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arXiv:1604.01730 · Modern Physics Letters B

Homogeneous one-dimensional Bose–Einstein condensate in the Bogoliubov’s regime

Elías Castellanos

Published Aug 16, 2016• 5 citations
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Abstract

We analyze the corrections caused by finite size effects upon the ground state properties of a homogeneous one-dimensional (1D) Bose–Einstein condensate. We assume from the very beginning that the Bogoliubov’s formalism is valid and consequently, we show that in order to obtain a well-defined ground state properties, finite size effects of the system must be taken into account. Indeed, the formalism described in the present paper allows to recover the usual properties related to the ground state of a homogeneous 1D Bose–Einstein condensate but corrected by finite size effects of the system. Finally, this scenario allows us to analyze the sensitivity of the system when the Bogoliubov’s regime is valid and when finite size effects are present. These facts open the possibility to apply these ideas to more realistic scenarios, e.g. low-dimensional trapped Bose–Einstein condensates.

DOI
10.1142/s0217984916503073

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We analyze the corrections caused by finite size effects upon the ground state properties of a homogeneous one-dimensional (1D) Bose–Einstein condensate. We assume from the very beginning that the Bogoliubov’s formalism is valid and consequently, we show that in order to obtain a well-defined ground state properties, finite size effects of the system must be taken into account. Indeed, the formalism described in the present paper allows to recover the usual properties related to the ground state of a homogeneous 1D Bose–Einstein condensate but corrected by finite size effects of the system. Finally, this scenario allows us to analyze the sensitivity of the system when the Bogoliubov’s regime is valid and when finite size effects are present. These facts open the possibility to apply these ideas to more realistic scenarios, e.g. low-dimensional trapped Bose–Einstein condensates.
We analyze the consequences caused by a deformed dispersion relation, suggested in several quantum gravity models, upon a bosonic gas. Concerning the ground state of the Bogoliubov space of this system, we deduce the corrections in the pressure, the speed of sound, and the corresponding healing length. Indeed, we prove that the corrections in the relevant thermodynamic properties associated with the ground state, defines a non trivial function of the density of particles and the deformation parameters, allowing us to constrain, in principle, the form of the modified energy-momentum dispersion relation.
In this work, we analyze the corrections obtained on a homogeneous one-dimensional Bose gas within the high densities limit by means of the non-regular representation of quantum mechanics introduced by the polymer quantization scheme. Thus, starting from the Bogoliubov formalism, we analyze the ground expectation value of the polymer momentum operator in terms of semiclassical states in order to obtain an analytic expression for the ground state energy of the N-body system, which allows us to solve the pathological behavior commonly associated with the one-dimensional Bose–Einstein condensation through the introduction of finite size effects characterized by the contribution of the polymer corrections. We also discuss the speed of sound in our polymer version of the Bose gas and the corresponding relative shift induced by the introduction of a minimum length parameter. Finally, we investigate the emergent superfluid behavior in our polymer model by implementing an appropriate Landau’s criterion. In this case, we are able to consequently analyze the changes in the critical velocity which defines the limit between the superfluid-condensate regions, thus deducing that the polymer length acts as a kind of pseudo-potential which induces a dissipationless flow associated with the superfluid phase even in the absence of self-interactions.
Abstract The finite-size effect plays a key role in one-dimensional Bose–Einstein condensation (BEC) of photons since such condensation cannot occur in the thermodynamic limit due to the linear dispersion relation of photons. However, since a divergence difficulty arises, the previous theoretical analysis of the finite-size effect often only gives the leading-order contribution. In this article, by using an analytical continuation method to overcome the divergence difficulty, we give an analytical treatment for the finite-size effect in BEC. We show that the deviation between experiment and theory becomes much smaller by taking into account the next-to-leading correction.
Bose–Einstein condensation in a finite one-dimensional atomic Bose gas trapped in an optical lattice is studied within Bogoliubov’s approximation and then beyond this approximation, within the static fluctuation approximation. A Bose–Hubbard model is used to construct the Hamiltonian of the system. The effect of the potential strength on the condensate fraction is explored at different temperatures; so is the effect of temperature on this fraction at different potential strengths. The role of the number of lattice points (the size effect) at constant number density (the filling factor) is examined; so is the effect of the number density on the condensate fraction. The results obtained are compared to other published results wherever possible.
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 investigate the validity of the Bogoliubov c-number approximation in the case of interacting Bose-gas in a homogeneous random media. To take into account the possible occurence of type III generalized Bose–Einstein condensation (i.e., the occurrence of condensation in an infinitesimal band of low kinetic energy modes without macroscopic occupation of any of them) we generalize the c-number substitution procedure to this band of modes with low momentum. We show that, as in the case of the one-mode condensation for translation-invariant interacting systems, this procedure has no effect on the exact value of the pressure in the thermodynamic limit, assuming that the c-numbers are chosen according to a suitable variational principle. We then discuss the relation between these c-numbers and the (total) density of the condensate.