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Title: Analytical Models for Cigar-Shaped Bose-Einstein Condensate under a Variety of External Confinements
Authors: Nath, A.
Keywords: Physics
Issue Date: 2015
Publisher: IIT Patna
Abstract: Bose-Einstein condensate (BEC) is a highly tunable coherent atomic system and it has become one of the mostly studied systems in the current literature. After its experimental discovery in the year 1995 for alkali metal gas, it has found enormous applications towards atom lasers, beam splitter, BEC on a chip, soliton, cavity quantum electrodynamics, phase transitions, atom interferometry, quantum information science, quantum metrology etc. BEC can be made quasi-one dimensional when it is under a spatially inhomogeneous external confinement. The dynamics become effectively one-dimensional and this cigar-shaped configuration is easily realizable and having a number of important features like generation and propagation of solitons, stability etc. Cigar-shaped BEC under various kinds of external confinement is established as a fascinating system to study various complex quantum phenomena. In general, the mean-field approximation is utilized to investigate the dynamics of cigar-shaped condensate by solving the one-dimensional Gross-Pit aevskii (GP) equation. In recent times, a large number of experiments have been performed in BEC for finding its applications towards science and technology. Compared to that, number of exact theoretical works are hugely lagging behind. The nonlinear nature of the GP equation makes it nontrivial to solve by exact analytical method. Inclusion of various external trapping potentials in the longitudinal direction makes it even more difficult to tackle, but is an essential requirement in the progress of this field. In this thesis, we emphasize on these aspects and construct the exact analytical models for cigar-shaped condensate under a variety of external confinements and study the system dynamics under different physically relevant scenarios. The thesis is divided into seven chapters in which chapters (2-6) are the research outcomes. The goal of this thesis is three fold: first, some specific external confinements are investigated, depending on the novelty and the need in the current literature. Bichromatic optical lattices (BOL), power law and double-well traps, and confinement for negative temperature are the specific examples, which the present thesis dealt with. Second, an analytical model is prescribed to incorporate more than one external traps, thereby unifying them in a single exact analytical technique. Third, an example of higher order nonlinearity is provided to reveal the condensate dynamics for some specific external confinements. Although, a detail description is provided in the beginning of each chapter, in the following, a brief abstract of each work (chapters: 2-6) is given in proper sequence. In Chapter 2, we provide an exact analytical model for the dynamics of a cigar-shaped BEC loaded in a bichromatic optical lattices. Although, a host of exact solution result from this novel method, we mainly concentrate on the solitonic excitations. We show that the trapping potential and its depth of lattice frustration can be varied by tuning the powers and the wavelengths of the two overlaying laser beams. Both attractive and repulsive regimes are thoroughly investigated. In the attractive domain, we obtain bright soliton, which reveals interesting variation with the depth of lattice frustration. Localization of the matter wave density is demonstrated as one of the applications in this regime. In the repulsive domain, dark soliton is obtained when the potential resembles optical lattice. With the appropriate tuning of the potential parameters, the dark soliton becomes modulated with oscillatory background and gradually transforms to bright solitary trains, each situated in the main lattice site. In the next Chapter (3), we solve 1D-GP equation for power-law trap which can be tuned to become double-well, depending upon either the presence of the chemical potential term or the presence of only the even powers. Moreover, the potential can be classified as asymmetric (symmetric) in the presence (absence) of odd powers. The consistency conditions governing the profile of the soliton is finally connected with the Ermakov-Pinney equation. The periodic and quasi-periodic collapses and revivals of the solitary wave are demonstrated for weak and attractive interactions without considering loss/gain. The temporal variation is demonstrated, where periodic and bi-periodic trajectories of the nonlinear excitations are depicted bounded by the potential well. Distance between the two potential wells and thus the position of the two maxima of the condensate density can also be controlled with time, which makes it a good candidate for quantum information processing. Chapter 4 emphasizes on the theoretical aspect of the recent experimental realization of negative temperature in BEC. The main goal is to create larger occupation in the high energy bound states. We propose a combination of expulsive and bichromatic optical lattices on cigar-shaped BEC with cubic interaction. By analytically calculating system's traveling variable, the exact form of the wavefunction, nonlinearity, gain/loss and phase for this system are found. The consistency condition is shown to map onto the Schr odinger equation, thereby enabling one to find the effect of a wide variety of temporal dependence, analytically. Lattice and oscillator frequencies are the key tuning factors for modulating the rate of axial compression and transport of nonlinear excitations with time. An addition of linear trap introduces an asymmetry in the potential, which allows atom distillation at negative temperature. In Chapter 5, a novel exact analytical technique is proposed which provides an exact analytical solution of 1D GP equation for a general form of the external confinement. This paves the way to investigate the system for a family of potential functions, unified as a physical parameter of the system. Explicit forms of the wavefunction are provided for Morse, P osch-Teller, and Toda-lattice potentials. These potential functions are not yet studied so far in the context of BEC. We also consider harmonic and double-well confinements to reproduce the results already existing in the literature. However, the method is not limited to these potential functions only and one can employ any other trap and give a try for getting the exact analytical solution. Each of these potentials can be investigated individually for specific physical application. The two most important factors for controlling the dynamics of a condensate are external trap and nonlinearity. The above applications encompass the first one for cubic nonlinearity. However, in some situations higher order nonlinearity is significant and worth investigating. Hence, in Chapter 6, we incorporate a quintic nonlinearity term in the 1D GP equation, along with a general form of the trapping potential. This particular scenario allows us to fetch the exact solutions for various external traps in presence of two- and three-body interactions, which compliments the unified model for cubic interaction. We provide the exact relation of the coefficients of cubic and quintic nonlinearities with the amplitude, phase and traveling coordinate of the nonlinear excitations. The exact form of the wave function is also furnished for completeness and future studies.
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