1. INTRODUCTION

Broadband time–energy entangled photons, owing to strong correlations in time and frequency, have been utilized in a range of quantum applications including communication [1–5], sensing [6–8], spectroscopy [9,10], positioning, and clock synchronization [11–14]. Multi-node quantum networks with the ability to distribute entanglement over local, metropolitan, and global scales will enable advances in distributed quantum information processing [8,15] for applications such as blind quantum computing, long-baseline interferometry, and secure communication. Metrology to gather precise position and timing information at network nodes will be crucial for scheduling and routing information flows. As local and metropolitan area quantum networks evolve from entanglement distribution [16–18] to heralded entanglement generation [19–21], precise knowledge of link latency will be critical to operations such as Bell state measurements.

Entanglement offers a distinguishing nonlocal resource [22–25] that can be exploited in quantum network architectures for a range of functions. Measurement of the joint temporal correlation of entangled photons through coincidence detection has enabled precision synchronization across remote sites [12,13,26]. Such temporal correlations are often measured by tagging the photon arrival times at remote sites using single-photon detectors (SPDs) and event timers. However, the precision of time-resolved correlation measurements using commercially available superconducting nanowire SPDs (SNSPDs) is limited to around 50 ps owing to detector jitter. While there have been demonstrations that go beyond these limits by either improving SPD jitter [27] or modifying event timing algorithms [26], nonlocal metrology with sensitivity independent of detector jitter and event timer resolution has thus far not been reported. While techniques such as Hong–Ou–Mandel [7,28] and Ramsey interferometry [29] are not limited by jitter from photodetection, they do not permit nonlocal, or distributed, sensing. Hong–Ou–Mandel interferometry invariably requires spatial overlap of photons and, therefore, careful balancing of the optical paths traversed by photons. In the case of Ramsey interferometry, nonlinear interaction between photons is crucial.

We overcome these limitations with an approach that maps changes in the joint spectral phase of a propagating biphoton wave packet to coincidence probability. Mapping information about the joint phase to the coincidence basis allows one to observe small changes in biphoton spectral phase and, therefore, in delay that would otherwise be unnoticeable owing to picosecond-scale jitter in SPDs. Our approach draws from prior work on the characterization of spectral phase coherence in high-dimensional frequency bin-entangled photons [30–34]. We similarly leverage electro-optic phase modulation for coherent frequency mixing, but rather than manipulating spectral phase to violate a Bell’s inequality, we instead measure changes in the spectral phase to determine the difference in signal–idler arrival for high-precision delay metrology. Owing to the use of sinusoidal radio frequency (RF) phase modulation, we measure a generalized delay—the difference in signal–idler arrival times relative to the phases of the local modulating RF waveforms. We showcase this unique capability in two kinds of precision measurements—one of link latency ($2\sigma = 0.04\;{\rm ps} $) with the relative RF phase held constant and one of RF phase offset ($2\sigma ={ 0.7^\circ}$) with static links. Critically, neither measurement requires local overlap of signal and idler, which propagate along separate paths with different lengths [cf. Fig. 2].

Section 2 presents an overview of our measurement approach, as well as theoretical analysis that relates biphoton delay and RF oscillator phase offset to measurements in the coincidence basis. In Section 3, we describe our setup and cover a series of measurements designed to showcase the versatility of our approach and compatibility with practical network environments. In particular, we acquire interferograms while changing path lengths in small increments with a motorized delay line. In addition to highlighting the sensitivity of the approach to small changes in path length, the shift in interferograms with changes in relative phase of the modulating RF waveforms is used to determine the RF phase offset. We also demonstrate uncompromised delay sensitivity even with several hundred meters of path length mismatch between optical paths. At the end of this section, we complement fine measurements of the difference in signal–idler arrival with time-of-flight (ToF) data from SPD time-tags for unambiguous measurement of absolute time delay. We conclude with Section 4, which provides a discussion of our results and explores the potential use in practical environments. Finally, in Supplement 1 (E), we highlight evidence for strong parallels between the second-order time correlation of biphotons and coincidence measurements based on our sensing approach.

2. OVERVIEW AND THEORY

A. Overview

The essence of the presented sensing scheme is the Fourier transform relationship between the biphoton wave function in the frequency and the delay bases [35,36], which is a direct consequence of entanglement. As depicted in Fig. 1(a), sidebands from the various input signal (idler) frequency bins are generated via phase modulation and made to overlap at the central signal (idler) frequency bin. These central bins are selected via spectral filters and routed to SPDs for coincidence detection, constituting a two-photon interference experiment where the coincidence probability is closely related to the inverse Fourier transform of the biphoton spectral amplitude function, cf. Eq. (8); hence our measurement provides information on the biphoton time correlation function [refer to the following Theory subsection and Supplement 1 (E) for details].

 

Fig. 1. (a-i) Frequency domain schematic of a nine-dimensional BFC. (a-ii) Illustration of phase modulation sidebands contributing to the bin pair $|0,0{\rangle _{{\rm SI}}}$, selected for coincidence detection. (b) Theoretical coincidence probability from equal-amplitude mixing of frequency bins plotted over one repetition period (${T_{{\rm rep}}}$) for BFCs with different dimensions. (c) Theoretical coincidence probability per photon pair for nine-dimensional BFC resulting from single-sine-wave phase modulation at different modulation depths. (d) Bessel mixing coefficients at modulation depth of 4.48 rad.

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The time variable involved in the Fourier transform takes the form of a difference in generalized delays, which is expressed here in a simplified form as $[({\tau _{\textit{S}}} + {\phi _{\textit{S}}}\omega _{{{\rm FSR}}}^{- 1}) - ({\tau _{\textit{I}}} + {\phi _{\textit{I}}}\omega _{{{\rm FSR}}}^{- 1})]$. Here, ${\tau _{\textit{S}}}$ and ${\phi _{\textit{S}}}$ refer to the arrival time of the signal photon and the phase of its corresponding RF modulation signal, respectively, both relative to a local clock; ${\tau _{\textit{I}}}$ and ${\phi _{\textit{I}}}$ refer to the analogous quantities for the idler photon, and $\omega_{{\rm FSR}}^{}$ refers to the spacing between the frequency bins of the biphoton frequency comb (BFC). Thus, the measured quantity in the presented scheme is the difference in the arrival times of the signal and idler photons, each in reference to the timing of their respective RF phase modulation. This is the quantity of interest for qubit or qudit rotations when time–frequency entanglement is distributed over a network, in particular, when quantum frequency processors (QFPs) [37,38] perform local operations at end nodes for teleportation or entanglement swapping, for example, a key parameter to be monitored and controlled is the photon arrival time relative to the RF phase modulation. While classical synchronization techniques are well explored [39–41], these approaches add to control overhead in quantum networks; our work raises the possibility of leveraging high-dimensional entangled photons themselves to sense critical timing information directly and provide synchronization data in-line in an entanglement distribution system.

B. Theory

Consider the state of a BFC of dimension ${2}{N} + {1}$, with a free spectral range (FSR) of $\omega _{{{\rm FSR}}}^{}$ represented as

The signal and idler frequency bins are routed to different optical links using a pulse shaper that is also employed to impart a spectral phase on the biphotons. The signal and idler traverse through delays ${\tau _{\textit{S}}}$ and ${\tau _{\textit{I}}}$, respectively, along their paths. Down the links, the biphotons are each phase modulated with RF sinusoids at modulation frequency ${\omega _{{{\rm RF}}}}$ equal to the bin spacing $\omega _{{{\rm FSR}}}^{}$. Finally, frequency bins centered at $\omega _0^{(S)} = {\omega _0} + {\Omega _0}$ and $\omega _0^{(I)} = {\omega _0} - {\Omega _0}$, (i.e., when $k = 0$) are selected using spectral filters and routed to different SPDs for coincidence detection.

The annihilation operator $\hat a_k^{(S)}$ ($\hat a_{- k}^{(I)}$) corresponding to the signal (idler) transforms into $\hat b_k^{(S)}$($\hat b_{- k}^{(I)}$) after traversing down the link (prior to phase modulation) as follows:

Down the links, after the transformation due to temporal phase modulation of the forms ${e^{- im\sin (\omega _{{{\rm FSR}}}^{}t + {\phi _{\textit{S}}})}}$ and ${e^{- im\sin (\omega _{{{\rm FSR}}}^{}t + {\phi _{\textit{I}}})}}$, respectively, applied in the signal and idler paths, annihilation operators $\hat c_k^{(S)}$ and $\hat c_{- k}^{(I)}$ corresponding to signal at the $k$th frequency bin and idler at the ${-}k$th frequency bin are given by

Based on the above formalism, the probability of measuring a coincidence count, ${\cal P}(\Delta \phi)$, between frequency bins at $\omega _0^{(S)}$ and $\omega _0^{(I)}$, per input photon pair, is given by the following equation:

Here, the mixing coefficient ${C_k} = {J_k}(m){J_{- k}}(m) = |{J_k}(m{)|^2}\def\LDeqbreak{}{e^{{\textit{ik}\pi}}}$ results from the phase modulation sidebands. For more details on the theory of incorporating frequency bin width and detection timing jitter, see Supplement 1 (A). We observe that the coincidence probability is sensitive to the differential biphoton delay $\tau$, the relative phase ${\phi _{{{\rm RF}}}}$ between the RF drive signals to the phase modulators, and the linear spectral phase increment ${\varphi _{{{\rm PS}}}}$ imparted by a pulse shaper. By collecting all the linear spectral phase terms, we can rewrite the coincidence probability per photon pair as

Figure 1(b) plots the coincidence probabilities over one period under equally weighted mixing (i.e., $|{C_k}| = 1$) and uniform probability amplitudes (i.e, ${\alpha _k} = 1/\sqrt {2N + 1}$) as a function of ${\tau _{{{\rm eff}}}}/{T_{{\rm rep}}}$. With higher dimensionality, the width of the trace in Fig. 1(b) decreases, and the maximum slope in the main lobe of the trace increases, offering better sensitivity with respect to changes in the delay and RF phase. Under this ideal equally weighted mixing scenario, the width of the coincidence trace is inversely proportional to the dimensionality ($2N + 1$). However, in actual experiments, the mixing coefficients are not equal; they have a Bessel function dependence on the phase modulation amplitude. To provide further insight, in Fig. 1(c), we plot theoretical coincidence traces for Bessel mixing coefficients as described in Eqs. (4)–(8) over modulation depths ranging from 0.5 to 6 rad (experimentally feasible using a single-phase modulator). Here we assume a nine-dimensional BFC, which coincides with our experiments, and retain equal probability amplitudes ${\alpha _k}$. We observe that the widths of the traces decrease with increasing modulation depth up to about 4 rad, while the coincidence probability at the peak remains roughly constant. For higher modulation depths, the peak coincidence probability decreases while the widths of the traces remain approximately constant in the considered range going up to 6 rad. We can understand these trends as follows. As the modulation depth increases, the phase modulators generate sidebands over a wider bandwidth, with decreased amplitude per sideband. As a result, the number of frequency bins contributing to the central ($k = 0$) bins selected for two-photon interference increases, leading to a higher effective dimensionality and narrower traces. However, for sufficiently high modulation depth ($m\gtrsim 4$), the number of sidebands exceeds the number of frequency bins in the initial BFC (nine in our example). Now the effective dimensionality is limited by the number of frequency bins, and the widths of the traces depend only on the distribution of the ${\pm}4$ sideband amplitudes. The increased modulation depth now results in an effective loss, since sidebands are generated outside the frequency space in which they can be used.

In our experiments, the depth of phase modulation on both photons is set to be ${\sim}4.48\,\,\rm rad$ [with magnitude of sideband intensities shown in Fig. 1(d)], serving to operate near optimal delay sensitivity setting for a nine-dimensional BFC with a constant input flux rate; see Supplement 1 (B) for more details.

3. EXPERIMENT AND RESULTS

A. Setup

In this section, we present some proof-of-concept experiments, demonstrating the key capabilities of our sensing approach. As sketched in Fig. 2, a continuous-wave (CW) pump laser at 778 nm is routed to a 2.1 cm long fiber-pigtailed periodically poled lithium niobite (PPLN) ridge waveguide to generate time–energy entangled photons under type-0 phase matching. The spontaneous parametric downconversion (SPDC) spectrum is centered around 1556 nm (${\omega _0} = 2\pi \times 192.7\,\,\rm THz$) and spans a bandwidth ${\gt}\;{5}\;{\rm THz}$. The 778 nm pump laser used in the setup has a linewidth of ${\sim}200\,\,\rm kHz$. We use a programmable pulse shaper (pulse shaper 1) to select frequency-correlated slices of spectral width ${\sim}\;288\,\,\rm GHz$ from the signal and idler spectra and route them to different arms. The centers of selected signal and idler spectral slices are offset from the SPDC center frequency by ${\Omega _0}/2\pi = 608\;{\rm GHz}$. The biphotons travel through different path lengths as they propagate along their respective optical links. Down the link, an electro-optic phase modulator (EOPM) is placed in each of the signal and idler arms. The EOPMs are driven by RF sinusoidal waveforms at modulation frequency ${\omega _{{{\rm RF}}}}/2\pi = 32\;{\rm GHz}$ and modulation depth $m \approx 4.48\,\,\rm rad$. In our experiment, the microwave drives for the EOPMs in the signal and idler arms are derived from a common RF oscillator. We note that nonlocality in our work connotes a lack of spatial overlap of the biphoton wave packet over the system being probed. Synchronization merely ensures a common timing reference for any nonlocal architecture. Several experimental architectures distributing biphotons to different nodes, such as those aimed at teleportation [42], dispersion cancellation [43,44], and ToF measurements for sensing [26], are termed nonlocal even though synchronization between remote detectors is required to meaningfully compare time-tags.

 

Fig. 2. Experimental setup. CW laser, continuous-wave laser; PPLN, periodically poled lithium niobate waveguide; ODL, optical delay line; EOPM, electro-optic phase modulator; SNSPD, superconducting nanowire single-photon detector.

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After phase modulation, frequency bins with spectral width of $\delta \Omega = 15\,\,\rm GHz$ are selected from the centers of the signal and idler spectra using two programmable filters (pulse shapers 2 and 3, respectively). The selected frequency bins are routed to SNSPDs for coincidence measurement. In our demonstration we use SNSPDs with a combined timing jitter of ${\sim}100\;{\rm ps} $ and an event timer to record coincidence time-tag histograms with a resolution up to ${\sim}1\,\,\rm ps$.

The phase modulation and spectral filtering operations effectively post-select a nine-dimensional BFC state, with FSR equal to the modulation frequency, i.e., $\omega _{{{\rm FSR}}}^{} = {\omega _{{{\rm RF}}}}$, and frequency bin width equal to $\delta \Omega$; see detailed theoretical analysis in Supplement 1 (A). The selected frequency bin pair contains coherent sideband contributions from all nine frequency bins accommodated in the biphoton spectra. Thus the probability of coincidence detection depends on both the spectral phase across the nine bin pairs and the modulation parameters.

Note that although the parameter settings specified above are used in the initial experiments, e.g., Figs. 3(a) and 3(b), we alter them in some of the later demonstrations. The total acquisition time of the coincidence histograms ($\Delta t$), the histogram time window ($\Delta T$) over which coincidences are integrated to plot the datapoints in the figures, bandwidth of the signal and idler spectrum ($\Delta \Omega$), offset from the SPDC center (${\Omega _0}$), modulation frequency ($\omega _{{{\rm FSR}}}^{}$), and the frequency bin width ($\delta \Omega$) used in all the experiments are tabulated in Supplement 1 (C). The dimensionality of the BFC (${d} = {9}$) and the phase modulation depth (${ m}\sim 4.48\,\,\rm rad$) are set to be the same across all experiments.

 

Fig. 3. Two-photon coincidence interferograms acquired as a function of change in group delay of the signal photon ${\tau _{\textit{S}}}$. Signal photon delay scanned by the ODL (a) over three repetition periods and (b) after each turn of the RF phase shifter control knob (phase shift ${\sim}{100^ \circ}$ per turn). (c) Overlaying three normalized coincidence interferograms acquired as follows: (i) by varying the slope of linear spectral phase on the signal bins using pulse shaper 1; (ii, iii) by scanning the signal photon delay using an ODL in the absence of additional dispersive fiber and after dispersion compensation in the presence of a dispersive spool in the signal arm, respectively. (d) Coincidences (acquired over 10 s) plotted as the signal photon delay is scanned in the presence of a dispersive spool in the signal arm, before and after dispersion compensation.

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B. Interferograms from Detuning Differential Biphoton Delay

Figure 3(a) shows coincidence counts measured when the delay of the signal photon ${\tau _s}$ is varied using the motorized optical delay line (ODL) [as shown in Fig. 2]. The data show a series of sharp peaks that repeat at ${\sim}31.25\,\,\rm ps$, corresponding to the period of the 32 GHz RF modulation. The full width at half maximum (FWHM) of the interferogram is equal to 2.8 ps, far below the ${\sim}100\,\,\rm ps$ timing jitter of our SNSPDs. We can clearly resolve subpicosecond delay steps especially at operating points situated close to the high-slope regions in the central lobe of the interferogram.The ODL is scanned in increments of 0.55 ps. The acquired-coincidence histogram is normalized with respect to the peak and fit with the theoretical probability in Eq. (8) considering bin pairs with uniform probability amplitudes (i.e., ${\alpha _k} = 1/\sqrt {2N + 1}$). The change to the group delay of the signal photon is denoted by $\Delta {\tau _{\textit{S}}}$ in the plots.

Figure 3(a) shows close agreement between the theoretical prediction and the experimental results. The dispersion accumulated by the biphotons (${\sim}15\,\,\rm m$ SMF in each arm) is ignored since it is negligible in broadening the interferogram in comparison to the statistical error in the delay.

In our next result, Fig. 3(b) plots a series of interferograms acquired at four different RF phase settings of the 32 GHz waveform modulating the signal photon. The shifts in the interferograms corroborate the sensitivity to relative RF phase as predicted by Eq. (8). The RF phase shifter is manually adjusted in steps of roughly ${\sim}{100^ \circ}$, and at each RF phase setting, the motorized ODL is scanned. Here, we estimate the relative RF shifts from a least squares fit of the interferograms to the theory. From Eq. (6), if the RF modulation phase shift of trace (1) with respect to trace (2) is $\phi _{{{\rm RF}}}^{(1)} - \phi _{{{\rm RF}}}^{(2)}$, then the resultant offset in the ODL setting corresponding to the peaks of traces (1) and (2) is

The RF phase settings corresponding to interferogram (1) relative to interferograms (2)–(4) are recovered to be : $\phi _{{{\rm RF}}}^{(1)} - \phi _{{{\rm RF}}}^{(2)}={ 102.0^ \circ} \pm {0.7^ \circ}$, $\phi _{{{\rm RF}}}^{(1)} - \phi _{{{\rm RF}}}^{(3)}={ 198.6^ \circ} \pm {0.6^ \circ}$, and $\phi _{{{\rm RF}}}^{(1)} - \phi _{{{\rm RF}}}^{(4)}={ 307.0^ \circ} \pm {0.6^ \circ}$; the reported 95% confidence intervals (${\pm}2\sigma$) incorporate the errors from the least squares fit of the interferograms and the resolution of the ODL.

Equation (7) predicts that the two-photon interferograms are equally sensitive to delays from changes in the physical path length and to delays generated without moving parts by application of a linear spectral phase. Figure 3(c) overlays one period of the interferogram from Fig. 3(a) over that measured by scanning the slope of the linear spectral phase on the signal frequency bins using pulse shaper 1 (while no phase is applied on idler bins). From Eq. (8), a spectral phase increment of ${\varphi _{\textit{S}}}$ applied to the signal frequency bins is associated with relative delay ${\tau _{{{\rm eff}}}}$ modulo ${T_{{\rm rep}}}$ given by ${\varphi _{\textit{S}}}\omega _{{{\rm FSR}}}^{- 1}$. This equivalence of relative delay with the linear spectral phase is evident from the closely matching traces in Fig. 3(c). The interferogram in our proposed scheme can thus be measured without requiring an ODL, by simply using a pulse shaper already employed in our setup for demultiplexing the signal and idler photons. Together, Figs. 3(a)–3(c) confirm the sensitivity of our two-photon interferograms to changes in the relative physical delays of signal and idler, to changes in the relative phases of the applied RF modulations, and to changes in applied linear spectral phases, as predicted by the $\Delta \phi$ expression in Eq. (7). Figure 3(c) also plots an interferogram acquired after compensating for the total dispersion in the setup when an SMF spool introducing ${\sim}1.5\,\,\unicode{x00B5}\rm s$ of delay is inserted in the signal arm. The details are covered in the following section.

C. Sub-ps Sensitivity over µs-Scale Delay Mismatch

The proposed scheme does not necessitate the precise balancing of path lengths traversed by the biphotons to observe the interferogram. The interference features can be observed as long as the imbalance in the signal and idler arms is within the biphoton coherence length. In our experiments, the coherence time is ${\sim}5\,\,\unicode{x00B5}\rm s$, dictated by the inverse of the CW pump linewidth, corresponding to ${\sim}1\,\,\rm km$ propagation length imbalance in standard fiber. A characteristic feature of the interference trace is its periodicity due to the sinusoidal phase modulation. One can measure small delay changes (modulo the modulation period) even with large delay offsets, because the delay changes are measured with respect to a periodic RF clock. In this section, we demonstrate sub-ps resolving capability despite large imbalances in the path lengths in the signal and idler arms. A ${\sim}313\,\,\rm m$ long SMF-28e spool is inserted in the signal path prior to modulation, which introduces ${\sim}1.5\,\,\unicode{x00B5}\rm s$ delay offset between the biphotons arriving at their respective detectors. Figure 3(d) shows the coincidences measured (in blue markers) at each ODL setting as delay in the signal path is scanned. The trace still achieves picosecond-scale delay sensitivity but is broadened and distorted due to dispersion. The total width of the interferogram at the half maximum points is 10.8 ps, close to four times that in the absence of the SMF spool.

By factoring in second-order dispersion into the theoretical coincidence probability, Eq. (8) becomes

Although dispersive broadening is undesirable, its effect can be accurately modeled, and it can be compensated for. We fit Eq. (10) to the interference pattern in Fig. 3(d) (blue) and estimate the sum total of dispersion ${\beta _2}({L_S} + {L_I})$ associated with the quadratic phase term to be ${\sim}- 7.4\,\,{\rm ps^2}$ (equivalent to a single photon traversing ${\sim}343\,\,\rm m$ of SMF-28e fiber with ${\beta _2} = - 2.16 \times {10^{- 2}}\,\,\rm p{s^2}/m$). This corresponds to the 313 m fiber spool plus an estimated ${\sim}15\,\,\rm m$ of fiber path for each signal and idler photon in the remainder of the apparatus. We compensate for the total estimated dispersion by applying an equal amount but opposite sign of quadratic spectral phase onto the signal photon bins using pulse shaper 1. The interferogram acquired after such compensation is overlayed onto Figs. 3(c) and 3(d). The normalized coincidences in Fig. 3(c) (blue) are restored almost identically to the interferogram obtained in the absence of the spool. The peak coincidences improve by a factor of 2.3 after dispersion compensation as shown in Fig. 3(d), close to the theoretically expected factor of 2.2.

Although here we compensate for dispersion prior to routing the photons to different paths, the dispersion can also be compensated for anywhere down the fiber links prior to modulation. Furthermore, in an effect known as nonlocal dispersion compensation [22,23], the total dispersion accumulated by the biphoton can be compensated for in only one of the links by controlling the spectral phase of either the signal or the idler photon.

Our results demonstrate the capability to achieve the original delay sensitivity of the interferogram despite a ${\sim}1.5\,\,\unicode{x00B5}\rm s$ mismatch between the biphoton delays. This highlights the important nonlocal delay sensing capabilty of our approach.

D. Unambiguous Delay Metrology

Although periodic modulation of biphotons allows for sensing relative delays despite a large imbalance in their path lengths, it comes at a price: the interferogram as a function of relative delay repeats every modulation period and restricts the unambiguous sensing range. One can circumvent this limitation by supplementing the interferogram with photon arrival time-tags recorded by the event timer after detection. In this section, we demonstrate a basic implementation that utilizes histograms generated by the event timer to determine the coarse delay, thereby removing the ambiguity from the periodic interferogram. As long as the resolution of the detectors and timing electronics is less than the repetition period, unambiguous sensing of delay can be performed. We demonstrate this scheme by comparing measurements performed with and without an SMF (FS P/N: SM-FCU-FCU-SX-FS-1M-PVC) of length $1.05 \pm 0.05\,\,\rm m$ in the idler arm of our setup prior to phase modulation (refer to Fig. 2). In this experiment, the modulation frequency is set to ${\omega _{{{\rm RF}}}} = \omega _{{{\rm FSR}}}^{} = 20\;{\rm GHz} $ to result in a repetition period (${T_{{\rm rep}}}$) of 50 ps. Note that the fiber inserted in the idler adds a delay (of ${\sim}5\,\,\rm ns$), which is ${\sim}100$ times larger than the RF modulation period.

We perform measurements with and without the additional fiber in the idler arm and adjust the ODL to position at an interferogram peak in each case—at which point time-tagger histograms of the difference between the signal and idler detection times are acquired. Positive values of the delay difference signify that the signal is detected later than the idler. Then the difference between the mean values of the time-tagger histograms should be an integer multiple of ${T_{{\rm rep}}}$, i.e., $k{T_{{\rm rep}}}$, where $k \in \mathbb{Z}$. We thus rely on the time-tagging electronics to disambiguate the value of the integer $k$. One can easily show that the effective delay that was inserted into the idler arm $\Delta {\tau _{\textit{I}}}$ should be equal to

 

Fig. 4. (a), (b) Two-photon coincidence interferograms recorded with and without an approximately 1 m long SMF in the idler arm, respectively. $\Delta {\tau _s}$ refers to the delay added into the signal arm via the motion of the ODL (scanned in steps of 0.4 ps), referenced to the same starting point ($\Delta {\tau _s} = 0$) for both interferograms. Solid lines: normalized theoretical coincidence probability fit to the experimental curves. (c) Coincidence time-tag histograms acquired when the ODL in the signal arm is set to the delay positions corresponding to the datapoints at peaks ${P_1}$ and ${P_2}$ from (a), and the datapoints at peaks ${P_3}$ and ${P_4}$ from (b). The time-bins in the histograms are 2 ps wide. Solid lines: Gaussian fit to time-tag histograms.

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Let ${\tau _h}({P_i})$ denote the mean value from the Gaussian fit of the histogram acquired at the interferogram peak ${P_i}$; the estimated means with the confidence intervals are tabulated in Supplement 1 (D). In our experiment, the difference ${\tau _h}({P_4}) - {\tau _h}({P_2})$ is obtained to be 5002 ps with a 95% confidence interval width $\lesssim 6\,\,\rm ps$. As previously stated, we expect this difference to be an integer multiple of the repetition period. Since the estimate of ${\tau _h}({P_4}) - {\tau _h}({P_2})$ is localized much tighter than ${T_{{\rm rep}}}$, it can be rounded to 100 ${T_{{\rm rep}}}$, i.e., the integer ${k_{{{P_4} - {P_1}}}} = 100$.

Furthermore, the difference between ODL settings at interferogram peaks ${P_i}$ and ${P_j}$ denoted by $\Delta \tau _{\textit{S}}^{(w)}({P_i}) - \Delta \tau _{\textit{S}}^{(w/o)}({P_j})$ are estimated from a weighted least squares fit of the interferograms to the theory and listed in Supplement 1 (D). For instance, $\Delta \tau _{\textit{S}}^{(w/)}({P_2}) - \Delta \tau _{\textit{S}}^{(w/o)}({P_4}) = 14.65 \pm 0.04$ ps, the error bars denoting the 95% confidence bounds (${\pm}2\sigma$). The effective delay due to the ${\sim}1\,\,\rm m$ SMF in the idler arm computed from Eq. (12) is $5014.65 \pm 0.04\,\,\rm ps$, consistent with the manufacturer specifications. The result is verified for different pairs of interferogram peaks as shown in Supplement 1 (D).

4. DISCUSSION AND CONCLUSION

In summary, we present a nonlocal sensing scheme with subpicosecond resolution that requires only off-the-shelf telecommunications equipment and resources expected to be staples of quantum networks. By interfering multiple frequency bins in a biphoton spectrum, we map the joint spectral phase accumulated by a propagating biphoton to a change in the probability of coincident events between each photon and its entangled counterpart. This is accomplished without spatial overlap of biphotons—thus enabling nonlocal delay metrology. We demonstrate precision sensing of RF phase shifts, as well as of relative biphoton delays, with ($2\sigma$) precision of ${\pm}{0.7^ \circ}$ and ${\pm}0.04\,\,\rm ps$, respectively. We use dispersion compensation to highlight compatibility with practical network environments and demonstrate that our approach can tolerate at least hundreds of meters path length imbalance (microsecond-scale time imbalance) between the optical links traversed by biphotons. Finally, we complement precision measurements of the relative biphoton delay with ToF measurement data from photon arrival time-tags for unambiguous measurement of absolute delay. In terms of use in practical quantum networks, our work can be leveraged for a range of critical functions—from tracking link latency and relative RF phase drifts, to distant clock synchronization and position verification [11,12,14,45]. There is potential to perform nonlocal clock synchronization [12] or to use the quantum signals for both time transfer and network protocols such as secret-key generation [46]. While we used programmable filters for demultiplexing spectral channels, in a deployed system, they can be easily replaced with coarse and dense wavelength division multiplexers.

The techniques presented here can be extended in multiple ways. Consider that distance measurements using classical dual frequency combs with slightly different repetition rates offer long-range sensing capability with high resolution [47,48]. Analogously, by applying two closely spaced RF frequencies to the modulator in our measurements, one can increase the interferogram repetition period to the inverse RF frequency separation, thereby realizing a large unambiguous range. In addition, our sensing approach is closely connected to measurement of the biphoton second-order time correlation as elucidated in Supplement 1 (E) and can potentially be exploited for quantum state tomography of biphotons. Finally, it may be possible to realize the quantum advantage offered by entanglement through global parameter estimation. If the overall detection efficiency can be sufficiently optimized, measurement of a linear combination of signal and idler delays (in our case, the difference between their individual delays) can achieve sensitivity beyond the shot noise limit [49,50].