 Switching-current distribution of a Josephson junction

A switching-current distribution of a Josephson junction records the bias electric current at the first detected transition to a sustained nonzero electric potential difference. Interpret it using a stipulated monotone ramp $dI/dt=r>0$, a reproducible initial trapped state, and an instantaneous escape hazard function $\Gamma (I)$. Neglect retrapping and readout delay. The survival function $S$ obeys $dS/dt=-\Gamma S$, so changing variable from time to electric current givesHere $p$ is a probability density function per unit electric current, not a rate per unit time. Division by the surviving population is essential: only trials still trapped can escape. A typical switching current is consequently not identical to the noiseless superconducting critical current.

For an independent illustrative check, choose constant ${\Gamma }_0=1000{\mathrm{s}}^{-1}$ and $r=1\mu \mathrm{A}{\mathrm{s}}^{-1}$. ThenHalf the trials have switched after $\ln 2/{\Gamma }_0=0.693147\mathrm{ms}$, corresponding to $0.693147\mathrm{nA}$ of ramp. Doubling $r$ doubles the mean increment without changing the underlying escape rate. Real junction rates vary sharply with bias; this example isolates the statistical conversion.

Repeating the preparation, measuring the ramp, resolving the voltage onset, and allowing cooling after each dissipative event are prerequisites for this interpretation. Missed events, finite detection delay, changing temperature and retrapping alter the measured distribution. Even a correctly reconstructed rate needs the independent thermal and spectral checks before identifying its escape mechanism.