We investigate the behavior of two coupled non-linear photonic cavities, in presence of inhomogeneous coherent driving and local dissipations. By solving numerically the quantum master equation, either by diagonalizing the Liouvillian superoperator or by using the approximated truncated Wigner approach, we extrapolate the properties of the system in a thermodynamic limit of large photon occupation. When the mean field Gross-Pitaevskii equation predicts a unique parametrically unstable steady-state solution, the open quantum many-body system presents highly non-classical properties and its dynamics exhibits the long lived Josephson-like oscillations typical of dissipative time crystals, as indicated by the presence of purely imaginary eigenvalues in the spectrum of the Liouvillian superoperator in the thermodynamic limit.Open many-body quantum systems [1-3] have become a major field of study over the last decade. The open nature is common to a vast class of modern experimental platforms in quantum science and technology, such as photonic systems [4], ultracold atoms [5-9], optomechanical systems [10][11][12][13] or superconducting circuits [14][15][16], for which driving and losses are omnipresent. Open quantum systems also display emergent physics, in particular dissipative phase transitions and topological phases [40][41][42][43][44][45].Several studies have highlighted the possibility for a continuous-wave driven-dissipative quantum system to reach a non-stationary state in the long time limit in which undamped oscillations arise spontaneously [46][47][48][49][50][51][52]. This phenomenon has been dubbed as boundary or dissipative time crystal (DTC), in analogy with the time crystals in some Hamiltonian systems [53]. Formally, DTCs are associated with the occurrence of multiple eigenvalues of the Liouvillian with vanishing real and finite imaginary part [54][55][56]. The experimental feasibility of DTC has been confirmed by their observation in phosphorousdoped silicon [57]. The research for further platforms showing this phenomenon is very active and important to understand the mechanisms behind the spontaneous breaking of the time-translation symmetry in open quantum many-body systems.One of the main difficulties in the realization of DTCs in real system is related to the fragility of this phase to external perturbations which affect the symmetric structure of the model. Indeed, in most of the cases considered so far, the engineering of the DTCs relies on the exploitation of certain symmetries (either manifest [48,52] or emergent [50]) in the Hamiltonian or in the dissipation mechanism, which can be hard to maintain in real driven-dissipative systems out of equilibrium.In this letter, we show that a DTC can arise in a simple system of two coupled photonic cavities, whose equation of motion does not preserve any symmetry but the timetranslation invariance. In a broad region of the parameter space, the dynamics of this system presents limit cycles associated to parametric instabilities [58], which can be regar...

We propose and demonstrate a versatile technique to measure the lifetime of the one-phonon Fock state using two-color pump-probe Raman scattering and spectrally resolved, time-correlated photon counting. Following pulsed laser excitation, the n=1 phonon Fock state is probabilistically prepared by projective measurement of a single Stokes photon. The detection of an anti-Stokes photon generated by a second, time-delayed laser pulse probes the phonon population with subpicosecond time resolution. We observe strongly nonclassical Stokes-anti-Stokes correlations, whose decay maps the single phonon dynamics. Our scheme can be applied to any Raman-active vibrational mode. It can be modified to measure the lifetime of n≥1 Fock states or the phonon quantum coherences through the preparation and detection of two-mode entangled vibrational states.

A single quantum of excitation of a mechanical oscillator is a textbook example of the principles of quantum physics. But mechanical oscillators, despite their pervasive presence in nature and modern technology, do not generically exist in an excited Fock state. In the past few years, careful isolation of gigahertz-frequency nanoscale oscillators has allowed experimenters to prepare such states at millikelvin temperatures. These developments illustrate the tension between the basic predictions of quantum mechanics-which should apply to all mechanical oscillators even at ambient conditions-and the extreme conditions required to observe those predictions. We resolve the tension by creating a single Fock state of a 40-THz vibrational mode in a crystal at room temperature and atmospheric pressure. After exciting a bulk diamond with a femtosecond laser pulse and detecting a Stokes-shifted photon, the Ramanactive vibrational mode is prepared in the Fock state j1i with 98.5% probability. The vibrational state is then mapped onto the anti-Stokes sideband of a subsequent pulse, which when subjected to a Hanbury Brown-Twiss intensity correlation measurement reveals the sub-Poisson number statistics of the vibrational mode. By controlling the delay between the two pulses, we are able to witness the decay of the vibrational Fock state over its 3.9-ps lifetime at ambient conditions. Our technique is agnostic to specific selection rules, and should thus be applicable to any Raman-active medium, opening a new general approach to the experimental study of quantum effects related to vibrational degrees of freedom in molecules and solidstate systems.

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