Optically levitating dielectrics in the quantum regime: Theory and protocols

We provide a general quantum theory to describe the coupling of light with the motion of a dielectric object inside a high-finesse optical cavity. In particular, we derive the total Hamiltonian of the system as well as a master equation describing the state of the center-of-mass mode of the dielectric and the cavity-field mode. In addition, a quantum theory of elasticity is used to study the coupling of the center-of-mass motion with internal vibrational excitations of the dielectric. This general theory is applied to the recent proposal of using an optically levitating nanodielectric as a cavity optomechanical system [see Romero-Isart et al., New J. Phys. 12, 033015 (2010); Chang et al., Proc. Natl. Acad. Sci. USA 107, 1005 (2010)]. On this basis, we also design a light-mechanics interface to prepare non-Gaussian states of the mechanical motion, such as quantum superpositions of Fock states. Finally, we introduce a direct mechanical tomography scheme to probe these genuine quantum states by time-of- flight experiments.

Figure 1

Schematic representation of the setup. A nanodielectric is confined by optical tweezers, providing a trapping frequency of ${\omega }_t$. The nanodielectric is placed inside an optical cavity with resonance frequency ${\omega }_c$, decay rate $\kappa$, and is driven by a laser at a frequency ${\omega }_L$.

Figure 2

Coordinates used to describe a position ${\mathbf{x}}^{\prime }$ within an arbitrary dielectric object given by ${\mathbf{x}}^{\prime }=\mathbf{r}+\mathrm{R̂}({\varphi }_1,{\varphi }_2,{\varphi }_3)(\mathbf{u}(\mathbf{x})+\mathbf{x})$, where $\mathbf{r}$ denotes the c.m., $\mathbf{u}(\mathbf{x})$ denotes a small displacement from the equilibrium position $\mathbf{x}$, and $R({\varphi }_1,{\varphi }_2,{\varphi }_3)$ denotes the Euler rotation matrix acting on the entire object.

Figure 3

Input-output dynamics after sending a one-photon pulse centered at $x_{\mathrm{in}}=5/\kappa$ from the cavity at $t=0$. A Gaussian pulse of width $\sigma =5.6\kappa$ is used. We plot the mean number of phonons in the mechanical system ${\mathrm{n̄}}_b(t)=|c_b(t)|{}^2$ (red solid line) and the mean number of cavity photons ${\mathrm{n̄}}_a(t)=|c_a(t)|{}^2$ (blue dashed line). We consider the strong-coupling regime $g=\kappa$, and tune the width of the pulse so that the maximum mean number of phonons is $~1/2$ (dotted gray line) at $t=t_h$.

Figure 4

Perfect state transfer of a $|0\rangle +|1\rangle$ photonic state by sending a Gaussian light pulse of width $\sigma =2\kappa /3$ from a distance $x_{\mathrm{in}}=10\kappa$ ($c=1$). We plot the time modulation of $g(t)/\kappa$ (solid red line) and $\langle b(t)\rangle$ (dashed blue line). After the modulation, when $g(t)=0$, one obtains that $\langle b\rangle =1/2$. This shows that the superposition state has been mapped to the mechanical system without requiring a measurement.

Figure 5

Schematic representation of a light-mechanics interface of teleportation in the bad-cavity limit. The cavity is driven by a blue-detuned laser which induces a two-mode squeezing interaction between the cavity mode and the mechanical mode. Being in the bad-cavity limit $\kappa >g$, the cavity photons, which are in a two-mode squeezed state $|\mathrm{TMS}\rangle$ with the mechanical phonons, rapidly leak out of the cavity. The output field is combined in a beam-splitter together with the non-Gaussian state to be teleported $|\psi \rangle {}_e$. A measurement of the output quadratures would realize the Bell measurement required for teleportation [69].

Figure 6

Schematic illustration of the time-of-flight protocol to perform full tomography of the mechanical state. The momentum operator at times $t_e$, which corresponds to the rotated phase quadrature $\chi ({\omega }_t{t}_e+\pi /2)$, is determined by measuring the position of the dielectric after some time of flight. By repeating the experiment at different times $t_e$, one can perform full tomography of the mechanical state.

Figure 7

The ratio between the decay rate of the cavity due to light scattering ${\kappa }_{\mathrm{sc}}$ [using Eq. (38) valid for spheres smaller than the wavelength] over the standard decay rate of a cavity of finesse $\mathcal{F}=5\times {10}^5$ is plotted as a function of the radius of the sphere. As can be observed, photon losses due to scattering dominate for objects larger than $150$ nm.

Figure 8

Final phonon occupation number using spheres of radius $R$ and for the parameters given in the text. Ground-state cooling is possible for spheres smaller than $R~200$ nm. This is obtained using the expressions valid for objects smaller than the wavelength.