Quantum Synchronization of a Driven Self-Sustained Oscillator

Abstract

Synchronization is a universal phenomenon that is important both in fundamental studies and in technical applications. Here we investigate synchronization in the simplest quantum-mechanical scenario possible, i.e., a quantum-mechanical self-sustained oscillator coupled to an external harmonic drive. Using the power spectrum we analyze synchronization in terms of frequency entrainment and frequency locking in close analogy to the classical case. We show that there is a step-like crossover to a synchronized state as a function of the driving strength. In contrast to the classical case, there is a finite threshold value in driving. Quantum noise reduces the synchronized region and leads to a deviation from strict frequency locking. PACS numbers: 05.45.Xt, Synchronization is an intriguing phenomenon exhibited by a wide range of physical, chemical, and biological systems [1]. The basic setting consists of coupled self-oscillating systems synchronizing their motion; examples include such different phenomena as orbital resonances in planetary motion or the rhythm of muscle cells in mammal hearts. A paradigmatic and widely studied model of synchronization is the Kuramoto model of coupled limit-cycle oscillators [2,3].The most fundamental scenario of classical synchronization is the frequency locking of a self-sustained oscillator which is externally driven by a harmonic force [1,4]. A self-sustained oscillator takes energy from a source, e.g. by negative damping, and can therefore maintain stable oscillatory motion and an undetermined phase in the presence of dissipation. If the oscillator is additionally driven by a harmonic force, there is a finite range of detuning for which the oscillator is frequencylocked to the drive, and noise can reduce or destroy this range of synchronization [1]. There are also regimes of frequency entrainment in which the oscillator frequency is pulled towards the drive frequency, but does not reach it. In this case, the frequency of the driven oscillator, the observed frequency ω obs , differs from both the natural frequency of the oscillator and the drive frequency. The simplest model exhibiting these effects is the van der Pol oscillator [1] which allows the analysis of the complex phenomenology of synchronization.Recently, the question if synchronization exists in quantum systems has attracted a lot of interest. There have been important first attempts to address this problem theoretically, from communities as diverse as trapped atomic ensembles, Josephson junctions, and nanomechanical systems [6][7][8][9][10][11][12][13][14][15][16]. Optomechanical systems [17] appear to offer a particularly promising approach. Recent experiments have reported classical synchronization of nanomechanical oscillators [18,19], the quantum many-body dynamics of an array of identical optomechanical cells has been predicted to show synchronized behavior [9], and quantitative measures for quantum synchronization based on the Heisenberg uncertainty principle have been applied to two and many coupled optomechani...

Synchronization of micromasers

“…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.…”

Quantum chimera states

“…In this paper we show that a simple quantum system, a closed chain of spins, offers quantum analogues of the chimera states if the couplings between the spins have the same structure that the couplings of the classical models. It is well known that spin chains can exhibit kinds of quantum disorder and of quantum chaos [8,9,10,11], and that quantum synchronization is related to the entanglement [12,13,14,15]. To involve a kind of chimera states, our model consists of a non-hermitian spin chain [16,17,19,22] which can be assimilated to a spin chain in contact with an environment.…”