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arXiv:1105.2884 · Europhysics Letters (EPL)

Converting Zitterbewegung oscillation to directed motion

Q. Zhang, Jiangbin Gong, C. H. Oh

Published Sep 15, 2011• 5 citations
arXivHTMLPDF

Abstract

Zitterbewegung (ZB) oscillation, namely, the jittering center-of-mass motion predicted by free-space Dirac (or Dirac-like) equations, has been studied in several different contexts. It is shown here that ZB can be converted to directed center-of-mass motion by a modulation of the Dirac-like equation, if the modulation is on resonance with the ZB frequency. Tailored modulation may also stop, re-launch or even reverse the directed motion of a wavepacket with negligible distortion. The predictions may be examined by current ZB experiments using trapped-ion systems.

DOI
10.1209/0295-5075/96/10004

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Zitterbewegung (ZB) oscillation, namely, the jittering center-of-mass motion predicted by free-space Dirac (or Dirac-like) equations, has been studied in several different contexts. It is shown here that ZB can be converted to directed center-of-mass motion by a modulation of the Dirac-like equation, if the modulation is on resonance with the ZB frequency. Tailored modulation may also stop, re-launch or even reverse the directed motion of a wavepacket with negligible distortion. The predictions may be examined by current ZB experiments using trapped-ion systems.
Zitterbewegung (ZB, the trembling motion) of free relativistic electrons in a vacuum in the presence of an external magnetic field is calculated. It is shown that the motion of an electron wave packet has intraband frequency components, corresponding to the classical cyclotron motion, and several interband frequency components corresponding to the Zitterbewegung. For a two-dimensional situation, the presence of a magnetic field makes the ZB motion stationary, i.e. not decaying in time. We show how to simulate the ZB in a magnetic field using trapped ions and laser excitations in the spirit of recently observed proof-of-principle ZB simulation by Gerritsma {\it et al.} Nature {\bf 463}, 68 (2010). It is demonstrated that, for the parameters of the Dirac equation simulated by the above experiment, the effect of a magnetic field on the Zitterbewegung is considerable.
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.
We seek the {\em immediate} description of chiral oscillations in terms of the trembling motion described by the velocity (Dirac) operator {\boldmathααα}. By taking into account the complete set of Dirac equation solutions which results in a free propagating Dirac wave packet composed by positive and negative frequency components, we report about the well-established {\em zitterbewegung} results and indicate how chiral oscillations can be expressed in terms of the well know quantum oscillating variables. We conclude with the interpretation of chiral oscillations as position very rapid oscillation projections onto the longitudinally decomposed direction of the motion.
Pulse walk-off in the process of sum frequency generation in a nonlinear χ (2) crystal is shown to be responsible for pulse jittering which is reminiscent to the Zitterbewegung (trembling motion) of a relativistic freely moving Dirac particle. An analytical expression for the pulse centre of mass trajectory is derived in the no-pump-depletion limit, and the numerical examples of Zitterbewegung are presented for sum frequency generation in periodically poled lithium niobate. The proposed quantum-optical analogy indicates that frequency conversion in nonlinear optics could provide an experimentally accessible simulator of the Dirac equation.
Zitterbewegung (ZB), the trembling of free relativistic electrons in a vacuum, could be simulated by a single trapped ion. We focus on the variations of ZB under different parity conditions and find no ZB in the case of odd or even parity. ZB occurs only for the admixture of the odd- and even-parity states. We also show that a similar role is played by the parity operator for the trapped ion in Fock-state representation and the space-inversion operator for a realistic relativistic electron. Although the ZB effect is invisible in a relativistic electron, preparation of the trapped ion in different parity states is a sophisticated job, which makes it possible to observe the parity-relevant ZB effects with currently available techniques.
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.