## Rigorous criterion for characterizing correlated multiphoton emissions

Optics Express, Vol. 18, Issue 7, pp. 7092-7100 (2010)

http://dx.doi.org/10.1364/OE.18.007092

Acrobat PDF (401 KB)

### Abstract

Strong correlation of photons, particularly in the single-photon regime, has recently been exploited for various applications in quantum information processing. Existing correlation measurements, however, do not fully characterize multi-photon correlation in a relevant context and may pose limitations in practical situations. We propose a conceptually rigorous, but easy-to-implement, criterion for detecting correlated multi-photon emission out of a quantum optical system, drawn from the context of wavefunction collapse. We illustrate the robustness of our approach against experimental limitations by considering an anharmonic optical system.

© 2010 Optical Society of America

## 1. Introduction

## 2. The criterion

### 2.1. Photon surge

_{-}𝓔

_{+}〉, where the operators 𝓔

_{±}correspond to the positive- and the negative-frequency part of the field. Let us assume that a quantum system can be described by a pure steady state ∣Ψ〉

*for simplicity, but our argument applies equally well to mixed states. If it has emitted*

_{s}*n*- 1 quanta, the wavefunction is collapsed to ∣Ψ

_{c}^{(n-1)}〉 =

^{n-1}

_{+}∣Ψ〉

*conditioned on these emissions. The detection rate for the succeeding*

_{s}*n*th photon is then given by

_{1}= 〈

_{-}

_{+}〉, which only characterizes the signal strength and has little to do with correlation. For n-photon correlation (

*n*> 1), once a photon is emitted, however, the next emission will immediately follow, thus the conditional rate 𝓡

_{2}must be large enough. In particular, we require 𝓡

_{2}to be larger than 𝓡

_{1}, i.e.

*g*

^{(2)}function. Extending the requirement to next emissions sequentially, we derive a set of

*surge*conditions

*k*= 2,…,

*n*.

### 2.2. Photon blockade

*k*= 2,…,

*n*is not sufficient to ensure

*n*-photon correlation, and importantly, one must also look at the next occurrences carefully. After the detection of

*n*photons, the succeeding emissions must be suppressed, which can be expressed as

*conditional*rates 𝓡

*, rather than the*

_{k}*bare*rates 〈

^{k}

_{-}

_{k}_{+}〉, as the former takes into account the correlation between adjacent emissions in a stronger sense. However, the resulting criterion does not require any conditional measurements. Instead, the quantities R

_{k,k-1}in Eqs. (2) and (3) simply involve various photon-coincidence rates and we particularly note that the numerator and the denominator are in the same order of the field strength. It is thus given in an experimentally desirable form, that is, insensitive to the quantum efficiency of photodetectors.

### 2.3. Measure of multi-photon correlation

*of*

_{n}*n*-photon correlation as

*N*

_{tr}is a truncated excitation number to be taken appropriate to a given situation. 𝓜

*quantifies the strength of the*

_{n}*n*-photon correlation by measuring the deviation of R

_{k,k-1}from unity in the surge and the blockade conditions of Eqs. (2) and (3), respectively, and returns a nonzero value only when all those conditions are satisfied. To experimentally obtain 𝓜

*for a given system, one first measures the bare*

_{n}*k*-photon coincidence rates 〈

^{k}

_{-}

^{k}

_{+}〉 for all

*k*= 1,…,

*N*

_{tr}. Then, each conditional rate R

_{k,k-1}defined by Eq. (2) is evaluated and plugged in to Eq. (4) to determine the value of 𝓜

*.*

_{n}### 2.4. Remarks

*x*: a general space-time point) [17

_{i}17. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. **130**, 2529–2539 (1963). [CrossRef]

*g*

^{(n)}is, however, rather limited and we particularly note that

*g*

^{(n)}compares the

*n*-photon coincidence rate (numerator) only with the single-photon counting rates (denominator). Large (small) value of

*g*

^{(n)}characterizes a bunching (antibunching) effect with no strict

*n*-photon correlation that can emerge even in a classical scattering system, e.g.

*g*

^{(n)}=

*n*! for a thermal light (Hanbury-Brown–Twiss effect [21

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

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

*g*

^{(2)}≫ 1 with no rigorous two-photon correlation will be shown below in Sec.3.

_{k,k-1}< 1 in Eq. (3), is thus a clear signature of nonclassicality. So our criteria of multiphoton correlation can be fulfiled only by nonclassical sources. We emphasize that, in the so called multiphoton antibunching criteria in [18

18. C. T. Lee, “Higher-order criteria for nonclassical effects in photon statistics,” Phys. Rev. A **41**1721–1723 (1990). [CrossRef] [PubMed]

19. D. N. Klyshko, “Observable signs of nonclassical light,” Phys. Lett. A , **213**7–15 (1996).
[CrossRef]

*P*function, thus lacking a clear interpretation as multiphoton correlation.

## 3. Application: Cavity QED system

### 3.1. Model

2. A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. **4**, 859–863 (2008). [CrossRef]

3. K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature **436**, 87–90 (2005). [CrossRef] [PubMed]

4. B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science **319**1062–1065 (2008). [CrossRef] [PubMed]

14. J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its √*n* nonlinearity in a cavity QED system,” Nature (London) **454**, 315–318 (2008). [CrossRef]

*ω*=

_{A}*ω*≡

^{c}*ω*

_{0}, where

*ω*is the qubit transition frequency and

_{A}*ω*the cavity resonance frequency. The qubit-cavity system at coupling strength

_{c}*g*is then described by the Hamiltonian

*σ*

_{±}and

*σ*are the Pauli pseudospin operators. The composite system has the ground state ∣0,

_{z}*g*〉 with the energy

*E*

_{0}= 0 and the polaritonic excited states

*ω*,

_{L}*n*-photon resonant absorption may occur [23

23. Y.-T Chough, H.-J. Moon, H. Nha, and K. An, “Single-atom laser based on multiphoton resonances at far-off resonance in the Jaynes-Cummings ladder,” Phys. Rev. A **63**, 013804 (2000). [CrossRef]

*γ*(2κ) is the qubit (cavity) decay rate and the interaction Hamiltonian

*δ*≡

*ω*

_{0}-

*ω*.

_{L}### 3.2. Correlation measures

*g*

^{(n)}of the output, the result may characterize the correlation of emitted photons to some extent, but not in a full rigorous sense. In particular,

*g*

^{(n)}(0) = 〈

*a*

^{†n}

*a*〉/〈

^{n}*a*

^{†}

*a*〉

*in Fig. 2(a) shows a large bunching at zero detuning*

^{n}*δ*= 0, which has nothing to do with genuine

*n*-photon correlation as we will clearly show below. Close inspection of photon statistics reveals that the system does exhibit some non-classical behavior at

*δ*= 0, e.g. the oscillation of conditional detection rate 𝓡

*which peaks at even number of*

_{k}*k*, but it is not a rigorous

*n*-photon correlation at any level

*n*in view of our criterion. To get rid of this “cumbersome” resonance effect observed at

*δ*= 0 that may overwhelm the other resonance peaks, Kubanek

*et al*. introduced the differential correlation function,

*C*

^{(2)}(0) = 〈

*a*

^{†2}

*a*

^{2}〉 - 〈

*a*

^{†}

*a*〉

^{2}, that measures the

*absolute*occurrence of two-photon excitation with respect to the single-photon excitation [12

12. A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-photon gateway in one-atom cavity quantum electrodynamics,” Phys. Rev. Lett. **101**, 203602 (2008). [CrossRef] [PubMed]

*g*

^{2}(0) in general, although it was instrumental to identify the second resonant peak in [12

12. A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-photon gateway in one-atom cavity quantum electrodynamics,” Phys. Rev. Lett. **101**, 203602 (2008). [CrossRef] [PubMed]

*n*-photon level,

*C*

^{(n)}(0) = 〈

*a*

^{†n}

*a*〉 - 〈

^{n}*a*†

*a*〉

*for*

^{n}*n*≥ 3, becomes hardly effective in identifying the higher-order peaks by the broadening effect in the realistic regime [Fig. 2 (a)–(c)].

### 3.3. Large driving field

*γ*/

*g*= 2κ/

*g*= 0.1. Due to the intensity-dependent broadening effect, the Glauber function

*g*

^{(n)}(0) no longer shows noticeable marks of resonance, except for the peak at

*δ*= 0 overwhelming the entire shape in the spectrum. On the other hand, our measure 𝓜

*identifies a clear signature of multi-photon correlations under the same condition. This capability would allow one to increase the pump strength to some extent, and thereby easing the difficulty of having to measure higher-order coincidence than*

_{n}*n*[i.e., blockade conditions in Eq. (3)] to identify

*n*-photon correlation in our method. Furthermore, we have also checked that other possible broadening effects, e.g. atomic motion in the cavity, do not degrade the capability of our criterion for characterizing multiphoton correlations. We attribute this robustness against experimental imperfections to the rigorous context established with the measure 𝓜

*.*

_{n}## 4. Conclusion

*n*is a clear signature of quantum nature of light field [24

24. Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, “Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations,” Phys. Rev. Lett. **64**, 2499–2502 (1990). [CrossRef] [PubMed]

25. H. J. Carmichael, P. Kochan, and B. C. Sanders, “Photon correlation spectroscopy,” Phys. Rev. Lett. **77**, 631–634 (1996). [CrossRef] [PubMed]

13. I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. **4**, 382–385 (2008). [CrossRef]

14. J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its √*n* nonlinearity in a cavity QED system,” Nature (London) **454**, 315–318 (2008). [CrossRef]

15. L. S. Bishop, J. M. Chow, J. Koch, A. A. Houck, M. H. Devoret, E. Thuneberg, S. M. Girvin, and R. J. Schoelkopf, “Nonlinear response of the vacuum Rabi resonance,” Nat. Phys. **5**, 105–109 (2009). [CrossRef]

3. K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature **436**, 87–90 (2005). [CrossRef] [PubMed]

4. B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science **319**1062–1065 (2008). [CrossRef] [PubMed]

12. A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-photon gateway in one-atom cavity quantum electrodynamics,” Phys. Rev. Lett. **101**, 203602 (2008). [CrossRef] [PubMed]

13. I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. **4**, 382–385 (2008). [CrossRef]

*n*-dependence despite experimental limitations. We anticipate that our conceptually rigorous approach can also be useful in addressing correlation effects in other quantum systems beyond optics.

## Acknowledgments

## References and links

1. | D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. |

2. | A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. |

3. | K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature |

4. | B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science |

5. | M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. |

6. | A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. |

7. | D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A |

8. | N. Na, S. Utsunomiya, L. Tian, and Y. Yamamoto, “Strongly correlated polaritons in a two-dimensional array of photonic crystal microcavities,” Phys. Rev. A |

9. | D. E. Chang, V. Gritsev, G. Morigi, V. Vuletić, M. D. Lukin, and E. A. Demler, “Crystallization of strongly interacting photons in a nonlinear optical fibre,” Nat. Phys. |

10. | A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. |

11. | L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A |

12. | A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-photon gateway in one-atom cavity quantum electrodynamics,” Phys. Rev. Lett. |

13. | I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. |

14. | J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its √ |

15. | L. S. Bishop, J. M. Chow, J. Koch, A. A. Houck, M. H. Devoret, E. Thuneberg, S. M. Girvin, and R. J. Schoelkopf, “Nonlinear response of the vacuum Rabi resonance,” Nat. Phys. |

16. | L. Horvath, B. C. Sanders, and B. F. Wielinga, “Multiphoton coincidence spectroscopy,” J. Opt. B: Quantum Semiclassic. Opt. |

17. | R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. |

18. | C. T. Lee, “Higher-order criteria for nonclassical effects in photon statistics,” Phys. Rev. A |

19. | D. N. Klyshko, “Observable signs of nonclassical light,” Phys. Lett. A , |

20. | H. J. Carmichael, R. J. Brecha, and P. R. Rice, “Quantum interference and collapse of the wavefunction in cavity QED,” Opt. Comm. |

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

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

23. | Y.-T Chough, H.-J. Moon, H. Nha, and K. An, “Single-atom laser based on multiphoton resonances at far-off resonance in the Jaynes-Cummings ladder,” Phys. Rev. A |

24. | Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, “Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations,” Phys. Rev. Lett. |

25. | H. J. Carmichael, P. Kochan, and B. C. Sanders, “Photon correlation spectroscopy,” Phys. Rev. Lett. |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(270.4180) Quantum optics : Multiphoton processes

(270.5290) Quantum optics : Photon statistics

(270.5580) Quantum optics : Quantum electrodynamics

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: February 3, 2010

Revised Manuscript: March 16, 2010

Manuscript Accepted: March 20, 2010

Published: March 23, 2010

**Citation**

Hyun-Gue Hong, Hyunchul Nha, Jai-Hyung Lee, and Kyungwon An, "Rigorous criterion for characterizing
correlated multiphoton emissions," Opt. Express **18**, 7092-7100 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-7-7092

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### References

- D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, "A single-photon transistor using nanoscale surface plasmons," Nat. Phys. 3, 807-812 (2007) [CrossRef]
- A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vuckovic, "Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade," Nat. Phys. 4, 859-863 (2008). [CrossRef]
- K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, "Photon blockade in an optical cavity with one trapped atom," Nature 436, 87-90 (2005). [CrossRef] [PubMed]
- B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, "A photon turnstile dynamically regulated by one atom," Science 319, 1062-1065 (2008). [CrossRef] [PubMed]
- M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, "Strongly interacting polaritons in coupled arrays of cavities," Nat. Phys. 2, 849-855 (2006). [CrossRef]
- A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, "Quantum phase transitions of light," Nat. Phys. 2, 856-861 (2006). [CrossRef]
- D. G. Angelakis, M. F. Santos, and S. Bose, "Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays," Phys. Rev. A 76, 031805(R) (2007). [CrossRef]
- N. Na, S. Utsunomiya, L. Tian, and Y. Yamamoto, "Strongly correlated polaritons in a two-dimensional array of photonic crystal microcavities," Phys. Rev. A 77, 031803(R) (2008). [CrossRef]
- D. E. Chang, V. Gritsev, G. Morigi, V. Vuletic, M. D. Lukin, and E. A. Demler, "Crystallization of strongly interacting photons in a nonlinear optical fibre," Nat. Phys. 4, 884-889 (2008). [CrossRef]
- A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, "Strongly interacting photons in a nonlinear cavity," Phys. Rev. Lett. 79, 1467-1470 (1997). [CrossRef]
- L. Tian and H. J. Carmichael, "Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom," Phys. Rev. A 46, R6801-R6804 (1992). [CrossRef] [PubMed]
- A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, "Two-photon gateway in one-atom cavity quantum electrodynamics," Phys. Rev. Lett. 101, 203602 (2008). [CrossRef] [PubMed]
- I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr and G. Rempe, "Nonlinear spectroscopy of photons bound to one atom," Nat. Phys. 4, 382-385 (2008). [CrossRef]
- J. M. Fink, M. G¨oppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, "Climbing the Jaynes-Cummings ladder and observing its √ n nonlinearity in a cavity QED system," Nature (London) 454, 315-318 (2008). [CrossRef]
- L. S. Bishop, J. M. Chow, J. Koch, A. A. Houck, M. H. Devoret, E. Thuneberg, S. M. Girvin, and R. J. Schoelkopf, "Nonlinear response of the vacuum Rabi resonance," Nat. Phys. 5, 105-109 (2009). [CrossRef]
- L. Horvath, B. C. Sanders, and B. F. Wielinga, "Multiphoton coincidence spectroscopy," J. Opt. B: Quantum Semiclassic. Opt. 1, 446-451 (1999). [CrossRef]
- R. J. Glauber, "The quantum theory of optical coherence," Phys. Rev. 130, 2529-2539 (1963). [CrossRef]
- C. T. Lee, "Higher-order criteria for nonclassical effects in photon statistics," Phys. Rev. A 41, 1721-1723 (1990). [CrossRef] [PubMed]
- D. N. Klyshko, "Observable signs of nonclassical light, " Phys. Lett. A 213, 7-15 (1996). [CrossRef]
- H. J. Carmichael, R. J. Brecha, and P. R. Rice, "Quantum interference and collapse of the wavefunction in cavity QED," Opt. Commun. 82, 73-79 (1991). [CrossRef]
- R. Hanbury-Brown and R. Twiss, "Correlation between photons in two coherent beams of light," Nature 177, 27-29 (1956). [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, 297-300 (2009). [CrossRef] [PubMed]
- Y.-T. Chough, H.-J. Moon, H. Nha, and K. An, "Single-atom laser based on multiphoton resonances at far-off resonance in the Jaynes-Cummings ladder," Phys. Rev. A 63, 013804 (2000). [CrossRef]
- Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, "Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations," Phys. Rev. Lett. 64, 2499-2502 (1990). [CrossRef] [PubMed]
- H. J. Carmichael, P. Kochan, and B. C. Sanders, "Photon correlation spectroscopy," Phys. Rev. Lett. 77, 631-634 (1996). [CrossRef] [PubMed]

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