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arXiv:2212.09388 · Physical Review A

Symmetries and synchronization blockade

Parvinder Solanki, Faraz Mohd Mehdi, Michal Hajdušek, Sai Vinjanampathy

Published Aug 18, 2023• 20 citations
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Abstract

Synchronization blockade refers to an interferometric cancellation of quantum synchronization. In this paper, we show how the choice of synchronization measure and Hamiltonian symmetries affect the discussion of synchronization blockade. Using counting principles, we prove a general theorem that synchronization blockade cannot be observed in an $N$-level system when the coherent state used to define the diagonal limit-cycle state is in the full $\text{SU}(N)$ group. We present several illustrative examples of synchronization blockade in multilevel systems and prove that information-theoretic measures of synchronization can also observe synchronization-blockade-like behavior by an appropriate choice of the set of limit-cycle states.

DOI
10.1103/physreva.108.022216

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Synchronization blockade refers to an interferometric cancellation of quantum synchronization. In this paper, we show how the choice of synchronization measure and Hamiltonian symmetries affect the discussion of synchronization blockade. Using counting principles, we prove a general theorem that synchronization blockade cannot be observed in an NNN-level system when the coherent state used to define the diagonal limit-cycle state is in the full SU(N)\text{SU}(N)SU(N) group. We present several illustrative examples of synchronization blockade in multilevel systems and prove that information-theoretic measures of synchronization can also observe synchronization-blockade-like behavior by an appropriate choice of the set of limit-cycle states.
We study the quantum synchronization of a single spin driven by an external semiclassical signal for spin numbers larger than S=1 , the smallest system to host a quantum self-sustained oscillator. The occurrence of interference-based quantum synchronization blockade is found to be qualitatively different for integer vs. half-integer spin number S . We explain this phenomenon as the interplay between the external signal and the structure of the limit cycle in the generation of coherence in the system. Moreover, we show that the same dissipative limit-cycle stabilization mechanism leads to very different levels of quantum synchronization for integer vs. half-integer S . However, by choosing an appropriate limit cycle for each spin number, comparable levels of quantum synchronization can be achieved for both integer and half-integer spin systems.
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 QQQ 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.
Limit-cycle oscillators are the basic building blocks for synchronization; yet, the notion of a quantum limit cycle has remained unclear. Here, we study quantum limit cycles and synchronization in the presence of continuous heterodyne measurement. The resulting quantum trajectories, i.e., time evolutions of the quantum state conditioned on the measurement outcome, make the quantum limit cycles apparent. We focus on the paradigmatic model of the quantum van der Pol oscillator and on two-level systems. Our work provides insights into limit cycles in quantum systems, emphasizing their similarity to classical limit cycles subject to noise. Additionally, we connect theoretical measures of quantum synchronization to quantities experimentally accessible via heterodyne detection.
We theoretically describe macroscopic quantum synchronization effects occurring in a network of all-to-all coupled quantum limit-cycle oscillators. The coupling causes a transition to synchronization as indicated by the presence of global phase coherence. We demonstrate that the microscopic quantum properties of the oscillators qualitatively shape the synchronization behavior in a macroscopically large network. Specifically, they result in a blockade of collective synchronization that is not expected for classical oscillators. Additionally, the macroscopic ensemble shows emergent behavior not present at the level of two coupled quantum oscillators.
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.
Quantum synchronization has been a subject of intensive research in the last decade. In this Letter, we propose a quantum Liénard system whose classical equivalent features two limit cycles to one of which the system will converge. In the quantum case, both limit cycles coexist in a single steady state. Each of these limit cycles localizes to a distinct phase if coupled to an external drive: one quantum state can thus be assigned two phases. Furthermore, coupling two such oscillators leads to the simultaneous appearance of synchronization and a synchronization blockade. To shed light on this apparent paradoxical result, we introduce finer measures of quantum synchronization.