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Many body effects in the transport of Bose Einstein Condensates

T. Ernst

Published Jan 1, 2006• 1 citation
OpenAlex

Abstract

The thesis investigates quantum phenomena beyond the Gross-Pitaevskii equation. Therefore we apply the Bogoliubov approach which results in an infinite number of coupled equations for which we developed a special truncation scheme. We also present numerical results for the transport of a Bose Einstein condensate through a quasi one-dimensional wave guide with scattering potentials.

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This thesis presents investigations on the interplay of coherence and many-body effects in the quasi one-dimensional transport of Bose-Einstein condensates (BEC) through scattering potentials. Such configurations can be realized with guided atom lasers that provide a coherent atomic beam. An exact theoretical description of the dynamics is out of reach due to the presence atom-atom interactions. Different levels of approximations are nevertheless possible with their strengths and weaknesses. The mean-field approximation, where the dynamics of the BEC is governed by the Gross-Pitaevskii equation, is most commonly used in the field of ultracold atoms. In this thesis the truncated Wigner method is used to go beyond the standard Gross-Pitaevskii description. This method is adapted in order to study the scattering of Bose-Einstein condensates in one-dimensional waveguides where atom-atom interactions and external potentials are nonvanishing only in a finite region of space. In this case, the truncated Wigner method is combined with the smooth exterior complex scaling method and incorporates quantum noise that originate from the vacuum fluctuations in the waveguide. Inelastic scattering is shown to play a major role in the resonant transport of BEC through a symmetric double potential barrier effectively forming an atomic quantum dot. Indeed, fully resonant transmission is prohibited and incoherent atoms as well as collective oscillations are detected in the transmitted beam. It is also shown that inelastic scattering destroys Anderson localization in the case of transport through disordered potentials. The classical (incoherent) ohmic transmission is recovered for finite atom-atom interactions. The validity of the truncated Wigner method is then assessed using the semiclassical van Vleck-Gutzwiller propagator in the Fock space of the many-body system. It is shown that the truncated Wigner method corresponds to the so-called diagonal approximation, and it is possible to identify the leading correction to the truncated Wigner results, which is provided by the so-called coherent backscattering (CBS) contribution. Coherent basckattering in Fock space is a genuine quantum many-body effect that lies beyond the reach of any mean-field approach. For the case of closed Bose-Hubbard models, the relevance of CBS is confirmed by numerically comparing the (classical) truncated Wigner evolution probabilities to the exact quantum probabilities in Bose-Hubbard models: While a CBS-induced enhancement of the return probability to the initial state is clearly seen in the exact quantum simulations of the bosonic many-body system, this enhancement is absent in the classical calculations. The magnitude and dependence of the CBS contribution on gauge fields, which break time-reversal invariance, is numerically confirmed. For the case of disordered open systems, it can be shown that this contribution as well as next-to leading order contributions vanish thereby confirming the validity of the truncated Wigner method.

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The thesis investigates quantum phenomena beyond the Gross-Pitaevskii equation. Therefore we apply the Bogoliubov approach which results in an infinite number of coupled equations for which we developed a special truncation scheme. We also present numerical results for the transport of a Bose Einstein condensate through a quasi one-dimensional wave guide with scattering potentials.
This thesis presents investigations on the interplay of coherence and many-body effects in the quasi one-dimensional transport of Bose-Einstein condensates (BEC) through scattering potentials. Such configurations can be realized with guided atom lasers that provide a coherent atomic beam. An exact theoretical description of the dynamics is out of reach due to the presence atom-atom interactions. Different levels of approximations are nevertheless possible with their strengths and weaknesses. The mean-field approximation, where the dynamics of the BEC is governed by the Gross-Pitaevskii equation, is most commonly used in the field of ultracold atoms. In this thesis the truncated Wigner method is used to go beyond the standard Gross-Pitaevskii description. This method is adapted in order to study the scattering of Bose-Einstein condensates in one-dimensional waveguides where atom-atom interactions and external potentials are nonvanishing only in a finite region of space. In this case, the truncated Wigner method is combined with the smooth exterior complex scaling method and incorporates quantum noise that originate from the vacuum fluctuations in the waveguide. Inelastic scattering is shown to play a major role in the resonant transport of BEC through a symmetric double potential barrier effectively forming an atomic quantum dot. Indeed, fully resonant transmission is prohibited and incoherent atoms as well as collective oscillations are detected in the transmitted beam. It is also shown that inelastic scattering destroys Anderson localization in the case of transport through disordered potentials. The classical (incoherent) ohmic transmission is recovered for finite atom-atom interactions. The validity of the truncated Wigner method is then assessed using the semiclassical van Vleck-Gutzwiller propagator in the Fock space of the many-body system. It is shown that the truncated Wigner method corresponds to the so-called diagonal approximation, and it is possible to identify the leading correction to the truncated Wigner results, which is provided by the so-called coherent backscattering (CBS) contribution. Coherent basckattering in Fock space is a genuine quantum many-body effect that lies beyond the reach of any mean-field approach. For the case of closed Bose-Hubbard models, the relevance of CBS is confirmed by numerically comparing the (classical) truncated Wigner evolution probabilities to the exact quantum probabilities in Bose-Hubbard models: While a CBS-induced enhancement of the return probability to the initial state is clearly seen in the exact quantum simulations of the bosonic many-body system, this enhancement is absent in the classical calculations. The magnitude and dependence of the CBS contribution on gauge fields, which break time-reversal invariance, is numerically confirmed. For the case of disordered open systems, it can be shown that this contribution as well as next-to leading order contributions vanish thereby confirming the validity of the truncated Wigner method.
We study the dynamics of repulsive and attractive Bose-Einstein condensates. While previous works used the Gross-Pitaevskii equation to calculate this process, we employ a kinetic dynamics approach similar to the famous Hartree-Fock-Bogoliubov equations. Therefore we are able to take into account excitations of the condensate as well as its depletion rate. We apply it to the transport of repulsive Bose-Einstein condensates and the scattering problem of a propagating soliton on an attractive potential, where previous works showed a sharp transition from total reflection to full transmission.
Recent experiments have been moving ever closer to the goal of creating cold quantum-degenerate Bose gases in one and two dimensional systems. In this thesis we investigate such systems by adapting the mean-field theories used so successfully to describe Bose-Einstein condensates in the three dimensional case. We find that the key to describing the low dimensional systems lies in an accurate description of the interactions within those systems. In three dimensions the interactions may be well approximated by a simple form in which atoms are assumed to interact in a vacuum. Many-body effects on the scattering then lead to only a small perturbation. However, in one and two dimensions we find that the inclusion of many-body effects on the scattering is crucial, even for a first order calculation. We use a many-body T-matrix to describe the interactions, and develop a simple approximation in the zero-temperature limit that involves the off-shell two-body T-matrix evaluated at a shifted interaction energy. We therefore derive expressions for the off-shell two-body T-matrix in one, two, and three dimensions, and so find expressions for the coupling parameters which describe interactions in one and two dimensional Bose gases. These lead to interaction terms in the one and two dimensional Gross-Pitaevskii equations which differ significantly from the three dimensional case. We present numerical solutions of the Gross-Pitaevskii equation in both one and two dimensions, and in one dimension we find that the results are in agreement with exact results in the extreme limits of the interaction strength. We also investigate higher order theories in the one-dimensional case, with a view to describing the cross-over from a weakly-interacting Bose-Einstein condensate to a quasicondensate as the interactions become stronger. Finally, we study the Bogoliubov-de Gennes excitation spectrum for a one dimensional condensate containing solitons. We show that it has an anomalous mode with a negative energy, and that this represents the collective motion of the soliton relative to the background condensate. (author)
We explore atom-laser-like transport processes of ultracold Bose-condensed atomic vapors in mesoscopic waveguide structures beyond the Gross-Pitaevskii mean-field theory. Based on a microscopic description of the transport process in the presence of a coherent source that models the outcoupling from a reservoir of perfectly Bose-Einstein condensed atoms, we derive a system of coupled quantum evolution equations that describe the dynamics of a dilute condensed Bose gas in the framework of the Hartree-Fock--Bogoliubov approximation. We apply this method to study the transport of dilute Bose gases through an atomic quantum dot and through waveguides with disorder. Our numerical simulations reveal that the onset of an explicitly time-dependent flow corresponds to the appearance of strong depletion of the condensate on the microscopic level and leads to a loss of global phase coherence.
A method is proposed for a self-consistent evaluation of the coupling constant in the Gross-Pitaevskii equation without involving a pseudopotential replacement. A renormalization of the coupling constant occurs due to medium effects and the trapping potential, e.g., in quasi-1D or quasi-2D systems. It is shown that a simplified version of the Hartree-Fock-Bogoliubov approximation leads to a variational problem for both the condensate and a two-body wave function describing the behavior of a pair of bosons in the Bose-Einstein condensate. The resulting coupled equations are free of unphysical divergences. Particular cases of this scheme that admit analytical estimations are considered and compared to the literature. In addition to the well-known cases of low-dimensional trapping, crossover regimes can be studied. The values of the kinetic, interaction, external, and release energies in low dimensions are also evaluated and contributions due to short-range correlations are found to be substantial.
We analyze the localization of a Bose-Einstein condensate in a one-dimensional bichromatic quasi-periodic optical-lattice potential by numerically solving the 1D Gross-Pitaevskii equation (1D GPE).We first derive the 1D Gross-Pitaevskii equation from the dimensional reduction of the 3D quantum field theory of interacting bosons obtaining two coupled differential equations (for axial wave fuction and space-time dependent transverse width) which reduce to the 1D Gross-Pitaevskii equation under strict conditions.Then, by using the 1D Gross-Pitaevskii equation we report the suppression of localization in the interacting Bose-Einstein condensate when the repulsive scattering length between bosonic atoms is sufficiently large.