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Modern Problems in Statistical Physics of Bose-Einstein Condensation and in Electrodynamics of Free Electron Lasers

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Modern Problems in Statistical Physics of Bose-Einstein Condensation and in Electrodynamics of Free Electron Lasers MODERN PROBLEMS IN STATISTICAL PHYSICS OF BOSE-EINSTEIN CONDENSATION AND IN ELECTRODYNAMICS OF FREE ELECTRON LASERS A Dissertation by KONSTANTIN DORFMAN Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY May 2009 Major Subject: Physics MODERN PROBLEMS IN STATISTICAL PHYSICS OF BOSE-EINSTEIN CONDENSATION AND IN ELECTRODYNAMICS OF FREE ELECTRON LASERS A Dissertation by KONSTANTIN DORFMAN Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Approved by: Chair of Committee, Vitaly Kocharovsky Committee Members, Glenn Agnolet Alexey Belyanin Peter Kuchment Head of Department, Edward Fry May 2009 Major Subject: Physics iii ABSTRACT Modern Problems in Statistical Physics of Bose-Einstein Condensation and in Electrodynamics of Free Electron Lasers. (May 2009) Konstantin Dorfman, B.S., Nizhny Novgorod State University; M.S., Texas A&M University Chair of Advisory Committee: Dr. Vitaly Kocharovsky In this dissertation, I have studied theoretical problems in statistical physics and electrodynamics of Bose particles, namely, mesoscopic effects in statistics of Bose- Einstein condensate (BEC) of atoms and electromagnetic waveguide effects of planar Bragg structures in Free Electron Lasers. A mesoscopic system of a trapped gas of Bose atoms is the most difficult for the theoretical analysis in quantum statistical physics since it cannot be studied by neither a quantum mechanics of the simple microscopic systems of one or very few atoms nor a standard statistical physics of the macroscopic systems that implies a thermodynamic limit. I present analytical formulas and numerical calculations for the moments and cumulants of BEC fluctuations in both ideal and weakly interacting gas. I analyze the universal scaling and structure of the BEC statistics in a mesoscopic ideal gas in the critical region. I present an exactly solvable Gaussian model of BEC in a degenerate interacting gas and its solution that confirms the universality and constraint-cut-off origin of the strongly non-Gaussian BEC statistics. I consider a two-energy-level trap with arbitrary degeneracy of an upper level and find an analytical solution for the condensate statistics in a mesoscopic ideal gas. I show how to model BEC in real traps by BEC in the two-level or three-level traps. I study wave propagation in the open oversized planar Bragg waveguides, in par- iv ticular, in a planar metal waveguide with corrugation. I show that a step perturbation in a corrugation phase provides a high selectivity over transverse modes. I present a new Free Electron Laser (FEL) amplifier scheme, in which the ra- diation is guided by the planar Bragg structure with slightly corrugated walls and a sheet electron beam is traveling at a significant angle to the waveguide axis. By means of nonlinear analysis, I demonstrate that the proposed scheme provides an effective mode filtration and control over the structure of the output radiation and allows one to achieve amplification up to 30 dB in the existing FEL machines. v To my beloved parents Evgeniy Dorfman and Tatiana Bystrova vi ACKNOWLEDGMENTS I would like to thank everybody who has inspired, encouraged, and helped me in my work towards Ph.D. although it is nearly impossible to come up with a complete list. In particular, I would like to express my gratitude to my advisor, Dr. Vitaly Kocharovsky, for his guidance and patience. I would like to thank the members of the Ph.D. committee, Dr. Glenn Agnolet, Dr. Alexey Belyanin, and Dr. Peter Kuchment, for their endurance and patience. The Physics Department of Texas A&M University is also gratefully acknowl- edged. Finally, I wish to thank my parents for their love, encouragement, understanding, and patience. vii TABLE OF CONTENTS CHAPTER Page I INTRODUCTION .......................... 1 A. Phenomenon of the Bose-Einstein condensation . 1 B. Theoretical models of BEC quantum statistics . 11 1. Standard theory of BEC and the problem of the mesoscopiceffects . .. .. 11 2. Micro-canonical, canonical and grand-canonical statistics ......................... 16 3. Exact recursion relation for the statistics of the number of condensed atoms in an ideal Bose gas . 19 4. Dynamical master equation approach . 21 5. Integral representation via the generalized Zeta function.......................... 25 C. Mesoscopicsystems . 27 D.PlanarBraggwaveguides . 29 E. Distributedfeedback . 32 F. Contents of the BEC studies (Chapters II-V) and brief discussion of the main results . 33 G. Contents of the FEL electrodynamics studies (Chapters VI-VII) and brief discussion of the main results . 37 II QUANTUM STATISTICS OF THE IDEAL AND WEAKLY INTERACTING BOSE GASES IN A BOX IN THE CANON- ICAL ENSEMBLE. ......................... 39 A. Canonical ensemble quasiparticle approach . 39 B. Characteristic function for the ground state occupation numberinanidealBosegas . 42 C. Constraint nonlinearity and many-body Fock space cut- offinthecanonicalensemble . 47 D. Properties of the spectrum of the box with the periodic boundaryconditions . 48 E. Multinomialexpansion . 52 F. Mesoscopic effects versus the thermodynamic limit . 56 viii CHAPTER Page III UNIVERSAL SCALING AND ORIGIN OF NON-GAUSSIAN BOSE-EINSTEIN CONDENSATE STATISTICS IN A MESO- SCOPIC IDEAL GAS ........................ 68 A. Constraint-cut-off mechanism of strong non-Gaussian BEC fluctuations. Universal structure and cut-off of the condensate occupation probability distribution . 68 B. Universal scaling and structure of the BEC order parameter ........................... 73 C. Universal scaling and sructure of all higher-order mo- ments and cumulants of the BEC fluctuations . 78 D. Exactly solvable Gaussian model of BEC statistics in a degenerate interacting gas demonstrating strongly non- Gaussian condensate fluctuations . 83 IV MESOSCOPIC BOSE-EINSTEIN CONDENSATION OF AN IDEAL GAS IN A TWO-LEVEL TRAP ............. 89 A. Exact solution for BEC in a two-level trap: cut-off neg- ativebinomialdistribution . 89 B. Continuous approximation: cut-off gamma distribution . 90 C. Modeling BEC in a real trap by BEC in a two-level trap . 91 V SADDLE-POINT METHOD FOR CONDENSED BOSE GASES 97 A. Review of the saddle-point method for a condensed Bosegasinaharmonictrap . 97 B. The development of the saddle-point method for a con- densedBosegasinabox . 100 C. Comparison with the numerical results . 104 VI ELECTROMAGNETIC MODES OF OPEN PLANAR BRAGG STRUCTURES ........................... 107 A. Dielectic layer with periodic modulation of dielectric constant ............................ 107 1. Dispersionrelation. 107 2. Eigenmodes........................ 108 3. Bragg waveguide with a defect of periodicity . 111 4. Large detuning from the cut-off frequency . 113 B. Planar Bragg waveguide with slightly corrugated plates . 117 ix CHAPTER Page VII FREE ELECTRON LASER AMPLIFIERS BASED ON PLA- NAR BRAGG WAVEGUIDES ................... 126 A. Free Electron Laser amplifier with Bragg waveguide . 126 B. Laser amplifier with Bragg waveguide . 129 C. Transverse current FEL amplifier . 134 D. Other possible amplification schemes based on Bragg waveguides........................... 143 VIII CONCLUSIONS ........................... 145 A. Summary of the main results on the BEC statistics . 145 B. Summary of the main results on the FEL electrodynamics 148 REFERENCES ................................... 151 APPENDIX A ................................... 157 APPENDIX B ................................... 159 APPENDIX C ................................... 161 APPENDIX D ................................... 164 VITA ........................................ 170 x LIST OF FIGURES FIGURE Page 1 Negative binomial distribution of the number of noncondensed atoms for an ideal Bose gas in a box with total number of particles N = 10000. ................................ 18 2 The scheme of an open Bragg waveguide. Wave propagates in the direction z which is transverse to the lattice vector ~h¯ = ~x(0)h¯, ¯ 2π h = d . .................................. 31 3 Unconstrained probability distribution of the number of noncon- (∞) densed atoms ρn (dashed line) and its cut offs (solid lines) for a ′ ′ ′ small number of atoms N < Nc (OA N , there is no condensate) and for a large number of atoms N > Nc (OAN, there is con- densate) for the trap with a given volume and temperature: the trap-size parameter in Eq. (3.2) is Nv = 100. ............. 49 4 Temperature scaling of the mean value of the ground state occu- pation fluctuations for an ideal Bose gas (grey lines) and weakly interacting Bose gas (black lines) for N = 100 obtained from thermodynamic limit expression in Eqs. (2.30) - (2.31) (dashed lines) compared with the multinomial expansion in Eq. (2.47) (dots) and with the exact recursion relation for an ideal gas in Eq. (2.51) (solid lines). The multinomial expansion result is al- most indistinguishable from the recursion relation as it is clearly seen in graphs. .............................. 62 5 Temperature scaling of the variance of the ground state occupa- tion fluctuations for an ideal Bose gas (grey lines) and weakly interacting Bose gas (black lines) for N = 100 according to Eqs. (2.30) - (2.31) (dashed lines), Eq. (2.47) (dots), and Eq. (2.51) (solid lines). ................................ 63 xi FIGURE Page 6 Temperature scaling of the third central moment µ3 of the ground state occupation fluctuations for an ideal Bose gas (grey lines) and weakly interacting
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