Synchronization in quantum systems has been largely driven by specific examples of frequency entrainment as well as mutual synchronization. Here we study quantum synchronization as a Liouville space perturbation theory. We begin by clarifying the role of centers, symmetries, and oscillating coherences in the context of quantum synchronization. We then analyze the eigenspectrum of the Liouville superoperator generating the dynamics of the quantum system and determine the conditions under which synchronization arises. We apply our framework to derive a powerful relationship between energy conservation, degeneracies, and synchronization in quantum systems. Finally, we demonstrate our approach by analyzing two mutually coupled thermal machines and prove that nondegenerate thermal networks cannot be simultaneously energy conserving and synchronous.
Synchronization in quantum systems has been largely driven by specific examples of frequency entrainment as well as mutual synchronization. Here we study quantum synchronization as a Liouville space perturbation theory. We begin by clarifying the role of centers, symmetries, and oscillating coherences in the context of quantum synchronization. We then analyze the eigenspectrum of the Liouville superoperator generating the dynamics of the quantum system and determine the conditions under which synchronization arises. We apply our framework to derive a powerful relationship between energy conservation, degeneracies, and synchronization in quantum systems. Finally, we demonstrate our approach by analyzing two mutually coupled thermal machines and prove that nondegenerate thermal networks cannot be simultaneously energy conserving and synchronous.
Synchronization is a collective phenomenon in which interacting nonlinear oscillators develop a persistent relation between their phases and frequencies. In quantum systems, its description requires connecting the classical dynamics of limit-cycle oscillators with quantum fluctuations, dissipation, and the notion of phase. In this tutorial review, we offer a systematic framework for analyzing synchronization in classical and quantum dynamical systems. Starting from amplitude-phase dynamics, we introduce successively topics of relevance, such as phase locking, the Adler equation, Arnold tongues, and the Kuramoto model, and use these concepts to analyze coupled oscillators and their stability. We then formulate the corresponding description for quantum limit-cycle oscillators using Lindblad master equations and phase-space methods, with explicit examples of two and many coupled oscillators. We further discuss noise, synchronization measures, and the extreme quantum regime, including synchronization in spin systems. The emphasis throughout is on analytical derivations and a practical methodology for identifying synchronized states, determining their stability, and characterizing synchronization in quantum systems. The review is designed as a self-contained guide from which we hope that students can learn the essential concepts and techniques, while researchers can readily find the analytical tools and methodological insights needed to approach new synchronization problems.
We show the emergence of spontaneous synchronization between a pair of detuned quantum oscillators within a harmonic network. Our model does not involve any nonlinearity, driving, or external dissipation, thus providing the simplest scenario for the occurrence of local coherent dynamics in an extended harmonic system. A sufficient condition for synchronization is established by building upon the Rayleigh normal mode approach to vibrational systems. Our results show that mechanisms favoring synchronization, even between oscillators that are not directly coupled to each other, are transient energy depletion and crosstalk. We also address the possible buildup of quantum correlations during synchronization and show that indeed entanglement may be generated in detuned systems, starting from uncorrelated states and without any direct coupling between the two oscillators.
The emergence of mutual synchronization between oscillators, largely studied in classical systems, is explored in quantum systems. After reviewing recent research about the quantum aspects of this phenomenon, we present results obtained in the case of coupled harmonic oscillators with different natural frequencies. The presence of different dissipation processes, described by common or separate heat baths, plays a crucial role in the dynamical properties of the system. We show that mutual synchronization phenomena may appear only for a common bath. Furthermore we show that the emergence of synchronization is related to robust quantum correlations between the oscillators. Synchronization phenomena appear indeed in correspondence with large plateau where mutual information and quantum discord decay slowly, showing for the first time a quantum feature of spontaneous synchronization.
We study the relationship between the collective phenomena of super- and subradiance and spontaneous synchronization of quantum systems. To this aim we revisit the case of two detuned qubits interacting through a pure dissipative bosonic environment, which contains the minimal ingredients for our analysis. By using the Liouville formalism, we are able to find analytically the ultimate connection between these phenomena. We find that dynamical synchronization is due to the presence of longstanding coherence between the ground state of the system and the subradiant state. We finally show that, under pure dissipation, the emergence of spontaneous synchronization and of subradiant emission occur on the same time scale. This reciprocity is broken in the presence of dephasing noise.
Can synchronization -- the widespread spontaneous emergence of coordinated dynamics in classical nonlinear systems -- also occur in the quantum regime? This question has recently sparked intense research into collective behavior and temporal self-organization in quantum systems. Typical quantum features such as the linearity of time evolution and the presence of quantum noise seem to hinder the appearance of synchrony at the microscopic level. At the same time, quantum coherence and quantum correlations may provide novel mechanisms for enhancing synchronization beyond its classical counterpart. We here survey recent theoretical and experimental advances in quantum synchronization, ranging from the characterization of synchronous oscillations and genuinely nonclassical forms of synchrony to many-body synchronization on quantum networks.
Strongly interacting quantum many-body systems are expected to thermalize, however, some evade thermalization due to symmetries. Quantum synchronization provides one such example of ergodicity breaking, but previous studies have focused on open systems. Here, motivated by the problem of ergodicity breaking in closed systems and the study of non-trivial dynamics, we investigate synchronization in a closed disordered Heisenberg spin chain. In the presence of large random disorder, strongly breaking the permutation symmetry of the system, we observe the emergence of spatial synchronization, where spins lock into locally synchronized patches. This behavior can be interpreted as a fragmentation of the global dynamical symmetry S+ into a collection of local dynamical symmetries, each characterized by a distinct frequency. In the weak-disorder regime, still without permutation symmetry, we show that the synchronization mechanism can be understood perturbatively within Krylov space. In the absence of disorder, the Krylov space associated with the dynamical symmetry S+ is two-dimensional. Introducing disorder couples this subspace to the remainder of the Krylov space. This coupling leads only to a second-order correction to the frequency of the dynamical symmetry, thereby preserving coherent oscillations despite the presence of small disorder. At stronger disorder, the perturbation modifies S+ so that it acquires a finite lifetime, providing an example of a transient dynamical symmetry.