# The return of a hysteretic Josephson junction to the zero-voltage state: I-V characteristic and quantum retrapping

## Abstract

We study the behavior of a hysteretic current-biased Josephson junction in the vicinity of its return to the zero-voltage states, with primary though not exclusive emphasis on the limit of weak damping (βJ≪1), and under the assumption that the zero-point and thermal energies are both small compared to Icφ0 so that fluctuation effects are important only very close to the return point. We consider in detail the resistively shunted junction (RSJ) and quasiparticle-tunneling models, and also make predictions for more general models. Denoting the value of imposed current I at which return to the zero-voltage state would take place in the absence of fluctuations by Ir, we study in particular (a) the dc current-voltage characteristic in the running state for I-Ir≪Ir, and (b) the first-passage-time statistics of the return to the zero-voltage state induced by both classical and quantum fluctuations. With regard to (b), we express our results in the form of a prediction of the width σ of the distribution of retrapping events as a function of imposed current; this prediction extends down to zero temperature and can be compared directly with the experimentally measured widths. Our two principal results are as follows: (a) In the running state, for I≪Ir, the current-voltage characteristic should be given quite generally by the formula (I-I r)/Ir =[(AV0/V)+B]exp-V0/V, where A and B are constants specific to the model, and V0 is a characteristic voltage which for the simplest models is given in the weak-damping limit by V0=ωJφ0+0(β2J) , with ωJ the junction plasma resonance frequency at zero current bias. (b) The square σ2 of the width of the retrapping distribution plotted as a function of I/Ir is given to within logarithmic factors by σ2(T)=const μf(T), where μ≡ℏωJ/Icφ0, the constant is of order 1, and f(T) is a function which tends to 1 as T→0 and is proportional to T in the limit of high T; it is computed explicitly for the RSJ model. We also suggest an explanation (other than lead effects) of the "forbidden voltage regions" which appear to be a characteristic of many high-quality junctions. We discuss the application of our results to the determination of the parameters of Josephson junctions necessary for the investigation of quantum effects on the macroscopic level.