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...

A superconducting circuit of parametric oscillators realizes a robust quantum optimizer with full connectivity and zero overhead.

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.

Precision spectroscopy of atomic and molecular ions offers a window to new physics, but is typically limited to species with a cycling transition for laser cooling and detection. Quantum logic spectroscopy has overcome this limitation for species with long-lived excited states. Here we extend quantum logic spectroscopy to fast, dipole-allowed transitions and apply it to perform an absolute frequency measurement. We detect the absorption of photons by the spectroscopically investigated ion through the photon recoil imparted on a co-trapped ion of a different species, on which we can perform efficient quantum logic detection techniques. This amplifies the recoil signal from a few absorbed photons to thousands of fluorescence photons. We resolve the line centre of a dipole-allowed transition in 40 Ca þ to 1/300 of its observed linewidth, rendering this measurement one of the most accurate of a broad transition. The simplicity and versatility of this approach enables spectroscopy of many previously inaccessible species. P recision optical spectroscopy of broad transitions provides information on the structure of molecules 1 , it allows tests of quantum electrodynamics 2 , and, through comparison with astrophysical data, probes for a possible variation of fundamental constants over cosmological scales 3,4 . Nuclear properties are revealed through isotope shift measurements 5-9 , or absolute frequency measurements 10,11 . Trapped ions are particularly well suited for such high precision experiments. The ions are stored in an almost field-free environment and can be laser-cooled to eliminate Doppler shifts. These features have enabled record accuracies in optical clocks [12][13][14][15] . For long-lived excited states such as in atoms with clock transitions, the electron-shelving technique amplifies the signal from a single absorbed photon by scattering many photons on a closed transition through selective optical coupling of one of the two spectroscopy states to a third electronic level 16 . The invention of quantum logic spectroscopy (QLS) 12,17 removed the need to detect the signal on the spectroscopically investigated ion (spectroscopy ion) by transferring the internal state information through a series of laser pulses to the co-trapped, so-called logic ion where the signal is observed via the electron-shelving technique. However, this original implementation of QLS requires long-lived spectroscopy states to implement the transfer sequence. For transitions with a short-lived excited state, spectroscopy of trapped ions is typically implemented through detection of scattered photons in laserinduced fluorescence 7,18-22 or detection of absorbed photons in laser absorption spectroscopy 23 . Neither of the two techniques reaches the fundamental quantum projection noise limit as in the electron-shelving technique 24 due to low light collection efficiency in laser-induced fluorescence and small atom-light coupling in laser absorption spectroscopy. In a variation of absorption spectroscopy, the detuning-dependent effect of...