# Classical and quantum interference in multiband optical Bloch oscillations 

Dipartimento di Fisica and Istituto di Fotonica e Nanotecnologie del CNR, Politecnico di Milano, Piazza L. da Vinci 32, I-20133 Milano, Italy 

Stefano Longhi ∗ 

Classical and quantum interference of light propagating in arrays of coupled waveguides and un- dergoing multiband optical Bloch oscillations (BOs) with negligible Zener tunneling is theoretically investigated. In particular, it is shown that Mach-Zehnder-like interference effects spontaneously arise in multiband BOs owing to beam splitting and subsequent beam recombination occurring in one BO cycle. As a noteworthy example of quantum interference, we discuss the doubling of inter- ference fringes in photon counting rates for a correlated photon pair undergoing two-band BOs, a phenomenon analogous to the manifestation of the de Broglie wavelength of an entangled biphoton state observed in quantum Mach-Zehnder interferometry. 

Since the pioneering works by Bloch, Zener and Wannier on the coherent electron dynamics in biased crystalline solids [1], it is well established that quantum particles in periodic potentials subjected to an external force do not delocalize but undergo a high-frequency oscillatory motion known as a Bloch oscillation (BO). Tunneling to higher-order bands, referred to as Zener tunneling (ZT) and observed for a strong external bias, is responsible for BO damping and broadening of Wannier-Stark resonances. Owing to detrimental dephasing and many-body effects, the experimental verification of BOs failed for many decades. Nowadays BOs are considered a rather universal wavy phenomenon and their observation has been reported in different physical systems for both quantum particles and classical waves. In the quantum realm, BOs have been observed for electrons in biased semiconductor superlattices [2] and for Bose-Einstein condensates in accelerating optical lattices [3]. Classical analogues of BOs, based on interference effects of classical waves in periodic media, have been proposed and experimentally observed using either optical [4–7] or acoustical [8] waves. Optical BOs in artificial materials or at the nanoscale have been also recently predicted [9]. Among various photonic structures, arrays of coupled waveguides with transverse refractive index gradients have provided an extremely rich laboratory tool to visualize the classical wave optic analogues of BOs [4, 6, 7] and related phenomena, such as the coherent superposition of BOs and ZT (the so- called Bloch-Zener oscillations [10]) occurring in binary structures and recently observed in circularly-curved femtosecond-laser-written waveguide arrays [11]. In another physical context, application of Bloch-Zener oscillations to matter-wave interferometry has been also proposed [12]. While there has been a lot of interest 

PACS numbers: 72.10.Bg, 72.90.+y, 42.82.Et, 42.50.Dv 

## I. INTRODUCTION 

in the propagation of light waves in complex photonic structures like lattices and superlattices, quasi-crystals, and even random lattices, most of previous works focused on propagation of classical light and used the photonic structures as ’classical simulators’ of electronic or matter wave systems in periodic or random potentials. However, it is known that coupled waveguides behave similarly to beam splitters (see, for instance, [13]) and may therefore show quantum interference effects when probed with nonclassical light. Demonstration of key quantum effects of nonclassical light in silicon-based based waveguide circuits has been recently reported in Ref.[14]. As shown in recent works [15, 16], nonclassical light consisting of only particle-like quanta propagating in waveguide arrays with a superimposed transverse refractive index gradient can also produce optical BOs. Remarkably, in addition to classical wave Bragg scattering the quantum nature of light introduces new quantum interference effects. In particular, two-photon Hong-Ou-Mandel quantum interference [17] has been predicted for pairs of correlated photons undergoing Bloch-Zener oscillations in binary arrays. Here ZT periodically mixes photons belonging to the two minibands of the array and thus acts like a beam splitter [15]. Recently, quantum corre- lations of photon pairs undergoing discrete diffraction in homogeneous arrays have been investigated as well, and their classical counterpart has been experimentally observed in Hanbury Brown-Twiss intensity correlation measurements [18]. The investigation of quantum interference effects in complex photonic structures such as homogeneous, inhomogeneous or even random photonic lattices has two main motivations. On the one hand, they may offer the possibility of engineering photon entanglement and of transporting nonclassical light [18, 19]; on the other hand, they enable to utilize optical-quantum analogies in the opposite direction, i.e. trying to observe quantum version of classical wave phenomena (such as the analogue of the Talbot effect in the second quantized setup [19]). It is the aim of this work to provide a comprehensive analysis of classical and quantum interference of light undergoing BOs in inhomogeneous waveguide arrays 

