Switching rates in Kerr resonator with two-photon dissipation and driving

TL;DR

The paper addresses the problem of quantifying switching between metastable states in a driven-dissipative Kerr cavity with two-photon drive and dissipation under finite detuning. It maps the quantum master equation to a Fokker-Planck equation in the complex $P$-representation and applies a multidimensional Kramers (Eyring–Kramers) framework to obtain an analytic expression for the switching rate $\Gamma = B\exp(-\delta\Phi)$, with explicit prefactor $B$ and barrier height $\delta\Phi$, validated against Liouvillian diagonalization. The analysis spans the dissipative limit ($U=0$) and finite Kerr nonlinearity ($U\neq 0$), revealing that detuning can monotonically increase the rate in the former but induce a nonmonotonic dependence with a finite detuning optimum $\Delta_{\text{opt}}$ in the latter; near criticality, Wigner-function arguments corroborate the rate behavior. The results offer practical guidelines for optimizing critical dissipative cat qubits and designing two-photon resonators for scalable bosonic quantum computing architectures, by clarifying how detuning and Kerr nonlinearity shape the activation barrier and tunneling dynamics.

Abstract

We analytically investigate the switching rate in a two-photon driven Kerr oscillator with finite detuning and two-photon dissipation. This system exhibits quantum bistability and supports a logical manifold for a bosonic qubit. Using Kramer's theory together with the $P$-representation, we derive an analytical expression for the bit-flip error rate within the potential-barrier approximation. The agreement is demonstrated between analytical calculations and numerical simulations obtained by diagonalization of the Liouvillian superoperator. In the purely dissipative limit, the switching rate increases monotonically with detuning, as the two metastable states approach each other in phase space. However, the exponential contribution to the bit-flip rate exhibits a nontrivial dependence on system parameters, extending beyond the naive scaling with the average photon number. In the presence of large Kerr nonlinearity, the switching rate becomes a nonmonotonic function of the detuning and reaches a minimum at a finite detuning. This effect arises because detuning lowers the activation barrier for weak nonlinearity but increases it for large ones, ensuring a minimum of the switching-rate at nonzero detuning. These results establish key conditions for optimizing the performance of critical cat qubits and are directly relevant for the design of scalable superconducting bosonic quantum architectures.

Figures (7)

• Figure 1: (a) The phase portrait of the semiclassical Eqs. \ref{['xp']}. Blue arrows indicate the drift. (b) The Wigner function as a function of the photonic quadratures $x$ and $p$. The red dots highlights location of the semi-classical solutions. In this calculation we set $G=6, \Delta=3,\eta=\sqrt{3}/2,U=1/2$.
• Figure 2: The switching rate, $\Gamma$, obtained by «small detuning approximation» \ref{['GammaSmallDetunings']} (green line) and numerical diagonalization of the Liouvillain superoperator (dots), vs the frequency detuning $\Delta$. The nonlinear coupling constant $\kappa_2=\eta+i U=|\kappa_2|\exp(i\theta)$ has a fixed modulus $|\kappa_2|=1$, but different values of the angle $\theta=\arctan(U/\eta)= \pi/2\cdot(0,0.25,0.5,0.66,0.83,0.9)$, as shown in the inset. An increase in Kerr nonlinearity $U$ (decrease in the two-photon dissipation $\eta$) corresponds to a color change of the dots from dark violet to bright orange. The two-photon pump rate is set to $G = 6$, respectively.
• Figure 3: The switching rate, $\Gamma$, obtained by «potential barrier approximation» \ref{['gamma']} (curves) and numerical diagonalization of the Liouvillain superoperator (dots), vs the frequency detuning $\Delta$. The nonlinear coupling constant $\kappa_2=\eta+i U=|\kappa_2|\exp(i\theta)$ has a fixed modulus $|\kappa_2|=1$, but different values of the angle $\theta=\arctan(U/\eta)=\pi/2\cdot(0,0.25,0.5,0.66,0.83,0.9)$, as shown in the inset. An increase in Kerr nonliniarity $U$ (decrease in the two-photon dissipation $\eta$) corresponds to a color change of the curve from dark violet to bright orange. The two-photon pump rate is set to $G = 6$, respectively.
• Figure 4: (a) Schematic dependence of the switching rate $\Gamma$ on detuning $\Delta$, for different values of $U/\eta$. The minimum of the green curve corresponds to $\Delta_{\rm {opt}}>0$. (b) The common logarithm of the switching rate, $\log_{10}\Gamma$, as a function of the frequency detuning $\Delta$ and the ratio $U/\eta$ is calculated using numerical diagonalization of the Liouvillian superoperator. The optimal detuning $\Delta_{\text{opt}}$ vs $U/\eta$, calculated numerically (red curve) and via the analytical expression \ref{['gamma']} (black dashed curve). The two-photon pump rate and modulus of the nonlinear coupling constant are set to $G = 6$, $|\kappa_2|=\sqrt{U^2+\eta^2}=1$, correspondingly.
• Figure 5: (a) The potential barrier height \ref{['Porential']} vs frequency detuning $\Delta$ for different values of the ratio $U/\eta=0$ (blue line), $U/\eta=1.4$ (orange line) and $U/\eta=6.3$ (green line). The two-photon pump rate and the modulus of the nonlinear coupling constant are set to $G = 6$, $|\kappa_2|=\sqrt{U^2+\eta^2}=1$, correspondingly. The dotted line indicates the detuning $\Delta=|\kappa_2|$, separating the region of small detunings ($\Delta<<|\kappa_2|$) from large detunings ($\Delta>>|\kappa_2|$), where the "potential barrier approximation" is valid. (b) The critical ratio $(U/\eta)_c$, defined in Eq. \ref{['CritRatio']}, vs the two-photon pump rate $G$. We consider the modulus of the nonlinear coupling constant $|\kappa_2|=\sqrt{U^2+\eta^2}=1$.