Dissipative quantum dynamics, phase transitions, and non-Hermitian random matrices

We explore the connections between dissipative quantum phase transitions and non-Hermitian random matrix theory. For this, we work in the framework of the dissipative Dicke model which is archetypal of symmetry-breaking phase transitions in open quantum systems. We establish that the Liouvillian describing the quantum dynamics exhibits distinct spectral features of integrable and chaotic character on the two sides of the critical point. We follow the distribution of the spacings of the complex Liouvillian eigenvalues across the critical point. In the normal and superradiant phases, the distributions are two-dimensional Poisson and that of the Ginibre unitary random matrix ensemble, respectively. Our results are corroborated by computing a recently introduced complex-plane generalization of the consecutive level-spacing ratio distribution. Our approach can be readily adapted for classifying the nature of quantum dynamics across dissipative critical points in other open quantum systems.

Figure 1

Scatter plot of the complex spectrum of the Liouvillian $\mathcal{L}$ of the dissipative Dicke model for $S=10$ and ${\omega }_{\mathrm{c}}={\omega }_{\mathrm{s}}=\kappa =1$ for which ${\lambda }^{*}=1/\sqrt{2}$. (a) Normal phase, $\lambda =0.2$. (b) Superradiant phase, $\lambda =1.0$. A stark difference in the structure of the spectrum above and below the critical point can be observed.

Figure 2

Level-spacing distribution of the complex spectrum of the Liouvillian $\mathcal{L}$ in (a) the normal phase with $\lambda =0.2$ and (b) the superradiant phase with $\lambda =1.0$. We find remarkable agreement with the 2D Poisson distribution $p_{2\mathrm{D}\text{-P}}\left(s\right)$ given in Eq. (3) and that of the GinUE RMT prediction $p_{\text{GinUE}}\left(s\right)$ given in Eq. (4) in the normal phase and in the superradiant phase, respectively.

Figure 3

The metric $\eta$ defined in Eq. (6) as we increase the coupling $\lambda$ from the normal phase to the superradiant phase. It shows the crossover of the complex-eigenvalue spacing distribution from integrable ($\eta \sim 0$) to RMT ($\eta \sim 1$) predictions. The crossover sharpens as we increase the system size. At $\lambda =0$ the dissipative cavity decouples from the spin. Hence, the spectrum is expected to display pathological statistics away from any universal behavior. This explains the observed discrepancies close to $\lambda =0$.

Figure 4

Scatter plot of the complex level-spacing ratio $z$ introduced in Eq. (7) for $S=10$ (a) in the normal phase, $\lambda =0.2$, and (b) in the superradiant phase, $\lambda =1.0$. The table gives $\langle cos\theta \rangle$ and $\langle r\rangle$ for a range of $\lambda$ values, along with their prediction from the 2D Poisson distribution and GinUE RMT.

Figure 5

Scatter plot of the Liouvillian spectrum of the dissipative Dicke model for different $n_{\mathrm{cutoff}}=20$, 30, and 40 in (a) the normal phase, $\lambda =0.2$, and (b) the superradiant phase, $\lambda =1.0$ ($S=10$).

Figure 6

$\langle cos\theta \rangle$ and $\langle r\rangle$ versus $n_{\mathrm{cutoff}}$, computed from the consecutive complex level-spacing ratio distribution of $z$ introduced in Eq. (7) ($S=10$).