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where the n -photon number state | f ( n $^{)}$〉 is defined by 

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The normalization conditions ∑ n | a$_{n}$ | 2 = 1 and ∫ dx$_{1}$dx$_{2}$...dx$_{n}$ | f ( n $^{)}$( x$_{1}$, x$_{2}$, ..., x$_{n}$, z ) | 2 = 1 are assumed. The Fock state | f ( n $^{)}$〉 defined by Eq.(26) is obtained from the vacuum state | 0 〉 by creating n photons with a spa- tial weighting function f ( n $^{)}$( x$_{1}$, x$_{2}$, ..., x$_{n}$, z ). The evo- lution equation for f ( n $^{)}$, obtained from the Schr¨ odinger equation and using the commutation relations of field operators, reads [15] 

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Owing to the bosonic nature of photons, solely symmet- ric functions f ( n ) should be considered. The simplest n - photon number state, denoted by | g 〉 $_{n}$, is obtained by as- suming f ( n ) = g ( x$_{1}$, z ) g ( x$_{2}$, z ) ...g ( x$_{n}$, z ), where the func- tion g ( x, z ) satisfies the classical wave equation (1) with the normalization ∫ dx | g ( x, z ) | 2 = 1. Note that in this case one has 

so that this quantum state describes the excitation of the array with a beam with a spatial profile g ( x, 0) at the entrance plane z = 0 and carrying n photons. More generally, for a given set of orthogonal and normalized functions g$_{1}$ ( x, z ), g$_{2}$ ( x, z ), ... that satisfy Eq.(1), one can construct the n -photon number state | g$_{1}$, g$_{2}$, ... 〉 n 1 ,n 2 ,... defined by 

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with n$_{1}$ + n$_{2}$ + ... = n . Such a quantum state describes the excitation of the array with a set of independent beams each carrying n$_{1}$ , n$_{2}$ ,... photons ( n = n$_{1}$ + n$_{2}$ + ... ). The classical picture of multiband BOs, described in previ- ous sections, is retrieved from the quantum model when the input beam is in a coherent state (classical light), i.e. when a superposition of Fock states with a Poisson distri- bution is considered [15]. Here we focus to the case where the input beam describes a nonclassical field. In partic- 

ular, let use suppose that the array is excited by a set of tilted beams g ( l $^{)}$( x, 0; κ$_{0}$ ) ( l = 0 , ± 1 , ± 2 , ... ), which have been introduced in Sec.III, and assume that g ( l ) is in a photon number state (a Fock state) carrying n$_{l}$ photons. After introduction of the creation operators ˆ a † l 

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