7 The quantum state of the system at the output plane, i.e. after one BO cycle, can be derived from Eqs.(14) and (29), and reads explicitly the quantum state of the system at the input plane of the array may be written as Note that, as g ( l $^{)}$( x, 0; κ$_{0}$ ) are orthogonal functions, the operators ˆ a † l and ˆ a$_{l}$ satisfy the commutation relations |Q ( z = 0) 〉 = 1 √$_{...n}$$_{0}$$_{!}$$_{n}$$_{1}$$_{!}$$_{n}$$_{2}$$_{!}$$_{....}$ ... ˆ a n 0 0 ˆ a n 1 1 ˆ a n 2 2 ... | 0 〉 ≡ | ..., n$_{0}$, n$_{1}$, n$_{2}$, ... 〉 . (31) [ˆ a$_{l}$, ˆ a † $_{ρ}$] = δ$_{l,ρ}$ , [ˆ a$_{l}$, ˆ a$_{ρ}$ ] = [ˆ a † $_{l}$, ˆ a † $_{ρ}$] = 0 . (32) |Q ( z = z$_{B}$ ) 〉 = 1 √$_{...n}$$_{0}$$_{!}$$_{n}$$_{1}$$_{!}$$_{n}$$_{2}$$_{!}$$_{....}$ ... ( ∑ ρ M 0 $_{,ρ}$ˆ a$_{ρ}$ ) n 0 ( ∑ ρ M 1 $_{,ρ}$ˆ a$_{ρ}$ ) n 1 ( ∑ ρ M 2 $_{,ρ}$ˆ a$_{ρ}$ ) n 2 ... | 0 〉 . (33) consider the two-band BO problem of Fig.3(a) discussed in Sec.III. Instead of using classical light (i.e. coherent states for the two beams g$_{1}$ and g$_{2}$ ), let us illuminate the array by correlated photon pairs generated by frequency- degenerate spontaneous parametric down-conversion and incident onto the array at the Bragg angles θ$_{1}$ = θ$_{B}$ and θ$_{2}$ = − θ$_{B}$ . The quantum state of light at the entrance plane of the array can be thus written as Note that, since the multiband BOs problem over one BO cycle admits of a formulation in terms of a scattering matrix [Eq.(14)], the quantum state at the output plane, as given by Eq.(33), is consistent with the result that one would obtain using the input-output operator formalism commonly adopted for linear quantum-optical networks (see, for instance, [33]). Equation (33) is at the basis of quantum interference and quantum entanglement observ- able in multiband BOs when the array is excited by pho- ton number states. As an example, we discuss in detail the doubling of interference fringes in photon counting rates for correlated photon pairs undergoing two-band BOs, a phenomenon analogous to the doubling of inter- ference fringes in photon correlation measurements ob- served in a Mach-Zehnder interferometer and attributed to the so-called de Broglie wavelength of the entangled biphoton state (see, for instance, [22]). To this aim, let us where | n$_{1}$ 〉 1 | n$_{2}$ 〉 2 denotes a n = n$_{1}$ + n$_{2}$ photon num- ber state with n$_{1}$ photons in the mode g$_{1}$ and n$_{2}$ pho- tons in the mode g$_{2}$ . Using Eqs.(16), (17) and (34), from Eqs.(31) and (33) it readily follows that the state of quan- tum field after one BO cycle is given by |Q ( z = 0) 〉 = | 1 〉 1 | 1 〉 $_{2}$, (34) |Q ( z = z$_{B}$ ) 〉 = √$_{2}$$_{S}$$_{11}$$_{S}$$_{21}$ | 2 〉 1 | 0 〉 2 + ( S$_{11}$S$_{22}$ + S$_{12}$S$_{21}$ ) | 1 〉 1 | 1 〉 2 $_{+}$√$_{2}$$_{S}$$_{12}$$_{S}$$_{22}$ | 0 〉 1 | 2 〉 $_{2}$. (35) Note that the counting rates R$_{11}$ = R$_{22}$ oscillate like ∼ cos(2Δ γ ), i.e. at twice the phase difference Δ γ , as op- posed to the classical first-order interference fringes [34], which oscillate like ∼ cos(Δ γ ) [see Eqs.(22) and (23)]. The doubling of the counting rate oscillation frequency versus the phase delay Δ γ is analogous to that observed in an ordinary Mach-Zehnder interferometer probed by correlated photon pairs and generally explained as a man- ifestation of the Broglie wavelength of the biphoton en- tangled state formed after the first beam splitter BS1 and probed at the second beam splitter BS2 of the interfer- ometer [22]. The joint probabilities R$_{11}$ and R$_{22}$ to find both photons in the same beam, either g$_{1}$ or g$_{2}$ , after one BO cycle are then given by Such relations give the two-photon counting rates that one would measure in an experiment [22]. Using Eq.(21) and assuming T = R = 1 / 2 (valid for incidence angles exactly tuned at the Bragg angles ± θ$_{B}$ ), one finally ob- tains R$_{11}$ = 2 | S$_{11}$ | $^{2}$| S$_{21}$ | 2 , R$_{22}$ = 2 | S$_{12}$ | $^{2}$| S$_{22}$ | $^{2}$. (36) R$_{11}$ = R$_{22}$ = 1 4 (1 − cos(2Δ γ )) . (37)