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We theoretically investigate the generation of quantum-correlated photon pairs through spontaneous four-wave mixing in chalcogenide As2S3 waveguides. For reasonable pump power levels, we show that such photonic-chip-based photon-pair sources can exhibit high brightness (≈1 × 109 pairs/s) and high correlation (≈100) if the waveguide length is chosen properly or the waveguide dispersion is engineered. Such a high correlation is possible in the presence of Raman scattering because the Raman profile exhibits a low gain window at a Stokes shift of 7.4 THz, though it is constrained due to multi-pair generation. As the proposed scheme is based on photonic chip technologies, it has the potential to become an integrated platform for the implementation of on-chip quantum technologies.
©2010 Optical Society of America
1. Introduction
When a pair of photons is generated spontaneously from vacuum noise fluctuations in a nonlinear material, the photons exhibit correlations in energy, time, and sometimes polarization, and are commonly referred to as a ‘quantum-correlated pair’. Such photon pairs can serve either as entangled photons, or as a simple heralded single-photon source where one photon acts as a signal to ‘herald’ the arrival of the other. The generation, manipulation and detection of quantum-correlated photon pairs constitute a leading approach to the implementation of quantum technologies [1–3]. In particular, recent demonstrations of qubit operation in quantum photonic circuits have achieved integrated logic gates [2,3]; however, these experiments relied on large-scale (bulk) optical elements to generate photon pairs, making such schemes physically unscalable, sensitive to instability, and impractical for applications outside the research laboratory. This is now driving the development of integrated solutions for photon pair sources.
Quantum-correlated photon pairs are conventionally generated by spontaneous parametric down-conversion in bulk χ (2) nonlinear crystals [4–6] or by χ (3) induced spontaneous four-wave mixing (SFWM) in silica fibers [1,7–10]. Nonlinear crystals have high nonlinearity, and can thus be efficient sources of photon pairs with only short interaction lengths; however, bulk nonlinear crystal photon pair sources do not integrate well with on-chip quantum circuits [2]. Silica-fiber photon pair sources are attractive because much less free-space optics is needed; however, the broadband Raman response of silica gives rise to uncorrelated photons through spontaneous Raman scattering (SpRS) [7,8]. The development of photonic crystal fiber technologies has made significant advances towards overcoming this issue, enabling phase matching in the normal dispersion regime, far from the Raman gain band [1,8–10], with dispersion engineering. However, this platform is unsuitable for on-chip integration due to the very low nonlinearity of silica.
There have been a number of attempts to develop integrated photon pair sources based on periodically poled waveguides using KTiOPO4 (PPKTP) [11] or lithium niobate (PPLN) [12]. These two materials provide a platform for generating photon pairs in integrated waveguides, but, moving forward, they are unlikely to allow integration of pump suppression and photon-pair wavelength division devices on a single chip. Since the silicon platform has been highly successful in integrated electronic circuits and silicon has promising optical properties (e.g., high nonlinearity and extremely narrow Raman gain bandwidth), it seems a natural option for integrated photon pair sources. Therefore photon pair sources based on silicon have also attracted attention [13–16]. However, the two-photon absorption-induced long-life-time free carriers in silicon result in free-carrier absorption (FCA), which may absorb one or both photons of each pair and degrade the source quality [13]. Thus there remains a need for a platform for correlated pair generation that supports integration and has no free carriers.
Chalcogenide As2S3 glass exhibits high third-order nonlinearity (100 times higher than silica), high transparency at infrared wavelengths, relatively narrow Raman response [17,18], negligible two-photon absorption, and significant photosensitivity [19]. The unique combination of these features has made chalcogenide attractive for developing next-generation integrated photonic chip platforms for ultrafast all-optical signal processing [19]. In particular, stimulated four-wave mixing (FWM) experiments in chalcogenide waveguides have demonstrated high parametric gain [20], indicating the potential for bright correlated photon pair generation based on SFWM. Since two-photon absorption is low, there are negligible induced free carriers in chalcogenide, and photon pairs do not suffer from FCA. The glass photosensitivity offers the feasibility of building optical devices such as Bragg gratings and arrayed waveguide gratings into the photon pair source for pump suppression and wavelength division, which is ideal for quantum circuit compatible integration.
In this paper, we theoretically investigate the generation of quantum-correlated photon pairs in the telecom band in dispersion engineered chalcogenide waveguides by SFWM in the anomalous dispersion regime. Note that it is not fundamental limits that restrict us to work in the anomalous dispersion regime, but, rather, practical considerations—As2S3 becomes highly lossy below about 1 μm. SFWM in the normal dispersion regime, say with a pump in the telecom band, would avoid Raman scattering effects but would generate photon pairs at wavelengths that are highly lossy in As2S3 and are out of the efficient detection range of infrared single-photon detectors. Instead, we work in the anomalous dispersion regime, with pump-signal shifts of a few THz, and must manage the effects of SpRS.
To guide practical waveguide design and photon pair generation, we study the generation rate of photon pairs by SFWM and that of noise photons by SpRS in waveguides with different dispersion profiles. Based on this, we propose a scheme to increase the pair correlation by minimizing SpRS noise through waveguide dispersion engineering, and give a design example of the optimum waveguide, thus paving the way for future implementations of integrated quantum technologies. We also provide a map of the biphoton probability density of the correlated-pair state, obtained from an extension of a quantum description of spontaneous processes in waveguides [21]. This biphoton description shows good agreement with the semi-classical analysis.
2. Theory
2.1 Photon generation rates
We focus on correlated photon pair generation from SFWM in the anomalous dispersion regime induced by a single continuous-wave (CW) pump launched in an As2S3 waveguide, as illustrated in Fig. 1 . Two pump photons at ω p are annihilated to create a signal photon (anti-Stokes at ω s) and an idler photon (Stokes at ω i = 2ω p–ω s) simultaneously in the same polarization mode as the pump photons if the phase-matching condition 2k p–k s–k i–2γP 0 = 0 is met [22]. Here k p, k s and k i are the propagation constants evaluated at ω p, ω s and ω i, respectively; γ = n 2 ω p/(cA eff) is the nonlinear coefficient with n 2 the nonlinear refractive index, c the light speed in vacuum and A eff the effective mode area. Finally P 0 is the pump power in the waveguide.
For correlated photon pair generation, the pump power is kept relatively low to prevent multi-pair generation [23,24]. In this limit, the SFWM photon pair flux spectral density, defined as the number of photons generated per unit frequency and per unit time, at frequency detuning ν generated over a propagation distance L is given by [25]
Here ν = (ω s–ω p)/2π = (ω p–ω i)/2π and β 2 = d2 β/dω 2 is the group velocity dispersion (GVD) parameter at ω p. The sinc function in Eq. (1) incorporates the phase-matching condition. In writing Eq. (1) we have neglected losses, which is not strictly appropriate in the chalcogenide glasses. We return to this issue in Section 2.2. Having in mind a typical experiment, where ~1 nm band-pass filters are used to spectrally filter the photon pairs, we integrate Eq. (1) over a 1 nm (Δν≈0.12 THz) bandwidth centered at ν, and obtain the photon pair generation rate by SFWMAs mentioned above, correlated photon pair sources based on silica fibers suffer from noise generated by SpRS because the SFWM gain band overlaps with the broad Raman gain band of silica glass [7,8]. As amorphous materials, chalcogenide glasses also have Raman gain in a certain bandwidth. The normalized Raman gain profile of As2S3 is reproduced in Fig. 2 together with that of silica [17,22]. The normalized Raman gain profile Im[h R(ν)] for each material is the imaginary part of the Fourier transform of its normalized Raman response function h R(t) [22]. The absolute strength of the Raman response contains the nonlinear coefficient γ (see Eq. (4)). In Fig. 2, the Raman gain data available for As2S3 only extends to 16 THz pump-Stokes frequency detuning, which is sufficient for our purposes.
Fig. 2 Raman gain profile for silica (broken line) and As2S3 (solid line). The inset shows the details of the low-Raman-gain window of chalcogenide.
Figure 2 shows that the normalized Raman response of As2S3 has a low-gain window at around 7.4 THz, as indicated by the arrow and magnified in the figure inset. This window is significant when the photon pairs are generated in the anomalous dispersion regime, as considered here, because the window provides a mechanism for minimizing the impact of SpRS noise. With this in mind, we investigate the impact of Raman noise on pair correlation. The SpRS photon flux spectral density in the idler channel can be calculated as [23,24]
where g R(ν) is the Raman gain spectrum of As2S3 glass, defined as [22]Here f R is the fractional contribution of the delayed Raman response; n th is the phonon population at frequency ν and at temperature T, which is described by the Bose-Einstein distributionwhere h is Planck’s constant, and k B is the Boltzmann constant. Similar to Eq. (2), the SpRS photon generation rate over a 1 nm bandwidth is given by2.2 Losses
While extraordinary in many ways, the chalcogenide glasses do exhibit non-negligible losses. The current recorded low loss for As2S3 waveguides is approximately α dB = 0.3 dB/cm (α dB = 4.343α) [26], a value which is probably close to the optimal possible. There is little point in using waveguides significantly longer than a distance 1/α, which for 0.3 dB/cm is 14.5 cm, since the intensity is too weak to contribute to the photon flux. Moreover unless, the loss will also impact the correlation of the signal and idler photons, since there is a significant possibility that at least one will be lost [16]. A complete theory of spontaneous quantum processes in lossy waveguides allowing calculation of the mixed photon states that would then arise would require a treatment in terms of density operators and the full apparatus of dissipative quantum systems. Such a treatment does not yet exist, although an analysis in the low loss regime that incorporates noise operators into the SFWM Heisenberg evolution equations has been presented [27]. Most recently an analysis has investigated quantum entanglement in lossy waveguides, but it deals with pair propagation in coupled waveguides rather than pair generation in channel waveguides [28]. Some of us plan to present a full theory to study pair generation in lossy channel waveguides in the near future [29].
In practice however, the significant insertion losses for waveguide experiments and the very low efficiencies for current single photon detectors (as low as a few percent) mean that the intrinsic waveguide losses are swamped by these external ones. In the present case therefore, we only consider waveguides with lengths and proceed to use results that neglect loss.
2.3 Figure of merit and quantum correlation
Using Eq. (2) and (6), we can define the figure of merit F comparing the rate of photons generated via SFWM (signal) to the rate of photons generated due to SpRS (noise) as
With the definition of Eq. (7), it is clear that the performance of photon pair sources can be improved by employing different materials or different experimental design with the same material. For example, there have been a number of attempts to increase this F in silica fibers: shifting the SFWM gain band far from the Raman gain band to reduce Im[h R(ν)] [1,8–10], and cooling the fiber to decrease n th [30]. Later in this paper we will discuss a scheme to increase the F in chalcogenide waveguides.Photon pair brightness and quantum correlation are the two standard measurements of the quality of photon pair sources. The photon pair brightness is determined by Eq. (2). The quantum correlation between signal and idler photons is defined as the ratio between true and accidental coincidences [23]. The figure of merit F defined by Eq. (7) describes the degree to which Raman noise affects the pair correlation. The other important factor inducing uncorrelated photons is multi-pair generation at high power [23,24], which constrains us from improving F by increasing power. A full description of pair correlation should include both effects, which can be quantified by the cross-correlation coefficient of signal and idler photons. Assuming identically shaped signal and idler filters, and perfect phase-matching (i.e., in the narrow bandwidth determined by the signal and idler filters, so that the sinc function in Eq. (1) is close to 1.), the pair correlation is given by [23,24]
where η = 1–f R + f R h R(ν). In the limit of a pure SFWM process (i.e., f R = 0 and g R = 0), the correlation value reaches its upper limitThe subscripts ‘Raman’ and ‘no Raman’ in Eqs. (8) and (9) denote the pair correlation with and without Raman noise, respectively. The pair correlation defined here has the same physical significance as the coincidence to accidental ratio (CAR) that is often used in experiments [8–16]. The difference is that the CAR in experiments takes the propagation loss of the signal and idler photons, dark counts and detector efficiencies into account. The theoretical pair correlation value is normally higher than the measured CAR, but provides a good indication of the intrinsic pair quality.3. Results and analysis
We first consider an existing waveguide structure that has been previously studied for classical FWM. The device has a width of 2 μm and a height of 850 nm with an etching depth of 350 nm [20]. This structure has been dispersion engineered such that only the TM0 mode is appropriate for efficient (classical) FWM. At 1550 nm the TM0 mode group-velocity dispersion (GVD) is D = 27 ps/nm/km which corresponds to β 2 = –3.43x10−26 s2/m. The nonlinear coefficient γ = 10 W−1m−1. As we discussed in Section 2.2, we focus on the limit of so that the loss effect is negligible. For the best record of loss α = 0.3 dB/cm, L = 2 cm is chosen in the following calculation so that αL = 0.138.
Using Eqs. (2) and (6), we calculate the expected SFWM photon pair and SpRS photon generation rates over a bandwidth of 1 nm at different pump power levels (i.e., γP 0 L) as a function of the pump-idler frequency detuning when the pump wavelength is set at 1550 nm. The results are plotted in Fig. 3(a) (blue curves for SFWM and red curves for SpRS). It can be seen that there is a certain SFWM photon generation bandwidth (~6 THz, the green area) that is determined by the phase-matching condition and the dispersion of the waveguide. Within this SFWM photon generation band, for a particular pump power, the photon pair generation rate from SFWM is approximately constant over the entire 6 THz pump-idler frequency detuning. This means that one can choose filters arbitrarily within this bandwidth according to the experimental requirements.
Fig. 3 (a) Photon generation rates by SFWM (blue) and SpRS (red) in a 2 cm-long waveguide. The green area shows the flat top of the SFWM photon pair generation band. (b) Pair generation rate and correlation at pump-idle frequency detuning where phase matching is perfect.
According to the Raman gain profile in Fig. 2, the normalized Raman response in As2S3 for small pump-idler detuning is similar to that in silica. As we can see in Fig. 3(a), in the idler photon channel, Raman photons contribute noise which will degrade the quality of photon pair sources. Figure 3(b) shows the SFWM photon pair generation rate (Eq. (2)) and photon pair correlation (Eq. (8)) as a function of γP 0 L when the phase matching is perfect (i.e., at pump-idler frequency detunings covered by the green area in Fig. 3(a)). It can be seen that we need to choose an appropriate pump power level to ensure a high photon pair generation rate and high correlation.
To avoid multi-pair generation degrading the generated photon pair correlation, it would be natural to lower the pump power. However, Fig. 3 indicates that Raman noise is significant in the pump-idler detuning range of <6 THz, which leads to a lower correlation than a similar device in silicon when this device is operated at low power. As we mentioned in Section 1, we cannot adopt the idea used in silica fiber design either, namely, to tailor the waveguide dispersion so that phase-matched SFWM occurs in the normal dispersion regime. However, as shown in Fig. 2 and Fig. 3, there is a spectral range in As2S3 where Raman noise is very low. If we can design the waveguide to broaden the flat top of the SFWM photon generation band so that perfect phase matching can be achieved for the SFWM idler photon generated around the pump-idler detuning of 7.4 THz, the pair correlation should increase due to the higher F defined by Eq. (7) even at a relatively low power. According to Eq. (1), the waveguide should be designed to allow broadband phase matching.
There are two ways to achieve this goal. One is to reduce the waveguide length so that the phase mismatching is negligible even at a relatively larger pump-idler frequency detuning. It can be seen from Fig. 3(a) that a 2 cm length is almost ideal because the SFWM photon generation rate in the 7.4 THz window is only slightly less than the flat top. Using a filter to select the idler photons in that window will certainly minimize the effect of Raman noise on pair correlation. For comparison, we plot the photon generation rate in a 14 cm-long and a 1 cm-long waveguide in Fig. 4 (a) and 4(b), respectively. Although the photon generation rate for the 14 cm-long waveguide will not be correct, as we neglect the loss, its shape will be essentially correct, indicating a much narrower pair generation bandwidth of ~2 THz that does not overlap with the low Raman gain window. On the other hand, our calculation shows that reducing the length to 1 cm extends the flat top gain region over the 7.4 THz window. The price for reducing the waveguide length is the need to increase the pump power to maintain the same pair brightness. For example, 1 W of power is required to keep γP 0 L = 0.1 when L = 1 cm. In this case, a CW pump would not be possible because the damage threshold of this device is 0.6 W of CW power [26]. A picosecond pulsed pump will be necessary for the experiment. For a pulse of 10 ps, the dispersion length in this device is a few kilometers and the walk-off length between pump, signal and idler photons is a few meters. So the 10 ps pulse can be treated as quasi-CW and the theory presented in Section 2 is still valid.
Fig. 4 Photon generation rates by SFWM (blue) and SpRS (red) in a (a) 14 cm and (b) 1 cm-long waveguide.
Another way is to design the waveguide structure to obtain a smaller dispersion at the pump wavelength 1550 nm. The dispersion of the waveguide can be tailored through designing the waveguide cross-section dimensions such as the width, height and etching depth (Fig. 5 inset) [31]. Using a commercial finite-element method package (RSoft FemSIM), we calculate the dispersion for waveguides with different cross-section dimensions. We find that the dispersion at 1550 nm is 13.5 ps/nm/km if the waveguide has a height of 920 nm and the same width and etching ratio as the preceding waveguide (i.e., etching depth = 350/850 × 920 nm = 379 nm). Figure 5 shows the calculated dispersion profiles for the TM0 mode of both 850 and 920 nm high waveguides. Similar to Fig. 3 and Fig. 4, Fig. 6 shows the SFWM photon pair and SpRS photon generation rates when D = 13.5 ps/nm/km. It can be seen that the flat top of the photon pair generation band extends over the low-Raman-gain window. In this case one can use a filter to extract idler photons in the low Raman noise region.
Fig. 5 The dispersion profiles of the TM0 mode of waveguides with a height of 850 and 920 nm. The inset is the schematic cross-section of a chalcogenide waveguide.
Fig. 6 Photon generation rates by SFWM (blue) and SpRS (red) in a 2 cm-long waveguide when D = 13.5 ps/nm/km.
To compare the pair correlation when D = 27 and 13.5 ps/nm/km, we plot the pair correlation ratio ρ Raman/ρ no Raman for the 2 cm-long waveguides with the height of 850 nm and 920 nm together in Fig. 7 ; The closer ρ Raman/ρ no Raman is to 1, the less impact Raman noise has on pair correlation. At low power (e.g., γP 0 L = 0.06), this ratio is enhanced by a factor of 2 after we tailor the dispersion of the waveguide from 27 to 13.5 ps/nm/km. This is because the dispersion engineering technique enables perfect phase matching for the idler photon generated in the low-Raman-gain window, which enhances the figure of merit F defined by Eq. (7). For each waveguide, when the power goes up, the ratio ρ Raman/ρ no Raman gets closer to 1 as well, but this is because the effect of multi-pair generation overwhelms that of Raman noise and makes the pair correlation even worse (see Fig. 3(b)). Thus, to improve the pair correlation in chalcogenide waveguides, we should first decrease the power to a level, at which the photon pair counts are sufficient and Raman noise is the dominant source of uncorrelated photons; next we should apply the dispersion engineering technique to enable perfect phase matched pair generation in the minimum Raman noise window.
Fig. 7 The pair correlation ratio ρ Raman/ρ no Raman, for the 2 cm-long waveguides with different dispersion profiles. Green for D = 13.5 ps/nm/km and red for D = 27 ps/nm/km.
We further characterize generated photon pairs resulting from this new waveguide design with a treatment adapted from [21] that describes the full quantum state of photons at the output of the waveguide and not just the probability of photon pair generation, for a given input coherent pump pulse in the undepleted pump approximation. In the limit of a low probability of pair production per pump pulse, the quantum state is specified through a normalized function of two variables known as the biphoton wave function, i.e.,
where φ(ν 1, ν 2) is the biphoton wave function. This function arises naturally within the theory and contains all of the information about the quantum correlated pair; that is, it can be used to calculate, for example, various correlations, a measure of the entanglement such as the Schmidt Number [32], or the width of the dip in a Hong-Ou-Mandel type experiment [1,4]. We note that the biphoton wave function multiplied by its complex conjugate, something we refer to as the biphoton probability density, is directly related to what is sometimes called the joint spectral intensity. Being a normalized function, the biphoton probability density maps out the likelihood of where in frequency space a correlated photon pair will be produced, neglecting the probability of whether or not a correlated photon pair is generated, and considering only cases when it is. Furthermore, integrating over one of the two frequencies and weighting the result by the total number of generated photon pairs, |ξ|2, per pump pulse, allows one to calculate the photon pair flux spectral density. Indeed, moving from a pump pulse to a theory involving a CW pump, where the pump field can be treated classically and self-phase modulation effects included easily, and carrying out this integration, Eq. (1) is easily reproduced. While the current fully quantum mechanical theory does not yet include the effects of self- and cross-phase modulation, we note that this effect is likely to be small in practice: for fixed, and very small, γP0L, and large enough detuning, ν, from the pump, the main contribution to the phase matching sinc function appearing in both Eq. (1) and the biphoton wave function, is from the term containing ν, and thus it is reasonable to neglect the self-phase modulation γP0L term anyway. In Fig. 8 , we plot the biphoton probability density resulting from a 10 ps intensity FWHM pulse incident on the 2 cm-long and 920 nm-high structure. Note how narrow it is along lines of ν r = (ν 1–ν 2)/2 compared to along the lines of ν t = ν 1 + ν 2. This is because a 10 ps pulse, while quite short compared to CW, is, as far as this structure is concerned, spectrally narrow enough to effectively ensure exact energy conservation as in the CW case. This is true in general for waveguide structures in which the sinc function of Eq. (1) is quite spectrally broad compared to the frequency spectrum of the pump pulse.4. Conclusion
We have quantified the quality of photon pairs generated by SFWM in chalcogenide As2S3 waveguides. We show that the pair correlation is sensitive to the Raman gain profile of the material at small pump-idler detuning and has an upper limit due to multi-pair generation. At a typical pump power level of γP 0 L = 0.1, such photonic-chip-based photon pair sources can exhibit high brightness (≈1 × 109 pairs/s). The pair correlation can be enhanced up to 100 either by reducing the waveguide length or by proper dispersion engineering to avoid the high Raman noise region. A length of 2 cm provides a good tradeoff between the lower losses and larger photon pair generation bandwidths afforded by shorter waveguides, and the lower power requirements afforded by longer waveguides. A picosecond pulsed pump is required for a short waveguide (e.g., 1 cm) to maintain high pair brightness without damaging the waveguide. As the proposed scheme is based on mature photonic chip technology, it has the potential to become an integrated platform for all on-chip quantum information processing applications.
Acknowledgments
We acknowledge the support of the Australian Research Council (ARC) Centre of Excellence program and the ARC Federation Fellowship Program. The Centre for Ultrahigh bandwidth Devices for Optical Systems (CUDOS) is an ARC Centre of Excellence. L. G. Helt and J. E. Sipe acknowledge the support from the National Sciences and Engineering Research Council of Canada (NSERC).
References and links
1. J. Fulconis, O. Alibart, J. L. O’Brien, W. J. Wadsworth, and J. G. Rarity, “Nonclassical interference and entanglement generation using a photonic crystal fiber pair photon source,” Phys. Rev. Lett. 99(12), 120501 (2007). [CrossRef] [PubMed]
2. A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320(5876), 646–649 (2008). [CrossRef] [PubMed]
3. J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulating multi-photon entanglement in waveguide quantum circuits,” Nat. Photonics 3(6), 346–350 (2009). [CrossRef]
4. C. K. Hong and L. Mandel, “Experimental realization of a localized one-photon state,” Phys. Rev. Lett. 56(1), 58–60 (1986). [CrossRef] [PubMed]
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. D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl,, M. Daniell, H. Weinfurter, and A. Zeilinger, “Experimental Quantum Teleportation,” Nature 390(6660), 575–579 (1997). [CrossRef]
7. X. Li, J. Chen, P. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications: Improved generation of correlated photons,” Opt. Express 12(16), 3737–3744 (2004). [CrossRef] [PubMed]
8. J. G. Rarity, J. Fulconis, J. Duligall, W. J. Wadsworth, and P. St. J. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express 13(2), 534–544 (2005). [CrossRef] [PubMed]
9. A. R. McMillan, J. Fulconis, M. Halder, C. Xiong, J. G. Rarity, and W. J. Wadsworth, “Narrowband high-fidelity all-fibre source of heralded single photons at 1570 nm,” Opt. Express 17(8), 6156–6165 (2009). [CrossRef] [PubMed]
10. M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express 17(6), 4670–4676 (2009). [CrossRef] [PubMed]
11. A. B. U’Ren, C. Silberhorn, K. Banaszek, and I. A. Walmsley, “Efficient conditional preparation of high-fidelity single photon states for fiber-optic quantum networks,” Phys. Rev. Lett. 93(9), 093601 (2004). [CrossRef] [PubMed]
12. O. Alibart, D. B. Ostrowsky, P. Baldi, and S. Tanzilli, “High-performance guided-wave asynchronous heralded single-photon source,” Opt. Lett. 30(12), 1539–1541 (2005). [CrossRef] [PubMed]
13. J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar, “Generation of correlated photons in nanoscale silicon waveguides,” Opt. Express 14(25), 12388–12393 (2006). [CrossRef] [PubMed]
14. H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and I. Sei-ichi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett. 91(20), 201108 (2007). [CrossRef]
15. K. Harada, H. Takesue, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, Y. Tokura, and S. Itabashi, “Generation of high-purity entangled photon pairs using silicon wire waveguide,” Opt. Express 16(25), 20368–20373 (2008). [CrossRef] [PubMed]
16. H. Ken-ichi, H. Takesue, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, Y. Tokura, and S. Itabashi, “Frequency and Polarization Characteristics of Correlated Photon-Pair Generation Using a Silicon Wire Waveguide,” IEEE J. Sel. Top. Quantum Electron. 16(1), 325–331 (2010). [CrossRef]
17. R. J. Kobliska and S. A. Solin, “Temperature Dependence of the Raman Spectrum and the Depolarization Spectrum of Amorphous As2S3,” Phys. Rev. B 8(2), 756–768 (1973). [CrossRef]
18. C. Xiong, E. Magi, F. Luan, A. Tuniz, S. Dekker, J. S. Sanghera, L. B. Shaw, I. D. Aggarwal, and B. J. Eggleton, “Characterization of picosecond pulse nonlinear propagation in chalcogenide As(2)S(3) fiber,” Appl. Opt. 48(29), 5467–5474 (2009). [CrossRef] [PubMed]
19. V. G. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. E. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express 15(15), 9205–9221 (2007). [CrossRef] [PubMed]
20. M. R. E. Lamont, B. Luther-Davies, D. Y. Choi, S. Madden, X. Gai, and B. J. Eggleton, “Net-gain from a parametric amplifier on a chalcogenide optical chip,” Opt. Express 16(25), 20374–20381 (2008). [CrossRef] [PubMed]
21. Z. Yang, M. Liscidini, and J. E. Sipe, “Spontaneous parametric down-conversion in waveguides: A backwards Heisenberg picture approach,” Phys. Rev. A 77(3), 033808 (2008). [CrossRef]
22. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, Elsevier, 2007), chap. 2, 8 & 10.
23. Q. Lin, F. Yaman, and G. P. Agrawal, “Photon-pair generation by four-wave mixing in optical fibers,” Opt. Lett. 31(9), 1286–1288 (2006). [CrossRef] [PubMed]
24. Q. Lin, F. Yaman, and G. P. Agrawa, “Photon-pair generation in optical fibers through four-wave mixing: Role of Raman scattering and pump polarization,” Phys. Rev. A 75(2), 023803 (2007). [CrossRef]
25. E. Brainis, “Four-photon scattering in birefringent fibers,” Phys. Rev. A 79(2), 023840 (2009). [CrossRef]
26. Private communication with Prof, Barry Luther-Davies at the Australian National University.
27. Q. Lin and G. P. Agrawal, “Silicon waveguides for creating quantum-correlated photon pairs,” Opt. Lett. 31(21), 3140–3142 (2006). [CrossRef] [PubMed]
28. A. Rai, S. Das, and G. S. Agarwal, “Quantum entanglement in coupled lossy waveguides,” Opt. Express 18(6), 6241–6254 (2010). [CrossRef] [PubMed]
29. L. G. Helt and J. E. Sipe, Quantum States of Generated Photons in Strongly Confined Lossy Waveguides (in preparation).
30. S. D. Dyer, M. J. Stevens, B. Baek, and S. W. Nam, “High-efficiency, ultra low-noise all-fiber photon-pair source,” Opt. Express 16(13), 9966–9977 (2008). [CrossRef] [PubMed]
31. M. R. Lamont, C. M. de Sterke, and B. J. Eggleton, “Dispersion engineering of highly nonlinear As(2)S(3) waveguides for parametric gain and wavelength conversion,” Opt. Express 15(15), 9458–9463 (2007). [CrossRef] [PubMed]
32. C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92(12), 127903 (2004). [CrossRef] [PubMed]
