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

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 $\Phi$. We provide Keldysh microscopic calculations and physical pictures for the voltage-$V$ dependence of the quartet current. Superconductivity is expected to be stronger at $\Phi/\Phi_0=0$ than at $\Phi/\Phi_0=1/2$. However, inversion $I_{q,c}(V,0)<I_{q,c}(V,1/2)$ is obtained in the critical current $I_{q,c}(V,\Phi/\Phi_0)$ on the quartet line in the voltage-$V$ ranges which match avoided crossings in the Floquet spectrum at $(V,\Phi/\Phi_0=0)$ 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, $\pi$-$0$ and $0$-$\pi$ cross-overs emerge in the current-phase relations as $V$ is further increased. The voltage-induced $\pi$-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.