Driven superconducting proximity effect in interacting quantum dots

We present a theory of nonequilibrium superconducting proximity effect in an interacting quantum dot induced by a time-dependent tunnel coupling between the dot and a superconducting lead. The proximity effect, which is established when the driving frequency fulfills a gate-voltage-dependent resonance condition, can be probed through the tunneling current into a weakly coupled normal lead. Furthermore, we propose to generate and manipulate coherent superpositions of quantum-dot states with electron numbers differing by two by applying pulsed oscillatory variations to the couplings between the dot and superconductors.

Figure 1

Schematic setups. The oscillation in the effective coupling between dot and superconductor can be realized by (a) a time-dependent phase difference between the superconductors or (b) varying the tunnel-coupling strength between the dot and superconductor.

Figure 2

Pair amplitude strength ${\Delta }_{\text{QD}}$ as a function of oscillation frequency $\Omega$ for different detunings $\delta$. The parameters are ${\Gamma }_{S1}=1k_{B}T$, ${\Gamma }_{S2}=3k_{B}T$, and $U=10k_{B}T$. Strong pairing is seen around the resonance points $\Omega \sim \pm \delta$. The apparent discontinuity at $\Omega =0$ is due to the failure of the rotating-wave approximation at small frequencies.

Figure 3

Normal-lead current as a function of frequency $\Omega$ for different values of $\delta$. The parameters are ${\Gamma }_{S1}=1k_{B}T$, ${\Gamma }_{S2}=3k_{B}T$, and $U=10k_{B}T$. The current shows resonance peaks at $\Omega =+\delta$ and $-\delta$ with widths ${\Gamma }_{S1}$ and ${\Gamma }_{S2}$, respectively.

Figure 4

The time-averaged current in the normal lead as a function of pulse duration $t_{\mathrm{on}}$. The inset shows the real-time variations of the dot charge when the driving is switched on and off periodically with durations $t_{\mathrm{on}}=0.45\pi /{\Gamma }_N$ and $t_{\mathrm{off}}=5/{\Gamma }_N$, respectively. The values of the parameters used for this figure are $U=10k_{B}T$, $\Omega =\delta =50k_{B}T$, ${\Gamma }_{S1}={\Gamma }_{S2}=2k_{B}T$, and ${\Gamma }_N=0.2k_{B}T$.

Figure 5

Real and imaginary components of the pairing strength ${\Delta }_{\text{QD}}$ as a function of frequency $\Omega$ for various values of tunnel couplings ${\Gamma }_N$ and ${\Gamma }_{S1}$. The inset of the lower panel shows the dependence of the imaginary part of ${\Delta }_{\text{QD}}$ on the relative value of the tunnel couplings ${\Gamma }_N/{\Gamma }_{S1}$ assuming $\Omega =\delta$ and ${\Gamma }_{S1}=0.1k_{B}T$. The values of parameters used in this figure are $U=10k_{B}T$ and $\delta =2U$. Here, the pairing strength around $\Omega \sim \delta$ is shown, and for negative values $\Omega \sim -\delta$ the plots can be obtained by mirror reflections with respect to 0 and with widths determined by ${\Gamma }_{S2}$.