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.
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.
The mesoscopic effects in the quantum trapped gases of the Bose atoms constitute\nthe main subject of the present thesis. These effects are the most difficult for\nthe theoretical analysis in the quantum statistical physics since they can’t be seen by\nneither a standard quantum mechanics of the simple microscopic systems of one or\nvery few atoms nor a standard statistical physics of the macroscopic systems that are\ninfinite in the bulk (thermodynamic) limit.\nMost of the experiments on the cold quantum gases performed in the last decade,\nstarting from the first demonstration of BEC in 1995, involve the mesoscopic systems\nof a finite number of atoms. The mesoscopic effects should manifest themselves most\nclearly and easily near a critical temperature of BEC; however, they could be observed\nalso above and below the critical temperature.\nHere I study the quantum and thermal fluctuations of the Bose-Einstein condensate\n(BEC) in a box with the periodic boundary conditions under a particle-number\nconstraint. The above constraint is the only reason for the BEC and is crucial for the\nmesoscopic effects in the BEC fluctuations, especially in the vicinity of the critical\ntemperature in the Bose gas.\nI employ the particle-number conserving operator formalism of Girardeau and\nArnowitt introduced in 1959 to analyze the canonical ensemble fluctuations. I present\nanalytical formulas and numerical calculations for the central moments of the ground\nstate occupation fluctuations in an ideal Bose gas in a box with a mesoscopic number of particles.\nI present the analysis of the BEC statistics both on a temperature at a fixed\nnumber of particles and on a number of particles at a fixed temperature. Both analyses\nare valid for the purpose of understanding the important mesoscopic effects near the\ncritical temperature. I emphasize the non-Gaussian nature of the fluctuations.\nThe presented formalism can be generalized to the case of a weakly interacting\nBose gas in a box in the framework of the Bogoliubov approximation. The work in\nthis direction is in progress but is not included in the present thesis.
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.
Many interesting physical systems, e.g. condensed matter systems, are hard to study. From the experimental point of view, in such systems most parameters are ''frozen" for a given sample. From the theoretical point of view, they often cannot be simply modelled from first principles. In contrast, laser-matter interactions are very well known theoretically, and a very high degree of control of laser beams is available for the experimentalist. In the last years, one learned to ''engineer" quantum hamiltonians using ultracold atoms interacting with electromagnetic fields. For example, laser beams may form an interference pattern that is ''seen" by the atoms as a periodic potential, thus creating an analog of a crystal. However, the analog has great advantages over the original system, as one has a larger control of its properties: One can vary the parameters of the potential by changing the properties of the laser beams, one can make a ''mesoscopic" quantum system by using a Bose-Einstein condensate, one can change it to a degenerate Fermi gas by using a fermionic isotope, one can control atom-atom interactions by using the so-called . Building on seminal ideas of Feynman, the notion of ''quantum simulator" -- a ''simpler" system able to mimic the physics of a complicate one -- recently became a reality. In a variety of situations of experimental interest, weakly interacting degenerate boson gases (Bose-Einstein condensates) are very well described by the Nonlinear Schödinger Equation with a cubic nonlinearity (called, in the present context, the {\it Gross-Pitaevskii equation}), which is much simpler than the underlying many-body problem. This opened a new field for both theoretical and experimental studies: The {\it nonlinear} quantum mechanics, where the familiar techniques like the spectral analysis in terms of eigenvalues and eigenvectors, which derive from the linearity of the usual Schödinger equation, cannot be used. The mathematical framework of this new quantum mechanics is still largely to be invented. Meanwhile, one can resort to numerical simulations -- and to experiments. In this talk, I will discuss two simple yet important examples illustrating these ideas. First, I will show how one can ''quantum simulate" with ultracold atoms the celebrated Anderson model, which describes quantum disordered systems. One can even enrich this system by adding particle-particle interactions in a controlled way. Second, I will show a simple model displaying ''quasiclassical" chaos, that is, chaotic behavior related to sensitivity to the initial conditions, which cannot exist in {\it linear} quantum mechanics. I will show that the observed behavior, although intrinsically quantum, can be described by the KAM theorem.
We analytically calculate the statistics of Bose-Einstein condensate (BEC) fluctuations in an interacting gas trapped in a three-dimensional cubic or rectangular box with the Dirichlet, fused or periodic boundary conditions within the mean-field Bogoliubov and Thomas-Fermi approximations. We study a mesoscopic system of a finite number of trapped particles and its thermodynamic limit. We find that the BEC fluctuations, first, are anomalously large and non-Gaussian and, second, depend on the trap's form and boundary conditions. Remarkably, these effects persist with increasing interparticle interaction and even in the thermodynamic limit-only the mean BEC occupation, not BEC fluctuations, becomes independent on the trap's form and boundary conditions.
The analytical theory of Bose–Einstein condensation of an ideal gas in mesoscopic systems has been briefly reviewed in application to traps with arbitrary shapes and dimension. This theory describes the phases of the classical gas and the formed Bose–Einstein condensate, as well as the entire vicinity of the phase transition point. The statistics and thermodynamics of Bose–Einstein condensation have been studied in detail, including their self-similar structure in the critical region, transition to the thermodynamic limit, effect of boundary conditions on the properties of a system, and nonequivalence of the description of Bose–Einstein condensation in different statistical ensembles. The complete classification of universality classes of Bose–Einstein condensation has been given.