Photon statistics characterization of a single-photon source

In a recent experiment, we reported the time-domain intensity noise measurement of a single-photon source relying on single-molecule fluorescence control. In this paper, we present data processing starting from photocount timestamps. The theoretical analytical expression of the time-dependent Mandel parameter Q(T) of an intermittent single-photon source is derived from ON↔OFF dynamics. Finally, source intensity noise analysis, using the Mandel parameter, is quantitatively compared with the usual approach relying on the time autocorrelation function, both methods yielding the same molecular dynamical parameters.

(a) Single-photon generation by pulsed excitation of a single fourlevel molecular system from the ground singlet state S 0 to a vibrationally excited sublevel of the singlet state S 1 . Solid arrows correspond to optical transitions, whereas dashed arrows depict non-radiative fast (ps) de-excitation. To emit a single photon per excitation pulse, the pulse duration δt must be much shorter than the radiative lifetime 1/ . (b) Absorption and fluorescence spectra of DiIC 18 (3) dye embedded in a thin polymer film, measured respectively with absorption spectrometer and spectrofluorimeter with 514 nm excitation wavelength. Note that in the SPS experiment, the 532 nm excitation wavelength is well separated from the dye's fluorescence emission which is centred at a wavelength around 570 nm.
[12]- [25], a molecular-based SPS presents several advantages. Firstly, it can be driven at room temperature with a relatively simple set-up, which achieves global efficiency exceeding 5% for single-photon production and detection. Secondly, since the molecular fluorescence lifetime is a few nanoseconds, a high repetition rate can potentially be used. Finally, background-emitted photon intensity level is extremely low for carefully prepared samples.
At the single-pulse timescale, the figure of merit of an SPS can be characterized by efficiency of delivering triggered photons to target and by the ratio of single-photon to multiphoton pulses [10]. In [11], we extended this analysis to measurement of SPS noise properties over a wide integration timescale range. In the detection scheme, complete statistical information is extracted from the 'photocount by photocount' record. We have shown that measured photon statistics strongly deviates from the Poisson law, clearly exhibiting non-classical features.
In this paper, we detail the steps of this work from realization of a molecular-based SPS to extensive statistical analysis of detected photons.

Principle of the experiment
As fluorescence light of a four-level single emitter is anti-bunched for a timescale of the order of the excited state radiative lifetime [14], [26]- [28], such systems can simply produce single photons on demand [9,10,13]. As summarized in figure 1, the molecule is pumped into a vibrational excited state by a short excitation pulse. It then quickly decays to the first electronic excited state by a non-radiative process [29]. The emission of a single fluorescence photon then coincides with radiative de-excitation towards the ground-state vibrational multiplicity, followed once again by fast non-radiative decay (figure 1). To emit more than one photon at a time, a molecule has to undergo a full excitation, emission and re-excitation cycle within the same excitation pulse. The probability of this occurrence is extremely small when the pulse duration is much shorter than the excited-state lifetime [9,10]. Following the theoretical analysis of Brouri et al [10], we chose a pulse duration of 150 fs, which makes this probability less than 5 × 10 −5 . This value is negligible in comparison with the one associated with parasitic light, such as residual fluorescence from the molecular host matrix. To get one fluorescent photon per excitation pulse, the repetition period must also be much longer than the excitedstate lifetime so as to ensure relaxation into the ground state before application of the next excitation pulse.

Experimental set-up
We use standard confocal microscopy techniques to perform selective excitation and detection of single-molecule light emission at room temperature [28]. This set-up allows one to readily achieve two required features for observation of non-classical photon statistics, namely good collection efficiency of emitted photons and high rejection of optical background noise.
The laser source, used for fluorescence excitation, is a fs-tunable titanium : sapphire (Ti : Sa) laser, frequency-doubled by single-pass propagation into a LiIO 3 non-linear crystal. The initial repetition rate of 82 MHz is divided by a pulse picker with frequency set to 2.05 MHz (pulsed excitation repetition period τ rep = 488 ns) to avoid surpassing the maximum electronics counting rate (see figure 2).
The excitation light, centred at 532 nm, is reflected by a dichroic mirror into an inverted microscope. It is focused on the sample with an oil-immersion high numerical aperture objective, leading to a spot diameter of 350 nm FWHM. The fluorescence light-red-shifted with respect to the excitation-is collected by the same objective, then transmitted through the dichroic mirror and finally focused inside a pinhole for the confocal configuration. After recollimation, residual excitation light is removed by a holographic notch filter.
The samples used in our experiment consist of cyanine molecules DiIC 18 (3). This dye choice was motivated by its fluorescence efficiency and photostability, with an emission spectrum well suited for detection using a silicium avalanche photodiode (see figure 1(b)). The dye molecules are embedded in a thin layer of PMMA deposited on a microscope coverplate by spin coating. The emitters are randomly distributed within the PMMA layer (thickness 30 nm) at an approximate concentration of 1 molecule per 10 µm 2 .
To ensure localization of a single emitter in the detection volume, we use a standard Hanbury Brown and Twiss set-up [30]. It consists of two single-photon-counting avalanche photodiodes (SPADs) placed on each side of a 50/50 non-polarizing beamsplitter. A start-stop technique with a time-to-amplitude converter allows us to build a coincidences histogram as a function of time delay between two consecutive photodetections on each side of the beamsplitter. Following the textbook experiment of Grangier and Aspect on quantum properties of singlephoton states [31], the absence of coincidence at zero delay gives clear evidence of single-photon emission [12].
We hence apply a simple three-step procedure as explained in [11]. We first raster-scan the sample at low energy per pulse ( 0.5 pJ) so as to map the fluorescence intensity and locate efficient emitters. We then put the excitation beam on a given emitter and measure Single molecules are excited by a frequency-doubled fs Ti : Sa laser at 532 nm. The laser is followed by PP, pulse picker; C, LiIO 3 non-linear χ (2) crystal; EO, ADP(NH 4 PO 4 ) electro-optic cell; P, linear polarizer; PZT, piezoelectric translation stage; Obj, oil-immersion microscope objective (60×, NA = 1.4); DM, dichroic mirror; PH, pinhole for confocal detection (30 µm diameter); NF, notch filter centred at 532 nm; BS, non-polarizing beamsplitter; SPAD, singlephoton silicium avalanche photodiode; TAC, time-to-amplitude converter; MA, multichannel analyser; TIA, time interval analyser (GuideTech, Model GT653); PC, computer.
the autocorrelation function at low excitation energy. We hence determine whether a single fluorophore or an aggregate of several molecules is excited. Once such preliminary identification has been achieved, the single molecule is excited at much higher power so as to ensure saturated emission [11].

Data acquisition
Once a single emitter is located, we switch the detection procedure from the start-stop method to a complete recording of photon arrival times. The properly normalized interval function c(τ) measured by the start-stop technique corresponds to the intensity autocorrelation function g (2) (τ) in the limit of short timescales and low detection efficiency [32]. However, to characterize more completely the statistical properties of the photon stream, one needs to test for correlations on timescales much longer than the excitation repetition period τ rep . In that case, the relationship between g (2) (τ) and c(τ) becomes more complicated [32,27]. Instead of solely inferring g (2) (τ) from c(τ) measurements, we have chosen to keep trace of the full range of dynamics by recording every photodetection time with a time interval analyser (TIA) computer board. From this set of photocounts moments (that we call timestamps), detected photons statistics can then be directly analysed over a wide range of timescales. Such a procedure avoids any mathematical bias in photon statistics analysis.
The total number of fluorescence photons that can be produced by a single molecule is limited at room temperature by its photostability [34]. Under weak CW excitation, a molecule of DiIC 18 (3) typically undergoes 10 6 excitation cycles before irreversible photobleaching occurs [35]. In our experiment, the excitation pulse energy is progressively ramped up to a maximum value of 5.6 pJ that ensures saturation of the S 0 → S 1 transition [11]. This energy ramp, realized using an electro-optic modulator, consists of a 50 ms linear rise followed by a plateau lasting 300 ms and linear decrease (figure 3). We found experimentally that applying such a procedure substantially improves molecular photostability, compared with an abrupt excitation. We then select the timestamps of photocounts that occurred during the plateau of the excitation. These events correspond to saturated emission of our molecular SPS. Our analysis 'photocount by photocount' then relies on determination of the number of detected photons in gated windows synchronized on Ti : Sa excitation pulses.
Data are first pre-processed over a discrete time grid. The excitation time base is reconstructed from the timestamps ensemble {t i } by applying a time filtering procedure described in appendix A. The gate duration is 30 ns, more than 10 times the typical radiative lifetime of the molecule in the PMMA layer. All records outside the time gates are rejected, slightly improving the signal-to-background ratio. Each timestamp t i is then attributed to a pulse p i in the time grid, and to each excitation pulse p, a number n (d) p = 0, 1, 2 of detected photons is Table 1. Single-pulse statistics of a molecular SPS, as obtained after numerical synchronization (see appendix A). These data will be referred to as (S). The total number of excitation pulses in the sequence is 325313, leading to a total of 0, 1 or 2 photon number events of 15138. The mean number of photons detected per pulse is n = 0.04653. No events with n (d) > 2 are observed due to deadtime in each detection channel. p }, as summarized in table 1. We next analyse the photon statistics of a data set extracted from SPS emission displayed in figure 3. We have selected photocounts recorded between the plateau beginning and molecular photobleaching clearly identified by a sudden drop in fluorescence emission. During this time, the molecule was excited at constant maximum pumping energy, yielding 15332 photodetection events for 325313 excitation pulses. The time filtering mentioned above is then applied keeping 15138 synchronous photocounts.

Single-pulse photon statistics
The single-pulse statistics, presented in table 1, are the direct outcome of photocounts acquisition. They correspond to the molecular emission displayed in figure 3. While our SPS photon statistics appear to differ from a classical Poissonian distribution, the influence of our experimental set-up on these measurements must be considered for accurate interpretation of these figures and for comparison with Poisson shotnoise reference.

Influence of deadtime
In the following, we make a distinction between the distribution of the photons produced by the source, denoted by script notation (P), and photocount statistics, for which we conserve the usual notation (P).
Due to existence of a 280 ns deadtime for each detection channel, a non-linear relationship exists between detected photon statistics and source photon statistics. Indeed, for a given excitation pulse, the number of detected photons in a 30-ns gated time window cannot exceed 2 if we use two avalanche photodiodes (APDs) operating in the photon counting regime compulsory for our experiment.
Denoting by P in (n) the photon number probability distribution of incoming light on the detection set-up, the non-linear tranformation relating this probability to the detected photon probability P(n = 0, 1, 2) is simply computed for 'ideal APDs'. By 'ideal APD', we mean that each photodiode clicks with 100% efficiency immediately upon receiving a photon, but that no more than one click can occur in a given repetition period. In the approach developed here, we consider the ideal APD case for the following reasons: • limited quantum efficiency of the APD (65% in our experiment) is included in an overall linear loss coefficient along with other linear losses of the detection chain. • deadtime is shorter than repetition period τ rep and much longer than pulse duration.
For detection with a single ideal APD, the relationship between the photocount and incoming light statistics is With our experimental detection scheme, random splitting of photons on two sides of 50/50 beamsplitter gives The relationship between P(n) and photon statistics P(n) in SPS emission comes from accounting for linear attenuation between SPS and detection. We call η the overall detection efficiency, which includes all linear propagation losses and photodetector quantum efficiency. P in is then related to P by the following binomial law: Combination of equations (2)-(5) leads to a direct analytical relation between P(n) and P(n = 0, 1, 2).
Note that the existence of such a saturation limit, due to detection deadtime, has no influence on photocount statistics of a perfect SPS, for which P(n 2) = 0, as long as the excitation repetition period is longer than the electronics deadtime. In contrast, for a 'real' source with background light, the number of detected multi-photon pulses is systematically underestimated. It leads to an artificial reduction of intensity noise in comparison with shotnoise reference.

Calibration with a coherent source
3.2.1. Coherent beam photocount statistics. SPS performance can be directly evaluated by comparing single-pulse photon statistics with those of a coherent source. This calibration takes into account the linear and non-linear effects of our detection set-up and permits accurate measurements of the multi-photon event probability reduction between single-photon and Poissonian sources. Table 2. Photocount probabilities P(n) for SPS (S), reference experimental coherent source (R) and theoretical coherent source (C), for which photocount statistics are affected by detection. The mean number n of detected photons per pulse is also given.
The photon number probability distribution for a coherent pulsed beam (C) is given by a Poisson law. According to equation (5), linear loss between the source and APDs change the mean photon number per pulse α to ηα, whereas the photon statistics remain Poissonian. The expected photocount statistics can then be calculated by applying non-linear transformations (equations (2)-(4)) to a Poissonian distribution of parameter ηα such distribution being termed as P C (n) in table 2.

Experimental calibration.
A strongly attenuated pulsed laser beam is used as experimental reference to mimic a pulsed coherent source. It is obtained by slightly detuning the Ti : Sa wavelength from the notch filter rejection band resulting in detection of residual pump light reflected from the sample. The photocount statistics of this experimental reference (R) are then compared both with the experimental SPS (S) and the calculated photocount distribution expected from a Poissonian source (C). To establish a valid comparison, calculated and experimental calibrations are determined for an almost identical mean number of photons detected per pulse. Table 2 shows that theoretical predictions for the coherent source are in good agreement with experimental calibrations, proving that our detection model accounts for all significant biases. We can therefore confidently interpret the molecular SPS photon statistics we measure. For our SPS, the number of two-photon pulses is 10 times smaller than the corresponding probability for a Poissonian source. As mentioned earlier, residual multi-photon pulses mostly result from background fluorescence light triggered by Ti : Sa excitation. Indeed, the wavelength of this parasitic light lies within the molecule's fluorescence band and therefore cannot be filtered out. Careful optimization of substrate purity as well as of the purity of the chemicals used in sample fabrication can probably lower background fluorescence.

Molecular SPS efficiency
Our molecular SPS emission can be modelled as the superposition of a perfect SPS and a coherent state of light. In this model, all sources of linear loss (production + collection + detection) are gathered as an overall efficiency η. For this perfect SPS (perf. SPS), the photon probability distribution of light impinging on the APD is then given by Background (backgnd.) emission is modelled by a coherent state of light with a mean number ηγ of detected photons per pulse. The corresponding photon probability distribution is then Applying equations (2)-(4) to the (perf. SPS + backgnd.) probability distribution leads to the following analytical expressions for the real SPS (S) photocounts statistics: Values for collection efficiency η and signal-to-background ratio 1/γ can be inferred from measured photocount statistics P S (see table 2). Using equations (11) for experimental values of P S (1) and P S (2), we find η 0.04456 and ηγ 2.02 × 10 −3 . This leads to a signal-tobackground ratio of 22, in good agreement with that measured by sample raster scan.

Single-pulse Mandel parameter
From a statistical point of view, there exist two main differences between experimental and ideal SPS: source overall efficiency <1, and finite ratio of single-to multi-photon pulses. Light produced by an ideal SPS consists of the periodic emission and detection of single photons with 100% efficiency. For such a source, there is therefore virtually no noise on the number of photon recorded during any integration time. In the following, we call this absence of intensity noise 'perfect intensity squeezing'. On the other hand, a real SPS yields non-ideal intensity squeezing [3].
It is then meaningful to assess SPS performance by measuring its intensity noise on the excitation repetition period τ rep timescale [11]. Such analysis requires evaluation of the singlepulse Mandel parameter Q [36]. This parameter characterizes deviation of photon statistics from Poissonian statistics for which Q = 0. Sub-Poissonian (resp. super-Poissonian) statistics correspond to Q < 0 (resp. Q > 0). For the distribution {n (d) p } of detected photon number, the Mandel parameter is defined by where n stands for the average value of {n (d) p } calculated over the ensemble {p} of excitation pulses. Note that an ideal SPS would yield Q = −1. Moreover, for any statistical distribution, the effect of linear attenuation can be straightforwardly evaluated: after linear attenuation η, a Mandel parameter Q 0 would be changed to ηQ 0 . This means that every statistical distribution converges towards Poissonian statistics under attenuation. This sensitivity to loss for measurements of non-zero Mandel parameters is similar to sensitivity observed in 'standard squeezing experiments' dealing with CW and pulsed laser beams [37]. In such experiments, photodetection is realized with photodiodes, the output current fluctuations reproducing the intensity fluctuations of the detected beam. Evidence for squeezing is then a measured reduced noise spectra with respect to a calibrated shotnoise reference.
For our molecular SPS, the Mandel parameter Q of the photocount statistics can be computed directly from single-pulse photocount probabilities From the data of table 2 we infer a Mandel parameter Q S = −0.04455 for the SPS. This negative value for Q confirms that our SPS indeed exhibits sub-Poissonnian statistics at the timescale τ rep . Since very few multi-photon events are observed, the value of Q S is almost only limited by the collection efficiency, which imposes a lower limit on Q: Q limit = −η = −0.04456. Our measurement of Q S can then be compared with a Poissonian reference measurement. Here again, statistical bias introduced by APD deadtime must be taken into account. From equations (6)- (8) and (13), we can derive the Mandel parameter of detected photons for a coherent source (C) of parameter α. Noticing that n C = 2(1 − e −α/2 ), we have As a consequence of photodetector deadtimes, a coherent source gives sub-Poissonian distribution of photodetection events. In our case, a coherent source with the same mean number n C = n = 0.04653 of detected photons per pulse as the SPS, would then yield Q C = −0.02327 > Q S . Despite this detection bias, our direct measurement of the Mandel parameter still yields a value for Q S that clearly departs from that of Poissonian statistics. This measured Mandel parameter is larger (in absolute value) than those measured in previous measurements by more than one order of magnitude [2,27,36].

SPS intensity fluctuations
Emission intermittency has been observed with most SPSs realized so far [13]- [15]. This effect decreases source efficiency and contributes to additional source of noise. Better understanding of physical processes responsible for intermittency would probably lead to significant improvement of current SPS devices. To characterize intermittency for our molecular SPS, we have investigated its influence on the photon statistics recorded with the time-resolved photon counting system. For a periodically trigerred SPS, this analysis is equivalent to studying the source intensity noise over a wide range of timescales, which is usually done in the frequency domain for 'standard squeezing experiments' [37], using a radio-frequency spectrum analyser.

Measuring intensity fluctuations: time-varying Mandel parameter Q(T )
The analysis performed on single-pulse photon statistics (see section 3.4) can be extended to multiple-pulses scale, allowing characterization of intensity fluctuations at any timescale greater than the pulse repetition period. To do this, we analyse fluctuations of the total number N(T ) of photons detected during an integration time T ≡ Mτ rep , which is a multiple of the repetition period. This analysis therefore corresponds to studying statistics of the photocounts number recorded during M successive pulses.
We then introduce the time-varying Mandel parameter Q(T ) [38], defined similarly to single-pulse Mandel parameter. To perform statistical analysis, we extend the procedure used in section 3. More precisely, the complete data {n (d) p } p=1,...,N corresponding to photocounts recorded during N consecutive excitation pulses are split in successive samples, each lasting T . We then obtain N sample = E[N /M] samples. We call N k (T ), the number of photocounts recorded during the kth sample. We then have The statistical average over these samples of duration T is denoted T , and we hence have Using these notations, the time-dependent Mandel parameter is given by which allows direct comparison of SPS noise properties with those of Poissonian light beam.

Intensity noise and intermittency in the molecular fluorescence: the ON-OFF model
To analyse our experimental results and link them to physical parameters of molecular fluorescence, we use a simple analytical model of molecular intermittency, in which we assume that the SPS can be ON or OFF. We call p the ON to OFF transition rate and q the OFF to ON one. These rates correspond to lifetimes τ on = 1/p and τ off = 1/q, respectively. [11] and other SPSs relying on fluorescence of a single emitter (e.g. single NV centres in diamond nanocrystals [14]), ON-OFF intermittency stems from the presence of a metastable non-fluorescent excited state in the energy level structure. Dynamics of ON-OFF behaviour can then be computed from the three-level structure shown in figure 4.

Physical interpretation of the ON-OFF model. For molecular SPS
• ON → OFF transition corresponds to relaxation from the optical excited state S 1 to the triplet state T 1 . For each excitation cycle, the probability P ISC of this intersystem crossing process is very small in the case of the DiIC 18 (3) molecule used in our experiment (P ISC 10 −4 ). Moreover, since singlet-triplet transitions occur exclusively from the excited state S 1 , the excitation repetition period must be considered in defining the source ON state lifetime τ on , which is then τ on = τ rep /P ISC , assuming saturated excitation regime. • OFF → ON transition consists simply of non-radiative decay from triplet T 1 to ground S 0 state. Note that the triplet level is metastable since selections rules forbid direct optical transition to the ground state. The triplet-state lifetime τ T = τ off = 1/q is therefore usually much longer than a typical fluorescent lifetime (in the case of DiIC 18 (3), τ T 200 µs [35]).

Dynamics of the ON-OFF system.
Under periodic pulsed excitation, the ON-OFF dynamics can be described using a discrete-time model. As transitions between ON and OFF states are random, we introduce a stochastic variable r k to account for the source state at instants t k = kτ rep . This parameter has value r k = 1 (resp. r k = 0) if the source is in the ON (resp. OFF) state at time t k . We then call u k the probability for the source to be in the ON state at time t k . As the SPS emits photons exclusively from the ON state and never from the OFF state, u k also corresponds to the photoemission probability at time t k . We assume that lifetimes τ on and τ off of the ON-OFF states are much larger than the repetition rate τ rep . Then, the ON→OFF transition probability is pτ rep and the OFF→ON transition is qτ rep . It follows that the state of the emitter at pulse k + 1 depends only on its state at pulse k. The recursion relation for the probability u k+1 of the source to be ON at time t k+1 is which leads to the general solution Stationary probabilities for the molecule to be either ON or OFF are then

Source intensity and Mandel parameter versus time.
According to our model, light emitted by the source is a succession of single photon pulses emitted at time t k = kτ rep with probability u k , corresponding to an intensity The recursive relation (19) for the ON-OFF model permits computation of statistical properties of source intensity I(t). In particular, we can derive the time-dependent Mandel parameter from the variance of the number N(T ) of photons emitted by the intermittent source during T = Mτ rep . Details of this calculation are given in appendix B.
Analytical expression of the Mandel parameter given by equation (B.13) can be simplified in the regime for which β = (p + q)τ rep 1, leading to the following Mandel parameter expression for a 'perfect' SPS with intermittency: Experimental measurements of the Mandel parameter are also affected by overall efficiency η < 1 (see section 3.3). Taking into account this limitation which is equivalent to linear loss, the Mandel parameter of the real source Q S (T) is given by

Experimental data analysis.
As shown in figure 5, our experimental data are well-fitted by equations (23) and (24) over more than four orders of magnitude in time. Setting the measured efficiency to η = 0.04456, the fit yields pτ rep = P ISC = 2.1 × 10 −4 and τ T = 250 µs, for the remaining two free parameters. These values are in good agreement with values given in [35]. Figure 5 clearly shows that source photon statistics differ on short and long timescales. On timescales shorter than 8τ rep , the Mandel parameter of the source Q(T ) is smaller than Q C , the theoretical value of the Mandel parameter for Poissonian light including the detection deadtime (broken horizontal line in figure 5). On this short timescale, the SPS's photocount statistics are those of non-classical light. On timescales larger than 10 µs, fluorescence intermittency due to the triplet state influences the photocount statistics by introducing excess of noise resulting in a positive value of the Mandel parameter.
The model developed here for a perfect intermittent SPS fits our experimental data with good accuracy. Indeed, apart from detection loss, other imperfections can be ignored or handled • Since the repetition period τ rep is much longer than the photodetection deadtime and since multi-photon events are extremely rare with our SPS,APD deadtime does not alter significantly the photocount statistics. The detection can be considered effectively linear and equation (24) remains valid in the presence of detection deadtimes. • High signal-to-background ratio means that background light does not contribute significantly to photocount statistics. It can therefore be neglected, as done implicitly in the model developed in this section. It can, moreover, be shown that addition of uncorrelated Poissonian background light of intensity B to the perfect SPS signal S is equivalent to loss. If we model the real source by the superposition of fluorescence background and light from a perfect SPS, then, introducing ρ ≡ S/(S + B), the Mandel parameter of the real source is simply given by Q S+B = ρQ S .

SPS intensity autocorrelation function.
As a consistency check for our study, the timedependent Mandel parameter analysis can be compared with a different approach using the intensity autocorrelation function g (2) [39], the measurement of which being at the heart of fluorescence correlation spectroscopy [40]. In particular, one can readily establish (see appendix C) that the area of the zero delay peak normalized to a coherent source, called g (2) (0), is related to the single-pulse Mandel parameter Q, by Q = n g (2) Due to technical limitations, it was however not possible to record simultaneously the timestamps and the histogram of time delays between consecutive photodetections. Nevertheless, g (2) (0) and Q can be extracted from the set of photocount timestamps to check the validity of equation (25), as shown in appendix C. From the measured single-pulse Mandel parameter, equation (25) predicts g (2) (0) = 0.085, whereas direct calculation of the autocorrelation function yields g (2) (0) = 0.082, in very good agreement with the 'predicted' value. These values of g (2) (0) and Q, measured at the level of a single pulse, are the signature of specific quantum features of the SPS. Molecular photodynamics on longer timescales (a few hundred µs to ms) can be inferred from measurements of the time dependance of the intensity autocorrelation function. Taking into account the specificity of the pulsed excitation regime, we introduce a discrete-time autocorrelation function for time delay × τ rep given by where n i is the number of detected photons in the ith excitation pulse, and is an integer. This discrete correlation function is directly related to the intensity autocorrelation function g (2) (τ) ≡ I(t)I(t + τ) / I(t) 2 usually measured with start-stop techniques [12]. It can be shown that the normalized area of the kth peak (with the 0th reference peak corresponding to τ = 0) of g (2) (τ) over a period τ rep is equal to G (2) The ON-OFF model developed in section 4 can be applied to calculate G (2) ( ): which coincides with the formula given in [15]. From our data {n (d) p }, we numerically compute G (2) ( ), varying from 1 to 1000. Note that the latter value is chosen because blinking occurs in a timescale range of 1000τ rep . Results of this G (2) ( ) calculation are displayed in figure 6. The experimental curve is fitted with equation (27), providing another way of measuring dynamical parameters of intermittent molecular SPS. This fit yields P ISC = 1.6 × 10 −4 and τ T = 180 µs, which are in good agreement with the values obtained in section 4.2.4 using Mandel parameter analysis.
Note that, on the short timescale, the statistical noise is higher on G (2) than on Q. This is due to the fact that G (2) is computed over fewer but bigger statistical samples.

Conclusion
We have realized an efficient triggered SPS relying on the temporal control of a single-molecule fluorescence. After a comparison with Poissonian coherent light pulses with the same mean number of photons per pulse, we have characterized intensity noise properties of this SPS in the time domain and photocounting regime.
From the record of every photocount timestamp, we calculate the second-order correlation function G (2)  specifically, we have modelled fluorescence intermittency by a two-state ON ↔ OFF dynamical process. By fitting a theoretical analytical expression of the Mandel parameter for an intermittent SPS, we obtained quantitative values for relevant molecular photodynamical parameters. Such a direct time-domain statistical analysis could give insight into molecular properties such as conformational changes [41,42], resonant energy transfer [43] or collective emission effects in multichromophoric systems [44].
With expected application to quantum cryptography, higher overall efficiency within a given emission spectral band should be reached so that SPSs can exhibit advantages over attenuated laser pulses [6,7]. In recent experiments, we coupled the fluorescence of a single emitter (a coloured centre in a diamond nanocrystal) to the single mode of a planar microcavity and observed a significant increase in spectral density of the emitted photons. These preliminary results are a promising realization of an efficient SPS well suited for open-air quantum key distribution.
postsynchronize the timestamps on an excitation timebase and then to build the set {n (d) p } of the number of detected photons for each excitation pulse p.
The pulsed excitation laser acts as a periodic trigger of emitted photons with repetition period τ rep 488 ns. An excitation laser pulse is emitted at time t start + pτ rep , where the pulse is indexed by the integer p. The parameter t start represents the pulse emission time taken as the first (p = 0) of the data. For each set of data, t start and τ rep must be determined because the repetition period of the Ti : Sa fs laser can fluctuate slightly between acquisitions. However, the laser repetition rate is stable over the typical acquisition duration (< s), and τ rep is therefore constant for a given data record.
Single-photon emission by the molecule occurs at each excitation pulse after a random time delay related to the molecule's excited state lifetime. Non-synchronous photocounts due to APD dark counts are rare, so almost all recorded photocounts are triggered by photons emitted by the molecule, with few by photons from residual fluorescence background. For these reasons, t start and τ rep can then be determined directly from recorded data.
The ith photocount timestamp t i can be expressed as where integer p i ∈ {1, . . . , N } indexes the laser pulse preceding detection of the ith photon, and data to be analysed lasts N repetition periods; δτ i is the time delay between excitation pulse and photocount timestamp (0 δτ i < τ rep ). Given a set of timestamps {t i }, relevant information can be equivalently represented by lists {p i } and {δτ i }, provided the value of τ rep is known accurately enough. Since the fluorescence lifetime of the molecule is much shorter than the laser repetition period, δτ i τ rep , as long as photocount i is not a dark count, which is rarely the case.
As the laser period τ rep is not known precisely, we first attempt to synchronize the data on the excitation timebase considered as a clock of period τ clock close to the expected laser period. We introduce a delay function parametrized with t start and τ clock , which gives for each timestamp t i the time delay between this timestamp and the corresponding top of the clock where E() stands for the integer part function and, if the clock period differs from the laser period, the drift of the time delay baseline with the pulse index p i is linear as can be seen in figure A.1(a), From the slope of the delay function baseline, we infer a new value for τ clock . Note that the first guess for τ clock is usually so far from the laser period that the delay function takes a saw-toothed shape, each jump corresponding to the delay reaching a multiple value of τ clock . As a consequence, only a linear fraction of the sample corresponding to a single saw tooth can be used at first. In further steps, estimation of τ rep improves and fewer jumps occur. Longer samples, corresponding to higher fit precision can then be processed. This procedure is repeated A time-filtering procedure is then used to eliminate all photocounts with time delay much longer than the molecular excited-state lifetime. To implement this filter, we use a time window of duration T window . From the set {p i }, we calculate the number n i of photons detected by the two photodiodes in the time interval [p i τ rep , p i τ rep + T window ]. The time window duration T window must be shorter than the laser period and much longer than the molecular excited-state lifetime 1/ so that the probability of discarding a 'real' photodetection event is negligible. The chosen time window duration T window = 30 ns meets these two conditions, considering 1/ 2.5 ns for DiIC 18 (3) dye. Note that the choice of a time window significantly shorter than the laser period has the advantage of filtering out our data from the majority of non-synchronous background photocounts, such as APD dark counts.
The processed data, now expressed as the table {n i , p i }, and shortened to {n (d) p } in the body of this paper, allows us to characterize the statistics of our source on timescales from τ rep 500 ns to ms. Brown and Twiss set-up. These data are usually acquired with a start-stop technique, relying on a time-to-amplitude converter and a multichannel analyser. In the limit of low collection efficiency and short timescale, it coincides with the second-order intensity correlation function g (2) [32].
The most usual statistical characterization of a SPS relies in the measurement of 'g (2) (0)' standing for the area of the g (2) function around zero time delay, normalized to the counts one would expect in the case of an equivalent coherent source. We have computed the histogram of the time delays from the set of timestamps on each detector to check for the consistency of the single-pulse Mandel parameter measurement with that of g (2) (0). Figure C.1 displays this computed histogram.