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arXiv:2405.05182 · Physical Review A

Quantum synchronization through the interference blockade

Tobias Kehrer, Tobias Nadolny, Christoph Bruder

Published Oct 4, 2024• 16 citations
arXivHTMLPDF

Abstract

Synchronization manifests itself in oscillators adjusting their frequencies and phases with respect to an external signal or another oscillator. In the quantum case, new features appear such as destructive interferences that can result in the suppression of phase locking. A three-level (spin-1) oscillator with equal gain and damping rates and subject to an external drive does not exhibit any 1:1 phase locking but 2:1 phase locking, i.e., its phase distribution features two maxima. This bistable locking at two opposite phases is a signature of the quantum interference synchronization blockade. An analogous behavior was found for two identical coupled spin-1 oscillators. In this work, we consider two coupled spin-1 oscillators and a drive applied to the first spin. This leads to two interference blockades between the drive and the first spin as well as between both spins. Although both interference blockades persist for strong drive and coupling strengths, remarkably, the undriven spin does show a 1:1 phase locking to the external drive. The magnitude of the locking is proportional to the drive strength if the drive strength is small. In other words, the undriven oscillator synchronizes to the external drive through both interference blockades while the blockades persist. For a chain of three coupled spin-1 oscillators, we find synchronization between the first and third spins mediated via the blockaded, second spin.

DOI
10.1103/physreva.110.042203

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Synchronization manifests itself in oscillators adjusting their frequencies and phases with respect to an external signal or another oscillator. In the quantum case, new features appear such as destructive interferences that can result in the suppression of phase locking. A three-level (spin-1) oscillator with equal gain and damping rates and subject to an external drive does not exhibit any 1:1 phase locking but 2:1 phase locking, i.e., its phase distribution features two maxima. This bistable locking at two opposite phases is a signature of the quantum interference synchronization blockade. An analogous behavior was found for two identical coupled spin-1 oscillators. In this work, we consider two coupled spin-1 oscillators and a drive applied to the first spin. This leads to two interference blockades between the drive and the first spin as well as between both spins. Although both interference blockades persist for strong drive and coupling strengths, remarkably, the undriven spin does show a 1:1 phase locking to the external drive. The magnitude of the locking is proportional to the drive strength if the drive strength is small. In other words, the undriven oscillator synchronizes to the external drive through both interference blockades while the blockades persist. For a chain of three coupled spin-1 oscillators, we find synchronization between the first and third spins mediated via the blockaded, second spin.
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.
We investigate the dynamics of a quadratically coupled system under the influence of an external drive applied to the second oscillator, where the coupling facilitates a high-order synchronization with phase-locking emerging in the form of 2:1 between the oscillators. Our analysis reveals a synchronization blockade in the first oscillator, characterized by the complete suppression of conventional 1:1 phase-locking with the drive. In contrast, we observe that the directly driven second oscillator synchronizes with the drive, showing 1:1 phase-locking but notably at second harmonic frequency. A classical mean-field analysis of the corresponding equations of motion reproduces this asymmetric phase-locking geometry which demonstrates that the phase-locking structure itself can be understood from the nonlinear classical dynamics. The quantum analysis, however, reveals the microscopic origin of the synchronization blockade. Furthermore, we show that the system exhibits mutual synchronization when both the oscillators satisfies the resonance condition, enabling coherent energy exchange facilitated by nonlinear quadratic coupling. The mutual synchronization shows synchronized regimes and also subtle suppression of synchronized regimes near resonance occurring due to spectral splitting of the energy states. Using perturbation analysis of the master equation within the low excitation subspace, we analyze steady-state phase distribution and synchronization measures, supported by population statistics and spectral responses. We also propose possible experimental realizations in trapped-ions and optomechanical setups. These findings highlight the crucial role of quadratic coupling in enabling nonclassical synchronization phenomena, offering deeper insights for quantum control strategies and the development of quantum information platforms.
We report a normal-mode induced synchronization blockade in coupled quantum van der Pol oscillators under the influence of external drive. In this mechanism, the coupling hybridizes the oscillator modes into spectrally split normal modes. The destructive interference between the transitions to these modes blocks synchronization. We find that this blockade can be controlled simply by tuning the coupling strength and detuning allowing dynamic manipulation of quantum synchronization through collective mode dynamics. We analyze the phase-locking behaviour using perturbation analysis. Further, by deriving steady-state probability amplitudes we show how the energy redistribution and spectral splitting forms the basis of the blockade. Our results might provide new insights into how synchronization can be controlled in quantum systems.
It is well known that a system with two or more levels exists a limit cycle and can be synchronized with an external drive when the system and the drive are directly coupled. One might wonder if a system can synchronize with the external drive when they are not coupled directly. In this paper, we examine this case by considering a composite system consisting of two coupled two-level quantum systems, one of which is driven by an external field, while another couples to the driven one. Due to the decoherence caused by environments, the composite system would stay in a mixed state, and an effective limit cycle is formed, so phase locking could occur. We find the phase locking phenomenon in the phase diagram characterized by Husimi QQQ function, and the synchronization can be generated consequently that we will refer to indirect synchronization. The SSS function defined in the earlier study can also be used to measure the strength of synchronization. We claim that indirect synchronization is possible. This result provides us with a method to synchronize a quantum system that coupled to its neighbour without interacting with external drive directly.
Classically, the tendency towards spontaneous synchronization is strongest if the natural frequencies of the self-oscillators are as close as possible. We show that this wisdom fails in the deep quantum regime, where the uncertainty of amplitude narrows down to the level of single quanta. Under these circumstances identical self-oscillators cannot synchronize and detuning their frequencies can actually help synchronization. The effect can be understood in a simple picture: Interaction requires an exchange of energy. In the quantum regime, the possible quanta of energy are discrete. If the extractable energy of one oscillator does not exactly match the amount the second oscillator may absorb, interaction, and thereby synchronization, is blocked. We demonstrate this effect, which we coin quantum synchronization blockade, in the minimal example of two Kerr-type self-oscillators and predict consequences for small oscillator networks, where synchronization between blocked oscillators can be mediated via a detuned oscillator. We also propose concrete implementations with superconducting circuits and trapped ions. This paves the way for investigations of new quantum synchronization phenomena in oscillator networks both theoretically and experimentally.
Synchronization transmission describes the emergence of coherence between two uncoupled oscillators mediated by their mutual coupling to an intermediate one. In classical star networks, such mediated coupling gives rise to remote synchronization--where nonadjacent leaf nodes synchronize through a nonsynchronous hub--and to explosive synchronization, characterized by an abrupt collective transition to coherence. In the quantum regime, analogous effects can arise from the interplay between 1:1 phase locking and 2:1 phase-locking blockade in coupled spin-1 particles. In this work, we investigate a star network composed of spin-1 particles. For identical oscillators, symmetric and asymmetric dissipation lead to distinct transmission behaviors: remote synchronization and quasi-explosive synchronization appear in different coupling regimes, a phenomenon absent in classical counterparts. For nonidentical networks, we find that at large detuning remote synchronization emerges in the weak-coupling regime and evolves into quasi-explosive synchronization as the coupling increases, consistent with classical star-network dynamics. These findings reveal the rich dynamical characteristics of mediated quantum synchronization and point toward new possibilities for exploring synchronization transmission in larger and more complex quantum systems.