## Measuring the dynamics of second-order photon correlation functions inside a pulse with picosecond time resolution |

Optics Express, Vol. 18, Issue 19, pp. 20229-20241 (2010)

http://dx.doi.org/10.1364/OE.18.020229

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### Abstract

We present a detailed discussion of a recently demonstrated experimental technique capable of measuring the correlation function of a pulsed light source with picosecond time resolution. The measurement involves a streak camera in single photon counting mode, which is modified such that a signal at a fixed repetition rate, and well defined energy, can be monitored after each pulsed laser excitation. The technique provides further insight into the quantum optical properties of pulsed light emission from semiconductor nanostructures, and the dynamics inside a pulse, on the sub-nanosecond time scale.

© 2010 OSA

## 1. Introduction

1. U. M. Titulaer and R. J. Glauber, “Correlation functions for coherent fields,” Phys. Rev. B **140**(3B), 676–682 (1965). [CrossRef]

2. L. Mandel, “Sub-Poissonian photon statistics in resonance fluorescence,” Opt. Lett. **4**(7), 205–207 (1979). [CrossRef] [PubMed]

3. 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]

*t*and

*τ*comparable to the coherence time of the light field [4

4. R. Loudon, “Non-classical effects in the statistical properties of light,” Rep. Prog. Phys. **43**(7), 913–949 (1980). [CrossRef]

5. H. Paul, “Photon antibunching,” Rev. Mod. Phys. **54**(4), 1061–1102 (1982). [CrossRef]

6. R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature **177**(4497), 27–29 (1956). [CrossRef]

7. H. J. Kimble, M. Dagenais, and L. Mandel, “Photon antibunching in resonance fluorescence,” Phys. Rev. Lett. **39**(11), 691–695 (1977). [CrossRef]

8. S. Strauf, K. Hennessy, M. T. Rakher, Y. S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, “Self-tuned quantum dot gain in photonic crystal lasers,” Phys. Rev. Lett. **96**(12), 127404 (2006). [CrossRef] [PubMed]

9. P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science **290**(5500), 2282–2285 (2000). [CrossRef] [PubMed]

11. Z. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, “Electrically driven single-photon source,” Science **295**(5552), 102–105 (2002). [CrossRef]

## 2. Experimental setup

12. 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]

13. F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconductors,” Nat. Phys. **5**(4), 267–270 (2009). [CrossRef]

14. 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]

### 2.1 General method of a streak camera

### 2.2 Optimization and customization

### 2.3 Data evaluation

12. 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]

*t*and

_{,}respectively. This information can be extracted from the single pictures as follows: At first all photons in all pictures are sorted to streak bins with a width of one streak. Detected photons between two streaks are discarded. Now the desired temporal resolution

12. 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]

*τ*. In the case of

*n*photons will be

*t*and

*τ*. Accordingly, a four-photon detection event, as depicted in the lower inset of Fig. 1, corresponds to twelve two-photon pairs.

*t*and

*t*and

*t*and

*t*-dependence of the correlation function. Calculating

### 2.4 Error sources and error correction

*R*of the photon size on the screen, which abruptly falls off step-like at distances, which are larger than

*R*of another photon are disregarded, any erroneous photon counts due to reconstruction problems are eliminated. However in this way also real photon detections are disregarded. Effectively, the detector size for detecting another photon after the first one went down, which simply modifies the normalization by a factor of

*A*is the size of a bin on the screen,

*w*is a weighting factor, which is needed, if the mean number of photon counts is not distributed equally along the width of a bin. If artificial dead pixels are used in measurements of higher-order correlation functions, it is necessary to consider the decreasing effective detector size inside a time bin accordingly after each photon detection. Fortunately the problem of erroneous photon reconstruction is usually only significant in analyzing a weak signal combined with a high electron gain and a fast horizontal blanking speed.

*t*, but they need to be taken into account when measuring

### 2.5 Characterizing streak camera performance

*τ-*dependence of the photon pair count rate and the sum of all photon pairs detected in all time bins will not depend on jitter. However, the mean photon count rate used for the normalization procedure might be broadened by the jitter. Assuming the simple case of a Gaussian pulse shape

*W*and Gaussian jitter

*τ*the cross-correlation between signal and background noise becomes dominant and

*τ*which are larger than approximately 1 ps, this effect vanishes. For a MCP gain value of 40 (corresponding to a MCP voltage of 1.55 kV) a clear dependence of the shape of the intensity autocorrelation on the photon counting threshold can be seen. The data set for a low photon counting threshold (black squares) shows a deeper dip and larger artificial bunching compared to the data set obtained for a higher photon counting threshold (red dots). Fits to these two data sets are shown as black and red lines. The black line corresponds to values of

*W*=1.42 ps,

*J*=1.81 ps and

*W*=1.42 ps,

*J*=1.38 ps and

*τ*and also explains the large jitter value because the center-of-gravity position of the intensity of these two photons on the screen is not too well defined. For the higher threshold value the jitter is significantly reduced and the underestimation of

*τ*. This shows that still two photons are sometimes counted as one, but the higher gain causes more pixels to contain intensity values above the threshold setting, allowing for an easier estimation of the center-of-gravity by the photon reconstruction algorithm. Taking single photon sources into account, one might be interested in whether a streak-camera approach allows for unperturbed measurements of photon statistics in terms of quantum efficiency and dark count. There is no general answer to this question as both can depend on signal wavelength and the experimental conditions. Our streak camera is equipped with a S-20 photocathode with a radiant sensitivity above 10 mA/W in the visible wavelength range. The P-46 phosphor emits approximately 90 photons per incident electron. The exact efficiency of the total camera therefore depends crucially on the wavelength and gain used, but will usually not exceed 10%.

*τ*will also pose a problem when trying to address the intensity correlation of short signals. Experimentally the most relevant quantities are the time-integrated equal time correlation functions of second and third order. If the influence of dark noise is small, the measured value

15. M. Aßmann, F. Veit, M. Bayer, S. Reitzenstein, S. Höfling, L. Worschech, and A. Forchel, “Ultrafast tracking of second-order photon correlations in the emission of quantum-dot microresonator lasers,” Phys. Rev. B **81**(16), 165314 (2010). [CrossRef]

## 3. Conclusion

## Acknowledgements

## References and links

1. | U. M. Titulaer and R. J. Glauber, “Correlation functions for coherent fields,” Phys. Rev. B |

2. | L. Mandel, “Sub-Poissonian photon statistics in resonance fluorescence,” Opt. Lett. |

3. | 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 |

4. | R. Loudon, “Non-classical effects in the statistical properties of light,” Rep. Prog. Phys. |

5. | H. Paul, “Photon antibunching,” Rev. Mod. Phys. |

6. | R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature |

7. | H. J. Kimble, M. Dagenais, and L. Mandel, “Photon antibunching in resonance fluorescence,” Phys. Rev. Lett. |

8. | S. Strauf, K. Hennessy, M. T. Rakher, Y. S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, “Self-tuned quantum dot gain in photonic crystal lasers,” Phys. Rev. Lett. |

9. | P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science |

10. | E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. |

11. | Z. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, “Electrically driven single-photon source,” Science |

12. | M. Assmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science |

13. | F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconductors,” Nat. Phys. |

14. | A. Hayat, A. Nevet, and M. Orenstein, “Ultrafast partial measurement of fourth-order coherence by HBT interferometry of upconversion-based autocorrelation,” Opt. Lett. |

15. | M. Aßmann, F. Veit, M. Bayer, S. Reitzenstein, S. Höfling, L. Worschech, and A. Forchel, “Ultrafast tracking of second-order photon correlations in the emission of quantum-dot microresonator lasers,” Phys. Rev. B |

16. | G. Li, T. C. Zhang, Y. Li, and J. M. Wang, “Photon statistics of light fields based on single-photon-counting modules,” Phys. Rev. A |

**OCIS Codes**

(030.5290) Coherence and statistical optics : Photon statistics

(270.4180) Quantum optics : Multiphoton processes

(320.7130) Ultrafast optics : Ultrafast processes in condensed matter, including semiconductors

(100.0118) Image processing : Imaging ultrafast phenomena

(140.3948) Lasers and laser optics : Microcavity devices

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: June 8, 2010

Revised Manuscript: July 9, 2010

Manuscript Accepted: July 16, 2010

Published: September 8, 2010

**Citation**

Marc Aßmann, Franziska Veit, Jean-Sebastian Tempel, Thorsten Berstermann, Heinrich Stolz, Mike van der Poel, Jørn M. Hvam, and Manfred Bayer, "Measuring the dynamics of second-order photon correlation functions inside a pulse with picosecond time resolution," Opt. Express **18**, 20229-20241 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-20229

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### References

- U. M. Titulaer and R. J. Glauber, “Correlation functions for coherent fields,” Phys. Rev. B 140(3B), 676–682 (1965). [CrossRef]
- L. Mandel, “Sub-Poissonian photon statistics in resonance fluorescence,” Opt. Lett. 4(7), 205–207 (1979). [CrossRef] [PubMed]
- 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]
- R. Loudon, “Non-classical effects in the statistical properties of light,” Rep. Prog. Phys. 43(7), 913–949 (1980). [CrossRef]
- H. Paul, “Photon antibunching,” Rev. Mod. Phys. 54(4), 1061–1102 (1982). [CrossRef]
- R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177(4497), 27–29 (1956). [CrossRef]
- H. J. Kimble, M. Dagenais, and L. Mandel, “Photon antibunching in resonance fluorescence,” Phys. Rev. Lett. 39(11), 691–695 (1977). [CrossRef]
- S. Strauf, K. Hennessy, M. T. Rakher, Y. S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, “Self-tuned quantum dot gain in photonic crystal lasers,” Phys. Rev. Lett. 96(12), 127404 (2006). [CrossRef] [PubMed]
- P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science 290(5500), 2282–2285 (2000). [CrossRef] [PubMed]
- E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
- Z. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, “Electrically driven single-photon source,” Science 295(5552), 102–105 (2002). [CrossRef]
- 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]
- F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconductors,” Nat. Phys. 5(4), 267–270 (2009). [CrossRef]
- 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]
- M. Aßmann, F. Veit, M. Bayer, S. Reitzenstein, S. Höfling, L. Worschech, and A. Forchel, “Ultrafast tracking of second-order photon correlations in the emission of quantum-dot microresonator lasers,” Phys. Rev. B 81(16), 165314 (2010). [CrossRef]
- G. Li, T. C. Zhang, Y. Li, and J. M. Wang, “Photon statistics of light fields based on single-photon-counting modules,” Phys. Rev. A 71(2), 023807 (2005). [CrossRef]

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