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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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