5 ( S$_{11}$ S$_{12}$ S$_{21}$ S$_{22}$ ) = ( T exp( − iγ$_{1}$ ) + R exp( − iγ$_{2}$ ) √$_{RT}$ exp( iα ) [exp( − iγ$_{1}$ ) − exp( − iγ$_{2}$ )] √$_{RT}$ exp( − iα ) [exp( − iγ$_{1}$ ) exp( − iγ$_{2}$ )] T exp( − iγ$_{2}$ ) + R exp( − iγ$_{1}$ ) ) . (21) way, the deviation of z$_{B}$ = λ/ ( F a ) from L = λ/ ( F$_{0}$a ) is negligible, i.e. the BO cycle is almost completed at the output of the arrayed region, whereas the change of Δ γ can be of the order of 2 π or larger as F a is typi- cally much larger than Δ E . Figures 4(a-d) show typical examples of two-band BOs observed in numerical simu- lations of Eq.(1) for the one-dimensional array of Fig.1. The length of the arrayed region is L = 23 mm, and the condition z$_{B}$ = L is attained for an applied refractive index gradient F$_{0}$ = 2 . 502 m − $^{1}$. For such a relatively low value of refractive index gradient, ZT from band 1 to band 2, and from band 2 to band 3, turns out to be neg- ligible. The array is excited by a broad Gaussian beam tilted at the Bragg angle θ$_{2}$ = − θ$_{B}$ , and the evolution of beam intensity | φ ( x, z ) | 2 along the sample is plotted for a few values of the refractive index gradient F close to F$_{0}$ . Beam splitting and beam recombination after a full BO cycle are clearly visible, as well as the change of the power levels in the two scattered output beams as the applied index gradient F is slightly varied (by a few percents) at around F$_{0}$ . The behavior of the fractional powers P$_{1}$ and P$_{2}$ carried by the two output beams versus the ratio F$_{0}$/F is depicted in Fig.4(e), clearly showing an oscillatory behavior with a visibility of about ∼ 90%. IV. QUANTUM INTERFERENCE IN MULTIBAND BLOCH OSCILLATIONS WITH NONCLASSICAL LIGHT In previous sections, multiband optical BOs and in- terference effects have been investigated in the frame- work of classical electromagnetic theory. Here we ex- tend the previous analysis to nonclassical states of light undergoing multiband BOs, and discuss relevant quan- tum interference effects similar to those observed in Mach-Zehnder-based quantum interferometry [20–23]. A second-quantization approach to study optical BOs in waveguide arrays, based on a procedure similar to that adopted in the quantum theory of solitons [31], has been developed in Ref.[15]. In this approach, the scalar wave equation for the classical field φ ( x, z ) [Eq.(1)] is written in Hamiltonian form assuming the paraxial spatial coor- dinate z as an independent variable, and a quantization procedure is then applied by introducing creation φ $^{†}$( x ) and annihilation φ ( x ) bosonic field operators (for details see [15, 32]). In the Schr¨ odinger picture, the quantum field is described by a quantum state |Q ( z ) 〉 which can be expanded in Fock space as |Q ( z ) 〉 = ∞ ∑ n =1 a$_{n}$ | f ( n $^{)}$〉 , (25) ond bands of the array, measured from their dc values [27]. Typically the value Δ E is of the order of the refrac- tive index change Δ n of the core region of waveguides from the dielectric substrate [see Fig.1(c)]. In practice, to vary Δ γ one can slightly change the index gradient F around the reference value F$_{0}$ = λ/ ( aL ) [30]. In this FIG. 3: (color online) (a) Schematic of two-band BOs in a waveguide array excited by two input broad beams tilted at the angles θ 1 ∼ θ B and θ 2 = θ 1 − 2 θ B ∼ − θ B . The paths followed by the wave packets φ 1 and φ 2 undergoing BOs map in real space the spectral shapes of the band dispersion curves E 1 ( κ ) and E 2 ( κ ), respectively. (b) Equivalent two-port Mach- Zehnder interferometer. F /F=0.93 0 F /F=1 0 (a) (c) x z inputGLYPH<.notdef>plane (z=0) outputGLYPH<.notdef>plane (z=L) L GLYPH<c102>$_{1}$ GLYPH<c102>$_{2}$ GLYPH<c113>$_{1}$ GLYPH<c113>$_{2}$ - beam g beam g 1 2 beam g$_{1}$ beam g$_{2}$ GLYPH<c68>GLYPH<c103> BS1 BS2 D1 D2 g$_{1}$ g$_{2}$ D1 D2 (b) array Classical Mach-Zehnder interferometry, which do not re- quire ZT, is thus expected to be observable in multiband BOs. As an example, for single beam excitation at the tilt angle θ$_{1}$ , i.e. for φ ( x, 0) = g$_{1}$ ( x, 0), the fractional light powers P$_{1}$ (Δ γ ) and P$_{2}$ (Δ γ ) of the two beams g$_{1}$ ( x, z$_{B}$ ) and g$_{2}$ ( x, z$_{B}$ ) leaving the array and measured by two pho- todetectors D1 and D2 [see Fig.3(a)], are readily calcu- lated from Eqs.(16), (17) and (21) as P$_{1}$ (Δ γ ) = | S$_{11}$ | 2 = T 2 + R 2 + 2 RT cos(Δ γ ) (22) P$_{2}$ (Δ γ ) = S$_{21}$ 2 = 2 RT (1 − cos(Δ γ )) . (23) Δ γ = 2 π F a ( a 2 π ∫ 2 π/a 0 dκ [ E ′ $_{2}$( κ ) − E ′ $_{1}$( κ ) ] ) ≡ 2 π Δ E F a (24) According to Eq.(10), the phase difference Δ γ is given by where Δ E is the distance between the first and sec-