Transient dynamics of a quantum dot embedded between two superconducting leads and a metallic reservoir

We study time-dependent subgap properties of a quantum dot (QD) embedded between two superconductors and another metallic lead, solving the Heisenberg equations of motion by the Laplace transform technique subject to the initial conditions. Focusing on the response of the system induced by a sudden coupling of the QD to external reservoirs, we analyze the transient currents and their differential conductance. We also derive analytical expressions for measurable quantities and find that they oscillate in time with the frequency governed by the QD coupling to superconducting reservoirs. Such quantum oscillations are damped due to relaxation processes caused by the normal lead, whereas their period is controlled by the phase difference $\varphi$ between the order parameters of superconducting leads (except the case $\varphi =\pi$, when all observables evolve to their stationary values without any oscillations). We also explore time-dependent development of the subgap quasiparticles and find their signatures measurable in the differential conductance. We evaluate (numerically and analytically) three typical time scales, characterizing the initial and large-time stages of the transient dynamics which in the asymptotic limit ($t\to \infty$) drives these subgap quasiparticles to the true Andreev states.

Figure 1

Schematic diagram for a quantum dot coupled with two superconducting leads (S1 and S2) and one normal (metallic) electrode (N).

Figure 2

Time evolution of the QD occupancies $n_{\uparrow }\left(t\right)$, $n_{\downarrow }\left(t\right)$ as a function of the phase difference $\varphi$ for ${\varepsilon }_{\sigma }=0$ (upper panel) and ${\varepsilon }_{\uparrow }=-{\varepsilon }_{\downarrow }=0.5$ (bottom panel). ${\mathrm{\Gamma }}_{S1}={\mathrm{\Gamma }}_{S2}={\mathrm{\Gamma }}_S=1$, ${\mathrm{\Gamma }}_N=0.1$, ${\mu }_N=0$, $|{\mathrm{\Delta }}_1|=|{\mathrm{\Delta }}_2|\to \infty$, $n_{\sigma }\left(0\right)=0$. The QD occupancies satisfy the relation $n_{\sigma }\left(\varphi \right)=n_{\sigma }(\varphi +2\pi )$ and for $t=\infty$ are symmetrical with regard to $\varphi =\pi$.

Figure 3

Energies of the subgap quasiparticles of the proximitized QD, $E_{\alpha \beta }$, (upper panel) and QD occupancies: $n_{\uparrow }$, $n_{\downarrow }$, and $n_{\downarrow }-n_{\uparrow }$ (solid black, broken red, and thick red curves, respectively) as a function of $\varphi$. The occupancies are obtained for $t=10$, $t=50$ (shifted down by 1.5) and for $t=200\mathrm{u}.\mathrm{t}.$ (shifted down by 3.0). The vertical black lines indicate characteristic points for $\varphi =\frac{2\pi }{3}$ and $\varphi =\frac{4\pi }{3}$, and the other parameters are ${\varepsilon }_{\uparrow }=-{\varepsilon }_{\downarrow }=0.5$, ${\mathrm{\Gamma }}_{S1}={\mathrm{\Gamma }}_{S2}={\mathrm{\Gamma }}_S=1$, ${\mathrm{\Gamma }}_N=0.02$, ${\mu }_N=0$.

Figure 4

The real (upper panel) and imaginary (bottom panel) parts of the QD induced pairing $\chi \left(t\right)=\langle c_{\downarrow }\left(t\right)c_{\uparrow }\left(t\right)\rangle$ as a function of time and the phase difference ${\phi }_1-{\phi }_2$ for ${\varepsilon }_{\sigma }=0$, ${\mathrm{\Gamma }}_{S_1}={\mathrm{\Gamma }}_{S_2}={\mathrm{\Gamma }}_S$, ${\mathrm{\Gamma }}_N=0.1$, and $n_{\sigma }\left(0\right)=0$. $\chi \left(t\right)$ satisfies the relation $\chi (t,\varphi )=\chi (t,\varphi +4\pi )$ and for $t=\infty$ is symmetrical about $\varphi =2\pi$. The bold green line for $t=40$ in the upper panel corresponds to the case ${\varepsilon }_{\uparrow }=-{\varepsilon }_{\downarrow }=0.5$.

Figure 5

The time dependent currents flowing between the QD and the normal lead, $j_{N\uparrow }\left(t\right)$ (upper panel), and between the QD and the superconducting lead, $j_{S1\uparrow }\left(t\right)$ (bottom panel), as a function of the phase difference ${\phi }_1-{\phi }_2$. The system parameters are: ${\varepsilon }_{\sigma }=0$, ${\mathrm{\Gamma }}_{S_1}={\mathrm{\Gamma }}_{S_2}={\mathrm{\Gamma }}_S$, ${\mathrm{\Gamma }}_N=0.1$, and $n_{\sigma }\left(0\right)=0$.

Figure 6

Time dependent currents flowing between the QD and the superconducting leads, $j_{S_1\uparrow }\left(t\right)$, $j_{S_2\uparrow }\left(t\right)$ as a function of the phase difference ${\phi }_1-{\phi }_2$ in the presence of the Zeeman splitting ${\varepsilon }_{\uparrow }=-{\varepsilon }_{\downarrow }=0.5$ (upper panel). In the bottom panel the asymptotic spin up currents (solid curves) and corresponding QD occupancies (broken curves) obtained for $t=\infty$ are shown for the Zeeman splitting: ${\varepsilon }_{\uparrow }=-{\varepsilon }_{\downarrow }=0,0.25,0.5,0.75$ and 1.0. The parameters are: ${\mathrm{\Gamma }}_{S_1}={\mathrm{\Gamma }}_{S_2}={\mathrm{\Gamma }}_S$, ${\mathrm{\Gamma }}_N=0.2$, and $n_{\sigma }\left(0\right)=0$.

Figure 7

The current $j_{S_2\uparrow }\left(t\right)$ as a function of the phase difference ${\phi }_1-{\phi }_2$ for different times: $t=5,20,60$ and 200 u.t. (from upper to bottom curves in both panels, respectively). The upper (bottom) panel corresponds to ${\mathrm{\Gamma }}_N=0.1\left(0.02\right)$ and the Zeeman splitting is ${\varepsilon }_{\uparrow }=-{\varepsilon }_{\downarrow }=0.5$, ${\mathrm{\Gamma }}_{S_1}={\mathrm{\Gamma }}_{S_2}={\mathrm{\Gamma }}_S$, and $n_{\sigma }\left(0\right)=0$. The curves for $t=20,60,200$ are shifted down by 0.5, 1.0, 1.5, respectively, for better visualization.

Figure 8

Time-dependent Andreev conductance (in $4e^2/h$ units) as a function of the bias voltage $\mu$ for the phase difference $\varphi ={\phi }_1-{\phi }_2=0$ (upper panel) and for $\varphi =0.85\pi$ (bottom panel). The other system parameters are the same as in Fig. 2 with ${\varepsilon }_{\sigma }=0$.

Figure 9

Positions of the quasiparticle maxima vs time and $\mu$ appearing in the differential conductance $G_{\sigma }(\mu ,t)$ for different ${\mathrm{\Gamma }}_N$ indicated in the legend and for superconducting phase difference $\varphi =0,\pi /2$ and $3\pi /4$, respectively (upper panel). The bottom panel shows the result for ${\mathrm{\Gamma }}_N=0.1$ where different time scales, ${\tau }_1$, ${\tau }_2$, and ${\tau }_f$ are indicated. For negative values of $\mu$ the results are symmetrical. The QD energy levels are: ${\varepsilon }_{\sigma }=0$ and ${\mathrm{\Gamma }}_{S_1}={\mathrm{\Gamma }}_{S_2}={\mathrm{\Gamma }}_S$.

Figure 10

The asymptotic conductance obtained for $t\to \infty$ as a function of $\mu$ and the phase difference $\varphi ={\phi }_1-{\phi }_2$. The contour lines correspond to $G=0.5$ and ${\mathrm{\Gamma }}_{S1}={\mathrm{\Gamma }}_{S2}={\mathrm{\Gamma }}_S$, ${\mathrm{\Gamma }}_N=0.75$.

Figure 11

Time-dependent Andreev conductance $G_{\uparrow }$ (in $4e^2/h$ units) as a function of the bias voltage $\mu$ for the phase difference $\varphi ={\phi }_1-{\phi }_2=0.85\pi$ and the Zeeman splitting ${\varepsilon }_{\uparrow }=-{\varepsilon }_{\downarrow }=0.5$. The upper (bottom) panel corresponds to ${\mathrm{\Gamma }}_N=0.1$ (${\mathrm{\Gamma }}_N=0.02$) and ${\mathrm{\Gamma }}_{S1}={\mathrm{\Gamma }}_{S2}={\mathrm{\Gamma }}_S$.

Figure 12

Phase-dependent Andreev conductance $G_{\uparrow }$ (left panels) and $G_{\downarrow }$ (right panels) (in units of $4e^2/h$) as a function of the bias voltage $\mu$ for $t=10,30$, and 100 u.t. (upper, middle, and bottom panels, respectively) after the quench. Other parameters: ${\varepsilon }_{\uparrow }=1$, ${\varepsilon }_{\downarrow }=0$, ${\mathrm{\Gamma }}_N=0.1$, ${\mathrm{\Gamma }}_{S1}={\mathrm{\Gamma }}_{S2}={\mathrm{\Gamma }}_S$. The lower curves in all panels correspond to $\varphi =0$ and each next upper curve is shifted up by $\frac{2\pi }{33}$, so the upper curves correspond to $\varphi =2\pi$. $E_{+-},E_{++},E_{--},E_{-+}$ are represented by corresponding solid lines in the upper panels: They show the localization of the ABS (for ${\mathrm{\Gamma }}_N=0$) on the $(\mu ,\varphi )$ plane.

Figure 13

Time-dependent differential conductance $G\equiv G_{\sigma }$ (expressed in units of $4e^2/h$) as a function the bias voltage $\mu$ obtained for the correlated quantum dot, assuming $\epsilon =-U/2$ and several values of $U$ as indicated. The phase difference is ${\phi }_1-{\phi }_2=0$ and ${\mathrm{\Gamma }}_N=0.1{\mathrm{\Gamma }}_S$. We assumed the initial QD occupancy $n_{\sigma }(t=0)=0$.