Nonequilibrium Bose-Einstein condensation

We investigate formation of Bose-Einstein condensates under nonequilibrium conditions using numerical simulations of the three-dimensional Gross-Pitaevskii equation. For this, we set initial random weakly nonlinear excitations and the forcing at high wave numbers and study propagation of the turbulent spectrum toward the low wave numbers. Our primary goal is to compare the results for the evolving spectrum with the previous results obtained for the kinetic equation of weak wave turbulence. We demonstrate existence of a regime for which good agreement with the wave turbulence results is found in terms of the main features of the previously discussed self-similar solution. In particular, we find a reasonable agreement with the low-frequency and the high-frequency power-law asymptotics of the evolving solution, including the anomalous power-law exponent $x^{*}\approx 1.24$ for the three-dimensional wave action spectrum. We also study the regimes of very weak turbulence when the evolution is affected by the discreteness of the Fourier space, and the strong turbulence regime when the emerging condensate modifies the wave dynamics and leads to formation of strongly nonlinear filamentary vortices.

Figure 1

Temporal evolution of (a) nonlinearity parameter $\eta =E_4/E_2$ [see Eqs. (4), (5), and (13) for the definitions], (b) the condensate fraction $M_0/M$.

Figure 2

Temporal evolution of spectra $n^{1\mathrm{D}}\left(k\right)$ vs $k$ at early stages of our numerical simulations. Plots (a)–(c) correspond to Run 1 ($0\le t{c}^{*}/{\xi }^{*}\le 1.93\times {10}^3$), Run 2 ($0\le t{c}^{*}/{\xi }^{*}\le 2.54\times {10}^3$), and Run 3 ($0\le t{c}^{*}/{\xi }^{*}\le 2.0\times {10}^3$), respectively. The vertical dotted line indicates the position of $k\xi \left(t\right)\sim 1$ at the latest moment of time. The various slopes shown here by dashed lines represent power-law fits in the interval $(1/\xi ,k_{f1})$ at the latest time. The dot-dashed line in (a) shows the $k^2$ scaling at small wave numbers.

Figure 3

Temporal evolution of spectra $n^{1\mathrm{D}}\left(k\right)$ at later stages of our numerical simulations. Plots (a)–(c) correspond to Run 1 ($2.03\times {10}^3\le t{c}^{*}/{\xi }^{*}\le 8.95\times {10}^3$), Run 2 ($3.05\times {10}^3\le t{c}^{*}/{\xi }^{*}\le 8.80\times {10}^3$), and Run 3 ($2.08\times {10}^3\le t{c}^{*}/{\xi }^{*}\le 3.64\times {10}^4$), respectively. The vertical dotted line indicates the position of $k\xi \left(t\right)\sim 1$. The various slopes shown here by dashed lines indicate power-law fits to the spectra at different times.

Figure 4

Plots of spatiotemporal spectra for Run 1 computed over the time interval $1.14\times {10}^3\le T{c}^{*}/{\xi }^{*}\le 1.37\times {10}^3$. (a) The full spectrum; (a1) and (a2) zoom at small and large wave numbers, respectively. The forcing range is $k_f{\xi }^{*}\in (6.3830,6.4321)$. During the above time interval $n^{1\mathrm{D}}\left(k\right)\sim k^{-0.58}$, see Fig. 2.

Figure 5

Plots of spatiotemporal spectra for Run 2. (a) ST spectra computed over the time-interval $2.45\times {10}^3\le T{c}^{*}/{\xi }^{*}\le 2.67\times {10}^3$. (a1) and (a2) show a zoom at small and large wave numbers, respectively. During the above time-interval $n^{1\mathrm{D}}\left(k\right)\sim k^{-0.8}$, see Fig. 2. (b) ST spectra computed over the time interval $6.52\times {10}^3\le T{c}^{*}/{\xi }^{*}\le 6.75\times {10}^3$; (b1) and (b2) show a zoom at small and large wave numbers, respectively. The forcing range is $k_f{\xi }^{*}\in (6.3830,6.4321)$.

Figure 6

Plots of spatiotemporal spectra for Run 3. (a) ST spectra computed over the time-interval $1.30\times {10}^3\le T{c}^{*}/{\xi }^{*}\le 2.14\times {10}^3$; (a1) and (a2) show a zoom at small and large wave numbers, respectively. During the above time-interval $n^{1\mathrm{D}}\left(k\right)\sim k^{-0.52}$, see Fig. 2. (b) ST spectra computed over the time-interval $2.80\times {10}^4\le T{c}^{*}/{\xi }^{*}\le 2.89\times {10}^4$; (b1) and (b2) show a zoom at small and large wave numbers, respectively. The forcing range is $k_f{\xi }^{*}\in (1.5990,1.6113)$.

Figure 7

Spectral broadening ${\delta }_{\omega }$ vs $k$ obtained from the spatiotemporal spectra for Runs: (a) $\mathrm{Run}1(1.14\times {10}^3\le t{c}^{*}/{\xi }^{*}\le 1.37\times {10}^3)$; (b) $\mathrm{Run}2(2.45\times {10}^3\le t{c}^{*}/{\xi }^{*}\le 2.67\times {10}^3)$; and (c) $\mathrm{Run}3(1.30\times {10}^3\le t{c}^{*}/{\xi }^{*}\le 2.14\times {10}^3)$. The vertical dotted line indicates the position of $k\xi \left(t\right)\sim 1$ at time: (a) $t{c}^{*}/{\xi }^{*}=1.3\times {10}^3$, (b) $t{c}^{*}/{\xi }^{*}=2.54\times {10}^3$, and (c) $t{c}^{*}/{\xi }^{*}=2.0\times {10}^3$. Shaded area indicates the approximate region over which power-law scaling are observed in Fig. 2. The dashed-dot lines represent the curve $\alpha k^2$, whereas solid lines represent $\mathrm{\Delta }\omega =2\alpha k$.

Figure 8

Isosurface plots of the condensate density for the runs: (a) Run 1 at $t{c}^{*}/{\xi }^{*}=8.5\times {10}^3$ for $\langle \rho \rangle /{\rho }^{*}=0.055$; (b) Run 2 at $t{c}^{*}/{\xi }^{*}=6.75\times {10}^3$ for $\langle \rho \rangle /{\rho }^{*}=0.023$; (c) Run 3 at $t{c}^{*}/{\xi }^{*}=2.89\times {10}^4\langle \rho \rangle /{\rho }^{*}=0.49$. Visualization is for the entire simulation domain $L^3={\left(2\pi \right)}^3$.

Figure 9

Temporal evolution of total density [top panel (a)-(c)] and different energy components [bottom panel (d)-(f)] for the runs Run 1, Run 2, and Run 3. $E_{\mathrm{kin}}^i$ and $E_{\mathrm{kin}}^c$ denote the incompressible and compressible components of the kinetic energy, respectively. $E_{\mathrm{q}}$ and $E_{\mathrm{int}}$ denote the quantum pressure energy and the interaction energy, respectively.

Figure 10

Temporal evolution of the healing length for Run 3.