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PHYSICAL REVIEW A
VOLUME 60, NUMBER 4
OCTOBER 1999
Effects of core scattering on level statistics in helium atoms in scaled external fields
K. Karremans, A. Kips, W. Vassen, and W. Hogervorst
Department of Physics and Astronomy, Vrije Universiteit, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands
共Received 23 December 1998兲
We recorded constant scaled-energy spectra of helium Rydberg atoms in both electric and magnetic fields in
a regime where the hydrogen atom behaves regularly. Jonckheere, Grémaud, and Delande 关Phys. Rev. Lett. 81,
2442 共1998兲兴 recently showed that the nearest-neighbor statistics of sufficiently highly excited states follows a
␦ -Poisson distribution, which depends on the quantum defect ␦ . We find global deviations from this ␦ -Poisson
model, which we attribute to the size of the effective Planck constant ប eff . Significant deviations at small level
spacings indicate that the description of level statistics with the ␦ -Poisson model is incomplete.
关S1050-2947共99兲50409-5兴
PACS number共s兲: 32.60.⫹i, 03.65.Sq, 05.45.⫺a, 32.80.Rm
Quantum systems showing chaotic behavior in the classical limit, sometimes referred to as quantum chaos, are the
subject of detailed theoretical and experimental studies
关1–5兴. Despite impressive progress in the last decade, a profound understanding of the connection between irregular
classical dynamics and quantum mechanics is still lacking.
The hydrogen atom in a magnetic field is a prototype system
to study this chaotic behavior. The classical dynamics in a
constant magnetic field undergoes a smooth transition from
regular motion to chaos when the energy is raised up to the
ionization limit. Experiments on Rydberg atoms and quantum R-matrix calculations have been used to search for quantum manifestations of chaos. Numerical investigations have
revealed a relation between quantum-level statistics and the
underlying classical dynamics. Nearest-neighbor spacings
共NNS’s兲 in spectra in the regular regime follow a Poisson
distribution, while in the chaotic regime they obey a Wigner
distribution 关6兴. A second connection between energy levels
in an absorption spectrum and the classical dynamics of the
atom has been provided by closed-orbit theory 关7兴. According to this theory each electron orbit that starts at and returns
to the nucleus shows up as an oscillation in the photoabsorption spectrum.
The scaling properties of Rydberg atoms in external fields
can be used to perform experiments under constant classical
conditions 关1兴. The Hamiltonian 关in cylindrical coordinates
( ␳ ,z, ␾ )兴 of a hydrogen atom in an external field, directed
along the z axis for M L ⫽0, is 共in a.u.兲
H⫽
p ␳2 ⫹p z2
2
⫺
1
冑␳
2
⫹z 2
⫹H ext .
共1兲
In a magnetic field H ext⫽ 81 ␥ 2 ␳ 2 and in an electric field
H ext⫽Fz ( ␥ and F are magnetic- and electric-field strengths
in a.u.兲. Applying a scaling transformation r̃⫽wr and p̃
⫽w ⫺1/2p, with the scaling parameter in the magnetic field
given by w⫽ ␥ 2/3 and in the electric field by w⫽F 1/2, a
scaled Hamiltonian results:
H̃⫽
p̃ 2␳ ⫹p̃ z2
2
⫺
1
冑˜␳ 2 ⫹z̃ 2
1050-2947/99/60共4兲/2649共4兲/$15.00
⫹H̃ ext⫽
H
w
.
共2兲
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The scaled Hamiltonian in a magnetic field, H̃ ext⫽ 81 ˜␳ 2 , and
in an electric field, H̃ ext⫽z̃, does not depend on the field
strength explicitly. Therefore the classical dynamics of the
system only depends on a single parameter, the scaled energy ␧⫽E/w. In the following we will discuss experiments
and numerical calculations at constant scaled energy. Although the classical dynamics of the system is fixed, the
quantum problem still depends on the field strength. This
follows directly from the commutation relation for position
and momentum in scaled coordinates:
关 r̃,p̃ 兴 ⫽iw 1/2⫽iប eff .
共3兲
This relation introduces the concept of an effective Planck
constant ប eff .
Because the scaled electric field Hamiltonian for hydrogen is separable the corresponding classical motion is always
regular. In case of many overlapping manifolds (␧⬎⫺2.5),
the energy levels are randomly distributed and the distance
of adjacent levels s follows a Poisson distribution: P(s)
⫽exp(⫺s). In the magnetic-field case, regular motion breaks
down around ␧⫽⫺0.5 and phase space becomes gradually
chaotic. In the regime ⫺0.75⭐␧⭐⫺0.50 the level distribution is Poissonian and does not depend on the value of ប eff
关6兴. The level statistics gradually changes to a Wigner distribution when the scaled energy is increased to
the region of complete chaotic motion (␧⬎⫺0.12): P(s)
⫽ 12 ␲ s exp(⫺ 41 ␲s2). In the regime of mixed regular and chaotic motion (⫺0.5⭐␧⭐⫺0.12) a heuristic Brody distribution, which involves a single parameter q that allows for a
smooth interpolation between a Poisson and Wigner function, provides good fitting results. This Brody parameter q
gives an indication for the chaotic fraction of phase space.
Experimentally the resolution required to reveal all spectral details has so far only been reached in nonhydrogenic
atoms. In this paper we present measurements and calculations of NNS’s in helium atoms in regimes where the corresponding hydrogen statistics is Poissonian. The influence of
the ionic core on the classical motion was observed for the
first time in diamagnetic helium experiments. In action spectra, i.e., Fourier transforms of scaled-energy spectra, nearly
all peaks corresponded to hydrogenic closed orbits 关8兴. AdR2649
©1999 The American Physical Society
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K. KARREMANS, A. KIPS, W. VASSEN, AND W. HOGERVORST
ditional peaks, which appeared at the sum action of two
closed orbits, could be attributed to the process of core scattering. In a fully semiclassical approach 关9兴, this ionic core
was represented by a model potential, which simulates the
quantum defect ␦ . This short-range potential generates dynamics that are fundamentally different from that in hydrogen. The classical trajectories become ergodic, independent
of the quantum defect. This theoretical result initiated a discussion as to whether core scattering gives rise to chaotic
behavior in a regime where the hydrogen system is regular.
The notion that the ionic core introduces chaotic motion was
supported by the observation of significant changes in the
level statistics as deduced from quantum R-matrix calculations; this was found in the integrable electric-field problem
as well 关2兴. In rubidium experiments in a magnetic field at
␧⫽⫺0.8, where a large quantum defect is involved, indeed a
Wigner-like distribution was found in the regime where hydrogen is completely regular 关3兴. In the case of small quantum defects in helium and lithium experiments, only minor
deviations from Poisson statistics were observed 关4,10兴. The
small deviation from Poisson statistics at ␧⫽⫺0.7, found in
a recent diamagnetic helium experiment ( ␦ ⫽0.068), allowed
for an observation of the transition to a Wigner distribution
in the chaotic regime (␧⫽⫺0.3) 关4兴. These experiments
demonstrated that for small quantum defects level statistics
differs only slightly from the hydrogen case, contradictory to
the significant changes in the classical dynamics predicted by
model potential calculations.
A quantum treatment of the ionic core within the framework of closed-orbit theory was recently introduced by
Dando et al. 关5兴. Experimental recurrence strengths in action
spectra of singlet helium atoms in a scaled electric field were
shown to be in excellent agreement with this theory 关11兴.
The recurrence strength of core scattered peaks in this approach not only depends on the value of the quantum defect
␦ but also on that of ប eff . A small quantum defect produces
weak modulations in the energy spectrum; modulations related to core-scattering processes decrease linearly with ប eff ,
which is faster than modulations connected to hydrogenic
orbits (⬀ 冑ប eff). Interestingly, in the semiclassical limit longrange modulations in nonhydrogenic atoms converge to the
hydrogenic ones.
In a recent publication 关12兴 the statistical behavior of nonhydrogenic atoms in a magnetic field, where the effect of the
ionic core is maximum 共i.e., for ␦ ⫽0.5兲, was investigated
numerically. At ␧⫽⫺0.5 the NNS of sufficiently highlyexcited states was found to follow a ‘‘semi-Poisson’’ distribution P(s)⫽4s exp(⫺2s). This distribution, intermediate
between Poisson and Wigner, is also a good approximation
of the statistical properties of highly excited states in scaled
electric fields. In the general case of a nonzero quantum defect the level distribution of a highly excited nonhydrogenic
atom in the regular regime was predicted to follow a
␦ -Poisson distribution:
P ␦共 s 兲 ⫽
冋 冉
冊 冉 冊册
1
s
s
exp ⫺
⫺exp ⫺
1⫺2 ␦
1⫺ ␦
␦
.
共4兲
This analytical prediction for sufficiently highly excited
states smoothly evolves from a semi-Poisson distribution for
␦ ⫽0.5 to a Poisson distribution for ␦ ⫽0.
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We have investigated the validity of Eq. 共4兲 experimentally and examined the statistical properties of helium Rydberg states in magnetic and electric fields. M L ⫽0 Rydberg
states between n⫽60 and 90 were excited with one photon
from a metastable state in a crossed laser-atomic beam experiment. The effect of different quantum defects is studied
by excitation from 2 1 S 共largest quantum defect ␦ s ⫽0.14) in
scaled electric-field experiments and from 2 3 S 共largest quantum defect ␦ p ⫽0.068) in scaled magnetic-field experiments.
In the magnetic field only odd-parity states are coupled by
the diamagnetic term, whereas the electric field couples all
orbital angular-momentum states. The excitation energy has
to be determined accurately during a constant scaled-energy
experiment. For this purpose a zero-field Rydberg transition
served as an absolute energy marker. In combination with
counting fringes of a 150-MHz etalon an on-line frequency
calibration was realized. The field strength was adjusted each
frequency step. In the electric-field experiments UV laser
light 共312 nm, 8 mW兲 was produced by intracavity
frequency-doubling of a cw rhodamine-B ring dye laser. The
electric field between two capacitor plates was directed along
the atomic beam. An absolute accuracy in ␧ of 0.004 was
achieved, mainly limited by uncertainties in the electric-field
calibration. In the magnetic-field experiment metastable 2 3 S
atoms were excited with light from a frequency-doubled
Coumarine-6 ring dye laser 共260 nm, 3 mW兲. The magnetic
field was aligned parallel to the atomic-beam direction to
minimize motional Stark effects. Adjusting the electromagnet 共up to 0.3 T兲 during the laser scan, an absolute accuracy
in ␧ of 0.0002 was obtained. The 2 3 S 1 →2 3 P 2 transition
was used for transverse cooling of the atoms to improve the
signal strength by a factor of 10 and to reduce the Doppler
width.
In the magnetic-field case, spectra at ␧⫽⫺0.7000(2)
were used 关4兴. The magnetic field was decreased from 0.30
to 0.25 T in several overlapping spectra covering a range in
principal quantum number n⫽79– 90. The experimental
range in ␥ ⫺1/3 (94– 108) corresponds to an average ប eff of
0.010. The experimental data are compared with numerical
R-matrix calculations for helium at constant scaled energy.
All energy levels were resolved in the experiment; however,
some peaks were lost in the background, while a few extra
peaks were induced by residual electric fields. When approaching the semiclassical limit (n→⬁) by increasing the
excitation energy, these stray electric fields induce line
broadening and the resolution becomes too low for level statistics.
For a small data set, statistical information may be lost
when a relatively large bin size in the NNS histograms must
be used. This problem is avoided using a cumulative NNS
distribution. The eigenenergies are converted to a w ⫺1 scale,
removing the variation in the density of states. The spacing
between adjacent levels in the experiment, normalized to the
mean level spacing, is represented in the cumulative NNS
distribution of Fig. 1. This distribution is in good agreement
with a quantum R-matrix calculation obtained in the same
energy range, included in Fig. 3. The ␦ -Poisson statistics
curve, intermediate between Wigner and Poisson statistics, is
derived from Eq. 共4兲 substituting the value for the quantum
defect ( ␦ ⫽0.068). These three curves are included in Fig. 1.
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EFFECTS OF CORE SCATTERING ON LEVEL . . .
FIG. 1. Experimental cumulative NNS distribution for triplet
helium ( ␦ ⫽0.068) in a scaled magnetic field at ␧⫽⫺0.7, compared
with the Poisson, ␦ -Poisson, and Wigner distributions.
The experimental distribution is close to ␦ -Poisson but
shows a significant shift in the direction of Wigner statistics.
The electric-field experiments were performed at ␧
⫽⫺2.35, where numerical calculations show that the level
distribution of the corresponding hydrogen system is Poissonian. Although excitation was to lower Rydberg states than
in the scaled magnetic-field experiment, the average value of
ប eff of 0.007 reflects that this system was closer to the semiclassical limit. Measuring several overlapping scans starting
at n⫽60, we recorded spectra in the F ⫺1/4 range from 130 to
150. The cumulative NNS distributions of the experimental
and numerical calculations in the same ប eff regime are presented in Fig. 2. The excellent agreement between experi-
FIG. 2. Experimental and calculated cumulative NNS distribution for singlet helium ( ␦ ⫽0.14) in a scaled electric field at ␧
⫽⫺2.35, compared with Poisson, ␦ -Poisson, and Wigner distributions. For small spacings (s⬍0.5) we observe a significant deviation from ␦ -Poisson statistics.
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FIG. 3. Cumulative NNS distribution for triplet helium ( ␦
⫽0.068) in a scaled magnetic field at ␧⫽⫺0.7 calculated for different values of ប eff : 0.010 ( ␥ ⫺1/3⫽94– 108) and 0.003 ( ␥ ⫺1/3
⫽350– 400) show the convergence towards a ␦ -Poisson distribution in the semiclassical limit. For small (s⬍0.25) spacings again a
deviation from ␦ -Poisson is observed.
ment and numerical calculations demonstrates that the experimental resolution was adequate. Due to the larger
quantum defect ( ␦ ⫽0.14) the ␦ -Poisson curve in the
electric-field case is shifted further away from the Poisson
curve than in the magnetic-field case of Fig. 1. For large
spacings (s⬎0.5) the statistical distribution of the experiment is in very good agreement with the ␦ -Poisson model.
For small spacings (s⬍0.2), however, deviations are clearly
visible: both distributions follow the Wigner curve.
We used numerical calculations for the scaled magneticfield case to investigate the dependence of spectral properties
on ប eff . In Fig. 3 we plot the calculated NNS distribution at
␧⫽⫺0.7 in the experimental range (ប eff⬃0.010) and compare it with a distribution evaluated for higher excited states
(ប eff⬃0.003). The statistics for ប eff⬃0.003 is in much better
agreement with the ␦ -Poisson model. For low excitation energy 共high value of ប eff), the increased effect of core scattering causes the shift toward Wigner statistics. This global
shift towards Wigner statistics for lowly excited states 共large
ប eff) was also found in previous R-matrix calculations 共for
␦ ⫽0.5). Jonckheere et al. 关12兴 carefully checked that their
calculated distributions converged to the ␦ -Poisson distribution, but did not relate this effect to the value of ប eff . The
analysis of our experiment indicates that ␦ and ប eff both are
relevant parameters for describing effects of the ionic core
on level statistics; these parameters are also used to describe
the process of core scattering in closed-orbit theory. A physical model connecting core scattering and quantum-level statistics is still lacking. The observed deviation from the
␦ -Poisson distribution in the magnetic-field experiment demonstrates that the core-scattering process depends on the
value of ប eff . In the semiclassical limit the recurrence amplitudes of sum orbits decrease faster than the amplitudes of
hydrogenic orbits. A slow ប eff-dependent change towards
Poisson statistics therefore may be expected, so the
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K. KARREMANS, A. KIPS, W. VASSEN, AND W. HOGERVORST
PRA 60
␦ -Poisson distribution appears somewhat as a surprise. The
analogy with the dynamics of a point-scatterer model, where
also a ␦ -Poisson distribution was found 关12兴, shows that this
reasoning is incomplete.
When we compare the calculated distribution for ប eff
⬃0.003 for small level spacings s⬍0.2 with the ␦ -Poisson
prediction from Eq. 共4兲, significant deviations are observed.
These deviations are less pronounced than in the electricfield case and reflect the smaller value of the quantum defect.
When we closely examine the calculated distributions in the
semiclassical limit for both quantum defects ␦ ⫽0.14 共Fig. 2兲
and 0.068 共Fig. 3兲 and compare them with the corresponding
␦ -Poisson statistics, we observe the same general trend. For
small level spacings the distribution is Wigner-like. A
␦ -Poisson distribution is only observed for larger spacings.
This type of behavior differs completely from the overall
deviations from ␦ -Poisson statistics found in previous ␦
⫽0.5 numerical calculations. Our data clearly show that the
␦ -Poisson model does not give a correct description of the
level statistics of nonhydrogenic atoms with small quantum
defects. This phenomenon is hitherto unexplained. Since the
␦ -Poisson model fails to reproduce all statistical properties
we tried to fit the distributions to a Brody curve. Again dif-
ficulties arise at small distances, which shows that the process of core scattering gives rise to a different type of statistics in the semiclassical limit.
To summarize, we have measured the level statistics in
scaled-energy experiments on helium atoms in an external
field, in the regime where the corresponding hydrogen statistics is Poissonian. The deviations from the Poisson distribution in the experimental statistics depend on both the quantum defect ␦ and on the effective Planck constant ប eff . The
overall shift from ␦ -Poisson to Wigner statistics in the
magnetic-field experiment is attributed to the value of the
effective Planck constant. For electric-field experiments the
experimental distribution for large level spacings is in good
agreement with this ␦ -Poisson statistics. For small spacings
significant deviations are found in both electric and magnetic
fields.
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We would like thank D. Delande and T. Jonckheere for
making available their numerical calculations. The authors
are indebted to Jacques Bouma for his excellent technical
assistance. Financial support from the Foundation for Fundamental Research on Matter 共FOM兲 is gratefully acknowledged.