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arXiv:2302.08866 · Physical Review Research

Geometric phase in quantum synchronization

Aaron Daniel, Christoph Bruder, Martin Koppenhöfer

Published Jun 23, 2023• 6 citations
arXivHTMLPDF

Abstract

We consider a quantum limit-cycle oscillator implemented in a spin system whose quantization axis is slowly rotated. Using a kinematic approach to define geometric phases in nonunitary evolution, we show that the quantum limit-cycle oscillator attains a geometric phase when the rotation is sufficiently slow. In the presence of an external signal, the geometric phase as a function of the signal strength and the detuning between the signal and the natural frequency of oscillation shows a structure that is strikingly similar to the Arnold tongue of synchronization. Surprisingly, this structure vanishes together with the Arnold tongue when the system is in a parameter regime of synchronization blockade. We derive an analytic expression for the geometric phase of this system, valid in the limit of slow rotation of the quantization axis and weak external signal strength, and we provide an intuitive interpretation for this surprising effect.

DOI
10.1103/physrevresearch.5.023182

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We consider a quantum limit-cycle oscillator implemented in a spin system whose quantization axis is slowly rotated. Using a kinematic approach to define geometric phases in nonunitary evolution, we show that the quantum limit-cycle oscillator attains a geometric phase when the rotation is sufficiently slow. In the presence of an external signal, the geometric phase as a function of the signal strength and the detuning between the signal and the natural frequency of oscillation shows a structure that is strikingly similar to the Arnold tongue of synchronization. Surprisingly, this structure vanishes together with the Arnold tongue when the system is in a parameter regime of synchronization blockade. We derive an analytic expression for the geometric phase of this system, valid in the limit of slow rotation of the quantization axis and weak external signal strength, and we provide an intuitive interpretation for this surprising effect.
Driven-dissipative quantum oscillators lock their phase, deform their limit cycles, and undergo dissipative phase transitions, yet these behaviors are read from unrelated quantities defined on the same steady-state density matrix. We show that a single geometric quantity organizes them. Winding the phase of the drive generates a closed loop of nonequilibrium steady states, and because the Liouvillian is covariant under number rotations, the kinematic mixed-state geometric phase of this loop reduces exactly to an eigensystem functional of a single steady state. Under weak driving, it is governed by the same nearest-neighbor coherences that produce phase locking and inherits the Arnold tongue of synchronization. Near the Hopf threshold, it registers the nonperturbative reorganization of the steady-state eigenvectors. And in the squeezing-driven Kerr resonator, it develops distinct signatures at the first and second order dissipative phase transitions. The geometric phase thus provides a unified and experimentally accessible characterization of steady-state reorganization.
Synchronization, a ubiquitous phenomenon in classical systems, has recently been extended to the quantum domain. Here, we show quantum synchronization of a bosonic mode exhibiting a Fock state-like limit cycle, manifesting as a steady state with a negative Wigner function. We demonstrate that this non-classical state can be phase-locked to an external drive, achieving synchronization within an Arnold tongue regime. We argue that synchronization is a dynamical property and fundamentally tied to the suppression of phase slips, which we show to occur with exponentially decreasing probability. We introduce a novel method to extract the phase slip rate from the Lindblad time evolution of the system. This work opens new avenues for understanding and manipulating non-classical synchronization dynamics.
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.
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.
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 study the synchronization of dissipatively coupled van der Pol oscillators in the quantum limit, when each oscillator is near its quantum ground state. Two quantum oscillators with different frequencies exhibit an entanglement tongue, which is the quantum analog of an Arnold tongue. It means that the oscillators are entangled in steady state when the coupling strength is greater than a critical value, and the critical coupling increases with detuning. An ensemble of many oscillators with random frequencies still exhibits a synchronization phase transition in the quantum limit, and we analytically calculate how the critical coupling depends on the frequency disorder. Our results can be experimentally observed with trapped ions or neutral atoms.