Genuine Quantum Signatures in Synchronization of Anharmonic Self-Oscillators

Abstract

We study the synchronization of a Van der Pol self-oscillator with Kerr anharmonicity to an external drive. We demonstrate that the anharmonic, discrete energy spectrum of the quantum oscillator leads to multiple resonances in both phase locking and frequency entrainment not present in the corresponding classical system. Strong driving close to these resonances leads to nonclassical steady-state Wigner distributions. Experimental realizations of these genuine quantum signatures can be implemented with current technology. DOI: 10.1103/PhysRevLett.117.073601 The synchronization of self-oscillators is a subject with great relevance to several natural sciences [1,2]. Its exciting frontiers include neuronal synchronization in the human brain [3,4] and stabilization of power-grid networks [5], as well as the engineering of high-precision clocks [6,7]. Recent advances in nanotechnology will enable experiments with large arrays of self-oscillators in the near future [8,9]. Whereas most research has focused on the classical domain, synchronization in the quantum regime [10] has become a very active topic. There has been much recent experimental progress with micro-and nanomechanical systems [11][12][13][14][15] [29,30].Studying a Van der Pol oscillator, the most prominent example of a self-oscillator, recent theoretical work characterized how synchronization quantitatively differs between its quantum and classical realization in phase locking [24,25] as well as in frequency entrainment [21,22]. While synchronization is hindered by quantum noise compared to the classical model [21,22], noise is less detrimental [24,25] than one would expect from a semiclassical description.In this Letter, we study self-oscillators for which both the damping and the frequency are amplitude dependent. We show that their synchronization behavior is qualitatively different in the quantum and the classical regime. Focusing on a Van der Pol oscillator with Kerr anharmonicity, we find two genuine quantum signatures. First, while the synchronization of one such oscillator to an external drive is maximal at one particular frequency classically, the corresponding quantum system shows a tendency to synchronize at multiple frequencies. Using perturbation theory in the drive strength, we demonstrate that these multiple resonances reflect the quantized anharmonic energy spectrum of the oscillator. We show that these features are observable in the phase probability distribution if the Kerr anharmonicity is large compared to the relaxation rates and the system is in the quantum regime; i.e., the limit cycle amplitudes are small. In the semiclassical limit, the energy spectrum becomes continuous, so that the resonances (and therefore the quantized energy spectrum) cannot be resolved. Using numerically exact simulations of the full quantum master equation, we find a second genuine quantum signature: For strong driving close to these resonances, the steady-state Wigner distribution exhibits areas of negative density; i.e., the steady state is nonclassica...

Synchronization of micromasers

“…Our assumptions mean that we need only consider the components ρ (p) n,m for which n, m ≫ p. We proceed by expanding the coefficients in (22) treating p/n, p/m, 1/n and 1/m as small quantities and keeping the lowest order (non-zero) contributions in each case so that we have…”

“…This is something that we will make extensive use of here, though it should be noted that the choice is by no means unique [18,22]. The relative phase distribution for the micromaser system is obtained by solving for the steady-state of the master equation (1) using standard numerical methods [40].…”

“…Synchronization has also been studied in quantum optical systems such as the laser, although generally focussing on regimes where approximate semiclassical descriptions work well [2,3]. In the last few years there has been considerable interest in studying the synchronization of oscillators and related systems [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] close to threshold or at low excitation levels where semiclassical approaches break down and fully quantum mechanical calculations are required. Recent theoretical work has explored different ways of quantifying synchronization in quantum oscillators [5,7,14,15,20], as well as investigating the connection between it and measures of correlation such as mutual information and entanglement [5,8,10,15,19].…”

“…Studies of synchronization effects in the quantum regime have largely concentrated on the behavior of simple model systems such as van der Pol oscillators [7,9,10,12,15,17,19,21] (together with closely related models [22]), though a number of other systems including atomic ensembles [13,16] and optomechanical oscillators [5,6,12,18] have also been investigated. In this article we investigate synchronization in a very different model system consisting of two weakly coupled micromasers.…”

Observing quantum synchronization blockade in circuit quantum electrodynamics

“…when the limit cycle steady states of the oscillators are quantum states with no classical analog. Previous work on quantum synchronization has focused mainly on theoretically identifying and characterizing differences between classical and quantum synchronization [7][8][9][10][11][12][13][14][15][16][17][18] and on potential applications of the latter [19][20][21]. Experimental observation of quantum synchronization phenomena is hindered by the stringent requirements of high quantum coherence and strong nonlinearities, both of which are also key requirements for quantum computation.…”