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Second-order number-conserving description of nonequilibrium dynamics in finite-temperature Bose-Einstein condensates

T. P. Billam, Peter Mason, SA Gardiner

Published Mar 27, 2013• 0 citations
Publisher
DOI
10.25405/ncl.34266120

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Bright solitary waves and non-equilibrium dynamics in atomic Bose-Einstein condensates

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In this thesis we investigate the static properties and non-equilibrium dynamics of bright solitary waves in atomic Bose-Einstein condensates in the zero-temperature limit, and we investigate the non-equilibrium dynamics of a driven atomic Bose-Einstein condensate at finite temperature. Bright solitary waves in atomic Bose-Einstein condensates are non-dispersive and soliton-like matter-waves which could be used in future atom-interferometry exper-iments. Using the mean-field, Gross-Pitaevskii description, we propose an exper-imental scheme to generate pairs of bright solitary waves with controlled velocity and relative phase; this scheme could form an important part of a future atom in-terferometer, and we demonstrate that it can also be used to test the validity of the mean-field model of bright solitary waves. We also develop a method to quanti-tatively assess how soliton-like static, three-dimensional bright solitary waves are; this assessment is particularly relevant for the design of future experiments. In reality, the non-zero temperatures and highly non-equilibrium dynamics occur-ring in a bright solitary wave interferometer are likely to necessitate a theoretical description which explicitly accounts for the non-condensate fraction. We show that a second-order, number-conserving description offers a minimal self-consistent treatment of the relevant condensate – non-condensate interactions at low temper-atures and for moderate non-condensate fractions. We develop a method to obtain a fully-dynamical numerical solution to the integro-differential equations of motion of this description, and solve these equations for a driven, quasi-one-dimensional test system. We show that rapid non-condensate growth predicted by lower-order descriptions, and associated with linear dynamical instabilities, can be damped by the self-consistent treatment of interactions included in the second-order descrip-tion. 1 Bright solitary waves and non-equilibrium dynamics in atomic Bose-Einstein condensates

•Jan 1, 2012•0 citations
While the Gross-Pitaevskii equation is well established as the canonical dynamical description of atomic Bose-Einstein condensates (BECs) at zero temperature, describing the dynamics of BECs at finite temperatures remains a difficult theoretical problem, particularly when considering low-temperature, nonequilibrium systems in which depletion of the condensate occurs dynamically as a result of external driving. In this paper, we describe a fully time-dependent numerical implementation of a second-order, number-conserving description of finite-temperature BEC dynamics. This description consists of equations of motion describing the coupled dynamics of the condensate and noncondensate fractions in a self-consistent manner, and is ideally suited for the study of low-temperature, nonequilibrium, driven systems. The $\ensuremath{\delta}$-kicked-rotor BEC provides a prototypical example of such a system, and we demonstrate the efficacy of our numerical implementation by investigating its dynamics at finite temperature. We demonstrate that the qualitative features of the system dynamics at zero temperature are generally preserved at finite temperatures, and predict a quantitative finite-temperature shift of resonance frequencies which would be relevant for, and could be verified by, future experiments.
We describe a number-conserving approach to the dynamics of Bose-Einstein condensed dilute atomic gases. This builds upon the works of Gardiner [Phys. Rev. A 56, 1414 (1997)] and Castin and Dum [Phys. Rev. A 57, 3008 (1998)]. We consider what is effectively an expansion in powers of the ratio of noncondensate to condensate particle numbers, rather than inverse powers of the total number of particles. This requires the number of condensate particles to be a majority, but not necessarily almost equal to the total number of particles in the system. We argue that a second-order treatment of the relevant dynamical equations of motion is the minimum order necessary to provide consistent coupled condensate and noncondensate number dynamics for a finite total number of particles, and show that such a second-order treatment is provided by a suitably generalized Gross-Pitaevskii equation, coupled to the Castin-Dum number-conserving formulation of the Bogoliubov--de Gennes equations. The necessary equations of motion can be generated from an approximate third-order Hamiltonian, which effectively reduces to second order in the steady state. Such a treatment as described here is suitable for dynamics occurring at finite temperature, where there is a significant noncondensate fraction from the outset, or dynamics leading to dynamical instabilities, where depletion of the condensate can also lead to a significant noncondensate fraction, even if the noncondensate fraction is initially negligible.
In this thesis we investigate the static properties and non-equilibrium dynamics of bright solitary waves in atomic Bose-Einstein condensates in the zero-temperature limit, and we investigate the non-equilibrium dynamics of a driven atomic Bose-Einstein condensate at finite temperature. Bright solitary waves in atomic Bose-Einstein condensates are non-dispersive and soliton-like matter-waves which could be used in future atom-interferometry exper-iments. Using the mean-field, Gross-Pitaevskii description, we propose an exper-imental scheme to generate pairs of bright solitary waves with controlled velocity and relative phase; this scheme could form an important part of a future atom in-terferometer, and we demonstrate that it can also be used to test the validity of the mean-field model of bright solitary waves. We also develop a method to quanti-tatively assess how soliton-like static, three-dimensional bright solitary waves are; this assessment is particularly relevant for the design of future experiments. In reality, the non-zero temperatures and highly non-equilibrium dynamics occur-ring in a bright solitary wave interferometer are likely to necessitate a theoretical description which explicitly accounts for the non-condensate fraction. We show that a second-order, number-conserving description offers a minimal self-consistent treatment of the relevant condensate – non-condensate interactions at low temper-atures and for moderate non-condensate fractions. We develop a method to obtain a fully-dynamical numerical solution to the integro-differential equations of motion of this description, and solve these equations for a driven, quasi-one-dimensional test system. We show that rapid non-condensate growth predicted by lower-order descriptions, and associated with linear dynamical instabilities, can be damped by the self-consistent treatment of interactions included in the second-order descrip-tion. 1 Bright solitary waves and non-equilibrium dynamics in atomic Bose-Einstein condensates