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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 16 — Aug. 12, 2013
  • pp: 19209–19218
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Spatial second-order interference of pseudothermal light in a Hong-Ou-Mandel interferometer

Jianbin Liu, Yu Zhou, Wentao Wang, Rui-feng Liu, Kang He, Fu-li Li, and Zhuo Xu  »View Author Affiliations


Optics Express, Vol. 21, Issue 16, pp. 19209-19218 (2013)
http://dx.doi.org/10.1364/OE.21.019209


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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

Two-photon interference is extensively studied in quantum optics and quantum information, among which, “Hong-Ou-Mandel (HOM) dip” or “Shih-Alley dip” is one of the most famous two-photon interference phenomena [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

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

]. Most of the reported experiments with HOM interferometers are in the temporal regime with entangled-photon pairs [4

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] .

9

9. Z. Y. Jeff Ou, Multi-Photon Quantum Interference (Springer Science+Business Media, LLC, 2007).

] (and references therein) or classical light [10

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

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] .

]. Recently, Kim 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] .

]. In their experiments, they observed two-photon anticorrelation when two detectors are in the symmetrical positions. Based on the conclusions demonstrated by Saleh 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] .

], two-photon anticorrelation should also be observed with thermal light in a HOM interferometer. However, it is predicted by Olivares 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

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] .

]. Hence it is important to understand what is the difference between the spatial second-order interference with thermal light and entangled-photon pairs in a HOM interferometer. In this paper, we will report the spatial second-order interference pattern with two independent pseudothermal light beams in a HOM interferometer and discuss the difference between the results with thermal light and the ones with entangled-photon pairs. We also employed our results to interpret some reported experiments with HOM interferometers.

2. Experiments

Our experimental setup is shown in Fig. 1. Two independent pseudothermal light sources are created by dividing a laser beam into two and sending them to two independent rotating ground glasses (RG), respectively [18

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

]. The rotating frequencies of RG1 and RG2 are 34.29Hz and 32.21Hz, respectively. The temporal coherence lengths of these two pseudothermal light beams are 83.9 μs and 90.8 μs, 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 μm 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 L1 and L2 are 50 mm. The distance between Lj and RGj is 80 mm (j = 1, and 2). All the distances between the source planes and the detection planes are 910 mm.

Fig. 1 HOM interferometer with two independent pseudothermal light beams. Laser: Single-mode cw laser with central wavelength λ = 780 nm. BS: 50:50 non-polarized beam splitter. HP: Half wave plate. M: Mirror. L: Lens. RG: Rotating ground glass. D: Single-photon detector. In the inset, Sj: Source j (j = 1, and 2). S′2: Virtual source image of S2 respect to BS2. The lengthes of S1 and S2 are l1 and l2, respectively. The distance between the middle points of S1 and S′2 is d. The distances between the source planes and the detector planes all equal z.

The experimental results are shown in Fig. 2. The measurement interval for each dot is 200 seconds and these four different sets of data are measured with the same experimental parameters except the polarization of one beam is changed in the last measurement. The polarizations of these two beams are orthogonal in the first three measurements (Figs. 2(a), 2(b) and 2(c)) and then they are changed to be parallel for the fourth measurement (Fig. 2(d)) by rotating HP in the experimental setup. Figure 2(a) is the normalized second-order coherence function of the field emitted by S1 when S2 is blocked. Figure 2(b) is the normalized second-order coherence function of the field emitted by S2 when S1 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 S1 and S′2 is calculated to be 1.5337 mm. Even if considering the lengths of S1 and S2 may not be exactly the same in the experiments, the theoretical simulations agree with the experimental results very well.

Fig. 2 The spatial second-order coherence functions of two independent pseudothermal light beams in a HOM interferometer. (a) and (b) are the second-order coherence functions of the fields emitted by S1 and S2 when the other source is blocked, respectively. (c) and (d) are second-order spatial coherence functions when the light beams emitted by S1 and S2 have orthogonal and parallel polarizations, respectively. The dots with error bars are experimental results and the red lines are theoretical simulations employed the following equations. The coordinates of all four experiments are the same. It is well-known that the two-photon spatial bunching peak of thermal light is observed when the two detectors are in the symmetric positions ((a) and (b)). Hence the second-order interference dip in (d) is also observed when the two detectors are in the symmetric positions. Please see text for detail.

The reason why we first measure the spatial Hanbury Brown and Twiss (HBT) bunching peak of individual thermal source is to assure the symmetrical positions of these two detectors. It is well-known that the spatial two-photon bunching peak of thermal light is observed when the two detectors are in the symmetric positions [19

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] .

]. Once one of the detector’s position is fixed, the symmetric position of the other detector is uniquely determined, no matter where the thermal source be (Figs. 2(a) or 2(b)) or more than one independent thermal sources are employed (Figs. 2(c) and 2(d)). In our experiments, we scan D1 to observe the second-order coherence functions in all four different conditions when the position of D2 is fixed.

3. Theory

Both classical and quantum theories can be employed to interpret the experimental results [20

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

, 21

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

]. In order to comparing the results between thermal light and entangled-photon pairs, we will employ two-photon interference based on Feynman’s path integral theory to interpret our experimental results [3

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

, 22

22. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, Inc., 1965).

, 23

23. Y. H. Shih, An Introduction to Quantum Optics (CRC Press, 2011).

]. We assume the intensity of these two thermal sources are the same. There are eight different ways for two photons to trigger a two-photon coincidence count in our experimental setup. For instance, there are two different ways when both photons are emitted by S2. The corresponding Feynman’s two-photon propagators are Aa21,b22 and Aa22,b21, respectively, where Aa21,b22 is the Feynman’s two-photon propagator that photon a goes through S2 and detected by D1 and photon b goes through S2 and detected by D2. All the eight different propagators are Aa21,b22, Aa22,b21, Aa11,b12, Aa12,b11, Aa22,b11, Aa21,b12, Aa11,b22, and Aa12,b21. If these eight different ways are indistinguishable, the second-order coherence function in the experimental setup is [22

22. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, Inc., 1965).

]
G(2)(r1,t1;r2,t2)=|i,j,m,n=1,2mnAaim,bjn|2,
(1)
where (rβ, tβ) is the space-time coordinate of photon detection event by Dβ (β = 1, and 2) and 〈...〉 is ensemble average. Aaim,bjn = eaim,bjn A′aim,bjn, where the extra phase, φaim,bjn, is the sum of the initial phases and the phase changes of photons a and b due to BS2. A′aim,bjn is two-photon propagation function. Supposing the phase changes of photons transmitted and reflected by BS are 0 and π/2, respectively [24

24. R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford Univ. Press, 2001).

], the extra phases of these eight Feynman’s two-photon propagators are shown in Table 1. Where φaβ is the initial phase of photon a emitted by Sβ (β = 1, and 2), other symbols are defined similarly. The phase changes due to BS1 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

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

].

Table 1. Extra phases correspond to different pathes

table-icon
View This Table

Substituting the extra phases into Eq. (1) and with straightforward derivation, it is easy to get
G(2)(r1,t1;r2,t2)=|expi(φa2+φb2+π2)(Aa21,b22+Aa22,b21)+expi(φa1+φb1+π2)(Aa11,b12+Aa12,b11)+expi(φa2+φb1)(Aa22,b11Aa21,b12)+expi(φa1+φb2)(Aa11,b22Aa12,b21)|2,
(2)
where the minus signs of the last two terms are due to 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

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

]. In general, there are 64 terms after finishing the modulus square. However, when S1 and S2 are two independent thermal light sources, all the terms including the phase φαj, φ′αj, φ″αj, and 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
G(2)(r1,t1;r2,t2)=|Aa21,b22+Aa22,b21|2+|Aa11,b12+Aa12,b11|2+|Aa22,b11Aa21,b12|2+|Aa11,b22Aa12,b21|2.
(3)
The Feynman’s two-photon propagators is multiplication of two one-photon propagators when these two photons are independent [22

22. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, Inc., 1965).

]
Aaim,bjn=AaimAbjn,
(4)
where A′bjn is the Feynman’s one-photon propagator corresponding to photon b goes to Dn via Sj[25

25. M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Westview Press, Colorado, 1995).

, 26

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] .

],
Abjn=expi(kjnrjnωtn)rjn.
(5)
Where kjn is the wave vector of photon emitted by Sj goes to Dn and rjn is the position vector from Sj to Dn. ω is the central frequency of the laser and tn is the photon detection time of Dn. 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.

Substituting Eqs. (4) and (5) into Eq. (3) and considering both sources have finite lengthes, the second-order coherence function can be calculated. The first and second terms on the right sides of Eq. (3) are ordinary two-photon spatial bunching peaks of thermal light, which is calculated in many textbooks of quantum optics (for instance, see [23

23. Y. H. Shih, An Introduction to Quantum Optics (CRC Press, 2011).

, 24

24. R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford Univ. Press, 2001).

]). The last two terms of Eq. (3) can be calculated with the same method by taking the equation,
±d2l2±d2+l2exp[ik(x1x2)xz]dx=lexp[ik(x1x2)(±d)2z]sinc[k(x1x2)l2z],
(6)
into consideration. It is straightforward the get the one-dimension second-order coherence function in the far field
G(2)(x1x2)=l12[1+sinc2πl1λz(x1x2)]+l22[1+sinc2πl2λz(x1x2)]+2l1l2{1cos[2πdλz(x1x2)]×sincπl1λz(x1x2)sincπl2λz(x1x2)},
(7)
where paraxial approximation has been employed to simplify the results. x1 and x2 are the transverse spatial coordinates of D1 and D2, 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 S1 and S2, respectively. The third term corresponds to the two-photon interference when one photon comes from each source, respectively.

Equations (3) and (7) can be employed to interpret above experimental results. It is easy to see when the sizes of two pseudothermal sources are different, the second-order coherence functions will be different even if the relative position of the sources, i.e., d, is fixed. In our experiments, we consider both the lengths of S1 and S2 equal l for simplicity, which is also our experimental condition.

When the polarizations of these two light beams are orthogonal, the last two terms in Eq. (3) should be changed into |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

22. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, Inc., 1965).

]. The normalized second-order coherence function in this condition can be expressed as
g(2)(x1x2)=1+12sinc2[πlλz(x1x2)],
(8)
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 S1, S2, 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 S1 or S2). One photon comes from each source respectively does not contribute to the peak for these two photons are distinguishable.

When the polarizations of the two beams are parallel, the normalized second-order coherence function can be simplified as
g(2)(x1x2)=1+12sinc2πlλz(x1x2)12cos[2πdλz(x1x2)]sinc2πlλz(x1x2).
(9)
When these two detectors are in the symmetrical positions, i.e., x1 = x2, 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.

When the relative distance between these two sources, 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)(x1x2) 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)(x1x2) 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

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] .

].

Fig. 3 Second-order coherence function vs. d when the photons emitted by S1 and S2 have parallel polarizations. Simulated parameters are the same as the ones in Fig. 2 except the visibility is ideal. Please refer to Fig. 1 for the meaning of d.

4. Discussions

In the above, we have successfully interpreted the observed spatial second-order interference patterns in our experiments. Comparing the observed second-order interference pattern shown in Fig. 2(d) in our experiments with Fig. 2(a) in Kim 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] .

], it is easy to find the main difference between these two is there is a constant background in our experiments while there is no background in the entangled-photon pair experiment. Based on the calculations above, the background two-photon coincidence counts correspond to the first and second terms on the right hand side of Eq. (7), which are the coincidence counts when both photons are emitted by the same source (either S1 or S2). If there is a way to eliminate these background coincidence counts, the spatial second-order interference pattern with thermal light in a HOM interferometer would be the same as the one with entangled-photon pairs. In fact, the background coincidence counts in our experiments can be eliminated by employing two thermal sources with different central frequencies and detecting the two-photon coincidence counts with sum frequency generation [12

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] .

] or two-photon absorption [27

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] .

]. The two-photon detection system can be specially designed that it only responds to the coincidence count that two photons comes from different sources [28

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] .

]. Hence both quantum and classical light sources can be employed to observe two-photon interference pattern without background. The only difference is that when quantum light source is employed, the entangled-photon pairs make sure there are only two photons and only the last two terms of Eq. (3) contribute to the final observed two-photon interference pattern [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

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

]. While in classical light source case, the special detection systems make sure only one photon from each source can trigger a two-photon coincidence count, which also means only the last two terms of Eq. (3) contribute to the observed two-photon interference pattern [12

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

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] .

]. Hence two-photon anticorrelation with thermal light beams in a HOM interferometer is observable when a special two-photon detection system as suggested above is employed. If normal two-photon detection systems just as the one in our experiments and the one in Brida 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] .

] are employed, no correlation will be observed when these two detectors are in the symmetrical positions. It is noteworthy that when the light is strong and PIN diodes are employed to detect the intensities, the background can be removed by employing DC-blocks after these two detectors and before multiplying these two signals [29

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

].

With our experimental and theoretical results, the spatial interference experiments in a HOM interferometer with entangled-photon pairs or thermal light can be understood better. For instance, Kim 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] .

].” This statement is not accurate. The true two-photon coincidence count comes from the same entangled-photon pair in the entangled-photon pair experiments. Changing the angle of the mirror in their experiments is equivalent to changing the relative position of these two detectors. When the two detectors are in the symmetrical positions, temporal HOM dip is always observed due to the destructive interference. When the relative distance between these two detector increases, the observed g(2)(x1x2) 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)(x1x2) 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.

As a second example, let us employ our results to analyze the experiment with thermal light in a HOM interferometer by Brida et al.[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] .

]. Based on the conclusion that there is no correlation when two thermal light beams are incident to a HOM interferometer, the authors give an interesting scenario that an illusionist can employ a third beam to distinguish whether there is a beam splitter or not when two thermal light beams are mixed. Their method is necessary only when the following conditions: (I) these two thermal sources have equal sizes and are symmetrical about the beam splitter, (II) the distances between the thermal sources and the detection planes are equal, (III) the polarizations of these two beams are parallel, are satisfied. In this special case, there is no correlation for all the relative positions of these two detectors when there is a beam splitter as shown in Fig. 3(a). Hence a third beam is needed to verify whether there is a beam splitter or not as suggested by Brida et al.[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] .

]. However, in most cases, the illusionist does not need the third beam to distinguish whether there is a beam splitter or not. For as verified in our experiments, when there is a beam splitter and the two detectors are not in the symmetrical positions, the correlation is usually not zero. When there is no beam splitter, the correlation is always zero, no matter whether the two detectors are in the symmetrical positions or not. Hence the illusionist can distinguish these two situations by measuring the correlation functions when these two detectors are not in the symmetrical positions.

5. Conclusions

In conclusion, we have observed the spatial second-order interference pattern with two independent pseudothermal light beams in a HOM interferometer. Two-photon interference theory based on Feynman’s path integral theory is employed to interpret our experiments. It is suggested that the background in the interference pattern can be eliminated by employing sum frequency generation or two-photon absorption. It is also predicted that g(2)(x1x2) 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

The authors acknowledge the helpful comments from the anonymous reviewers. This project is supported by the National Basic Research Program of China (973 Program) under Grant No. 2009CB623306, International Science & Technology Cooperation Program of China under Grant No. 2010DFR50480, and the Fundamental Research Funds for the Central Universities.

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. 59, 2044–2046 (1987) [CrossRef] [PubMed] .

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. 61, 2921–2914 (1988) [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] .

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. 68, 2421–2414 (1992) [CrossRef] [PubMed] .

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. 91, 083601 (2003) [CrossRef] [PubMed] .

7.

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] .

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 77, 053822 (2008) [CrossRef] .

9.

Z. Y. Jeff Ou, Multi-Photon Quantum Interference (Springer Science+Business Media, LLC, 2007).

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] .

11.

R. Kaltenbaek, B. Blauensteiner, M. Zukowski, M. Aspelmeyer, and A. Zeilinger, “Experimental Interference of Independent Photons,” Phys. Rev. Lett. 96, 240502 (2006) [CrossRef] [PubMed] .

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] .

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] .

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] .

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] .

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] .

18.

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

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] .

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] .

22.

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, Inc., 1965).

23.

Y. H. Shih, An Introduction to Quantum Optics (CRC Press, 2011).

24.

R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford Univ. Press, 2001).

25.

M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Westview Press, Colorado, 1995).

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] .

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] .

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

  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]
  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.61, 2921–2914 (1988). [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. A45, 7729–7739 (1992). [CrossRef] [PubMed]
  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.68, 2421–2414 (1992). [CrossRef] [PubMed]
  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.91, 083601 (2003). [CrossRef] [PubMed]
  7. 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]
  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. A77, 053822 (2008). [CrossRef]
  9. Z. Y. Jeff Ou, Multi-Photon Quantum Interference (Springer Science+Business Media, LLC, 2007).
  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. B6, 100–103 (1989). [CrossRef]
  11. R. Kaltenbaek, B. Blauensteiner, M. Zukowski, M. Aspelmeyer, and A. Zeilinger, “Experimental Interference of Independent Photons,” Phys. Rev. Lett.96, 240502 (2006). [CrossRef] [PubMed]
  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]
  13. 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]
  14. 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]
  15. 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]
  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]
  18. W. Martienssen and E. Spiller, “Coherence and fluctuation in light beams,” Am. J. Phys.32, 919–926 (1964). [CrossRef]
  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]
  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]
  22. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, Inc., 1965).
  23. Y. H. Shih, An Introduction to Quantum Optics (CRC Press, 2011).
  24. R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford Univ. Press, 2001).
  25. M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Westview Press, Colorado, 1995).
  26. 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]
  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]
  29. We thank the anonymous reviewer for bringing this method into our attention.

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