Theory of a Josephson junction detector of non-Gaussian noise

The measurement of higher order cumulants of the current noise generated by a nonlinear mesoscopic conductor using a Josephson junction as on-chip detector is theoretically investigated. The paper addresses the regime where the noise of the mesoscopic conductor initiates activated escape of the Josephson detector out of the zero-voltage state, which can be observed as a voltage rise. It is shown that the deviations from Johnson-Nyquist noise can mostly be accounted for by an effective temperature which depends on the second noise cumulant of the conductor. The deviations from Gaussian statistics lead to rather weak effects and essentially only the third cumulant can be measured exploiting the dependence of the corrections to the rate of escape from the zero-voltage state on the direction of the bias current. These corrections vanish as the bias current approaches the critical current. The theory is based on a description of irreversible processes and fluctuations in terms of state variables and conjugate forces. This approach, going back to work by Onsager and Machlup, is extended to account for non-Gaussian noise, and it is shown that the thermodynamically conjugate force to the electric charge plays a role similar to the counting field introduced in more recent approaches to describe non-Gaussian noise statistics. The theory allows one to obtain exact results for the rate of escape in the weak noise limit for all values of the damping strength of the Josephson detector. Also the feedback of the detector on the noise generating conductor is fully taken into account by treating both coupled mesoscopic devices on an equal footing.

Figure 1

Circuit diagram of a Josephson junction with critical current $I_c$ and capacitance $C$ biased by a voltage source $V$ via a resistor $R$.

Figure 2

Circuit diagram of a Josephson junction with critical current $I_c$ and capacitance $C$ biased in a twofold way. The branch to the right puts an Ohmic resistor $R_B$ in series with the junction and is biased by the voltage $V_B$. The branch to the left is biased by a voltage $V_N$ and $R_N$ is a noise generating nonlinear element, specifically a normal state tunnel junction with tunneling resistance $R_N$.

Figure 3

(Color online) The scaled dimensionless energy $ϵ$ is shown as a function of $\psi$ for $\gamma =0.25$. The energy decreases as the trajectory moves back and forth in the potential $\varsigma \left(\psi \right)$ depicted as a gray line.

Figure 4

(Color online) $W$ is depicted as a function of $\tilde{\gamma }$ (straight line). Also shown are the results of Ref. 17 for vanishing damping, Eq. (120) (dot), and in the strong damping limit, Eq. (121) (dotted line). The approximate result of Ref. 16 is depicted as a dashed line.