Limit Cycles and Synchronization Go Quantum

Reduced basis surrogate models provide an efficient way of mapping phase diagrams of strongly correlated many-body quantum systems. The method relies on using the exact solutions at select parameter values to construct a low-dimensional basis, from which observables can be efficiently and reliably computed throughout parameter space. Here we show that this method can be generalized to driven-dissipative Markovian systems allowing efficient calculations of observables in the transient and steady states. A subsequent distillation of the reduced basis vectors according to their explained variances allows for an unbiased exploration of the most pronounced parameter dependencies indicative of phase boundaries in the thermodynamic limit.

Despite the individual systems having characteristics of chaos, coupling these systems often results in ordered, predictable, collective dynamics. Inspired by this phenomenon, we investigate transverse-field quantum Ising chains coupled by an Ising chain with dissipation. We observe synchronization of periodic oscillations of the spins in the $xy$-plane induced by the Ising chain, which undergoes dissipative dynamics. We have also analyzed the spectral properties of the Lindbladian, and we have found non-trivial eigenstates with purely imaginary eigenvalues. We identify the origin of this dynamics to these non-trivial states. This model is experimentally accessible in near-term cold atom experiments.

In Josephson photonic devices, Cooper pairs tunnel inelastically across a dc-biased Josephson junction, creating excitations in microwave cavities. Due to the inherent nonlinearity of this drive, these devices can create nontrivial quantum states and serve as a versatile source of quantum microwaves. Since the power supply is provided by a "battery," the system constitutes a self-sustained oscillator that is vulnerable to perturbations. In particular, shot noise disturbs the oscillation phase of the system, leading to degraded quantum states and a broadened emission spectrum. To counter this issue, an ac-signal added on top of the dc-bias can stabilize the oscillator's phase and restore the desired quantum state. To model the system's dynamics, we use a modified number-resolved master equation that captures the statistics of Cooper pair current and provides a nonlinear feedback mechanism. This approach yields a Fokker-Planck equation for an effective "phase particle" in a tilted-washboard potential, offering a simple description of the system's dynamics and revealing phase-slips. Within this model, the synchronization of several quantum states, such as single- or two-mode squeezed states, can be studied, as well as the mutual synchronization of two or more devices.

The Hepp-Lieb-Dicke model is ubiquitous in cavity quantum electrodynamics, describing spin-cavity coupling which does not conserve excitation number. Using “bad cavities'' which couple to dissipative environments, the dynamics can be made manifestly nonequilibrium allowing for engineering of interesting spin-only models. In this work, we discuss the spin-only dynamics of a variation of the open Dicke model which realizes mediated nonreciprocal interactions between spin species (Landini et al. Phys. Rev. Lett. 2018) and has been shown to exhibit a novel nonreciprocal phase transition (Fruchart et al. Nature 2021) associated with a Z2 parity-like symmetry to dynamic limit-cycle states (Chiacchio et al. Phys. Rev. Lett. 2023). In particular, we go beyond adiabatic elimination and, instead, employ a Redfield master equation to obtain an effective description of the spin-only system. We analyze the behavior of this model at the mean-field level and beyond, appealing to the permutation symmetry of the master equation to perform exact numerical diagonalization.

Limit cycles in classical systems are closed phase-space trajectories to which the system converges regardless of its initial state. Their quantum counterparts have been proposed for open quantum systems, exhibiting steady-state phase-space representations with ring-like structures of stable radius but no phase preference. The synchronization of such quantum systems manifest, e.g., in the localization of the phase of the steady state to an external drive. Unlike in classical systems, quantum synchronization can exhibit coherence cancellations, leading to a synchronization blockade. In this work, we propose a quantum system whose classical analogue features two limit cycles. In the classical analogue, the system can end up in either one of the limit cycles, defined by their basins of attraction and choice of initial states. In the quantum system, both limit cycles coexist independently of the initial state, i.e., the Wigner function of the steady state features two rings. Adding an external drive to a single oscillator, its limit cycles localize to distinct phases, exhibiting different synchronization behaviors within the same system. Furthermore, we demonstrate that coupling two such twin limit-cycle oscillators leads to simultaneous synchronization and synchronization blockades between different limit cycles of oscillator A and B.

A synchronization transition of tunneling systems in glasses has been proposed on a phenomenological basis [1], to explain the surprising transition to a phase with strongly enhanced magneto sensitivity, observed in the low-temperature electric permittivity of multicomponent glasses [2]. Even though, meanwhile, alternative theories to explain these experimental results have been developed, in particular the theory of tunneling systems coupled to nuclear quadrupole moments [3], it remains an intriguing and unresolved problem whether tunneling systems can synchronize. Here, we will give an update on the progress of work on this and related problems of disordered long range coupled quantum systems[3,4]. [1] S. Kettemann, P. Fulde, P. Strehlow, Correlated Persistent Tunneling Currents in Glasses, Phys. Rev. Lett. 83, 4325 (1999). [2] P. Strehlow, C. Enss, S. Hunklinger, Evidence for a phase transition in glasses at very low temperature: a macroscopic quantum state of tunneling systems?, Phys. Rev. Lett. 80, 5361 (1998). [3] Y. Mohdeb, J. Vahedi, S. Haas, R. N. Bhatt, S. Kettemann, Global Quench Dynamics and the Growth of Entanglement Entropy in Disordered Spin Chains with Tunable Range Interactions, Phys. Rev. B Letters 108, L140203 (2023). [4| S. Kettemann, Bond Disordered Antiferromagnetic Quantum Spin Chains with Long Range Interactions, https://arxiv.org/abs/2501.07298, submitted to PRL (2025).

Time crystals are non-equilibrium phases of matter with broken time-translational symmetry. These novel phases of matter are classified as discrete time crystals and continuous-time crystals (CTC) depending on the type of time-translational symmetry broken. Strong measurements restrict the dynamics of finite dimensional systems, leading to effective unitary dynamics in the Zeno limit. In this work (PhysRevLett.130.150401), we demonstrate how competition between strong measurement and thermodynamic limit could result in qualitative changes in steady-state properties. In particular, we consider a spin-star system with strong continuous monitoring of the central spin and show that the system exhibits a time-translation symmetry-breaking phase transition resulting in a continuous time crystal. Above a critical value of measurement strength, the magnetization of the thermodynamically large ancilla spins, along with the central spin, develops limit-cycle oscillations.

We investigate the phase transitions and dynamics in a generalized open Dicke trimer, considering a three-component spinor Bose-Einstein condensate (BEC) interacting with a single-mode cavity field via complex couplings. Owing to the component-dependent couplings among the BECs mediated by the cavity field, and the underlying triangular geometry of the three components, both nonreciprocity and geometric frustration can arise. Our study uncovers the competition and cooperation between the two. At low coupling strengths, nonreciprocity drives chiral dynamics with limit cycles associated with the parity-time symmetry breaking. At high coupling strengths, triangular frustration dominates, giving rise to six steady states, each featuring a distinctive self-organization pattern in the BECs. Such frustrated condensates in a multicomponent order-parameter system present a striking phenomenon. At intermediate coupling, these effects synergize, leading to chiral dynamics encircling a hexagonal structure formed by the frustrated states, a signature for experimental observation. The introduce of geometric frustration enriches the study of nonreciprocal physics, particularly in quantum many-body systems, meriting further experimental and theoretical investigations.

We study a transversely laser-driven atomic BEC coupled to an optical cavity, where atom interactions are mediated by laser-pump and cavity photons. This has been realized experimentally using different setups and parameters [1][2]. In these setups the atoms can form a superradiant crystal which supports constructive interference of scattered laser photons above a threshold determined by the driving frequency and intensity. While a minimal model describing this transition is the two-state Dicke model, here, we perform a full mean-field analysis of the system, including all relevant 2D momentum states and the cavity field. With this description we uncover phases and dynamics of the atom-cavity system that are neither captured by the simplified two-level Dicke model nor by a 1D description. We map out the complete phase diagram depending on pump strength and cavity detuning, and provide an in depth understanding of the instabilities that are linked to the emergence of spatio-temporal patterns. We find parameter regimes of mean-field bistability, and regimes where the atom-cavity dynamics forms chaotic trajectories and limit cycles triggered by density-wave resonances. [1] Baumann, K., Guerlin, C., Brennecke, F. \& Esslinger, T. Dicke quantum phase transition with a superfluid gas in an optical cavity. Nature 464, 1301–1306 (2010). [2] Klinder, J., Keßler, H., Wolke, M., Mathey, L. \& Hemmerich, A. Dynamical phase transition in the open Dicke model. Proc. Natl. Acad. Sci. (U.S.A.) 112, 3290–3295 (2015).

We show that limit cycle systems in Langevin bath exhibit uncertainty in observables that define the limit-cycle plane, and maintain a positive lower bound. The uncertainty-bound depends on the parameters that determine the shape and periodicity of the limit cycle. In one dimension, we use the framework of canonical dissipative systems to construct the limit cycle, whereas in two dimensions, particle in central potentials with radial-dissipation provide us natural examples. We show that, the position-momenta uncertainty of particle in a central potential is larger than half the magnitude of the angular momentum (conserved) of the particle. We also investigate how uncertainties, which are absent in deterministic systems, increase with time when the systems are attached to a bath and eventually cross the lower bound before reaching the steady state.

Extending classical synchronization to the quantum domain is of great interest both from the fundamental physics point of view and with view toward quantum technology applications. This work characterizes a cold atom platform, namely 87Rb atoms in a magneto-optical trap (MOT), that allows for spin degrees of freedom to be synchronized. With the F = 1 ground state hyperfine manifold serving as the spin-1 system, effective coherent couplings within the ground state manifold as well as incoherent loss and gain are realized by coupling to auxiliary states. Several synchronization measures are contrasted, both for the spin-1 system as well as the spin-1/2 and higher-spin systems. An effective master equation, which is obtained by integrating out the auxiliary states, is benchmarked using numerical simulations and perturbation theory. Experimentally realistic parameter regimes are identified.

The Tavis-Cummings and Dicke models serve as fundamental frameworks in the study of light-matter interactions, describing the collective coupling of an ensemble of emitters to a single cavity mode. They provide a framework for exploring cooperative symmetry-breaking phenomena and critical behavior in cavity QED systems. In this work, we explore a driven-dissipative Tavis-Cummings model that undergoes a transition from a stationary (melted) phase to a non-stationary (time crystal) phase, stabilized by the interplay of drive and dissipation. We further extend this framework by introducing a two-component spin ensemble with asymmetric cavity couplings, engineered through a controlled phase difference between spin species. Remarkably, we find that non-reciprocal light-matter interactions significantly lower the threshold for time crystal formation. Moreover, this non-reciprocity enables the time crystal phase to persist even in the presence of detuning, an effect that is absent without asymmetric cavity coupling— thus making the time crystal phase more robust.

Quantum phase-locking is typically explored within the framework of continuous time dynamics. In this study, we consider two qudits undergoing the same discrete unitary evolution, driven by the repeated application of a selected unitary gate. Their individual complex evolutions can generally be decomposed into multiple Rabi cycles with initially different phases. We introduce a simple random unitary mechanism designed to asymptotically phase-lock one an arbitrary chosen pair of corresponding qudit Rabi cycles. The structure of all these mechanisms will be presented and we will discuss how they can be combined to asymptotically achieve a predetermined phase pattern between selected pairs of qubit Rabi cycles. This includes the full synchronization of their complex evolutions. Given that the open system dynamics responsible for phase-locking inevitably lead to decoherence, we also discuss the memory effects associated with these mechanisms and explore their potential applications in synchronizing the internal clocks of two quantum walkers, along with the consequences of such synchronization.

We provide a quantum description of quasiperiodic attractors—limit tori—in driven-dissipative Kerr cavities. Using Liouvillian spectral analysis and quantum trajectory unraveling, we identify distinctive quantum signatures of these attractors and examine their emergence, stability, and decay across the quantum-to-classical crossover. Remarkably, this quantum melting follows universal power-law scaling, revealing a nonequilibrium dynamical universality class. Our results bridge classical attractor theory and Liouvillian spectral theory, offering a new perspective on topological phase-space structures and chaos theory. These insights establish a framework for understanding emergent many-body attractors in open quantum systems. References: - Quantum melting of limit tori in driven-dissipative cavities and its universality (Soon on arXiv) - https://journals.aps.org/pra/abstract/10.1103/PhysRevA.101.033839 - https://journals.aps.org/pra/abstract/10.1103/PhysRevA.105.053530 - https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.134.060401

Information-theoretic quantities have received significant attention as system-independent measures of correlations in many-body quantum systems. In particular, mutual information has been proposed as a universal (system-independent) order parameter of synchronization [1]. In this work, we present a method to determine the macroscopic behavior of the steady-state multipartite mutual information between $N$ coupled oscillators undergoing Markovian evolution that is invariant under permutations of oscillators. We show that the scaling of mutual information becomes extensive with system size when the oscillators synchronize, so that the system exhibits limit cycles in the mean field dynamics. In contrast, it is subextensive in the unsynchronized phase, when the system relaxes to a fixed point. We illustrate the applicability of our method using the driven-dissipative Lipkin-Meshkov-Glick model. [1] Phys. Rev. A 91, 012301 (2015)

In the field of superconducting electronics, the on-chip generation of AC radiation is essential for further advancements. Although a Josephson junction can emit AC radiation from a purely DC voltage bias, the coherence of this radiation is significantly limited by Johnson-Nyquist noise. We relate this limitation to the thermodynamic uncertainty relation (TUR) in the field of stochastic thermodynamics. Recent findings indicate that the thermodynamic uncertainty relation can be broken by a classical pendulum clock. We demonstrate how the synchronization-induced violation of the TUR can be used as a design principle for radiation sources by showing that a superconducting clock circuit emits coherent AC radiation from a DC bias.

Criteria for the existence of unique steady states have been established in the literature concernig properties of either the Kossakowski matrix C or the Lindblad operators. A new framework based on operator algebras and graph theory will be presented for examining NESS uniqueness.

Incoherent stochastic processes added to unitary dynamics are typically deemed detrimental since they are expected to diminish quantum features such as superposition and entanglement. Instead of exhibiting energy-conserving persistent coherent motion, the dynamics of such open systems feature, in most cases, a steady state, which is approached in the long-time limit from all initial conditions. This can, in fact, be advantageous as it offers a mechanism for the creation of robust quantum correlations on demand without the need for fine-tuning. Here, we show this for a system consisting of two coherently coupled bosonic modes, which is a paradigmatic scenario for the realization of quantum resources such as squeezed entangled states. Rather counterintuitively, the mere addition of incoherent hopping, which results in a statistical coupling between the bosonic modes, leads to steady states with robust quantum entanglement and enables the emergence of persistent coherent self-sustained limit cycles.

Spontaneous phase synchronization is a riveting ubiquitous phenomenon observed in a great range of both classical and quantum systems. Based on the thorough analysis of the simplest two-qubit case, two major principles of asymptotic synchronization, respectively generally phase-locking were identified in continuous open systems. Namely, synchronization through decoherence-free subspace preservation and apt combination of symmetric and antisymmetric attractor contributions resulting in synchronized asymptotics. Their generalization to finite-dimensional systems and possible applications to networks with bipartite interactions shall be presented.