## Abstract

We present classical and quantum studies of the scattering dynamics of Rydberg electron wave packets from the electronic core of alkali atoms. In quantum systems an ideal state for studying such effects is an angularly localized wave packet in which the primary effect of the scattering is to cause precession. The scattering is enhanced by the application of an external dc electric field. We calculate and animate the field-induced dynamics of both hydrogenic and alkali wave packets and compare them to classical atomic models. We find that in alkali systems the scattered wavefunction can be divided into two components: one whose nearly hydrogenic behavior is due to quantum interference near the core, and another which exhibits the orbital precession found in classical models of nonhydrogenic atoms.

© Optical Society of America

## 1. Introduction

Although wave packet studies in Rydberg atoms were initially motivated by a desire to
explore the classical limit of the hydrogen atom^{1}, recent work has shown that such wave packets also provide an excellent arena for
examining non-hydrogenic effects in multi-electron atoms^{2,3}. In atoms with two or more valence electrons, the possibility of exciting
multi-electron wave packets is very intriguing and represents a considerable challenge
both theoretically and experimentally. But even in atoms with a single valence electron
there are interesting multi-electron effects due to the presence of the non-Coulombic
core. In this paper we will show that the behavior of wave packet states in such atoms
can reveal the classical dynamics underlying scattering from complex cores.

Recent treatments of core scattering in alkali atoms have used a semiclassical analysis
of the scaled energy levels to model nonhydrogenic effects in applied fields^{4–6}. The aim of these studies was to gain a better understanding of non-integrable
quantum systems whose analogs are classically chaotic. The standard semiclassical
techniques rely on the summation of contributions from hydrogenic closed orbits and
include both Coulomb-scattered and core-scattered orbits. These calculations have
provided excellent agreement with the full quantum treatments for both the diamagnetic^{4,5} and Stark^{6} cases.

Our approach concentrates on the dynamics of the wavefunction instead of the spectrum.
By preparing the electron initially in an angularly localized wave packet^{7} or elliptic state^{1}, we can isolate the core scattering effects from the dynamics associated with radial^{8} or Stark wave packets^{2}. Elliptic states have a strong correspondence with classical theory^{9} and this allows us to compare the alkali atom results with the well-known
classical and quantum dynamics of the hydrogen atom. With only minor modifications of
existing experimental techniques^{2,7} it should be possible to very directly examine the dynamics of an individual
scattering event.

## 2. Stark dynamics of elliptic states in hydrogen

We begin with a brief review of the dynamics of a classical hydrogen atom in a dc
electric field^{10–12}. We will use atomic units throughout the paper. The interesting field-induced
dynamics can be separated from the Kepler motion by examining slowly varying orbital
parameters such as the angular momentum **L** = **r** ×
**p** and the Runge-Lenz vector **A** = **p** ×
**L** - **r**/*r*. Like the angular momentum, the
Runge-Lenz vector is a constant of the field-free motion. It lies antiparallel to the
major axis of the orbit and has a magnitude equal to the orbital eccentricity
*∊*.

The Hamiltonian for the system including the dc electric field **E** is

Choosing *V*(*r*) = -1/*r*, the hydrogen
Coulomb potential, we can form the equations of motion for **L** and
**A**, and average them over a Kepler period *T _{K}*. The remaining adiabatic evolution of the averaged classical parameters is
given by

$${\u3008\dot{\mathbf{L}}\u3009}_{{\mathit{T}}_{K}}=\frac{3{n}^{2}}{2}{\u3008\mathbf{A}\u3009}_{{T}_{K}}\times \mathbf{E}\left(t\right),$$

where *n* corresponds to the classical field-free energy
*ε* = -1/(2*n*
^{2}). Although the field does induce a classical Stark shift in the energy (for
a recent discussion see Ref. 12) we will focus on
the dynamics of the orbit.

Equations (2) are easily solved for a dc
electric field in the *xy*-plane of the classical orbit giving sinusoidal
solutions for *L _{z}*,

*A*and

_{x}*A*. This analytic result allows us to interpret the orbital deformation in the trajectory shown in Fig. 1a. The frequency of this oscillation is given by the Stark frequency ${\omega}_{s}=\frac{3}{2}\mathit{nE}$ and we show the corresponding oscillation of the angular momentum in Fig. 2.

_{y}Quantum mechanically we can see identical behavior in the dynamics of an elliptic state
in a dc electric field. The analysis of the classical electron interacting with the
field did not depend on the exact initial position and momentum, but rather on the
initial orbital parameters. The time averages in Eqs. (2) can be replaced by ensemble averages and the results can now be
interpreted in terms of an ensemble of electrons rather than in terms of one isolated
electron. A quantum-mechanical analog of such a classical ensemble traveling in an
elliptic orbit is an elliptic state^{9}. Formally an elliptic state is a coherent state of the rotation group in three
dimensions, SO(3), whose generators are two components of the scaled Runge-Lenz
operator, *nÂ _{x}* and

*nÂ*, and a component of the angular momentum,

_{y}*L̂*. These states can be well-described classically and, although they represent stationary states, the spatial distributions of elliptic state wavefunctions closely resemble the corresponding classical ellipses. In this paper the dynamics of the wavefunction will result from an external electric field. To avoid confusion we point out that there are

_{z}*superpositions*of elliptic states from neighboring

*n*manifolds which produce dynamic field-free wave packets known as elliptic orbit wave packets. For the purposes of this paper, we feel that the use of a single elliptic state better illustrates the effect of core scattering.

The circular states |*n*,*n* - 1,
±(*n* - 1)) are the only states in a given *n*
manifold which are both elliptic states and field-free eigenstates. The animation in
Fig. 3a shows the time evolution of a
circular state of hydrogen in a weak dc electric field (manifolds of different n are not
mixed). The calculation was performed by integrating SchrÖdinger’s
equation in the presence of a dc electric field using hydrogenic dipole moments. The
three dimensional probability distribution is projected onto the xy plane. The
parameters were chosen to match the classical orbit in Fig. 1. The resulting deformation of the wavefunction is identical to the
oscillations of the classical ellipse in Fig. 1a
and it can be shown that the wavefunction remains in an elliptic state throughout its
evolution. The dynamics of the expectation value of the angular momentum is sinusoidal
as shown by the dashed curve in Fig. 4a

## 3. Stark dynamics of elliptic states in alkali atoms

When non-Coulombic effects are introduced into the potential
*V*(*r*) the classical Stark dynamics is quite
different as shown in Fig 1b. We use a model potential^{16} for sodium which includes long range polarizability terms as well as short range
core effects

where *α _{d}* is the dipole core polarizability,

*α′*is the dynamical quadrupole polarizability of the core and the parameters

_{q}*a*,

*b*,

*c*, and

*r*are chosen to match sodium spectroscopic data

_{c}^{16}.

It is well-known^{13} that additions to the Coulomb core *V*(*r*) = -
1/*r* + *ϕ*(*r*) lead to a
precession of the Runge-Lenz vector and hence the major axis of the orbital ellipse

This precession of the orbit is analogous to the precession caused by perturbations in
celestial potentials such as the gravitational correction that leads to the advance of
the perihelion of Mercury^{14,15}. Under field-free conditions the precession rate of the atomic orbit is constant
and for nearly circular orbits it is negligible. However, for highly eccentric orbits,
the electron closely approaches the core at its inner turning point causing the orbit to
precess. In Fig. 1b we show the effect of this
precession on the Stark dynamics^{15}. Initially, the behavior is nearly hydrogenic and the orbital parameters begin to
vary sinusoidally. However, as the eccentricity increases, the core effects become
prominent causing a rapid precession of the orbit. As the highly eccentric orbit
precesses its interaction with the field changes and when the orbit has rotated by an
angle of nearly 180° the field now causes the magnitude of the angular momentum
to increase and the precessional rate becomes negligible once again. This precessional
mechanism prevents the angular momentum from changing sign as it does in hydrogen and
from the solid curve in Fig. 2 we see that this
effect places a lower limit on the angular momentum.

For a field of 800 V/cm the rapid precession occurred in only a few Kepler periods but its duration is inversely proportional to the field strength. In weaker fields the period of rapid precession may last many Kepler periods and the electron undergoes many near-collisions with the core. As we will see in the following quantum mechanical simulations, these multiple scattering events in weak fields produce a larger scattered component in the wavefunction.

Although the long-range polarizability terms in Eqn. 3 do produce nonhydro-genic effects, we find that they are not
essential for a qualitative description of the classical dynamics of the electron^{17}. It is the short range exponential term that provides the dominant
core-scattering effect that we see in Fig.
1b.

To examine core effects quantum mechanically we integrate Schrodinger’s equation
in a weak dc electric field using sodium dipole moments^{18} for the *n* = 15 manifold. We also include the
16*s* and 16*p* states whose large quantum defects
place them within the *n* = 15 manifold.

The animation in Fig. 3b shows the results of this calculation for an initial circular state. The behavior is initially hydrogenic but as the eccentricity increases we see evidence of the underlying classical precession. However, part of the wavefunction appears to continue to behave hydrogenically and after half of a Stark period these two parts of the wavefunction interfere in counter-propagating circular states. During the second collision with the core, the non-hydrogenic part of the wavefunction is rescattered and after one full Stark period the total wavefunction is almost entirely in the original circular state. The time dependence of the expectation value of the angular momentum (see Fig. 4a) behaves like neither hydrogen or classical sodium.

## 4. Stark dynamics of core-scattered wave packets

In order to understand the physics underlying this effect we will divide the
wavefunction at *t* = 0.5*T _{s}* into two wave packets: one containing only states for which

*m*< 0 (this part of the wave function did not scatter from the core) and the other containing the remaining

*m*≥ 0 states (this represents the scattered portion of the total wavefunction). Experimentally this could be accomplished through the use of selective field ionization of the wave packet at

*t*= 0.5

*T*.

_{s}Before renormalization, the total population in the *m* < 0 states was
approximately 48% with the remaining 52% scattered into the *m* ≥
0 states. The ratio between the scattered and unscattered parts of the wave function
varies as a function of the strength of the electric field. The dependence is
approximately logarithmic on the Stark period as shown in Fig 5.

The reason for such a division of the wavefunction is revealed in the animations shown in Fig. 6. The behavior of the wave packets is strikingly different in the two cases.

The *m* < 0 wave packet in Fig.
6b behaves nearly hydrogenically except near the core where, unlike hydrogen,
there is never appreciable population. Because of this interference near the core the
*m* < 0 wave packet experiences much reduced core scattering
resulting in hydrogenic behavior. The wave packet consisting of *m*
≥ 0 states displays the rapid precession characteristic of the classical sodium
model and is almost entirely scattered from the core. As in the classical model, this
scattering process lasts only a few Kepler periods as the wavefunction precesses by
approximately 180°.

In Fig. 4b we show the evolution of the angular momenta of the two wave packets discussed above. The difference in the scattering behavior is clearly evident and shows all of the features of the classical behavior shown in Fig. 2. The the finite extent of the quantum wave function smoothes the minima of the angular momentum of the scattered wave packet. We also see the shortening of the Stark period in the scattered case.

## 5. Summary

We have presented a classical perspective of core-scattering in alkali atoms by using the Stark dynamics of angularly localized wave packets to visualize core effects. Although the scattering of these wave packets has a classical interpretation there is not a direct correspondence between the classical and quantum theories as there is in hydrogen. The phase shifts associated with the quantum defects of the low angular momentum alkali states produce interference which leads to behavior containing features of both classical sodium and classical hydrogen. The use of angularly-localized wave packets reveals how quantum phase affects the dynamics underlying scattering and allows us to understand the behavior suggested by the studies of semiclassical scaled energy spectra.

## 6. Acknowledgments

We would like to acknowledge Z. D. Gaeta and J. Bromage for useful discussions, and the Army Research Office for support of this research.

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