ψ ( ρ, z, t ) = [ S$_{11}$ ( z ) r$_{1}$ ( t − z v$_{g}$ ) + S$_{21}$ ( z ) r$_{2}$ ( t − z v$_{g}$ )] u$_{1}$ ( ρ ) + + [ S$_{12}$ ( z ) r$_{1}$ ( t − z v$_{g}$ ) + S$_{22}$ ( z ) r$_{2}$ ( t − z v$_{g}$ u$_{2}$ ( ρ ) + ψ$_{S}$ (16) 4 where ψ$_{S}$ = ψ$_{S}$ ( ρ, z, t ) is the fractional part of the field tunnelled into the slab waveguide. In particular, let us consider two coherent pulses with the same envelope but phase reversed and delayed by an interval δ , i.e. r$_{1}$ ( t ) = r ( t ) and r$_{2}$ ( t ) = − r ( t − δ ). In this case, in- terference effects, leading to light trapping in waveguides W$_{1}$ and W$_{2}$ and full suppression of leakage into the slab waveguide S ( ψ$_{S}$ ≃ 0), occurs for δ = 0 or, approxi- mately, for a delay δ much smaller than the characteris- tic pulse duration τ$_{c}$ . For δ larger than τ$_{c}$ , the two wave packets are not overlapped and behave as independent beams, leading to fractional decay. The latter scenario is also observed if the two pulses are temporally overlapped ( δ = 0) but incoherent, i.e. their phase difference changes randomly in time. it readily follows that the Hamilton equations ψ$_{z}$ = ( δH/δ Π), Π$_{z}$ = − ( δH/δψ ) yield Eq.(1) and its com- plex conjugate, so that Π is canonically conjugated to ψ . Quantization is then accomplished by replac- ing the classical fields ψ and Π with the operators ˆ ψ ( x, y, t ) and ˆ Π = i ℏ ˆ ψ $^{†}$( x, y, t ) satisfying the commu- tation relations [ ˆ ψ ( ρ, t ) , ˆ ψ $^{†}$( ρ $^{′}$, t$^{′}$)] = δ ( ρ − ρ $^{′}$) δ ( t − t $^{′}$) and [ ˆ ψ ( ρ, t ) , ˆ ψ ( ρ $^{′}$, t$^{′}$)] = [ ˆ ψ $^{†}$( ρ, t ) , ˆ ψ $^{†}$( ρ $^{′}$, t$^{′}$)] = 0, where we have set ρ = ( x, y ) . By introducing the spectral decom- position ˆ ψ ( ρ, t ) = (2 π ) − 1 / 2 ∫ d Ω ˆ φ ( ρ, Ω) exp( − i Ω t ), the second-quantized Hamiltonian operator reads H = ∫ dxdydt H with density H = − i 2 β (Π$_{x}$ ψ$_{x}$ + Π$_{y}$ ψ$_{y}$ ) − 1 v$_{g}$ Π ψ$_{t}$ − iV ( x, y )Π ψ, (17) classical paraxial wave equation (1) in Hamiltonian form assuming the paraxial spatial coordinate z as an indepen- dent variable [26]. Introducing the new field Π = i ℏ ψ † and the Hamiltonian H = dxdydt with density III. QUANTUM DESCRIPTION AND NONCLASSICAL EFFECTS A. Quantization procedure ˆ H = ℏ ∫ dρd Ω { 1 2 β ( ˆ φ † $_{x}$ˆ φ$_{x}$ + ˆ φ † $_{y}$ˆ φ$_{y}$ ) + [ V ( ρ ) − Ω v$_{g}$ ] ˆ φ † ˆ φ } . (18) [ ˆ φ ( ρ, Ω) , ˆ φ $^{†}$( ρ $^{′}$, Ω $^{′}$) ] = δ ( ρ − ρ $^{′}$) δ (Ω − Ω $^{′}$) , [ ˆ φ ( ρ, Ω) , ˆ φ ( ρ $^{′}$, Ω $^{′}$) ] = [ ˆ φ $^{†}$( ρ, Ω) , ˆ φ $^{†}$( ρ $^{′}$, Ω $^{′}$) ] = 0 . (19) Note that in the spectral domain the following commuta- tion relations hold for the operators ˆ φ ( ρ, Ω) and ˆ φ $^{†}$( ρ, Ω) Note also that the field energy U [Eq.(2)] corresponds to the operator ˆ U = ℏ ω ∫ dρd Ω ˆ φ $^{†}$( ρ, Ω) ˆ φ ( ρ, Ω). In the Schr¨ odinger picture, the quantum field is described by a vector state |Q ( z ) 〉 which evolves according to whereas the operators ˆ φ ( ρ, Ω) do not evolve with z [30]. The state |Q〉 can be expanded in Fock space as |Q〉 = ∑ $_{n}$a$_{n}$ | f ( n $^{)}$( q , Ω , z ) 〉 , where the n -photon number state | f ( n $^{)}$〉 is defined by (see, for instance, [26]) i $_{ℏ}$d |Q〉 dz = ˆ H |Q〉 (20) | f ( n $^{)}$〉 = ∫ d q d $_{Ω}$f ( n $^{)}$( q , Ω , z ) √$_{n}$$_{!}$ ˆ φ $^{†}$( ρ$_{1}$, Ω$_{1}$) .... ˆ φ $^{†}$( ρ$_{n}$, Ω$_{n}$) | 0 〉 (21) and where we have set q = ( ρ$_{1}$, ρ$_{2}$, ... ), Ω = (Ω$_{1}$ , Ω$_{2}$ , ... ). The normalization conditions ∑ n | a$_{n}$ | 2 = 1 and ∫ d q d Ω | f ( n $^{)}$( q , Ω , z ) | 2 = 1 are also assumed, which To describe propagation of nonclassical light in the coupled waveguide system, the classical paraxial wave field ψ in Eq.(1) or, similarly, the classical c-numbers c$_{1}$ , c$_{2}$ and c$_{k}$ in Eqs.(5-7), have to be replaced by quantum- mechanical operators satisfying suitable commutation re- lations, and different quantization procedures may be adopted. A first approach, which is well suited when the classical problem is formulated in terms of coupled-mode equations (5-7), is the input-output operator formalism commonly used for linear quantum-optical networks, ei- ther in the Heisenberg or in the Schr¨ odinger pictures (see, for instance, [17, 25]). A second approach, suited when the classical problem in Eq.(1) is formulated as a prop- agative (rather than as an initial-value) problem, is to adopt a quantization procedure for the classical field ψ as an evolution in space (rather than in time). In the Schr¨ odinger picture, this leads to an evolution in space of a many-photon probability amplitude. Such a phe- nomenological approach, which will be adopted in the following analysis, has received a growing use and appre- ciation in quantum theories of optical solitons [26–28]; its consistency with standard canonical quantization proce- dure has been discussed in Ref.[27]. Similar procedures have been also developed to study, in the Heisenberg pic- ture, paraxial propagation of nonclassical light and ap- plied to problems of quantum imaging (see, for instance, [29]). The quantization procedure consists in writing the