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arXiv:quant-ph/0404084 · Open Systems & Information Dynamics

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

M. Turek, P. Rozmej

Published Dec 1, 2004• 0 citations
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Abstract

Time evolution of radial wave packets built from the eigenstates of Dirac equation for a hydrogenic system is considered. Radial wave packets are constructed from the states of different n quantum numbers and the same lowest angular momentum. In general they exhibit a kind of breathing motion with dispersion and (partial) revivals. Calculations show that for some particular preparations of the wave packet one can observe interesting effects in spin motion, coming from inherent entanglement of spin and orbital degrees of freedom. These effects manifest themselves through some oscillations in the mean values of spin operators and through changes of spatial probability density carried by upper and lower components of the wave function. It is also shown that the characteristic time scale of predicted effects (called T ls ) is much smaller for radial wave packets than in other cases, reaching values comparable to (or even less than) the time scale for the wave packet revival.

DOI
10.1007/s11080-004-6632-4

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Time evolution of radial wave packets built from the eigenstates of Dirac equation for a hydrogenic system is considered. Radial wave packets are constructed from the states of different n quantum numbers and the same lowest angular momentum. In general they exhibit a kind of breathing motion with dispersion and (partial) revivals. Calculations show that for some particular preparations of the wave packet one can observe interesting effects in spin motion, coming from inherent entanglement of spin and orbital degrees of freedom. These effects manifest themselves through some oscillations in the mean values of spin operators and through changes of spatial probability density carried by upper and lower components of the wave function. It is also shown that the characteristic time scale of predicted effects (called T ls ) is much smaller for radial wave packets than in other cases, reaching values comparable to (or even less than) the time scale for the wave packet revival.
The time evolution of wave packets (WPs) 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 almost 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 also show that the motion is universal for all hydrogenic systems with an arbitrary value of the atomic number Z .
The motion of circular wave packets for one electron in a central Coulomb field with a high atomic number Z is calculated. The wave packet is defined in terms of solutions of the Dirac equation in order to take into account all possible relevant effects, in particular the spin-orbit potential. A time scale is defined within which spin dynamics must be taken into account, mainly in atoms with high Z. Within this time scale there exists a mechanism of collapses and revivals of the spin, already shown by the authors for harmonic oscillator potential and called the spin-orbit pendulum. Though this effect does not have the exact periodicity of the simpler model, the wave packet spatial motion is nevertheless quite similar.
We study the dynamics of a circular-orbit wave packet1 and find that it has both classical and quantum features. A circular-orbit wave packet consists of aligned, standard hydrogenic eigenstates of maximum angular momentum (l = in = n − 1), superimposed with a Gaussian weighting function. We calculate the uncertainty products in all three degrees of freedom and find that, as the quantum numbers of the constituent eigenstates increase, the uncertainty products tend toward h/2. The classical aspect of the dynamics displays itself during revivals, when a well-localized wave packet travels along a Kepler orbit, obeying the classical equations of motion. Spreading and revivals (including fractional revivals2) of the wave packet, which are manifestations of its quantum nature, are related to the nonlinearity of the hydrogen atom Hamiltonian. We study fractional revivals of the wavepacket in detail.
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.
The revival structure and evolution of Rydberg wave packets are studied on a time scale much greater than the revival time trevt_{\rm rev}trev​. We find a new level of revival structure and periodic motion different from that of the known fractional revivals. The new sequence of revivals culminates with the formation of a wave packet that more closely resembles the initial packet than does the full revival at time trevt_{\rm rev}trev​. We refer to such a revival as a superrevival. We also show that an initial radial wave packet may be described as a type of squeezed state known as a radial squeezed state. Our results apply not only for hydrogenic wave packets, but for wave packets in alkali-metal atoms as well in the context of quantum defect theory.
We have calculated the time evolution of the density matrix reduced to the spin degree of freedom for a particle in a spherical harmonic oscillator with a constant spin-orbit interaction. We have studied two classes of spatial wave functions in the initial state: pure orbital states \ensuremath{\Vert}lm〉 and coherent states. When the initial spin is a pure state, its evolution in time produces a mixed state during a time called the collapse time. This behavior of the spin is similar to that observed in the Jaynes-Cummings model. A similar dynamics is produced for a broad range of geometrical initial conditions for the spin as well as for the wave packet. The system is exactly recurrent but we have also found the existence of approximate recurrences corresponding to a reversal of the spin direction. We have tentatively applied our result to mimic the evolution of the spin in the H atom. In this case only a few spin-orbit partners play a role and, after a long time, produce a state where the admixture is not complete. The average quantum behavior is always very different from that of a classical particle having a spin as a vector of constant length. The increase of the number 2S+1 of spin-orbit partners changes the time scale of the collapse by the factor \ensuremath{\surd}S , and a strong mixing is always observed.