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International Journal of Modern Physics B

KLEIN–GORDON FIELDS AND BOSE–EINSTEIN CONDENSATES: THERMAL BATH CONTRIBUTIONS

Elías Castellanos, Tonatiuh Matos

Published Apr 11, 2013• 13 citations
Publisher

Abstract

We analyze the consequences caused by a thermal bath upon the properties associated to the symmetry breaking of scalar fields with one-loop correction potential. Concerning the nonrelativistic regime associated with the aforementioned system, we calculate the shift in the condensation temperature caused by the thermal bath, assuming a harmonic oscillator type potential. We prove that the shift in the condensation temperature depends on the associated scale of the system. In addition, we obtain bounds associated to the scale that could lead to relevant corrections on the condensation temperature under typical conditions.

DOI
10.1142/s0217979213500604

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We analyze the consequences caused by a thermal bath upon the properties associated to the symmetry breaking of scalar fields with one-loop correction potential. Concerning the nonrelativistic regime associated with the aforementioned system, we calculate the shift in the condensation temperature caused by the thermal bath, assuming a harmonic oscillator type potential. We prove that the shift in the condensation temperature depends on the associated scale of the system. In addition, we obtain bounds associated to the scale that could lead to relevant corrections on the condensation temperature under typical conditions.
We deduce the relation between the critical temperature associated to the symmetry breaking of scalar fields with one--loop correction potential immersed in a thermal bath and the condensation temperature of the aforementioned system, assuming a harmonic oscillator type potential. We show that these two temperatures are related through the \emph{scale} associated to the system. In this aim, we infer the order of magnitude for the \emph{scale} as a function of the corresponding healing length, in order to give a criterium to compare both temperatures. Additionally, we prove that the condensation temperature is independent of the thermal bath within the semiclassical approximation, for a positive coupling constant, assuming that the thermal bath contribution is the lowest energy associated to the system.
Starting with a scalar field in a thermal bath and using the one-loop quantum correction potential, we rewrite the Klein-Gordon equation in its thermodynamical representation and study the behavior of this scalar field due to temperature variations in the equations of motion. We find the generalization of a Gross-Pitaevskii–like equation for a relativistic Bose gas with finite temperature, the corresponding thermodynamic and viscosity expressions, and an expression for the postulate of the first law of the thermodynamics for this BECs. We also propose that the equations obtained might help to explain at some level the phase transition of a Bose-Einstein Condensate in terms of quantum field theory in a simple way.
In this work the Klein–Gordon equation for a complex scalar field with U(1) symmetry endowed in a mexican-hat scalar field potential with thermal and electromagnetic contributions is written as a Gross–Pitaevskii (GP)-like equation. This equation is interpreted as a charged generalization of the GP equation at finite temperatures found in previous works. Its hydrodynamical representation is obtained and the corresponding thermodynamical properties are derived and related to measurable quantities. The condensation temperature in the non-relativistic regime associated with the aforementioned system within the semiclassical approximation is calculated. Also, a generalized equation for the conservation of energy for a charged bosonic gas is found when electromagnetic fields are introduced, and it is studied how under certain circumstances its breaking of symmetry can give some insight on the phase transition of the system not just into the condensed phase but also on other related systems
We rewrite the complex Klein-Gordon (KG) equation with a mexican-hat scalar field potential in a thermal bath with one loop contribution as a new Gross-Pitaevskii (GP)-like equation. We interpret it as a charged and finite temperature generalization of the GP equation. We find its hydrodynamic version as well and using it, we derive the corresponding thermodynamics. We obtain a generalized first law for a charged Bose-Einstein Condensate (BEC). We translate the breaking of the U(1) local symmetry of the KG field into the new version of the GP equation and demonstrate that this symmetry breaking corresponds to a phase transition of the gas into a BEC, and show the conditions for which this system naturally becomes superfluid and/or superconductor.
We optimize the classical field approximation of the version described by M. Brewczyk, M. Gajda, and K. Rz\k{a}\ifmmode \dot{z}\else \.{z}\fi{}ewski [J. Phys. B 40, R1 (2007)] for the oscillations of a Bose gas trapped in a harmonic potential at nonzero temperatures, as experimentally investigated by Jin et al. [Phys. Rev. Lett. 78, 764 (1997)]. Similar to experiment, the system response to external perturbations strongly depends on the initial temperature and the symmetry of perturbation. While for lower temperatures the thermal cloud follows the condensed part, for higher temperatures the thermal atoms oscillate rather with their natural frequency, whereas the condensate exhibits a frequency shift toward the thermal cloud frequency (m=0m=0m=0 mode) or in the opposite direction (m=2m=2m=2 mode). In the latter case, for temperatures approaching critical, we find that the condensate begins to oscillate with the frequency of the thermal atoms, as in the m=0m=0m=0 mode. A broad range of frequencies of the perturbing potential is considered.
We show various aspects of finite size effects on Bose-Einstein condensation(BEC). In the first section we introduce very briefly the BEC of harmonically trapped ideal Bose gas. In the second section we theoretically argued that Bose-Einstein(B-E) statistics needs a correction for finite system at ultralow temperatures. As a corrected statistics we introduced a Tsallis type of generalized B-E statistics. The condensate fraction calculated with this generalized B-E statistics, is satisfied well with the experimental result. In the third section we show how to apply the scaling theory in an inhomogeneous system like harmonically trapped Bose condensate at finite temperatures. We calculate the temperature dependence of the critical number of particles by a scaling theory within the Hartree-Fock approximation and find that there is a dramatic increase in the critical number of particles as the condensation point is approached. Our results support the experimental result which was obtained well below the condensation temperature. In the fourth section we concentrate on the thermodynamic Casimir force on the Bose-Einstein condensate. We explored the temperature dependence of the Casimir force.