Intensity fluctuations in steady-state superradiance

Alkaline-earth-metal-like atoms with ultranarrow optical transitions enable superradiance in steady state. The emitted light promises to have an unprecedented stability with a linewidth as narrow as a few millihertz. In order to evaluate the potential usefulness of this light source as an ultrastable oscillator in clock and precision metrology applications, it is crucial to understand the noise properties of this device. In this paper, we present a detailed analysis of the intensity fluctuations by means of Monte Carlo simulations and semiclassical approximations. We find that the light exhibits bunching below threshold, is to a good approximation coherent in the superradiant regime, and is chaotic above the second threshold.

Figure 1

Schematic of $N$ two-level atoms (black dots) in a single-mode cavity field. The atoms are being incoherently repumped. The output of the cavity field is monitored by two detectors $D_1$ and $D_2$ with a variable time delay $\tau$ between them.

Figure 2

Time evolution of the intracavity intensity during a single Monte Carlo trajectory (a) for $N=10$ atoms and $w=5{\Gamma }_c$. The black tick marks indicate cavity-decay events. Panel (b) shows the binning of the decay times subsequent to $t_i$, which leads to the histogram $n_{i,j}(\Delta t)$ used in the calculation of $g^{(2)}(\tau )$ (see text for explanation).

Figure 3

Second-order intensity correlation $g^{(2)}(0)$ as a function of the repump rate. The green symbols (solid circles) show the Monte Carlo results including the statistical errors for $N=10$ atoms. The blue solid line shows the analytical result of Eq. (17) for $N=10$ atoms, the purple dashed line is for $N=100$ atoms, and the yellow dashed-dotted line is for $N=1000$ atoms. The gray dashed lines at $g^{(2)}(0)=2$ and $g^{(2)}(0)=1$ are for orientation.

Figure 4

(a) Inversion $\langle {\mathrm{\sigma ̂}}_z^{(1)}\rangle$, (b) spin-spin correlation $\langle {\mathrm{\sigma ̂}}_{+}^{(1)}{\mathrm{\sigma ̂}}_{-}^{(2)}\rangle$, and (c) spin-spin correlation $\langle {\mathrm{\sigma ̂}}_z^{(1)}{\mathrm{\sigma ̂}}_z^{(2)}\rangle -\langle {\mathrm{\sigma ̂}}_z^{(1)}\rangle {}^2$ as a function of pump strength for $N=10$ (blue solid line), $N=100$ (purple dashed line), and $N=1000$ atoms (yellow dashed-dotted line).

Figure 5

Second-order correlation $g^{(2)}(\tau )$ for $N=10$ atoms (a) for the subradiant regime for weak pumping $w=0.25{\Gamma }_c$, (b) for the superradiant regime with $w=5.0{\Gamma }_c$, and (c) for the strong pumping regime with $w=100{\Gamma }_c$. Note the different scales on the time axis for each figure. The purple (solid) line in (c) is the thermal light result.