Nonequilibrium Andreev bound states population in short superconducting junctions coupled to a resonator

Inspired by recent experiments, we study a short superconducting junction of length $L\ll \xi$ (coherence length) inserted in a dc-SQUID containing an ancillary Josephson tunnel junction. We evaluate the nonequilibrium occupation of the Andreev bound states (ABS) for the case of a conventional junction and a topological junction, with the latter case of ABS corresponding to a Majorana mode. We take into account small phase fluctuations of the Josephson tunnel junction, acting as a damped LC resonator, and analyze the role of the distribution of the quasiparticles of the continuum assuming that these quasiparticles are in thermal distribution with an effective temperature different from the environmental temperature. We also discuss the effect of strong photon irradiation in the junction leading to a nonequilibrium occupation of the ABS. We systematically compare the occupations of the bound states and the supercurrents carried by these states for conventional and topological junctions.

Figure 1

Model of the system. (a) The dc-SQUID is formed by two junctions. The left junction, having a phase difference $\chi$ between the two superconducting leads, is a Josephson tunnel junction with energy $E_{\mathrm{J}}$. The right junction, having a phase difference $\varphi$, is a short superconducting junction (SSJ). The SQUID is penetrated by a magnetic flux $\phi +\delta \phi \left(t\right)$. (b) For small phase difference fluctuations, the Josephson junction behaves as a damped LC resonator with capacitance $C_{\mathrm{J}}$ and inductance $L_{\mathrm{J}}={\mathrm{\Phi }}_0^2/\left(4{\pi }^2{E}_{\mathrm{J}}\right)$. A resistance $R$ accounts for a finite damping on this resonator.

Figure 2

Possible transitions in the SSJ. For both the topological and the conventional junction, there are four transition rates (solid blue arrows) that change the fermionic parity of the ABS in a conventional (Andreev energy $E_{\mathrm{A}}$) and a topological junction (Andreev energy $E_{\mathrm{M}}$), respectively. The processes with out-rates ${\mathrm{\Gamma }}_{\mathrm{out},i}$ empty the bound state, while the processes with in-rates ${\mathrm{\Gamma }}_{\mathrm{in},i}$ fill the bound state. The index $i=1,2$ labels direct transitions between the discrete state and the continuum with or without involving the ground state. For the conventional junction, there are additional parity-conserving processes with rates ${\mathrm{\Gamma }}_{\mathrm{in}/\mathrm{out},\mathrm{AA}}$ between the ground state $E=0$ and the ABS $E_{\mathrm{A}}$ (dashed red arrows), which are absent for the topological junction since the bound states are nondegenerate.

Figure 3

Sketch of the SSJ of length $L$ formed by two superconductors (S) separated by a small normal (N) region. In S, there is an excitation gap of $2\mathrm{\Delta }$ around the Fermi energy $E=0$, and there is a superconducting phase difference of $\phi$ across the junction, which is controlled by the dc magnetic flux. For energies $E>\mathrm{\Delta }$, we have propagating QPs whose wave functions are obtained by calculation of $s$ scattering states ($s=1,2,3,4$) labeling the incident QP (see main text). In general, each incident QP produces four outgoing QPs due to normal or Andreev reflection.

Figure 4

Stationary nonequilibrium bound-state occupation in SSJ as a function of phase difference $\phi$ and Josephson plasma frequency ${\omega }_0$. (a) MBS occupation $n_{\mathrm{M}}$ in topological junctions. The black solid lines separating regions (I) and (II) is given by $\hslash {\omega }_0=\mathrm{\Delta }+E_{\mathrm{M}}$ and by $\hslash {\omega }_0=\mathrm{\Delta }-E_{\mathrm{M}}$ for the regions (III) and (IV). The black dashed line separating regions (II) and (III) is $\hslash {\omega }_0=\mathrm{\Delta }$. The upper bound for region (I) is given by $\hslash {\omega }_0=2\mathrm{\Delta }+E_{\mathrm{M}}$. (b) ABS occupation $n_{\mathrm{A}}$ for finite transmission $\mathcal{T}=0.99$ in conventional junctions. The black solid line separating regions (I) and (II) is given by $\hslash {\omega }_0=\mathrm{\Delta }+E_{\mathrm{A}}$ and by $\hslash {\omega }_0=\mathrm{\Delta }-E_{\mathrm{A}}$ for the regions (II)/(III) and (IV). The black dashed line separating regions (II) and (III) is given by ${\mathrm{\Gamma }}_{\mathrm{out},\mathrm{AA}}^{\mathrm{res}}\approx 2{\mathrm{\Gamma }}_{\mathrm{in},1}^{\mathrm{res}}$ in the limit ${\mathrm{\Gamma }}_{\mathrm{out},\mathrm{AA}}^{\mathrm{res}},{\mathrm{\Gamma }}_{\mathrm{in},1}^{\mathrm{res}}\gg {\mathrm{\Gamma }}_{\mathrm{out},2}^{\mathrm{res}}$. The upper bound for region (I) is given by $\hslash {\omega }_0=2\mathrm{\Delta }+E_{\mathrm{A}}$. Common parameters: $\gamma =0.001\mathrm{\Delta }$, $k_{\mathrm{B}}{T}_{\mathrm{qp}}=0.1\mathrm{\Delta }$, $T_{\mathrm{env}}=0$.

Figure 5

Sketch of the contributions to the transition rates ${\mathrm{\Gamma }}_{\mathrm{in},1/\mathrm{out},2}^{\mathrm{res}}$ at $T_{\mathrm{env}}=0$, which determine the absolute value of the rates depending on the value of the Josephson plasma frequency ${\omega }_0$. QPs in the continuum with energies $E>\mathrm{\Delta }$ are occupied by the effective function $f_{\mathrm{eff}}\left(E\right)=f\left(E\right)\sqrt{E^2-{\mathrm{\Delta }}^2}/\mathrm{\Delta }$, which is a combination of matrix elements, density of states in a superconductor and the Fermi-Dirac distribution. Most of the continuum QPs are located closely above the gap in an energy range approximately proportional to $T_{\mathrm{qp}}$. The effective spectral function ${\chi }_{\mathrm{eff}}\left(E\right)=\chi \left(E\right)\mathrm{\Delta }/E$ is related to the absorption peak of the resonator shifted to values ${\omega }_0\pm E_{\mathrm{M}}$ according to ${\chi }_{\mathrm{eff}}(E\mp E_{\mathrm{M}})$ for the rate ${\mathrm{\Gamma }}_{\mathrm{in},1/\mathrm{out},2}^{\mathrm{res}}$. The maximal shifts ${\omega }_0\pm \mathrm{\Delta }$ can be achieved at a phase difference $\phi =0$, while the minimal shifts are at $\phi =\pi$. The cases (a), (b), (c), and (d) correspond to the regions (II), (I), (III), and (IV), respectively, defined in Fig. 4.

Figure 6

Supercurrent $I_{\mathrm{M}}$ carried by Majorana bound states in short topological superconducting junctions as a function of phase difference $\phi$ for different Josephson plasma frequencies ${\omega }_0$. Parameters are the same as for Fig. 4. Current is plotted in units of $I_0=e\mathrm{\Delta }/\hslash$.

Figure 7

Supercurrent $I_{\mathrm{A}}$ carried by Andreev bound states in short conventional superconducting junctions as a function of phase difference $\phi$ for Josephson plasma frequencies: (a) $\hslash {\omega }_0\le 1.1\mathrm{\Delta }$ and (b) $\hslash {\omega }_0\ge 1.3\mathrm{\Delta }$. Parameters are the same as for Fig. 4. Current is plotted in units of $I_0=e\mathrm{\Delta }/\hslash$.

Figure 8

Stationary average bound-state occupation in short superconducting junctions as a function of phase difference $\phi$ and microwave frequency $\mathrm{\Omega }$ for different Josephson plasma frequencies ${\omega }_0$. Majorana bound-state occupation $n_{\mathrm{M}}$ at (a) $\hslash {\omega }_0=\mathrm{\Delta }$ and (c) $\hslash {\omega }_0=0.2\mathrm{\Delta }$. Andreev bound-state occupation $n_{\mathrm{A}}$ for $\mathcal{T}=0.99$ and ${\gamma }_{\mathrm{A}}=(2/43)\mathrm{\Delta }$ at (b) $\hslash {\omega }_0=\mathrm{\Delta }$ and (d) $\hslash {\omega }_0=0.2\mathrm{\Delta }$. Common parameters: $\gamma =0.001\mathrm{\Delta }$, $k_{\mathrm{B}}{T}_{\mathrm{qp}}=0.1\mathrm{\Delta }$, $T_{\mathrm{env}}=0$, and ${(\delta \phi /\lambda )}^2={10}^{-5}$.

Figure 9

Supercurrent as a function of phase difference $\phi$ for different microwave frequencies $\mathrm{\Omega }$ in the presence of the resonator with plasma frequency ${\omega }_0=0.2\mathrm{\Delta }/\hslash$. (a) Current $I_{\mathrm{M}}$ carried by Majorana bound states in short topological superconducting junctions. (b) Current $I_{\mathrm{A}}$ carried by Andreev bound states in short conventional superconducting junctions. Parameters are the same as for Figs. 8 and 8, respectively. Current is plotted in units of $I_0=e\mathrm{\Delta }/\hslash$.