Stochastic projected Gross-Pitaevskii equation

We have achieved a full implementation of the stochastic projected Gross-Pitaevskii equation for a three-dimensional trapped Bose gas at finite temperature. Our work advances previous applications of this theory, which have only included growth processes, by implementing number-conserving scattering processes. We evaluate an analytic expression for the coefficient of the scattering term and compare it to that of the growth term in the experimental regime, showing that the two coefficients are comparable in size. We give an overview of the numerical implementation of the deterministic and stochastic terms for the scattering process, and use simulations of a condensate excited into a large amplitude breathing mode oscillation to characterize the importance of scattering and growth processes in an experimentally accessible regime. We find that in such nonequilibrium regimes the scattering can dominate over the growth, leading to qualitatively different system dynamics. In particular, the scattering causes the system to rapidly reach thermal equilibrium without greatly depleting the condensate, suggesting that it provides a highly coherent energy-transfer mechanism.

Figure 1

The different interparticle scattering processes arising from each term of the SPGPE (4), where ${\epsilon }_{\mathrm{cut}}$ specifies the boundary between C and I. The I region is static, parametrized by the temperature ($T$) and chemical potential $\mu$. (a) Nonlinear mixing interaction between modes in the $\mathbf{C}$ region. (b) Growth process in which both energy and number are transferred between C and I. The time-reversed process also occurs, leading to particle loss in the C region. (c) Scattering process in which energy is transferred between C and I.

Figure 2

The growth ($\overline{\gamma }$) and scattering ($\overline{\mathcal{M}}$) coefficients, and their ratio, as a function of $\mu /k_{B}T$. Reservoir parameters $T,\mu ,{\epsilon }_{\mathrm{cut}}$ are determined from the Hatree-Fock parameter estimation scheme from Ref. [18]. The parameters found ensure $T$ can be varied without altering the total atom number $N$, and critical temperature $T_c\left(N\right)$. Along each curve the temperature varies from $T=T_c$ to $T=0.4T_c$, where $T_c$ corresponds to a total atom number of $N=1\times {10}^4$ (black), $N=1\times {10}^5$ (dashed gray), $N=1\times {10}^6$ (gray).

Figure 3

Results of the energetically damped PGPE, comparing the density with the scattering effective potential where $\overline{\mathcal{M}}=0.005$, at representative times given in units of trap cycles ($2\pi /\overline{\omega }$). Column densities are shown in the left column, with arrows indicating the direction of the breathing motion and flow of current at each time. The scattering potential, $V_M$ [light blue (light gray) curve], and a radial slice of density in energy units, the local $\mathbf{C}$-region interaction energy ${u\left|\psi \right|}^2$ [red (black) curve], are shown in the right column. At $t=0$ we show $V_M^A$ (circles), the analytical scattering potential for a Gaussian wave function found from (34). $V_M^{HA}$ (crosses) is (34) found using the harmonic approximation in Eq. (36).

Figure 4

C-field energy per particle as a function of time (in trap cycles), for the scattering SPGPE evolution of an initial Gaussian wave function with $N_{\mathbf{C}}=1\times {10}^4$ atoms. Solid lines: scattering SPGPE evolution including the noise, at temperature of $T=10\hslash \overline{\omega }/k_B$ for a range of $\overline{\mathcal{M}}$. Dashed lines: evolution with the deterministic scattering term, neglecting the noise, for a range of $\overline{\mathcal{M}}$. The initial energy ($E_0$) determined analytically from (32) agrees with the numerical calculation. For the quiet simulations, the ground-state energy agrees with the Gross-Pitaevskii equation ground-state energy labeled $E_G$. For the noisy simulations, the occupation of thermally excited C-region modes raises the equilibrium energy.

Figure 5

$N_{\mathbf{C}}$, $K_{\mathbf{C}}$, and $\langle r^2\rangle$ as a function of time, for the evolution of a Gaussian wave function using the SPGPE [light blue (light gray) solid line], scattering SPGPE [red (black) dashed line], and simple growth SPGPE (black solid line). We show the time evolution for each method until equilibrium has been reached. In the inset we focus on the early time dynamics, over which the SPGPE and scattering SPGPE reach equilibrium at a much faster rate than the simple growth SPGPE. The SPGPE and scattering SPGPE results for $K_{\mathbf{C}}$ and $\langle r^2\rangle$ are virtually indistinguishable.

Figure 6

Condensate fraction found from the Penrose-Onsager criterion, as a function of time for the same system as in Fig. 5. The SPGPE [light blue (light gray) solid line] and scattering SPGPE [red (black) dashed line] simulations are shown in the inset, and the simple growth (black solid line) simulations are shown in the main figure.