Quasiparticle trapping, Andreev level population dynamics, and charge imbalance in superconducting weak links

We present a comprehensive theoretical framework for the Andreev bound-state population dynamics in superconducting weak links. Contrary to previous works, our approach takes into account the generated nonequilibrium distribution of the continuum quasiparticle states in a self-consistent way. As application of our theory, we show that the coupling of the superconducting contact to environmental phase fluctuations induces a charge imbalance of the continuum quasiparticle population. This imbalance is due to the breaking of the left-right symmetry in the rates connecting continuum quasiparticles and the Andreev bound-state system, and causes a quasiparticle current on top of the Josephson current in a ring geometry. We evaluate the phase dependence of the quasiparticle current for realistic choices of the model parameters. Our theory also allows one to analyze the quantum coherent evolution of the system from an arbitrary initial state.

Figure 1

Schematic illustration of the SQUID geometry considered in this work, where the SAC is embedded into a ring containing a conventional Josephson junction that generates environmental phase fluctuations. The Andreev bound states are depicted in the right panel. For details, see text.

Figure 2

Schematic illustration of the rate equation dynamics (see text). Direct transitions (solid arrows) connect the Andreev level ground state, ${|-\rangle }_A$, to the excited state ${|+\rangle }_A$. Transitions to the two degenerate odd-parity states (dashed arrows), ${|0\rangle }_A$ and ${|2\rangle }_A$, are mediated through quasiparticle continuum states with energy $\left|E\right|\ge \Delta$, which we indicate by a blue box.

Figure 3

Self-consistent solution of Eq. (3.28) for the steady-state Andreev level occupation probabilities in a transparent SAC, $\mathcal{T}=1$. Here, $P_{\pm }$ refers to the even-parity Andreev levels, with ${|-\rangle }_A$ being the ground state, and $P_0=P_2$ to the degenerate pair of odd-parity states. These results have been obtained for plasma frequency $\Omega =0.5\Delta$, quasiparticle-photon coupling $\lambda =\sqrt{E_C/4\Omega }=0.1$, environmental temperature $T_{\mathrm{env}}=0.2\Delta$, quasiparticle temperature $T_{\mathrm{qp}}=0.2\Delta$, channel length $L={\xi }_0$, Ohmic damping constant ${\eta }_d=0.01\Delta$, and ${\tau }_{\mathrm{qp}}\Delta ={10}^5$ (weak quasiparticle relaxation). The inset shows the case $\Omega =0.2\Delta$, $T_{\mathrm{env}}=0.5\Delta$, and $T_{\mathrm{qp}}=0.01\Delta$, where all other parameters are as in the main panel.

Figure 4

Transition rates ${\Gamma }_{\mathrm{in}}$ and ${\Gamma }_{\mathrm{out}}$ (in units of $\Delta /\hslash$) vs $E_A/\Delta$ on a semilogarithmic scale. ${\Gamma }_{\mathrm{in}}$ describes the rate for entering the odd-parity sector, and ${\Gamma }_{\mathrm{out}}$ is the decay rate of odd-parity states. Parameters are as in Fig. 3. The inset shows the rates for parameters as in the inset of Fig. 3.

Figure 5

Main: Induced quasiparticle current $I_{\mathrm{qp}}$ (in units of $e\Delta /\hslash$) vs $E_A/\Delta$ for varying ${\tau }_{\mathrm{qp}}\Delta ={10}^5,{10}^4$ and ${10}^3$ from bottom to top; other parameters are as in the main panel of Fig. 3. Insets: Continuum quasiparticle distributions $n_{L,R}\left(E\right)$ vs $E/\Delta$ for two $E_A/\Delta$ values and ${\tau }_{\mathrm{qp}}\Delta ={10}^5$. For $E<0$, the distribution functions follow by using the electron-hole symmetry relation $n_R(-E)=1-n_L\left(E\right)$. Dotted curves indicate the corresponding equilibrium Fermi distributions.

Figure 6

Main: Quasiparticle current $I_{\mathrm{qp}}$ (in $e\Delta /\hslash )$ and accumulated charge $Q_{\mathrm{qp}}$ (in units of e) vs $E_A/\Delta$ for the parameters in the inset of Fig. 3, i.e., $\Omega =0.2\Delta ,T_{\mathrm{env}}=0.5\Delta$ and $T_{\mathrm{qp}}=0.01\Delta$. The insets show the continuum quasiparticle distributions, $n_{R,L}\left(E\right)$, for two different $E_A$ values. In contrast to the case studied in Fig. 5, the induced quasiparticle current is now significant for the whole $E_A$ range, and exhibits a sign change for $E_A\simeq \Omega$.