Qubit decoherence in dissipative two-photon resonator: real-time instantons and Wigner function

TL;DR

This paper analyzes quantum bistability and decoherence in a detuned, two-photon driven-dissipative cavity with two-photon loss. It derives the exact stationary Wigner function by exploiting a hidden time-reversal symmetry, and develops a weak-dissipation (WKB) phase-space potential Φ that yields a compact steady-state description via $W_0 \approx e^{-\Phi}$. By connecting the Wigner formulation to the real-time Keldysh instanton framework, it shows that the quantum field along the instanton trajectory equals the gradient of the effective potential, and obtains a closed-form expression for the decoherence rate: $\ln \Gamma \approx -\frac{2\sqrt{G^2-\Delta^2}}{\eta} + \frac{2\Delta}{\eta}\arctan\left(\frac{\sqrt{G^2-\Delta^2}}{\Delta}\right)$, with special limits at $\Delta=0$ and near $\Delta=G$. This work unifies steady-state phase-space structure with dynamical quantum activation, providing a coherent framework for quantum metastability in driven-dissipative nonlinear resonators and informing the design of bosonic qubits and cat-code architectures.

Abstract

We study the quantum dynamics of a single bosonic cavity subject to two-photon driving and two-photon dissipation in the presence of finite detuning. Exploiting a hidden time-reversal symmetry, the Wigner representation and the WKB method, we introduce an effective phase-space potential for description of the steady state. It reveals two attracting points, which are metastable due to quantum fluctuations. By employing the Keldysh real-time path integral formalism, we compute the instanton trajectory governing the quantum activation process between these attractors and establish a fundamental connection with the Wigner representation. This relation unifies the steady-state phase-space description with dynamical quantum activation processes. We also derive an analytical expression for the decoherence rate of the system. Our work provides a coherent theoretical framework for analyzing quantum bistability, metastability, and decoherence in driven-dissipative nonlinear resonators, with direct implications for the design of bosonic qubits and quantum information processing.

Figures (4)

• Figure 1: (a,b) The minus logarithm of the stationary Wigner function, $-\ln(W_0)$, vs the photonic quadratures $x=\sqrt{2}\rm{Re}(\alpha)$ and $p=\sqrt{2}\rm{Im}(\alpha)$. It is obtained from (a) the exact solution \ref{['PsiExact']} and (b) the WKB method \ref{['WKBFinal']}. The red dots highlights location of the semi-classical fixed point \ref{['semicl']} . The system's parameters are set to $G=10, \Delta=7,\eta=1$.
• Figure 2: (a,b) The Wigner distribution vs the photonic quadratures $x=\sqrt{2}\rm{Re}(\alpha)$ and $p=\sqrt{2}\rm{Im}(\alpha)$. It is obtained from (a) the exact solution \ref{['PsiExact']}-\ref{['Decomp']} and (b) the effective potential approximation \ref{['Phi']}. The red dots highlights location of the semi-classical fixed point \ref{['semicl']} . The white lines shows the brunch-cuts of the effective potential \ref{['Phi']}. The system's parameters are set to $G=10, \Delta=7,\eta=1$.
• Figure 3: The phase portrait of the system in the classical subspace where the quantum field $\chi$ is zero. The instanton trajectory consists of two parts colored blue and red. The blue line schematically illustrates the instanton trajectory, starting from the classical fixed point and ending at the saddle point. Along this trajectory the system explores a quantum four-dimensional phase space where quantum field has nonzero value. The red line shows how the system descends from the saddle point to a second attractive point in the classical subspace.
• Figure 4: The dependence of the logarithm of the switching rate on the frequency detuning $\Delta$, plotted for different values of two-photon pump: $G=7$ (green curve), $G=6$ (orange curve), $G=5$ (blue curve), and for the two-photon pumping rate parameter $\eta=1$.