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Maximal length of trapped one-dimensional Bose-Einstein condensates

Uwe R. Fischer

Published Jan 1, 2004• 7 citations
OpenAlex

Abstract

I discuss a Bogoliubov inequality for obtaining a rigorous bound on the maximal axial extension of inhomogeneous one-dimensional Bose-Einstein condensates. An explicit upper limit for the aspect ratio of a strongly elongated, harmonically trapped Thomas-Fermi condensate is derived. The question on the existence and dimension dependence of off-diagonal long-range order, for an infinitely extended system of scalar uncharged bosons, was conclusively answered by Hohenberg 1. He made use of the fact that in the thermodynamic limit of an infinite system of interacting particles of bare mass m, the relation (we set � = kB = 1 in what follows, where kB is the Boltzmann constant) Nk ≥ − 1 mT 2 k2 n0

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I discuss a Bogoliubov inequality for obtaining a rigorous bound on the maximal axial extension of inhomogeneous one-dimensional Bose-Einstein condensates. An explicit upper limit for the aspect ratio of a strongly elongated, harmonically trapped Thomas-Fermi condensate is derived. The question on the existence and dimension dependence of off-diagonal long-range order, for an infinitely extended system of scalar uncharged bosons, was conclusively answered by Hohenberg 1. He made use of the fact that in the thermodynamic limit of an infinite system of interacting particles of bare mass m, the relation (we set � = kB = 1 in what follows, where kB is the Boltzmann constant) Nk ≥ − 1 mT 2 k2 n0
The Hohenberg theorem on the absence of Bose-Einstein condensation (BEC) in homogeneous systems of space dimensions D< 2D<~2D< 2 is based on a well-known Bogoliubov inequality. Applied to an assembly trapped in a harmonic potential we show that the Bogoliubov inequality rules out BEC in all dimensions at finite temperatures. However, this conflicting result with both theory and experiment disappears when the effect of the order parameter is properly taken into account in the boson-field commutation relations. For a hard-sphere Bose gas the theory is consistent with the expansion of the condensate when a positive scattering length is increased, as well as the collapse of the condensate when the sign of the scattering length is reversed and it reaches a minimum critical value.
We consider the non-interacting Bose gas of NNN bosons in dimension d≥3d\geq 3d≥3 in a trap in a mean-field setting with a vanishing factor aNa_NaN​ in front of the kinetic energy. The choice aN=N−2/da_N=N^{-2/d}aN​=N−2/d is the semi-classical setting and was analysed in great detail in a special, interacting case in Deuchert and Seiringer (2021). Using a version of the well-known Feynman--Kac representation and a further representation in terms of a Poisson point process, we derive precise asymptotics for the reduced one-particle density matrix, implying off-diagonal long-range order (ODLRO, a well-known criterion for Bose--Einstein condensation) for aNa_NaN​ above a certain threshold and non-occurrence of ODLRO for aNa_NaN​ below that threshold. In particular, we relate the condensate and its total mass to the amount of particles in long loops in the Feynman--Kac formula, the order parameter that Feynman suggested in Feynman (1953). For aN≪N−2/da_N\ll N^{-2/d}aN​≪N−2/d, we prove that all loops have the minimal length one, and for aN≫N−2/da_N\gg N^{-2/d}aN​≫N−2/d we prove 100 percent condensation and identify the distribution of the long-loop lengths as the Poisson--Dirichlet distribution.
We consider the non-interacting Bose gas of N bosons in dimension d≥3 in a trap in a mean-field setting with a vanishing factor aN in front of the kinetic energy. The choice aN=N−2∕d is the semi-classical setting and was analysed in great detail in a special, interacting case in [DS21]. Using a version of the well-known Feynman–Kac representation and a further representation in terms of a Poisson point process, we derive precise asymptotics for the reduced one-particle density matrix, implying off-diagonal long-range order (ODLRO, a well-known criterion for Bose–Einstein condensation) for aN above a certain threshold and non-occurrence of ODLRO for aN below that threshold. In particular, we relate the condensate and its total mass to the amount of particles in long loops in the Feynman–Kac formula, the order parameter that Feynman suggested in [Fe53]. For aN≪N−2∕d, we prove that all loops have the minimal length one, and for aN≫N−2∕d we prove 100 percent condensation and identify the distribution of the long-loop lengths as the Poisson–Dirichlet distribution.
Corrections to the zero-temperature Thomas-Fermi description of a dilute interacting condensed Bose-Einstein gas confined in an isotropic harmonic trap arise due to the presence of a boundary layer near the condensate surface. Within the Bogoliubov approximation, the various contributions to the ground-state condensate energy all have terms of order ${R}^{\ensuremath{-}4}\mathrm{ln}R$ and ${R}^{\ensuremath{-}4},$ where R is the number-dependent dimensionless condensate radius in units of the oscillator length $\sqrt{\ensuremath{\Elzxh}/m{\ensuremath{\omega}}_{0}}.$ The zero-order hydrodynamic density-fluctuation amplitudes are extended beyond the Thomas-Fermi radius through the boundary layer to provide a uniform description throughout all space. The first-order correction to the excitation frequencies is shown to be of order ${R}^{\ensuremath{-}4}.$