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Physical review. B./Physical review. B

Exotic Zitterbewegung-like motion of wave packets near exceptional points

Weimin Ye, Jianlong Liu, Yang Gao, Shuang Zhang

Published Aug 19, 2022• 3 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.106.l060303

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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.
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.
The appearance of topological singularities, namely exceptional points (EPs) is an intriguing feature of parameter-dependent open quantum or wave systems. EPs are the special type of nonHermitian degeneracies where two (or more) eigenstates of the underlying system coalesce. In this paper, we present a three level non-Hermitian Hamiltonian which hosts three interacting eigenstates. The matrix elements are optimized in such a way that the intermediate eigenstate interacts with both the other states and the underlying system hosts at least two different second order EPs. The impact of quasi-static parameter variation along a cyclic contour around the embedded EPs on the dynamics of interacting eigenvalues is well investigated in the context of cascaded state conversion. Such dynamics of the eigenvalues shows a clear signature of the third order EP with a combined effect of both the second order EPs. Moreover, we examine the accumulation of phases around the identified EPs and study the hallmark of phase exchange during cascaded state conversions accompanied by the parametric encirclement of the third order EP.
Eigenstates of a non-Hermitian system exist on complex Riemannian manifolds, with multiple sheets connecting at branch cuts and exceptional points (EPs). These eigenstates can evolve across different sheets-a process that naturally corresponds to state permutation. Here, we report the first experimental realization of non-Abelian permutations in a three-state non-Hermitian system. Our approach relies on the stroboscopic encircling of two different exceptional arcs (EAs), which are smooth trajectories of order-2 EPs appearing from the coalescence of two adjacent states. The non-Abelian characteristics are confirmed by encircling the EAs in opposite sequences. A total of five non-trivial permutations are experimentally realized, which together comprise a non-Abelian group. Our approach provides a reliable way of investigating non-Abelian state permutations and the related exotic winding effects in non-Hermitian systems.
Non-Hermitian systems exhibit a variety of unique features rooted in the presence of exceptional points (EPs). The distinct topological structure in the proximity of an EP gives rise to counterintuitive behaviors absent in Hermitian systems, which emerge after encircling the EP either quasistatically or dynamically. However, experimental exploration of EP encirclement in quantum systems, particularly those involving high-order EPs, remains challenging due to the difficulty of coherently controlling more degrees of freedom. In this work, we experimentally investigate the eigenvalue braiding and state transfer arising from the encirclement of EP in a three-dimensional non-Hermitian quantum system using superconducting circuits. We characterize the second- and third-order EPs through the coalescence of eigenvalues. Then we reveal the topological structure near the EP3 by quasistatically encircling it along various paths with three independent parameters, which yields the eigenvalue braiding described by the braid group B 3 . Additionally, we observe chiral state transfer between three eigenstates under a fast driving scheme when no EPs are enclosed, while time-symmetric behavior occurs when at least one EP is encircled. Our findings offer insights into understanding non-Hermitian topological structures and the manipulation of quantum states through dynamic operations.
Non-Hermitian (NH) systems have attracted great attention due to their exotic phenomena beyond Hermitian domains. Here we study the wave-packet dynamics in general one-dimensional NH lattices and uncover several unexpected phenomena. The group velocity of a wave packet during the time evolution in such NH lattices is not only governed by the real part of the band structure but also by its imaginary part. The momentum also evolves due to the imaginary part of the band structure, which can lead to a self-induced Bloch oscillation in the absence of external fields. Furthermore, we discover the wave-packet dynamics can exhibit disorder-free NH jumps even when the energy spectra are entirely real. Finally, we show that the NH jumps can lead to both positive and negative temporal Goos--Hänchen shifts at the edge.