## Spatial second-order interference of pseudothermal light in a Hong-Ou-Mandel interferometer |

Optics Express, Vol. 21, Issue 16, pp. 19209-19218 (2013)

http://dx.doi.org/10.1364/OE.21.019209

Acrobat PDF (1475 KB)

### Abstract

The spatial second-order interference of two independent pseudothermal light beams in a Hong-Ou-Mandel interferometer is studied experimentally and theoretically. The similar cosine modulation in the second-order coherence function as the one with entangled-photon pairs in a Hong-Ou-Mandel interferometer is observed. Two-photon interference based on Feynman’s path integral theory is employed to interpret the results. The experimental results and theoretical simulations agree with each other very well.

© 2013 osa

## 1. Introduction

1. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurent of subpicosecond time intervals betweens two photons by interference,” Phys. Rev. Lett. **59**, 2044–2046 (1987) [CrossRef] [PubMed] .

3. L. Mandel, “Quantum effects in one-photon and two-photon interference,” Rev. Mod. Phys. **71**, S274–282 (1999) [CrossRef] .

4. P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “Observation of a ’quantum eraser’: a revival of coherence in a two-photon interference experiment,” Phys. Rev. A **45**, 7729–7739 (1992) [CrossRef] [PubMed] .

10. Z. Y. Ou, E. C. G. Gage, B. E. Magill, and L. Mandel, “Fourth-order interference technique for determining the coherence time of a light beam,” J. Opt. Soc. Am. B **6**, 100–103 (1989) [CrossRef] .

13. H. Chen, T. Peng, S. Karmakar, Z. D. Xie, and Y. H. Shih, “Observation of anticorrelation in incoherent thermal light fields,” Phys. Rev. A **84**, 033835 (2011) [CrossRef] .

*et al.*observed cosine modulation in the spatial second-order coherence function in a HOM interferometer with entangled-photon pairs [14

14. H. Kim, O. Kwon, W. Kim, and T. Kim, “Spatial two-photon interference in a Hong-Ou-Mandel interferometer,” Phys. Rev. A **73**, 023820 (2006) [CrossRef] .

*et al.*that a number of spatial properties of entangled-photon pairs are analogous to those of ordinary photons generated by incoherent sources [15

15. B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A **62**, 043816 (2000) [CrossRef] .

*et al.*and experimentally verified by the same group that there is no correlation when two thermal light beams are incident to a HOM interferometer and these two detectors are in the symmetrical positions [16

16. S. Olivares and M. G. A. Paris, “Fidelity matters: the birth of entanglement in the mixing of gaussian states,” Phys. Rev. Lett. **107**, 170505 (2011) [CrossRef] [PubMed] .

17. G. Brida, I. P. Degiovanni, M. Genovese, A. Meda, S. Olivares, and M Paris, “The illusionist game and hidden correlations,” Phys. Scr. **T153**, 014006 (2013) [CrossRef] .

## 2. Experiments

18. W. Martienssen and E. Spiller, “Coherence and fluctuation in light beams,” Am. J. Phys. **32**, 919–926 (1964) [CrossRef] .

*μ*s and 90.8

*μ*s, respectively. The laser employed in our experiment is single-mode continuous wave laser with central wavelength 780 nm and frequency bandwidth 200 kHz (Newport, SWL-7513). The two-photon coincidence counts are measured by single-photon detectors (PerkinElmer, SPCM-AQRH-14-FC) and two-photon coincidence counting system (Becker & Hickl GmbH, SPC630), which are not shown in the figure. Two identical single-mode fibers with 5

*μ*m diameters are employed to couple the photons into the two single-photon detectors, respectively. The two-photon coincidence count time window is 61.6 ns. A half wave plate is used to control the polarization of one beam so that we can measure the second-order coherence functions when the polarizations of these two beams are orthogonal and parallel, respectively. Both focal lengths of the lens L

_{1}and L

_{2}are 50 mm. The distance between L

*j*and RG

*j*is 80 mm (

*j*= 1, and 2). All the distances between the source planes and the detection planes are 910 mm.

_{1}when S

_{2}is blocked. Figure 2(b) is the normalized second-order coherence function of the field emitted by S

_{2}when S

_{1}is blocked. Both the spatial coherence lengths at the detection planes of these two sources are calculated to be 1.15 mm. Thus both the sizes of these two thermal sources are 0.6172 mm. Figures 2(c) and 2(d) are measured when the polarizations of these two light beams are orthogonal and parallel, respectively. The red lines in Figs. 2(c) and 2(d) are theoretical simulations by employing following derived Eqs. (8) and (9), respectively. The measured value of

*g*

^{(2)}(0) in Fig. 2(c) is 1.39 ± 0.07, which is less than the theoretical maximum value 1.5. The measured value of

*g*

^{(2)}(0) in Fig. 2(d) is 1.00 ± 0.05. The distance between the middle points of S

_{1}and S′

_{2}is calculated to be 1.5337 mm. Even if considering the lengths of S

_{1}and S

_{2}may not be exactly the same in the experiments, the theoretical simulations agree with the experimental results very well.

19. R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of
light,” Nature(London) **177**, 27–29 (1956);
“A test of a new type of stellar interferometer on
sirius,” Nature(London) **178**, 1046–1048 (1956) [CrossRef] .

_{1}to observe the second-order coherence functions in all four different conditions when the position of D

_{2}is fixed.

## 3. Theory

20. R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. **131**, 2766–2788 (1963) [CrossRef] .

21. E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. **10**, 277–279 (1963) [CrossRef] .

3. L. Mandel, “Quantum effects in one-photon and two-photon interference,” Rev. Mod. Phys. **71**, S274–282 (1999) [CrossRef] .

_{2}. The corresponding Feynman’s two-photon propagators are

*A*

_{a21,b22}and

*A*

_{a22,b21}, respectively, where

*A*

_{a21,b22}is the Feynman’s two-photon propagator that photon

*a*goes through S

_{2}and detected by D

_{1}and photon

*b*goes through S

_{2}and detected by D

_{2}. All the eight different propagators are

*A*

_{a21,b22},

*A*

_{a22,b21},

*A*

_{a11,b12},

*A*

_{a12,b11},

*A*

_{a22,b11},

*A*

_{a21,b12},

*A*

_{a11,b22}, and

*A*

_{a12,b21}. If these eight different ways are indistinguishable, the second-order coherence function in the experimental setup is [22] where (

**r**

*,*

_{β}*t*) is the space-time coordinate of photon detection event by D

_{β}*(*

_{β}*β*= 1, and 2) and 〈...〉 is ensemble average.

*A*=

_{aim,bjn}*e*

^{iφaim,bjn}

*A′*, where the extra phase,

_{aim,bjn}*φ*, is the sum of the initial phases and the phase changes of photons

_{aim,bjn}*a*and

*b*due to BS

_{2}.

*A′*is two-photon propagation function. Supposing the phase changes of photons transmitted and reflected by BS are 0 and

_{aim,bjn}*π*/2, respectively [24], the extra phases of these eight Feynman’s two-photon propagators are shown in Table 1. Where

*φ*is the initial phase of photon

_{a}_{β}*a*emitted by S

*(*

_{β}*β*= 1, and 2), other symbols are defined similarly. The phase changes due to BS

_{1}do not affect the results, for the rotating ground glasses give random phases to the scattered photons [13

13. H. Chen, T. Peng, S. Karmakar, Z. D. Xie, and Y. H. Shih, “Observation of anticorrelation in incoherent thermal light fields,” Phys. Rev. A **84**, 033835 (2011) [CrossRef] .

18. W. Martienssen and E. Spiller, “Coherence and fluctuation in light beams,” Am. J. Phys. **32**, 919–926 (1964) [CrossRef] .

*π*phase difference between these two pathes, respectively [1

1. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurent of subpicosecond time intervals betweens two photons by interference,” Phys. Rev. Lett. **59**, 2044–2046 (1987) [CrossRef] [PubMed] .

3. L. Mandel, “Quantum effects in one-photon and two-photon interference,” Rev. Mod. Phys. **71**, S274–282 (1999) [CrossRef] .

_{1}and S

_{2}are two independent thermal light sources, all the terms including the phase

*φ*,

_{αj}*φ′*,

_{αj}*φ″*, and

_{αj}*φ′″*(

_{αj}*α*=

*a*, and

*b; j*= 1, and 2) will disappear, for the ensemble average of these terms equal zero, respectively. Eq. (2) can be simplified as

*A′*is the Feynman’s one-photon propagator corresponding to photon

_{bjn}*b*goes to D

*via S*

_{n}*[25, 26*

_{j}26. J. B. Liu and G. Q. Zhang, “Unified interpretation for second-order subwavelength interference based on Feyn-mans path-integral theory,” Phys. Rev. A **82**, 013822 (2010) [CrossRef] .

**k**is the wave vector of photon emitted by S

_{jn}*goes to D*

_{j}*and*

_{n}**r**is the position vector from S

_{jn}*to D*

_{j}*.*

_{n}*ω*is the central frequency of the laser and

*t*is the photon detection time of D

_{n}*. In our experiments, the bandwidth of the laser is 200 kHz and we only measure spatial correlation, thus we will drop the temporal part in the following calculations.*

_{n}*x*

_{1}and

*x*

_{2}are the transverse spatial coordinates of D

_{1}and D

_{2}, respectively. Other symbols are defined in Fig. 1. The first and second terms on the right of Eq.(7) correspond to typical HBT spatial bunching of the fields emitted by S

_{1}and S

_{2}, respectively. The third term corresponds to the two-photon interference when one photon comes from each source, respectively.

*i.e.*,

*d*, is fixed. In our experiments, we consider both the lengths of S

_{1}and S

_{2}equal

*l*for simplicity, which is also our experimental condition.

*A′*

_{a22,b11}|

^{2}+ |

*A′*

_{a21,b12}|

^{2}and |

*A′*

_{a11,b22}|

^{2}+ |

*A′*

_{a12,b21}|

^{2}, respectively. For these different ways to trigger a two-photon coincidence count are distinguishable. Probabilities instead of probability amplitudes should be added to get the final probability distribution [22]. The normalized second-order coherence function in this condition can be expressed as where the maximum ratio between the bunching peak and background is 1.5. The reason why it is less than 2 is because the coincidence counts in the background consist of two photons come from S

_{1}, S

_{2}, and one photon comes from each source, respectively. While the coincidence counts in the bunching peak only consist of both two photons come from one source (either S

_{1}or S

_{2}). One photon comes from each source respectively does not contribute to the peak for these two photons are distinguishable.

*i.e.*,

*x*

_{1}=

*x*

_{2},

*g*

^{(2)}(0) always equals 1 for all different relative positions of these two sources. This is due to the constructive interference when both photons are from the same source cancels the destructive interference when one photon comes from each source, which correspond to the second and third terms on the right hand side of Eq. (9), respectively.

*i.e.*,

*d*, does not equal zero, there is a cosine modulation in the spatial second-order coherence function. We also analyze how the cosine modulation changes with

*d*. Figure 3 shows the simulated results of

*g*

^{(2)}(

*x*

_{1}−

*x*

_{2}) for different values of

*d*. The parameters are the same as the ones in the experiments except the visibility is ideal in simulation. There is no correlation peak or dip when

*d*= 0, for

*g*

^{(2)}(

*x*

_{1}−

*x*

_{2}) always equals 1. As

*d*increases, the period of the modulation decreases. However,

*g*

^{(2)}(0) always equals 1 for different values of

*d*, which is consistent with the conclusions given by Olivares

*et. al.*[16

16. S. Olivares and M. G. A. Paris, “Fidelity matters: the birth of entanglement in the mixing of gaussian states,” Phys. Rev. Lett. **107**, 170505 (2011) [CrossRef] [PubMed] .

17. G. Brida, I. P. Degiovanni, M. Genovese, A. Meda, S. Olivares, and M Paris, “The illusionist game and hidden correlations,” Phys. Scr. **T153**, 014006 (2013) [CrossRef] .

## 4. Discussions

*et al.*’s paper with entangled photon pairs [14

14. H. Kim, O. Kwon, W. Kim, and T. Kim, “Spatial two-photon interference in a Hong-Ou-Mandel interferometer,” Phys. Rev. A **73**, 023820 (2006) [CrossRef] .

12. R. Kaltenbaek, J. Lavoie, D. N. Biggerstaff, and K. J. Resch, “Quantum-inspired interferometry with chirped laser pulses,” Nat. Phys. **4**, 864–868 (2008) [CrossRef] .

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

28. A. Nevet, A. Hayat, P. Ginzburg, and M. Orenstein, “Indistinguishable photon pairs from independent true chaotic sources,” Phys. Rev. Lett. **107**, 253601 (2011) [CrossRef] .

1. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurent of subpicosecond time intervals betweens two photons by interference,” Phys. Rev. Lett. **59**, 2044–2046 (1987) [CrossRef] [PubMed] .

**71**, S274–282 (1999) [CrossRef] .

12. R. Kaltenbaek, J. Lavoie, D. N. Biggerstaff, and K. J. Resch, “Quantum-inspired interferometry with chirped laser pulses,” Nat. Phys. **4**, 864–868 (2008) [CrossRef] .

28. A. Nevet, A. Hayat, P. Ginzburg, and M. Orenstein, “Indistinguishable photon pairs from independent true chaotic sources,” Phys. Rev. Lett. **107**, 253601 (2011) [CrossRef] .

*et al.*’s experiment [17

17. G. Brida, I. P. Degiovanni, M. Genovese, A. Meda, S. Olivares, and M Paris, “The illusionist game and hidden correlations,” Phys. Scr. **T153**, 014006 (2013) [CrossRef] .

*et al.*claimed by changing the angle of the mirror in their experiments, the observed temporal HOM dip can be changed to a peak. Hence they concluded that “both the fermionic and bosonic properties of the twin photons can be found [14

14. H. Kim, O. Kwon, W. Kim, and T. Kim, “Spatial two-photon interference in a Hong-Ou-Mandel interferometer,” Phys. Rev. A **73**, 023820 (2006) [CrossRef] .

*g*

^{(2)}(

*x*

_{1}−

*x*

_{2}) are modulated by a cosine function as shown in Fig. 2(d) in our experiments. If we fix the relative position of these two detectors when

*g*

^{(2)}(

*x*

_{1}−

*x*

_{2}) gets its maximums, just as Kim

*et al.*did in their experiments, temporal HOM dip can be changed into a peak. However, this is not fermionic properties of entangled-photon pairs. It is bosonic properties of entangled-photon pairs.

## 5. Conclusions

*g*

^{(2)}(

*x*

_{1}−

*x*

_{2}) is a constant when the two sources have equal size and are symmetrical about the 50:50 non-polarized BS and photons emitted by these two thermal sources have the same polarization.

## Acknowledgments

## References and links

1. | C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurent of subpicosecond time intervals betweens two photons by interference,” Phys. Rev. Lett. |

2. | Y. H. Shih and C. O. Alley, “New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. |

3. | L. Mandel, “Quantum effects in one-photon and two-photon interference,” Rev. Mod. Phys. |

4. | P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “Observation of a ’quantum eraser’: a revival of coherence in a two-photon interference experiment,” Phys. Rev. A |

5. | A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Dispersion cancellation in a measurement of the single-photon propagation velocity in glass,” Phys. Rev. Lett. |

6. | M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Demonstration of dispersion-canceled quantum-optical coherence tomography,” Phys. Rev. Lett. |

7. | S. P. Walborn, A. N. de Oliveira, S. Pádua, and C. H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. |

8. | O. Cosme, S. Pádua, F. A. Bovino, A. Mazzei, F. Sciarrino, and F. De Martini, “Hong-Ou-Mandel interferometer with one and two photon pairs,” Phys. Rev. A |

9. | Z. Y. Jeff Ou, |

10. | Z. Y. Ou, E. C. G. Gage, B. E. Magill, and L. Mandel, “Fourth-order interference technique for determining the coherence time of a light beam,” J. Opt. Soc. Am. B |

11. | R. Kaltenbaek, B. Blauensteiner, M. Zukowski, M. Aspelmeyer, and A. Zeilinger, “Experimental Interference of Independent Photons,” Phys. Rev. Lett. |

12. | R. Kaltenbaek, J. Lavoie, D. N. Biggerstaff, and K. J. Resch, “Quantum-inspired interferometry with chirped laser pulses,” Nat. Phys. |

13. | H. Chen, T. Peng, S. Karmakar, Z. D. Xie, and Y. H. Shih, “Observation of anticorrelation in incoherent thermal light fields,” Phys. Rev. A |

14. | H. Kim, O. Kwon, W. Kim, and T. Kim, “Spatial two-photon interference in a Hong-Ou-Mandel interferometer,” Phys. Rev. A |

15. | B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A |

16. | S. Olivares and M. G. A. Paris, “Fidelity matters: the birth of entanglement in the mixing of gaussian states,” Phys. Rev. Lett. |

17. | G. Brida, I. P. Degiovanni, M. Genovese, A. Meda, S. Olivares, and M Paris, “The illusionist game and hidden correlations,” Phys. Scr. |

18. | W. Martienssen and E. Spiller, “Coherence and fluctuation in light beams,” Am. J. Phys. |

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

20. | R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. |

21. | E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. |

22. | R. P. Feynman and A. R. Hibbs, |

23. | Y. H. Shih, |

24. | R. Loudon, |

25. | M. E. Peskin and D. V. Schroeder, |

26. | J. B. Liu and G. Q. Zhang, “Unified interpretation for second-order subwavelength interference based on Feyn-mans path-integral theory,” Phys. Rev. A |

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

28. | A. Nevet, A. Hayat, P. Ginzburg, and M. Orenstein, “Indistinguishable photon pairs from independent true chaotic sources,” Phys. Rev. Lett. |

29. | We thank the anonymous reviewer for bringing this method into our attention. |

**OCIS Codes**

(260.3160) Physical optics : Interference

(270.2500) Quantum optics : Fluctuations, relaxations, and noise

(270.5290) Quantum optics : Photon statistics

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: June 11, 2013

Revised Manuscript: July 5, 2013

Manuscript Accepted: July 10, 2013

Published: August 6, 2013

**Citation**

Jianbin Liu, Yu Zhou, Wentao Wang, Rui-feng Liu, Kang He, Fu-li Li, and Zhuo Xu, "Spatial second-order interference of pseudothermal light in a Hong-Ou-Mandel interferometer," Opt. Express **21**, 19209-19218 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-16-19209

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### References

- C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurent of subpicosecond time intervals betweens two photons by interference,” Phys. Rev. Lett.59, 2044–2046 (1987). [CrossRef] [PubMed]
- Y. H. Shih and C. O. Alley, “New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett.61, 2921–2914 (1988). [CrossRef] [PubMed]
- L. Mandel, “Quantum effects in one-photon and two-photon interference,” Rev. Mod. Phys.71, S274–282 (1999). [CrossRef]
- P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “Observation of a ’quantum eraser’: a revival of coherence in a two-photon interference experiment,” Phys. Rev. A45, 7729–7739 (1992). [CrossRef] [PubMed]
- A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Dispersion cancellation in a measurement of the single-photon propagation velocity in glass,” Phys. Rev. Lett.68, 2421–2414 (1992). [CrossRef] [PubMed]
- M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Demonstration of dispersion-canceled quantum-optical coherence tomography,” Phys. Rev. Lett.91, 083601 (2003). [CrossRef] [PubMed]
- S. P. Walborn, A. N. de Oliveira, S. Pádua, and C. H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett.90, 143601 (2003). [CrossRef] [PubMed]
- O. Cosme, S. Pádua, F. A. Bovino, A. Mazzei, F. Sciarrino, and F. De Martini, “Hong-Ou-Mandel interferometer with one and two photon pairs,” Phys. Rev. A77, 053822 (2008). [CrossRef]
- Z. Y. Jeff Ou, Multi-Photon Quantum Interference (Springer Science+Business Media, LLC, 2007).
- Z. Y. Ou, E. C. G. Gage, B. E. Magill, and L. Mandel, “Fourth-order interference technique for determining the coherence time of a light beam,” J. Opt. Soc. Am. B6, 100–103 (1989). [CrossRef]
- R. Kaltenbaek, B. Blauensteiner, M. Zukowski, M. Aspelmeyer, and A. Zeilinger, “Experimental Interference of Independent Photons,” Phys. Rev. Lett.96, 240502 (2006). [CrossRef] [PubMed]
- R. Kaltenbaek, J. Lavoie, D. N. Biggerstaff, and K. J. Resch, “Quantum-inspired interferometry with chirped laser pulses,” Nat. Phys.4, 864–868 (2008). [CrossRef]
- H. Chen, T. Peng, S. Karmakar, Z. D. Xie, and Y. H. Shih, “Observation of anticorrelation in incoherent thermal light fields,” Phys. Rev. A84, 033835 (2011). [CrossRef]
- H. Kim, O. Kwon, W. Kim, and T. Kim, “Spatial two-photon interference in a Hong-Ou-Mandel interferometer,” Phys. Rev. A73, 023820 (2006). [CrossRef]
- B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A62, 043816 (2000). [CrossRef]
- S. Olivares and M. G. A. Paris, “Fidelity matters: the birth of entanglement in the mixing of gaussian states,” Phys. Rev. Lett.107, 170505 (2011). [CrossRef] [PubMed]
- G. Brida, I. P. Degiovanni, M. Genovese, A. Meda, S. Olivares, and M Paris, “The illusionist game and hidden correlations,” Phys. Scr.T153, 014006 (2013). [CrossRef]
- W. Martienssen and E. Spiller, “Coherence and fluctuation in light beams,” Am. J. Phys.32, 919–926 (1964). [CrossRef]
- R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature(London)177, 27–29 (1956); “A test of a new type of stellar interferometer on sirius,” Nature(London)178, 1046–1048 (1956). [CrossRef]
- R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev.131, 2766–2788 (1963). [CrossRef]
- E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett.10, 277–279 (1963). [CrossRef]
- R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, Inc., 1965).
- Y. H. Shih, An Introduction to Quantum Optics (CRC Press, 2011).
- R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford Univ. Press, 2001).
- M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Westview Press, Colorado, 1995).
- J. B. Liu and G. Q. Zhang, “Unified interpretation for second-order subwavelength interference based on Feyn-mans path-integral theory,” Phys. Rev. A82, 013822 (2010). [CrossRef]
- F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconuctors,” Nat. Phys.5, 267–270 (2009). [CrossRef]
- A. Nevet, A. Hayat, P. Ginzburg, and M. Orenstein, “Indistinguishable photon pairs from independent true chaotic sources,” Phys. Rev. Lett.107, 253601 (2011). [CrossRef]
- We thank the anonymous reviewer for bringing this method into our attention.

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