Cavity-induced switching between localized and extended states in a noninteracting Bose-Einstein condensate

We study an ultracold atom-cavity coupling system, which had been implemented in an experiment to display weak light nonlinearity [S. Gupta, K. L. Moore, K. W. Murch, and D. M. Stamper-Kurn, Phys. Rev. Lett. 99, 213601 (2007)]. The model is described by a noninteracting Bose-Einstein condensate contained in a Fabry-Pérot optical resonator, in which two incommensurate standing-wave modes are excited and thus form a quasiperiodic optical lattice potential for the atoms. Special emphasis is paid to the variation of the atomic wave function induced by the cavity light field. We show that bistability between the atomic localized and extended states can be generated under appropriate conditions.

Figure 1

Schematic diagram showing the system under consideration.

Figure 2

Superlattice potential formed by the cavity light field for $\beta =0.5$, $U_t=10$, and $U_0{\left|\alpha \right|}^2=1$. The stripes indicate the first and second Bloch bands of the primary lattice formed by the trap mode. The perturbation introduced by the probe mode is indicated by the dashed line. The label 0(1) indicates the deep (shallow) site in a supercell.

Figure 3

The steady-state intensity of the probe mode ${\left|{\alpha }_{ss}\right|}^2$ vs $\eta /\kappa$ using the two-mode treatment (red-dashed line) and the self-consistent imagninary-time propagation (black-dotted line). The insets show ${\left|\psi \left(x\right)\right|}^2$ (solid line) vs the optical potential (dashed line) inside one period for two different pump amplitudes located at the two sides of the bistable region. The horizontal black-dashed line indicates the value of ${\left|{\alpha }_{ss}\right|}^2$ corresponding to $\Delta /J=2$. Parameters: $N={10}^4$, $\kappa =200{\omega }_R$, ${\delta }_c=0$, $U_0=0.2{\omega }_R$, $U_t=10{\omega }_R$.

Figure 4

(a) Steady-state intensity of the probe mode ${\left|{\alpha }_{ss}\right|}^2$ and (b) inversion $z$ vs cavity-pump detuning ${\delta }_c$. The red-dashed lines are the result from the two-mode treatment while the black-dotted lines result from the self-consistent imaginary time propagation. The parameters are the same as Fig. 3 except that $\eta =120{\omega }_R$.

Figure 5

Evolution of population imbalance $z$ and intensity of the probe mode ${\left|\alpha \right|}^2$ with time $t$ for $\eta /\kappa =0.5$ (upper panel) and $1.5$ (lower panel). The other parameters are the same as those in Fig. 3. The black-solid lines are the results calculated with numerical method (i), and the red-dashed lines are with method (ii). The dash-dotted lines are given by Eq. (8).

Figure 6

(a) Pump amplitude, (b) population imbalance, and (c) probe intensity vs time when the pump amplitude is swept up. The parameters are the same as those in Fig. 3. The black-solid lines indicate the result gotten with method (i) while the red-dashed lines use method (ii).

Figure 7

Same as Fig. 6, except that the pump amplitude is swept down.

Figure 8

Bistability between atomic extended and localized states. (a) Intracavity photon number for the probe mode ${\left|{\alpha }_{ss}\right|}^2$ and (b) inverse participation ratio $P^{-1}$ plotted as function of $\eta /\kappa$. Atomic density (in log scale) and its momentum distribution for the (c), (d) lower bistable branch and the (e), (f) upper bistable branch at $\eta /\kappa =1.5$. The parameters are the same as those in Fig. 3.

Figure 9

Pump amplitude and probe intensity vs time when the pump amplitude is (a), (c) swept up and (b), (d) down. The parameters are the same as those used in Fig. 8.

Figure 10

Evolution of the inverse participation ratio $P^{-1}$ when the pump amplitude is swept up (the uppermost panel). The lower panels show the atomic density (in log scale) (left panel) and its momentum distribution (right panel). The rows marked with A, B, and C correspond to ${\omega }_{R}t=100,250$, and 350, respectively.

Figure 11

Same as Fig. 10, except that the pump amplitude is swept down. Rows A, B, and C correspond to ${\omega }_{R}t=100$, 250, and 400, respectively.