Second-order coherence of superradiance from a Bose-Einstein condensate

We have measured the two-particle correlation function of atoms from a Bose-Einstein condensate participating in a superradiance process, which directly reflects the second-order coherence of the emitted light. We compare this correlation function with that of atoms undergoing stimulated emission. Whereas the stimulated process produces correlations resembling those of a coherent state, we find that superradiance, even in the presence of strong gain, shows a correlation function close to that of a thermal state, just as for ordinary spontaneous emission.

Figure 1

(a) Sketch of the experiment. A 9-G magnetic field $\mathbf{B}$ applied along the $y$ axis defines the quantization axis. The excitation beam propagates with an angle of ${10}^{\circ }$ (not shown) relative to the $x$ axis and its polarization is linear, with the same angle relative to the $z$ axis. After emission, the atoms fall 46 cm to a position-sensitive microchannel plate (MCP). The atom cloud forms a sphere with enhanced occupation of the endfire modes. (b) Atomic level scheme. The atoms, initially in the $2{}^3{S}_1,m=+1$ state, are excited to the $2{}^3{P}_0$ state. From there, they can decay with equal branching ratios to the three sublevels of the ground state. We detect only the atoms which scatter into the $m=0$ state.

Figure 2

Momentum distribution of scattered atoms in the $yz$ plane (containing the emission dipole). Both panels show the distribution in the $yz$ plane, integrated between $k_x=\pm 0.1k_{\mathrm{rec}}$ and summed over 100 shots. See the Supplemental Information for a cut in the $xz$ plane [35]. Left: Excitation laser applied 500 $\mu$s after the trap switch-off. Only the radiation pattern for a $y$-polarized dipole is visible. Right: Excitation laser applied immediately after the trap switch-off. Strong superradiance is visible in the vertical, endfire modes.

Figure 3

Angular distribution of scattered atoms in the $yz$ plane (containing the emission dipole) for different values of the delay $\tau$ before the excitation pulse. From bottom to top: light-gray (green) circles correspond to $\tau =500$ $\mu$s; dark-gray (blue) circles, to $\tau =200\mu$s; and light-gray (red) circles, to $\tau =0\mu$s. Data for $\tau =0$ and 500 $\mu$s are the same as those shown in Fig. 2. Images were integrated along the $x$ axis between $\pm 0.1k_{\mathrm{rec}}$, and only atoms lying inside a shell with inner radius 0.8 $k_{\mathrm{rec}}$ and outer radius 1.2 $k_{\mathrm{rec}}$ were taken into account. The delays $\tau =0$, 200, and 500 $\mu$s correspond to peak densities of $\approx 8$, 2, and $0.4\times {10}^{18}{\mathrm{m}}^{-3}$ and to aspect ratios of 10, 5, and 2.5, respectively. The endfire modes are located at $\pm \pi /2$. The half-width at half-maximum of the highest peak is 0.14 rad. Error bars are shown, denoting the $68\%$ confidence interval.

Figure 4

Correlation functions along the (a) $z$ and (b) $y$ axis for $\tau \approx 0$. Darker (blue) circles correspond to superradiant peaks (defined by $|k_z|>0.95k_{\mathrm{rec}}$). Lighter (orange) circles correspond to atoms from the scattering sphere away from the superradiant peaks (defined by $|k_z|<0.92k_{\mathrm{rec}}$). Solid lines are Gaussian fits constrained to approach unity at large separation. Filled gray circles correspond to a fraction of the initial condensate transferred to the $m=0$ state via a stimulated Raman transfer. The dashed gray line shows unity. Error bars denote the $68\%$ confidence interval.