Abstract
The stochastic projected Gross-Pitaevskii equation (SPGPE) is a finite-temperature theory of Bose-Einstein condensate dynamics, utilizing a classical-field approach to describing the dynamics of a low-energy coherent subspace in contact with an incoherent thermal reservoir. Interactions with the reservoir are encapsulated in two distinct processes, known as number damping and energy damping. Historically the energy-damping process has received little attention, at least in part due to difficulty in finding a sufficiently efficient and accurate numerical algorithm for solving the SPGPE. Rooney et al. [Phys. Rev. E 89, 013302 (2014)] developed such an algorithm, providing new opportunities to investigate the energy-damping process. In this work we provide an analytic treatment of the energy-damping process for a range of systems using functional Ito calculus.
We perform a linear fluctuation analysis on the SPGPE for a homogeneous system. We find the dispersion relation to be the usual Bogoliubov dispersion with an additional momentum dependent damping rate. The damping rate is a combination of number- and energy-damping terms with distinct momentum dependence, revealing that each damping process is dominant for excitations of different length scales. We also find the spectra of density and phase fluctuations in momentum space, again with distinct momentum dependence associated with the two damping processes. These results demonstrate that there always exists a regime of length scales where one can expect energy-damping to be the dominant process.
We derive stochastic Ehrenfest relations (SER) from the SPGPE using projected functional change of variables. The SER take the form of stochastic differential equations for matter wave moments of the system and include drift and diffusion terms corresponding to the number- and energy-damping processes. We find that for a well-chosen cutoff the SER can often admit approximate analytic solutions. Analytic solutions of the SER for a quasi-1D Thomas-Fermi system show good agreement with simulations of the 1D SPGPE.
Next, we characterise the centre of mass equilibrium properties of a 3D harmonically trapped finite temperature system in the Thomas-Fermi regime using the SER for position, momentum, and particle number. We find that the energy-damping process is generally dominant over the number-damping process when considering position and momentum fluctuations. Considering the stochastic evolution of particle number in this system, we find that the steady-state fluctuation properties depend only on the number-damping process. This suggests that the effects of each reservoir interaction can be measured individually in the same system, allowing for experimental detection of the energy-damping reservoir interaction.
We finish by considering the motion of a singly-charged quantum vortex confined to a disc trap in an energy-damped system. We find a stochastic differential equation for the dissipative motion of the vortex. It is revealed that the presence of noise can cause a stabilizing effect on the vortex, extending its lifetime. We also consider the presence of a rotating thermal cloud, a scenario that leads to the vortex becoming the energetically favourable state. This allows for analysis of steady-state stochastic properties of the vortex motion.