## Efficiency vs. multi-photon contribution test for quantum dots |

Optics Express, Vol. 22, Issue 4, pp. 4789-4798 (2014)

http://dx.doi.org/10.1364/OE.22.004789

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### Abstract

The development of linear quantum computing within integrated circuits demands high quality semiconductor single photon sources. In particular, for a reliable single photon source it is not sufficient to have a low multi-photon component, but also to possess high efficiency. We investigate the photon statistics of the emission from a single quantum dot with a method that is able to sensitively detect the trade-off between the efficiency and the multi-photon contribution. Our measurements show, that the light emitted from the quantum dot when it is resonantly excited possess a very low multi-photon content. Additionally, we demonstrated, for the first time, the non-Gaussian nature of the quantum state emitted from a single quantum dot.

© 2014 Optical Society of America

## 1. Introduction

1. P. Grangier, G. Roger, and A. Aspect, “Experimental evidence for a photon anticorrelation effect on a beam splitter: A new light on single-photon interferences,” Europhys. Lett. **1**, 173–179 (1986). [CrossRef]

2. S. Scheel, “Single-photon sources-an
introduction,” J. Mod. Opt. **56**, 141–160 (2009). [CrossRef]

3. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature **409**, 46–52 (2001). [CrossRef] [PubMed]

4. J. L. O’Brien, “Optical quantum computing,” Science **318**, 1567–1570 (2007). [CrossRef]

6. T. Jennewein, M. Barbieri, and A. White, “Single-photon device requirements for operating linear
optics quantum computing outside the post-selection basis,” J. Mod.
Opt. **58**, 276–287 (2011). [CrossRef]

9. R. L. Hudson, “When is the Wigner quasi-probability density non-negative?,” Rep. Math. Phys. **6**, 249–252 (1974). [CrossRef]

10. A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. **87**,050402 (2001). [CrossRef] [PubMed]

*η*. Namely, if

*η*< 0.5 the measured Wigner function is positive, even though the photon source may give a perfectly antibunched intensity autocorrelation measurement. Therefore, the Wigner function has never been reconstructed for solid-state single photon sources, mainly because the collection and detection efficiencies are very unfavourable.

11. R. Filip and L. Mišta Jr., “Detecting quantum states with a positive Wigner function beyond mixtures of Gaussian states,” Phys. Rev. Lett. **106**,200401 (2011). [CrossRef] [PubMed]

## 2. Quantum dot setup

15. H. Jayakumar, A. Predojević, T. Huber, T. Kauten, G. S. Solomon, and G. Weihs, “Deterministic photon pairs and coherent optical control of a single quantum dot,” Phys. Rev. Lett. **110**,135505 (2013). [CrossRef] [PubMed]

15. H. Jayakumar, A. Predojević, T. Huber, T. Kauten, G. S. Solomon, and G. Weihs, “Deterministic photon pairs and coherent optical control of a single quantum dot,” Phys. Rev. Lett. **110**,135505 (2013). [CrossRef] [PubMed]

16. T. Flissikowski, A. Betke, I. A. Akimov, and F. Henneberger, “Two-photon coherent control of a single quantum dot,” Phys. Rev. Lett. **92**,227401 (2004). [CrossRef] [PubMed]

15. H. Jayakumar, A. Predojević, T. Huber, T. Kauten, G. S. Solomon, and G. Weihs, “Deterministic photon pairs and coherent optical control of a single quantum dot,” Phys. Rev. Lett. **110**,135505 (2013). [CrossRef] [PubMed]

**110**,135505 (2013). [CrossRef] [PubMed]

## 3. NG criterion

11. R. Filip and L. Mišta Jr., “Detecting quantum states with a positive Wigner function beyond mixtures of Gaussian states,” Phys. Rev. Lett. **106**,200401 (2011). [CrossRef] [PubMed]

*p*

_{1}and the error rate (multi-photon contribution)

*p*

_{2+}of the single photon source, which directly apply to the use of single photon sources for linear optical quantum computing [6

6. T. Jennewein, M. Barbieri, and A. White, “Single-photon device requirements for operating linear
optics quantum computing outside the post-selection basis,” J. Mod.
Opt. **58**, 276–287 (2011). [CrossRef]

*p*

_{0}, single photon

*p*

_{1}, and multi-photon terms

*p*

_{2+}, to the emitted exciton signal, [17

17. M. Ježek, I. Straka, M. Mičuda, M. Dušek, J. Fiurášek, and R. Filip, “Experimental test of the quantum non-Gaussian character of a heralded single-photon state,” Phys. Rev. Lett. **107**,213602 (2011). [CrossRef]

17. M. Ježek, I. Straka, M. Mičuda, M. Dušek, J. Fiurášek, and R. Filip, “Experimental test of the quantum non-Gaussian character of a heralded single-photon state,” Phys. Rev. Lett. **107**,213602 (2011). [CrossRef]

*p*

_{0}, can be expressed as

*R*

_{0}is the total number of counts (”singles”) in the biexciton signal, while

*R*

_{1}

*,*

_{A}*R*

_{1}

*are total numbers of the two-fold coincidences between the biexciton signal and either exciton signals, respectively.*

_{B}*R*

_{2}is the total number of the three-fold coincidences. The lower bound estimator of the single photon contribution is given by where

*T*is the splitting ratio of the beamsplitter used in the measurements. The contribution

*p*

_{2+}of the multi-photon terms can be estimated as

*p*

_{0},

*p*

_{1},

*p*

_{2+}) using this method because we are interested in calculating

*p*

_{1}as the lower bound estimator,

*p*

_{1,est}. The lower bound estimation takes into account the splitting ratio of the beamsplitter. In this way we avoid that the number of coincidences is artificially modified by an unbalanced beamsplitter.

*W*, that the given state is not a mixture of Gaussian states

*ρ*∉

*𝒢*, where

*𝒢*is the set of all mixtures of Gaussian states [17

17. M. Ježek, I. Straka, M. Mičuda, M. Dušek, J. Fiurášek, and R. Filip, “Experimental test of the quantum non-Gaussian character of a heralded single-photon state,” Phys. Rev. Lett. **107**,213602 (2011). [CrossRef]

**107**,213602 (2011). [CrossRef]

*r*[18] is used to parametrize the curve with the displacement

*d*given by

*d*

^{2}= (e

^{4}

*− 1)/4 [17*

^{r}**107**,213602 (2011). [CrossRef]

*W*is the directed distance between the measured point (

*p*

_{0},

*p*

_{1,est}) and the NG boundary. Δ

*W*> 0 indicates that the measured state is non-Gaussian. The results are shown in Fig. 2(b) and Tables 1 and 2.

## 4. Discussion

*p*

_{0}prevails. The statistical uncertainties for

*p*

_{0},

*p*

_{1}, and

*p*

_{2}+ were determined from the Poissonian statistics of the recorded events. In Tables 1 and 2 the last column gives the value of the witness, Δ

*W*, in units of the standard deviation. For example, the result given in the first row of Table 2 indicates a non-Gaussian state, (+), which is situated 2.63 standard deviations away from the boundary. In Fig. 2(b), the states of light in the white area are non-Gaussian and produced by a source incompatible with only quadratic non-linear processes.

*W*, is positive for any coincidence window that is smaller than the repetition period of the laser pulses. For example, for a coincidence window of 10 ns the measured state exceeds the Gaussian boundary by 2.63 standard deviations (green circle in Fig. 2(b)). Extending the coincidence window to 10.24 ns, which includes the beginning of the consecutive pulse, we find the measured state to move towards the boundary of the Gaussian states. Further extension of the coincidence window places the state in the region where we cannot distinguish it from a coherent mixture of Gaussian states (shown in Fig. 2(b) in yellow circles).

*p*

_{1}will be reduced and eventually detector dark counts will dominate and form the noise floor. As a result for a decreasing coincidence window our data show a tendency of approaching the NG boundary, but cannot cross it. The overall measurement time in this case was close to 8 hours and the average single count rates in the heralding (biexciton) channel

*S*

_{0}and the signal (exciton) channels

*S*

_{1}

*and*

_{A}*S*

_{1}

*were {5.4, 358, and 196} kcounts per second, respectively. The beamsplitter ratio for these measurements was*

_{B}*T*=0.64(1) and the detector dark count rate was 500 counts per second. Given these parameters we estimate a noise floor of 0.41 three-fold coincidences in eight hours for the smallest chosen coincidence window of 1.54 ns. This noise floor, which is depicted for the various coincidence windows as an orange dashed line in Fig. 2(b), ultimately limits the sensitivity of the measurement system.

_{mm}*p*

_{2+}to zero, but also yields a statistical error that is much larger than the NG boundary itself providing only an inconclusive result. This exemplifies that our measurements and analysis do take into account the entire system of source, collection, and detection and when we observe a non-Gaussian state it is a direct observation, not an extrapolation.

*S*

_{0},

*S*

_{1}

*and*

_{A}*S*

_{1}

*} = {37, 20, and 17} kcounts per second with the respective beamsplitter ratio of*

_{B}*T*=0.54(1). The measurement time was 3 hours. The use of resonant excitation gives an excellent suppression of multi-photon events, as evident in Fig. 2(a). For resonant excitation we estimate the noise floor to be 0.04 three-fold coincidences per 8 hour measurement. This threshold is lower than the one we obtain in above-band excitation due to the different nature of the excitation. Namely, resonant excitation produces predominantly cascaded emissions therefore the heralding efficiency and thus

_{sm}*p*

_{1}is a bit higher than with above-band excitation despite the lower overall exciton detection rates.

## 5. Comparison to autocorrelation

**110**,135505 (2013). [CrossRef] [PubMed]

**110**,135505 (2013). [CrossRef] [PubMed]

*p*

_{ac}_{1}/

*p*

_{1}=0.3×0.6=0.18. In this laser-triggering regime, the blinking severely deteriorates the efficiency of the source and tailors its photon statistics. Consequently, the detected state cannot be distinguished from the coherent mixture of Gaussian states.

*g*

_{2}(0) as 2×[1−

*p*

_{0}−

*p*

_{1}]/[2×(1−

*p*

_{0})−

*p*

_{1}]

^{2}and the Grangier’s [1

1. P. Grangier, G. Roger, and A. Aspect, “Experimental evidence for a photon anticorrelation effect on a beam splitter: A new light on single-photon interferences,” Europhys. Lett. **1**, 173–179 (1986). [CrossRef]

*α*=

*R*

_{0}

*R*

_{2}/

*R*

_{1}

_{A}R_{1}

*, which is equal to*

_{B}*g*

_{2}(0) for a symmetric beamsplitter. For the biexciton triggered measurement in resonant excitation we obtain

*g*

_{2}(0)=0.0010(10). The same measurement triggered on the laser pulse arrival gives

*g*

_{2}(0)=0.041(2). The later result is comparable with the autocorrelation parameter extracted from the traditional intensity autocorrelation measurement of 0.031(2). Additionally, it is possible to reconstruct a biexciton triggered intensity autocorrelation measurement. Such a measurement yields autocorrelation parameter 0.0029(29).

*p*

_{1}in above-band excitation for various excitation powers. The results are shown as gray dots in Fig. 3(a). For comparison the dashed line shows the

*p*

_{1}value we obtained in resonant excitation. An interesting feature is that

*p*

_{1}decreases with excitation power, reaching its minimum at the biexciton line saturation. We believe that this happens because the biexciton is re-excited directly from the exciton state before the system has reached the ground state and thus could not emit an exciton photon. Therefore the ratio of emitted exciton to biexciton photons is reduced. This is ultimately limits how strongly quantum dot-based pair sources can be driven incoherently.

## 6. Pulsed down-conversion

20. A. Predojević, S. Grabher, and G. Weihs, “Pulsed Sagnac source of polarization entangled photon
pairs,” Opt. Express **20**, 25022–25029 (2012). [CrossRef]

*p*

_{1}of the down-conversion source was 14% and in the experiment we gradually attenuated the idler beam to simulate losses comparable to the quantum dot case. The results show that for comparable overall efficiency and for the range of operating parameters used, the down-conversion source emission contains a higher proportion of multi-photon events than the emission from a single quantum dot. We do not generalize this statement to all types of down-conversion sources and their range of operating parameters. For example, the multi-photon contribution in continuous-wave sources depends on parameters like the size of the coincidence window, which in pulsed-pumping regime does not play a significant role.

## 7. Conclusion

11. R. Filip and L. Mišta Jr., “Detecting quantum states with a positive Wigner function beyond mixtures of Gaussian states,” Phys. Rev. Lett. **106**,200401 (2011). [CrossRef] [PubMed]

22. J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. **89**,137903 (2002). [CrossRef] [PubMed]

23. J. Niset, J. Fiurášek, and N. J. Cerf, “No-Go theorem for Gaussian quantum error correction,” Phys. Rev. Lett. **102**,120501 (2009). [CrossRef] [PubMed]

25. A. Mari and J. Eisert, “Negative quasi-probability as a resource for quantum computation,” Phys. Rev. Lett. **109**,230503 (2012). [CrossRef]

26. A. Dousse, J. Suffczyski, A. Beveratos, O. Krebs, A. Lematre, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature **466**, 217–220 (2010). [CrossRef] [PubMed]

3. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature **409**, 46–52 (2001). [CrossRef] [PubMed]

4. J. L. O’Brien, “Optical quantum computing,” Science **318**, 1567–1570 (2007). [CrossRef]

27. 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**, 135–174 (2007). [CrossRef]

## Acknowledgments

## References and links

1. | P. Grangier, G. Roger, and A. Aspect, “Experimental evidence for a photon anticorrelation effect on a beam splitter: A new light on single-photon interferences,” Europhys. Lett. |

2. | S. Scheel, “Single-photon sources-an
introduction,” J. Mod. Opt. |

3. | E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature |

4. | J. L. O’Brien, “Optical quantum computing,” Science |

5. | M. Varnava, D. E. Browne, and T. Rudolph, “How good must single photon sources and detectors be for efficient linear optical quantum computation?,” Phys. Rev. Lett. |

6. | T. Jennewein, M. Barbieri, and A. White, “Single-photon device requirements for operating linear
optics quantum computing outside the post-selection basis,” J. Mod.
Opt. |

7. | U. Leonhardt, |

8. | R. Glauber, |

9. | R. L. Hudson, “When is the Wigner quasi-probability density non-negative?,” Rep. Math. Phys. |

10. | A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. |

11. | R. Filip and L. Mišta Jr., “Detecting quantum states with a positive Wigner function beyond mixtures of Gaussian states,” Phys. Rev. Lett. |

12. | Z. Y. Ou, S. F. Pereira, and H. J. Kimble, “Quantum noise reduction in optical amplification,” Phys. Rev. Lett. |

13. | H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goler, K. Danzmann, and R. Schnabel, “Observation of squeezed light with 10-dB quantum-noise reduction,” Phys. Rev. Lett. |

14. | Y. Takeno, M. Yukawa, H. Yonezawa, and A. Furusawa, “Observation of −9 dB quadrature squeezing with improvement of phase stability in homodyne measurement,” Optics Express |

15. | H. Jayakumar, A. Predojević, T. Huber, T. Kauten, G. S. Solomon, and G. Weihs, “Deterministic photon pairs and coherent optical control of a single quantum dot,” Phys. Rev. Lett. |

16. | T. Flissikowski, A. Betke, I. A. Akimov, and F. Henneberger, “Two-photon coherent control of a single quantum dot,” Phys. Rev. Lett. |

17. | M. Ježek, I. Straka, M. Mičuda, M. Dušek, J. Fiurášek, and R. Filip, “Experimental test of the quantum non-Gaussian character of a heralded single-photon state,” Phys. Rev. Lett. |

18. | D. F. Walls and G. J. Milburn, |

19. | 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 |

20. | A. Predojević, S. Grabher, and G. Weihs, “Pulsed Sagnac source of polarization entangled photon
pairs,” Opt. Express |

21. | O. Kuzucu and F. N. C. Wong, “Pulsed Sagnac source of narrow-band polarization entangled
photons,” Opt. Express |

22. | J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. |

23. | J. Niset, J. Fiurášek, and N. J. Cerf, “No-Go theorem for Gaussian quantum error correction,” Phys. Rev. Lett. |

24. | V. Veitch, C. Ferrie, D. Gross, and J. Emerson, “Negative quasi-probability as a resource for quantum computation,” New. J. Phys |

25. | A. Mari and J. Eisert, “Negative quasi-probability as a resource for quantum computation,” Phys. Rev. Lett. |

26. | A. Dousse, J. Suffczyski, A. Beveratos, O. Krebs, A. Lematre, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature |

27. | 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. |

**OCIS Codes**

(000.1600) General : Classical and quantum physics

(270.0270) Quantum optics : Quantum optics

(270.5290) Quantum optics : Photon statistics

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: January 2, 2014

Revised Manuscript: February 11, 2014

Manuscript Accepted: February 13, 2014

Published: February 21, 2014

**Citation**

Ana Predojević, Miroslav Ježek, Tobias Huber, Harishankar Jayakumar, Thomas Kauten, Glenn S. Solomon, Radim Filip, and Gregor Weihs, "Efficiency vs. multi-photon contribution test for quantum dots," Opt. Express **22**, 4789-4798 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-4-4789

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### References

- P. Grangier, G. Roger, A. Aspect, “Experimental evidence for a photon anticorrelation effect on a beam splitter: A new light on single-photon interferences,” Europhys. Lett. 1, 173–179 (1986). [CrossRef]
- S. Scheel, “Single-photon sources-an introduction,” J. Mod. Opt. 56, 141–160 (2009). [CrossRef]
- E. Knill, R. Laflamme, G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001). [CrossRef] [PubMed]
- J. L. O’Brien, “Optical quantum computing,” Science 318, 1567–1570 (2007). [CrossRef]
- M. Varnava, D. E. Browne, T. Rudolph, “How good must single photon sources and detectors be for efficient linear optical quantum computation?,” Phys. Rev. Lett. 100,060502 (2008). [CrossRef] [PubMed]
- T. Jennewein, M. Barbieri, A. White, “Single-photon device requirements for operating linear optics quantum computing outside the post-selection basis,” J. Mod. Opt. 58, 276–287 (2011). [CrossRef]
- U. Leonhardt, Measuring the Quantum State of Light (Cambridge University, 1997).
- R. Glauber, Quantum Theory of Optical Coherence (Wiley-VCH, 2007).
- R. L. Hudson, “When is the Wigner quasi-probability density non-negative?,” Rep. Math. Phys. 6, 249–252 (1974). [CrossRef]
- A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87,050402 (2001). [CrossRef] [PubMed]
- R. Filip, L. Mišta, “Detecting quantum states with a positive Wigner function beyond mixtures of Gaussian states,” Phys. Rev. Lett. 106,200401 (2011). [CrossRef] [PubMed]
- Z. Y. Ou, S. F. Pereira, H. J. Kimble, “Quantum noise reduction in optical amplification,” Phys. Rev. Lett. 70, 3239–3242 (1993). [CrossRef] [PubMed]
- H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goler, K. Danzmann, R. Schnabel, “Observation of squeezed light with 10-dB quantum-noise reduction,” Phys. Rev. Lett. 100,033602 (2008). [CrossRef] [PubMed]
- Y. Takeno, M. Yukawa, H. Yonezawa, A. Furusawa, “Observation of −9 dB quadrature squeezing with improvement of phase stability in homodyne measurement,” Optics Express 15, 4321–4327 (2007). [CrossRef]
- H. Jayakumar, A. Predojević, T. Huber, T. Kauten, G. S. Solomon, G. Weihs, “Deterministic photon pairs and coherent optical control of a single quantum dot,” Phys. Rev. Lett. 110,135505 (2013). [CrossRef] [PubMed]
- T. Flissikowski, A. Betke, I. A. Akimov, F. Henneberger, “Two-photon coherent control of a single quantum dot,” Phys. Rev. Lett. 92,227401 (2004). [CrossRef] [PubMed]
- M. Ježek, I. Straka, M. Mičuda, M. Dušek, J. Fiurášek, R. Filip, “Experimental test of the quantum non-Gaussian character of a heralded single-photon state,” Phys. Rev. Lett. 107,213602 (2011). [CrossRef]
- D. F. Walls, G. J. Milburn, Quantum Optics (Springer, 2008).
- C. Santori, D. Fattal, J. Vučković, G. S. Solomon, E. Waks, Y. Yamamoto, “Submicrosecond correlations in photoluminescence from InAs quantum dots,” Phys. Rev. B 69,205324 (2004). [CrossRef]
- A. Predojević, S. Grabher, G. Weihs, “Pulsed Sagnac source of polarization entangled photon pairs,” Opt. Express 20, 25022–25029 (2012). [CrossRef]
- O. Kuzucu, F. N. C. Wong, “Pulsed Sagnac source of narrow-band polarization entangled photons,” Opt. Express 15, 15377–15386 (2007).
- J. Eisert, S. Scheel, M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89,137903 (2002). [CrossRef] [PubMed]
- J. Niset, J. Fiurášek, N. J. Cerf, “No-Go theorem for Gaussian quantum error correction,” Phys. Rev. Lett. 102,120501 (2009). [CrossRef] [PubMed]
- V. Veitch, C. Ferrie, D. Gross, J. Emerson, “Negative quasi-probability as a resource for quantum computation,” New. J. Phys 14,113011 (2012). [CrossRef]
- A. Mari, J. Eisert, “Negative quasi-probability as a resource for quantum computation,” Phys. Rev. Lett. 109,230503 (2012). [CrossRef]
- A. Dousse, J. Suffczyski, A. Beveratos, O. Krebs, A. Lematre, I. Sagnes, J. Bloch, P. Voisin, P. Senellart, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010). [CrossRef] [PubMed]
- P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007). [CrossRef]

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