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Two-photon interference with non-identical photons
Optics Communications 354 (2015) 79–83
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Two-photon interference with non-identical photons Jianbin Liu a,n, Yu Zhou b, Huaibin Zheng a,b, Hui Chen a, Fu-li Li b, Zhuo Xu a a Electronic Materials Research Laboratory, Key Laboratory of the Ministry of Education, and International Center for Dielectric Research, Xi'an Jiaotong University, Xi'an 710049, China b MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter, Department of Applied Physics, Xi'an Jiaotong University, Xi'an 710049, China
art ic l e i nf o
a b s t r a c t
Article history: Received 21 April 2015 Accepted 28 May 2015 Available online 1 June 2015
Two-photon interference with non-identical photons is studied based on the superposition principle in Feynman's path integral theory. The second-order temporal interference pattern is observed by superposing laser and pseudothermal light beams with different spectra. The reason why there is two-photon interference for photons of different spectra is that non-identical photons can be indistinguishable for the detection system when Heisenberg's uncertainty principle is taken into account. These studies are helpful to understand the second-order interference of light in the language of photons. & 2015 Elsevier B.V. All rights reserved.
Keywords: Two-photon interference Second-order temporal beating Interference with photons of different colors
1. Introduction Two-photon interference is essential to understand the foundations of quantum mechanics [1,2], which has been studied extensively [3,4] since the observation of two-photon bunching of thermal light [5]. In Feynman's book, he employed an intuitive way to calculate the interference of particles based on the superposition principle. Assuming there are more than one different alternatives to trigger a detection event, there is interference when these alternatives are in principle indistinguishable, and there is no interference when these alternatives are distinguishable [6]. For instance, there are two different alternatives for two photons to trigger a two-photon coincidence count in the scheme shown in Fig. 1. The first alternative is photon A goes to D1 and photon B goes to D2. The second alternative is photon A goes to D2 and photon B goes to D1. These two alternatives are indistinguishable if photons A and B are identical, since it is impossible to tell which photon is detected by which detector in a two-photon coincidence count. Based on the superposition principle in Feynman's path integral theory [6], there is two-photon interference when photons A and B are identical. When photons A and B are non-identical, are these two alternatives to trigger a two-photon coincidence count in Fig. 1 distinguishable? The answer is yes in classical physics since the measurement accuracy can be arbitrarily high without influencing the measuring results. In quantum mechanics, the answer depends n
Corresponding author. E-mail address: [email protected] (J. Liu).
http://dx.doi.org/10.1016/j.optcom.2015.05.072 0030-4018/& 2015 Elsevier B.V. All rights reserved.
on the properties of photon and detection scheme due to Heisenberg's uncertainty principle [7]. Photons can be different in frequency, polarization, momentum, angular momentum and so on. We will take photons of different spectra for example to study two-photon interference with non-identical photons. Two-photon interference with photons of different spectra was first reported by Ou and Mandel, in which they observed beating between nondegenerate parametric down-conversion beams with central wavelengths at 680 nm and 725 nm, respectively [8]. In the same year, Ou et al. reported a similar beating experiment with blue and green laser light beams [9]. The second-order interference with photons of different spectra was further studied with photons generated by spontaneous parametric down conversion [10–16], single-photon sources [17], laser and single-photon source [18], chirped laser pulses [19], and spontaneous Raman process [20], etc. These experiments can be categorized into two groups. One group is that the two light beams of different spectra are incident to both input ports of a Hong–Ou–Mandel (HOM) interferometer [8–16,20]. The other group is that two light beams are incident to the two input ports of a HOM interferometer, respectively [17–19]. The first group of experiments [8–16,20] can be easily understood in quantum mechanics. By adding more pathes, there can be indistinguishablealternatives for non-identical photons to trigger a two-photon coincidence count. Things become interesting for the second group of experiments [17–19], which is similar as the scheme in Fig. 1. The second-order interference patterns were observed in these experiments [17–19], just like the ones with extra pathes [8–16,20]. In a recent paper, Raymer et al. suggests that by employing an active, frequencyshifting beam splitter, one can observe two-photon interference
80
J. Liu et al. / Optics Communications 354 (2015) 79–83
Fig. 1. Two-photon interference scheme. A and B are two photons. BS is a 1:1 nonpolarizing beam splitter. D1 and D2 are two single-photon detectors. CC is twophoton coincidence count detection system.
between photons of different colors [21]. In this paper, we will show that there can be two-photon interference with photons of different colors with a normal beam splitter. Photons emitted by laser and pseudothermal light sources with different spectra are employed to verify our predictions. These studies are helpful to understand the physics of two-photon interference and applications of two-photon interference in measurement-device-independent quantum key distribution with weak coherent light pulses [22–27].
2. Theory and experiments The scheme in Fig. 2 is employed to discuss the second-order interference of laser and pseudothermal light beams with different frequency spectra. A single-mode continuous-wave 780 nm laser is divided into two beams by a 1:1 non-polarizing beam splitter (BS1). One of the beams is incident to a rotating ground glass (RG) to simulate thermal light [28]. The other light beam is incident to
an acoustooptic modulator (AOM) to change frequency. These two light beams are superposed at the second 1:1 non-polarizing beam splitter (BS2) and then sent to a 1:1 non-polarizing fiber beam splitter (FBS) to measure the second-order temporal interference pattern. P1 and P2 are two linearly polarizers. M1 and M2 are two mirrors. L1 and L2 are two identical lens with focus length of 100 mm and the distance between them is 200 mm. AOM is at the confocal point of L1 and L2. H is a pinhole to block the laser light that does not change frequency after passing through AOM. L3 and L4 are two identical lens with focus length of 50 mm. W is a halfwave plate. Sa and Sb are point pseudothermal and laser light sources, respectively. D1 and D2 are two single-photon detectors. CC is two-photon coincidence count detection system. The distance between L3 and the collector of FBS is equal to the distance between L4 and the collector of FBS via BS2. The optical lengths between the laser and detectors via M1 are both 4.24 m. Both quantum and classical theories can be employed to calculate the second-order interference of classical light [29,30]. Recently, we have employed Feynman's path integral theory to discuss the second-order subwavelength interference of light [31], the relationship between the first-order and second-order interference patterns [32], and the second-order interference between laser and thermal light [33–35]. These studies indicate that the advantage of this method is not only simple, but also offers a unified interpretation for the second-order interference of classical and nonclassical light. In fact, Feynman himself had employed path integral theory to discuss the first-order interference of particles in a Young's double-slit interferometer [6] and two-photon bunching of thermal light [36]. Fano employed the same method to discuss the second-order interference of photons emitted by two independent atoms [37]. We will employ two-photon interference based on the superposition principle in Feynman's path integral theory to discuss the second-order interference of laser and pseudothermal light beams with different spectra in Fig. 2. There are three different cases to trigger a two-photon coincidence count in Fig. 2. The first one is both photons are emitted by Sa. The second one is both photons are emitted by Sb. The third one is one photon is emitted by Sa and the other photon is emitted by Sb. For simplicity, the half wave plate is removed and the intensities of light beams emitted by Sa and Sb are assumed to be
Fig. 2. The second-order interference of laser and pseudothermal light beams with different spectra. Laser: 780 nm single-mode laser with bandwidth of 200 kHz. P: Polarizer. BS: 1:1 non-polarizing beam splitter. W: half wave plate. RG: Rotating ground glass. S: Light source. L: Lens. M: Mirror. AOM: Acoustooptic modulator. H: Pinhole. FBS: 1:1 non-polarizing fiber beam splitter. D: Single-photon detector. CC: two-photon coincidence count detection system. See text for details.
J. Liu et al. / Optics Communications 354 (2015) 79–83
equal in the calculation. The ratio between the probabilities for these three different cases is 1:1:2. In the first case, both photons are emitted by pseudothermal light source and there are two different alternatives to trigger a two-photon coincidence count. One is the first photon goes to D1 and the second photon goes to D2. The second alternative is the first photon goes to D2 and the second photon goes to D1 [5]. In the second case, both photons are emitted by laser source, Sb. There is only one alternative since these two exchanging alternatives are identical [38]. In the third case, there are two alternatives. One is the photon emitted by Sa goes to D1 and the photon emitted by Sb goes to D2. The other alternative is the photon emitted by Sa goes to D2 and the photon emitted by Sb goes to D1. If all the different alternatives are indistinguishable, the two-photon probability distribution in Fig. 2 is [6]
→ → G (2)( r1 , t1; r2 , t2) = 〈|[eiφaAKa1ei(φaB + π /2)Ka2 + ei(φaA + π /2)Ka2eiφaB Ka1] + eiφbAKb1ei(φbB + π /2)Kb2 + 2[eiφaAKa1ei(φbB + π /2)Kb2 + eiφbB Kb1ei(φaA + π /2)Ka2]|2 〉,
(1)
where φαβ is the phase of photon β emitted by source α. Kαγ is the probability amplitude that photon emitted by source α goes to detector γ (α ¼a and b, β ¼A and it B, γ ¼1 and 2). The extra phase π/2 is due to the photon reflected by the fiber beam splitter will gain an extra phase comparing to the transmitted one [39]. 〈⋯〉 is ensemble average. The three lines on the righthand side of Eq. (1) correspond to three different cases to trigger a two-photon coincidence count, respectively. The coefficient, 2, comes from the ratio between the probabilities of these three cases. Since the phases of photons emitted by Sa are randomized by the rotating ground glass, the phases of photons emitted by Sa and Sb are independent. Eq. (1) can be simplified as
→ → G (2)( r1 , t1; r2 , t2) = 〈|Ka1Ka2 + Ka2Ka1|2 〉 + 〈|Kb1Kb2|2 〉 + 2〈|Ka1Kb2 + Kb1Ka2|2 〉.
(2)
For a point light source, Feynman's photon propagator is [40]
Kαγ =
→ → exp[ − i( kαγ · rαγ − 2πναγtαγ )] rαγ
,
(3)
→ which is the same as Green function in classical optics [41]. kαγ and → rαγ are the wave and position vectors of the photon emitted by Sα and detected at Dγ , respectively. rαγ = |rαγ| is the distance between Sα and Dγ . ναγ and tαγ are the frequency and time for the photon that is emitted by Sα and detected at Dγ , respectively (α ¼a and b, γ ¼ 1 and 2). Substituting Eq. (3) into Eq. (2) and with similar calculations as the one in Refs. [31–35,42], it is straightforward to have onedimension temporal two-photon probability distribution as
G (2)(t1 − t2) ∝ 7 + 2 sinc2[π Δνa(t1 − t2)] + 4 cos[2π Δνab(t1 − t2)]sinc[π Δνa(t1 − t2)],
(4)
where paraxial and quasi-monochromatic approximations have been employed to simplify the calculations. The positions of D1 and D2 are the same in order to concentrate on the temporal part. Δνa is the frequency bandwidth of pseudothermal light. Δνab is the difference between the mean frequencies of pseudothermal and laser light. The first term on the righthand side of Eq. (4) is background coming from all three lines on the righthand side of
81
Eq. (2). The second term on the righthand side of Eq. (4) is twophoton bunching of pseudothermal light coming from the first line on the righthand side of Eq. (2). The last term of Eq. (4) is twophoton interference which corresponds to the last line of Eq. (2). When the mean frequencies of these two superposed light beams are different, two-photon temporal beating can be observed. The beating can only be observed within the second-order coherence time of pseudothermal light. The second-order coherence time is measured to be 51 μs, which is much larger than the beating period (E5 ns). The maximum visibility of the second-order interference pattern in Eq. (4) is 4/9, which is consistent with the conclusion that the maximal visibility of the second-order interference pattern with two independent classical light beams is 50% [43]. The single-photon counting rates of D1 and D2 are both about 50 000 c/s, which means on average there is only 1.41 × 10−3 photon in the experimental setup at one time. Our experiments are done at single photon's level. The measured temporal second-order interference patterns are shown in Fig. 3. CC is two-photon coincidence counts for 600 s. t1 − t2 is the time difference between the photon detection events of D1 and D2. The black squares are raw data without subtracting any background and red lines are sine fittings. Fig. 3(a), (c) and (d) is taken when the frequency shift of AOM is 206.80 MHz and Fig. 3(b) is taken when the frequency shift is 217.83 MHz. Two-photon beatings are observed in Fig. 3(a) and (b) when the polarizations of the superposed laser and pseudothermal light beams are parallel. The fitted beating frequencies are 206.0270.08 MHz and 218.1570.07 MHz, respectively, which are consistent with the measured frequency shifts. When the polarizations of laser and pseudothermal light beams are orthogonal and P2 is removed, the photons in these two light beams are distinguishable by their polarizations. Two-photon beating disappears as shown in Fig. 3(c). When we put P2 back and set the axis at 45°, the orthogonally polarized photons become indistinguishable. Two-photon beating is observed again in Fig. 3(d). No matter the polarizations of the photons emitted by these two sources are parallel or orthogonal, the single photon counting rates of D1 and D2 are constant, which means the first-order interference pattern cannot be observed by either D1 or D2. Although the photons are originated from the same single-mode laser, the phases of photons emitted by Sa and Sb are independent and the frequencies of the photons emitted by these two sources are different. Our experiments can be repeated with independent laser and thermal light beams as long as the spectrum of each light beam does not change during the measurement.
3. Discussions In the last section, we have observed the second-order temporal interference pattern with laser and pseudothermal light beams and employed Feynman's path integral theory to interpret the experimental results. The frequency bandwidth of the employed single-mode laser is 200 kHz, which is much less than the difference between the mean frequencies of the superposed light beams. There is no overlap between the spectra of the photons emitted by Sa and Sb. It is possible to distinguish which photon is detected by which detector by putting a frequency filter in front of each detector. For instance, we can put filters centered at νa and νb in front of D1 and D2, respectively. The filter νβ only lets photon with frequency at νβ ± 100 kHz passes (β ¼a and b). There is only one alternative to trigger a two-photon coincidence count, which is photon emitted by Sa goes to D1 and photon emitted by Sb goes to D2. There is no two-photon interference when there are frequency filters in front of the detectors.
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Fig. 3. The observed second-order temporal interference patterns. CC is two-photon coincidence counts for 600 s. t1 − t2 is the time difference between the photon detection events of D1 and D2. The black squares are raw data without subtracting any background and red lines are sine fittings. See text for details. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
The second-order temporal interference patterns are observed when photons of different spectra are incident to the two input ports of a HOM interferometer, respectively [17–19], which means there is two-photon interference. There is two-photon interference only if the different alternatives to trigger a two-photon coincidence count are indistinguishable [6]. Based on the theoretical results above, the second-order temporal beating is originated from the interference of the probability amplitudes corresponding to the last two alternatives in Eq. (1). The first one is photon emitted by Sa goes to D1 and photon emitted by Sb goes to D2. The second alternative is photon emitted by Sa goes to D2 and photon emitted by Sb goes to D1. These two different alternatives are indistinguishable only if the photons emitted by these two sources are indistinguishable for the detection system. The reasons why photons of different spectra can be indistinguishable are as follows. Heisenberg's uncertainty principle is ΔνΔt ≥ 1 in the measurement of photon [7], where Δν and Δt are the frequency and time measurement uncertainties, respectively. Photon is usually detected by photoelectric effect. Forrester et al. proved that the time delay between the photon absorption and electron release must be significantly less than 10 10 s [45], which can be treated as the time measurement uncertainty of photon detection based on photoelectric effect. Based on Heisenberg's uncertainty principle, the frequency measurement uncertainty must be greater than 1/Δt in the photon detection process. Photons with frequency difference less than 10 GHz are indistinguishable for the twophoton coincidence count detection system based on photoelectric effect. The values of time measurement uncertainty of photon detection are different for different measurement schemes. If a twophoton detection system with very short time measurement uncertainty is employed, two-photon interference with photons of different colors can be observed with a normal beam splitter. Let us take photons at 532 nm and 633 nm for example. The frequency difference between them is 5.65 × 105 GHz, which is much larger than 10 GHz. These photons are distinguishable for the detection system based on photoelectric effect and no second-order
interference pattern can be observed. The time measurement uncertainty of two-photon absorption is at femtosecond range [46]. Photons with frequency difference less than 106 GHz are indistinguishable for the detection system. With two-photon absorption as the two-photon detection system, one is able to observe two-photon interference with photons of different colors in the scheme of Fig. 1. When single-photon sources instead of classical light sources are employed in Fig. 2, there are only two possible alternatives to trigger a two-photon coincidence count, which is given by the last line of Eq. (2). If there is a way to eliminate all the terms in Eq. (2) except the last line, the observed two-photon interference pattern with classical light will be the same as the one with single-photon sources. If sum frequency generation that responds to νa + νb (νa ≠ νb) is employed as two-photon detection system [47,48], only the last line of Eq. (2) is left for two-photon probability distribution, where νa and νb are the central frequencies of photons emitted by Sa and Sb, respectively. The visibility of twophoton interference pattern with classical light can exceed 50% [44]. In the measurement-device-independent quantum key distribution with weak coherent light pulses [22–27], it is important to get high visibility two-photon interference pattern to ensure successful distillation of the final key. In Yuan's experiments, the highest visibility of the observed two-photon interference pattern is 46% [27]. It is interesting to study measurement-device-independent quantum key distribution with sum frequency generation as the detection scheme, since the two-photon interference pattern with visibility of 85.4% was observed with laser light pulses [44].
4. Conclusions In this paper, we have employed Feynman's path integral theory to discuss two-photon interference with non-identical photons emitted by laser and thermal light sources. It is concluded that
J. Liu et al. / Optics Communications 354 (2015) 79–83
there are two-photon interference for photons of different spectra if the detection system cannot distinguish them. The second-order temporal interference pattern is observed by superposing laser and pseudothermal light beams with different spectra in our experiments. These results are helpful to understand the physics of two-photon interference and the applications of two-photon interference in quantum information.
Acknowledgment The authors wish to thank D. Wei for the help on the AOM. This project is supported by National Science Foundation of China (No. 11404255), Doctoral Fund of 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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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Two-photon interference with non-identical photons Jianbin Liu a,n, Yu Zhou b, Huaibin Zheng a,b, Hui Chen a, Fu-li Li b, Zhuo Xu a a Electronic Materials Research Laboratory, Key Laboratory of the Ministry of Education, and International Center for Dielectric Research, Xi'an Jiaotong University, Xi'an 710049, China b MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter, Department of Applied Physics, Xi'an Jiaotong University, Xi'an 710049, China
art ic l e i nf o
a b s t r a c t
Article history: Received 21 April 2015 Accepted 28 May 2015 Available online 1 June 2015
Two-photon interference with non-identical photons is studied based on the superposition principle in Feynman's path integral theory. The second-order temporal interference pattern is observed by superposing laser and pseudothermal light beams with different spectra. The reason why there is two-photon interference for photons of different spectra is that non-identical photons can be indistinguishable for the detection system when Heisenberg's uncertainty principle is taken into account. These studies are helpful to understand the second-order interference of light in the language of photons. & 2015 Elsevier B.V. All rights reserved.
Keywords: Two-photon interference Second-order temporal beating Interference with photons of different colors
1. Introduction Two-photon interference is essential to understand the foundations of quantum mechanics [1,2], which has been studied extensively [3,4] since the observation of two-photon bunching of thermal light [5]. In Feynman's book, he employed an intuitive way to calculate the interference of particles based on the superposition principle. Assuming there are more than one different alternatives to trigger a detection event, there is interference when these alternatives are in principle indistinguishable, and there is no interference when these alternatives are distinguishable [6]. For instance, there are two different alternatives for two photons to trigger a two-photon coincidence count in the scheme shown in Fig. 1. The first alternative is photon A goes to D1 and photon B goes to D2. The second alternative is photon A goes to D2 and photon B goes to D1. These two alternatives are indistinguishable if photons A and B are identical, since it is impossible to tell which photon is detected by which detector in a two-photon coincidence count. Based on the superposition principle in Feynman's path integral theory [6], there is two-photon interference when photons A and B are identical. When photons A and B are non-identical, are these two alternatives to trigger a two-photon coincidence count in Fig. 1 distinguishable? The answer is yes in classical physics since the measurement accuracy can be arbitrarily high without influencing the measuring results. In quantum mechanics, the answer depends n
Corresponding author. E-mail address: [email protected] (J. Liu).
http://dx.doi.org/10.1016/j.optcom.2015.05.072 0030-4018/& 2015 Elsevier B.V. All rights reserved.
on the properties of photon and detection scheme due to Heisenberg's uncertainty principle [7]. Photons can be different in frequency, polarization, momentum, angular momentum and so on. We will take photons of different spectra for example to study two-photon interference with non-identical photons. Two-photon interference with photons of different spectra was first reported by Ou and Mandel, in which they observed beating between nondegenerate parametric down-conversion beams with central wavelengths at 680 nm and 725 nm, respectively [8]. In the same year, Ou et al. reported a similar beating experiment with blue and green laser light beams [9]. The second-order interference with photons of different spectra was further studied with photons generated by spontaneous parametric down conversion [10–16], single-photon sources [17], laser and single-photon source [18], chirped laser pulses [19], and spontaneous Raman process [20], etc. These experiments can be categorized into two groups. One group is that the two light beams of different spectra are incident to both input ports of a Hong–Ou–Mandel (HOM) interferometer [8–16,20]. The other group is that two light beams are incident to the two input ports of a HOM interferometer, respectively [17–19]. The first group of experiments [8–16,20] can be easily understood in quantum mechanics. By adding more pathes, there can be indistinguishablealternatives for non-identical photons to trigger a two-photon coincidence count. Things become interesting for the second group of experiments [17–19], which is similar as the scheme in Fig. 1. The second-order interference patterns were observed in these experiments [17–19], just like the ones with extra pathes [8–16,20]. In a recent paper, Raymer et al. suggests that by employing an active, frequencyshifting beam splitter, one can observe two-photon interference
80
J. Liu et al. / Optics Communications 354 (2015) 79–83
Fig. 1. Two-photon interference scheme. A and B are two photons. BS is a 1:1 nonpolarizing beam splitter. D1 and D2 are two single-photon detectors. CC is twophoton coincidence count detection system.
between photons of different colors [21]. In this paper, we will show that there can be two-photon interference with photons of different colors with a normal beam splitter. Photons emitted by laser and pseudothermal light sources with different spectra are employed to verify our predictions. These studies are helpful to understand the physics of two-photon interference and applications of two-photon interference in measurement-device-independent quantum key distribution with weak coherent light pulses [22–27].
2. Theory and experiments The scheme in Fig. 2 is employed to discuss the second-order interference of laser and pseudothermal light beams with different frequency spectra. A single-mode continuous-wave 780 nm laser is divided into two beams by a 1:1 non-polarizing beam splitter (BS1). One of the beams is incident to a rotating ground glass (RG) to simulate thermal light [28]. The other light beam is incident to
an acoustooptic modulator (AOM) to change frequency. These two light beams are superposed at the second 1:1 non-polarizing beam splitter (BS2) and then sent to a 1:1 non-polarizing fiber beam splitter (FBS) to measure the second-order temporal interference pattern. P1 and P2 are two linearly polarizers. M1 and M2 are two mirrors. L1 and L2 are two identical lens with focus length of 100 mm and the distance between them is 200 mm. AOM is at the confocal point of L1 and L2. H is a pinhole to block the laser light that does not change frequency after passing through AOM. L3 and L4 are two identical lens with focus length of 50 mm. W is a halfwave plate. Sa and Sb are point pseudothermal and laser light sources, respectively. D1 and D2 are two single-photon detectors. CC is two-photon coincidence count detection system. The distance between L3 and the collector of FBS is equal to the distance between L4 and the collector of FBS via BS2. The optical lengths between the laser and detectors via M1 are both 4.24 m. Both quantum and classical theories can be employed to calculate the second-order interference of classical light [29,30]. Recently, we have employed Feynman's path integral theory to discuss the second-order subwavelength interference of light [31], the relationship between the first-order and second-order interference patterns [32], and the second-order interference between laser and thermal light [33–35]. These studies indicate that the advantage of this method is not only simple, but also offers a unified interpretation for the second-order interference of classical and nonclassical light. In fact, Feynman himself had employed path integral theory to discuss the first-order interference of particles in a Young's double-slit interferometer [6] and two-photon bunching of thermal light [36]. Fano employed the same method to discuss the second-order interference of photons emitted by two independent atoms [37]. We will employ two-photon interference based on the superposition principle in Feynman's path integral theory to discuss the second-order interference of laser and pseudothermal light beams with different spectra in Fig. 2. There are three different cases to trigger a two-photon coincidence count in Fig. 2. The first one is both photons are emitted by Sa. The second one is both photons are emitted by Sb. The third one is one photon is emitted by Sa and the other photon is emitted by Sb. For simplicity, the half wave plate is removed and the intensities of light beams emitted by Sa and Sb are assumed to be
Fig. 2. The second-order interference of laser and pseudothermal light beams with different spectra. Laser: 780 nm single-mode laser with bandwidth of 200 kHz. P: Polarizer. BS: 1:1 non-polarizing beam splitter. W: half wave plate. RG: Rotating ground glass. S: Light source. L: Lens. M: Mirror. AOM: Acoustooptic modulator. H: Pinhole. FBS: 1:1 non-polarizing fiber beam splitter. D: Single-photon detector. CC: two-photon coincidence count detection system. See text for details.
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equal in the calculation. The ratio between the probabilities for these three different cases is 1:1:2. In the first case, both photons are emitted by pseudothermal light source and there are two different alternatives to trigger a two-photon coincidence count. One is the first photon goes to D1 and the second photon goes to D2. The second alternative is the first photon goes to D2 and the second photon goes to D1 [5]. In the second case, both photons are emitted by laser source, Sb. There is only one alternative since these two exchanging alternatives are identical [38]. In the third case, there are two alternatives. One is the photon emitted by Sa goes to D1 and the photon emitted by Sb goes to D2. The other alternative is the photon emitted by Sa goes to D2 and the photon emitted by Sb goes to D1. If all the different alternatives are indistinguishable, the two-photon probability distribution in Fig. 2 is [6]
→ → G (2)( r1 , t1; r2 , t2) = 〈|[eiφaAKa1ei(φaB + π /2)Ka2 + ei(φaA + π /2)Ka2eiφaB Ka1] + eiφbAKb1ei(φbB + π /2)Kb2 + 2[eiφaAKa1ei(φbB + π /2)Kb2 + eiφbB Kb1ei(φaA + π /2)Ka2]|2 〉,
(1)
where φαβ is the phase of photon β emitted by source α. Kαγ is the probability amplitude that photon emitted by source α goes to detector γ (α ¼a and b, β ¼A and it B, γ ¼1 and 2). The extra phase π/2 is due to the photon reflected by the fiber beam splitter will gain an extra phase comparing to the transmitted one [39]. 〈⋯〉 is ensemble average. The three lines on the righthand side of Eq. (1) correspond to three different cases to trigger a two-photon coincidence count, respectively. The coefficient, 2, comes from the ratio between the probabilities of these three cases. Since the phases of photons emitted by Sa are randomized by the rotating ground glass, the phases of photons emitted by Sa and Sb are independent. Eq. (1) can be simplified as
→ → G (2)( r1 , t1; r2 , t2) = 〈|Ka1Ka2 + Ka2Ka1|2 〉 + 〈|Kb1Kb2|2 〉 + 2〈|Ka1Kb2 + Kb1Ka2|2 〉.
(2)
For a point light source, Feynman's photon propagator is [40]
Kαγ =
→ → exp[ − i( kαγ · rαγ − 2πναγtαγ )] rαγ
,
(3)
→ which is the same as Green function in classical optics [41]. kαγ and → rαγ are the wave and position vectors of the photon emitted by Sα and detected at Dγ , respectively. rαγ = |rαγ| is the distance between Sα and Dγ . ναγ and tαγ are the frequency and time for the photon that is emitted by Sα and detected at Dγ , respectively (α ¼a and b, γ ¼ 1 and 2). Substituting Eq. (3) into Eq. (2) and with similar calculations as the one in Refs. [31–35,42], it is straightforward to have onedimension temporal two-photon probability distribution as
G (2)(t1 − t2) ∝ 7 + 2 sinc2[π Δνa(t1 − t2)] + 4 cos[2π Δνab(t1 − t2)]sinc[π Δνa(t1 − t2)],
(4)
where paraxial and quasi-monochromatic approximations have been employed to simplify the calculations. The positions of D1 and D2 are the same in order to concentrate on the temporal part. Δνa is the frequency bandwidth of pseudothermal light. Δνab is the difference between the mean frequencies of pseudothermal and laser light. The first term on the righthand side of Eq. (4) is background coming from all three lines on the righthand side of
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Eq. (2). The second term on the righthand side of Eq. (4) is twophoton bunching of pseudothermal light coming from the first line on the righthand side of Eq. (2). The last term of Eq. (4) is twophoton interference which corresponds to the last line of Eq. (2). When the mean frequencies of these two superposed light beams are different, two-photon temporal beating can be observed. The beating can only be observed within the second-order coherence time of pseudothermal light. The second-order coherence time is measured to be 51 μs, which is much larger than the beating period (E5 ns). The maximum visibility of the second-order interference pattern in Eq. (4) is 4/9, which is consistent with the conclusion that the maximal visibility of the second-order interference pattern with two independent classical light beams is 50% [43]. The single-photon counting rates of D1 and D2 are both about 50 000 c/s, which means on average there is only 1.41 × 10−3 photon in the experimental setup at one time. Our experiments are done at single photon's level. The measured temporal second-order interference patterns are shown in Fig. 3. CC is two-photon coincidence counts for 600 s. t1 − t2 is the time difference between the photon detection events of D1 and D2. The black squares are raw data without subtracting any background and red lines are sine fittings. Fig. 3(a), (c) and (d) is taken when the frequency shift of AOM is 206.80 MHz and Fig. 3(b) is taken when the frequency shift is 217.83 MHz. Two-photon beatings are observed in Fig. 3(a) and (b) when the polarizations of the superposed laser and pseudothermal light beams are parallel. The fitted beating frequencies are 206.0270.08 MHz and 218.1570.07 MHz, respectively, which are consistent with the measured frequency shifts. When the polarizations of laser and pseudothermal light beams are orthogonal and P2 is removed, the photons in these two light beams are distinguishable by their polarizations. Two-photon beating disappears as shown in Fig. 3(c). When we put P2 back and set the axis at 45°, the orthogonally polarized photons become indistinguishable. Two-photon beating is observed again in Fig. 3(d). No matter the polarizations of the photons emitted by these two sources are parallel or orthogonal, the single photon counting rates of D1 and D2 are constant, which means the first-order interference pattern cannot be observed by either D1 or D2. Although the photons are originated from the same single-mode laser, the phases of photons emitted by Sa and Sb are independent and the frequencies of the photons emitted by these two sources are different. Our experiments can be repeated with independent laser and thermal light beams as long as the spectrum of each light beam does not change during the measurement.
3. Discussions In the last section, we have observed the second-order temporal interference pattern with laser and pseudothermal light beams and employed Feynman's path integral theory to interpret the experimental results. The frequency bandwidth of the employed single-mode laser is 200 kHz, which is much less than the difference between the mean frequencies of the superposed light beams. There is no overlap between the spectra of the photons emitted by Sa and Sb. It is possible to distinguish which photon is detected by which detector by putting a frequency filter in front of each detector. For instance, we can put filters centered at νa and νb in front of D1 and D2, respectively. The filter νβ only lets photon with frequency at νβ ± 100 kHz passes (β ¼a and b). There is only one alternative to trigger a two-photon coincidence count, which is photon emitted by Sa goes to D1 and photon emitted by Sb goes to D2. There is no two-photon interference when there are frequency filters in front of the detectors.
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J. Liu et al. / Optics Communications 354 (2015) 79–83
Fig. 3. The observed second-order temporal interference patterns. CC is two-photon coincidence counts for 600 s. t1 − t2 is the time difference between the photon detection events of D1 and D2. The black squares are raw data without subtracting any background and red lines are sine fittings. See text for details. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
The second-order temporal interference patterns are observed when photons of different spectra are incident to the two input ports of a HOM interferometer, respectively [17–19], which means there is two-photon interference. There is two-photon interference only if the different alternatives to trigger a two-photon coincidence count are indistinguishable [6]. Based on the theoretical results above, the second-order temporal beating is originated from the interference of the probability amplitudes corresponding to the last two alternatives in Eq. (1). The first one is photon emitted by Sa goes to D1 and photon emitted by Sb goes to D2. The second alternative is photon emitted by Sa goes to D2 and photon emitted by Sb goes to D1. These two different alternatives are indistinguishable only if the photons emitted by these two sources are indistinguishable for the detection system. The reasons why photons of different spectra can be indistinguishable are as follows. Heisenberg's uncertainty principle is ΔνΔt ≥ 1 in the measurement of photon [7], where Δν and Δt are the frequency and time measurement uncertainties, respectively. Photon is usually detected by photoelectric effect. Forrester et al. proved that the time delay between the photon absorption and electron release must be significantly less than 10 10 s [45], which can be treated as the time measurement uncertainty of photon detection based on photoelectric effect. Based on Heisenberg's uncertainty principle, the frequency measurement uncertainty must be greater than 1/Δt in the photon detection process. Photons with frequency difference less than 10 GHz are indistinguishable for the twophoton coincidence count detection system based on photoelectric effect. The values of time measurement uncertainty of photon detection are different for different measurement schemes. If a twophoton detection system with very short time measurement uncertainty is employed, two-photon interference with photons of different colors can be observed with a normal beam splitter. Let us take photons at 532 nm and 633 nm for example. The frequency difference between them is 5.65 × 105 GHz, which is much larger than 10 GHz. These photons are distinguishable for the detection system based on photoelectric effect and no second-order
interference pattern can be observed. The time measurement uncertainty of two-photon absorption is at femtosecond range [46]. Photons with frequency difference less than 106 GHz are indistinguishable for the detection system. With two-photon absorption as the two-photon detection system, one is able to observe two-photon interference with photons of different colors in the scheme of Fig. 1. When single-photon sources instead of classical light sources are employed in Fig. 2, there are only two possible alternatives to trigger a two-photon coincidence count, which is given by the last line of Eq. (2). If there is a way to eliminate all the terms in Eq. (2) except the last line, the observed two-photon interference pattern with classical light will be the same as the one with single-photon sources. If sum frequency generation that responds to νa + νb (νa ≠ νb) is employed as two-photon detection system [47,48], only the last line of Eq. (2) is left for two-photon probability distribution, where νa and νb are the central frequencies of photons emitted by Sa and Sb, respectively. The visibility of twophoton interference pattern with classical light can exceed 50% [44]. In the measurement-device-independent quantum key distribution with weak coherent light pulses [22–27], it is important to get high visibility two-photon interference pattern to ensure successful distillation of the final key. In Yuan's experiments, the highest visibility of the observed two-photon interference pattern is 46% [27]. It is interesting to study measurement-device-independent quantum key distribution with sum frequency generation as the detection scheme, since the two-photon interference pattern with visibility of 85.4% was observed with laser light pulses [44].
4. Conclusions In this paper, we have employed Feynman's path integral theory to discuss two-photon interference with non-identical photons emitted by laser and thermal light sources. It is concluded that
J. Liu et al. / Optics Communications 354 (2015) 79–83
there are two-photon interference for photons of different spectra if the detection system cannot distinguish them. The second-order temporal interference pattern is observed by superposing laser and pseudothermal light beams with different spectra in our experiments. These results are helpful to understand the physics of two-photon interference and the applications of two-photon interference in quantum information.
Acknowledgment The authors wish to thank D. Wei for the help on the AOM. This project is supported by National Science Foundation of China (No. 11404255), Doctoral Fund of 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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