6 where the n -photon number state | f ( n $^{)}$〉 is defined by | f ( n $^{)}$〉 = ∫ dx$_{1}$dx$_{2}$...dx$_{n}$ f ( n $^{)}$( x$_{1}$, x$_{2}$, ...x$_{n}$, z ) √$_{n}$$_{!}$ × × ˆ φ $^{†}$( x$_{1}$ ) ˆ φ $^{†}$( x$_{2}$ ) ... ˆ φ $^{†}$( x$_{n}$ ) | 0 〉 . (26) The normalization conditions ∑ n | a$_{n}$ | 2 = 1 and ∫ dx$_{1}$dx$_{2}$...dx$_{n}$ | f ( n $^{)}$( x$_{1}$, x$_{2}$, ..., x$_{n}$, z ) | 2 = 1 are assumed. The Fock state | f ( n $^{)}$〉 defined by Eq.(26) is obtained from the vacuum state | 0 〉 by creating n photons with a spa- tial weighting function f ( n $^{)}$( x$_{1}$, x$_{2}$, ..., x$_{n}$, z ). The evo- lution equation for f ( n $^{)}$, obtained from the Schr¨ odinger equation and using the commutation relations of field operators, reads [15] iλ ∂f ( n ) ∂z = n ∑ l =1 [ − λ 2 2 n$_{s}$ ∂ 2 ∂x 2 l + V ( x$_{l}$ ) − F x$_{l}$ ] f ( n $^{)}$. (27) Owing to the bosonic nature of photons, solely symmet- ric functions f ( n ) should be considered. The simplest n - photon number state, denoted by | g 〉 $_{n}$, is obtained by as- suming f ( n ) = g ( x$_{1}$, z ) g ( x$_{2}$, z ) ...g ( x$_{n}$, z ), where the func- tion g ( x, z ) satisfies the classical wave equation (1) with the normalization ∫ dx | g ( x, z ) | 2 = 1. Note that in this case one has | g 〉 n = 1 √$_{n}$$_{!}$ (∫ dxg ( x, z ) ˆ φ $^{†}$( x ) n | 0 〉 , (28) 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 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 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- ˆ a † l = ∫ dxg ( l $^{)}$( x, 0; κ$_{0}$ ) ˆ φ $^{†}$( x ) , (30) FIG. 4: (color online) Classical Mach-Zehnder interferometry based on two-band BOs. (a),(b),(c) and (d) show the evolu- tion of light intensity distribution along the optical structure, excited at the input plane by a broad Gaussian beam tilted at θ 2 = − θ B , for a few values of F 0 /F . The arrayed structure has a length L = 23 mm and it is comprised between the two dotted horizontal lines. The index gradient F 0 , at which the array length L is exactly equal to one BO cycle z B , is given by F 0 = 2 . 502 m − $^{1}$. In (e) the numerically-computed fractional powers P 1 and P 2 carried by the two output beams (right- and left-diffracted beams, respectively) are plotted as functions of the ratio F 0 /F . F F $_{0}$/ 0.9 0.95 1 1.05 1.1 0 0.5 1 F /F=0.97 0 F /F=0.93 0 F /F=1 0 F /F=1.03 0 -300 300 -300 300 xGLYPH<.notdef>( m) GLYPH<c109> xGLYPH<.notdef>( m) GLYPH<c109> zGLYPH<.notdef>(mm) zGLYPH<.notdef>(mm) 0 30 0 30 (a) (b) (c) (d) fractionalGLYPH<.notdef>power P$_{2}$ P$_{1}$ (e) | g$_{1}$, g$_{2}$, ... 〉 n 1 ,n 2 ,... = 1 √$_{n}$$_{1}$$_{!}$ (∫ dxg$_{1}$ ( x, z ) ˆ φ $^{†}$( x ) n 1 × 1 √$_{n}$$_{2}$$_{!}$ (∫ dxg$_{2}$ ( x, z ) ˆ φ $^{†}$( x ) n 2 × .... | 0 〉 , (29)