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Fragmentation of Bloch oscillations in quasiperiodic waveguide arrays

Bloch oscillations (BOs) are periodic motions of waves confined in the periodic lattices superimposed by a tilted potential. It is generally believed that their formation is closely related to the Bragg scattering at the edges of the Brillouin zone. This has been observed in a variety of contexts. However, in quasiperiodic lattices, for example the one-dimensional Harper model, the translational symmetry is broken by quasiperiodicity; hence, the concept of the Brillouin zone does not apply anymore and only pseudogaps can exist. Here, we study the propagation of light beams in one-dimensional quasiperiodic waveguide arrays with an application of linearly growing potential. In such lattices, we find that BOs can survive in the quasiperiodic environment but the evolution undergoes fragmentation in both real and momentum spaces, which contrasts directly with the characteristic BOs in fully periodic lattices. We elucidate this phenomenon by relating it to the basic properties of quasiperiodic lattices.