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.
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.
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.
Natural and engineered classical systems are replete with examples of synchronization, understood as the adjustment of rhythms of physical systems. Such synchronization is at the heart of the stability of several classical technologies, such as mechanical bridges and electrical networks. Given the advent of quantum simulation and computation technologies, it is natural to study a quantum analogue of synchronization and explore novel applications. This review surveys synchronization in few and many-body quantum systems, measures that quantify them, and their applications to quantum technologies.
Motivated by the spin self-rephasing recently observed in an atomic clock, we introduce a simple dynamical model to study the competition between dephasing and synchronization. Two spins
S are taken to be initially parallel and in the plane perpendicular to an inhomogeneous magnetic field
Δ that tends to dephase them. In addition, the spins are coupled by exchange interaction
J that tries to keep them locked. The analytical solution of the classical dynamics shows that, there is a phase transition to a synchronized regime for sufficiently large exchange interaction
J>Δ compared to the inhomogeneity. The quantum dynamics is solved analytically in four limits -- large/small
J/Δ and large/small
S -- and numerically in between. In sharp contrast to the classical case, the quantum solution features very rich
S-dependent multiscale dynamics. For any finite
S, there is no synchronization but a crossover around
J=Δ between two regimes. The synchronization transition is only recovered when
S→∞, approaching the classical solution in a non-trivial way. Quantum effects therefore suppress the synchronization transition.