Phase matching as a gate for photon entanglement.

Phase matching is shown to provide a tunable gate that helps discriminate entangled states of light generated by four-wave mixing (FWM) in optical fibers against uncorrelated photons originating from Raman scattering. Two types of such gates are discussed. Phase-matching gates of the first type are possible in the normal dispersion regime, where FWM sidebands can be widely tuned by high-order dispersion management, enhancing the ratio of the entangled-photon output to the Raman noise. The photon-entanglement gates of the second type are created by dual-pump cross-phase-modulation-induced FWM sideband generation and can be tuned by group-velocity mismatch of the pump fields.

Four-wave mixing (FWM) in optical fibers provides a compact, tunable, and efficient source of quantum states of light1234. Modern fiber technologies lend a vast parameter space to tailor such states56, helping tune their entanglement degree and enabling the generation of factorable photon states7. Specifically, photonic crystal fibers (PCFs)8, where the dispersion and nonlinearity can be managed by fiber design engineering9, have been shown to enable photon-pair generation within a broad range of pump wavelengths1234510, offering a unique platform for fiber-based quantum communication and information technologies. Highly birefringent fibers11, including specifically designed PCFs8, have been found to be instrumental in the generation of polarization-entangled photon pairs, opening the ways toward multipartite entanglement12. Frequency conversion via FWM has been demonstrated as a method of ultralow-noise of individual- and entangled-photon-state translation13. When combined with appropriate single-mode filtering, FWM in optical fibers can serve as a source of single photons with a high degree of quantum indistinguishability14, offering an advantageous framework for quantum information processing, quantum metrology, and quantum key distribution.

Raman scattering has long been recognized as a major physical factor that limits the performance of fiber-optic sources of quantum states of light15. Light fields propagating through optical fibers inevitably interact with optical phonons, accumulating noise due to the Raman scattering1617. This noise limits soliton squeezing in optical fibers15 and degrades the performance of fiber-based sources of nonclassical light, including fiber-optic sources of entangled photon pairs1617.

In a broader context of classical nonlinear optics, the interplay between FWM and Raman scattering gives rise to a vast variety of nonlinear-optical field evolution scenarios. In optical fibers, FWM effects have been shown18 to dominate over stimulated Raman scattering (SRS) as long as phase matching is satisfied for the FWM process. While for narrowband input fields, provided by pico- and nanosecond input pulses, well-resolved FWM and SRS signatures can often be isolated in broadened output spectra1920, femtosecond laser pulses tend to undergo more complicated temporal and spectral transformations, where the FWM dynamics is intertwined with SRS effects, giving rise to octave-spanning supercontinua2122, as well as frequency-shifting2324 and self-compressing1825 soliton transients. In nonlinear Raman spectroscopy26 and microscopy27, FWM is manifested as a coherent nonresonant background, which generally limits the sensitivity of imaging and spectroscopic measurements, but in certain schemes can also serve as a local oscillator, facilitating a heterodyning of the coherent Raman signal28. The nonresonant FWM background in nonlinear Raman spectroscopy and microscopy can be efficiently suppressed by using properly optimized delay times29, polarization geometries3031, pulse shapes32, and phase profiles33 of the pump, Stokes, and probe pulses.

Here, we demonstrate that the Raman noise can be radically reduced in fiber-optic FWM-based photon entanglement generation through carefully tailored phase matching, which provides a tunable gate that helps discriminate entangled photon pairs against uncorrelated photons originating from Raman scattering. Two types of phase-matching gates will be considered. Phase-matching gates of the first type are possible, as shown below in this paper, in the normal dispersion regime, where FWM sidebands can be widely tuned by high-order dispersion management, enhancing the ratio of the entangled-photon output to the SpRS noise. The photon-entanglement gates of the second type are created by dual-pump cross-phase-modulation-induced FWM sideband generation and can be tuned by group-velocity mismatch of the pump fields.

Four-wave mixing as a source of entangled photon pairs

We consider a generic 2ωp = ωs + ω a FWM process where two pump photons of the same frequency, ωp, give rise to idler and signal photons (also referred to hereinafter as the Stokes and anti-Stokes photons) with frequencies ωs and ω a. In the undepleted-pump approximation, the Hamiltonian HFWM that describes all the FWM processes coupling these fields is quadratic34 in the Stokes and anti-Stokes field creation and annihilation operators and a (j = s and a for the Stokes and anti-Stokes fields), defined in such a way as to satisfy the commutation relations . In the Heisenberg picture, the solution to the evolution equations for these operators,  = , a, can be written in the input–output form as35 , and , with μ(z) and ν(z) being the transfer functions.

In the case when both pump photons are taken from the same pump field, ωp1 = ω2p = ωp and 2ωp = ωs + ωa, and the FWM Hamiltonian34 is , where P0 is the peak power of the pump, γ is the Kerr nonlinearity coefficient, δ = Δβ/2 + γP0, Δβ = βs + βa − 2βp, and β, β, and β are the propagation constants of the pump, Stokes, and anti-Stokes fields, the solution for and a reduces to12

where μ(z) = cos(κz) + i(δ/κ)sin(κz), ν(z) = i(γP0/κ)sin(κz),κ2 = δ2 − (γP0)2.

With a and defined by Eqs (1) and (2), the expectation value for the photon number is 〈n(z)〉 = |ν(z)|2. For a two-mode input vacuum state , the FWM-sideband output is in the squeezed state35

Quantum correlations between the Stokes and anti-Stokes photons are quantified in terms of the cross-correlation coefficient34 . For a pure, Raman-noise-free two-mode squeezed-state FWM output12, and , with c = (μ*)−1(ν/μ*), we find and II = 〈n〉〈n〉 = |ν|4, leading to the following expression for the cross-correlation coefficient: ρ0(z) = |μ(z)|2/|ν(z)|2.

Raman-effect-induced degradation of photon-pair correlations

The Raman effect is included in the model of FWM sideband generation through the inertial part of the nonlinear-optical response18 and through the term in the evolution equations for a and with the Hermitian noise source operator m15 defined in such a way that , where gR(Ω) is the Raman gain. With the Raman effect added, the spectral density S(z) of the Stokes (j = s) and anti-Stokes (j = a) photon flux is17 , where , , , , η = 1 + f(R1 + R2 − 1), fR is the Raman fraction of the nonlinear refractive index, R1 and R2 are the Fourier transforms of the isotropic and anisotropic parts of the Raman response, such that g = 2γfIm(R1 + R2), r(z) = n(Ω), r(z) = −n(Ω) − 1, Ω = ω − ωp, n(Ω) = [exp(ħ|Ω|/θ) − 1]−1 is the thermal photon number, θ = kBT, kB is the Boltzmann constant, and T is the temperature.

Unlike FWM, which can generate strongly correlated Stokes and anti-Stokes photon pairs as a part of the two-mode squeezed-state FWM output of Eq. (3), spontaneous Raman scattering gives rise to uncorrelated Stokes and anti-Stokes photons, which follow a thermal distribution of phonon population nth(Ω). As a result, the Raman noise decreases correlations between the Stokes and anti-Stokes photons. The degree of this correlation degradation, however, strongly depends on the phase mismatch Δβ. This dependence, as shown below in this paper, helps discriminate entangled states of light generated by FWM against uncorrelated photons originating from Raman scattering.

Phase matching

To understand the significance of phase matching for correlated photon-pair generation, we first consider the case of large phase mismatch, |Δβ| ≫ γP0. In this regime, the spectral density of the photon-pair flux is given by S(z) = F(z) + S(Ω, z), with the FWM part of the flux, , controlled by the signature sinc(Δβz/2) = [sin(Δβz/2)]/(Δβz/2) phase-mismatch factor. When |Δβz| ≫ 1 and γP0z is kept small, γP0z ≪ 1, to avoid an excessive degradation of photon-pair correlations as dictated by ρ0(z) = |μ(z)|2/|ν(z)|2, the Raman noise dominates over the FWM photon-pair flux, giving rise to uncorrelated Stokes and anti-Stokes photons with ρ(z) ≪ 1.

In the opposite limit, when δ is small, FWM sidebands are strongly coupled, giving rise to correlated Stokes and anti-Stokes photons. Moreover, the FWM parametric gain is at its maximum at δ = 0, providing the highest efficiency of FWM sideband generation. Within the FWM parametric gain band, i.e., for δ < γP0, the solutions for a and are given by Eqs (1) and (2) with μ(z) = cosh(gz) + i(δ/g)sinh(gz), , and g2 = (γP0)2 − δ2. At the center of the FWM parametric gain band, δ = 0, the two-mode squeezed-state output is , where .

The cross-correlation coefficient of Raman-noise-contaminated Stokes and anti-Stokes photon pairs in the γP0z ≪ 1 and δ = 0 regime is given by17

When the Raman noise is negligible, f ≪ 1, Eq. (4) reduces to the expression for the Raman-noise-free cross-correlation coefficient ρ0(z) written in the same approximation, i.e., with δ = 0 and γP0z ≪ 1, leading to ρ0(z) ≈ (γP0z)−2. The choice of the nonlinear phase φnl = γP0z is thus a tradeoff between the photon flux I, which increases with φnl as |ν(z)|2, and the correlation between the Stokes and anti-Stokes photons, which decreases with φnl even in the absence of the Raman noise as |μ(z)|2/|ν(z)|2.

Discriminating correlated photon pairs against the Raman noise

We quantify the time–energy entanglement36 of the Stokes and anti-Stokes photons in terms of the fringe visibility V = ρ/(ρ + 2) of a two-photon interference pattern, which can be measured, e.g., with the use of an unbalanced Mach–Zehnder interferometer3738. Figure 1 shows the parameter V plotted as a function of the frequency Ω/(2π) = (ω − ωp)/(2π) for Stokes and anti-Stokes photons generated through pure FWM with f = 0 (red line), as well as through FWM with the Raman noise (blue line). For the highest efficiency of photon-pair generation, FWM is assumed to be ideally phase-matched in both cases, δ = 0. The nonlinear phase shift is kept small, γP0z = 0.1, to provide a low-n output, which helps avoid an excessive degradation of photon-pair correlations. Parameters of the Raman noise are chosen in such a way as to mimic the Raman effect in silica fibers1718: fR = 0.18 and the peak Raman gain gR0 = 6.2 10–12 cm/W.

Figure 1

Two-photon interference fringe visibility V as a function of the frequency Ω/(2π) = (ω − ωp)/(2π) for Stokes and anti-Stokes photons generated through the phase-matched 2ωp = ωs + ω a FWM with δ = 0, T = 300 K, and γP0z = 0.1.

(red line) pure FWM with f = 0 and (blue line) FWM with the Raman noise, fR = 0.18.

To understand the influence of Raman scattering on quantum correlations between the Stokes and anti-Stokes photons as a function of the frequency at which phase matching δ = 0 is achieved, it is instructive to isolate the spectral density of the Raman noise1517 S(Ω, z) = s(Ω, z)P0z, where s(Ω) = |g|[n(Ω) + H(−Ω)] and H(Ω) is the Heaviside step function. Figure 1a shows the spectral density of the Raman noise sR(Ω) for fused silica at T = 300 K with the Raman gain profile gR(Ω) as specified by Stolen et al.39. As can be seen from this plot, sR(Ω) is symmetric with respect to Ω = 0, sR(Ωs) ≈ sR(Ωa), only for low Ω (|Ω|/(2π) < 1 THz for T = 300 K in Fig. 2a), where |Ω|/(2π) < kT/ħ. In this region, nth ∝ 1/|Ω|, giving rise to constant low-frequency noise. For |Ω|/(2π) > kT/ħ, the Stokes sideband is much more prone to the Raman noise than its anti-Stokes counterpart. The sR(Ω) profile features a broad peak at Ω/(2π) ≈ 13 THz (Fig. 2a), rolling off by more than an order of magnitude for |Ω|/(2π) > 20–30 THz.

Figure 2

(a) The spectral density of the Raman noise sR(Ω) for fused silica at T = 300 K. (b) The coherence length lc = π/|2δ| for 2ωp = ωs + ω a FWM with P0 = 27 W calculated as a function of the pump frequency and the Stokes/anti-Stokes wavelengths λs,a = 2πc/ωs,a.

This behavior of the spectral density of the Raman noise is crucial for understanding the properties of the time–energy entanglement of the Stokes and anti-Stokes photons as quantified by the fringe visibility V. In Fig. 1, we plot the V parameter as a function of the frequency Ω/(2π) for the phase-matched 2ωp = ωs + ω a, δ = 0, and γP0z = 0.1 FWM process with (blue line) and without (red line) the Raman effect. As one would expect from the spectral profile of sR(Ω), for low Ω, the Raman noise dramatically reduces the entanglement of the Stokes and anti-Stokes photons. Indeed, for Ω/(2π) ranging from approximately 1 to 15 THz, the two-photon interference fringe visibility is very low, V < 0.1. In this range, Raman scattering imposes severe limitations on fiber sources of quantum states of light.

As the spectral intensity of the Raman noise decreases beyond Ω/(2π) > 20–30 THz, the time–energy entanglement of the Stokes and anti-Stokes photons becomes stronger, approaching, for Ω/(2π) > 35–40 THz, the Stokes–anti-Stokes entanglement in pure phase-matched FWM (cf. the blue and red curves in Fig. 1). The entanglement of the Stokes and anti-Stokes outputs of FWM can thus be radically enhanced if the high-Ω FWM photons could be selected with an appropriate spectral filtering.

Four-wave mixing with a single pump

We are going to show now that such a filter can be provided by finely tuned phase matching in optical fibers. Photonic-crystal fibers, where dispersion can be tailored by fiber structure engineering89, thus enabling a fine adjustment of FWM phase matching, are ideally suited for this purpose40. As an example, we consider a PCF with zero group-velocity-dispersion (GVD) wavelength λz ≈ 800 nm and a dispersion profile similar to that provided by a family of commercial, NL-800-series PCFs. Fibers of this type have been shown2140 to enable highly efficient parametric FWM pumped by a 760–820 nm Ti: sapphire laser output.

FWM gives rise to parametric sideband generation when the wave number K of a harmonic perturbation of a cw solution of the relevant wave equation has a nonzero imaginary part. When β2 = ∂2β/∂ω2 < 0 and higher order dispersion terms involving β = ∂β/∂ω with even k ≥ 4 are negligible, the dispersion relation for K is written as18

where q = βΩ/k!. FWM parametric sideband generation is thus possible for any Ω meeting the inequality Ω2 < . The maximum gain is achieved at the frequency , exactly where the phase matching , equivalent to δ = 0, is satisfied.

Figure 3 compares the phase-matching frequency Ωpm calculated by numerically solving the equation δ = 0 for FWM with P0 = 27 W in a fiber with the dispersion of an NL-2.4-800 PCF (solid line) with the approximate solution Ωpm ≈  (red dashed curve). As can be seen from this comparison, the approximation Ωpm ≈ Ω0 provides a highly accurate prediction for the frequency of phase matching everywhere in the anomalous-GVD range except a narrow region near the zero-GVD frequency ωz, which corresponds to in Fig. 3.

Figure 3

The phase-matching frequency Ωpm as a function of (ωp and ωz are the pump and zero-GVD frequencies) calculated by numerically solving the equation δ = 0 for FWM with P0 = 27 W in a fiber with the dispersion of an NL-2.4-800 PCF (solid line) versus the approximate solutions Ω ≈ (2γP0/|β2|)1/2 in the anomalous-dispersion range (red dashed curve) and Ωpm ≈ (−12β2/β4)1/2 in the normal-dispersion range (green dashed curve).

In Fig. 2b, we present a typical map of the coherence length lc = π/|2δ| for 2ωp = ωs + ω a FWM with P0 = 27 W calculated as a function of the pump frequency and the Stokes/anti-Stokes wavelengths λs,a = 2πc/ωs,a. As an important universal tendency, the FWM phase-matching maps and, hence, the maps of the FWM gain look drastically different for the normal- and anomalous-GVD regions (Figs 2b and 3). When the wavelength of the pump with a peak power P0 lies in the region of anomalous GVD, where β2 < 0, a simple phase matching is possible for parametric FWM processes, giving rise to two δ = 0 phase-matching branches (Figs 2b, 3) with the centers of these parametric gain bands separated from ωp by a small spectral interval of .

In the region of normal dispersion, on the other hand, the equation is no longer solvable in the class of real Ω0 as β2 > 0. Still, the δ = 0 phase matching is possible due to high-order dispersion, giving rise to two phase-matching branches that lie much further away from ωp (Figs 2b, 3). In particular, when high-order dispersion terms β with k > 4 are negligible, the dispersion equation for K is written as1841

Parametric sideband generation is now possible when and . Provided that , that is, when the fourth-order dispersion effects can be treated as a small correction within the bandwidth ΔΩ ≈ (2γP0/β2)1/2, the upper bound of the parametric gain band is given by . FWM sideband generation is thus confined to a narrow gain band , whose bandwidth is on the order of .

As can be seen in Fig. 3, the approximation (green dashed curve) agrees very well with the frequency of phase matching found by numerically solving the δ = 0 equation (solid line in Fig. 3) everywhere in the normal-GVD range except a small region near the zero-GVD wavelength. This closed-form approximate expression for Ωpm drastically simplifies the design of fiber sources of entangled photon pairs. Specifically, with set at just a few terahertz, the entanglement degree of Stokes and anti-Stokes photons, as can be seen from Fig. 1, is increased by more than an order of magnitude. Indeed, with  ≈ 0.7 THz, FWM phase matching is achieved at Ωpm ≈ 40 THz (Fig. 3). The two-photon interference fringe visibility for Ω ≈ 40 THz, as can be seen from Fig. 1, is V ≈ 0.91, which is more than an order of magnitude higher than the V value for Ω ≈ 15 THz. Moreover, with  ≈ 7.5 THz, which corresponds to a pump wavelength λp = 2πc/ωp ≈ 710 nm in the case of a fiber with λz ≈ 800 nm, we find Ωpm ≈ 96 THz (Fig. 3). For a fiber at T ≈ 25 °C, sideband photons with such a frequency correspond to ħ|Ω|/(kT) ≈ 16. The thermal photon number is exponentially small in this regime, n ≈ exp(−ħ|Ω|/θ), providing a strong suppression of the Raman noise in the photon-pair output.

Four-wave mixing with a dual pump

In dual-pump FWM, cross-phase modulation (XPM) tends to induce energy transfer from one of the pump fields to the sidebands of the other pump1842, giving rise to an exponential buildup of sidebands ω1,2 ± Ω around the central frequency ω (k = 1, 2) of each of the pump fields. The domains of this XPM-induced parametric gain and their central frequencies Ω0 are defined by the dispersion equation4243

where , , , P, ω, u, and βp are the peak power, the central frequency, the group velocity, and the propagation constants of the first (k = 1) and second (k = 2) pump fields, and γ is the nonlinear coefficient at the frequency ω.

The buildup of XPM-induced sidebands ω1,2 ± Ω is controlled by the gain g = 2 ImK, which can be found by solving the quartic equation (7). With Θ = 0, the solution to this equation reduces to , where k = 1, 2. Each of these solutions is equivalent to the solution of Eq. (5), corresponding to a decoupled parametric sideband generation by each of the pump fields.

In a more general scenario, Θ ≠ 0, the two pump fields and their sidebands are coupled by XPM. Both the gain bands and the gain controlling the buildup of XPM-induced sidebands can be tuned in this scheme by varying the frequencies and the peak powers of both pump fields, as well as by tailoring fiber dispersion and nonlinearity. As a typical example, Fig. 4 shows the XPM-induced parametric gain g calculated by numerically solving Eq. (7) as a function of Ω and σ for the conditions of experiments4445, which have demonstrated efficient XPM-induced sideband generation in a PCF pumped by a two-color field consisting of the 1.25 μm Cr: forsterite laser output and its second harmonic. The fundamental-frequency output of a Cr: forsterite laser in this scheme falls in the range of anomalous GVD, with β21 ≈ −0.115 ps2/m, while the second-harmonic pump lies in the region of normal GVD, with β22 ≈ 0.016 ps2/m. The nonlinear coupling constant is γ1P1 + γ2P2 ≈ 5 cm−1.

Figure 4

The XPM-induced parametric gain g calculated by numerically solving Eq. (7) as a function of Ω and σ for fiber pumped by a two-color field consisting of the 1.25 μm Cr: forsterite laser output and its second harmonic with β21 ≈ − 0.115 ps2/m, β22 ≈ 0.016 ps2/m, and γ1P1 + γ2P2 ≈ 5 cm−1.

The g(Ω,σ) map in Fig. 4 exhibits two clearly resolved parametric gain bands. The high-frequency band is seen to shift almost linearly with the group-velocity mismatch (GVM) of the pump fields , while the low-frequency band is largely independent of the GVM. The former gain band is of special interest for the generation of entangled photon pairs, as it delivers photons with large frequency offsets Ω, thus helping reduce the flux of uncorrelated photons due to the Raman effect.

Both the low- and high-frequency parametric bands seen in Fig. 4 have been studied earlier2028 by means of numerical analysis of Eq. (7). As an important empirical result, such an analysis confirms that, for sufficiently large σ, the frequency shift of the high-Ω gain band grows linearly with the GVM σ of the pump pulses. We show below in this section that some of the key properties of XPM-induced FWM gain bands can be qualitatively understood in terms of phase matching, thus suggesting physically transparent design rules for fiber sources of entangled photon pairs.

With this goal in mind, we set γ1 ≈ γ2 = γ and approximate the propagation constants of the ω1 + Ω and ω2 − Ω sidebands as β ≈ β ± Ω/u + β2Ω2/2 + 2γ(P1 + P2). The phase-matching condition for the ω1 + ω2 = (ω1 + Ω) + (ω2 − Ω) XPM-coupled FWM sideband generation is then written as

The solution to this equation is

In the case of low pump peak powers, 2γ(P1 + P2) ≪ σ2/|β21 + β22|, Eq. (9) gives

With γ(P1 + P2) = 0 and β21 ≈ β22 = β2, Eq. (10) fully recovers the empirical result of the earlier numerical studies2028, Ωpm ≈ σ/|β2|. In a more general case of nonzero, but low nonlinearity, σ2/|β21 + β22| ≫ γ(P1 + P2) ≠ 0 and β21 ≠ β22, the frequency shift of the considered parametric gain band, as can be seen from Eq. (10), is still a linear function of σ. In particular, for the parameters of the fiber and the pump in Fig. 4, |β21 + β22| ≈ 0.1 ps2/m and 2γ|β21 + β22|(P1 + P2) ≈ 100 ps2/m2, the approximation of Eq. (10) is valid for GVMs σ > 10 ps/m. Specifically, for σ ≈ 30 ps/m, Eq. (10) predicts Ωpm/(2π) ≈ 100 THz, which agrees very well with numerical calculations in Fig. 4. For sideband photons with Ω/(2π) ≈ 100 THz in a fiber at T ≈ 25 °C, ħ|Ω|/(kT) ≈ 16, and the thermal photon number is exponentially small, n ≈ exp(−ħ|Ω|/θ), leading to a strong suppression of the Raman noise in the photon-pair output.

In the context of correlated photon-pair generation, Eq. (9) provides a closed-form approximate expression that radically simplifies the design of fiber sources of entangled photon pairs. Specifically, with |β21 + β22|/2 ≈ 0.01 ps2/m, σ ≈ 2.5 ps/m, and γ(P1 + P2) ≪ 2σ2/|β21 + β22|, the maximum gain of XPM-induced sideband generation in dual-pump FWm is achieved at Ωpm ≈ 40 THz. At this frequency, the time–energy photon-pair entanglement parameter, V ≈ 0.91, is more than an order of magnitude higher (Fig. 1) than the V parameter for Ω ≈ 15 THz.

Notably, with γ(P1 + P2) ≪ σ2/|β21 + β22|, the frequency shift of the high-frequency gain band in XPM-induced FWM sideband generation, as can be seen from Eqs (9) and (10), is almost independent of the pump peak power. The flux of FWM photons can thus be adjusted to avoid photon-pair correlation degradation (see Section 5), independently of the frequency of FWM photon pairs Ω, which helps discriminate correlated FWM photon pairs against uncorrelated Raman photon pairs. As dual-pump FWM offers a vast variety of polarization and spatial-mode arrangements for multiple sideband generation in optical fibers12, GVM-controlled phase-matching filter in such schemes is ideally suited for low-noise multipartite photon entanglement creation.

In its general, polarization-nondegenerate version, the dual-pump FWM scheme considered in this section gives rise to multiple sideband pairs, which can be coupled to each other by the Kerr-type optical nonlinearity18. The effect that the resulting correlations have on the quantum properties of sideband pairs is, however, drastically different from the effects induced by the Raman scattering. While the Raman-induced sidebands are not correlated as they build up from the noise that follows a thermal distribution of phonon population nth(Ω), the manifold of FWM processes in orthogonal polarization modes of the fiber give rise to strongly correlated Stokes and anti-Stokes photon pairs, enabling the generation of multipartite entanglement. Indeed, when the peak power of both pump fields in dual-pump FWM is P0 and the input is a four-mode vacuum state, , involving two modes qj (j = 1, 2) of Stokes and anti-Stokes (q = s, a) vacuum fields, the FWM-sideband four-mode output in the gz ≪ 1 regime, as shown in the earlier work12, is in the squeezed state , where η = 1 − iγP0z, Θ = iγP0z/(2η), and . Such states, as elegantly demonstrated by McKinstrie et al.12, display distinctly identifiable signatures of multipartite entanglement.

Conclusion

We have shown that phase matching can provide a tunable gate that helps discriminate entangled states of light generated by four-wave mixing in optical fibers against uncorrelated photons originating from Raman scattering. Two types of such gates are discussed. Phase-matching gates of the first type are possible in the normal dispersion regime, where FWM sidebands can be widely tuned by high-order dispersion management, enhancing the ratio of the entangled-photon output to the Raman noise. The photon-entanglement gates of the second type are created by dual-pump cross-phase-modulation-induced FWM sideband generation and can be tuned by group-velocity mismatch of the pump fields.

Additional Information

How to cite this article: Zheltikov, A. M. Phase matching as a gate for photon entanglement. Sci. Rep. 7, 46115; doi: 10.1038/srep46115 (2017).

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