Phase synchronization inside a superradiant laser

Superradiant lasers may soon achieve state-of-the-art frequency purity, with linewidths of 1 mHz or less. In a superradiant (or bad-cavity) laser, coherence is primarily stored in the atomic gain medium instead of the optical field. This phase storage is characterized by spontaneous quantum synchronization of the optical dipole moments of each atom. To observe this synchronization, we create two independent superradiant atomic ensembles lasing in a single optical cavity and observe the dynamics of phase realignment, collective power enhancement, and steady-state frequency locking. This work introduces superradiant ensembles as a testbed for fundamental study of quantum synchronization as well and informs research on narrow linewidth superradiant lasers.

Figure 1

Experimental diagram and Raman lasing energy levels. (a) Two spatially distinct beams (red, blue) dress an ensemble of laser-cooled atoms inside an optical cavity, defining the two superradiant ensembles $a$ and $b$. Repumping beams (green) are also applied transverse to the cavity. (b) Dressing beams induce Raman decay from $\left|\uparrow \right.〉$ to $\left|\downarrow \right.〉$. Both emitted photon frequencies (wavy lines) are within the linewidth $\kappa$ of a single-cavity mode. The repumping laser returns atoms back to $\left|\uparrow \right.〉$ via single-particle repumping at rate $W$.

Figure 2

Healing of an instantaneous phase error between optical dipoles. (a) Timing diagram and visualization of atomic Bloch vectors. Before time $t=0$ the two dipoles interact and synchronize. At $t=0$, the dressing laser phase ${\alpha }_b$ is jumped by ${90}^{\circ }$. The ensembles' interaction begins to heal the relative phase error. At $t=T_{\text{evol}}$, dressing laser $a$ is turned off (${\mathrm{\Omega }}_a\to 0$) so that only ensemble $b$ radiates into the cavity. The difference $\mathrm{\Delta }\overline{\psi }$ in the phases of the radiated light in the gray windows before $t=0$ and after $t=T_{\text{evol}}$ indicates the change in the optical dipole phase $\mathrm{\Delta }{\varphi }_b=\mathrm{\Delta }\overline{\psi }$. The upper panels provide cartoon visualizations of phasors representing the radiated fields (red for $a$, blue for $b$, purple for the sum) and Bloch vectors. (b) Light phase change $\mathrm{\Delta }\overline{\psi }$ vs evolution time $T_{\text{evol}}$. The solid and open points correspond to experiments with dipole ratios $R_d=(1.5,4.0)$, respectively. Vertical solid and dashed lines show the characteristic time scale of the respective single-atom repumping rates for the two data sets $W^{-1}=(0.77,1.6) \mu \mathrm{s}$ corresponding to (solid, open) data. The solid and dashed curves are the results of a numerical mean-field model for the respective data (red for ensemble $a$, blue for ensemble $b$).

Figure 3

Total power output vs detuning for the data shown in Fig. 4. Vertical dashed lines are at the repumping rate $\pm W/2\pi$. The horizontal dashed line is the predicted maximum synchronized output power.

Figure 4

Spectrograms of light emitted from two superradiant ensembles. The vertical axis is the Fourier frequency of each power spectrum and the horizontal axis is the detuning of the dressing lasers $\delta$. The power (color scale) is normalized to the maximum power across the entire spectrogram. (a) Each power spectrum displayed here represents the mean of five power spectra at each $\delta$. Collective dipoles are roughly balanced with $N_a/N_b=0.6$ and ${\gamma }_a/{\gamma }_b=0.8$. (b) Asymmetric operating conditions: $N_a/N_b=1.1$ and ${\gamma }_a/{\gamma }_b=1.6$.