## Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber

Optics Express, Vol. 15, Issue 22, pp. 14870-14886 (2007)

http://dx.doi.org/10.1364/OE.15.014870

Acrobat PDF (828 KB)

### Abstract

We study theoretically the generation of photon pairs by spontaneous four-wave mixing (SFWM) in photonic crystal optical fiber. We show that it is possible to engineer two-photon states with specific spectral correlation (“entanglement”) properties suitable for quantum information processing applications. We focus on the case exhibiting no spectral correlations in the two-photon component of the state, which we call factorability, and which allows heralding of single-photon pure-state wave packets without the need for spectral post filtering. We show that spontaneous four wave mixing exhibits a remarkable flexibility, permitting a wider class of two-photon states, including ultra-broadband, highly-anticorrelated states.

© 2007 Optical Society of America

## 1. Introduction

1.
See, for example, the review by
P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. **79**, 135–174 (2007). [CrossRef]

2. S. E. Harris, M. K. Oshman, and R. L. Byer, “Observation of Tunable Optical Parametric Fluorescence,” Phys. Rev. Lett. **18**, 732–734 (1967). [CrossRef]

4. M. G. Raymer, J. Noh, K. Banaszek, and I. A. Walmsley, “Pure-state single-photon wave-packet generation by parmametric down-conversion in a distributed microcavity,” Phys. Rev. A **72**, 023825 (2005). [CrossRef]

5. 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**, 093601 (2004). [CrossRef] [PubMed]

6. K. Banaszek, A. B. U’Ren, and I. A. Walmsley, “Generation of correlated photons in controlled spatial modes by downconversion in nonlinear waveguides,” Opt. Lett. **26**, 1367–1369 (2001). [CrossRef]

7. J. Fan and A. Migdall, “A broadband high spectral brightness fiber-based two-photon source,” Opt. Express **15**, 2915–2920 (2007). [CrossRef] [PubMed]

8. J. Rarity, J. Fulconis, J. Duligall, W. Wadsworth, and P. St. J. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express **13**, 534–544 (2005). [CrossRef] [PubMed]

9. J. Fan and A. Migdall, “Generation of cross-polarized photon pairs in a microstructure fiber with frequency-conjugate laser pump pulses,” Opt. Express **13**, 5777–5782 (2005). [CrossRef] [PubMed]

10. X. Li, J. Chen, P. Voss, J. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications: Improved generation of correlated photons,” Opt. Express **12**, 3737–3744 (2004). [CrossRef] [PubMed]

11. W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A **64**, 063815 (2001). [CrossRef]

12. P. Russell, “Photonic Crystal Fiber,” Science **299**, 358–362 (2003). [CrossRef] [PubMed]

*all*degrees of freedom, resulting in factorable two-photon states.

13. M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. **14**, 983–985 (2002). [CrossRef]

7. J. Fan and A. Migdall, “A broadband high spectral brightness fiber-based two-photon source,” Opt. Express **15**, 2915–2920 (2007). [CrossRef] [PubMed]

8. J. Rarity, J. Fulconis, J. Duligall, W. Wadsworth, and P. St. J. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express **13**, 534–544 (2005). [CrossRef] [PubMed]

15. R. Jiang, R. Saperstein, N. Alic, M. Nezhad, C. J. McKinstrie, J. Ford, S. Fainman, and S. Radic, “Parametric wavelength conversion from conventional near-infrared to visible band,” IEEE Photon. Technol. Lett. **18**, 2445–2447 (2006). [CrossRef]

16. J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber,” Opt. Lett. **28**, 2225–2227 (2003). [CrossRef] [PubMed]

17. M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Modulational instabilities in dispersion-flattened fibers,” Phys. Rev. E **52**, 1072–1080 (1995). [CrossRef]

*ψ*〉 = ∣0〉

_{s}∣0〉

_{i}+

*κ*∣1〉

_{s}∣1〉

_{i}, in the photon number basis with

*s, i*indicating signal and idler modes respectively. We call this property “factorability,” which corresponds to the absence of correlations between the frequencies (and momenta) of the idler and signal photons. In physical terms, factorability implies that no information about, say, the idler photon (apart from its existence) can be extracted from the detection of the signal photon, or vice-versa. To date, such factorable states have been produced only in PDC at a single wavelength using a particular crystal having special dispersion properties [18], and other techniques could in principle be used to extend possible operation wavelengths [4

4. M. G. Raymer, J. Noh, K. Banaszek, and I. A. Walmsley, “Pure-state single-photon wave-packet generation by parmametric down-conversion in a distributed microcavity,” Phys. Rev. A **72**, 023825 (2005). [CrossRef]

19. Z.D. Walton, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Polarization-Entangled Photon Pairs with Arbitrary Joint Spectrum” Phys. Rev. A **70**, 052317 (2004) [CrossRef]

20. J. P. Torres, F. Macia, S. Carrasco, and L. Torner, “Engineering the frequency correlations of entangled two-photon states by achromatic phase matching” Opt. Lett. **30**, 314 (2005) [CrossRef] [PubMed]

21. O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. Kärtner, Phys. Rev. Lett.94, 083601 (2005) [CrossRef] [PubMed]

23. A. B. U’Ren, R. Erdmann, M. De la Cruz, and I. A. Walmsley, ”Generation of two-photon states with an arbitrary degree of entanglement via nonlinear crystal superlattices,” Phys. Rev. Lett. **97**, 223602 (2006). [CrossRef] [PubMed]

## 2. Spontaneous four wave mixing theory

*χ*

^{(3)}. In this process, two-photons from pump fields

*E*

_{1}and

*E*

_{2}are jointly annihilated to create a photon pair comprised of one photon in the signal mode,

*E*^

_{s}, and one photon in the idler mode

*E*^

_{i}. We assume that all fields propagate in the fundamental spatial mode of the fiber. This assumption is justified if the fiber core radius is small enough that it only supports the fundamental mode, or alternatively if only this fundamental mode is excited. Following a standard perturbative approach [26], the two-photon state produced by spontaneous four-wave mixing in an optical fiber of length

*L*can be shown to be given by [27

27. J. Chen, X. Li, and P. Kumar, “Two-photon-state generation via four-wave mixing in optical fibers,” Phys. Rev. A **72**, 033801 (2005). [CrossRef]

*κ*is a constant which represents the generation efficiency (linearly proportional to the fiber length, electric field amplitude for each of the pump fields and dependent on the relative polarizations of the pump and created pair fields) and

*F*(

*ω*,

_{s}*ω*) is the joint spectral amplitude function (JSA), which describes the spectral entanglement properties of the generated photon pair

_{i}*α*

_{1,2}(

*ω*), and the phase mismatch function Δ

*k*(

*ω*

_{1},

*ω*,

_{s}*ω*), that in the case where the two pumps, signal and idler are co-polarized, is given by

_{i}*P*

_{1}and

*P*

_{2}, characterized by the nonlinear parameters

*γ*

_{1}and

*γ*

_{2}, which depend on specific fiber used and pump wavelength[29, 30

30. C. J. McKinstrie, H. Kogelnik, and L. Schenato, “Four-wave mixing in a rapidly-spun fiber,” Opt. Express **15**, 8516–8534 (2006). This paper also reviews scalar and vector FWM in strongly-birefringent and randomly-birefringent fibers. [CrossRef]

*F*(

*ω*,

_{s}*ω*) is equal to a product of two functions,

_{i}*F*(

*ω*,

_{s}*ω*) =

_{i}*S*(

*ω*)

_{s}*I*(

*ω*), where the functions

_{i}*S*(

*ω*) and

*I*(

*ω*) depend only on the signal and idler frequencies respectively.

*α*

_{1,2}(

*ω*) as Gaussian functions with bandwidth

*σ*

_{1,2}respectively, it is possible to obtain an expression for the joint spectral amplitude in closed analytical form. Expanding

*k*(

*ω*) in a first-order Taylor series about frequencies

_{μ}*ω*

_{μ}^{0}for which perfect phase-matching is attained (where

*μ*= 1,2,

*s, i*), and defining the detunings

*ν*=

_{s}*ω*-

_{s}*ω*

_{s}^{0}and

*ν*=

_{i}*ω*-

_{i}*ω*

_{i}^{0}, the approximate phase mismatch Δ

*k*is defined by

_{lin}*k*

^{(0)}, given by Eq.(3) evaluated at the frequencies

*ω*

_{μ}^{0}, must vanish to guarantee phase-matching at these center frequencies. The coefficients

*T*are given by

_{μ}*T*=

_{μ}*τ*+

_{μ}*τ*

_{p}*σ*

_{1}

^{2}/(

*σ*

_{1}

^{2}+

*σ*

_{2}

^{2}), where

*τ*represent group-velocity mismatch terms between the pump centered at frequency

_{μ}*ω*

_{2}

^{0}and the generated photon centered at the frequency

*ω*

_{μ}^{0}, and

*τ*is the group velocity mismatch between the two pumps

_{p}*k*

_{μ}^{(n)}(

*ω*) =

*d*/

^{n}k_{μ}*dω*∣

^{n}_{ω=ωμ0}. Note that this approach requires

*apriori*knowledge, for given pump fields, of the signal and idler frequencies (

*ω*

_{s}^{0}and

*ω*

_{i}^{0}) at which perfect phase-matching is achieved. These frequencies can be determined by solving (for example numerically) the condition Δ

*k*

^{(0)}= 0 (see Eq.(3)). It can be shown that within the linear approximation, the integral in Eq. (2) can be carried out analytically, yielding

*α*(

*ν*,

_{s}*ν*) is derived from the pump spectral amplitudes for the two individual pump fields through the integral in Eq. (2), and is given by

_{i}*ϕ*(

*ν*,

_{s}*ν*) describes the phase-matching properties in the fiber. For degenerate pumps (where

_{i}*α*

_{1}(

*ω*) =

*α*(

*ω*), it may be shown that the phase-matching function is given by

*L*Δ

*k*is given in Eq. (4) with

_{lin}*τ*= 0. For non-degenerate pumps,

_{p}*τ*≠ 0 and ϕ (

_{p}*ν*,

_{s}*ν*) = Φ(

_{i}*B*;

*L*Δ

*k*), with

_{lin}*z*) is the error function, the parameter

*B*is defined as

*B*= (

*σ*

_{1}

^{2}+

*σ*

_{2}

^{2})

^{1/2}/(

*σ*

_{1}

*σ*

_{2}

*τ*), and

_{p}*M*is a normalization coefficient. In the present paper, we concentrate on the important class of factorable states for which

*F*(

*ω*,

_{s}*ω*) =

_{i}*S*(

*ω*)

_{s}*I*(

*ω*).

_{i}## 3. Phase and group-velocity matching properties of photonic crystal fibers

31. K. P. Hansen, “Dispersion flattened hybrid-core nonlinear photonic crystal fiber,” Opt. Express **11**, 1503–1509 (2003). [CrossRef] [PubMed]

32. T. A. Birks, J. C. Knight, and P. St. J. Russell. “Endlessly single-mode photonic crystal fiber,” Opt. Lett. **22**, 961–963 (1997). [CrossRef] [PubMed]

*n*by

_{eff}*k*(

*ω*) =

*n*(

_{eff}*ω*)

*ω*/

*c*. We adopt a step-index model, where the core has radius

*r*, its index is that of fused silica

*n*(

_{s}*ω*), and the cladding index is calculated as

*n*(

_{clad}*ω*) =

*f*+ (1 -

*f*)

*n*(

_{s}*ω*), where

*f*is the air-filling fraction. This fiber dispersion model has been shown to be accurate for

*f*from 0.1 to 0.9 according to Ref [33

33. G. K. L. Wong, A. Y. H. Chen, S. W. Ha, R. J. Kruhlak, S. G. Murdoch, R. Leonhardt, J. D. Harvey, and N. Y. Joly, “Characterization of chromatic dispersion in photonic crystal fibers using scalar modulation instability,” Opt. Express **13**, 8662–8670 (2005). [CrossRef] [PubMed]

*r*,

*f*} parameter space.

*γ*

_{1}=

*γ*

_{2}=

*γ*) is illustrated in Fig. 1, for a specific fiber with

*r*= 0.67

*μ*m,

*f*= 0.52 and length

*L*= 30cm. Fig. 1(a) shows the pump envelope function plotted as a function of

*ω*and

_{s}*ω*(see Eq.(7)) where we have assumed that the pump is centered at 723nm, has a bandwidth of 1nm, the incident power is 5W and

_{i}*γ*= 70km

^{-1}W

^{-1}. Fig. 1(b) shows the phasematching function (see Eq.(8)) for this choice of parameters. Note that the phasematched region forms a strip on {

*ω*,

_{s}*ω*)} space, oriented at an angle

_{i}*θ*= - arctan(

_{si}*τ*/

_{i}*τ*) = -55° with respect to the

_{i}*ω*axis. Fig. 1(c) shows the resulting joint spectral intensity ∣

_{s}*F*(

_{lin}*ω*,

_{s}*ω*)∣

_{i}^{2}(see Eq.(6)). It is apparent from Fig. 1 that the properties of the two photon state are determined by the i) relative orientations and ii) widths of the strips representing the phasematching and pump envelope functions. In this paper we explore how the interplay of the various design parameters can lead to two-photon states with engineered spectral entanglement properties.

*k*(

*ω*,

_{p}*ω*, 2

_{s}*ω*-

_{p}*ω*) = 0) for degenerate pumps versus

_{s}*ω*and

_{s}*ω*gives for each pump frequency the expected signal and idler central frequencies

_{p}*ω*

_{s}^{0}and

*ω*

_{i}^{0}. Such a phase-matching contour is illustrated in Fig. 2 for

*r*= 0.616

*μ*m and

*f*= 0.6, with pump power 30W and

*γ*= 70km

^{-1}W

^{-1}, where the generated frequencies are expressed as detunings from the pump frequency Δ

_{s,i}, =

*ω*-

_{s,i}*ω*. Note that energy conservation implies that Δ

_{p}_{s}= -Δ

_{i}. PCFs often have two ZDWs, where one can be as low as 500 nm (in comparison with 1270 nm for bulk silica). The phase-matching contours take the form of closed loops, with inner branches near the pump frequency and outer branches that can be hundreds of nanometers from the pump wavelength. Note that four wave mixing relying on outer-branch phasematching has been observed in previous work, in the context of classical non-linear optics [15

15. R. Jiang, R. Saperstein, N. Alic, M. Nezhad, C. J. McKinstrie, J. Ford, S. Fainman, and S. Radic, “Parametric wavelength conversion from conventional near-infrared to visible band,” IEEE Photon. Technol. Lett. **18**, 2445–2447 (2006). [CrossRef]

*λ*

_{zd1}= 0.668

*μ*m and

*λ*

_{zd2}= 1.132

*μ*m). The power-induced phase modulation terms in Δ

*k*split the trivial Δ

_{s}= Δ

_{i}= 0 branch, leading to frequency-distinct inner-branch signal/idler frequencies close to the pump (used for SFWM in [13

13. M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. **14**, 983–985 (2002). [CrossRef]

15. R. Jiang, R. Saperstein, N. Alic, M. Nezhad, C. J. McKinstrie, J. Ford, S. Fainman, and S. Radic, “Parametric wavelength conversion from conventional near-infrared to visible band,” IEEE Photon. Technol. Lett. **18**, 2445–2447 (2006). [CrossRef]

16. J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber,” Opt. Lett. **28**, 2225–2227 (2003). [CrossRef] [PubMed]

17. M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Modulational instabilities in dispersion-flattened fibers,” Phys. Rev. E **52**, 1072–1080 (1995). [CrossRef]

*ω*,

_{s}*ω*} space has contours of equal amplitude which have negative unit slope in all of {

_{i}*ω*,

_{s}*ω*} space. In contrast, the contours of the phase-matching function

_{i}*ϕ*(

*ω*,

_{s}*ω*) are characterized by a slope in {

_{i}*ω*,

_{s}*ω*} space given by

_{i}*θ*= - arctan(

_{si}*T*/

_{s}*T*) (with

_{i}*T*given according to Eq. (5)). The relationship between the slope

_{μ}*θ*

_{Δp}of the curve at a point in {Δ

_{s,i}

*ω*} space to the resulting phasematching function in {

_{p}*ω*,

_{s}*ω*} space is

_{i}*θ*

_{Δp}= 45° -

*θ*. For example,

_{si}*θ*= 45° corresponds to zero slope on the contour in Fig. 2(a) or

_{Si}*θ*

_{Δp}= 0. The colored background in Fig. 2 indicates the slope of the phase-matching contour, or orientation angle, rangingfrom

*θ*= -90° to

_{si}*θ*= +90°, indicated in blue and red respectively.

_{si}*θ*of the phase-matching contour. If the phase-matching contour is given by a closed loop (which is true for most fibers, i.e. range of values of {

_{si}*r*,

*f*}, of interest), all phase-matching orientation angles

*θ*are possible, controlled by the pump frequency. Thus, for certain relative orientations and widths of these two functions, it becomes possible to generate factorable two-photon states. It can be shown that a factorable state is possible if

_{si}*θ*= 45°, or

_{si}*T*= -

_{s}*T*, are of particular interest. For these states, in the degenerate pumps case, a factorable, symmetric state is guaranteed if

_{i}*k*

^{(1)}(

*ω*) <

_{s}*k*

^{(1)}(

*ω*) <

_{p}*k*

^{(1)}(

*ω*), or

_{i}*k*

^{(1)}(

*ω*) <

_{i}*k*

^{(1)}(

*ω*) <

_{p}*k*

^{(1)}(

*ω*) must be satisfied. The condition in Eq. (11) constrains the bandwidths. Thus, the region in {

_{s}*ω*,

_{s}*ω*} space in which factorability is possible is bounded by the conditions

_{i}*T*= 0 and

_{s}*T*= 0.

_{i}*ω*, must propagate at a higher group velocity than one of the generated photons at

_{p}*ω*/2, which can be interpreted as anomalous group-velocity dispersion. In practice, this can be achieved for certain materials, within a restricted spectral region (usually with PDC in the infrared) in a type-II process, where the polarization of the pump is orthogonal to that of one of the generated photons [18]. In the case of SFWM, because the frequencies (in the degenerate pump regime) will in general obey

_{p}*ω*<

_{s}*ω*<

_{p}*ω*or

_{i}*ω*<

_{i}*ω*<

_{p}*ω*, no such anomalous group-velocity dispersion is needed, making it much more straightforward to fulfill the group-velocity matching conditions required for factorable photon pair generation.

_{s}*θ*. In particular, symmetric factorable two-photon states, for which the signal and idler photons have identical spectral widths, are possible if

_{si}*θ*= 45°. In this case symmetric group-velocity matching 2

_{si}*k*

_{p}^{(1)}=

*k*

_{s}^{(1)}+

*k*

_{i}^{(1)}(or equivalently

*T*= -

_{s}*T*) is attained, and the phase-matching contours are oriented so that its contours have unit slope. The frequency values that fulfill this condition are represented by the dashed line. The pump frequency that permits symmetric factorable states can be determined from the intersection of the phase-matching contour with the group-velocity matching contour.

_{i}*θ*= 0° or

_{si}*θ*= 90° . This is the case of asymmetric group-velocity matching, for which

_{si}*k*

_{p}^{(1)}=

*k*

_{s}^{(1)}or

*k*

_{p}^{(1)}=

*k*

_{i}^{(1)}(or equivalently

*T*= 0 or

_{s}*T*= 0) is required. In this case the phase-matching contours are oriented parallel to the

_{i}*ω*or

_{s}*ω*axes. In addition, Eq. (11) leads to the condition that

_{i}*T*≫ 1/

_{i}*σ*(for

*T*= 0) or

_{s}*T*≫ 1/

_{s}*σ*(for

*T*= 0).

_{i}## 4. Co-polarized fields and degenerate pumps: symmetric factorable states

## 5. Co-polarized fields and non-degenerate pumps: symmetric factorable states

*ω*

_{1}

^{0}and

*ω*

^{0}

_{2}) and their bandwidths (

*σ*

_{1}and

*σ*

_{2}); in what follows we exploit both of these aspects of non-degeneracy. The orientation of the phase-matching function (see Eqs. (4) and (9)) in {

*ω*,

_{s}*ω*} space is determined by the angle

_{i}*θ*= - arctan(

_{si}*T*/

_{s}*T*). Note that the orientation of the phase-matching function depends on

_{i}*σ*

_{1}and

*σ*

_{2}(through

*T*and

_{s}*T*), while the shape of its profile and its width also depend on

_{i}*σ*

_{1}and

*σ*

_{2}(through parameter

*B*). A considerable simplification results by imposing the condition

*σ*

_{1}≪

*σ*

_{2}. In this case,

*T*reduces to the corresponding degenerate-pump values

_{μ}*τ*(with

_{μ}*μ*=

*s,i*). Furthermore, parameter

*B*reduces to 1/

*σ*

_{1}

*τ*), so that the phase-matching function width exhibits no dependence on the broader pump bandwidth (

_{p}*σ*

_{2}), while the effective pump envelope function depends only on

*σ*

_{2}. This de-coupling of

*σ*

_{1}and

*σ*

_{2}translates into a more straightforward exploration of possible fiber geometries for the generation of states with specific spectral entanglement properties.

*mean*pump frequency i.e. Δ

_{s,i}=

*ω*- (

_{s,i}*ω*

_{1}

^{0}+

*ω*

_{2}

^{0})/2; note that energy conservation implies that Δ

_{s}= -Δ

_{i}. Figure 4 illustrates the phase-matching properties for a specific non-degenerate geometry, with

*r*= 0.601

*μ*m,

*f*= 0.522 and

*ω*

_{2}

^{0}= 1.508 × 10

^{15}rad/sec (that is,

*λ*

_{2}

^{0}= 2

*πc*/

*ω*

_{2}

^{0}= 1250 nm). The black solid curve represents the resulting phase-matching contour. The two straight lines represent trivial phase-matching branches (in which the created photons are degenerate with the two pumps), while the loop represents the non-trivial phase-matching branch. Depending on the fiber geometry, the non-trivial branch may become large enough to overlap the trivial branches, or may shrink down to a single point. For the former case, if self-phase modulation becomes appreciable, this contour splits into three distinct loops (rather than two as for degenerate pumps). The colored background represents the orientation angle

*θ*ranging from -90° in blue to 90° in red.

_{si}*θ*= 45° which implies

_{si}*T*= -

_{s}*T*and ii) the spectral widths of the phase-matching function

_{i}*ϕ*(

*ω*,

_{s}*ω*) and the pump envelope function

_{i}*α*(

*ω*,

_{s}*ω*) are equal to each other. From Eqs. (5) it is straightforward to show that the first condition is satisfied if the four fields satisfy

_{i}*σ*

_{1}≪

*σ*

_{2}, the right-hand side of Eq. (12) vanishes, leading to a condition that is identical in form to that obtained for degenerate pumps (where the broader pump now plays the role of the degenerate pump). In Fig. 4, pairs {Δ

_{s}, (

*ω*

_{p1}} that satisfy this generalized group-velocity matching condition are represented by a dashed line. Note that the position and shape of this contour in general depend on

*σ*

_{1}and

*σ*

_{2}; however, in the case

*σ*

_{1}≪

*σ*

_{2}, the contour becomes decoupled from these bandwidths. Intersection points between the two contours determine the center frequency for pump field 1 that satisfies the requisite group-velocity matching, in addition to phase-matching. The values of

*r*and

*f*are selected so that

*θ*= 45° is satisfied

_{si}*and*so that the broadband pump frequency

*ω*

_{1}

^{0}takes a certain desired value (in this case,

*λ*

_{1}

^{0}= 2

*πc*/

*ω*

_{1}

^{0}= 0.625

*μ*m). This leads to the generated signal and idler wavelengths 736nm and 960 nm. The fiber length, along with the two bandwidths (subject to

*σ*

_{1}≪

*σ*

_{2}) are selected so as to match the widths of the pump envelope and phasematching functions. For the specific geometry shown, these values are

*L*= 25cm, while

*σ*

_{1}and

*σ*

_{2}are given in terms of the corresponding FWHM bandwidths: Δ

*λ*

_{1}= 1.51 nm and Δ

*λ*

_{2}= 0.12 nm.

*α*(

*ω*,

_{s}*ω*). Figure 5(b) shows the phase-matching function

_{i}*ϕ*(

*ω*,

_{s}*ω*) (see Eq.(9)) for which

_{i}*B*= 1.73. Figure 5(c) shows the resulting joint spectral intensity which exhibits a factorable character, obtained using the linear-dispersion approximation (see Eq. (4)). For comparison, Fig. 5(d) shows the joint spectral intensity obtained by numerical integration of Eq.(2), exhibiting excellent agreement with Fig. 5(c). This source leads to a numerically-obtained state purity of 0.89. As in the case of a factorable symmetric state obtained with degenerate pumps (see Fig. 3), the purity can be increased by filtering out the sidelobes in the joint spectral intensity at small cost in terms of collected flux. Two separate narrowband rectangular-profile spectral filters for the signal and idler modes with equal frequency bandwidth of 12.90 × 10

^{12}rad s

^{-1}(corresponding to wavelength widths Δ

*λ*≈ 3.7nm and Δ

_{S}*λ*≈ 6.3nm) increases the purity to 0.98, while reducing the flux by ≲ 5.9%.

_{i}## 6. Cross-polarized fields and degenerate pumps: asymmetric factorable states

_{n}≈ 1 × 10

^{-5}, for normal PCF, to Δ

_{n}≈ 7 × 10

^{-3}[34

34. A. Ortigosa-Blanch, A. Diez, M. Delgado-Pinar, J. L. Cruz, and Miguel V. Andres, “Ultrahigh birefringent nonlinear microstructured fiber,” IEEE Photon. Technol. Lett. **16**, 1667–1669 (2004). [CrossRef]

*xx*→

*xx*,

*xy*→

*xy*,

*xx*→

*yy*, where

*x*and

*y*can be any two linear orthogonal polarizations. Photon pair generation with frequency separated pumps in the

*xy*→

*xy*process was analyzed in 2004 [36

36. C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express **12**, 4973–4979 (2004). [CrossRef] [PubMed]

9. J. Fan and A. Migdall, “Generation of cross-polarized photon pairs in a microstructure fiber with frequency-conjugate laser pump pulses,” Opt. Express **13**, 5777–5782 (2005). [CrossRef] [PubMed]

*xx*→

*yy*, 2

*ω*→

_{p}*ω*+

_{i}*ω*). The first realization of birefringent FWM was a demonstration of this process in regular fibers [37

_{s}37. R. H. Stolen, M. A. Bosch, and C. Lin, “Phase matching in birefringent fibers,” Opt. Lett. **6**, 213–215 (1981). [CrossRef] [PubMed]

38. R. J. Kruhlak, G. K. L. Wong, J. S. Y. Chen, S. G. Murdoch, R. Leonhardt, J. D. Harvey, N. Y. Joly, and J. C. Knight, “Polarization modulation instability in photonic crystal fibers,” Opt. Lett. **31**, 1379–1381 (2006). [CrossRef] [PubMed]

13. M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. **14**, 983–985 (2002). [CrossRef]

*k*from the individual wavevectors as in Eq. (14). It was experimentally observed in [38

38. R. J. Kruhlak, G. K. L. Wong, J. S. Y. Chen, S. G. Murdoch, R. Leonhardt, J. D. Harvey, N. Y. Joly, and J. C. Knight, “Polarization modulation instability in photonic crystal fibers,” Opt. Lett. **31**, 1379–1381 (2006). [CrossRef] [PubMed]

*n*is approximately independent of frequency and we note that

*ω*is relatively constant in comparison to

_{p}*k*(

_{x}*ω*). Consequently the birefringent contribution to the wavevector mismatch functions much like the power phase-modulation term in Eq. (3), introducing in the Δ

_{p}*k*= 0 curve a splitting of the Δ

_{s}= 0 solution (creating the inner branch in Fig. 2). Unlike the power term, the birefringence can create a large splitting and can be either positive or negative.

*f*but different core diameters

*d*and

*d*+ Δ

*d*, simulating the birefringence [33

33. G. K. L. Wong, A. Y. H. Chen, S. W. Ha, R. J. Kruhlak, S. G. Murdoch, R. Leonhardt, J. D. Harvey, and N. Y. Joly, “Characterization of chromatic dispersion in photonic crystal fibers using scalar modulation instability,” Opt. Express **13**, 8662–8670 (2005). [CrossRef] [PubMed]

38. R. J. Kruhlak, G. K. L. Wong, J. S. Y. Chen, S. G. Murdoch, R. Leonhardt, J. D. Harvey, N. Y. Joly, and J. C. Knight, “Polarization modulation instability in photonic crystal fibers,” Opt. Lett. **31**, 1379–1381 (2006). [CrossRef] [PubMed]

*k*(

*ω*, Δ

_{p}_{s,i}) = 0) and group-velocity matching (

*T*= 0) curves are plotted for a PCF fiber with ZDWs at 790 nm and 1404 nm, where

_{s,i}*d*= 1.75

*μ*m and

*f*= 0.43. Three pairs of curves are plotted for Δ

*d*= -0.001,0,0.001

*μ*m, resulting in a birefringence of Δ

*n*= - 3 × 10

^{-5},0,3 × 10

^{-5}(where Δ

*n*=

*n*-

_{y}*n*). The corresponding shift in the ZDW is -0.07,0,0.07 nm and

_{x}*k*

_{x}^{(1)}-

*k*

_{y}^{(1)}= -0.5,0,0.5 ps/km, respectively. The segments of the Δ

*k*= 0 curve bounded by intersections with the

*T*= 0 curve are regions where factorability is possible. For both positive and negative Δ

_{s,i}*n*the outer branch intersections remain approximately the same as in the copolarized degenerate case. In addition, negative Δ

*n*creates two intersections (Points C and D) in between the two ZDWs. In between these points factorability is possible. Likewise, positive Δ

*n*also creates two intersections (Points E and F). However, now it is outside the region bounded by the intersections that factorability is possible. The splitting of the phase-matching (Δ

*k*= 0) solution from the Δ

_{s,i}= 0 line in the Δ

*n*< 0 case is similar to that from the power-induced phase modulation term 2

*γP*in the phase matching equation Eq. (3), but is an order of magnitude larger. Unlike power-induced inner branch FWM, birefringence allows for phase-matching beyond the Raman peak and is relatively insensitive to pump laser power fluctuations. In contrast, the Δ

*n*> 0 case is qualitatively different from either of these since 2

*γP*is always positive in silica. Moreover, it allows for factorable state generation over an unprecedented pump wavelength range, essentially anywhere outside the ZDWs.

*k*= 0 and

*T*= 0 curves is plotted as a function of the Δ

_{s, i}*n*at 800 nm. Thus for each Δ

*n*the figure indicates the pump wavelength

*λ*that creates an asymmetric factorable state (

_{p}*θ*= 0° or

_{si}*θ*= 90°), as well as the center wavelengths

_{si}*λ*and

_{s}*λ*of the generated photons. At these points, the pump bandwidth must satisfy condition

_{i}*T*≫

_{s,i}*σ*

^{-1}but is otherwise arbitrary. The shaded regions indicate the

*λ*range in which factorability is possible (including the symmetric state) if the pump bandwidth is set according to Eq. (11).

_{p}39. S. G. Murdoch, R. Leonhardt, and J. D. Harvey, “Polarization modulation instability in weakly birefringent fibers,” Opt. Lett. **20**, 866–868 (1995). [CrossRef] [PubMed]

40. Q. Lin, F. Yaman, and G. P. Agrawal, “Photon-pair generation by four-wave mixing in optical fibers,” Opt. Lett. **31**, 1286–1288 (2006). [CrossRef] [PubMed]

*g*is reduced by as much as an order of magnitude in the axis orthogonal to the strong pump pulse [14]. However, the relevant

_{Raman}*χ*

^{(3)}for birefringent pair production is also reduced by a factor of three compared to co-polarized FWM [29]. Tripling the power P to compensate, Ref. [41

41. Q. Lin, F. Yaman, and G. P. Agrawal, “Photon-pair generation in optical fibers through four-wave mixing: Role of Raman scattering and pump polarization,” Phys. Rev. A **75**, 023803 (2007). [CrossRef]

*I*is the intensity of signal or idler mode. They found that at the Raman peak,

*ρ*(0) was 7 for co-polarized SFWM as compared to 60 for cross-polarized SFWM, indicating that the cross-polarized case can be a higher quality photon-pair source.

_{c}## 7. Co-polarized fields: ultra-broadband two-photon states

## 8. Conclusions

4. M. G. Raymer, J. Noh, K. Banaszek, and I. A. Walmsley, “Pure-state single-photon wave-packet generation by parmametric down-conversion in a distributed microcavity,” Phys. Rev. A **72**, 023825 (2005). [CrossRef]

23. A. B. U’Ren, R. Erdmann, M. De la Cruz, and I. A. Walmsley, ”Generation of two-photon states with an arbitrary degree of entanglement via nonlinear crystal superlattices,” Phys. Rev. Lett. **97**, 223602 (2006). [CrossRef] [PubMed]

10. X. Li, J. Chen, P. Voss, J. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications: Improved generation of correlated photons,” Opt. Express **12**, 3737–3744 (2004). [CrossRef] [PubMed]

48. H. Takesue and K. Inoue, “1.5-μm band quantum-correlated photon pair generation in dispersion-shifted fiber: suppression of noise photons by cooling fiber,” Opt. Express **13**, 7832–7839 (2005). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. |
See, for example, the review by
P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. |

2. | S. E. Harris, M. K. Oshman, and R. L. Byer, “Observation of Tunable Optical Parametric Fluorescence,” Phys. Rev. Lett. |

3. | A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. |

4. | M. G. Raymer, J. Noh, K. Banaszek, and I. A. Walmsley, “Pure-state single-photon wave-packet generation by parmametric down-conversion in a distributed microcavity,” Phys. Rev. A |

5. | 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. |

6. | K. Banaszek, A. B. U’Ren, and I. A. Walmsley, “Generation of correlated photons in controlled spatial modes by downconversion in nonlinear waveguides,” Opt. Lett. |

7. | J. Fan and A. Migdall, “A broadband high spectral brightness fiber-based two-photon source,” Opt. Express |

8. | J. Rarity, J. Fulconis, J. Duligall, W. Wadsworth, and P. St. J. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express |

9. | J. Fan and A. Migdall, “Generation of cross-polarized photon pairs in a microstructure fiber with frequency-conjugate laser pump pulses,” Opt. Express |

10. | X. Li, J. Chen, P. Voss, J. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications: Improved generation of correlated photons,” Opt. Express |

11. | W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A |

12. | P. Russell, “Photonic Crystal Fiber,” Science |

13. | M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. |

14. | R. H. Stolen, “Fundamentals of Raman amplification in fibers,” in |

15. | R. Jiang, R. Saperstein, N. Alic, M. Nezhad, C. J. McKinstrie, J. Ford, S. Fainman, and S. Radic, “Parametric wavelength conversion from conventional near-infrared to visible band,” IEEE Photon. Technol. Lett. |

16. | J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber,” Opt. Lett. |

17. | M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Modulational instabilities in dispersion-flattened fibers,” Phys. Rev. E |

18. | Peter J. Mosley, Jeff S. Lundeen, Brian J. Smith, Ian A. Walmsley, Piotr Wasylczyk, Alfred B. U’Ren, and Christine Silberhorn, in Coherence and Quantum Optics IX, (Kluwer Academic/Plenum, New York) (accepted). |

19. | Z.D. Walton, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Polarization-Entangled Photon Pairs with Arbitrary Joint Spectrum” Phys. Rev. A |

20. | J. P. Torres, F. Macia, S. Carrasco, and L. Torner, “Engineering the frequency correlations of entangled two-photon states by achromatic phase matching” Opt. Lett. |

21. | O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. Kärtner, Phys. Rev. Lett.94, 083601 (2005) [CrossRef] [PubMed] |

22. | A.B. U’Ren, K. Banaszek, and I. A. Walmsley, “Photon engineering for quantum information processing” Quantum Information and Computation |

23. | A. B. U’Ren, R. Erdmann, M. De la Cruz, and I. A. Walmsley, ”Generation of two-photon states with an arbitrary degree of entanglement via nonlinear crystal superlattices,” Phys. Rev. Lett. |

24. | V. Giovanetti, S. Lloyd, and L. Maccone, “Quantum-enhanced positioning and clock synchronization,” Nature |

25. | M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Demonstration of dispersion-canceled quantum-optical coherence tomography,” Phys. Rev. Lett. |

26. | L. Mandel and E. Wolf, |

27. | J. Chen, X. Li, and P. Kumar, “Two-photon-state generation via four-wave mixing in optical fibers,” Phys. Rev. A |

28. | J. Chen, K. F. Lee, and P. Kumar R, “Quantum theory of degenerate χ |

29. | G. P. Agrawal, |

30. | C. J. McKinstrie, H. Kogelnik, and L. Schenato, “Four-wave mixing in a rapidly-spun fiber,” Opt. Express |

31. | K. P. Hansen, “Dispersion flattened hybrid-core nonlinear photonic crystal fiber,” Opt. Express |

32. | T. A. Birks, J. C. Knight, and P. St. J. Russell. “Endlessly single-mode photonic crystal fiber,” Opt. Lett. |

33. | G. K. L. Wong, A. Y. H. Chen, S. W. Ha, R. J. Kruhlak, S. G. Murdoch, R. Leonhardt, J. D. Harvey, and N. Y. Joly, “Characterization of chromatic dispersion in photonic crystal fibers using scalar modulation instability,” Opt. Express |

34. | A. Ortigosa-Blanch, A. Diez, M. Delgado-Pinar, J. L. Cruz, and Miguel V. Andres, “Ultrahigh birefringent nonlinear microstructured fiber,” IEEE Photon. Technol. Lett. |

35. | A. L. Berkhoer and V. E. Zakharov, “Self-excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP |

36. | C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express |

37. | R. H. Stolen, M. A. Bosch, and C. Lin, “Phase matching in birefringent fibers,” Opt. Lett. |

38. | R. J. Kruhlak, G. K. L. Wong, J. S. Y. Chen, S. G. Murdoch, R. Leonhardt, J. D. Harvey, N. Y. Joly, and J. C. Knight, “Polarization modulation instability in photonic crystal fibers,” Opt. Lett. |

39. | S. G. Murdoch, R. Leonhardt, and J. D. Harvey, “Polarization modulation instability in weakly birefringent fibers,” Opt. Lett. |

40. | Q. Lin, F. Yaman, and G. P. Agrawal, “Photon-pair generation by four-wave mixing in optical fibers,” Opt. Lett. |

41. | Q. Lin, F. Yaman, and G. P. Agrawal, “Photon-pair generation in optical fibers through four-wave mixing: Role of Raman scattering and pump polarization,” Phys. Rev. A |

42. | Such a state is typically referred to as highly entangled, but one should keep in mind that the large vacuum component of the state renders this “entanglement” useful only in a post-selection experiment. |

43. | K. A. O’Donnell and A. B. U’Ren, “Observation of ultrabroadband, beamlike parametric downconversion,” Opt. Lett. |

44. | L. Zhang, A. B. U’Ren, R. Erdmann, K. A. O’Donnell, C. Silberhorn, K. Banaszek, and I. A. Walmsley, “Generation of highly entangled photon pairs for continuous variable Bell inequality violation,” J. Mod. Opt. |

45. | R. Jiang, N. Alic, C. J. McKinstrie, and S. Radic, “Two-pump parametric amplifier with 40 dB of equalized gain over a bandwidth of 50 nm,” Proc. OFC2007, paper OWB2. |

46. | J. M. Chavez Boggio, J. D. Marconi, S. R. Bickham, and H. L. Fragnito, “Spectrally flat and broadband double-pumped fiber optical parametric amplifiers,” Opt. Express |

47. | S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, Q. Lin, and G. P. Agrawal, “Record performance of parametric amplifier constructed with highly nonlinear fibre,” Electron. Lett. |

48. | H. Takesue and K. Inoue, “1.5-μm band quantum-correlated photon pair generation in dispersion-shifted fiber: suppression of noise photons by cooling fiber,” Opt. Express |

**OCIS Codes**

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: August 16, 2007

Revised Manuscript: September 19, 2007

Manuscript Accepted: September 20, 2007

Published: October 26, 2007

**Citation**

K. Garay-Palmett, H. J. McGuinness, Offir Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U'ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, "Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber," Opt. Express **15**, 14870-14886 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-22-14870

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### References

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- S. E. Harris, M. K. Oshman, and R. L. Byer, "Observation of Tunable Optical Parametric Fluorescence," Phys. Rev. Lett. 18, 732-734 (1967). [CrossRef]
- A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice and M. G. Raymer, "Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion, " Laser Phys. 15, 146-161 (2005).
- M. G. Raymer, J. Noh, K. Banaszek and I. A. Walmsley, "Pure-state single-photon wave-packet generation by parmametric down-conversion in a distributed microcavity," Phys. Rev. A 72, 023825 (2005). [CrossRef]
- 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, 093601 (2004). [CrossRef] [PubMed]
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- J. Fan and A. Migdall, "A broadband high spectral brightness fiber-based two-photon source," Opt. Express 15, 2915-2920 (2007). [CrossRef] [PubMed]
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- J. Fan and A. Migdall, "Generation of cross-polarized photon pairs in a microstructure fiber with frequencyconjugate laser pump pulses," Opt. Express 13, 5777-5782 (2005). [CrossRef] [PubMed]
- X. Li, J. Chen, P. Voss, J. Sharping, and P. Kumar, "All-fiber photon-pair source for quantum communications: Improved generation of correlated photons," Opt. Express 12, 3737-3744 (2004). [CrossRef] [PubMed]
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- M. B. Nasr, B. E. A. Saleh, A. V. Sergienko and M. C. Teich, "Demonstration of dispersion-canceled quantumoptical coherence tomography," Phys. Rev. Lett. 91, 083601 (2003). [CrossRef] [PubMed]
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- J. Chen, K. F. Lee and P. Kumar R, "Quantum theory of degenerate |(3) two-photon state," e-print arXiv:quantph/ 0702176v1.
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- C. J. McKinstrie, H. Kogelnik and L. Schenato, "Four-wave mixing in a rapidly-spun fiber," Opt. Express 15, 8516-8534 (2006). This paper also reviews scalar and vector FWM in strongly-birefringent and randomlybirefringent fibers. [CrossRef]
- K. P. Hansen, "Dispersion flattened hybrid-core nonlinear photonic crystal fiber," Opt. Express 11, 1503-1509 (2003). [CrossRef] [PubMed]
- T. A. Birks, J. C. Knight and P. St. J. Russell. "Endlessly single-mode photonic crystal fiber," Opt. Lett. 22, 961-963 (1997). [CrossRef] [PubMed]
- G. K. L. Wong, A. Y. H. Chen, S. W. Ha, R. J. Kruhlak, S. G. Murdoch, R. Leonhardt, J. D. Harvey and N. Y. Joly, "Characterization of chromatic dispersion in photonic crystal fibers using scalar modulation instability," Opt. Express 13, 8662-8670 (2005). [CrossRef] [PubMed]
- A. Ortigosa-Blanch, A. Diez, M. Delgado-Pinar, J. L. Cruz and MiguelV. Andres, "Ultrahigh birefringent nonlinear microstructured fiber," IEEE Photon. Technol. Lett. 16, 1667-1669 (2004). [CrossRef]
- A. L. Berkhoer and V. E. Zakharov, "Self-excitation of waves with different polarizations in nonlinear media," Sov. Phys. JETP 31, 486-493 (1970).
- C. J. McKinstrie and S. Radic, "Phase-sensitive amplification in a fiber," Opt. Express 12, 4973-4979 (2004). [CrossRef] [PubMed]
- R. H. Stolen, M. A. Bosch and C. Lin, "Phase matching in birefringent fibers," Opt. Lett. 6, 213-215 (1981). [CrossRef] [PubMed]
- R. J. Kruhlak, G. K. L. Wong, J. S. Y. Chen, S. G. Murdoch, R. Leonhardt, J. D. Harvey, N. Y. Joly and J. C. Knight, "Polarization modulation instability in photonic crystal fibers," Opt. Lett. 31, 1379-1381 (2006). [CrossRef] [PubMed]
- S. G. Murdoch, R. Leonhardt and J. D. Harvey, "Polarization modulation instability in weakly birefringent fibers," Opt. Lett. 20, 866-868 (1995). [CrossRef] [PubMed]
- Q. Lin, F. Yaman and G. P. Agrawal, "Photon-pair generation by four-wave mixing in optical fibers," Opt. Lett. 31, 1286-1288 (2006). [CrossRef] [PubMed]
- Q. Lin, F. Yaman and G. P. Agrawal, "Photon-pair generation in optical fibers through four-wave mixing: Role of Raman scattering and pump polarization," Phys. Rev. A 75, 023803 (2007). [CrossRef]
- Such a state is typically referred to as highly entangled, but one should keep in mind that the large vacuum component of the state renders this "entanglement" useful only in a post-selection experiment.
- K. A. O’Donnell and A. B. U’Ren, "Observation of ultrabroadband, beamlike parametric downconversion," Opt. Lett. 32, 817-819 (2007). [CrossRef] [PubMed]
- L. Zhang, A. B. U’Ren, R. Erdmann, K. A. O’Donnell, C. Silberhorn, K. Banaszek and I. A.Walmsley, "Generation of highly entangled photon pairs for continuous variable Bell inequality violation," J. Mod. Opt. 54, 707-719 (2007). [CrossRef]
- R. Jiang, N. Alic, C. J. McKinstrie and S. Radic,"Two-pump parametric amplifier with 40 dB of equalized gain over a bandwidth of 50 nm, " Proc. OFC 2007, paper OWB2.
- J. M. Chavez Boggio, J. D. Marconi, S. R. Bickham and H. L. Fragnito, "Spectrally flat and broadband doublepumped fiber optical parametric amplifiers," Opt. Express 15, 5288-5309 (2007). [CrossRef]
- S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, Q. Lin and G. P. Agrawal, "Record performance of parametric amplifier constructed with highly nonlinear fibre," Electron. Lett. 39, 838-839 (2003). [CrossRef]
- H. Takesue and K. Inoue, "1.5- m band quantum-correlated photon pair generation in dispersion-shifted fiber: suppression of noise photons by cooling fiber," Opt. Express 13, 7832-7839 (2005). [CrossRef] [PubMed]

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