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 1 . M. Nauenburg , C. R. Stroud Jr. , and Yeazell J. , “ The classical limit of an atom
,” Sci. Am. 270 , 24 – 29 (
1994 ). }, recent work has shown that such wave packets also provide an excellent arena for
examining non-hydrogenic effects in multi-electron atoms^{2 2 . C. Raman , T. C. Weinacht , and P. H. Bucksbaum , “ Stark wave packets viewed with half-cycle
pulses ,” Phys. Rev. A , 55
, R3995 – R3998 ( 1997
). [CrossRef] ,3 3 . D. W. Schumacher , B. J. Lyons , and T. F. Gallagher , “ Wave packets in perturbed Rydberg systems
,” Phys. Rev. Lett. , 78
, 4359 – 4362 ( 1997
). [CrossRef] }. 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 4 . B. Hupper , J. Main , and G. Wunner , “ Nonhydrogenic Rydberg atoms in a magnetic
field: A rigorous semiclassical approach ”,
Phys. Rev. A , 53 , 744 –
759 ( 1966 ). [CrossRef] }. 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 4 . B. Hupper , J. Main , and G. Wunner , “ Nonhydrogenic Rydberg atoms in a magnetic
field: A rigorous semiclassical approach ”,
Phys. Rev. A , 53 , 744 –
759 ( 1966 ). [CrossRef] ,5 5 . P. A. Dando , T. S. Monteiro , D. Delande , and K. T. Taylor , “ Role of core-scattered closed orbits in
nonhydrogenic atoms ,” Phys. Rev. A ,
54 , 127 – 138 (
1996 ). [CrossRef] [PubMed] } and Stark^{6 6 . M. Courtney , N. Spellmeyer , H. Jiao , and D. Kleppner , “ Classical, semiclassical and quantum
dynamics in the lithium Stark spectra ,” Phys.
Rev. A , 51 , 3604 –
3620 ( 1995 ); M. Courtney and D. Kleppner , “ Core-induced chaos in diamagnetic lithium
,” Phys. Rev. A 53 , 178 – 191 (
1996 ). [CrossRef] [PubMed] } 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 7 . J. Yeazell and C. R. Stroud Jr. , “ Rydberg-atom wave packets localized in
the angular variables ,” Phys. Rev. A 35 , 2806 – 2809 (
1987 ); “ Observation of spatially
localized atomic electron wave packets ,” Phys.
Rev. Lett. , 60 , 1494 –
1497 ( 1988 ). [CrossRef] [PubMed] } or elliptic state^{1 1 . M. Nauenburg , C. R. Stroud Jr. , and Yeazell J. , “ The classical limit of an atom
,” Sci. Am. 270 , 24 – 29 (
1994 ). }, we can isolate the core scattering effects from the dynamics associated with radial^{8 8 . R. Bluhm and V. A. Kostelecky , ” Long-term evolution and revival structure
of Rydberg wave packets for hydrogen and alkali-metal atoms
,” Phys. Rev. A , 51 , 4767
– 86 ( 1995 ). [CrossRef] [PubMed] } or Stark wave packets^{2 2 . C. Raman , T. C. Weinacht , and P. H. Bucksbaum , “ Stark wave packets viewed with half-cycle
pulses ,” Phys. Rev. A , 55
, R3995 – R3998 ( 1997
). [CrossRef] }. Elliptic states have a strong correspondence with classical theory^{9 9 . J. Gay , D. Delande , and A. Bommier , “ Atomic quantum states with maximum
localization on classical elliptical orbits ,”
Phys. Rev. A 39 , 6587 ( 1989 ). [CrossRef] [PubMed] } 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 2 . C. Raman , T. C. Weinacht , and P. H. Bucksbaum , “ Stark wave packets viewed with half-cycle
pulses ,” Phys. Rev. A , 55
, R3995 – R3998 ( 1997
). [CrossRef] ,7 7 . J. Yeazell and C. R. Stroud Jr. , “ Rydberg-atom wave packets localized in
the angular variables ,” Phys. Rev. A 35 , 2806 – 2809 (
1987 ); “ Observation of spatially
localized atomic electron wave packets ,” Phys.
Rev. Lett. , 60 , 1494 –
1497 ( 1988 ). [CrossRef] [PubMed] } 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 10 . N. Bohr , “ On the quantum theory of line spectra
,” in Niels Bohr, Collected Works ,
edited by J. Nielsen ( North-Holland Pub. Co., New York
, 1976 ), Vol. 3 . }. 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
Figure 1: Numerical simulations of a classical electron trajectory showing Stark
oscillations in a dc electric field. The electron is initially in a circular orbit
in the xy-plane with n = 15 and
L_{z} = |L| = 15. An 800 V/cm dc electric field is in the
x-direction and the resulting Stark period T_{s} is approximately 84 Kepler periods. The first few orbits are colored red
for clarity and the trajectory is followed for slightly less than half of a Stark
period. a) The trajectory for a pure Coulomb potential. The electron backscatters
in the linear obit and nearly retraces the original trajectory in the opposite
direction. b) The trajectory in the sodium model potential. Note the precession of
the highly eccentric orbit.
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
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 12 . A. Hooker , C. H. Greene , and W. Clark , “ Classical examination of the Stark effect
in hydrogen ,” Phys. Rev. A ,
55 , 4609 – 4612 ( 1997
). [CrossRef]
) we will focus on
the dynamics of the orbit.
Figure 2: The classical dynamics of the angular momentum for the trajectories shown in
Fig. 1. The evolution is followed for two
complete Stark periods. The dashed curve is hydrogen and the solid curve is for
the sodium model potential. Note that the sodium curve has a shorter Stark period
because the scattering occurs at a non-zero value of the angular momentum.
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. 4a3. 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 16 . J. Pascale , R. E. Olson , and C.O. Reinhold , “ State-selective capture in collisions
between ions and ground- and excited-state alkali-metal atoms
,” Phys. Rev. A 42 , 5305 – 5314 (
1990 ). [CrossRef] [PubMed] } for sodium which includes long range polarizability terms as well as short range
core effects
where α_{d} is the dipole core polarizability, α′_{q} is the dynamical quadrupole polarizability of the core and the parameters
a, b, c, and r_{c} are chosen to match sodium spectroscopic data^{16 16 . J. Pascale , R. E. Olson , and C.O. Reinhold , “ State-selective capture in collisions
between ions and ground- and excited-state alkali-metal atoms
,” Phys. Rev. A 42 , 5305 – 5314 (
1990 ). [CrossRef] [PubMed] }.
It is well-known^{13 13 . See, for example R. J. Finkelstein , “ Nonrelativistic Mechanics
,” ( W. A. Benjamin, Inc., Reading, MA
, 1973 ) 298 . } 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
Figure 3: An animation comparing the quantum Stark dynamics in hydrogen and sodium. In both
cases the electron is initially in a circular state in the xy plane with
n = 15 and 〈
L_{z}〉 = 14 with an 800 V/cm dc electric field in the x-direction. a)
In hydrogen the quantum behavior is identical to the classical Stark evolution
shown in
Fig. 1a. b) The behavior in
sodium is more complex. Note how part of the wave function behaves hydrogenically
while the remainder shows the rapid precession of the non-Coulombic classical
orbits in
Fig. 1b.
[
Media 1]
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 14 . See, for example H. Goldstein , “ Classical Mechanics
,” ( Addison-Wesley, Reading, MA ,
1981 ) 509 . ,15 15 . T. P. Hezel , C. E. Burkhardt , M. Ciocca , and J. J. Leventhal , “ Classical view of the properties of
Rydberg atoms: Application of the correspondence principle
,” Am. J. Phys. , 60 , 329
– 335 ( 1992 ). [CrossRef] }. 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 15 . T. P. Hezel , C. E. Burkhardt , M. Ciocca , and J. J. Leventhal , “ Classical view of the properties of
Rydberg atoms: Application of the correspondence principle
,” Am. J. Phys. , 60 , 329
– 335 ( 1992 ). [CrossRef] }. 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 17 . J. N. Bardsley , “ Pseudopotentials in atomic and molecular
physics ,” Case Studies in Atomic Physics
, 4 299 – 368 ( 1974 ).
}. It is the short range exponential term that provides the dominant
core-scattering effect that we see in
Fig.
1b.
Figure 4: The quantum mechanical evolution of the expectation value of the z-component of
the angular momentum for the wavefunction dynamics shown in the animations (
Figs. 3 and
6). The evolution is followed for one Stark period. a)
〈
L_{z}(
t)〉 for the states shown in
Fig. 3. The dashed curve is hydrogen and the
solid curve is sodium. b) The evolution for core-scattered wave packets shown in
Fig. 6. The dashed is the wave packet
given initially by
m < 0 states and the solid curve is the
wave packet given initially by
m ≥ 0 states.
Figure 5: The ratio of the scattered wavefunction (m > 0) to the
unscattered wavefunction (m ≤ 0) is plotted as a function
of the Stark period. Classically for larger Stark periods the electron spends more
time in the low angular momentum orbit. Quantum mechanically this extended period
of scattering leads to an increase in the scattered fraction of the
wavefunction.
To examine core effects quantum mechanically we integrate Schrodinger’s equation
in a weak dc electric field using sodium dipole moments^{18 18 . M. L. Zimmerman , M. G. Littman , M. M. Kash , and D. Kleppner , “ Stark structure of the Rydberg states of
alkali-metal atoms ,” Phys. Rev. A ,
20 , 2251 – 2275 (
1979 ). [CrossRef] } for the n = 15 manifold. We also include the
16s and 16p 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 = 0.5T_{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 =
0.5T_{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.
Figure 6: An animation comparing the quantum Stark dynamics of the core-scattered wave
packet (initially
m ≥ 0 states) and the
“hydrogenic” wave packet (initially
m < 0
states). In both cases the dc electric field strength is 800 V/cm in the
x-direction. a) This wave packet isolates the effect of core scattering and its
behavior is very similar to the classical trajectory in
Fig. 1b. b) Interference near the core minimizes core
scattering and leads to nearly hydrogenic behavior.
[
Media 2]
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.