Synchronization is a phenomenon where interacting particles lock their motion and display non-trivial dynamics. Despite intense efforts studying synchronization in systems without clear classical limits, no comprehensive theory has been found. We develop such a general theory based on novel necessary and sufficient algebraic criteria for persistently oscillating eigenmodes (limit cycles) of time-independent quantum master equations. We show these eigenmodes must be quantum coherent and give an exact analytical solution for all such dynamics in terms of a dynamical symmetry algebra. Using our theory, we study both stable synchronization and metastable/transient synchronization. We use our theory to fully characterise spontaneous synchronization of autonomous systems. Moreover, we give compact algebraic criteria that may be used to prove absence of synchronization. We demonstrate synchronization in several systems relevant for various fermionic cold atom experiments.
Synchronization is a phenomenon where interacting particles lock their motion and display non-trivial dynamics. Despite intense efforts studying synchronization in systems without clear classical limits, no comprehensive theory has been found. We develop such a general theory based on novel necessary and sufficient algebraic criteria for persistently oscillating eigenmodes (limit cycles) of time-independent quantum master equations. We show these eigenmodes must be quantum coherent and give an exact analytical solution for all such dynamics in terms of a dynamical symmetry algebra. Using our theory, we study both stable synchronization and metastable/transient synchronization. We use our theory to fully characterise spontaneous synchronization of autonomous systems. Moreover, we give compact algebraic criteria that may be used to prove absence of synchronization. We demonstrate synchronization in several systems relevant for various fermionic cold atom experiments.
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.
Recently, it was shown that dissipative quantum systems with three or more levels are able to synchronize to an external signal. Some researchers have stated that this is not possible for two-level systems, as those systems lack a stable limit cycle in unperturbed dynamics, but others have demonstrated, under a different definition of synchronization, that it is possible in qubits in models which include other elements. We show how a quantum two-level system can be understood as containing a valid limit cycle as the starting point of synchronization and that it can synchronize its dynamics to an external weak signal. This is demonstrated by analytically solving the Lindblad equation of a two-level system coupled to an environment, determining the steady state. This is a mixed state with contributions from many pure states, each of which provides a valid limit cycle. We show that this is sufficient to phase lock the dynamics to a weak external signal, hence clarifying synchronization in two-level systems. We use the Husimi Q representation to analyze the synchronization region, defining a synchronization measure which characterizes the strength of the phase locking. Also, we study the stability of the limit cycle and its deformation with the strength of the signal in terms of the components of the Bloch vector of the system. Finally, we generalize a model of the three-level system to illustrate how the stationary fixed point of that model can be changed into a limit cycle similar to the one that we describe for the two-level system.
From the motion of hurricanes to the cooling of a cup of tea, dissipative dynamics are ubiquitous in nature. Unfortunately, when studying lattice-based strongly correlated quantum systems, a common focus of experiments and highly relevant for material science, the curse of dimensionality prevents us from directly solving the exact equations describing the dissipative motion. We instead often use approximations and numerical analysis, as few analytical techniques currently exist which can provide useful insights. This thesis studies the constraints that non-abelian algebraic structures place on the dynamics of such non-equilibrium many-body quantum systems. In doing so, we develop novel algebraic methods that provide previously absent analytic insight into a range of problems. We apply these new techniques to study dynamical phases of quantum matter that are of both theoretical and experimental interest. We study how simple algebraic structures imply persistent non-stationarity in dissipative many-body quantum systems. These results enable us to develop a theory of quantum synchronisation based on symmetries and to explore a novel class of time crystals where order is induced by noise. These dynamical phases of matter have been gaining recent interest for their insights into thermalisation and their potential application in quantum technologies. We therefore use our results to explore experimentally realisable models. We also use the general principles of prethermalisation to propose a new theory to potentially explain recent experiments studying light-induced superconductivity in terms of eta-pairing and approximate symmetries. We discuss the merits and weaknesses of our proposal in relation to the experimental evidence. We also indicate techniques that, within our theory, stabilise superconducting states by harnessing the quantum Zeno effect For certain systems, known as integrable, there are enough symmetries to allow for exact solutions using the Bethe ansatz. We demonstrate that Bethe ansatz techniques can be extended and applied to non-equilibrium problems. As an example, we use this method to analytically study the dynamical phase transitions of an XXZ spin chain with boundary loss.
Spontaneous macroscopic quantum synchronization is an emergent phenomenon where an ensemble of quantum oscillators achieves global phase coherence through the interplay of interaction and dissipation. To illuminate this phenomenon, we study an ensemble of two-level systems (TLS) and establish its associated nonlinear quantum master equation, for which self-consistent analytical solutions of quantum synchronization can be obtained. The trajectories on the Bloch sphere vividly illustrate how dissipation and interaction drive the system toward a synchronized state. We present a phase diagram for macroscopic synchronization as a function of interaction strength and the gain-to-damping ratio. Furthermore, we demonstrate full synchronization and partial synchronization between two groups of TLS with different natural frequencies. This work establishes ensemble of TLS as a remarkable system for understanding spontaneous quantum synchronization.
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 slow-decaying 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.
Abstract We investigate the long-time behavior of quantum N -level systems that are coupled to a Markovian environment and subject to periodic driving. As our main result, we obtain a general algebraic condition ensuring that all solutions of a periodic quantum master equation with Lindblad form approach a unique limit cycle. Quite intuitively, this criterion requires that the dissipative terms of the master equation connect all subspaces of the system Hilbert space during an arbitrarily small fraction of the cycle time. Our results provide a natural extension of Spohn’s algebraic condition for the approach to equilibrium to systems with external driving. Moreover, our theory leads to a rigorous condition for the emergence of dissipative discrete time crystals and covers also classical, periodically modulated Markov jump processes.