5 beam profile αg ( x, y, 0). More generally, for a nonclassi- cal state |Q〉 obtained by an arbitrary superposition of photon number states | g 〉 n with amplitudes a$_{n}$ , one can readily show that the expectation value of ˆ φ $^{†}$( x, y ) ˆ φ ( x, y ) yields the classic wave optics intensity distribution, namely 〈Q| ˆ φ $^{†}$( x, y ) ˆ φ ( x, y ) |Q〉 = 〈 n 〉| g ( x, y, z ) | $^{2}$, where 〈 n 〉 = ∑ $_{n}$n | a$_{n}$ | 2 is the mean photon number of the in- put beam. The quantum aspects of nonclassical light for single beam excitation may be revealed when the statis- tics of photons trapped in waveguides W$_{1}$ and W$_{2}$ are considered. For example, let us consider excitation of waveguide W$_{1}$ in its fundamental mode at the input plane z = 0, so that g ( x, y, 0) = u$_{1}$ ( x, y ), and let us compare the statistics of photons that remain in waveguide W$_{1}$ when the input beam is a photon number state (nonclas- sical light) or a coherent state (classical light). According to the analysis of Sec.II.B, the wave amplitude g evolves according to [see Eq.(15)] ensure that 〈Q|Q〉 = 1. Note that ˆ U | f ( n $^{)}$〉 = n ℏ ω | f ( n $^{)}$〉 , i.e. the Fock state | f ( n $^{)}$〉 is obtained from the vacuum state | 0 〉 by creating n photons with space-frequency weighting function f ( n $^{)}$. The evolution equation for the weighting function f ( n ) is obtained by substitut- ing Eqs.(18) and (21) into the Scr¨ odinger equation (20) and using the commutation relations of field operators [Eqs.(19)]. One then obtains Owing to the bosonic nature of photons, solely symmetric functions f ( n ) should be considered. i ∂f ( n ) ∂z = n ∑ l =1 [ − 1 2 β ( ∂ 2 ∂x 2 l + ∂ 2 ∂y 2 l ) + V ( ρ$_{l}$ ) − Ω$_{l}$ v$_{g}$ ] f ( n $^{)}$. (22) g ( ρ, z ) = S$_{11}$ ( z ) u$_{1}$ ( ρ ) + S$_{12}$u$_{2}$ ( ρ ) + S$_{13}$ ( z ) θ ( ρ, z ) , (26) ˆ a † 1 ≡ ∫ dxdyu$_{1}$ ( x, y ) ˆ φ $^{†}$( x, y ) (27) ˆ a † 2 ≡ dxdyu$_{2}$ ( x, y ) ˆ φ $^{†}$( x, y ) (28) ˆ a † 3 ≡ ∫ dxdyθ ( x, y, z ) ˆ φ $^{†}$( x, y ) (29) where | S$_{11}$ | 2 + | S$_{12}$ | 2 + | S$_{13}$ | 2 = 1 and θ ( ρ, z ) defines the normalized spatial profile of the field tunnelled into the slab waveguide. If we introduce the operators the commutation relations [ˆ a$_{i}$, ˆ a † $_{k}$] = δ$_{i,k}$ and [ˆ a † $_{i}$, ˆ a † $_{k}$] = [ˆ a$_{i}$, ˆ a$_{k}$ ] = 0 ( i, k = 1 , 2 , 3) hold. Assuming that the vector state |Q〉 is given by a superposition of photon number states | g 〉 n with amplitudes a$_{n}$ , one can write Let us consider first the propagation of monochro- matic fields, so that in Eqs.(18), (19), (21) and (22) we may disregard integration over different spectral com- ponents Ω and use a single renormalized bosonic cre- ation operator ˆ φ $^{†}$( ρ ) at frequency Ω = 0 satisfying the commutation relations [31] [ ˆ φ ( ρ ) , ˆ φ $^{†}$( ρ $^{′}$)] = δ ( ρ − ρ $^{′}$) and [ ˆ φ ( ρ ) , ˆ φ ( ρ $^{′}$)] = [ ˆ φ $^{†}$( ρ ) , ˆ φ $^{†}$( ρ $^{′}$)] = 0 , which replace Eq.(19). The simplest n -photon number state, denoted by | g 〉 $_{n}$, is obtained by assuming in Eq.(22) f ( n $^{)}$( q , z ) = g ( ρ$_{1}$, z ) g ( ρ$_{2}$, z ) ...g ( ρ$_{n}$, z ), where the function g ( ρ, z ) sat- isfies the classical wave optics equation B. Nonclassical effects with monochromatic beams ig$_{z}$ = − 1 2 β ( g$_{xx}$ + g$_{yy}$ ) + V ( x, y ) g (23) with the normalization ∫ dxdy | g ( x, y, z ) | 2 = 1. In this case one has The joint photon distribution P ( n$_{1}$ ; n$_{2}$, n$_{3}$ ; z ) to find n$_{1}$ photons in waveguide W$_{1}$, n$_{2}$ photons in waveguide W$_{2}$ and n$_{3}$ photons in the slab waveguide S is given by |Q〉 = ∞ ∑ n =0 a$_{n}$ √$_{n}$$_{!}$ ( S$_{11}$a † 1 + S$_{12}$a † 2 + S$_{13}$a † 3 ) $^{n}$| 0 〉 . (30) | g 〉 n = 1 √$_{n}$$_{!}$ (∫ dxdyg ( x, y, z ) ˆ φ $^{†}$( x, y ) n | 0 〉 . (24) Physically, this state describes the excitation of the op- tical system with an n -photon number state input beam with a spatial profile g ( x, y, 0) at the entrance plane z = 0. The classic wave optics description of light prop- agation, discussed in Sec.II, is attained by considering a superposition of photon number states | g$_{n}$ 〉 with a Pois- sonian distribution with c-number α , i.e. the coherent state | g ; α 〉 coh defined by One can readily show that the coherent state | g ; α 〉 coh is an eigenstate of the field annihilation operator ˆ φ ( x, y ) with eigenvalue αg ( x, y, z ), i.e. ˆ φ ( x, y ) | g ; α 〉 coh = αg ( x, y, z ) | g ; α 〉 $_{coh}$, where g ( x, y, z ) evolves according to the classical wave equation (23). Therefore the expecta- tion value coh 〈 g ; α | ˆ φ ( x, y ) | g ; α 〉 coh = αg ( x, y, z ) yields the classical solution of the wave equation (23) for an input | g ; α 〉 coh = ∞ ∑ n =0 exp( −| α | $^{2}$/ 2) α n n ! (∫ dρg ( ρ, z ) ˆ φ $^{†}$( ρ ) n | 0 〉 . (25) P$_{1}$ ( n$_{1}$ ; z ) = ∞ ∑ n = n 1 | a$_{n}$ | 2 ( n n$_{1}$ ) κ n 1 (1 − κ ) n − n 1 (33) | n$_{1}$, n$_{2}$, n$_{3}$ 〉 ≡ 1 √$_{n}$$_{1}$$_{!}$$_{n}$$_{2}$$_{!}$$_{n}$$_{3}$$_{!}$ ˆ a † n 1 1 ˆ a † n 2 2 ˆ a † n 3 3 | 0 〉 . (32) P ( n$_{1}$, n$_{2}$, n$_{3}$ ; z ) = |〈 n$_{1}$, n$_{2}$, n$_{3}$ |Q〉| 2 (31) where we have set The explicit expression of P ( n$_{1}$, n$_{2}$, n$_{3}$ ; z ) can be ob- tained by a double binomial expansion of the opera- tor ( S$_{11}$a † 1 + S$_{12}$a † 2 + S$_{13}$a † $_{3}$) n entering in Eq.(30). The marginal photon distribution P$_{1}$ ( n$_{1}$, z ) to find n$_{1}$ pho- tons in waveguide W$_{1}$ is then obtained as P$_{1}$ ( n$_{1}$ ; z ) = ∑ ∞ n 2 ,n 3 =0 P ( n$_{1}$, n$_{2}$, n$_{3}$ ; z ), and reads explicitly (see, for instance, [17])