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arXiv:2511.07140 · Physical review. B./Physical review. B

Synchronizing microwave cQED limit-cycle oscillators

Cecilie Hermansen, Jens Paaske

cond-mat.mes-hallcond-mat.str-elquant-phSubmitted Nov 10, 2025• Announced Nov 11, 2025• 0 citations• Revised Nov 18, 2025 (v2)
v1 · Nov 10, 2025v2 · Nov 18, 2025
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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.

Journal reference
Phys. Rev. B 113, 075434 (2026)
DOI
10.1103/k45q-jmcb
Comments
Modified and corrected figs.2,4,5,8

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It is well known from classical physics that weakly coupled self-sustained oscillators may spontaneously lock their phases. Just like classical synchronization is known to break down due to noise induced phase slips, we show here how the synchronization of continuous variable quantum systems breaks down by proliferation of quantum phase slips. Within a Keldysh path integral formulation of limit cycles, we analyze the phase dynamics and show how, in spite of strong phase correlations, quantum phase slips degrade the actual phase locking. This approach also allows us to address non-Markovian effects on the synchronization of limit cycles, which we illustrate explicitly for superconducting resonators coupled via a voltage biased double quantum dot.
Abstract Synchronization is a widespread phenomenon encountered in many natural and engineered systems with nonlinear classical dynamics. How synchronization concepts and mechanisms transfer to the quantum realm and whether features are universal or platform specific are timely questions of fundamental interest. They can be studied in superconducting electrical circuits which provide a well-established platform for nonlinear quantum dynamics. Here, we consider a Josephson-photonics device, where a dc-biased Josephson junction creates (non-classical) light in a microwave cavity. The combined quantum compound constitutes a self-sustained oscillator: a system susceptible to synchronization. This is due to the inherent effect of an in-series resistance, which realizes an autonomous feedback mechanism of the charge transport on the driving voltage. Accounting for the full counting statistics of transported charge not only yields phase diffusion, but allows us to describe phase locking to an ac-signal and the mutual synchronization of two such devices. Thereby one can observe phase stabilization leading to a sharp emission spectrum as well as unique charge transport statistics revealing shot noise induced phase slips. Two-time perturbation theory is used to obtain a reduced description of the oscillators phase dynamics in form of a Fokker–Planck equation in generalization of classical synchronization theories.
We present a theoretical study of a hybrid circuit-quantum electrodynamics system composed of two semiconducting charge-qubits confined in a microwave resonator. The qubits are defined in terms of the charge states of two spatially separated double quantum dots (DQDs) which are coupled to the same photon mode in the microwave resonator. We analyse a transport setup where each DQD is attached to electronic reservoirs and biased out-of-equilibrium by a large voltage, and study how electron transport across each DQD is modified by the coupling to the common resonator. In particular, we show that the inelastic current through each DQD reflects an indirect qubit–qubit interaction mediated by off-resonant photons in the microwave resonator. As a result of this interaction, both charge qubits stay entangled in the steady (dissipative) state. Finite shot noise cross-correlations between currents across distant DQDs are another manifestation of this nontrivial steady-state entanglement.
We have realized a hybrid solid-state quantum device in which a single-electron semiconductor double quantum dot is dipole coupled to a superconducting microwave frequency transmission line resonator. The dipolar interaction between the two entities manifests itself via dispersive and dissipative effects observed as frequency shifts and linewidth broadenings of the photonic mode respectively. A Jaynes-Cummings Hamiltonian master equation calculation is used to model the combined system response and allows for determining both the coherence properties of the double quantum dot and its interdot tunnel coupling with high accuracy. The value and uncertainty of the tunnel coupling extracted from the microwave read-out technique are compared to a standard quantum point contact charge detection analysis. The two techniques are found to be consistent with a superior precision for the microwave experiment when tunneling rates approach the resonator eigenfrequency. Decoherence properties of the double dot are further investigated as a function of the number of electrons inside the dots. They are found to be similar in the single-electron and many-electron regimes suggesting that the density of the confinement energy spectrum plays a minor role in the decoherence rate of the system under investigation.
We study quantum van der Pol oscillator in which single-photon pumping competes with a nonlinear two-photon dissipative channel to produce a quantum limit cycle, and the question of how two such oscillators with non-identical intrinsic frequencies synchronize in the sense of phase locking when coupled dissipatively to one another. For a single oscillator, starting from a Lindblad description of the dynamics, we derive the exact Wigner-function evolution equation, and show that its Fokker--Planck approximation admits in the appropriate regime an equivalent classical Itô stochastic description that reproduces remarkably well the full Lindblad dynamics. Notably, the stochastic approach provides an efficient means of capturing system dynamics in the semiclassical regime, where a large number of energy eigenstates contribute and direct numerical simulations become computationally demanding. Restricting to the three lowest Fock states and using a parity symmetry, we obtain in closed form the stationary-state density matrix and its Wigner-function representation that displays pronounced nonclassical features. For two non-identical oscillators, we study the full Lindblad dynamics numerically and characterize synchronization through the stationary-state order parameter as a function of frequency mismatch and dissipative coupling. We find that increasing the effective nonlinear dissipative strength suppresses the synchronized region. Restricting the two oscillators to the three-level qutrit subspace, a parity symmetry reduces the stationary-state problem to an exact 414141-dimensional linear system, allowing to compute the synchronization order parameter. In this reduced description, results in the limit of large dissipation agree closely with those from the full dynamics, while exhibiting in the limit of small dissipation a double-peak synchronization structure accompanied by a phase reversal.
Abstract We explore the dissipative dynamics of nonlinearly driven oscillator systems tuned to resonances where multiple excitations are generated. Such systems are readily realised in circuit QED systems combining Josephson junctions with a microwave cavity and a drive achieved either through flux or voltage bias. For resonances involving three or more photons the system undergoes a sequence of two closely spaced dynamical transitions (the first one discontinuous and the second continuous) as the driving is increased leading to steady states that form complex periodic structures in phase space. In the vicinity of the transitions the system displays interesting bistable behaviour: we find that coherent effects can lead to surprising oscillations in the weight of the different dynamical states in the steady state of the system with increasing drive. We show that the dynamics is well-described by a simple effective rate model with transitions between states localised at different points in the phase space crystal. The oscillations in the weights of the dynamical states is reflected in corresponding oscillations in a time-scale that describes transitions between the states.