Excitable quantum systems: the bosonic avalanche laser

Abstract

We investigate the dynamics of a lasing system driven by a current of bosonic (quasi-)particles via a dissipative three-mode mixing process. A semi-classical analysis of this system predicts distinct dynamical regimes, where both the cavity mode and the gain medium can undergo lasing transitions. Of particular interest is an intermediate self-pulsing phase that exhibits the characteristics of an excitable system and converts random input signals into separated, quasi-periodic pulses at the output. By performing exact Monte-Carlo simulations, we extend this analysis into the quantum regime and show that despite being dominated by huge bosonic particle number fluctuations, this effect -- reminiscent of coherence resonance -- survives even for rather low average photon numbers. Our system thus represents an intriguing model of an excitable quantum many-body system, with practical relevance for quantum detectors or autonomous quantum machines. As an illustration, we discuss the realization of this system with superconducting quantum circuits and its application as a number-resolved avalanche detector for microwave photons.

Figures (12)

• Figure 1: Sketch of a bosonic avalanche laser. Bosonic (quasi-) particles are injected randomly into the system with rate $\gamma_{\textsl{g}}$ and transition down a ladder of $N\gg1$ equidistant energy levels by emitting a photon of frequency $\omega_c$ into the lasing cavity at each step. In turn, a high cavity occupation number accelerates the dissipative bosonic current, which, under certain conditions, can produce a semi-regular periodic signal at the output. See text for more details.
• Figure 2: Mean-field phase diagram of the bosonic avalanche laser, as obtained from the solution of Eq. \ref{['eq:MF_casc']} (we used an initial seed amplitude $\alpha_c(t=0)=\sqrt{10}$ here, but the results are independent of the specific initial value). The different phases are distinguished by their characteristic dynamics, shown in the panels on the left for $\gamma_{\textsl{g}}/\Gamma=2,12,40$ and $\kappa_c/\Gamma=20$, and by the order parameters $\alpha_c$ and $n_{\rm stag}$ in the steady state. Here, $|\alpha_c|>0$ indicates lasing of the cavity mode and $n_{\rm stag}\neq0$ condensation of the ladder modes garbe_bosonic_2024 into a staggered density pattern, which is shown in the panel below for $\gamma_{\textsl{g}}/\Gamma=5$ and $\kappa_c/\Gamma=40$. In the self-pulsing phase (orange), no stationary state is reached. The other parameters used in these plots are $\kappa_\ell/\Gamma=10$, $\zeta=1/2$ and $N=10$.
• Figure 3: Origin of self-pulsing. The panel on the bottom right shows the time evolution of the mode occupation numbers $n_p$ over a few cycles for $\kappa_c/\Gamma=\kappa_\ell/\Gamma=20$ and $\gamma_{\textsl{g}}/\Gamma=10$. The corresponding population of the cavity mode, $n_c$, is shown on top. In each cycle, the system evolves through three distinct phases, which are illustrated by the corresponding sketches on the left. The inset shows the rescaled pulsing period $\tau$ obtained from mean-field simulations. The various symbols represent different combinations of $\kappa_c/\kappa_\ell=5,25$ and $\Gamma/\kappa_\ell=0.05,5$, keeping $\kappa_\ell$ fixed. Additional parameter combinations are shown in Appendix \ref{['sec:appendix_period']}. The colors correspond to different values of $N=10$ (red) and $N=20$ (blue). Upon rescaling, all curves collapse to the same universal behavior. For all plots, $\zeta=1/2$.
• Figure 4: (a) Typical stochastic trajectories of the cavity photon number $n_c$ (same parameters as the mean-field results shown in Fig. \ref{['fig:phasediag']}). (b) Plot of the noise spectrum for different values of $\gamma_{\textsl{g}}$, which characterizes both the average pumping rate and the strength of the intrinsic bosonic shot-noise. (c) The coherence parameter $\beta$ defined in Eq. \ref{['eq:beta']} is plotted as a function of $\gamma_{\textsl{g}}/\kappa_c$, where the stars represent the numerically evaluated values and the solid line is a guide to the eye. A maximum of $\beta$ is found around $\gamma_{\textsl{g}}/\kappa_c\approx 1$. For all plots, we have assumed $N=10$, $\kappa_\ell/\Gamma=20$, and $\zeta=1/2$.
• Figure 5: Inter-spike interval statistics. (a) Illustration of the fitting procedure used to identify pulses in the output of the lasing cavity. The actual trajectories obtained from Monte-Carlo simulations (blue lines) are overlapped with the fitted Lorentzian curves (red lines), whose maxima lie at times $t_{\rm b}^{(i)}$. For these examples, we have assumed the ratios $\gamma_{\rm g}/\kappa_c=0.3,0.6,1$ (from top to bottom). (b) Histogram of the intervals $\tau_i$ between successive bursts, for the same three $\gamma_{\rm g}/\kappa_c$ values. (c) Plot of the ratio between the mean burst interval and its standard deviation as a function of the pump strength. All other parameters in the plots are the same as in Fig. \ref{['fig:SCR']}.