Relativistic precession and spin dynamics of an elliptic Rydberg wave packet

Abstract

Time evolution of wave packets built from the eigenstates of the Dirac equation for a hydrogenic system is considered. We investigate the space and spin motion of wave packets which, in the non-relativistic limit, are stationary states with a probability density distributed uniformly along the classical, elliptical orbit (elliptic WP). We show that the precession of such a WP, due to relativistic corrections to the energy eigenvalues, is strongly correlated with the spin motion. We show also that the motion is universal for all hydrogenic systems with an arbitrary value of the atomic number Z.PACS numbers: 03.65.P, 03.65, 03.65.S The detailed study of the time evolution of quantum wave packets (WPs) in simple atomic or molecular systems has been the object of growing attention for more than ten years [1]. Most of the previous theoretical studies were done in non-relativistic framework. In the field of relativistic quantum mechanics most of the efforts have been focused on the problem of the interaction between the atoms and a mixture of static fields with, most of the time, intense laser fields [2][3][4][5][6][7][8][9][10][11]. Under these conditions the use of a relativistic theory is fully justified since the external field is then able to bring considerable energy to the WP. For isolated atoms, however, the use of relativistic dynamics is more questionable, if the WP is followed or observed only during a short period of time. In ref.[12] relativistic wave packets, corresponding to circular orbits, have been constructed for hydrogenic atoms with a large Z, and propagated over time according to the Dirac equation. Particular attention was paid to the spin collapse event, i.e. to the maximum entanglement, in the course of time, of the spin degree of freedom with the spatial ones. This phenomenon was first shown to take place for a WP in a harmonic oscillator with a spin-orbit force [13], where it is periodic. For this reason it has been called the spin-orbit pendulum. In the Dirac equation with a Coulomb ¶ Research supported by KBN grants No. 5 P03B 010 20 and 5 P03B 104 21

Spin-Orbit Entanglement in Time Evolution of Radial Wave Packets in Hydrogenic Systems

“…The main relativistic effect is the appearance of the new time scale due to the spin-orbit coupling. As shown above this time scale can be much smaller for the radial WP than for previously discussed cases of circular [12] or elliptic [13] WP. This fact implies that in principle experimental observations of some spin-orbit effects may become possible with existing techniques.…”

“…σ z t = n |w n | 2 a 2 − b 2 (2l + 1)(2l − 1) (2l + 1) 2 − b 2 8l (2l + 1) 2 cos (ω n t) , (13) where ω n = [(E + nl − E − nl )/ ]. The terms containing cos (ω n t) and sin (ω n t) indicate that in the beginning one can expect the spin precession followed by the spin collapse implied by nonlinear dependence of frequencies ω n on n. This behaviour is clearly seen in the Fig.…”

“…Such approach was used by us in [12], where we investigated circular wave packets and in [13], where we considered elliptic wave packets, both in hydrogenic systems. Results of [12,13,14] show that contributions given by small components of wave functions are negligibly small. Therefore for time scales of motion, relevant for our purposes, one can safely apply an approximation in which the time evolution is calculated with non-relativistic wave functions and relativistic energies.…”

“…The most natural framework for our considerations is using relativistic wave packets built from the solutions of Dirac equation for hydrogenic systems. Such approach was used by us in [12], where we investigated circular wave packets and in [13], where we considered elliptic wave packets, both in hydrogenic systems. Results of [12][13][14] show that contributions given by small components of wave functions are negligible small.…”