Precision optical displacement measurements using biphotons

We propose and examine the use of biphoton pairs, such as those created in parametric down-conversion or four-wave mixing, to enhance the precision and the resolution of measuring optical displacements by position-sensitive detection. We show that the precision of measuring a small optical beam displacement with this method can be significantly enhanced by the correlation between the two photons, given the same optical mode. The improvement is largest if the correlations between the photons are strong, and falls off as the biphoton correlation weakens. More surprisingly, we find that the smallest resolvable parameter of a simple split detector scales as the inverse of the number of biphotons for small biphoton number (“Heisenberg scaling”), because the Fisher information diverges as the parameter to be estimated decreases in value. One usually sees this scaling only for systems with many entangled degrees of freedom. We discuss the transition for the split-detection scheme to the standard quantum limit scaling for imperfect correlations as the biphoton number is increased. An analysis of an $N$-pixel detector is also given to investigate the benefit of using a higher resolution detector. The physical limit of these metrology schemes is determined by the uncertainty in the birth zone of the biphoton in the nonlinear crystal.

Figure 1

Schematic for a biphoton displacement measurement. Photons from a laser source are converted into correlated pairs in a colinear type-II spontaneous parametric down-conversion (SPDC) process (i.e., one in which two photons of orthogonal polarization are generated), for example, with a BBO crystal. Uncorrelated short wavelength photons (green solid line) are filtered out, while the horizontally polarized (red line) and vertically polarized (blue line) photons pass through to a movable mirror. The shift $d$ of the mirror from the origin displaces the optical beam, and is the small, unknown parameter being measured with this apparatus. Each photon in the pair is detected separately at a position-sensitive detector placed at the output ports of a polarizing beam splitter (PBS). By using coincidences between each detector it is possible to discard spurious events caused by lossy optics or imperfections in the PBS.

Figure 2

The probability distribution (2) as a function of $x_1$ and $x_2$. Clockwise from the upper left we have increasing correlation with $\epsilon /\sigma =1$, 0.75, 0.5, and 0.1. As expected, $x_1$ and $x_2$ are increasingly anticorrelated with decreasing $\epsilon$.

Figure 3

Net signal for biphotons after 10 000 independent events for split detection. The net signal here is analogous to the net steps from the origin in a random walk, with the probabilities of two steps backward, two steps forward, and zero steps per trial given by Eq. (12). The solid blue signal represents uncorrelated photon pairs, while the red dashed line represents entangled photons with $\epsilon /\sigma =0.01$.

Figure 4

A comparison of the Fisher information (in arbitrary units) due to biphoton pairs incident on detectors with different numbers of pixels $N$, using the numerical value $\sigma =1$ and $d=0.05$. For each curve, the information for uncorrelated photons is given by the value of the curve for $\epsilon /\sigma =1$. The dotted black line represents perfect spatial resolution calculated by Eq. (3), the solid blue curve represents split detection ($N=2$), the red dashed curve represents a 10-pixel detector, and the case of 50 pixels is given by the purple dot-dashed curve.

Figure 5

The number of events needed to achieve an SNR of 1 under split detection as a function of the parameter $d$ (in arbitrary units) for different values of $\epsilon /\sigma$. From top to bottom the curves represent $\epsilon /\sigma =1$ (i.e., uncorrelated), 0.5, 0.1, 0.05, and 0 (i.e., perfect correlations). As expected from Eq. (33), Heisenberg scaling can only be observed for numbers of events much less than $\sigma /\epsilon$, e.g., the $\epsilon /\sigma =0.01$ curve only matches the Heisenberg curve for $\nu <\sim 100$.