Abstract The paper concerns spontaneous asymptotic phase-locking and synchronization in two-qubit systems undergoing continuous Markovian evolution described by Lindbladian dynamics with normal Lindblad operators. Using analytic methods, all phase-locking-enforcing mechanisms within the given framework are obtained and classified. Detailed structures of their respective attractor spaces are provided and used to explore their properties from various perspectives. Amid phase-locking processes those additionally enforcing identical stationary parts of both qubits are identified, including as a special case the strictest form of synchronization conceivable. A prominent basis is presented which reveals that from a physical point of view two main types of phase-locking mechanisms exist. The ability to preserve information about the initial state is explored and an upper bound on the amplitude of oscillations of the resulting phase-locked dynamics is established. Permutation symmetry of both asymptotic states and phase-locking mechanisms is discussed. Lastly, the possibility of entanglement production playing the role of a phase-locking witness is rebutted by three analytically treatable examples.
Abstract The paper concerns spontaneous asymptotic phase-locking and synchronization in two-qubit systems undergoing continuous Markovian evolution described by Lindbladian dynamics with normal Lindblad operators. Using analytic methods, all phase-locking-enforcing mechanisms within the given framework are obtained and classified. Detailed structures of their respective attractor spaces are provided and used to explore their properties from various perspectives. Amid phase-locking processes those additionally enforcing identical stationary parts of both qubits are identified, including as a special case the strictest form of synchronization conceivable. A prominent basis is presented which reveals that from a physical point of view two main types of phase-locking mechanisms exist. The ability to preserve information about the initial state is explored and an upper bound on the amplitude of oscillations of the resulting phase-locked dynamics is established. Permutation symmetry of both asymptotic states and phase-locking mechanisms is discussed. Lastly, the possibility of entanglement production playing the role of a phase-locking witness is rebutted by three analytically treatable examples.
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.
Abstract In nature, instances of synchronisation abound across a diverse range of environments. In the quantum regime, however, synchronisation is typically observed by identifying an appropriate parameter regime in a specific system. In this work we show that this need not be the case, identifying conditions which, when satisfied, guarantee that the individual constituents of a generic open quantum system will undergo completely synchronous limit cycles which are, to first order, robust to symmetry-breaking perturbations. We then describe how these conditions can be satisfied by the interplay between several elements: interactions, local dephasing and the presence of a strong dynamical symmetry—an operator which guarantees long-time non-stationary dynamics. These elements cause the formation of entanglement and off-diagonal long-range order which drive the synchronised response of the system. To illustrate these ideas we present two central examples: a chain of quadratically dephased spin-1s and the many-body charge-dephased Hubbard model. In both cases perfect phase-locking occurs throughout the system, regardless of the specific microscopic parameters or initial states. Furthermore, when these systems are perturbed, their nonlinear responses elicit long-lived signatures of both phase and frequency-locking.
Abstract We discuss the generation and the long-time persistence of entanglement in open two-qubit systems whose reduced dissipative dynamics is not a priori engineered but is instead subjected to filtering and Markovian feedback. In particular, we analytically study (1) whether the latter operations may enhance the environment capability of generating entanglement at short times and (2) whether the generated entanglement survives in the long-time regime. We show that, in the case of particularly symmetric Gorini–Kossakowski–Sudarshan–Lindblad it is possible to fully control the convex set of stationary states of the two-qubit reduced dynamics, therefore the asymptotic behaviour of any initial two-qubit state. We then study the impact of a suitable class of feed-back operations on the considered dynamics.
We study quantum van der Pol oscillator in which single-photon pumping competes with a nonlinear two-photon dissipative channel to produce a quantum limit cycle, and the question of how two such oscillators with non-identical intrinsic frequencies synchronize in the sense of phase locking when coupled dissipatively to one another. For a single oscillator, starting from a Lindblad description of the dynamics, we derive the exact Wigner-function evolution equation, and show that its Fokker--Planck approximation admits in the appropriate regime an equivalent classical Itô stochastic description that reproduces remarkably well the full Lindblad dynamics. Notably, the stochastic approach provides an efficient means of capturing system dynamics in the semiclassical regime, where a large number of energy eigenstates contribute and direct numerical simulations become computationally demanding. Restricting to the three lowest Fock states and using a parity symmetry, we obtain in closed form the stationary-state density matrix and its Wigner-function representation that displays pronounced nonclassical features. For two non-identical oscillators, we study the full Lindblad dynamics numerically and characterize synchronization through the stationary-state order parameter as a function of frequency mismatch and dissipative coupling. We find that increasing the effective nonlinear dissipative strength suppresses the synchronized region. Restricting the two oscillators to the three-level qutrit subspace, a parity symmetry reduces the stationary-state problem to an exact 41-dimensional linear system, allowing to compute the synchronization order parameter. In this reduced description, results in the limit of large dissipation agree closely with those from the full dynamics, while exhibiting in the limit of small dissipation a double-peak synchronization structure accompanied by a phase reversal.
In a dissipative regime, we study the properties of several qubits coupled to a driven resonator in the framework of a Jaynes-Cummings model. The time evolution and the steady state of the system are numerically analyzed within the Lindblad master equation, with up to several million components. Two semi-analytical approaches, at weak and strong (semiclassical) dissipations, are developed to describe the steady state of this system and determine its validity by comparing it with the Lindblad equation results. We show that the synchronization of several qubits with the driving phase can be obtained due to their coupling to the resonator. We establish the existence of two different qubit synchronization regimes: In the first one, the semiclassical approach describes well the dynamics of qubits and, thus, their quantum features and entanglement are suppressed by dissipation and the synchronization is essentially classical. In the second one, the entangled steady state of a pair of qubits remains synchronized in the presence of dissipation and decoherence, corresponding to the regime non-existent in classical synchronization.