Quantum Many-Body Dynamics in Optomechanical Arrays

We study the nonlinear driven dissipative quantum dynamics of an array of optomechanical systems. At each site of such an array, a localized mechanical mode interacts with a laser-driven cavity mode via radiation pressure, and both photons and phonons can hop between neighboring sites. The competition between coherent interaction and dissipation gives rise to a rich phase diagram characterizing the optical and mechanical many-body states. For weak intercellular coupling, the mechanical motion at different sites is incoherent due to the influence of quantum noise. When increasing the coupling strength, however, we observe a transition towards a regime of phase-coherent mechanical oscillations. We employ a Gutzwiller ansatz as well as semiclassical Langevin equations on finite lattices, and we propose a realistic experimental implementation in optomechanical crystals.

Figure 1

Example implementation of an optomechanical array: A two-dimensional snowflake optomechanical crystal [38, 39] supports localized optical and mechanical modes around defect cavities. Here, we propose arranging them in a superstructure, forming the array. The insets show electric field $\overrightarrow{E}$ and displacement field $\overrightarrow{u}$ of an isolated defect cavity (obtained from finite element simulations). Due to the finite overlap between modes of neighboring sites [29], photons and phonons can hop through the array, see Eq. (2). A wide laser beam drives the optical modes of the array continuously and the reflected light is read out.

Figure 2

Loss of the photon blockade for increasing optical coupling in an array of optomechanical cavities. (a) The equal time photon correlation function shows antibunching ($g^{(2)}(0)<1$) and bunching ($g^{(2)}(0)>1$) as a function of detuning $\Delta$ and optical coupling strength $J$. The smallest values of $g^{(2)}(0)$ are found for a detuning ${\Delta }_0=-g_0^2/\Omega$. (b) When increasing the coupling $J$ while keeping the intracavity photon number constant, i.e., along the dashed line in panel (a), the photon blockade is lost (black solid line). For a smaller driving power (blue solid line, ${\alpha }_L=5\times {10}^{-5}\kappa$), anti-bunching is more pronounced and the behavior is comparable to that of a nonlinear cavity (dashed line). The hatched area in (a) outlines a region where a transition towards coherent mechanical oscillations has set in. $\kappa =0.3\Omega$, ${\alpha }_L=0.65\kappa$, $g_0=0.5\Omega$, $\Gamma =0.074\Omega$.

Figure 3

Transition from the incoherent to the synchronized (coherent) phase: (a) Mechanical coherence $r$ [Eq. (4)] as a function of laser detuning $\Delta$ and mechanical coupling $K$. At weak coupling, the self-oscillations are incoherent, $r=0$, due to quantum noise. When increasing the coupling strength, the systems shows a sharp transition towards the ordered regime, where the mechanical oscillations are phase coherent, $r>0$. (b),(c) Modulus of the density matrix elements (in Fock space) and Wigner density of the collective mechanical state in the incoherent (b) and the coherent regime (c), as marked in (a). (d) Mechanical coherence $r$ as a function of coupling strength $K$ along the dashed line in (a). The dotted line shows the optical readout of coherence, i.e., the oscillating component of the photon number $⟨{\widehat{a}}^{†}\widehat{a}⟩$, proportional to the intensity of the reflected beam and thus directly accessible in experiment. $g_0=\kappa =0.3\Omega$, ${\alpha }_L=1.1\kappa$, $\Gamma =0.074\Omega$.

Figure 4

Langevin dynamics on finite lattices: (a) Correlations $C(d=|i-j|)=|⟨e^{i{\phi }_i}{e}^{-i{\phi }_j}⟩|$ in a $30\times 30$ optomechanical array. Quasi-long-range order sets in for sufficiently large coupling strengths. $K=\{0.09,0.105,0.107,0.12,0.15\}\Omega$. (b) Correlations over a distance of $d=14$ as a function of mechanical coupling strength $K$ for a square lattice ($z=4$, squares), a hexagonal lattice ($z=6$, triangles) slightly below the mean-field result (circles). (c) Coupling threshold as a function of quantum parameter $g_0/\kappa$ (squares: square lattice, empty (filled) circles: semiclassical (quantum) mean-field approach). $\Delta +g_0^2/\Omega =0.34$, $g_0=0.1\kappa$ in (a),(b), $g_0{\alpha }_L=0.33\kappa$ in (c), other parameters as in Fig. 3.