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...

Optomechanical systems driven by an effective blue detuned laser can exhibit self-sustained oscillations of the mechanical oscillator. These self-oscillations are a prerequisite for the observation of synchronization. Here, we study the synchronization of the mechanical oscillations to an external reference drive. We study two cases of reference drives: (1) An additional laser applied to the optical cavity; (2) A mechanical drive applied directly to the mechanical oscillator. Starting from a master equation description, we derive a microscopic Adler equation for both cases, valid in the classical regime in which the quantum shot noise of the mechanical self-oscillator does not play a role. Furthermore, we numerically show that, in both cases, synchronization arises also in the quantum regime. The optomechanical system is therefore a good candidate for the study of quantum synchronization.Comment: 9 pages, 6 figure

Coupling two or more self-oscillating systems may stabilize their zero-amplitude rest state, therefore quenching their oscillation. This phenomenon is termed "amplitude death." Well known and studied in classical self-oscillators, amplitude death was only recently investigated in quantum self-oscillators [Ishibashi and Kanamoto, Phys. Rev. E 96, 052210 (2017)2470-004510.1103/PhysRevE.96.052210]. Quantitative differences between the classical and quantum descriptions were found. Here, we demonstrate that for quantum self-oscillators with anharmonicity in their energy spectrum, multiple resonances in the mean phonon number can be observed. This is a result of the discrete energy spectrum of these oscillators, and is not present in the corresponding classical model. Experiments can be realized with current technology and would demonstrate these genuine quantum effects in the amplitude death phenomenon.

A Cooper pair splitter consists of two quantum dots side-coupled to a conventional superconductor. Usually, the quantum dots are assumed to have a large charging energy compared to the superconducting gap, in order to suppress processes other than the coherent splitting of Cooper pairs. In this work, in contrast, we investigate the limit in which the charging energy is smaller than the superconducting gap. This allows us, in particular, to study the effect of a Zeeman field comparable to the charging energy. We find analytically that in this parameter regime the superconductor mediates an interdot tunneling term with a spin symmetry determined by the Zeeman field. Together with electrostatically tunable quantum dots, we show that this makes it possible to engineer a spin triplet state shared between the quantum dots. Compared to previous works, we thus extend the capabilities of the Cooper pair splitter to create entangled nonlocal electron pairs.

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