We propose a system for observing the correlated phase dynamics of two mesoscopic ensembles of atoms through their collective coupling to an optical cavity. We find a dynamical quantum phase transition induced by pump noise and cavity output-coupling. The spectral properties of the superradiant light emitted from the cavity show that at a critical pump rate the system undergoes a transition from the independent behavior of two disparate oscillators to the phase-locking that is the signature of quantum synchronization. PACS numbers: 05.45.Xt, 42.50.Lc, 37.30.+i, 64.60.Ht Synchronization is an emergent phenomenon that describes coupled objects spontaneously phase-locking to a common frequency in spite of differences in their natural frequencies [1]. It was famously observed by Huygens, the seventeenth century clock maker, in the antiphase synchronization of two maritime pendulum clocks [2]. Dynamical synchronization is now recognized as ubiquitous behavior occurring in a broad range of physical, chemical, biological, and mechanical engineering systems [1,3,4].Theoretical treatments of this phenomenon are often based on the study of phase models [5,6], and as such have been applied to an abundant variety of classical systems, including the collective blinking of fireflies, the beating of heart cells, and audience clapping. The concept can be readily extended to systems with an intrinsic quantum mechanical origin such as nanomechanical resonators [7,8], optomechanical arrays [9], and Josephson junctions [10,11]. When the number of coupled oscillators is large, it has been demonstrated that the onset of classical synchronization is analogous to a thermodynamic phase transition [12] and exhibits similar scaling behavior [13].Recently, there has been increasing interest in exploring manifestations in the quantum realm. Small systems have been considered, e.g., one qubit [14] and two qubits [15] coupled to a quantum dissipative driven oscillator, two dissipative spins [16], two coupled cavities [17], and two micromechanical oscillators [18,19]. Connections between quantum entanglement and synchronization have been revealed in continuous variable systems [19]. It has been shown that quantum synchronization may be achieved between two canonically conjugate variables [20]. Since the phenomenon is inherently non-equilibrium, all of these systems share the common property of competition between coherent and incoherent driving and dissipative forces.In this paper, we propose a modern-day realization of the original Huygens experiment [2]. We consider the synchronization of two active atomic clocks coupled to a common single-mode optical cavity. It has been predicted that in the regime of steady-state superradiance [21-24] a neutral atom lattice clock could produce an ultracoherent optical field with a quality factor (ratio of frequency to linewidth) that approaches 10 18 . We show that two such clocks may exhibit a dynamical phase transition [26][27][28][29] from two disparate oscillators to quantum phase-locked dynamics. ...

We show that open quantum systems of two-level atoms symmetrically coupled to a single-mode photon field can be efficiently simulated by applying a SU(4) group theory to quantum master equations. This is important since many foundational examples in quantum optics fall into this class. We demonstrate the method by finding exact solutions for many-atom open quantum systems such as lasing and steady state superradiance.

As an alternative to state-of-the-art laser frequency stabilisation using ultra-stable cavities, it has been proposed to exploit the non-linear effects from coupling of atoms with a narrow transition to an optical cavity. Here we have constructed such a system and observed non-linear phase shifts of a narrow optical line by strong coupling of a sample of strontium-88 atoms to an optical cavity. The sample temperature of a few mK provides a domain where the Doppler energy scale is several orders of magnitude larger than the narrow linewidth of the optical transition. This makes the system sensitive to velocity dependent multi-photon scattering events (Dopplerons) that affect the cavity field transmission and phase. By varying the number of atoms and the intra-cavity power we systematically study this non-linear phase signature which displays roughly the same features as for much lower temperature samples. This demonstration in a relatively simple system opens new possibilities for alternative routes to laser stabilization at the sub 100 mHz level and superradiant laser sources involving narrow line atoms. The understanding of relevant motional effects obtained here has direct implications for other atomic clocks when used in relation with ultranarrow clock transitions.