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arXiv:1711.05053 · Journal of Physics A Mathematical and Theoretical

Zitterbewegung and symmetry switching in Klein’s four-group

L. Chotorlishvili, P Zięba, Igor E. Tralle, A Ugulava

Published Nov 13, 2017• 5 citations
arXivHTMLPDF

Abstract

Abstract Zitterbewegung is the exotic phenomenon associated either with relativistic electron–positron rapid oscillation or to electron–hole transitions in narrow gap semiconductors. In the present work, we enlarge the concept of Zitterbewegung and show that trembling motion may occur due to dramatic changes in the symmetry of the system. In particular, we exploit a paradigmatic model of quantum chaos, the quantum mathematical pendulum (universal Hamiltonian). The symmetry group of this system is Klein’s four-group that possesses three invariant subgroups. The energy spectrum of the system parametrically depends on the height of the potential barrier, and contains degenerate and non-degenerate areas, corresponding to the different symmetry subgroups. Change in the height of the potential barrier switches the symmetry subgroup and leads to trembling motion. We analyzed mean square fluctuations of the velocity operator and observed that trembling is enhanced in highly excited states. We observed a link between the phenomena of trembling motion and the uncertainty relations of noncommutative operators of the system.

DOI
10.1088/1751-8121/aa9a2c

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Abstract Zitterbewegung is the exotic phenomenon associated either with relativistic electron–positron rapid oscillation or to electron–hole transitions in narrow gap semiconductors. In the present work, we enlarge the concept of Zitterbewegung and show that trembling motion may occur due to dramatic changes in the symmetry of the system. In particular, we exploit a paradigmatic model of quantum chaos, the quantum mathematical pendulum (universal Hamiltonian). The symmetry group of this system is Klein’s four-group that possesses three invariant subgroups. The energy spectrum of the system parametrically depends on the height of the potential barrier, and contains degenerate and non-degenerate areas, corresponding to the different symmetry subgroups. Change in the height of the potential barrier switches the symmetry subgroup and leads to trembling motion. We analyzed mean square fluctuations of the velocity operator and observed that trembling is enhanced in highly excited states. We observed a link between the phenomena of trembling motion and the uncertainty relations of noncommutative operators of the system.
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.
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.
The Klein-Gordon equation is used to calculate the Zitterbewegung (ZB, trembling motion) of spin-zero particles in the absence of fields and in the presence of an external magnetic field. Both Hamiltonian and wave formalisms are employed to describe ZB and their results are compared. It is demonstrated that if one uses wave packets to represent particles, then the ZB motion has a decaying behavior. It is also shown that the trembling motion is caused by an interference of two subpackets composed of positive- and negative-energy states, which propagate with different velocities. In the presence of a magnetic field, the quantization of the energy spectrum results in many interband frequencies contributing to ZB oscillations and the motion follows a collapse-revival pattern. In the limit of nonrelativistic velocities, the interband ZB components vanish and the motion is reduced to cyclotron oscillations. The exact dynamics of a charged Klein-Gordon (KG) particle in the presence of a magnetic field is described on an operator level. The trembling motion of a KG particle in the absence of fields is simulated using a classical model proposed by Morse and Feshbach---it is shown that a variance of a Gaussian wave packet exhibits ZB oscillations.
The Mathieu-Schrödinger equation, which describes the behavior of a quantum pendulum, depending on the value of the parameter l (pendulum filament length), can have the symmetry of the Klein's four-group or its invariant subgroups. The paper shows that the mean values of z-components of the angular momentum of nondegenerate quantum states (the symmetry region of the four-group) tend to zero and their root mean square fluctuations are non-zero. Consequently, in this region of parameter values, the fluctuations overlap the mean values of the angular momentum and they become indistinguishable. Therefore, it can be argued that if, with an increase in the parameter, the system goes into a non-degenerate state, then after the inversion of the parameter change and the transition to the region of degenerate states, the initial states will not be restored. This behavior of the average values of angular momenta is caused by the combined actions of two factors: discontinuous change in the system at the points of change of its symmetry and the presence of quantum fluctuations in nondegenerate states.
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, a force-free trembling motion first predicted for relativistic fermions like electrons, was an unexpected consequence of the Dirac equation's unification of quantum mechanics and special relativity. Though the oscillatory motion's large frequency and small amplitude have precluded its measurement with electrons, zitterbewegung is observable via quantum simulation. We engineered an environment for 87 Rb Bose–Einstein condensates where the constituent atoms behaved like relativistic particles subject to the one-dimensional Dirac equation. With direct imaging, we observed the sub-micrometre trembling motion of these clouds, demonstrating the utility of neutral ultracold quantum gases for simulating Dirac particles.