content_image/392450.jpg content_image/392451.jpg content_image/392452.jpg content_image/392456.jpg content_image/392457.jpg content_image/392461.jpg content_image/392462.jpg
the quantum state of the system at the input plane of the array may be written as 

<img src='content_image/392450.jpg'>

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 

<img src='content_image/392451.jpg'>

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 

<img src='content_image/392452.jpg'>

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 

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 

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 

<img src='content_image/392456.jpg'>

<img src='content_image/392457.jpg'>

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 

<img src='content_image/392461.jpg'>

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 

<img src='content_image/392462.jpg'>

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]. 

