Phase diagram for injection locking a superradiant laser

We study experimentally and theoretically the response of a superradiant or bad-cavity laser to an applied coherent drive. We observe two forms of synchronization (injection locking) between the superradiant ensemble and the applied drive: one attractive and one repulsive in nature. In the region of repulsion, the atomic spin state that stores laser coherence undergoes three-dimensional dynamics, as opposed to a two-parameter description of the electric field in a traditional good-cavity laser. We derive a phase diagram of predicted behavior and experimentally measure the response of the system across various trajectories therein.

Figure 1

Experimental setup and level diagram. (a) Atoms interact with both the externally applied drive (gray) and the intracavity field generated by their collective emission (blue and red). The superradiant laser responds primarily at two frequencies, the drive frequency ${\omega }_d$ and a self-lasing frequency ${\omega }_{\ell }$. (b) The characteristic frequencies are displayed in a level diagram, and all lie within one cavity mode of width $\kappa$. The Raman laser system is approximated as a two-level laser incoherently repumped through intermediate optically excited states (not shown) at rate $W$. $W$ is also the primary source of broadening of the lasing transition (shown as broadening of $\left|\downarrow \right.〉$). In this work, the ratio of atomic and optical linewidths is $W/\kappa \approx 5\times {10}^{-2}$ to $5\times {10}^{-3}\ll 1$, placing the system deep into the bad-cavity or superradiant regime. The state $\left|\uparrow \right.〉$ is a dressed state consisting of a ground hyperfine state of Rb coupled nonresonantly to an optically excited state as described in [1, 2, 3, 17]. The applied drive couples $\left|\downarrow \right.〉$ and $\left|\uparrow \right.〉$ with an on-resonance Rabi frequency ${\Omega }_d$.

Figure 2

The predicted phase diagram for the driven superradiant laser is shown in a plane defined by the applied drive strength ${\Omega }_d$ and the drive detuning ${\delta }_d$, normalized to the repumping rate $W$, which is fixed to $NC\gamma /2$ here. The regions are first divided by the number of distinct emission frequencies [(1) or (2)]. Region (2) is further divided by the frequency shift of the self-lasing component at ${\omega }_{\ell }$, which can be attracted (2A) or repelled (2R) from the applied drive frequency ${\omega }_d$. When ${\Omega }_d<0.2\times W$, the laser synchronizes by smoothly coalescing in frequency with the drive (dashed line). For larger drives, the self-lasing component remains distinct and is quenched. The two trajectories (black arrows) refer to the two parameter trajectories explored by the data in Fig. 3.

Figure 3

Experimental observation of coalescing attractive (a) and repulsive (b) synchronizations. (Left) 2D spectrograms are taken with fixed drive strength ${\Omega }_d$ as the detuning of the drive ${\delta }_d$ is varied along the representative vertical trajectories in Fig. 2. Darker colors indicate higher power in a frequency bin (i.e., PSD). The red line indicates the expected self-lasing trajectory in the absence of an applied drive. (Right) Two panels show the frequency shift ${\delta }_{\ell }={\omega }_{\ell }-{\omega }_{{\ell }_0}$ between the lasing frequency and the lasing frequency when no drive is present. In each region, we qualitatively identify attraction and repulsion by the sign of ${\delta }_{\ell }$ and label and color each region similarly to the phase diagram in Fig. 2. The behaviors follow the prediction for their respective trajectories across the phase diagram, which are represented by vertical lines in Fig. 2.

Figure 4

Synchronization and gain saturation. (a) Measured gain and phase response of the superradiant laser at the drive frequency ${\omega }_d$. At large drive detunings, the response displays linear small-signal gain (red fit to perturbative model overlayed). The gain saturates (gray shaded region) at small detunings as the laser approaches the synchronization transition. (b) The same gain and phase response are represented in a phasor picture. Points of small signal gain lie along the straight line, and the region of saturation is approximately described by a curve of maximum stimulated electric field (inner curve). (c) The stimulated output powers $P_s$ and $P_{\ell }$, at the self-lasing frequency ${\omega }_{\ell }$ (red) and at the drive frequency ${\omega }_d$ (blue), respectively, are displayed as the laser is driven across a repulsive synchronization transition at approximately ${\Omega }_d/2\pi \approx 40$ kHz. Theoretical predictions (solid lines) show good agreement with the data.

Figure 5

Quantitative small-signal gain measurements. (a) We measure the total transmitted power at the drive frequency ${\omega }_d$ and define power gain $G$ as the measured transmitted power normalized to the drive power transmitted through the cavity on resonance with no atoms present. For these measurements, ${\delta }_d$ is scanned over a frequency range greater than the cavity linewidth $\kappa$. The data are fit to the model in Eq. (B2) (red line). Note that here, the cavity resonance marked by the vertical solid line is a few MHz higher than the atomic resonance. (b) From fits to data such as (a), we plot the variation of the fitted gain coefficient $\alpha$ (see text) vs $W$. The prediction $\alpha ={\gamma }_{\perp }^2$ is shown in gray. The width of the gray band corresponds to the uncertainty in an independent calibration of the repumping rate $W$.

Figure 6

Bloch vector interpretation of the phase diagram. (a) The types of behavior for the driven superradiant laser can be characterized by a phase diagram. The characteristic rates that determine the lasing behavior are drive Rabi frequency ${\Omega }_d$, detuning ${\delta }_d$, and repumping rate $W$. The regions correspond to the number of distinct emission frequencies (1 or 2) and the frequency shift (attraction or repulsion) of the carrier (A and R, respectively). The behavior of the synchronization (a), attraction (b), and repulsion (c) configurations are shown in a Bloch sphere picture. In the frame of the atomic transition frequency ${\omega }_a$, the drive is represented by a rotation ${\vec{\Omega }}_d$, with an orientation that rotates along the dashed green trajectory at frequency ${\delta }_d$. In the unsynchronized case, this modulates the Bloch vector (red vector), causing drift toward or away from the drive, with the average precession of the Bloch vector indicated in each case via the large blue and red arrows. In the synchronized case, the Bloch vector follows the drive all the way around the sphere.