Andreev Bound-State Dynamics in Quantum-Dot Josephson Junctions: A Washing Out of the $0\text{−}\pi$ Transition

We consider a Josephson junction formed by a quantum dot connected to two bulk superconductors in the presence of Coulomb interaction and coupling to both an electromagnetic environment and a finite density of electronic quasiparticles. In the limit of a large superconducting gap we obtain a Born-Markov description of the relevant Andreev bound-states dynamics. We calculate the current-phase relation and we find that the experimentally unavoidable presence of quasiparticles can dramatically modify the $0\text{−}\pi$ standard transition picture. We show that photon-assisted quasiparticle absorption allows the dynamic switching from the 0 to the $\pi$ state and vice versa, washing out the $0\text{−}\pi$ transition predicted by purely thermodynamic arguments. Spectroscopic signatures of Andreev bound-states broadening are investigated by considering microwave irradiation.

Figure 1

(a) Representation of the Josephson junction formed, for instance, by a carbon nanotube quantum dot bridging two superconductors. (b) Schematics of the transitions between the Andreev bound states induced by incoherent ${\mathrm{\Gamma }}_{ij}$ and coherent $\mathrm{\Omega }$ perturbations.

Figure 2

Average current $⟨I⟩$ as a function of Coulomb repulsion $U$. The black dashed curve results from the thermodynamic arguments in the absence of driving. The full red (dashed blue) curve is the numerical solution of the Floquet master equation in the presence (absence) of ac driving. The red dash-dotted curve is the outcome of the optical Bloch equations in the RWA approximation. Inset: the corresponding plain (dashed) numerical curves for the populations ${\rho }_{++},{\rho }_{--}$, and $2{\rho }_{\sigma \sigma }$ of the Andreev bound states in the presence (absence) of ac driving. The parameters describing the Josephson junction are common to both plots: $\mathrm{\Delta }/\hslash \mathrm{\Gamma }=10.0,{\epsilon }_0/\hslash \mathrm{\Gamma }=-1.5,{\epsilon }_1/\hslash \mathrm{\Gamma }=0.25,\varphi =\pi /2,\omega /\mathrm{\Gamma }=2.5,k_B{T}_{\text{EM}}/\mathrm{\Delta }=0,k_B{T}_{\text{qp}}/\mathrm{\Delta }=1/20$, and $\gamma {\mathrm{\Gamma }}^2=1.4\times {10}^{-4}$.