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arXiv:1803.05180 · Communications in Mathematical Physics

Bose–Einstein Condensation in a Dilute, Trapped Gas at Positive Temperature

Andreas Deuchert, Robert Seiringer, Jakob Yngvason

Published Aug 30, 2018• 26 citations
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Abstract

We consider an interacting, dilute Bose gas trapped in a harmonic potential at a positive temperature. The system is analyzed in a combination of a thermodynamic and a Gross–Pitaevskii (GP) limit where the trap frequency $${\omega}$$ , the temperature T, and the particle number N are related by $${N \sim (T / \omega)^{3} \to\infty}$$ while the scattering length is so small that the interaction energy per particle around the center of the trap is of the same order of magnitude as the spectral gap in the trap. We prove that the difference between the canonical free energy of the interacting gas and the one of the noninteracting system can be obtained by minimizing the GP energy functional. We also prove Bose–Einstein condensation in the following sense: The one-particle density matrix of any approximate minimizer of the canonical free energy functional is to leading order given by that of the noninteracting gas but with the free condensate wavefunction replaced by the GP minimizer.

DOI
10.1007/s00220-018-3239-0

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We consider an interacting, dilute Bose gas trapped in a harmonic potential at a positive temperature. The system is analyzed in a combination of a thermodynamic and a Gross–Pitaevskii (GP) limit where the trap frequency ω{\omega}ω , the temperature T, and the particle number N are related by N∼(T/ω)3→∞{N \sim (T / \omega)^{3} \to\infty}N∼(T/ω)3→∞ while the scattering length is so small that the interaction energy per particle around the center of the trap is of the same order of magnitude as the spectral gap in the trap. We prove that the difference between the canonical free energy of the interacting gas and the one of the noninteracting system can be obtained by minimizing the GP energy functional. We also prove Bose–Einstein condensation in the following sense: The one-particle density matrix of any approximate minimizer of the canonical free energy functional is to leading order given by that of the noninteracting gas but with the free condensate wavefunction replaced by the GP minimizer.
Abstract We consider a dilute, homogeneous Bose gas at positive temperature. The system is investigated in the Gross–Pitaevskii limit, where the scattering lengthais so small that the interaction energy is of the same order of magnitude as the spectral gap of the Laplacian, and for temperatures that are comparable to the critical temperature of the ideal gas. We show that the difference between the specific free energy of the interacting system and the one of the ideal gas is to leading order given by 4πa(2ϱ2−ϱ02)4 \pi a \left( 2 \varrho ^2 - \varrho _0^2 \right) 4πa(2ϱ2−ϱ02​) 4πa2ϱ2-ϱ02 . Here ϱ\varrho ϱ ϱ denotes the density of the system and ϱ0\varrho _0ϱ0​ ϱ0 is the expected condensate density of the ideal gas. Additionally, we show that the one-particle density matrix of any approximate minimizer of the Gibbs free energy functional is to leading order given by the one of the ideal gas. This in particular proves Bose–Einstein condensation with critical temperature given by the one of the ideal gas to leading order. One key ingredient of our proof is a novel use of the Gibbs variational principle that goes hand in hand with the c-number substitution.
We evaluate the zero temperature properties and Bose-Einstein condensation (BEC) of hard sphere bosons in a harmonic trap using Quantum Monte Carlo methods. The chief purpose of this work is to go beyond the dilute limit, to test the limits of the Gross-Pitaevskii equation and related mean field approximations and to explore the role of interactions in determining the zero temperature properties of the trapped Bose gas. The density is increased by adjusting both the number of trapped bosons, N , and the s-wave scattering length, a, to study systems from the highly dilute region corresponding to early experiments in trapped BEC [2, 3] up to liquid He densities and beyond. Rather than relying on the perturbative Bogoliubov approximation (which assumes a large condensate fraction) to describe the condensate, we use the one-body density matrix (OBDM) formulation of BEC so that the properties of the condensate for systems with arbitrarily large interactions may be studied. In this formulation of BEC, condensate properties are obtained by diagonalizing the OBDM and obtaining the corresponding single particle “natural orbitals” and their occupation numbers for the system. The condensate wave function and condensate fraction are then obtained from the single particle orbital(s) with macroscopic occupation (N0 >> 1). Within this framework, we calculate the effects of interactions and increased density on the ground state energy, the density profile, the momentum distribution, the condensate fraction and condensate “wave-function” and several other properties.
The mean-field properties of finite-temperature Bose-Einstein gases confined in spherically symmetric harmonic traps are surveyed numerically. The solutions of the Gross-Pitaevskii (GP) and Hartree-Fock-Bogoliubov (HFB) equations for the condensate and low-lying quasiparticle excitations are calculated self-consistently using the discrete variable representation, while the most high-lying states are obtained with a local-density approximation. Consistency of the theory for temperatures through the Bose condensation point Tc{T}_{c}Tc​ requires that the thermodynamic chemical potential differ from the eigenvalue of the GP equation; the appropriate modifications lead to results that are continuous as a function of the particle interactions. The HFB equations are made gapless either by invoking the Popov approximation or by renormalizing the particle interactions. The latter approach effectively reduces the strength of the effective scattering length asc,{a}_{\mathrm{sc}},asc​, increases the number of condensate atoms at each temperature, and raises the value of Tc{T}_{c}Tc​ relative to the Popov approximation. The renormalization effect increases approximately with the log of the atom number, and is most pronounced at temperatures near Tc.{T}_{c}.Tc​. Comparisons with the results of quantum Monte Carlo calculations and various local-density approximations are presented, and experimental consequences are discussed.
We consider a homogeneous Bose gas in the Gross-Pitaevskii limit at temperatures that are comparable to the critical temperature for Bose-Einstein condensation. Recently, an upper bound for the grand canonical free energy was proved in Boccato et al. (SIAM J Math Anal 56(2):2611-2660, 2024) capturing two novel contributions. First, the free energy of the interacting condensate is given in terms of an effective theory describing the probability distribution of the number of condensed particles. Second, the free energy of the thermally excited particles equals that of a temperature-dependent Bogoliubov Hamiltonian. We extend this result to a more general class of interaction potentials, including interactions with a hard core. Our proof follows a different approach than the one in Boccato et al. (SIAM J Math Anal 56(2):2611-2660, 2024): We model microscopic correlations between the particles by a Jastrow factor and exploit a cancellation in the computation of the energy that emerges due to the different length scales in the system.
We prove the following results. (i) One-dimensional Bose gases which interact via unscaled integrable pair interactions and are confined in an external potential increasing faster than quadratically undergo a complete generalized Bose-Einstein condensation (BEC) at any temperature, in the sense that a macroscopic number of particles are distributed on a o(N) number of one-particle states. (ii) In a one dimensional harmonic trap the replacement of the oscillator frequency ω by ω ln N/N gives rise to a phase transition at a ≡ �ωβ = 1 in the noninteracting gas. For a < 1 the limit distribution of n0/N a is exponential and 〈n0〉/N a → 1. For a> 1 there is BEC with a condensate density 〈n0〉/N → 1 − a −1. For a ≥ 1, (lnN/N)(n0 − 〈n0〉) is asymptotically distributed following Gumbel’s law. For any a> 0 the free energy is −(π 2 /6βa)N / lnN + o(N/ln N), with no singularity at a = 1. (iii) In Model (ii) both above and below the critical temperature the gas undergoes a complete generalized BEC, thus providing a coexistence of ordinary and generalized condensates below the critical point. (iv) Adding an interaction 〈UN 〉 = o(N ln N) to Model (ii) we prove that a complete generalized BEC occurs for any β> 0.
We investigate the thermodynamics of a Bose gas interacting with repulsive forces and confined in a harmonic trap. We show that the relevant parameters of the system (temperature, number NNN of atoms, harmonic oscillator length, deformation of the trap, sss-wave scattering length) fix its thermodynamic behavior in the large NNN limit through two dimensionless scaling parameters. These are the reduced temperature t=T/Tc0t\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}T/{T}_{c}^{0}t=T/Tc0​ and the ratio $\ensuremath{\eta}$ between the T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 value of the chemical potential, evaluated in the Thomas-Fermi limit, and the critical temperature Tc0{T}_{c}^{0}Tc0​ of the noninteracting model. The scaling functions relative to the condensate fraction, energy, chemical potential, and moment of inertia are calculated within the Popov approximation.