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Annalen der Physik

Spin depolarization of spreading and propagating wave packets

H. M. Krenzlin, K. W. Kehr

Published Jan 1, 1997• 1 citation
Publisher

Abstract

Abstract Spin relaxation of spin‐1/2 quantum particles on linear chains with random Larmor frequencies is studied. Wave packets are prepared with initial spin polarization in the propagation direction and with either zero, or finite group velocity. The discrete Schrödinger‐Pauli equation is solved numerically. Exponential spin relaxation is observed in a restricted time range. The decay constant depends on the width of the distribution of the Larmor frequencies and on the group velocity. No dependence on the initial halfwidth of the wave packet is found.

DOI
10.1002/andp.19975090405

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Abstract Spin relaxation of spin‐1/2 quantum particles on linear chains with random Larmor frequencies is studied. Wave packets are prepared with initial spin polarization in the propagation direction and with either zero, or finite group velocity. The discrete Schrödinger‐Pauli equation is solved numerically. Exponential spin relaxation is observed in a restricted time range. The decay constant depends on the width of the distribution of the Larmor frequencies and on the group velocity. No dependence on the initial halfwidth of the wave packet is found.
The usual textbook discussions of the velocity and spreading of a wave packet are based on special cases and approximations which have a limited range of validity. In this paper more general relations are obtained by considering the time dependence of the moments of a wave packet. Both a packet of real sinusoidal plane waves (a real pulse, e.g., sound) and a packet of complex exponential plane waves (e.g., the quantum-mechanical wave function of a free particle) are considered. The results are similar if the real wave packet contains no standing wave components. It is shown that the centroid of a wave packet moves at a constant velocity which is a weighted average of the group velocity. The spreading of the wave packet depends on the product of the variance of the group velocity and the square of the time interval before or after the time of minimum packet width. The packet moves and spreads like a collection of particles in free flight, in which the velocities of the particles are the group velocities of the wave components of the packet.
We study the dynamics of a particle in continuous time and space, the displacement of which is governed by an internal degree of freedom (spin). In one definite limit, the so-called quantum random walk is recovered but, although quite simple, the model possesses a rich variety of dynamics and goes far beyond this problem. Generally speaking, our framework can describe the motion of an electron in a magnetic sea near the Fermi level when linearization of the dispersion law is possible, coupled to a transverse magnetic field. Quite unexpected behaviours are obtained. In particular, we find that when the initial wave packet is fully localized in space, the J z angular momentum component is frozen; this is an interesting example of an observable which, although it is not a constant of motion, has a constant expectation value. For a non-completely localized wave packet, the effect still occurs although less pronounced, and the spin keeps for ever memory of its initial state. Generally speaking, as time goes on, the spatial density profile looks rather complex, as a consequence of the competition between drift and precession, and displays various shapes according to the ratio between the Larmor period and the characteristic time of flight. The density profile gradually changes from a multimodal quickly moving distribution when the scattering rate is small, to a unimodal standing but flattening distribution in the opposite case.
In the Majorana equation for particles with arbitrary spin, the wave packet is due not only to the uncertainty affecting position and momentum but also to the infinite components with decreasing mass forming the Majorana spinor. In this paper we prove that such components contribute to increase the spreading of the wave packet. Moreover, as occurs in the time propagation of the Dirac wave packet, also in that of Majorana the Zitterbewegung takes place, but it shows a peculiar fine structure. Finally, the group velocity always remains subluminal and the contributions due to the infinite components decrease progressively increasing the spin.
We study the dynamics of a particle in continuous time and space, the displacement of which is governed by an internal degree of freedom (spin). In one definite limit, the so-called quantum random walk is recovered but, although quite simple, the model possesses a rich variety of dynamics and goes far beyond this problem. Generally speaking, our framework can describe the motion of an electron in a magnetic sea near the Fermi level when linearisation of the dispersion law is possible, coupled to a transverse magnetic field. Quite unexpected behaviours are obtained. In particular, we find that when the initial wave packet is fully localized in space, the JzJ_{z}Jz​ angular momentum component is frozen; this is an interesting example of an observable which, although it is not a constant of motion, has a constant expectation value. For a non-completely localized wave packet, the effect still occurs although less pronounced, and the spin keeps for ever memory of its initial state. Generally speaking, as time goes on, the spatial density profile looks rather complex, as a consequence of the competition between drift and precession, and displays various shapes according to the ratio between the Larmor period and the characteristic time of flight. The density profile gradually changes from a multimodal quickly moving distribution when the scatttering rate is small, to a unimodal standing but flattening distribution in the opposite cas case.