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arXiv:1110.3114 · Europhysics Letters (EPL)

Finite temperature and dissipative corrections to the Gross-Pitaevskii equation from λΦ 4 one-loop contributions

T. Matos, Abril Suárez

Published Nov 22, 2011• 25 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1209/0295-5075/96/56005

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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.
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.
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 introduce a time-dependent projected Gross-Pitaevskii equation to describe a partially condensed homogeneous Bose gas, and find that this equation will evolve randomized initial wave functions to equilibrium. We compare our numerical data to the predictions of a gapless, second order theory of Bose-Einstein condensation [S. A. Morgan, J. Phys. B 33, 3847 (2000)], and find that we can determine a temperature when the theory is valid. As the Gross-Pitaevskii equation is nonperturbative, we expect that it can describe the correct thermal behavior of a Bose gas as long as all relevant modes are highly occupied. Our method could be applied to other boson fields.
So far much theoretical consideration of experiments on the Bose-Einstein condensation (BEC) of alkali-metal atoms in harmonic traps is based on the Gross-Pitaevskii (GP) equation. In this paper, we attempt to formulate the BEC in the language of quantum field theory and to estimate the quantum and thermal fluctuation effects, which are neglected in the approximation using the GP equation. First, the formulation at zero temperature is developed, and then it is extended to the finite-temperature case by means of thermofield dynamics. We treat the zero-energy mode with care, so that the canonical commutation relations hold. As a result, an infrared divergence appears, but it can be renormalized into the observed condensate number. Numerical calculations are performed. For illustration, the corrections at one-loop level to the original GP equation are given. We also calculate numerically the effects of quantum and thermal fluctuations on the distribution of condensed atoms.
So far many theoretical considerations for experiments of Bose-Einstein condensation (BEC) of alkali atoms in harmonic traps are based on the Gross-Pitaevskii (GP) equation. In this report, we attempt to formulate the BEC in the language of quantum field theory in order to go beyond the approximation of GP equation. First the formulation at zero-temperature is presented, and then it is extended to finite-temperature case by means of Thermo Field Dynamics. Numerical calculations, which are inevitable even in the unperturbative formulation of our approach, are performed. For illustration, the corrections at one-loop level to the original GP equation are given. We also calculate the effects of quantum and thermal fluctuations to the distribution of condensed atoms numerically.
We present a detailed derivation of the quantum and quantum–thermal effective action for non-relativistic systems, starting from the single-particle case and extending to the Gross–Pitaevskii (GP) field theory for weakly interacting bosons. In the single-particle framework, we introduce the one-particle-irreducible (1PI) effective action formalism, taking explicitly into account the choice of the initial quantum state, its saddle-point plus Gaussian-fluctuation approximation, and its finite-temperature extension via Matsubara summation, yielding a clear physical interpretation in terms of zero-point and thermal contributions to the Helmholtz free energy. The formalism is then applied to the GP action, producing the 1PI effective potential at zero and finite temperature, including beyond-mean-field Lee–Huang–Yang and thermal corrections. We discuss the gapless and gapped Bogoliubov spectra, their relevance to equilibrium and non-equilibrium regimes, and the role of regularization. Applications include the inclusion of an external potential within the local density approximation, the derivation of finite-temperature Josephson equations, and the extension to D-dimensional systems, with particular attention to the zero-dimensional limit. This unified approach provides a transparent connection between microscopic quantum fluctuations and effective macroscopic equations of motion for Bose–Einstein condensates.