We investigate a classical phase-space approach of matter-wave propagation based on the truncated Wigner equation (TWE). We show that such a description is suitable for ideal matter waves in quadratic time-dependent confinement as well as for harmonically trapped Bose-Einstein condensates in the Thomas-Fermi regime. In arbitrary interacting regimes, the TWE combined with the moment method yields the low-energy spectrum of a condensate as predicted by independent variational methods. TWE also gives the right breathing-mode frequency for long-ranged interactions decaying as $1/{r}^{2}$ in three dimensions and for a contact potential in two dimensions. Quantum signatures, beyond the TWE, may only be found in the condensate dynamics beyond the regimes of classical phase-space propagation identified here.
We investigate a classical phase-space approach of matter-wave propagation based on the truncated Wigner equation (TWE). We show that such a description is suitable for ideal matter waves in quadratic time-dependent confinement as well as for harmonically trapped Bose-Einstein condensates in the Thomas-Fermi regime. In arbitrary interacting regimes, the TWE combined with the moment method yields the low-energy spectrum of a condensate as predicted by independent variational methods. TWE also gives the right breathing-mode frequency for long-ranged interactions decaying as 1/r2 in three dimensions and for a contact potential in two dimensions. Quantum signatures, beyond the TWE, may only be found in the condensate dynamics beyond the regimes of classical phase-space propagation identified here.
This work considers the semi-classical description of two applications involving cold atoms. This is, on one hand, the behavior of a BOSE-EINSTEIN condensate in hybrid systems, i.e. in contact with a microscopic object (carbon nanotubes, fullerenes, etc.). On the other, the evolution of phase space distributions in matter wave interferometers utilizing ray tracing methods was discussed. For describing condensates in hybrid systems, one can map the GROSS-PITAEVSKII equation, a differential equation in the complex-valued macroscopic wave function, onto a system of two differential equations in density and phase. Neglecting quantum dispersion, one obtains a semiclassical description which is easily modified to incorporate interactions between condensate and microscopical object. In our model, these interactions comprise attractive forces (CASIMIR-POLDER forces) and loss of condensed atoms due to inelastic collisions at the surface of the object. Our model exhibited the excitation of sound waves that are triggered by the object’s rapid immersion, and spread across the condensate thereafter. Moreover, local particle loss leads to a shrinking of the bulk condensate. We showed that the total number of condensed particles is decreasing potentially in the beginning (large condensate, strong mean field interaction), while it decays exponentially in the long-time limit (small condensate, mean field inetraction negligible). For representing the physics of matter wave interferometers in phase space, we utilized the WIGNER function. In semi-classical approximation, which again consists in ignoring the quantum dispersion, this representation is subject to the same equation of motion as classical phase space distributions, i.e. the LIOUVILLE equation. This implies that time evolution of theWIGNER function follows a phase space flow that consists of classical trajectories (classical transport). This means, for calculating a time-evolved distribution, one has know the initial distribution and one has to solve the classical equations of motion. Concerning the initial distribution, we have studied a stationary solution of the nonlinear LIOUVILLE equation, the LAMBERT density. We saw that it agrees very well with results from singleparticle quantum mechanics as well as the MAXWELL-BOLTZMANN distribution in the weakly interacting limit. Likewise, in the strongly interacting limit, familiar results of the THOMAS-FERMI approximation are recovered. A distribution that is first prepared in a trap and then released can be described quite conveniently in terms of WIGNER functions. However, propagation in optical potentials associated to the interferometer elements (beam splitter, Pi-half-pulse) do not satisfy the condition of the semiclassical approximation. Nevertheless, one finds discrete before-after mappings that describe the effect of these elements on incident distributions. This leads to several channels of phase space propagation which relate to the interferometer paths and interferences between them. The formalism for WIGNER functions in an interferometer can be translated straightforwardly into a ray tracing algorithm. As mentioned above, this algorithm solves the classical equations of motion and computes time-evolved distributions, using values of the initial distribution. This procedure, in contrast to most analytical solutions, does not require the HAMILTON function to be quadratic (e.g. free propagation, const. acceleration, harmonic oscillator). We compared simulation results to analytic expressions in case of freely propagating GAUSS distribution. They showed perfect agreement, especially for the functional dependence of wave length and contrast of interference fringes on the laser pulse timings.
We study the truncated Wigner method applied to a weakly interacting spinless Bose-condensed gas which is perturbed away from thermal equilibrium by a time-dependent external potential. The principle of the method is to generate an ensemble of classical fields ψ( r ) which samples the Wigner quasi-distribution function of the initial thermal equilibrium density operator of the gas, and then to evolve each classical field with the Gross–Pitaevskii equation. In the first part of the paper we improve the sampling technique over our previous work (Sinatra et al 2000 J. Mod. Opt. 47 2629–44) and we test its accuracy against the exactly solvable model of the ideal Bose gas. In the second part of the paper we investigate the conditions of validity of the truncated Wigner method. For short evolution times it is known that the time-dependent Bogoliubov approximation is valid for almost pure condensates. The requirement that the truncated Wigner method reproduces the Bogoliubov prediction leads to the constraint that the number of field modes in the Wigner simulation must be smaller than the number of particles in the gas. For longer evolution times the nonlinear dynamics of the noncondensed modes of the field plays an important role. To demonstrate this we analyse the case of a three-dimensional spatially homogeneous Bose-condensed gas and we test the ability of the truncated Wigner method to correctly reproduce the Beliaev–Landau damping of an excitation of the condensate. We have identified the mechanism which limits the validity of the truncated Wigner method: the initial ensemble of classical fields, driven by the time-dependent Gross–Pitaevskii equation, thermalizes to a classical field distribution at a temperature T class which is larger than the initial temperature T of the quantum gas. When T class significantly exceeds T a spurious damping is observed in the Wigner simulation. This leads to the second validity condition for the truncated Wigner method, T class − T ≪ T , which requires that the maximum energy max of the Bogoliubov modes in the simulation does not exceed a few k B T .
We introduce an approximate phase-space technique to simulate the quantum dynamics of interacting bosons. With the future goal of treating Bose-Einstein condensate systems, the method is designed for systems with a natural separation into highly occupied (condensed) modes and lightly occupied modes. The method self-consistently uses the Wigner representation to treat highly occupied modes and the positive-P representation for lightly occupied modes. In this method, truncation of higher-derivative terms from the Fokker-Planck equation is usually necessary. However, at least in the cases investigated here, the resulting systematic error, over a finite time, vanishes in the limit of large Wigner occupation numbers. We tested the method on a system of two interacting anharmonic oscillators, with high and low occupations, respectively. The hybrid method successfully predicted atomic quadratures to a useful simulation time 60 times longer than that of the positive-P method. The truncated Wigner method also performed well in this test. For the prediction of the correlation in a quantum nondemolition measurement scheme, for this same system, the hybrid method gave excellent agreement with the exact result, while the truncated Wigner method showed a large systematic error.
Abstract We study the mean-field phase-space distribution of the ground state of an anisotropic, harmonically trapped Bose–Einstein condensate at zero temperature. First, we derive the Fourier transform of the ground-state wave function in the Thomas–Fermi limit in d dimensions. Second, we obtain analytical approximations for the Thomas–Fermi Wigner function in one and three dimensions with a factorization ansatz of the coherence function. These approximate expressions are compared with exact numerical simulations of the Gross–Pitaevskii equation.
This thesis presents investigations on the interplay of coherence and many-body effects in the quasi one-dimensional transport of Bose-Einstein condensates (BEC) through scattering potentials. Such configurations can be realized with guided atom lasers that provide a coherent atomic beam. An exact theoretical description of the dynamics is out of reach due to the presence atom-atom interactions. Different levels of approximations are nevertheless possible with their strengths and weaknesses. The mean-field approximation, where the dynamics of the BEC is governed by the Gross-Pitaevskii equation, is most commonly used in the field of ultracold atoms. In this thesis the truncated Wigner method is used to go beyond the standard Gross-Pitaevskii description. This method is adapted in order to study the scattering of Bose-Einstein condensates in one-dimensional waveguides where atom-atom interactions and external potentials are nonvanishing only in a finite region of space. In this case, the truncated Wigner method is combined with the smooth exterior complex scaling method and incorporates quantum noise that originate from the vacuum fluctuations in the waveguide. Inelastic scattering is shown to play a major role in the resonant transport of BEC through a symmetric double potential barrier effectively forming an atomic quantum dot. Indeed, fully resonant transmission is prohibited and incoherent atoms as well as collective oscillations are detected in the transmitted beam. It is also shown that inelastic scattering destroys Anderson localization in the case of transport through disordered potentials. The classical (incoherent) ohmic transmission is recovered for finite atom-atom interactions. The validity of the truncated Wigner method is then assessed using the semiclassical van Vleck-Gutzwiller propagator in the Fock space of the many-body system. It is shown that the truncated Wigner method corresponds to the so-called diagonal approximation, and it is possible to identify the leading correction to the truncated Wigner results, which is provided by the so-called coherent backscattering (CBS) contribution. Coherent basckattering in Fock space is a genuine quantum many-body effect that lies beyond the reach of any mean-field approach. For the case of closed Bose-Hubbard models, the relevance of CBS is confirmed by numerically comparing the (classical) truncated Wigner evolution probabilities to the exact quantum probabilities in Bose-Hubbard models: While a CBS-induced enhancement of the return probability to the initial state is clearly seen in the exact quantum simulations of the bosonic many-body system, this enhancement is absent in the classical calculations. The magnitude and dependence of the CBS contribution on gauge fields, which break time-reversal invariance, is numerically confirmed. For the case of disordered open systems, it can be shown that this contribution as well as next-to leading order contributions vanish thereby confirming the validity of the truncated Wigner method.
We present a generalized model for the evolution of atomic wave-packets in matter-wave interferometers. This method provides an efficient tool for analyzing the performance of atomic interferometers using atom clouds prepared in a trap as a Bose-Einstein condensate (BEC) or as a thermal ensemble. Predictions of the model for dynamic properties such as wave-packet size and phase are in excellent agreement with explicit numerical solutions of the non-linear Gross-Pitaevskii equations and enable fast calculations of interferometric performance in regimes where full numerical solutions become impractical. As a starting point, the static Thomas-Fermi (TF) approximation for a BEC in a harmonic trap is generalized to the whole range of atom-atom interaction strengths: from non-interacting atoms (low densities) to the standard TF limit (high atomic densities, as long as the condensate approximation still holds). In particular, this generalization allows a good estimation of atomic cloud properties along the transition from a three-dimensional to a quasi-one-dimensional BEC in an elongated trap. We then develop a theoretical model of wave-packet evolution in time-dependent conditions. The model is applicable for a wide range of dynamical problems involving evolution in time-dependent potentials and in a changing mean-field atomic repulsion due to splitting and separation of wave-packets. We use the model for studying two effects that influence interferometric coherence: imperfect spatial recombination in a two-state interferometer (the so-called "Humpty-Dumpty effect") and phase diffusion due to number uncertainty in the two interferometer arms, which was previously studied thoroughly only for interferometric schemes where the BECs in the two arms stay trapped (for example, in a double-well potential).