Critical Temperature of a Trapped Bose Gas: Comparison of Theory and Experiment

We apply the projected Gross-Pitaevskii equation (PGPE) formalism to the experimental problem of the shift in critical temperature T-c of a harmonically confined Bose gas as reported in Gerbier , Phys. Rev. Lett. 92, 030405 (2004). The PGPE method includes critical fluctuations and we find the results differ from various mean-field theories, and are in best agreement with experimental data. To unequivocally observe beyond mean-field effects, however, the experimental precision must either improve by an order of magnitude, or consider more strongly interacting systems. This is the first application of a classical field method to make quantitative comparison with experiment.

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 behavior of the specific heat.

Calculation of the microcanonical temperature for the classical Bose field

The ergodic hypothesis asserts that a classical mechanical system will in time visit every available configuration in phase space. Thus, for an ergodic system, an ensemble average of a thermodynamic quantity can equally well be calculated by a time average over a sufficiently long period of dynamical evolution. In this paper, we describe in detail how to calculate the temperature and chemical potential from the dynamics of a microcanonical classical field, using the particular example of the classical modes of a Bose-condensed gas. The accurate determination of these thermodynamics quantities is essential in measuring the shift of the critical temperature of a Bose gas due to nonperturbative many-body effects.

First-principles quantum dynamics in interacting Bose gases: I. The positive P representation

The performance of the positive P phase-space representation for exact many- body quantum dynamics is investigated. Gases of interacting bosons are considered, where the full quantum equations to simulate are of a Gross-Pitaevskii form with added Gaussian noise. This method gives tractable simulations of many-body systems because the number of variables scales linearly with the spatial lattice size. An expression for the useful simulation time is obtained, and checked in numerical simulations. The dynamics of first-, second- and third-order spatial correlations are calculated for a uniform interacting 1D Bose gas subjected to a change in scattering length. Propagation of correlations is seen. A comparison is made with other recent methods. The positive P method is particularly well suited to open systems as no conservation laws are hard-wired into the calculation. It also differs from most other recent approaches in that there is no truncation of any kind.

One-dimensional interacting anyon gas: Low-energy properties and haldane exclusion statistics

The low-energy properties of the one-dimensional anyon gas with a delta-function interaction are discussed in the context of its Bethe ansatz solution. It is found that the anyonic statistical parameter and the dynamical coupling constant induce Haldane exclusion statistics interpolating between bosons and fermions. Moreover, the anyonic parameter may trigger statistics beyond Fermi statistics for which the exclusion parameter alpha is greater than one. The Tonks-Girardeau and the weak coupling limits are discussed in detail. The results support the universal role of alpha in the dispersion relations.

Three-body recombination of ultracold Bose gases using the truncated Wigner method

We apply the truncated Wigner method to the process of three-body recombination in ultracold Bose gases. We find that within the validity regime of the Wigner truncation for two-body scattering, three-body recombination can be treated using a set of coupled stochastic differential equations that include diffusion terms, and can be simulated using known numerical methods. As an example we investigate the behavior of a simple homogeneous Bose gas, finding a very slight increase of the loss rate compared to that obtained by using the standard method.

Equation of state of a superfluid Fermi gas in the BCS-BEC crossover

We present a theory for a superfluid Fermi gas near the BCS-BEC crossover, including pairing fluctuation contributions to the free energy similar to that considered by Nozieres and Schmitt-Rink for the normal phase. In the strong coupling limit, our theory is able to recover the Bogoliubov theory of a weakly interacting Bose gas with a molecular scattering length very close to the known exact result. We compare our results with recent Quantum Monte Carlo simulations both for the ground state and at finite temperature. Excellent agreement is found for all interaction strengths where simulation results are available.

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 (2002 J. Phys. B: At. Mol. Opt. Phys. 35 1555). This derivation does not rely on the concept of local energy and momentum conservation and is based on a quasiclassical Wigner function representation of a 'high temperature' master equation for a Bose gas, which includes only modes below an energy cut-off ER 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, provides 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.

Properties of the stochastic Gross-Pitaevskii equation: finite temperature Ehrenfest relations and the optimal plane wave representation

We present Ehrenfest relations for the high temperature stochastic Gross-Pitaevskii equation description of a trapped Bose gas, including the effect of growth noise and the energy cutoff. A condition for neglecting the cutoff terms in the Ehrenfest relations is found which is more stringent than the usual validity condition of the truncated Wigner or classical field method-that all modes are highly occupied. The condition requires a small overlap of the nonlinear interaction term with the lowest energy single particle state of the noncondensate band, and gives a means to constrain dynamical artefacts arising from the energy cutoff in numerical simulations. We apply the formalism to two simple test problems: (i) simulation of the Kohn mode oscillation for a trapped Bose gas at zero temperature, and (ii) computing the equilibrium properties of a finite temperature Bose gas within the classical field method. The examples indicate ways to control the effects of the cutoff, and that there is an optimal choice of plane wave basis for a given cutoff energy. This basis gives the best reproduction of the single particle spectrum, the condensate fraction and the position and momentum densities.

Many-body quantum dynamics of polarization squeezing in optical fibers

We report new experiments that test quantum dynamical predictions of polarization squeezing for ultrashort photonic pulses in a birefringent fiber, including all relevant dissipative effects. This exponentially complex many-body problem is solved by means of a stochastic phase-space method. The squeezing is calculated and compared to experimental data, resulting in excellent quantitative agreement. From the simulations, we identify the physical limits to quantum noise reduction in optical fibers. The research represents a significant experimental test of first-principles time-domain quantum dynamics in a one-dimensional interacting Bose gas coupled to dissipative reservoirs.

Three-dimensional solitons in coupled atomic-molecular Bose-Einstein condensates

We present a theoretical analysis of three-dimensional (3D) matter-wave solitons and their stability properties in coupled atomic and molecular Bose-Einstein condensates (BECs). The soliton solutions to the mean-field equations are obtained in an approximate analytical form by means of a variational approach. We investigate soliton stability within the parameter space described by the atom-molecule conversion coupling, the atom-atom s-wave scattering, and the bare formation energy of the molecular species. In terms of ordinary optics, this is analogous to the process of sub- or second-harmonic generation in a quadratic nonlinear medium modified by a cubic nonlinearity, together with a phase mismatch term between the fields. While the possibility of formation of multidimensional spatiotemporal solitons in pure quadratic media has been theoretically demonstrated previously, here we extend this prediction to matter-wave interactions in BEC systems where higher-order nonlinear processes due to interparticle collisions are unavoidable and may not be neglected. The stability of the solitons predicted for repulsive atom-atom interactions is investigated by direct numerical simulations of the equations of motion in a full 3D lattice. Our analysis also leads to a possible technique for demonstrating the ground state of the Schrodinger-Newton and related equations that describe Bose-Einstein condensates with nonlocal interparticle forces.

Einstein-Podolsky-Rosen correlations via dissociation of a molecular Bose-Einstein condensate

Recent experimental measurements of atomic intensity correlations through atom shot noise suggest that atomic quadrature phase correlations may soon be measured with a similar precision. We propose a test of local realism with mesoscopic numbers of massive particles based on such measurements. Using dissociation of a Bose-Einstein condensate of diatomic molecules into bosonic atoms, we demonstrate that strongly entangled atomic beams may be produced which possess Einstein-Podolsky-Rosen (EPR) correlations in field quadratures in direct analogy to the position and momentum correlations originally considered by EPR.

First-principles quantum dynamics in interacting Bose gases II: stochastic gauges

First principles simulations of the quantum dynamics of interacting Bose gases using the stochastic gauge representation are analysed. In a companion paper, we showed how the positive-P representation can be applied to these problems using stochastic differential equations. That method, however, is limited by increased sampling error as time evolves. Here, we show how the sampling error can be greatly reduced and the simulation time significantly extended using stochastic gauges. In particular, local stochastic gauges (a subset) are investigated. Improvements are confirmed in numerical calculations of single-, double- and multi-mode systems in the weak-mode coupling regime. Convergence issues are investigated, including the recognition of two modes by which stochastic equations produced by phase-space methods in general can diverge: movable singularities and a noise-weight relationship. The example calculated here displays wave-like behaviour in spatial correlation functions propagating in a uniform 1D gas after a sudden change in the coupling constant. This could in principle be tested experimentally using Feshbach resonance methods.