Strong effects of weak ac driving in short superconducting junctions

We study a short multichannel superconducting junction subject to dc and ac phase biases. The ac modulation changes the occupation of the Andreev bound states formed at the constriction by transitions between bound states and the continuum. In a short junction, the nonequilibrium Andreev bound-state population may relax through processes that conserve parity of the occupation number on the same bound state and processes that do not conserve it. We argue that the parity-conserving processes occur on a much faster time scale. In this case, even a weak driving may lead to a large nonequilibrium quasiparticle population scaling with the number of channels and results in a large deviation of the supercurrent from its equilibrium value. We show that this effect is accompanied by a quasiparticle current which may lead to a measurable charge imbalance in the vicinity of the junction. Furthermore, we study the time evolution of the supercurrent after switching off the ac drive. On a time scale where parity relaxation is negligible, the supercurrent relaxes to a stationary nonequilibrium state. Finally, we briefly outline the regime of ultraweak driving where the ac-induced processes occur on a time scale comparable to that of parity relaxation.

Figure 1

Energy diagram of the multichannel weak link as a function of the stationary phase difference $\varphi$. The collection of red curves indicates the ABS energies below the gap. The blue area represents the continuum of free quasiparticle states above the gap. The arrows show the processes that change the ABS occupations. The ionization process, occurring with rate ${\Gamma }_{\text{I}}$, promotes a quasiparticle in a bound state to the continuum. The refill process (${\Gamma }_{\text{R}})$ breaks a Cooper pair and occupies a bound state while releasing a second quasiparticle into the continuum. In addition to these ac-induced processes, phase fluctuations induce the annihilation of two quasiparticles in the same bound state, the rate of which is ${\Gamma }_{\text{A}}$. Channels with a bound-state energy above the threshold $\Omega -\Delta$ are empty, while states below this threshold can be populated, $n_A^{\text{st}}\ne 0$, due to the ac-induced refill process, ${\Gamma }_{\text{R}}$.

Figure 2

The ionization (blue lines) and refill (red lines) rates as functions of the driving frequency $\Omega$ for the transparencies $T=0.5,0.75,1$ (dotted, dashed, and solid lines, respectively). The phase bias is $\varphi =2\pi /3$.

Figure 3

The supercurrent $I_S$ as a function of the stationary phase $\varphi$ for a diffusive junction (a), (c), (e) and a double junction with $T_c=0.8$ (b), (d), (f). We show the supercurrent for $\delta \varphi /{\varphi }_q=0.1$ (dotted line), $\delta \varphi /{\varphi }_q=1$ (thick dashed line), and $\delta \varphi /{\varphi }_q=10$ (solid line). The equilibrium supercurrent in the absence of an ac drive is shown as the thin dashed line. The driving frequencies are $\Omega =2\Delta$ (a), (b), $\Omega =1.9\Delta$ (c), (d), and $\Omega =1.7\Delta$ (e), (f).

Figure 4

The supercurrent $I_S$ as a function of driving power (a), (c) and as a function of driving frequency (b), (d). Panels (a) and (b) show the results for the diffusive junction, while in (c) and (d) we display the result of the double junction. In (a) and (c) the curves are shown for $\varphi =0.95\pi$ and the driving frequencies $\Omega =1.7\Delta ,1.9\Delta ,2\Delta$ (dotted, dashed, and solid lines, respectively). In (b) and (d) the curves are drawn for the driving power ${\left(\delta \varphi \right)}^2=1000{\left({\varphi }_q\right)}^2$ and for the different phases $\varphi =0.75\pi ,0.85\pi ,0.95\pi$ (dotted, dashed, and solid lines, respectively).

Figure 5

The charge transfer $\dot{q}$ as a function of the stationary phase $\varphi$ for the diffusive junction (a),(c),(e) and for the double junction (b),(d),(f). We see that in the diffusive junction the charge transfer can change its sign when refill processes dominate (see main text). The parameters are the same as in Fig. 3.

Figure 6

The charge transfer $\dot{q}$ in a diffusive junction as a function of the driving power ${\left(\delta \varphi \right)}^2$. Here $\varphi =8\pi /10$ (solid lines) and $\varphi =7\pi /10$ (dashed lines). The driving frequency is $\Omega =2\Delta$. The inset is a magnification of the main figure to show the charge imbalance inversion for weak driving.

Figure 7

The annihilation rate ${\Gamma }_{\text{A}}$ as a function of $T$ for the stationary phases $\varphi =\pi /3,2\pi /3,0.9\pi$ (dotted, dashed, and solid lines, respectively). Note that in the figure the rate is divided by ${sin}^4(\varphi /2)$. As the rate itself scales with this factor, its values for phases $\varphi$ significantly different from $\pi$ are suppressed.

Figure 8

The time evolution of the supercurrent for the diffusive junction. Panels (a) and (c) show the time evolution during the ac pulse, where the supercurrent departs from the equilibrium value $I_S^{\text{eq}}$ and eventually reaches the stationary value for the driven system $I_S^{\text{st}}$ on a time scale $1/\left[\Delta {\left(\delta \varphi \right)}^2\right]$. The equilibrium current $I_S^{\text{eq}}$ and the stationary driven current $I_S^{\text{st}}$ are shown as the upper/lower dashed lines, respectively. The driving frequency is $\Omega =1.9\Delta$. Panels (b) and (d) show the time evolution of the supercurrent once the pulse is switched off, on the time scale of the decay of the doubly occupied ABS. In panels (a) and (b) [(c) and (d)], the stationary phase bias is $\varphi =\pi /3(2\pi /3)$. The supercurrent decays with the respective rates $\langle {\Gamma }_{\text{A}}\rangle =1.22\times {10}^{-3}\Delta {\left({\varphi }_q\right)}^2$ (b) and $\langle {\Gamma }_{\text{A}}\rangle =4.20\times {10}^{-2}\Delta {\left({\varphi }_q\right)}^2$ (d). As the singly occupied ABS does not decay on this time scale, the value of the supercurrent decays to the new stationary value $I_S^{\text{long}}$ in the long-time limit [see Eq. (23)], which is indicated as the dotted line.

Figure 9

The time evolution of the supercurrent [Figs. 8 and 8] after switching off the ac pulse on a logarithmic scale (solid line). For a stationary phase $\varphi =\pi /3$ (a), the time evolution can be described by an exponential decay for short times (thin dashed line) and an algebraic $1/\sqrt{t}$ decay for long times (dotted line). For $\varphi =2\pi /3$ (b), there is an additional time scale appearing due to the decay of the low-$T$ channels, captured by the two-exponential fit (thick dashed line), with $I_S^{\text{long}}-I_S\left(t\right)=a_{\text{fast}}{e}^{-{\Gamma }_{\text{fast}}t}+a_{\text{slow}}{e}^{-{\Gamma }_{\text{slow}}t}$, where ${\Gamma }_{\text{fast}}\approx 1.275\langle {\Gamma }_A\rangle$ and ${\Gamma }_{\text{slow}}\approx 0.205\langle {\Gamma }_A\rangle$.

Figure 10

The time evolution of the supercurrent for the double junction, after the ac pulse $(\Omega =1.9\Delta )$ in a linear scale (a) and a logarithmic scale (b). In (b), both the long-term (dotted line) and short-term (dashed line) asymptotics are exponential decays. The stationary phase bias is $\varphi =2\pi /3$.

Figure 11

The supercurrent $I_S$ as a function of the phase. This figure shows the transition between the weak driving (red) to ultraweak driving (blue) limit. From top to bottom the curves are shown for ${\left(\delta \varphi \right)}^2=\{\frac{1}{64},\frac{1}{9},\frac{1}{3},1,3,9,64\}\times 16\frac{{\Gamma }_{\text{P}}}{\Delta }$. The driving frequency is $\Omega =2\Delta$. As the driving power decreases, the current approaches its equilibrium value (dashed line).