Semiclassical theory of energy diffusive escape in a Duffing oscillator

Motivated by recent experimental progress to readout quantum bits implemented in superconducting circuits via the phenomenon of dynamical bifurcation, transitions between steady orbits in a driven anharmonic oscillator, the Duffing oscillator, are analyzed. In the regime of weak dissipation a consistent diffusion equation in the semiclassical limit is derived to capture the intimate relation between finite tunneling and reflection and bath induced quantum fluctuations. From the corresponding steady-state distribution an analytical expression for the switching probability is obtained. It is shown that a reduction of the transition rate due to finite reflection at the phase-space barrier is overcompensated by an increase due to environmental quantum fluctuations that are specific for diffusion processes over dynamical barriers. The scaling behavior of the rate is discussed and it is revealed that close to the bifurcation threshold the escape dynamics enters an overdamped domain such that the quantum-mechanical energy scale associated with friction even exceeds the thermal energy scale.

Figure 1

(Color online) The Hamiltonian function [from Eq. (6)] in the rotating frame for $\alpha =1/27$. The energy is scaled with ${\omega }_{d}M\delta \omega L^2$. The minimum (m) and the maximum (a) in the figure are the stable sates with low and high amplitudes, respectively, separated by a marginal state (b).

Figure 2

The energies $E_m$ (solid), $E_b$ (dotted), and $E_a$ (dashed) of the minimum, of the marginal state, and of the maximum as functions of $\alpha$. The barrier height $V_b\equiv E_b-E_m$ is maximal for $\alpha =0$ and vanishes for $\alpha =4/27$. The energies are scaled with $L^2{\omega }_{d}M\delta \omega$.

Figure 3

Escape rate [Eq. (55)] normalized to the classical rate as a function of the bifurcation parameter $\alpha$ for $\hslash {\omega }_d\beta /\left(2\pi \right)=0.01$ (solid), $\hslash {\omega }_d\beta /\left(2\pi \right)=0.05$ (dashed), and $\hslash {\omega }_d\beta /\left(2\pi \right)=0.1$ (dotted). For all the lines, we use $\beta L^2{\omega }_{d}M\delta \omega =40$, $\delta \omega =0.1{\omega }_0$, and the dimensionless friction constant $\beta \tilde{\gamma }M{L}^2\delta \omega =0.1$.

Figure 4

Escape rate [Eq. (60)] normalized to the classical rate as a function of $\alpha$ for $\hslash {\omega }_d\beta /\left(2\pi \right)=0.01$ (solid), $\hslash {\omega }_d\beta /\left(2\pi \right)=0.05$ (dashed), and $\hslash {\omega }_d\beta /\left(2\pi \right)=0.1$ (dotted). For all the lines, we use $\delta \omega =0.1{\omega }_0$ and $\beta L^2{\omega }_{d}M\delta \omega =40$.

Figure 5

The dimensionless barrier height $V_b/\left(L^2{\omega }_{d}M\delta \omega \right)$ (solid line), the well frequency ${\omega }_m/{\omega }_d$ (dotted line), and the barrier frequency ${\omega }_b/{\omega }_d$ (dashed line) as functions of the bifurcation parameter $\alpha$ for $\delta \omega /{\omega }_d=0.1$.

Figure 6

Typical behavior of the effective friction ${\gamma }_b$. For sufficiently large $\alpha$, where ${\gamma }_b<{\omega }_b,\kappa /\hslash$, the rate expressions [Eq. (63)] for weak friction, and its extension to somewhat larger friction [Eq. (66)] apply. In the range ${\omega }_b<{\gamma }_b<\kappa /\hslash$ the dynamics of the escape process in the barrier range follows a classical Smoluchowski equation, which turns into a quantum Smoluchowski equation for ${\gamma }_b>{\omega }_b,\kappa /\hslash$.