Extended Gross-Pitaevskii description for the dynamics of thermal dilute Bose gases

We introduce a technique to simulate the dynamics of a thermal dilute Bose gas. With the method of the nonequilibrium statistical operator, a correction to the Gross-Pitaevskii equation (GPE) is derived. This modification of the GPE describes spontaneous scattering processes. It regularizes the ultraviolet catastrophe and captures the dynamics of the particles at all energies, including the tail of the momentum distribution. We apply our model to the simulation of an evaporative cooling quench of a two-dimensional Bose gas.

Figure 1

Schematic representation of (a) the standard approaches [1] to describe the thermal Bose gas vs (b) the approach adopted in this work.

Figure 2

(a) Time evolution of the condensate fraction after a sudden removal of the high-energy particles with $k>k_{\mathrm{kn}}$. (b) The final momentum distribution $n\left(k\right)$ in a semilogarithmic scale. (c) $k^{2}n\left(k\right)$ in a linear scale. The separation momentum $k_s$ is indicated by the vertical gray line. For comparison, the analytical result (15) is shown with the black full line for a temperature $T=28gn$; the black dashed line shows the BE distribution that neglects interactions. The red full line shows the classical field limit of Eq. (15), where the red dashed line shows the classical field equipartition. (d) The first-order coherence function. System parameters: size $L_x=L_y=13\xi$, where $\xi =\sqrt{1/\left(gnm\right)}$ is the healing length at $T=0$. The “knife momentum” was $k_{\mathrm{kn}}\xi =9.3$ and the total number of particles was $N=7000$. The simulation was performed on a $128\times 128$ grid, corresponding to a momentum space cutoff of $K=30{\xi }^{-1}$.

Figure 3

The final momentum distribution for a (a) lower and (b) higher value of the separation momentum $k_s$, indicated with the arrow.