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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 

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According to Eq.(10), the phase difference Δ γ is given by 

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where Δ E is the distance between the first and sec- 

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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 

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 

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