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Chinese Physics B

Non-Hermitian Zitterbewegung Effect

Jiahao Wu, Han‐Jie Zhu, Wu-Ming Liu

Published Oct 9, 2026• 0 citations
Publisher

Abstract

Abstract We investigate the general form of the Zitterbewegung (ZB) effect in non-Hermitian systems for a multi-band Hamiltonian depending on momentum, where the ZB effect is the multifrequency oscillatory motion of the quasiparticle. We propose a real non-Hermitian Luttinger Hamiltonian that describes holes near the top of the valence band in common zinc-blende and diamond-structured semiconductors (GaAs, Si, Ge, etc.) and enables the observation of the ZB effect. By evaluating the dynamical evolution of light and heavy holes, we demonstrate that non-Hermiticity acts as an amplifier for ZB oscillations. This amplification, rooted in the complex energy spectrum, allows us to predict the maximal enhancement of transient ZB oscillations within the short-time physical window. An experimental scheme is proposed to observe the ZB effect by measuring the motion of the holes. Our findings offer a new perspective on the transport properties of open quantum systems and semiconductor carrier transport.

DOI
10.1088/1674-1056/aeb244

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Abstract We investigate the general form of the Zitterbewegung (ZB) effect in non-Hermitian systems for a multi-band Hamiltonian depending on momentum, where the ZB effect is the multifrequency oscillatory motion of the quasiparticle. We propose a real non-Hermitian Luttinger Hamiltonian that describes holes near the top of the valence band in common zinc-blende and diamond-structured semiconductors (GaAs, Si, Ge, etc.) and enables the observation of the ZB effect. By evaluating the dynamical evolution of light and heavy holes, we demonstrate that non-Hermiticity acts as an amplifier for ZB oscillations. This amplification, rooted in the complex energy spectrum, allows us to predict the maximal enhancement of transient ZB oscillations within the short-time physical window. An experimental scheme is proposed to observe the ZB effect by measuring the motion of the holes. Our findings offer a new perspective on the transport properties of open quantum systems and semiconductor carrier transport.
Zitterbewegung (ZB) is a phenomenon in relativistic quantum systems where the electron wave packet exhibits a trembling or oscillating behavior during its motion, caused by its interaction or coupling with the negative energy state. To directly observe ZB in electronic systems is difficult, due to the challenges associated with the small amplitude of the motion which is of the order of Compton wavelength. Photonic systems offer an alternative paradigm. We exploit the concept of pseudo parity-time (pseudo ) symmetry to study ZB in non-Hermitian quantum systems implemented as an experimentally feasible optical waveguide array. In particular, the non-Hermitian Hamiltonian is realized through evanescent coupling among the waveguides to form a one-dimensional lattice with periodic modulations in gain and loss along the guiding direction. As the modulation frequency is changed, we obtain a number of phenomena including periodically suppressed ZB trembling, spatial energy localization, and Hermitian-like ZB oscillations. We calculate phase diagrams indicating the emergence of different types of dynamical behaviors of the relativistic non-Hermitian quantum system in an experimentally justified parameter space. We provide numerical results and a physical analysis to explain the distinct dynamical behaviors revealed by the phase diagrams. Our findings provide a deeper understanding of both the relativistic ZB phenomenon and non-Hermitian pseudo- systems, with potential applications in controlling/harnessing light propagation in waveguide-based optical systems.
Nonorthogonality of eigenstates is a fundamental characteristic of non-Hermitian systems. It is expected that nonorthogonality-induced effects could be enhanced in the vicinity of the exceptional points (EPs) where the multiple eigenstates coalesce into one. Here, focusing on the wave-packet motions in non-Hermitian systems, we find that a wave packet near the EPs could exhibit new types of Zitterbewegung- (ZB-) like motion arising from the nonorthogonality of eigenstates. The observed ZB-like motion of a wave packet centered around a single eigenstate contradicts the notion that ZB motion only occurs for a wave packet centered around a superposed state. We further show the existence of a ZB-like motion in the form of an interesting helical trajectory of the wave packet on the incident plane, stemming from the imaginary part of group velocities. These exotic ZB-like motions may provide great insight into nonorthogonal eigenstates and complex eigenfrequencies of non-Hermitian systems.
In the Dirac theory for the motion of free relativistic electrons, highly oscillatory components appear in the time evolution of physical observables such as position, velocity, and spin angular momentum. This effect is known as zitterbewegung. We present a theoretical analysis of rather different Hamiltonians with gapped and/or spin-split energy spectrum (including the Rashba, Luttinger, and Kane Hamiltonians) that exhibit analogs of zitterbewegung as a common feature. We find that the amplitude of oscillations of the Heisenberg velocity operator v(t)\mathbf{v}(t)v(t) generally equals the uncertainty for a simultaneous measurement of two linearly independent components of v\mathbf{v}v. It is also shown that many features of zitterbewegung are shared by the simple and well-known Landau Hamiltonian, describing the dynamics of two-dimensional (2D) electron systems in the presence of a magnetic field perpendicular to the plane. Finally, we also discuss the oscillatory dynamics of 2D electrons arising from the interplay of Rashba spin splitting and a perpendicular magnetic field.
Bloch oscillation (BO), i.e., the oscillatory motion of a quantum particle in a periodic potential, is one of the most striking effects of coherent quantum transport in 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 real space is described by a trajectory in the 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.
Non-Hermitian physics has emerged as a rapidly advancing field of research, revealing a range of novel phenomena and potential applications. Traditional non-Hermitian Hamiltonians are typically simulated by constructing asymmetric couplings or by introducing dissipation and gain to realize non-Hermitian systems. The quadratic bosonic system (QBS) with squeezing interaction is intrinsically Hermitian; however, its dynamical evolution matrix in both real and momentum spaces is non-Hermitian. Based on this, applying a field-operator transformation xp to the dynamical evolution matrix yields quadrature nonreciprocal transmission between the x and p operators. This nonreciprocal characteristic can be utilized in signal amplifiers. On the other hand, within the Bogoliubov-de Gennes framework in momentum space, one can observe non-Hermitian topological phenomena such as point-gap topology and the non-Hermitian skin effect, both induced by spectra with nonzero winding numbers. Additionally, QBS can be employed to realize non-Hermitian Aharonov-Bohm cages and to extend non-Bloch band theory. Previous studies in non-Hermitian physics have largely concentrated on classical systems. The influence of non-Hermitian properties on quantum effects remains a key issue awaiting exploration and has evolved into a research direction at the interface of non-Hermitian and quantum physics.
Non-Hermitian systems can exhibit unique quantum phases without any Hermitian counterparts. For example, the latest theoretical studies predict a new surprising phenomenon that bulk bands can localize and dissipate prominently at the system boundary, which is dubbed the non-Hermitian edge burst effect. Here we realize a one-dimensional non-Hermitian Su-Schrieffer-Heeger lattice with bulk translation symmetry implemented with a photonic quantum walk. Employing time-resolved single-photon detection to characterize the chiral motion and boundary localization of bulk bands, we determine experimentally that the dynamics underlying the non-Hermitian edge burst effect is due to the interplay of non-Hermitian skin effect and imaginary band gap closing. This new non-Hermitian physical effect deepens our understanding of quantum dynamics in open quantum systems.