Synchronizing microwave cQED limit-cycle oscillators

TL;DR

This work presents a microscopic quantum electrodynamics framework for driven, dissipative electron–photon hybrids formed by voltage-biased double quantum dots coupled to microwave resonators. By deriving a nonequilibrium Keldysh action and performing a controlled expansion to fourth order in the electron–photon coupling, it reveals a Hopf bifurcation to a quantum limit cycle in a single mode and demonstrates robust phase and frequency synchronization for two coupled modes via the same DQD. The analysis combines saddle-point solutions, Fokker–Planck fluctuations, and exact Lindblad master equation benchmarks in the infinite-bias limit, highlighting how nonlinearities and dissipative coupling enable quantum synchronization beyond linear level attraction. These results provide a microscopic route to quantum Stuart–Landau dynamics in cQED and suggest experimentally accessible regimes for observing quantum limit cycles and their synchronization in DQD–resonator platforms.

Abstract

Self-sustained oscillators play a central role in the stabilization and synchronization of complex dynamical systems. A number of different physical systems are currently being investigated to clarify the importance of such active components in the quantum realm. Here we explore the properties of a driven dissipative electron-photon hybrid system based on superconducting microwave resonators coupled resonantly to a voltage-biased double quantum dot (DQD). First, we establish a Hopf bifurcation at a critical value of the electron-photon coupling, beyond which an effective negative friction sustains steady limit-cycle oscillations of individual resonators. Second, we show that two such limit-cycle resonators coupled via the same voltage-biased DQD synchronize for small enough frequency detuning. A nonlinear photon Keldysh action is derived by perturbation theory in the effective circuit fine-structure constant, and the limit-cycle dynamics is analyzed in terms of resulting saddle-point, and Fokker-Planck equations. In the Markovian limit of infinite bias voltage, these results are shown to agree well with the solution of a corresponding Lindblad master equation for the DQD resonator system.

Figures (10)

• Figure 1: Voltage-biased double quantum dot with each dot coupled capacitively to individual superconducting single-mode microwave resonators.
• Figure 2: Real and imaginary parts of the retarded, and imaginary part of the Keldysh photon self energy plotted in panels (a, c, e) against DQD detuning for $V=10\omega_0$, and in panels (b, d, f) against bias voltage for $\varepsilon=(\sqrt{3}/2)\omega_{0}$. In both cases our $L/R$-symmetric parameter set implies that $\Pi^\text{R}_{LL}=\Pi^\text{R}_{RR}$ and $\Pi^\text{R}_{LR}=\Pi^\text{R}_{RL}.$ The dashed gridline in panel (e) shows the resonance condition, $\varepsilon=(\sqrt{3}/2)\omega_{0}$, and in (f) the threshold condition for the bias voltage, $V=\omega_0.$ The width of the transition region around the dashed gridline is set by $\Gamma$.
• Figure 3: Real and imaginary parts of the photon interaction vertices appearing in the fourth order contribution to the effective action plotted in panels (a, c) against DQD detuning for $V=10\omega_0$, and in panels (b, d) against bias voltage for $\varepsilon=(\sqrt{3}/2)\omega_{0}$. The dashed gridline in (c) shows the resonance condition, $\varepsilon=(\sqrt{3}/2)\omega_{0}$, and in (d) the threshold condition for the bias voltage, $V=\omega_0.$ The width of the transition region around the dashed gridline is set by $\Gamma$.
• Figure 4: Limit cycle radius, $r_\ast$, determined respectively by the saddle-point equations, $r_\text{sp}$, or the value at which the Wigner function attains its maximum, $r_\text{W}$, and plotted together with the root mean square of $r$, calculated from the Wigner function. The inset shows a zoom in on the bifurcation region, $g\sim g_c$, revealing a slight difference in the predicted values for the critical coupling, $g_c$ (gridline marks $\tilde{\kappa}=0$).
• Figure 5: Wigner function (blue) overlaid by the limit-cycle radius, $r_{\ast}$, determined from the saddle-point equations (orange) [(a), (c), (e)] and radial probability distribution [(b), (d), (f)] obtained by solving the FP equation without/with ($P_0/P$) the nonlinear terms ($\Lambda_{2/5}$), plotted alongside $P_W=W(\bar{\phi}/\sqrt{2},\phi/\sqrt{2},t)/2$ with $\phi=r e^{i\theta}$. From top to bottom the coupling strength increases as $g/\omega_0=[0.01, 0.042, 0.05]$. The dashed gridline in panel (f) marks the maximum of $P_0$, i.e. the saddle-point value of the limit-cycle radius, $r_{\ast}$.