## Third-order antibunching from an imperfect single-photon source |

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)}(*τ*_{1},*τ*_{2}), 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)}(*τ*_{1},*τ*_{2}).

© 2014 Optical Society of America

## 1. Introduction

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]

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]

*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. 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]

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 *n*th-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]

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]

*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

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]

*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

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. 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]

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]

## 2. What *g*^{(2)} and *g*^{(3)} tell us about photon probabilities *P*(*n*)

*t*, and the

*τ*’s are time delays.

*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)}(

*τ*

_{1}

*,τ*

_{2}).

*g*

^{(2)}[0], can be interpreted as the autocorrelation of the pulse train in Fig. 1, 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]

*k*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

^{th}*k*-dependence of the number operator and use

*g*

^{(2)}[0]. We can rewrite this in terms of the per-pulse photon probabilities aswhere

*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: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*(1) >>

*P*(2) >>

*P*(

*n*>2), as is true for many low-efficiency sources, this leads to the familiar approximation

*P*(1) >>

*P*(

*n*>1) and

*P*(3) >>

*P*(

*n*>3), this can be approximated as

*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

*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*= 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.

_{rep}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]

## 4. How well do these correlation measurements approximate *g*^{(2)} and *g*^{(3)}?

*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].

*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.

*P*

_{01}(

*click*,

*click*) is the probability that both detectors 0 and 1 click during the same pulse;

*P*

_{0}(

*click*) is the probability that detector 0 clicks during a pulse, independent of whether detector 1 clicks; and

*P*

_{1}(

*click*) is the probability that detector 1 clicks during a pulse, independent of whether detector 0 clicks.

*γ*

^{(2)}[0] in terms of the photon probability distribution of the incident light. Following the usual conventions [22,25

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

*R*and transmittance

*T*=

*R*-1, and the detection inefficiencies (along with any other optical losses) are modeled as beamsplitters with transmittance

*η*

_{0}and

*η*

_{1}in front of detectors with 100% detection efficiency, as shown in Fig. 3(a). The joint probability distribution of

*n*

_{0}and

*n*

_{1}photons arriving at detectors D

_{0}and D

_{1}, respectively, is thenWe can calculate

*γ*

^{(2)}[0] by summing the relevant probabilities for all photon events that cause the two detectors to either click or not click:We can rewrite this expression using expectation values of the incident photon probability distribution

*P*(

*n*):

*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: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*,

*η*

_{0}and

*η*

_{1}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) yieldsregardless 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), thenEven if these conditions are not met directly from the source, they can be satisfied by introducing additional attenuation before the HBT setup.

*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,

*η*

_{0}≠

*η*

_{1}. 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.

*P*

_{012}(

*click*,

*click*,

*click*) is the probability that detectors 0, 1 and 2 all click during the same pulse;

*P*

_{0}(

*click*) is the probability that detector 0 clicks during a pulse, independent of whether detectors 1 or 2 click; and

*P*

_{1}(

*click*) and

*P*

_{2}(

*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*

^{(2)}[0], the measured quantity is a good approximation to the true third-order coherence,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

*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

*τ*

_{1}= 0,

*τ*

_{2}= 0 and

*τ*

_{1}=

*τ*

_{2}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

*g*

^{(2)}[0]. To minimize the contribution of background counts, we use an integration width of 4.1 ns =

*T*/3, and ignore the histogram counts in the hashed regions.

_{rep}*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.

^{2}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

*τ*

_{1}=

*τ*

_{2}= 0 is divided by the average number of counts in 162 peaks where

*τ*

_{1}and

*τ*

_{2}are near, but not equal to, zero.

*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

**82**(7), 071101 (2011). [CrossRef] [PubMed]

*g*

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

*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

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]

*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].

*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.

*g*

^{(2)}[0], we might expect that state emitted by the quantum dot could be modeled as

*g*

^{(2)}[0] could then be described by increasing the magnitude of

*c*

_{2}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.

*μ*and zero photons with probability 1-

_{s}*μ*, and 2) a background with Poissonian photon number statistics and mean photon number

_{s}*μ*. The probability that this source produces

_{b}*m*photons iswhere

*P*(

_{s}*m*) is the probability that the single-photon source emits

*m*photons, and

*P*(

_{b}*m*-

*k*) is the probability that the background emits

*m*-

*k*photons. Explicitly writing in the individual distributions yieldsSubstituting

*P*(

*m*) into Eqs. (7) and (12), we find where we have defined

*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.

*g*

^{(2)}[0] and

*g*

^{(3)}[0,0]: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.

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

*g*

^{(2)}increases as pump power increases.

*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. 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]

## 6. Cross-correlation measurements

*λ*

_{0}. The monochromator is then tuned so that the other SPAD detects photons at a series of other fixed wavelengths

*λ*

_{1}. The cross-correlation between photons at these two wavelengths is recorded and analyzed, and the zero delay value,

*g*

^{(2)}[0] is found in Sections 3 and 5.

*λ*

_{1}=

*λ*

_{0}, we find

## 7. Summary

*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.

*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*-order coincidence rates grow as efficiency to the

^{th}*n*power, so the time required to carry out the measurements can decrease dramatically as system efficiencies increase.

^{th}## Acknowledgments

## References and links

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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 |

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 |

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. |

32. | M. Yamaguchi, T. Asano, and S. Noda, “Third emission mechanism in solid-state nanocavity quantum electrodynamics,” Rep. Prog. Phys. |

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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