Abstract
Photon Bose–Einstein condensates are characterised by a quite weak interaction, so they behave nearly as an ideal Bose gas. Moreover, since the current experiments are conducted in a microcavity, the longitudinal motion is frozen out and the photon gas represents effectively a two-dimensional trapped gas of massive bosons. In this paper we focus on a harmonically confined ideal Bose gas in two dimensions, where the anisotropy of the confinement allows for a dimensional crossover. If the confinement in one direction is strong enough so that this squeezed direction is frozen out, then only one degree of freedom survives and the system can be considered to be quasi-one dimensional. In view of an experimental set-up we work out analytically the thermodynamic properties for such a system with a finite number of photons. In particular, we focus on examining the dimensional information which is contained in the respective thermodynamic quantities.
Highlights
The question of Bose-Einstein condensation in lower dimensions got already tackled quite early in the post-war era of physics
In this paper we present an analytical description of the dimensional crossover from 1D to 2D for an ideal Bose gas in terms of a dimensional expansion, see Eq (5)
We find the same structure for all investigated thermodynamic quantities, such as the critical particle number, the condensate fraction, and the specific heat, namely that the 1D expression gets corrected by terms yielding the 2D result
Summary
The question of Bose-Einstein condensation in lower dimensions got already tackled quite early in the post-war era of physics. In the thermodynamic limit they showed for a trapping potential, which is stronger confining than a box in the sense of a monomial spatial dependence ∼ xα, that a condensation in 2D can occur, whilst in a 1D setting a potential more confining than a quadratic potential is necessary Soon after this a full quantum mechanical follow-up study [4] revealed that for the harmonically trapped 1D-Bose gas with a finite number of particles BEC is possible. The question remains how to define and how to determine the effective dimension of the gas when changing from an isotropic harmonic confinement to a highly anisotropic confinement giving rise for a dimensional crossover from a two-dimensional gas to a quasi-1D gas To this end, we work out how the thermodynamic quantities change as a function of the trap-aspect ratio.
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