6 | S$_{11}$ ( z ) | $^{2}$] n$_{0}$ . The binomial photon distribution highlights the particle-like behavior of photons undergoing the de- cay from waveguide W$_{1}$ and it is analogous to that cre- ated by a beam splitter when excited by a photon num- ber state in one port, and the vacuum state in the other one (see, for instance, [19]). The photons in the initially excited waveguide behave like independent classical par- ticles and tunnel into the other waveguides is ruled by a Bernoulli trial (a coin toss) with a cumulative tunneling probability given by 1 − | S$_{11}$ ( z ) | $^{2}$. When the two waveguides W$_{1}$ and W$_{2}$ are simultaneously excited by two nonclassical independent beams, quan- tum signatures of light propagation resulting from quan- tum interference can be detected by photon coincidence measurements. To describe propagation of independent beams, let us notice that for a given set of normalized and orthogonal solutions g$_{1}$ ( ρ, z ), g$_{2}$ ( ρ, z ), ... to the clas- sic wave equation (23), one can construct the n -photon number state From Eq.(33) it follows that the marginal photon distri- bution P$_{1}$ ( n$_{1}$ ; z ) remains Poissonian with mean photon number 〈 n$_{1}$ 〉 = n$_{0}$ | S$_{11}$ ( z ) | 2 that decays along the prop- agation distance z according to the classical decay law given by Eq.(12). Conversely, if the input beam is a pho- ton number state, i.e. a$_{n}$ = δ$_{n,n}$$_{0}$ , from Eq.(33) it follows that the marginal photon distribution P$_{1}$ ( n$_{1}$ ; z ) is given by the binomial distribution | a$_{n}$ | 2 = n n 0 exp( − n$_{0}$ ) n ! . (34) where κ ≡ | S$_{11}$ ( z ) | $^{2}$. Similar expressions are obtained for the marginal photon distributions P$_{2}$ ( n$_{2}$, z ) and P$_{3}$ ( n$_{3}$, z ) for waveguides W$_{2}$ and S after setting κ = | S$_{12}$ ( z ) | 2 and κ = | S$_{13}$ ( z ) | $^{2}$, respectively. Let us now suppose that the input beam is a coherent state with mean photon number 〈 n 〉 = n$_{0}$ and Poissonian distribution P$_{1}$ ( n$_{1}$ ; z ) = ( n$_{0}$ n$_{1}$ ) κ n 1 (1 − κ ) n 0 − n 1 ( n$_{1}$ ≤ n$_{0}$ ) (35) [ P$_{1}$ ( n$_{1}$ ; z ) = 0 for n$_{1}$ > n$_{0}$ ] with photon mean 〈 n$_{1}$ 〉 = n$_{0}$ | S$_{11}$ ( z ) | 2 and variance 〈 Δ n 2 1 〉 = | S$_{11}$ ( z ) | $^{2}$[1 − | g$_{1}$, g$_{2}$, ... 〉 n 1 ,n 2 ,... = 1 √$_{n}$$_{1}$$_{!}$$_{n}$$_{2}$$_{!}$$_{....}$ (∫ dρg$_{1}$ ( ρ, z ) ˆ φ $^{†}$( ρ ) n 1 × (∫ dρg$_{2}$ ( ρ, z ) ˆ φ $^{†}$( ρ ) n 2 × ... | 0 〉 (36) which describes excitation of the optical system with a set of independent beams with spatial profiles g$_{1}$ , g$_{2}$ ,... carrying n$_{1}$ , n$_{2}$ ,... photons ( n = n$_{1}$ + n$_{2}$ + ... ). In par- ticular, let us assume that at the input plane the waveg- uides W$_{1}$ and W$_{2}$ are excited in their fundamental modes by single photon number states, i.e. g$_{1}$ ( ρ, 0) = u$_{1}$ ( ρ ), g$_{2}$ ( ρ, 0) = u$_{2}$ ( ρ ) and propagation distance z is given by where ˆ a 3 is defined in Eq.(29). It can be readily shown that the expectation value of the field intensity for the two-photon state (40) is given by the incoherent super- position |Q〉 = ( S$_{11}$ ˆ a † 1 + S$_{12}$ ˆ a † 2 + S$_{13}$ ˆ a † 3 ) ( S$_{21}$ ˆ a † 1 + S$_{22}$ ˆ a † 2 + S$_{23}$ ˆ a † 3 ) | 0 〉 (40) † | Q ( z = 0) 〉 = | u$_{1}$, u$_{2}$ 〉 1 , 1 = ˆ a † $_{1}$ˆ a † 2 | 0 〉 , (37) 〈Q| ˆ φ $^{†}$( ρ ) ˆ φ ( ρ ) |Q〉 = | g$_{1}$ ( ρ, z ) | 2 + | g$_{2}$ ( ρ, z ) | $^{2}$, (41) the absence of interference being due to the lack of a defi- nite phase relationship between the two photons (see, for instance, [33]). The two independent beams which excite the two waveguides W$_{1}$ and W$_{2}$ behave, therefore, as two incoherent classical fields, and therefore according to the analysis of Sec.II.B half of the light power tunnels, on average, into the slab waveguide S for propagation dis- tances z ≫ l$_{d}$ . As the operators ˆ a † $_{1}$, ˆ a † 2 and ˆ a † 3 create photons in waveguides W$_{1}$, W$_{2}$ and W$_{3}$, respectively [see Eqs.(27-29)], from Eq.(40) the joint photon distribution P ( n$_{1}$, n$_{2}$, n$_{3}$ ; z ) can be readily calculated, and its nonva- nishing terms are explicitly given by with S$_{23}$ = S$_{13}$ for symmetry reasons [32], S$_{12}$ = S$_{21}$ , S$_{11}$ = S$_{22}$ [see Eqs.(12-14)], and | S$_{11}$ | $^{2}$+ | S$_{12}$ | $^{2}$+ | S$_{13}$ | 2 = 1 for power conservation. From Eqs.(36), (38) and (39) it follows that the state vector of the system at the generic where the operators ˆ a † 1 and ˆ a † 2 are defined by Eqs.(27) and (28). According to the classical analysis of Sec.II.B, the waves g$_{1}$ and g$_{2}$ evolve according to g$_{1}$ ( ρ, z ) = S$_{11}$ ( z ) u$_{1}$ ( ρ ) + S$_{12}$ ( z ) u$_{2}$ ( ρ ) + S$_{13}$ ( z ) θ ( ρ, z ) (38) g$_{2}$ ( ρ, z ) = S$_{21}$ ( z ) u$_{1}$ ( ρ ) + S$_{22}$ ( z ) u$_{2}$ ( ρ ) + S$_{23}$ ( z ) θ ( ρ, z ) (39)