This letter investigates a hybrid quantum system combining cavity quantum electrodynamics and optomechanics. The Hamiltonian problem of a photon mode coupled to a two-level atom via a Jaynes-Cummings coupling and to a mechanical mode via radiation pressure coupling is solved analytically. The atom-cavity polariton number operator commutes with the total Hamiltonian leading to an exact description in terms of tripartite atom-cavity-mechanics polarons. We demonstrate the possibility to obtain cooling of mechanical motion at the single-polariton level and describe the peculiar quantum statistics of phonons in such unconventional regime. [4][5][6]. In superconducting circuits, strong coupling and control of the mechanical motion at the quantum level have also been demonstrated [7]. Today, the maturity of solid-state quantum devices appears thus promising to bridge QED and optomechanics. The physical interaction at play in QED results in a resonant coupling linear in the photon field operators (Jaynes-Cummings Hamiltonian), while in optomechanics a non-linear radiation pressure term couples two off-resonant photonic and mechanical modes. A rich physics is expected in systems that would merge these distinct physical features.The basic principle of inserting a two-level artificial atom in an optomechanical setting was discussed in classical terms for fine tuning of dispersive and dissipative optomechanical interactions [8]. The coupling of an optomechanical cavity to an atom motion [9] or to collective excitations of an ensemble of atoms [10] was also discussed, resulting in the physical situation of two linearly coupled harmonic oscillators. In that case the anharmonic internal structure of a single atom and its corresponding nonlinear dynamics, a key feature of cavity and circuit QED, is absent. Since optomechanical systems progressively move towards regimes where single photon coupling exceeds dissipation [11-16] a growing interest is emerging for hybrid systems where artificial atoms, photons and phonons would all be strongly coupled at the quantum level.In this letter, we investigate the physics of a hybrid quantum system where a cavity photon mode is coupled to an artificial two-level atom and to a mechanical resonator. We describe analytically the polaron excitations of this tripartite system and determine the dynamics in presence of losses and driving. We show atomassisted cooling of mechanical motion down to the single atom-cavity polariton level and reveal unusual mechanical amplification. Last, we demonstrate the emergence of phonon antibunching in such tripartite quantum systems.As illustrated in Fig.1, we consider a joint system where a confined photon mode is coupled both to a twolevel artificial atom and to a mechanical resonator. Our system combines the usual Jaynes-Cummings (JC) coupling of cavity (circuit) QED architectures [1] and the nonlinear coupling of optomechanics [17]. We thus consider the total Hamiltonian ( = 1):whereσ x,y,z are Pauli matrices for the two-level system (σ ± being the ladder ...

Nano-and micro-optomechanical systems / Nano-et micro-résonateurs optomécaniques Classical and quantum theory of photothermal cavity cooling of a mechanical oscillator Théorie classique et quantique du refroidissement photothermique en cavité d'un oscillateur mécanique