Spin correlations as a probe of quantum synchronization in trapped-ion phonon lasers

Abstract

We investigate quantum synchronization theoretically in a system consisting of two cold ions in microtraps. The ions' motion is damped by a standing-wave laser while also being driven by a blue-detuned laser which results in self-oscillation. Working in a nonclassical regime, where these oscillations contain only a few phonons and have a sub-Poissonian number variance, we explore how synchronization occurs when the two ions are weakly coupled using a probability distribution for the relative phase. We show that strong correlations arise between the spin and vibrational degrees of freedom within each ion and find that when two ions synchronize their spin degrees of freedom in turn become correlated. This allows one to indirectly infer the presence of synchronization by measuring the ions' internal state.Introduction. Two macroscopic self-oscillators synchronize when their relative phase locks to a fixed value [1]. Important studies of synchronization effects were carried out using lasers [2], with arrays of Josephson junctions [3] and over the last few years much attention has been devoted to exploring synchronization in micromechanical oscillators [4]. Recently, theoretical work has begun to explore synchronization in the quantum regime [5][6][7][8][9][10][11][12][13][14]: the formation of a relative phase preference between two (or more) weakly coupled quantum oscillators operating in a regime far from the classical correspondence limit. Differences between classical and quantum predictions for the synchronization of van der Pol oscillators have been identified in the case where the oscillators are only weakly excited [5]. Nevertheless, many important questions about quantum synchronization remain open, such as how it should be quantified and how it can best be probed experimentally.Cold ions in microtraps provide a natural platform for exploring synchronization in the quantum regime [5]. The generation of self-oscillations in the motional state of ions, phonon lasing, has already been observed [15]. Furthermore, precise control of trapping potentials of the individual ions can now be achieved with microtraps [16] allowing the vibrational frequencies of individual ions and the coupling between different ions to be tuned. Here, we investigate synchronization in two trapped-ion phonon lasers which are pumped in a similar way to that demonstrated in recent experiments [15].We identify a parameter regime where phonon lasing of an individual ion occurs with just a few quanta leading to a nonclassical state of the phonons and investigate the emergence of synchronization in this regime when a weak interion coupling is introduced (weak as it is the slowest time scale in the system). Our model includes two of the electronic levels of the ions used in the pumping process (which we refer to as "spin"), allowing us to uncover strong correlations which arise between the electronic and vibrational degrees of freedom of the individual ions. We study the degree of synchronization as the strength and detuning of the pumping lase...

Discrete Time-Crystalline Order in Cavity and Circuit QED Systems

“…In fact, we find even richer dynamical phases for other κ, such as higher-order period multipling and asymmetric limit-cycle pairs [68]. These phases can unambiguously be diagnosed by a measure of synchronization [71][72][73] and can systematically be understood by employing bifurcation theory [74][75][76][77][78][79].…”

“…2 in the main text. This approach is inspired by the recent studies on synchronization [71][72][73] in nonlinear classical and quantum systems. With the help of this tool, we identify several novel dynamical phases beyond those discussed in the main text.…”

Synchronization of micromasers

“…It is also helpful to be able to characterize the emergence of a phase preference using a single number. Starting from the relative phase distribution, one can simply extract the size of the peak relative to the uniform distribution [14],…”

“…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]. Detailed comparisons have also been made between the predictions of quantum models and those of related semiclassical or classical descriptions [7,12].…”

Synchronizing quantum harmonic oscillators through two-level systems

“…Mutual information and specific correlations have been used, not only as a signature of dynamical alignment of quantum systems, but also to evaluate the degree of synchronization [14][15][16].…”