Feasibility study towards comparison of the g^(2)(0) measurement in the visible range

This work reports on the pilot study, performed by INRIM, NPL and PTB, on the measurement of the g^(2)(0) parameter in the visible spectral range of a test single-photon source based on a colour centre in diamond. The development of single-photon sources is of great interest to the metrology community as well as the burgeoning quantum technologies industry. Measurement of the g^(2)(0) parameter plays a vital role in characterising and understanding single-photon emission. This comparison has been conducted by each partner individually using its own equipment at INRIM laboratories, which were responsible for the operation of the source

The typical parameter employed to test the properties of a SPS is the second order correlation function (or Glauber function) defined as g (2) (τ = 0) = I(t)I(t + τ ) where I(t) is the intensity of the optical field. In the regime of low photon flux, this parameter has been shown to be substantially equivalent to the parameter α introduced by Grangier et al [33], which is experimentally measured as the ratio between the coincidence probability at the output of a Hanbury Brown and Twiss (HBT) interferometer [34], typically implemented by a 50:50 beam-splitter connected to two non photon-number-resolving detectors, and the product of the click probabilities at the two detectors, i.e.:

Metrologia
Feasibility study towards comparison of the g (2) (0) measurement in the visible range (Some figures may appear in colour only in the online journal) Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
where P C , P A , P B are, respectively, the coincidence and click probabilities at the outputs A, B of an HBT interferometer. This identity holds strictly for very low value of the clickprobabilities P A and P B (namely much less than 0.1), i.e. for very faint SPSs, while it is only approximately verified for brighter sources. Due to the equivalence between g (2) (0) and α in the regime typical of quantum optics experiments, all experimental measurements of g (2) (0) in the relevant literature are actually measurements of α, since the two parameters are used substantially without distinction in this community. This work presents a systematic study of the α measurement for a SPS in the pulsed regime, with the purpose of developing a measurement procedure and an analysis of the uncertainty to provide an unbiased value of the measurand which is independent of the experimental apparatus used, ultimately producing an estimate unaffected by the non-ideal behavior of the physical systems. Consensus on such a procedure would produce great benefits for the metrology community, enabling the development of SPS characterization techniques that are robust enough for practical measurement services. The results reported in this work were obtained during a pilot study performed by INRIM, NPL and PTB. This is a precursor to organising an international comparison on the g (2) (0) measurement, which would pave the way for the realization of a mutual recognition agreement on the calibration of key elements for forthcoming optical quantum technologies, such as SPSs and single-photon detectors. This comparison was hosted at INRIM from October 16 to October 29 2017 and was composed of two joint measurements of α on the same emitter: one performed by INRIM and PTB and the other one by INRIM and NPL. This procedure was adopted because it allows the results of two measuring devices operating simultaneously to be compared. Measurements on the same source at different times can yield slightly different results, since the imperfectly reproducible alignment of the source can lead to a different amount of noise coupled to the detection system. An SPS based on a nitrogen-vacancy centre excited in the pulsed regime, emitting single photons in the spectral range from 650 nm to 750 nm was used as a source.
An analogous effort to establish a proper procedure for the measurement of the g (2) function of a telecom heralded SPS (in continuous regime) can be found in [35].

Measurement technique
With regards to equation (2), probabilities P C , P A , P B are estimated as the ratio between the total number of the corresponding events versus the number of excitation pulses during the experiment, i.e. P x = N x /(Rt acq )(x = C, A, B), where R is the excitation rate and t acq is the total acquisition time. The value of the measurand is independent from the total efficiencies (η A , η B ) of individual channels (including detection and coupling efficiency), optical losses and splitting ratio since The value of the parameter from the experimental data, corrected for the contribution of the background coincidences (due, for example, to stray light or residual excitation light), can be estimated as follows: where P Cbg , P Abg , P Bbg are, respectively, the coincidence and click probabilities of background photons, calculated analogously to their counterparts P C , P A , P B . Figure 1 shows the typical chronogram of the behaviour of a pulsed SPS obtained by sampling the coincidence events at the two outputs of an HBT interferometer. The coincidence probability has been estimated as the ratio between the total number of events in the chronogram falling in a fixed temporal  figure 1) and the total number of excitation pulses occurring in the acquisition time. The product P A P B , corresponding to the probability of accidental coincidences, has been evaluated by integrating the events occurring in an equal interval around the subsequent peak ('c' interval in figure 1) not showing antibunching (always divided by the number of pulses). In fact, those coincidence events (amounting to N ξ ) are related to independent events (coincidences between single photons emitted after two subsequent laser pulses and detected by detector A and detector B respectively) and thus P ξ − P bg = (N ξ − N bg )/(R * t acq ).
The parameter to be estimated is thus: where ( N i being the coincidence events sampled in the ith channel) k w is the number of bins corresponding to the chosen coincidence window w, N bg is the estimated background due to spurious coincidences (the number of events in the 'a' interval in figure 1) and T is the excitation period (expressed in bins).
In figure 1 two backflash peaks [36][37][38] can be observed on either side of the central peak. These are due to secondary photon emission that arises from the avalanche of charge carriers that occurs in one of the two detectors in the HBT interferometer as a photon is absorbed and that are afterwards detected from the other detectors. To avoid overestimating α, these peaks must not be included in the coincidence window.
The presence of the backflash peaks prevented us to estimate P A , P B directly from the counts of the two detectors, since we were forced to consider a coincidence window smaller than the NV-center emission time window (of the order of tens of nanoseconds, i.e. at least three NV lifetimes). For this reason we estimated P A P B consistently with the coincidences measured at time 0. The probability of observing a coincidence in the autocorrelation window around the peak at 400 ns can be underestimated by the presence of the coincidence counts between 0 and 400 ns, because of detectors and electronics dead-time. Due to the extremely low level of counts in this interval, we have estimated that this correction is negligible within the declared probability uncertainty. Figure 2 shows the experimental setup: a laser-scanning confocal microscope whose signal is split by a 50:50 beamsplitter and connected to two measurement devices, i.e. two single-photon sensitive HBT interferometers. Note that, according to the model described in section 2, the value of α measured by the two HBT interferometers is independent of optical losses and splitting ratio at the beam-splitter. The excitation light, produced by a pulsed laser (48 ps FWHM, 560 pJ per pulse) emitting at 532 nm with a repetition rate R = 2.5 MHz was focused by a 100× oil-immersion objective on the nano-diamond (ND) sample hosting an SPS based on a single NV center of negative charge, with emission in a broad spectral band starting approximately at 630 nm and ending at 750 nm (λ ZPL = 638 nm) [17]. The optical filters used were a notch filter at 532 nm and two long-pass filters (FEL600 and FEL650). The photoluminescence signal (PL), thus occurring in a 650 nm-750 nm spectral range, was collected by a multimode fibre and split by a 50:50 beam-splitter (BS). As stated above, each end of the BS was connected to a separate HBT setup used for the joint measurement. In particular: The detailed description of the sample fabrication and preparation is reported elsewhere [39].

Results
Each measurement consisted of 10 runs each of 500 s acquisition time. The total coupling rate, accounting for limited SPS quantum efficiency, collection angle, optical losses   Sens. coeff.

Unc. contribution
and detection efficiency (excluding the splitting ratio of the detector-tree), has been estimated as the ratio between the counting rate of the detectors (summing over all four of them) and the excitation rate, yielding η TOT = (1.76 ± 0.01)%. The coincidence window w considered for evaluating the reported α was w = 16 ns. By repeating the analysis for different temporal widths w it was observed that the results were consistent as long as the backflash peaks were not included in the coincidence window (see figure 5). Figures 3 and 4 show the distributions of the α exp values measured by each partner; the continuous line indicates the mean value and the dashed lines draw a 1-σ confidency band around the mean value. Tables 1-4 report the uncertainty budgets associated with the measurements. The summary of the results of the joint measurement is presented in where the correlation coefficient ρ xy is defined as

Dependence on the coincidence window
To prove that the estimation of α is independent of the choice of the time interval of integration, we performed an analysis of the values of the measurand obtained by varying the coincidence window w. The results are shown in figure 5, demonstrating that, as long as the backflash peaks are not included in the integration, the estimate is consistent independently of w.

Conclusions
A pilot study on the characterization of a pulsed-pumped test SPS based on a NV centre in nanodiamonds was performed by INRIM, NPL and PTB and hosted by INRIM. This study will greatly benefit the single-photon metrology community, as  well as rapidly-growing quantum-technology-related industries. The main result of this study was the development of a standardized measurement technique as well as an uncertainty estimation procedure. The validity of the technique (systemindependent and unaffected by the non-ideality of the apparatus) is demonstrated by the results obtained, yielding for all the participants estimated values of g (2) (0) that are compatible within the uncertainty (k = 2).

Acknowledgments
This work was funded by: project EMPIR 14IND05 MIQC2, EMPIR 17FUN06 SIQUST, EMPIR 17FUN01 BECOME (the EMPIR initiative is co-funded by the European Union

Appendix. Lifetime estimation
The mean lifetime associated with the source has been estimated by numerically fitting the coincidence histograms (as in figure 1) via the single-exponential function [41][42][43] where a corresponds to the number of background coincidences, b is a normalization factor, δ 0n is the Dirac Delta, c is the number of excited emitters, n is the excitation pulse number, ∆t is the excitation period and, finally, d accounts for the lifetime (convoluted with the detectors' jitter) of the center. Figure A1 shows the results of the lifetime estimation independently performed by the partners. Each value in the plot represents the mean of the results of 10 fits (one for each experimental run performed by one partner). Averaging the results, it is obtained the value t LIFE = (15.34 ± 0.08) ns.