Figure 1

(a) Superradiant emission mapping the collective atomic coherence, represented by a Bloch vector $\vec{J}$ with phase $\varphi$ and equatorial projection $J_{\perp }$, onto the phasor representing the emitted light field defined by a phase $\psi$ and amplitude $A$. (b) The experimental setup. ${}^{87}$Rb atoms are trapped in a 1D optical lattice (dashed, orange) within an optical cavity. Optical repumping light is applied perpendicular to the cavity axis. The emitted light is detected in heterodyne, then demodulated using a direct digital synthesis frequency reference to obtain both field quadratures $I\left(t\right)$ and $Q\left(t\right)$ and calculate the light's amplitude $A$ and phase $\psi$. (c) An energy-level diagram of a superradiant Raman laser. The Raman dressing laser detuned from an intermediate state $|i\rangle$ induces optical decay at rate $\gamma$ between ground hyperfine states $|\uparrow \rangle$ and $|\downarrow \rangle$. The atoms are incoherently repumped back to $|\uparrow \rangle$ at rate $w$. When the dressing and repumping lasers are off, the atoms remain in a superposition of $|\uparrow \rangle$ and $|\downarrow \rangle$, and the quantum phase $\varphi \left(t\right)$ that evolves at $f_{\mathrm{hf}}$. (d) Example data from a Ramsey-like sequence in a superradiant laser. Superradiant emission continuously maps $\varphi \left(t\right)$ onto $\psi \left(t\right)$. We measure a differential light phase $\psi \left(T\right)-\psi \left(0\right)$ over a free evolution time $T$.Reuse & Permissions

Figure 2

(a) Example time trace of the emitted light amplitude with an evolution time $T=0.4$ ms. $A\left(0^{-}\right)$ and $A\left(T^{+}\right)$ are average amplitudes measured in the blue time windows before and after the evolution time. The characteristic rise time $t_{80}$ is the time when the amplitude has returned to 80$\%$ of the steady-state value. (b) Left: The decay of $J_{\perp }$ as estimated by (blue squares) the ratio $A\left(T^{+}\right)/A\left(0^{-}\right)$ and (blue line) a fit to loss of contrast $c\left(t\right)$ of traditional Ramsey microwave spectroscopy. (b) Right: Measured (red circles) and predicted (line) recovery time $t_{80}$. The solid curve is the prediction accounting for the low-pass filter that was applied to the data. The band around the data indicates 1 standard deviation (s.d.) on each side of the data point. We attribute the fluctuations observed at short times to finite measurement precision. Each point is the average of 20 trials. (c) Standard deviation of the phase accumulated in time $T$ as estimated by (diamonds) superradiant mapping and (circle) population mapping. The lines are predictions based on the observed loss of contrast $c\left(T\right)$.Reuse & Permissions

Figure 3

Demonstration of nondemolition superradiant measurements enabling a hybrid active-passive oscillator for two different duty cycles shown in (a) and (b). The measured light phase $\psi \left(t\right)$ is shown when the superradiance is switched on (black). The periods of evolution time, when the superradiance is switched off, are gray regions (random phase data not shown). The blue and red exponential curves in (a) correspond to the ideal optimal estimate weight functions before and after an evolution period, respectively, calculated with our experimental parameters of $N$, ${\Gamma }_c$, and $q$. (b) The emission amplitude (red) returns to the steady-state value (dashed line) during active oscillation, reflecting restoration of the coherence $J_{\perp }$.Reuse & Permissions