We report two-photon interference experiments performed with correlated photon pairs generated via spontaneous four-wave mixing in a Doppler-broadened atomic ensemble involving the 5S1/2-5P3/2-5D5/2 transition of 87Rb atoms. When two photons with different wavelengths are incident on a polarization-based Michelson interferometer, two kinds of two-photon superposition states, the frequency-entangled state and dichromatic path-entangled state depending on whether the two photons are in different paths or in the same path, are probabilistically generated within the interferometer arms. Hong-Ou-Mandel-type interference fringes resulting from the frequency-entangled state are observed over the range of the single-photon coherence length, following introduction of a coarse path-length difference between the two interferometer arms and employing phase randomization. When the interferometer is highly phase-sensitive without phase randomization, a phase super-resolved fringe arising from the dichromatic path-entangled state is observed, both with and without the accompanying one-photon interference fringes.
© 2017 Optical Society of America
Realization of an efficient source of single-photon and entangled photon pairs is one of the key requirements for the study of modern quantum mechanics as well as for implementing entanglement-based photonic quantum technologies [1–4]. During the last three decades, a variety of techniques have been employed to develop correlated photon-pair sources with high brightness and low noise, including spontaneous parametric down-conversion (SPDC) in birefringent nonlinear media [5–8], spontaneous four-wave mixing (SFWM) in nonlinear fibers or waveguides [9–11], radiative decay of an exciton and biexciton in semiconductors , and SFWM in hot or cold atomic systems [13–18]. To date, SPDC has been one of the most widely used methods for the generation of the correlated photon pairs because of its robustness, low noise, and high brightness. However, the weak point of photon pairs from SPDC sources is their broader spectral bandwidth and, thus, very short coherence time. Recently, photon pairs obtained from an atomic ensemble have attracted considerable interest in the field of quantum information science and technology, because of their very narrow spectral bandwidth, which is essential for realization of atom-photon interface for quantum memory [19–21].
Characterization of the generated photons is usually performed by direct measurement involving coincidence counting and two-photon interference (TPI) experiments. In practice, observation of the TPI fringes can provide important information aiding exploration of the spectral and temporal correlation properties of the two-photon states. In the field of experimental quantum optics, quantum interference of correlated photons is regarded as important evidence for the nonclassical nature of light fields. A number of TPI experiments have been performed for various kinds of interferometric schemes to study the nonclassical and nonlocal properties of the two-photon quantum state as well as to characterize the generated photon pairs [22–24]. Nevertheless, TPI experiments employing correlated photons generated in atomic systems have rarely been conducted. However, some experimental studies of TPI have been performed indirectly through use of a time-resolved two-photon counting method, because of the very long coherence lengths of the photons from atomic systems [25–29].
In an optical interferometry setup involving correlated photons, when two non-identical photons are incident on a balanced beam splitter (BS), two kinds of two-photon states containing different individual properties are probabilistically generated in the two BS-output ports . Here, “non-identical photons” means that the two incident photons are distinguishable in terms of their internal/external degrees of freedom based on features such as their polarization, arrival time, frequency, and spatial property. Of the two output two-photon states, one is the path-entangled state in which the two photons are in the same spatial mode of the interferometer arms like the N00N state [31,32], and the other is a two-photon state in which the two photons are on different paths, which is related to the Hong-Ou-Mandel (HOM) interference [33,34]. Recently, we have demonstrated the TPI of temporally separated photons with degenerate and nondegenerate wavelengths obtained from SPDC in a nonlinear crystal .
In this paper, we experimentally demonstrate observations of the two-photon quantum interference effects of nondegenerate photons generated via the SFWM process in a Doppler-broadened ladder-type 87Rb atomic ensemble. This work shows TPI with high-visibility for the first time by taking advantage of the high photon pair generation rate in a Doppler-broadened warm atomic vapor cell. When the two kinds of two-photon states are generated within the polarization-based Michelson interferometer with equal probability, TPI fringes (including the HOM-type spatial beating interference and the path-entangled-state interference) are observed with and without implementation of phase randomization between the two interferometer arms. As a result, the phase-insensitive HOM-type interference fringes oscillating with the beat frequency are measured over the range of the single-photon coherence length. In particular, the phase super-resolved TPI fringe of the dichromatic path-entangled state is also measured without the accompanying one-photon interference; this is due to a rather long beat-period between the two nondegenerate-wavelength photons, even though one-photon interferences with different wavelengths persist.
2. Generation of nondegenerate photon pairs via SFWM
Highly bright nondegenerate photon pairs are generated via an SFWM process in a Doppler-broadened ladder-type atomic ensemble in which the 5S1/2−5P3/2−5D5/2 transition lines of 87Rb atoms are utilized . Figure 1(a) shows the experimental setup used to generate two temporally correlated photons with different wavelengths of 776 (signal) and 780 nm (idler). Figure 1(b) shows the ladder-type energy diagram in which the pump and coupling lasers interact with the 5S1/2–5P3/2 and 5P3/2–5D5/2 transitions, respectively. The wavelength difference between the pump (780 nm) and coupling (776 nm) lasers is approximately 4 nm. The detuning frequency (δ) of both lasers from resonance is set to 1 GHz for reduction of the uncorrelated fluorescence. The signal and idler photons are emitted from the atomic vapor cell in the phase-matched directions at an angle of 1.3° relative to the input laser directions, and subsequently coupled into the single-mode fiber. Uncorrelated fluorescence and laser beams are effectively removed from the correlated photons using interference filters, etalon filters, and linear polarizers.
The generated photon pairs are characterized by obtaining the coincidence counts using two single-photon detectors (SPCM-AQ4C, Perkin Elmer) and, also, by measuring the temporal correlation properties with a time-correlated single-photon counter (PicoHarp 300, PicoQuant). In this study, the single and coincidence counting rates, approximately 200 and 10 kHz, respectively, were obtained for 1-mW pump power and 10-mW coupling power. The heralding efficiency for both the signal and idler photons was calculated to be approximately 5%, with the filtering efficiency being neglected. For the auto- and cross-correlation functions measured as functions of the arrival time delay from the atomic vapor cell to the single-photon detectors, the maximum values were estimated to be 1.50 ± 0.03 and 1.43 ± 0.04 for individual 776- and 780-nm photons, respectively, and 206 for correlated photons. The high peak value of the cross-correlation function implies that the generated photon pair has a strong temporal correlation. The full-width at half maximum (FWHM) value of the cross-correlation function was approximately 1.48 ns, corresponding to the spectral bandwidth caused by Doppler broadening of 540 MHz. Note that a slight discrepancy between the coherence time and reciprocal bandwidth can be caused by experimental conditions such as the cell temperature and δ.
3. Two-photon interference of nondegenerate photon pairs
Figure 2(a) shows the experimental setup used to observe the TPI effect of the nondegenerate photon pairs from the atomic ensemble. We utilized a polarization-based Michelson interferometer involving two temporally correlated photons with different wavelengths of 776 and 780 nm. The two photons were guided from the source part to the interferometer through a single mode fiber, as shown in Fig. 1(a). Two nondegenerate photons with orthogonal polarization were mixed at polarizing beam splitter 1 (PBS1) and substantially mixed into a single spatial mode for the interferometer input port. Then, the two-photon polarization components were equally divided along two paths by PBS2 through a half-wave plate (HWP), the axis of which was oriented at 22.5°. Two quarter-wave plates (QWPs) with axed oriented at 45° were also placed on the two interferometer arms to rotate the polarization direction. For this case, the two-photon state within the interferometer is expressed in the formEquation (1) describes the two kinds of two-photon states: one is the dichromatic path-entangled state for which two nondegenerate photons are on the same path, and the other is the frequency-entangled state, which is related to the HOM interference of nondegenerate photons for the case in which the two photons are on different paths. In the experimental setup, these two two-photon states were coexistent with equal probability within the interferometer arms . The two photons from the PBS2-output port were passed through the HWP with axis oriented at 22.5° and coupled into single-mode fiber couplers 1 and 2 (FC1 and FC2, respectively), which were placed at the PBS-3 output ports, in order to erase the polarization information. Subsequently, the two photons from the FC outputs were directed toward the single-photon detectors through single-mode fibers. Note that, in our experiments, frequency post-selection was not necessary to observe a TPI fringe involving two distinguishable wavelengths.
We performed two kinds of TPI experiments. First, the interference fringe was measured for coincidence counting of the nondegenerate photons when phase randomization between the two interferometer arms was employed; this phase randomization was achieved using a PZT actuator and through application of an alternating-current (AC) voltage. In this case, the interference effects of one-photon interferences as well as the phase-sensitive path-entangled state could be eliminated; therefore, the HOM-type TPI fringe only was measured as a function of . Consequently, the spatial-beating fringe resulting from the frequency-entangled state was observed in the coincidence counting rates. The fringe-oscillation period corresponded to the beat frequency between the two nondegenerate photons. Second, the TPI effect was observed without actuating the PZT. In this case, the two two-photon states shown in Eq. (1) contributed simultaneously to the interference. We measured the single and coincidence counting rates as functions of a small , which was comparable to the wavelengths at the two positions at which the sum of the one-photon interference fringe patterns reveals the node and antinode, respectively.
When two nondegenerate photons with orthogonal polarizations were incident on the interferometer, two single-photon detectors D1 and D2, which were connected to FC1 and FC2, respectively, recorded the two one-photon interference fringes simultaneously with modulation frequencies and . Here, , , , and . As a result, the single counting rates measured at the two detectors revealed the one-photon interference fringes containing the sum- and difference-frequency modulation terms, and , as follows:Equation (3) contains two interference terms for the phase-insensitive HOM-type interference of the frequency-entangled state and the phase-sensitive interference of the dichromatic path-entangled state employing nondegenerate photon pairs. The spacing of the one- and two-photon fringe envelopes corresponding to the beat period between two nondegenerate-wavelengths photons with wavelengths of 775.98 and 780.24 nm is approximately 142 μm.
3.1 Two-photon interference of frequency-entangled state
Figure 2(b) shows a typical TPI fringe apparent in the coincidence counting rates when the relative phase between the two interferometer arms was randomized to eliminate the phase-sensitive interference effect of the path-entangled state as well as the one-photon interference. The measured single- and coincidence counting rates at the two single-photon detectors were approximately 70 and 1.36 kHz, respectively, for a 5-ns coincidence window and zero path-length difference between the two arms. The filled circles represent the experimental data for 10 repetitions at each position and the solid line is the theoretical fit. The error bars represent the square roots of the measured coincidence counting rates. The fringe visibility was measured to be 0.50 ± 0.01 from the sinusoidal fitting, which is the highest limited value when the interference of the path-entangled state is deactivated by the phase randomization. The fringe spacing corresponds to the beat-period between the two nondegenerate-wavelengths photons, which was found to be 142.07 ± 0.12 μm.
To observe the full TPI pattern over the range of the single-photon coherence length, one of the mirrors, M2, was moved from the balanced to the noninterfering position. The normalized coincidences measured at the four M2 positions are shown as functions of in Figs. 3(a)-3(d). From the measured coincidence counting rates for the coarse path-length delay positions, we plotted the full interferogram of the oscillatory two-photon interference fringes of the frequency-entangled state, as shown in Fig. 3(e); hence, the spectral/temporal property of the measured single-photon wavepacket was explored. Here, the envelope function corresponding to the spectral properties of the detected photons is Gaussian; this is thought to be because of a Doppler-broadened atomic system was used as the photon source. The filled squares represent the normalized coincidences at the given M2 positions and the gray area depicts a simulation result considering the Gaussian-shaped wavepacket with a 1-ns coherence time. Figure 3(f) shows the TPI fringe visibility as a function of the coarse . The solid line represents the Gaussian fit to the data points with an FWHM value of 151.18 ± 2.55 mm, which corresponds to a single-photon spectral bandwidth of approximately 2 pm.
3.2 Two-photon interference of dichromatic path-entangled state
If the interferometer is highly phase-sensitive without phase randomization, the two two-photon states in Eq. (1) contribute simultaneously to the interference effect with equal probability. In this case, one-photon interference fringes for the two nondegenerate photons with orthogonal polarizations are simultaneously recorded by each detector, therefore, the interference fringes contain sinusoidal modulation terms of and , simultaneously, as shown in Eq. (2). Consequently, the one-photon interference fringes at the two detectors are almost cancelled by the product of the sum- and difference-frequency terms in the vicinity of the node positions. Moreover, the TPI fringe arising from the frequency-entangled state has a rather long oscillation period compared to the single-photon wavelength. As a result, a clear phase super-resolved TPI fringe can be measured at the node positons of the periodic one-photon beat fringes, without the accompanying one-photon interference and the oscillatory fringe from the frequency-entangled state. The TPI fringe accompanying the one-photon oscillatory fringes is also observed in the vicinity of the antinode positions. This feature implies that the phase super-resolved interference fringe can be obtained without preparing the path-entangled N00N state only in the interferometer. Note that, previously TPI experiments with nondegenerate photons from a broadband SPDC source have been performed in a standard Mach-Zehnder interferometer [35,36].
Figure 4 shows the experimental results for observations of the one- and TPI fringes measured without phase randomization. One-photon interference fringes obtained by varying the coarse between the two arms were observed at the two detectors with the beat period envelope and the sum of the one-photon fringes with oscillation frequencies and , as shown in Fig. 4(a). Here, the spacing between the neighboring nodes (antinodes) of the fringe envelope corresponds to the beat-frequency period between the two nondegenerate photons, and the phase-sensitive oscillation period is identical to the average wavelength of the two single photons. Figure 4(b) shows the measured coincidence interferograms in which the fringe modulation has the sum- and difference-frequency terms. The filled squares and circles in Figs. 4(a) and 4(b) represent the measured single and coincidence counts obtained by varying the coarse by 0.32 μm, and the gray areas represent the simulation results for the phase-resolved oscillatory one- and TPI fringes. The normalized single and coincidence counts measured at the node and antinode positions of the one-photon interference fringes are shown in Figs. 4(c)-4(f). The coincidence fringe visibilities were found to be 0.31 ± 0.01 (node position) and 0.97 ± 0.02 (antinode position), respectively, and the fringe period was estimated to be 390.68 ± 1.45 nm from the sinusoidal fitting of the measured data points. The one-photon fringe visibilities at the antinode position were 0.97 ± 0.01 and 0.96 ± 0.01, respectively. Slight modulations of the single counting rates at the node position were also observed with visibilities of 2.53 ± 0.26% and 2.81 ± 0.29%. Theoretically, the one-photon fringe visibility is expressed in the form, ; thus it is expected to be 2.21% for a 1-μm relative between the two arms. At the antinode position, the phase super-resolved TPI fringe remained apparent, and was also accompanied by the one-photon interference fringes with modulation frequencies of and . When we consider the phase resolution only, this feature can be obtained with classically correlated photons; however, the coincidence counting rates are considerably higher than the accidental coincidences for the given 5-ns coincidence window. Here, it is worth emphasizing the fact that the phase super-resolved TPI fringe could be obtained without employing the path-entangled N00N state only. Thus, our experiment demonstrated that a phase super-resolved TPI fringe is observed without one-photon fringe modulation due to a rather long beat-period and is also obtained even when perfect one-photon interferences exist in the interferometer.
Next, we considered the effect of spectral filtering on the measured one-photon interference fringes, as well as the TPI fringes at both the node and antinode positions. Two interference filters centered on 780 and 776 nm with 3-nm bandwidth were placed in front of FC1 and FC2, respectively. In this case, the two detectors D1 and D2 recorded one-photon interference fringes with individual oscillation frequencies and , without beating. Here, and , because the two input photons had well-defined wavelengths and polarizations. Further, instead of the expression given in Eq. (3), the coincidence counting probability was expressed in the form,Figs. 5(a) and (b). The periodic coincidence peaks correspond to the beat-period between the two nondegenerate photons; however, the fringe shape differs considerably from that shown in Fig. 4(b). Similar to the results shown in Figs. 4(c)-4(f), one- and two-photon interference fringes were measured for two M2 positions, corresponding to the balanced and half-beat-period positions, as shown in Figs. 5(c)-5(f). Note that the TPI fringes were always accompanied by perfect one-photon interference fringes. However, the fringe shapes and periods at the two M2 positions differed completely from each other. Moreover, the phase super-resolved fringe with oscillation period of 390.01 ± 0.51 nm was observed at the half-beat-period position only. This feature is also distinct from that shown in Figs. 4(d) and 4(f). These results imply that the TPI fringe pattern of correlated photons is strongly affected by the one-photon interference effects, although the TPI effect itself is considered to be a phenomenon independent from any one-photon interferences. Note that the one-photon fringe is invisible at each detector in many TPI experiments with correlated photons, because the two individual photon-counting events cancel at the single detector, even though one-photon self-interference persists.
In summary, we have experimentally demonstrated the observation of two-photon interference effects in a polarization-based Michelson interferometer employing nondegenerate photon pairs, which were generated via an SFWM process in a Doppler-broadened atomic ensemble utilizing the 5S1/2-5P3/2-5D5/2 transitions of the 87Rb atomic ensemble. Two kinds of two-photon states consisting of nondegenerate photon pairs were probabilistically generated within the two interferometer arms, i.e., the frequency-entangled state and the dichromatic path-entangled state. HOM-type interference fringes revealing spatial quantum beating were observed over the range of the single-photon coherence length, following introduction coarse path-length difference between the two interferometer arms. These experiments were performed by utilizing a phase-randomization technique to eliminate the phase-sensitive interference effect of the path-entangled two-photon state. The TPI fringe of the frequency-entangled state was observed with a very high visibility of 0.50 ± 0.01. This fringe exhibited an oscillation period of 142.07 ± 0.12 μm, which corresponds to the beat-frequency period between the two nondegenerate photons. From the measured TPI fringes for several coarse path-length differences, the single-photon coherence length was estimated to be 0.30 ± 0.01 m, which corresponds to a spectral bandwidth of 2 pm.
When the interferometer was highly phase-sensitive, the two kinds of two-photon states simultaneously and equally contributed to the TPI effects. In this case, the phase super-resolved interference fringe arising from the dichromatic path-entangled state was observed without the accompanying fringe oscillations due to either the frequency-entangled state or one-photon interferences; this behavior was due to a rather long beat period and cancellation between the two one-photon interference fringes of the nondegenerate photons at the node position. The two-photon interference effect was observed for the two-photon coincidence counting involving almost perfect one-photon interferences at the antinode position; however, the counting level was considerably higher than the accidental coincidences caused by multiphoton emission from the highly bright photon-pair source. In addition, the effect of spectral filtering on TPI fringe was investigated to observe the TPI effect accompanied by one-photon interference fringes. The present results will facilitate a consistent understanding of multiphoton interference phenomena involving nondegenerate photon pairs employing highly bright photon pairs with moderate spectral bandwidth generated from a Doppler-broadened atomic ensemble.
KIST Institutional Program (2E26681); National Research Foundation of Korea (NRF) (2015R1A2A1A05001819, 2016R1D1A1B03936222).
References and links
1. B. Lounis and M. Orrit, “Single-photon sources,” Rep. Prog. Phys. 68(5), 1129–1179 (2005). [CrossRef]
2. A. Migdall, S. Polyakov, J. Fan, and J. Bienfang, Single-Photon Generation and Detection: Physics and Applications (Academic Press, 2013).
3. Z. S. Yuan, X. H. Bao, C. Y. Lu, J. Zhang, C. Z. Peng, and J. W. Pan, “Entangled photons and quantum communication,” Phys. Rep. 497(1), 1–40 (2010). [CrossRef]
4. J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3(12), 687–695 (2009). [CrossRef]
5. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75(24), 4337–4341 (1995). [CrossRef] [PubMed]
6. P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60(2), 773–776 (1999). [CrossRef]
7. S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18(2), 155–160 (2002). [CrossRef]
8. C. E. Kuklewicz, M. Fiorentino, G. Messin, F. N. C. Wong, and J. H. Shapiro, “High-flux source of polarization-entangled photons from a periodically poled KTiOPO4 parametric down-converter,” Phys. Rev. A 69(1), 013807 (2004). [CrossRef]
10. H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S. Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett. 91(20), 201108 (2007). [CrossRef]
11. J. W. Silverstone, D. Bonneau, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, V. Zwiller, G. D. Marshall, J. G. Rarity, J. L. O’Brien, and M. G. Thompson, “On-chip quantum interference between silicon photon-pair sources,” Nat. Photonics 8(2), 104–108 (2014). [CrossRef]
12. R. M. Stevenson, R. J. Young, P. Atkinson, K. Cooper, D. A. Ritchie, and A. J. Shields, “A semiconductor source of triggered entangled photon pairs,” Nature 439(7073), 179–182 (2006). [CrossRef] [PubMed]
14. A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423(6941), 731–734 (2003). [CrossRef] [PubMed]
15. B. Srivathsan, G. K. Gulati, B. Chng, G. Maslennikov, D. Matsukevich, and C. Kurtsiefer, “Narrow band source of transform-limited photon pairs via four-wave mixing in a cold atomic ensemble,” Phys. Rev. Lett. 111(12), 123602 (2013). [CrossRef] [PubMed]
16. R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Correlated photon pairs generated from a warm atomic ensemble,” Phys. Rev. A 82(5), 053842 (2010). [CrossRef]
17. C. Shu, P. Chen, T. K. A. Chow, L. Zhu, Y. Xiao, M. M. T. Loy, and S. Du, “Subnatural-linewidth biphotons from a Doppler-broadened hot atomic vapour cell,” Nat. Commun. 7, 12783 (2016). [CrossRef] [PubMed]
20. A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photonics 3(12), 706–714 (2009). [CrossRef]
21. K. Heshami, D. G. England, P. C. Humphreys, P. J. Bustard, V. M. Acosta, J. Nunn, and B. J. Sussman, “Quantum memories: emerging applications and recent advances,” J. Mod. Opt. 63(20), 2005–2028 (2016). [CrossRef] [PubMed]
22. P. Hariharan and B. C. Sanders, “Quantum phenomena in optical interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1996).
23. G. Jaeger and A. V. Sergienko, “Multi-photon quantum interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2001).
24. J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012). [CrossRef]
25. Z.-S. Yuan, Y.-A. Chen, S. Chen, B. Zhao, M. Koch, T. Strassel, Y. Zhao, G. J. Zhu, J. Schmiedmayer, and J. W. Pan, “Synchronized independent narrow-band single photons and efficient generation of photonic entanglement,” Phys. Rev. Lett. 98(18), 180503 (2007). [CrossRef] [PubMed]
26. C. Liu, J. F. Chen, S. Zhang, S. Zhou, Y.-H. Kim, M. M. T. Loy, G. K. L. Wong, and S. Du, “Two-photon interferences with degenerate and nondegenerate paired photons,” Phys. Rev. A 85(2), 021803 (2012). [CrossRef]
27. P. Qian, Z. Gu, R. Cao, R. Wen, Z. Y. Ou, J. F. Chen, and W. Zhang, “Temporal purity and quantum interference of single photons from two independent cold atomic ensembles,” Phys. Rev. Lett. 117(1), 013602 (2016). [CrossRef] [PubMed]
28. X. Guo, Y. Mei, and A. N. D. S. Du, “Testing the Bell inequality on frequency-bin entangled photon pairs using time-resolved detection,” Optica 4(4), 388–392 (2017). [CrossRef]
29. T. Jeong, Y.-S. Lee, J. Park, H. Kim, and H. S. Moon, “Quantum interference between autonomous single-photon sources from Doppler-broadened atomic ensemble,” Optica 4(10), 1167–1170 (2017). [CrossRef]
32. H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49(14–15), 2325–2338 (2002). [CrossRef]
35. T. S. Larchuk, R. A. Campos, J. G. Rarity, P. R. Tapster, E. Jakeman, B. E. A. Saleh, and M. C. Teich, “Interfering entangled photons of different colors,” Phys. Rev. Lett. 70(11), 1603–1606 (1993). [CrossRef] [PubMed]
36. Y. H. Shih, A. V. Sergienko, M. H. Rubin, T. E. Kiess, and C. O. Alley, “Two-photon interference in a standard Mach-Zehnder interferometer,” Phys. Rev. A 49(5), 4243–4246 (1994). [CrossRef] [PubMed]