Standard quantum limit of angular motion of a suspended mirror and homodyne detection of a ponderomotively squeezed vacuum field

Compared to the quantum noise in the measurement of the translational motion of a suspended mirror using laser light, the quantum noise in the measurement of the angular motion of a suspended mirror has not been investigated intensively despite its potential importance. In this article, an expression for the quantum noise in the angular motion measurement is explicitly derived. The expression indicates that one quadrature of the vacuum field of the first-order Hermite-Gaussian mode of light causes quantum sensing noise and the other causes quantum back-action noise, or in other words the first-order vacuum field is ponderomotively squeezed. It is also shown that the Gouy phase shift the light acquires between the mirror and the position of detection of the light corresponds to the homodyne angle. Therefore, the quantum back-action noise can be canceled and the standard quantum limit can be surpassed by choosing the appropriate position of detection analogously to the cancellation of quantum radiation pressure noise by choosing an appropriate homodyne angle.

Figure 1

Schematic picture of the configuration of the measurement of the angular motion of a mirror. The transverse displacement and tilt of the incident light measured at the waist and the mirror are shown. The position of the mirror is $z=-Z_1$ in the coordinates of the incident light or $z=Z_1$ in the coordinates of the reflected light. The tilt angle of the mirror is ${\theta }_{\mathrm{mir}}$. The reflected light is measured at $z=Z_{\mathrm{meas}}$. COC represents the center of curvature of the mirror. $R\equiv R(-Z_1)=-R\left(Z_1\right)$.