Temperature dependence of the density and excitations of dipolar droplets

Droplet states of ultracold gases which are stabilized by fluctuations have recently been observed for dipolar and two component Bose gases. These systems present a novel form of equilibrium where an instability at the mean field level is arrested by higher order correlations making the droplet states sensitive probes of fluctuations. In a recent paper, we argued that thermal fluctuations can play an important role for droplets even at low temperatures where the non-condensed density is much smaller than the condensate density. We used the Hartree-Fock-Bogoliubov theory together with local density approximation for fluctuations to obtain a generalized Gross Pitaevskii (GP) equation and solved it with a Gaussian variational ansatz to show that the transition between the low density and droplet states can be significantly modified by the temperature. In this paper, we first solve the same GP equation numerically with a time splitting spectral method to check the validity of the Gaussian variational ansatz. Our numerical results are in good agreement with the Gaussian ansatz for a large parameter regime and show that the density of the gas is most strongly modified by temperature near the abrupt transition between a pancake shaped cloud and the droplet. For cigar shaped condensates, as in the recent Er experiments, the dependence of the density on temperature remains quite small throughout the smooth transition. We then consider the effect of temperature on the collective oscillation frequencies of the droplet using both a time dependent Gaussian variational ansatz and real time numerical evolution. We find that the oscillation frequencies depend significantly on the temperature close to the transition for the experimentally relevant temperature regime ($\simeq 100$nK).