Optimal orientation detection of an anisotropic dipolar scatterer

The angular orientation of an anisotropic scatterer with cylindrical symmetry in a linearly polarized light field represents an optomechanical librator. Here, we propose and theoretically analyze an optimal measurement scheme for the two angular degrees of freedom of such a librator. The imprecision–back-action product of this scheme reaches the Heisenberg uncertainty limit. Furthermore, we propose and analyze a realistic measurement scheme and show that, in the absence of spinning motion around the symmetry axis, measurement-based ground-state cooling of the rotational degrees of freedom of an anisotropic point scatterer levitated in an optical trap is feasible.

Figure 1

(a) Asymmetric scatterer with cylindrical symmetry depicted as a dumbbell composed of two touching spheres. The orientation of the dumbbell is described by the orientation of its axis of symmetry, illustrated by the dashed arrow. (b) Illustration of the coordinate system. The space-fixed laboratory frame is described by the Cartesian coordinate axes $x, y$, and $z$. The orientation of the dumbbell (shown as the dashed arrow) can be described by the orientation of its symmetry axis relative to the space-fixed Cartesian axes. Small deviations of this symmetry axis from the Cartesian $x$ axis can be described by the rotation angle $\delta$ around the $z$ axis and the rotation angle $\epsilon$ around the $-y$ axis. (c) Illustration of optimal detection scheme for the angle $\epsilon$. The dumbbell is driven by an $x$-polarized field, exemplarily illustrated in red as a beam of light traveling along the $z$ axis. The tilt by $\epsilon$ gives rise to an induced dipole moment along $z$. The field ${\mathit{E}}_{\text{sc}}$ generated by that $z$-oriented dipole (illustrated in green) populates the mode ${\mathit{u}}^{\left(z\right)}$. For optimal detection, that scattered field is mixed with a strong local oscillator field ${\mathit{E}}_{\text{lo}}^{\epsilon }$ in the same dipolar radiation mode (illustrated in yellow). The field intensity is measured on a $4\pi$ detector, illustrated as the dashed black line.

Figure 2

Realistic detection scheme. A dumbbell (long axis along $x$) is trapped in an $x$-polarized field. The scattered field is collected in the forward direction and mixed with a strong $y$-polarized local oscillator (LO) before it is measured on a detector split into two halves along the $xz$ plane. The sum signal gives access to the orientation angle $\delta$, while the difference signal is proportional to the angle $\epsilon$ of the dumbbell.

Figure 3

Detection efficiencies ${\eta }_{\delta }$ (blue solid line) and ${\eta }_{\epsilon }$ (orange dashed line) of our realistic detection scheme according to Eqs. (15) and (17) as a function of the numerical aperture (NA) of the collection lens. The black dashed horizontal line marks $\eta =1/9$, the detection efficiency required to feedback cool to a phonon occupation number $n=1$.