  Quantitative numerical analyses of interacting dilute Bose-Einstein
condensates are most often based on semi-classical 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 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 finitebaryon 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
non-relativistic 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/-species models.
