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Changing correlation into anticorrelation by superposing thermal and laser light Jianbin Liu,1,* Yu Zhou,2 Fu-Li Li,2 and Zhuo Xu1 1
Electronic Materials Research Laboratory, Key Laboratory of the Ministry of Education & International Center for Dielectric Research, Xi’an Jiaotong University, Xi’an 710049, China 2 MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter, Department of Applied Physics, Xi’an Jiaotong University, Xi’an 710049, China *Corresponding author: [email protected] Received March 18, 2014; revised April 26, 2014; accepted May 11, 2014; posted May 15, 2014 (Doc. ID 208479); published June 13, 2014 Correlation can be changed into anticorrelation by superposing thermal and laser light with the same frequency and polarization. Two-photon interference theory is employed to interpret this phenomenon. An experimental scheme is designed to verify the theoretical predictions by employing pseudothermal light to simulate thermal light. The experimental results are consistent with the theoretical results. © 2014 Optical Society of America OCIS codes: (030.5290) Photon statistics; (260.3160) Interference. http://dx.doi.org/10.1364/JOSAA.31.001481
1. INTRODUCTION
2. THEORY
Two-photon correlation and anticorrelation are defined by the relationship between two-photon coincidence count probability and accidental two-photon coincidence count probability. When the two-photon coincidence count probability is greater than the accidental two-photon coincidence count probability, these photon detection events are correlated. These events are anticorrelated when the contrary condition is true [1,2]. When two detectors are at the symmetrical positions, photon anticorrelation was first observed in a Hong–Ou–Mandel (HOM) interferometer with entangled photon pairs generated by spontaneous parametric downconversion [3,4]. Later, similar phenomena were observed with other nonclassical light [5–7] and nonclassical light mixed with classical light [8]. Kaltenbaek et al. observed two-photon anticorrelation with classical light in an HOM interferometer [9,10]. However, nonlinear sum-frequency generation was employed as a twophoton detection system in their experiments. One may wonder if it is possible to observe photon anticorrelation with classical light and a normal two-photon detection system. The answer is yes. In fact, it was pointed out by Belinsky and Klyshko [11] that “any interference effects of first and second (in intensity) have close classical analogs with the same monoharmonic interference pattern and differing only in the visibility,” which had been confirmed in the spatial second-order interference of two independent light beams [12] and ghost imaging [13]. In one of our recent studies, we have observed the spatial second-order interference pattern between thermal and laser light in an HOM interferometer and anticorrelation [14]. In this paper, we will study the temporal second-order interference between thermal and laser light in an HOM interferometer and discuss how correlation is changed into anticorrelation by mixing thermal light with laser light. These results are helpful to understand the physics of the second-order interference of light.
The normalized second-order coherence function or the degree of second-order coherence [15],
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g2 r1 ; t1 ; r2 ; t2
G2 r1 ; t1 ; r2 ; t2 ; G r1 ; t1 G1 r2 ; t2 1
(1)
will be employed to discuss the second-order correlation of light in the following part, where G2 r1 ; t1 ; r2 ; t2 is the second-order coherence function at space-time coordinates r1 ; t1 and r2 ; t2 . G1 r1 ; t1 and G1 r2 ; t2 are the first-order coherence functions at r1 ; t1 and r2 ; t2 , respectively [16,17]. Based on the definitions of correlation and anticorrelation above, it is easy to conclude that these two events are correlated when g2 r1 ; t1 ; r2 ; t2 is greater than 1. When g2 r1 ; t1 ; r2 ; t2 is equal to 1, these two events are independent. When g2 r1 ; t1 ; r2 ; t2 is less than 1, these two events are anticorrelated. When a thermal light beam is incident to a beam splitter, it is well-known that the photon detection events are correlated on condition that these two detectors are at the symmetrical positions [18,19]. It is equivalent to thermal light in a Hanbury Brown and Twiss (HBT) interferometer as shown in Fig. 1(a). There are two different ways to superpose thermal and laser light at a beam splitter as shown in Figs. 1(b) and 1(c), respectively. We assume the frequency and polarization of thermal and laser light are the same. Photons emitted by these two light beams are indistinguishable. In Fig. 1(b), laser light is incident to the beam splitter at the same port as thermal light, which is equivalent to an HBT interferometer with two input beams [18,19]. In Fig. 1(c), laser light is incident to the beam splitter at the adjacent port of thermal light, which is equivalent to an HOM interferometer [3]. In the following part, we will calculate the temporal second-order coherence functions for these two different situations. © 2014 Optical Society of America
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L T
D1 BS
D2
T
D1 T
L
D1
BS
BS
D2 (a)
where Δν is the frequency bandwidth of thermal light. Photon anticorrelation cannot be observed in this situation, for g2 12 t1 ; t2 in Eq. (4) cannot be less than 1. With the same method, the second-order coherence function in Fig. 1(c) can be expressed as
D2 (b)
π
G2 r1 ; t1 ; r2 ; t2 P 2t hjeiφta φtb 2 Ata1;tb2 Ata2;tb1 j2 i
(c)
π
P 2l hjeiφla φlb 2 Ala1;lb2 Ala2;lb1 j2 i
Fig. 1. Thermal and laser light beams mixed at a beam splitter. T, thermal light; L, laser light; BS, 50∶50 nonpolarized beam splitter; D, single-photon detector.
Both classical and quantum theories can be employed to calculate optical phenomena involving classical light in a linear optical system [16,17,20]. We will employ two-photon interference theory in our calculation, for quantum theory is valid for both classical and nonclassical light [21]. There are three different ways for two photons to trigger a two-photon coincidence count in Fig. 1(b). The first is that both photons come from thermal light. The second is that both photons come from laser light. The last is that a photon comes from thermal light, and the other photon comes from laser light. Based on the superposition principle [22], the secondorder coherence function in Fig. 1(b) can be expressed as [23]
2P t P l hjeiφta φlb Ata1;lb2 − Ata2;lb1 j2 i;
(5)
where the minus sign in the last term of Eq. (5) is due to a π phase difference between Ata1;lb2 and Ata2;lb1 [3,12]. The temporal second-order coherence function in Fig. 1(c) can be simplified as [13,14,24] 2 2 2 g2 12 t1 ; t2 P t 1 sinc πΔνt1 − t2 P l × 1
2P t P l 1 − sinc2 πΔνt1 − t2 :
(6)
When P 2t − 2P t P l is less than 0, g2 12 t1 ; t2 in Eq. (6) is less than 1. Hence photon anticorrelation can be observed in the situation shown in Fig. 1(c). Substituting P l x and P t 1 − x (x ∈ 0; 1) into Eq. (6), it is easy to calculate
π
G2 r1 ; t1 ; r2 ; t2 P 2t hjeiφta φtb 2 Ata1;tb2 Ata2;tb1 j2 i
2 2 g2 12 t1 ; t2 1 3x − 4x 1sinc πΔνt1 − t2 ;
π
P 2l hjeiφla φlb 2 Ala1;lb2 Ala2;lb1 j2 i π
2P t P l hjeiφta φlb 2 Ata1;lb2 Ata2;lb1 j2 i; (2) where P t and P l (P t ≥ 0, P l ≥ 0, and P t P l 1) are the probabilities of the detected photon coming from thermal and laser light, respectively. h…i means ensemble average. φtα and φlα are the initial phases of photon α (α a, and b) coming from thermal and laser light, respectively. Ata1;lb2 is the two-photon probability amplitude that photon a coming from thermal light goes to detector 1 and photon b coming from laser light goes to detector 2, which is equal to the product of these two singlephoton probability amplitudes [22]. Other symbols in Eq. (2) are defined similarly. π∕2 is the phase difference of one photon reflected by a beam splitter compared with the transmitted one [15]. In order to simplify the calculation of the temporal second-order coherence function, we only consider the case when these two detectors are at the symmetrical positions, i.e., r1 r2 . In this condition, the first-order coherence function at the jth detector (j 1, and 2) is G1 rj ; tj hjP t eiφt Atj P l eiφl Alj j2 i;
where x is the ratio between the intensities of laser and total light beams. As x increases from 0 to 1, correlation can be changed into anticorrelation.
3. EXPERIMENT We have designed an experimental scheme to verify our theoretical predictions, which is shown in Fig. 2. A single-mode continuous wave laser with a central wavelength at 780 nm and frequency bandwidth of 200 kHz is divided into two equal portions by a nonpolarized beam splitter (BS1 ). One beam is incident to a rotating ground glass (RG) after passing through a convex lens (L1 ) to simulate thermal light [25]. The other beam is expanded by another identical lens (L2 ) to ensure that the intensity of the laser beam is approximately constant across the measurement range. The focus lengths of L1 and L2 are both 50 mm. The distance between L1 and RG is 83 mm. The distances between the lens and detector planes
M2
(3)
in which φt and φl are the initial phases of photons emitted by thermal and laser sources, respectively. Due to the relative phase between photons emitted by these two sources is randomly changing with time, the first-order coherence function in Eq. (3) is a constant. Substituting Eqs. (2) and (3) into Eq. (1), the normalized temporal second-order coherence function can be simplified as [13,14,24]
(7)
Laser
L2
BS 1
D1 M1 L 1 RG
BS 2 D2
2 2 2 g2 12 t1 ; t2 P t 1 sinc πΔνt1 − t2 P l × 1
2P t P l 1 sinc2 πΔνt1 − t2 1 P 2t 2P t P l sinc2 πΔνt1 − t2 ;
(4)
Fig. 2. Experimental setup for the second-order interference of pseudothermal and laser light beams in an HOM interferometer. Laser, single-mode continuous wave laser; BS, 50:50 nonpolarized beam splitter; M, mirror; L, lens; RG, rotating ground glass; D, single-photon detector.
Liu et al.
all equal 825 mm. The temporal second-order coherence length of pseudothermal light is measured to be 78.5 ns. The spatial second-order coherence length of pseudothermal light is calculated to be 55.1 μm by noting the size of the laser light beam on the ground glass is 10.5 mm, and the distance between the source and the detection planes is 742 mm. Two single-mode fibers are coupled to D1 and D2 to collect photons, respectively. The diameter of the collecting single-mode fiber is 5 μm, which is much less than the spatial second-order coherence length of pseudothermal light in the experiments. During the experiments, the single-photon counting rates of D1 and D2 are kept at the level of about 20,000 counts per second. Figure 3 shows the normalized temporal second-order coherence functions for different ratios between the intensities of laser and total light beams when these two detectors are at the symmetrical positions. g2 12 t1 − t2 is the normalized second-order coherence function when these two singlephoton detection events are at r1 ; t1 and r2 ; t2 , respectively. t1 − t2 is the time difference between these two photon detection events within a two-photon coincidence count. The black squares are the experimental results without subtracting any background. The red lines are the theoretical fitting curves by employing Eq. (7). There are six groups of experimental results in Fig. 3, and the measurement time for each group is 120 s. The same temporal second-order coherence length, 78.5 ns, is employed in all the six fitting processes. The only difference among these fittings is the visibility.
4. DISCUSSION The reasons why the correlation is changed into anticorrelation are as follows. The contribution of two-photon coincidence counts in Fig. 1(c) consists of three parts. The first term on the right-hand side of Eq. (6) corresponds to correlation with a background. The second term only contributes to a background. The third term corresponds to anticorrelation with a background. When the ratio between the
Fig. 3. Temporal second-order interference of pseudothermal and laser light beams in an HOM interferometer when these two detectors are at symmetrical positions. g2 12 t1 − t2 is the normalized temporal second-order coherence function when these two single-photon detection events are at r1 ; t1 and r2 ; t2 , respectively. t1 − t2 is the time difference between these two photon detection events within a twophoton coincidence count. x is the ratio between the intensities of laser and total light beams. (See text for details.)
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intensities of laser and total light beams is small, the first term will dominate. Hence correlation is observed when x is in the regime of [0,1/3). When x equals 1/3, the contribution of correlation from the first term and anticorrelation from the third term are equal. These events are independent in this case. When x is in the regime of (1/3,1), the contribution from the third term is greater than the one from the first term. Anticorrelation will be observed in this condition. These photon detections are independent when x equals 1, for it is a laser light beam in an HBT interferometer [16,17]. 2 In order to study how g12 0 changes with the ratio between the intensities of laser and total light beams, Eq. (7) can be simplified as 2 g12 0 3x2 − 4x 2
(8)
by setting t1 t2 . g2 12 0s for different ratios between the intensities of laser and total light beams are shown in Fig. 4. g2 12 0 is the normalized second-order coherence function when these two photon detections are at the symmetrical positions with zero time difference. The red curve is the theoretical fitting of the experimental results by employing Eq. (8), where the ratios among the three coefficients are fixed to be 3∶ − 4∶2 in the fitting process. The reasons why the theoretical and experimental results are different in Fig. 4 are that x is not a constant during one measurement and g2 12 0 is less than 2 for pseudothermal light in an HBT interferometer [14]. However, it is obvious that g2 12 0 can be less than 1 in our experiments, which means anticorrelation is observed when these two photon detections are at the same space-time coordinates. It is worth noting that we also measured the second-order auto-correlation function when laser and thermal light are input to a beam splitter as well [26]. It is found that the secondorder autocorrelation function of classical light is always greater or equal to unity, while the cross-correlation function with classical light can be less than unity [27]. There is no rule that prevents the second-order cross-correlation function of classical light be less than unity. Hence the observed
Fig. 4. Normalized second-order coherence function versus the ratio between the intensities of laser and total light beams. g2 12 0 is the normalized second-order coherence function when these two photon detections are at the symmetrical positions with zero time difference. x is the ratio between the intensities of laser and total light beams. As x increases, g2 12 0 will decrease to a minimum and then increase. (See text for details).
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anticorrelation in this experiment does not mean we have observed a nonclassical phenomenon with classical light.
5. CONCLUSION In conclusion, we have observed anticorrelation with a normal two-photon detection system and classical light in an HOM interferometer. By changing the ratio between the intensities of laser and total light beams, correlation can be changed into anticorrelation. Two-photon interference theory is employed to interpret our experiments, which is helpful to understand the physics of the second-order interference of light.
ACKNOWLEDGMENTS This project is supported by the Doctoral Fund of the Ministry of Education of China (no. 20130201120013), the 111 project of China (no. B14040), and the Fundamental Research Funds for the Central Universities.
REFERENCES P. Graingier, 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). 2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995). 3. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987). 4. Y. H. Shih and C. O. Alley, “New type of Einstein-PodolskyRosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. 61, 2921–2924 (1988). 5. J. Beugnon, M. P. A. Jones, J. Dingjan, B. Darquié, G. Messin, A. Browaeys, and P. Grangier, “Quantum interference between two single photons emitted by independently trapped atoms,” Nature 440, 779–782 (2006). 6. R. Kaltenbaek, B. Blauensteiner, M. Żukowski, M. Aspelmeyer, and A. Zeilinger, “Experimental interference of independent photons,” Phys. Rev. Lett. 96, 240502 (2006). 7. P. Maunz, D. L. Moehring, S. Olmschenk, K. C. Younge, D. N. Matsukevich, and C. Manroe, “Quantum interference of photon pairs from two remote trapped atomic ions,” Nat. Phys. 3, 538–541 (2007). 1.
Liu et al. 8. A. J. Bennett, R. B. Patel, C. A. Nicoll, D. A. Ritchie, and A. J. Shields, “Interference of dissimilar photon sources,” Nat. Phys. 5, 715–717 (2009). 9. R. Kaltenbaek, J. Lavoie, D. N. Biggerstaff, and K. J. Resch, “Quantum-inspired interferometry with chirped laser pulses,” Nat. Phys. 4, 864–868 (2008). 10. R. Kaltenbaek, J. Lavoie, and K. J. Resch, “Classical analogous of two-photon quantum interference,” Phys. Rev. Lett. 102, 243601 (2009). 11. A. V. Belinsky and D. N. Klyshko, “Interference of classical and non-classical light,” Phys. Lett. A 166, 303–307 (1992). 12. L. Mandel, “Quantum effects in one-photon and two-photon interference,” Rev. Mod. Phys. 71, S274–S282 (1999). 13. Y. H. Shih, An Introduction to Quantum Optics: Photons and Biphoton Physics (CRC Press, 2011). 14. J. B. Liu, Y. Zhou, F. L. Li, and Z. Xu, “The second-order interference between laser and thermal light,” Europhys. Lett. 105, 64007 (2014). 15. R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford University, 2001). 16. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963). 17. R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963). 18. R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956). 19. R. Hanbury Brown and R. Q. Twiss, “A test of a new type of stellar interferometer on sirius,” Nature 178, 1046–1048 (1956). 20. E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279 (1963). 21. J. H. Shapiro and R. W. Boyd, “The physics of ghost imaging,” Quantum Inf. Process 11, 949–993 (2012). 22. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965). 23. J. B. Liu, Y. Zhou, W. T. Wang, R. F. Liu, K. He, F. L. Li, and Z. Xu, “Spatial second-order interference of pseudothermal light in a Hong-Ou-Mandel interferometer,” Opt. Express 21, 19209– 19218 (2013). 24. Y. Zhou, J. Simon, J. B. Liu, and Y. H. Shih, “Third-order correlation function and ghost imaging of chaotic thermal light in the photon counting regime,” Phys. Rev. A 81, 043831 (2010). 25. W. Martienssen and E. Spiller, “Coherence and fluctuation in light beams,” Am. J. Phys. 32, 919–926 (1964). 26. J. B. Liu, Y. Zhou, W. T. Wang, F. L. Li, and Z. Xu, “Experimental study of the second-order coherence of partially polarized thermal light,” Opt. Commun. 317, 18–23 (2014). 27. M. M-Tehrani and L. Mandel, “Intensity fluctuations in a twomode ring laser,” Phys. Rev. A 17, 694–700 (1978).
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Changing correlation into anticorrelation by superposing thermal and laser light Jianbin Liu,1,* Yu Zhou,2 Fu-Li Li,2 and Zhuo Xu1 1
Electronic Materials Research Laboratory, Key Laboratory of the Ministry of Education & International Center for Dielectric Research, Xi’an Jiaotong University, Xi’an 710049, China 2 MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter, Department of Applied Physics, Xi’an Jiaotong University, Xi’an 710049, China *Corresponding author: [email protected] Received March 18, 2014; revised April 26, 2014; accepted May 11, 2014; posted May 15, 2014 (Doc. ID 208479); published June 13, 2014 Correlation can be changed into anticorrelation by superposing thermal and laser light with the same frequency and polarization. Two-photon interference theory is employed to interpret this phenomenon. An experimental scheme is designed to verify the theoretical predictions by employing pseudothermal light to simulate thermal light. The experimental results are consistent with the theoretical results. © 2014 Optical Society of America OCIS codes: (030.5290) Photon statistics; (260.3160) Interference. http://dx.doi.org/10.1364/JOSAA.31.001481
1. INTRODUCTION
2. THEORY
Two-photon correlation and anticorrelation are defined by the relationship between two-photon coincidence count probability and accidental two-photon coincidence count probability. When the two-photon coincidence count probability is greater than the accidental two-photon coincidence count probability, these photon detection events are correlated. These events are anticorrelated when the contrary condition is true [1,2]. When two detectors are at the symmetrical positions, photon anticorrelation was first observed in a Hong–Ou–Mandel (HOM) interferometer with entangled photon pairs generated by spontaneous parametric downconversion [3,4]. Later, similar phenomena were observed with other nonclassical light [5–7] and nonclassical light mixed with classical light [8]. Kaltenbaek et al. observed two-photon anticorrelation with classical light in an HOM interferometer [9,10]. However, nonlinear sum-frequency generation was employed as a twophoton detection system in their experiments. One may wonder if it is possible to observe photon anticorrelation with classical light and a normal two-photon detection system. The answer is yes. In fact, it was pointed out by Belinsky and Klyshko [11] that “any interference effects of first and second (in intensity) have close classical analogs with the same monoharmonic interference pattern and differing only in the visibility,” which had been confirmed in the spatial second-order interference of two independent light beams [12] and ghost imaging [13]. In one of our recent studies, we have observed the spatial second-order interference pattern between thermal and laser light in an HOM interferometer and anticorrelation [14]. In this paper, we will study the temporal second-order interference between thermal and laser light in an HOM interferometer and discuss how correlation is changed into anticorrelation by mixing thermal light with laser light. These results are helpful to understand the physics of the second-order interference of light.
The normalized second-order coherence function or the degree of second-order coherence [15],
1084-7529/14/071481-04$15.00/0
g2 r1 ; t1 ; r2 ; t2
G2 r1 ; t1 ; r2 ; t2 ; G r1 ; t1 G1 r2 ; t2 1
(1)
will be employed to discuss the second-order correlation of light in the following part, where G2 r1 ; t1 ; r2 ; t2 is the second-order coherence function at space-time coordinates r1 ; t1 and r2 ; t2 . G1 r1 ; t1 and G1 r2 ; t2 are the first-order coherence functions at r1 ; t1 and r2 ; t2 , respectively [16,17]. Based on the definitions of correlation and anticorrelation above, it is easy to conclude that these two events are correlated when g2 r1 ; t1 ; r2 ; t2 is greater than 1. When g2 r1 ; t1 ; r2 ; t2 is equal to 1, these two events are independent. When g2 r1 ; t1 ; r2 ; t2 is less than 1, these two events are anticorrelated. When a thermal light beam is incident to a beam splitter, it is well-known that the photon detection events are correlated on condition that these two detectors are at the symmetrical positions [18,19]. It is equivalent to thermal light in a Hanbury Brown and Twiss (HBT) interferometer as shown in Fig. 1(a). There are two different ways to superpose thermal and laser light at a beam splitter as shown in Figs. 1(b) and 1(c), respectively. We assume the frequency and polarization of thermal and laser light are the same. Photons emitted by these two light beams are indistinguishable. In Fig. 1(b), laser light is incident to the beam splitter at the same port as thermal light, which is equivalent to an HBT interferometer with two input beams [18,19]. In Fig. 1(c), laser light is incident to the beam splitter at the adjacent port of thermal light, which is equivalent to an HOM interferometer [3]. In the following part, we will calculate the temporal second-order coherence functions for these two different situations. © 2014 Optical Society of America
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L T
D1 BS
D2
T
D1 T
L
D1
BS
BS
D2 (a)
where Δν is the frequency bandwidth of thermal light. Photon anticorrelation cannot be observed in this situation, for g2 12 t1 ; t2 in Eq. (4) cannot be less than 1. With the same method, the second-order coherence function in Fig. 1(c) can be expressed as
D2 (b)
π
G2 r1 ; t1 ; r2 ; t2 P 2t hjeiφta φtb 2 Ata1;tb2 Ata2;tb1 j2 i
(c)
π
P 2l hjeiφla φlb 2 Ala1;lb2 Ala2;lb1 j2 i
Fig. 1. Thermal and laser light beams mixed at a beam splitter. T, thermal light; L, laser light; BS, 50∶50 nonpolarized beam splitter; D, single-photon detector.
Both classical and quantum theories can be employed to calculate optical phenomena involving classical light in a linear optical system [16,17,20]. We will employ two-photon interference theory in our calculation, for quantum theory is valid for both classical and nonclassical light [21]. There are three different ways for two photons to trigger a two-photon coincidence count in Fig. 1(b). The first is that both photons come from thermal light. The second is that both photons come from laser light. The last is that a photon comes from thermal light, and the other photon comes from laser light. Based on the superposition principle [22], the secondorder coherence function in Fig. 1(b) can be expressed as [23]
2P t P l hjeiφta φlb Ata1;lb2 − Ata2;lb1 j2 i;
(5)
where the minus sign in the last term of Eq. (5) is due to a π phase difference between Ata1;lb2 and Ata2;lb1 [3,12]. The temporal second-order coherence function in Fig. 1(c) can be simplified as [13,14,24] 2 2 2 g2 12 t1 ; t2 P t 1 sinc πΔνt1 − t2 P l × 1
2P t P l 1 − sinc2 πΔνt1 − t2 :
(6)
When P 2t − 2P t P l is less than 0, g2 12 t1 ; t2 in Eq. (6) is less than 1. Hence photon anticorrelation can be observed in the situation shown in Fig. 1(c). Substituting P l x and P t 1 − x (x ∈ 0; 1) into Eq. (6), it is easy to calculate
π
G2 r1 ; t1 ; r2 ; t2 P 2t hjeiφta φtb 2 Ata1;tb2 Ata2;tb1 j2 i
2 2 g2 12 t1 ; t2 1 3x − 4x 1sinc πΔνt1 − t2 ;
π
P 2l hjeiφla φlb 2 Ala1;lb2 Ala2;lb1 j2 i π
2P t P l hjeiφta φlb 2 Ata1;lb2 Ata2;lb1 j2 i; (2) where P t and P l (P t ≥ 0, P l ≥ 0, and P t P l 1) are the probabilities of the detected photon coming from thermal and laser light, respectively. h…i means ensemble average. φtα and φlα are the initial phases of photon α (α a, and b) coming from thermal and laser light, respectively. Ata1;lb2 is the two-photon probability amplitude that photon a coming from thermal light goes to detector 1 and photon b coming from laser light goes to detector 2, which is equal to the product of these two singlephoton probability amplitudes [22]. Other symbols in Eq. (2) are defined similarly. π∕2 is the phase difference of one photon reflected by a beam splitter compared with the transmitted one [15]. In order to simplify the calculation of the temporal second-order coherence function, we only consider the case when these two detectors are at the symmetrical positions, i.e., r1 r2 . In this condition, the first-order coherence function at the jth detector (j 1, and 2) is G1 rj ; tj hjP t eiφt Atj P l eiφl Alj j2 i;
where x is the ratio between the intensities of laser and total light beams. As x increases from 0 to 1, correlation can be changed into anticorrelation.
3. EXPERIMENT We have designed an experimental scheme to verify our theoretical predictions, which is shown in Fig. 2. A single-mode continuous wave laser with a central wavelength at 780 nm and frequency bandwidth of 200 kHz is divided into two equal portions by a nonpolarized beam splitter (BS1 ). One beam is incident to a rotating ground glass (RG) after passing through a convex lens (L1 ) to simulate thermal light [25]. The other beam is expanded by another identical lens (L2 ) to ensure that the intensity of the laser beam is approximately constant across the measurement range. The focus lengths of L1 and L2 are both 50 mm. The distance between L1 and RG is 83 mm. The distances between the lens and detector planes
M2
(3)
in which φt and φl are the initial phases of photons emitted by thermal and laser sources, respectively. Due to the relative phase between photons emitted by these two sources is randomly changing with time, the first-order coherence function in Eq. (3) is a constant. Substituting Eqs. (2) and (3) into Eq. (1), the normalized temporal second-order coherence function can be simplified as [13,14,24]
(7)
Laser
L2
BS 1
D1 M1 L 1 RG
BS 2 D2
2 2 2 g2 12 t1 ; t2 P t 1 sinc πΔνt1 − t2 P l × 1
2P t P l 1 sinc2 πΔνt1 − t2 1 P 2t 2P t P l sinc2 πΔνt1 − t2 ;
(4)
Fig. 2. Experimental setup for the second-order interference of pseudothermal and laser light beams in an HOM interferometer. Laser, single-mode continuous wave laser; BS, 50:50 nonpolarized beam splitter; M, mirror; L, lens; RG, rotating ground glass; D, single-photon detector.
Liu et al.
all equal 825 mm. The temporal second-order coherence length of pseudothermal light is measured to be 78.5 ns. The spatial second-order coherence length of pseudothermal light is calculated to be 55.1 μm by noting the size of the laser light beam on the ground glass is 10.5 mm, and the distance between the source and the detection planes is 742 mm. Two single-mode fibers are coupled to D1 and D2 to collect photons, respectively. The diameter of the collecting single-mode fiber is 5 μm, which is much less than the spatial second-order coherence length of pseudothermal light in the experiments. During the experiments, the single-photon counting rates of D1 and D2 are kept at the level of about 20,000 counts per second. Figure 3 shows the normalized temporal second-order coherence functions for different ratios between the intensities of laser and total light beams when these two detectors are at the symmetrical positions. g2 12 t1 − t2 is the normalized second-order coherence function when these two singlephoton detection events are at r1 ; t1 and r2 ; t2 , respectively. t1 − t2 is the time difference between these two photon detection events within a two-photon coincidence count. The black squares are the experimental results without subtracting any background. The red lines are the theoretical fitting curves by employing Eq. (7). There are six groups of experimental results in Fig. 3, and the measurement time for each group is 120 s. The same temporal second-order coherence length, 78.5 ns, is employed in all the six fitting processes. The only difference among these fittings is the visibility.
4. DISCUSSION The reasons why the correlation is changed into anticorrelation are as follows. The contribution of two-photon coincidence counts in Fig. 1(c) consists of three parts. The first term on the right-hand side of Eq. (6) corresponds to correlation with a background. The second term only contributes to a background. The third term corresponds to anticorrelation with a background. When the ratio between the
Fig. 3. Temporal second-order interference of pseudothermal and laser light beams in an HOM interferometer when these two detectors are at symmetrical positions. g2 12 t1 − t2 is the normalized temporal second-order coherence function when these two single-photon detection events are at r1 ; t1 and r2 ; t2 , respectively. t1 − t2 is the time difference between these two photon detection events within a twophoton coincidence count. x is the ratio between the intensities of laser and total light beams. (See text for details.)
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intensities of laser and total light beams is small, the first term will dominate. Hence correlation is observed when x is in the regime of [0,1/3). When x equals 1/3, the contribution of correlation from the first term and anticorrelation from the third term are equal. These events are independent in this case. When x is in the regime of (1/3,1), the contribution from the third term is greater than the one from the first term. Anticorrelation will be observed in this condition. These photon detections are independent when x equals 1, for it is a laser light beam in an HBT interferometer [16,17]. 2 In order to study how g12 0 changes with the ratio between the intensities of laser and total light beams, Eq. (7) can be simplified as 2 g12 0 3x2 − 4x 2
(8)
by setting t1 t2 . g2 12 0s for different ratios between the intensities of laser and total light beams are shown in Fig. 4. g2 12 0 is the normalized second-order coherence function when these two photon detections are at the symmetrical positions with zero time difference. The red curve is the theoretical fitting of the experimental results by employing Eq. (8), where the ratios among the three coefficients are fixed to be 3∶ − 4∶2 in the fitting process. The reasons why the theoretical and experimental results are different in Fig. 4 are that x is not a constant during one measurement and g2 12 0 is less than 2 for pseudothermal light in an HBT interferometer [14]. However, it is obvious that g2 12 0 can be less than 1 in our experiments, which means anticorrelation is observed when these two photon detections are at the same space-time coordinates. It is worth noting that we also measured the second-order auto-correlation function when laser and thermal light are input to a beam splitter as well [26]. It is found that the secondorder autocorrelation function of classical light is always greater or equal to unity, while the cross-correlation function with classical light can be less than unity [27]. There is no rule that prevents the second-order cross-correlation function of classical light be less than unity. Hence the observed
Fig. 4. Normalized second-order coherence function versus the ratio between the intensities of laser and total light beams. g2 12 0 is the normalized second-order coherence function when these two photon detections are at the symmetrical positions with zero time difference. x is the ratio between the intensities of laser and total light beams. As x increases, g2 12 0 will decrease to a minimum and then increase. (See text for details).
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anticorrelation in this experiment does not mean we have observed a nonclassical phenomenon with classical light.
5. CONCLUSION In conclusion, we have observed anticorrelation with a normal two-photon detection system and classical light in an HOM interferometer. By changing the ratio between the intensities of laser and total light beams, correlation can be changed into anticorrelation. Two-photon interference theory is employed to interpret our experiments, which is helpful to understand the physics of the second-order interference of light.
ACKNOWLEDGMENTS This project is supported by the Doctoral Fund of the Ministry of Education of China (no. 20130201120013), the 111 project of China (no. B14040), and the Fundamental Research Funds for the Central Universities.
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