Distinguishing phases using the dynamical response of driven-dissipative light-matter systems

We present a peculiar transition triggered by infinitesimal dissipation in the interpolating Dicke-Tavis-Cummings model. The model describes a ubiquitous light-matter setting using a collection of two-level systems interacting with quantum light trapped in an optical cavity. In a previous work [Phys. Rev. Lett. 120, 183603 (2018)], dissipation was shown to extend a normal phase (dark state) into new regions of the model's parameter space. Harnessing Keldysh's action formalism to compute the response function of the light, we show that the normal phase does not merely spread but encompasses a transition between the old and the dissipation-stabilized regimes of the normal phase. This transition, however, solely manifests in the dynamical fluctuations atop the empty cavity through stabilization of an excited state of the closed system. Consequently, we reveal that the fluctuations flip from being particlelike to holelike across this transition. This inversion is also accompanied by the behavior of the Liouvillian eigenvalues akin to exceptional points. Our work forges the way to discovering transitions in a wide variety of driven-dissipative systems and is highly pertinent for current experiments.

Figure 1

Dissipative interpolating Dicke-Tavis-Cummings model [17], cf. Eq. (1). (a) Steady-state phase diagram: normal phases (white) are stable in the bottom-left (NP) and upper-right (e-NP) quadrants. The orange dotted cut passes through the NP and e-NP phases via the superradiant one (gray region) and is used to show the eigenvalue behavior, see Fig. 2. (b) Cavity dynamical response function over the NP (dashed dark-green) and e-NP (light-green), evaluated at ${\lambda }_{1,2}$ in (a), respectively. The response always presents four peaks appearing at paired frequencies (arrows indicate unresolved small peaks). Remarkably, there is a soft-mode peak inversion between the two NPs. The non-Lorentzian shape of the peaks and the zero-response point (marked with $★$) are attributed to a Fano resonance [34]. In all plots, ${\omega }_c={\omega }_a,\kappa /{\omega }_c=0.1,{\lambda }_1=(0.35,0.29),{\lambda }_2=(0.65,0.59)$.

Figure 2

Closed IDTC, cf. Eq. (1). (a) Mean-field energy landscape as a function of the real and imaginary parts of the cavity field $\alpha =〈a〉$ at representative points of the parameter space, cf. Fig. 1. The fluctuation eigenfrequencies ${\omega }_1$ and ${\omega }_2$ are schematically identified as the azimuthal and radial frequencies of fluctuations around the NP, respectively. Across the NP $\to$ e-NP transition, ${\omega }_2$ changes sign and is ill defined at the transition, i.e., at ${\lambda }_c$. (b) Real (solid) and imaginary (dashed) parts of the fluctuation eigenfrequencies [cf. Eqs. (2) and (4)] along the orange dotted cut line in Fig. 1. The eigenfrequencies $\pm {\omega }_1$ (cyan, orange lines) do not show qualitative changes. The eigenfrequencies $\pm {\omega }_2$ (blue, red lines) instead become imaginary at ${\lambda }_c$, signaling the transition $\mathrm{NP}\to \mathrm{SP}$ (shaded region) followed by a return to real values when ${\lambda }_x,{\lambda }_y>{\lambda }_c$, marking $\mathrm{SP}\to \mathrm{e}$-NP. Blue and cyan lines are associated with particlelike fluctuations ($d{s}^2>0$) [cf. Eq. (5)]. Red and orange lines are associated with holelike fluctuations ($d{s}^2<0$).

Figure 3

Dark laser and its response. (a) Schematic energy potentials of particle- and holelike fluctuations of the normal modes in the NP (left) and e-NP (right)[cf. Eqs. (2), (4), and (5)]. Blue (red) wiggly lines refer to blue (red) detuned response processes. Solid (dashed) parabolic potentials with companying ladders of levels indicate normal-mode fluctuations around NP with a positive (negative) norm [cf. Eq. (5)]. Solid arrows mark the associated process, i.e., upward blue (downward red) corresponds to creation of positive (negative) excitations while downward blue (upward red) to annihilation of such excitations. Orange arrows mark the effect of dissipation in reducing the populations of the modes. (b) Top: Real (dashed lines) and imaginary (solid lines) parts of the soft-mode Liouvillian eigenvalues. Bottom: The extracted FWHM of the positive peak. Parameters along the dashed (white) line in (d). (c) Calculated dynamical response along the $\mathrm{NP}\to \mathrm{e}$-NP transition. The positive and negative soft-mode peaks invert alongside a region where they overlap. (d) Dynamical phase diagram of the dissipative IDTC for $\kappa /\omega =0.1$. Regions I, III correspond to the NP, e-NP, respectively. Region II is the region where the imaginary parts of the soft mode coalesce to zero.

Figure 4

Cavity dynamical response function over the SP, ${\lambda }_{SP}=(0.7,0.2)$. The response presents four peaks appearing at paired frequencies (arrow indicates unresolved small peak). Positive (negative) frequencies correspond to positive (negative) peaks, as in the NP [cf. Fig. 1 main body], ${\omega }_c={\omega }_a,\kappa /{\omega }_c=0.1$.