Circumventing spontaneous Raman noise in a correlated photon pair source

The on-demand generation of single photons is keenly sought in quantum optics. There are a multitude of photon generation schemes, including atom-like sources^1–4 and heralded photon pair sources based on nonlinear optics. Of the latter, the two most common schemes are spontaneous parametric down conversion (SPDC)^5–7 and spontaneous four-wave mixing (SFWM).^8–10 

Due to the strength of the χ⁽²⁾ nonlinearity as compared with χ⁽³⁾, SPDC sources typically require lower pump powers than SFWM sources, and consequently exhibit negligible noise from competing χ⁽³⁾ processes. However, most optical materials do not have a χ⁽²⁾ response as they are inversion-symmetric. Utilising the universal χ⁽³⁾ response allows for a greater number of materials and platforms to be used in heralded single photon sources, and integrates well with current telecommunication networks. In particular, silica and silicon allow for near infra-red generation and thus efficient coupling to standard SMF-28 fiber.^8,9

Many of these χ⁽³⁾ materials, such as fused silica, are amorphous and therefore exhibit broadband spontaneous Raman scattering (SpRS) due to inhomogeneous broadening of the Raman transitions. In the quantum regime, this corresponds to emission of uncorrelated single photons. In a typical amorphous degenerate SFWM source, the strong pump field produces these uncorrelated Raman photons over a broad energy range. This noise often overlaps with the desired frequency range of the generated pairs¹¹ (see Fig. 1(a)). Without due care, one may generate many more Raman photons than correlated pairs.¹² 

There are several existing techniques for mitigating Raman noise in amorphous SFWM sources,¹³ including dispersion engineering waveguides such that the produced pairs lie in a window of low Raman photon production.^12,14,15 Another method is to use non-Raman active materials, such as monatomic gasses.¹⁶ However, these are subject to material and engineering constraints that may be challenging to implement, or difficult to integrate with existing technologies.

Here we propose a new method which sidesteps the issue of SpRS in amorphous materials. Pumping a χ⁽³⁾ material strongly at or near the third harmonic, ω_p ≈ 3ω_s, three photons can be spontaneously generated at the third sub-harmonic ω_s, which we call the fundamental, with low probability. When unstimulated, this process is inefficient,¹⁷ and so authors seeking three-photon generation tend to use cascaded χ⁽²⁾ processes,^18–20 or parallel pair emission, for example, in the source used by Broome et al.²¹ As we are aiming for pair generation, here we add a weak coherent field near the fundamental frequency ω_s to seed the process. This leaves the desired photon pair accessible, with energy conservation dictated by ω_p = ω₁ + ω₂ + ω_s. For a seed exactly at the fundamental, the pair of photons lie on either side of the seed, as seen in Fig. 1(b). We may consider this process “seeded three photon down conversion” (STPDC). The principal advantage of this scheme is that the Stokes band of the Raman spectrum lies between the generated pairs and the pump frequency. The bandwidth of the spontaneous Raman response is typically ∼10 THz, whereas the fundamental and third harmonic fields involved at optical frequencies are separated by ∼400 THz. This large spectral separation ensures low Raman noise in the signal band.

To describe this process and all of the multimode physics involved, we follow the formalism outlined by Yang, Liscidini, and Sipe.²² We look for solutions to the first order Schrödinger equation

where |ψ_in〉 describes the coherent state input in the pump and seed modes, and is vacuum in the photon pair bands. As the nonlinearity outside the interaction length is zero, we are free to extend the integration limits to infinity, t₀ → −∞ and t₁ → ∞. The relevant interaction Hamiltonian is given by

Here $D ˆ ( i ) ( r , t )$ are the components of the vector displacement operator, written in the interaction picture associated with the linear Hamiltonian $H ˆ L = ∑ γ ∫dk ħ ω γ , k a ˆ γ , k † a ˆ γ , k$ (the summation is over the different modes γ, and the vacuum term is disregarded). The symbol Γ⁽³⁾ is a rank four tensor, related to the standard third order susceptibility tensor²² (see below).

To treat the STPDC process, we express the displacement field operator as a sum over modes γ,

where $a ˆ γ , k$ and $b ˆ γ , k$ are the usual bosonic annihilation operators for mode γ and wavenumber k, D_γ,k and F_γ,k are mode functions, and h.c. denotes the Hermitian conjugate. We separate the expansion into low (⁠$a ˆ γ , k$⁠) and high (⁠$b ˆ γ , k$⁠) frequency bands, with k_B a wavenumber between the two. For uniform waveguides, the field mode functions may be decomposed into a transverse mode function and a longitudinal plane wave,

On substituting Eq. (3) into Eq. (2), and considering only one mode per band, we keep only terms involving the annihilation of one photon in the high frequency band and the creation of three photons in the low frequency band, and their conjugates. While a full quantum analysis including SPM and XPM is beyond the scope of this paper, a classical analysis suggests that they are insignificant for the system introduced below and so we neglect them here for simplicity. The interaction Hamiltonian can then be expressed as

where Δk = k₄ − k₃ − k₂ − k₁, Δω_k = ω_k4 − ω_k3 − ω_k2 − ω_k1, $α 2$ and $β 2$ are the average numbers of photons in the input classical seed and pump pulses, respectively, and ϕ_s(k), ϕ_p(k) are their spectral profiles, localised in k. The effective mode coupling area satisfies

where $Γ i j m n ( 3 ) = χ i j m n ( 3 ) / ( ϵ 0 n k 1 2 n k 2 2 n k 3 2 n k 4 2 )$⁠, the refractive index is abbreviated as n(x, y; ω_ki) = n_ki, and the nonlinear susceptibility has been decomposed into a transverse and longitudinal part $χ i j m n ( 3 ) ( r ) = χ i j m n ( 3 ) ( x , y ) s ( z )$⁠. Additionally, the typical size of a nonvanishing component of χ⁽³⁾(x, y) is denoted as $χ ¯ ( 3 )$⁠, and $n ¯$ represents the mean group index, both introduced solely for convenience.²² We associate k₁ and k₂ with the generated pairs, k₃ with the seed and k₄ with the pump. Note that the second form of Eq. (6) in vector notation holds if the material is isotropic, and both definitions for the effective area account for fields of arbitrary polarisation. The phasematching condition for STPDC in Eq. (5) is captured in the spatial Fourier transform of the longitudinal nonlinearity profile s(z), that is, $s ( k ) = ∫ − ∞ ∞ dz s ( z ) e − i k z$⁠.

We use Eq. (5) to find the first order solution as given by Eq. (1) and transform from k to ω. This introduces factors of the group velocity v_g which account for the density of states in frequency.²² The integration over all time yields $∫ − ∞ ∞ e − i Δ ω k t =2πδ ( Δ ω k ) ,$ allowing further integration over one frequency. Now the state can be described as |ψ_out〉 ≈ |vac〉 + η|II〉, where η is a normalisation factor, and the biphoton state is described by

The joint spectral amplitude (JSA) is

This is our main result. It fully describes the biphoton state for STPDC, and from Eq. (8) one can calculate the rate of photon pair production as well as arbitrary expectation values. In particular, normalising the biphoton state imposes the normalisation of the JSA $∫d ω 1 d ω 2 Φ ( ω 1 , ω 2 ) 2 =1$⁠. From Eq. (7), this allows the physical interpretation of $η 2$ as the probability of pair production per pump pulse.

Taking the seed and pump fields to be Gaussians, $ϕ ̄ j ( ω ) = τ j / π 1 / 4 e − τ ( ω − ω j ) 2 / 2$⁠, with sufficiently long pulse durations τ_s and τ_p such that the seed and pump can be considered Dirac delta-like, the pair production possibility can be approximated as

where L is the interaction length, and the nonlinear parameter is

The effective area $A$ has been evaluated for ω = ω_s and ω_1,2 = ω_s. The powers $P s =ħ ω s α 2 / τ s$ and $P p =ħ ω p β 2 / τ p$ are nominal average pulse powers, related to the time averaged power by the duty cycle of a high repetition rate laser.

As a proof-of-principle, we consider a system where we expect to be able to phasematch this process. The phasematching and efficient conversion of one-third harmonic generation (OTHG) or backwards third harmonic generation has been studied by Grubsky and Savchenko²³ and further refined by Zhang et al.,²⁴ demonstrating how to phasematch the process in air-clad fused silica microfiber. This microfiber shares the same refractive index as the core of standard fiber. By tuning the width of the fiber, the high frequency HE₂₁ mode can be phasematched with the low frequency HE₁₁ mode. In this system, a classical analysis shows SPM and XPM effects on phase matching to be insignificant for pump powers below 6 kW.

Setting the pump to the common frequency-doubled laser wavelength 532 nm, we solve for these modes exactly and find that $n eff HE 21 ( ω p ) = n eff HE 11 ( ω s )$ when the diameter of the fiber is 0.790 μm. The effective area then is $A=4.7μ m 2$⁠, with the third order susceptibility χ⁽³⁾ = 2.5 × 10⁻²² m²/V². We take an interaction length L = 10 cm typical of fiber tapers. We envisage modest pump configurations, typical of current mode-locked green sources, as shown in Table I. Note that to manage the fast walk-off, the seed pulse duration is on the order of nanoseconds. With this set of parameters, the pair production probability as given by Eq. (9) is $η 2 =0.019$ per pulse. Without relying on these coarse approximations, imagining a 2 nm bandwidth bandstop filter to suppress the seed, and assuming Gaussian pulses, the equivalent numerical result derived from Eq. (8) yields $η 2 =0.012$ per pulse, across a bandwidth of 1.9 THz. This is sufficient for producing an effective photon source, for a pump laser with a typical 100 MHz repetition rate. The resulting joint spectral intensity (JSI) is shown in Fig. 2. It has a Schmidt number²⁵ of K = 149.1, found from the singular value decomposition of the JSA. This indicates that the state is highly correlated, with an unheralded second order correlation function²⁶ of g⁽²⁾(0) = 1.0067.

Being a four-wave mixing process with four distinct fields, the parameter space for STPDC is large, and there is room to engineer desirable JSAs. For example, decreasing the pump duration to τ_p = 1 ps increases the Schmidt number to 411.2 (g⁽²⁾(0) = 1.0024), decreases the rate to $η 2 =0.0004$ pairs per pulse, and increases the generation bandwidth to 5.8 THz. Doubling L to 20 cm increases the Schmidt number to 275.9 (g⁽²⁾(0) = 1.0036), decreases the generation bandwidth to 1.8 THz, but increases the rate to $η 2 =0.023$ pairs per pulse. As a final example, a fourfold increase in the group velocity dispersion decreases the rate to $η 2 =0.005$ pairs per pulse (as we would expect from Eq. (9)), decreases the generation bandwidth to 0.9 THz, and also decreases the Schmidt number to 71.2 (g⁽²⁾(0) = 1.014).

What about the Raman problem? As a first approximation, we follow Refs. 11 and 26, working in the quasi-CW limit and expressing the Raman photon flux as $I u R =Δ ν u P p L g R ( Δ ) [ ρ + 1 / ( e h Δ / ( k B T ) − 1 ) ] /A$⁠, where u = s, i, the detuning from the seed is $2πΔ= ω − ω s$⁠, and Δν_u is a filter bandwidth. The factor ρ = 1 for the Stokes process and is zero for the anti-Stokes process. The Stokes channel coincides with the idler photon band (ω < ω_s) and the anti-Stokes channel coincides with the signal photon (ω > ω_s). The Raman gain g_R(Δ) is taken directly from Ref. 27. In this limit, the generated SFWM and STPDC spectral densities are

where P_sp is the single-pump power, and γ′ is defined similarly to Eq. (10), with $A ′ =84.0μ m 2$⁠, $n g ′ ( ω s ) =1.463$⁠, and $β 2 ′ ( ω s ) =−26.2 ps 2 km − 1$⁠. As the spontaneous Raman bandwidth is approximately 40 THz, and the pump and seed fields are separated by roughly 400 THz, we do not expect there to be any measurable contribution from the pump to the Raman noise in the pair generation band.

Defining the signal to noise ratio (SNR) as $SNR= I u / I u R$⁠, we define a figure of merit (FOM) to be the SNR for STPDC in microfiber over the SNR of SFWM in SMF-28. As $I SFWM ∝ P sp 2$ and I^STPDC ∝ P_pP_s, for a fair comparison of the rates we take the single pump power to be $P sp = P s P p$⁠. The FOM is then expressed

To keep $η 2 ≪1$ (recall Eq. (9)), we fix P_p at 2 kW with a pulse duration of 1 ns (200 mW average power at a repetition rate of 100 kHz), and plot the peak spectral densities (Eqs. (11) and (12)), and FOM (Eq. (13)) as functions of P_s. The spectral densities themselves are plotted for a representative seed power of 1 W (the vertical line in Fig. 3(a)) in Fig. 3(b). Figure 3(a) shows a considerably improved SNR as compared with SFWM for a range of pumping configurations. Figure 3(b) demonstrates a reduction in the production of Raman photons and, critically, that the STPDC signal lies well above the SpRS of the STPDC seed, whereas the SFWM signal is buried beneath the SpRS from the SFWM pump.

Evaluating the two processes for a common fiber, the ratio between P_sp and P_s predicts a 100-fold decrease in Raman noise. If we compare STPDC in microfiber with SFWM in SMF-28, taking into account that the Raman gain increases in magnitude for smaller areas, we find that the gain in microfiber is larger by a factor of $A ′ /A=17.0$⁠, and so the ratio that determines the suppression is approximately $P sp A/ ( P s A ′ ) =5.9$⁠. In other systems that support third harmonic generation, for example, in photonic crystal fiber,²⁸ one might expect to achieve or exceed the suppression given by P_sp/P_s. The freedom to separate the strong pump from the pair generation, inherent to STPDC, is what drives this improvement.

We have proposed and demonstrated the feasibility of the third order photon pair generation method we call seeded three-photon down conversion (STPDC). The process can be phasematched and with realistic pump requirements produce pairs at rates sufficiently high for an effective heralded single photon source. With the strongest pump field spectrally distinct from the generated pairs, it sidesteps the issue of noise from spontaneous Raman scattering. In microfiber, we calculate an improvement in the ratio of photon pairs to uncorrelated Raman photons as compared with standard SFWM in SMF-28, for various pumping configurations.

This work was supported in part by the ARC Centre for Ultrahigh bandwidth Devices for Optical Systems (CUDOS) (Project No. CE110001018). We thank M. J. Collins for useful discussions.