Supersensitive measurement of angular displacements using entangled photons

We show that the use of path-entangled states of photons, having nonzero orbital angular momentum (OAM), increases the resolution and sensitivity of angular-displacement measurements performed using an interferometer. In the ideal case of maximally path-entangled states, the resolution of angular-displacement measurements increases by a factor of $\mathit{Nl}$, while the uncertainty in the measurement of angular displacements scales as $1/\mathit{Nl}$, where $N$ is the number of entangled photons, half of which carry, on average, an OAM of $+l\hslash$ per photon and the other half carry an OAM of $-l\hslash$ per photon. We analyze measurement schemes for two- and four-photon entangled states produced by parametric down-conversion and, by employing a 4$\times$4 matrix formalism to study the propagation of entangled OAM modes, obtain explicit expressions for the resolution and sensitivity in these schemes. These results constitute an improvement over what could be obtained with $N$ nonentangled photons carrying an orbital angular momentum of $\left|l\right|\hslash$ per photon.

Figure 1

Scheme for angular-displacement measurement using $N$ independent photons. The photons have an orbital angular momentum of $l\hslash$ per photon and are detected by the detector $D_A$ in mode $a$. DP stands for Dove prism and BS denotes beam splitter.

Figure 2

Angular-displacement measurement using $N$ entangled photons. The photons in modes $g$ and $h$ are in the maximally entangled state and have an orbital angular momentum of $\left|l\hslash \right|$ per photon. The detector $D$ is an $N$-photon detector.

Figure 3

Scheme for supersensitive measurement of angular displacement using entangled photons. $\theta$ is the angle of rotation of the Dove prism in mode $g$. $D_a$ and $D_b$ are detectors set to detect photons with OAM mode indices $\pm l$ only.

Figure 4

Transformation of OAM modes when they pass through (a) a beam splitter, (b) a pair of mirrors, and (c) a Dove prism.

Figure 5

Plot of $\Delta {\theta }_4$ as a function of $\theta$ for $\left|l\right|=1$. The solid horizontal line is the plot of $\Delta {\theta }_4^{(i)}$, as defined in Eq. (6), for four independent photons. The dashed horizontal line is the minimum attainable uncertainty with four entangled photons, as shown by Eq. (11).