Three “Universal” Mesoscopic Josephson Effects

Abstract

Abstract. A recent theory is reviewed for the sample-to-sample fluctuations in the critical current of a Josephson junction consisting of a disordered point contact or microbridge. The theory is based on a relation between the supercurrent and the scattering matrix in the normal state. The root-mean-square amplitude rms I c of the critical current I c at zero temperature is given by rms I c ≃ e∆ 0 /h, up to a numerical coefficient of order unity (∆ 0 is the energy gap). This is the superconducting analogue of "Universal Conductance Fluctuations" in the normal state. The theory can also be applied to a ballistic point contact, where it yields the analogue of the quantized conductance, and to a quantum dot, where it describes supercurrent resonances. All three phenomena provide a measurement of the supercurrent unit e∆ 0 /h, and are "universal" through the absence of a dependence on junction parameters.

Linear-scaling source-sink algorithm for simulating time-resolved quantum transport and superconductivity

“…We can go a little bit further and look at the structure of the bound states that carry the supercurrent. They are given by the stationary condition 13,30,31 ,…”

Transfer matrix approach to the persistent current in quantum rings: Application to hybrid normal-superconducting rings

“…Our technique to compute the persistent current through mesoscopic normal/superconducting rings is based on a combination of the approach to the Josephson current across an SNS junction based on the analytical continuation of the quasiparticle S-matrix to the imaginary axis [32][33][34][35] with the transfer matrix (TM) approach to transport in mesoscopic systems 39 , particularly well-suited to account for periodic boundary conditions in quantum rings. The key feature of our approach is that it eventually leads to an exact, closed-form formula for the groundstate energy of the system at a given Φ and, therefore, for the persistent current across the ring.…”

“…Our Eq. (13) is the analog, for a superconducting ring, of the equation giving the Josephson current across an SNS junction in terms of the single-particle S-matrix, analytically continued to the imaginary axis 32,34 . In this respect, it is expected by analogy to have a wide range of applicability, either as it stands, as an exact, closed form formula for I[Φ], or as a starting point to account, for instance, for the effect of disorder (in analogy to what is done in Ref.…”

“…(32,34), an important point to stress is that both matrices are block-factorizable, with the factorization corresponding to the possibility of regarding the system as a sequence of homogeneous regions separated by interfaces. Indeed, the TM for a one-dimensional system comes out to be simply the ordered product of the matrices corresponding to the homogeneous regions and of the ones corresponding to the interfaces, taken in the appropriate sequence.…”

“…In this respect, our method can be regarded as an adapted version of the technique developed in Refs. [32][33][34][35] to compute the dc Josephson current across an SNS-junction in terms of the single-quasiparticle S-matrix of the junction, which has been successfully applied to a number of cases of interest, such as the Josephson current through a chaotic Josephson junction 36 , the critical supercurrent in the quantum spin-Hall effect 37 , and the current across a long Josephson junction 35,38 . In fact, our method combines the idea of deforming the integration path to the imaginary axis with the idea of using the transfer matrix, rather than the S-matrix, to recover the quasiparticle scattering dynamics of the system.…”