Reservoir engineering and dynamical phase transitions in optomechanical arrays

We study the driven-dissipative dynamics of photons interacting with an array of micromechanical membranes in an optical cavity. Periodic membrane driving and phonon creation result in an effective photon-number-conserving nonunitary dynamics, which features a steady state with long-range photonic coherence. If the leakage of photons out of the cavity is counteracted by incoherent driving of the photonic modes, we show that the system undergoes a dynamical phase transition to the state with long-range coherence. A minimal system, composed of two micromechanical membranes in a cavity, is studied in detail, and it is shown to be a realistic setup where the key processes of the driven-dissipative dynamics can be seen.

Figure 1

Bosonic modes in two bands of a superlattice. Coherent driving of neighboring sites from the ground to the upper band and incoherent decay is represented by straight and wavy arrows, respectively. The whole array supports a chain of $\Lambda$ systems, each defined by a triplet of neighboring modes, two of which belong to the ground band. Phase locking takes place in the shaded area as a result of the combined action of the chain of $\Lambda$ systems.

Figure 2

(a) OM setup for implementing engineered dissipation processes for photons. Three optical modes ${\widehat{a}}_{1,2}$ and $\widehat{c}$ are delimited by two membranes (vertical thick lines) at positions $X_1$ and $X_2$ and the two fixed end mirrors of the optical cavity. The modes are tunnel coupled with amplitude $J$ and their frequencies are modulated by oscillations of the membranes with frequency ${\Omega }_{\mathrm{M}}$ around their equilibrium positions. (b) The resulting energy level diagram assuming degenerate modes ${\widehat{a}}_1$ and ${\widehat{a}}_2$ and a slightly higher frequency ${\omega }^{\prime }\approx \omega +{\Omega }_{\mathrm{M}}$ for the intermediate mode $\widehat{c}$ (see text for more details). (c) and (d) Alternative implementations using an array of coupled microtoriodal cavities or coupled OM crystal cavities, respectively. In both cases, the optical fields are confined inside the structures, coupled among each other via their evanescent fields and interact with localized mechanical vibrations of the structures via optical gradient forces.

Figure 3

Spectrum of the light radiated from the ${\widehat{a}}_2$ mode. The ${\widehat{a}}_1$ mode is coherently pumped with strength ${\varepsilon }_1=0.1\tilde{\Gamma }$, all the modes decay with amplitude $\kappa =0.1\tilde{\Gamma }$. The OM driving is $\mathcal{E}=0.5\tilde{\Gamma }$ (solid line), $0.01\tilde{\Gamma }$ (dashed line), with $\tilde{J}=0.2\tilde{\Gamma }$. The system is first evolved to the steady state at $t_0\tilde{\Gamma }=50.0$ and subsequently to $t\tilde{\Gamma }=500.0$ to compute the two-times correlation functions with sufficient spectral resolution.

Figure 4

Main panel: steady-state populations $n_{\ell }$ for the symmetric (solid line), antisymmetric (short dashed line), and excited (long dashed line) modes, in the presence of coherent pumping of the antisymmetric mode with detuning ${\Delta }_1$. Inset: peak value of $n_{\ell }$ at detuning ${\Delta }_1=\tilde{J}=2.0\tilde{\Gamma }$ with finite thermal occupation $n_{\mathrm{th}}$ of the phononic reservoir. The strength of the coherent pumping is ${\varepsilon }_{\ell }=2.0\kappa$, with $\mathcal{E}=2.0\tilde{\Gamma }$ and $\tilde{\Gamma }=10.0\kappa$.

Figure 5

Schematic representation of the system under consideration. An array of membranes (thick vertical lines), grouped into doublets at positions $X_{2\ell -1}$ and $X_{2\ell }$, within an optical cavity, define two sets of quantized localized light modes ${\widehat{a}}_{\ell }$ and ${\widehat{c}}_{\ell }$. In Sec. 3, we discuss the case in which only two membranes are present.

Figure 6

Time evolution of the photonic population in the excited [panel (a)] and lower [panel (b)] levels, and of the relative phase [panel (c)] between the semiclassical field on neighboring ground sites, computed according to the average-number-conserving semiclassical EOM. $L=12$ sites (corresponding to $2L$ membranes) are used with periodic boundary conditions. The OM driving is $0.2\tilde{\Gamma }$ with detuning $-10.0\tilde{\Gamma }$.

Figure 7

Time evolution of the coherent fraction ${\left|\alpha \right|}^2/n$ (left panel) and second-order correlation function $g^{\left(2\right)}$ at zero delay (right panel), with incoherent pumping ${\rm Y}=0.5\kappa$, for three different values of the rescaled driving strength ${\Gamma }^{\prime }\simeq 0.5\kappa$ (long dashed line), ${\Gamma }^{\prime }\simeq 1.5\kappa$ (short dashed line), and ${\Gamma }^{\prime }=2.5\kappa$ (solid line), at zero temperature $n_{\mathrm{th}}=0.0$. The system is perturbed for $t\kappa \lesssim 1.0$ by a weak coherent pumping field.

Figure 8

Dependence of the coherent fraction ${\left|\alpha \right|}^2/n$ on the rescaled driving strength ${\Gamma }^{\prime }$, at zero temperature $n_{\mathrm{th}}=0.0$, for ${\rm Y}=0.5\kappa$. The system is initialized in the vacuum state, perturbed for $t\kappa \lesssim 1.0$ by a weak coherent pumping field. Stroboscopic plots of the profile at increasing times (dotted, dashed-dotted, short dashed, long dashed, and solid lines) are shown to converge towards a steady-state profile.

Figure 9

Properties of the steady state as the strength $\mathcal{E}$ of the OM driving is increased. The ratio between the number of particles in the symmetric and antisymmetric modes (dashed line, left axis) increases, while the trace of the separable approximation ${\rho }_{\mathrm{sep}}$ of Eq. (B1) to the full density matrix (empty circles, right axis) approaches unity exponentially (solid line). We use ${\varepsilon }_1=0.2\tilde{\Gamma }$, ${\Delta }_1=\tilde{J}=0.2\tilde{\Gamma }$, $\kappa =0.1\tilde{\Gamma }$, and $\Delta =2.0\tilde{\Gamma }$. The state has been evolved to $t_{\max }\tilde{\Gamma }=10.0$. The truncation of the Fock space is $n_{\max }=5$ for the modes ${\widehat{a}}_{\ell }$ and $n_{\max }=1$ for the mode $\widehat{c}$, which is negligibly populated.