Two-level trap model of BEC in an ideal gas

Konstantin E. Dorfman

In this dissertation, I have studied theoretical problems in statistical physics and\nelectrodynamics of Bose particles, namely, mesoscopic effects in statistics of Bose-\nEinstein condensate (BEC) of atoms and electromagnetic waveguide effects of planar\nBragg structures in Free Electron Lasers.\nA mesoscopic system of a trapped gas of Bose atoms is the most difficult for\nthe theoretical analysis in quantum statistical physics since it cannot be studied by\nneither a quantum mechanics of the simple microscopic systems of one or very few\natoms nor a standard statistical physics of the macroscopic systems that implies a\nthermodynamic limit.\nI present analytical formulas and numerical calculations for the moments and\ncumulants of BEC fluctuations in both ideal and weakly interacting gas.\nI analyze the universal scaling and structure of the BEC statistics in a mesoscopic\nideal gas in the critical region. I present an exactly solvable Gaussian model of BEC\nin a degenerate interacting gas and its solution that confirms the universality and\nconstraint-cut-off origin of the strongly non-Gaussian BEC statistics.\nI consider a two-energy-level trap with arbitrary degeneracy of an upper level\nand find an analytical solution for the condensate statistics in a mesoscopic ideal gas.\nI show how to model BEC in real traps by BEC in the two-level or three-level traps.\nI study wave propagation in the open oversized planar Bragg waveguides, in particular, in a planar metal waveguide with corrugation. I show that a step perturbation\nin a corrugation phase provides a high selectivity over transverse modes.\nI present a new Free Electron Laser (FEL) amplifier scheme, in which the radiation\nis guided by the planar Bragg structure with slightly corrugated walls and\na sheet electron beam is traveling at a significant angle to the waveguide axis. By\nmeans of nonlinear analysis, I demonstrate that the proposed scheme provides an\neffective mode filtration and control over the structure of the output radiation and\nallows one to achieve amplification up to 30 dB in the existing FEL machines.

Mark Edwards

S. B. Weiss, Michael J. Banks, J. Janis +1 more

Vitaly V. Kocharovsky, Vl. V. Kocharovsky

We find the universal structure and scaling of the Bose-Einstein condensation (BEC) statistics and thermodynamics (Gibbs free energy, average energy, heat capacity) for a mesoscopic canonical-ensemble ideal gas in a trap with an arbitrary number of atoms, any volume, and any temperature, including the whole critical region. We identify a universal constraint-cutoff mechanism that makes BEC fluctuations strongly non-Gaussian and is responsible for all unusual critical phenomena of the BEC phase transition in the ideal gas. The main result is an analytical solution to the problem of critical phenomena. It is derived by, first, calculating analytically the universal probability distribution of the noncondensate occupation, or a Landau function, and then using it for the analytical calculation of the universal functions for the particular physical quantities via the exact formulas which express the constraint-cutoff mechanism. We find asymptotics of that analytical solution as well as its simple analytical approximations which describe the universal structure of the critical region in terms of the parabolic cylinder or confluent hypergeometric functions. The obtained results for the order parameter, all higher-order moments of BEC fluctuations, and thermodynamic quantities perfectly match the known asymptotics outside the critical region for both low and high temperature limits. We suggest two- and three-level trap models of BEC and find their exact solutions in terms of the cutoff negative binomial distribution (which tends to the cutoff gamma distribution in the continuous limit) and the confluent hypergeometric distribution, respectively. Also, we present an exactly solvable cutoff Gaussian model of BEC in a degenerate interacting gas. All these exact solutions confirm the universality and constraint-cutoff origin of the strongly non-Gaussian BEC statistics. We introduce a regular refinement scheme for the condensate statistics approximations on the basis of the infrared universality of higher-order cumulants and the method of superposition and show how to model BEC statistics in the actual traps. In particular, we find that the three-level trap model with matching the first four or five cumulants is enough to yield remarkably accurate results for all interesting quantities in the whole critical region. We derive an exact multinomial expansion for the noncondensate occupation probability distribution and find its high-temperature asymptotics (Poisson distribution) and corrections to it. Finally, we demonstrate that the critical exponents and a few known terms of the Taylor expansion of the universal functions, which were calculated previously from fitting the finite-size simulations within the phenomenological renormalization-group theory, can be easily obtained from the presented full analytical solutions for the mesoscopic BEC as certain approximations in the close vicinity of the critical point.

Sergey V. Tarasov, Vl. V. Kocharovsky, Vitaly V. Kocharovsky

We calculate the Bose-Einstein-condensate (BEC) occupation statistics vs the interparticle interaction in a dilute gas with a nonuniform condensate in a box trap within the Bogoliubov approach. The results are compared against the previously found BEC-occupation statistics in (i) an ideal gas and (ii) a weakly interacting gas with a uniform condensate. In particular, we reveal and explicitly describe an appearance of a nontrivial transition from the ideal gas to the Thomas-Fermi regime. The results include finding the main regimes of the BEC statistics---the anomalous non-Gaussian thermally dominated fluctuations and the Gaussian quantum-dominated fluctuations---as well as a crossover between them and their manifestations in a mesoscopic system. Remarkably, we show that the effect of the boundary conditions, imposed at the box trap, on the BEC fluctuations does not vanish in the thermodynamic limit of a macroscopic system even in the presence of the interparticle interactions. Finally, we discuss a challenging problem of an experimental verification of the theory of the BEC fluctuations addressing a much deeper level of the many-body statistical physics than usually studied quantities related to the mean condensate occupation.

Alfred Hammond, L. Lavoine, Alessio Recati +2 more