Quantum Signatures in Quench from Chaos to Superradiance

The driven-dissipative Dicke model features normal, superradiant, and lasing steady states that may be regular or chaotic. We report quantum signatures of chaos in a quench protocol from the lasing states. Within the framework of a classical mean-field perspective, once quenched, the system relaxes either to the normal or to the superradiant state. Quench from chaos, unlike quench from a regular lasing state, exhibits erratic dependence on control parameters. In the quantum domain, this sensitivity implies an effect that is similar to universal conductance fluctuations.

Figure 1

Quantum fluctuations in QFC. In the classical (mean-field) limit, the outcome of the measurement (blue line) is binary and erratically depends on the parameter that controls the preparation protocol (in our demonstration it is the preparation time $t_{\mathrm{prep}}$). In the semiclassical (truncated Wigner) approximation, this erratic dependence is smoothed away (black line). The measured $⟨Q⟩$ reflects the relative volume of the basin that leads to the $Q=1$ attractor. In the proper quantum treatment, the outcome (red line) manifests fluctuations that arise from interference of trajectories. However, any mesoscopic system eventually relaxes, such that for $t=\infty$ the expectation value $⟨Q⟩$ reflects a thermal equilibrium that does not depend on the initial preparation.

Figure 2

Steady-state phase diagram. The vertical axis is the $\tilde{g}/g$ ratio that reflects coherent pumping. In (a) the horizontal axis is the normalized incoherent collective pumping. We assume $\mathrm{\Omega }=\mathcal{E}=1$ and $g=2$, while $\kappa =2$ and ${\gamma }_c=0.5$. The label ${\mathrm{NS}}^{*}$ indicates a stable all-atom-excited state. The labels “LC” and “Chaos” indicate a regular limit cycle and a chaotic lasing state, respectively. With vanishing dissipation, bistability appears for $\tilde{g}/g\le 0.5$, and the energy landscape has three attractors (NS and two SR fixed points), while with finite dissipation this range is shifted. (b) Dependence of the bistability region on $\kappa$, for $g=2$, while $f_c={\gamma }_c=0$. The symbols are based on numerical analysis, while the lines are based on stability analysis (see Supplemental Material [20]).

Figure 3

Relaxation toward $\mathrm{NS}/\mathrm{SR}$ attractors. we start with all the atoms in the ground state, while $n\sim 0$. (a),(c),(e) $\tilde{g}/g=0.75$, and the relaxation is toward the SR state. (b),(d),(f) $\tilde{g}/g=0.48$, and the relaxation is toward NS-SR bistability. The other parameters are $g=2$, $\kappa =2$, ${\gamma }_c=0.5$, and $f_c=0.04$. In the quantum simulation, we have $N=16$ atoms (meaning $\ell =8$) and use $N_b=80$ truncation for the bosonic mode. The semiclassical results of (a),(b) and the quantum results of (c),(d) are compared in (e),(f). The waiting time up to the measurement is $t=t_m=20$. The solid black line is the semiclassical distribution, while the dashed red line is the quantum distribution. The classical SR fixed points are marked by horizontal dashed lines in (a)–(d) and by arrowheads in (e),(f). Note the $n=0$ peak at (f).

Figure 4

The prepared state. The system is prepared in a nondissipative chaotic state with $g=1$ and $\tilde{g}=0.48$. This is done by launching a coherent state with $s_z=s_x=1/\sqrt{8}$, and $s_y=0$, while $n\approx 0$, followed by a long waiting time $50<t_{\mathrm{prep}}<1000$. In the quantum simulation, we have $N=16$ atoms (meaning $\ell =8$) and use $N_b=80$ truncation for the bosonic mode. (a) The quantum Husimi distribution of the prepared state in the $[\mathrm{Re}(a)-\mathrm{Im}(a)]$ plane at $t=t_{\mathrm{prep}}=50$. On top we display the corresponding cloud of classical points. The latter are color coded based on the postquench outcome: blue for those that belong to the NS basin and red and magenta for those of the SR basins. (b) The associated $s_z=0$ Poincaré section (the $s_y,a>0$ branch) projected on the $(n-s_x)$ plane, with added blue, red, and magenta circles that indicate the attractors. For the quench we assumed $g=2$, but kept the same $\tilde{g}/g$, with dissipation parameters $\kappa =2$ and ${\gamma }_c=0.5$ and with incoherent pumping $f_c=0.04$.

Figure 5

QFC contrasted with nonchaotic dependence. The outcome of a quench versus the control parameter $t_{\mathrm{prep}}$. The preparation assumes dissipation-free dynamics. (a),(c) For a quench from a $g=0.1$ quasiregular; (b),(d) for a quench from $g=1$ chaos. (a),(b) The prequench dynamics of $s_x(t)$. The inset displays the associated classical power spectrum (function of $|\omega |$). (c),(d) The dependence of the quench outcome on $t_{\mathrm{prep}}$. The quench parameters are the same as in Fig. 3, with measurement time $t_m=2$. Solid blue and black lines are the classical and semiclassical, respectively, while the dashed red line is the quantum. (d) The fluctuations that were caricatured in Fig. 1. The solid, dotted, and dash-dotted blue lines in (d) are representative trajectories of the semiclassical cloud in Fig. 4 exhibiting uncorrelated fluctuations.

Figure 6

Suppression of quantum fluctuations. (a) The dependence of $⟨n(t_m)⟩$ on the control parameter $t_{\mathrm{prep}}$ for several values of $t_m$. For $t_m=0$ it is merely the conventional calculation of $⟨n⟩$ versus $t$ for the dissipation-free system. For $t_m=\infty$ (in practice, $t_m=10$) it is formally a measurement of the final equilibrium state. The intermediate value $t_m=2$, which has been used in Fig. 5, reflects the outcome of a realistic measurement protocol. It exhibits the fluctuations that were caricatured in Fig. 1. The inset shows the variance ${\sigma }_n$ of those fluctuations versus $t_m$. The partial correlation between $⟨n(t_m)⟩$ and $⟨n⟩$ is inspected in (b), where the data points (symbols) of (a) are connected by thin lines. The thick lines are based on a semiclassical procedure that is explained in the main text. The departure of the data points from the latter is due to relaxation.