Limit Cycles and Synchronization Go Quantum

Synchronization in nonlinear oscillatory systems extends from classical to quantum domains, where quantum effects, such quantum fluctuations, discrete energy level, quantum measurements, and quantum entanglement introduce new challenges. In this talk, we present a phase dynamics approach, which has been developed in the fields of classical nonlinear dynamics, to quantum limit-cycle oscillations and quantum synchronization. First, we formulate a semiclassical phase-reduction theory for quantum limit-cycle oscillators, extending the classical phase-reduction framework. This allows us to derive a one-dimensional phase equation that captures synchronization properties in the semiclassical regime, where quantum noise is small. Second, we introduce fully quantum-mechanical definition of the asymptotic phase for quantum nonlinear oscillators, which naturally extends the definition of the asymptotic phase for classical oscillatory systems from the Koopman-operator viewpoint. Our quantum asymptotic phase is generally applicable in the strong quantum regime and serves as a fundamental quantity for characterizing quantum nonlinear oscillatory dynamics. Also, other studies based on the analysis of classical nonlinear dynamics to quantum systems has been introduced

Synchronization is one of the most intriguing collective phenomena in nature. It is intimately related to diverse fields ranging from engineering, biology, mathematics and physics [1]. Intriguingly, synchronization is related to pattern formation, and it is one example of self-organization in manybody systems under non-equilibrium conditions [2]. Most of the developments in the theory of synchronization are based on nonlinear dynamical systems. It is therefore an open question if quantum signatures or fingerprints of synchronization appear in quantum systems, whose dynamics is inherently linear. Recently, there has been an onset of interest in the investigation of quantum synchronization, with impressive developments providing measures of synchronization in diverse systems such as quantum Van der Pol and Stuart Landau oscillators [3-8]. In this talk I will discuss quantum fingerprints of synchronization induced by a classical drive that undergoes a synchronization transition. Specifically, I consider a one-dimensional spin chain locally driven by a classical Kuramoto model. I initialize the phases of the Kuramoto model randomly. As a result, the spin chain does not have any spatial symmetries. When the Kuramoto undergoes a synchronization transition, there are emergent spatial and temporal symmetries that are quantum fingerprints of classical synchronization [9]. At the end of my talk I will discuss how the Kuramoto model can induce emergent topological behavior in the spin chain. I envision our results to open a new direction of research on self-organization of quantum manybody systems driven by classical synchronization. [1] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series (Cambridge University Press, 2001). [2] M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys. 65, 851 (1993). [3] T. E. Lee and H. R. Sadeghpour, Quantum synchronization of quantum van der pol oscillators with trapped ions, Phys. Rev. Lett. 111, 234101 (2013). [4] S. Walter, A. Nunnenkamp, and C. Bruder, Quantum synchronization of a driven self-sustained oscillator, Phys. Rev. Lett. 112, 094102 (2014). [5] V. M. Bastidas, I. Omelchenko, A. Zakharova, E. Scholl, and T. Brandes, Quantum signatures of chimera states, Phys. Rev. E 96, 052210 (2017). [6] S. Sonar, M. Hajdusek, M. Mukherjee, R. Fazio, V. Vedral, S. Vinjanampathy, and L.-C. Kwek, Squeezing enhances quantum synchronization, Phys. Rev. Lett. 120, 163601 (2018). [7] C. W. Wachtler and G. Platero, Topological synchronization ¨ of quantum van der pol oscillators, Phys. Rev. Res. 5, 023021 (2023). [8] C. W. Wachtler and J. E. Moore, Topological quantum synchronization of fractionalized spins, Phys. Rev. Lett. 132, 196601 (2024). [9] V. M Bastidas, arXiv:2406.17062 (2024).

In this talk I will present a method to quantize classical non-Hamiltonian systems. We call this method cascade quantization [1]. It covers an expansive set of dynamical systems---defined by (dx/dt,dy/dt)=(f(x,y),g(x,y)) with f(x,y) and g(x,y) given by arbitrary polynomials. Cascade quantization maps every such classical system directly to a quantum mechanical generator of time evolution in the form of a Lindbladian. As such, it allows every classical system whose f(x,y) and g(x,y) are analytic to be quantized with arbitrary precision. Cascade quantization is straightforward to use and I will illustrate its application on some well-known examples from classical nonlinear dynamics. The quantization of non-Hamiltonian systems has a long history, dating as far back as the 1930s. Since then, there has been a quest to bring non-Hamiltonian systems into the fold of canonical quantization, a program which continues to live on (see Appendix A of Ref.[1] and the references therein). I will try to explain why this research program has not seen much progress despite receiving sustained inputs from various authors. References [1] A. Chia, W.-K. Mok, L.-C. Kwek, and C. Noh, Quantization of nonlinear non-Hamiltonian systems, arXiv:2503.06939.

Time crystals are non-equilibrium phases of matter with broken time-translation symmetry. Their existence, initially conjectured by Wilczek, has been further developed theoretically and confirmed experimentally in several synthetic systems. They display robust coherence and correlations making them appealing candidates for application in various quantum technologies. I will discuss two possible applications in quantum sensing and in the realization of clocks. Time crystals can become advantageous when used for sensing extremely weak AC-fields. In such systems, collective interactions stabilize their dynamics against noise making them robust enough to protocol imperfections. I will also discuss how spontaneous breaking of time translational invariance may be also important in realizing autonomous clocks.

Continuous time crystals, i.e., nonequilibrium phases with a spontaneously broken continuous time-translational symmetry, have been studied and recently observed in the long-time dynamics of open quantum systems. Here, we investigate a lattice of interacting three-level systems and find two distinct time-crystal phases that cannot be described within mean-field theory. Remarkably, one of them emerges only in the presence of quantum fluctuations. Our findings extend explorations of continuous time-translational symmetry breaking in dissipative systems beyond the classical phenomenology of periodic orbits in a low-dimensional nonlinear system. The proposed model applies directly to the laser-driven dynamics of interacting Rydberg states in neutral atom arrays and suggests that the predicted time-crystal phases are observable in such experiments.

Dissipative time crystals are many-body systems that show peristent collective oscillations in the thermodynamic limit, thereby breaking the time-translation symmetry. They can appear in spin systems, when the ????_2 symmetry of the Hamiltonian is broken by the environment, and the square of total spin operator ????^2 is conserved. In this contribution, I relax the latter condition by using a dissipation with Lindblad operators power-law-decaying in space, and show that time-translation-symmetry breaking persists when the decay exponent obeys 0

Computational problems may be solved by realizing physics systems that can simulate them. Here we present a new system of coupled lasers in a modified degenerate cavity that is used to solve difficult computational tasks. The degenerate cavity possesses a huge number of degrees of freedom (300,000 modes in our system), that can be coupled and controlled with direct access to both the x-space and k-space components of the lasing mode. Placing constraints on these components are mapped on different computational minimization problems. Due to mode competition, the lasers select the mode with minimal loss to find the solution. We demonstrate this ability for simulating XY spin systems and finding their ground state, for phase retrieval, for imaging through scattering medium, and more.

I will present how to engineer dissipative systems of trapped ions such as to realize a number of archetypal models of dissipative dynamics, in particular focussing on the harmonic oscillator motion of the ions. This will include systems with limit-cycles, as well as subspaces protected from various noise channels which can be used for quantum error correction. I will illustrate how non-linearities can be realized, both through the use of ancillary spin systems as well as through the form of the laser-ion interaction. Implementing such models on multi-ion systems should enable realisations of coupled non-linear oscillators subject to dissipation, providing a benchmark system for studies of synchronised oscillators in the quantum regime.

A bondary time-crystal is an open many-body system that features a transition between a stationary and an oscillatory phase. It can be realised within dense atomic gases or atoms held in a cavity. The fact that the system is open allows one to continuously monitor its quantum trajectories and to analyze their dependence on parameter changes. This enables the realization of a sensing device whose performance we investigate as a function of the monitoring time and system size. The best achievable sensitivity turns out to follows the standard quantum limit in time and Heisenberg scaling in the particle number. This theoretical scaling can be achieved in the oscillatory time-crystal phase and it is rooted in emergent quantum correlations. A central challenge is, however, to tap this capability in a measurement protocol that is experimentally feasible. We show that the standard quantum limit can be surpassed by cascading two time crystals, where the quantum trajectories of one time crystal are used as input for the other one. We will also comment on the possibility of obtaining such quantum enhancement with imperfect measurements of two-time photon correlations. [1] A. Cabot, F. Carollo and I. Lesanovsky, Continuous Sensingand Parameter Estimation with the Boundary Time Crystal, Phys. Rev. Lett. 132, 050801 (2024) [2] A. Cabot, F. Carollo and I. Lesanovsky, Quantum enhanced parameter estimation with monitored quantum nonequilibrium systems using inefficient photo detection, arXiv:2503.21753 (2025)

Driven-dissipative quantum systems can undergo transitions from stationary to dynamical phases, reflecting the emergence of collective non-equilibrium behavior. We study such a transition in a Bose-Einstein condensate coupled to an optical cavity and develop a cavity-assisted Bragg spectroscopy technique to resolve its collective modes. We observe dissipation-induced synchronization at the quasiparticle level, where two roton-like modes coalesce at an exceptional point. This reveals how dissipation microscopically drives collective dynamics and signals a precursor to a dynamical phase transition.

I will report on the experimental implementation of quantum synchronization in systems of qubits and nonlinear oscillators. In particular, I will discuss the occurrence of noise-induced synchronization in a chain of superconducting qubits and the observation entangled synchronized oscillations in a many-body system. I will additionally present the realization of a quantum van der Pol oscillator using a single trapped ion, the measurement of its quantum limit cycle, as well as the demonstration of its synchronization with an external drive.

Self-sustained oscillators are known to be able to synchronize under even a weak interaction, and this is often exemplified by the paradigmatic van del Pol oscillator driven by negative friction. The question of whether and how such entrainment plays out for quantum oscillators has recently been addressed theoretically, and here we propose an experimental realization of such a system using two microwave resonators coupled resonantly to a voltage-biased double quantum dot with normal metallic leads. First, we demonstrate that each individual resonator exhibits the Keldysh saddle-point dynamics of a Duffing van der Pol oscillator, the non-linearity provided by the coupling to the electron system and the negative friction arising from the large voltage. Quantum fluctuations and phase-space diffusion are captured by an effective Fokker-Planck equation as well as a Lindblad master equations. Second, we show that two resonators coupled to the same DQD are capable of synchronization of both their frequency and phase.

We investigate a dissipative phase transition in a driven nonlinear quantum oscillator, specifically the quantum van der Pol (QvdP) oscillator with squeezed drive, which exhibits two distinct time-crystalline orders: a discrete time crystal (DTC) and an incommensurate time crystal (ITC). The system is analyzed in both the laboratory and rotating frames, revealing a correspondence between the ITC in the laboratory frame and a continuous time crystal (CTC) in the rotating frame, as well as between the DTC and a parity-broken stationary phase. The transition between these regimes is mediated by an exceptional point (EP) in the Liouvillian spectrum, which serves as a spectral signature of the nonequilibrium phase transition.

We develop and study an atom-only description of the Dicke model with time-periodic couplings between atoms and a dissipative cavity mode. The cavity mode is eliminated, giving rise to effective atom-atom interactions and dissipation. We use this effective description to analyze the dynamics of the atoms that undergo a transition to a dynamical superradiant phase with macroscopic coherences in the atomic medium and the light field. Using Floquet theory in combination with the atom-only description we provide a precise determination of the phase boundaries and of the dynamical response of the atoms. From this we can predict the existence of dissipative time crystals that show a subharmonic response with respect to the driving frequency. We show that the atom-only theory can describe the relaxation into such a dissipative time crystal and that the damping rate can be understood in terms of a cooling mechanism.

Quantum synchronization is often studied using a master equation, typically the Lindblad equation. This gives the time evolution of the density matrix. If we study the response to a periodic external signal, synchronization is detected by a steady state density matrix that is in phase with the external signal. However, the system will normally only partially synchronize with the degree of synchronization being hard to quantify. This is to some extent similar to the synchronization of a classical system in the presence of external noise, but in that case we can measure the degree of synchronization by the average frequency of the system as compared to the frequency of the external signal. For a quantum system, we apply the quantum trajectory approach, which gives a stochastic trajectory, and measures the average frequency of such trajectories. In this way, we can study the synchronization process in more detail than when using the master equation. We illustrate this by studying the synchronization of a two-level system to an external signal, and the mutual synchronization of two two-level systems.

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).

Here I will give a general algebraic theory of synchronization in both open and closed time independent many-body systems based on the recently introduced theory of dynamical symmetries. Closed locally interacting many-body systems can act as their own baths thus inducing synchronization that spreads inside a light cone given by a Lieb-Robinson bound. I will use Krylov Liouvillian approaches to demonstrate the existence of a phase transition between a regime where synchronization is present and absent controlled by onsite disorder. More specifically, this new kind of transition can be associated with a level crossing in the complex eigenfrequencies of the Krylov Liouvillian and the associated transient dynamical symmetries. An example of a disordered spin chain will be studied. References: N. Loizeau, B. Buca, D. Sels. arXiv preprint arXiv:2503.07403 B. Buca. Phys. Rev. X 13, 031013 (2023). B Buca, C Booker, D Jaksch. SciPost Physics 12 (3), 097 (2022) D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, E. Altman. Phys. Rev. X 9, 041017 (2019). B Buca, J Tindall, D Jaksch. Nat. Comm. 10 (1), 1730 (2019)

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)

Superradiant lasers, which consist of incoherently driven atoms coupled to a lossy cavity, are a promising source of coherent light with an extremely narrow linewidth. In this talk, I will discuss how the narrow linewidth arises through a Kuramoto-like synchronization transition from an incoherent to a phase-locked state. When a fraction of the atoms is not driven, nonreciprocal interactions emerge, i.e., a competition between alignment and antialignment of the atomic dipoles. These nonreciprocal interactions, which are usually studied in the context of active matter, result in a frequency shift of the superradiant laser. Our work shows that concepts of synchronization and active matter offer insights into the quantum-optical model of a superradiant laser.

I will consider two very different open quantum systems that can synchronize to an external periodic signal. Specifically, I will analyze a squeezed quantum van der Pol oscillator [1,2] and a central spin model subject to periodic resetting [3]. The former has an intuitive classical limit from which the synchronization dynamics can be well understood. In contrast, the considered spin model has no well defined thermodynamic or classical limit. I will show that both systems can display subharmonic synchronization to an external periodic signal. Despite their different nature, I will show that this phenomenon displays the same spectral features in both cases, namely the opening of a spectral gap between several low lying excitations and the rest [1,2,3]. Hence, the regime of subharmonic synchronization manifests as a metastable dynamical regime in which, after a short transient, the systems display a very long plateau of oscillations locked to a fraction of the signal frequency. Eventually, these oscillations decay out due to the effects of quantum fluctuations. By applying the theory of quantum metastability, I will show that this eventual relaxation corresponds to a classical stochastic (activation) process between the different phases at which these systems can lock. Finally, I will comment on connections of these phenomena with discrete time crystals in dissipative quantum systems. [1] A. Cabot, G. L. Giorgi, R. Zambrini, New J. Phys. 23, 103017 (2021). [2] A. Cabot, G. L. Giorgi, R. Zambrini, PRX Quantum 5, 030325 (2024). [3] A. Cabot, F. Carollo, I. Lesanovsky, Phys. Rev. B 106, 134311 (2022).

Among the most iconic features of classical dissipative dynamics are persistent limit-cycle oscillations and critical slowing down at the onset of such oscillations, where the system relaxes purely algebraically in time. On the other hand, quantum systems subject to generic Markovian dissipation decohere exponentially in time, approaching a unique steady state. Here we show how coherent limit-cycle oscillations and algebraic decay can emerge in a quantum system governed by a Markovian master equation as one approaches the classical limit, illustrating general mechanisms using a single-spin model and a two-site lossy Bose-Hubbard model. In particular, we demonstrate that the fingerprint of a limit cycle is a slowdecaying branch with vanishing decoherence rates in the Liouville spectrum, while a power-law decay is realized by a spectral collapse at the bifurcation point. We also show how these are distinct from the case of a classical fixed point, for which the quantum spectrum is gapped and can be generated from the linearized classical dynamics. S Dutta, S Zhang, M Haque, Physical Review Letters 134, 050407 (2025)

I will consider two very different open quantum systems that can synchronize to an external periodic signal. Specifically, I will analyze a squeezed quantum van der Pol oscillator [1,2] and a central spin model subject to periodic resetting [3]. The former has an intuitive classical limit from which the synchronization dynamics can be well understood. In contrast, the considered spin model has no well defined thermodynamic or classical limit. I will show that both systems can display subharmonic synchronization to an external periodic signal. Despite their different nature, I will show that this phenomenon displays the same spectral features in both cases, namely the opening of a spectral gap between several low lying excitations and the rest [1,2,3]. Hence, the regime of subharmonic synchronization manifests as a metastable dynamical regime in which, after a short transient, the systems display a very long plateau of oscillations locked to a fraction of the signal frequency. Eventually, these oscillations decay out due to the effects of quantum fluctuations. By applying the theory of quantum metastability, I will show that this eventual relaxation corresponds to a classical stochastic (activation) process between the different phases at which these systems can lock. Finally, I will comment on connections of these phenomena with discrete time crystals in dissipative quantum systems. [1] A. Cabot, G. L. Giorgi, R. Zambrini, New J. Phys. 23, 103017 (2021). [2] A. Cabot, G. L. Giorgi, R. Zambrini, PRX Quantum 5, 030325 (2024). [3] A. Cabot, F. Carollo, I. Lesanovsky, Phys. Rev. B 106, 134311 (2022).