1. Introduction

Single-photon sources are an important component of many emerging research fields, ranging from quantum communications and optical quantum computing to fundamental tests of quantum mechanics and high-precision metrology [1]. One common approach to generating single photons is through use of a single quantum emitter, whether it be an atom, ion, quantum dot, or defect center in a bulk crystal. Single photons can also be derived from processes (such as parametric downconversion or four-wave mixing) that produce photons in pairs; detecting one photon of the pair heralds the presence of the other photon, typically in a different spatial, spectral or polarization mode [1].

The standard metric for determining the quality of a single-photon source is the second-order temporal coherence, g⁽²⁾(τ), evaluated at zero time delay (τ = 0). In the ideal case of a source that never emits more than one photon at a time, g⁽²⁾(0) = 0. Unfortunately, some finite probability of multi-photon emission appears to be unavoidable in real-world single-photon sources [1]. Many sources allow a trade-off of higher efficiency—and thus higher photon count rates—in exchange for higher g⁽²⁾(0). For any given application, one would like to know how to optimize this trade-off. Unfortunately, although g⁽²⁾(0) ≠ 0 indicates the presence of multi-photon emission, the measured value provides incomplete information about these higher photon number events. As a result, the maximum value of g⁽²⁾(0) that an application can tolerate is not always possible to determine without additional information about the photon number probability distribution.

Measurements of higher-order coherences (g⁽ⁿ⁾, where n >2) offer one way of obtaining more insight into the details of multi-photon emission. High-order measurements have been used in investigations of a variety of physical systems. Much work has focused on the physics underlying threshold behavior of both macroscopic [2–6] and microcavity [7–9] lasers. High-order measurements have also been used to identify molecular aggregates [10] and non-Gaussian scattering processes [11], to observe time asymmetries in photons emitted by a driven, strongly coupled atom-cavity system [12], to enhance the resolution of ghost imaging [13–15], and to explore the statistical properties of photon pair sources [16] and exciton-polariton condensates [17]. Simultaneous measurements of g⁽²⁾ and g⁽³⁾ have been used to observe how a coherent light source can be transformed into one with sub-Poissonian statistics after selective one-photon absorption by a single quantum dot [18], and to reconstruct the relative magnitudes of thermal, coherent and single-photon modes that have been added incoherently [19].

The simple start-stop timers typically used to measure g⁽²⁾(τ) are not adequate to fully capture the dynamics of high-order coherences, since these coherences are functions of multiple time delays. Early high-order measurements recorded values only at zero delay, or for some subset of the full multi-dimensional delay space. Recently, the availability of fast, multi-channel time-tagging electronics, combined with ongoing increases in computing power, have enabled the multi-dimensional measurements necessary to characterize g⁽ⁿ⁾ at all delays [9,12,14,18,20,21].

Previous measurements of high-order coherences have generally explored the behavior of bunched photons (g⁽ⁿ⁾ > 1) from thermal or pseudo-thermal (often referred to as “chaotic”) sources, and how these are distinguished from coherent sources, which obey Poissonian statistics (g⁽ⁿ⁾ = 1). Here, we explore a different regime, measuring second- and third-order coherences of a quantum-dot single-photon source and presenting experimental demonstrations of antibunching in third-order coherence for both continuous-wave (CW) and pulsed excitation. In the case of pulsed excitation, we present a quantitative analysis of how coherences measured with click detectors relate to photon number distributions. We further show that the magnitude of the pulsed third-order antibunching is consistent with an ideal single-photon source added incoherently to a background having a Poissonian photon number distribution. Spectrally resolved photon correlation measurements indicate that the photons in this background are uncorrelated with those from the quantum dot.

2. What g⁽²⁾ and g⁽³⁾ tell us about photon probabilities P(n)

The second- and third-order temporal coherences for a CW, stationary source can be defined in terms of the creation (${\widehat{a}}^{†}$) and annihilation ($\widehat{a}$) operators as [22]

For pulsed sources—which are inherently not stationary—it is convenient to define discrete versions of these expressions [23]

The zero-delay value of the discrete second-order coherence, g⁽²⁾[0], can be interpreted as the autocorrelation of the pulse train in Fig. 1, while g⁽²⁾ [1] can be interpreted as the cross-correlation of each pulse with its nearest subsequent neighbor. The zero-delay value can be written

 

Fig. 1 Optical pulse train illustrating the labeling scheme used in the discrete notation. The repetition time of the source, T_rep, is set by that of the excitation laser.

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Similarly, the zero-delay value of the pulsed, third-order coherence can be written

Thus, to leading order, we expect g⁽²⁾[0] to contain information about the relative probability of two-photon emission, and g⁽³⁾[0,0] to contain information about the three-photon emission probability. If g⁽²⁾[0] = 0, then the zero-delay value of all higher-order coherences will also be zero. But since no real-world single-photon source is ideal, there is nearly always more information to be gained by performing higher-order measurements.

3. Measuring second- and third-order temporal correlations

The single-photon source is a self-assembled InGaAs quantum dot (QD) embedded in a planar microcavity that has been etched into a rectangular mesa containing a small number of dots. The upper and lower mirrors of the cavity are distributed Bragg reflectors composed of alternating layers of GaAs and AlAs, and the etched mesa has transverse dimensions of ~1.5 mm × ~10 mm. The sample is held at a temperature of ~5 K close to the window of a closed-cycle optical cryostat. The source is optically excited with either a CW or pulsed pump laser, and the emission is spectrally filtered to block the pump and transmit only the emission from a single transition in the quantum dot, at a wavelength of ~959.5 nm. As Fig. 2 shows, filtering is achieved with three elements in succession: a dichroic beamsplitter, a piezo-tunable Fabry-Perot etalon (bandwidth ~0.06 nm FWHM; free spectral range ~3 nm), and a fixed bandpass filter (center wavelength ~960 nm; bandwidth ~10 nm). The CW pump is a HeNe laser with λ = 633 nm, and the pulsed pump is a Ti:Sapphire laser with center wavelength of ~780 nm, pulse duration of ~1 ps, and repetition time T_rep = 12.3 ns. Each laser incoherently pumps the QD by first exciting carriers above the band gap of the wetting layer and the surrounding GaAs.

 

Fig. 2 Experimental geometry. As the dashed box on the left indicates, the “source” is defined to include the pump laser, quantum dot (QD) sample, and all optics for collection and filtering. Obj. is a microscope objective. The tunable filter is a piezo-tunable Fabry-Perot etalon with a bandwidth of ~0.06 nm and free spectral range of ~1.5 nm; this filter is tuned to line up with the peak QD emission line at a wavelength of ~959.5 nm. The fixed-bandpass filter has a nominal center wavelength of 960 nm and bandwidth of ~10 nm. The “measurement” side is a modified Hanbury Brown-Twiss interferometer, consisting of two ~50:50 beamsplitters (BS_a and BS_b), three single-photon avalanche diodes (SPADs 0, 1 and 2), and multi-channel time-stamping electronics.

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To measure the second- and third-order coherences, we use a modified Hanbury Brown-Twiss (HBT) geometry with two beamsplitters followed by three Si single-photon avalanche diodes (SPADs). Time-tagging electronics record the arrival times of all detected photons at the three detectors. We post-process the time-tagged data to obtain multi-start, multi-stop correlation histograms between combinations of two or three detectors [20].

For CW measurements, the correlation histograms are normalized by the number of counts expected in each time bin for completely uncorrelated events: the second- and third-order normalization factors are r₀r₁ΔτT and r₀r₁r₂(Δτ)²T, respectively, where r_i is the measured count rate on detector i, Δτ is the histogram bin width, T is the integration time. The g⁽²⁾(τ) data are averages of the three measured two-channel correlation histograms: 0-1, 0-2 and 1-2. Third-order coherences, g⁽³⁾(τ₁,τ₂), are measured as a function of two delays: τ₁ (τ₂) is the time delay between a photon detected by SPAD 0 and a photon registered by SPAD 1 (SPAD 2). The normalization procedure for pulsed measurements is discussed in section 5.2.

4. How well do these correlation measurements approximate g⁽²⁾ and g⁽³⁾?

A standard HBT interferometer employing one beamsplitter and two detectors will accurately measure g⁽²⁾, provided the two detectors are ideal photon number-resolving devices (see section 5.9 of [22], for example). Remarkably, measurement accuracy is unaffected by loss—whether the loss occurs before or after the beamsplitter. Measurements are similarly unaffected by unbalanced interferometer arms, whether resulting from a beamsplitter with a splitting ratio different from 50:50 or from two detectors with unequal efficiencies [22].

However, most experimental implementations of HBT interferometry—including those reported here—employ “click” detectors, which only register whether photons are detected, but do not provide information on the number of detected photons. Click detectors can approximately determine the number of incident photons only when the probability of two or more photons being incident on the detector is far less than the one-photon probability.

To explore the conditions under which an HBT interferometer with click detectors correctly measures the second-order coherence in a pulsed experiment, we define the measured discrete correlation function at zero delay:

Next we evaluate γ⁽²⁾[0] in terms of the photon probability distribution of the incident light. Following the usual conventions [22,25,26], the beamsplitter is assumed to be lossless with reflectance R and transmittance T = R-1, and the detection inefficiencies (along with any other optical losses) are modeled as beamsplitters with transmittance η₀ and η₁ in front of detectors with 100% detection efficiency, as shown in Fig. 3(a). The joint probability distribution of n₀ and n₁ photons arriving at detectors D₀ and D₁, respectively, is then

 

Fig. 3 Idealized experimental configuration considered for the calculations presented in Section 4 for (a) γ⁽²⁾[0] and (b) γ⁽³⁾[0,0]. Each detector D_i (i = 0,1,2) shown here is an ideal “click” detector with unity detection efficiency. Non-unity detection efficiency η_i is modeled by placing a beamsplitter with transmittance η_i in front of each detector. The reflectances (R,R_a,R_b) and transmittances (T,T_a,T_b) of the other beamsplitters are equal to the modulus squared of the corresponding field reflection and transmission coefficients.

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In general, γ⁽²⁾[0] is not equal to g⁽²⁾[0], but under appropriate experimental conditions the two quantities can approximate each other quite well. To explore these conditions, it is illustrative to write out the first few terms of each factor in the numerator and denominator:

Although the coincidence probability is maximized for R = T = ½, because of the way γ⁽²⁾[0] is normalized, it can still give a good approximation to g⁽²⁾[0] even when R ≠ T or when the detection efficiencies are not matched, η₀ ≠ η₁. Because loss does not change g⁽²⁾, even though it alters the photon probabilities P(n), the measurement fidelity can be improved by introducing additional loss to the system, provided the additional loss is the same for all spatial, spectral and polarization modes.

We can extend this analysis to the modified three-detector HBT setup in Fig. 3(b) by defining

5. Experimental results: g⁽²⁾ and g⁽³⁾

5.1 CW excitation

Figure 4 shows measured coherences for the quantum-dot source pumped with a CW laser. The second-order coherence exhibits the well-known antibunching behavior of a single-photon source, albeit not a very good one, with g⁽²⁾(0) = 0.44 ± 0.01. The third-order data display more complex features. The three valleys that intersect the origin along lines at τ₁ = 0, τ₂ = 0 and τ₁ = τ₂ correspond to two of the three SPADs detecting photons simultaneously. g⁽³⁾ reaches a minimum value of ~0.5 along each of these valleys, close to the measured g⁽²⁾(0), as expected. At the origin, where all three SPADs must detect photons simultaneously to register a coincidence, the value is further reduced to g⁽³⁾(0,0) = 0.15 ± 0.03. Far from the origin and away from the valleys, g⁽³⁾ ≈1.

 

Fig. 4 Measured coherence of the single-photon source with CW excitation. g⁽²⁾(τ) is shown as green circles in the upper panel. Third-order measurements in the upper panel are cross-sections of the full data set in the lower panel: blue triangles (red squares) lie along the diagonal running from the upper left to lower right (lower left to upper right) in the lower figure. The histogram time bin width (Δτ) is 100 ps for g⁽²⁾ and 500 ps for g⁽³⁾. The weak, ~1.8 ns-period oscillations visible g⁽²⁾ are due to periodic intensity fluctuations in the pump, which is a two-mode HeNe laser.

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5.2. Pulsed excitation

Figure 5 shows the measured histograms from the pulsed source for a series of pump powers. The dark blue areas under the peaks show the regions used for computing the peak areas, which in turn are used to estimate g⁽²⁾[0]. To minimize the contribution of background counts, we use an integration width of 4.1 ns = T_rep/3, and ignore the histogram counts in the hashed regions. g⁽²⁾[0] is then taken as the number of counts in the zero-delay peak divided by the average number of counts in the first nine peaks on each side of zero delay.

 

Fig. 5 Measured second-order correlation histograms for the source at four different excitation powers, illustrating the degradation of antibunching with increasing pump power. The values in red are the time-averaged pump power for each measurement.

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Third-order data are plotted in Fig. 6, where the height of each bar is proportional to the number of counts in a 4.1 × 4.1 ns² area of the histogram centered on each peak. The qualitative features are quite similar to those in the CW data, with antibunched valleys appearing where two of the three SPADs detect photons from the same pump pulse, and the strongest antibunching occurring at the origin. To find g⁽³⁾[0,0], the number of histogram counts in the peak centered at τ₁ = τ₂ = 0 is divided by the average number of counts in 162 peaks where τ₁ and τ₂ are near, but not equal to, zero.

 

Fig. 6 Measured coherences at zero delay for a pulsed pump laser at three average excitation powers: (a) 800 nW, (b) 1.4 μW and (c) 3.3 μW.

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Figure 7(a) illustrates the trade-off between count rate and g⁽²⁾[0] that can be made by changing the pump power. At the lowest pump powers, g⁽²⁾[0] is close to 0.1, but with a detected count rate < 100 kHz. Much higher count rates can be achieved by increasing the pump power, but g⁽²⁾[0] quickly approaches unity for a count rate of ~300 kHz. Qualitatively similar trade-offs can be made with many other single-photon sources [1], and g⁽²⁾[0] is the standard metric for placing bounds on the multi-photon emission probability, as detailed in Eq. (9).

 

Fig. 7 (a) Measured zero-delay coherences (left axis) and total detected count rate (right axis) as a function of time-averaged pump power. The count rate is the sum of the detected rate on the three individual SPADs. (b) Probabilites of the source emitting 1, 2 and 3 photons per pulse, computed from the data in (a).

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The uncertainties in g⁽²⁾[0] and g⁽³⁾[0,0] are estimated assuming standard N^1/2 counting statistics, where N is the number of counts in each histogram bin or peak. The uncertainties on g⁽²⁾[0] are typically ~10⁻³; thus, the error bars are not visible because they are smaller than the data symbols. The total uncertainty is likely somewhat higher due to fluctuating count rates (and hence coincidence rates) during the data run. However, the very low triples rates precluded us from quantifying this source of uncertainty, as we did in [20]. Systematic errors may also result because the measurements were performed over the course of several days, during which time the alignment of the pump beam with respect to the quantum dot likely varied somewhat.

Figure 7(b) shows the values of P(n) inferred from the coherence measurements. P(1) is estimated by taking the total count rate and dividing it by the product of the repetition rate of the pump laser (82 MHz) and the SPAD detection efficiency (~0.20 at λ = 960 nm). P(2) and P(3) are then estimated using Eqs. (20) and (23). The data show that the condition P(1) >> P(2) >> P(3) is easily satisfied for all pump powers, giving us confidence that our experiment accurately measures g⁽²⁾[0] and g⁽³⁾[0,0].

Figure 8(a) plots the data in Fig. 7(a) in a different format, showing g⁽³⁾[0,0] as a function of g⁽²⁾[0]. The green triangles and black squares in Fig. 8(b) are the results of similar measurements on two other quantum dots, located in two different etched mesas. The relationship between g⁽³⁾[0,0] and g⁽²⁾[0] is similar for all three QDs.

 

Fig. 8 Zero-delay third-order coherence plotted as a function of zero-delay second-order coherence. Data in (a) are for the same quantum dot as in Figs. 4–7. Green triangles and black squares in (b) denote measurements for two quantum dots located in other mesas on the same wafer. For the open magenta circles in (b), the excitation spot is spatially centered on QD1, but the spectral filter is tuned to be resonant with the cavity mode (magenta arrow in Fig. 9). The blue curves show the prediction of the model described in the text. The gray × ’s in (a) denote the predicted values for one- and two-photon Fock states and a coherent state, $|\alpha 〉$.

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If we had measured only g⁽²⁾[0], we might expect that state emitted by the quantum dot could be modeled as $|\psi 〉=c_0|0〉+c_1|1〉+c_2|2〉$. The pump power-dependence of g⁽²⁾[0] could then be described by increasing the magnitude of c₂ at higher pump powers. This might be the case if, for example, there were a biexcitonic transition close enough in energy to the excitonic transition to be transmitted through the bandpass filters. However, we would expect g⁽³⁾[0,0] = 0 for such a state, which is clearly contradicted by our third-order data.

Another approach to modeling the light emitted by the QD is to treat it as an incoherent mixture of two independent sources: 1) an ideal single-photon source that never emits two or more photons, emitting one photon with probability μ_s and zero photons with probability 1-μ_s, and 2) a background with Poissonian photon number statistics and mean photon number μ_b. The probability that this source produces m photons is

Combining Eqs. (26) and (27) yields an expression for the relationship between g⁽²⁾[0] and g⁽³⁾[0,0]:

Further confirmation of the validity of this model can be gleaned from the quantum dot emission spectra shown in Fig. 9. For low pump power, we observe a strong single QD emission line centered at ~959.5 nm, along with several other weak “background” features at other wavelengths. This background appears to be associated with other light sources inside the mesa that are spectrally filtered by the cavity. If the structure were a small cylindrical micropillar, this cavity-filtered emission would have a Lorentzian spectral shape [27]; the more complicated spectral structure observed here can be ascribed to the rectangular shape of the mesa. As pump power is increased, the intensity of the QD emission line increases somewhat, but the background features grow even faster. At the highest pump power shown, the magnitude of this background is comparable to the QD emission line. Given that a larger fraction of photons originates from the background at higher pump power, it is not surprising that g⁽²⁾ increases as pump power increases.

 

Fig. 9 (Solid curves, left axis) Unfiltered photoluminescence (PL) spectra of the quantum-dot source for three different pulsed pump laser powers. The black arrow labeled “QD” denotes the approximate central wavelength (λ₀ = 959.5 nm) of the ~0.06 nm-wide tunable bandpass filter used for coherence measurements of the quantum dot in Figs. 4–7. The magenta arrow at 959.2 nm denotes the filter tuning for the “cavity” g⁽²⁾ and g⁽³⁾ measurements in Fig. 8(b). The green squares (right axis) show the results of second-order cross-correlation measurements, with one detector measuring photons at the fixed wavelength λ₀ = 959.5 nm, and the wavelength of the other detector (λ₁) swept across the spectrum.

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Coherence measurements of this cavity emission are performed by tuning the bandpass filter to ~959.2 nm center wavelength. Results for two pump powers are shown as open magenta circles in Fig. 8(b): both g⁽²⁾[0] and g⁽³⁾[0,0] are approximately unity, as expected for photons with a Poissonian number distribution. (We might expect this cavity emission to behave as a thermal source, but, as is common in HBT measurements [28,29], this thermal light source appears to obey Poissonian statistics because the coherence time of the bunching is much shorter than the detector timing jitter.)

Even though this simple model is consistent with our data, it would be disingenuous to argue that this QD is an ideal single-photon source, unless we could experimentally demonstrate a way to remove the background emission.

6. Cross-correlation measurements

The model described above assumes that the cavity emission is uncorrelated with the QD emission. Prior measurements of quantum dots embedded in microcavities have shown that even when the cavity emission itself obeys Poissonian statistics, photons emitted at the cavity wavelengths can be strongly anticorrelated with photons emitted by the QD [30,31]. This cavity feeding of photons from a QD into off-resonant cavity modes can result from dephasing via acoustic phonons or from the presence of a quasi-continuum of multi-excitonic states, among other processes [30–33].

To investigate whether the cavity and QD can be described as independent emitters for the sample used here, we performed spectrally resolved cross-correlation measurements between cavity and QD emission using the experimental geometry in Fig. 10. One SPAD detects photons at a fixed wavelength, at the center of the QD emission peak, λ₀. The monochromator is then tuned so that the other SPAD detects photons at a series of other fixed wavelengths λ₁. The cross-correlation between photons at these two wavelengths is recorded and analyzed, and the zero delay value, $g_{{\lambda }_0{\lambda }_1}^{\left(2\right)}[0]$, is determined similar to the way g⁽²⁾[0] is found in Sections 3 and 5.

 

Fig. 10 Experimental geometry for spectrally resolved cross-correlation measurements between photons at two different wavelengths. λ₀ is fixed at the QD emission wavelength, while λ₁ is scanned across the spectrum of cavity and QD emission. BS is a nonpolarizing beamsplitter, and the tunable and fixed filters are the same as in Fig. 2.

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The green squares in Fig. 9. show the results of these cross-correlation measurements for a pump power of 1.6 μW. When λ₁ = λ₀, we find $g_{{\lambda }_0{\lambda }_1}^{\left(2\right)}[0]=0.52\pm 0.01$; as expected, this is fairly close to the value of $g^{\left(2\right)}[0]=0.58\pm 0.01$ in Fig. 7(a). for a pump power of 1.7 μW. (Inexact agreement is not terribly surprising, given that the monochromator passes a slightly different bandwidth than tunable bandpass filter, and that day-to-day variations in alignment of the pump beam to the quantum dot are difficult to avoid entirely.)

As λ₁ is tuned away from the QD peak, $g_{{\lambda }_0{\lambda }_1}^{\left(2\right)}[0]$ approaches unity, indicating that most emission at other wavelengths is uncorrelated with emission from the QD peak. The magenta arrow shows the tuning at which we measured g⁽²⁾[0] ≈1 in Fig. 8(b). Taken together, these two measurements substantiate our claim that the emission of this source can be described by the sum of two independent emitters: one a near-ideal single photon source, and the other a background obeying Poissonian statistics.

The lone exception to this uncorrelated behavior is a second antibunched dip centered at ~960.1 nm, coinciding with a weak emission peak visible in the photoluminescence spectra. This antibunching indicates that photons are less likely to be emitted at both the primary QD wavelength and this secondary wavelength after excitation by a single pump pulse. As a result, this peak at 960.1 nm is probably another emission line from the same QD, possibly a differently charged exciton. However, given the small number of photons emitted at this wavelength, in addition to its spectral separation from the main QD peak, it is unlikely that many photons emitted from this line will pass through the bandpass filter when tuned to the main QD peak.

Although this result of cavity-filtered emission being uncorrelated with QD emission is different from some prior results that demonstrated cavity feeding from off-resonant quantum dots [30,31], those prior measurements used cavities with mode volumes of less than ~(λ/n)³ that were likely to contain only one QD. Here, by contrast, the comparatively large rectangular mesa has a mode volume >> (λ/n)³ and is likely to contain several QDs.

7. Summary

The results in this work demonstrate that measuring coherences higher than second order are a powerful way of obtaining information about the details of multi-photon emission, even for a single-photon source that is not particularly notable for its efficiency or for its lack of multi-photon emission. The new information from g⁽³⁾ that was not available in g⁽²⁾ could be used to construct a more accurate picture of the internal physics of the quantum dot or to better determine the suitability of this source in a particular quantum information application.

In the future, as single-photon sources increase in efficiency, it is likely that the condition P(1) >> P(2) >> P(3) >> P(n>3) will become more difficult to satisfy. Thus, it will become even more important to measure higher order coherences. Fortunately, n^th-order coincidence rates grow as efficiency to the n^th power, so the time required to carry out the measurements can decrease dramatically as system efficiencies increase.