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

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 on average, and that the periodic wave packet motion of the particle in the real space is described by a trajectory in 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.

Tight-binding lattices with an oscillating imaginary gauge field

“…Recently, an optical implementation of the Hatano-Nelson model with an artificial maginary gauge field, based on a chain of coupled optical microrings with tailored gain and loss regions, was suggested [13] and the phenomenon of non-Hermitian transparency was disclosed [14]. In such previous studies [6,[11][12][13][14][15] the imaginary gauge field was considered stationary. However, it is well known that in ordinary tight-binding Hermitian quantum models oscillating electric and/or magnetic fields can deeply change the hopping dynamics via Peierls' substitution with important applications to coherent quantum state storage, dynamic decoupling and decoherence control (see, for instance, [16] and references * stefano.longhi@polimi.it therein).…”

Non-Hermitian tight-binding network engineering

“…To implement such an Hamiltonian, we assume ω/κ = 7, θ/κ = 0.2 and tune the values of U , σ 1 and σ 2 according to Eqs. (34)(35)(36), namely σ 1 /κ = 3.2 + 1.05i, σ 2 /κ = 0.2 + 1.05i and U/κ 11 − 45i. To check the fidelity of the synthesized Hamiltonian, in Fig.3(e) we compare the numerically-computed evolution of the occupation probability P (t) = |c 0 (t)| 2 with the initial condition c n (0) = δ n,0 , as obtained by the exact Lee Hamiltonian with complex coupling [Eq.…”

“…For example, in optics non-Hermitian tight-binding networks with complex on-site potentials are readily implemented by evanescent coupling of light modes trapped in optical waveguides or resonators with optical gain and loss in them, while the realization of controllable non-Hermitian coupling constants is a much less trivial task. However, complex hopping amplitudes play an important role for the observation of a wide variety of phenomena that have been disclosed in recent works [24,[29][30][31][32][33][34]. These include incoherent control of non-Hermitian Bose-Hubbard dimers [24], self-sustained emission in semi-infinite non-Hermitian systems at the exceptional point [29], optical simulation of PT -symmetric quantum field theories in the ghost regime [30,31], invisible defects in tight-binding lattices [32], non-Hermitian bound states in the continuum [33], and Bloch oscillations with trajectories in complex plane [34].…”

Super Bloch oscillation in aPTsymmetric system

“…It is well known that the acceleration theorem explains Bloch oscillation semi-classically for Hermitian lattice. An extension of the the acceleration theorem to nonHermitian lattices was discussed and Bloch oscillation for the generalized non-Hermitian Hatano and Nelson was specifically studied [18]. Another theoretical paper investigated Bloch-Zener oscillation for locally PT symmetric system [19].…”