The radius of the highest and the lowest tubes are respectively 0.102 m and 0.056 m. Source rack and shield container are shifted by 0.3 m toward negative X coordinates. The product to be irradiated is transported by carriers suspended from conveyor. When not in use, sources are stored in a cylindrical shield container (radius=0.39 m, height=0.7 m) of lead. The nominal activity (reported August 2004) was 1.82 ±0.18 PBq [49224 Ci]. Fig. 1 presents a schematic view of the irradiation facility. Dose measurements were carried out using Red Perspex and Gammachrome dosimeters in order to validate the Monte Carlo calculations. These routine dosimeters are polymethyl methacrylate (PMMA) with an overall uncertainty of ±6%, at a 95% confidence level, in the range of 5-50 kGy and 0.1-3 kGy respectively (ISO/ASTM 51276). The determination of absorbed dose was carried out indirectly through spectrophotometric evaluation (Spectronic Genesys 5 UV-VIS spectrophotometer + Kafer KMF30 thickness gauge + Aer’ODE software [14]) of the specific absorbance. Dosimeters were calibrated against Alanine/EPR at the Laboratory of Dosimetry of AERIAL [15] and absorbed doses are referenced against water. Dose measurements were carried out using three PMMA dosimeters per measurement point to reduce errors.

2.2.

Structure and Operation of the GEANT4 Code System

GEANT4 is a toolkit that simulates accurately the passage of particles through matter. It contains a complete range of functionalities including tracking, geometry, physics models and hits. The physics processes offered cover a comprehensive range, including electromagnetic, hadronic and optical processes, a large set of long-lived particles, materials and

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Figure 1. Schematic representation of the Tunisian gamma irradiation facility: (S)source rack; (C) conveyor; 1, control room; 2, dosimetry laboratory; 3, other laboratory; 4, maintenance group; 5, inlet storage; 6, outlet storage; 7, carrier and walls. elements, over a wide energy range. The toolkit has been created exploiting software engineering and object-oriented technology and implemented in the C++ programming language. It has been used in application in particle physics, nuclear physics, accelerator design, space engineering, medical physics [16] and with this work [13, 17, 18], in cylindrical gamma irradiator design. To carry out dose distribution measurements and to accomplish the main topics of the present work, “dummy”products in the density range of 0 .1 - 1.0 g/cm3 were filled into a carrier of dimensions 70 × 108 × 162 cm3 . Then, three different height of carrier of 162, 192 and 222 cm for all the density range were simulated, in order to calculate the efficiency and the capacity of the irradiation facility associated to the corresponding two parameters density and volume. Simulation exercises were carried out using the GEANT4 Monte Carlo code of CERN version 8.2 under a RedHat 9.0 workstation running with 512 MB RAM and a 2 .80 GHz CPU. A specific Fortran 77 code uses the dose calculated by the model and sums them in an appropriate way in order to reproduce plant parameters and critical density values. The usual way followed in GEANT4 applications includes the definition of the geometry associated to the physical arrangement, the generation of primary events according to the simulated experiment, the registration of particles and physics processes involved and the event manipulation in order to render statistics and data output. GEANT4 provides the abstract interface for eight user classes (see Fig. 2). The concrete implementation and registration of these classes are mandatory in three cases, optional in the other five. The three mandatory user class bases are: G4VUserDetectorConstruction, G4VUserPhysicsList and G4VUserPrimaryGeneratorAction; whereas optional ones are: G4UserRunAction, G4UserEventAction, G4UserStackingAction,

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G4UserTrackingAction and G4UserSteppingAction. The default geometry is constructed in DetectorConstruction class, but all of its parameters can be modified interactively via the commands defined in the DetectorMessenger class. Air gas, Lead, Water, Cobalt, Zirconium, Stainless steel and Concrete materials are defined firstly, then assigned to volumes. The setup consists of an experimental hall, which is the world volume, made of air gas and having a box shape, including all walls of the irradiation chamber made of concrete mixture at the corresponding positions. Having all of them cylindrical shapes, the eight 60 Co sources and the shield container were placed at considered positions. Each 60 Co source, consists of cobalt material encapsulated in zirconium followed by stainless steel. We note that the material simulated in the irradiation chamber is the air gas, whose pressure, temperature and density could be modified according to the requirements. Dimension, composition and density of the product were established within this class. For simulation purposes, each dosimeter was considered as a sphere filled with water. Due to the statistical nature of the Monte Carlo method, the accuracy of the results will depend on the number of histories run. Generally, the statistical uncertainties are proportional to the inverse square root of the number of histories. Thus, to cut uncertainties in half, it is necessary to run four times as many histories. Also, for given cutoff energies, the computer time for a history is slightly more than linear in the energy of the incident particle [6]. Statistical tests showed that the most appropriate radius of these spheres is 4 cm, which gives low statistical errors (not more than 5%) for a sufficient total number of generated photons, and don’t effect the non stochastic property of the dose definition. The particle’s type and the physic processes which are used in this GEANT4 application are set in PhysicsList class. All the so called ’electromagnetic processes’ are introduced for gamma, electrons and positrons. In this application the code simulates the transport of photons inside the irradiation cell media. The detailed physics treatment for photons interactions was used (photoelectric effect, Compton effect, pair production). The generation of electrons from photons is considered as well. G4MultipleScattering, G4eIonisation and G4eBremsstrahlung GEANT4 classes are also taken into account for electron and positron particles. An important data member of this class is the DefaultCutValue which define the production threshold of secondary particles, corresponding to the stopping range of the particles; in our code the production cutoffs were set to 1mm for photons(2.902 keV equivalent energy in water) and for e+ /e− (338.695/347.138 keV equivalent energy in water). Interactively mode can activate or not all the processes one by one. After a random choice of a 60 Co source (within eight possibilities), we randomly generate photons from the concerned source and we generate the momentum direction, isotropically, in all the cell space. After a well definition of the primary particle parameters to generate, such as ParticleDefinition, ParticlePosition, ParticleEnergy and ParticleMomentumDirection, we collect the energy deposited in the dosimeters and we convert it to absorbed dose rate. We note that the two decay energies 1.17 and 1.33 MeV of 60 Co were considered as well. The type of particle and its energy are set in the PrimaryGeneratorAction class, and can be changed via the G4 build-in commands of ParticleGun class. A RUN is a set of events.

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The Visualization Manager is set in the main program. The initialization of the drawing is done via a macro file, which is automatically read from the main program when running in interactive mode. The tracks are drown at the end of event and erased at the end of the run. Mandatory and optional user classes described above are constructed and set to the G4RunManager object. Then, the ”Initialize()” instruction of G4RunManager is invoked. Some user interface commands for setting verbosity are applied via hardly coded ”ApplyCommand()” instruction. The ”BeamOn()” instruction triggers the event loop. The argument of this section of the program is the number of event in the run. The computer running time was chosen in such away that the statistical errors were always less than 5%.

Figure 2. Mandatory and some optionally classes needed by a GEANT4-based application.

3.

Experimental Benchmarks

In the following subsections, experimental dose rates were given with error bars corresponding to ±6% uncertainty associated with the use of the PMMA detectors.

3.1.

Special Points within the Cell

An experiment to measure the dose rate at seven choosen points within the cell, locally distributed at a hight of 160 cm (see figure 3), was performed at the National Center for Nuclear Sciences and Technologies (CNSTN). Fig. 4 illustrates a comparison between measured and calculated dose rate of the seven special points. As expected, a completely different dose rate gradient was obtained.

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Figure 3. Special points locations within the cell. Figure 5 shows typical gamma spectra for point P7 with a total number of 50 million primary events. All special points have the same shape as P7, but differ in amplitude. The obtained spectrum clearly reveals the presence of 60 Co peaks corresponding to unscattered photons. The photon spectra amplitudes increase in the following order: P1, P5, P3, (P4 and P6), P2 and P7, which is in accordance with dose rate values in Figure 4. Taking into account that the source rack is shifted by 0.3 m in the negative X direction, we can arrange special points locations from the farthest to the nearest close to the source rack in the following way: P1, (P3 and P5), (P4 and P6), P2 and P7. Photons scattered by walls of the cell contribute more to the P3 photon spectra than to P5, due to the closer position of P3 to the cell corner. The photon spectra at points P4 and P6 are similar, as a result of similarity of different origin contributions to the absorbed photons.

3.2.

Transversal Dose Rate Distribution

Figure 6 presents the comparison between simulated and experimental transversal dose rate distributions, showing the same distribution behaviors along Z axis and the agreement satisfactions. Error bars, corresponding to the 5% statistical errors of simulation data, were not shown in this figure for clarity purpose. Furthermore, the offset seen between the experimental and simulation data explain errors of experimental positioning of dosimeters.

3.3.

Longitudinal Dose Rate Distribution

Finally, Figure. 7 illustrates the differences between the calculated and simulated doses, at the middle plane of the source, clearly shows the correct use of the code. Such satisfactory

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Figure 4. Simulated and experimental dose rate at special points within the cell. agreement confirms that the simulation can be used to predict dose distribution in the useful space of the irradiation cell.

3.4.

3D Dose Rate Mapping of a Phantom Product

In order to check the capability of GEANT4 to reproduce the 3D dose rate mapping of a given phantom product, a carrier was loaded with sawdust material corresponding to weight percentage of 77.6% C and 22.4% O. The package has a volume of 70 × 108 × 162 cm3 and 0.11 g/cm3 bulk density. Loaded with conveyor, shifted by 150 cm from the source and placed at 77 cm from the floor, the carrier is divided into four vertical rows. Each row has

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Figure 5. Photon spectra in air at P7 special point.

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0,25 Simulated data

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Figure 6. Calculated and measured dose rate along the Z axis distributions. six product boxes. There are 6 different levels. The lowest level is designated by level 1; the highest level is designated by level 6; the other levels, in the middle, are designated by level 2, 3, 4 and 5. In order to achieve a good dose homogeneity in the product, one complete irradiation run consists of two cycles. At the end of the first cycle, the carrier leaves the irradiation room to the storage area, where the total package will be rotated by 180 degrees around Z axis. Then, the second irradiation cycle starts, using the same dwell time. For this experiments, the irradiation run time was 42 hours. The average dose rates at one row are shown in Figure 8. From these results it is clear that the dose distribution in the product is symmetric relative to the middle plane of the 60 Co sources, at 157 cm from the floor. This confirms the behavior of the distribution of the dose rate in air, from a distance of 77 cm to 239 cm, as demonstrated in Figure 6. The dose uniformity in a given box is defined as the ratio between maximum and minimum dose rates in the twelve dosimeters in the box. Experimental and simulated dose uniformities have a mean value of about 1.6. Both experiment and simulation indicated that the maximum doses are obtained on the center of the front and the back vertical planes facing the source between levels 3 and 4, while the minimum dose was found at the four corners of the middle vertical plane of the carrier. Maximum dose rates, for simulation and experimental dosimetry, are 7.80 and 8.20 Gy/min, respectively. Whereas, minimum dose rates, for simulation and experimental dosimetry, varies between 4.18-4.24 and 4.31-4.58 Gy/min, respectively. Taking into account ±6% of uncertainty for routine dosimeters, the results are in excellent agreement. The information, of minimum and maximum dose zones in a specified loading pattern, will be used to select monitoring locations for routine processing.

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Figure 7. Experimental and simulated dose rate ratio of 32 different points at the horizontal middle plane of the source rack, with error bars.

Figure 8. Calculated and measured average dose rates at one row in the dummy boxes by level.

4.

Applications of GEANT4

After the last section, we can conclude that that the code was used in a correct way and the GEANT4 code was a suitable tool that can be used in the gamma irradiation processing field. In the following subsections, we will summarize two applications of such code at the 60 Co facility of the CNSTN.

4.1.

Optimization of Some Parameters of the Treatment Process

In order to study the effect of the product volume on the uniformity of the dose rate distribution within products, three different volumes were chosen. All of them have a width of 108 cm and a depth of 70 cm. With respect to the geometry of the irradiation cell and to product box height, the dose distribution in the current product carrier of 162 cm of height

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Figure 9. Experimental and simulation isodose distributions within sawdust product: (a)experimental front plane, (b)simulation front plane, (c)experimental middle plane, (d)simulation middle plane.

(volume1) was compared to the two other situations of height 192 cm (volume2) and 226 cm (volume3) . The efficiency of an irradiation facility for a given carrier volume (1, 2 or 3 described above) and a product bulk density (0.1 - 1.0 g/cm3 ), is defined as the ratio of the total absorbed energy by the product and the total emitted energy from the 60 Co source. The results of the calculations are illustrated in Table 1. Notice that, in our case, four carriers can be loaded and irradiated at the same time without great discrepancy in dose uniformity values. On other hand, and as a second main parameter for characterization of the facility, the Table 1. Plant parameters corresponding to carrier volumes 1, 2 and 3: (E) efficiency in %; (T) throughput product in m3 /7000 h. Density (g/cm3 ) 0.1 0.2 0.3 0.4 0.5 0.7 1.0

Volume 1 E T 6.21 544.13 10.08 478.78 14.01 443.60 17.28 361.91 19.97 299.08 27.37 204.83 31.49 100.53

Volume 2 E T 6.23 452.68 11.55 476.27 16.03 368.19 19.74 334.27 22.77 257.61 22.23 299.08 31.20 106.81

Volume 3 E T 6.95 468.73 12.87 444.85 17.82 348.81 21.92 295.39 25.26 232.53 30.1 159.59 34.37 98.02

throughput product or the capacity of the irradiation plant derived from minimum dose rate values. Using calculated data and applying 7000 annual working hours and 25 kGy mini-

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mum dose, for a given volume and density of product, the throughput values are tabulated in Table 1. From this table, we conclude that the actual carrier dimensions (those of volume 1) were correctly studied by the manufacturer. So, using these dimensions and covering the density range of 0.1-1.0 g/cm 3 , we generally obtain the highest throughput product and the lowest efficiency values for all the studied cases, compared to those of volume 2 and volume 3. Remark that all the previously calculated commissioning parameters (efficiency and capacity) correspond to a 3700 TBq(100 kCi) source activity.

4.2.

Improvement of Dose Uniformity

The second application of such code consists on the study of the possibility of the dose uniformity improvement goal. To achieve good dose uniformity in the irradiated product, three procedures were planned, as shown in Fig. 10. - P(a): zz’ axis rotation of the carrier by 180 degrees, which separate two half irradiation cycles. - P(b): consists of four consecutive steps: after the completion of one irradiation step, levels 1, 2 and 3 are interchanged with 4, 5 and 6 levels, respectively, and the second step starts; then the carrier was rotated around zz’ axis in the third step; in the fourth step, levels 4, 5 and 6 are interchanged with 1, 2 and 3, respectively. - P(c): consists of four consecutive steps: after the completion of one irradiation step, front rows are interchanged with back ones, and the second step starts; then the carrier was rotated around zz’ axis in the third step; in the fourth step, front rows are interchanged with back ones. In order to study a given product bulk density, we consider the ”global dose uniformity” in a given carrier as the ratio between their maximum and minimum dose rates and the ”level dose uniformity” as the ratio between maximum and minimum dose rates in the corresponding level. Figures 11, 12 and 13 illustrate the dose uniformity distributions, fitted with a second order polynomial curve vs the density of “dummy”product following the three uniformity improvement procedures P(a), P(b) and P(c), for volumes 1, 2 and 3. These figures indicate that critical density values of volumes 1, 2 and 3 are 0 .40 ± 0.01, 0.50 ± 0.02 and 0.60 ± 0.03 g/cm3 , respectively. If the carrier height increased, P(b) gives the same uniformity values for a given density, due to the fact that exchanging levels of product compensate the influence of increasing height in terms of transversal energy distribution within carrier. While, P(c), and more faster P(a), increased the uniformity values for a given density. In other words, if we study the effect of the width, instead of the height, we could obtain the same curves of uniformity for P(c), whereas, P(b) curves increased, and more faster P(a) curves. Notice that P(b) and P(c) are more efficient than P(a) in the uniformity improvement goal since they include the last one in their definition.

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Figure 10. Schematic view of dose uniformity improvement procedures: (a) zz’ axis rotation; (b) exchange levels; (c) exchange rows. B: box; gamma rays direction.

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Figure 11. Uniformity ratio as a function of product density for procedures P(a), P(b) and P(c). Carrier dimensions of volume 1: 70 × 108 × 162 cm3 .

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Figure 12. Uniformity ratio as a function of product density for procedures P(a), P(b) and P(c). Carrier dimensions of volume 2: 70 × 108 × 192 cm3 . 10 Pa

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Figure 13. Uniformity ratio as a function of product density for procedures P(a), P(b) and P(c). Carrier dimensions of volume 3: 70 × 108 × 222 cm3 .

5.

Conclusion

The highly selective set of comparisons, presented within this chapter, of GEANT4 calculations with experimental data demonstrates that the code can be used as a predictive simulation tool in the research field of gamma irradiation processing. The excellent agree-

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ment seen between the calculated and the measured dose rates allow us to apply such code in order to determine the efficiency and the throughput product characteristic parameters of a given 60 Co girradiation facility, firstly, and then to study the possibility to improve the dose distribution uniformity within irradiated products by means of a convenient procedure.

References [1] Herring, C.M., Saylor, M.C., 1993. Sterilisation with radioisotopes. In: Morrissey, R.F., Phillips, G.B.(Eds.), Sterilization Technology - A Practical Guide for Manufacturers and Users of Health Care Products. Van Nostrand Reinhold, New York, pp.196-217. [2] Farrar IV, H., 1995. Placement of dosimeters and radiation-sensitive indicators. Radiat. Phys. Chem. 46(4-6), pp.1353-1357. [3] ISO, 1995. ISO-11137. Sterilisation of health care products- Requirements for validation and routine control- Radiation sterilisation. International Organisation for Standardisation, Case Postal 56, CH-1211, Geneve 20, Switzerland. [4] Kovacks, A., et al.,1998. Dosimetry commissioning of the gamma irradiation facility ”ROBO”. Radiat. Phys. Chem. 52 pp 585-589. [5] Briesmeister, J. 1997. MCNP - A general Monte Carlo N-Particles Transport Code , LA 1265-M, Version 4B, Los Alamos National Laboratory, Los Alamos, New Mexico, USA. [6] Nelson, W.R., Rogers, D.W., 1988. Structure and operation of the EGS4 code system . In: Jenkins, T.M., Nelson, W.R., Rindi, A.(Eds.), Monte Carlo transport of electrons and photons. Plenum Press, New York, pp.287-305. [7] Weiss, D.E., Johnson, W.C., Kensek, R.P., 1997. Dose distribution in tubing irradiated by electron beam: Monte Carlo simulation and measurements. Radiat.Phys.Chem. 50(5), 475-485. [8] Agostinelli, S., et al., 2003. Geant4 - a simulation toolkit. NIM A 506, 250-303. [9] BABAR Computing Group (D.H. Wright et al.), CHEP-2003-TUMT006, May 2003, 7pp. Proceedings of the International Conference CHEP’03, La Jolla, California, 2003, e-Print Archive: hep-ph/0305240. [10] ATLAS Liquid Argon HEC Collaboration (B.Dowler et al.), Nucl. Instr. and Meth. A 482 (2002)94. [11] Arce, P., et al., Nucl. Instr. and Meth. A 502(2003)687. [12] Rodrigues, P., et al., 2004. Application of GEANT4 radiation transport toolkit to dose calculations in anthropomorphic phantoms. Applied Radiation and Isotopes 61, pp 1451-1461.

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[13] Kadri, O., et al., 2005. Monte Carlo studies of the Tunisian gamma irradiation facility using GEANT4 code. To be published in App. Radiat. and Isot. [14] Kuntz, F., Fung, R., and Strasser, A., 1999. A dedicated tool for the quality assurance in the field of radiation processing. Proceedings of the 11th International Meeting on Radiation Processing, Melbourne, Austria. [15] www.aerial-crt.com [16] Lazaro D., et al., 2004. Validation of the GATE Monte Carlo simulation platform for modeling a CsI(Tl) scintillation camera dedicated to small-animal imaging. Phys. Med. Biol. 49 pp 271-285. [17] Kadri, O., et al., 2005. Monte Carlo improvement of dose uniformity in gamma irradiation processing using the GEANT4 code. Nucl. Instr. and Meth. B. 239(4) pp 391-398. [18] Kadri, O., et al., 2005. Monte Carlo improvement of dose uniformity in gamma irradiation processing using the GEANT4 code. Nucl. Instr. and Meth. B. 239(4) pp 391-398.

In: Theoretical Physics Research Developments Editors: J. P. Sullivan and A. L. Montey

ISBN 978-1-61209-446-5 © 2011 Nova Science Publishers, Inc.

Chapter 6

STUDY OF ATOMIC ENTANGLEMENT IN MULTIMODE CAVITY OPTICS 1

Papri Saha1, N. Nayak2 and A.S. Majumdar3 Department of Computer Science, Derozio Memorial College, Kolkata, India 2 Department of Applied Physics, Birla Institute of Technology, Mesra, Ranchi-835215, India 3 S.N. Bose National Centre for Basic Sciences, Block-JD, Sector-3, Salt Lake City, Kolkata-700098, India

Abstract In this work the resonant interaction between two two-level atoms and melectromagnetic modes in a cavity is considered. We have considered an ideal cavity that is cavity losses are assumed to be zero. Such a system can be achieved in a microwave cavity with pairs of Rydberg atoms having long lifetime of the order of 0.2 seconds. Entanglement dynamics between two atoms is also examined. In particular, we compare dynamical variations for different cavity modes as well as for different cavity photon numbers. The collapse and revival of entanglement is exhibited by varying the atom-photon interaction times.

I. Introduction The ability to entangle large numbers of atoms that interact with the electromagnetic field for long periods of time has received a great deal of attention in recent years [1-6]. Quantum dynamics of two-level systems (qubits), such as spins in a magnetic field, Rydberg atoms or Cooper Pair Boxes, coupled to a single mode of an electromagnetic cavity are of considerable interest in connection with NMR studies of atomic nuclei [7], Cavity Quantum Electrodynamics [8] and Quantum Computing [9] respectively. The simplest model, which captures the salient features of the relevant physics in these fields, is the Jaynes-Cummings model (JCM) [10] for the one-qubit case and its generalization for multi qubit systems known as Tavis and Cummings model (TCM) [11].

76

Papri Saha, N. Nayak and A. S. Majumdar

Until recently, the quantum light-atom entanglement has been mostly considered at the level of a single mode-single atom interaction which is a natural setting for qubit-type quantum information processing, but an efficient interface between propagating light beams and atoms is yet to be demonstrated. Theoretical efforts have been devoted to control the entanglement because it has been realized that the controlled manipulation of such resources has its practical importance in the actual quantum information processing. Bipartite entanglement in the context of JCM has been widely studied [13-15]. The entanglement sharing in the two-atom TCM coupled to a single mode of the electromagnetic field has been described by various authors [12]. Most of the above works deal with a single mode cavity. Although they are widely studied models, confining the photons to a single mode of the cavity can be very difficult experimentally. Higher mode effects may arise when one includes the cavity losses, and moreover, multimode fields have been conjectured to give rise to significant improvement in terms of the steady-state temperature when many cavity modes are driven simultaneously [16] and also in the case of the two-photon Doppler effect [17]. Hence, study of bipartite entanglement in TCM in a multimode cavity is not out of place. In a recent paper, one of us [18] has analyzed the so-called ‘collapse and revival’ in the atomic population of a single atom in a multimode cavity. It has been shown there some interesting deviation of the atomic population revival times occur compared to a single-mode dynamics. This indicates that the bipartite entanglement in TCM in a multimode cavity could show a different dynamics compared to the single-mode case [19]. In a recent related work, the death and rebirth of entanglement of two non-identical two-level atoms in a two-mode cavity field has been studied [20]. The motivation of this paper is to study the dynamics of entanglement between the atoms (or qubits) in m-mode cavity fields. This paper is organized as follows: A general introduction to the Hamiltonian of our system along with the parameters used to study population inversion and atomic entanglement is provided in Section II. In Sec III, we derive the equations that govern the dynamics of the composite atoms-field system. Entanglement dynamics of a bipartite system in a single mode field will also be studied in this section. Here we also compare the atomic population dynamics between single atom and two-atom case. In Section IV we study the entanglement dynamics for the m-mode radiation field. Finally, we summarize our results in Section V.

II. Atom-Field Interaction Hamiltonian The Tavis-Cummings model (TCM) [11] describes the simplest fundamental interaction between a single mode of the quantized electro-magnetic field and a collection of l atoms under the usual two-level and rotating wave approximations. The two- atom ( l =2) TCM (lower state operators l

Sz = ∑ j =1

[( a

b

a

) −(b b ) ] j

j

2

, transition frequency ω ) can be represented by the spin

a

, upper state

,

S + = ( a b )1 + ( a b )2 S − = ( b a ,

) +(b a ) 1

2

(1)

Study of Atomic Entanglement in Multimode Cavity Optics

77

These are usually Rydberg levels having long lifetime of the order of 0.2 seconds. It follows that the resonant interaction of a m − mode quantized field with two two-level atoms is described by the Hamiltonian H = H 0 + H1 ,

(2)

m

H 0 = ∑ hωak+ ak + hωS z k =1

m

(

,

H1 = h ∑ g k S + ak + S − ak+

(3)

)

(4)

k =1

It is convenient to work in the interaction picture, defined as

H int = e iH 0t / h H1e − iH 0t / h

(5)

We have assumed resonant interaction between both the atoms with the cavity modes. However, the different cavity modes would be different practically as they depend on various factors such as wave vector, transverse variation, polarization, number of photons in each mode etc. Using the Baker-Hausdorff formula this Hamiltonian reduces to m ⎛ m ⎞ H int = h⎜ S + ∑ g k ak + S − ∑ g k ak+ ⎟ k =1 ⎝ k =1 ⎠

(6)

For the interaction of atoms with a quantized radiation field, the unitary time-evolution operator is given by

⎛ − iH int t ⎞ U (t ) = exp⎜ ⎟ ⎝ h ⎠ where the interaction picture Hamiltonian

(7)

H int is given by Eq. (6). The atom-field state at

time t in terms of the state at time t = 0 is given by

ψ (t ) = U (t )ψ (0) We take atoms initially in the ground state number states, i.e.,

ψ (0) =

b1 , b2

(8) and the field as a superposition of

∑ c (0)c (0)...c (0) b , b , n , n ,..., n

1 n1 ,n2 ,...,nm

2

m

1

2

1

2

m

,

(9)

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Papri Saha, N. Nayak and A. S. Majumdar

where

nm represents the number of photons in mode m .

The population dynamics of the atoms can be studied in terms of the function W (t ) defined as

W (t ) =

∑ ( a , a , n , n ,..., n

n1 ,n2 ,...,nm

1

2

1

2

m

ψ (t ) − b1 , b2 , n1 , n2 ,..., nm ψ (t ) 2

2

) (10)

that gives the probability of finding both the atoms in the excited state minus the probability of atoms in the ground state. We use the state

ψ (t )

to obtain the measure of entanglement for the two atoms. The

Concurrence and Entanglement of Formation [21,22] for a bipartite density operator given by

⎛ 1 + 1 − C 2 (ρ ) ⎞ atom ⎟ E F (ρ atom ) = h⎜ ⎟ ⎜ 2 ⎠ ⎝ For the wave function

ψ (t )

ρ atom is

(11)

of atom-field system, the two-atom density operator

ρ atom (t ) is obtained by tracing over field states and is given by ρ atom (t ) =

where,

ψ (t )





n , n2 ,...nm ψ (t ) ψ (t ) n1 , n2 ,...nm

1 n1 ,n2 ,...,nm =0

(12)

is given by Eq. (8) and, C is called the concurrence defined as

(

C (ρ atom ) = max 0, λ1 − λ2 − λ3 − λ4 where the

),

(13)

* λi are the eigenvalues of ρ atom (σ y ⊗ σ y )ρ atom (σ y ⊗ σ y ) in descending order,

and h is the binary entropy function.

III. Population Dynamics and Atomic Entanglement for a Single Mode Field Given an initial state, we time evolve the system according to the dynamics governed by Eq. (8) and then trace over the field state. We use the following derivations

Study of Atomic Entanglement in Multimode Cavity Optics m ⎛ m ⎞ ⎜ S + ∑ g k ak + S − ∑ g k ak+ ⎟ k =1 ⎝ k =1 ⎠

79

2p

= ∑ ( a b )i ( a b ) j 2 p−1 ∑ g k ak i≠ j

(∑ g a ∑ g a + ∑ g a ∑ g a ) ∑ g a k

k

k

+ k

k

p −1

+ k

k

k

k

∑ ( b a ) ( b a ) 2 ∑ g a (∑ g a ∑ g a + ∑ g a ∑ g a ) ∑ g a p −1

i

+ i≠ j

+ k

k

j

k

k

k

+ k

k

p −1

+ k

k

∑ ( a b ) ( b a ) 2 (∑ g a ∑ g a + ∑ g a ∑ g a ) p −1

i

+ i≠ j

∑( a

k

j

k

a )i ( a a ) j 2 p−1 ∑ g k ak

+ i≠ j

k

+ k

+ k

k

k

k

k

p

k

k

k

+ k

k

p −1

+ k

k

k

k

∑ ( b b ) ( b b ) 2 ∑ g a (∑ g a ∑ g a + ∑ g a ∑ g a ) ∑ g a p −1

i

+ i≠ j

∑( a

+ i≠ j

a )i ( b b ) j 2 p−1

+ k

k

j

k

k

k

+ k

m ⎛ m ⎞ ⎜ S + ∑ g k ak + S − ∑ g k ak+ ⎟ k =1 ⎝ k =1 ⎠

k

k

+ k

+ k

k

k

k

k

k

i≠ j

p

k

(14)

(∑ g a ∑ g a + ∑ g a ∑ g a ) k

k

k

+ k

k

+ k

k

p

k

∑ ( b b ) ( a b ) 2 (∑ g a ∑ g a + ∑ g a ∑ g a ) ∑ g a

k

(∑ g a ∑ g a + ∑ g a ∑ g a ) ∑ g a

+ k

p

+ i≠ j

∑( a

+ i≠ j

k

j

a )i ( b a ) j 2 p

k

k

k

k

k

+ k

k

+ k

k

+ k

+ k

p

k

k

k

p

k

k

k

∑ ( b b ) ( b a ) 2 ∑ g a (∑ g a ∑ g a + ∑ g a ∑ g a ) p

+ i≠ j

i

j

k

2 p +1

= ∑ ( a a )i ( a b ) j 2 p ∑ g k ak i

+ k

p −1

+ k

k

(∑ g a ∑ g a + ∑ g a ∑ g a ) k

+ k

(∑ g a ∑ g a + ∑ g a ∑ g a ) ∑ g a k

k

k

+ k

k

k

k

+ k

k

+ k

k

k

p

(15)

to write the wave function of this atom-field state as

ψ (t ) = ψ 1 (t ) + ψ 2 (t ) + ψ 3 (t ) + ψ 4 (t )

,

(16)

ψ (t )

s’ are given as per the situations of the system. where i On using the Eq. (7), (8), (9), (14), (15) and (16), we obtain for single mode field (assuming g1 = g )

ψ 1 (t ) = ∑ x1 n

a1 , a2 , n1

(17)

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Papri Saha, N. Nayak and A. S. Majumdar

ψ 2 (t ) = ∑ x2 n

b1 , b2 , n1

ψ 3 (t ) = −i ∑ x3

b1 , a2 , n1

(19)

ψ 4 (t ) = −i ∑ x4

a1 , b2 , n1

(20)

n

n

here,

(18)

xi ’s are given by x1 = cn1 +2 (0 )

x2 = cn1 (0 )

(n1 + 1)(n1 + 2) 2n1 + 3

(

(

) )

4n1 + 6 − 1

)

)

1 n1 cos gt 4n1 − 2 + n1 − 1 2n1 − 1

(

n1 + 1 sin gt 4n1 + 2 4n1 + 2

x3 = x4 = cn1 +1 (0)

Hence, the

(cos(gt

(21)

(22)

) (23)

ρ atom (t ) of the two two-level atoms is expressed as,

ρ atom (t ) = ∑ x12 a1 , a2 a1 , a2 + ∑ x1 x2 a1 , a2 b1 , b2 + i ∑ x1 x3 a1 , a2 b1 , a2 + i ∑ x1 x4 a1 , a2 a1 , b2 + ∑ x2 x1 b1 , b2 a1 , a2 + ∑ x2 b1 , b2 b1 , b2 2

+ i ∑ x2 x3 b1 , b2 b1 , a2 + i ∑ x2 x4 b1 , b2 a1 , b2 − i ∑ x3 x1 b1 , a2 a1 , a2 − i ∑ x3 x2 b1 , a2 b1 , b2 + ∑ x3 b1 , a2 b1 , a2 + ∑ x3 x4 b1 , a2 a1 , b2 2

− i ∑ x4 x1 a1 , b2 a1 , a2 − i ∑ x4 x2 a1 , b2 b1 , b2 + ∑ x4 x3 a1 , b2 b1 , a 2 + ∑ x4 a1 , b2 a1 , b2 2

(24)

Study of Atomic Entanglement in Multimode Cavity Optics

81

In figures (Figure 1 and Figure 2) we show the time evolution of the atomic inversion, defined in Eq. (10) as the probability W of finding atom (atoms) in the excited state minus the probability of finding the atom (atoms) in the ground state. We notice in Figure 1 that the peaks of the revival times agree with the expression t j g = 2 jπ

n for the j th revival [23].

We further notice in Figure 2 that W (t ) in the two-atom Tavis-Cummings case follow a similar dynamical pattern. This is due to the initial condition given in Eq.(9) that both atoms in their upper states at t = 0 [19]. Such a pattern is again reflected in the entanglement properties of the two atoms as described below.

Figure 1. Atomic inversion for single atom case (a)=50, (b) =25. Note that the curve (a) is shifted by (+1) for clarity.

Figure 2. Atomic inversion for two atoms (a) =50, (b) =25.Curve (a) is shifted as in Figure 1.

The entanglement of formation E of the two-atom mixed state

ρ atom (t ) may then be

calculated using Eq.(24) and is plotted in Figure 3. At certain points of time the entanglement collapses to zero and the two atoms are completely disentangled, while for large time scales revivals of the two atoms entanglement takes place and thus the two atoms behave again as a collective entity, rather than as two individual particles. Gea-Banacloche [24] has shown disentanglement between atom and the field during the course of their interaction in the

82

Papri Saha, N. Nayak and A. S. Majumdar

Jaynes-Cummings model. In the present case, the two atoms interact individually with the common field and in the process the atoms get entangled. The momentarily disentanglement of the two atoms is thus due to the reflection of atom-field disentanglement shown in Ref [24]. 0.7 0.6

(c)

0.5 (b)

0.4 E

0.3 0.2 0.1

(a)

0 0

10

20

30

40

50

gt

Figure 3. Evolution of entanglement for both atoms initially in the ground state and the field (single mode) in an initial coherent state with less photon numbers. (a) solid curve, =0.5; (b) dotted curve, =2.5; and (c) dot-dashed curve, =5.

IV. Entanglement of Two Atoms through Multimode ( m > 1 ) Dynamics We now consider the general multimode case. Let us make the following assumption in successive calculations, i.e.,

cnk (0) = 0, ∀k ,1 ≤ k ≤ m

and

nk = 0,1

(25)

A coherent state radiation field with high average photon number mostly meets this condition. Our analysis shall center on radiation fields in the above-mentioned conditions. The wave function for the atom-field system for multimode radiation is given by

ψ 1 (t ) = ψ 2 (t ) =

∑x

1 n1 ,n2 ,...,nm

∑x

2 n1 ,n2 ,...,nm

ψ 3 (t ) = −i

∑x

3 n1 ,n2 ,...nm

a1 , a2 , n1 , n2 ,..., nm

(26)

b1 , b2 , n1 , n2 ,..., nm

(27)

b1 , a2 , n1 , n2 ,..., nm

(28)

Study of Atomic Entanglement in Multimode Cavity Optics

ψ 4 (t ) = −i

where,

∑x

4 n1 ,n2 ,...nm

83

a1 , b2 , n1 , n2 ,..., nm

(29)

xi ’s are given by

⎛ ⎜ gp ⎜ G m m ⎜ x1 = 2∑ ⎜ cn p + 2 (0)∏ cnk +1 (0 ) p =1 ⎜ k =1 k≠ p ⎜ ⎜⎜ ⎝ ⎛ ⎜ gp ⎜ G m m ⎜ ⎜ x2 = 2∑ cn p (0)∏ cnk +1 (0 ) p =1 ⎜ k =1 k≠ p ⎜ ⎜⎜ ⎝

⎛ ⎜g n p + 1⎜ p ⎜G ⎝

m

np + 2 + ∑ k =1 k≠ p

⎞ ⎟ nk + 1 ⎟ ⎟ ⎠ × − 1 + cos Gt A 1

gk G

(

A1

⎛ ⎜g np ⎜ p ⎜G ⎝

m

np + ∑ k =1 k≠ p

gk G

A2

⎛ ⎞ ⎛ m ⎜ ⎟ gp ⎜ ⎜ cn p +1 (0 )⎜ ∏ cnk (0 )⎟ ∑ p =1 ⎜ ⎟G ⎜ kk =≠1p ⎠ ⎝ ⎝ x3 = x4 = A3 m

⎞ ⎟ nk + 1 ⎟ ⎟ ⎠ × − 1 + cos Gt A 2

(

(

(

))

⎞ ⎟ ⎟ (31) ⎟ m ⎟ + ∏ cnk (0 ) ⎟ k =1 ⎟ ⎟⎟ ⎠

))

⎞ ⎟ np + 1 ⎟ ⎟ ⎠ × sin Gt A 3

(

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎟ ⎠ (30)

) (32)

where, 2 2 ⎛ ⎛⎛ g ⎞⎞ ⎜ ⎜ ⎜ p n + 1 + g k n + 1 ⎞⎟ + ⎛⎜ g p n + 2 + g k n ⎞⎟ ⎟ ⎟ p k p k m ⎟ ⎜G ⎟ ⎟⎟ ⎜ ⎜ ⎜⎝ G G G ⎠ ⎝ ⎠ ⎜∑⎜ ⎟⎟ k = 1 g p gk ⎜ k≠ p ⎜ ⎟⎟ (n p + 1)(nk + 1) + (n p + 2)(nk ) ⎟⎟ ⎜ ⎜⎝ + 2 G 2 ⎠ ⎜ ⎟ 2 2 ⎜ ⎟ ⎛⎛ g ⎞ g g ⎞ ⎛g ⎞ ⎜ m c2 −m+1 ⎜ ⎜⎜ i ni + 1 + j n j ⎟⎟ + ⎜⎜ i ni + j n j + 1 ⎟⎟ ⎟ ⎟ ⎜⎝ G G G ⎜+ ⎠ ⎟ ⎟ ⎠ ⎝G ⎜ ⎟ ⎟ ⎜ k =∑ 1 g g ⎜ ⎟ ⎟ i j ⎜ i< j ⎜ + 2 (ni + 1)(n j ) + (ni )(n j + 1) ⎟ ⎟ 2 ⎜ i, j≠ p ⎝ G ⎠ ⎠ A1 = ⎝ 1≤i , j≤m

(

)

(

)

(33)

84

Papri Saha, N. Nayak and A. S. Majumdar 2 2 ⎛ ⎛⎛ g ⎞ ⎞ ⎜ ⎜ ⎜ p n + g k n ⎞⎟ + ⎛⎜ g p n − 1 + g k n + 1 ⎞⎟ ⎟ ⎟ p k ⎟ p k ⎜ ⎟ ⎟ ⎟ ⎜ m ⎜ ⎜⎝ G G G ⎠ ⎝G ⎠ ⎜∑⎜ ⎟ ⎟ g p gk ⎜ kk =≠1p ⎜ ⎟ ⎟ (n p )(nk ) + (n p − 1)(nk + 1) ⎟ ⎟ ⎜ ⎜⎝ + 2 G 2 ⎠ ⎟ ⎜ 2 2 ⎜ ⎛⎛ g ⎞⎟ gj gj ⎞ ⎛ ⎞ g i i ⎜ ⎜ m c2 −m+1 ⎜⎜ ni + 1 + n j ⎟⎟ + ⎜⎜ ni + n j + 1 ⎟⎟ ⎟ ⎟ ⎜⎝ G G G ⎜+ ⎠ ⎝G ⎠ ⎟⎟ ∑ ⎜ ⎟⎟ ⎜ k =1 ⎟⎟ ⎜ i< j ⎜⎜ + 2 g i g j (n + 1)(n ) + (n )(n + 1) ⎟⎟ i j i j 2 ⎜ 1i ≤, ji≠, jp≤m ⎝ G ⎠⎠ A2 = ⎝

(

)

(

)

2 ⎛ ⎛ ⎜ m c ⎜ ⎛⎜ g i n + 1 + g j n ⎞⎟ + ⎛⎜ g i n + g j j ⎟ i ⎜ ⎜ 2 ⎜ ⎜⎝ G i G G ⎠ ⎝G ⎜ ∑ ⎜ gi g j ⎜ ik 0 ∗ E-mail

address: [email protected], Tel: 0680-2282065, Fax: 0680 - 2243322

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S. N. Jena, H. H. Muni, P. K. Mohapatra et al.

was found quite successful in describing the static electromagnetic properties such as magnetic moments, the axial vector coupling constant ratios for beta decay processes and the charge radii of octet baryons as well as the electromagnetic form factors of nucleons. This model was adapted to study successfully the mass and decay constant of the (qq) ¯ pion [3], S-state mass splittings of the mesons of ss, ¯ cc¯ and bb¯ systems and ground state mass splittings of the heavy non-self-conjugate mesons in strange, charm and bottom flavour sectors [4]. This model was also employed to explain reasonably well the mass spectrum of octet baryons [5] taking into account the contributions due to the colour-electric and magnetic energies arising out of the residual one-gluon-exchange (OGE) interaction alongwith that due to the residual quark-pion coupling arising out of the requirement of the chiral symmetry and that due to necessary center-of-mass (c.m.) motion. In view of the remarkable success obtained with such a model we intend to see its applicability in the present work to the study of radiative(M1) transitions among the vector (V) and pseudoscalar (P) mesons.Our main purpose in this work is in a way to make a parameter-free calculation of the radiative decay rates of mesons. Therefore we, first of all, apply our model to the study of the ground state mass splittings of the non-selfconjugate heavy mesons in the strange, charm and bottom flavour sectors to set the potential parameters and the quark masses so that the model gets ready with no further adjustable parameters and is then used to calculate the radiative decay rates of mesons in light and heavy flavour sectors. In fact a large number of both theoretical and experimental work has been done on the allowed radiative transitions among the ordinary low-lying vector (V) and pseudoscalar (P) mesons. By now experimental results [6] are available with a broad range of accuracies for almost all kinematically allowed transitions of the type V (J p = 1− ) → P(J P = 0−) + γ and P(J P = 0− ) → V (J P = 1− ) + γ, which proceed through magnetic dipole (M1) transitions. A considerable number of theoretical papers based on the constituent quark model study [7-11] within the framework of the naive quark model, the phenomenological Lagrangian approach [12-16] within SU(3) unitary symmetry scheme, and more recently the bag model approach [17-19] as well as the potential model approach [20] taking into account the basic features of QCD have appeared in the literature providing a large amount of understanding in this area. Still none of the theoretical or phenomenological description of one-photon radiative transitions among the low-lying vector and pseudoscalar meson can successfully account for all the experimentally observed decay widths. Most of these descriptions usually fail to explain some specific transitions such as K ∗ → Kγ and V → πγ. The study of radiative transitions among the low-lying vector (V) and pseudoscalar (P) mesons has been considered as a very useful testing ground for various phenomenological quark models, which provide effective methods of investigation for such low-lying hadronic phenomena for which a rigorous field-theoretic formulation with a first principles application of QCD has not so far been possible. Therefore, to gain a clear understanding of these radiative transitions there have been several attempts by various authors starting with the pioneering work of Gell-Mann and others [7-16,21-28]. Taking advantage of the aspect that there exists an apparent dichotomy between the (qq) ¯ structure and the Goldstone boson facet of the pion, Singer and Miller [29] have considered three different dynamical mechanisms such as:

Radiative Transitions of Mesons in an Independent-Quark Potential Model

99

i. photon emission by quarks; ii. photon emission by pion cloud; iii. transitions of a vector bag to a photon accompanied by pion emission, which contribute either singly or in combination However, following this approach the decay rates for V → πγ in the cloudy bag model (CBM) were found to be rather overestimated. In order of throw further light on the physical picture of the controversial pion, static calculation of the partial decay widths of several possible M1 transitions in the light flavour sector was prepared earlier within the conventional framework with the help of different potential models of relativistic independent quarks [30-33]. But these calculations were based on the assumption of moderate momentum transfer as a result of which some amount of uncertainty was found to creep into these model predictions. Therefore further attempts have been made [34,35] to minimize the uncertainty in the predictions of these models by introducing some momentum dependence into the calculation in the form of the recoil of daughter meson in going beyond the static approximation. In fact these models were found to improve the results for the radiative decays of light flavoured mesons. However, until a clear understanding of the intriguing nature of pion is available through some complete theoretical formulation, various alternate schemes of relativistic potential models can be persued to investigate the radiative decays of the mesons. The present work deals with an attempt in this direction where we go beyond static approximation with an aim to provide a satisfactory study of radiative decay widths of light and heavy mesons in the framework of an independent quark model with a square-root confining potential. The potential model used in the present work has a Lorentz structure with equally mixed scalar and vector parts in the square-root form.The advantage of taking such a Lorentz structure is that the Dirac equation can be converted into a Schrodinger-type equation that leads to a simplification of the analysis made in the present work. The implication of such a Lorentz structure has been discussed by Magyari in earlier works [36]. He has shown that the linearly rising potential, the oscillator like potential and the logarithmic potential considered as the fourth component Vv(r) of a Lorentz vector can yield a relativistically consistent confinement of quarks, if it is assumed that the surrounding gluon field generates simultaneously an accompanying scalar interaction Vs(r), which in the rest frame of hadrons is formally identical to to Vv(r). In the present work we are interested for a general formulation incorporating to some extent the recoil effect in order to have a more realistic calculation beyond the static approximation. Since a decay process occurs physically in the definite momentum eigen states of the participating mesons, such a calculation of the radiative decay widths requires suitable expressions for the initial- and final-state mesons reflecting appropriate momentum distribution of the constituent quark and antiquark in their corresponding spin-flavour configuration. We would like to use this generalized approach to calculate the radiative decay widths in the light as well as heavy-flavour sector which would take into account more effectively some momentum-dependent effects due to recoil.

100

2.

S. N. Jena, H. H. Muni, P. K. Mohapatra et al.

Basic Formalism

In a meson, the quark-confining interaction, which is believed to be generated by the nonperturbative multigluon mechanism, cannot possibly be calculated theoretically from first principles of QCD. Therefore, from a phenomenological point of view, the present model assumes that the quark and/or antiquark in a meson core is independently confined by an average potential Vq(r)in the form Vq (r) = (1 + γo )V (r)

(2)

where V (r) = a3/2 r1/2 +Vo with a > 0 To a first approximation, the confining part of the interaction is believed to provide the zeroth order quark dynamics inside the meson core through the quark Lagrangian density   i µ↔ 0 Lq (x) = ψ¯ q(x) γ ∂ µ − mq −Vq(r) ψq (x) (3) 2 The normalized wave function satisfying the Dirac equation derivable from Lq0 (x) as  γ Eq −~γ.~p − mq −Vq (~r) ψq (~r) = 0



0

can be written in two component form as     ψA (~r) i f nl j (r)/r = Nnl j y (~r) ψnl j (~r) = ~σ.ˆr gnl j (r)/r l jm ψB (~r)

(4)

(5)

where normalized spin angular part is yl jm (~r) =



ml ,ms



 1 s l, ml , , ms | jm j Ylml χm 1/2 2

(6)

and Nnl j is the overall normalization constant. The reduced radial part of fnl j (r) of the upper component of Dirac spinor ψnl j (~r) satisfies the equation     l(l + 1) 00 (7) fnl j(r) = 0 f nl j (r) + λnl Enl − mq − 2V (r) − r2 where λnl = Enl + mq . The present model in principle, provide the quark orbital ψnl j (~r) and zeroth order binding energies of the confined quark for various possible eigenmodes through equations (5)(7). However, in this calculation of the S-matrix element for radiative transitions within a static approximation, only the lowest eigenmode corresponding to the ground state of the involved mesons would be relevant and one would be concerned only with the quark orbitals in the 1S1/2 states. Therefore taking f1s(r) = fq (r), E1s = Eq and λ1s = λq for ground state quarks, eq.(7) can be transformed into a convenient dimensionless form as fq” (ρ) + (εq − ρ1/2 ) fq (ρ) = 0

(8)

Radiative Transitions of Mesons in an Independent-Quark Potential Model where ρ =

r roq

101

2

is a dimensionless variable with r0q = (2λq a3/2)− 5 and εq =



λq 16a6

1/5

(Eq − mq − 2V0 )

(9)

Following the discussions given in our earlier work [1,2] the basic eigenvalue equation (8) can be easily solved yielding εq = 1.8418. Then the ground state individual quark binding energy Eq in zeroth order is obtainable from the eigenvalue condition eq.(9). The positive and negative energy quark orbitals for definite quantum numbers n, l, j in terms of which the quark field operators can be expanded whenever required in the present application of the model can be written as   i f nl j (r)/r (+) (ˆr) (10) y ψnl j (~r) = Nnl j ~σ.ˆrgnl j (r)/r l, j,m j   i~σ.ˆrgnl j (r)/r (−) ψnl j (~r) = Nnl j (−1) j+m j−l yl, j,m j (ˆr) (11) f nl j (r)/r The quark orbitals in the lowest eigenmodes, which are only relevant for the description of mesons in their ground-states are obtainable from eqs. (10) and (11) in a simple form as   φq (~r) (+) χλ (12) ψqλ (~r) = Nq r 0 −i ~σ.ˆ λq φq (~r) (−)

ψqλ (~r) = Nq

 ~σ.ˆr λq

 .φ0q (~r)

−i.φq (~r)

χ¯ λ

(13)

  f (r) where φq (~r) = √i4π 1sr and the component spinors χλ and the χ¯ λ denotes χ ↑= 10 , χ ↓= 01   0 , χ¯ ↓= 0i respectively. and χ¯ ↑= −i Finally the overall normalization constant Nq of ψq in eqs. (12) and (13) can be obtained in a simplified form as 5(Eq + mq ) (14) Nq2 = 6Eq + 4mq − 2V0 This provides a brief outline of the potential model and its conventions which is to be adapted in the present investigation of the radiative transitions involving light and heavy mesons.

3.

Energy Corrections to the Ground-State Meson Masses

The quark binding energy Eq obtained from energy eigen-value condition (9) corresponding to the ground state meson leads to the mass of meson core in zeroth order as 0 = ∑ Eq m0M = EM

(15)

q

In order to explain the physical mass of the meson in the ground state one needs to take into account, the other possible contributions due to residual interactions such as the

102

S. N. Jena, H. H. Muni, P. K. Mohapatra et al.

one-gluon exchange interaction at short distances and the quark-pion interaction arising out of the requirement of chiral symmetry preservation at flavour SU(2) level. There will also be further correction necessary due to c.m. motion. However, for the meson in the heavy flavour sector with the pionic corrections, being minimal, one can consider only the correction due to c.m. motion and that due to the one-gluon exchange interaction. Therefore in this paper where we first of all need to apply our model to the study of the physical mass spectrum of heavy non-self-conjugate mesons in their ground states, we give a brief account of these corrections as follows.

3.1.

Center-of-mass Correction

In this model there would be a sizeable spurious contribution to the energy Eq from the motion of the center of mass of the quark-antiquark system. Unless this aspect is duly accounted for, the concept of the independent motion of the quarks inside the meson core will not lead to a physical meson state of definite momentum. Although there is still some controversy on this subject, we follow the technique adapted by Bartelski et al.[37] and Eich et al. [38] which is just one way of accounting for the center-of-mass motion. Following their prescription, a ready estimate of the center-of-mass momentum ~PM can be obtained as     ~PM2 = ∑ ~p2q (16) q

where < ~p2q > is the average value of the square of the individual quark momentum taken over 1S1/2 single-quark states and is given in this model as   (Eq + mq )(Eq − mq − 2V0 )(5Eq − 3V0 + 2mq ) ~p2q = 7(3Eq −V0 + 2mq )

(17)

In the same way, one can find the expression for the c.m corrected mass of the bare meson core as   02 (18) EM = (EM − ∑ ~p2q )1/2 q

which provides the necessary c.m correction to the energy as   ∆EM

=

0 EM − EM

=



02 EM

>

1/2

0 − EM

(19)

q

c.m

3.2.

−∑

< ~p2q

One-Gluon Exchange Correction

The individual quark and antiquark inside the meson core are considered to be experiencing the only force coming from the average effective potential Vq (r) in equation (2). All that remains inside the meson core is hopefully the weak one-gluon exchange interaction provided by the interaction Lagrangian density

L1g(x) = ∑ Jiµa (x)Aaµ (x) a

(20)

Radiative Transitions of Mesons in an Independent-Quark Potential Model

103

µa

where Aaµ (x) are the 8-vector gluon fields and Ji (x) is the ith -quark colour current. Since at small distances, the quarks should be almost free, it is reasonable to calculate the energy shift in the mass spectrum arising out of the quark interaction energy due to their coupling to the coloured gluons, using a first-order perturbation theory. Such an approach leads to the colour-electric and colour-magnetic energy shifts, e,m  e  m  = ∆EM + ∆EM (21) ∆EM g

g

g

which, according to our earlier work [5] on this model are e    αc a a ∆EM = ∑ ∑ λi λ j Ni2N 2j Iiej π a g i, j 

∆EM

m g

4αc =− ∑ 3π i< j







λai λaj (~σi .~σ j )

a

where λai , λaj are the usual Gellmann SU(3) matrices and αc = Iiej = with Fie

1 = 2 2λi



4(Ei −V0 )λi − k

and Iimj

=

Z ∞

Z ∞ 0



2

dkk

2



0

Ni2 N 2j m I λi λ j i j g2c 4π .

dkFie (k)Fje (k) 

j0 (|~k|ri)

(23)

  1/2 ~ − 2λi a ri j0 (|k|ri)



j0 (|~k|ri )

(22)



j0 (|~k|r j )

(24)

(25)

where j0 (|~k|ri) represents the zeroth order spherical Bessel function and the double angular brackets denote the expectation values with respect to the radial angular part φq (~r) of the quark wave function. Now taking into account, the specific quark flavour and spin con  2  = 16 figurations in various ground-state mesons and using the relations, ∑a λai 3 and   = − 16 ∑a λai λaj 3 for mesons, one can in general write the energy correction due to i6= j

one-gluon exchange as



∆EM

e g



∆EM

m g

= αc ∑ ai j Tiej

(26)

= αc ∑ bi j Timj

(27)

i, j

i, j

where ai j and bi j are the numerical co-efficients depending upon each meson and the terms Tie.m j are (Ei − mi − 2V0 )(E j + m j ) 100 Ie (28) Tiej = 3π (3Ei + 2mi −V0 )(3E j − 5V0 − 2m j ) i j

104

S. N. Jena, H. H. Muni, P. K. Mohapatra et al. Timj =

1 200 Im 9π (3Ei + 2mi −V0 )(3E j + 2m j −V0 ) i j

(29)

Here, it should be noted that the colour-electric contribution for the meson masses vanishes when the constituent quark and antiquark masses in a meson core are equal, whereas it is non-zero otherwise. In this type of model the degeneracy among the meson is essentially removed through the strong spin-spin interaction energy in the colour magnetic part only. For non-self-conjugate mesons of the Qq¯ system (Q and q being heavy and light quark respectively) the coefficients are obtained as aQQ = 1, aqq = 1, aQq = −2, bQQ = 0 = bqq and bQq =



2 for triplet states , −6 for singlet states ,

(30)

The integral expressions for Iie,m j in eqs. (23) and (25) can be evaluated with the help of e,m a standard numerical method and these values would yield the terms Ti j from eqs. (28) and (29) which would ultimately enable one to compute the energy corrections (∆EM )e.m g due to one-gluon exchange through eqs. (26) and (27).

3.3.

The Physical Mass of the Ground-State Meson

0 We find that the zeroth-order mass m0M or EM of a ground-state meson, arising out of the binding energies of the constituent quarks confined independently by the phenomenological average potential Vq (r), that presumably represents the dominant non-perturbative multigluon interactions must be subjected to certain corrections due to spurious c.m. motion and the residual quark-gluon interactions. Treating these corrections, as though they are of the same order of magnitude one can obtain the physical mass of a ground-state meson as 0 + (∆EM )c.m + (∆EM )eg + (∆EM )m mM = E M g

(31)

where (∆EM )cm is the energy associated with the spurious c.m. motion [eq. (19)], (∆EM )eg and (∆EM )m g are the colour electric and colour magnetic interaction energies respectively, arising out of the residual one-gluon exchange process [(eqs. (26) and (27))].

4.

Effective Momentum Distribution Function

Since decay of a meson physically occurs between the momentum eigen states of the participating mesons, a more exact field theoretic calculation should have the initial- and finalmeson states represented as appropriate momentum wave packets reflecting their respective constituent quark-antiquark momentum distribution. Although the bound quark and the antiquark inside the meson are in definite energy states having no definite momenta, one can always obtain a momentum probability amplitude by suitable momentum space projection of the corresponding bound quark or antiquark orbital derivable in the model as in eqs.(12)(13). Then using a momentum profile function constructed suitably from quark-antiquark

Radiative Transitions of Mesons in an Independent-Quark Potential Model

105

momentum probability amplitude, one can represent a meson M(q1 q¯2 ) in its momentum state (~P) as momentum wave packet. Taking such wave packets for the initial and final meson states in the radiative decay processes, one can evaluate the S-matrix elements leading to the field theoretic calculation of the decay rates. We represent the state of a meson M(q1 q¯2 ) with an arbitrary momentum ~P and spin projection SV as √  Z 3 M M(~P), SV q ζ (λ , λ ) = ∑ q1 q2 1 2 d~p1 d~p2 δ(3) (~p1 +~p2 − ~P) N(~P) λ1 λ2 εSV  † † ˜ (32) × GM (~p1 ,~p2 )bq1 (~p1 , λ1 )bq2 (~p2 , λ2 ) 0 p1 , λ1 ) and b˜ †q2 (~p2 , λ2 )|0 > are respectively, the quark and antiquark creation where b†q1 (~ SU(6)-spin flavour coefficients for the operators. ζM q1 q2 (λ1 , λ2 ) stands for the appropriate √ pseudoscalar meson M(q1 q¯2 ). The factor 3 is due to the effective color singlet configuration of the meson. N(~P) represents the overall normalization factor, which can be expressed in an integral form as Z (33) N(~P) = d~p|GM (~p, ~P −~p)|2 This is obtainable from the meson-state normalization considered here in the form as < M(~P)|M(~ P0) >= δ(3) (~P − ~P0 )

(34)

Finally, GM (~p1 ,~p2 ) in eq. (32) provides the effective momentum distribution amplitude for the quark and antiquark inside the meson. In an independent particle picture of the present model, GM (~p1 ,~p2 ) can be expressed in terms of individual momentum distribution p1 ) and G˜ q2 (~p2 ) of the quark q1 and antiquark q¯ 2 respectively, for which amplitudes Gq1 (~ we take ansatz that the effective momentum distribution amplitude is simply a geometric mean of the individual momentum probability amplitudes of the constituent particles giving q p1 )G˜ q2 (~ p2 ) (35) GM (~p1 ,~p2 ) = Gq1 (~ p1 ) is the straightforward extension of the idea of Margolis and Mendel [39]. Here Gq1 (~ p2 ) can be obtained by a suitable momentum space projection of the bound as well as G˜ q2 (~ (+) quark and antiquark orbitals ψqλ (~r) in eq. (12) corresponding to the lowest eigenmode. If (+) Gq (~p, λ, λ0 )is the amplitude of a bound quark in its eigemmodes ψqλ (~r) for being found in 0 a state of definite momentum ~p and spin projection λ , then [40] Uq† (~p, λ0 ) Z (+) d~rψqλ (~r)e−i~p.~r Gq (~p, λ, λ0) = p 2E p

(36)

q where E p = ~p2 + m2q and Uq (~p, λ0 ) is the usual free Dirac spinor which are normalized according to the relations Uq† (~p, λ1 )Uq (~p, λ2 ) = 2Eq δλ1 λ2 = Vq† (~p, λ1 )Vq (~p, λ2 )

106

S. N. Jena, H. H. Muni, P. K. Mohapatra et al. U¯ q (~p, λ1 )Uq(~p, λ2 ) = 2mq δλ1 λ2 = V¯q (~p, λ1 )Vq (~p, λ2 )

(37)

and

∑ Uq(~p, λ)U¯ q(~p, λ) = (γµ pµ + mq ) λ

∑ Uq(~p, λ)V¯q(~p, λ) = (γµ pµ − mq )

(38)

λ

For simplification of eq. (36) in the present independent quark model, we take the (+) radial part of the bound quark orbital ψqλ (~r) in Gaussian form like harmonic potential wave functions so that   2 r (39) e−αq r /2 f q (r) = Aq r0q   1 2 . This yields with normalization constant Aq given by Aq = 32πr2 and αq = 18 r0q oq

Gq (~p, λ, λ0 ) = Gq (~p)δλλ0 where Gq (~p) =



Nq λq

  √ 3 1/4  2 E p + mq 1/2 −p 2π 2αq (E + E ) e p q α3q Ep

(40)

(41)

Thus Gq (~p) essentially provides the momentum probability amplitude for a quark q (+) in its eigenmode ψqλ (~r) to have a definite ~p inside the meson. In a similar manner one can obtain the momentum probability amplitude G˜ q (~p) for an antiquark in its eigenmode (−) ψqλ (~r) to realize that, for like flavors, G˜ q (~p) = G∗q (~p)

5.

(42)

Radiative Decay Widths beyond Static Approximation

In this section we perform a more realistic calculation of the radiative decay widths for several energetically allowed mesonic transitions of the type A → Bγ among several vector mesons V and their corresponding pseudoscalar mesons P in the light as well as heavy flavour sector. With the knowledge of the effective momentum profile function and corresponding momentum wave-packets for initial- and final- meson states as given by the expression in eq. (32) one can calculate the S-matrix elements leading to the field theoretic calculation of the radiative decay rates. Now assuming that A → Bγ transitions are predominately single-vertex processes governed mainly by the photon emission from independently confined quark or anti-quark inside the meson [figs. 1(a) and 1(b)], the S-matrix element in the configuration space can be written as   Z ¯ q (x)γµ ψq (x)Aµ (x) |A > (43) SBA =< Bγ| − ie d 4 xT ∑ eq ψ q

Radiative Transitions of Mesons in an Independent-Quark Potential Model

107

Figure 1. Lower order graphs contributing to mesonic M1 transitions. Here the quark and photon field expansions are taken in the usual form   Z 1 d~p −ipx † ipx ˜ p + bq (~p, λ)Vq(~p, λ)e bq (~p, λ)Uq (~p, λ)e ψq (x) = ∑ 3/2 2E p λ (2π) 1 Aµ (x) = ∑ 3/2 δ (2π)

Z

  d~ K −iKx † iKx √ + a (~ K , δ)e K , δ) a(~ K , δ)e εµ (~ 2K0

(44)

eq. (43) can be simplified [34] and can be expressed in the rest frame of the decaying meson A as r   α (4) ˆ A ) Q(~P, ~K ) − Q( ˜ ~P, K ~) δ (P + K − OM (45) SBA = i K0 where P and K represent the four momentum of the daughter meson B and the emitted ~ 0 |, ~K ) respectively. MA stands for the K ) and K ≡ (K0 = |K photon given by P ≡ (EB , ~P = −~ ˜ ~ decaying meson mass and Oˆ ≡ (1, 0, 0, 0). Finally Q(~ K ) and Q( K ) are found in the forms: Q(~P, ~ K) =

˜ ~P, ~ Q( K) =

eq1 A ζ (λ1 λ2 )ζBq1 q2 (λ01 λ2 ) e q1 q2 Z GA (~p, −~p)GB (~P +~p, −~p)C(~p0λ01 ,~pλ1 , ~ K δ) q × d~p 4E1 E1P NA (0)NB1 (~P)



eq2 A ζ (λ1 λ2 )ζBq1 q2 (λ1 λ02 ) e q1 q2 Z ˜ pλ2 , ~P − ~pλ0 , ~K δ) GA (~p, −~p)GB (~P +~p, −~p)C(~ 2 q × d~p 4E2 E2P NA (0)NB2 (~P)

(46)



(47)

108

S. N. Jena, H. H. Muni, P. K. Mohapatra et al. q q Here Ei = ~p2 + m2qi and EiP = (~P +~p)2 + m2qi , i = 1, 2. We must mention that in extracting the correct δ-function relating to the energy conservation at the photon-hadron vertex, we have used the usual approximation [40,41] in setting (E1 + E2 ) ' MA and (E1K + E2 ) ' (E1 + E2k ) ' EB . Now making use of the explicit form of the Dirac spionrs U(~p, λ) and V (~p, λ) eqs. (46) and (47) can be further simplified as eq Q(~p, ~K ) = ∑ 1 ζAq1 ,q2 (λ1 , λ2 )ζBq1 ,q2 (λ01 , λ2 ) e ˜ p, ~K) = ∑ eq2 ζAq ,q (λ1 , λ2 )ζBq ,q (λ01 , λ2 ) Q(~ 1 2 e 1 2 ~ × ε(K, δ) and where, ~κ = K K) = Jq1 (~P, ~



N¯ A (0)N¯ B1 (~P)



Jq2 (~P, ~K ) =

−1/2 Z

E1 + mq 1 4E1 E1K (E1K + mq1 )



d~pχ†λ0 (~σ.~κ)χλ1 , Jq1 (~P, ~ K)

(48)

d~pχ†λ0 (~σ.~κ)χλ2 , Jq2 (~P, ~ K)

(49)

1

Z

2

d~pGA (~ p1 −~p)GB(~p − ~ K , −~p) 1/2

−1/2 Z

N¯ A (0)N¯ B2 (~P)



Z

E2 + mq 2 4E2 E2K (E2K + mq2 )

d~pGA (−~p,~p)GB (−~p,~p − ~ K) 1/2

(50)

In view of the four-δ function appearing in eq.(45) the recoil momentum ~P of the product meson B is obviously implied to be −~ K, which has been qincorporated in eq. (50). ~ )2 + m2q . ConsiderTherefore, EiK in the above expressions for i = 1, 2 stands for (~p − K i

ing the fact that all possible directions of quark and photon momenta are ultimately included K)2 in the in the calculations, it would be a good approximation to take (~p − ~ K)2 = (~p)2 + (~ integrands of the expressions in eq. (50). The same approximation also becomes useful in K) evaluating the meson-state normalization factors using eq.(33). Then the integrals Jq1 (~P, ~ ~ ~ and Jq2 (P, K ) in eq. (50) can be effectively expressed as  −1/2 Z ~ ~ ¯ ¯ Jqi (K) = NA (0)NBi (P)

∞ 0

2 d pp2 Xi (~p, ~ K)e−βp

(51)

where β = 12 ( α1q + α1q ) and 1

2

N¯ A (0) = N¯ Bi (~P) =

Z ∞ 0

Z ∞ 0

d pp2 RA (p)e−βp

2

d pp2 RBi (p, K)e−βp

(52) 2

when   mq j 1/2 ) RA (p) = Π2j=1 (E j + Eq j )(1 + Ej

(53)

Radiative Transitions of Mesons in an Independent-Quark Potential Model   (EiK + Eqi ) Ei (Eik + mqi ) 1/2 RBi (p, K) = RA (p) (Ei + Eqi ) Eik (Ei + mqi )

109 (54)

 1/4 (Ei + mqi )(EiK + Eqi )2 RA (p) Xi(p, K) = 3 2 (EiK + mqi )(Ei + Eqi )2 Ei EiK We must also mention the mixing angle conventions followed here for vector and pseudovector mesons. It is well known that for vector √ mesons ω andφ, the mixing angle is quite close to the ideal nonet value ΘV0 = arcsin(1/ 3) ≈ 35.5◦ for which the quark-flavour com¯ However, in view of the fact that the no ¯ and ω0 = √12 (uu¯ + d d). binations are φ0 = −(ss) net-forbidden φ → ρπ and φ → πγ decays do occur, there is a small departure from the ideal mixing angle ΘV0 . The deviation δV = (ΘV0 − ΘV ) can be obtained from the requirement ΘV = 39◦ by the quadratic mass formula [6]. Hence the flavour- contents of physical φ and ω meson states can be expressed as  0    φ φ sin δV cos δV (55) = − sin δV cos δV ω0 ω The mixing angle for the pseudoscalar η and η0 can also be obtained either from the quadratic mass formula [6] as Θ p = −10.1◦ or from the two photon decays of η and η0 as Θ p = −9.5◦ ± 2.0◦ [42]. Defining the purely strange and non-strange flavour compo¯ (uu+d ¯ d) ¯ and ηns = √2 respectively, we can nents in the pseudoscalar(η,η0)-sector as ηs = −(ss) √ identically express the flavour contents of η and η0 in terms of δ p = [arcsin(1/ 3) − Θ p ] as      ηs η cosδ p sin δ p (56) = − sin δ p cosδ p ηns η0 √ ◦ ¯ − 2ss)/2 = 45 when η = (u u ¯ + d d ¯ and Then the perfect mixing is realized with δ p √ ¯ have the same strange flavour content. However, it must be noted η0 = (uu¯ + d d¯+ 2ss)/2 that measurement [43] of Γη (2γ) leads to a mixing angle Θ p = −17.6◦ + 3.6◦. This would therefore imply a possible range of value for δ p between 45◦ and 56◦. Therefore, for convenience we would express the appropriate matrix elements in terms of δV and δ p . Now following the mixing angle conventions as discussed above and specifying the appropriate spin-flavor coefficients ζM q1 q2 (λ1 λ2 ) for the pseudoscalar meson states and vectormeson states of different spin projections SV = (±1, 0), eqs. (48) and (49) can be further simplified so as to reduce eq. (45) to a form r α (3) ~ ~ δ (P, K )δ(EB + K0 − MA )¯µBA (~ K )κSV (57) SBA = K0 where κSV , for SV = (±1, 0) stands for the following combinations expressed separately for V → Pγ and P → V γ type transitions:   √ (58) κSV (V → Pγ) = ± (κ1 ± iκ2 / 2, κ3   √ κSV (P → V γ) = ± (κ1 ∓ iκ2 )/ 2, κ3

(59)

110

S. N. Jena, H. H. Muni, P. K. Mohapatra et al.

Nevertheless, the summation over the photon polarization index δ and the meson spin Sv yields a general relation (60) ∑ |κSV |2 = 2K¯ 2 δ,SV

¯ in eq. (57) is related to the transition magnetic moment which can be Finally, µ¯ BA (K) K) and Jq2 (~ K). Now summing over the photon poexpressed appropriately in terms of Jq1 (~ larization index δ and the final meson spin appropriately while averaging over initial meson spin when necessary, the partial decay widths for the transition A → Bγ can be obtained as Γ(A → Bγ) =



Z

dKdP|SBA |2 /[V T /(2π)3]

(61)

δ,SV

Then for V → Pγ and P → V γ processes, we can realize the partial decay widths in the standard form in terms of the outgoing photon energy given by (MA2 − MB2 ) K¯ = 2MA 2 q 4 ¯ ¯ E p (K)/M µ ¯ ( K) Γ(V → Pγ) = αK¯ 3 A PV 3 2 q ¯ ¯ EV (K)/M µ ¯ ( K) Γ(P → V γ) = 4αK¯ 3 P VP

(62)

(63) (64)

p ¯ One must note that the phase space factor E p (K)/M A in eq. (63) is arising here out of the argument factorization of energy δ-function appearing in eq. (57). Therefore, before taking this factor so seriously, one must compare the expression (5.22) with the ones obtainable from a formal relativistic calculation at mesonic level. In fact, starting with a relativistic effective interaction of the form µPV εµνλσ ∂µ Aν (x)∂λVσ(x)P(x),

(65)

where Aν (x),Vσ(x) and P(x) are, respectively, the fields of photon, vector meson and pseudoscalar meson, one can arrive at an expression for Γ(V → Pγ) in terms of the transition K )/MA as found in eq. (63). moment µPV without the mesonic level phase space factor EP (~ One can also do the same thing by considering the covariant expansion µ |V (P) >= µPV εµνλσ εν (P + P0 )λ (P − P0 )σ < P(P0)|Jem

(66)

in eq. (43) together with the appropriate relativistic phase space in eq. (63). This is therefore a pathological problem common to all such models attempting to explain hadronic level decay in terms of its constituent level dynamics considered in zeroth order. q Hence it requires

EB appropriate corrective measures for eliminating this spurious phase MA at the mesonic level. For the transitions involving a photon energy much less compared to the decaying meson mass, this factor is ¯ K¯ EB (K) = 1− '1 (67) MA MA

Radiative Transitions of Mesons in an Independent-Quark Potential Model

111

¯ ¯ as the corresponding transition moment hence setting EBM(AK) ' 1 and interpreting µ¯ BA (K) as has been done by many authors [20,24-28] in the past may be justified only for static calculations. But such a prescription is not, is general, correct when one wants to look beyond the static calculation. However, there is a clear possibility of explicit cancelation of this phase space factor taken appropriately alongwith the contribution of the quark spinors. This has been shown explicitly in the works of Altomari [44] and also has been discussed in a recent work by O’Donnell and Tung [45] within the score of their model based on the description of the meson states in the loose binding approximation in terms of relative momentum wave function of the constituent quarks in the non-relativistic Gaussian forms. In view of this observation, we prefer here to push back this phase factor from the mesonic ¯ under the same approximation K) describing µ¯ BA (K) level to the quark level integrals Jqi (~ with which it was extracted out through the energy argument of the δ-function. Hence

q



¯ A ) ' EiK + E j EB (K/M E1 + E2

1/2

(68)

K ) in eq. (51) and the resulting when i 6= j = 1, 2 are included in the quark level integrals Jqi (~ ~ K). integrals are denoted as Iqi (K) which are used for calculation of transition moments µBA (~ ¯ Since we have considered flavour SU(2) symmetry with mu = md 6= ms, the integral Iqi (K) ¯ = Iu (K). ¯ corresponding to any particular transition can provide Id (K) In one considers a heavy meson with a heavy quark qi and a light antiquark q¯ 2 , then the ¯ when evaluated for very large quark mass mq1 yield integrals Iqi (K) ¯ ¯ ~ −(1/2) ¯ = [2NA (0)NB1 (P)] × Iq1 (K) mq 1

Z ∞ 0

 mq2 1/2 −βp2 d p p (Eq1 + mq1 )(E2 + Eq2 )(1 + ) e E2 (69) 2



and Z ∞ mq [2N¯ A (0)N¯ B2 (~P)]−(1/2) × d p p2 (Eq1 + mq1 )(E2 + Eq2 )(1 + 2 )1/2 mq 1 E2 0  1/4  1/2 2 (E2 + mq2 )(E2k + Eq2 ) E2k + mq1 2 × e−βp (70) 3 2 E2 + mq 1 (E2k + mq2 )(E2 + Eq2 ) E2 E2k

¯ = Iq2 (K)

¯ indicates that in case of a heavy meson the The integral expression (69) for Iq1 (K) contribution of the electromagnetic current of the heavy quark to the decay amplitude gives rise to a characteristic magnetic moment suppression factor 1 /mq1 for M1 transitions. Then the partial decay widths for the transition A → Bγ are expressed as 2  4 ¯3 ¯ Γ(V → Pγ) = αK µPV (K) 3 2  3 ¯ ¯ Γ(P → V γ) = 4αK µV P (K)

(71)

¯ for each radiative transition of meson in the light as where the transition moments µBA (K) ¯ and well as heavy flavor sector can now find separate explicit expressions in terms of Iq1 (K)

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S. N. Jena, H. H. Muni, P. K. Mohapatra et al.

¯ in the following manner, Iq2 (K) ¯ = 1 Iu (K) ¯ (i)µπρ(K) 3 ¯ = cos δV Iu (K) ¯ (ii)µπω(K) ¯ ¯ (iii)µ πφ(K) = sin δV Iu (K) ¯ = sin δ p Iu (K) ¯ (iv)µηρ(K) ¯ = [cosδV sin δ p Iu (K) ¯ + 2 sin δV cos δ p Is (K)]/3 ¯ (v)µηω(K) ¯ ¯ ¯ (vi)µηφ(K) = [sin δV cos δ p Iu (K) − 2 cos δV cosδ p Is (K)]/3 ¯ = [sin δV cos δ p Iu (K) ¯ + 2 cos δV sin δ p Is (K)]/3 ¯ (vii)µη0φ (K) ¯ = [cosδV cos δ p Iu (K) ¯ − 2 sin δV sin δ p Is (K)]/3 ¯ (viii)µωη0 (K) ¯ ¯ (ix)µρη0 (K) = cosδ p Iu (K) ¯ = [2Iu (K) ¯ − Is (K)]/3 ¯ (x)µK + K ∗+ (K) ¯ ¯ ¯ (xi)µK 0K ∗0 (K) = −[Iu (K) + Is (K)]/3 ¯ ¯ ¯ (xii)µD+D∗+ (K) = [2Ic(K) − Iu (K)]/3 ¯ = 2[Ic (K) ¯ + Iu (K)]/3 ¯ (xiii)µD0D∗0 (K) ¯ ¯ ¯ (K) = [2Ic(K) − Is (K)]/3 (xiv)µD+s D∗+ s ¯ = [2Iu (K) ¯ − Ib (K)]/3 ¯ (xv)µB+ B∗+ (K) ¯ ¯ ¯ (xvi)µB0 B∗0 (K) = −[Iu (K) + Ib (K)]/3 ¯ ¯ ¯ (K) = −[Is(K) + Ib (K)]/3 (xvii)µB0s B∗0 s ¯ = [2Ic(K) ¯ − Ib (K)]/3 ¯ (K) (xviii)µB+c B∗+ c

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In the above expression√δV and δ p are the mixing angle deviations from the ideal mixing such that δM = (arcsin(1/ 3) − ΘM ).

6.

Results and Discussion

In this section we calculate the partial decay widths of several energetically possible M1 transitions among the vector and pseudoscalar mesons in a broken SU(3) flavour symmetry by using the formalism developed in the previous sections. The calculation primarily involves the potential parameters(a, b),quark masses(mu = md , ms , mc and mb ) and the mixing angles(δV , δ p ). Here, we begin by determining the potential parameters and quark masses by applying the present model to study the ground state hyperfine splittings of non-selfconjugate heavy mesons in the strange, charm and bottom flavour sectors.

A. Hyperfine Splittings of Non-self-conjugate Mesons We find that a suitable choice of potential parameters (a,V0) = (0.454, −0.465) GeV and the quark masses (mu = md , ms ) = (0.225, 0.433) GeV (mc, mb ) = (1.462, 4.8184) GeV

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Radiative Transitions of Mesons in an Independent-Quark Potential Model

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which yield the individual quark binding energies Eq as (Eu = Ed , Es ) = (0.5901, 0.7128) GeV (Ec, Eb ) = (1.5305, 4.6809) GeV

(74)

0 ∗ provide the ground state masses for (D∗, D), (D∗s , Ds), (B∗, B), (B0∗ S , Bs ) and (Bc , Bc ) mesons in close agreement with the available experimental values. In the present phenomenological fit, we consider only the gluonic corrections (the pionic corrections being minimal) to the binding energy corrected first by the contribution arising out of the c.m. motion. For this e,m fit we evaluate the values of < ~PM2 > in eq. (16) and Ti j in eqs. (28) and (29), which are necessary for computing respectively the c.m and OGE corrections. These values are displayed in Table-1. We find here that the description of QCD mass splittings require a reasonable value of the strong coupling constant αc = 0.1805 which is quite consistent with the idea of treating one-gluon exchange effects in lowest order perturbation theory. The results so obtained for the meson masses in the charm, strange and b-flavour sectors are presented in table-2 in comparison with experimental values. The calculated values of the energy contributions considered here are also displayed in table-2.

Table 1. The Calculated values of < ~PM2 > in MeV 2 and Tie,m j in MeV required respectively for c.m. and OGE corrections. mesons Qq¯ D(cu) ¯ DS (cs) ¯ B(bu) ¯ BS (bs) ¯ Bc (bc) ¯

2 >