Optical Bloch Oscillations and Zener Tunneling with Nonclassical Light

A quantum theory of optical Bloch oscillations and Zener tunneling (ZT) in arrays of coupled waveguides is theoretically presented, and the particlelike behavior of photons undergoing ZT is highlighted. In singly-periodic arrays excited by a photon-number-state input beam, each photon behaves as a classical particle which independently undergoes a coin-toss ZT event with a probability described by classical Zener theory. In binary arrays, excitation with two tilted beams enables us to observe the Hong-Ou-Mandel interference for two photons undergoing Bloch-Zener oscillations.

Figure 1

(a) Schematic of a singly periodic waveguide array. (b) Refractive index profile. (c) Band diagram. (d) Damped optical BO (pseudocolor map of $|\psi (x,z){|}^2$) for a broad Gaussian input beam at normal incidence (index gradient $F=18.46{\mathrm{m}}^{-1}$, $\lambda =1.44\mu \mathrm{m}$). (e) Fractional light power $Z_1(z)$ trapped in the first band of the array.

Figure 2

(a) Schematic of a binary array made of a sequence of alternating wide and narrow waveguides. (b) Refractive index profile. (c) Band diagram. (d) Plane-wave excitation coefficients $B_n$ for low-order bands versus incidence angle $\theta =\lambda \kappa /(2\pi )$, normalized to the Bragg angle ${\theta }_B=\lambda /(2a)$. (e) and (f) BZ oscillations when the array is excited by a broad Gaussian beam at either ${\theta }_1=-0.8{\theta }_B$ [in (e)] and ${\theta }_2=1.2{\theta }_B$ [in (f)]. $F=3.6{\mathrm{m}}^{-1}$, $\lambda =1.44\mu \mathrm{m}$.

Figure 3

(a) Numerically computed joint photon probability in the binary array of Fig. 2 excited by two-photon beams in the geometry of Fig. 2a. (b) Pseudocolor map of the expectation value $⟨{\widehat{\varphi }}^{†}(x)\widehat{\varphi }(x)⟩$.