Kolmogorov Turbulence Defeated by Anderson Localization for a Bose-Einstein Condensate in a Sinai-Oscillator Trap

We study the dynamics of a Bose-Einstein condensate in a Sinai-oscillator trap under a monochromatic driving force. Such a trap is formed by a harmonic potential and a repulsive disk located in the center vicinity corresponding to the first experiments of condensate formation by Ketterle and co-workers in 1995. We allow that the external driving allows us to model the regime of weak wave turbulence with the Kolmogorov energy flow from low to high energies. We show that in a certain regime of weak driving and weak nonlinearity such a turbulent energy flow is defeated by the Anderson localization that leads to localization of energy on low energy modes. This is in a drastic contrast to the random phase approximation leading to energy flow to high modes. A critical threshold is determined above which the turbulent flow to high energies becomes possible. We argue that this phenomenon can be studied with ultracold atoms in magneto-optical traps.

Figure 1

Classical time evolution of average energy $⟨E⟩$ and its standard deviation $\sigma$ for $f=0.4$. The data are obtained from $10^4$ trajectories with random initial conditions at $⟨E⟩=1$ and $\sigma =0.5$. Top panel: $⟨E(t)⟩$ and $\sigma (t)$ are shown by black and red (gray) curves, respectively. Bottom panels show probability distribution of trajectories $\rho (E,t)$ for (a) $t=10$, 50 [blue (black), orange (gray) curves] and (b) $t=500$, 1000 [yellow (gray), violet (black) curves]. Vertical dashed lines in main panels mark snapshot times corresponding to bottom panels.

Figure 2

Time evolution of $M$ (top panel) and energy $E$ (bottom panel) for GPE (2) averaged over time intervals $\mathrm{\Delta }t=1$. The initial state is the ground state of (2) at $\beta =0$, $f=0$ [see Fig. 5(a) in Ref. 28]. Both panels show the cases of $f=0.4$, $\beta =0$ (black solid lines), $f=0.4$, $\beta =1.5$ [red (gray) dotted lines], $f=0.4$, $\beta =5$ [orange(gray) dashed lines], $f=2$, $\beta =0$ [blue (gray) dot-dashed lines].

Figure 3

Top panel shows $M$ as a function of driven force $f$ for linear case ($\beta =0$). Bottom panels show probability distribution ${\rho }_k$, averaged over time interval $\mathrm{\Delta }t=5$, as a function of eigenenergies $E_k$ with $t=10$ in black solid lines, $t=50$ in the red (gray) dashed lines, and $t=250$ in blue (gray) dotted lines. Left, center, and right bottom panels show the cases of $f=0.4$, 2, and 3, respectively [highlighted with orange (gray) circles in top panel].

Figure 4

Same as in bottom panels of Fig. 3 for $f=0.3$, $\beta =1.5$ (left panel); $f=0.5$, $\beta =5$ (center panel); $f=1$, $\beta =5$ (right panel).

Figure 5

Number of modes $M$ shown by color (grayness) in the plane of parameters $f$ and $\beta$ (average is done in the time intervals $100\le t\le 150$ and $250\le t\le 300$ in left and right panels, respectively). The approximate separation of KAM or insulator phase (KAM) and delocalized turbulent or metallic phase (TB) is shown by the white line (3).