Bloch oscillations in non-Hermitian lattices with trajectories in complex plane

TL;DR

This paper explores Bloch oscillations in non-Hermitian lattices, revealing that their trajectories are complex and involve reshaping and breathing of wave packets, extending the semiclassical model to complex energy bands.

Contribution

It demonstrates that the acceleration theorem applies only on average in non-Hermitian lattices and introduces the concept of complex trajectories for Bloch oscillations.

Findings

Bloch oscillations in non-Hermitian lattices involve complex trajectories.

Wave packets exhibit reshaping and breathing during oscillations.

The acceleration theorem holds on average for complex energy bands.

Abstract

Bloch oscillations (BOs), i.e. the oscillatory motion of a quantum particle in a periodic potential, are one of the most striking effects of coherent quantum transport in the matter. In the semiclassical picture, it is well known that BOs can be explained owing to the periodic band structure of the crystal and the so-called 'acceleration' theorem: since in the momentum space the particle wave packet drifts with a constant speed without being distorted, in real space the probability distribution of the particle undergoes a periodic motion following a trajectory which exactly reproduces the shape of the lattice band. In non-Hermitian lattices with a complex (i.e. not real) energy band, extension of the semiclassical model is not intuitive. Here we show that the acceleration theorem holds for non-Hermitian lattices with a complex energy band only {\it on average}, and that the periodic wave packet motion of the particle in the real space is described by a trajectory in {\it complex} plane, i.e. it generally corresponds to reshaping and breathing of the wave packet in addition to a transverse oscillatory motion. The concept of BOs involving complex trajectories is exemplified by considering two examples of non-Hermitian lattices with a complex band dispersion relation, including the Hatano-Nelson tight-binding Hamiltonian describing the hopping motion of a quantum particle on a linear lattice with an imaginary vector potential and a tight-binding lattice with imaginary hopping rates.