Limit Cycles and Synchronization Go Quantum

The quantum $O(n)$ model is a paradigmatic example of a many-body quantum systems which exibith interesting equilibrium and out-of-equilibrium features. We show that the model is completely integrable in the large $n$ limit regarless of the dimension $d$, already at the lattice model. As a consequence qe are able to characterize the dynamics and the spectrum of the model in the thermodynamic limit: in partivular we find that the system theramalizes toward a Generalized Gibbs Enseble ensemble in the Quantum Field Theory limit in which the lattice space is sent to zero, while persistent, self-susatained oscillations arises at the lattice level, when the momentum cutoff is kept finite.

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.

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 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.

We investigate semiclassical dynamics of a coupled atom-photon interacting system described by a dimer of anisotropic Dicke model in the presence of photon loss, exhibiting a rich variety of non-linear dynamics. Based on symmetries and dynamical classification, we characterize and chart out various dynamical phases in a phase diagram. A key feature of this system is the multistability of different dynamical states, particularly the coexistence of various superradiant phases as well as limit cycles. Remarkably, this dimer system manifests self-trapping phenomena, resulting in a photon population imbalance between the cavities. Additionally, we identify a unique class of oscillatory dynamics, namely, the self-trapped limit cycle, hosting the self-trapping of photons, which may give rise to the formation of an intriguing time crystalline phase. The absence of stable dynamical phases leads to the onset of chaos, which is diagnosed using the saturation value of the decorrelator dynamics. The quantum mechanical signatures of such fascinating phenomena are also investigated using the stochastic wave function method.

Quantum synchronization effects imply that synchronization in quantum systems qualitatively differs from synchronization in classical systems. These effects can arise due to interference or the discreteness of energy levels, and are typically discussed in the context of a single driven quantum oscillator or two coupled oscillators. Here, we show that quantum effects in synchronization also occurs in macroscopically large networks. Additionally, these ensembles exhibit emergent behavior not present at the level of two coupled quantum oscillators.

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.

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)

We investigate the quantum dynamics of limit cycles and tori in driven-dissipative bosonic systems, focusing on their emergence, stability, and distinctive quantum signatures. Using a minimal model of two coupled nonlinear photonic cavities subject to local driving and dissipation, we examine the transition from classical to quantum dynamics through the quantum master equation, diagonalization of the Liouvillian superoperator, and quantum trajectory unraveling. In the high photon occupation limit, limit cycles and tori are well-characterized periodic and quasiperiodic attractors, respectively. Quantum effects give rise to novel features, arising from the interplay between coherence, dissipation, and noise, which manifest in the Liouvillian spectrum. These insights provide a framework for understanding emergent quantum many-body attractors in open quantum systems. References: - Universal scaling and spectral encoding of quasiperiodicity in open quantum systems (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

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.

Optically levitating dielectric nanoparticles in ultra-high vacuum, where their motion can be cooled into the deep quantum regime, provides a promising platform for force and torque sensing and for high-mass tests of quantum physics. In this talk I will discuss recent results on the coupled dynamics of co-levitated nanoparticles interacting via optical binding and via electrostatic forces. I will show how non-reciprocal interactions [1,2] and mechanical entanglement [3,4] between two particles can be generated and observed by controlling the light fields suspending them. [1] Rieser, Ciampini, Rudolph, Kiesel, Hornberger, Stickler, Aspelmeyer, and Delić, Science 377, 987 (2022) [2] Reisenbauer, Rudolph, Egyed, Hornberger, Zasedatelev, Abuzarli, Stickler, and Delić, Nat. Phys. (2024) [3] Rudolph, Delić, Aspelmeyer, Hornberger, and Stickler, Phys. Rev. Lett. 129, 193602 (2022) [4] Rudolph, Delić, Hornberger, and Stickler, 133, 233603 (2024)

Mutual synchronization phenomena in open, non-driven quantum systems have been extensively studied through two distinct frameworks: the persistently oscillating and synchronized dynamics of local observables, and the concentrated phase-space distributions of non-equilibrium steady states (NESS) without any dynamics. The relationship between these two characterizations of quantum synchronization is not yet fully understood. In this work, we bridge the gap between these concepts in systems with strong dynamical symmetries. While strong dynamical symmetry results in persistently oscillating observables, its explicit breaking typically leads to a NESS characterized by phase synchronization. By examining key examples of observable-based synchronization, we elucidate how the oscillating observables can help identify the basis in which the corresponding phase synchronization is present. Additionally, we discuss potential extensions of our findings to a broader class of systems.