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arXiv:cond-mat/0406630 · Physical Review A

Classical-field approximation for cold weakly interacting bosons without free parameters

Łukasz Zawitkowski, Mirosław Brewczyk, Mariusz Gajda, Kazimierz Rza̧żewski

Published Sep 28, 2004• 19 citations
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Abstract

The classical-field approximation to cold weakly interacting bosons allows for a unified treatment of condensed and uncondensed parts of the system. Until now, however, the quantitative predictions were limited by a dependence of the results on a grid chosen for numerical implementation of the method. In this paper, we propose replacing this unphysical ambiguity by an additional postulate: the temperature of the gas at thermal equilibrium should be the same as that of an ideal Bose gas with the same fraction of condensed atoms. As it turns out, with this additional assumption, nearly all atoms are within the classical fields, thus the method applies to the whole system.

DOI
10.1103/physreva.70.033614

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The classical-field approximation to cold weakly interacting bosons allows for a unified treatment of condensed and uncondensed parts of the system. Until now, however, the quantitative predictions were limited by a dependence of the results on a grid chosen for numerical implementation of the method. In this paper, we propose replacing this unphysical ambiguity by an additional postulate: the temperature of the gas at thermal equilibrium should be the same as that of an ideal Bose gas with the same fraction of condensed atoms. As it turns out, with this additional assumption, nearly all atoms are within the classical fields, thus the method applies to the whole system.
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.
A classical field approximation to the finite temperature microcanonical thermodynamics of weakly interacting Bose gases is applied to the idealized case of atoms confined in a box with periodic boundary conditions. The whole isolated system is described by a set of classical amplitudes. We show that if the system is in equilibrium the classical amplitudes follow the equipartition theorem which allows us to assign a unique temperature to each given energy state. We also analyse the spectral properties of classical amplitudes and obtain nonperturbative results for the chemical potential as well as temperature-dependent Bogoliubov frequencies and their damping rates. Where available, we make comparison with the analytical estimates of these quantities.
The main goal of this paper is to provide a deeper physical understanding of the classical field method—a method that is widely and successfully used for description of the atomic Bose–Einstein condensates at finite temperatures. To this end, we start with a genuine nonlinear classical system—interacting oscillators. We investigate the dynamics and show that the system approaches thermal equilibrium. We show that extra assumptions are necessary to interpret the results in quantum terms. The quantum interpretation is inevitably related to the cut-off of high momenta modes. A proper choice of the cut-off momentum is equivalent to the introduction of the Planck constant. The presented approach sheds more light on the foundations of the classical field method.
We present a semiclassical treatment of one-dimensional many-body quantum systems in equilibrium, where quantum corrections to the classical field approximation are systematically included by a renormalization of the classical field parameters. Our semiclassical approximation is reliable in the limit of weak interactions and high temperatures. As a specific example, we apply our method to the interacting Bose gas and study experimentally observable quantities, such as correlation functions of bosonic fields and the full counting statistics of the number of particles in an interval. Where possible, our method is checked against exact results derived from integrability, showing excellent agreement.
Quantum systems consisting of many particles are both interesting from the physics and the mathematics perspective. One aspect that makes them fascinating from the physics point of view is that they can exhibit quantum effects on macroscopic scales such as Bose-Einstein condensation. This phenomenon, first predicted 100 years ago by Bose [19] and Einstein [44], occurs when a dilute gas of bosons is cooled down to very low temperatures. Below a certain critical temperature a condensate forms, meaning that a macroscopic fraction of the particles is in the same one-particle state, and thus, quantum phenomena become relevant on length scales much larger than the individual atom. The occurrence of Bose-Einstein condensation is also related to superfluidity as suggested in [88]. Bose-Einstein condensation has been achieved experimentally independently by the groups of Cornell and Wieman [3] and Ketterle [40] earning them a Nobel prize in 2001. While Bose and Einstein analyzed theoretically the non-interacting gas, an effective theory describing weakly interacting Bose gases was proposed by Bogoliubov in 1947 with the goal of explaining the occurrence of superfluidity in liquid helium [18]. Using assumptions such as that condensation indeed occurs, his theory yields predictions for the low-energy states of the system. Seen from the mathematics perspective, describing the Bose gas is an interesting challenge due to the strong correlations between particles introduced by the interaction making this case much harder to study than the non-interacting one. In particular, a proof that Bose-Einstein condensation occurs in the thermodynamic limit for continuous systems is still out of reach. However, there has been a lot of work regarding the rigorous analysis of bosonic systems in various relevant regimes. This thesis contains three new contributions establishing rigorous results on bosonic systems in different, physically relevant settings. It is based on the articles [37, 27, 31]. The first result concerns the time-evolution of initially trapped Bose gases in the ultra-dilute Gross-Pitaevskii regime. Under some assumptions on the initial data satisfied in particular by a class of states that are expected to approximate the ground state in a trap, we derive a quasi-free approximation that is close to the true solution in L2L^2L2-norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for sufficiently low densities as predicted in the physics literature first by Wu and subsequently others. Full-text embargoed until: 2026-01-01\textit{Full-text embargoed until: 2026-01-01}Full-text embargoed until: 2026-01-01
We develop a semiclassical field method for the study of the weakly interacting Bose gas at finite temperature which, contrary to the usual classical field model, does not suffer from an ultraviolet cutoff dependence. We apply the method to the study of thermal vortices in spatially homogeneous, two-dimensional systems. We present numerical results for the vortex density and the vortex pair distribution function. Insight in the physics of the system is obtained by comparing the numerical results with the predictions of simple analytical models. In particular, we calculate the activation energy required to form a vortex pair at low temperature.