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Direct generation of frequency-bin entangled photons via two-period quasi-phase-matched parametric downconversion

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Abstract

We report a simple scheme for direct generation of frequency-bin entangled photon pairs via spontaneous parametric downconversion. Our fabricated nonlinear optical crystal with two different poling periods can simultaneously satisfy two different, spectrally symmetric nondegenerate quasi-phase-matching conditions, enabling the direct generation of entanglement in two discrete frequency-bin modes. Our produced photon pairs exhibited Hong-Ou-Mandel interference with high-visibility beating oscillations— a signature of two-mode frequency-bin entanglement. Moreover, we demonstrate deterministic entanglement-mode conversion from frequency-bin to polarization modes, with which our source can be more versatile for various quantum applications. Our scheme can be extended to direct generation of high-dimensional frequency-bin entanglement, and thus will be a key technology for frequency-multiplexed optical quantum information processing.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Photons and their entanglement are key resources for quantum information processing (QIP) including quantum teleportation, quantum computations, and quantum communication. In particular, polarization entangled photons generated via spontaneous parametric downconversion (SPDC) [1] have been successfully used in many proof-of-principle demonstrations of QIP protocols [2, 3].

Meanwhile, SPDC photon pairs often have entanglement in another degree of freedom; due to energy conservation in the SPDC process, downconverted photons can be naturally entangled in frequency. This frequency entanglement is undesirable in polarization-encoded QIP, in which photons need to be indistinguishable from each other except in polarization states for utilizing two-photon interference [4], a key effect in two-qubit quantumgates [5]. To meet such requirements spectral engineering techniques [6–12] have been developed for eliminating frequency entanglement. However, while only two orthogonal states can be used in polarization encoding, a number of modes can be extracted in the frequency domain, and therefore frequency encoding can be efficiently used for high-dimensional QIP applications such as high-capacity quantum communications [13–15] and mode-multiplexed photonic state engineering [16–18].

In this paper, we consider an entangled state with n discrete frequency modes:

|FBn=j=1n1n|ωj|ωnj+1,
where |ωj denotes a single-photon state having center frequency ωj and a bandwidth much narrower than the peak frequency separation ωjωjj. This “frequency-bin” entangled state is more useful than continuous frequency entanglement in the context of QIP that in general requires the manipulation of dicrete modes of photons (although broadband continuous frequency entanglement is also a key resource in some important applications such as clock synchronizations [19], quantum optical coherence tomography [20, 21], and probing nonlinear optical process [22]). Experimental observation of two-mode frequency-bin entanglement was first reported by Ou and Mandel in 1988 [23]; however, the behavior of frequency-bin entanglement was extracted from continuous frequency entanglement by postselective projection measurements via two-photon coincidence detections. Such postselective schemes cannot directly distribute entanglement over remote parties for quantum communication and networking. Similar postselective schemes have been demonstrated in later works [24–28]. Non-postselective, direct generation of two-mode frequency-bin entanglement have been demonstrated by using SPDC with interferometers [29–31] and noncollinear phase-matching conditions [32]. However, for experimental simplicity, system stability, and compatibility with integrated optics, a single-pass, collinear SPDC is more desirable. Recently, Xie et al., [33] have demonstrated remarkable 17-mode frequency-bin entanglement, but that experiment used a resonator cavity for spectral filtering that may result in limited transmissions and generation rates.

Here, we report a simple scheme for direct generation of two-mode frequency-bin entangled photons at telecom wavelengths. A nonlinear optical crystal with two poling periods simultaneously satisfies two different quasi-phase-matching (QPM) conditions [34, 35], directly generating frequency-bin entanglement between orthogonally polarized photons. Therefore, our scheme does not need either spectral filtering, postselective measurements, or interferometers. Our scheme can be extended for generating larger-mode frequency-bin entanglement by introducing additional poling periods for higher-dimensional QIP applications. We also demonstrate entanglement-mode conversion from frequency-bin to polarization modes, allowing us to choose an appropriate entanglement degree of freedom for different applications.

 figure: Fig. 1

Fig. 1 Schematic diagram of the generation of frequency-bin entangled photon pairs. (a) The illustration of two-period QPM crystal. (b) Frequency-bin entanglement generation. (c) Polarization entanglement generation demonstrated in [34]. PBS: polarizing beamsplitter, DM: dichroic mirror. (d) Theoretical tuning curves for a PPLN crystal with our designed poling periods (Λ1= 9.25 μm and Λ2 = 9.50 μm) and a pump wavelength of 775 nm.

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2. Principle of operation

Figure 1 shows a schematic diagram of the frequency-bin entanglement generation. A nonlinear crystal has two sequential poling structures with the period Λ1 and Λ2 over the lengths L1 and L2, respectively (see Fig. 1(a)). In general, SPDC has the highest conversion efficiency when the pump and downconverted (signal and idler) modes satisfy energy and momentum conservation

Δω=ωpωsωi=0,
Δk=kpkski2πΛ1(2)=0,
where ωl and kl are angular frequency and the wavenumber of the pump (l=p), signal (l=s), and idler (l=i) modes, respectively. In our scheme, the two poling periods are determined to generate othrogonally polarized, spectrally symmetric two-photon states via type-II collinear QPM conditions; Λ1 and Λ2 are, respectively, designed to produce |H,ω1|V,ω2 and |H,ω2|V,ω1, where |H(V),ω1(2) denotes a single-photon state having horizontal (vertical) polarization and frequency ω1(2). Thus, a two-photon state emitted from the two-period crystal can form a superposition state
|ψ=12(|H,ω1|V,ω2+eiϕ|V,ω1|H,ω2),
where ϕ denotes the relative phase: ϕ=2π(1/Λ11/Λ2)L2 after the two-period crystal, but this phase can be manipulated by introducing birefringence and/or dispersive media [34], as will be demonstrated. As shown in Fig. 1(b), by spatially separating photons in the state |ψ in terms of polarization, e.g., with a polarizing beamsplitter (PBS), one can obtain a two-mode frequency-bin entangled state,
|ψf=12(|ω1A,H|ω2B,V+eiϕ|ω2A,H|ω1B,V),
where A and B represent the spatial modes of photons. Thus, our scheme can directly generate frequency-bin entanglement via single-pass collinear SPDC without any spectral filtering or postselective measurements. Moreover, our scheme can be extended to the generation of arbitrary n

-mode frequency-bin entanglement with a crystal having n different poling periods in which Λi produces |H,ωi|V,ωni+1. As has been reported in [34] and as shown in Fig. 1(c), by splitting collinear photons in |ψ with a dichroic mirror (DM), a nondegenerate polarization entangled state can be obtained:

|ψp=12(|HA,ω1|VB,ω2+eiϕ|VA,ω1|HB,ω2).
Note that |ψf(|ψp) can be deterministically changed back to |ψ by recoupling two photons with a PBS(DM): therefore, as will be demonstrated, the entanglement in frequency mode can also be deterministically transferred to polarization mode, and vice versa. Figure 1(d) shows theoretical tuning curves, i.e., peak emission wavelengths of the signal and idler modes versus the crystal temperature, for a periodically-poled LiNbO3 (PPLN) crystal with our designed poling periods (Λ1 =9.25 μm and Λ2 = 9.50 μm). For a pump wavelength of 775 nm we find that the photon emission wavelengths from the two different poling periods are matched at a crystal temperature of 166°C; however, as will be described, we experimentally found the matched emission spectra at 120 °C in our fabricated crystal [34]. The disagreement may be caused by a slight difference between the refractive indices in our LN crystal and those from the Sellmeier equations [36, 37] used in our calculations.

 figure: Fig. 2

Fig. 2 Illustration of the experimental setup for (a) generation and detection of frequency-bin entangled photons and (b) detection of polarization entanglement. PPLN: periodically poled lithium niobate, PBS: polarizing beamsplitter, QWP: quarter-wave plate, HWP: half-wave plate, SMF: single-mode fiber, SPD: single-photon detector, DM: dichroic mirror, PA: polarization analyzer. Each PA consists of a QWP, a HWP, and a PBS.

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3. Experiment

Our experimental setup for generation and characterization of frequency-bin entangled photons is illustrated in Fig. 2(a). We designed and fabricated [34] a PPLN crystal with Λ1 = 9.25 μm and Λ2 = 9.50 μm and L1,L2=L20 mm, so that two different QPM conditions have the same photon-pair generation probability. The crystal is pumped by second harmonic (wavelength of 775 nm) of an amplified external cavity diode laser with a repetition rate of 4 MHz and a pump pulse width of 2.5 ns, much longer than the LN crystal length to eliminate the temporal distinguishability of produced photons. The pump beam was focused into the crystal with a waist radius of ∼ 80 μm. The crystal temperature was kept at 120.0 °C with an accuracy of 0.01 °C where the two different QPM conditions generated the same peak wavelengths at 1506 nm and 1594 nm with a ∼1.4 nm bandwidth.

The frequency entanglement of the produced photons was characterized by Hong-Ou-Mandel interference (HOMI) [4]: In general, a photon pair produced in distinguishable spectral modes (e.g., |ω1A|ω2B) does not exhibit HOMI (unless the coincidence detection time window is much shorter than the reciprocal of the frequency detuning δω=ω1ω2, as demonstrated in [38]). However, despite having distinguishable spectral modes, frequency-bin entangled photons, an equal coherent superposition of |ω1A|ω2B and |ω2A|ω1B, are perfectly indistinguishable as a two-photon joint state, and therefore can exhibit HOMI. In other words, HOMI experiments can be a direct test of entanglement of photons generated in discrete frequency modes.

Our experimental setup can measure HOMI for photons in two orthogonal polarization modes rather than spatial modes used in conventional HOMI experiments [4]. In a polarization-mode Michelson interferometer, our collinear SPDC photons are first split by PBS1 and thereby converted to |ψf in Eq. (5). After being reflected by mirrors the photons come back to PBS1 with a time delay τ between the two spatial modes of frequency-bin entangled photons. This variable time delay τ introduces a phase ϕ=δωτ to see HOMI over the two-photon coherence length, as well as to compensate a group delay between H- and V-polarized photons [34]. A half-wave plate (HWP) rotated at 22.5° and PBS2 work together as a 50:50 beam splitter for H/V polarization modes. The photons coupled into single-mode fibers (SMFs) are detected by a pair of single photon detectors (SPD, ID Quantique id201). Coincidence events of the two SPDs are recorded by a time interval analyzer (EG&G 9308) with a coincidence window of 2.5 ns. In our experimental setup with the pump power of 9 mW, typical single and coincidence count rates after the PBS2 are 1.7×104/s and 8.5×102/s, respectively. With those single and coincidence rates, we estimate that the average system transmission (including SPDs’ ∼25% and ∼18% detection efficiencies at 1506 nm and 1594 nm) is ∼5%.

 figure: Fig. 3

Fig. 3 Characterization of the frequency-bin entanglment. (a) Observed HOM interference for (a) -3 psτ 3 ps and (b) -0.5 psτ 0.5 ps. (c) Reconstructed density matrix. Error bars were calculated from Poissonian photon-counting statistics.

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4. Frequency-bin entanglement

Figure 3(a) shows the result of HOMI for our produced photons and Fig. 3(b) is a close-up view of Fig. 3(a) around τ = 0. Here, we set the origin (τ = 0) where we observed the lowest coincidence rate. The observed high-visibility sinusoidal oscillations are a clear signature of the frequency-bin entanglement. The solid curve is the best fit to the experimental data with a fitting function [29]

I(τ)={N2{1Vcos (δωτ)(1|ττc|)}for|τ|τcN2for|τ|>τc
where the N and V represent the coincidence count rate and interference visibility, respectively. τc denotes the full width at half maximum of the triangular envelope predicted from a convolution of two-photon rectangular temporal amplitudes [39]. We obtained δω/2π = 11.5± 0.5 THz and τc=2.40±0.03 ps (and corresponding single-photon bandwidth of 1.47±0.11 nm) in excellent agreement with our single-photon spectral measurements. Our observed interference visibility is V=93.4±1.0%, which is mainly limited by the distinguishability of the two photon wave packets due to slightly different peak amplitudes and group delays in the two different QPM conditions; our predicted upper limit accounting for the different group delays is V= 95.4% [34]. We note, however, that a perfect indistinguishability can be achieved by adjusting the ratio L1/L2 and poling duty cycles [12] to control group delays and effective nonlinearities in a LN crystal, respectively. The degradation of visibility may also arise from slight misalignment of the Michelson interferometer, multi-photon-pair emissions, and dark counts in the SPDs.

In order to quantify the entanglement in our produced state, we estimated its density matrix in two-frequency-bin space. By the method introduced in [29] a frequency-bin mode density matrix can be parametrized as

ρF=p|ω1ω2ω1ω2|AB+(1p)|ω2ω1ω2ω1|AB+V2(eiϕ|ω1ω2ω2ω1|AB+eiϕ|ω2ω1ω1ω2|AB),
where p (0p1) represents the probability of |ω1A|ω2B. Here we assume that frequency correlated modes (i.e., |ω1A|ω1B or |ω2A|ω2B) have no amplitudes, recalled by the energy conservation in Eq. (1). Thus, we simply obtained p=0.516±0.012 from the ratio of the coincidence count rates of |ω1A|ω2B and |ω2A|ω1B with a setup in Fig. 2b that can perform spectral projection measurements by a DM. Additionally, using the phase tunability by the Michelson interferometer, we selected ϕ = 0 (and thus the only real part of our reconstructed density matrix has non-zero values). Our reconstructed density matrix is shown in Fig. 3(c). The fidelity to the maximally entangled state |ψf (for ϕ = 0) and the concurrence of ρF are F=96.7±0.1% and C=0.934±0.005, respectively. We note that this highly entangled state was obtained without spectral filtering.

 figure: Fig. 4

Fig. 4 Measured polarization-mode density matrices after the entanglement-mode transfer for τ = (a) 0 fs, (b) 47 fs, and (c) -20 fs.

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5. Entanglement mode conversion and symmetry of photon pairs

The entanglement in our produced photons can be deterministically transferred from frequency-bin to polarization domains. Figure 2(b) illustrates our setup (a similar setup has been used for direct generation of polarization entanglement [34]). As described above, after the creation and phase adjustment of the frequency-bin entangled state, the Michelson interferometer transmits the photons into the same output port of PBS1. The collinear photons are then split by the DM with respect to the photon’s frequency (i.e., frequency-mode projection measurements), deterministically converted to nondegenerate polarization entangled photons |ψp as shown in Eq. (6). Our produced two-photon polarization states are analyzed by incorporating coincidence detections with polarization analyzers (PAs), each of which consists of a quarter-wave plate, a half-wave plate, and a PBS).

Figure 4(a)--4(c)show our reconstructed density matrices via maximum likelihood quantum state tomography [40] for τ= 0 fs, 47 fs, and -20 fs, where the HOMI in Fig. 3(b) shows dark, bright, and intermediate fringes, respectively. While all of the three density matrices have large populations in |HA|VB and |VA|HB, phases of off-diagonal elements clearly change with τ. Using these results we investigate the relations of the HOMI and the phase of polarization entangled states. In HOMI experiments using a non-polarizing 50:50 beamsplitter [4] two-photon states behave differently due to the different symmetry of the spatial wave functions [41]; spatially (anti-)symmetric two-photon states are spatially (anti-)bunched after the beamsplitter. For an entangled two-photon state, such spatial (anti-)symmetry can be indirectly introduced by (anti-)symmetrized states in an entanglement degree of freedom, since the overall wave function needs to be symmetric in bosonic systems [42]. In this context, our HOMI experiment using a polarization-mode splitter shown in Fig. 2(a) can be interpreted as a test of the polarization-mode symmetry, which can be introduced by the phase of the frequency-bin entanglement adjusted by τ.

Table 1 summarizes characteristics of the reconstructed density matrices together with the normalized coincidence rates and phases of the frequency-bin entanglement measured by our HOMI experiment. For τ= 0 fs, where we observed two-photon bunching (i.e., the lowest coincidence rate), the polarization state has a high fidelity (F=95.6%) to the symmetric state (i.e., ϕ = 0), while the polarization state is closely matched (with F=95.9%) to the anti-symmetric state (ϕ=π) for τ= 47 fs, where anti-bunching (i.e., the highest coincidence rate) was observed. Those observed relations are consistent with our predictions discussed above. For the three different time delays within a beating oscillation period, we obtained comparable values of the fidelities (F 95%) and concurrences (C 0.93) to those of the frequency-bin entangled state, since the envelope of the HOMI is sim27 times wider than the beating oscillation period.

Tables Icon

Table 1. Characteristics of the polarization-mode density matrices. τ, time delay in the Michelson interferometer; I(τ)/N, normalized coincidence count rate in the HOMI; ϕ, phase of the frequency entangled state predicted from the HOMI; F, state fidelity to the ideal density matrix |ψp for ϕ; C, concurrence.

6. Conclusion

In conclusion, we have demonstrated the direct generation of frequency-bin entangled photons, using nondegenerate, two-period QPM-SPDC. Our proposed scheme has produced high-quality frequency-bin entanglement with a collinear, single-pass SPDC, not requiring spectral filters or postselective measurements. Frequency-bin entanglement of our produced photons was confirmed by measuring high-visibility beating oscillations in HOMI. Moreover, we demonstrated deterministic entanglement-mode conversion from frequency-bin to polarization modes, and verified the correlation of the symmetry of polarization states and the bunching and anti-bunching properties of the HOMI for the frequency-bin entangled photons. Our scheme can be extended to the generation of entanglement in n frequency-bin modes by periodically poling a QPM crystal with n different periods. Such high-dimensional frequency entanglement will be a key resource for high capacity quantum communications as well as for frequency-multiplexed QIP applications. Our multi-period QPM scheme is also compatible with using waveguide structures to enhance the generation rates for practical QIP applications.

Funding

Japan Society for the Promotion of Science (JSPS) (17GS1204).

Acknowledgment

We thank Mark Sadgrove and So-Young Baek for helpful discussions.

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References

  • View by:

  1. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Physical Review Letters 75, 4337–4341 (1995).
    [Crossref] [PubMed]
  2. K. Edamatsu, “Entangled Photons: Generation, Observation, and Characterization,” Japanese Journal of Applied Physics 46, 7175–7187 (2007).
    [Crossref]
  3. J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, and A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Reviews of Modern Physics 84, 777–838 (2012).
    [Crossref]
  4. C. Hong, Z. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Physical Review Letters 59, 2044–2046 (1987).
    [Crossref] [PubMed]
  5. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001).
    [Crossref] [PubMed]
  6. W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Physical Review A 64, 063815 (2001).
    [Crossref]
  7. A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, “Shaping the waveform of entangled photons,” Physical Review Letters 99, 243601 (2007).
    [Crossref]
  8. P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric downconversion: a recipe for purity,” New Journal of Physics 10, 093011 (2008).
    [Crossref]
  9. P. G. Evans, R. S. Bennink, W. P. Grice, T. S. Humble, and J. Schaake, “Bright source of spectrally uncorrelated polarization-entangled photons with nearly single-mode emission,” Physical Review Letters 105, 253601 (2010).
    [Crossref]
  10. M. Yabuno, R. Shimizu, Y. Mitsumori, H. Kosaka, and K. Edamatsu, “Four-photon quantum interferometry at a telecom wavelength,” Physical Review A 86, 010302 (2012).
    [Crossref]
  11. F. Kaneda, K. Garay-Palmett, A. B. U’Ren, and P. G. Kwiat, “Heralded single-photon source utilizing highly nondegenerate, spectrally factorable spontaneous parametric downconversion,” Optics Express 24, 10733–10747 (2016).
    [Crossref] [PubMed]
  12. C. Chen, C. Bo, M. Y. Niu, F. Xu, Z. Zhang, J. H. Shapiro, and F. N. C. Wong, “Efficient generation and characterization of spectrally factorable biphotons,” Optics Express 25, 7300–7312 (2017).
    [Crossref] [PubMed]
  13. L. Olislager, J. Cussey, A. Nguyen, P. Emplit, S. Massar, J. M. Merolla, and K. Huy, “Frequency-bin entangled photons,” Physical Review A 82, 013804 (2010).
    [Crossref]
  14. T. Zhong, H. Zhou, R. D. Horansky, C. Lee, V. B. Verma, A. E. Lita, A. Restelli, J. C. Bienfang, R. P. Mirin, T. Gerrits, S. W. Nam, F. Marsili, M. D. Shaw, Z. Zhang, L. Wang, D. Englund, G. W. Wornell, J. H. Shapiro, and F. N. C. Wong, “Photon-efficient quantum key distribution using time–energy entanglement with high-dimensional encoding,” New Journal of Physics 17, 022002 (2015).
    [Crossref]
  15. M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New Journal of Physics 17, 033033 (2015).
    [Crossref]
  16. K. McCusker and P. G. Kwiat, “Efficient Optical Quantum State Engineering,” Physical Review Letters 103, 163602 (2009).
    [Crossref] [PubMed]
  17. F. Kaneda, B. G. Christensen, J. J. Wong, H. S. Park, K. T. McCusker, and P. G. Kwiat, “Time-Multiplexed Heralded Single-Photon Source,” Optica 2, 1010–1013 (2015).
    [Crossref]
  18. F. Kaneda, F. Xu, J. Chapman, and P. G. Kwiat, “Quantum-memory-assisted multi-photon generation for efficient quantum information processing,” Optica 4, 1034–1037 (2017).
    [Crossref]
  19. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced positioning and clock synchronization,” Nature 412, 417–419 (2001).
    [Crossref] [PubMed]
  20. M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Physical Review Letters 100, 183601 (2008).
    [Crossref] [PubMed]
  21. M. Okano, H. H. Lim, R. Okamoto, N. Nishizawa, S. Kurimura, and S. Takeuchi, “m resolution two-photon interference with dispersion cancellation for quantum optical coherence tomography,” Scientific Reports 5, 18042 (2015).
    [Crossref]
  22. M. Nakatani, R. Shimizu, and K. Koshino, “Up-conversion dynamics for temporally entangled two-photon pulses,” Physical Review A 83, 013824 (2011).
    [Crossref]
  23. Z. Ou and L. Mandel, “Observation of spatial quantum beating with separated photodetectors,” Physical Review Letters 61, 54–57 (1988).
    [Crossref] [PubMed]
  24. J. Rarity and P. Tapster, “Two-color photons and nonlocality in fourth-order interference,” Physical Review A 41, 5139 (1990).
    [Crossref]
  25. Y. Shih and A. Sergienko, “Observation of quantum beating in a simple beam-splitting experiment: Two-particle entanglement in spin and space-time,” Physical Review A 50, 2564–2568 (1994).
    [Crossref]
  26. X. Li, L. Yang, X. Ma, L. Cui, Z. Y. Ou, and D. Yu, “All-fiber source of frequency-entangled photon pairs,” Physical Review A 79, 033817 (2009).
    [Crossref]
  27. Q. Zhou, W. Zhang, C. Yuan, Y. Huang, and J. Peng, “Generation of 1.5 μm discrete frequency-entangled two-photon state in polarization-maintaining fibers,” Optics letters 39, 2109–2112 (2014).
    [Crossref]
  28. Q. Zhou, S. Dong, W. Zhang, L. You, Y. He, W. Zhang, Y. Huang, and J. Peng, “Frequency-entanglement preparation based on the coherent manipulation of frequency nondegenerate energy-time entangled state,” Journal of the Optical Society of America B-Optical Physics 31, 1801–1806 (2014).
    [Crossref]
  29. S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete Tunable Color Entanglement,” Physical Review Letters 103, 253601 (2009).
    [Crossref]
  30. R.-B. Jin, R. Shimizu, M. Fujiwara, M. Takeoka, R. Wakabayashi, T. Yamashita, S. Miki, H. Terai, T. Gerrits, and M. Sasaki, “Simple method of generating and distributing frequency-entangled qudits,” Quantum Science and Technology 1, 015004 (2016).
    [Crossref]
  31. R.-B. Jin, R. Shiina, and R. Shimizu, “Quantum manipulation of biphoton spectral distributions in a 2D frequency space toward arbitrary shaping of a biphoton wave packet,” Optics Express 26, 21153–21158 (2018).
    [Crossref]
  32. H. Kim, H. J. Lee, S. M. Lee, and H. S. Moon, “Highly efficient source for frequency-entangled photon pairs generated in a 3rd-order periodically poled MgO-doped stoichiometric LiTaO3 crystal,” Optics letters 40, 3061–3064 (2015).
    [Crossref]
  33. Z. Xie, T. Zhong, S. Shrestha, X. A. Xu, J. Liang, Y. X. Gong, J. C. Bienfang, A. Restelli, J. H. Shapiro, F. N. C. Wong, and C. W. Wong, “Harnessing high-dimensional hyperentanglement through a biphoton frequency comb,” Nature Photonics 9, 536–543 (2015).
    [Crossref]
  34. W. Ueno, F. Kaneda, H. Suzuki, S. Nagano, A. Syouji, R. Shimizu, K. Suizu, and K. Edamatsu, “Entangled photon generation in two-period quasi-phase-matched parametric down-conversion,” Optics Express 20, 5508–5517 (2012).
    [Crossref]
  35. H. Herrmann, X. Yang, A. Thomas, A. Poppe, W. Sohler, and C. Silberhorn, “Post-selection free, integrated optical source of non-degenerate, polarization entangled photon pairs,” Optics Express 21, 27981 (2013).
    [Crossref]
  36. M. V. Hobden and J. Warner, “The temperature dependence of the refractive indices of pure lithium niobate,” Physics Letters 22, 243–244 (1966).
    [Crossref]
  37. D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, ne, in congruent lithium niobate,” Optics letters 22, 1553–1555 (1997).
    [Crossref]
  38. T.-M. Zhao, H. Zhang, J. Yang, Z.-R. Sang, X. Jiang, X.-H. Bao, and J.-W. Pan, “Entangling Different-Color Photons via Time-Resolved Measurement and Active Feed Forward,” Physical Review Letters 112, 103602 (2014).
    [Crossref]
  39. M. Rubin, D. Klyshko, Y. Shih, and A. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Physical Review A 50, 5122–5133 (1994).
    [Crossref]
  40. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Physical Review A 64, 052312 (2001).
    [Crossref]
  41. K. Wang, “Quantum theory of two-photon wavepacket interference in a beamsplitter,” Journal of Physics B: Atomic, Molecular and Optical Physics 39, R293–R324 (2006).
    [Crossref]
  42. A. Fedrizzi, T. Herbst, M. Aspelmeyer, M. Barbieri, T. Jennewein, and A. Zeilinger, “Anti-symmetrization reveals hidden entanglement,” New Journal of Physics 11, 103052 (2009).
    [Crossref]

2018 (1)

R.-B. Jin, R. Shiina, and R. Shimizu, “Quantum manipulation of biphoton spectral distributions in a 2D frequency space toward arbitrary shaping of a biphoton wave packet,” Optics Express 26, 21153–21158 (2018).
[Crossref]

2017 (2)

F. Kaneda, F. Xu, J. Chapman, and P. G. Kwiat, “Quantum-memory-assisted multi-photon generation for efficient quantum information processing,” Optica 4, 1034–1037 (2017).
[Crossref]

C. Chen, C. Bo, M. Y. Niu, F. Xu, Z. Zhang, J. H. Shapiro, and F. N. C. Wong, “Efficient generation and characterization of spectrally factorable biphotons,” Optics Express 25, 7300–7312 (2017).
[Crossref] [PubMed]

2016 (2)

F. Kaneda, K. Garay-Palmett, A. B. U’Ren, and P. G. Kwiat, “Heralded single-photon source utilizing highly nondegenerate, spectrally factorable spontaneous parametric downconversion,” Optics Express 24, 10733–10747 (2016).
[Crossref] [PubMed]

R.-B. Jin, R. Shimizu, M. Fujiwara, M. Takeoka, R. Wakabayashi, T. Yamashita, S. Miki, H. Terai, T. Gerrits, and M. Sasaki, “Simple method of generating and distributing frequency-entangled qudits,” Quantum Science and Technology 1, 015004 (2016).
[Crossref]

2015 (6)

F. Kaneda, B. G. Christensen, J. J. Wong, H. S. Park, K. T. McCusker, and P. G. Kwiat, “Time-Multiplexed Heralded Single-Photon Source,” Optica 2, 1010–1013 (2015).
[Crossref]

T. Zhong, H. Zhou, R. D. Horansky, C. Lee, V. B. Verma, A. E. Lita, A. Restelli, J. C. Bienfang, R. P. Mirin, T. Gerrits, S. W. Nam, F. Marsili, M. D. Shaw, Z. Zhang, L. Wang, D. Englund, G. W. Wornell, J. H. Shapiro, and F. N. C. Wong, “Photon-efficient quantum key distribution using time–energy entanglement with high-dimensional encoding,” New Journal of Physics 17, 022002 (2015).
[Crossref]

M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New Journal of Physics 17, 033033 (2015).
[Crossref]

M. Okano, H. H. Lim, R. Okamoto, N. Nishizawa, S. Kurimura, and S. Takeuchi, “m resolution two-photon interference with dispersion cancellation for quantum optical coherence tomography,” Scientific Reports 5, 18042 (2015).
[Crossref]

H. Kim, H. J. Lee, S. M. Lee, and H. S. Moon, “Highly efficient source for frequency-entangled photon pairs generated in a 3rd-order periodically poled MgO-doped stoichiometric LiTaO3 crystal,” Optics letters 40, 3061–3064 (2015).
[Crossref]

Z. Xie, T. Zhong, S. Shrestha, X. A. Xu, J. Liang, Y. X. Gong, J. C. Bienfang, A. Restelli, J. H. Shapiro, F. N. C. Wong, and C. W. Wong, “Harnessing high-dimensional hyperentanglement through a biphoton frequency comb,” Nature Photonics 9, 536–543 (2015).
[Crossref]

2014 (3)

Q. Zhou, W. Zhang, C. Yuan, Y. Huang, and J. Peng, “Generation of 1.5 μm discrete frequency-entangled two-photon state in polarization-maintaining fibers,” Optics letters 39, 2109–2112 (2014).
[Crossref]

Q. Zhou, S. Dong, W. Zhang, L. You, Y. He, W. Zhang, Y. Huang, and J. Peng, “Frequency-entanglement preparation based on the coherent manipulation of frequency nondegenerate energy-time entangled state,” Journal of the Optical Society of America B-Optical Physics 31, 1801–1806 (2014).
[Crossref]

T.-M. Zhao, H. Zhang, J. Yang, Z.-R. Sang, X. Jiang, X.-H. Bao, and J.-W. Pan, “Entangling Different-Color Photons via Time-Resolved Measurement and Active Feed Forward,” Physical Review Letters 112, 103602 (2014).
[Crossref]

2013 (1)

H. Herrmann, X. Yang, A. Thomas, A. Poppe, W. Sohler, and C. Silberhorn, “Post-selection free, integrated optical source of non-degenerate, polarization entangled photon pairs,” Optics Express 21, 27981 (2013).
[Crossref]

2012 (3)

W. Ueno, F. Kaneda, H. Suzuki, S. Nagano, A. Syouji, R. Shimizu, K. Suizu, and K. Edamatsu, “Entangled photon generation in two-period quasi-phase-matched parametric down-conversion,” Optics Express 20, 5508–5517 (2012).
[Crossref]

M. Yabuno, R. Shimizu, Y. Mitsumori, H. Kosaka, and K. Edamatsu, “Four-photon quantum interferometry at a telecom wavelength,” Physical Review A 86, 010302 (2012).
[Crossref]

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, and A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Reviews of Modern Physics 84, 777–838 (2012).
[Crossref]

2011 (1)

M. Nakatani, R. Shimizu, and K. Koshino, “Up-conversion dynamics for temporally entangled two-photon pulses,” Physical Review A 83, 013824 (2011).
[Crossref]

2010 (2)

P. G. Evans, R. S. Bennink, W. P. Grice, T. S. Humble, and J. Schaake, “Bright source of spectrally uncorrelated polarization-entangled photons with nearly single-mode emission,” Physical Review Letters 105, 253601 (2010).
[Crossref]

L. Olislager, J. Cussey, A. Nguyen, P. Emplit, S. Massar, J. M. Merolla, and K. Huy, “Frequency-bin entangled photons,” Physical Review A 82, 013804 (2010).
[Crossref]

2009 (4)

K. McCusker and P. G. Kwiat, “Efficient Optical Quantum State Engineering,” Physical Review Letters 103, 163602 (2009).
[Crossref] [PubMed]

X. Li, L. Yang, X. Ma, L. Cui, Z. Y. Ou, and D. Yu, “All-fiber source of frequency-entangled photon pairs,” Physical Review A 79, 033817 (2009).
[Crossref]

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete Tunable Color Entanglement,” Physical Review Letters 103, 253601 (2009).
[Crossref]

A. Fedrizzi, T. Herbst, M. Aspelmeyer, M. Barbieri, T. Jennewein, and A. Zeilinger, “Anti-symmetrization reveals hidden entanglement,” New Journal of Physics 11, 103052 (2009).
[Crossref]

2008 (2)

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Physical Review Letters 100, 183601 (2008).
[Crossref] [PubMed]

P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric downconversion: a recipe for purity,” New Journal of Physics 10, 093011 (2008).
[Crossref]

2007 (2)

A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, “Shaping the waveform of entangled photons,” Physical Review Letters 99, 243601 (2007).
[Crossref]

K. Edamatsu, “Entangled Photons: Generation, Observation, and Characterization,” Japanese Journal of Applied Physics 46, 7175–7187 (2007).
[Crossref]

2006 (1)

K. Wang, “Quantum theory of two-photon wavepacket interference in a beamsplitter,” Journal of Physics B: Atomic, Molecular and Optical Physics 39, R293–R324 (2006).
[Crossref]

2001 (4)

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Physical Review A 64, 052312 (2001).
[Crossref]

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001).
[Crossref] [PubMed]

W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Physical Review A 64, 063815 (2001).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced positioning and clock synchronization,” Nature 412, 417–419 (2001).
[Crossref] [PubMed]

1997 (1)

D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, ne, in congruent lithium niobate,” Optics letters 22, 1553–1555 (1997).
[Crossref]

1995 (1)

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Physical Review Letters 75, 4337–4341 (1995).
[Crossref] [PubMed]

1994 (2)

Y. Shih and A. Sergienko, “Observation of quantum beating in a simple beam-splitting experiment: Two-particle entanglement in spin and space-time,” Physical Review A 50, 2564–2568 (1994).
[Crossref]

M. Rubin, D. Klyshko, Y. Shih, and A. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Physical Review A 50, 5122–5133 (1994).
[Crossref]

1990 (1)

J. Rarity and P. Tapster, “Two-color photons and nonlocality in fourth-order interference,” Physical Review A 41, 5139 (1990).
[Crossref]

1988 (1)

Z. Ou and L. Mandel, “Observation of spatial quantum beating with separated photodetectors,” Physical Review Letters 61, 54–57 (1988).
[Crossref] [PubMed]

1987 (1)

C. Hong, Z. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Physical Review Letters 59, 2044–2046 (1987).
[Crossref] [PubMed]

1966 (1)

M. V. Hobden and J. Warner, “The temperature dependence of the refractive indices of pure lithium niobate,” Physics Letters 22, 243–244 (1966).
[Crossref]

Aspelmeyer, M.

A. Fedrizzi, T. Herbst, M. Aspelmeyer, M. Barbieri, T. Jennewein, and A. Zeilinger, “Anti-symmetrization reveals hidden entanglement,” New Journal of Physics 11, 103052 (2009).
[Crossref]

Bao, X.-H.

T.-M. Zhao, H. Zhang, J. Yang, Z.-R. Sang, X. Jiang, X.-H. Bao, and J.-W. Pan, “Entangling Different-Color Photons via Time-Resolved Measurement and Active Feed Forward,” Physical Review Letters 112, 103602 (2014).
[Crossref]

Barbieri, M.

A. Fedrizzi, T. Herbst, M. Aspelmeyer, M. Barbieri, T. Jennewein, and A. Zeilinger, “Anti-symmetrization reveals hidden entanglement,” New Journal of Physics 11, 103052 (2009).
[Crossref]

Bennink, R. S.

P. G. Evans, R. S. Bennink, W. P. Grice, T. S. Humble, and J. Schaake, “Bright source of spectrally uncorrelated polarization-entangled photons with nearly single-mode emission,” Physical Review Letters 105, 253601 (2010).
[Crossref]

Bienfang, J. C.

T. Zhong, H. Zhou, R. D. Horansky, C. Lee, V. B. Verma, A. E. Lita, A. Restelli, J. C. Bienfang, R. P. Mirin, T. Gerrits, S. W. Nam, F. Marsili, M. D. Shaw, Z. Zhang, L. Wang, D. Englund, G. W. Wornell, J. H. Shapiro, and F. N. C. Wong, “Photon-efficient quantum key distribution using time–energy entanglement with high-dimensional encoding,” New Journal of Physics 17, 022002 (2015).
[Crossref]

Z. Xie, T. Zhong, S. Shrestha, X. A. Xu, J. Liang, Y. X. Gong, J. C. Bienfang, A. Restelli, J. H. Shapiro, F. N. C. Wong, and C. W. Wong, “Harnessing high-dimensional hyperentanglement through a biphoton frequency comb,” Nature Photonics 9, 536–543 (2015).
[Crossref]

Bo, C.

C. Chen, C. Bo, M. Y. Niu, F. Xu, Z. Zhang, J. H. Shapiro, and F. N. C. Wong, “Efficient generation and characterization of spectrally factorable biphotons,” Optics Express 25, 7300–7312 (2017).
[Crossref] [PubMed]

Boyd, R. W.

M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New Journal of Physics 17, 033033 (2015).
[Crossref]

Carrasco, S.

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Physical Review Letters 100, 183601 (2008).
[Crossref] [PubMed]

Ceré, A.

A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, “Shaping the waveform of entangled photons,” Physical Review Letters 99, 243601 (2007).
[Crossref]

Chapman, J.

Chen, C.

C. Chen, C. Bo, M. Y. Niu, F. Xu, Z. Zhang, J. H. Shapiro, and F. N. C. Wong, “Efficient generation and characterization of spectrally factorable biphotons,” Optics Express 25, 7300–7312 (2017).
[Crossref] [PubMed]

Chen, Z.-B.

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, and A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Reviews of Modern Physics 84, 777–838 (2012).
[Crossref]

Christensen, B. G.

Cui, L.

X. Li, L. Yang, X. Ma, L. Cui, Z. Y. Ou, and D. Yu, “All-fiber source of frequency-entangled photon pairs,” Physical Review A 79, 033817 (2009).
[Crossref]

Cussey, J.

L. Olislager, J. Cussey, A. Nguyen, P. Emplit, S. Massar, J. M. Merolla, and K. Huy, “Frequency-bin entangled photons,” Physical Review A 82, 013804 (2010).
[Crossref]

Dong, S.

Q. Zhou, S. Dong, W. Zhang, L. You, Y. He, W. Zhang, Y. Huang, and J. Peng, “Frequency-entanglement preparation based on the coherent manipulation of frequency nondegenerate energy-time entangled state,” Journal of the Optical Society of America B-Optical Physics 31, 1801–1806 (2014).
[Crossref]

Edamatsu, K.

M. Yabuno, R. Shimizu, Y. Mitsumori, H. Kosaka, and K. Edamatsu, “Four-photon quantum interferometry at a telecom wavelength,” Physical Review A 86, 010302 (2012).
[Crossref]

W. Ueno, F. Kaneda, H. Suzuki, S. Nagano, A. Syouji, R. Shimizu, K. Suizu, and K. Edamatsu, “Entangled photon generation in two-period quasi-phase-matched parametric down-conversion,” Optics Express 20, 5508–5517 (2012).
[Crossref]

K. Edamatsu, “Entangled Photons: Generation, Observation, and Characterization,” Japanese Journal of Applied Physics 46, 7175–7187 (2007).
[Crossref]

Emplit, P.

L. Olislager, J. Cussey, A. Nguyen, P. Emplit, S. Massar, J. M. Merolla, and K. Huy, “Frequency-bin entangled photons,” Physical Review A 82, 013804 (2010).
[Crossref]

Englund, D.

T. Zhong, H. Zhou, R. D. Horansky, C. Lee, V. B. Verma, A. E. Lita, A. Restelli, J. C. Bienfang, R. P. Mirin, T. Gerrits, S. W. Nam, F. Marsili, M. D. Shaw, Z. Zhang, L. Wang, D. Englund, G. W. Wornell, J. H. Shapiro, and F. N. C. Wong, “Photon-efficient quantum key distribution using time–energy entanglement with high-dimensional encoding,” New Journal of Physics 17, 022002 (2015).
[Crossref]

Evans, P. G.

P. G. Evans, R. S. Bennink, W. P. Grice, T. S. Humble, and J. Schaake, “Bright source of spectrally uncorrelated polarization-entangled photons with nearly single-mode emission,” Physical Review Letters 105, 253601 (2010).
[Crossref]

Fedrizzi, A.

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete Tunable Color Entanglement,” Physical Review Letters 103, 253601 (2009).
[Crossref]

A. Fedrizzi, T. Herbst, M. Aspelmeyer, M. Barbieri, T. Jennewein, and A. Zeilinger, “Anti-symmetrization reveals hidden entanglement,” New Journal of Physics 11, 103052 (2009).
[Crossref]

Fejer, M. M.

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Physical Review Letters 100, 183601 (2008).
[Crossref] [PubMed]

Fujiwara, M.

R.-B. Jin, R. Shimizu, M. Fujiwara, M. Takeoka, R. Wakabayashi, T. Yamashita, S. Miki, H. Terai, T. Gerrits, and M. Sasaki, “Simple method of generating and distributing frequency-entangled qudits,” Quantum Science and Technology 1, 015004 (2016).
[Crossref]

Garay-Palmett, K.

F. Kaneda, K. Garay-Palmett, A. B. U’Ren, and P. G. Kwiat, “Heralded single-photon source utilizing highly nondegenerate, spectrally factorable spontaneous parametric downconversion,” Optics Express 24, 10733–10747 (2016).
[Crossref] [PubMed]

Gauthier, D. J.

M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New Journal of Physics 17, 033033 (2015).
[Crossref]

Gerrits, T.

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P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric downconversion: a recipe for purity,” New Journal of Physics 10, 093011 (2008).
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L. Olislager, J. Cussey, A. Nguyen, P. Emplit, S. Massar, J. M. Merolla, and K. Huy, “Frequency-bin entangled photons,” Physical Review A 82, 013804 (2010).
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M. Okano, H. H. Lim, R. Okamoto, N. Nishizawa, S. Kurimura, and S. Takeuchi, “m resolution two-photon interference with dispersion cancellation for quantum optical coherence tomography,” Scientific Reports 5, 18042 (2015).
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M. Okano, H. H. Lim, R. Okamoto, N. Nishizawa, S. Kurimura, and S. Takeuchi, “m resolution two-photon interference with dispersion cancellation for quantum optical coherence tomography,” Scientific Reports 5, 18042 (2015).
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M. Okano, H. H. Lim, R. Okamoto, N. Nishizawa, S. Kurimura, and S. Takeuchi, “m resolution two-photon interference with dispersion cancellation for quantum optical coherence tomography,” Scientific Reports 5, 18042 (2015).
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L. Olislager, J. Cussey, A. Nguyen, P. Emplit, S. Massar, J. M. Merolla, and K. Huy, “Frequency-bin entangled photons,” Physical Review A 82, 013804 (2010).
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Z. Ou and L. Mandel, “Observation of spatial quantum beating with separated photodetectors,” Physical Review Letters 61, 54–57 (1988).
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X. Li, L. Yang, X. Ma, L. Cui, Z. Y. Ou, and D. Yu, “All-fiber source of frequency-entangled photon pairs,” Physical Review A 79, 033817 (2009).
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M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New Journal of Physics 17, 033033 (2015).
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T.-M. Zhao, H. Zhang, J. Yang, Z.-R. Sang, X. Jiang, X.-H. Bao, and J.-W. Pan, “Entangling Different-Color Photons via Time-Resolved Measurement and Active Feed Forward,” Physical Review Letters 112, 103602 (2014).
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J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, and A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Reviews of Modern Physics 84, 777–838 (2012).
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Peng, J.

Q. Zhou, W. Zhang, C. Yuan, Y. Huang, and J. Peng, “Generation of 1.5 μm discrete frequency-entangled two-photon state in polarization-maintaining fibers,” Optics letters 39, 2109–2112 (2014).
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Q. Zhou, S. Dong, W. Zhang, L. You, Y. He, W. Zhang, Y. Huang, and J. Peng, “Frequency-entanglement preparation based on the coherent manipulation of frequency nondegenerate energy-time entangled state,” Journal of the Optical Society of America B-Optical Physics 31, 1801–1806 (2014).
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H. Herrmann, X. Yang, A. Thomas, A. Poppe, W. Sohler, and C. Silberhorn, “Post-selection free, integrated optical source of non-degenerate, polarization entangled photon pairs,” Optics Express 21, 27981 (2013).
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S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete Tunable Color Entanglement,” Physical Review Letters 103, 253601 (2009).
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Restelli, A.

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Figures (4)

Fig. 1
Fig. 1 Schematic diagram of the generation of frequency-bin entangled photon pairs. (a) The illustration of two-period QPM crystal. (b) Frequency-bin entanglement generation. (c) Polarization entanglement generation demonstrated in [34]. PBS: polarizing beamsplitter, DM: dichroic mirror. (d) Theoretical tuning curves for a PPLN crystal with our designed poling periods ( Λ 1 = 9.25 μm and Λ2 = 9.50 μm) and a pump wavelength of 775 nm.
Fig. 2
Fig. 2 Illustration of the experimental setup for (a) generation and detection of frequency-bin entangled photons and (b) detection of polarization entanglement. PPLN: periodically poled lithium niobate, PBS: polarizing beamsplitter, QWP: quarter-wave plate, HWP: half-wave plate, SMF: single-mode fiber, SPD: single-photon detector, DM: dichroic mirror, PA: polarization analyzer. Each PA consists of a QWP, a HWP, and a PBS.
Fig. 3
Fig. 3 Characterization of the frequency-bin entanglment. (a) Observed HOM interference for (a) -3 ps τ 3 ps and (b) -0.5 ps τ 0.5 ps. (c) Reconstructed density matrix. Error bars were calculated from Poissonian photon-counting statistics.
Fig. 4
Fig. 4 Measured polarization-mode density matrices after the entanglement-mode transfer for τ = (a) 0 fs, (b) 47 fs, and (c) -20 fs.

Tables (1)

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Table 1 Characteristics of the polarization-mode density matrices. τ, time delay in the Michelson interferometer; I ( τ ) / N, normalized coincidence count rate in the HOMI; ϕ, phase of the frequency entangled state predicted from the HOMI; F, state fidelity to the ideal density matrix | ψ p for ϕ; C, concurrence.

Equations (8)

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| FB n = j = 1 n 1 n | ω j | ω n j + 1 ,
Δ ω = ω p ω s ω i = 0 ,
Δ k = k p k s k i 2 π Λ 1 ( 2 ) = 0 ,
| ψ = 1 2 ( | H , ω 1 | V , ω 2 + e i ϕ | V , ω 1 | H , ω 2 ) ,
| ψ f = 1 2 ( | ω 1 A , H | ω 2 B , V + e i ϕ | ω 2 A , H | ω 1 B , V ) ,
| ψ p = 1 2 ( | H A , ω 1 | V B , ω 2 + e i ϕ | V A , ω 1 | H B , ω 2 ) .
I ( τ ) = { N 2 { 1 V cos  ( δ ω τ ) ( 1 | τ τ c | ) } for | τ | τ c N 2 for | τ | > τ c
ρ F = p | ω 1 ω 2 ω 1 ω 2 | A B + ( 1 p ) | ω 2 ω 1 ω 2 ω 1 | A B + V 2 ( e i ϕ | ω 1 ω 2 ω 2 ω 1 | A B + e i ϕ | ω 2 ω 1 ω 1 ω 2 | A B ) ,

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