Anderson localization with second quantized fields in a coupled array of waveguides

Optical waveguide arrays: quantum effects and PT symmetry breaking

“…The time required for the emergence of the steady-state profile is inversely proportional to the strength of the disorder. In another sharp contrast, the typical strength of disorder in (weakly conducting) electronic materials is v d E F whereas in waveguide arrays, the disorder strength can be comparable to the tunneling, v d ∼ C [15,16].…”

“…In the limit N → ∞, this equation can be exactly solved by using Fourier transform [16] and we get the following expression for the timeevolution operator G(t) = exp[−iHt/ ] in the site-index space,…”

“…The off-diagonal disorder randomly modulates the tunneling C i → C i + v i where v i is a zero-mean random variable with variance v dt . We use uniformly distributed random variables to ensure that the modulated tunneling rates remain strictly positive, although the results are independent of the type of distribution used as long as any such distribution has zero mean and identical variance [16,66]. The resultant intensity distribution is averaged over multiple M ∼ 10 4 realizations to ensure that the final results are independent of the number of disorder realizations and the probability distribution of the site or tunneling disorder.…”

“…It is straightforward to obtain the eigenvalues λ ± and (right) eigenvectors |± R of the Hamiltonian (16). We remind the reader that since the matrix H is not Hermitian, its left-eigenvectors and right-eigenvectors are not Hermitian conjugates of each other [70,71].…”

“…A variation in the index of refraction or the tunneling amplitude, both of which can be introduced easily, permit the modeling of a tight-binding Hamiltonian with site or bond disorders respectively. Due to this versatility, many quantum and condensed matter phenomena -Bloch oscillations [12,13], Dirac zitterbewegung [14], and increased intensity fluctuations [15,16] of light undergoing Anderson localization [17][18][19] -have been theoretically predicted to occur or experimentally observed in waveguide arrays. They have been used to investigate solitonic solutions that arise due to nonlinearities in the dielectric response [20,21].…”