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arXiv:0812.1332 · Physical Review A

Critical properties of a trapped interacting Bose gas

A. Bezett, P. B. Blakie

Published Mar 12, 2009• 31 citations
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Abstract

We develop a practical theoretical formalism for studying the critical properties of a trapped Bose-Einstein condensate using the projected Gross-Pitaevskii equation. We show that this approach allows us to investigate the behavior of the correlation length, condensate mode, and its number fluctuations about the critical point. Motivated by recent experiments [Donner et al., Science 315, 1556 (2007)], we calculate the critical exponent for the correlation length of the trapped system. We observe finite-size effects in our results when the correlation length becomes comparable to the Ginzburg length. We extend the Binder cumulant to the trapped system and discuss an experimental method for measuring number fluctuations.

DOI
10.1103/physreva.79.033611

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We develop a practical theoretical formalism for studying the critical properties of a trapped Bose-Einstein condensate using the projected Gross-Pitaevskii equation. We show that this approach allows us to investigate the behavior of the correlation length, condensate mode, and its number fluctuations about the critical point. Motivated by recent experiments [Donner et al., Science 315, 1556 (2007)], we calculate the critical exponent for the correlation length of the trapped system. We observe finite-size effects in our results when the correlation length becomes comparable to the Ginzburg length. We extend the Binder cumulant to the trapped system and discuss an experimental method for measuring number fluctuations.
The phase transition of Bose-Einstein condensation was studied in the critical regime, where fluctuations extend far beyond the length scale of thermal de Broglie waves. We used matter-wave interference to measure the correlation length of these critical fluctuations as a function of temperature. Observations of the diverging behavior of the correlation length above the critical temperature enabled us to determine the critical exponent of the correlation length for a trapped, weakly interacting Bose gas to be nu = 0.67 +/- 0.13. This measurement has direct implications for the understanding of second-order phase transitions.
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Calculation of g2g_2g2​ for a Finite Temperature Gas
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Bose–Einstein condensation has been realized as dilute atomic vapors. This achievement has generated immense interest in this field. This article review of recent theoretical research into the properties of trapped dilute-gas Bose–Einstein condensates. Among these properties, stability of Bose–Einstein condensates confined in traps is mainly discussed. Static properties of the ground state are investigated by using the variational method. The analysis is extended to the stability of two-component condensates. Time-development of the condensate is well-described by the Gross–Pitaevskii equation which is known in nonlinear physics as the no nlinear Schrödinger equation. For the case that the inter-atomic potential is effectively attractive, a singularity of the solution emerges in a finite time. This phenomenon which we call collapse explains the upper bound for the number of atoms in such condensates under traps.
Bose–Einstein condensation represents a quintessential manifestation of quantum mechanics at macroscopic scales, where a dilute gas of bosonic atoms undergoes a phase transition to form a coherent matter wave. This study presents a comprehensive theoretical investigation of collective excitations and critical phenomena in interacting Bose–Einstein condensates (BECs). Beginning with the Gross–Pitaevskii equation , we develop the Bogoliubov theory of excitations yielding the spectrum , where is the free-particle energy and characterizes contact interactions. The spectrum reveals phonon behavior at low momenta with sound velocity , transitioning to free-particle dispersion at momenta exceeding the inverse healing length . Collective modes in harmonically trapped condensates are analyzed in the Thomas–Fermi regime, yielding breathing mode frequency and quadrupole frequency . Damping mechanisms including Landau damping ( ) and Beliaev damping ( ) are characterized. The Lee–Huang–Yang correction extends the equation of state beyond mean field. Critical phenomena near are analyzed within the 3D XY universality class, with critical exponents , , and the interaction-induced shift with . All predictions show excellent agreement with experimental measurements on ultracold atomic gases.