content_image/385795.jpg content_image/385797.jpg content_image/385794.jpg content_image/385796.jpg
in the regime of multiband excitation and negligible ZT, i.e. in a complementary regime of that previously considered in Refs.[12, 15]. In spite of the absence of interband transitions, it is shown that Mach-Zehnder- like interference effects do spontaneously occur for a multiband excitation of the array at the input plane. In this regime, the input beam breaks up into two (or more) wave packets belonging to different bands of the array which follow distinct paths in the real space and recombine after a full BO cycle. Wave and photon interference effects after a full BO cycle may be thus observed by classical and nonclassical light illumination, even in absence of ZT. In particular, it is shown that array excitation with correlated photon pairs tilted at the Bragg angles enables to observe quantum interference patterns in two-photon correlation measurements analogous to those occurring in quantum Mach-Zehnder interferometry as a manifestation of the de Broglie wavelength of entangled photon states [20–23]. 

## II. WAVE-OPTICS MODEL OF MULTIBAND BLOCH OSCILLATIONS IN A WAVEGUIDE ARRAY 

The starting point of our analysis is provided by a rather standard wave optics model describing BOs of monochromatic light waves at carrier frequency ω = 2 πc$_{0}$/λ propagating in a weakling guiding one- dimensional waveguide array [Fig.1(a)], with a periodic refractive index profile n ( x ) and with a superimposed transverse refractive index gradient F x [7, 15, 24]. In the paraxial approximation, the slow evolution of a scalar field component φ ( x, z ) along the paraxial z direction is governed by the Schr¨ odinger-like wave equation (see, for instance, [24]) 

where λ = λ/ (2 π ) is the reduced wavelength, n$_{s}$ is the substrate refractive index, V ( x ) = n$_{s}$ − n ( x ) accounts for the periodic modulation of the refractive index with spa- tial periodicity a [ V ( x + a ) = V ( x )], and F is the super- imposed transverse index gradient. The array is assumed to be confined in the region 0 < z < L [see Fig.1(a)] and array excitation is accomplished at the input plane z = 0. Since V ( x ) = F = 0 for z < 0 and z > L , usual paraxial diffraction occurs before and after the light beam enters and leaves the arrayed structure. The sample length L is typically chosen to be equal to the characteristic period z$_{B}$ of BOs at a reference value F = F$_{0}$ of index gradient, i.e. L = z$_{B}$ where [24] 

In absence of the transverse index gradient ( F = 0), the modes of the arrayed structure are the Bloch 

<img src='content_image/385795.jpg'>

<img src='content_image/385797.jpg'>

<img src='content_image/385794.jpg'>

states ϕ$_{n}$ ( x, κ ) with corresponding dispersion curves E$_{n}$ ( κ ), where n = 1 , 2 , 3 , ... is the band index, κ is the Bloch wave number (quasi momentum) which varies in the first Brillouin zone − π/a < κ ≤ π/a , and the orthogonal and normalization conditions 〈 ϕ$_{n}$ ′ ( x, κ $^{′}$) | ϕ$_{n}$ ( x, κ ) 〉 = δ$_{n,n}$ ′ δ ( κ − κ $^{′}$) hold. ϕ$_{n}$ ( x, κ ) and E$_{n}$ ( κ ) are found as eigenfunctions and eigenvalues of the problem H $_{0}$ϕ$_{n}$ ( x, κ ) = E$_{n}$ ( κ ) ϕ$_{n}$ ( x, κ ) with pe- riodic Hamiltonian H 0 = − [ λ $^{2}$/ (2 n$_{s}$ )] ∂ 2 xx + V ( x ). Ac- cording to the Bloch theorem, one can write ϕ$_{n}$ ( x, κ ) = u$_{n}$ ( x, κ ) exp( iκx ), where the periodic part u$_{n}$ ( x, κ ) of the Bloch state can be expanded as a Fourier series 

<img src='content_image/385796.jpg'>

with coefficients θ$_{n,l}$ ( κ ). Owing to the orthogonal and normalization conditions of Bloch states, the matrix A ≡ θ$_{n,l}$ turns out to be unitary, i.e. A − 1 = A $^{†}$. In place of the Fourier coefficients θ$_{n,l}$ ( κ ), defined in the 

