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arXiv:0801.4466 · Journal of Physics B Atomic Molecular and Optical Physics

2PI nonequilibrium versus transport equations for an ultracold Bose gas

Alexander Branschädel, Thomas Gasenzer

Published Jun 23, 2008• 15 citations
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Abstract

The far-from-equilibrium dynamics of an ultracold, one-dimensional Bose gas is studied. The focus is set on the comparison between the solutions of fully dynamical evolution equations derived from the two-particle irreducible (2PI) effective action and their corresponding kinetic approximation in the form of Boltzmann-type transport equations. It is shown that during the time evolution of the gas a kinetic description which includes non-Markovian memory effects in a gradient expansion becomes valid. The timescale at which this occurs is shown to exceed significantly the timescale at which the system's evolution enters a near-equilibrium drift period where a fluctuation–dissipation relation is found to hold and which would seem to be predestined for the kinetic approximation.

DOI
10.1088/0953-4075/41/13/135302

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The far-from-equilibrium dynamics of an ultracold, one-dimensional Bose gas is studied. The focus is set on the comparison between the solutions of fully dynamical evolution equations derived from the two-particle irreducible (2PI) effective action and their corresponding kinetic approximation in the form of Boltzmann-type transport equations. It is shown that during the time evolution of the gas a kinetic description which includes non-Markovian memory effects in a gradient expansion becomes valid. The timescale at which this occurs is shown to exceed significantly the timescale at which the system's evolution enters a near-equilibrium drift period where a fluctuation–dissipation relation is found to hold and which would seem to be predestined for the kinetic approximation.
We investigate the conditions under which quantum fluctuations are relevant for the quantitative interpretation of experiments with ultracold Bose gases. This requires to go beyond the description in terms of the Gross-Pitaevskii and Hartree-Fock-Bogoliubov mean-field theories, which can be obtained as classical (statistical) field-theory approximations of the quantum many-body problem. We employ functional-integral techniques based on the two-particle irreducible (2PI) effective action. The role of quantum fluctuations is studied within the nonperturbative 2PI 1∕N1∕\mathcal{N}1∕N expansion to next-to-leading order. At this accuracy level memory integrals enter the dynamic equations, which differ for quantum and classical statistical descriptions. This can be used to obtain a classicality condition for the many-body dynamics. We exemplify this condition by studying the nonequilibrium evolution of a one-dimensional Bose gas of sodium atoms, and discuss some distinctive properties of quantum versus classical statistical dynamics.
We extend our earlier work on the nonequilibrium dynamics of a Bose-Einstein condensate initially loaded into a one-dimensional optical lattice. From the two-particle-irreducible (2PI) closed-time-path (CTP) effective action for the Bose-Hubbard Hamiltonian we derive causal equations of motion that treat mean-field effects and quantum fluctuations on an equal footing. We demonstrate that these equations reproduce well-known limits when simplifying approximations are introduced. For example, when the system dynamics admits two-time separation, we obtain the Kadanoff-Baym equations of quantum kinetic theory, and in the weakly interacting limit, we show that the local equilibrium solutions of our equations reproduce the second-order corrections to the self-energy of the type originally derived by Beliaev. The derivation of quantum kinetic equations from the 2PI-CTP effective action not only checks the viability of the formalism but also shows it to be a tractable framework for going beyond standard Boltzmann equations of motion.
We present kinetic equations that describe the evolution of O(N)-symmetric real scalar quantum fields out of thermal equilibrium in a systematic nonperturbative approximation scheme. This description starts from the 1/N-expansion of the 2PI effective action to next-to-leading order, which includes scattering and memory effects. From this starting point one is lead to evolution equations for the propagator, which are nonlocal in time. Numerical solutions showed that the propagator depends only very slightly on the center coordinates already after moderate times, and that correlations between earlier and later times are suppressed exponentially, which causes an effective memory loss. Exploiting these two observations, we combine a first order gradient expansion with a Wigner transformation to derive our kinetic equations, which are local in time, from the nonlocal evolution equations. In contrast to standard descriptions based on loop expansions, our kinetic equations remain valid even for nonperturbatively large fluctuations. Additionally, employing a quasi-particle approximation, we eventually arrive at a generalized Boltzmann equation.
In this thesis, far-from-equilibrium dynamics of fermionic quantum gases is discussed utilising functional quantum field theoretical methods. Employing the Schwinger-Keldysh path integral, real-time Schwinger-Dyson/Kadanoff-Baym dynamic equations for the two-point correlation functions are derived from the two-particle irreducible (2PI) effective action. For two specific models, these dynamic equations are investigated further. (a) For an N-fold spin-degenerate ultra-cold Fermi gas, non-perturbative approximation schemes based on either a loop or a 1/N expansion of the 2PI effective action are presented. Adopting these approximations, the long-time evolution of a homogeneous Fermi gas with N=2 after an initial preparation far from thermal equilibrium is thoroughly studied in one spatial dimension. Depending on the total energy, the gas is found to evolve into thermal as well as non-thermal states, the latter becoming manifest in violating the fluctuation-dissipation relation. (b) A similar 1/N expansion is derived for the SU(N) symmetric Kondo lattice model. At leading order, the mean-field dynamic equations of the U=0 Anderson model are recovered. At next-to-leading order (NLO), both spin-flip and direct interactions between localised atoms and conduction band atoms are taken into account non-perturbatively into the dynamic equations. This allows future studies of possibly existing novel phases in coupling regimes where the Kondo screening and RKKY-type interactions are competing.
The dynamical evolution of a Bose-Einstein condensate trapped in a one-dimensional lattice potential is investigated theoretically in the framework of the Bose-Hubbard model. The emphasis is set on the far-from-equilibrium evolution in a case where the gas is strongly interacting. This is realized by an appropriate choice of the parameters in the Hamiltonian, and by starting with an initial state, where one lattice well contains a Bose-Einstein condensate while all other wells are empty. Oscillations of the condensate as well as noncondensate fractions of the gas between the different sites of the lattice are found to be damped as a consequence of the collisional interactions between the atoms. Functional integral techniques involving self-consistently determined mean fields as well as two-point correlation functions are used to derive the two-particle-irreducible (2PI) effective action. The action is expanded in inverse powers of the number of field components N\mathcal{N}N, and the dynamic equations are derived from it to next-to-leading order in this expansion. This approach reaches considerably beyond the Hartree-Fock-Bogoliubov mean-field theory, and its results are compared to the exact quantum dynamics obtained by Rey et al. [Phys. Rev. A 69, 033610 (2004)] for small atom numbers.
A dynamical many-body theory is presented which systematically extends beyond mean-field and perturbative quantum-field theoretical procedures. It allows us to study the dynamics of strongly interacting quantum-degenerate atomic gases. The nonperturbative approximation scheme is based on a systematic expansion of the two-particle irreducible effective action in powers of the inverse number of field components. This yields dynamic equations which contain direct scattering, memory, and ``off-shell'' effects that are not captured by the Gross-Pitaevskii equation. This is relevant to account for the dynamics of, e.g., strongly interacting quantum gases atoms near a scattering resonance, or of one-dimensional Bose gases in the Tonks-Girardeau regime. We apply the theory to a homogeneous ultracold Bose gas in one spatial dimension. Considering the time evolution of an initial state far from equilibrium we show that it quickly evolves to a nonequilibrium quasistationary state and discuss the possibility to attribute an effective temperature to it. The approach to thermal equilibrium is found to be extremely slow.