Critical Temperature of an Interacting Bose Gas in a Generic Power-Law Potential

Abstract

We investigate the critical temperature of an interacting Bose gas confined in a trap described by a generic isotropic power-law potential. We compare the results with respect to the non-interacting case. In particular, we derive an analytical formula for the shift of the critical temperature holding to first order in the scattering length. We show that this shift scales as N n 3(n+2) , where N is the number of Bosons and n is the exponent of the power-law potential.Moreover, the sign of the shift critically depends on the power-law exponent n. Finally, we find that the shift of the critical temperature due to finite-size effects vanishes as N − 2n 3(n+2) .

Mean-field analysis of Bose–Einstein condensation in general power-law potentials

“…Note that equation 13would yield the exact critical temperature (within the local-density approximation), if f (cr) MF (x) were replaced by the exact scaled density distribution. In general, the mean-field critical temperature for a given set of system parameters can only be determined numerically by calculating f (cr) MF (x) from equation (12) and evaluating the integral I ∞ 0 (q, η). For small q = a/λ T , however, an analytic approximation can be derived.…”

“…Let us now consider equation (12) in the limit of large x. If x 4q ζ(3/2) − f (cr) MF (x) , we can perform a Taylor expansion of the right-hand side around x, i.e.…”

“…The mean-field theory of T c in general power-law traps has been the subject of several recent studies in the literature [10][11][12][13]. In spite of the interest in this topic and the progress made so far, none of these works provides a complete picture of the problem in the sense that it also accurately covers the quasi-homogeneous limit.…”

Stability analysis of a Bose–Einstein condensate trapped in a generic potential

“…(2.2) in order to study its relation to the dynamical structure analyzed in the previous section. The properties of a condensate trapped in a spherical power law potential have been extensively analyzed, see for instance, [29,32,41,42,43] and references therein. To this aim, let us propose a particularly simple Hartree-Fock type spectrum, in the semi-classical approximation.…”

Modified bosonic gas trapped in a generic 3-dim power law potential

“…It is noteworthy to mention that the use of these generic potentials, opens the possibility to adiabatically cool the system in a reversible way, by changing the shape of the trap [20]. The analysis of a Bose-Einstein condensate in the ideal case, weakly interacting, and with a finite number of particles, trapped in different potentials show that the main properties associated with the condensate, and in particular the condensation temperature, strongly depends on the trapping potential under consideration [22][23][24][25][26][27][28][29][30][31][32][33][34][35]. Additionally, the characteristics of the potential (in particular, the parameter that defines the shape of the potential) has a strong impact on the dependence of the condensation temperature with the number of particles (or the associated density).…”