We present a stochastic matter field equation for an interacting many-body Bose system in equilibrium at ultracold finite temperature. Moreover, the proposed equation can be used for non-equilibrium dynamics on phenomenological grounds. This stochastic differential equation is based on a field phase space representation reflecting the underlying canonical density operator (fixed particle number N). Remarkably, it allows for an efficient numerical implementation. We apply our canonical c-field method to interference experiments with quasi-one dimensional Bose gases. Crucial features of these interference patterns are reproduced very well and also statistical properties in terms of distributions, e.g. for the contrasts, agree well with experimental results.
We present a stochastic matter field equation for an interacting many-body Bose system in equilibrium at ultracold finite temperature. Moreover, the proposed equation can be used for non-equilibrium dynamics on phenomenological grounds. This stochastic differential equation is based on a field phase space representation reflecting the underlying canonical density operator (fixed particle number N). Remarkably, it allows for an efficient numerical implementation. We apply our canonical c-field method to interference experiments with quasi-one dimensional Bose gases. Crucial features of these interference patterns are reproduced very well and also statistical properties in terms of distributions, e.g. for the contrasts, agree well with experimental results.
We present a novel quantum stochastic evolution equation for a matter field describing the canonical state of a weakly interacting ultracold Bose gas. In the ideal gas limit our approach is exact. This numerically very stable equation suppresses high-energy fluctuations exponentially, which enables us to describe condensed and thermal atoms within the same formalism. We present applications to ground-state occupation and fluctuations, density profile of ground-state and thermal cloud, and ground-state number statistics. Our main aim are spatial coherence properties which we investigate through the determination of interference contrast and spatial density correlations. Parameters are taken from actual experiments.
In this thesis, we apply the stochastic Gross-Pitaevskii equation, using the formulation of Stoof, to the study of weakly interacting, finite temperature atomic Bose gases, both in and out of equilibrium. Explicitly maintaining the effects of phase and density fluctuations, this state-of-the-art method is well suited to the study of low dimensional Bose systems. As such, we consider the non-equilibrium dynamics of dark matter wave solitons, oscillating within harmonically trapped, one-dimensional phase-fluctuating condensates, through numerical simulations of the stochastic Gross-Pitaevskii equation. In treating the damping due to higher lying thermal modes of the system, we include the HartreeFock contribution of the equilibrium atomic density, which is a novel feature of our work. A stochastic approach is found to be essential in capturing the shot-to-shot variation in soliton trajectories observed experimentally. Neglecting stochastic effects, we find the well-known dissipative Gross-Pitaevskii equation to recover the average dynamics of the stochastic model, on retaining the microscopically justified form for the damping of this approach. Such a coupling is found to be necessary in modelling the dissipative nature of finite temperature soliton dynamics. Furthermore, we undertake a comparison between the finite temperature equilibrium state of the stochastic Gross-Pitaevskii equation, and that predicted within a number conserving Bogoliubov approach. Through a series of numerical experiments, we compare the equilibrium properties of a harmonically trapped Bose gas, considering our results in light of those of independent theoretical treatments, and additionally assessing the sensitivity of the numerical implementation of the former method to the numerical grid spacing. In the regime in which thermal effects dominate, the stochastic Gross-Pitaevskii approach proves to be a more consistent means of generating such equilibrium states.
The quantum dynamics and grand canonical thermodynamics of many-mode (one-, two-, and three-dimensional) interacting Bose gases are simulated from first principles. The model uses a lattice Hamiltonian based on a continuum second-quantized model with two-particle interactions, external potential, and interactions with an environment, with no further approximations. The interparticle potential can be either an (effective) delta function as in Bose-Hubbard models, or extended with a shape resolved by the lattice. Simulations are of a set of stochastic equations that in the limit of many realizations correspond exactly to the full quantum evolution of the many-body systems. These equations describe the evolution of samples of the gauge P distribution of the quantum state, details of which are developed. Conditions under which general quantum phase-space representations can be used to derive stochastic simulation methods are investigated in detail, given the criteria: 1) The simulation corresponds exactly to quantum mechanics in the limit of many trajectories. 2) The number of equations scales linearly with system size, to allow the possibility of efficient first-principles quantum mesoscopic simulations. 3) All observables can be calculated from one simulation. 4) Each stochastic realization is independent to allow straightforward use of parallel algorithms. Special emphasis is placed on allowing for simulation of open systems. In contrast to typical Monte Carlo techniques based on path integrals, the phase-space representation approach can also be used for dynamical calculations. Two major (and related) known technical stumbling blocks with such stochastic simulations are instabilities in the stochastic equations, and pathological trajectory distributions as the boundaries of phase space are approached. These can (and often do) lead to systematic biases in the calculated observables. The nature of these problems are investigated in detail. Many phase-space distributions have, however, more phase-space freedoms than the minimum required for exact correspondence to quantum mechanics, and these freedoms can in many cases be exploited to overcome the instability and boundary term problems, recovering an unbiased simulation. The stochastic gauge technique, which achieves this in a systematic way, is derived and heuristic guidelines for its use are developed. The gauge P representation is an extension of the positive P distribution, which uses coherent basis states, but allows a variety of useful stochastic gauges that are used to overcome the stability problems. Its properties are investigated, and the resulting equations to be simulated for the open interacting Bose gas system are derived. The dynamics of the following many-mode systems are simulated as examples: 1) Uniform one-dimensional and two-dimensional Bose gases after the rapid appearance of significant two-body collisions (e.g. after entering a Feshbach resonance). 2) Trapped bosons, where the size of the trap is of the same order as the range of the interparticle potential. 3) Stimulated Bose enhancement of scattered atom modes during the collision of two Bose-Einstein condensates. The grand canonical thermodynamics of uniform one-dimensional Bose gases is also calculated for a variety of temperatures and collision strengths. Observables calculated include first to third order spatial correlation functions (including at finite interparticle separation) and momentum distributions. The predicted phenomena are discussed. Improvements over the positive P distribution and other methods are discussed, and simulation times are analyzed for Bose-Hubbard lattice models from a general perspective. To understand the behavior of the equations, and subsequently optimize the gauges for the interacting Bose gas, single- and coupled two-mode dynamical and thermodynamical models of interacting Bose gases are investigated in detail. Directions in which future progress can be expected are considered. Lastly, safeguards are necessary to avoid biased averages when exponentials of Gaussian-like trajectory distributions are used (as here), and these are investigated.
We present a new exact method to numerically compute the thermodynamical properties of an interacting Bose gas in the canonical ensemble. As in our previous paper (Carusotto I, Castin Y and Dalibard J 2001 Phys. Rev. A 63 023606), we write the density operator ρ as an average of Hartree dyadics | N :ϕ 1 ⟩⟨ N :ϕ 2 | and we find stochastic evolution equations for the wavefunctions ϕ 1,2 such that the exact imaginary-time evolution of ρ is recovered after averaging over noise. In this way, the thermal equilibrium density operator can be obtained for any temperature T . The method is then applied to study the thermodynamical properties of a homogeneous one-dimensional N -boson system: although Bose-Einstein condensation cannot occur in the thermodynamical limit, a macroscopic occupation of the lowest mode of a finite system is observed at sufficiently low temperatures. If k B T >>µ, the main effect of interactions is to suppress density fluctuations and to reduce their correlation length. Different effects such as a spatial antibunching of the atoms are predicted for the opposite k B T ⩽µ regime. Our exact stochastic calculations have been compared with existing approximate theories.
We review phase-space techniques based on the Wigner representation that provide an approximate description of dilute ultra-cold Bose gases. In this approach the quantum field evolution can be represented using equations of motion of a similar form to the Gross–Pitaevskii equation but with stochastic modifications that include quantum effects in a controlled degree of approximation. These techniques provide a practical quantitative description of both equilibrium and dynamical properties of Bose gas systems. We develop versions of the formalism appropriate at zero temperature, where quantum fluctuations can be important, and at finite temperature where thermal fluctuations dominate. The numerical techniques necessary for implementing the formalism are discussed in detail, together with methods for extracting observables of interest. Numerous applications to a wide range of phenomena are presented.
In this thesis we use the stochastic Gross-Pitaevskii equation (SGPE), a finite temperature model for weakly interacting ultracold Bose gases which fully incorporates density and phase fluctuations, to demonstrate ab initio quantitative agreement with a number of quasi-one-dimensional experiments. To achieve this, we propose and numerically solve a quasi-one-dimensional form of the SGPE, supported by a self-consistent treatment of radially-excited thermal modes. The quasi-one-dimensional stochastic Gross-Pitaevskii equation provides an accurate finite temperature description of the dynamical equilibrium of the lowenergy axial modes of a Bose gas, assumed to be highly populated and thus treated within the ‘classical field’ approximation. This treatment allows to selfconsistently account for transverse, quasi-one-dimensional effects, which makes it a valid model in the regime where the chemical potential, μ, is approximately equal to a few times the transverse excitation energy, ~ω⊥. In the regime where the thermal energy, kBT , is also comparable to or larger than ~ω⊥, the transverse excited states play an increasingly important role, and are treated here as onedimensional independent Bose gases at static equilibrium. Firstly, we demonstrate that this is an excellentmodel for ab initio investigation of equilibrium properties, such as density profiles and density fluctuations. This is shown by accurately reproducing the in situ density profiles recently obtained in the experiments of Trebbia et al. [Phys. Rev. Lett. 97, 250403 (2006)] and van Amerongen et al. [Phys. Rev. Lett. 100, 090402 (2008)], and the density fluctuation data reported by Armijo et al. [Phys. Rev. Lett. 105, 230402 (2010)]. Moreover, we perform an ab initio analysis of the temperature dependence of the phase coherence of finite temperature, quasi-one-dimensional Bose gases measured in the experiments of Richard et al. [Phys. Rev. Lett. 91, 010405 (2003)] and Hugbart et al. [Eur. Phys. J. D 35, 155 (2005)]. We find very good agreement across the entire observed temperature range in both experiments, and improve upon previous theoretical modelling of the latter.