Amplification of Angular Rotations Using Weak Measurements

We present a weak measurement protocol that permits a sensitive estimation of angular rotations based on the concept of weak-value amplification. The shift in the state of a pointer, in both angular position and the conjugate orbital angular momentum bases, is used to estimate angular rotations. This is done by an amplification of both the real and imaginary parts of the weak-value of a polarization operator that has been coupled to the pointer, which is a spatial mode, via a spin-orbit coupling. Our experiment demonstrates the first realization of weak-value amplification in the azimuthal degree of freedom. We have achieved effective amplification factors as large as 100, providing a sensitivity that is on par with more complicated methods that employ quantum states of light or extremely large values of orbital angular momentum.

Figure 1

Experimental setup. A light beam from HeNe laser working at 632.8 nm is coupled into a SMF and the output is then collimated. The beam is sent to a phase-only SLM and then to a $4f$ optical system containing a spatial filter in the Fourier plane. A polarization state is prepared by means of a polarizer and a HWP. A Dove prism, a HWP and two QWPs are placed inside the Sagnac interferometer that uses a polarizing beam splitter (PBS). The DP induces a small rotation between the counterpropagating beams; this is the weak perturbation. The QWPs together with the HWP induce a geometric phase between the $H$ and $V$ polarized beams. After postselection, measurements of angular rotations and OAM spectra are performed to access the real or imaginary part of the weak value.

Figure 2

Amplification of angular displacements using real weak values. (a)–(e) show simulations of our scheme for $\mathrm{\Delta }\varphi =1.2°$, different postselection angles (PA) and amplification factors (Amp). (f)–(j) show experimental evidence of our protocol under the same conditions.

Figure 3

Measured OAM power spectra of $|{\mathrm{\Psi }}_p⟩$ without postselection (blue) and with postselection (green) demonstrating the shift in $⟨\ell ⟩$ due to $\Im ({\sigma }_w)$ for (a) ${\eta }_{\varphi }=11.4°$, $\gamma /2=6°$ and (b) ${\eta }_{\varphi }=13.7°$ and $\gamma /2=5°$. The angle $\theta /2$ equals 5° for all the cases. Histograms represent measured data, while lines represent theoretically predicted shifts. (c) OAM centroid shift $\mathrm{\Delta }⟨\ell ⟩$ for various measured OAM power spectra plotted against the imaginary WV amplification factor, $\Im ({\sigma }_w)/2{\eta }_{\varphi }^2$. Dots represent data, while the line is the theoretical linear curve predicted by Eq. (5).