Role of Quasiparticles in an Electric Circuit with Josephson Junctions

Although Josephson junctions can be viewed as highly nonlinear impedances for superconducting quantum technologies, they also possess internal dynamics that may strongly affect their behavior. Here, we construct a computational framework that includes a microscopic description of the junction (full fledged treatment of both the superconducting condensate and the quasiparticles) in the presence of a surrounding electrical circuit. Our approach generalizes the standard resistor capacitor Josephson model to arbitrary junctions (including, e.g., multiterminal geometries and/or junctions that embed topological or magnetic elements) and arbitrary electric circuits treated at the classical level. By treating the superconducting condensate and quasiparticles on equal footings, we capture nonequilibrium phenomena such as multiple Andreev reflection. We show that the interplay between the quasiparticle dynamics and the electrical environment leads to the emergence of new phenomena. In a $RC$ circuit connected to single channel Josephson junction, we find out-of-equilibrium current-phase relations that are strongly distorted with respect to the (almost sinusoidal) equilibrium one, revealing the presence of the high harmonic ac Josephson effect. In an $RLC$ circuit connected to a junction, we find that the shape of the resonance is strongly modified by the quasiparticle dynamics: close to resonance, the current can be smaller than without the resonator. Our approach provides a route for the quantitative modeling of superconducting-based circuits.

Figure 1

Upper-left inset: simulated circuit, with an $RC$ biased Josephson junction. Main panel: result of the $RC\text{−}\mathrm{BdG}$ simulation. Dashed line: voltage $R{I}_0$ applied by the generator versus time $t$. $I_0$ is raised and decreased slowly to keep the system quasiadiabatic. Blue line: voltage $V(t)$ measured across the junction. Upper-right inset: enlarged view of main curve revealing the oscillations due to the ac Josephson effect.

Figure 2

RC-BdG model. Bottom panel: bias current $I_0$ versus the average voltage across the junction $\overline{V}$ for an underdamped oscillator $Q\approx 1.7$. Dotted lines: various asymptotes of the RCJ model; see text. Dashed line: pure BdG model without the environment. Upper panels: out-of-equilibrium current-phase relations at four different points of the $I_0-\overline{V}$ curve. The dotted line corresponds to the equilibrium current-phase relation of the pure junction.

Figure 3

$RLC\text{−}\mathrm{BdG}$ model. Upper inset: schematic of the $RLC$ circuit. Main panel: voltage $V_J(t)$ across the junction (blue line) versus time $t$ for a linear voltage ramp in $V_0(t)$ (red dashed line). $Q=20$, ${\omega }_0=\mathrm{\Delta }$, and $R=3h/2e^2\simeq 38.7\mathrm{k}\mathrm{\Omega }$. Bottom inset: enlarged view of the main curve showing the ac Josephson effect oscillations. The resonance of the $RLC$ circuit is visible for $e{V}_J=\hslash {\omega }_0/2$ and $e{V}_J=\hslash {\omega }_0/4$

Figure 4

Current-voltage relation for four different resonator frequencies ${\omega }_0/\mathrm{\Delta }=1/4,2/3,1$, and 1.4. Dashed line: $I_{\mathrm{MAR}}(V)$ in the absence of environment. Thin color lines: $RLC\text{−}\mathrm{BdG}$ simulations. Dotted lines: improved RLCJ model. Bottom panels: enlarged views of the main figure.