Finite Temperature Effects in the Condensates of Dilute Atomic Gases

Abstract

The stationary state solutions and dynamics of Bose-Einstein condensates (BECs) at T = 0 are well described by the Gross-Pitaevskii (GP) equation. BECs of dilute atomic gases have been experimentally achieved at ultracold temperatures of the orders of 10ð€€€9 K. To include the effects of finite temperature on these condensates one needs to generalize the GP equation. We report here the development of the Hartree-Fock-Bogoliubov theory with the Popov (HFB-Popov) approximation for trapped twocomponent BECs (TBECs). It is a gapless theory and satisfies the Hugenholtz-Pines theorem. The method is particularly well suited to examine the evolution of the lowlying energy excitation spectra at T = 0 and T 6= 0. Apart from the two Goldstone modes corresponding to each of the species in quasi-1D TBEC, we show that the third Goldstone mode, which emerges at phase-separation due to softening of the Kohn mode, persists to higher interspecies interaction for density profiles where one component is surrounded on both sides by the other component. These are termed as sandwich type density profiles. This is not the case with symmetry-broken density profiles where one species is entirely to the left and the other is entirely to the right which we refer to as side-by-side density profiles. However, the third Goldstone mode which appears at phase-separation gets hardened when the confining potentials have separated trap centers. This hardening increases with the increase in the separation of the trap centers in which the TBECs have been confined. Furthermore, we demonstrate the existence of mode bifurcation near the critical temperature. We also examine the role of thermal fluctuations in quasi-1D TBECs of dilute atomic gases. In particular, we use this theory to probe the impact of non-condensate atoms to the phenomenon of phase-separation in TBECs. We demonstrate that, in comparison to T = 0, there is a suppression in the phase-separation of the binary condensates at T 6= 0. This arises from the interaction of the condensate atoms with the thermal cloud. We also show that, when T 6= 0 it is possible to distinguish the phase-separated case from miscible from the trends in the correlation function. However, this is not the case at T = 0. In a BEC, a soliton enhances the quantum depletion which is sufficient enough to induce dynamical instability of the system. For phase-separated TBECs with a dark soliton in one of the components, two additional Goldstone modes emerge in the excitation spectrum. We demonstrate that when the anomalous mode collides with a higher energy mode it renders the solitonic state oscillatory unstable. We also report soliton induced change in the topology of the density profiles of the TBEC at phase-separation. For quasi-2D BECs, at T = 0, we show that with the transformation of a harmonically to toroidally trapped BECs, the energy of the Kohn modes gets damped. This is examined for the case when the radial angular frequencies of the trap are equal. The other instance, when the condensate is asymmetric, the degeneracy of the modes gets lifted. The variation in the anisotropy parameter is accompanied by the damping of the modes, the quasiparticle modes form distinct family of curves; each member being different from the other by the principal quantum number n. When T 6= 0, with the production of a toroidally trapped BEC, the maxima of the thermal density tends to coincide with the maxima of the condensate density profiles. This is different from the case of a harmonically trapped BEC in which due to the presence of repulsive interaction between the atoms, the thermal density gets depleted where the condensate atoms are the highest.