9 research outputs found
Numerical method for evolving the Projected Gross-Pitaevskii equation
In this paper we describe a method for evolving the projected
Gross-Pitaevskii equation (PGPE) for a Bose gas in a harmonic oscillator
potential. The central difficulty in solving this equation is the requirement
that the classical field is restricted to a small set of prescribed modes that
constitute the low energy classical region of the system. We present a scheme,
using a Hermite-polynomial based spectral representation, that precisely
implements this mode restriction and allows an efficient and accurate solution
of the PGPE. We show equilibrium and non-equilibrium results from the
application of the PGPE to an anisotropic trapped three-dimensional Bose gas.Comment: 12 pages, 5 figures. To appear in Phys. Rev. E. Convergence results
added, a few minor changes made and typos fixe
Quantum turbulence in condensate collisions: an application of the classical field method
We apply the classical field method to simulate the production of correlated
atoms during the collision of two Bose-Einstein condensates. Our
non-perturbative method includes the effect of quantum noise, and provides for
the first time a theoretical description of collisions of high density
condensates with very large out-scattered fractions. Quantum correlation
functions for the scattered atoms are calculated from a single simulation, and
show that the correlation between pairs of atoms of opposite momentum is rather
small. We also predict the existence of quantum turbulence in the field of the
scattered atoms--a property which should be straightforwardly measurable.Comment: 5 pages, 3 figures: Rewritten text, replaced figure
Quantum turbulence and correlations in Bose-Einstein condensate collisions
We investigate numerically simulated collisions between experimentally
realistic Bose-Einstein condensate wavepackets, within a regime where highly
populated scattering haloes are formed. The theoretical basis for this work is
the truncated Wigner method, for which we present a detailed derivation, paying
particular attention to its validity regime for colliding condensates. This
paper is an extension of our previous Letter [A. A. Norrie, R. J. Ballagh, and
C. W. Gardiner, Phys. Rev. Lett. 94, 040401 (2005)] and we investigate both
single-trajectory solutions, which reveal the presence of quantum turbulence in
the scattering halo, and ensembles of trajectories, which we use to calculate
quantum-mechanical correlation functions of the field
Classical Region of a Trapped Bose Gas
The classical region of a Bose gas consists of all single-particle modes that
have a high average occupation and are well-described by a classical field.
Highly-occupied modes only occur in massive Bose gases at ultra-cold
temperatures, in contrast to the photon case where there are highly-occupied
modes at all temperatures. For the Bose gas the number of these modes is
dependent on the temperature, the total number of particles and their
interaction strength. In this paper we characterize the classical region of a
harmonically trapped Bose gas over a wide parameter regime. We use a
Hartree-Fock approach to account for the effects of interactions, which we
observe to significantly change the classical region as compared to the
idealized case. We compare our results to full classical field calculations and
show that the Hartree-Fock approach provides a qualitatively accurate
description of classical region for the interacting gas.Comment: 6 pages, 5 figures; updated to include new results with interaction
Microcanonical temperature for a classical field: application to Bose-Einstein condensation
We show that the projected Gross-Pitaevskii equation (PGPE) can be mapped
exactly onto Hamilton's equations of motion for classical position and momentum
variables. Making use of this mapping, we adapt techniques developed in
statistical mechanics to calculate the temperature and chemical potential of a
classical Bose field in the microcanonical ensemble. We apply the method to
simulations of the PGPE, which can be used to represent the highly occupied
modes of Bose condensed gases at finite temperature. The method is rigorous,
valid beyond the realms of perturbation theory, and agrees with an earlier
method of temperature measurement for the same system. Using this method we
show that the critical temperature for condensation in a homogeneous Bose gas
on a lattice with a UV cutoff increases with the interaction strength. We
discuss how to determine the temperature shift for the Bose gas in the
continuum limit using this type of calculation, and obtain a result in
agreement with more sophisticated Monte Carlo simulations. We also consider the
behaviour of the specific heat.Comment: v1: 9 pages, 5 figures, revtex 4. v2: additional text in response to
referee's comments, now 11 pages, to appear in Phys. Rev.
Simulations of thermal Bose fields in the classical limit
We demonstrate that the time-dependent projected Gross-Pitaevskii equation
derived earlier [Davis, et al., J. Phys. B 34, 4487 (2001)] can represent the
highly occupied modes of a homogeneous, partially-condensed Bose gas. We find
that this equation will evolve randomised initial wave functions to
equilibrium, and compare our numerical data to the predictions of a gapless,
second-order theory of Bose-Einstein condensation [S. A. Morgan, J. Phys. B 33,
3847 (2000)]. We find that we can determine the temperature of the equilibrium
state when this theory is valid.
Outside the range of perturbation theory we describe how to measure the
temperature of our simulations. We also determine the dependence of the
condensate fraction and specific heat on temperature for several interaction
strengths, and observe the appearance of vortex networks. As the
Gross-Pitaevskii equation is non-perturbative, we expect that it can describe
the correct thermal behaviour of a Bose gas as long as all relevant modes are
highly occupied.Comment: 15 pages, 12 figures, revtex4, follow up to Phys. Rev. Lett. 87
160402 (2001). v2: Modified after referee comments. Extra data added to two
figures, section on temperature determination expande
The stochastic Gross-Pitaevskii equation II
We provide a derivation of a more accurate version of the stochastic
Gross-Pitaevskii equation, as introduced by Gardiner et al. (J. Phys. B
35,1555,(2002). The derivation does not rely on the concept of local energy and
momentum conservation, and is based on a quasi-classical Wigner function
representation of a "high temperature" master equation for a Bose gas, which
includes only modes below an energy cutoff E_R that are sufficiently highly
occupied (the condensate band). The modes above this cutoff (the non-condensate
band) are treated as being essentially thermalized. The interaction between
these two bands, known as growth and scattering processes, provide noise and
damping terms in the equation of motion for the condensate band, which we call
the stochastic Gross-Pitaevskii equation. This approach is distinguished by the
control of the approximations made in its derivation, and by the feasibility of
its numerical implementation.Comment: 24 pages of LaTeX, one figur