Coupling rotational and translational motion via a continuous measurement in an optomechanical sphere

We consider a measurement of the position of a spot painted on the surface of a trapped nano-optomechanical sphere. The measurement extracts information about the position of the spot and in doing so measures a combination of the orientation and position of the sphere. The quantum backaction of the measurement entangles and correlates these two degrees of freedom. Such a measurement is not available for atoms or ions and provides a mechanism to probe the quantum mechanical properties of trapped optomechanical spheres. In performing simulations of this measurement process we also test a numerical method introduced recently by Rouchon and collaborators [H. Amini, M. Mirrahimi, and P. Rouchon, in Proceedings of the 50th IEEE Conference on Decision and Control (CDC, 2011), pp. 6242–6247; P. Rouchon and J. F. Ralph, Phys. Rev. A 91, 012118 (2015)] for solving stochastic master equations. This method guarantees the positivity of the density matrix when the Lindblad operators for all simultaneous continuous measurements are mutually commuting. We show that it is both simpler and far more efficient than previous methods.

Figure 1

Schematic representation of the dielectric sphere with fluorescent “spot.” The position of the spot along the $x$ axis is a combination of the position of the center of the sphere along the $x$ axis, $X$, and the angle of the rotation of the sphere in the $xy$ plane, $\theta$. The spot is shown larger than it would be in an experiment.

Figure 2

Accuracy comparison for Rouchon's stochastic integration method (solid blue line) versus the Euler-Milstein method (dashed red line) for sphere radius $R=1.75\mathrm{nm}$ and $k=0.005{\omega }_t$. Other parameter values for the optical tweezers and the dielectric sphere are given in the text.

Figure 3

Average negativity for dielectric sphere versus ratio between the rotational and translational energies, $E_{\mathrm{Rot}}/E_{Tr}$ for $k=0.005{\omega }_t$ and (inset) average negativity versus radius of the dielectric sphere $R$. Error bars shown are $1\sigma$ fluctuations about the mean value. The mean and standard deviation were calculated in the steady state (after 100 oscillator cycles) and over 200 individual realizations for each point. Other parameter values for the optical tweezers and the dielectric sphere are given in the text.

Figure 4

Average mutual information (crosses) and average negativity (circles) for a dielectric sphere as a function of time for $k=0.005{\omega }_t$. Sphere radius is fixed to be $R=1.75\mathrm{nm}$ so that $E_{\mathrm{Rot}}/E_{\mathrm{Tr}}\simeq 1$, and averages are calculated over 500 realizations with (dashed red line) and without the thermal environment (solid blue line).

Figure 5

Average purity for dielectric sphere as a function of time for $k=0.005{\omega }_t$. Sphere radius is fixed to be $R=1.75\mathrm{nm}$ so that $E_{\mathrm{Rot}}:E_{\mathrm{Tr}}\simeq 1$. Averages are calculated with (dashed red line) and without the thermal environment (solid blue line). (Inset) The purity of an example trajectory, with and without the thermal environment.

Figure 6

Average energy for dielectric sphere as a function of time for $k=0.005{\omega }_t$. Sphere radius is fixed to be $R=1.75\mathrm{nm}$ so that $E_{\mathrm{Rot}}:E_{\mathrm{Tr}}\simeq 1$. Average energies are calculated with the thermal environment, translational (dashed red line) and rotational (dashed red circles); and without the thermal environment, translational (solid blue line) and rotational (dotted blue pluses).

Figure 7

Average mutual information (crosses) and average negativity (circles) for a dielectric sphere as a function of time with $k=0.005{\omega }_t$ for $t<50T_{\mathrm{osc}}$ (measurement “on”) and $k=0$ for $t>50T_{\mathrm{osc}}$ (measurement “off”). Sphere radius is fixed to be $R=1.75\mathrm{nm}$ so that $E_{\mathrm{Rot}}:E_{\mathrm{Tr}}\simeq 1$, and averages are calculated over 500 realizations with the thermal environment.