Inversion in a four-terminal superconducting device on the quartet line. Ⅱ. Quantum dot and Floquet theory

摘要

In this paper, we consider a quantum dot connected to four superconducting terminals biased at opposite voltages on the quartet line. The grounded superconductor contains a loop threaded by the magnetic flux Φ. We provide Keldysh microscopic calculations and physical pictures for the voltage-V dependence of the quartet current. Superconductivity is expected to be stronger at Φ/Φ_0 = 0 than at Φ/Φ_0 = 1/2. However, inversion I_(q,c)(V,0) ＜I_(q,c)(V,1/2) is obtained in the critical current I_(q,c)(V,Φ/Φ_0) on the quartet line in the voltage-V ranges which match avoided crossings in the Floquet spectrum at (V,Φ/Φ_0= 0) but not at (V,1/2). A reduction in I_(q,c) appears in the vicinity of those avoided crossings, where Landau-Zener tunneling produces dynamical quantum mechanical superpositions of the Andreev bound states. In addition, π-0 and 0-π crossovers emerge in the current-phase relations as V is further increased. The voltage-induced π shift is interpreted as originating from the nonequilibrium Floquet populations produced by voltage biasing. The numerical calculations reveal that the inversion is robust against strong Landau-Zener tunneling and many levels in the quantum dot. Our theory provides a simple "Floquet level and population" mechanism for inversion tuned by the bias voltage V, which paves the way towards more realistic models for the recent Harvard group experiment where the inversion is observed.