Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group

Efficient photon triplet generation in integrated nanophotonic waveguides

Open Access Open Access

Abstract

Generation of entangled photons in nonlinear media constitutes a basic building block of modern photonic quantum technology. Current optical materials are severely limited in their ability to produce three or more entangled photons in a single event due to weak nonlinearities and challenges achieving phase-matching. We use integrated nanophotonics to enhance nonlinear interactions and develop protocols to design multimode waveguides that enable sustained phase-matching for third-order spontaneous parametric down-conversion (TOSPDC). We predict a generation efficiency of 0.13 triplets/s/mW of pump power in TiO2-based integrated waveguides, an order of magnitude higher than previous theoretical and experimental demonstrations. We experimentally verify our device design methods in TiO2 waveguides using third-harmonic generation (THG), the reverse process of TOSPDC that is subject to the same phase-matching constraints. We finally discuss the effect of finite detector bandwidth and photon losses on the energy-time coherence properties of the expected TOSPDC source.

© 2016 Optical Society of America

1. Introduction

Optical quantum technologies [1] often rely on nonlinear optical crystals pumped with coherent radiation to generate light with non-classical properties, including entanglement. Entangled photons have applications in many fields, such as quantum cryptography [2], quantum metrology [3], quantum imaging [4, 5], quantum spectroscopy [6], and quantum information processing [7]. Many experimental efforts aim towards the generation, manipulation, and detection of entangled photons using nanoscale optical devices that provide a compact and scalable platform to perform quantum optics experiments on a single chip [1].

Entangled states of light containing three or more photons are currently built interferometrically starting from entangled photon pairs or single photons [2], with entangled pairs typically generated via spontaneous parametric down-conversion (SPDC) [8], biexciton decay [9], and spontaneous four-wave mixing [10]. Tripartite entanglement can be used for quantum secret sharing [11] and measurement-based quantum computing [12]. The production of entangled photon triplets via cascaded SPDC has been demonstrated [13–15], with generation efficiencies on the order of 10−2 triplets/s/mW pump power [14]. Third-order spontaneous parametric down conversion (TOSPDC), when one pump photon splits into three signal photons, can also generate entangled photon triplets. The temporal coherence properties of TOSPDC photon triplets are different from cascaded SPDC sources [16].

TOSPDC represents an experimental challenge [17] due to low third-order nonlinearities (χ(3)) in typical optical materials. Moreover, phase-matching becomes increasingly difficult to satisfy for widely-spaced wavelengths. Recently, nanoscale optical devices have emerged as a promising route to enhance nonlinear-optical processes, mostly due to high field intensity from sub-wavelength confinement of the electromagnetic field [1]. In addition, sustained phase-matching for widely-spaced wavelengths can be achieved using strong dispersion in non-fundamental waveguide modes. These remarkable improvements over bulk materials indicate that integrated nanophotonic waveguides can enable efficient TOSPDC.

In this work, we propose a protocol to optimize efficient generation of entangled photon triplets via TOSPDC in nanophotonic waveguides. We discuss in detail how to optimize key design parameters, such as phase-matching and mode overlap between visible pump and infrared signal modes, while also considering the impacts of photon losses. Our proposed design protocol can be applied to many waveguide materials and geometries. We validate our design protocol experimentally by demonstrating THG, the reverse of TOSPDC, in nanoscale TiO2 waveguides. Titanium dioxide (TiO2) is a promising material platform for efficient TOSPDC [18–22]. Pumping with a visible laser at λp = 420–450 nm or 510–520 nm generates photons in the O and C telecommunication bands, allowing integration with silicon photonics, existing telecommunications infrastructure, and efficient single photon detection [23]. We finally calculate the temporal coherence properties of TOSPDC sources in coincidence measurements, taking into account finite detector bandwidths and propagation losses.

2. Device design

TOSPDC in bulk nonlinear optical materials has previously been ruled out as a viable source of entangled photons due to the inherent difficulty of achieving phase-matching across nearly two octaves, low χ(3) nonlinearities, and weak light confinement over long interaction lengths [17]. TOSPDC in nano-scale silica fibers was proposed theoretically by Corona et al. [24]. The predicted photon triplet generation efficiency in 10-cm long silica fibers, ignoring the impact of photon losses, is 1.9 × 10−2 triplets/s/mW of pump power [24, 25], which is comparable to current experimental demonstrations [13,14]. However, silica fibers cannot be easily integrated to produce on-chip photon triplet sources due to incompatibility with CMOS fabrication techniques and low index contrast with commonly-used cladding materials.

We propose to use an alternative material, TiO2, for its high refractive index, high χ(3) nonlinearity, and CMOS compatibility [26, 27]. These characteristics make TiO2 an excellent platform for an on-chip, integrated photon triplet source.

Optimizing the efficiency of photon triplet generation in nanophotonic waveguides requires four major considerations: phase-matching, mode overlap, interaction length, and power coupled into the phase-matched pump mode. In the rest of this section, we define criteria to optimize these factors and achieve the largest triplet generation efficiency. We consider photon losses due to their impact on conversion efficiency, optimal device length, and signal spectrum. Using optimized parameters for a TiO2 nanoscale waveguide, we predict its performance for TOSPDC under realistic conditions, including photon losses.

2.1. Higher-order mode phase-matching

According to Fermi’s golden rule [17], the transition rate for the conversion of a pump photon of frequency ωp into three signal photons at frequencies denoted as ωs, ωr and ωi is strongly suppressed for any combination of frequencies that does not satisfy the energy conservation constraint ωp = ωr + ωs + ωi. The probability of generating signal photons that satisfy energy conservation is maximal for those combinations that also satisfy the phase-matching or momentum-conservation rule, kp = kr + ks + ki, where km is the wavevector of each photon in mode m = {p, s, r, i}. In this work, we consider nanoscale ridge waveguides (Fig. 1(b)), for which the propagation direction of guided modes is along the waveguide. In this case, momentum conservation reduces to a scalar identity for the magnitude of the wavenumbers of propagating modes k = 2πneff/λ, where neff is the effective refractive index of the signal mode at wavelength λ.

 figure: Fig. 1

Fig. 1 (a) Schematic of higher-order mode phase matching using calculated dispersion data for a fully-etched 550 × 360 nm TiO2 waveguide with SiO2 cladding. Shifting the pump mode from λp to 3λp gives a point of intersection where the effective indices of the TM00 signal and TM02 pump modes are the same. Phase matching is achieved at λp = 532 nm. Guided modes reach a cutoff wavelength when their effective index is equal to the cladding SiO2 index (gray region). (b) Schematic of the waveguide cross section and tunable parameters. (c) 3D schematic of an integrated device with input and output coupling.

Download Full Size | PPT Slide | PDF

The wavelength dependence of the effective index neff(λ) in guided modes is determined by the waveguide geometry, dispersion properties of the core and cladding materials, and mode order. In the optical regime, the refractive index for a transparent material is typically larger at the shortest wavelengths, monotonically decreasing with increasing wavelength. For example, in TiO2 there is an index mismatch of 0.17 between a 532-nm pump and 1596-nm signal wavelength, which leads to a 1.6-μm coherence length. Additionally, for a given wavelength λ, higher-order modes have a lower effective index than the fundamental mode. In Fig. 1(a), we demonstrate these two points by plotting the effective index for two modes (i.e., the TM00 fundamental [signal] and TM02 higher-order [pump] modes) of a TiO2 waveguide with a 550-nm core width and a 360-nm thickness, with an SiO2 cladding. By employing a higher-order mode at the shorter wavelength, we can match effective indices for highly disparate wavelengths [22, 28].

We define the perfect phase matching (PPM) point as the pump wavelength λp0 for which neffs(3λp0)=neffs(λp0), where neffs is the effective index of the signal mode of interest and neffp is the pump index. At the PPM point, the phase-matching condition reads kp0=3ks0, since all three signal photon frequencies are degenerate. For the case of a monochromatic pump at the PPM-condition, the signal photons can be generated away from ωp/3, so long as energy conservation is maintained. A finite phase-mismatch will result from group velocity dispersion (GVD) in the signal mode ( Ds=2ks/ωs2|ωs=ωs0), limiting the bandwidth. The signal bandwidth at PPM is thus given by ΔPPM, corresponding to the FWHM of the function sinc2kL/2), which determines the triplet generation rate as a function of finite phase mismatch (see Section 2.3):

ΔPPM=4πL|Ds|
where L is the nonlinear interaction length. The relative intensities of the signal photons that are emitted for the PPM case are visualized in Fig. 2(a), where the idler (ωr, ωi) vs. signal-frequency (ωs) intensity-plot shows a single peak centered at the degenerate signal frequency. The black curve in Fig. 2(d) shows the corresponding intensity of all three signal photons as a function of frequency.

 figure: Fig. 2

Fig. 2 We visualize the joint spectral intensity as an intensity plot of frequency versus frequency for several values of the pump detuning Δλp from the PPM wavelength. Panels (a), (b) and (c) correspond to Δp = 0, Δp = −2 nm and Δp = −0.5 nm, respectively. We assume a waveguide length L = 2 mm and use dispersion values for the phase matching point shown in Fig. 1(a). Fixing one of the three signal wavelengths (ωs) on the horizontal axis, the two other signal wavelengths (ωr and ωi) are given by the two values on the vertical axis that intersect the ellipse. Phase matching is achieved along the perimeter of the ellipse and the thickness is determined by the pump and signal interaction length, based on Eq. (1). By collapsing the plots in (a), (b), and (c) onto the horizontal axis, we generate the spectra shown in (d). All three spectra are normalized by the total conversion efficiency. Panel (e) shows signal spectrum bandwidth as a function of pump detuning Δωp. The dashed black curve gives the bandwidth defined by Eq. (2). Red and blue curves are numerically obtained signal bandwidths for waveguide lengths L = 2 mm and L =20 mm, respectively. Panel (f) zooms near the PPM point.

Download Full Size | PPT Slide | PDF

In TOSPDC, the pump wavelength can be blue-detuned away from the PPM point while still satisfying energy and momentum conservation efficiently for normal signal-mode dispersion. This produces non-degenerate photon triplets in energy. Energy and momentum conservation cannot simultaneously be satisfied for a red-detuned pump with normal dispersion in the signal mode. Satisfying energy and momentum conservation rules for 1D propagation results in an under-constrained system of equations with two free parameters. One is fixed by choosing the pump frequency ωp = 2πc/λp. We can choose the remaining degree of freedom to be one of the signal frequencies ωm, with m = {s, r, i}.

We refer to the range of signal frequencies over which energy and momentum conservation is fulfilled for TOSPDC as the signal bandwidth δs (derived in Appendix A), given by:

δs=2(23)1/2[2(vp1vs1)DsΔp+(DpDs)Δp2]1/2,
where Δp=ωpωp0 and Δm=ωmωp0/3, with m = {s, r, i}, are the pump and signal detunings from the PPM point. ωp0 is the pump frequency that gives PPM. vp and vs are the group velocities of the pump and signal modes at the PPM point, and Dp and Ds are their group velocity dispersion.

For a long interaction length L and a monochromatic pump with detuning Δp, the signal frequencies that satisfy energy and momentum conservation simultaneously can be represented in a (ωs, ωr)-plane by the ellipse (derived in Appendix A):

Δs2+Δr2ΔrΔsΔp(Δr+Δs)=Ap,
where Ap=(3/16)δs2Δp2. In Figs. 2(b) and 2(c), we plot the relative intensity of emitted signal photons in a frequency-frequency plane for cases where the pump is slightly detuned from the PPM point. The highest intensities occur at frequencies satisfying the ellipse equation above. The photon triplet wavefunction has a non-vanishing amplitude for signal frequencies outside the ellipse in Eq. (3); however these frequencies are strongly suppressed because of poor phase matching (Figs. 2(b) and 2(c)).

The ellipse width δs grows with a scaling (Δp)1/2 for pump detuning 0<Δp<Δpc=2(vsvp)/(vpvsDp) and with a scaling (Δp)1 for greater pump detuning. Maximizing signal Ds and minimizing vsvp and Dp minimizes signal bandwidth for finite phase mismatch. This provides a method to minimize signal bandwidth in the case that fabrication variations introduce finite phase mismatch. For fixed pump detuning Δp, the signal bandwidth δs in Eq. (2) corresponds to the distance between the highest and lowest points in the ellipse. The corresponding spectrum consists of a two-peak structure with intensity maxima ocurring roughly at frequencies ωp0/3±δs/2 (Fig. 2(d)). The spectra in Fig. 2 are obtained numerically by computing the triplet generation rate from Eq. (6) in Section 2.3.

In Fig. 2(e), we plot the signal bandwidth δs as a function of pump detuning Δp for the TiO2 waveguide parameters from Fig. 1. The bandwidth is obtained from numerical spectra (as in Figs. 2a, b, and c) computed at 1/10-th the peak maximum, for the interaction lengths L = 2 mm and L = 20 mm. As the pump-signal interaction length increases, the spectral amplitude of the three-photon state (Section 2.3) approaches a delta function in the phase mismatch Δk = kpkrkski. In this limit, the bandwidth δs in Eq. (2) accurately matches the numerical bandwidth. The discrepancy between analytical and numerically calculated bandwidth is greatest at Δp = 0, with a minimum bandwidth δs = 54 × 1012 rad/s for L = 2 mm and δs = 14 × 1012 rad/s for L = 20 mm. As we discuss in Section 2.3, photon loss due to scattering and absorption reduces the effective interaction length, thus broadening the signal.

2.2. Effective nonlinearity and modal overlap

Once phase matching and energy conservation are satisfied for the interacting pump and signal modes, the efficiency of TOSPDC is determined by the effective nonlinearity [24]:

γ=3χ(3)ωp4ε0c2n2η,
where n is the material index and χ(3) is the effective third-order susceptibility at the pump frequency. We can define η as the sum of mode overlap components ηijkl[Aeffijkl]1=Epi*EsjEsldxdy between pump and signal electric field components [24] corresponding to all non-zero susceptibility tensor elements χijkl(3). The electric fields in this calculation are normalized. The non-zero tensor elements are determined by the material crystal structure and orientation.

Our TiO2 is deposited on a thick thermal oxide and is thus polycrystalline with randomly oriented grains of diameter smaller than 50 nm. The lack of long-range order and the small grain size allows us to treat the material as effectively isotropic. Using the methods above we typically find an effective interaction area Aeff on the order of 10−10 m2 for our TiO2 waveguides. The details on how to obtain the nonlinear overlaps ηijkl are given in Appendix B.

We calculate the effective nonlinearity γ in Eq. (4) for a pump and signal mode pair using mode profiles calculated with a commercial finite-difference eigenmode solver [29]. Because a significant part of the field in nanoscale waveguides is evanescent, we must consider the components of the mode profile that are in the waveguide core and cladding independently and utilize the corresponding nonlinear tensors.

Direct measurements of χ(3) are difficult because of multi-photon absorption and other competing nonlinear interactions, especially at photon wavelengths below the half band gap; therefore, we use the nonlinear index [30]

n2=3χ(3)4n02ε0c
to parameterize the magnitude of the third-order nonlinearity. We use the value n2 = 4.65 × 10−19 m2/W for the TiO2 nonlinearity at 532 nm. This value is obtained from the bandgap scaling of n2(λ) [31] (see Appendix B for details).

2.3. Triplet generation rate in lossy waveguides

The rate of direct generation of photon triplets R3 from TOSPDC in the absence of photon losses has been calculated in previous work [16, 24]. Findings in Refs. [16, 24] can be qualitatively summarized by the expression R3ζ̃Np(0) L, where Np(0) is the number of pump photons entering the waveguide and ζ̃γ2 is the conversion efficiency per unit length, and L the waveguide length.

In the presence of photon losses, the triplet generation rate no longer increases linearly with L. We account for scattering losses by introducing pump and signal intensity attenuation coefficients αp and αs, respectively. We emphasize that an optimal length Lopt must exist such that the waveguide is long enough to generate a large number of triplets, but short enough so that a large fraction of complete triplets reaches the end when losses are taken into consideration. Extending the analysis in Ref. [24], we can express the triplet generation rate R3 in the presence of losses for a continuous wave (cw) pump as (derivation in Appendix C)

R3=2233h¯c3np3π2[ωp0]2γ2L2P(ω0n02(ω)g(ω0))3e(αp+3αs)L/2dνrdνs|Φ(νr,νs)|2,
where the spectral amplitude Φ(νr, νs) is defined in Appendix C in terms of the frequency-dependent phase mismatch Δk(νr, νs) and a loss mismatch parameter Δα = (αp − 3αs). P is the pump power and ω0=ωp0/3. The integration variables νr and νs are signal detunings from ωp0/3.

We can compare the triplet generation rate from Eq. (6), which takes into account the dispersion properties of the signal mode around the phase matching point, with a simplified expression that ignores dispersion (derivation in Appendix C)

N30LdN3(z)dz=ζ˜Np0αp3αs(e3αsLeαpL),
where ζ̃ quantifies the conversion efficiency. The device length that optimizes N3 is thus given by Lopt = [1/(αp − 3αs)] ln(αp/3αs), for Δα = (αp − 3αs) ≠ 0.

In Fig. 3(a), we plot the normalized signal intensity in the waveguide as a function of the waveguide length L, for different amounts of photon loss in TiO2 waveguides. The simplified model that ignores dispersion (Eq. (7)) is an excellent approximation for the results obtained by integrating the squared spectral amplitude |Φ(νr, νs)|2 (Eq. (6)) near points of PPM. In Fig. 3(b), we plot the optimal device length Lopt, as a function of the pump and signal loss parameters αp and αs. Highlighted is the parameter region for the waveguides considered in this work.

 figure: Fig. 3

Fig. 3 (a) Signal intensity as a function of device length for pump and signal loss values, respectively, of (i) 16 and 4 dB/cm, (ii) 16 and 12 dB/cm, (iii) 28 and 4 dB/cm, and (iv) 28 and 12 dB/cm, representing losses on the lower and upper extremes for TiO2 devices. Circles show the full quantum prediction, including signal dispersion, from Eq. (6) and lines show the prediction from Eq. (7). All curves are normalized by the maximum signal. (b) Ideal waveguide length plotted as a function of pump and signal losses. Current polycrystalline anatase TiO2 waveguide losses give optimal device lengths Lopt = 1 – 4 mm. White denotes Lopt > 8 mm.

Download Full Size | PPT Slide | PDF

3. Device optimization

For efficient device operation, we maximize the triplet generation rate R3 in Eq. (6). Given an optimal device length Lopt, we must thus minimize the phase mismatch Δk for the desired combination of pump and signal frequencies. This will produce narrow-bandwidth photon triplets as shown in Fig. 2(a) and 2(d). We must also maximize the effective nonlinearity γ by choosing a waveguide geometry that enhances the electric field overlap between pump and signal modes within a material with a high χ(3) coefficient. We design TiO2 waveguides with rectangular core geometry that achieve γ = 1100 W−1km−1 while maintaining single mode operation at 1596 nm. In Section 3.1 we describe the optimization protocol used and in Section 3.2 we verify the design method with a demonstration of THG.

3.1. Optimization protocol

We design our devices for λp = 532 nm because high-power pump lasers with narrow bandwidth, high stability, and low cost are readily available at this wavelength. Our design protocol can be extended to other wavelengths and material platforms depending on the desired signal photon wavelength and available pump laser wavelength. Designing for a particular pump wavelength requires precise control of the waveguide dimensions [22, 32, 33].

We illustrate in Fig. 1(b) the geometry of our TiO2 waveguides. The design parameters that can be tuned include: the waveguide core material, bottom cladding or substrate material, top cladding material, waveguide width, waveguide height, etch fraction, and the waveguide sidewall angle (θ). Due to the current state of the art in TiO2 waveguide fabrication, we focus on fully etched, symmetric waveguide geometries with 90° sidewall angles.

We use a commercial finite-difference eigenmode solver [29] to complete our device design and optimization. For a given device geometry, we calculate all propagating modes and their dispersion properties at λp and 3λp, using measured materials properties. We then use calculated mode dispersion properties to calculate the phase mismatch Δk for all mode pairs consisting of visible pump and IR signal. The mode profiles and the χ(3) nonlinearity of the core and cladding materials are used to calculate the mode overlap and γ for all mode pairs (see Appendix B). This process is repeated for all device geometries to minimize phase mismatch and maximize γ.

We complete a sweep for rectangular waveguides, varying the waveguide height in the range 200 – 400 nm and width in the range 400 – 600 nm. This range of dimensions is chosen to ensure single mode operation at 3λp, meaning only one TE and one TM guiding mode are supported by the waveguide. To quickly assess different waveguide geometries we introduce a figure of merit that describes the effective spectral density of signal photons

=γ2δs,
where the minimum signal bandwidth min{δs} = ΔPPM is given by the PPM bandwidth in Eq. (1) for degenerate TOSPDC. For each set of waveguide dimensions, we find the mode pair (pump and signal) with the lowest phase mismatch that has non-zero effective nonlinearity. In Fig. 4, we show the effective nonlinearity γ and figure of merit for the best phase matched mode pair with the fundamental IR signal mode, as a function of waveguide dimensions. Regions that are colored in black for the TM signal γ do not support a TM signal mode. Discontinuities in the plots of γ arise when the pump mode with the lowest phase mismatch changes. The regions with high in Fig. 4 are further restricted by the fact that the conversion efficiency drops significantly for negative phase mismatch Δk < 0 (black regions) because normal dispersion in the signal modes at the signal frequencies makes simultaneous phase matching and energy conservation impossible. Rapid broadening of the signal spectrum when shifting waveguide dimensions away from regions that achieve PPM greatly reduces .

 figure: Fig. 4

Fig. 4 (top) The effective nonlinearity γ as a function of waveguide width and thickness for the best phase-matching point. The highest γ = 1100 W−1km−1 is achieved in a 600 × 300 nm waveguide for TE signal and a 600 × 400 nm waveguide for TM signal. However, these dimensions do not achieve phase-matching. Lines mark regions with high which do achieve phase-matching. (bottom) Figure of merit as a function of waveguide dimensions. The best combination of phase matching and γ are achieved at 600 × 245 nm (γ = 908 W−1km−1) for TE signal and 550 × 360 nm (γ = 674 W−1km−1) for TM signal.

Download Full Size | PPT Slide | PDF

We extract the optimal waveguide dimensions using the maximum values of (Table 1). Choosing a region where a high figure of merit is maintained for a larger range of waveguide dimensions can reduce the negative impacts of fabrication tolerances on device performance in experimental demonstrations. The same figure of merit can be used to optimize other waveguide parameters and geometries.

Tables Icon

Table 1. Waveguide Parameters for Phase-matching Regions With High Figure of Merit . es is the Signal Mode Polarization and γ the Effective Nonlinearity.

3.2. Experimental validation of the protocol

The reverse process of TOSPDC, THG, has been demonstrated in nanoscale polycrystalline anatase TiO2 waveguides by Evans et al. [22]. Because the same energy conservation, phase matching and mode overlap constraints govern both processes, we perform similar experimental validation of our design methods by means of THG measurements.

The waveguide (780 × 244 nm) is designed to have multiple phase-match points within the TE-polarized pump bandwidth. Using the techniques from Sections 2.1 and 2.2, we calculate phase match points with γ = 253 and 304 W−1km−1 at 513.3 and 523.0 nm, respectively. In Fig. 5, we plot the expected relative intensities of THG based on the measured input pump spectrum, calculated mode dispersion, and calculated γ of both mode pairs, showing close agreement with experimental results. The experimental phase-match points are within 1.5 nm of the calculated phase match points and the measured THG signal is broader by approximately 3 nm. This can be explained by variations in waveguide width of ±5 nm and thickness of ±2 nm along the length of the waveguide. These variations are within the measured fabrication tolerances of waveguide width and roughness of the film used to fabricate the device. This demonstration experimentally shows that the device design methods outlined in this paper can be used to optimize devices for THG and, consequently, TOSPDC.

 figure: Fig. 5

Fig. 5 Experimental demonstration of THG in an integrated waveguide with two phase-matching points within the infrared pump bandwidth (black curve). The calculated THG signal (dashed curve) shows agreement with the measured THG signal (solid green curve). The inset shows a top-down image of scattered THG signal from the waveguide.

Download Full Size | PPT Slide | PDF

3.3. Realistic device performance

Combining the results of the design sweeps presented in Figs. 3 and 4 with measured losses, we estimate TOSPDC device performance. We have previously measured losses as low as 4 dB/cm in polycrystalline anatase TiO2 waveguides in the telecommunications wavelengths [20, 26] and estimate losses of 20–30 dB/cm at 532 nm. With this amount of losses, the optimal device length is Lopt = 2.2 – 2.8 mm. We estimate a photon triplet conversion efficiency of 0.1 – 0.13 triplets/s/mW of pump power and maximum generation rate of 130 – 160 triplets/s. This rate takes into account an optimal end-fire coupling efficiency of 20.6% calculated for the 550 × 360 nm waveguide and higher-order pump mode described in Fig. 1 and an input pump power of 1250 mW at 532 nm, corresponding to the measured damage threshold of our TiO2 devices. We note that the TE signal phase-matching point in a 600 × 245 nm waveguide does have a higher γ, however, the photon triplet generation rate would be lower due to a maximum end-fire coupling efficiency of only 10.5% into the higher-order pump mode. We use a nonlinear index of n2 = 4.65 × 10−19 m2/W. Film losses as low as 3 dB/cm in the telecommunications wavelengths and 15 dB/cm in the visible wavelengths have been measured using prism coupling techniques [20]. By optimizing the fabrication, we can reduce losses to the limit of the film losses, increasing the optimal device length to 3.7 mm and the maximum triplet generation efficiency to 0.17 triplets/s/mW of pump power, over an order of magnitude higher than previous theoretical and experimental results [13,14,24,25,34]. This highlights the importance of reducing photon losses in waveguide-based devices.

4. Temporal coherence of TOSPDC sources

Second-order spontaneous parametric down conversion (SPDC) generates photon pairs that possess quantum correlations in energy and time degrees of freedom. This energy-time entanglement can be exploited in quantum communication protocols [35, 36]. Our TOSPDC sources are expected to exhibit tripartite energy-time entanglement. However, the type of encoding of quantum information that can be exploited for quantum communication protocols with our source ultimately depends on the spectral and temporal coherence properties of the down-converted signal [37]. In this section we therefore discuss the coherence properties of TOSPDC sources for realistic detector and loss parameters.

The temporal coherence of a triplet source can be characterized in three-photon coincidence detection experiments, which gives access to the third-order intensity correlation function [16]

G(3)(x1,x2,x3)=E()(x1)E()(x2)E()(x3)E(+)(x3)E(+)(x2)E(+)(x1),
where Ê(+)(xj) is an operator describing the propagating electric field at the j-th detector, with Ê(−) = [Ê(+)]. The field at the detector is given by a wavepacket of the form Ê(+)(xj) = ∫ dωfj(ω)â(ω)exp[− iωxj], where â(ω) is a free-space bosonic operator. We assume a Gaussian detector filter function fj(ω)=f0exp[(ωωj)2/2σj2] [8]. f0 is a normalization constant and ωj is the center frequency of the j-th spectral filter. For simplicity, we assume that all filters have the same bandwidth σk = σ.

The G(3)(x1, x2, x3) correlation function is proportional to the probability of detecting a photon at the space-time location x1 = t1r1/c, followed by a detection event at location x2 = t2r2/c and another one at x3 = t3r3/c. In this notation, the j-th detector fires at time tj at a distance rj from the TOSPDC output. This triple coincidence signal can be computed for an output triplet state |Ψ3〉 as

G(3)(x1,x2,x3)=|0|E(+)(x3)E(+)(x2)E(+)(x1)|Ψ3|2.
We are interested in G(3) for a triplet state generated by a narrow band cw pump near the perfect phase matching point (PPM), defined in Section 2.1. For a pump photon with fixed frequency ωp0 and wavenumber kp0, the triplet state |Ψ3〉 is a continuous wavepacket describing three signal photons having frequencies (ωr, ωs, ωi) that satisfy energy and momentum conservation. The triplet wavepacket is characterized by a joint spectral amplitude function Φ(ωr, ωs, ωi), whose imaginary part depends on the amount of pump and signal losses. We refer the reader to Appendix C for more details on this point. Signal photon frequencies that are detuned from the PPM point, lead to a phase mismatch Δk ≠ 0. For the triplet state |Ψ3〉 described in Appendix C, we numerically evaluate the G(3) function by expanding the frequency mismatch Δk up to second order in signal detunings from PPM.

In Fig. 6 we plot the computed G(3) signals with time delays in units of the timescale τ0=DsL/2, where L is the waveguide length and Ds is the group velocity dispersion (GVD) of the signal guiding mode. For a typical optimized TiO2 waveguide with L = 2 mm and Ds = 5.36 × 10−3 ps2/mm, the characteristic coherence time is τ0 = 73 fs. The signal depends on the bandwidth σ of the detectors, given here in units of 1/τ0, as well as the loss mismatch parameter Δα ≡ (αp − 3αs), which characterizes the frequency-time decoherence of the triphoton state. For our 2 mm long waveguides, we have a small loss mismatch ΔαL ≈ 0.08. The mirror symmetry around τ12 = τ32 is due to the fact that all photons in a triplet have the same GVD in mode-degenerate TOSPDC. The FHWM of the G(3) function along the cut τ12 = −τ32 is roughly equals to τ0 for large detector bandwidths στ0 ≫ 1. Small bandwidths στ0 ≪ 1 broaden the G(3) function such that highly correlated photons can be detected within a wide range of time delays. This behaviour resembles the two-photon coincidence signals for second-order SPDC sources [8]. Increasing the loss mismatch Δα also broadens the G(3) function, but in a qualitatively different way. Photon losses directly broaden the joint spectral amplitude Φ(ωr, ωs, ωi), which characterizes the wavepacket structure of the triplet state |Ψ3〉 in the frequency domain. This broadening is independent of the time-domain field representation of |Ψ3〉 at the detectors, the so-called triphoton wavepacket ψ(x1, x2, x3) ≡ 〈0|E(+)(x3)E(+)(x2)E(+)(x1) |Ψ3〉, which involves shaping the triplet state |Ψ3〉 by the detector filters.

 figure: Fig. 6

Fig. 6 Triple and two-photon coincidence signals for detectors placed at equal distances from the TOSPDC output. Time delays τ12 = t1t2 and τ32 = t3t2 are given in units of the characteristic timescale τ0=DsL/2. Panels (a) and (b) are G(3) functions for the detector filter bandwidths σ = 0.2ν0 and σ = 5ν0, respectively, with a loss mismatch parameter ΔαL ≡ (αp − 3αs) L = 0.1. Panels (d) and (e) correspond to the G(3) functions for ΔαL = 1 and ΔαL = 10, respectively, with a detector bandwidth σ = 5ν0. Panel (c) is the G(2) function for several values of σ with fixed ΔαL = 0.1 and (f) is the G(2) signal for several values of ΔαL with fixed σ = 5ν0. Frequency is in units of ν0 = 1/τ0, L is the waveguide length and Ds is the group velocity dispersion (GVD) of the signal guiding mode. Plots are normalized to their maximum values.

Download Full Size | PPT Slide | PDF

We finally consider a scenario where only two photons of the TOSPDC source are detected in coincidence measurements. In other words, we discard the information about the third photon. The resulting coincidence signal is proportional to the correlation function G(2)(x1, x2), the second order analogue of G(3). For the TOSPDC triplet state |Ψ3〉, this signal can be written as

G(2)(x1,x2)=dω3|0|a(ω3)E(+)(x2)E(+)(x1)|Ψ3|2.
We compute the G(2) correlation function as described in Appendix D for a source pumped at the PPM point, for several values of the detector bandwidth σ and loss mismatch Δα. The results are shown in Fig. 6 (panels c and f). The two-photon correlation function has qualitatively the same behaviour as the the G(3) function with σ and Δα. Its width is also on the order of 1/τ0 for στ0 ≫ 1. These results show that unlike an SPDC process, where detection of one photon in a photon pair destroys any useful entanglement [8], frequency-time entanglement in a TOSPDC source is robust against single photon losses [38].

5. Conclusion and outlook

We provide design principles for practical on-chip sources of photon triplets generated via TOSPDC and discuss the non-unitary propagation of the photon triplet state in the presence of pump and signal photon losses. Including the impact of losses is critical, especially when operating at visible wavelengths where surface roughness and fabrication imperfections are closer to the size of a single wavelength and material absorption increases close to the band edge of the waveguide core material. The design methods discussed in this work can be applied broadly to various device geometries, loss conditions, and material platforms. This will facilitate experimental demonstrations of spontaneously generated photon triplets and development of a platform for commercially viable sources.

We use TiO2 to illustrate our design protocol. TiO2 is a promising material platform for an integrated entangled photon triplet source due to its transparency across the visible and telecommunication wavelengths as well as its high linear and nonlinear refractive indices. We discuss the mean signal flux and prospects for detecting continuous-variable entanglement of the photon triplet quantum state in the presence of photon loss. We calculate triplet generation rates on the order of 10−1 triplets/s/mW of pump power, which exceeds the triplet generation efficiencies obtained by cascading entangled photon pairs by an order of magnitude. Our work thus sets the stage for the development of on-chip photon triplet sources for applications in photonic quantum technologies.

Improvements to the proposed waveguide design and current fabrication techniques can greatly enhance generation rates and accelerate experimental efforts. Generating photon triplets within an integrated micro-ring resonator or resonant cavity can provide an additional constraint on the spectrum of generated photons, enabling greater spectral control of the output signal via the Purcell effect. Optimization of fabrication processes, including resist reflow techniques and new etchless fabrication techniques [39], can reduce waveguide losses. Photonic crystals and slot waveguides can be used to greatly enhance the effective nonlinearity of integrated photonic devices [40].

Device performance can also benefit from improvements in material deposition. Film optimization through epitaxial growth has the potential to dramatically reduce scattering losses due to grain boundaries, increasing the achievable optimal device length and conversion efficiency. Rutile TiO2 is unexplored as a photonic device platform although other studies have shown extraordinarily high nonlinearity at visible wavelengths in this material [41]. Alternatively, exploring other material platforms, for example silicon nitride or diamond, may enable better devices despite their lower nonlinearity due to the trade-offs between propagation losses, nonlinearity, and damage threshold.

Several experimental considerations need to be taken into account for successful measurements of photon triplets. Although a pulsed pump would enable gated detection schemes important for many applications, competing intensity-dependent nonlinear interactions, such as multi-photon absorption and self- and cross-phase modulation, would limit the usable pulse energy and attainable photon triplet generation rate. Secondly, spectral filtering at the device output is necessary to reduce noise from residual pump photons. Lastly, as in all experimental demonstrations, the rate of detected photon triplets is lower than the generated rate due to losses at the device output, detector efficiency, and the detection scheme for measuring 3-photon coincidences. These factors, coupled with the required fabrication precision for realizing an efficient device, highlight the challenges associated with an experimental demonstration.

In summary, we have developed a design protocol to optimize the efficiency of TOSPDC in nanoscale waveguides. We illustrate the scheme using TiO2 photonic chips, but the method is general and can be applied to any material platform. Entangled photon triplets generated this way can find applications in quantum tasks that can benefit from states with non-Gaussian quantum statistics, serve as a starting point to build large entangled states of light for quantum information purposes, or to develop novel spectroscopic tools.

A. Signal bandwidth for mode-degenerate TOSPDC

In this Appendix we derive Eq. (2) of the main manuscript, for the signal bandwidth Δs. We expand the signal and pump wavenumbers around the perfect phase-matching (PPM) point. We allow the signal and pump frequencies to be detuned from this crossing by the quantities Δm=ωmωp0/3 with m = {s, r, i}. The phase-matching condition at the PPM point reads kp0=3ks0. Expanding the pump and signal wavenumbers around PPM with respect to the pump and signal detunings, the condition (kpkskrki) = 0 yields

(Δs2+Δr2+Δi2)=2(vp1vs1)DsΔp+(DpDs)Δp2rp2
where vmm/dkp and Dmd2km/dωm2 are the mode group velocity and group velocity dispersion (GVD) of the m-th photon at the PPM point. Because the three signal photons belong to the same guided mode, they have the same values of vs and Ds, defined at ωp0/3.

We use energy conservation to simplify the linear terms with respect to the signal detunings. Eq. (12) shows that for mode-degenerate and frequency non-degenerate TOSPDC, phase matching can be satisfied by the signal detunings Δm that lie on the surface of a sphere with radius rp, subject to the linear constraint Δp = Δs + Δr + Δi, due to energy conservation. We define dimensionless signal detunings xm = Δmp, for m = {r, s}, and the parameter ap=rp2/(2Δp2)1 (in Eq. 3 we use Ap=apΔp2). The points satisfying both energy and momentum conservation form the ellipse

xs2+xr2xrxsxrxs=ap.
The signal bandwidth is therefore given by the distance between the critical points xs=1±(1+ap)/3. Inserting the expression for ap and inserting the original frequency variables we can write the signal bandwidth as
δs23×2rp=2(23)1/2[2(vp1vs1)DsΔp+(DpDs)Δp2]1/2,
which exhibits a crossover in the scaling (Δp)α from α = 1/2 to α = 1 roughly at the pump detuning |Δp| = 2|vsvp|/(vsvpDp).

B. Overlap calculation with nonlinear susceptibility tensor elements

In this Appendix, we provide a generalized method of calculating overlap η and effective nonlinearity γ that can be applied to any crystal symmetry and orientation, as well as a derivation of the symmetry conditions that apply in polycrystalline anatase TiO2. We calculate the effective nonlinearity of TOSPDC for phase-matched pump and signal modes using the definition

γ=3χ(3)ωp4ε0c2n2η,
where n is the material index, χ(3) is the nonlinear susceptibility and η is the overlap between the modes. Crystal symmetry is taken into account to determine relative magnitudes of χ(3) tensor elements. We discuss considerations relevant to determining the strength of the nonlinearity at the pump wavelength.

The overlap between phase-matched pump and signal modes propagating along the z-direction is calculated using the x, y, z electric field components of each mode as

ηijkl=dxdyEpi*EsjEskEsl[|Ep(x,y)|2dxdy]1/2[|Es(x,y)|2dxdy]3/2,
where Epi(x,y) and Esj(x,y) are the components of the pump and signal mode electric fields. The denominator in Eq. (16) normalizes the pump and signal fields such that they satisfy the condition ∬ |E|2 dxdy = 1. Figure 7 gives an example of a pair of modes with non-zero overlap.

 figure: Fig. 7

Fig. 7 Examples of (a) a signal mode power density profile and (b) pump mode power density profile. These mode profiles are for a TM00 signal mode and TM02 pump mode phase matched at 532 nm in a 550 × 360 nm TiO2 waveguide with SiO2 cladding (first presented in Fig. 1 of the main text).

Download Full Size | PPT Slide | PDF

Spatial distributions of the x, y, z E-field components of each mode in Eq. (16) are calculated using simulations [29]. The subscripts i, j, k, l correspond to the Cartesian indices for crystal axes along which the E-field is polarized. In our notation, the first index refers to the pump polarization and the subsequent three indices refer to the polarization of the signal modes. In the degenerate TOSPDC case, all three signal modes have the same E-field distribution.

To illustrate the calculation of overlap terms for a specific χ(3) tensor element, we first consider the simplest case of an E-field and a single crystal oriented along the same axes (Figs. 8(a) and 8(b)). We use indices x, y, z for the E-field coordinate axes and i, j, k, l for the crystal axes. In this case, the χiijj contribution to the TOSPDC process is determined by the field overlap component ηiijj=Epx*EsxEsyEsydxdy. For the same electric field orientation but a different crystal orientation shown in Fig. 8(c), the contribution of the χiijj term is calculated by projecting the E-field onto the crystal axes, such that e.g. Ei = Ex cosθ + Ey sinθ and Ej = −Ex sinθ + Ey cosθ and Ek = Ez, and then computing ηiijj=Epi*EsiEsjEsjdxdy. Because the crystal symmetry determines the nonlinearity, projecting the E-field onto the crystal axes allows us to calculate the total nonlinear overlap with the fewest number of unique terms, ηijkl.

 figure: Fig. 8

Fig. 8 Examples of a) E-field orientation, b) crystal axes orientation and c) crystal axes orientation with a rotation by an arbitrary angle in the xy plane.

Download Full Size | PPT Slide | PDF

There are 81 possible permutations of the field polarizations in the overlap calculation for ηijkl. Their contribution to the total overlap is determined by the magnitude of the corresponding nonlinear susceptibility tensor element, χijkl. The material crystal structure of the waveguide determines which χijkl terms are non-zero and their relative magnitudes. As an example, anatase TiO2 has tetragonal symmetry and belongs to the crystal class 4/mmm. Our anatase TiO2 films polycrystalline with randomly oriented grains much smaller than the wavelength (λs/30 and λp/10). As a result, an incoming field along the x-direction polarizes the medium along all orthogonal crystal axes. Furthermore, this response is indistinguishable from the polarization of the medium when the incoming field is along y or z. Therefore, we can treat the material as effectively isotropic. The relationships between non-zero isotropic tensor elements can be found in Ref. [30] and are as follows:

jjkk=kkjj=kkii=iikk=iijj=jjii;jkkj=kjjk=kiik=ikki=ijji=jiijjkjk=kjkj=kiki=ikik=ijij=jiji;iiii=jjjj=kkkk=iijj+ijij+ijji

For a material containing large grains with n crystal orientations, the E-field projection can be performed on each of the n crystal axes to identify the overlap terms of interest for each non-zero χ(3) term. Considering the small grain sizes and the assumption of isotropic crystal symmetry in our waveguide medium, we can significantly simplify this process. The relations in Eq. (17) show that there are three independent χ(3) variables: iijj, ijij and ijji. Furthermore, when we consider the degenerate TOSPDC case, we find that all three signal modes are the same; therefore, with a given pump index, we can arbitrarily arrange the signal indices and obtain the same overlap:

ηiijj=ηijij=ηijji.
This leads to three important conclusions: (i) all overlap terms of interest can be calculated by considering a single crystal axis orientation; (ii) using the relations in Eq. (18) we can simplify the number of overlap terms we calculate from 21 to 9 unique terms; and, (iii) we find that χiiii = χjjjj = χkkkk = 3χiijj, meaning that the contribution of the diagonal nonlinear tensor elements χiiii, χjjjj and χkkkk is three times stronger than the contribution generated by the non-zero off-diagonal terms. Note that there are three times as many off-diagonal χ(3) terms, so that each of the 9 unique overlap terms makes an equal contribution to the total overlap.

Based on the symmetry of the χ(3) tensor, we can also gain an intuition for which combinations of mode profiles yield non-zero overlap and contribute to the effective nonlinearity. In the case of polycrystalline TiO2, each of the x, y, and z components appears an even number of times in each non-zero χ(3) tensor element. The mode profile of the x, y, and z components of the field must therefore be even, meaning that sign of Ex, Ey, and Ez is symmetrical about the x and y axes. Alternatively, an even number of odd field profiles must be present to yield non-zero overlap.

Nanoscale waveguides often have a significant evanescent field, requiring that nonlinear overlaps take into account different χ(3) nonlinearities in the core and cladding regions. In this case, the integration in Eq. (16) is carried out over the area of the core, top cladding, and bottom cladding separately. These values are then used to calculate γ in each region separately using Eq. (15). The total effective nonlinearity of the waveguide will be the sum γ = γcore + γtopcladding + γbottomcladding.

In the main text we note that the χ(3) nonlinearity is parametrized using the nonlinear index

n2=3χ(3)4n02ε0c,
which is often measured using Z-scan and self-phase modulation techniques, both of which also rely on third-order nonlinear interactions. In order to produce photon triplets in the telecommunications band via TOSPDC, the pump wavelength is in the visible. Z-scan and self-phase modulation measurements are challenging in the presence of two-photon absorption and other effects prevalent below the half-bandgap wavelength [41, 42]. For this reason, we have applied a bandgap scaling approach of n2 [31] to nonlinearities measured in polycrystalline anatase TiO2 films for the range of wavelengths λ = 800 – 1560 nm [19]. The estimated value of n2 can be considered as a lower bound because the model in Ref. [31] is known to underestimate the magnitude of the nonlinearity at photon energies above three-quarters of the bandgap. As a result, our calculated nonlinearities for a 532-nm pump wavelength and the corresponding photon triplet generation rates can be considered a lower bound on expected device performance.

In conclusion, we find the total overlap in polycrystalline anatase TiO2 by summing 9 unique overlap terms calculated using Eq. (16). The effective nonlinearity is calculated, in turn, using the total overlap as an input parameter in Eq. (4) of the main text. Our approach for determining the effective interaction area and effective nonlinearity by assuming that our material has isotropic symmetry can be used for other polycrystalline materials with grains on the order of λ/10 or smaller. In addition, the general approach describes how to determine the effective nonlinearity for arbitrary crystal symmetries, crystal orientations and electric field orientations.

C. Triplet generation rate in presence of propagation losses

In this appendix we derive Eqs. (6) and (7) of the main text. We note that the effects of multi-photon absorption are ignored in this analysis and justify this assumption for the waveguide parameters presented in the main text.

We use a formalism that extends the approach in Refs. [16, 24]. The starting point is the expression for the TOSPDC light-matter interaction Hamiltonian

H^1=3ε0χ(3)4dVE^p(+)(r,t)E^r()(r,t)E^s()(r,t)E^i()(r,t)+H.c.
in terms of the electric field operators E^(+)(r,t)=iA(x,y)δkk(ω)exp[i(kzωt)]a^(t), where (ω)=h¯ω/πε0n2(ω). ε0 is the vacuum permittivity, n(ω) the refractive index, and is Planck’s constant. The label p refers to the pump field, and {r, s, i} refer to the signal fields. δk = 2π/LQ is the mode spacing defined by the quantization length LQ, A(x, y) characterizes the transverse spatial distribution of the field, taken to be normalized and frequency-independent.

We describe the pump mode as a strong classical field of the form

Ep(+)(r,t)=A0Ap(x,y)dωpβ(ωp)e[i(kp(ωp)zωpt)],
and consider the pump Gaussian amplitude β(ωp)=21/4/(π1/4σ)exp[(ωpωp0)2/σ2] where ωp0 is the pump carrier frequency and σ the pump pulse bandwidth. In general, A0 is related to the peak pump power, but for a Gaussian pump amplitude, it is related to the average pump power P by A0=P/πε0cnpR, with R the pulse repetition rate and np=n(ωp0).

In the absence of propagation losses, first-order perturbation theory with respect to the interaction Hamiltonian allows us to write the output signal state as [8]

|Ψ=[1ih¯t0tdtH1(t)]|0=|0+λ|Ψ3.
The photon triplet state can be written as
|Ψ3=dωpβ(ωp)ks,kr,ki(ωr)(ωs)(ωi)[t0tdtdzϕ(t,z)a^(kr)a^(ks)a^(ki)|0],
where ωm = ω(km) for m = {s, r, i}. We have defined the frequency-dependent function ϕ(t, z) = exp[iΔωt]exp[iΔkz], where Δk = (kpkrkski) is the phase mismatch and Δω = (ωr + ωs + ωiωp) the energy mismatch. The spatial integration is carried out over the waveguide length L. The photon triplet state amplitude λ is given by
λ=3ε0χ(3)A04h¯Aeff(δk)3/2,
where the effective area is the inverse of η given in Eq. (16).

In order to include the propagation losses, we model them phenomenologically through an exponential decay of the pump and signal fields. This approach ignores the probability of observing signal modes with one or two photons as a result of photon scattering. Such conditional states are highly unlikely to occur because the signal modes are found in the vacuum |0〉 with a near unit probability because λ ≪ 1. We are interested in the part of the output state that involves three photons. If the interaction takes place at position z within a waveguide that spans z = [−L/2, L/2], the corresponding amplitude is decreased by a factor: e−(αp/2)(z+L/2) e−3(αs/2)(L/2−z), where we have used the attenuation coefficients αp and αs (in units of inverse length) to describe pump and signal photon loss, respectively. Therefore, the photon triplet state in Eq. (23) needs to be modified in the presence of photon losses to read

|Ψ3=dωpβ(ωp)kr,ks,ki(ωr)(ωs)(ωi)×[t0tdtdzϕ(t,z)e(αp3αs)z/2e(αp3αs)L/4×a^(kr)a^(ks)a^(ki)|0].

Because the propagation times inside the waveguide are much longer than 1/Δω, we carry out the time integration by setting t0 = −∞ and t = ∞ and obtain the energy conservation rule dtϕ(z,t)=2πeiΔkzδ(ωpωrωsωi). Carrying out the spatial integration including the loss terms, imposes the phase-matching condition associated to momentum conservation, which gives the triplet state

|Ψ3=(2πλL(δk)3)e(αp+3αs)L/4krkskiΦ(ωr,ωs,ωi)a^(ωr)a^(ωs)a^(ωi)|0,
where the condition ωp = ωr + ωs + ωi holds. We have defined the spectral amplitude function
Φ(ωr,ωs,ωi)=β(ωp)(ωr)(ωs)(ωi)sinc[{Δk+iΔα}L/4],
where the function sinc(x) = sin(x)/x is highly peaked at x = 0. We have defined the loss mismatch Δα = (αp − 3αs).

The signal intensity outside the waveguide is proportional to the expectation value 〈Ψ3|Ê(−)(t)Ê(+)(t)|Ψ3〉, with electric field operator E^(+)(t)iδkk(ω)exp[iωt]a^(k), that includes all possible wavevectors k. The detected intensity must be averaged over a specific time interval. The Fourier components of the intensity sum up separately. Since the signal intensity is proportional to the number of photons, we have N3Ψ3|ka^(k)a^(k)|Ψ3. Accordingly, the number of triplets per second (generalizing Eq. (19) in Ref. [24]) can be written as

R3=RΨ3|ka^(k)a^(k)|Ψ3=R(2πλL)2(32)δk3e(αp+3αs)L/2dωrdωsdωig(ωr)g(ωs)g(ωi)|Φ(ωr,ωs,ωi)|2
where g(ωi) = [∂k/∂ω]ω=ωi is a factor resulting from the change of variable from k to ω and R is the repetition rate of the pump. To further simplify the expression for R3, we expand the phase mismatch Δk to second order in the signal detuning νm from the point of PPM ωm0=ωp/3, assuming that the pump frequency is fixed at the PPM point. The zeroth and first order terms in the expansion vanish, giving
Δk=k(ωs)+k(ωr)+k(ωi)k(ωp)=Ds2[νr2+νs2+(νs+νr)2],
where we have eliminated one of the signal detunings (νi) from the integration using the energy conservation rule. Ds2k/∂ω2 is the group velocity dispersion (GVD) of the signal mode, which is the same for the all frequencies in mode-degenerate TOSPDC.

We note that for a Gaussian pump, the peak pump power P0 is related to the average pump power P and to the repetition rate R by P0=Pσ/2πR and that |dωpβ(ωp)|2=2πσ. The dependence on the pump bandwidth σ disappears from the factor outside the integrals characterizing the rate R3 in Eq. (28), so we can easily take the limit σ → 0 corresponding to continuous wave (cw) pumping to obtain

R3=2232h¯c3np3π2(ωp0)2γ2L2p(ω0n02(ω)g(ω0))3×e(αp+3αs)L/2dνrdνs|Φ(νr,νs)|2.
In the integration we used the approximation (ω0 + νr) = (ω0 + νs) = (ω0νrνs) = (ω0), with ω0 = ωp/3. For notational convenience, we made the replacement
(2πλL)232δk3=32(2π)2ε03c3np3h¯2(ωp0)22γ2L2P|dωpβ(ωp)|2,
where γ is the effective nonlinearity from Eq. (15).

In the main text, we also introduced Eq. (7) for estimating the triplet generation rate with losses, not taking into account mode dispersion. The expression can be derived as follows. The number of complete triplets that reach the end of the waveguide after being generated in the segment of the waveguide from z to z + Δz can be estimated as

dN3(z)=ζ˜Np(z)e3αs(Lz)Δz=ζ˜Np0eαpze3αs(Lz)dz,
where Np0 is the number of pump photons at the input facet of the waveguide. ζ̃ quantifies the conversion efficiency. Integrating the expression above from z = 0 to z = L gives the total number of generated triplets
N30LdzN3(z)=ζ˜Np0αp3αs(e3αsLeαpL).
Maximizing this expression with respect to L gives the optimal length Lopt.

Due to strong light confinement within integrated waveguides and the potential to use high pump powers with materials that have a high damage threshold, two-photon absorption must be considered even for continuous wave pump sources. Multi-photon absorption is described by the relationship

dI(z)dz=α(1)I(z)α(2)[I(z)]2
Where α(1) is single photon loss, α(2) is the two-photon absorption coefficient, and I(z) is the light intensity in the waveguide as a function of position. We can solve Eq. (34) to calculate light intensity as a function of z
I(z)=αI0/(α(1)+α(2)I0)eα(1)zα(2)I0/(α(1)+α(2)I0)
Where I0 is the initial light intensity inside the waveguide. Intensity relates to the number of photons in a waveguide propagating mode by N0 = I0Amode/h̄ωp where Amode is the cross-sectional area of the mode. We can rewrite Eq. (32) to include the effect of two-photon absorption
dN3(z)=ζ˜AmodeIp0h¯ωpαp(1)/(αp(1)+αp(2)Ip0)eαp(1)zαp(2)Ip0/(αp(1)+αp(2)Ip0)e3αs(1)(Lz)dz.

We can use the ratio of Eq. (36) and (32) to gain insight under which conditions two-photon absorption has a significant impact on device performance. We define an arbitrary threshold

αp(2)Ip0αp(1)(1eαp(1)Lopt)<0.2
below which the impact of two-photon absorption decreases the photon triplet generation rate by less than 10%. For our proposed TiO2 devices, this corresponds to αp(1)>10dB/cm, a maximum pump power of approximately 1 W, and αp(3)=3cm/GW (determined by band-gap scaling [30] of two-photon absorption measured in the wavelength range 780 – 1560 nm [19]).

D. Coincidence detection signals

The temporal coherence of a triplet source is characterized by the third-order intensity correlation function

G(3)(x1,x2,x3)=Ψ3|E()(x1)E()(x2)E()(x3)E(+)(x3)E(+)(x2)E(+)(x1)Ψ3|.
where |Ψ3〉 is the triplet state defined in Eq. (26). For waveguides much longer than a typical signal wavelength, we can replace the wave vector summations by frequency integrals. Energy conservation allows us to eliminate one of the signal variables from the integration. We can also absorb physical parameters like the cw pump amplitude, the effective waveguide area and the material nonlinear susceptibility into a normalization factor factor that depends on the waveguide length L. We assume a narrow cw pump at fixed frequency ωp. The triplet state is completely characterized by the spectral amplitude Φ(ωr, ωs) in Eq. (27). The loss mismatch parameter Δα ≡ (αp − 3αs) introduces a frequency-dependent imaginary part to the spectral amplitude, which broadens the triplet state in the frequency domain. The G(3) function in Eq. (38) can be written in simplified form as
G3(x1,x2,x3)=|ψ1(x1,x2,x3)|2,
were ψ(x1, x2, x3) = 〈0|E(+)(x3)E(+)(x2)E(+)(x1) |Ψ3〉 is the so-called triphoton state [16], which can be written as
ψ(x1,x2,x3)×dω1dω2dω3dωrdωsf1(ω1)f2(ω2)f3(ω3)eiω1x1iω2x2iω3x3×Φ(ωr,ωs)0|a(ω1)a(ω2)a(ω3)a^(ωr)a^(ωs)a^(ωpωrωs)|0.
We assume Gaussian detector filters fj(ω)=f0exp[(ωωj)2/2σj2], where f0 is a normalization constant and ωj is the center frequency of the j-th spectral filter. All filters are assumed to have the same bandwidth σk = σ. We ignore the frequency-independent normalization factor of the state.

The correlation function 〈0|a(ω1)a(ω2)a(ω3)a(ωr)a(ωs)a(ωi)|0〉 in Eq. (40) contains six terms corresponding to permutations of the frequency variables. We can interpret each term by labelling a photon in the triplet source by its frequency. There are thus 3! = 6 ways for these photons to reach three photodetectors. If the optical paths from the triplet source to each detector are indistinguishable, then all six frequency permutations give the same contribution to the field correlation function. It thus suffices to compute a single permutation to study the triple coincidence signal. We choose ω1 = ωr, ω2 = ωs, and ω3 = ωpωrωs to obtain

ψ(x1,x2,x3)=eiωpx3dωrdωsf(ωr)f(ωs)f(ωpωrωs)Φ(ωrωs)eiωr(x1x3)iωs(x2x3).
We expand the frequencies as ωm = Ωm + νm, with m = {r, s, i}, around central signal frequencies Ωm that satisfy the energy conservation rule ωp = ωr + ωs + ωi at the PPM point. In what follows we consider that the spectral filter f(ωm) is centered at the frequency Ωm. Expanding the wavenumbers to second order in the frequency detunings νm around the PPM point gives a phase mismatch Δk as in Eq. (29). For mode-degenerate down-conversion the group velocities are equal for the three photons, making the linear term vanish by energy conservation for a cw pump. We can rewrite the integrals in terms in terms of frequency detunings to obtain
G(3)(x1,x2,x3)=|dνrdνsfr(νr)fs(νs)fi(νr+νs)Φ(νr,νs)eiνr(x1x3)iνs(x2x3)|2.
The numerical evaluation of the integral is simplified by rescaling the detunings and delay times as νn0 and xn0, where τ0=DsL/2 and ν0 = 1/τ0 are the characteristic frequency and timescale associated with a waveguide of length L and group velocity dispersion Ds of the signal guiding mode.

The two-photon coincidence detection signal is proportional to the second-order correlation function

G(2)(x1,x2)=E^()(x1)E^()(x2)E^(+)(x2)E^(+)(x1),
which can be written by inserting the resolution of the identity in the Fock basis as
G(2)(x1,x2)=dω3|0|a(ω3)E(+)(x2)E(+)(x1)|Ψ3|2,
with a wavepacket amplitude given by
0|a(ω3)E(+)(x2)E(+)(x1)|Ψ3=dω1dω2dωrdωsΦ(ωr,ωs)f(ω1)f(ω2)eiω1x1iω2x2×0|a(ω3)a(ω1)a(ω2)a(ωr)a(ωs)a(ωpωrωs)|0.
The evaluation of the vacuum correlation function gives 2! = 4 non-zero terms associated with the commutation of the operators a(ω1) and a(ω2) with a(ωr) and a(ωs). These these four terms can be made equal to each other by exchanging integration variables appropriately when the optical paths to the detectors are equal. We thus evaluate a single representative term, corresponding to ω1 = ωr and ω2 = ωs to obtain
G(2)(x1,x2)=dν3|dνrG(νr,νr+ν3)f1(νr)f2(νr+ν3)eiνr(x1x2)|2.

Acknowledgments

MGM and FH are co-first authors of the manuscript. CCE proposed to use TiO2 for TOSPDC. MGM, CCE, FH and OR developed the description of phase matching in integrated devices. MGM, SG-N and OR developed the mode overlap calculations and device design methods. OR fabricated waveguides and MGM, SG-N, and OR completed measurements of THG. FH and GGG developed the quantum theory of lossy TOSPDC. FH performed stochastic wavefunction simulations and derived the entanglement witness for non-degenerate TOSPDC. EM and AA-G supervised the research and manuscript preparation. All authors approved the final copy of the manuscript. Benjamin Franta and Philip Muñoz provided feedback on the manuscript throughout its development. We acknowledge support from the NSF Division of Physics under contract PHY-1415236, the Harvard Quantum Optics Center, the Natural Sciences and Engineering Research Council of Canada, the NSF Graduate Research Fellowship Program (DGE1144152), and Harvard CNS. FH thanks support from DTRA HDTRA 1-100109946 and the CONICYT programs PAI 79140030 and Iniciación 11140158. CCE acknowledges support from the Kavli Institute at Cornell for Nanoscale Science.

References and links

1. J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009). [CrossRef]  

2. Z.-S. Yuan, X.-H. Bao, C.-Y. Lu, J. Zhang, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Phys. Rep. 497, 1–40 (2010). [CrossRef]  

3. V. Giovannetti and S. Lloyd, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011). [CrossRef]  

4. L. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. 4, S176 (2002). [CrossRef]  

5. M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-a. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013). [CrossRef]  

6. M. G. Raymer, A. H. Marcus, J. R. Widom, and D. L. P. Vitullo, “Entangled photon-pair two-dimensional fluorescence spectroscopy,” J. Phys. Chem. B 117, 15559–15575 (2013). [CrossRef]   [PubMed]  

7. T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464, 45 (2010). [CrossRef]   [PubMed]  

8. Y. Shih, “Entangled biphoton source–roperty and preparation,” Rep. Prog. Phys. 66, 1009–1044 (2003). [CrossRef]  

9. A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaitre, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010). [CrossRef]   [PubMed]  

10. 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: At. Mol. Opt. Phys. 75, 023803 (2007). [CrossRef]  

11. A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004). [CrossRef]   [PubMed]  

12. M. Gimeno-Segovia, P. Shadbolt, D. E. Browne, and T. Rudolph, “From three-photon Greenberger-Horne-Zeilinger states to ballistic universal quantum computation,” Phys. Rev. Lett. 115, 020502 (2015). [CrossRef]   [PubMed]  

13. H. Hubel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature 466, 601–603 (2010). [CrossRef]   [PubMed]  

14. L. K. Shalm, D. R. Hamel, Z. Yan, C. Simon, K. J. Resch, and T. Jennewein, “Three-photon energy–time entanglement,” Nat. Phys. 9, 19–22 (2012). [CrossRef]  

15. D. R. Hamel, L. K. Shalm, H. Hübel, A. J. Miller, F. Marsili, V. B. Verma, R. P. Mirin, S. W. Nam, K. J. Resch, and T. Jennewein, “Direct generation of three-photon polarization entanglement,” Nat. Photonics 8, 801–807 (2014). [CrossRef]  

16. M. Chekhova, O. Ivanova, V. Berardi, and A. Garuccio, “Spectral properties of three-photon entangled states generated via three-photon parametric down-conversion in a χ(3) medium,” Phys. Rev. A: At. Mol. Opt. Phys. 72, 023818 (2005). [CrossRef]  

17. K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” C. R. Phys. 8, 206–220 (2007). [CrossRef]  

18. C. C. Evans, J. D. B. Bradley, E. A. Marti-Panameno, and E. Mazur, “Mixed two- and three-photon absorption in bulk rutile (TiO2) around 800 nm,” Opt. Express 20, 3118–3128 (2012). [CrossRef]   [PubMed]  

19. C. C. Evans, K. Shtyrkova, J. D. B. Bradley, O. Reshef, E. Ippen, and E. Mazur, “Spectral broadening in anatase titanium dioxide waveguides at telecommunication and near-visible wavelengths,” Opt. Express 21, 18582–18591 (2013). [CrossRef]   [PubMed]  

20. J. D. B. Bradley, C. C. Evans, J. T. Choy, O. Reshef, P. B. Deotare, F. Parsy, K. C. Phillips, M. Loncar, and E. Mazur, “Submicrometer-wide amorphous and polycrystalline anatase TiO2 waveguides for microphotonic devices,” Opt. Express 20, 23821–23831 (2012). [CrossRef]   [PubMed]  

21. J. T. Choy, J. D. B. Bradley, P. B. Deotare, I. B. Burgess, C. C. Evans, E. Mazur, and M. Loncar, “Integrated TiO2 resonators for visible photonics,” Opt. Lett. 37, 539–541 (2012). [CrossRef]   [PubMed]  

22. C. C. Evans, K. Shtyrkova, O. Reshef, M. Moebius, J. D. B. Bradley, S. Griesse-Nascimento, E. Ippen, and E. Mazur, “Multimode phase-matched third-harmonic generation in sub-micrometer-wide anatase TiO2 waveguides,” Opt. Express 23, 7832 (2015). [CrossRef]   [PubMed]  

23. M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: single-photon sources and detectors,” Rev. Sci. Instrum. 82, 71101–71125 (2011). [CrossRef]  

24. M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Third-order spontaneous parametric down-conversion in thin optical fibers as a photon-triplet source,” Phys. Rev. A: At. Mol. Opt. Phys. 84, 033823 (2011). [CrossRef]  

25. S. Richard, K. Bencheikh, B. Boulanger, and J. A. Levenson, “Semiclassical model of triple photons generation in optical fibers,” Opt. Lett. 36, 3000–3002 (2011). [CrossRef]   [PubMed]  

26. O. Reshef, K. Shtyrkova, M. G. Moebius, S. Griesse-Nascimento, S. Spector, C. C. Evans, E. Ippen, and E. Mazur, “Polycrystalline anatase titanium dioxide micro-ring resonators with negative thermo-optic coefficient,” J. Opt. Soc. Am. B: Opt. Phys. 32, 2288–2293 (2015). [CrossRef]  

27. K. Preston, B. Schmidt, and M. Lipson, “Polysilicon photonic resonators for large-scale 3D integration of optical networks,” Opt. Express 15, 17283 (2007). [CrossRef]   [PubMed]  

28. I. A. Bufetov, M. V. Grekov, K. M. Golant, E. M. Dianov, and R. R. Khrapko, “Ultraviolet-light generation in nitrogen-doped silica fiber,” Opt. Lett. 22, 1394 (1997). [CrossRef]  

29. Z. Zhu and T. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express 10, 853–864 (2002). [CrossRef]   [PubMed]  

30. R. Boyd, Nonlinear Optics, 3rd ed. (Elsevier, Burlington, USA, 2008).

31. M. Sheik-Bahae, D. Hagan, and E. Van Stryland, “Dispersion and band-gap scaling of the electronic kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65, 96–99 (1990). [CrossRef]   [PubMed]  

32. T. Lee, N. G. R. Broderick, and G. Brambilla, “Resonantly enhanced third harmonic generation in microfiber loop resonators,” J. Opt. Soc. Am. B: Opt. Phys. 30, 505–511 (2013). [CrossRef]  

33. B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009). [CrossRef]  

34. S. Krapick and C. Silberhorn, “Analysis of photon triplet generation in pulsed cascaded parametric down-conversion sources,” arXiv.org/abs/1506.07655 (2015).

35. N. Gisin and R. Thew, “Quantum communication,” Nat. Photonics 1, 165–171 (2007). [CrossRef]  

36. J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594 (1999). [CrossRef]  

37. I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007). [CrossRef]   [PubMed]  

38. J. Wen and M. Rubin, “Distinction of tripartite Greenberger-Horne-Zeilinger and W states entangled in time (or energy) and space,” Phys. Rev. A: At. Mol. Opt. Phys. 79, 025802 (2009). [CrossRef]  

39. C. C. Evans, C. Liu, and J. Suntivich, “Low-loss titanium dioxide waveguides and resonators using a dielectric lift-off fabrication process,” Opt. Express 23, 11160 (2015). [CrossRef]   [PubMed]  

40. M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211–219 (2004). [CrossRef]  

41. Y. Watanabe, M. Ohnishi, and T. Tsuchiya, “Measurement of nonlinear absorption and refraction in titanium dioxide single crystal by using a phase distortion method,” Appl. Phys. Lett. 66, 3431 (1995). [CrossRef]  

42. S. K. Das, C. Schwanke, A. Pfuch, W. Seeber, M. Bock, G. Steinmeyer, T. Elsaesser, and R. Grunwald, “Highly efficient THG in TiO2 nanolayers for third-order pulse characterization,” Opt. Express 19, 16985–16995 (2011). [CrossRef]   [PubMed]  

References

  • View by:

  1. J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
    [Crossref]
  2. Z.-S. Yuan, X.-H. Bao, C.-Y. Lu, J. Zhang, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Phys. Rep. 497, 1–40 (2010).
    [Crossref]
  3. V. Giovannetti and S. Lloyd, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
    [Crossref]
  4. L. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. 4, S176 (2002).
    [Crossref]
  5. M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-a. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
    [Crossref]
  6. M. G. Raymer, A. H. Marcus, J. R. Widom, and D. L. P. Vitullo, “Entangled photon-pair two-dimensional fluorescence spectroscopy,” J. Phys. Chem. B 117, 15559–15575 (2013).
    [Crossref] [PubMed]
  7. T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464, 45 (2010).
    [Crossref] [PubMed]
  8. Y. Shih, “Entangled biphoton source–roperty and preparation,” Rep. Prog. Phys. 66, 1009–1044 (2003).
    [Crossref]
  9. A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaitre, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
    [Crossref] [PubMed]
  10. 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: At. Mol. Opt. Phys. 75, 023803 (2007).
    [Crossref]
  11. A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
    [Crossref] [PubMed]
  12. M. Gimeno-Segovia, P. Shadbolt, D. E. Browne, and T. Rudolph, “From three-photon Greenberger-Horne-Zeilinger states to ballistic universal quantum computation,” Phys. Rev. Lett. 115, 020502 (2015).
    [Crossref] [PubMed]
  13. H. Hubel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature 466, 601–603 (2010).
    [Crossref] [PubMed]
  14. L. K. Shalm, D. R. Hamel, Z. Yan, C. Simon, K. J. Resch, and T. Jennewein, “Three-photon energy–time entanglement,” Nat. Phys. 9, 19–22 (2012).
    [Crossref]
  15. D. R. Hamel, L. K. Shalm, H. Hübel, A. J. Miller, F. Marsili, V. B. Verma, R. P. Mirin, S. W. Nam, K. J. Resch, and T. Jennewein, “Direct generation of three-photon polarization entanglement,” Nat. Photonics 8, 801–807 (2014).
    [Crossref]
  16. M. Chekhova, O. Ivanova, V. Berardi, and A. Garuccio, “Spectral properties of three-photon entangled states generated via three-photon parametric down-conversion in a χ(3) medium,” Phys. Rev. A: At. Mol. Opt. Phys. 72, 023818 (2005).
    [Crossref]
  17. K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” C. R. Phys. 8, 206–220 (2007).
    [Crossref]
  18. C. C. Evans, J. D. B. Bradley, E. A. Marti-Panameno, and E. Mazur, “Mixed two- and three-photon absorption in bulk rutile (TiO2) around 800 nm,” Opt. Express 20, 3118–3128 (2012).
    [Crossref] [PubMed]
  19. C. C. Evans, K. Shtyrkova, J. D. B. Bradley, O. Reshef, E. Ippen, and E. Mazur, “Spectral broadening in anatase titanium dioxide waveguides at telecommunication and near-visible wavelengths,” Opt. Express 21, 18582–18591 (2013).
    [Crossref] [PubMed]
  20. J. D. B. Bradley, C. C. Evans, J. T. Choy, O. Reshef, P. B. Deotare, F. Parsy, K. C. Phillips, M. Loncar, and E. Mazur, “Submicrometer-wide amorphous and polycrystalline anatase TiO2 waveguides for microphotonic devices,” Opt. Express 20, 23821–23831 (2012).
    [Crossref] [PubMed]
  21. J. T. Choy, J. D. B. Bradley, P. B. Deotare, I. B. Burgess, C. C. Evans, E. Mazur, and M. Loncar, “Integrated TiO2 resonators for visible photonics,” Opt. Lett. 37, 539–541 (2012).
    [Crossref] [PubMed]
  22. C. C. Evans, K. Shtyrkova, O. Reshef, M. Moebius, J. D. B. Bradley, S. Griesse-Nascimento, E. Ippen, and E. Mazur, “Multimode phase-matched third-harmonic generation in sub-micrometer-wide anatase TiO2 waveguides,” Opt. Express 23, 7832 (2015).
    [Crossref] [PubMed]
  23. M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: single-photon sources and detectors,” Rev. Sci. Instrum. 82, 71101–71125 (2011).
    [Crossref]
  24. M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Third-order spontaneous parametric down-conversion in thin optical fibers as a photon-triplet source,” Phys. Rev. A: At. Mol. Opt. Phys. 84, 033823 (2011).
    [Crossref]
  25. S. Richard, K. Bencheikh, B. Boulanger, and J. A. Levenson, “Semiclassical model of triple photons generation in optical fibers,” Opt. Lett. 36, 3000–3002 (2011).
    [Crossref] [PubMed]
  26. O. Reshef, K. Shtyrkova, M. G. Moebius, S. Griesse-Nascimento, S. Spector, C. C. Evans, E. Ippen, and E. Mazur, “Polycrystalline anatase titanium dioxide micro-ring resonators with negative thermo-optic coefficient,” J. Opt. Soc. Am. B: Opt. Phys. 32, 2288–2293 (2015).
    [Crossref]
  27. K. Preston, B. Schmidt, and M. Lipson, “Polysilicon photonic resonators for large-scale 3D integration of optical networks,” Opt. Express 15, 17283 (2007).
    [Crossref] [PubMed]
  28. I. A. Bufetov, M. V. Grekov, K. M. Golant, E. M. Dianov, and R. R. Khrapko, “Ultraviolet-light generation in nitrogen-doped silica fiber,” Opt. Lett. 22, 1394 (1997).
    [Crossref]
  29. Z. Zhu and T. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express 10, 853–864 (2002).
    [Crossref] [PubMed]
  30. R. Boyd, Nonlinear Optics, 3rd ed. (Elsevier, Burlington, USA, 2008).
  31. M. Sheik-Bahae, D. Hagan, and E. Van Stryland, “Dispersion and band-gap scaling of the electronic kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65, 96–99 (1990).
    [Crossref] [PubMed]
  32. T. Lee, N. G. R. Broderick, and G. Brambilla, “Resonantly enhanced third harmonic generation in microfiber loop resonators,” J. Opt. Soc. Am. B: Opt. Phys. 30, 505–511 (2013).
    [Crossref]
  33. B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
    [Crossref]
  34. S. Krapick and C. Silberhorn, “Analysis of photon triplet generation in pulsed cascaded parametric down-conversion sources,” arXiv.org/abs/1506.07655 (2015).
  35. N. Gisin and R. Thew, “Quantum communication,” Nat. Photonics 1, 165–171 (2007).
    [Crossref]
  36. J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594 (1999).
    [Crossref]
  37. I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
    [Crossref] [PubMed]
  38. J. Wen and M. Rubin, “Distinction of tripartite Greenberger-Horne-Zeilinger and W states entangled in time (or energy) and space,” Phys. Rev. A: At. Mol. Opt. Phys. 79, 025802 (2009).
    [Crossref]
  39. C. C. Evans, C. Liu, and J. Suntivich, “Low-loss titanium dioxide waveguides and resonators using a dielectric lift-off fabrication process,” Opt. Express 23, 11160 (2015).
    [Crossref] [PubMed]
  40. M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211–219 (2004).
    [Crossref]
  41. Y. Watanabe, M. Ohnishi, and T. Tsuchiya, “Measurement of nonlinear absorption and refraction in titanium dioxide single crystal by using a phase distortion method,” Appl. Phys. Lett. 66, 3431 (1995).
    [Crossref]
  42. S. K. Das, C. Schwanke, A. Pfuch, W. Seeber, M. Bock, G. Steinmeyer, T. Elsaesser, and R. Grunwald, “Highly efficient THG in TiO2 nanolayers for third-order pulse characterization,” Opt. Express 19, 16985–16995 (2011).
    [Crossref] [PubMed]

2015 (4)

M. Gimeno-Segovia, P. Shadbolt, D. E. Browne, and T. Rudolph, “From three-photon Greenberger-Horne-Zeilinger states to ballistic universal quantum computation,” Phys. Rev. Lett. 115, 020502 (2015).
[Crossref] [PubMed]

C. C. Evans, K. Shtyrkova, O. Reshef, M. Moebius, J. D. B. Bradley, S. Griesse-Nascimento, E. Ippen, and E. Mazur, “Multimode phase-matched third-harmonic generation in sub-micrometer-wide anatase TiO2 waveguides,” Opt. Express 23, 7832 (2015).
[Crossref] [PubMed]

O. Reshef, K. Shtyrkova, M. G. Moebius, S. Griesse-Nascimento, S. Spector, C. C. Evans, E. Ippen, and E. Mazur, “Polycrystalline anatase titanium dioxide micro-ring resonators with negative thermo-optic coefficient,” J. Opt. Soc. Am. B: Opt. Phys. 32, 2288–2293 (2015).
[Crossref]

C. C. Evans, C. Liu, and J. Suntivich, “Low-loss titanium dioxide waveguides and resonators using a dielectric lift-off fabrication process,” Opt. Express 23, 11160 (2015).
[Crossref] [PubMed]

2014 (1)

D. R. Hamel, L. K. Shalm, H. Hübel, A. J. Miller, F. Marsili, V. B. Verma, R. P. Mirin, S. W. Nam, K. J. Resch, and T. Jennewein, “Direct generation of three-photon polarization entanglement,” Nat. Photonics 8, 801–807 (2014).
[Crossref]

2013 (4)

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-a. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

M. G. Raymer, A. H. Marcus, J. R. Widom, and D. L. P. Vitullo, “Entangled photon-pair two-dimensional fluorescence spectroscopy,” J. Phys. Chem. B 117, 15559–15575 (2013).
[Crossref] [PubMed]

C. C. Evans, K. Shtyrkova, J. D. B. Bradley, O. Reshef, E. Ippen, and E. Mazur, “Spectral broadening in anatase titanium dioxide waveguides at telecommunication and near-visible wavelengths,” Opt. Express 21, 18582–18591 (2013).
[Crossref] [PubMed]

T. Lee, N. G. R. Broderick, and G. Brambilla, “Resonantly enhanced third harmonic generation in microfiber loop resonators,” J. Opt. Soc. Am. B: Opt. Phys. 30, 505–511 (2013).
[Crossref]

2012 (4)

2011 (5)

V. Giovannetti and S. Lloyd, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: single-photon sources and detectors,” Rev. Sci. Instrum. 82, 71101–71125 (2011).
[Crossref]

M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Third-order spontaneous parametric down-conversion in thin optical fibers as a photon-triplet source,” Phys. Rev. A: At. Mol. Opt. Phys. 84, 033823 (2011).
[Crossref]

S. Richard, K. Bencheikh, B. Boulanger, and J. A. Levenson, “Semiclassical model of triple photons generation in optical fibers,” Opt. Lett. 36, 3000–3002 (2011).
[Crossref] [PubMed]

S. K. Das, C. Schwanke, A. Pfuch, W. Seeber, M. Bock, G. Steinmeyer, T. Elsaesser, and R. Grunwald, “Highly efficient THG in TiO2 nanolayers for third-order pulse characterization,” Opt. Express 19, 16985–16995 (2011).
[Crossref] [PubMed]

2010 (4)

Z.-S. Yuan, X.-H. Bao, C.-Y. Lu, J. Zhang, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Phys. Rep. 497, 1–40 (2010).
[Crossref]

T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464, 45 (2010).
[Crossref] [PubMed]

H. Hubel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature 466, 601–603 (2010).
[Crossref] [PubMed]

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaitre, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
[Crossref] [PubMed]

2009 (3)

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[Crossref]

J. Wen and M. Rubin, “Distinction of tripartite Greenberger-Horne-Zeilinger and W states entangled in time (or energy) and space,” Phys. Rev. A: At. Mol. Opt. Phys. 79, 025802 (2009).
[Crossref]

2007 (5)

N. Gisin and R. Thew, “Quantum communication,” Nat. Photonics 1, 165–171 (2007).
[Crossref]

I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
[Crossref] [PubMed]

K. Preston, B. Schmidt, and M. Lipson, “Polysilicon photonic resonators for large-scale 3D integration of optical networks,” Opt. Express 15, 17283 (2007).
[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: At. Mol. Opt. Phys. 75, 023803 (2007).
[Crossref]

K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” C. R. Phys. 8, 206–220 (2007).
[Crossref]

2005 (1)

M. Chekhova, O. Ivanova, V. Berardi, and A. Garuccio, “Spectral properties of three-photon entangled states generated via three-photon parametric down-conversion in a χ(3) medium,” Phys. Rev. A: At. Mol. Opt. Phys. 72, 023818 (2005).
[Crossref]

2004 (2)

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[Crossref] [PubMed]

M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211–219 (2004).
[Crossref]

2003 (1)

Y. Shih, “Entangled biphoton source–roperty and preparation,” Rep. Prog. Phys. 66, 1009–1044 (2003).
[Crossref]

2002 (2)

L. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. 4, S176 (2002).
[Crossref]

Z. Zhu and T. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express 10, 853–864 (2002).
[Crossref] [PubMed]

1999 (1)

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594 (1999).
[Crossref]

1997 (1)

1995 (1)

Y. Watanabe, M. Ohnishi, and T. Tsuchiya, “Measurement of nonlinear absorption and refraction in titanium dioxide single crystal by using a phase distortion method,” Appl. Phys. Lett. 66, 3431 (1995).
[Crossref]

1990 (1)

M. Sheik-Bahae, D. Hagan, and E. Van Stryland, “Dispersion and band-gap scaling of the electronic kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65, 96–99 (1990).
[Crossref] [PubMed]

Agrawal, G. P.

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: At. Mol. Opt. Phys. 75, 023803 (2007).
[Crossref]

Ali-Khan, I.

I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
[Crossref] [PubMed]

Bachor, H.-a.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-a. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

Bao, X.-H.

Z.-S. Yuan, X.-H. Bao, C.-Y. Lu, J. Zhang, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Phys. Rep. 497, 1–40 (2010).
[Crossref]

Bencheikh, K.

S. Richard, K. Bencheikh, B. Boulanger, and J. A. Levenson, “Semiclassical model of triple photons generation in optical fibers,” Opt. Lett. 36, 3000–3002 (2011).
[Crossref] [PubMed]

K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” C. R. Phys. 8, 206–220 (2007).
[Crossref]

Berardi, V.

M. Chekhova, O. Ivanova, V. Berardi, and A. Garuccio, “Spectral properties of three-photon entangled states generated via three-photon parametric down-conversion in a χ(3) medium,” Phys. Rev. A: At. Mol. Opt. Phys. 72, 023818 (2005).
[Crossref]

Beveratos, A.

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaitre, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
[Crossref] [PubMed]

Bloch, J.

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaitre, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
[Crossref] [PubMed]

Bock, M.

Boulanger, B.

S. Richard, K. Bencheikh, B. Boulanger, and J. A. Levenson, “Semiclassical model of triple photons generation in optical fibers,” Opt. Lett. 36, 3000–3002 (2011).
[Crossref] [PubMed]

K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” C. R. Phys. 8, 206–220 (2007).
[Crossref]

Bowen, W. P.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-a. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[Crossref] [PubMed]

Boyd, R.

R. Boyd, Nonlinear Optics, 3rd ed. (Elsevier, Burlington, USA, 2008).

Bradley, J. D. B.

Brambilla, E.

L. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. 4, S176 (2002).
[Crossref]

Brambilla, G.

T. Lee, N. G. R. Broderick, and G. Brambilla, “Resonantly enhanced third harmonic generation in microfiber loop resonators,” J. Opt. Soc. Am. B: Opt. Phys. 30, 505–511 (2013).
[Crossref]

Brendel, J.

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594 (1999).
[Crossref]

Broadbent, C. J.

I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
[Crossref] [PubMed]

Broderick, N. G. R.

T. Lee, N. G. R. Broderick, and G. Brambilla, “Resonantly enhanced third harmonic generation in microfiber loop resonators,” J. Opt. Soc. Am. B: Opt. Phys. 30, 505–511 (2013).
[Crossref]

Brown, T.

Browne, D. E.

M. Gimeno-Segovia, P. Shadbolt, D. E. Browne, and T. Rudolph, “From three-photon Greenberger-Horne-Zeilinger states to ballistic universal quantum computation,” Phys. Rev. Lett. 115, 020502 (2015).
[Crossref] [PubMed]

Bufetov, I. A.

Burgess, I. B.

Chekhova, M.

M. Chekhova, O. Ivanova, V. Berardi, and A. Garuccio, “Spectral properties of three-photon entangled states generated via three-photon parametric down-conversion in a χ(3) medium,” Phys. Rev. A: At. Mol. Opt. Phys. 72, 023818 (2005).
[Crossref]

Choy, J. T.

Corcoran, B.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[Crossref]

Corona, M.

M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Third-order spontaneous parametric down-conversion in thin optical fibers as a photon-triplet source,” Phys. Rev. A: At. Mol. Opt. Phys. 84, 033823 (2011).
[Crossref]

Daria, V.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-a. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

Das, S. K.

Deotare, P. B.

Dianov, E. M.

Douady, J.

K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” C. R. Phys. 8, 206–220 (2007).
[Crossref]

Dousse, A.

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaitre, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
[Crossref] [PubMed]

Eggleton, B. J.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[Crossref]

Eisaman, M. D.

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: single-photon sources and detectors,” Rev. Sci. Instrum. 82, 71101–71125 (2011).
[Crossref]

Elsaesser, T.

Evans, C. C.

C. C. Evans, C. Liu, and J. Suntivich, “Low-loss titanium dioxide waveguides and resonators using a dielectric lift-off fabrication process,” Opt. Express 23, 11160 (2015).
[Crossref] [PubMed]

C. C. Evans, K. Shtyrkova, O. Reshef, M. Moebius, J. D. B. Bradley, S. Griesse-Nascimento, E. Ippen, and E. Mazur, “Multimode phase-matched third-harmonic generation in sub-micrometer-wide anatase TiO2 waveguides,” Opt. Express 23, 7832 (2015).
[Crossref] [PubMed]

O. Reshef, K. Shtyrkova, M. G. Moebius, S. Griesse-Nascimento, S. Spector, C. C. Evans, E. Ippen, and E. Mazur, “Polycrystalline anatase titanium dioxide micro-ring resonators with negative thermo-optic coefficient,” J. Opt. Soc. Am. B: Opt. Phys. 32, 2288–2293 (2015).
[Crossref]

C. C. Evans, K. Shtyrkova, J. D. B. Bradley, O. Reshef, E. Ippen, and E. Mazur, “Spectral broadening in anatase titanium dioxide waveguides at telecommunication and near-visible wavelengths,” Opt. Express 21, 18582–18591 (2013).
[Crossref] [PubMed]

C. C. Evans, J. D. B. Bradley, E. A. Marti-Panameno, and E. Mazur, “Mixed two- and three-photon absorption in bulk rutile (TiO2) around 800 nm,” Opt. Express 20, 3118–3128 (2012).
[Crossref] [PubMed]

J. T. Choy, J. D. B. Bradley, P. B. Deotare, I. B. Burgess, C. C. Evans, E. Mazur, and M. Loncar, “Integrated TiO2 resonators for visible photonics,” Opt. Lett. 37, 539–541 (2012).
[Crossref] [PubMed]

J. D. B. Bradley, C. C. Evans, J. T. Choy, O. Reshef, P. B. Deotare, F. Parsy, K. C. Phillips, M. Loncar, and E. Mazur, “Submicrometer-wide amorphous and polycrystalline anatase TiO2 waveguides for microphotonic devices,” Opt. Express 20, 23821–23831 (2012).
[Crossref] [PubMed]

Fan, J.

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: single-photon sources and detectors,” Rev. Sci. Instrum. 82, 71101–71125 (2011).
[Crossref]

Fedrizzi, A.

H. Hubel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature 466, 601–603 (2010).
[Crossref] [PubMed]

Furusawa, A.

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

Garay-Palmett, K.

M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Third-order spontaneous parametric down-conversion in thin optical fibers as a photon-triplet source,” Phys. Rev. A: At. Mol. Opt. Phys. 84, 033823 (2011).
[Crossref]

Garuccio, A.

M. Chekhova, O. Ivanova, V. Berardi, and A. Garuccio, “Spectral properties of three-photon entangled states generated via three-photon parametric down-conversion in a χ(3) medium,” Phys. Rev. A: At. Mol. Opt. Phys. 72, 023818 (2005).
[Crossref]

Gatti, A.

L. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. 4, S176 (2002).
[Crossref]

Gimeno-Segovia, M.

M. Gimeno-Segovia, P. Shadbolt, D. E. Browne, and T. Rudolph, “From three-photon Greenberger-Horne-Zeilinger states to ballistic universal quantum computation,” Phys. Rev. Lett. 115, 020502 (2015).
[Crossref] [PubMed]

Giovannetti, V.

V. Giovannetti and S. Lloyd, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

Gisin, N.

N. Gisin and R. Thew, “Quantum communication,” Nat. Photonics 1, 165–171 (2007).
[Crossref]

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594 (1999).
[Crossref]

Golant, K. M.

Gravier, F.

K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” C. R. Phys. 8, 206–220 (2007).
[Crossref]

Grekov, M. V.

Griesse-Nascimento, S.

O. Reshef, K. Shtyrkova, M. G. Moebius, S. Griesse-Nascimento, S. Spector, C. C. Evans, E. Ippen, and E. Mazur, “Polycrystalline anatase titanium dioxide micro-ring resonators with negative thermo-optic coefficient,” J. Opt. Soc. Am. B: Opt. Phys. 32, 2288–2293 (2015).
[Crossref]

C. C. Evans, K. Shtyrkova, O. Reshef, M. Moebius, J. D. B. Bradley, S. Griesse-Nascimento, E. Ippen, and E. Mazur, “Multimode phase-matched third-harmonic generation in sub-micrometer-wide anatase TiO2 waveguides,” Opt. Express 23, 7832 (2015).
[Crossref] [PubMed]

Grillet, C.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[Crossref]

Grunwald, R.

Hagan, D.

M. Sheik-Bahae, D. Hagan, and E. Van Stryland, “Dispersion and band-gap scaling of the electronic kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65, 96–99 (1990).
[Crossref] [PubMed]

Hage, B.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-a. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

Hamel, D. R.

D. R. Hamel, L. K. Shalm, H. Hübel, A. J. Miller, F. Marsili, V. B. Verma, R. P. Mirin, S. W. Nam, K. J. Resch, and T. Jennewein, “Direct generation of three-photon polarization entanglement,” Nat. Photonics 8, 801–807 (2014).
[Crossref]

L. K. Shalm, D. R. Hamel, Z. Yan, C. Simon, K. J. Resch, and T. Jennewein, “Three-photon energy–time entanglement,” Nat. Phys. 9, 19–22 (2012).
[Crossref]

H. Hubel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature 466, 601–603 (2010).
[Crossref] [PubMed]

Howell, J. C.

I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
[Crossref] [PubMed]

Hubel, H.

H. Hubel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature 466, 601–603 (2010).
[Crossref] [PubMed]

Hübel, H.

D. R. Hamel, L. K. Shalm, H. Hübel, A. J. Miller, F. Marsili, V. B. Verma, R. P. Mirin, S. W. Nam, K. J. Resch, and T. Jennewein, “Direct generation of three-photon polarization entanglement,” Nat. Photonics 8, 801–807 (2014).
[Crossref]

Ippen, E.

Ivanova, O.

M. Chekhova, O. Ivanova, V. Berardi, and A. Garuccio, “Spectral properties of three-photon entangled states generated via three-photon parametric down-conversion in a χ(3) medium,” Phys. Rev. A: At. Mol. Opt. Phys. 72, 023818 (2005).
[Crossref]

Janousek, J.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-a. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

Jelezko, F.

T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464, 45 (2010).
[Crossref] [PubMed]

Jennewein, T.

D. R. Hamel, L. K. Shalm, H. Hübel, A. J. Miller, F. Marsili, V. B. Verma, R. P. Mirin, S. W. Nam, K. J. Resch, and T. Jennewein, “Direct generation of three-photon polarization entanglement,” Nat. Photonics 8, 801–807 (2014).
[Crossref]

L. K. Shalm, D. R. Hamel, Z. Yan, C. Simon, K. J. Resch, and T. Jennewein, “Three-photon energy–time entanglement,” Nat. Phys. 9, 19–22 (2012).
[Crossref]

H. Hubel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature 466, 601–603 (2010).
[Crossref] [PubMed]

Joannopoulos, J. D.

M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211–219 (2004).
[Crossref]

Khrapko, R. R.

Knittel, J.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-a. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

Krapick, S.

S. Krapick and C. Silberhorn, “Analysis of photon triplet generation in pulsed cascaded parametric down-conversion sources,” arXiv.org/abs/1506.07655 (2015).

Krauss, T. F.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[Crossref]

Krebs, O.

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaitre, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
[Crossref] [PubMed]

Ladd, T. D.

T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464, 45 (2010).
[Crossref] [PubMed]

Laflamme, R.

T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464, 45 (2010).
[Crossref] [PubMed]

Lam, P. K.

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[Crossref] [PubMed]

Lance, A. M.

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[Crossref] [PubMed]

Lee, T.

T. Lee, N. G. R. Broderick, and G. Brambilla, “Resonantly enhanced third harmonic generation in microfiber loop resonators,” J. Opt. Soc. Am. B: Opt. Phys. 30, 505–511 (2013).
[Crossref]

Lemaitre, A.

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaitre, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
[Crossref] [PubMed]

Levenson, A.

K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” C. R. Phys. 8, 206–220 (2007).
[Crossref]

Levenson, J. A.

Lin, Q.

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: At. Mol. Opt. Phys. 75, 023803 (2007).
[Crossref]

Lipson, M.

Liu, C.

Lloyd, S.

V. Giovannetti and S. Lloyd, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

Loncar, M.

Lu, C.-Y.

Z.-S. Yuan, X.-H. Bao, C.-Y. Lu, J. Zhang, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Phys. Rep. 497, 1–40 (2010).
[Crossref]

Lugiato, L.

L. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. 4, S176 (2002).
[Crossref]

Marcus, A. H.

M. G. Raymer, A. H. Marcus, J. R. Widom, and D. L. P. Vitullo, “Entangled photon-pair two-dimensional fluorescence spectroscopy,” J. Phys. Chem. B 117, 15559–15575 (2013).
[Crossref] [PubMed]

Marsili, F.

D. R. Hamel, L. K. Shalm, H. Hübel, A. J. Miller, F. Marsili, V. B. Verma, R. P. Mirin, S. W. Nam, K. J. Resch, and T. Jennewein, “Direct generation of three-photon polarization entanglement,” Nat. Photonics 8, 801–807 (2014).
[Crossref]

Marti-Panameno, E. A.

Mazur, E.

C. C. Evans, K. Shtyrkova, O. Reshef, M. Moebius, J. D. B. Bradley, S. Griesse-Nascimento, E. Ippen, and E. Mazur, “Multimode phase-matched third-harmonic generation in sub-micrometer-wide anatase TiO2 waveguides,” Opt. Express 23, 7832 (2015).
[Crossref] [PubMed]

O. Reshef, K. Shtyrkova, M. G. Moebius, S. Griesse-Nascimento, S. Spector, C. C. Evans, E. Ippen, and E. Mazur, “Polycrystalline anatase titanium dioxide micro-ring resonators with negative thermo-optic coefficient,” J. Opt. Soc. Am. B: Opt. Phys. 32, 2288–2293 (2015).
[Crossref]

C. C. Evans, K. Shtyrkova, J. D. B. Bradley, O. Reshef, E. Ippen, and E. Mazur, “Spectral broadening in anatase titanium dioxide waveguides at telecommunication and near-visible wavelengths,” Opt. Express 21, 18582–18591 (2013).
[Crossref] [PubMed]

J. D. B. Bradley, C. C. Evans, J. T. Choy, O. Reshef, P. B. Deotare, F. Parsy, K. C. Phillips, M. Loncar, and E. Mazur, “Submicrometer-wide amorphous and polycrystalline anatase TiO2 waveguides for microphotonic devices,” Opt. Express 20, 23821–23831 (2012).
[Crossref] [PubMed]

J. T. Choy, J. D. B. Bradley, P. B. Deotare, I. B. Burgess, C. C. Evans, E. Mazur, and M. Loncar, “Integrated TiO2 resonators for visible photonics,” Opt. Lett. 37, 539–541 (2012).
[Crossref] [PubMed]

C. C. Evans, J. D. B. Bradley, E. A. Marti-Panameno, and E. Mazur, “Mixed two- and three-photon absorption in bulk rutile (TiO2) around 800 nm,” Opt. Express 20, 3118–3128 (2012).
[Crossref] [PubMed]

Migdall, A.

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: single-photon sources and detectors,” Rev. Sci. Instrum. 82, 71101–71125 (2011).
[Crossref]

Miller, A. J.

D. R. Hamel, L. K. Shalm, H. Hübel, A. J. Miller, F. Marsili, V. B. Verma, R. P. Mirin, S. W. Nam, K. J. Resch, and T. Jennewein, “Direct generation of three-photon polarization entanglement,” Nat. Photonics 8, 801–807 (2014).
[Crossref]

Mirin, R. P.

D. R. Hamel, L. K. Shalm, H. Hübel, A. J. Miller, F. Marsili, V. B. Verma, R. P. Mirin, S. W. Nam, K. J. Resch, and T. Jennewein, “Direct generation of three-photon polarization entanglement,” Nat. Photonics 8, 801–807 (2014).
[Crossref]

Moebius, M.

Moebius, M. G.

O. Reshef, K. Shtyrkova, M. G. Moebius, S. Griesse-Nascimento, S. Spector, C. C. Evans, E. Ippen, and E. Mazur, “Polycrystalline anatase titanium dioxide micro-ring resonators with negative thermo-optic coefficient,” J. Opt. Soc. Am. B: Opt. Phys. 32, 2288–2293 (2015).
[Crossref]

Monat, C.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[Crossref]

Monroe, C.

T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464, 45 (2010).
[Crossref] [PubMed]

Moss, D. J.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[Crossref]

Nakamura, Y.

T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464, 45 (2010).
[Crossref] [PubMed]

Nam, S. W.

D. R. Hamel, L. K. Shalm, H. Hübel, A. J. Miller, F. Marsili, V. B. Verma, R. P. Mirin, S. W. Nam, K. J. Resch, and T. Jennewein, “Direct generation of three-photon polarization entanglement,” Nat. Photonics 8, 801–807 (2014).
[Crossref]

O’Brien, J. L.

T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464, 45 (2010).
[Crossref] [PubMed]

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

O’Faolain, L.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[Crossref]

Ohnishi, M.

Y. Watanabe, M. Ohnishi, and T. Tsuchiya, “Measurement of nonlinear absorption and refraction in titanium dioxide single crystal by using a phase distortion method,” Appl. Phys. Lett. 66, 3431 (1995).
[Crossref]

Pan, J.-W.

Z.-S. Yuan, X.-H. Bao, C.-Y. Lu, J. Zhang, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Phys. Rep. 497, 1–40 (2010).
[Crossref]

Parsy, F.

Peng, C.-Z.

Z.-S. Yuan, X.-H. Bao, C.-Y. Lu, J. Zhang, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Phys. Rep. 497, 1–40 (2010).
[Crossref]

Pfuch, A.

Phillips, K. C.

Polyakov, S. V.

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: single-photon sources and detectors,” Rev. Sci. Instrum. 82, 71101–71125 (2011).
[Crossref]

Preston, K.

Ramelow, S.

H. Hubel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature 466, 601–603 (2010).
[Crossref] [PubMed]

Raymer, M. G.

M. G. Raymer, A. H. Marcus, J. R. Widom, and D. L. P. Vitullo, “Entangled photon-pair two-dimensional fluorescence spectroscopy,” J. Phys. Chem. B 117, 15559–15575 (2013).
[Crossref] [PubMed]

Resch, K. J.

D. R. Hamel, L. K. Shalm, H. Hübel, A. J. Miller, F. Marsili, V. B. Verma, R. P. Mirin, S. W. Nam, K. J. Resch, and T. Jennewein, “Direct generation of three-photon polarization entanglement,” Nat. Photonics 8, 801–807 (2014).
[Crossref]

L. K. Shalm, D. R. Hamel, Z. Yan, C. Simon, K. J. Resch, and T. Jennewein, “Three-photon energy–time entanglement,” Nat. Phys. 9, 19–22 (2012).
[Crossref]

H. Hubel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature 466, 601–603 (2010).
[Crossref] [PubMed]

Reshef, O.

Richard, S.

Rubin, M.

J. Wen and M. Rubin, “Distinction of tripartite Greenberger-Horne-Zeilinger and W states entangled in time (or energy) and space,” Phys. Rev. A: At. Mol. Opt. Phys. 79, 025802 (2009).
[Crossref]

Rudolph, T.

M. Gimeno-Segovia, P. Shadbolt, D. E. Browne, and T. Rudolph, “From three-photon Greenberger-Horne-Zeilinger states to ballistic universal quantum computation,” Phys. Rev. Lett. 115, 020502 (2015).
[Crossref] [PubMed]

Sagnes, I.

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaitre, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
[Crossref] [PubMed]

Sanders, B. C.

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[Crossref] [PubMed]

Schmidt, B.

Schwanke, C.

Seeber, W.

Senellart, P.

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaitre, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
[Crossref] [PubMed]

Shadbolt, P.

M. Gimeno-Segovia, P. Shadbolt, D. E. Browne, and T. Rudolph, “From three-photon Greenberger-Horne-Zeilinger states to ballistic universal quantum computation,” Phys. Rev. Lett. 115, 020502 (2015).
[Crossref] [PubMed]

Shalm, L. K.

D. R. Hamel, L. K. Shalm, H. Hübel, A. J. Miller, F. Marsili, V. B. Verma, R. P. Mirin, S. W. Nam, K. J. Resch, and T. Jennewein, “Direct generation of three-photon polarization entanglement,” Nat. Photonics 8, 801–807 (2014).
[Crossref]

L. K. Shalm, D. R. Hamel, Z. Yan, C. Simon, K. J. Resch, and T. Jennewein, “Three-photon energy–time entanglement,” Nat. Phys. 9, 19–22 (2012).
[Crossref]

Sheik-Bahae, M.

M. Sheik-Bahae, D. Hagan, and E. Van Stryland, “Dispersion and band-gap scaling of the electronic kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65, 96–99 (1990).
[Crossref] [PubMed]

Shih, Y.

Y. Shih, “Entangled biphoton source–roperty and preparation,” Rep. Prog. Phys. 66, 1009–1044 (2003).
[Crossref]

Shtyrkova, K.

Silberhorn, C.

S. Krapick and C. Silberhorn, “Analysis of photon triplet generation in pulsed cascaded parametric down-conversion sources,” arXiv.org/abs/1506.07655 (2015).

Simon, C.

L. K. Shalm, D. R. Hamel, Z. Yan, C. Simon, K. J. Resch, and T. Jennewein, “Three-photon energy–time entanglement,” Nat. Phys. 9, 19–22 (2012).
[Crossref]

Soljacic, M.

M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211–219 (2004).
[Crossref]

Spector, S.

O. Reshef, K. Shtyrkova, M. G. Moebius, S. Griesse-Nascimento, S. Spector, C. C. Evans, E. Ippen, and E. Mazur, “Polycrystalline anatase titanium dioxide micro-ring resonators with negative thermo-optic coefficient,” J. Opt. Soc. Am. B: Opt. Phys. 32, 2288–2293 (2015).
[Crossref]

Steinmeyer, G.

Suffczynski, J.

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaitre, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
[Crossref] [PubMed]

Suntivich, J.

Symul, T.

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[Crossref] [PubMed]

Taylor, M. A.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-a. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

Thew, R.

N. Gisin and R. Thew, “Quantum communication,” Nat. Photonics 1, 165–171 (2007).
[Crossref]

Tittel, W.

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594 (1999).
[Crossref]

Tsuchiya, T.

Y. Watanabe, M. Ohnishi, and T. Tsuchiya, “Measurement of nonlinear absorption and refraction in titanium dioxide single crystal by using a phase distortion method,” Appl. Phys. Lett. 66, 3431 (1995).
[Crossref]

U’Ren, A. B.

M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Third-order spontaneous parametric down-conversion in thin optical fibers as a photon-triplet source,” Phys. Rev. A: At. Mol. Opt. Phys. 84, 033823 (2011).
[Crossref]

Van Stryland, E.

M. Sheik-Bahae, D. Hagan, and E. Van Stryland, “Dispersion and band-gap scaling of the electronic kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65, 96–99 (1990).
[Crossref] [PubMed]

Verma, V. B.

D. R. Hamel, L. K. Shalm, H. Hübel, A. J. Miller, F. Marsili, V. B. Verma, R. P. Mirin, S. W. Nam, K. J. Resch, and T. Jennewein, “Direct generation of three-photon polarization entanglement,” Nat. Photonics 8, 801–807 (2014).
[Crossref]

Vitullo, D. L. P.

M. G. Raymer, A. H. Marcus, J. R. Widom, and D. L. P. Vitullo, “Entangled photon-pair two-dimensional fluorescence spectroscopy,” J. Phys. Chem. B 117, 15559–15575 (2013).
[Crossref] [PubMed]

Voisin, P.

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaitre, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
[Crossref] [PubMed]

Vuckovic, J.

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

Watanabe, Y.

Y. Watanabe, M. Ohnishi, and T. Tsuchiya, “Measurement of nonlinear absorption and refraction in titanium dioxide single crystal by using a phase distortion method,” Appl. Phys. Lett. 66, 3431 (1995).
[Crossref]

Wen, J.

J. Wen and M. Rubin, “Distinction of tripartite Greenberger-Horne-Zeilinger and W states entangled in time (or energy) and space,” Phys. Rev. A: At. Mol. Opt. Phys. 79, 025802 (2009).
[Crossref]

White, T. P.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[Crossref]

Widom, J. R.

M. G. Raymer, A. H. Marcus, J. R. Widom, and D. L. P. Vitullo, “Entangled photon-pair two-dimensional fluorescence spectroscopy,” J. Phys. Chem. B 117, 15559–15575 (2013).
[Crossref] [PubMed]

Yaman, F.

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: At. Mol. Opt. Phys. 75, 023803 (2007).
[Crossref]

Yan, Z.

L. K. Shalm, D. R. Hamel, Z. Yan, C. Simon, K. J. Resch, and T. Jennewein, “Three-photon energy–time entanglement,” Nat. Phys. 9, 19–22 (2012).
[Crossref]

Yuan, Z.-S.

Z.-S. Yuan, X.-H. Bao, C.-Y. Lu, J. Zhang, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Phys. Rep. 497, 1–40 (2010).
[Crossref]

Zbinden, H.

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594 (1999).
[Crossref]

Zhang, J.

Z.-S. Yuan, X.-H. Bao, C.-Y. Lu, J. Zhang, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Phys. Rep. 497, 1–40 (2010).
[Crossref]

Zhu, Z.

Appl. Phys. Lett. (1)

Y. Watanabe, M. Ohnishi, and T. Tsuchiya, “Measurement of nonlinear absorption and refraction in titanium dioxide single crystal by using a phase distortion method,” Appl. Phys. Lett. 66, 3431 (1995).
[Crossref]

C. R. Phys. (1)

K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” C. R. Phys. 8, 206–220 (2007).
[Crossref]

J. Opt. B: Quantum Semiclassical Opt. (1)

L. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. 4, S176 (2002).
[Crossref]

J. Opt. Soc. Am. B: Opt. Phys. (2)

O. Reshef, K. Shtyrkova, M. G. Moebius, S. Griesse-Nascimento, S. Spector, C. C. Evans, E. Ippen, and E. Mazur, “Polycrystalline anatase titanium dioxide micro-ring resonators with negative thermo-optic coefficient,” J. Opt. Soc. Am. B: Opt. Phys. 32, 2288–2293 (2015).
[Crossref]

T. Lee, N. G. R. Broderick, and G. Brambilla, “Resonantly enhanced third harmonic generation in microfiber loop resonators,” J. Opt. Soc. Am. B: Opt. Phys. 30, 505–511 (2013).
[Crossref]

J. Phys. Chem. B (1)

M. G. Raymer, A. H. Marcus, J. R. Widom, and D. L. P. Vitullo, “Entangled photon-pair two-dimensional fluorescence spectroscopy,” J. Phys. Chem. B 117, 15559–15575 (2013).
[Crossref] [PubMed]

Nat. Mater. (1)

M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211–219 (2004).
[Crossref]

Nat. Photonics (6)

V. Giovannetti and S. Lloyd, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-a. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[Crossref]

N. Gisin and R. Thew, “Quantum communication,” Nat. Photonics 1, 165–171 (2007).
[Crossref]

D. R. Hamel, L. K. Shalm, H. Hübel, A. J. Miller, F. Marsili, V. B. Verma, R. P. Mirin, S. W. Nam, K. J. Resch, and T. Jennewein, “Direct generation of three-photon polarization entanglement,” Nat. Photonics 8, 801–807 (2014).
[Crossref]

Nat. Phys. (1)

L. K. Shalm, D. R. Hamel, Z. Yan, C. Simon, K. J. Resch, and T. Jennewein, “Three-photon energy–time entanglement,” Nat. Phys. 9, 19–22 (2012).
[Crossref]

Nature (3)

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaitre, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
[Crossref] [PubMed]

H. Hubel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature 466, 601–603 (2010).
[Crossref] [PubMed]

T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464, 45 (2010).
[Crossref] [PubMed]

Opt. Express (8)

C. C. Evans, J. D. B. Bradley, E. A. Marti-Panameno, and E. Mazur, “Mixed two- and three-photon absorption in bulk rutile (TiO2) around 800 nm,” Opt. Express 20, 3118–3128 (2012).
[Crossref] [PubMed]

C. C. Evans, K. Shtyrkova, J. D. B. Bradley, O. Reshef, E. Ippen, and E. Mazur, “Spectral broadening in anatase titanium dioxide waveguides at telecommunication and near-visible wavelengths,” Opt. Express 21, 18582–18591 (2013).
[Crossref] [PubMed]

J. D. B. Bradley, C. C. Evans, J. T. Choy, O. Reshef, P. B. Deotare, F. Parsy, K. C. Phillips, M. Loncar, and E. Mazur, “Submicrometer-wide amorphous and polycrystalline anatase TiO2 waveguides for microphotonic devices,” Opt. Express 20, 23821–23831 (2012).
[Crossref] [PubMed]

K. Preston, B. Schmidt, and M. Lipson, “Polysilicon photonic resonators for large-scale 3D integration of optical networks,” Opt. Express 15, 17283 (2007).
[Crossref] [PubMed]

C. C. Evans, K. Shtyrkova, O. Reshef, M. Moebius, J. D. B. Bradley, S. Griesse-Nascimento, E. Ippen, and E. Mazur, “Multimode phase-matched third-harmonic generation in sub-micrometer-wide anatase TiO2 waveguides,” Opt. Express 23, 7832 (2015).
[Crossref] [PubMed]

Z. Zhu and T. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express 10, 853–864 (2002).
[Crossref] [PubMed]

S. K. Das, C. Schwanke, A. Pfuch, W. Seeber, M. Bock, G. Steinmeyer, T. Elsaesser, and R. Grunwald, “Highly efficient THG in TiO2 nanolayers for third-order pulse characterization,” Opt. Express 19, 16985–16995 (2011).
[Crossref] [PubMed]

C. C. Evans, C. Liu, and J. Suntivich, “Low-loss titanium dioxide waveguides and resonators using a dielectric lift-off fabrication process,” Opt. Express 23, 11160 (2015).
[Crossref] [PubMed]

Opt. Lett. (3)

Phys. Rep. (1)

Z.-S. Yuan, X.-H. Bao, C.-Y. Lu, J. Zhang, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Phys. Rep. 497, 1–40 (2010).
[Crossref]

Phys. Rev. A: At. Mol. Opt. Phys. (4)

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: At. Mol. Opt. Phys. 75, 023803 (2007).
[Crossref]

M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Third-order spontaneous parametric down-conversion in thin optical fibers as a photon-triplet source,” Phys. Rev. A: At. Mol. Opt. Phys. 84, 033823 (2011).
[Crossref]

M. Chekhova, O. Ivanova, V. Berardi, and A. Garuccio, “Spectral properties of three-photon entangled states generated via three-photon parametric down-conversion in a χ(3) medium,” Phys. Rev. A: At. Mol. Opt. Phys. 72, 023818 (2005).
[Crossref]

J. Wen and M. Rubin, “Distinction of tripartite Greenberger-Horne-Zeilinger and W states entangled in time (or energy) and space,” Phys. Rev. A: At. Mol. Opt. Phys. 79, 025802 (2009).
[Crossref]

Phys. Rev. Lett. (5)

M. Sheik-Bahae, D. Hagan, and E. Van Stryland, “Dispersion and band-gap scaling of the electronic kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65, 96–99 (1990).
[Crossref] [PubMed]

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594 (1999).
[Crossref]

I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
[Crossref] [PubMed]

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[Crossref] [PubMed]

M. Gimeno-Segovia, P. Shadbolt, D. E. Browne, and T. Rudolph, “From three-photon Greenberger-Horne-Zeilinger states to ballistic universal quantum computation,” Phys. Rev. Lett. 115, 020502 (2015).
[Crossref] [PubMed]

Rep. Prog. Phys. (1)

Y. Shih, “Entangled biphoton source–roperty and preparation,” Rep. Prog. Phys. 66, 1009–1044 (2003).
[Crossref]

Rev. Sci. Instrum. (1)

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: single-photon sources and detectors,” Rev. Sci. Instrum. 82, 71101–71125 (2011).
[Crossref]

Other (2)

S. Krapick and C. Silberhorn, “Analysis of photon triplet generation in pulsed cascaded parametric down-conversion sources,” arXiv.org/abs/1506.07655 (2015).

R. Boyd, Nonlinear Optics, 3rd ed. (Elsevier, Burlington, USA, 2008).

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1 (a) Schematic of higher-order mode phase matching using calculated dispersion data for a fully-etched 550 × 360 nm TiO2 waveguide with SiO2 cladding. Shifting the pump mode from λp to 3λp gives a point of intersection where the effective indices of the TM00 signal and TM02 pump modes are the same. Phase matching is achieved at λp = 532 nm. Guided modes reach a cutoff wavelength when their effective index is equal to the cladding SiO2 index (gray region). (b) Schematic of the waveguide cross section and tunable parameters. (c) 3D schematic of an integrated device with input and output coupling.
Fig. 2
Fig. 2 We visualize the joint spectral intensity as an intensity plot of frequency versus frequency for several values of the pump detuning Δλp from the PPM wavelength. Panels (a), (b) and (c) correspond to Δp = 0, Δp = −2 nm and Δp = −0.5 nm, respectively. We assume a waveguide length L = 2 mm and use dispersion values for the phase matching point shown in Fig. 1(a). Fixing one of the three signal wavelengths (ωs) on the horizontal axis, the two other signal wavelengths (ωr and ωi) are given by the two values on the vertical axis that intersect the ellipse. Phase matching is achieved along the perimeter of the ellipse and the thickness is determined by the pump and signal interaction length, based on Eq. (1). By collapsing the plots in (a), (b), and (c) onto the horizontal axis, we generate the spectra shown in (d). All three spectra are normalized by the total conversion efficiency. Panel (e) shows signal spectrum bandwidth as a function of pump detuning Δωp. The dashed black curve gives the bandwidth defined by Eq. (2). Red and blue curves are numerically obtained signal bandwidths for waveguide lengths L = 2 mm and L =20 mm, respectively. Panel (f) zooms near the PPM point.
Fig. 3
Fig. 3 (a) Signal intensity as a function of device length for pump and signal loss values, respectively, of (i) 16 and 4 dB/cm, (ii) 16 and 12 dB/cm, (iii) 28 and 4 dB/cm, and (iv) 28 and 12 dB/cm, representing losses on the lower and upper extremes for TiO2 devices. Circles show the full quantum prediction, including signal dispersion, from Eq. (6) and lines show the prediction from Eq. (7). All curves are normalized by the maximum signal. (b) Ideal waveguide length plotted as a function of pump and signal losses. Current polycrystalline anatase TiO2 waveguide losses give optimal device lengths Lopt = 1 – 4 mm. White denotes Lopt > 8 mm.
Fig. 4
Fig. 4 (top) The effective nonlinearity γ as a function of waveguide width and thickness for the best phase-matching point. The highest γ = 1100 W−1km−1 is achieved in a 600 × 300 nm waveguide for TE signal and a 600 × 400 nm waveguide for TM signal. However, these dimensions do not achieve phase-matching. Lines mark regions with high which do achieve phase-matching. (bottom) Figure of merit as a function of waveguide dimensions. The best combination of phase matching and γ are achieved at 600 × 245 nm (γ = 908 W−1km−1) for TE signal and 550 × 360 nm (γ = 674 W−1km−1) for TM signal.
Fig. 5
Fig. 5 Experimental demonstration of THG in an integrated waveguide with two phase-matching points within the infrared pump bandwidth (black curve). The calculated THG signal (dashed curve) shows agreement with the measured THG signal (solid green curve). The inset shows a top-down image of scattered THG signal from the waveguide.
Fig. 6
Fig. 6 Triple and two-photon coincidence signals for detectors placed at equal distances from the TOSPDC output. Time delays τ12 = t1t2 and τ32 = t3t2 are given in units of the characteristic timescale τ 0 = D s L / 2. Panels (a) and (b) are G(3) functions for the detector filter bandwidths σ = 0.2ν0 and σ = 5ν0, respectively, with a loss mismatch parameter ΔαL ≡ (αp − 3αs) L = 0.1. Panels (d) and (e) correspond to the G(3) functions for ΔαL = 1 and ΔαL = 10, respectively, with a detector bandwidth σ = 5ν0. Panel (c) is the G(2) function for several values of σ with fixed ΔαL = 0.1 and (f) is the G(2) signal for several values of ΔαL with fixed σ = 5ν0. Frequency is in units of ν0 = 1/τ0, L is the waveguide length and Ds is the group velocity dispersion (GVD) of the signal guiding mode. Plots are normalized to their maximum values.
Fig. 7
Fig. 7 Examples of (a) a signal mode power density profile and (b) pump mode power density profile. These mode profiles are for a TM00 signal mode and TM02 pump mode phase matched at 532 nm in a 550 × 360 nm TiO2 waveguide with SiO2 cladding (first presented in Fig. 1 of the main text).
Fig. 8
Fig. 8 Examples of a) E-field orientation, b) crystal axes orientation and c) crystal axes orientation with a rotation by an arbitrary angle in the xy plane.

Tables (1)

Tables Icon

Table 1 Waveguide Parameters for Phase-matching Regions With High Figure of Merit . es is the Signal Mode Polarization and γ the Effective Nonlinearity.

Equations (46)

Equations on this page are rendered with MathJax. Learn more.

Δ PPM = 4 π L | D s |
δ s = 2 ( 2 3 ) 1 / 2 [ 2 ( v p 1 v s 1 ) D s Δ p + ( D p D s ) Δ p 2 ] 1 / 2 ,
Δ s 2 + Δ r 2 Δ r Δ s Δ p ( Δ r + Δ s ) = A p ,
γ = 3 χ ( 3 ) ω p 4 ε 0 c 2 n 2 η ,
n 2 = 3 χ ( 3 ) 4 n 0 2 ε 0 c
R 3 = 2 2 3 3 h ¯ c 3 n p 3 π 2 [ ω p 0 ] 2 γ 2 L 2 P ( ω 0 n 0 2 ( ω ) g ( ω 0 ) ) 3 e ( α p + 3 α s ) L / 2 d ν r d ν s | Φ ( ν r , ν s ) | 2 ,
N 3 0 L d N 3 ( z ) d z = ζ ˜ N p 0 α p 3 α s ( e 3 α s L e α p L ) ,
= γ 2 δ s ,
G ( 3 ) ( x 1 , x 2 , x 3 ) = E ( ) ( x 1 ) E ( ) ( x 2 ) E ( ) ( x 3 ) E ( + ) ( x 3 ) E ( + ) ( x 2 ) E ( + ) ( x 1 ) ,
G ( 3 ) ( x 1 , x 2 , x 3 ) = | 0 | E ( + ) ( x 3 ) E ( + ) ( x 2 ) E ( + ) ( x 1 ) | Ψ 3 | 2 .
G ( 2 ) ( x 1 , x 2 ) = d ω 3 | 0 | a ( ω 3 ) E ( + ) ( x 2 ) E ( + ) ( x 1 ) | Ψ 3 | 2 .
( Δ s 2 + Δ r 2 + Δ i 2 ) = 2 ( v p 1 v s 1 ) D s Δ p + ( D p D s ) Δ p 2 r p 2
x s 2 + x r 2 x r x s x r x s = a p .
δ s 2 3 × 2 r p = 2 ( 2 3 ) 1 / 2 [ 2 ( v p 1 v s 1 ) D s Δ p + ( D p D s ) Δ p 2 ] 1 / 2 ,
γ = 3 χ ( 3 ) ω p 4 ε 0 c 2 n 2 η ,
η i j k l = d x d y E p i * E s j E s k E s l [ | E p ( x , y ) | 2 d x d y ] 1 / 2 [ | E s ( x , y ) | 2 d x d y ] 3 / 2 ,
j j k k = k k j j = k k i i = i i k k = i i j j = j j i i ; j k k j = k j j k = k i i k = i k k i = i j j i = j i i j j k j k = k j k j = k i k i = i k i k = i j i j = j i j i ; i i i i = j j j j = k k k k = i i j j + i j i j + i j j i
η i i j j = η i j i j = η i j j i .
n 2 = 3 χ ( 3 ) 4 n 0 2 ε 0 c ,
H ^ 1 = 3 ε 0 χ ( 3 ) 4 d V E ^ p ( + ) ( r , t ) E ^ r ( ) ( r , t ) E ^ s ( ) ( r , t ) E ^ i ( ) ( r , t ) + H . c .
E p ( + ) ( r , t ) = A 0 A p ( x , y ) d ω p β ( ω p ) e [ i ( k p ( ω p ) z ω p t ) ] ,
| Ψ = [ 1 i h ¯ t 0 t d t H 1 ( t ) ] | 0 = | 0 + λ | Ψ 3 .
| Ψ 3 = d ω p β ( ω p ) k s , k r , k i ( ω r ) ( ω s ) ( ω i ) [ t 0 t d t d z ϕ ( t , z ) a ^ ( k r ) a ^ ( k s ) a ^ ( k i ) | 0 ] ,
λ = 3 ε 0 χ ( 3 ) A 0 4 h ¯ A eff ( δ k ) 3 / 2 ,
| Ψ 3 = d ω p β ( ω p ) k r , k s , k i ( ω r ) ( ω s ) ( ω i ) × [ t 0 t d t d z ϕ ( t , z ) e ( α p 3 α s ) z / 2 e ( α p 3 α s ) L / 4 × a ^ ( k r ) a ^ ( k s ) a ^ ( k i ) | 0 ] .
| Ψ 3 = ( 2 π λ L ( δ k ) 3 ) e ( α p + 3 α s ) L / 4 k r k s k i Φ ( ω r , ω s , ω i ) a ^ ( ω r ) a ^ ( ω s ) a ^ ( ω i ) | 0 ,
Φ ( ω r , ω s , ω i ) = β ( ω p ) ( ω r ) ( ω s ) ( ω i ) sinc [ { Δ k + i Δ α } L / 4 ] ,
R 3 = R Ψ 3 | k a ^ ( k ) a ^ ( k ) | Ψ 3 = R ( 2 π λ L ) 2 ( 3 2 ) δ k 3 e ( α p + 3 α s ) L / 2 d ω r d ω s d ω i g ( ω r ) g ( ω s ) g ( ω i ) | Φ ( ω r , ω s , ω i ) | 2
Δ k = k ( ω s ) + k ( ω r ) + k ( ω i ) k ( ω p ) = D s 2 [ ν r 2 + ν s 2 + ( ν s + ν r ) 2 ] ,
R 3 = 2 2 3 2 h ¯ c 3 n p 3 π 2 ( ω p 0 ) 2 γ 2 L 2 p ( ω 0 n 0 2 ( ω ) g ( ω 0 ) ) 3 × e ( α p + 3 α s ) L / 2 d ν r d ν s | Φ ( ν r , ν s ) | 2 .
( 2 π λ L ) 2 3 2 δ k 3 = 3 2 ( 2 π ) 2 ε 0 3 c 3 n p 3 h ¯ 2 ( ω p 0 ) 2 2 γ 2 L 2 P | d ω p β ( ω p ) | 2 ,
d N 3 ( z ) = ζ ˜ N p ( z ) e 3 α s ( L z ) Δ z = ζ ˜ N p 0 e α p z e 3 α s ( L z ) d z ,
N 3 0 L d z N 3 ( z ) = ζ ˜ N p 0 α p 3 α s ( e 3 α s L e α p L ) .
d I ( z ) d z = α ( 1 ) I ( z ) α ( 2 ) [ I ( z ) ] 2
I ( z ) = α I 0 / ( α ( 1 ) + α ( 2 ) I 0 ) e α ( 1 ) z α ( 2 ) I 0 / ( α ( 1 ) + α ( 2 ) I 0 )
d N 3 ( z ) = ζ ˜ A mode I p 0 h ¯ ω p α p ( 1 ) / ( α p ( 1 ) + α p ( 2 ) I p 0 ) e α p ( 1 ) z α p ( 2 ) I p 0 / ( α p ( 1 ) + α p ( 2 ) I p 0 ) e 3 α s ( 1 ) ( L z ) d z .
α p ( 2 ) I p 0 α p ( 1 ) ( 1 e α p ( 1 ) L opt ) < 0.2
G ( 3 ) ( x 1 , x 2 , x 3 ) = Ψ 3 | E ( ) ( x 1 ) E ( ) ( x 2 ) E ( ) ( x 3 ) E ( + ) ( x 3 ) E ( + ) ( x 2 ) E ( + ) ( x 1 ) Ψ 3 | .
G 3 ( x 1 , x 2 , x 3 ) = | ψ 1 ( x 1 , x 2 , x 3 ) | 2 ,
ψ ( x 1 , x 2 , x 3 ) × d ω 1 d ω 2 d ω 3 d ω r d ω s f 1 ( ω 1 ) f 2 ( ω 2 ) f 3 ( ω 3 ) e i ω 1 x 1 i ω 2 x 2 i ω 3 x 3 × Φ ( ω r , ω s ) 0 | a ( ω 1 ) a ( ω 2 ) a ( ω 3 ) a ^ ( ω r ) a ^ ( ω s ) a ^ ( ω p ω r ω s ) | 0 .
ψ ( x 1 , x 2 , x 3 ) = e i ω p x 3 d ω r d ω s f ( ω r ) f ( ω s ) f ( ω p ω r ω s ) Φ ( ω r ω s ) e i ω r ( x 1 x 3 ) i ω s ( x 2 x 3 ) .
G ( 3 ) ( x 1 , x 2 , x 3 ) = | d ν r d ν s f r ( ν r ) f s ( ν s ) f i ( ν r + ν s ) Φ ( ν r , ν s ) e i ν r ( x 1 x 3 ) i ν s ( x 2 x 3 ) | 2 .
G ( 2 ) ( x 1 , x 2 ) = E ^ ( ) ( x 1 ) E ^ ( ) ( x 2 ) E ^ ( + ) ( x 2 ) E ^ ( + ) ( x 1 ) ,
G ( 2 ) ( x 1 , x 2 ) = d ω 3 | 0 | a ( ω 3 ) E ( + ) ( x 2 ) E ( + ) ( x 1 ) | Ψ 3 | 2 ,
0 | a ( ω 3 ) E ( + ) ( x 2 ) E ( + ) ( x 1 ) | Ψ 3 = d ω 1 d ω 2 d ω r d ω s Φ ( ω r , ω s ) f ( ω 1 ) f ( ω 2 ) e i ω 1 x 1 i ω 2 x 2 × 0 | a ( ω 3 ) a ( ω 1 ) a ( ω 2 ) a ( ω r ) a ( ω s ) a ( ω p ω r ω s ) | 0 .
G ( 2 ) ( x 1 , x 2 ) = d ν 3 | d ν r G ( ν r , ν r + ν 3 ) f 1 ( ν r ) f 2 ( ν r + ν 3 ) e i ν r ( x 1 x 2 ) | 2 .

Metrics

Select as filters


Select Topics Cancel
© Copyright 2022 | Optica Publishing Group. All Rights Reserved