Stochastic Dynamics of a Josephson Junction Threshold Detector

We generalize the stochastic path integral formalism by considering Hamiltonian dynamics in the presence of general Markovian noise. Kramers’ solution of the activation rate for escape over a barrier is generalized for non-Gaussian driving noise in both the overdamped and underdamped limit. We apply our general results to a Josephson junction detector measuring the electron counting statistics of a mesoscopic conductor. The activation rate dependence on the third current cumulant includes an additional term originating from the backaction of the measurement circuit.

Figure 1

Josephson junction (JJ) threshold detector: Simplified electrical circuit with the JJ (marked with an $X$) and mesoscopic system inserted. The driving noise of the system creates fluctuations of the center node voltage $V$, that can activate the JJ out of its supercurrent state, into its running dissipative state until it is recaptured.

Figure 2

The normalized potential threshold $\Delta U/E_J$ and dimensionless factors $D_1$ and $D_2$, are plotted versus the normalized current bias $\mathcal{J}={\Phi }_0(J_S+J_B)/E_J$. The inset shows the phase space plot of a trajectory for weak damping (thick line) which leads to the escape, and the separatrix (dashed line). The turning points ${\varphi }_0$ are shown by the dots.