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J. Phys. B: At. Mol. Opt. Phys. 31 (1998) L489–L495. Printed in the UK
PII: S0953-4075(98)89238-9
LETTER TO THE EDITOR
Classical support for non-dispersive two-electron wave
packets in the driven helium atom
Peter Schlagheck and Andreas Buchleitner
Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany
Received 14 November 1997
Abstract. We present numerical evidence for the existence of a classically non-ionizing, highly
correlated two-electron configuration which emerges from the frozen planet configuration of
helium exposed to an electromagnetic field of linear polarization. A static electric field parallel
to the polarization axis prevents this configuration from transverse ionization. We argue that
the corresponding nonlinear resonance island in 12D phase space supports nondispersive wave
packets of the helium atom in combined oscillating and static electric fields.
The classical dynamics of the three-body Coulomb problem, and, more specifically, of
doubly excited states of the helium atom, has provided various surprises during recent years.
The most prominent of those has probably been the discovery of a new, highly correlated
and globally stable electronic configuration, the ‘frozen planet’ [1, 2]. The appeal of this
configuration resides in its counterintuitive, asymmetric character where both electrons are
located on the same side of the atomic nucleus. This is in pronounced contrast to other
doubly excited modes such as the Wannier, the asymmetric stretch and the Langmuir orbit,
with one electron on either side of the nucleus [3, 4]. In addition, the frozen planet is
the most robust of all known two-electron configurations, in the sense that it occupies
a large volume in phase space and remains stable over a wide range of values of the
nuclear charge Z [5]. Extensive semiclassical and quantum studies have shown that this
nonlinear resonance gives rise to doubly excited autoionizing states of the real helium atom
with extremely long lifetimes [5, 6]. Therefore, we have by now strong evidence that
the globally destructive character of the interelectronic repulsion, typically leading to the
immediate autoionization of the classical helium atom, simultaneously creates islands of
classically stable motion which have their exact quantum mechanical counterparts. Due
to their origin in the nonlinear coupling of the electronic degrees of freedom through the
electron–electron interaction these states can legitimately be considered as prototypes of
correlated electronic motion in two-electron systems.
In the present letter, we shall address the question of how such a strongly correlated
electronic configuration responds to an external perturbation. More precisely, we consider
the classical dynamics of the frozen planet configuration exposed to an electromagnetic
field linearly polarized along its symmetry axis. We shall see that the combination of
the interelectronic repulsion and of the periodic external perturbation gives rise to fieldinduced nonlinear resonances which emerge from the frozen planet island. These correspond
to classically bounded, regular and nonionizing motion of the electrons in a collinear
configuration. To guarantee the global stability of this doubly excited mode, i.e. to maintain
c 1998 IOP Publishing Ltd
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Letter to the Editor
the strongly correlated character of the electronic configuration within the driving field, with
respect to deviations from collinearity, an additional static electric field has to be applied.
Consequently, let us start with the Hamiltonian of the driven helium atom in the presence
of an additional, parallel static electric field of strength Fst . Employing atomic units, and
for an infinite mass of the nucleus, it is written as:
H =
p2
p21
2
1
2
+ 2− − +
+ (Fem cos(ωt) + Fst )(x1 + x2 ),
2
2
r1
r2
|r1 − r2 |
(1)
with Fem and ω the amplitude and frequency of the oscillating field, in the dipole
approximation. Note that the explicit time dependence of the problem leaves only the
projection of the total angular momentum on the field polarization axis as a conserved
quantity. Thus, the driven dynamics explores a 12D phase space (extended by the time t).
Before we proceed with the numerical solution of the equations of motion generated by
H , we make use of the Hamiltonian’s classical scaling properties: The equations of motion
remain invariant under the substitution
H := λH,
t := λ−3/2 t,
r := λ−1 r,
F := λ2 F,
p := λ1/2 p,
ω := λ3/2 ω,
(2)
with λ > 0 an arbitrary scalar. For our present purposes, λ = |H0 |−1 , H0 being the field-free
energy of the initial state of the atom, is a suitable choice. Hence, we shall consider the
scaled dynamics for H0 = −1, as familiar from published results on the classical dynamics
of the bare helium atom [3].
The numerical integration of the classical dynamics is performed after regularizing the
equations of motion by a twofold Kustaanheimo–Stiefel transformation [7, 8]. The latter
allows for a complete regularization of those singularities of the equations of motion which
correspond to the encounter of an electron and the nucleus. The triple collision between both
electrons and the nucleus which cannot be regularized does not occur for the configuration
we are considering here.
It should be noted that the Kustaanheimo–Stiefel transformation is a skilful canonical
coordinate transformation which exactly accounts for the various Coulomb singularities in
the dynamics which are essential for the mere existence of the frozen planet configuration
in the bare atom [2, 9]. The equations of motion in the new coordinates therefore provide
an exact description of the problem at hand. They are not to be confused with wide
spread ‘regularization’ schemes which simply replace the Coulomb terms in the original
Hamiltonian by a ‘soft-core’, non-singular potential and therefore replace the actual physical
problem by an approximation [10] (see also [9]).
Let us first consider the collinear frozen planet configuration along the polarization axis
of the oscillating field, for zero static electric field strength. Under this restriction, the
classical phase space reduces to five dimensions, and it is in fact possible to construct a
Poincaré surface of section by fixing the inner electron’s position to x2 = 0 and the field’s
phase ωt to a constant value (for fixed initial field-free energy H0 ; details will be given
elsewhere). The response of the outer electron will be most pronounced when the driving
frequency is near-resonant with its characteristic frequency around the equilibrium position
or with higher harmonics thereof. Due to the KAM theorem [11], we can expect that the
external driving will create nonlinear resonance islands within the elliptic island defining the
unperturbed frozen planet. Indeed, figures 1(a), (d)–(f) show Poincaré surfaces of section
for the one-dimensional frozen planet configuration under external driving for four different
values of the driving frequency at given field amplitude and fixed phase ωt = 0 of the
field. For low frequencies (ω = 0.05 in figure 1(a)), which are resonant with a slow
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L491
Figure 1. Poincaré surface of section (spanned by the position and the momentum of the outer
electron) of the strongly driven frozen planet, collinear with the field polarization axis, for
fixed scaled driving field amplitude Fem , zero static field Fst = 0, and four different values of
the driving frequency (in scaled units, see (2)): ω = 0.05, Fem = 0.001 (a)–(c); ω = 0.08,
Fem = 0.001 (d); ω = 0.1, Fem = 0.001 (e); ω = 0.2, Fem = 0.002 (f). In (a)–(c), the phase
of the driving field is scanned through ωt = 0 (a), ωt = π/2 (b), ωt = π (c), to highlight the
distinction between the intrinsic and the field-induced resonance. ωt = 0 in (d)–(f).
oscillation of large amplitude of the outer electron around the frozen planet equilibrium, we
observe two well separated resonance islands within a largely chaotic phase space (the same
picture shows almost completely integrable dynamics in the absence of external driving;
see, e.g., [5]). The one centred around smaller x-values represents those frozen planet tori
which have not been destroyed by the external field, whereas the smaller one at larger xvalues corresponds to a nonlinear resonance between the external driving and the resonant
unperturbed frozen planet motion. As the field frequency is increased to higher values, the
relative size of the frozen planet island decreases whereas the field-induced island increases.
At frequency ω = 0.08 (figure 1(d)), the field-induced island clearly dominates the phase
space structure, only a tiny stable island corresponding to the unperturbed frozen planet is
left. As one increases the driving frequency beyond the maximum frequency ωm ' 0.09 of
the electronic motion within the frozen planet island (with n2 ' 1.5 the radial action of the
inner electron for H0 = −1, see [6]), the external field can no longer drive the fundamental
of the electronic motion in the frozen planet’s island but only its higher harmonics. Hence,
only higher-order resonances can be observed for such frequencies, e.g. the 2:1 resonance
(where two field cycles correspond to one electronic oscillation) for ω = 0.1 in figure 1(e),
and the 3:1 resonance for ω = 0.2 in figure 1(f). An increase of the driving field amplitude
Fem beyond a value of Fc ' 0.01 . . . 0.1. completely destroys the frozen planet for the
values of ω considered here.
The intrinsic and the field-induced island can be distinguished by their temporal
dynamics, e.g. by comparison of Poincaré sections for different values of the phase
ωt. Figures 1(a)–(c) show that the intrinsic island is practically invariant as the phase
evolves, whereas the field-induced resonance traces closed orbits around the frozen planet’s
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Letter to the Editor
equilibrium position. Hence, an electron trapped in the field-induced resonance island is
locked by the field on a periodic frozen planet orbit around the equilibrium position.
Let us now abandon our point of departure of a collinear configuration of both electrons
along the x-axis, but consider small deviations from the collinear arrangement, i.e. small
components of the position and the momentum of the electrons transverse to the field
polarization axis (we keep Fst = 0). For vanishing driving field strength, such a deviation
is known to lead to stable, confined motion [5]. Due to the increased number of phase
space dimensions as compared to the collinear case (12 instead of 5), it is no longer
possible to illustrate the motion by means of a Poincaré surface of section. Consequently,
we probed the phase space neighbourhood of the collinear configuration by choosing
initial conditions homogeneously distributed over the collinear phase space volume, with
a transverse component of the outer electron’s position which was varied within the range
y = 10−5 to 1 (the y-axis being perpendicular to the field polarization axis). (The
information obtained by this approach was verified to be the same if the configuration
was displaced in a different way, e.g. by a transverse momentum component.)
Typically, for field strengths leading to a clear separation of the intrinsic and the fieldinduced stability island (see figure 1), we observe that the outer electron drifts away from
the field polarization axis and flips to the opposite side of the nucleus, followed by the
inner electron. As long as the characteristic angular correlation [2] with the inner electron
can be sustained, the outer electron can continue flipping sides, followed by the inner
electron, and remains bounded. However, depending on the choice of the outer electron’s
initial coordinate along the polarization axis, the angular correlation between inner and
outer electron may be lost after as few as 50–100 field cycles, and ionization is then almost
immediate, due to the electron–electron interaction. Figure 2 shows the x- and the ycoordinate of a typical trajectory, started from an x-position which corresponds to the fieldinduced stability island in the one-dimensional configuration, see figure 1(a). The trajectory
ionizes after approximately 32 field cycles, where ionization is defined by the outer electron
leaving a region of radius R = 100 around the nucleus, with positive effective energy
Figure 2. Time dependence of the trajectory for ω = 0.05 and Fem = 0.001 (cf figures 1(a)–(c))
where the initial position of the outer electron is x(t = 0) = 13 (i.e. at the centre of the 1:1
resonance island) and y(t = 0) = 0.01 (all quantities in scaled units, see (2)). The initial
position of the inner electron is such that the total initial field-free energy H0 equals −1. (a)
x-coordinate of the outer electron; (b) x-coordinate of the inner electron; (c) y-coordinate of
the outer electron; (d) y-coordinate of the inner electron. Note the clear correlation between the
inner and the outer electron’s motion as long as the configuration remains bounded.
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L493
(p1 + A)2 /2 − 1/r1 + 1/|r1 − r2 | (A the vector potential). A similar scenario applies for
initial conditions with x-coordinates corresponding to the intrinsic resonance island.
Hence, the observed ionization of the outer electron is the consequence of a two-step
process: first, the driving field destroys the correlation between the motion of the outer and
the inner electron, leading then to immediate ionization due to the interelectronic potential.
Let us also mention here that, notwithstanding the instability and subsequent ionization
observed in figure 2, there do exist initial conditions off the x-axis defining trajectories
which (i) do not flip neither ionize but remain close to the x-axis for at least 104 field
cycles, or (ii) do flip but remain bounded over propagation times as long as 104 driving
field cycles. The former correspond to high-order field-induced resonances N :M, with
N, M 6= 1. However, their phase space volume is very small and they therefore should
become relevant only for very high atomic excitations. The latter turn out to be scattered
over the phase space adjacent to the collinear configuration according to a complicated
pattern. A discussion in terms of chaotic scattering seems more appropriate and will be
presented elsewhere.
Since the frozen planet tends to be destroyed by the flipping, which means that the
stability islands of the collinear configuration are not globally stable against ionization, it is
highly desirable to prevent the outer electron from changing sides. Note that the observed
flipping of the outer and simultaneously of the inner electron (figure 2) is reminiscent of the
precession of the Kepler ellipse of a hydrogen Rydberg state under external driving [12].
In that case we know that the Rydberg dynamics can be confined to the vicinity of the
polarization axis (and made effectively one dimensional) by the application of an additional
static electric field parallel to the polarization axis of the oscillating field (exerting on the
electron a force directed away from the nucleus), with a field strength Fst of about 10% of
the electromagnetic field’s amplitude [13]. Indeed, the same recipe turns out to apply for
the frozen planet: application of a weak electric field (weak for the outer, loosely bound
electron) is of negligible effect for the inner electron, which already moves along a strongly
polarized orbit and is additionally tightly bound to the nucleus. The outer electron, on the
other hand, is prevented from flipping for appropriately tuned Fst and not too large Fem ,
Figure 3. (a) Unstable trajectory in configuration space for ω = 0.2, Fem = 0.002, Fst = 0 (cf
figure 1(f)) where the initial position of the outer electron is x(t = 0) = 9.6 (i.e. at the centre
of the 3:1 resonance island) and y(t = 0) = 0.01 (all quantities in scaled units, see (2)). The
initial position of the inner electron is such that the total initial field-free energy H0 equals −1.
(Total propagated time in the figure: 90 field cycles; ionization time: 1000 field cycles.) (b)
Stabilized trajectory in configuration space for ω = 0.2, Fem = 0.002, Fst = −0.0002, with
the same initial conditions as in (a) (total propagated time in the figure: 130 field cycles). The
stabilizing effect of the static electric field is obvious (note the difference in the x- and y-scale
in (a) and (b), respectively!). The configuration does not ionize during interaction times as long
as 30 000 field cycles, the maximum propagation time employed in the present study.
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Letter to the Editor
and the field-induced stability islands shown in figures 1(d)–(f) become stable (the island
in figure 1(a) can only be stabilized for slightly lower Fem ). As an illustration, figure 3(a)
shows the motion of the inner and the outer electron for initial conditions with x-values
corresponding to the 3:1 resonance in figure 1(f), with Fem = 0.002 and Fst = 0. As
for the case considered in figure 2, the electrons deviate from the x-axis and flip to the
opposite side of the nucleus, leading to ionization after the relatively long interaction time
of 1000 field cycles. Exactly the same initial conditions and field parameters are used
in figure 3(b), except for a finite Fst = −0.0002. Clearly, the motion is confined to the
neighbourhood of the field axis, the interelectronic angular correlation is maintained, and
this quasi one-dimensional configuration remains bounded over the maximum integration
time of 30 000 field cycles used in our present calculations. Note that the phase space
structure along the field axis (figure 1) is essentially the same for figures 3(a) and (b).
Our choice of the 3:1 resonance of figure 1(f) for the comparison in figure 3 is due
to the observation that, out of the induced stability islands of figure 1, it is this particular
nonlinear resonance which allows for the largest deviations from collinearity in the presence
of a static electric field, and hence it has the largest volume in 12D phase space. Again
from the extensive studies on strongly driven atomic hydrogen, it is established by now that
such a global nonlinear stability island gives rise to non-dispersive wave packet eigenstates
in the corresponding quantum problem [14], provided the island’s volume is large enough
to support a single quantum state. A preliminary estimation based on the semiclassical
EBK criterion [4], which provides an estimation of the atomic excitation necessary to
resolve the classical phase space structure, gives n2 ' 100 for the inner electron’s principal
quantum number in order to fully localize a wave packet eigenstate in the 3:1 resonance
island. However, one should note that the EBK criterion is generally too restrictive as far
as the mere existence of quantum states anchored to a classical resonance is concerned.
The Langmuir orbit, for example, is seen in the quantum spectrum already at quantum
numbers n ' 10, whereas the EBK condition implies n ' 500 [4, 15]. Consequently, the
above estimation n2 ' 100 suggests that it is not unreasonable to expect the manifestation of
nondispersive two-electron wave packets in the quantum spectrum of strongly driven helium
at even lower quantum numbers. Present experiments on doubly excited, helium-like states
of Ba [1] realize excitations with n2 ' 40. . .80. For n2 = 50, for example, the laboratory
values of driving field strengths and frequencies needed for the preparation of the above
two-electron wave packets would be given by
Fem ' 8 V cm−1 ,
ω/(2π ) ' 35 GHz,
(3)
Fst ' 0.8 V cm−1 ,
by virtue of the above scaling laws (2). These are typical orders of magnitude which
are encountered in current state of the art experiments on the microwave excitation and
ionization of single-electron atomic Rydberg states [16].
To conclude, our results suggest that the combination of the electron–electron interaction
and an external oscillating field, together with a static electric field, allows for the
construction of nondispersive two-electron wave packets which oscillate around the
equilibrium position of the unperturbed frozen planet configuration. Such states should
be accessible from bare frozen planetary states, at readily available driving frequencies and
amplitudes. Their stability with respect to ionization can be controlled by the strength of
the static electric field.
AB acknowledges partial financial support by the EU HCM network no ERB CHRX
CT940470 and the hospitality of Ken Taylor and his group at Queen’s University of Belfast
during parts of the time that this work was performed. CPU time has been provided by the
RZG.
Letter to the Editor
L495
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