Early stage of superradiance from Bose-Einstein condensates

We investigate the dynamics of matter and optical waves at the early stage of superradiant Rayleigh scattering from Bose-Einstein condensates. Our analysis is within a spatially dependent quantum model which is capable of providing analytic solutions for the operators of interest. The predictions of the present model are compared to the predictions of a closely related mean-field model, and we provide a procedure that allows one to calculate quantum expectation values by averaging over semiclassical solutions. The coherence properties of the outgoing scattered light are also analyzed, and it is shown that the corresponding correlation functions may provide detailed information about the internal dynamics of the system.

Figure 1

Atomic density profiles of the two side modes, according to the quantum model (black solid line) and the MF model (red dashed line), at $\Gamma \tau =6$. The left column shows the weak-pulse regime and the right column the strong-pulse regime.

Figure 2

Evolution of the side-mode populations in the quantum (black solid line) and the MF (red dashed line) model. Note that the quantities are plotted on a log scale. The dot-dashed line marks the time when the snapshots in Fig. 1 were taken.

Figure 3

Comparison of the predictions for the photon density within the BEC in the quantum (black solid line) and the MF (red dashed line) model. The snapshots correspond to $\Gamma \tau =6$, in the weak-pulse ($\Gamma =1$) and the strong-pulse ($\Gamma =100$) regimes.

Figure 4

Comparison of predictions for the number of photons which have left the BEC as a function of scaled time $\Gamma \tau$ in the quantum (black solid line) and the MF (red dashed line) model. (Top) Weak-pulse regime ($\Gamma =1$); (bottom) strong-pulse regime ($\Gamma =100$). The dot-dashed line marks the time when snapshots in Fig. 3 are taken.

Figure 5

Example of photon and atom densities obtained through averaging over randomly seeded MF solutions. (a) Snapshot of the photon density; (b) atomic density of the $(-1)$ mode. The quantum solution is shown as solid black lines, while averages over 20 and 2000 trajectories are shown in red dashed and blue dotted lines. Parameters: $\Gamma =1$ and $\tau =2$.

Figure 6

Time evolution of the ratio of backward to forward-recoiling atoms as a function of scaled time in the quantum (black solid line) and the MF (red dashed line) model for the weak-pulse (a) and the strong-pulse (b) regimes.

Figure 7

Behavior of the photon intensity correlation $g^{(2)}(\tau ,\tau +T)$ as a function of $\tau$ for various fixed delays $T$ as given in the figures. The dashed lines in the top panel give the same function neglecting the $(-1)$ mode.

Figure 8

Integrated correlation function ${\mathrm{g̃}}^{(2)}(T)$ as a function of the delay time $T$ for the weak- (top) and strong- (bottom) pulse regimes. The dashed line in the top figure is the same function neglecting backward-recoiling atoms.