Quantum-limited estimation of the frequency shift between two interfering photons by time sampling of their quantum beats

TL;DR

This work introduces a time-resolved two-photon interferometric scheme to estimate the frequency difference $\Delta\omega$ between two non-entangled photons by sampling arrival times after a 50:50 beam splitter. The method achieves quantum-limited precision, saturating the Cramér-Rao bound with roughly $10^3$ measurements, and attributes the metrological advantage to the photons’ coherence time $\tau$ rather than direct frequency resolution. The analysis derives the Fisher information $F_\nu(\Delta\omega)$ and the quantum Fisher information $H(\Delta\omega)$, showing that for $\nu=1$ the classical FI matches the quantum limit, while for general $\nu$ the FI scales with $\tau$ and the degree of indistinguishability. A key finding is that time-resolved sampling yields information across a broad range of $\Delta\omega$, outperforming non-resolving schemes, especially at large frequency differences, and is robust to partial indistinguishability. Potential applications include vibrometry, biological-material characterization, and optical coherence tomography, leveraging coherence-time as the primary resource and relaxing detector-frequency-resolution requirements.

Abstract

We present a sensing scheme for estimating the frequency difference of two non-entangled photons. The technique consists of time-resolving sampling measurements at the output of a beam splitter. With this protocol, the frequency shift between two photons can be estimated with the ultimate precision achievable in nature, overcoming the limits in precision and the range of detection of frequency-resolving detectors employed in standard direct measurements of the frequencies. The sensitivity can be increased by increasing the coherence time of the photons. We show that, already with $\sim 1000$ sampling measurements, the Cramér-Rao bound is saturated independently of the value of the difference in frequency.

Figures (5)

• Figure 1: Sensing scheme. Two photons, with central frequencies $\omega_1$ and $\omega_2$, described in the temporal domain in \ref{['eq:input']} enter in the input channel of a 50:50 beam splitter (BS). Then, their temporal delay is resolved with two detectors $D_1$ and $D_2$. Here, for example, three repetitions of the experiment are illustated, in which the temporal delays $\Delta t_A$, $\Delta t_B$, $\Delta t_C$, are recorded. These delays, correspond to a bunching event, a coincidence event and a bunching event, respectively.
• Figure 2: Plot of the probability distribution in eq. \ref{['eq:probs']}. The temporal distribution $\left\vert\psi\left( t \right)\right\vert^2$ is considered Gaussian with unitary variance. This yields a Gaussian envelope $C\left( \Delta t \right)$, represented by the black line. The choice of the variance fixed a natural scale for $\Delta t$ and for $\Delta\omega=4/\tau$
• Figure 3: Contributions $f_\nu\left( \Delta\omega ; \Delta t \right)$ in the Fisher information as a function of the sampled time delay $\Delta t$ for different values of the parameter $\nu$. The plot is for a Gaussian distribution in time of the two photons $\left\vert \psi\left( t \right)\right\vert^2$ with variance $\tau$. The variance fixes a natural scale for $\Delta t$ and for $\Delta\omega=4/\tau_{\Delta t}$. The solid lines represent the envelope of $f_\nu\left( \Delta\omega; \Delta t \right)$ and show the contribution of the Fisher information that is significant or negligible.
• Figure 4: Simulation of the variance normalized with respect to the Cramér-Rao bound for $\nu=1$ (top figure), and $\nu=0.7$ (bottom figure), for values of $\sigma\Delta\omega=0.5,1,3$, respectively represented with a circle, a rhombus and a square marker. In the inset it is shown the simulation of the Expected value of $\Delta\omega$ normalized. The simulation has been executed by repeating for each point the measure process $10^4$ times. It is possible to show that even for an amount of data $N\sim 10^3$ the Cramér-Rao bound is saturated, and the bias is inferior to the $1\%$.
• Figure 5: Plot of Fisher information $F_\nu^G\left( \Delta\omega \right)$ in Eq. \ref{['eq:fig']} and $F_\nu^{G,NR}\left( \Delta\omega \right)$ in Eq. \ref{['eq:fignr']} vs $1/\Delta\omega$, for different values of the parameter $\nu=1,0.9,0.8$. The non resolving Fisher information reaches the Quantum Cramér-Rao bound at $\nu=1$ and for $\Delta\omega\ll 1/\tau$. Instead, for $\Delta\omega\gg 1/\tau$, the non-resolving approach cannot retrieve any information of the parameter $\Delta\omega$.