Supercurrent and Andreev bound state dynamics in superconducting quantum point contacts under microwave irradiation

We present an extensive theoretical analysis of the supercurrent of a superconducting point contact of arbitrary transparency in the presence of a microwave field. Our paper is mainly based on two different approaches: a two-level model that describes the dynamics of the Andreev bound states in these systems and a fully microscopic method based on the Keldysh–Green function technique. This combination provides both a deep insight into the physics of irradiated Josephson junctions and quantitative predictions for arbitrary range of parameters. The main predictions of our analysis are: (i) for weak fields and low temperatures, the microwaves can induce transitions between the Andreev states, leading to a large suppression of the supercurrent at certain values of the phase; (ii) at strong fields, the current-phase relation is strongly distorted and the corresponding critical current does not follow a simple Bessel-function-like behavior; and (iii) at finite temperatures, the microwave field can enhance the critical current by means of transitions connecting the continuum of states outside the gap region and the Andreev states inside the gap. Our paper is of relevance for a large variety of superconducting weak links as well as for the proposals of using the Andreev bound states of a point contact for quantum computing applications.

Figure 1

The current-phase relation in the adiabatic approximation for (a) $\tau =0.2$ and (b) $\tau =0.95$. The different curves correspond to different values of $\alpha$ as indicated in the graphs. The current is given in units of $I_0=e\Delta /\hslash$, where $\Delta$ is the value of the superconducting gap at $T=0$. (c) The zero-temperature critical current as a function of $\alpha$ for three different values of the transmission $\tau$. Notice that critical current is normalized by its value in the absence of microwaves.

Figure 2

(a)–(c) Zero-temperature supercurrent, in units of $I_0=e\Delta /\hslash$, as a function of the phase for $\tau =0.95$, $\alpha =0.15$ and three different values of the microwave frequency, as indicated in the upper part of the graphs. The solid lines correspond to the numerical results obtained with the two-level model, while the dashed lines is the result obtained with the adiabatic approximation.(d)–(f) Time-averaged-occupation of the upper ABS for the cases shown in the upper panels.

Figure 3

(a) CPR in the two-level model, obtained from Eq. (33) (solid line) and numerics based on Eq. (19) (dotted line). Parameters are $\omega =0.6\Delta$, $\tau =0.95$, and $\alpha =0.05$. The $n=3$ resonance is not included in the analytical approximation. Inset: close-up of the second resonance. (b) Time-averaged population ${\overline{p}}_{+}$ of the upper Andreev state for the same parameters.

Figure 4

Four examples of the zero-temperature CPR for $\tau =0.95$ obtained from the microscopic model (solid lines), the two-level model (dashed lines), and the adiabatic approximation (dotted lines). The parameters characterizing the microwave field are indicated in the different panels.

Figure 5

(a) The CPR for $\alpha =0.1$, $\hslash \omega =0.6\Delta$, and two values of the transmission coefficient, $\tau =0.8$ and $\tau =0.6$. (b) The CPR for $\hslash \omega =0.3\Delta$, $\tau =0.95$, and two values of $\alpha$, 0.2 and 0.6. In both panels the solid lines correspond to the microscopic theory and the dashed lines to the two-level model.

Figure 6

The zero-temperature CPR for $\hslash \omega =0.3\Delta$ and two values of the transmission: (a) $\tau =0.95$ and (b) $\tau =0.8$. The different curves correspond to different values of $\alpha$, as indicated in the graphs.

Figure 7

The CPR for $\hslash \omega =0.6\Delta$, $\alpha =0.1$, $\tau =0.95$, and two different temperatures: (a) $k_{B}T=0.2\Delta$ and (b) $k_{B}T=0.3\Delta$. The solid lines in both panels correspond to the results of the microscopic theory and the dashed lines to the supercurrent in the absence of microwaves ($\alpha =0$).

Figure 8

The local DOS at the contact in the absence of microwaves, as defined in Eq. (52), as a function of energy for $\tau =0.95$, $\varphi =3\pi /4$, and $\eta ={10}^{-3}\Delta$. The lower arrows represent the microwave-induced transitions between the continuum part of the spectrum and the ABSs which are responsible for the supercurrent enhancement at finite temperatures. The upper arrow indicates the resonant transition between the ABSs, which suppresses the supercurrent.

Figure 9

The dc Josephson current as a function of the frequency $\omega$ of the microwave field for a fixed value of the phase $\varphi =\pi /2$, and $\alpha =0.1$, $\tau =0.95$, $k_{\mathrm{B}}T=0.4\Delta$, and $\eta ={10}^{-3}\Delta$. The solid line shows the exact numerical result while the dashed line shows the result obtained from Eq. (53). The dotted line shows the value of the current in the absence of the microwave field.

Figure 10

The critical current as a function of $\alpha$ for $\hslash \omega =0.6\Delta$. The different curves correspond to different values of the temperature and the transmission as indicated in the graphs. The solid lines correspond to the exact results, while the dashed lines show the results of the adiabatic approximation. In the three panels the critical current has been normalized by $e\Delta \left(T\right)/\hslash$, where $\Delta \left(T\right)$ is the gap at the corresponding temperature.

Figure 11

The zero-temperature CPR for a point contact consisting of three channels with transmissions $\tau =0.17,0.6,0.97$, for $\alpha =0.1$ and $\hslash \omega =0.6\Delta$. The dashed lines show the contribution of each channel, while the solid line corresponds to the total current.