J. Plasma Fusion Res. SERIES, Vol.7 (2006) 64-67

The Stark Effect and Translational Control of Hydrogen Molecules
YAMAKITA Yoshihiro1, 2 , PROCTER Simon R.2 and SOFTLEY Timothy P.2
1 Department of Chemistry, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan
2 Department of Chemistry, University of Oxford, Chemistry Research Laboratory, Mansfield Road, Oxford, OX1 3TA, United Kingdom（Received: 5 October 2004 / Accepted: 16 January 2006）

Abstract
The Stark eﬀect is studied for the gerade Rydberg states (principal quantum number n ≥ 16) of molecularhydrogen by using vacuum ultraviolet/ultraviolet two-photon excitation and detecting ions produced by delayed field ionization. Experiment and theory show that the nd Rydberg states with N + = 2, J = 1, M J = 0
are long-lived, and have predominant contributions to the intensities of the Stark manifolds (the quantumnumbers are defined as N + : rotational angular momentum of the ion core, J: total angular momentum, M J :quantization-axis projection of J). The deflection of a supersonic beam of Rydberg H 2 molecules is demonstrated by using the inhomogeneous dipole electric field. With the dipole rods mounted parallel to the beam
direction, the high-field-seeking and low-field-seeking Stark states are deflected towards and away from thedipole, respectively.
Keywords:
Rydberg state, Stark eﬀect, vacuum ultraviolet photoexcitation, cold molecule
1．Introduction
This paper reports on the demonstration of the deflection of Rydberg H2 molecules in a supersonic beamand high-resolution spectroscopy of the Stark eﬀect in
the Rydberg states of H2 . Controlling the translationalmotion of atoms and molecules in the gas phase is ofinterest for experimental methods in studies on cold
collisions and Doppler-reduced spectroscopy in the gas
phase. It is connected with the research area of cold
molecules, following the successful development oflaser cooling methods for atoms. However the opticalcycling methodology used in the laser cooling cannot
be extended to molecules due to complicated internalenergy levels in molecules.
One of the alternative approaches for coolingmolecules is the use of the Stark eﬀect which takes
place in a molecule with a significant electric dipole
moment in the ground state. An electric-field gradientopposing a molecular beam direction can create decelerating (or accelerating) forces F = −∇E on molecules
in so-called low-field-seeking states (high-field-seekingstates). For ND3 molecules translational temperaturesof ∼25 mK have been achieved with respect to the laboratory frame [1].
The method presented in this paper can be regarded
as an extension of this approach. Since the Rydberg
states are highly susceptible to electric fields, the Stark
eﬀect in Rydberg states leads to a significant force onRydberg atoms [2,3] and molecules [4]. The advantagesof the Rydberg-Stark control method include that it canbe applied to any non-polar, non-magnetic moleculein principle and that it does not need a large electricfield of order 100 kV/cm. However the disadvantages
to be overcome include the complicated energy-levelstructures of the Rydberg series converging to diﬀerent
ro-vibrational levels of the ion, and the short lifetimes
due to autoionization, predissociation and radiative processes. In this paper we report on the spectroscopic results for the Stark eﬀect in Rydberg states of H2 anddiscuss the behavior in electric fields such as field ionization properties, state mixing, and decay processes.
Spectral comparisons between experiment and theory
allow us to understand the Rydberg-Stark control experiment quantitatively.
2．Method
The experimental apparatus has been described indetail elsewhere [5,6]. Briefly the Rydberg states
are excited by two-photon, two-color (110.81 nm and∼295 nm) excitation via a selected rotational level of theB1 Σ+u intermediate state by using nano-second pulsed
dye lasers (∼ 6 ns). The vacuum ultraviolet (VUV) radiation is generated by non-resonant frequency tripling
in a cell filled with rare gases (Kr/Ar=1100/ 200 mbar).
Both laser beams are focused into a vacuum chamber
C 2006 by The Japan Society of Plasma

Science and Nuclear Fusion Research

Corresponding author’s e-mail: tim.softley@chemistry.ox.ac.uk

64

The Stark Eﬀect and Translational Control of Hydrogen Molecules

perpendicularly to each other and also to a supersonicmolecular beam.
In the measurement of the Stark eﬀect, a homogeneous electric field is applied across a pair of grids V1and V2 (15 mm separation), as shown in Fig. 4 (a) (i).
The Rydberg states are excited between V1 and V2 ,and then field ionized by a pulsed field of magnitude∼5000 V/cm when they enter the region between grids
V2 and V3 (10 mm separation). Laser polarizations are
set perpendicular to the electric field, defining the final state to be M J = 0, ±2 for excitation from an initial state with J = 0, M J = 0. Deflection experiments
are performed with a pair of rods mounted parallel tothe molecular beam (diameter 0.9 mm, length 20 mm,
center-to-center distance 4 mm). The Rydberg states are
field ionized, and detected as ions with a position sensitive detector after travelling through a ∼ 30 cm field-freeregion.
Theoretical calculations are performed by diagonal= H
0 + eFz withization of the eﬀective Hamiltonian H+ +a basis set of the form {v N nlJM J } [7-9]. The previously reported values for the quantum defects [7-9], andthe ionization thresholds I(0) = 124417.507(10) cm−1[10] and I(2) = 124591.743 cm−1 are used in the calculations.

1000 V/cm

Ion Signal (arb. units)

800 V/cm
600 V/cm
400 V/cm
200 V/cm
100 V/cm
0 V/cm1617201825124100

1242001920302135124300

402223 24 25 30

nd0124400

3540124500

nd2124600

Energy (cm-1)

Fig. 1 Pulsed ﬁeld ionization spectra of H2 obtained in
the presence of various homogeneous electricﬁelds in the range 0 – 1000 V/cm shown as a function of the sum of the VUV and UV excitation energies. The ionizing pulsed ﬁeld is applied aftera delay of 3.7 s. The dotted line represents theclassical saddle point energy of the N + = 0 threshold.

(17d2)1

Energy (cm-1)

1242003．Results and Discussion
At zero field, the Rydberg states accessible are thens and nd series converging to the N + = 0 and 2 vibrotational level (ν+ = 0) of the H+2 ion since the electronic wavefunction of H+2 (X2 Σ+g ) has a gerade symmetry. The selection rule ∆J = 0, ±1 restricts the final
states of the present two-photon excitation to the (ns0) 0 ,(ns2)2 , (nd0)2 , (nd2)0 , (nd2)1 , (nd2)2 , and (nd4)2 states[notated (nlN + ) J ].
Fig. 1 (bottom) shows the Rydberg excitation spectrum of H2 recorded at 0 V/cm. Most of the observed
peaks are assignable to the nd series converging to theN + = 0 and 2 thresholds as shown in the figure. It isnoted that the N + = 2 series are observed as strongly
as the N + = 0 series although autoionization is energetically possible for the N + = 2 series. Since theions produced by rapid autoionization or direct ionization are eliminated by a small delayed discrimination
field V2 ≈ 20 V/cm, the observed peaks are undoubtedly ascribed to the Rydberg states which survive a delay time of 3.7 µs. Applying the Stark field makes all
the N + = 0 Rydberg states disappear very easily (seeFig. 1); even at 100 V/cm the N + = 0 Rydberg states are
hardly noticeable except for a weak broad band around124350 cm−1 . If we examine the appearance onsets for
the Stark manifolds, they seem to be correlated with the(16d2)1124150

124100
01000

2000
3000
Electric Field (V/cm)

4000

5000

Fig. 2 Stark map including N + = 0 (n = 15–17) MJ = 0states. The dotted line indicates the position of
the classical saddle point energy.

classical saddle point energy associated with the N + = 0threshold at IE(0) = 124417.507(10) cm−1 (dotted linein Fig. 1). Thus the Stark manifolds of the N + = 2
threshold are likely to be field ionized in the presenceof the static electric fields through interaction with thecontinuum associated with the N + = 0 threshold.
Fig. 2 shows the Stark map calculated for the N + =2, M J = 0 states. The Stark map showing how theenergy levels vary with electric field is used to predict
the force exerted on the molecule in an inhomogeneous
field (see below). Fig. 3 represents simulated Starkspectra. Although the selection rule ∆M J = ±1 for theσ-polarization for two photons allows the final stateswith M J = 0, ±2, spectral simulations for the M J = ±2states do not fit the observed spectra at all. The exper-65

The Stark Eﬀect and Translational Control of Hydrogen Molecules

regions of the manifolds in addition to the main
prominent series. Considering the coupling scheme
J = N + + l (i.e., M J = MN+ +ml ) in the present case, it
is reasonable to suppose that the three fine structure
series in the N + = 2, M J = 0 manifold are due to unequal contributions from ml = 0, 1, and 2 which result in diﬀerent quantum defects, intensities and lifetimes. Such a feature was discussed in the previous
multichannel quantum defect theory study [12].
4. It might seem contradictory that the classical saddle
point energy is located at the middle of the n = 16
manifold for a field of 5000 V/cm in comparison tothe fact that the entire n = 16 manifold is observed at800 V/cm and 1000 V/cm. However it should be recalled that the excited Stark sub-levels traverse to thezero-field Rydberg states when the Rydberg states

(b)

(a)
1000 V/cm

Intensity (arb. units)

800 V/cm

600 V/cm

400 V/cm

200 V/cm

x 0.1

0 V/cmx 0.1

124140 124150 124160 124170

124150 124160 124170 124180

Energy (cm-1)

Fig. 3 Comparison between (a) experimental and (b) calculated Stark spectra of the N + = 2, n = 16, MJ = 0Stark manifold. The simulation has been convoluted with a Gaussian laser bandwidth with fullwidth at half maximum of 0.5 cm–1 .

imental spectra are fairly well reproduced only if thetransition probability to the (nd2)1 , M J = 0 states is assumed to be non-zero and to the other optically-allowed
states zero (Fig. 3). From comparisons between experiment and theory the following can be mentioned:
1. At 0 V/cm and 200 V/cm, the calculated (16d2)1
band is more than ten times stronger than the experimental one. The discrepancy may seem to be due todecay by predissociation, but the (16d2)1 states giverise to no peaks in the previous ion-detected Rydberg excitation spectra [8]. Therefore the weak intensities are caused by other reasons. It is noted that
intensities become increasingly small on going fromhigh Rydberg states of n ≈ 65 to lower states, whichis in contrast to the general tendency of oscillator
strengths in atomic Rydberg states [11].
2. Some of the sub-levels are not observed in the lowwavenumber side of the n = 16 Stark manifold in
the presence of fields of 400 V/cm and 600 V/cm. A
similar behavior is also observed for the n = 17 Stark
manifold [6]. Thus the absence of these states is unlikely to be caused by an ionization mechanism near
the threshold of field ionization.
3. The missing sub-levels regain their intensities at800 V/cm and 1000 V/cm (Fig. 3), and the intensities of the entire set of sub-levels show a good
agreement with calculation. Two fine structure series are noticeable in the low- and high-wavenumber

Fig. 4 (a) Schematics of the apparatus, and (b) Rydberg excitation spectra of H2 recorded in the presence of (i) a homogeneous electric ﬁeld 800 V/cm
and (ii) an inhomogeneous dipole ﬁeld ∼800 V/cm.
Signals were detected utilizing a position sensitive
detector (PSD). (c) Images showing the distributions of the H2 Rydberg molecules after deﬂectionin the inhomogenesous ﬁeld (ii) for 7 s. The Rydberg levels labeled A-G are excited by scanningwavelength of the UV laser respectively. The Rydberg states are ﬁeld ionized in a ﬁeld 5000 V/cm
between the grids after 10 – 11 s ﬂight time.

66

The Stark Eﬀect and Translational Control of Hydrogen Molecules

enter the field-free region between V2 and V3 before an ionizing electric field is applied. Therefore
most of the populated sub-levels are at once merged
into the field-free eigenstates, presumably diabatically, and then field ionized through zig-zag adiabatic ionization passages shown in Fig. 2.
Figure 4 presents the results of the deflection experiments of Rydberg H2 . The Rydberg states of n = 16
and 17 are excited in the presence of an inhomogeneous
dipole field (Fig. 4 (a) (ii)). In the Stark spectrum in Fig.
4 (b) (ii) the extreme Stark sub-levels are broadened bythe inhomogeneity of the field and a finite excitation
volume compared to that recorded in the homogeneous
field (Fig. 3 and Fig. 4 (b) (i)). The images labeled AG in Fig. 4 (c) correspond to the deflected Rydberg H2molecules which are initially populated in Stark states
A-G (Fig. 4 (b)), respectively. The supersonic beam interacts with the field along a length of ∼20 mm (see Fig.
4 (a) (ii)). The dipole field is switched oﬀ before being ionized at ∼ 28 mm away from the excitation point.
The trajectories of the molecules are monitored usingion-imaging and time-of-flight measurements. The oﬀaxis displacement (deflection angle) is estimated to be+1.3 mm (+2.7◦ ) and -0.6 mm (+1.2◦ ) for the low-fieldseeking and high-field-seeking states, respectively. The
square shapes represent the intersections of VUV andUV lasers which are magnified by non-planar electricfields in the vicinity of the grids and the long flight
length for H2 ions (∼ 30 cm). The maximum magnitudeof a force in the present experiment is estimated to be
1.0 × 10−19 N from the gradient of the dipole field andthe slopes of the Stark map.

priate Rydberg states and pulsed electric fields. We believe the present experiment will initiate methodological development for controlling molecular motion in thegas phase.
Acknowledgements
This work is financially supported by the EPSRC.
One of the authors (Y.Y.) gratefully acknowledges theRamsay Memorial Fellowships Trust for a Fellowship,
the Daiwa Foundation and the Matsuo Foundation for
financial aid, and the MEXT for a Grant-in-Aid forYoung Scientists (A) (No. 14703003).
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4．Conclusion
The experimental spectra contain the details of thedecay processes. The long-lived (nd2)1 states are likely
to be field ionized into the N + = 0 continuum. The finestructure is clearly resolved for the n = 16 manifold
at fields greater than 800 V/cm. Controlling Rydberg
molecules can be realized by critically choosing appro-67