Cavity-immune spectral features in the pulsed superradiant crossover regime

Lasing in the bad cavity regime has promising applications in precision metrology due to the reduced sensitivity to cavity noise originating from cavity length fluctuations. Here we investigate the spectral properties and phase behavior of pulsed lasing on the ${}^{1}S_0\text{−}{}^{3}P_1$ line of ${}^{88}\mathrm{Sr}$ in a mK thermal ensemble, as first described by S. A. Schäffer et al. [Phys. Rev. A 101, 013819 (2020)]. The system operates in a regime where the Doppler-broadened atomic transition linewidth is several times larger than the cavity linewidth. We find that for some detunings of the cavity resonance, the influence of the cavity noise on the peak lasing frequency can be eliminated to first order despite the system not being deep in the bad cavity regime. Experimental results are compared to a model based on a Tavis-Cummings Hamiltonian, which enables us to investigate the interplay between different thermal velocity classes as the underlying mechanism for the reduction in cavity noise. These velocity-dependent dynamics can occur in pulsed lasing and during the turn-on behavior of lasers in the superradiant crossover regime.

Figure 1

(a) Schematic of the experimental system showing a thermal ensemble of strontium atoms partially overlapping with the cavity mode. Coherent pumping prepares the atoms in ${}^{3}P_1$ and allows subsequent emission of a lasing pulse into the cavity mode. Light couples out of both ends of the cavity with the cavity linewidth $\kappa$. At one end the cavity output is overlapped with a reference laser beam, and the beat signal is detected. (b) Level scheme for ${}^{88}\mathrm{Sr}$ showing the cooling and lasing transitions. Wavelength ($\lambda$) and natural transition linewidths ($\gamma$) are indicated.

Figure 2

(a) Beat signal of a lasing pulse at a cavity detuning of 1 MHz. (b) Intensity spectrogram obtained from Fourier transforming the beat signal with a Gaussian window ($\sigma =400$ ns). (c) Lasing intensity spectrum obtained from Fourier transforming with increasing window sizes, starting at $t=0$. ${\mathrm{\Delta }}_{le}=0$ corresponds to the atomic transition frequency. Minima in emitted power are marked by green dashed lines.

Figure 3

Time evolution of the cavity output power (a) and cavity field phase (b) for three simulated laser pulses after a pump pulse (on during the red shaded interval). The three instances have the same ensemble parameters but show random variation. (a) The laser pulses have random delays and are followed by small intensity oscillations when the cavity is on atom resonance. The blue intervals mark every even number oscillation of the pulse with shortest delay. (b) The phase is random after the pump pulse, then evolves and stabilizes as the laser power builds up. For every intensity oscillation the phase flips sign. (c) Calculating $sin\left(\varphi \right)·\sqrt{P_{out}}$ illustrates how a demodulated beat signal looks when measured in experiment. (d) Moving mean of three experimentally measured demodulated beat signals (shaded: raw data). We see qualitative agreement with the simulated behavior.

Figure 4

Time evolution of the cavity output power (a) and cavity field phase (b) for two simulated laser pulses after a pump pulse (on during the red shaded interval) at a cavity detuning of 1 MHz. (a) Oscillations in the output power are pronounced when the cavity is detuned. The blue areas mark every even number oscillation of the pulse with shortest delay. (b) The phase evolution during the laser pulses in a rotating reference frame at ${\omega }_c$. The sloped behavior is related to pulling of the lasing frequency away from the cavity resonance and toward the atomic transition frequency. (c) Calculated and (d) moving mean of experimental demodulated beat signals (shaded: raw data). Purple intervals mark every second oscillation in the experimentally measured output power.

Figure 5

(a) Dynamics of the output power from one side of the cavity in a simulation, for a cavity detuning of 1 MHz. (b) Normalized lasing intensity spectrum as a function of window length in the simulation (starting at $t=0$). As the intensity oscillations become included in the window, the peaks in the spectrum split up into multiple branches, and the dominant frequency is pulled toward the resonance frequency of the stationary atoms. Minima in output power are marked by green dashed lines. (c) The final spectrum at $t=8\mu \mathrm{s}$ (black) compared to relevant line shapes in the system (dashed blue: stationary atom, dash-dotted green: ensemble Doppler profile, red: cavity).

Figure 6

Intensity spectra from many lasing pulses for varying cavity detuning: (a) simulations, (b) experiments. Each column in the images constitutes the spectrum of a single simulated or measured lasing pulse. The spectra are normalized such that the peak of each column equals one. The dotted gray lines indicate the center of gravity of the lasing spectrum, which has a minimal slope of 0.25 at a cavity detuning of $\pm 1.2$ MHz in panel (a). The dashed green lines refer to the simulation in Fig. 7 which illustrates the dynamics in this regime where location of the spectral peak is independent of cavity detuning.

Figure 7

Simulated time evolution of the rate of emission (yellow) and absorption (blue) of cavity photons, depending on the velocity of atoms along the cavity axis. The dynamics are shown for a cavity detuning of 300 kHz. The dotted green line shows where the Doppler shift equals the cavity detuning on the right axis, and the red line shows the spectral peak frequency in comparison, with a window starting at $t=0$.

Figure 8

The sensitivity of the lasing peak frequencies found in experiments and simulations for different cavity detunings. Purple points show experimental frequency stability with error bars indicating $1\sigma$ uncertainty. The curves show simulations with added ${\sigma }_{jitter}$ of 30 kHz (dark gray), 100 kHz (orange), and 300 kHz (light gray). Variations in atom number and pumping efficiency cause additional variations in the peak frequencies found in experiments when compared to simulations. The fluctuations are minimal for a cavity detuning near $+300$ kHz and $-300$ kHz.