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states), such system realizes in optics a simple analogue of destructive Fano interference with suppression of light leakage from the channel waveguides (analogous to two discrete states) into the slab waveguide (the analogue of the continuum). When the description of light propaga- tion is made at the quantum level, photon pair excitation reveals strictly quantum features, such as the tendency of photons to bunch when decaying into the continuum. 

## II. THE OPTICAL ANALOGUE OF POPULATION TRAPPING IN THE CONTINUUM: BASIC MODEL AND CLASSIC WAVE OPTICS DESCRIPTION 

Let us consider propagation of quasi-monochromatic and paraxial light wave packets at carrier frequency ω = 2 πc$_{0}$/λ in a weakling guiding dielectric structure, with optical axis z and refractive index profile n ( x, y ), composed by two straight and parallel single-mode and equal channel waveguides W$_{1}$ and W$_{2}$, side-coupled to a slab waveguide S as shown in Fig.1(a). In the parax- ial, weak guidance and quasi-monochromatic approxima- tions, the electric field can be written as E ( x, y, z, t ) ∝ ψ ( x, y, z, t ) exp( iβz − iωt )+ ψ $^{†}$( x, y, z, t ) exp( − iβz + iωt ), where the envelope ψ varies slowly with respect to z and t over one wavelength λ and one optical cycle 2 π/ω . Ne- glecting nonlinearities and group-velocity dispersion, the slow evolution of the electric field envelope ψ ( x, y, z, t ) along the paraxial z direction is governed by the scalar wave equation 

<img src='content_image/417585.jpg'>

where β = ( ω/c$_{0}$ ) n$_{s}$ is the reference propagation constant, n$_{s}$ is the substrate refractive index, v$_{g}$ = ( dβ/dω ) − 1 is the group velocity of light, and V ( x, y ) = β [ n$_{s}$ − n ( x, y )] /n$_{s}$ . The electric field envelope ψ has been normalized such that the cycle-averaged total energy of the electromagnetic field (see, e.g., [24]) is given by 

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At the entrance plane z = 0, light is typically injected into either one, or in both, the channel waveguides W$_{1}$ and W$_{2}$, in the form of either monochromatic waves or wave packets with spatial profiles tailored to match their fundamental modes. Owing to evanescent coupling with the slab waveguide S, light leakage into the contin- uous set of modes of the slab is generally found, how- ever for certain geometric settings a trapping state may exist, corresponding to destructive interference between different tunneling paths into the continuum [9]. At a classical level, light propagation in the waveguides is 

## A. The model 

well captured in the framework of a coupled mode equa- tion approach, as previously shown in Ref.[9]. After in- troduction of the spectral decomposition ψ ( x, y, z, t ) = (2 π ) − 1 / 2 ∫ d Ω φ ( x, y, z, Ω) exp( − i Ω t ), from Eq.(1) it fol- lows that the spectral amplitude φ ( x, y, z, Ω) satisfies the wave equation 

Let us indicate by u$_{1}$ ( ρ ) and u$_{2}$ ( ρ ) the fundamental modes of waveguides W$_{1}$ and W$_{2}$, and by u$_{k}$ ( ρ ) the con- tinuous set of modes of the slab waveguide S with the normalization conditions ∫ dρ | u$_{1}$ ( ρ ) | 2 = ∫ dρ | u$_{2}$ ( ρ ) | 2 = 1, ∫ dρu$_{k}$ ( ρ ) u ∗ k ′ ( ρ ) = δ ( k − k $^{′}$) and ∫ dρu$_{1}$ ( ρ ) u ∗ $_{2}$( ρ ) ≃ ∫ dρu$_{1}$ ( ρ ) u ∗ $_{k}$( ρ ) ≃ ∫ dρu$_{2}$ ( ρ ) u ∗ $_{k}$( ρ ) ≃ 0, where ρ ≡ ( x, y ). Let us search for a solution to Eq.(3) in the form 

<img src='content_image/417578.jpg'>

The evolution equations of modal amplitudes c$_{1}$ , c$_{2}$ and c$_{k}$ then read [9] 

where: g$_{1}$ ( k ) is the coupling amplitude between modes u$_{1}$ and u$_{k}$ ; g$_{2}$ ( k ) is the coupling amplitude between modes u$_{2}$ and u$_{k}$ ; and Δ β ( k ), Δ β$_{0}$ are the propagation constant shifts of modes u$_{k}$ and u$_{1}$$_{,}$$_{2}$ , respectively, from the refer- ence value β . In their present form, Eqs.(5-7) are analo- gous to the dynamical equations describing the quantum mechanical decay of two bound states W$_{1}$ and W$_{2}$ into a common continuum S [see Fig.1(b)], provided that the temporal dynamics of the quantum mechanical problem is replaced by the paraxial propagation in space of light waves. The two bound states, with the same energy, are embedded in the continuum provided that Δ β$_{0}$ falls in- side the continuous spectrum Δ β ( k ), a condition which is satisfied whenever the refractive index change Δ n$_{S}$ in the slab waveguide is smaller than the index change Δ n$_{g}$ in the two channel waveguides (see [9] for more details). Note that, if waveguides W$_{1}$ and W$_{2}$ are symmetrically placed at opposite sides from the slab waveguide S as shown in Fig.1(a), the coupling coefficients g$_{1}$ ( k ) and g$_{2}$ ( k ) are the same, i.e. g$_{2}$ ( k ) = g$_{1}$ ( k ) ≡ g ( k ), and in this case a trapped state does exist, as discussed in Ref.[9]. Although trapped states in the continuum may also ex- ist when the waveguides W$_{1}$ and W$_{2}$ are horizontally displaced or placed at different distances from the slab waveguide [9], in this work we will consider the simplest symmetric case shown in Fig.1(a). 

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