Quantum-limited optical delay sensing across an enhanced dynamic range by frequency-resolving two-photon interference

TL;DR

The paper tackles the limited dynamic range of two-photon-interference-based optical delay sensing by introducing frequency-resolved sampling of photon inner modes with high-resolution time-of-flight spectroscopy. By measuring both the frequency difference $Δω$ and the coincidence/bunching outcomes, the authors construct a maximum-likelihood estimator for the relative delay $Δt$ that remains informative over delays far beyond the coherence time, approaching the quantum limit. They demonstrate more than a twentyfold expansion in dynamic range and substantial precision gains—down to sub-femtosecond levels in synthetic experiments—while using either independent photon sources or a single high-rate source. The method enables scan-free, nanometre-scale depth sensing over millimetre-scale ranges and has potential applications in biological/nanomaterial imaging, integrated photonics, and high-precision time transfer.

Abstract

Optical sensing schemes that rely on two-photon interference provide a powerful platform for precision metrology, although they are inherently constrained by a trade-off between dynamic range and measurement precision. To overcome this limitation, we sample the frequencies of two interfering photons, which extends the sensitivity in the time domain. This enhances the dynamic range of optical delay estimation by up to twenty times compared to the non-resolved estimates. We demonstrate this approach with independent photon sources and show the behaviour of finite frequency resolving detectors. This technique enables scan-free nanometre resolution depth sensing over a millimetre-scale range, with applications in biological and nanomaterial imaging.

Figures (8)

• Figure 1: (a) Two-photon interference setup. Two photons are input on a 50:50 beam splitter (BS) while a translation stage scans the path length delay $x$. Detectors placed on each exit port of the beam splitter can be either bucket detectors or frequency-resolving detectors. (b) The two-photon frequency-resolved coincidence outcome probability, Eq. \ref{['FRout']} is plotted as a function of the time delay $\Delta t$ and frequency difference $\Delta \omega$. Integrating over all frequencies results in the familiar HOM-dip along the non-resolved axis.
• Figure 2: (a) Two-photon interference over the entire delay range, measuring coincidence counts from a single source (2-fold) over a 20 mm range and converted to units of time in picoseconds. Insets (i-v) show the joint spectral intensity (JSI) for $[0, 6, 16, 30,60]$ ps delays. The (2-fold) marginal for the JSI at 30 ps is presented in (b). The fringe visibility is found to be 0.78, which is equivalent to $\eta$. Indistinguishability $\eta$ as a function of delay is plotted in (c), extracted by fitting the marginals to the theoretical distribution in Eq. \ref{['FRout']}, for both the single source (2-fold) and independent sources (4-fold). The solid and dashed lines are fitted using a spline and follow a linear trend after the maximum $\eta$ value.
• Figure 3: Accuracy of estimated delays using (a) non-resolved (NR) methods and (b) the frequency resolved (FR) method with a single source (filled) or independent source (not filled). Two-fold estimates use $r = 1000$ repeats, each using $N = 1000$ samples, while the four-fold points use $r=100$ estimates with $N = 100$ samples. Error bars are calculated from one standard deviation of the set of estimates $\{\widetilde{\Delta t}_i \}^r_{i=1}$ rounds.
• Figure 4: (a) The experimental metrological information $\mathscr{F}_{exp}$ for the 2-fold estimates with frequency-resolved (FR) and non-resolved (NR) measurements is plotted against the prepared delay. The non-resolved Fisher information is plotted as a solid orange line using Eq. \ref{['NR_FI']} and $\eta=0.98$, displaying a peak at around 1 coherence length, while the frequency-resolved Fisher information is plotted as a solid blue line using Eq.\ref{['finiteFI']} and displays an extended sensitivity range. The blue dashed horizontal line is the QFI with value 0.52 ($\text{ps}^{-2}$). (b) The metrological advantage is plotted for each estimate in logarithmic scale using the ratio of the resolved and non-resolved mean squared error, $\mathrm{MSE_{NR}/MSE_{FR}}$.
• Figure SM1: Experimental layout with two sources. A 775 nm pulsed Ti:Sapphire laser with 1.3 ps duration and 80 MHz repetition rate pumps two photon pair sources. The sources comprise of a aperiodicity-poled KTP crystal housed in a Sagnac configuration, generating spectrally degenerate Gaussian profile photon pairs at 1550 nm. One photon from each pair is separated using dichroic polarising beam splitters (dPBS) and coupled to a single-mode fibre (SMF) to be detected imediately on superconducting nanowire detectors (SNSPD) and time-stamped on Hydra harp time tagger. The other pair from each source is then coupled to SMF and launched into free space onto a 50:50 beam splitter. The polarisation of each independent photon is prepared using a linear plate polariser set to Horizontal to match the polarisation degree of freedom. The Spectral component is also matched by calibrating the temperature of the crystal ovens to yield the highest independent-HOM visibility, found to be 91%. Each path exiting the beam splitter is coupled to 50km SMF to induce chromatic dispersion and delay the photon packets in time. A time-of-flight spectrometer converts the time difference between the arrival of a photon and the arrival of the Ti:Sapphire sync pulse to estimate the frequency component after experience dispersion. Finally each path is multiplexed into two spatial modes to act as a pseudo number-resolving detectors.