Thermodynamics of a Bose-Einstein condensate with weak disorder

G. M. Falco, A. Pelster, and R. Graham

Phys. Rev. A 75, 063619 – Published 19 June 2007

Abstract

We consider the thermodynamics of a homogeneous superfluid dilute Bose gas in the presence of weak quenched disorder. Following the zero-temperature approach of Huang and Meng, we diagonalize the Hamiltonian of a dilute Bose gas in an external random $δ$ -correlated potential by means of a Bogoliubov transformation. We extend this approach to finite temperature by combining the Popov and the many-body $T$ -matrix approximations. This approach permits us to include the quasiparticle interactions within this temperature range. We derive the disorder-induced shifts of the Bose-Einstein critical temperature and of the temperature for the onset of superfluidity by approaching the transition points from below, i.e., from the superfluid phase. Our results lead to a phase diagram consistent with that of the finite-temperature theory of Lopatin and Vinokur which was based on the replica method, and in which the transition points were approached from above.

Received 5 February 2007

DOI:https://doi.org/10.1103/PhysRevA.75.063619

Authors & Affiliations

G. M. Falco, A. Pelster, and R. Graham

Fachbereich Physik, Universität Duisburg-Essen, Campus Duisburg, Lotharstrasse 1, 47057 Duisburg, Germany

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Vol. 75, Iss. 6 — June 2007

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Images

Figure 1
(a) Temperature-dependent scattering length as a function of the rescaled temperature $T ∕ T_{c}^{0}$ calculated numerically from Eq. (58) (solid line) and from the analytic expression of Eq. (62) (dashed line). The dotted line shows the asymptotic limit described by Eq. (63) valid in the Popov region for temperatures not too close to $T_{c}^{0}$ . (b) Temperature-dependent scattering length as a function of the rescaled temperature $T ∕ T_{c}^{0}$ near $T_{c}^{0}$ . The lower line shows the numerical result from Eq. (58) compared to the asymptotic expression given in Eq. (64) described by the upper line. In both pictures the gas parameter of the Bogoliubov theory has been chosen such as ${(n a^{3})}^{1 ∕ 3} \sim 0.01$ .Reuse & Permissions
Figure 2
The lower and the upper dashed lines describe the condensate density $n_{0}$ and the scattering length $a_{T}$ obtained by solving self-consistently Eqs. (62, 65) in absence of disorder, i.e., when $R = 0$ . In this case, both quantities vanish at $T = T_{c}^{0}$ and no shift of the critical temperature is induced [7, 20]. The gas parameter of the Bogoliubov theory has been chosen the same as in Fig. 1. The solid lines show the solution for finite disorder. We see that the temperature at which they vanish is shifted with respect to the critical temperature of the ideal gas. In order to make clearly visible the effect we have considered in the figure the case $R_{0}^{'} = 0.5$ . However, for such a value the condensate depletion due to the disorder at zero temperature is already $\sim 0.4$ of the total density. Note also that, in contrast with Popov theory, for each value of $R_{0}^{'}$ the curve for $n_{0}$ obtained from the many-body $T$ -matrix approximation exhibits a second-order phase transition.Reuse & Permissions
Figure 3
The solid line indicates the critical temperature $T_{s}$ of the normal to superfluid transition as a function of the dimensionless disorder parameter $R_{0}^{'}$ obtained by solving self-consistently Eq. (62), Eq. (65), and Eq. (68) for ${(n a^{3})}^{1 ∕ 3} = 0.01$ . The upper dashed line is obtained by approximating Eq. (62) with the asymptotic solution of Eq. (64) which is valid when $T_{s} ≃ T_{c}$ for $R_{0}^{'} ⩽ {(n a^{3})}^{1 ∕ 6}$ . In contrast, the lower dashed line describes the regime $R_{0}^{'} ⩾ {(n a^{3})}^{1 ∕ 6}$ where $T_{s} ⪡ T_{c}$ and Eq. (62) can be approximated with Eq. (63). The comparison between these three curves describes clearly how the temperature effects in the many-body $T^{MB}$ -matrix determine the crossover from the quadratic regime to the linear regime for the dependence of $T_{s} ∕ T_{c}^{0}$ on the disorder coupling constant $R_{0}^{'}$ . In addition, the short-dashed line shows the effects of neglecting the self-consistency in the equation of state (65), where the condensate density is approximated with the ideal gas result $n_{0} = n [1 - {(T_{s} ∕ T_{c}^{0})}^{3 ∕ 2}]$ . For $R_{0}^{'} ⩾ {(n a^{3})}^{1 ∕ 6}$ the short-dashed line coincides with the analytical result of the Popov theory given by Eq. (53).Reuse & Permissions

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