Exceptional phase transition in a single Kerr-cat qubit

Abstract

Exceptional points in non-Hermitian quantum systems give rise to novel genuine quantum phenomena. Recent explorations of exceptional-point-induced quantum phase transitions have extended from discrete-variable to continuous-variable-encoded quantum systems. However, quantum phase transitions driven by Liouvillian exceptional points (LEPs) in continuous-variable platforms remain largely unexplored. Here, we construct and investigate a Liouvillian exceptional structure based on a driven-dissipative Kerr-cat qubit. Through numerical simulations, we reveal a quantum phase transition occurring at the LEP characterized by a sudden change in dynamical behavior from underdamped oscillations to overdamped relaxations as visualized via Wigner functions and Bloch sphere trajectories. Notably the negativity of the Wigner function serves as a direct signature of genuine quantum coherence unattainable in conventional single-qubit non-Hermitian systems. Furthermore, we introduce the phase difference between the off-diagonal elements of the Liouvillian eigenmatrices as a novel parameter to quantify the transition. Our results establish the Kerr-cat qubit as a novel continuous-variable setting for exploring dissipative quantum criticality and intrinsic non-Hermitian physics.

Figures (5)

• Figure 1: Eigenspectrum of the Liouvillian matrix $\mathcal{L}_{\mathrm{matrix}}$. Panels (a) and (b) depict the real and imaginary parts, respectively, with colors denoting different eigenvalues (excluding the steady-state eigenvalue $E_1=0$). The solid green curve in (a) indicates LEP2s in the $\Delta$–$\alpha$ parameter space at fixed single-photon loss $\kappa$.
• Figure 2: Time evolution of (a) $\langle X\rangle$ and (b) $\langle Y\rangle$ under different detunings $\Delta$. For $|\Delta| > \Delta_{\text{LEP2}}$, damped oscillations with exponential envelopes are observed. When $|\Delta| < \Delta_{\text{LEP2}}$, the system is overdamped without oscillation. At the critical point $|\Delta| = \Delta_{\text{LEP2}}$, critical damping occurs, leading to the fastest convergence to the steady state. The parameters are $\kappa/2\pi=10$ kHz, $K/2\pi=6.7$ MHz, $P/2\pi=15.5$ MHz and $\kappa_{\phi}=0$.
• Figure 3: The phase difference ($\varphi$) between the off-diagonal elements of $\rho_3$ (solid blue line) and $\rho_4$ (dashed orange line).
• Figure 4: Bloch representation of quantum state evolution and the corresponding Wigner functions. Evolution trajectories for (a) $|\Delta|=3\Delta_{\text{LEP2}}$ and (b) $|\Delta|=0.5\Delta_{\text{LEP2}}$. All trajectories are confined to the XY-plane. The inset at the upper right shows a top-down view of the trajectory, with its color corresponding to the evolution time. (a) For $|\Delta|>\Delta_{\text{LEP2}}$, the trajectory spirals inward toward the origin (steady state). (b) For $|\Delta|<\Delta_{\text{LEP2}}$, the trajectory converges monotonically toward the origin without spiraling. Panels (c) and (d) show the corresponding Wigner functions at different evolution times. (c) When $|\Delta|>\Delta_{\text{LEP2}}$, interference fringes periodically emerge and vanish before reaching steady state. (d) When $|\Delta|<\Delta_{\text{LEP2}}$, interference fringes appear only transiently before each steady state is achieved. The parameters are $\kappa/2\pi=10$ kHz, $K/2\pi=6.7$ MHz, $P/2\pi=15.5$ MHz and $\kappa_{\phi}=0$.
• Figure 5: Validation of the Liouvillian dynamics. Color represents the fidelity $F(\rho_{\text{H}},\rho_{\text{L}})$ between states evolved under the Lindblad master equation of Eq. (\ref{['master eq']}) and the Liouvillian approximation of Eq. (\ref{['Liouvillian dynamics']}). Here, we set $K/2\pi = 6.7$ MHz, $P/2\pi = 15.5$ MHz, $\kappa/2\pi = 10$ kHz, and $\kappa_{\phi}=0$.