P H Y S I C A L R E V I E W L E T T E R S
Scaled-Energy Spectroscopy of the Rydberg-Stark Spectrum of Helium: Influence of Exchange on Recurrence Spectra Department of Physics, Wesleyan University, Middletown, Connecticut 06459
We report the first observation of the manifestation of exchange effects in recurrence spectra. Using
fast-atom – laser-beam scaled-energy spectroscopy, we have measured the Rydberg-Stark recurrencespectrum of the two spin forms of helium in strong fields.
excitation of metastable 1s2s 1,3S states. We observe pronounced modulations in the triplet spectrumdue to interference between hydrogenic and core-scattered combination orbits.
are phase correlated with repeating hydrogenic orbits and not present in singlets or in hydrogen. [S0031-9007(98)06521-1]
A complete theoretical treatment of nonhydrogenic
of the spectrum and observation of trajectory evolution
atoms in external fields remains a challenge.
as a function of scaled energy. Using this information,
mental progress has provided substantial information on
we introduce a new analysis technique of integrating
Rydberg atoms in external fields [1]. Application of the
recurrence strength of an orbit over its scaled energy
semiclassical description, in the form of closed orbit the-
range. This permits study of the behavior of orbit types
ory, has been quite successful and is the basis for the
independent of launching angle, and that occur at very
field of recurrence spectroscopy [2].
orbit theory treats nonhydrogenic effects with quantum
In our experiment the absorption spectrum is measured
defects. Recently, nonhydrogenic effects have been in-
at fixed scaled energy ´ EF21͞2 over a range of n,
cluded through core-scattered waves [3] and by an ex-
where F is the applied electric field strength and E the
plicit core potential resulting in classical core scattering
total energy of the electron. In this case, the classical
[4]. These extensions have demonstrated that core scatter-
Hamiltonian of the system is not dependent on the energy
ing gives rise to combination orbits not seen in hydrogen.
of the electron and electric field separately, but rather on
To date, experimental studies of Rydberg-Stark recurrence
scaled energy only [2]. Recording spectra at fixed scaled
spectra have used alkali atoms which contain closed shell
energy maintains the same classical dynamics at all points
ionic cores and have one spin state [5,6]. The present
work extends experiment to open shell ionic cores and
The physical picture associated with recurrence spec-
their role in producing combination orbits. In addition,
troscopy is based on closed orbit theory [2]. When a pho-
open shell cores provide polarized spin-state dependent
ton is absorbed by an atom in an electric field, the electron
scattering systems that permit experimental access to the
becomes a near-zero-energy Coulomb outgoing wave.
After sufficient distance, the wave propagates semiclas-
The exchange interaction is isolated experimentally
sically and follows classical trajectories. Trajectories that
by measuring recurrence spectra versus spin state. This
are turned back by the combined Coulomb and external
provides a tool to examine the role of the Pauli exclusion
electric field eventually become waves that interfere with
principle in the semiclassical theory and in the dynamics
outgoing waves provided they return near the nucleus.
of Rydberg systems, where it is not well understood.
This “atomic interferometer” gives structure to the pho-
Closed orbits are sensitive to the spin-dependent exchange
interaction since the Rydberg electron returns to the
It can be shown from closed orbit theory that the
nucleus and the charge clouds of the electrons overlap.
In helium, when the Rydberg electron samples the core,
it moves as if it experiences an attraction or repulsion
depending on the relative orientation of the two spins. Ck,j͑´͒ sin͑2pj ˆSkF21͞4 2 ak,j͒ ,
This introduces a coupling between the spatial variables
of the Rydberg electron and the spin state of the atom.
How this is revealed in the Stark recurrence spectrum of
the prototypical two-electron helium atom is a focus of
Extensive data have been obtained for m 1 states at
60 different scaled energies in strong fields ͑20 , n ,
Df0 is the flat background when the electric field is not
30͒ up to scaled action 15. This allows a global view
present. This semiclassical result is exact for hydrogen
P H Y S I C A L R E V I E W L E T T E R S
and a good approximation for other Rydberg systems.
metastable states formed have long lifetimes compared
It applies to all orbits that are not on the field axis.
to their transit time. Ions that do not undergo charge
We neglect those on axis because of the centrifugal
transfer are removed from the beam by an electric
barrier for m 1 states. The second term is a sum
field and monitored. The beam of metastables enters a
over all closed orbits k, including repetitions j. Ck,j͑´͒
37-cm drift region that contains an electric field transverse
is the recurrence amplitude and its square the recurrence
to it, produced by a pair of plates separated by 1.0 cm.
strength. It incorporates trajectory stability, the geometry
In this region the metastables are excited to Stark-
of a transition, and the quantum defect. The sinusoidal
Rydberg states by a counterpropagating collinear 20-Hz
part arises from phase differences between outgoing and
uv dye-laser beam produced by frequency doubling the
returning waves and is described by the classical scaled
output of a Nd:YAG-pumped dye laser. Excitation energy
Sk adjusted by a constant ak,j. mk is the Maslov
below zero-field ionization limit ranged from 274.18
index and Dk is the phase shift caused by the core and
to 121.84 cm21. Each constant scaled-energy spectrum
is related to the quantum defect. Dk 0 for hydrogen.
covers 52.34 cm21, and was recorded in a continuous
Each closed orbit provides a sinusoidal variation to the
scan lasting ϳ15 min. The range of applied field was
scaled absorption spectrum. Consequently, the Fourier
130 V cm21 to 2168 V cm21. Linearly polarized light
transform power spectrum with respect to F21͞4, referred
was oriented perpendicular to the field, exciting m 1
to as the recurrence spectrum, yields discrete peaks at
scaled actions of classical closed orbits, their repetitions,
After the interaction region the beam enters a field
and their combinations. The amplitude of a peak at a
ionizer of 7.0 kV cm21, oriented along the beam axis,
scaled action is the coherent sum of orbit contributions to
that ionizes Rydberg atoms and accelerates resulting He1
that action within experimental resolution. Combination
ions into an analyzer for detection by a channeltron. The
orbits are created by scattering of one orbit into another
modulated ion signal is amplified, boxcar averaged, and
by the core and do not exist in hydrogen. The action
recorded. To maintain a fixed scale energy ´ EF21͞2,
of combination orbits is approximately the sum of the
the computer monitors the laser wavelength and adjusts
the electric field, while recording data. For a fixed scaled
Nonhydrogenic effects are incorporated within closed
energy, spectral data are recorded as a function of F21͞4
orbit theory through quantum defects, affecting the am-
and a Fourier transform is applied to obtain the recurrence
plitude and phase of a classical closed orbit [2]. Ne-
glecting relativistic effects, the quantum defect for helium
Contributions to uncertainty in scaled energy arise from
separates into two parts d dc 6 dex: one from electron
knowledge of electric field strength and laser energy. The
exchange 6dex and a direct Coulomb part dc, identical
applied voltage is accurate to ,1%, and spacing between
for both spin states, due to screening and polarization [7].
field plates is known to ϳ1023 cm. Laser wavelength is
The spin dependence of the quantum defect is contained
calibrated from field free spectra. Uncertainty in wave-
in the 6 sign: 1 for triplets, 2 for singlets. This term,
length is due primarily to reproducibility of a wavelength
due to the requirement of antisymmetrization of the total
setting, accurate to ,0.04 cm21. These effects conspire
wave function imposed by the Pauli exclusion principle,
to produce an uncertainty in the value of scaled energy of
is responsible for spin-dependent effects in our recurrence
about 0.2%. Laser linewidth is about 0.4 cm21.
spectra. dex 0, to first order, for alkali atoms. Since
Recurrence spectra have been measured for singlet and
1 $ m there is no s-state character in our spectra and the
triplet helium for scaled energy from 22.0 (classical
main contribution to the quantum defect comes from the
ionization threshold) to 23.5, and up to ˆ
p-state quantum defect, which is dominated by dex. This
S ϳ0.1. In Fig. 2 we present an experimental triplet
is seen in Table I, where we list relevant quantum defects
recurrence map. Three sequences of peaks are identified
Figure 1 shows the experiment. A beam of metastable
1s2s 1,3S helium atoms is prepared by starting with a He1beam.
Ions are produced in an ion source, extracted,
focused, accelerated to 4 keV, and passed through avelocity filter.
charge transfer collisions in a potassium vapor cell. The
l 0, 1 quantum defects for triplet and singlet
Experiment: (a) ionizer, ( b) channeltron, (c) deflector,
(d ) prisms, and (e) doubler. The scaling ´ EF21͞2 betweenStark plates and dye laser is shown.
P H Y S I C A L R E V I E W L E T T E R S
Combination orbits in III are more complex and comefrom both I and II, with double core scattering possible. Changes in the spectrum due to the presence of interlop-ing combination orbits are dramatic in the triplet case, in-troducing a strong peak height modulation in sequences IIand III as seen in Fig. 2 (see also Fig. 3).
We examine the data of Fig. 2 using a new recurrence
spectroscopy analysis technique. Rather than comparingresults at fixed scaled energy, which provides only a localview of an orbit at a specific launching angle, we focuson global properties, independent of launching angle, byintegrating recurrence strength in each constituent orbit ofa particular type. This allows identification of spin-state-dependent behavior in terms of hydrogenic orbit types,and comparison of orbits that do not exist at the samescaled energy. In Fig. 3 we present total strength of II
Experimental recurrence map of Rydberg-Stark spec-
(III) orbits versus hydrogen-orbit period ratio tu͞ty, with
trum of m 1 triplet helium. I, II, and III indicate the classical
two (three) time repeating orbits indicated by arrows.
closed orbit sequences. Solid dots are above the 9 orbit and
Also shown is the result of a simple quantum hydrogen
cutting through the map at different angles, labeled I,II, and III. In hydrogen, the sequences result from in-dividual orbits and their repetitions, and are classified[2,5] into types according to their period ratios tu͞tyin semiparabolic coordinates where u ͑r 1 z͒1͞2 andy ͑r 2 z͒1͞2. Sequences I, II, and III correspond totu͞ty i͞i 1 1, i͞i 1 2, i͞i 1 3, with i an integer. Sequence II orbits consist of two alternating types: irre-ducible nonrepeating orbits and reducible repeating orbits. A sequence II reducible orbit, for example, 12 , repeats se-
quence I 6 orbit two times. Sequence III consists of re-
ducible and irreducible orbits also, with reducible orbitsrepeating I orbits three times and separated by two irre-ducible orbits.
For m 1 helium, the Stark structure should look hy-
drogenic. We find this to be true; however, two of thehydrogen recurrence sequences (II and III) are modified,containing modulations in peak strength not present in hy-drogen. Recent experimental work on the Stark struc-ture of lithium found nonhydrogenic behavior in m 0states, identifying new peaks in the spectrum at loca-tions far from hydrogen, and due to combination orbitsfrom core scattering [5]. In the present case combina-tion orbits occur very close to hydrogenic peaks, modi-fying the hydrogenic spectra because of interference andproducing the novel structure shown in Fig. 2. Combi-nation orbits in II arise from two different orbit pairsin I.
Combination orbits that interfere with repeating
i͞i 1 1 3 2 hydrogen orbits are i 2 1͞i i 1 1͞i 1 2and i 2 2͞i 2 1 i 1 2͞i 1 3, where indicates [4]core scattering from one hydrogenic orbit into another.
Integrated experimental recurrence strength versus
For example, at ´ 22.7, the 2 3 9 orbit recurs at
hydrogen orbit type. Arrows indicate repeating orbits. Closed
triangles: triplet helium; open circles: singlet helium. Hydrogen
Sk 8.187 and interferes with the 8 10 orbit recur-
calculation is shown as a dashed line. Sk 8.177 and the 7 11 orbit at ˆS
( b) Sequence III. Vertical scales are the same.
P H Y S I C A L R E V I E W L E T T E R S
calculation [8], normalized at the recurrence strength of
A recent theoretical Letter [10] on diffractive orbits of
the singlet 9 repeating orbit, included to show lack
Rydberg atoms in external fields provides new insight into
the classical interpretation of quantum defects. Our work
the recurrence strength’s dramatic dependence on spin
provides stimulus for investigation of the classical inter-
state. Overall, singlets and hydrogen behave similarly. In
pretation of the exchange interaction and illuminates this
the triplet case strong modulation in recurrence strength
intriguing aspect of classical-quantum correspondence.
In summary, the measurement of different spin states
(repeating) orbit recurrence strength is favored at lower
in our experiment demonstrates the influence of exchange
(higher) scaled action. In III, triplet modulation grows in
strength with repeating orbits always favored.
(i) pronounced modulation in the recurrence map of triplet
The origin of the selectivity to hydrogenic orbit type
helium, not present in singlet helium nor in hydrogen, and
(primitive or repeating) of the modulations can be ex-
(ii) strong phase correlation of the modulation with re-
plained within the context of closed orbit theory. In the
peating hydrogenic orbits, producing suppression of recur-
vicinity of a hydrogenic recurrence peak there are sev-
rence strength in second repetitions and enhancement of
eral combination orbit peaks. The phase of the interfer-
strength in third repetitions. These effects result from the
S, where F21͞4 is the average
interference between hydrogenic orbits and core-scattered
combination orbits, and demonstrate the sensitivity of re-
distance between peaks. Orbit calculations indicate that
currence strength to the spatial symmetry of the wave
S is usually very small, of order 0.01. However, since
function imposed by the Pauli exclusion principle. Incor-
the value of F21͞4 is proportional to n, it has a consid-
poration of an effective potential to model the influence of
erable value, about 60, so the total phase shift can be of
the exchange interaction in the semiclassical framework is
order p. The closer a combination recurrence is to a hy-
drogenic peak, the smaller the phase of the interference.
We are grateful to R. Jensen, V. Kondratovich, and
Consequently, the main cause of the observed interference
J. Shaw for valuable discussions, to D. Cullinan and
effects in the recurrence spectrum is contained in D ˆ
H. Flores for help in the lab, and special thanks to Sam
S has a regular pattern which depends on
(i) the hydrogenic orbit type (primitive or repeating) and(ii) the recurrence sequence (type II or III orbits). In-spection shows that clusters of nearby combination peaksare much closer to primitive hydrogenic orbits than to re-peating ones and have a regular dependence of the mini-mal distance between hydrogenic and combination peaks
[1] T. F. Gallagher, Rydberg Atoms (Cambridge University
on the type of hydrogenic orbit. Denoting D ˆ
Press, Cambridge, England, 1994); Rydberg States of
the shortest distance between repeating and combination
Atoms and Molecules, edited by R. F. Stebbings and
orbits in sequence II, it can be shown [9] that this dis-
F. B. Dunning (Cambridge University Press, Cambridge,England, 1983).
tance in sequence III is ͑ 3 ͒d, and that the comparable dis-
[2] J. Gao and J. B. Delos, Phys. Rev. A 49, 869 (1994).
tances for primitive orbits are ͑ 1 ͒d and ͑ 1 ͒d. It is the
[3] P. A. Dando et al., Phys. Rev. Lett. 74, 1099 (1995); Phys.
Rev. A 54, 127 (1996).
tions which correlate with hydrogenic orbits. The ampli-
[4] B. Hüpper et al., Phys. Rev. Lett. 74, 2650 (1995); Phys.
tude of the oscillations can be estimated with the aid of
Rev. A 53, 744 (1996).
scattering theory as Abs ͑e2pim 2 1͒2 ϳ 4p2m2 0.19,
[5] M. Courtney et al., Phys. Rev. Lett. 73, 1340 (1994); Phys.
for a quantum defect m 0.07. This value is in ac-
Rev. Lett. 74, 1538 (1995); M. Courtney et al., Phys. Rev.
cord with our measurements, which vary between 20%
A 51, 3604 (1995).
and 50% (see Fig. 3), since several combination orbits
[6] U. Eichmann et al., Phys. Rev. Lett. 61, 2438 (1988). [7] H. A. Bethe and E. Salpeter, Quantum Mechanics of One-
interfere with different phases. A detailed closed orbit
and Two-Electron Atoms (Plenum, New York, 1977).
theory calculation of the experimentally observed modu-
[8] Levels were calculated using Ref. [7], with oscillator
lations must include explicit phase counting of all relevant
strengths from D. A. Harmin, Phys. Rev. A 24, 2491
combination orbits and integration over a range of scaled
energy. A thorough theoretical analysis of this type will
[9] V. Kondratovich (private communication).
[10] P. A. Dando et al., Phys. Rev. Lett. 80, 2797 (1998).

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