J. Plasma Fusion Res. SERIES, Vol.7 (2006) 68-72

The Stark Effect and Field Ionization in Triplet
Rydberg States of Helium Atoms
TAKAHASHI Risako, YAMAKITA Yoshihiro∗ , HORI Nobutaka, OHNO Koichi
Department of Chemistry, Graduate School of Science, Tohoku University,
Sendai 980-8578, Japan（Received: 5 October 2004 / Accepted: 9 February 2006）

Abstract
The DC Stark eﬀect is studied for high Rydberg states (principal quantum number n ≥ 15) of helium atoms
in the triplet spin state. A signal is detected through pulsed field ionization by an electric field F = 6000 V/cm
with a slew rate of 4.0 × 1010 Vcm−1 s−1 . Stark manifolds are clearly observed in the region below the classicalsaddle point energy of the Stark field and above the adiabatic pulsed field ionization limit. A detailed Stark
map is calculated for magnetic quantum numbers m = 0 and |m| = 1 states based on matrix diagonalization
to explain pulsed-field ionization dynamics. Ionization of the Stark states is well explained by the adiabatic
ionization passages.
Keywords:
Stark eﬀect, Rydberg state, metastable helium, field ionization, avoided crossing
1．Introduction
The Stark eﬀect can be utilized for controlling
the translational motion of gas-phase atoms [1] andmolecules [2] based on a similar principle to that of
the Stern-Gerlach experiment. In particular, an electric field gradient opposing the molecular translation
can take away translational energy from the molecule
when it is switched oﬀ abruptly. Indeed Meijer and coworkers have demonstrated deceleration and trapping(25 mK) of neutral ND3 molecules using fast-switched
multi-stage electric fields [2]. Electric-field deceleration is therefore possible for neutral atoms and nonpolar molecules in Rydberg states, since high Rydberg
states gives rise to large Stark shifts in energy. Actually molecular control of deflection and deceleration has
been demonstrated for a supersonic beam of non-polar,
non-magnetic neutral H2 [3].
This paper summarizes spectroscopic measurementson the He atom with n ≥ 15 and |m| = 0, 1, and describes the behavior of these states in static and pulsed
electric fields. The Stark eﬀect in He has been extensively studied for more than 50 years, and basicallyestablished by numerous early studies [4-6], and also
by recent studies based on scaled-energy spectroscopy[7,8]. However, to our best knowledge, the reported
experimental data are limited to either a narrow rangeof large n values (n ≈ 40) [5,6], or diﬀerent experimental conditions in terms of, e.g., field strength andtunablity of lasers [4]. Hence experimental data are de-sired for appropriate ranges of n and F, in the unscaled
axes of energy and field, and simple theoretical calculations are of practical use rather than sophisticated theoretical treatments, such as WKB-quantum defect theory [4,6]. In this paper, we investigate an experimental”Stark map” for a range of ca. 15 ≤ n ≤ 20, and discuss temporal behavior in time-dependent fields using
matrix diagonalization of eﬀective Hamiltonian [5,9].
2．Experiment
A schematic overview for the experimental apparatus
is shown in Fig. 1. A metastable helium atom beam is
produced by a continuous discharge source [10]. A setof three parallel fine grids is placed in the excitationV
800 nsV1
0

V2
Photoexcitationt 1 = t 2 =150 ns
t1 t2 t1
ionizing
field
6000 V

0V
He *
V2
V1

t

V 2 :Stark field( 0~-1600 V)

He + MCPs

Deflector

Nd:YAG
laser
355nm
Dye Laser

Discharge
Nozzle0V-500V
260-264 nm

Doubler
BBO

Fig. 1 Experimental apparatus. A shape of ionizing
pulsed ﬁeld is presented in the inset.
C 2006 by The Japan Society of Plasma

Science and Nuclear Fusion Research

Corresponding author’s e-mail: yy@qpcrkk.chem.tohoku.ac.jp

68The Stark Eﬀect and Field Ionization in Triplet Rydberg States of Helium Atoms

region with separation of 1.0 and 1.5 cm respectively.
Grid 2 applies a homogeneous DC Stark field of 0 to–1600 V/cm (inset, Fig. 1). The metastable atoms arephoto-excited to high Rydberg states by a UV pulsed
laser in the presence of the Stark field. Grid 1 generates
an ionizing pulsed field of magnitude 6000 V/cm withrespect to the Stark voltage, which is controlled by afast high-voltage pulse generator (DEI PVX-4130). It
increases from 0 to 6000 V/cm in 150 ns with a delay
time of 800 ns, keeps a constant voltage for 150 ns, anddecreases in 150 ns. Spectra are recorded by monitoring
the ions as a function of laser frequency.
A UV beam (λ = 259 − 264 nm, 10 Hz, 6 ns) is
provided by a frequency-doubled dye laser pumped
by a Nd:YAG laser. The frequency is resonant with
one-photon transitions from the intermediate 23 S state(159856.07760 cm−1 above the ground state [11]) to theRydberg series converging on the ionization threshold(IE = 198310.7723 cm−1 ). The wavelength of the UV
light is calibrated with an iodine cell. The polarizationof the laser can be set either parallel or perpendicular
to the Stark field, which defines the space quantization
axis z. From the intermediate 23 S state, the Rydberg
states with m = 0 or |m| = 1 are accessible by the parallel or perpendicular polarizations respectively.

reproduces the experimental values with an accuracy
0.075 cm−1 . The intensity distribution in the regionof n ≥ 18 reflects the ionization rate and oscillator strength. All the transitions are recorded throughpulsed-field ionization in a total duration 450 ns (see inset, Fig. 1) with a delay time of 800 ns. The Rydberg
states with larger n can be field ionized for a longer interval due to the rise time of the field. Moreover theionization rate dramatically decreases as n decreases inan electric field. The oscillator strength for the ns-n ptransition decreases as n p energy increases since thenodes of the bound np wavefunction become close together with increasing energy [6,13]. In the region below n = 18 the intensity decreases towards zero as itgoes to n = 15. This is likely due to the slow ionization
rates in a short time window of the ionizing pulse forthe lower-n Rydberg states.
4.2 Stark spectroscopy and field ionization dynamics

Figure 3 (b) presents the Stark spectra of the m = 0,
1 snp3 P Rydberg states recorded in homogeneous electric fields of 0 – 1000 V/cm. At field larger than 400
V/cm, additional peaks are observed in the lowwavenumber sides of the Stark manifolds. These peaks
originate from l = 0 states and gain their intensities by
mixing with l = 1 states. The magnitude of mixing canbe recognized as the increase of their intensities relative to the Stark manifolds. The mixing of diﬀerent lstates can be viewed more quantitatively in Stark maps
(Fig. 3 (a)). Separation width at avoided crossings of
the upper and lower states diﬀers significantly between
the cases of m = 0 and |m| = 1 (Fig. 4 (a)), which iscommonly found in alkali-metal atoms [9]. The width
which is approximately twice the magnitude of the oﬀdiagonal matrix elements nlm|z|n l m  in Eq. (2) islarger in the m = 0 states, where the lobe of the electronic wavefunction of a Rydberg electron extends tothe same z-direction, than in the m  0 states in which
there are degenerate wavefunctions pointing to various
directions. The width is estimated to be 1 cm−1 and0.3 cm−1 at the first avoided crossing between n = 15
and 16 extreme states.
Field ionization takes place from the position of theapplied Stark field towards the right-hand side of the
figures, where Stark states couple to the continuum.
It is commonly known that a Stark sub-level populated traverses adiabatically or diabatically dependingon the slew rate of the ionizing pulsed field [14]. The
mechanism determines the threshold for field ionization. If field ionization is adiabatic, it goes along zigzag passages and the threshold would be approximately

3．Calculation
We employ a matrix diagonalization treatment whichis essentially the same as those for alkali metals [9] except that the formalism of Edmonds et al. [12] is used toevaluate the radial integral in Eq. (2). The Hamiltonian
is of the form
(1)
H = H0 + eFz,
where the matrix elements of z are given asvlm|z|v l m  = δm,m δl,l±1 vl|r|v l  × lm| cos θ|l m ,(2)where eﬀective quantum number v = n − µl . The diagonal matrix elements of the zero-field Hamiltonian are
given by
vlm|H0 |vlm = IE −RHe,(n − µl )2(3)

where RHe is the Rydberg constant for the helium atom.
The values for the quantum defects µl are available inRef. [11].
4．Results and discussion
4.1 Zero-field spectroscopy

Figure 2 shows the zero-field spectrum of the 1snp3 P
Rydberg series of He; the levels resolved are in therange n = 15 – 58 and a single quantum defect [11]69The Stark Eﬀect and Field Ionization in Triplet Rydberg States of Helium Atoms

263

261

260

Ion Signal (arb. units)

262

n=15

197800161719790018

19

20 21 22 23 2425

198000198100-1
Energy (cm )3035198200198300Fig. 2 Zero-ﬁeld spectrum of the 1sn p3 P, |m| = 1 Rydberg series of He.

√
at the classical saddle point energy, 6.12 F(cm−1 ) below the ionization energy, where F is given in V/cm
(Figs. 3 (a) and 4 (a)). Probability of the diabatic transition calculated for the slew rate of the ionizing pulse
S = dF/dt = 4.0×1010 Vcm−1 s−1 is PD = 1.9×10−13 in
the |m| = 1 case, assuming energy change for the above
mentioned two states is linear. Therefore diabatic ionization is highly unlikely, and the Rydberg states of bothm = 0 and |m| = 1 overwhelmingly ionize through the
adiabatic passages.
Comparisons between experimental and calculated
spectra are given in Figs. 3 and 4 for the m = 0 and|m| = 1 states, respectively. First the experimental intensity distributions of the Stark manifolds are in reasonable agreements with the calculated ones, of which
intensities are borrowed from the optically-allowed np
states. It follows that the transition intensities reflect
predominantly the magnitude of mixing with the np
states. The Stark manifolds show asymmetric distributions until they become nearly complete hydrogenic
manifolds.
Second it is remarkable that the observed numberof sub-levels in the n = 16 Stark manifolds is fewer
than the calculated ones (low-energy sides in Fig. 4 (c)).
This is likely to be connected with the ionization threshold, which is determined by the relation between the
adiabatic ionization passage and the classical saddlepoint energy. In Figs. 3 (a) and 4 (a) the lowest Stark
sub-levels which can be field ionized are shown as boldlines. In the case of m = 0, the lowest sub-level in
the n = 16 manifold which is associated with 16s statemeets the saddle point energy at 6000 V/cm. On thecontrary, the lowest sub-level of the n = 16, |m| = 1
manifold is located far below the energy. The saddle
point energy for the m  0 states is also lifted by a few% from those for m = 0 states, owing to the centrifugal
barrier of a Rydberg electron [13]. Hence the solid zigzag line depicted in Fig. 4 (a) marks the lowest states
which can ionize by the field, and precisely corresponds
to the observed threshold for ionization.
4.3 Ionization rate and tunnelling ionization

The ionizing field employed in the present measurements holds its peak for 150 ns, therefore the ionization
rate is required to exceed Γ = 4.7 × 106 s−1 for the states
to be detected. On the other hand, the ionization rate
for the H atom shows that the rate for the highest extreme sub-levels is much smaller than those for the lowest states [15]. However, in the non-hydrogenic atom,
Stark states can mix with each other, and the amountof mixing with the fast-ionizing sub-level would eﬀectively determine the ionization rate. Observation of then = 15 Rydberg state (Fig. 2) is likely to be connectedwith tunnelling ionization, since it lies below the ionization threshold.
5．Conclusions
The experimental spectra are fairly well explained bya simple calculation based on matrix diagonalization of
the eﬀective Hamiltonian. Possibility for the diabatic
ionization at the avoided crossings is negligibly small
for a field of a slew rate S = 4.0 × 1010 Vcm−1 s−1 . A
70

The Stark Eﬀect and Field Ionization in Triplet Rydberg States of Helium Atoms

(a)

(b)

(c)

1980001000 V/cm
1000 V/cm

18( l >2)18p
18s
197950800 V/cm

800 V/cm

17s

Ion signal (arb. units)

Energy (cm-1)

17( l ->2)
17p

19790016( l ->2)
16p
16s
19785015( l >2)
15p

600 V/cm

600 V/cm

400 V/cm
400 V/cm

0 V/c m0 V/c m197800
15s

0.5
n =16

0

1000

2000

3000

4000

5000

6000

197850

0.1
n =16

17
197900Electric Field (Vcm-1)

197950 197850

17197900197950

Energy (cm-1)

Fig. 3 (a) Stark map for m = 0 Rydberg states. Comparison between (b) experimental and (c) calculated spectra of n = 16
and 17 Stark manifolds. The simulation has been convoluted with a Gaussian bandwidth (FWHM) of 0.5 cm–1 . The
grey region represents the continuum beyond the classical saddle point energy.

(a)

(b)

(c)

1000 V/cm

1000 V/cm

19800018( l >2)
18p
197950Ion signal (arb. units)

Energy (cm-1)

800
V/cm

800
V/cm

17( l >2)17p
19790016( l >2)
16p

197850

600
V/cm

600
V/cm

400
V/cm

400
V/cm

15( l >2)15p
1978000
0.5 V/cm

0 V/c m
n =16

0

1000

2000

3000

4000

5000

6000

197850197900

17

0.1

n =16

197950 1 97850

19790017197950

Energy (cm-1)

Electric Field (Vcm-1)

Fig. 4 (a) Stark map for |m| = 1 Rydberg states. (b) Experimental and (c) calculated spectra. See also the caption to Fig. 3.

pulsed electric field of a height 6000 V/cm ionizes Rydberg states adiabatically, whereas state n = 15 is most
likely to ionize through tunnelling ionization. Some of
the Stark sub-levels of the n = 16 manifold are not observed in the |m| = 1 case. These sub-levels are cut-

oﬀ by the ionization threshold, which corresponds tothe increased height of the classical saddle point energy
due to the centrifugal barrier. The high Rydberg statesstudied here are triplet states and radiatively decay tothe metastable 23 S state of which lifetime is 4200 s.

71

The Stark Eﬀect and Field Ionization in Triplet Rydberg States of Helium Atoms

Therefore if decelerating and trapping triplet Rydberg
He atoms is possible, it could lead to new experimental
techniques for future studies involving a translationallycold bunch of the metastable He (23 S).

[5]
[6]
[7]

Acknowledgements
This work was financially supported by the MEXT
through a Grant-in-Aids for Young Scientists (A) (No14703003), and by the Matsuo Foundation through a
research grant. The authors thank Professor T. P. Softley
for encouraging discussions.

[8]
[9]
[10]

[1]
[2]
[3]

[4]

References
T. Breeden and H. Metcalf, Phys. Rev. Lett. 47,
1726 (1981).
H.L. Bethlem and G. Meijer, Int. Rev. Phys.
Chem. 22, 73 (2003).
Y. Yamakita, S.R. Procter, A.L. Goodgame, T.P.
Softley and F. Merkt, J. Chem. Phys. 121, 1419
(2004) and references therein.
W. van der Water, D.R. Mariani and P.M. Koch,Phys. Rev. A 30, 2399 (1984).

[11]
[12]
[13]
[14]

[15]

72

C.T.W. Lahaye and W. Hogervorst, Phys. Rev. A
39, 5658 (1989).
A. Nussenzweig, E.E. Eyler, T. Bergeman and E.
Pollack, Phys. Rev. A 41, 4944 (1990).
R.V. Jensen, H. Flores-Rueda, J.D. Wright, M.L.
Keeler and T.J. Morgan, Phys. Rev. A 62, 053410(2000) and references therein.
C.T.W. Lahaye and W. Hogervorst, Phys. Rev. A
58, 3043 (1998).
M.L. Zimmerman, M.G. Littman, M.M. Kash and
D. Kleppner, Phys. Rev. A 20, 2251 (1979).
K. Ohno, T. Takami, K. Mitsuke and T. Ishida, J.
Chem. Phys. 94, 2675 (1991).
W.C. Martin, Phys. Rev. A 36, 3575 (1987).
A.R. Edmonds, R. Pullen, J. Picart and N. Tran
Minh, J. Phys. B 12, 2781 (1979).
T.F. Gallagher, Rydberg Atoms (Cambridge University Press, Cambridge, 1994).
J.R. Rubbmark, M.M. Kash, M.G. Littman and
D. Kleppner, Phys. Rev. A 23, 3107 (1981) andreferences therein.
R.J. Damburg and V.V. Kolosov, J. Phys. B 12,2637 (1979).