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arXiv:1506.03682

Oscillation and synchronization of two quantum van der Pol oscillators

Lisa Morgan, Haye Hinrichsen B

Published Jun 11, 2015• 0 citations
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Abstract

The synchronization properties of two self-sustained quantum oscillators are studied in the Wigner representation. Instead of considering the quantum limit of the quantum van-der-Pol master equation we derive the quantum master equation directly from a suitable Hamiltonian. Moreover, the oscillators are coupled in incorporating an additional phase factor which shows up in the mutual correlations.

DOI
10.48550/arxiv.1506.03682

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The synchronization properties of two self-sustained quantum oscillators are studied in the Wigner representation. Instead of considering the quantum limit of the quantum van-der-Pol master equation we derive the quantum master equation directly from a suitable Hamiltonian. Moreover, the oscillators are coupled in incorporating an additional phase factor which shows up in the mutual correlations.
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 414141-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.
Recently, several studies have investigated synchronization in quantum-mechanical limit-cycle oscillators. However, the quantum nature of these systems remained partially hidden, since the dynamics of the oscillator's phase was overdamped and therefore incoherent. We show that there exist regimes of underdamped and even quantum-coherent phase motion, opening up new possibilities to study quantum synchronization dynamics. To this end, we investigate the Van der Pol oscillator (a paradigm for a self-oscillating system) synchronized to an external drive. We derive an effective quantum model which fully describes the regime of underdamped phase motion and additionally allows us to identify the quality of quantum coherence. Finally, we identify quantum limit cycles of the phase itself.
We implement nonlinear anharmonic interaction in the coupled van der Pol oscillators to investigate the quantum synchronization behavior of the systems. We study the quantum synchronization in two oscillator models, coupled quantum van der Pol oscillators and anharmonic self-oscillators. We demonstrate that the considered systems exhibit a high-order synchronization through coupling in both classical and quantum domains. We show that, due to the anharmonicity of the nonlinear interaction between the oscillators, the system exhibits phonon blockade in the phase-locking regime, which is a pure nonclassical effect and has not been observed in the classical domain. We also demonstrate that, for coupled anharmonic oscillators, the system shows a multiple resonance phase-locking behavior due to nonlinear interaction. We point out that the synchronization blockade arises due to strong anticorrelation between the oscillators, which leads to phonon antibunching in the same parametric regime. In the anharmonic oscillator case we illustrate the simultaneous occurrence of bunching and antibunching effects as a consequence of simultaneous negative and positive correlation between the anharmonic oscillators. We examine the aforementioned characteristic features in the frequency entrainment of the oscillators using a power spectrum where one can observe normal mode splitting and the Mollow triplet in the strong coupling regime. Finally, we propose a possible experimental realization for the considered system in the trapped ion and optomechanical setting.
We introduce a quantum spin van der Pol (vdP) oscillator as a prototypical model of quantum spin-based limit-cycle oscillators, which coincides with the quantum optical vdP oscillator in the high-spin limit. The system is described as a noisy limit-cycle oscillator in the semiclassical regime at large spin numbers, exhibiting frequency entrainment to a periodic drive. Even in the smallest spin-1 case, mutual synchronization, Arnold tongues, and entanglement tongues in two dissipatively coupled oscillators, and collective synchronization in all-to-all coupled oscillators are clearly observed. The proposed quantum spin vdP oscillator will provide a useful platform for analyzing quantum spin synchronization.
Classical self-sustained oscillators, which generate periodic motion without periodic external forcing, are ubiquitous in science and technology. The realization of nonclassical self-oscillators is an important goal of quantum physics. We here present the experimental implementation of a quantum van der Pol oscillator, a paradigmatic autonomous quantum driven-dissipative system with nonlinear damping, using a single trapped atom. We demonstrate the existence of a quantum limit cycle in phase space in the absence of a drive and the occurrence of quantum synchronization when the nonlinear oscillator is externally driven. We additionally show that synchronization can be enhanced with the help of squeezing perpendicular to the direction of the drive and, counterintuitively, linear dissipation. We also observe the bifurcation to a bistable phase-space distribution for large squeezing. Our results pave the way for the exploration of self-sustained quantum oscillators and their application to quantum technology.