Permanent Directional Heat Currents in Lattices of Optomechanical Resonators

We study the phonon dynamics in lattices of optomechanical resonators where the mutually coupled photonic modes are coherently driven and the mechanical resonators are uncoupled and connected to independent thermal baths. We present a general procedure to obtain the effective Lindblad dynamics of the phononic modes for an arbitrary lattice geometry, where the light modes play the role of an effective reservoir that mediates the phonon nonequilibrium dynamics. We show how to stabilize stationary states exhibiting directional heat currents over arbitrary distance, despite the absence of thermal gradient and of direct coupling between the mechanical resonators.

Figure 1

(a) Schematic representation of a single optomechanical cell and its nearest-neighbor optical couplings. ${\widehat{a}}_{\ell }$ (${\widehat{b}}_{\ell }$) is the optical (mechanical) mode of index $\ell$. (b) Ring of optomechanical disk resonators. Each site is optically driven with a phase that varies as $\ell \varphi$, being $\ell$ the site number. Optical modes are coupled while mechanical ones are not.

Figure 2

Left panel: absolute value of the effective coherent coupling rate $J_p^{(+)}$ between mechanical sites as a function of their distance $p$ and the nonlinear detuning $\tilde{\mathrm{\Delta }}$. The hatched regions correspond to negative values. Three horizontal section cuts (labeled as A, B, C) are plotted in the right panel. Here $J/{\gamma }_c=2$.

Figure 3

Gain ${\mathrm{\Gamma }}_{-k}(-{\omega }_m)$ (dashed) and loss ${\mathrm{\Gamma }}_k(+{\omega }_m)$ rates induced by the engineered reservoir for various $J/{\gamma }_c$. $\tilde{\mathrm{\Delta }}=+{\omega }_m-J$ for the gain rate and $\tilde{\mathrm{\Delta }}=-{\omega }_m-J$ for the loss rate.

Figure 4

Absolute value of the predicted net permanent heat current $|Q_C|$ in units of ${\omega }_m\times {\gamma }_m$ around a ring of cavity-coupled optomechanical resonators as a function of the intercavity coupling and the detuning. The hatched regions correspond to negative values. The system is unstable in the gray region.

Figure 5

(a) Contributions $Q_{\ell \to \ell +p}={\omega }_m{\sum }_{\ell }⟨{\widehat{𝚥}}_{\ell \to \ell +p}⟩$ and net directional heat flow $Q_C={\sum }_{p<\lceil L/2\rceil }{Q}_{\ell \to \ell +p}$ along the dash-dotted line of Fig. 4 as predicted by our effective theory (lines) and mean field (circles). (b) Sketch of the two leading contributions in (a). (c) Gradual triggering of off-diagonal coherence $g_{\ell ,\ell +p}^{(1)}=⟨{\widehat{d}}_{\ell }^{†}{\widehat{d}}_{\ell +p}⟩/(⟨{\widehat{d}}_{\ell }^{†}{\widehat{d}}_{\ell }⟩⟨{\widehat{d}}_{\ell +p}^{†}{\widehat{d}}_{\ell +p}⟩{)}^{1/2}$ along the lowest anti-Stokes sideband $\tilde{\mathrm{\Delta }}=-J-{\omega }_m$.