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In this paper, we consider a quantum dot connected to four superconductingterminals biased at opposite voltages on the quartet line. The groundedsuperconductor contains a loop threaded by the magnetic flux $\Phi$. We provideKeldysh microscopic calculations and physical pictures for the voltage-$V$dependence of the quartet current. Superconductivity is expected to be strongerat $\Phi/\Phi_0=0$ 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 whichmatch avoided crossings in the Floquet spectrum at $(V,\Phi/\Phi_0=0)$ but notat $(V,1/2)$. A reduction in $I_{q,c}$ appears in the vicinity of those avoidedcrossings, where Landau-Zener tunneling produces dynamical quantum mechanicalsuperpositions of the Andreev bound states. In addition, $\pi$-$0$ and$0$-$\pi$ cross-overs emerge in the current-phase relations as $V$ is furtherincreased. The voltage-induced $\pi$-shift is interpreted as originating fromthe nonequilibrium Floquet populations produced by voltage biasing. Thenumerical calculations reveal that the inversion is robust against strongLandau-Zener tunneling and many levels in the quantum dot. Our theory providesa simple ``Floquet level and population'' mechanism for inversion tuned by thebias voltage $V$, which paves the way towards more realistic models for therecent Harvard group experiment where the inversion is observed.

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