Quantum-limited estimation of the difference between photonic momenta via spatially resolved two-photon interference

Abstract

We present a quantum sensing protocol for three-dimensional estimation of the difference between the momenta of two photons based on spatially resolved interferometric sampling measurements. The protocol attains ultimate quantum precision in the simultaneous estimation of the components of the relative momentum for any values of the parameters already with $\sim 2000$ sampling measurements and a bias below $1\%$. These results identify 3D spatially resolved two-photon interference as an efficient tool for multi-parameter quantum sensing, with potential applications in high-precision 3D localization, refractometry, and near-field calibration in free-space quantum technologies.

Figures (4)

• Figure 1: Three-parameter quantum sensing scheme for the three-parameter estimation of the components of the relative momenta of two input photons. Two photons, $P_1$ and $P_2$, with different colors and approximately identical distribution in their transverse positions and emission times, interfere on a balanced beam splitter from two different angulations. In order to show the relative angulation of the two photons, we also plotted the symmetric image $P'_1$ of the first photon with respect to the plane of the beam splitter. The two-photon input state is represented by Eq. \ref{['maineq:input']}. After the beam splitter, their relative transverse position and temporal delay are measured in the near-field by the cameras $C_1$ and $C_2$.
• Figure 2: Plot of the output probability distribution in Eq. \ref{['maineq:prob']}. For simplicity, the plot is made by imposing $\nu=1$ (top figures) and $\nu=0.7$ (bottom figures), $\gamma=1$ and $\vert\vec{\kappa}\vert=4$. As is it evident in Eq. \ref{['maineq:prob']}, the probability distribution presents a beating in the direction of $\vec{\kappa}$ that depends on $\vec{\kappa}\cdot\vec{ \rho}$ and on a Gaussian envelope that is function of $\vert\vec{ \rho}\vert^2=l^2+(l^{\perp})^2=(\vert\vec{\rho}\cdot\vec{ \kappa}\vert^2+\vert\vec{\rho}\times\vec{ \kappa}\vert^2)/\vert\vec{\kappa}\vert^2$, where the vectors $\vec{ \rho}$ and $\vec{\kappa}$ are defined in Eq. \ref{['eq:reparameters']}. Therefore, although the beatings oscillate only as a function of $l$, its modulation changes in both $l$ and $l^\perp$.
• Figure 3: Contributions of the Fisher information matrix density $f_{\nu}\left( l;\vert\vec{\kappa}\vert,\theta,\phi \right)$ in Eq \ref{['eq:densityfi']} (normalized by the quantum Fisher information matrix $Q(\vert\vec{\kappa}\vert,\theta,\phi)$ in Eq. \ref{['eq:Q']}) as a function of $l=\vec{ \rho}\cdot\vec{\kappa}/\vert\vec{\kappa}\vert$ for different values of the distinguishability parameter $\nu=1,0.9$ for photons with Gaussian spatial distributions $\vert\psi_i(\vec{r})\vert^2$, $i=1,2$, with $\vec{r}=(x,y,ct)$, where $t$ is the detection time, $c$ is the speed of light, and $(x,y)$ is the transverse position resolved by the cameras. For simplicity, we impose $\vert\vec{\kappa}\vert=4$.
• Figure 4: Simulations of the variance for the estimation of the parameters $\vert\vec{\kappa}\vert,\theta,\phi$ normalized with respect to the Cramér-Rao bound as a function of the number $N$ of experimental iterations for the distinguishability values $\nu=0.7$ (left) and $\nu=0.8$ (right) and the three values of the 3D parameter to estimate: $(\vert\vec{\kappa}\vert,\theta,\phi)=(3,\pi/5,\pi/4),(4,\pi/4,\pi/3),(5,\pi/3,\pi/5)$. In the insets we show the simulations of the expected value of the estimators $(\tilde{\vert\vec{\kappa}\vert},\tilde{\theta},\tilde{\phi})$ normalized to the actual parameter values, showing the unbiasedness of the estimators. We show that the function $1+A/N$ (continuous, dashed and dotted lines, respectively) fit the points of the main figures, where $A$ is the coefficient of the correction term of the order $1/N$ in the variance normalized to the Cramér-Rao bound. From the main figure, we can see that the terms of the variance normalized of the order $O(1/N^2)$ are negligible. The term of order $1/N$ corresponds to $1\%$ of the Cramér-Rao bound for $N\simeq2000$.