#http://pra.aps.org/pdf/PRA/v88/i3/e031601
In this Rapid Communication we study the transient dynamics of a Bose superfluid subsequent to an interaction quench. Essential for equilibration is a source of dissipation which we include following the approach of Caldeira and Leggett. Here we solve the equations of motion exactly by integrating out an environmental bath. We thereby derive precisely the time dependent density correlation functions with the appropriate analytic and asymptotic properties. The resulting structure factor exhibits the expected damping and thereby differs from that of strict
Bogoliubov theory. These damped sound modes, which reflect the physics beyond mean-field approaches, are characterized and the structure factors are found to compare favorably with experiment.

The focus on out-of-equilibrium quantum dynamics and
dissipation has intensified with recent experiments in cold
atoms. These atomic systems afford access to nonequilibrium behavior whose conceptual basis is relevant to such
diverse fields as cosmology and quark-gluon plasmas. Many
experimental probes of, for example, nonequilibrium super-
fluidity are not available in condensed matter counterparts;
among these are a sudden change of the interaction strength
or “quench” [1,2]. Because of this unique atomic physics
laboratory, it is important to develop accessible theoretical
tools which should impact a variety of different fields and
problems.
The goal of this Rapid Communication is to exploit the
seminal work of Leggett and Caldeira] to address
out-of-equilibrium dynamics in a Bose superfluid following a
quench. Because in LC theory the dynamics and correlation
functions are exactly solvable (without requiring complex and
nontransparent numerics), we are able to derive a dissipative
Bogoliubov–de Gennes theory. Damping plays an
important role in insuring thermalization to a new equilibrium
state.

This formalism is applied to address experimental studies of
two-dimensional (2D) Bose gases. In these experiments,
one has direct access to the equal-time density correlation
functions, or structure factor Sk(t). A key experimental observation was that the quench appears to excite acoustic waves,
which interfere in both the spatial and temporal domains,
leading to Sakharov oscillations. A simple Bogoliubov-level
theory was applied to analyze these experiments (see also [5]),
while leaving a few experimental features, such as damping in
the Sakharov oscillations, unexplained. Here, in investigating
the physics of dissipation, we re-enforce the observation of
oscillatory behavior by incorporating the presence of damping
in the data analysis. Our study supports the earlier observation
of oscillatory sound modes for some range of k and t in Sk(t). It
also provides insights into why the simplest Bogoliubov-based
scheme is more inadequate in situations in which the coupling
constant g is suddenly increased. Moreover, our formalism for
treating such dissipation, should be a first step in developing

tools for elucidating the dynamics of cold gases which go
beyond the simplest mean-field theories of the steady state.
In the Leggett-Caldeira approach one effectively
parametrizes the damping by introducing a bath of quantum
oscillators. In this way we split the total system into two parts:
the quantum system where dissipation occurs (say the Bose
superfluid at the Bogoliubov level) and a so-called environment. Evidence for universality suggests that the particular
description of the bath will not affect the essential features of
the dissipative process. The latter is often modeled presuming
Ohmic dissipation. As discussed the Hamiltonian is
then quadratic and one can solve the equations of motion
exactly by effectively integrating out the environment. One
similarly introduces a (single parameter) representation of
“noise” or dissipation into fermionic superfluids via time
dependent Ginsburg-Landau theory.

The approach of this Rapid Communication is to be
contrasted with past work in the literature. These include
previous studies of damping of the condensate collective
modes [12,13], based on the so-called Beliaev and Landau
damping processes. Theoretical studies of their temperature
dependence are based on equilibrium techniques (see, e.g.,
[14,15]). These equilibrium approaches are not appropriate
after a quench where the system is far from equilibrium.
Here one has to resort to alternative approaches. Methods
include stochastic versions of a Gross-Pitaevski equation
(SGPE) [16–18] which are presumed to describe the combined
condensate as well as low-lying excitations. Reinforcing
the approach of this Rapid Communication, the similarity
with the Leggett-Caldeira approach has been noted [19].
However, numerical solution of the SGPE restricts its range of
application to rather high temperature. An alternative Keldysh
nonequilibrium formalism, which may be contemplated, is
problematic, as unbounded secular terms and other complications appear in simple perturbative schemes and one has to use
more involved field theory techniques.

The bath of the LC approach, which will be parametrized by
a single (Ohmic) parameter, is viewed as representing multiple
dissipative processes, some of which have antecedents in
Landau and Beliaev processes as well as direct interparticle
interactions, not involving the condensate. We, parenthetically,
note that other sources of dissipation which enter into the
actual experiments cannot be ruled out, as cold atom systems
are subject to lasers and other probes and are not truly isolated.
Throughout this Rapid Communication we neglect trap effects;
our focus is on reasonably short times where the trap geometry
is not important.