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Science in China. Series G, Physics, mechanics & astronomy/Science in China. Series G, Physics, Mechanics & Astronomy

Quantum simulation of ‘zitterbewegung' in a single trapped ion under conditions of parity-keeping and parity-breaking

Liu Ta

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

The motional trembling(‘zitterbewegung’)of a relativistic electron governed by Dirac equation was originally predicted by Schr¨odinger in the early days of quantum mechanics and simulated in a recent experiment with a single trapped ultracold ion.We investigate stable and instable confinements of a single trapped ion in a Paul trap under different conditions relevant to parity.Since our treatment involves neither restriction of Lamb-Dicke limit nor rotating-wave approximation,we may demonstrate different quantum dynamics of the single trapped ion in a wide range of the trapping parameters.We discuss the origin of the zitterbewegung which is relevant to the stability of the ion trapping.

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The motional trembling(‘zitterbewegung’)of a relativistic electron governed by Dirac equation was originally predicted by Schr¨odinger in the early days of quantum mechanics and simulated in a recent experiment with a single trapped ultracold ion.We investigate stable and instable confinements of a single trapped ion in a Paul trap under different conditions relevant to parity.Since our treatment involves neither restriction of Lamb-Dicke limit nor rotating-wave approximation,we may demonstrate different quantum dynamics of the single trapped ion in a wide range of the trapping parameters.We discuss the origin of the zitterbewegung which is relevant to the stability of the ion trapping.
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, 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 present a method of simulating the Dirac equation in 3+1 dimensions for a free spin-1/2 particle in a single trapped ion. The Dirac bispinor is represented by four ionic internal states, and position and momentum of the Dirac particle are associated with the respective ionic variables. We show also how to simulate the simplified 1+1 case, requiring the manipulation of only two internal levels and one motional degree of freedom. Moreover, we study relevant quantum-relativistic effects, like the Zitterbewegung and Klein's paradox, the transition from massless to massive fermions, and the relativistic and nonrelativistic limits, via the tuning of controllable experimental parameters.
One-electron 3+13+13+1 and 2+12+12+1 Dirac equations are used to calculate the motion of a relativistic electron in a vacuum in the presence of an external magnetic field. First, calculations are carried on an operator level and exact analytical results are obtained for the electron trajectories which contain both intraband frequency components, identified as the cyclotron motion, as well as interband frequency components, identified as the trembling motion, Zitterbewegung (ZB). Next, time-dependent Heisenberg operators are used for the same problem to compute average values of electron position and velocity employing Gaussian wave packets. It is shown that the presence of a magnetic field and the resulting quantization of the energy spectrum has pronounced effects on the electron ZB: it introduces intraband frequency components into the motion, influences all the frequencies, and makes the motion stationary (not decaying in time) in case of the 2+12+12+1 Dirac equation. Finally, simulations of the 2+12+12+1 Dirac equation and the resulting electron ZB in the presence of a magnetic field are proposed and described employing trapped ions and laser excitations. Using simulation parameters achieved in recent experiments of Gerritsma and coworkers, we show that the effects of the simulated magnetic field on ZB are considerable and can certainly be observed.
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.
Ultracold atoms which are subject to ultra-relativistic dynamics are investigated. By using optically induced gauge potentials we show that the dynamics of the atoms is governed by a Dirac-type equation. To illustrate this we study the trembling motion of the centre of mass for an effective two-level system, historically called Zitterbewegung. Its origin is described in detail, where in particular the role of the finite width of the atomic wave packets is seen to induce a damping of both the centre of mass dynamics and the dynamics of the populations of the two levels.