Cavity Optomechanics of Levitated Nanodumbbells: Nonequilibrium Phases and Self-Assembly

Levitated nanospheres in optical cavities open a novel route to study many-body systems out of solution and highly isolated from the environment. We show that properly tuned optical parameters allow for the study of the nonequilibrium dynamics of composite nanoparticles with nonisotropic optical friction. We find optically induced ordering and nematic transitions with nonequilibrium analogs to liquid crystal phases for ensembles of dimers.

Figure 1

(a) Levitated dimers composed of dielectric nanospheres in a laser driven optical two-mode cavity with laser strengths ${\Omega }_{1,2}$, a decay rate $\kappa$, and an additional harmonic trap $V_{\mathrm{trap}}$. The orientation of the dimers is denoted $\varphi$, their equilibrium separation $x_0$, and their frequency $\nu$. The direct pair interaction between the nanospheres leads to liquid crystal phases with rich dimer patterns. (b) The particles are in contact with a thermal bath with coupling $\gamma$. The cavity-particle interaction results in strong additional dissipation in one spatial direction only, which acts as an effective zero temperature bath ${\gamma }_{\mathrm{opt}}$. (c) Comparison of the relevant time scales: We assume that ${\gamma }_{\mathrm{opt}}$ is faster than $\gamma$ and find three qualitatively different regimes, which give rise to different orientations, depending on the relative timescales.

Figure 2

Steady-state probabilities of the orientation $\varphi$ as a function of the dimer frequency from numerical integration of the second order Langevin equations (5) for cooling rate ratios ${\gamma }_{\mathrm{opt}}/\gamma =10$ (a), ${\gamma }_{\mathrm{opt}}/\gamma =100$ (b), and ${\gamma }_{\mathrm{opt}}/\gamma =1000$ (c). The comparison of the analytic expressions (6) for small frequencies [(d) dashed lines] and numerical results [(d) full lines] show good agreement over a wide range of frequencies. (e) Analytic results from Eq. (6) for a range of frequencies with ${\gamma }_{\mathrm{opt}}/\gamma =100$ predicts the transition from $\varphi =0$ orientation to $\varphi =\pi /2$. For larger frequencies the model breaks down as it predicts $\varphi =\pi /2$ for large $\nu$ (e). In the rigid-rotor regime, the analytic approximation Eq. (8) is in agreement with the numerical simulation as shown for various cooling rates $r_c={\gamma }_{\mathrm{opt}}/\gamma$ (f).

Figure 3

(a) Ensemble of dimers with frequency $\nu =200\gamma$ [intermediate regime (ii)] in thermal equilibrium are uniformly orientated (inset a). (b) The nonisotropic optical friction induces orientational ordering of individual dimers [inset (b)] and in addition, many-body nematic ordering along the direction of cooling. (c) The nonequilibrium phase analog to a liquid crystal phase characterized by the pair-correlation function $g(x)$. The rigid rotors align orthogonal to cooling with additional nematic order in the direction of cooling (black line) and liquid order in the $y$ direction (red line). (d) Mixture of dimers self-assembles into a crystal with dimer-orientations according to the frequencies $\nu$.