Frequency Combs in a Lumped-Element Josephson-Junction Circuit

We investigate the dynamics of a microwave-driven Josephson junction capacitively coupled to a lumped-element $LC$ oscillator. In the regime of driving where the Josephson junction can be approximated as a Kerr oscillator, this minimal nonlinear system has been previously shown to exhibit a bistability in phase and amplitude. In the present study, we characterize the full phase diagram and show that besides a parameter regime exhibiting bistability, there is also a regime of self-oscillations characterized by a frequency comb in its spectrum. We discuss the mechanism of comb generation which appears to be different from those studied in microcavity frequency combs and mode-locked lasers. We then address the fate of the comblike spectrum in the regime of strong quantum fluctuations, reached when nonlinearity becomes the dominant scale with respect to dissipation. We find that the nonlinearity responsible for the emergence of the frequency combs also leads to its dephasing, leading to broadening and ultimate disappearance of sharp spectral peaks. Our study explores the fundamental question of the impact of quantum fluctuations for quantum systems which do not possess a stable fixed point in the classical limit.

Figure 1

(a) Schematic representation of the two-mode system. (b) Mode and drive frequencies. (c) Lumped-element circuit QED implementation of (a).

Figure 2

Phase diagram in $\eta -{\mathrm{\Delta }}_{db}$ space. The possible phases are listed in the top-left table. (a) Phase diagram for $g<(\kappa /2)$ (here, $g=(\kappa /4)$). (b) Phase diagram for $g>(\kappa /2)$ (here, $g=2\kappa$). (c) through (f) Plots of $|\overline{\beta }{|}^2$ against $\eta$ showing the change in the $S$ curve as ${\mathrm{\Delta }}_{db}$ is swept across the four dynamical regions in (b). Green (purple) segments of the $S$ curve depict unstable (stable) $|\overline{\beta }{|}^2$ values.

Figure 3

Numerically simulated phase diagram in $\eta -{\mathrm{\Delta }}_{db}$ space for $g=2\kappa >(\kappa /2)$. Surface plot shows spectral spacing $\mathrm{\Delta }\omega$ obtained from $S_{\alpha }[\omega ]$; the “island” of multifrequency solutions overlaps perfectly with the analytically predicted unstable region below (shaded green). Plots (a) through (c) show $S_{\alpha }[\omega ]$ at the correspondingly labeled points on the phase diagram.

Figure 4

$|S_a[\omega ]|$, log scale, (a) within, and (b) outside the limit cycle region, as a function of increasing nonlinearity $\mathrm{\Lambda }$ from left to right. Master equation (ME), SDE simulations (PP), and stable regime linearized spectra (Lin.) are shown.