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arXiv:1107.3983 · Physical Review A

Long-term cyclotron dynamics of relativistic wave packets: Spontaneous collapse and revival

V. Ya. Demikhovskiǐ, Galina M. Maksimova, А. А. Перов, A. V. Telezhnikov

Published Feb 3, 2012• 12 citations
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Abstract

In this work we study the effects of collapse and revival, as well as the zitterbewegung (ZB) phenomenon, for the relativistic electron wave packets, which are a superposition of the states with quantum numbers sharply peaked around some Landau level ${n}_{0}$ of the order of few tens. The probability densities as well as average velocities of the packet center and the average spin components were calculated analytically and their evolution is visualized. Our computations demonstrate that due to the dephasing of the states for times larger than the cyclotron period the initial wave packet (which includes the states with the positive energy only) loses the spatial localization so that the evolution can no longer be described classically. However, at the half-revival time $t={T}_{R}/2$ its reshaping takes place first. It is shown that the behavior of the wave packet containing the states of both energy bands (with ${E}_{n}>0$ and ${E}_{n}<0$) is more complicated. At short times of a few classical periods such a packet splits into two parts, which rotate with cyclotron frequency in the opposite directions and meet each other every one-half of the cyclotron period. At these moments their wave functions have a significant overlap that leads to ZB, and at the time of fractional revival each of two subpackets is decomposed into few packet fractions. However, at $t={T}_{R}$ each of the two subpackets (with positive or negative energy) restores at various points of the cyclotron orbit, which makes reshaping of the initial wave packet impossible, entirely unlike the wave packet which consists of states with energies ${E}_{n}>0$ only. The obtained results can be useful for the description of electromagnetic radiation and absorption in relativistic plasma on astrophysics objects, where superhigh magnetic field has a value of the order ${10}^{8}--{10}^{9}$T, as well as for interpretation of experiments with trapped ions.

DOI
10.1103/physreva.85.022105

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In this work we study the effects of collapse and revival, as well as the zitterbewegung (ZB) phenomenon, for the relativistic electron wave packets, which are a superposition of the states with quantum numbers sharply peaked around some Landau level n0{n}_{0}n0​ of the order of few tens. The probability densities as well as average velocities of the packet center and the average spin components were calculated analytically and their evolution is visualized. Our computations demonstrate that due to the dephasing of the states for times larger than the cyclotron period the initial wave packet (which includes the states with the positive energy only) loses the spatial localization so that the evolution can no longer be described classically. However, at the half-revival time t=TR/2t={T}_{R}/2t=TR​/2 its reshaping takes place first. It is shown that the behavior of the wave packet containing the states of both energy bands (with En>0{E}_{n}>0En​>0 and En<0{E}_{n}<0En​<0) is more complicated. At short times of a few classical periods such a packet splits into two parts, which rotate with cyclotron frequency in the opposite directions and meet each other every one-half of the cyclotron period. At these moments their wave functions have a significant overlap that leads to ZB, and at the time of fractional revival each of two subpackets is decomposed into few packet fractions. However, at t=TRt={T}_{R}t=TR​ each of the two subpackets (with positive or negative energy) restores at various points of the cyclotron orbit, which makes reshaping of the initial wave packet impossible, entirely unlike the wave packet which consists of states with energies En>0{E}_{n}>0En​>0 only. The obtained results can be useful for the description of electromagnetic radiation and absorption in relativistic plasma on astrophysics objects, where superhigh magnetic field has a value of the order 108−−109{10}^{8}--{10}^{9}108−−109T, as well as for interpretation of experiments with trapped ions.
The article discusses the properties of time evolution of wave packets in a few systems. Dynamics of wave packet motion for Rydberg atoms with the hierarchy of collapses and revivals is briefly reviewed. The main part of the paper focuses on the new mechanism of quantum reccurrences in wave packet dynamics. This mechanism can occur (in principle) in any physical system with strong enough spin-orbit interaction. We discuss here the SPIN_ORBIT PENDULUM effect that consists in different motions of subpackets possessing different spin fields and results in oscillations of a fraction of average angular momentum between spin and ordinary subspaces. The evolution of localized wave packet into toroidal objects and backwards (for other class of initial conditions) is also subject to discussion.
We report an observation of the long-term evolution of a radially localized electronic wave packet formed by the coherent superposition of Rydberg states of atomic potassium. Initially, the wave packet can be described classically. Subsequent dephasing of the discrete states in the superposition leads to a loss of spatial localization so that the evolution can no longer be described classically. However, the wave packet revives at a later time. Theory and experiment show good agreement including an accurate measurement of the phase shift of the wave packet on revival.
This paper begins with an examination of the revival structure and long-term evolution of Rydberg wave packets for hydrogen. We show that after the initial cycle of collapse and fractional or full revival, which occurs on the time scale trev{\mathit{t}}_{\mathrm{rev}}trev​, a new sequence of revivals begins. We find that the structure of the new revivals is different from that of the fractional revivals. The new revivals are characterized by periodicities in the motion of the wave packet with periods that are fractions of the revival time scale trev{\mathit{t}}_{\mathrm{rev}}trev​. These long-term periodicities result in the autocorrelation function at times greater than trev{\mathit{t}}_{\mathrm{rev}}trev​ having a self-similar resemblance to its structure for times less than trev{\mathit{t}}_{\mathrm{rev}}trev​. The new sequence of revivals culminates with the formation of a single wave packet that more closely resembles the initial wave packet than does the full revival at time trev{\mathit{t}}_{\mathrm{rev}}trev​, i.e., a superrevival forms. Explicit examples of the superrevival structure for both circular and radial wave packets are given. We then study wave packets in alkali-metal atoms, which are typically used in experiments. The behavior of these packets is affected by the presence of quantum defects that modify the hydrogenic revival time scales and periodicities. Their behavior can be treated analytically using supersymmetry-based quantum-defect theory. We illustrate our results for alkali-metal atoms with explicit examples of the revival structure for radial wave packets in rubidium.
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.
In this work we study the dynamics of free three-dimensional relativistic Gaussian wave packets with different spin polarizations. We analyze the connection between the symmetry of initial state and the dynamical characteristics of moving particle. The corresponding solutions of the Dirac equation having different types of symmetry were evaluated analytically and numerically and after that the electron probability densities, as well as the spin densities were visualized. The average values of velocity of the packet center and the average spin were calculated analytically, and the parameters of transient Zitterbewegung in different directions were obtained. These results can be useful for the interpretation of future experiments with trapped ions.
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.