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arXiv:1810.12984 · Journal of Physics A Mathematical and Theoretical

Phase-space representations of thermal Bose–Einstein condensates

King Lun Ng, Rodney Polkinghorne, Bogdan Opanchuk, P. D. Drummond

Published Nov 6, 2018• 9 citations
arXivHTMLPDF

Abstract

Abstract Phase-space methods allow one to go beyond the mean-field approximation to simulate the quantum dynamics of interacting fields. Here, we obtain a technique for initializing either Wigner or positive-P phase-space simulations of Bose–Einstein condensates with quantum states at a finite temperature. As a means to calculate the initial states, we introduce the idea of a nonlinear chemical potential, which removes the zero-momentum phase-noise divergences of Bogoliubov theory to give a diagonal Hamiltonian. The resulting steady-state quantum theory is then directly applicable to the calculations of initial conditions for quantum simulations of BEC dynamics using phase-space techniques. These methods allow efficient and scalable simulation of large Bose–Einstein condensates. We suggest that nonlinear chemical potentials may have a general applicability to cases of broken symmetry.

DOI
10.1088/1751-8121/aaeeb1

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Abstract Phase-space methods allow one to go beyond the mean-field approximation to simulate the quantum dynamics of interacting fields. Here, we obtain a technique for initializing either Wigner or positive-P phase-space simulations of Bose–Einstein condensates with quantum states at a finite temperature. As a means to calculate the initial states, we introduce the idea of a nonlinear chemical potential, which removes the zero-momentum phase-noise divergences of Bogoliubov theory to give a diagonal Hamiltonian. The resulting steady-state quantum theory is then directly applicable to the calculations of initial conditions for quantum simulations of BEC dynamics using phase-space techniques. These methods allow efficient and scalable simulation of large Bose–Einstein condensates. We suggest that nonlinear chemical potentials may have a general applicability to cases of broken symmetry.
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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-PPP 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-PPP 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.
We introduce the study of dynamical quantum noise in Bose-Einstein condensates through numerical simulation of stochastic partial differential equations obtained using phase-space representations. We derive evolution equations for a single trapped condensate in both the positive-PPP and Wigner representations and perform simulations to compare the predictions of the two methods. The positive-PPP approach is found to be highly susceptible to the stability problems that have been observed in other strongly nonlinear, weakly damped systems. Using the Wigner representation, we examine the evolution of several quantities of interest using from a variety of choices of initial state for the condensate and compare results to those for single-mode models.
In this tutorial we present an introduction to some theoretical methods of quantum field theory applied to the description of a trapped Bose–Einstein condensate. First of all, we give a brief account of the main characteristics of the phenomenon of condensation and present the many-body Hamiltonian of the system. We outline some of the most important approaches used in the characterization of a condensed Bose gas, including the mean-field theory and the Hartree–Fock–Bogoliubov method. Finally we illustrate the use of these techniques addressing some important issues in quantum atom optics. We characterize the quantum state of a Bose–Einstein condensate (BEC) at zero temperature. We also describe a process of Beliaev coupling between quasiparticles using a method that includes terms beyond the usual Bogoliubov approach.
We propose an efficient stochastic method to implement numerically the Bogolubov approach to study finite-temperature Bose-Einstein condensates. Our method is based on the Wigner representation of the density matrix describing the non-condensed modes and a Brownian motion simulation to sample the Wigner distribution at thermal equilibrium. Allowing it to sample any density operator Gaussian in the field variables, our method is very general and it applies both to the Bogolubov and to the Hartree-Fock Bogolubov approach, in the equilibrium case as well as in the time-dependent case. We think that our method can be useful to study trapped Bose-Einstein condensates in two or three spatial dimensions without rotational symmetry properties, as in the case of condensates with vortices, where the traditional Bogolubov approach is difficult to implement numerically due to the need to diagonalize very big matrices.
Quantitative numerical analyses of interacting dilute Bose-Einstein condensates are most often based on semiclassical approximations. Since the complex-valued field-theoretic action of the Bose gas does not offer itself to the direct application of standard Monte Carlo techniques, simulations beyond their scope by now almost exclusively rely on quantum-mechanical techniques. Here we explore an alternative approach based on a Langevin-type sampling in an extended state space, known as the complex Langevin (CL) algorithm. While the use of the CL technique has a long-standing history in high-energy physics, in particular in the simulation of QCD at finite baryon density, applications to ultracold atoms are still in their infancy. Here we examine the applicability of the CL approach for a one- and two-component, three-dimensional nonrelativistic Bose gas in thermal equilibrium, below and above the Bose-Einstein phase transition. By comparison with analytic descriptions at the Gaussian level, including Bogoliubov and Hartree-Fock theory, we find that the method allows computing reliably and efficiently observables in the regime of experimentally accessible parameters. Close to the transition, quantum corrections lead to a shift of the critical temperature which we reproduce within the numerical range known in the literature. With this work, we aim to provide a first test of CL as a potential out-of-the-box tool for the simulation of experimentally realistic situations, including trapping geometries and multicomponent (or multiple species) models.