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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 3 — Feb. 10, 2014
  • pp: 3244–3260
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Third-order antibunching from an imperfect single-photon source

Martin J. Stevens, Scott Glancy, Sae Woo Nam, and Richard P. Mirin  »View Author Affiliations


Optics Express, Vol. 22, Issue 3, pp. 3244-3260 (2014)
http://dx.doi.org/10.1364/OE.22.003244


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Abstract

We measure second- and third-order temporal coherences, g(2)(τ) and g(3)(τ1,τ2), of an optically excited single-photon source: an InGaAs quantum dot in a microcavity pedestal. Increasing the optical excitation power leads to an increase in the measured count rate, and also an increase in multi-photon emission probability. We show that standard measurements of g(2) provide limited information about this multi-photon probability, and that more information can be gained by simultaneously measuring g(3). Experimental results are compared with a simple theoretical model to show that the observed antibunchings are consistent with an incoherent addition of two sources: 1) an ideal single-photon source that never emits multiple photons and 2) a background cavity emission having Poissonian photon number statistics. Spectrally resolved cross-correlation measurements between quantum-dot and cavity modes show that photons from these two sources are largely uncorrelated, further supporting the model. We also analyze the Hanbury Brown-Twiss interferometer implemented with two or three “click” detectors, and explore the conditions under which it can be used to accurately measure g(2)(τ) and g(3)(τ1,τ2).

© 2014 Optical Society of America

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

1. M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82(7), 071101 (2011). [CrossRef] [PubMed]

]. 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

1. M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82(7), 071101 (2011). [CrossRef] [PubMed]

].

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

1. M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82(7), 071101 (2011). [CrossRef] [PubMed]

]. 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

2. F. Davidson, “Measurements of photon correlations in a laser beam near threshold with time-to-amplitude converter techniques,” Phys. Rev. 185(2), 446–453 (1969). [CrossRef]

6

6. J. F. Dynes, Z. L. Yuan, A. W. Sharpe, O. Thomas, and A. J. Shields, “Probing higher order correlations of the photon field with photon number resolving avalanche photodiodes,” Opt. Express 19(14), 13268–13276 (2011). [CrossRef] [PubMed]

] and microcavity [7

7. J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460(7252), 245–249 (2009). [CrossRef] [PubMed]

9

9. D. Elvira, X. Hachair, V. B. Verma, R. Braive, G. Beaudoin, I. Robert-Philip, I. Sagnes, B. Baek, S. W. Nam, E. A. Dauler, I. Abram, M. J. Stevens, and A. Beveratos, “Higher-order photon correlations in pulsed photonic crystal nanolasers,” Phys. Rev. A 84, 061802 (2011).

] lasers. High-order measurements have also been used to identify molecular aggregates [10

10. A. G. Palmer 3rd and N. L. Thompson, “Molecular aggregation characterized by high order autocorrelation in fluorescence correlation spectroscopy,” Biophys. J. 52(2), 257–270 (1987). [CrossRef] [PubMed]

] and non-Gaussian scattering processes [11

11. P.-A. Lemieux and D. J. Durian, “Investigating non-Gaussian scattering processes by using nth-order intensity correlation functions,” J. Opt. Soc. Am. A 16(7), 1651–1664 (1999). [CrossRef]

], to observe time asymmetries in photons emitted by a driven, strongly coupled atom-cavity system [12

12. M. Koch, C. Sames, M. Balbach, H. Chibani, A. Kubanek, K. Murr, T. Wilk, and G. Rempe, “Three-photon correlations in a strongly driven atom-cavity system,” Phys. Rev. Lett. 107(2), 023601 (2011). [CrossRef] [PubMed]

], to enhance the resolution of ghost imaging [13

13. I. N. Agafonov, M. V. Chekhova, T. Sh. Iskhakov, and L.-A. Wu, “High-visibility intensity interference and ghost imaging with pseudo-thermal light,” J. Mod. Opt. 56(2-3), 422–431 (2009). [CrossRef]

15

15. Y. Zhou, J. Liu, J. Simon, and Y. Shih, “Resolution enhancement of third-order thermal light ghost imaging in the photon counting regime,” J. Opt. Soc. Am. B 29(3), 377 (2012). [CrossRef]

], and to explore the statistical properties of photon pair sources [16

16. A. Hayat, A. Nevet, and M. Orenstein, “Ultrafast partial measurement of fourth-order coherence by HBT interferometry of upconversion-based autocorrelation,” Opt. Lett. 35(5), 793–795 (2010). [CrossRef] [PubMed]

] and exciton-polariton condensates [17

17. T. Horikiri, P. Schwendimann, A. Quattropani, S. Hing, A. Forchel, and Y. Yamamoto, “Higher order coherence of exciton-polariton condensates,” Phys. Rev. B 81(3), 033307 (2010). [CrossRef]

]. 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

18. M. Bajcsy, A. Rundquist, A. Majumdar, T. Sarmiento, K. Fischer, K. G. Lagoudakis, S. Buckley, and J. Vuckovic, “Non-classical three-photon correlations with a quantum dot strongly coupled to a photonic-crystal nanocavity,” arXiv:1307.3601 (2013).

], and to reconstruct the relative magnitudes of thermal, coherent and single-photon modes that have been added incoherently [19

19. E. A. Goldschmidt, F. Piacentini, I. Ruo Berchera, S. V. Polyakov, S. Peters, S. Kuck, G. Brida, I. P. Degiovanni, A. L. Migdall, and M. Genovese, “Mode reconstruction of a light field by multi-photon statistics,” Phys. Rev. A 88(1), 013822 (2013). [CrossRef]

].

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

9. D. Elvira, X. Hachair, V. B. Verma, R. Braive, G. Beaudoin, I. Robert-Philip, I. Sagnes, B. Baek, S. W. Nam, E. A. Dauler, I. Abram, M. J. Stevens, and A. Beveratos, “Higher-order photon correlations in pulsed photonic crystal nanolasers,” Phys. Rev. A 84, 061802 (2011).

,12

12. M. Koch, C. Sames, M. Balbach, H. Chibani, A. Kubanek, K. Murr, T. Wilk, and G. Rempe, “Three-photon correlations in a strongly driven atom-cavity system,” Phys. Rev. Lett. 107(2), 023601 (2011). [CrossRef] [PubMed]

,14

14. Y. Zhou, J. Simon, J. Liu, and Y. Shih, “Third-order correlation function and ghost imaging of chaotic thermal light in the photon counting regime,” Phys. Rev. A 81(4), 043831 (2010). [CrossRef]

,18

18. M. Bajcsy, A. Rundquist, A. Majumdar, T. Sarmiento, K. Fischer, K. G. Lagoudakis, S. Buckley, and J. Vuckovic, “Non-classical three-photon correlations with a quantum dot strongly coupled to a photonic-crystal nanocavity,” arXiv:1307.3601 (2013).

,20

20. M. J. Stevens, B. Baek, E. A. Dauler, A. J. Kerman, R. J. Molnar, S. A. Hamilton, K. K. Berggren, R. P. Mirin, and S. W. Nam, “High-order temporal coherences of chaotic and laser light,” Opt. Express 18(2), 1430–1437 (2010). [CrossRef] [PubMed]

,21

21. L. Ma, M. T. Rakher, M. J. Stevens, O. Slattery, K. Srinivasan, and X. Tang, “Temporal correlation of photons following frequency up-conversion,” Opt. Express 19(11), 10501–10510 (2011). [CrossRef] [PubMed]

].

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 (a^) and annihilation (a^) operators as [22

22. R. Loudon, The Quantum Theory of Light, Third Edition (Oxford University, Oxford, 2000).

]
g(2)(τ)=a^(t)a^(t+τ)a^(t+τ)a^(t)a^(t)a^(t)2,
(1)
g(3)(τ1,τ2)=a^(t)a^(t+τ1)a^(t+τ2)a^(t+τ2)a^(t+τ1)a^(t)a^(t)a^(t)3,
(2)
where 〈 〉 indicates an average over time, t, and the τ’s are time delays.

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

23. C. Santori, D. Fattal, J. Vučković, G. S. Solomon, E. Waks, and Y. Yamamoto, “Submicrosecond correlations in photoluminescence from InAs quantum dots,” Phys. Rev. B 69(20), 205324 (2004).

]
g(2)[l]=a^[k]a^[k+l]a^[k+l]a^[k]a^[k]a^[k]a^[k+l]a^[k+l],
(3)
g(3)[l,m]=a^[k]a^[k+l]a^[k+m]a^[k+m]a^[k+l]a^[k]a^[k]a^[k]a^[k+l]a^[k+l]a^[k+m]a^[k+m],
(4)
where k, l and m take on integer values denoting pulse number, and the angled brackets indicate an average over k. The square brackets are used here to distinguish the discrete (pulsed) forms g(2)[l] and g(3)[l,m] from the continuous forms g(2)(τ) and g(3)(τ12).

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
Fig. 1 Optical pulse train illustrating the labeling scheme used in the discrete notation. The repetition time of the source, Trep, is set by that of the excitation laser.
, while g(2) [1

1. M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82(7), 071101 (2011). [CrossRef] [PubMed]

] can be interpreted as the cross-correlation of each pulse with its nearest subsequent neighbor. The zero-delay value can be written
g(2)[0]=n^k(n^k1)n^k2,
(5)
where the number operator n^k=a^[k]a^[k] measures the number of photons in the kth pulse. Pulse-to-pulse variations in the quantum state of light can be described in a probabilistic manner using the density matrix; thus, we can drop the explicit k-dependence of the number operator and use n^=a^a^. The expectation values in Eq. (5) can then be computed by tracing over the product of the operators and the density matrix:
g(2)[0]=Tr{ρ^n^(n^1)}(Tr{ρ^n^})2.
(6)
When the density matrix is written in the photon-number state basis, only its diagonal terms contribute to g(2)[0]. We can rewrite this in terms of the per-pulse photon probabilities as
g(2)[0]=n=0n(n1)P(n)[n=0nP(n)]2=2P(2)+6P(3)+μ2=2P(2)+6P(3)+[P(1)+2P(2)+]2,
(7)
where P(n) is the probability of having n photons in each pulse and μ is the mean photon number per pulse. This expression can be used to bound the probability of multi-photon events:
P(n>1)=n=2P(n)12n=0n(n1)P(n).
(8)
As a result, the measured g(2)[0] can be used to place an upper limit on the multi-photon probability [24

24. E. Waks, C. Santori, and Y. Yamamoto, “Security aspects of quantum key distribution with sub-Poisson light,” Phys. Rev. A 66(4), 042315 (2002). [CrossRef]

]:
P(n>1)12μ2g(2)[0].
(9)
If P(1) >> P(2) >> P(n>2), as is true for many low-efficiency sources, this leads to the familiar approximation

g(2)[0]2P(2)[P(1)]2.
(10)

Similarly, the zero-delay value of the pulsed, third-order coherence can be written
g(3)[0,0]=n^k(n^k1)(n^k2)n^k3,
(11)
and expanded as
g(3)[0,0]=6P(3)+24P(4)+μ3.
(12)
If P(1) >> P(n>1) and P(3) >> P(n>3), this can be approximated as

g(3)[0,0]6P(3)[P(1)]3.
(13)

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
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 (BSa and BSb), three single-photon avalanche diodes (SPADs 0, 1 and 2), and multi-channel time-stamping electronics.
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 Trep = 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.

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

20. M. J. Stevens, B. Baek, E. A. Dauler, A. J. Kerman, R. J. Molnar, S. A. Hamilton, K. K. Berggren, R. P. Mirin, and S. W. Nam, “High-order temporal coherences of chaotic and laser light,” Opt. Express 18(2), 1430–1437 (2010). [CrossRef] [PubMed]

].

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 r0r1r2τ)2T, respectively, where ri 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

22. R. Loudon, The Quantum Theory of Light, Third Edition (Oxford University, Oxford, 2000).

], 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

22. R. Loudon, The Quantum Theory of Light, Third Edition (Oxford University, Oxford, 2000).

].

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:
γ(2)[0]P01(click,click)P0(click)P1(click).
(14)
P01(click,click) is the probability that both detectors 0 and 1 click during the same pulse; P0(click) is the probability that detector 0 clicks during a pulse, independent of whether detector 1 clicks; and P1(click) is the probability that detector 1 clicks during a pulse, independent of whether detector 0 clicks.

γ(2)[0]=n=2[1(1η0R)n(1η1T)n+(1η0Rη1T)n]P(n)[n=1[1(1η0R)n]P(n)][n=1[1(1η1T)n]P(n)].
(17)

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:
γ(2)[0]=2P(2)+6P(3)(112η0R12η1T)+[P(1)+2P(2)(112η0R)+][P(1)+2P(2)(112η1T)+],
(18)
comparison to Eq. (7) reveals that the first terms in the numerator and in each factor of the denominator of γ(2)[0] are exactly equal to those in g(2)[0]. For higher photon numbers, R, T, η0 and η1 appear as correction factors, indicating that the higher-photon-number terms tend to be underestimated by the click detectors. In the limit of very low detection efficiencies, these corrections to the higher-photon-number terms become negligible. Applying l’Hôpital’s rule to Eq. (17) yields
limη0,η10γ(2)[0]=g(2)[0].
(19)
regardless of the values of R and T. Alternatively, if the source has very low multi-photon generation probabilities so that P(1) >> P(2) >> P(n>2), then
γ(2)[0]2P(2)[P(1)]2g(2)[0].
(20)
Even if these conditions are not met directly from the source, they can be satisfied by introducing additional attenuation before the HBT setup.

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 RT or when the detection efficiencies are not matched, η0η1. 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
γ(3)[0,0]P012(click,click,click)P0(click)P1(click)P2(click),
(21)
where P012(click,click,click) is the probability that detectors 0, 1 and 2 all click during the same pulse; P0(click) is the probability that detector 0 clicks during a pulse, independent of whether detectors 1 or 2 click; and P1(click) and P2(click) are similarly defined. Applying a procedure similar to that used for the second-order measurement, one can show that γ(3)[0,0] approximates g(3)[0,0] when the three detectors’ efficiencies are low. Writing out the first few terms in each factor for the pulsed, zero delay correlation yields
γ(3)[0,0]=6P(3)+24P(4)(112η0Ra12η1TaRb12η2TaTb)+[P(1)+2P(2)(112η0Ra)+][P(1)+2P(2)(112η1TaRb)+][P(1)+2P(2)(112η2TaTb)+].
(22)
Under conditions similar to those for γ(2)[0], the measured quantity is a good approximation to the true third-order coherence,
γ(3)[0,0]6P(3)[P(1)]3g(3)[0,0],
(23)
provided that P(1) >> P(n>1) and P(3) >> P(n>3). As with γ(2)[0], the values of beamsplitter reflectance and transmittance only affect the third-order measurement result by altering the coincidence rate.

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

5.1 CW excitation

Figure 4
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.
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.

5.2. Pulsed excitation

Figure 5
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.
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 = Trep/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.

Third-order data are plotted in Fig. 6
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.
, 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.

Figure 7(a)
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).
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

1. M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82(7), 071101 (2011). [CrossRef] [PubMed]

], and g(2)[0] is the standard metric for placing bounds on the multi-photon emission probability, as detailed in Eq. (9).

The uncertainties in g(2)[0] and g(3)[0,0] are estimated assuming standard N1/2 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

20. M. J. Stevens, B. Baek, E. A. Dauler, A. J. Kerman, R. J. Molnar, S. A. Hamilton, K. K. Berggren, R. P. Mirin, and S. W. Nam, “High-order temporal coherences of chaotic and laser light,” Opt. Express 18(2), 1430–1437 (2010). [CrossRef] [PubMed]

]. 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)
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. 47. 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, |α.
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.

If we had measured only g(2)[0], we might expect that state emitted by the quantum dot could be modeled as |ψ=c0|0+c1|1+c2|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 μs and zero photons with probability 1-μs, and 2) a background with Poissonian photon number statistics and mean photon number μb. The probability that this source produces m photons is
P(m)=k=0Ps(m)Pb(mk),
(24)
where Ps(m) is the probability that the single-photon source emits m photons, and Pb(m-k) is the probability that the background emits m-k photons. Explicitly writing in the individual distributions yields
P(m)=[μsμbm1(m1)!+(1μs)μbmm!]eμb.
(25)
Substituting P(m) into Eqs. (7) and (12), we find
g(2)[0]=1+2r(1+r)2
(26)
g(3)[0,0]=1+3r(1+r)3,
(27)
where we have defined rμs/μb as the average ratio of the number of photons emitted by the single-photon source to the number of photons emitted by the Poissonian background. If r = 0, then all photons originate from the Poissonian background and g(2)[0] = g(3)[0,0] = 1. If r = 19, then only 5% of the photons originate from the Poissonian background, g(2)[0] = 0.0975, and g(3)[0,0] = 7.25 × 10−3. A g(2)[0] of 0.01 would imply a source where approximately one out of 200 photons is from the background.

Combining Eqs. (26) and (27) yields an expression for the relationship between g(2)[0] and g(3)[0,0]:
g(3)[0,0]=1+3r(1+r)(1+2r)g(2)[0].
(28)
This expression is plotted as a blue curve in Fig. 8(a) and (b) for r ranging from 0 (upper right) to ∞ (lower left); within our experimental uncertainties, this model agrees reasonably well with the data.

Further confirmation of the validity of this model can be gleaned from the quantum dot emission spectra shown in Fig. 9
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. 47. 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.
. 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

27. S. Reitzenstein and A. Forchel, “Quantum dot micropillars,” J. Phys. D Appl. Phys. 43(3), 033001 (2010). [CrossRef]

]; 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.

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

28. P. R. Tapster and J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. 45(3), 595–604 (1998). [CrossRef]

,29

29. B. Blauensteiner, I. Herbauts, S. Bettelli, A. Poppe, and H. Hübel, “Photon bunching in parametric down-conversion with continuous-wave excitation,” Phys. Rev. A 79(6), 063846 (2009). [CrossRef]

], 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

30. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007). [CrossRef] [PubMed]

,31

31. M. Winger, T. Volz, G. Tarel, S. Portolan, A. Badolato, K. J. Hennessy, E. L. Hu, A. Beveratos, J. Finley, V. Savona, and A. Imamoğlu, “Explanation of photon correlations in the far-off-resonance optical emission from a quantum-dot-cavity system,” Phys. Rev. Lett. 103(20), 207403 (2009). [CrossRef] [PubMed]

]. 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

30. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007). [CrossRef] [PubMed]

33

33. M. Florian, P. Gartner, A. Steinhoff, C. Gies, and F. Jahnke, “Coulomb-assisted cavity feeding in the non-resonant optical emission from a quantum dot,” arXiv:1308.2080 (2013).

].

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
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.
. 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, gλ0λ1(2)[0], is determined similar to the way g(2)[0] is found in Sections 3 and 5.

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 gλ0λ1(2)[0]=0.52±0.01; as expected, this is fairly close to the value of g(2)[0]=0.58±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, gλ0λ1(2)[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

30. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007). [CrossRef] [PubMed]

,31

31. M. Winger, T. Volz, G. Tarel, S. Portolan, A. Badolato, K. J. Hennessy, E. L. Hu, A. Beveratos, J. Finley, V. Savona, and A. Imamoğlu, “Explanation of photon correlations in the far-off-resonance optical emission from a quantum-dot-cavity system,” Phys. Rev. Lett. 103(20), 207403 (2009). [CrossRef] [PubMed]

], 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, nth-order coincidence rates grow as efficiency to the nth power, so the time required to carry out the measurements can decrease dramatically as system efficiencies increase.

Acknowledgments

We thank A. Beveratos, J. Bienfang, T. Gerrits, K. Silverman, A. Migdall and L. K. Shalm for helpful discussions.

References and links

1.

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82(7), 071101 (2011). [CrossRef] [PubMed]

2.

F. Davidson, “Measurements of photon correlations in a laser beam near threshold with time-to-amplitude converter techniques,” Phys. Rev. 185(2), 446–453 (1969). [CrossRef]

3.

R. F. Chang, V. Korenman, C. O. Alley, and R. W. Detenbeck, “Correlations in light from a laser at threshold,” Phys. Rev. 178(2), 612–621 (1969). [CrossRef]

4.

M. Corti and V. Degiorgio, “Intrinsic third-order correlations in laser light near threshold,” Phys. Rev. A 14(4), 1475–1478 (1976). [CrossRef]

5.

Y. Qu, S. Singh, and C. D. Cantrell, “Measurements of higher order photon bunching of light beams,” Phys. Rev. Lett. 76(8), 1236–1239 (1996). [CrossRef] [PubMed]

6.

J. F. Dynes, Z. L. Yuan, A. W. Sharpe, O. Thomas, and A. J. Shields, “Probing higher order correlations of the photon field with photon number resolving avalanche photodiodes,” Opt. Express 19(14), 13268–13276 (2011). [CrossRef] [PubMed]

7.

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460(7252), 245–249 (2009). [CrossRef] [PubMed]

8.

M. Assmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science 325(5938), 297–300 (2009). [CrossRef] [PubMed]

9.

D. Elvira, X. Hachair, V. B. Verma, R. Braive, G. Beaudoin, I. Robert-Philip, I. Sagnes, B. Baek, S. W. Nam, E. A. Dauler, I. Abram, M. J. Stevens, and A. Beveratos, “Higher-order photon correlations in pulsed photonic crystal nanolasers,” Phys. Rev. A 84, 061802 (2011).

10.

A. G. Palmer 3rd and N. L. Thompson, “Molecular aggregation characterized by high order autocorrelation in fluorescence correlation spectroscopy,” Biophys. J. 52(2), 257–270 (1987). [CrossRef] [PubMed]

11.

P.-A. Lemieux and D. J. Durian, “Investigating non-Gaussian scattering processes by using nth-order intensity correlation functions,” J. Opt. Soc. Am. A 16(7), 1651–1664 (1999). [CrossRef]

12.

M. Koch, C. Sames, M. Balbach, H. Chibani, A. Kubanek, K. Murr, T. Wilk, and G. Rempe, “Three-photon correlations in a strongly driven atom-cavity system,” Phys. Rev. Lett. 107(2), 023601 (2011). [CrossRef] [PubMed]

13.

I. N. Agafonov, M. V. Chekhova, T. Sh. Iskhakov, and L.-A. Wu, “High-visibility intensity interference and ghost imaging with pseudo-thermal light,” J. Mod. Opt. 56(2-3), 422–431 (2009). [CrossRef]

14.

Y. Zhou, J. Simon, J. Liu, and Y. Shih, “Third-order correlation function and ghost imaging of chaotic thermal light in the photon counting regime,” Phys. Rev. A 81(4), 043831 (2010). [CrossRef]

15.

Y. Zhou, J. Liu, J. Simon, and Y. Shih, “Resolution enhancement of third-order thermal light ghost imaging in the photon counting regime,” J. Opt. Soc. Am. B 29(3), 377 (2012). [CrossRef]

16.

A. Hayat, A. Nevet, and M. Orenstein, “Ultrafast partial measurement of fourth-order coherence by HBT interferometry of upconversion-based autocorrelation,” Opt. Lett. 35(5), 793–795 (2010). [CrossRef] [PubMed]

17.

T. Horikiri, P. Schwendimann, A. Quattropani, S. Hing, A. Forchel, and Y. Yamamoto, “Higher order coherence of exciton-polariton condensates,” Phys. Rev. B 81(3), 033307 (2010). [CrossRef]

18.

M. Bajcsy, A. Rundquist, A. Majumdar, T. Sarmiento, K. Fischer, K. G. Lagoudakis, S. Buckley, and J. Vuckovic, “Non-classical three-photon correlations with a quantum dot strongly coupled to a photonic-crystal nanocavity,” arXiv:1307.3601 (2013).

19.

E. A. Goldschmidt, F. Piacentini, I. Ruo Berchera, S. V. Polyakov, S. Peters, S. Kuck, G. Brida, I. P. Degiovanni, A. L. Migdall, and M. Genovese, “Mode reconstruction of a light field by multi-photon statistics,” Phys. Rev. A 88(1), 013822 (2013). [CrossRef]

20.

M. J. Stevens, B. Baek, E. A. Dauler, A. J. Kerman, R. J. Molnar, S. A. Hamilton, K. K. Berggren, R. P. Mirin, and S. W. Nam, “High-order temporal coherences of chaotic and laser light,” Opt. Express 18(2), 1430–1437 (2010). [CrossRef] [PubMed]

21.

L. Ma, M. T. Rakher, M. J. Stevens, O. Slattery, K. Srinivasan, and X. Tang, “Temporal correlation of photons following frequency up-conversion,” Opt. Express 19(11), 10501–10510 (2011). [CrossRef] [PubMed]

22.

R. Loudon, The Quantum Theory of Light, Third Edition (Oxford University, Oxford, 2000).

23.

C. Santori, D. Fattal, J. Vučković, G. S. Solomon, E. Waks, and Y. Yamamoto, “Submicrosecond correlations in photoluminescence from InAs quantum dots,” Phys. Rev. B 69(20), 205324 (2004).

24.

E. Waks, C. Santori, and Y. Yamamoto, “Security aspects of quantum key distribution with sub-Poisson light,” Phys. Rev. A 66(4), 042315 (2002). [CrossRef]

25.

R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. 34(6-7), 709–759 (1987). [CrossRef]

26.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).

27.

S. Reitzenstein and A. Forchel, “Quantum dot micropillars,” J. Phys. D Appl. Phys. 43(3), 033001 (2010). [CrossRef]

28.

P. R. Tapster and J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. 45(3), 595–604 (1998). [CrossRef]

29.

B. Blauensteiner, I. Herbauts, S. Bettelli, A. Poppe, and H. Hübel, “Photon bunching in parametric down-conversion with continuous-wave excitation,” Phys. Rev. A 79(6), 063846 (2009). [CrossRef]

30.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007). [CrossRef] [PubMed]

31.

M. Winger, T. Volz, G. Tarel, S. Portolan, A. Badolato, K. J. Hennessy, E. L. Hu, A. Beveratos, J. Finley, V. Savona, and A. Imamoğlu, “Explanation of photon correlations in the far-off-resonance optical emission from a quantum-dot-cavity system,” Phys. Rev. Lett. 103(20), 207403 (2009). [CrossRef] [PubMed]

32.

M. Yamaguchi, T. Asano, and S. Noda, “Third emission mechanism in solid-state nanocavity quantum electrodynamics,” Rep. Prog. Phys. 75(9), 096401 (2012). [CrossRef] [PubMed]

33.

M. Florian, P. Gartner, A. Steinhoff, C. Gies, and F. Jahnke, “Coulomb-assisted cavity feeding in the non-resonant optical emission from a quantum dot,” arXiv:1308.2080 (2013).

OCIS Codes
(030.5260) Coherence and statistical optics : Photon counting
(270.1670) Quantum optics : Coherent optical effects
(270.5290) Quantum optics : Photon statistics

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: November 21, 2013
Manuscript Accepted: January 23, 2014
Published: February 4, 2014

Virtual Issues
April 21, 2014 Spotlight on Optics

Citation
Martin J. Stevens, Scott Glancy, Sae Woo Nam, and Richard P. Mirin, "Third-order antibunching from an imperfect single-photon source," Opt. Express 22, 3244-3260 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-3-3244


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References

  1. M. D. Eisaman, J. Fan, A. Migdall, S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82(7), 071101 (2011). [CrossRef] [PubMed]
  2. F. Davidson, “Measurements of photon correlations in a laser beam near threshold with time-to-amplitude converter techniques,” Phys. Rev. 185(2), 446–453 (1969). [CrossRef]
  3. R. F. Chang, V. Korenman, C. O. Alley, R. W. Detenbeck, “Correlations in light from a laser at threshold,” Phys. Rev. 178(2), 612–621 (1969). [CrossRef]
  4. M. Corti, V. Degiorgio, “Intrinsic third-order correlations in laser light near threshold,” Phys. Rev. A 14(4), 1475–1478 (1976). [CrossRef]
  5. Y. Qu, S. Singh, C. D. Cantrell, “Measurements of higher order photon bunching of light beams,” Phys. Rev. Lett. 76(8), 1236–1239 (1996). [CrossRef] [PubMed]
  6. J. F. Dynes, Z. L. Yuan, A. W. Sharpe, O. Thomas, A. J. Shields, “Probing higher order correlations of the photon field with photon number resolving avalanche photodiodes,” Opt. Express 19(14), 13268–13276 (2011). [CrossRef] [PubMed]
  7. J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460(7252), 245–249 (2009). [CrossRef] [PubMed]
  8. M. Assmann, F. Veit, M. Bayer, M. van der Poel, J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science 325(5938), 297–300 (2009). [CrossRef] [PubMed]
  9. D. Elvira, X. Hachair, V. B. Verma, R. Braive, G. Beaudoin, I. Robert-Philip, I. Sagnes, B. Baek, S. W. Nam, E. A. Dauler, I. Abram, M. J. Stevens, A. Beveratos, “Higher-order photon correlations in pulsed photonic crystal nanolasers,” Phys. Rev. A 84, 061802 (2011).
  10. A. G. Palmer, N. L. Thompson, “Molecular aggregation characterized by high order autocorrelation in fluorescence correlation spectroscopy,” Biophys. J. 52(2), 257–270 (1987). [CrossRef] [PubMed]
  11. P.-A. Lemieux, D. J. Durian, “Investigating non-Gaussian scattering processes by using nth-order intensity correlation functions,” J. Opt. Soc. Am. A 16(7), 1651–1664 (1999). [CrossRef]
  12. M. Koch, C. Sames, M. Balbach, H. Chibani, A. Kubanek, K. Murr, T. Wilk, G. Rempe, “Three-photon correlations in a strongly driven atom-cavity system,” Phys. Rev. Lett. 107(2), 023601 (2011). [CrossRef] [PubMed]
  13. I. N. Agafonov, M. V. Chekhova, T. Sh. Iskhakov, L.-A. Wu, “High-visibility intensity interference and ghost imaging with pseudo-thermal light,” J. Mod. Opt. 56(2-3), 422–431 (2009). [CrossRef]
  14. Y. Zhou, J. Simon, J. Liu, Y. Shih, “Third-order correlation function and ghost imaging of chaotic thermal light in the photon counting regime,” Phys. Rev. A 81(4), 043831 (2010). [CrossRef]
  15. Y. Zhou, J. Liu, J. Simon, Y. Shih, “Resolution enhancement of third-order thermal light ghost imaging in the photon counting regime,” J. Opt. Soc. Am. B 29(3), 377 (2012). [CrossRef]
  16. A. Hayat, A. Nevet, M. Orenstein, “Ultrafast partial measurement of fourth-order coherence by HBT interferometry of upconversion-based autocorrelation,” Opt. Lett. 35(5), 793–795 (2010). [CrossRef] [PubMed]
  17. T. Horikiri, P. Schwendimann, A. Quattropani, S. Hing, A. Forchel, Y. Yamamoto, “Higher order coherence of exciton-polariton condensates,” Phys. Rev. B 81(3), 033307 (2010). [CrossRef]
  18. M. Bajcsy, A. Rundquist, A. Majumdar, T. Sarmiento, K. Fischer, K. G. Lagoudakis, S. Buckley, and J. Vuckovic, “Non-classical three-photon correlations with a quantum dot strongly coupled to a photonic-crystal nanocavity,” arXiv:1307.3601 (2013).
  19. E. A. Goldschmidt, F. Piacentini, I. Ruo Berchera, S. V. Polyakov, S. Peters, S. Kuck, G. Brida, I. P. Degiovanni, A. L. Migdall, M. Genovese, “Mode reconstruction of a light field by multi-photon statistics,” Phys. Rev. A 88(1), 013822 (2013). [CrossRef]
  20. M. J. Stevens, B. Baek, E. A. Dauler, A. J. Kerman, R. J. Molnar, S. A. Hamilton, K. K. Berggren, R. P. Mirin, S. W. Nam, “High-order temporal coherences of chaotic and laser light,” Opt. Express 18(2), 1430–1437 (2010). [CrossRef] [PubMed]
  21. L. Ma, M. T. Rakher, M. J. Stevens, O. Slattery, K. Srinivasan, X. Tang, “Temporal correlation of photons following frequency up-conversion,” Opt. Express 19(11), 10501–10510 (2011). [CrossRef] [PubMed]
  22. R. Loudon, The Quantum Theory of Light, Third Edition (Oxford University, Oxford, 2000).
  23. C. Santori, D. Fattal, J. Vučković, G. S. Solomon, E. Waks, Y. Yamamoto, “Submicrosecond correlations in photoluminescence from InAs quantum dots,” Phys. Rev. B 69(20), 205324 (2004).
  24. E. Waks, C. Santori, Y. Yamamoto, “Security aspects of quantum key distribution with sub-Poisson light,” Phys. Rev. A 66(4), 042315 (2002). [CrossRef]
  25. R. Loudon, P. L. Knight, “Squeezed light,” J. Mod. Opt. 34(6-7), 709–759 (1987). [CrossRef]
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