Droplet states of ultracold gases which are stabilized by fluctuations have recently been observed in dipolar and two-component Bose gases. These systems present an alternate 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 noncondensed density is much smaller than the condensate density. We used the Hartree-Fock-Bogoliubov theory together with the 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 ($\simeq 100\mathrm{nK}$).

中文翻译：

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偶极液滴密度和激发的温度依赖性

最近在偶极和两组分玻色气体中观察到通过波动稳定的超冷气体的液滴状态。这些系统提供了另一种形式的平衡，其中平均场水平的不稳定性被高阶相关性所抑制，从而使液滴状态成为波动的敏感探测器。在最近的一篇论文中，我们认为，即使在非凝结密度远小于凝结物密度的低温下，热涨落也可能对液滴起重要作用。我们使用Hartree-Fock-Bogoliubov理论以及波动的局部密度近似来获得广义的Gross-Pitaevskii（GP）方程，并用高斯变率ansatz对其进行求解，以表明低密度和液滴状态之间的过渡可以温度明显改变。在本文中，我们首先使用时间分裂谱方法对相同的GP方程进行数值求解，以检验高斯变率ansatz的有效性。我们的数值结果与高斯ansatz在较大的参数范围内非常吻合，并且表明气体的密度受煎饼形云与液滴之间突然转变附近的温度影响最大。对于雪茄状的冷凝物，如最近的Er实验所示，在整个平稳过渡过程中，密度对温度的依赖性仍然很小。然后，我们使用时间相关的高斯变率ansatz和实时数值演化来考虑温度对液滴集体振荡频率的影响。我们发现，振荡频率在很大程度上取决于与实验相关的温度范围接近转变的温度（$\simeq 100\mathrm{nK}$）。