Simultaneous, full characterization of a single-photon state using semiconductor quantum-dot light

As single-photon sources become more mature and are used more often in quantum information, communications and measurement applications, their characterization becomes more important. Single-photon-like light is often characterized by its brightness, and two quantum properties: the single-photon composition and the photon indistinguishability. While it is desirable to obtain these quantities from a single measurement, currently two or more measurements are required. Here, we simultaneously determine the brightness, the single photon purity, the indistinguishability, and the statistical distribution of Fock states to third order for a quantum light source. The measurement uses a pair of two-photon (n = 2) number-resolving detectors. n>2 number-resolving detectors provide no additional advantage in the single-photon characterization. The new method extracts more information per experimental trial than a conventional measurement for all input states, and is particularly more e cient for statistical mixtures of photon states. Thus, using this n=2, number- resolving detector scheme will provide advantages in a variety of quantum optics measurements and systems.


INTRODUCTION
Single-photon light is a central element of emerging quantum information systems such as quantum repeaters [1, 2] and bosonic logic [3][4][5][6]. This nonclassical light is also used in quantum measurement protocols. Such protocols offer advantages over classical measurement protocols for classical properties [7,8], such as in accuracy and sensitivity [9][10][11] and, clearly quantum measurement protocols are essential to access quantum properties.
Light has been traditionally characterized by its coherence properties through a series of normalized Glauber functions g (n) , where 2n is the field correlation order [12]. The brightness is given by the unnormalized g (1) function (historically denoted G (1) [12]) and the normalized second-order correlation function g (2) gives the likelihood of two-photon correlations. When two photon correlations are nonzero (g (2) = 0), as is often the case, it is necessary to evaluate higher order correlations [13,14]. In general, to measure the photon-state statistics to n th order, normalized n th -order correlations could be measured using a single, appropriately fast n th -order number resolving detector, if such a detector were available (see Fig. 1a) [15][16][17]. Alternatively, n single-photon detectors can be used with beam-splitters in place of an n number resolving detector [18]. For example, a n = 2 number-resolving detector can be replaced by two single-photon detectors and a beam-splitter (Fig. 1b), and photon detections between the two detectors can be correlated [19], as discussed in the next section.
In many quantum information applications; for instance, quantum repeater and bosonic sampling, the single photon state must also be indistinguishable. The indistinguishability is measured by interfering replicas of the photon state, sampled at different times or positions [20], and can be measured with an unbalanced interferometer (see Fig. 1c). However, in an unbalanced interferometer scheme, the result must account for the single-photon nature of the light [21], requiring additional information. Thus, to fully characterize the quantum state of the source-the photon number state and the indistinguishability-at least two distinct set-ups and measurements are required. Besides the obvious inefficiencies in changing set-ups and acquiring separate measurement results, the quantities defining the photon state are never evaluated together.
Here, we show a method to simultaneously access brightness, number-state statistics and indistinguishability. This ensures that these quantities are derived from the same measurement set, and that all aspects of the source and measurement are identical. A main feature of this simulantaneous second-order correlation (Sg2) approach is the use of n = 2 single spatialmode input  [22,23], and they will likely be commercially available in the near future. There are other important features of Sg2. Using a pair of two-photon number resolving detectors, one can also simultaneously provide a complete characterization of the photon state to third order. We also show that the photon-number resolved measurement intrinsically collects more information about an input photonic state than a similar measurement made with conventional detectors. Thus, the Sg2 method is more efficient than a combination of conventional measurements, and becomes even more efficient as the single-photon purity degrades. Finally, when we substitute SPADs and beam-splitters for number-resolving detectors, the Sg2 layout mimics a simple linear optical circuit. The Sg2 scheme can be used to model such circuits or incorporated within them for local metrology testing.

SECOND-ORDER CORRELATIONS
The use of two detectors and a beam splitter to measure the second-order normalized correlation function goes back to measurements by Hanbury Brown and Twiss [19], and we denoted it g (2) HBT . In the g HBT measurement [19], a single spatial mode is incident on one port of a beamsplitter and two detectors measure coincidences in the two output ports to assess if more than one photon is present. If τ is the difference in detection times for the two detectors, g HBT [τ = 0] < 1 is the hallmark of a quantum state. One value of g (2) HBT , g HBT [0] = 0, represents a unique state. For this case, the non-vacuum component of the light is comprised of only single photons [14,24]. 1 > g (2) HBT [0] > 0 signifies non-classical light with some multi-photon component, and the greater the value of g (2) HBT the higher the proportion of the multi-photon component in the source [25]. Thus, for a source that is expected to provide single photons, g HBT is used as a metric for single photon purity. This is not the purity of the n = 1 number state, since the vacuum component is not represented HBT , but the purity against n > 1 population.
A second-order correlation function can also be used to characterize the second-order interference of two input fields, and for non-entangled inputs determines their indistinguishability. Using an interferometer, replicas of the same field can be used, and when used is often referred to as Hong, Ou, and Mandel (HOM) interferometry [20], denoted g HOM depends on the single-photon purity, and additional characterization is required for complete evaluation. One option is to directly determine the single-photon purity through g HBT [26]. Alternatively, a second measurement [21] or series of measurements [27] can be made in which the indistinguishablity is controllably varied; for example, by varying the polarization difference in each arm, and thus indirectly accessing the single photon purity [21].  consists of an unbalanced interferometer, followed by two beam splitters (BS3, BS4) and four single-photon detectors. Each BS -two-detector pair combination emulates a two-photon number-resolving photon detector.
We operate under pulsed excitation, and use the notation g (2) [j] to denote the normalized integrated correlations between two detections j pulses apart. In the Supplemental Material, we derive functions representing the single-photon purity g HBT [0] and the indistinguishability C. C ranges from 0 for perfectly distinguishable photons to 1 for perfectly indistinguishable photons. The results of these derivations are: where d is the delay in the unbalanced interferometer. ζ accounts for temporal instabilities, in particular the source spectral jitter, and is discussed below. g (2) cross [0] are autoand cross-correlations of the two output fields. While the Supplemental Material accounts for non-ideal beamsplitters in the interferometer, in Eqn. 1 we assume the ideal case of 50:50 beamsplitters. In previous non-number resolving two-detector schemes, just one function, cross is measured. Since we now have two functions, both the single photon purity and the indistinguishability can be simultaneously extracted from these quantities. Because pairs of non-number resolving detectors are used, we simultaneously measure six of these secondorder correlation functions: two are auto-correlations of the output field (g CD ), and four are cross-correlations (g BD ), where A, B, C and D denote the 4 detectors in Fig. 2. We average them to form Discrete solid-state emitters can shift in energy with time, leading to spectral jitter [28][29][30]. It is accounted for in Eqn. 1 by the function ζ, also discussed in the Supplemental Material. If present, this jitter degrades the indistinguishably, but will not degrade the single-photon purity. ζ is assumed to be a decaying exponential function of the form, where k refers to the number of pulses separating the generated photons, τ 1 is the characteristic lifetime of the jitter here measured in pulse periods, and ζ 0 is the value of ζ at zero delay.

EXPERIMENTAL REALIZATION OF SG2 CORRELATION MEASUREMENTS
To demonstrate the Sg2 measurement we use photons emitted from a single InAs quantum dot (QD). Technical details about the QD device, how it is excited and how collection is made is in the Supplemental. It is an emerging source of bright, single photon light [31].
Using the set-up in Fig. 2a, we measure the normalized second-order auto-and crosscorrelations for each detector combination, as in Eqn. 2. We normalize by the product of single-count probabilities p l p m , where l and m are the relevant detectors. The result is shown in Fig. 3.
Correlations are grouped into two categories; g We can quantitatively extract single photon purity and the indistinguishability from the data in Fig. 3. First, using Eqn. 1 we can determine the single photon purity through HBT [0]. The result is shown in blue in Fig. 4 where we plot the average of the unnormalized autocorrelations, uncorrected for the source jitter, i.e., with ζ(k)=1. This data captures the additional dynamics associated with jitter in the QD photon frequency on a longer time scale than the QD decay. Here, g Determining the photon indistinguishability follows in a straight-forward manner for the data in Fig. 3 using Eqn. 1. We find C = 0.61(1). Instead of directly determining C, in many situations it is more convenient to associate the fringe visibility V with indistinguishability, particularly when a variable controlling indistinguishably is continuously varied; for instance, the polarization [27]. For completeness, we determine it here using only one measurement set. We calculate V using Eqn. S1 in the Supplemental Material, where it is determined directly from g To determine the indistinguishability in the traditional way, the value of g HBT needs to be known and a second measurement with the interferometer determines g HOM . Here the indistinguishability is found from Ref. [21] where the probability of coalescence is C = 1+g (2) HOM when the HWP makes the two paths distinguishable.

SIMULTANEOUS FULL PROBABILITY DISTRIBUTION MEASUREMENT
To fully characterize the photon number distribution, higher order correlation measurements are generally required [14,37]. Characterization to at least third order is necessary.
Using the standard HBT-type measurement with two single-photon detectors, second-order correlations determine photon number statistics to second order (N=2) [38]. Using the Sg2 scheme, we can determine a probability distribution of the photon number states up to N=3.
In the QD light source used, a photon should be emitted from the light source every laser pump cycle (2 × 76 MHz); however, the system is only pumped to 70 % of saturation. We determine the photon count rate at the fiber exiting our cryostat (just before the interferometer input) to be 3.08 · 10 6 cts/s, indicating the source efficiency (p 1 ) exiting the fiber Sg2 Raw Autocorrelation HBT [0] can be found. is 0.029. p 1 is calculated based on the detector efficiencies and transmission through the interferometer, and these were measured to be 0.65 %. p 0 = 1−p 1 is then directly calculated to be 0.97. p 2 can be determined from g (2) HBT [0] = 2p 2 /p 2 1 [37], and p 2 = 2.9 · 10 −5 . The uncertainties on the above values are dominated by the long-term fluctuations in the set-up which we estimate to be 10 %. Zero 3 or 4 photon coincidences were measured for a trial number of 1.82 · 10 13 , giving an upper limit of p 3 of 2.1 · 10 −6 . The assumption of p 0 p 1 p 2 p N >2 holds, and the QD emission is described by a mixed state with a density matrix ||p|| = 0.97p 0,0 + 0.029p 1,1 + (2.9 · 10 −5 )p 2,2 + i>2 0p i,i (+2.1 · 10 −6 ), where p n,m = |f n f m |, and n, m are elements of the density matrix. We note that the off-diagonal elements of p are expected to be zero (i.e., no coherence between different number states).

CHARACTERIZING THE EFFICIENCY
In characterizing the efficiency of the Sg2 measurement, we first prove that the underlying photon-number resolving detection at the outputs of the unbalanced MZI provides more information about the input state than a conventional method. To do so, we compare the scaling of the variances, σ 2 , in measuring C for each measurement (i.e., per trial) of the photon-number resolved method and the conventional method. While we vary the indistinguishability and purity of the source, we assume that the purity is first known with certainty, since in the traditional approach it must be determined from a separate measurement. Thus, the ratio of variances is σ 2 (C trad ) σ 2 (C Sg2 ) = 2(g HBT + 1) (g (2) where subscripts trad and Sg2 denote the two interferometer measurements with conventional or photon-number resolving detectors, respectively; see Fig. 5a. σ 2 (C trad )/σ 2 (C Sg2 ) ≥ 1 for all physically meaningful g HBT and C, thus proving that the new method reduces the uncertainty in C faster than the traditional method. The advantage is maximum at g HBT [0] = 1, C = 0, and reduces to unity (i.e. the two methods become identical) at g (2) HBT = 0, C = 1. For the source investigated in this work, σ 2 (C trad )/σ 2 (C Sg2 ) = 1.28, indicating that the photon-number resolving detection provides a 28 % faster measurement than a traditional one.
Next, we compare our Sg2 method to one of the popular two-step methods [39,40] for characterizing a single photon source, introduced by Santori, et al. [26] which combines separate measurements of g (2) HBT and C. The efficiency is assessed by calculating the variances associated with g (2) HBT and C, and then using a scoring function of the form σ 2 g (2) [0] + σ 2 (C). The ratio of scoring functions is shown in Fig. 5b, where it is assumed that in a Santori two-step method the total measurement time is evenly split between HBT and interferometer measurements. The Sg2 is more efficient most of the time except for those very near to perfect single-photon purity and indistinguishability. For light with g (2) HBT [0] = 0.5, which is often used as a threshold for a discrete number-state source, the method is ≈ 7 times faster when the indistinguishability is 0.5. The maximum ratio is equal to 11.125, and the ratio for our source is equal to ≈ 2.5. This is because in a traditional method only a fraction of the total measurement can be used to determine single-photon purity, because the rest of the measurement should be used to determine indistinguishability. In addition, as established earlier, a photon-number resolved interferometer measurement reduces uncertainties faster than a traditional interferometer measurement. However, in the interferometer methods, the variance σ 2 g We would like to point out that other approaches exist. For example, a visibility measurement can be made using two conventional MZI measurements in co-and cross-polarization configurations to measure g (2) HBT [0] and C. Comparing the Sg2 method with this visibility method, the Sg2 approach is always more efficient.
Finally, since n = 2 number resolving detectors provide a more efficient measurement method for second-order correlation measurements, perhaps higher order number resolving detectors would provide further improvements? This is not the case; no further efficiency benefit is provided by n > 2 number resolving detectors. Using n > 2 number-resolving detectors would make higher order Glauber correlation measurements more efficient. However, they would not improve the second-order correlation measurements, specifically g HBT [0] and C, and here assumes that g HBT [0] is known from a separate measurement. (b) Scaling of the scoring function HBT and C. Comparison between the Sg2 method and the method where two independent HBT and HOM measurements are used with two non-number resolving detectors [26], expressed as ratios of uncertainty scoring functions, see text. Larger scale-bar numbers indicate a larger efficiency advantage of the Sg2 method. White dot: this source.

CONCLUSIONS
Using the light emission from a semiconductor QD structure as a test light source, we have demonstrated a new measurement approach allowing simultaneous measurement of the brightness, single photon purity and photon indistinguishability. The approach uses an interferometer and 2 two-photon number-resolving detectors, here simulated by four detectors. The simultaneous second-order correlation (Sg2) approach proposed here eliminates any variation in source and experiment that may be present in independent measurements, and in nearly all cases does so with reduced uncertainty. The Sg2 measurement is especially efficient when the nonclassical light has less than ideal single-photon properties. Finally, while n = 2 number resolving detectors are the critical elements in this approach, higher-order number resolving detectors do not offer improved efficiency, although they would be useful to characterize higher order number states.
The Sg2 measurement scheme can be evaluated using a boson sampling approach, as seen in the equivalent circuit in Fig. 2b. Boson sampling is the partial sampling of a bosonic circuit of an array of inputs and beam-splitters. An important parameter in evaluating this circuit is the matrix permanent. For simple systems like the one here, completely solving for the permanent of the matrix is trivial, but for large unitaries the solution of the permanent is difficult, in the P complexity class [41], and boson sampling of the unitary can offer a tractable solution [3][4][5][6]42]. The unitary matrix of this circuit, Fig. 2c, can be used with the boson sampling model to determine the g HBT [0] and C values which most closely matches the Fock-state distribution [5] found for the QD source. The g (2) HBT [0] and C determined by the boson sampling model that best matches Fig. 3 is within the uncertainty of those determined by the Sg2 approach. This Sg2 scheme could be incorporated into complex photonic circuits to assess the second-order correlation properties of the light.
Number-resolving detectors, as well as quantum-dot based single-photon sources like the one used here are emerging technologies that will likely have a strong impact in quantum measurement, experiments and systems. While we have shown that N=2 number resolving detectors will advance the characterization of quantum light, we believe they will also improve a diverse set of quantum optics experiments, for instance the boson sampling class of problems discussed above. We hope this work helps to further motivate such efforts.
This work was partially supported by the NSF PFC@JQI and the Army Research Laboratory. We thank P.

THE SEMICONDUCTOR LIGHT SOURCE
The nonclassical light source used is made from a semiconductor quantum dot (QD). The QD is made using the molecular-beam epitaxy crystal growth process. The QD portion of the sample is formed in MBE without lithography by the lattice-mismatch strain between InAs and the GaAs substrate. The QD density in the sample is approximately 10 QDs per µm 2 . The QDs are located at the center anti-node of a 4-λ planar distributed Bragg reflector (DBR) microcavity with 15.5 lower and 10 upper DBR pairs of GaAs and AlAs; the cavity mode is centered at a wavelength, λ ≈ 920 nm. A schematic showing how the sample is pumped and light extracted is shown in Fig. S1 S1. The QD sample is in a 5K cryostat. It is excited through a cleaved edge by 895 nm laser light via a fiber glued to a cleaved sample edge. Light is collected from the top of the sample with lens that is fiber coupled. QD light in the fiber is coupled to an interferometer using a fiber beam splitter (FBS), with polarization controllers (PC), and is filter with volume Bragg gratings (VBG). The cryostat side of the interferometer is fiber (black), the section containing the VBG and beyond is free space (red).
cleaved [110] sample face to couple the excitation laser into the guided mode of the DBR cavity [S1]. Collection of light is made vertically from the top of the sample. The sample is maintained at approximately 5 K in a cryostat. Excitation via the side-bonded fiber is by a mode-locked Ti:sapphire laser with a repetition rate of 76.1 MHz and approximately 8 ps pulse duration. However, the laser repetition rate is doubled so that the pulse spacing at the sample is 6.6 ns. This side excitation is made with 895 nm light that is absorbed into the roughened quantum well region intrinsic to the QD formation process, known as the wetting layer.

SECOND-ORDER INTENSITY CORRELATIONS USING NUMBER-RESOLVING DETECTORS
To characterize the second-order correlation properties of the source with conventional (non-number resolving) detectors, two independent measurements are required. In one possible scheme, an unbalanced Mach Zehnder interferometer (MZI) is used with a polarizer in one arm. Measurements are recorded when the second-order interference is switched on and off by setting the polarization difference between the two arms to 0 and π/2, thus measuring the second-order interference visibility. The visibility can also be measured by a discrete, step-wise adjustment of the polarization [S2]. An alternative method again uses an unbalanced MZI in a maximal interference configuration, supplemented with an separate Hanbury-Brown Twiss (HBT) measurement, i.e., a second-order correlation measurement with only one input to the beamsplitter.
Two important properties determined by these second-order correlation functions are the single photon purity, g (2) HBT (the purity against multi-photon state content > 1) and the photon indistinguishability, C. g HBT = 0 for a pure single photon source and g (2) HBT = 1 for a purely uncorrelated (Poissonian) source. C = 1 for fully indistinguishable photons and C = 0 for fully distinguishable ones. The physical meaning of C = 1 is that single photons will coalesce into a bi-photon state with unity probability when simultaneously arriving at two input arms of a beamsplitter. For the two conventional detectors with perfect 50/50 MZI beamsplitters, these properties are related by: .
where g HOM,(//,⊥) [0] in Eqn. S1 comes directly from the derivation of C in Ref. [S3], and where C = 1 + g HBT [0] −2g HOM,// , and correctly, C = 0. If there is maximum coalescence, there is maximum interference in the MZI and a cross correlation, g HOM,// [0], would give 0 except that the light source may not be a pure single photon source. If it is not a pure single photon source, a nonzero measure will occur but in the equation for C it is accounted for by g HBT [0]. V in Eqn. S1 is the visibility. Determining indistinguishability from V is the method traditionally used by many groups. However, as shown in Section V, it is the least efficient method to characterize a single-photon state due to uncertainty scaling and lost information.
We can make an autocorrelation measurement and a cross-correlation of the two fields, both taken at the output of the MZI. Because coalescence of indistinguishable fields at the beamsplitter produces bi-photon states, we can take advantage of number-resolving detectors in place of the conventional detectors to gain improved statistics. For the two output fields there are two autocorrelations (g (2) auto [j]), one for each number-resolving detector, and one crosscorrelation (g (2) cross [j]), where j is the time difference of detection events at the two detectors. Here, we use a pair of conventional detectors in a beamsplitter configuration to represent such a number-resolving detector (Fig. S2), and there are two g auto [0] in Eqn. S2 is the autocorrelation after the MZI and does not directly evaluate the single-photon purity of the source, while g HBT [0] is a traditional autocorrelation without the MZI. We note that these expressions are general for any detectors at the MZI output.
For the four detector configuration we obtain two auto correlation terms and four cross correlation terms: (S10)