Table Of Content+
High-resolution spectroscopy of He using
2
Rydberg-series extrapolation and Zeeman-decelerated
supersonic beams of metastable He
2
7
1
∗
Paul Jansen, Luca Semeria, Fr´ed´eric Merkt
0
2
Laboratory of Physical Chemistry, ETH Zurich, CH-8093 Zurich, Switzerland
n
a
J
3
1
Abstract
]
h
p Recently,high-resolutionspectroscopyofslowbeamsofmetastableheliummol-
- ∗
m ecules (He ) generated by multistage Zeeman deceleration was used in combi-
2
e nation with Rydberg-series extrapolation techniques to obtain the lowest rota-
h
c tionalintervalinthemolecularheliumionataprecisionof18MHz[Jansenetal.
.
s
c Phys. Rev. Lett. 115(13)(2015)133202], limitedbythetemporalwidthofthe
i
s Fourier-transform-limited laser pulses used to record the spectra. We present
y
h here an extension of these measurements in which we have (1) measured higher
p
+
[ rotational intervals of He2 , (2) replaced the pulsed UV laser by a cw UV laser
1 and improved the resolution of the spectra by a factor of more than five, and
v
6 (3) studied MJ redistribution processes in regions of low magnetic fields of the
4
Zeeman decelerator and shown how these processes can be exploited to assign
6
3 ′′ ′′ ∗
transitions originating from specific spin-rotational levels (N ,J ) of He .
0 2
.
1 Keywords: High-Resolution Spectroscopy, Molecular Rydberg States, Zeeman
0
Deceleration, Metastable Helium Molecules.
7
1
2015 MSC: 00-01, 99-00
:
v
i
X
1. Introduction
r
a
Translationally cold samples of molecules offer interesting perspectives for
high-resolution spectroscopy. The long measurement times that are possible
∗
[email protected]
Preprint submitted to Journal of Molecular Spectroscopy January 16, 2017
with such samples and the reduced Doppler widths are ideally suited for pre-
cision measurements of transition frequencies in molecules, and such measure-
mentsarebeginningtoberelevantinthecontextoftestsofthestandardmodel
of particle physics and some of its extensions [1]. Precision measurements in
+ +
few-electron, light molecules such as H , H and He are used as tests of
2 2 2
ab initio quantum-chemical calculations which aim at an exact solution of the
Schr¨odingerequationandarigorousdeterminationofrelativisticandquantum-
electrodynamics(QED)corrections[2,3,4,5,6]. Inthesemolecules,thevelocity
of the electrons is relatively low, and nonrelativistic quantum-electrodynamics
turnsouttobeparticularlysuccessful. Inthisapproach,theenergyisexpressed
asaseriesexpansioninpowersofthefine-structureconstantα, whichisamea-
sure of the classical electron speed
E(α)=E(0)+α2E(2)+α3E(3)+α4E(4)+O(α5). (1)
The terms in different powers of α are associated with different contributions
to the overall energy. E(0) contains the Born-Oppenheimer energy including
adiabatic and nonadiabatic interactions, while α2E(2) and α3E(3) represent the
relativistic and leading-order QED corrections. Higher powers of α are associ-
ated with higher-order QED corrections. Recent calculations for the molecular
hydrogen ion include relativistic and QEDcorrections up to terms proportional
to α6 and report an accuracy of 2kHz for the first vibrational intervals of H +
2
+
[3] and HD [6]. The most accurate calculations of the energies in H , HD, and
2
D include full corrections up to terms proportional to α3 as well as the dom-
2
inant one-loop contribution of the α4 term [4, 5]. The reported uncertainties
are less than 30MHz and the calculated and experimental results agree within
+
this uncertainty [7, 8]. The best calculations of the rovibrational levels of He
2
[9] have an accuracy of about 120MHz, sufficient to reproduce the energy-level
structuremeasuredinearlierexperiments[10,11],althoughtheydonotinclude
relativistic and radiative corrections.
Few-electron diatomic molecules present experimental challenges for high-
resolution studies of their spectra, and experimental data on their energy-level
2
structures are scarce. Indeed, the symmetric isotopomers do not have a per-
manent electric dipole moment, which implies that these species do not have a
pure rotational nor a rovibrational spectrum. Spectroscopic data on the molec-
ular helium cation are limited to rotational and vibrational transitions in the
asymmetric 3He4He+ isotopomer reported by Yu et al. [10, 11] and microwave
transitions between highly excited vibrational levels of the electronic ground
state and the lowest vibrational levels of the first electronically excited states
+
in He by Carrington et al. [12]. The only experimental data available on
2
thelow-lyingrovibrationallevelsof4He + havebeenobtainedbyphotoelectron
2
spectroscopy [13] and from the Rydberg spectrum of He using Rydberg-series
2
extrapolation techniques [14, 15, 13, 16, 17].
The work presented in this article is devoted to measurements of the energy
+
level structure of He by high-resolution spectroscopy of Rydberg states of
2
He and extrapolation of the Rydberg series [13, 16, 17]. The strength of our
2
+
approach to study He relies on the facts that (1) its energy levels structure is
2
obtainedbyextrapolationofallowedelectronictransitionsofHe intheultravi-
2
olet (UV) range of the electromagnetic spectrum, (2) the initial state of He we
2
use, the metastable a3Σ+ state (called He∗ hereafter), can easily be generated
u 2
∗
in supersonic beams, (3) He has a magnetic moment of two Bohr magneton,
2
∗
which makes it possible to decelerate He beams to low velocities in the lab-
2
oratory reference frame using the technique of multistage Zeeman deceleration
∗
[18, 19], and (4) the electronic spectrum of He is well known and information
2
on low-lying Rydberg states and fine-structure intervals facilitates the interpre-
tation of spectra of high Rydberg states. We believe that, in the long term,
these advantages will enable us to reach a higher precision and accuracy than
currently possible in H [7, 8, 20].
2
ThespectrumofHe hasbeeninvestigatedexhaustivelyafteritsfirstdetec-
2
tionin1913,independentlybyCurtis[21]andGoldstein[22]. Mostinformation
about the Rydberg states of He has been obtained with classical emission
2
grating spectroscopy in the extensive measurements of Ginter and coworkers
[23, 24, 25, 26, 27, 28, 29, 30, 15]. In addition, low-lying Rydberg states have
3
been investigated using infrared emission [31] and absorption [32] spectroscopy,
laser-induced fluorescence spectroscopy [33], Fourier-transform emission spec-
troscopy [34, 35, 36, 37], laser absorption spectroscopy [38, 39, 40, 41], optical
heterodyne concentration-modulation spectroscopy [42], and infrared emission
spectroscopy from proton-irradiated cryogenic helium gas [43, 44]. Highly ac-
∗
curate measurements of the fine structure in the lowest rotational states of He
2
(ν′′ = 0) have been performed by Lichten et al. using molecular-beam radio-
frequency (r.f.) spectroscopy [45, 46, 47]. Bjerre and coworkers [40, 48, 41]
employed laser-r.f. double-resonance spectroscopy to extend the measurements
of the fine-structure intervals to higher rotational and vibrational states. Focsa
et al. [36]performedaglobalfitoninfraredandr.f. datatoobtainaconsistent
set of molecular constants for the six lowest excited electronic states of He .
2
Thestructureofthisarticleisasfollows: Afteranoverviewofcurrentknowl-
∗
edge on He , on the triplet Rydberg states of He , and on the ground state of
2 2
+
He in section 2, we summarize our experimental approach in section 3. The
2
experimentalresultsare presentedinsection4anda briefsummaryisprovided
in the conclusions section.
+
2. Energy levels of He and He : general considerations
2 2
Thevan-der-Waalsinteractionbetweentwoheliumatomsintheir1S ground
0
state is extremely weak and gives rise to a very shallow potential-energy well
for the ground state of He , with a depth in the order of 10−3cm−1 [49] and
2
a single bound rovibrational state with a mean internuclear distance of almost
5 nm [50]. In contrast, He + in its X+2Σ+ ground state is covalently bound,
2 u
+
with a well depth of almost 2.5eV [9]. The strongly bound nature of the He
2
ground state implies the existence of singlet and triplet Rydberg series of He .
2
Many Rydberg states are known for He that all belong to series converging on
2
the X+2Σ+ electronic ground state of He + [51, 52, 15]. With the exception of
u 2
thesingleboundleveloftheelectronicgroundstate,allboundstatesofHe are
2
Rydberg states, so that He can be regarded as a Rydberg molecule [53].
2
4
∗
2.1. Metastable helium molecules He
2
The lowest Rydberg states of He , the a3Σ+ state, is metastable, with a
2 u
calculated radiative lifetime of 18s [54], because radiative decay to the ground
electronic state is spin forbidden. Its long lifetime and the ease with which it
∗
can be produced in electric discharges make He an ideal initial state to study
2
the electronic spectrum and the photoionization of He , as demonstrated in the
2
∗
numerous studies cited in the introduction. The triplet nature of He gives rise
2
to a magnetic moment of two Bohr magneton and thus to an electron-Zeeman
effectthatcanbeexploitedtoslowdownsupersonicbeamsofHe bymultistage
2
Zeeman deceleration [19, 17] (see also Sections 3 and 4).
The generalized Pauli principle requires the total wavefunction to be sym-
metric under exchange of the two bosonic 4He2+ (I = 0) nuclei, so that only
rotational states for which the quantum number N (N is the quantum number
associatedwiththetotalangularmomentumexcludingspin)isoddareallowed
instatesofΣ+ symmetry, suchasthea3Σ+ stateof4He andtheX+2Σ+ state
u u 2 u
+
of He . The rotational and fine structure in the vibrational ground state of
2
∗
He canbedescribedbyaneffectiveHamiltonian[55,56]appropriatetoHund’s
2
case (b) molecules in electronic states of Σ symmetry
H =B N⃗2−D N⃗4+H N⃗6+ 2λ (3S2−S⃗2)+γ S⃗⋅N⃗, (2)
0 0 0 3 0 z 0
where B is the rotational constant, D and H are the quartic and sextic cen-
0 0 0
trifugal distortion constants, λ is the spin-spin interaction constant, γ is the
0 0
spin-rotation interaction constant, and N⃗ and S⃗ are the total angular momen-
tum excluding spin and the total electron spin, respectively. The spin-spin and
spin-rotation interactions split each rotational state N into three fine-structure
componentswithtotalangularmomentumquantumnumberJ =N,N±1. Lev-
els of the same J value but N values differing by 2 mix under the influence of
the spin-spin interaction. Matrix elements of Eq. (2) can be found in Ref. [56].
The fine-structure splittings of the three lowest rotational states in He (a3Σ+
2 u
ν =0) are shown on the left-hand side of Fig. 1.
5
212.06 J” N”
212.04
212.02 4 5
212.00
6
211.98
211.96 5
211.94
1) 75.88
-m 75.86
c
) ( 75.84 2 3
c
h 75.82
( 4
y/ 75.80
g
er 75.78 3
n
e 75.76
0.06 0 1
0.04
0.02 2
0
1
-0.02
-0.04
-0.06
0 10 20 30 40 50
magnetic-field strength (mT)
Figure 1: Fine structure (left panel) and Zeeman effect (right panel) of the N′′ =1,3, and 5
statesofHe∗. Thelow-field-seekingmagneticsublevelsoftheJ′′=N′′+1manifoldareshown
2
inredandstatesoftheJ′′=N′′ andN′′−1areshowningrey.
To treat the effects of an external magnetic field, one needs two additional
terms in the Hamiltonian
g µ g µ
H =− e BS⃗⋅B⃗ − R BN⃗⋅B⃗, (3)
Z h̵ h̵
where g ≈ −2.00232 and g denote the electron and rotational g-factors, re-
e R
spectively. The second term on the right-hand side of Eq. (3) represents the
coupling of the overall rotation of the molecule to the external magnetic field.
This coupling is four orders of magnitude weaker than the coupling of the elec-
tronspintothefieldandplaysanegligibleroleforthedecelerationexperiments
described below. On the right-hand side of Fig. 1, we show the eigenvalues of
the combined Hamiltonian as a function of the magnetic-field strength for the
∗
three lowest rotational states in He . The Zeeman effect in higher rotational
2
6
states is almost identical to that for N = 5 because the fine structure at zero
field only slowly changes with N at high N values and the rotational spacings
are much larger than the Zeeman shifts, even for magnetic-field strengths of
several Tesla [19].
+
2.2. He and He Rydberg states
2 2
In the X+2Σ+ electronic ground state, the rotational structure of He + can
u 2
also be described by Eq. (2). However, only the spin-rotation interaction [last
termontheright-handsideofEq. (2)]contributestothefinestructure,because
thereisonlyoneunpairedelectron. Thespin-rotationinteractionintheX+2Σ+
u
+ +
state of He splits each rotational state N into two fine-structure states with
2
J+ = N+ ± 1. In the following, we denote quantum numbers and molecular
2
constants of the a3Σ+ state of He and the X+2Σ+ state of He + with double
u 2 u 2
primed symbols and a “+” superscript, respectively, to avoid confusion. The
splittingbetweenthetwoJ+ statesofagivenN+ valueisgivenbyγ+(N++1),
0 2
+
where γ is the spin-rotation coupling constant of the vibrational ground state
0
of He + (X+2Σ+ ν+ =0).
2 u
+ +
Although γ has not been measured for He yet, one can estimate it to be
0 2
≈−3MHzfromtheknownspin-rotationcouplingconstantin3He4He+ [11]and
thefactthatγ+ scalesasµ−1 [57], whereµisthereducedmassofthemolecule.
0
+ +
Toresolvethetwofine-structurelevelsoftherotationalstatesofHe withN
2
values greater than 9 and 51, an experimental resolution of 25 and 150MHz,
respectively, would be required.
The energy-level structure of low-lying Rydberg states of He is adequately
2
describedbyHund’sangular-momentumcouplingcase(b)andthoroughanaly-
ses of many of these low-lying states have been reported (see references cited in
the introduction). Rydberg states of high principal quantum number are more
conveniently described by Hund’s angular-momentum coupling case (d). The
∗
RydbergseriesofHe thatcanbeaccessedfromHe bysingle-photonexcitation
2 2
are npσ 3Σ+ and npπ 3Π± in Hund’s case (b) notation and npN+ in Hund’s
g g N
case (d) notation. The two angular-momentum coupling schemes are related
7
by a unitary angular-momentum frame transformation with which rotational-
electronic interactions are treated in the realm of multichannel quantum-defect
theory[58]. Thehigh-nstatesweusetoextrapolatetheseriesanddeterminethe
+
rotational energy-level structure of He are best described in Hund’s angular-
2
momentum coupling case (d), as illustrated schematically in Fig. 2. Three np
+ + +
series converge on each rotational level N of He , with N values of N and
2
N+ ±1 (N⃗ = N⃗+ +(cid:96)⃗, with (cid:96) = 1 for p series). The series with N = N+ have
pure Hund’s case (b) npπ 3Π− character and those with N =N+±1 and N >1
g
have mixed npσ 3Σ+ and npπ 3Π+ character.
g g
+
Because N, and not N , is the good quantum number, levels of the same
+
N value that converge on different rotational states of He interact, giving
2
rise to spectral perturbations below and to rotational autoionization above the
N+ = N − 1 thresholds. These interactions, indicated by horizontal arrows
+
in Fig. 2, couple series differing in N by 2 and need to be accounted for in
the extrapolation of the Rydberg series. The best way to do so is by using
multichannel quantum-defect theory as implemented by Jungen [58]. In our
+
determinationofthelowestrotationalintervalofHe weextrapolatedtheseries
2
using the quantum defect parameters of the np triplet states of He reported
2
by Sprecher et al. [16]. If the positions of the rotational levels of the cation are
not known with sufficient precision, as is the case for the series converging to
the N+ = 11 and 13 ionic levels discussed in Section 4, the series limits can be
extrapolated in first approximation using Rydberg’s formula
hcR
hcν˜ =E (He∗)+E (He +)− He2 , (4)
n(cid:96) I 2 rv 2 (n−δ )2
(cid:96)
where ν˜ represents the spectral position of the Rydberg states of principal
n(cid:96)
and orbital angular-momentum quantum numbers n and (cid:96), respectively, and
quantum defect δ . The quantities E (He∗), E (He +) and R represent the
(cid:96) I 2 rv 2 He2
∗ +
adiabatic ionization energy of He , the rovibrational energy of the He ion
2 2
core,andthemass-correctedRydbergconstantforHe ,respectively. Rydberg’s
2
+
formula adequately describes the noninteracting npN series and also gives
N=N+
goodextrapolationresultsfortheothertwoseriesifstatesofveryhighnvalues
8
34550
N+ = 5
34500
34450
34400 N+ = 3
) 34350
1
-m N+ = 1
34300
c
(
ue 34250 N = 0 1 2 2 3 4 4 5 6
l
va ~ν
m 55
250
r 4
te 200 ~ν N” = 5 56
35
150 ~ν
33
100 2
4
~
50 ~ν ν13N” = 3 3
11
0
0 2
N” = 1 1
∗
Figure2: Energy-leveldiagramshowingtherotationallevelsofHe andthetripletnpRydberg
2
+
states of He that converge to the three lowest rotational levels of He . The positions of
2 2
purerotationallevelsofthemetastablestatearemarkedbydashedhorizontallines. Thespin-
rotation fine structure is exaggerated for clarity. Solid and dashed arrows indicate optically
allowed transitions and channel interactions, respectively. Rapidly autoionizing levels are
drawninblue.
are used in the extrapolation.
The spin-spin coupling of the triplet Rydberg states of He scales with n−3
2
andbecomesnegligibleathighnvalues. Thespin-rotationinteractionisprimar-
ily that of the ion core, so that the fine structure converges to the spin-rotation
+
splitting of He at high values of n (see Fig. 6 of Ref. [59]).
2
Therotationalselectionrulesforsingle-photonexcitationfromHe∗tonpσ3Σ+
2 g
and npπ3Π± Rydberg states are given by N −N′′ =0,±1. Combined with the
g
∆(cid:96) = ±1 selection rule for transitions between Rydberg states, the overall se-
lection rule for single-photon excitation from the a3Σ+ state to Rydberg levels
u
converging to the X+2Σ+ state is given by ∆N = N+ −N′′ = 0,±2. Transi-
u
9
∗
tionsfromHe tonpRydberglevelscanthereforebeunambiguouslylabeledas
2
N′′ →npN+. Transitions to Rydberg states with ∆N =0,−2, and +2 are re-
J′′ N
′′ ′′ ′′
ferred to as Q(N )-type, O(N )-type, and S(N )-type transitions, respectively.
The Q-type transitions are far stronger than the O- and S-type transitions be-
+
cause the He ion core is left with an electron hole of mainly s character after
2
excitation of He to np Rydberg states [60]. However, O-type transitions can
2
gainintensitythroughrotationalchannelinteractionsbetweenlevelsconverging
on different rotational states of the ion discussed above and indicated by the
horizontal dashed arrows in Fig. 2. The observation of these O-type transitions
inHe isessentialforthedeterminationoftherelativepositionsofenergylevels
2
in both the metastable state and ion ground state.
3. Experimental setup and procedure
A schematic view of the laser systems and the experimental setup is shown
in Fig. 3. In the experiments, we use both a pulsed UV laser system with
a near Fourier-transform-limited bandwidth of 150MHz [depicted in Fig. 3a)]
andacontinuous-wave(cw)single-modeUVlaserwithabandwidthof1.5MHz
[depicted in Fig. 3c)]. To generate the pulsed UV radiation, the cw output of a
ring dye laser with a wavelength around 580nm is pulse amplified in dye cells
pumped with the second harmonic of a neodymium-doped yttrium-aluminium-
garnet (Nd:YAG) laser and frequency doubled in a beta-barium-borate (BBO)
crystal. The pulse amplified fundamental radiation is frequency calibrated to
an accuracy of 20MHz (1σ) with a wave meter and a fraction of the cw output
of the ring dye laser is used to record the laser-induced-fluorescence spectrum
of I , as desribed in Ref. [17]. The difference between the frequencies of the
2
cw and pulse-amplified outputs of the ring-dye laser is used to quantify the
effects of the frequency chirp arising in the pulse-amplification process. The
experiments involving cw UV radiation made use of the cw output of the ring
laser, which was frequency doubled in an external cavity. In the experiments
involvingcwradiation,onlytherelativefrequencywasmeasuredwithanetalon
10
