On Measuring the Electron Electric Dipole Moment in Trapped Molecular Ions
ABSTRACT Trapped diatomic molecular ions could prove to be a sensitive probe for a
permanent electron electric dipole moment (eEDM). We propose to use a ground or
metastable $^3\Delta_1$ level, due to its high polarizability and large EDM
enhancement factor. Ions allow for simple trapping and long interrogation
times, but require a time-varying electric bias field in order to probe the
eEDM. We discuss experimental design as well as challenges in performing a
precision spectroscopic measurement in rapidly time-varying electric fields.
Article: The Physics of Time Reversal[show abstract] [hide abstract]
ABSTRACT: The notion that fundamental equations governing the motions of physical systems are invariant under the time reversal transformation (T) has been an important, but often subliminal, element in the development of theoretical physics. It serves as a powerful and useful tool in analyzing the structure of matter at all scales, from gases and condensed matter to subnuclear physics and the quantum theory of fields. The assumption of invariance under T was called into question, however, by the 1964 discovery that a closely related assumption, that of CP invariance (where C is charge conjugation and P is space inversion), is violated in the decay of neutral K mesons. In The Physics of Time Reversal, Robert G. Sachs comprehensively treats the role of the transformation T, both as a tool for analyzing the structure of matter and as a field of fundamental research relating to CP violation. For this purpose he reformulates the definitions of T, P, and C so as to avoid subliminal assumptions of invariance. He summarizes the standard phenomenology of CP violation in the K-meson system and addresses the question of the mysterious origin of CP violation. Using simple examples based on the standard quark model, Sachs summarizes and illustrates how these phenomenological methods can be extended to analysis of future experiments on heavy mesons. He notes that his reformulated approach to conventional quantum field theory leads to new questions about the meaning of the transformations in the context of recent theoretical developments such as non-Abelian gauge theories, and he suggests ways in which these questions may lead to new directions of research.American Journal of Physics 01/1987; · 0.78 Impact Factor
arXiv:1008.2997v3 [physics.atom-ph] 18 Jul 2011
High-resolution spectroscopy on trapped molecular ions in rotating electric fields:
A new approach for measuring the electron electric dipole moment
A.E. Leanhardt,2J.L. Bohn,1H. Loh,1P. Maletinsky,3E.R. Meyer,1L.C. Sinclair,1R.P. Stutz,1and E.A. Cornell1, ∗
1JILA, National Institute of Standards and Technology and University of Colorado,
and Department of Physics, University of Colorado, Boulder, Colorado 80309-0440, USA
2Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA
3Physics Department, Harvard University, Cambridge, Massachusetts, 02138, USA
(Dated: July 19, 2011)
High-resolution molecular spectroscopy is a sensitive probe for violations of fundamental symme-
tries. Symmetry violation searches often require, or are enhanced by, the application of an electric
field to the system under investigation. This typically precludes the study of molecular ions due to
their inherent acceleration under these conditions. Circumventing this problem would be of great
benefit to the high-resolution molecular spectroscopy community since ions allow for simple trap-
ping and long interrogation times, two desirable qualities for precision measurements. Our proposed
solution is to apply an electric field that rotates at radio frequencies. We discuss considerations for
experimental design as well as challenges in performing precision spectroscopic measurements in
rapidly time-varying electric fields. Ongoing molecular spectroscopy work that could benefit from
our approach is summarized. In particular, we detail how spectroscopy on a trapped diatomic molec-
ular ion with a ground or metastable3∆1 level could prove to be a sensitive probe for a permanent
electron electric dipole moment (eEDM).
Keywords: High-resolution spectroscopy, radio-frequency, fundamental symmetries, Stark and Zeeman inter-
actions, molecular ions
A.High-Resolution Molecular Spectroscopy as a Probe of Fundamental Physics
The quest to verify the most basic laws of nature, and then to search for deviations from them, is an ongoing challenge
at the frontier of precision metrology. To this end, high resolution spectroscopy experiments have made significant
contributions overthe years. For example, the coupling strengths and transition energies between atomic and molecular
levels are predominantly determined by the electromagnetic interaction. However, the Standard Model does include
fundamental processes, e.g. the weak interaction , which have spectroscopic signatures that are both theoretically
calculable and experimentally detectable. Parity-violating transition amplitudes, forbidden by the electromagnetic
interaction but allowed in the presence of the weak interaction, have been calculated and measured in atomic cesium [2,
3] and ytterbium  with sufficient precision to test electroweak theory at the ∼ 1% level. In addition, high-resolution
molecular spectroscopy experiments are underway to probe parity violation in chiral polyatomic molecules [5–8] and
to probe nuclear spin-dependent parity violation in diatomic molecules [9, 10]. Looking outside of the Standard
Model, precision molecular spectroscopy experiments have been designed to search for time-variation of fundamental
constants, such as the electron-to-proton mass ratio [11–15] and the fine structure constant [11, 16], as well as to
search for simultaneous parity and time-reversal symmetry violation in the form of permanent electric dipole moments
In most cases, atoms and molecules that are either neutral or ionic can be studied in an effort to observe the same
underlying physics; however, typically there are technical advantages to selecting one system over the other. Systems
of neutral, as opposed to ionic, particles are attractive for precision spectroscopic studies due to the relative ease
of constructing high-flux neutral particle beams, the relatively weak interactions between neutral particles, and the
lack of coupling between the translational motion of neutral particles and external electromagnetic fields. Conversely,
charged particles are favored due to the relative ease of constructing ion traps and the long interrogation times
that come with studying trapped particles. Indeed, some of the most stringent tests of the Standard Model have
been performed using trapped ions [31–34], and spectroscopy on trapped molecular ions is of fundamental interest
for studying interstellar chemistry [35–37]. Looking to combine the techniques of ion trapping and high-resolution
∗Electronic address: email@example.com; URL: http://jilawww.colorado.edu/bec/CornellGroup/
TABLE I: Theoretical predictions of the electron electric dipole moment, de. Current listings are taken from Ref. , which
extracted the numbers from Refs. [57, 58].
CP Violating Model
Left-right symmetric models
Lepton flavor changing models
|de| [e cm]
|de| < 10−38
|de| < 10−27
10−28< |de| < 10−26
10−28< |de| < 3 × 10−27
10−29< |de| < 10−26
molecular spectroscopy, several research groups are working to develop experimental platforms for studying ensembles
of trapped molecular ions [38–43].
The additional degrees-of-freedom afforded to molecular systems, in comparison with simple atomic systems, pro-
vide additional interaction mechanisms and correspondingly more routes for experimental investigation. For example,
molecular levels are inherently more sensitive to applied electric fields due to the presence of nearby states of opposite
parity, e.g. rotational levels and/or Λ-doublet levels. On the surface, this means that the Stark shifts observed in
molecular spectra will be significantly larger than the corresponding shifts to atomic transitions. More fundamen-
tally, this means that in relative weak electric fields the quantum eigenstates of an atomic system are still dominated
by a single parity eigenstate, while the quantum eigenstates of molecular systems asymptotically approach an equal
admixture of even and odd parity eigenstates. There are several classes of atomic and molecular symmetry violation
experiments where larger Stark mixing amplitudes give rise to larger signals. For example, the parity violation signals
already attained in atomic systems [2–4] are expected to be exceeded by the next-generation of experiments using
polarized diatomic molecules [9, 10]. Similarly, in experiments designed to search for permanent electric dipole mo-
ments, the expected signal size scales with the ability to thoroughly mix parity eigenstates and increases dramatically
when going from atoms to diatomic molecules [44–46].
Herein lies the conundrum for symmetry violation searches using trapped molecular ions: the electric field required
to fully polarize the molecules will interfere with the electromagnetic fields necessary for trapping the ions with the
likely result of accelerating the ions out of the trap. Our solution to this problem is to apply an electric field that
rotates at radio frequencies. Under these conditions, the ions will still accelerate, however they will undergo circular
motion similar to charged particles in a Penning trap [31–33] or storage ring [47–52]. The nuances of performing
high-resolution electron spin resonance spectroscopy in this environment will be the main focus of this work, with the
ultimate goal of demonstrating that such an experiment on the valence electrons in a ground or metastable3∆1level
could prove to be a sensitive probe for a permanent electron electric dipole moment (eEDM).
B.Motivation for Electric Dipole Moment Searches
The powerful techniques of spin resonance spectroscopy, as applied to electrons, muons, nuclei, and atoms, have
made possible exquisitely precise measurements of electric and magnetic dipole moments. These measurements in
turn represent some of the most stringent tests of existing theory, as well as some of the most sensitive probes for new
particle physics. As an example, the recent improved measurement of the electron’s magnetic moment  agrees with
the predictions  of quantum electrodynamics out to four-loop corrections. Compared to the electron work, muonic
g-2 measurements [53, 54] are less accurate but are nonetheless more sensitive (due to the muon’s greater mass) to
physics beyond the Standard Model. Digging a new-physics signal out of the muon g-2 measurement is made difficult
by uncertainty in the hadronic contributions to the Standard Model prediction . One of the primary motivations
for experimental searches for electric dipole moments (EDM) is the absence of such Standard Model backgrounds to
complicate the interpretation of these studies. In the case of the electron, for example, the Standard Model predicts
an electric dipole moment less than 10−38e cm . The natural scale of the electron electric dipole moment (eEDM)
predicted by supersymmetric models is 10−29to 10−26e cm [57–59] (Table I). The current experimental limit is
|de| < 1.6 × 10−27e cm . With predictions of new physics separated by nine orders of magnitude from those
of “old” physics, and with the current experimental situation such that a factor-of-ten improvement in sensitivity
would carve deeply into the predictions of supersymmetry, an improved measurement of the eEDM is a tempting
experimental goal. In this paper we will describe an ongoing experiment that we believe will be able to improve on
the existing experimental upper limit for an eEDM by a factor of thirty in a day of integration time.
C.A Brief Overview of the JILA Experiment
Our JILA eEDM experiment will be based on electron spin resonance (ESR) spectroscopy in a sample of trapped
diatomic molecular ions. We will use an Λ-doubled molecular state that can be polarized in the lab frame with a
lab frame electric field of only a few volts/cm. The very large internal electric field of the molecule, coupled with
relativistic effects near the nucleus of a heavy atom, will lead to a large effective electric field, Eeff, on the electron
spin. Confining the molecules in a trap leads to the possibility of very long coherence times and therefore high
sensitivity. Trapping of neutral molecules has been experimentally realized recently, but it remains an extremely
difficult undertaking. Conversely, trapping of molecular ions is straight forward to implement with long-established
On the face of it, measuring the electric dipole moment of a charged object is problematic. Even for a relatively
polarizable object like a molecule, one must apply sufficient electric field to mix energy eigenstates of opposite parity.
This field will cause the ion to accelerate in the lab-frame and limit trapping time. We will circumvent this problem
via the application of a rotating electric bias field, which will drive the ion in a circular orbit. The rotation rate will
be slow enough that the molecule’s polarization can adiabatically follow the electric field, but rapid enough that the
orbit diameter is small compared to the trap size. The ESR spectroscopy will be performed in the rotating frame. We
note that this approach is conceptually related to efforts measuring electric dipole moments of charged particles in
storage rings [47–52], but in our case the radius of the circular trajectory will be measured in millimeters, not meters.
Precision spectroscopy in time-varying fields can be afflicted with novel sources of decoherence and systematic error,
which will be discussed in Secs. IV, V, and VI.
D. A Comparative Survey of Ongoing Experimental Work
The primary purpose of this section will be to review experimental searches for eEDM. We will make no attempt to
survey the rapidly increasing diversity of low-energy  and astrophysical searches for physics beyond the Standard
Model. A subset of that broad area of endeavor is the search for permanent electric dipole moments (EDMs), and
a subset within that focuses on electrons (eEDMs). For comparative surveys of the discovery potential of various
EDM studies see [63–67], we summarize here by saying that from the point of view of new physics, experiments on
leptons provide physics constraints complementary to those on diatomic atoms and to those directly on bare nucleons
and nuclei. As for the lepton experiments, there is work on the tau lepton , on muons [51, 52] and of course
on electrons as discussed in some detail below. The current best neutron EDM measurement was done at ILL ;
there are ongoing neutron EDM searches [69–71]. Beam-line measurements on bare nucleons are envisioned .
The current best atomic dipole measurement is an experiment is in the diamagnetic species, Hg, by the Washington
group . Many other groups are looking for EDMs in diamagnetic (that is, net electron spin S = 0), ground-state
electronic levels in Hg , Xe [74–77], Rn , Yb  and Ra [67, 80–82]. Experiments on diamagnetic atoms
(with net electron spin S = 0) are sensitive to new physics predominantly via the nucleonic contribution to the Schiff
moment of the corresponding atomic nucleus. Higher-order contributions from eEDM contribute to the atomic EDM
of S = 0 atoms , but these are probably too small to provide a competitive eEDM limit.
For 20 years the most stringent limits on the eEDM have been the atomic-beam experiments of Commins’ group
at Berkeley [60, 61, 84]. That work set a standard against which one can compare ongoing and proposed experiments
to improve the limit. Here is a brief survey of ongoing experiments of which we are aware.
For evaluating the sensitivity of an eEDM experiment the key figure-of-merit is Eeffτ√N, where Eeffis the effective
electric field on the unpaired electron, τ is the coherence time of the resonance, and N is the number of spin-flips
that can be counted in some reasonable experimental integration time, for instance one week. The statistics-limited
sensitivity to the eEDM is just the inverse of our figure-of-merit. We will discuss the three terms in order.
The conceptually simplest version of an eEDM experiment would simply be to measure the spin-flip frequency of
a free electron in an electric field Elab, ωd= deElab, where deis the electric dipole moment of the electron . Alas,
a free electron in a large electric field would not stay still long enough for one to make a careful measurement of its
spin-flip frequency; in practice all eEDM experiments involve heavy atoms with unpaired electron spins. An applied
laboratory electric field distorts the atomic wavefunction, and the eEDM contribution to the atomic spin-flip frequency
ωdis enhanced by relativistic effects occurring near the high-Z nucleus [86, 87], so that ωd= deEeff, where the effective
electric field Eeffcan be many times larger than the laboratory electric field Elab. The enhancement factor is roughly
proportional to Z3although details of the atomic structure come into play such that the enhancement factors for
thallium (Z = 81) and cesium (Z = 55) are −585  and +114 , respectively. Practical DC electric fields in
a laboratory vacuum are limited by electric breakdown to about 105V/cm. The Commins experiment used a very
high-Z atom, thallium, and achieved an Eeff of about 7 × 107V/cm . There have been proposed a number of
experiments in cesium [90–92] that expect to achieve Eeffof about 107V/cm. A completed experiment at Amherst 
achieved Eeff= 4.6 × 105V/cm in Cs by using Elab= 4 kV/cm.
It was pointed out by Sandars [44–46] that much larger Eeff can be achieved in polar diatomic molecules. In these
experiments, the atomic wavefunctions of the high-Z atom are distorted by the effects of a molecular bond, typically
to a much lighter partner atom, rather than by a laboratory electric field. One still applies a laboratory electric field,
but it need be only large enough to align the polar molecule in the lab frame. The Imperial College group  is
working with YbF, for which the asymptotic value of Eeffis 26 GV/cm [22, 94–99]. The Yale group [17–19] uses PbO,
with an asymptotic value of Eeff ≃ 25 GV/cm [20, 21, 100]. The Oklahoma group  has proposed to work with
PbF, which has a limiting value of Eeff≃ 29 GV/cm [25, 26]. The ACME collaboration  will use ThO, with Eeff
≃ 100 GV/cm . The Michigan group is working with WC, with Eeff≃ 54 GV/cm . We will discuss candidate
molecules for our experiment in Sec. IIB; we anticipate having an Eeffof around 25 to 90 GV/cm [28, 30, 101].
After Eeff, the next most important quantity for comparison is the coherence time τ, which determines the linewidth
in the spectroscopic measurement of ωd. In Commins’ beams experiment, τ was limited by transit time to 2.4 ms.
Future beams experiments may do better with a longer beam line , or with a decelerated beam . Groups
working in laser-cooled cesium anticipate coherence times of around 1 s, using either a fountain  or an optical trap
[91, 92]. The PbO experiment has τ limited to 80 µs by spontaneous decay of the metastable electronic level in which
they perform their ESR. Coherence in ThO experiment will be limited by the excited-state lifetime to 2 ms . A
now discontinued experiment at Amherst  achieved τ = 15 ms in a vapor cell with coated walls and a buffer gas.
The JILA experiment will work with trapped ions. The mechanisms that will limit the coherence time in our trapped
ions are discussed in Secs. IV and V. We anticipate a value in the vicinity of 300 ms.
The quantity Eeffconverts a hypothetical value of deinto a frequency ωd, and τ sets the experimental linewidth of
ωd. The final component of the overall figure-of-merit is
state sensitivity, and low background counts, determines the fractional precision by which we can split the resonance
line. Since we have defined N as the number of spin flips counted, detection efficiency is already folded into the
quantity. Vapor-cell experiments such as those at Amherst or Yale can achieve very high values of effective N, atomic
beams machines are usually somewhat lower, and molecular beams usually lower yet (due to greater multiplicity of
thermally occupied states.) Atomic fountains and atomic traps have still lower count rates, but the worst performers
in this category are ion traps. The JILA experiment may trap as few as 100 ions at a time, and observe only 4
transitions in a second.
The discussion above is summarized in Table II. To improve on the experiment of Commins, it is necessary to
do significantly better in at least one of the three main components of the figure-of-merit. The various ongoing or
proposed eEDM experiments can be sorted into categories according to the component or components in which they
represent a potential improvement over the Commins’ benchmark. The prospects of large improvements in both τ
and Eeffput JILA’s experiment in its own category. This combination means that our resonance linewidth, expressed
in units of a potential eEDM shift, will be 105times narrower than was Commins’. Splitting our resonance line by
even a factor of 100 could lead to an improved limit on the eEDM. This is an advantage we absolutely must have,
because by choosing to work with trapped, charged molecules, we have guaranteed that our count rate,˙N, will be far
smaller than those of any of the competing experiments.
We note that there are in addition ongoing experiments attempting to measure the eEDM in solid-state systems [103–
106]. These experiments may also realize very high sensitivity, but because they are not strictly speaking spectroscopic
measurements, it is not easy to compare them to the other proposals by means of the same figure-of-merit.
Finally, atoms with diamagnetic ground states may have S ?= 0 metastable states amenable to an eEDM search
. Closely spaced opposite parity states in Ra can give rise to an Eeff on the electron spin larger  than in Tl
or Cs, but very short coherence times  may make complicate efforts  to measure the eEDM in Ra.
√N, which, assuming good initial polarization, good final-
A brief overview on the molecular level structure where the eEDM will be measured and on how the measurement
will be performed is given below in Sec. II. Some aspects of the experimental design, including production of molecular
ions and ion trapping will be covered in Sec. III. Difficulties in performing precision spectroscopy in time-varying
and inhomogeneous electric and magnetic fields will be discussed in Sec. IV. This will include discussions of trap
imperfections, stray magnetic fields, and effects of rotating bias fields. Experimental chops used to minimize systematic
errors will also be explained. In Sec. V, the effects on spin coherence time and systematic errors of ion-ion collisions
will be investigated. An estimate for experimental sensitivity to the eEDM will be given in Sec. VI. The Appendix
gives a listing of variables used throughout the paper and a sample set of experimental parameters.
TABLE II: Figure-of-merit comparison between several recently completed and ongoing eEDM experiments. For ongoing
experiments these numbers are subject to change and are often order-of-magnitude estimates. For the JILA entry M is Hf, Th,
or Pt and x is H or F.
GroupRefs. Species Elab [V/cm] Eeff [V/cm]
Berkeley Tl1.23 × 105
Amherst Cs4 × 103
Imperial [22, 23] YbF8.3 × 103
Oklahoma[24–26]PbF7 × 104
7 × 107
4.6 × 105
2.5 × 1010
1.3 × 1010
2.9 × 1010
5.4 × 1010
3 − 9 × 1010
2.4 × 10−3
1.5 × 10−2
8 × 10−5
2 × 10−3
0.2 − 1
II.MOLECULAR STRUCTURE AND THE BASIC SPECTROSCOPIC IDEA
A. Molecular Notation
As we prepare this paper, we have not made a final decision as to which molecule we will use. For reasons discussed
below, the main candidates are diatomic molecular ions Mx+, where M = Hf, Pt, or Th and x = H or F. In the case
of molecules such as HfF+, ab initio methods [30, 101] enable us to determine that the3∆ state is well described
by a set of Hund’s case (a) quantum numbers: J,S,Σ,Λ,Ω,MJ,e/f. Here J is the sum of electronic plus rotational
angular momentum, S the total electronic spin angular momentum, Σ the projection of S onto the molecular axis,
Λ the projection of L, the electronic orbital angular momentum, onto the molecular axis, and Ω the projection of J
onto the molecular axis. In a case (a)3∆ molecule |Ω| can take the values one, two or three. MJ is the projection of
J along the quantization axis and the labels e/f specify the parity of the molecular state.
In addition to these quantum numbers, the experiment will be concerned with the nuclear spin quantum number
I, the total angular momentum quantum number F, given by the vector sum of J and I, and mF the projection of
F along the quantization axis. Throughout this paper we shall assume a total nuclear spin of I = 1/2, the nuclear
spin of fluorine or hydrogen. This leads to the values F = 3/2 and F = 1/2 for the states of experimental interest.
B.Choosing a Molecule
In selecting a molecular ion for this experiment we have several criteria. First, we want a simple spectrum. Ideally,
we would like the supersonic expansion to be able to cool the molecules into a single internal quantum state so that
every trapped molecule could contribute to the contrast of the spectroscopic transition. Failing that, we want to
minimize the partition function by using a molecule with a large rotational constant, most likely a diatomic molecule
with one of its atoms being relatively light. Small or vanishing nuclear spin is to be preferred, as are atoms with only
one abundant isotope. Second, we need to be able to make the molecule. This requirement favors more deeply bound
molecules and is the main reason we anticipate working with fluorides rather than hydrides. Third, the molecule
should be polarizable with a small applied electric field, i.e. it should have a relatively small Λ-doublet splitting, ωef.
Fourth, and most important, the molecule should have unpaired electron spin that experiences a large value of Eeff.
These latter two requirements would appear to be mutually exclusive: a small Λ-doublet splitting requires a
large electronic orbital angular momentum, which prohibits good overlap with the nucleus required for a large Eeff.
Fortunately, working with two valance electrons in a triplet state allows us to satisfy our needs. One valance electron
can carry a large orbital angular momentum making the molecule easily polarizable, while the other can carry zero
orbital angular momentum giving it good overlap with the nucleus and generating a large Eeff. This concept was
detailed by some of us in Ref.  and for the3∆1state of interest here, the two valance electrons occupy molecular
σ and δ orbitals. Our calculations, as well as those of Ref. , indicate that in the3∆1state of ThF+and HfF+
we should expect ωef<
∼2π × 40 kHz with Eeff≈ 90 GV/cm for ThF+and Eeff≈ 30 GV/cm for HfF+[28, 101].
C.|Ω| = 1 vs. |Ω| = 3
We mention one final valuable feature we look for in a candidate molecule: a small magnetic g-factor, so as to
reduce the vulnerability to decoherence and systematic errors arising from magnetic fields. To the extent that spin-
orbit mixing does not mix other |Ω| = 1 states into a nominally3∆1molecular level, it will have a very small magnetic
moment, a feature shared by PbF in the2Π1/2state . This is because Σ = −Λ/2, and because the spin g-factor
is ∼2 times the orbital g-factor. Under these conditions, the contributions of the electronic spin and orbital angular
momentum to the net molecular magnetic dipole moment nominally cancel. In HfF+, the magnetic moment of a
stretched magnetic sublevel level of the3∆1, J = 1 rotational ground state is about 0.05 µB. This is a factor of 20
less than the magnetic moment of ground state atomic cesium. In the3∆3 level, on the other hand, the magnetic
moment in the stretched zeeman level is 4.0 µB. The |Ω| = 3 state may nonetheless be of scientific interest. The3∆1
and3∆3levels have Eeffequal in magnitude but opposite in sign. If one could accurately measure the science signal, ωd,
in the3∆3level despite its larger sensitivity to magnetic field background (and despite its shorter spontaneous-decay
lifetime), the comparison with the3∆1result would allow one to reject many systematic errors.
D.|Ω| = 1, J = 1 Λ-doublet
Since we have not made a final decision as to which molecule we will use, and also because we have yet to measure
the hyperfine constants of our candidate molecules, the discussion of level schemes in this section will be qualitative
in nature, usually emphasizing general properties shared by all the molecules we are investigating. To simplify the
discussion, we will specialize to discussing spectroscopy within the J = 1 rotational manifold of a molecular3∆1level.
For Hunds’ case (a) molecular levels with |Λ| ≥ 1, each rotational level is a Λ-doublet, that is, it consists of two
closely spaced levels of opposite parity. We can think of the even (odd) parity level as the symmetric (antisymmetric)
superposition of the electronic angular momentum lying predominantly parallel and antiparallel to the molecular axis
[Fig. 1(a)]. The parity doublet is split by the Λ-doubling energy ωef. A polar diatomic molecule will have a permanent
electric dipole moment,?dmf, aligned along the internuclear axis ˆ n, but in states of good parity, there will be vanishing
expectation value ?ˆ n? in the lab frame. An applied laboratory electric field, Erot, will act on dmf to mix the states
of good parity. In the limit of dmfErot≫ ωef, energy eigenstates will have nonvanishing ?ˆ n? in the lab frame. More
to the point, Ω, a signed quantity given by the projection of the electron angular momentum on the molecular axis,
(?L+?S)· ˆ n, can also have a nonzero expectation value [Fig. 1(b)]. Heuristically, it is the large electric fields developed
internal to the molecule, along ˆ n, that gives rise to the large value of Eeff that the electron spin can experience in
polar molecules. In the absence of the Λ-doublet mechanism for polarizing the molecule, a much larger field would
be necessary, dmfElab≫ 2Be, to mix rotational states with splitting typically twice the rotational constant Be. For
HfF+, we estimate ωefwill be 2π×10 kHz, whereas Bewill be about 2π×10 GHz. For a dipole moment dmf= 4.3 D,
mixing the Λ-doublet levels will take a field well under 1 V/cm, whereas “brute force” mixing of rotational levels
would require around 10 kV/cm. For an experiment on trapped ions, the smaller electric fields are essential.
In the context of their eEDM experiment on the a3Σ1level in PbO, DeMille and his colleagues have explored in
some detail [17–19] the convenient features of an |Ω| = 1, J = 1 state, especially with respect to the suppression of
systematic error. Our proposal liberally borrows from those ideas. In a molecule with at least one high-Z atom,3∆1
states will be very similar to the a3Σ1state of PbO, but with typically smaller values of ωefand much smaller values
of magnetic g-factor. Singly charged molecules with spin triplet states will necessarily have an odd-Z atom, and thus
the unavoidable complication of hyperfine structure, not present in PbO.
In Fig. 1 we present the3∆1, J = 1 state with hyperfine splitting due to the fluorine I=1/2 nucleus. A key feature
is the existence of two near-identical pairs of mF-levels with opposite parity. As seen in Fig. 1(b), an external electric
field, Erot, mixes these opposite parity states to yield pairs of mF-levels with opposite sign of Eeff  relative to
the external field. Fig. 1(c) shows the effect of a rotating magnetic bias field, parallel with the electric field, applied
to break a degeneracy as described in Sec. IVD below. Note that any two levels connected by arrows in Fig. 1(c)
transform into each other under time reversal. Time reversal takes mF → −mF, Ω → −Ω, and B → −B, where B
is the magnetic field. If we measure the resonant frequency for the transition indicated by the solid (or dashed) line
once before and once after inverting the direction of the magnetic field, time reversal invariance tells us the difference
between the two measurements should be zero. In the presence of an eEDM, which violates time-reversal invariance,
this energy difference Wu(B) − Wu(−B) will give 2deEeff. As well, under the same magnetic field the transitions
indicated by the solid and dashed lines should be degenerate, if the magnetic g-factors are identical for the states
involved . With non-zero eEDM the energy difference Wu− Wlalso gives 2deEeff.
Potential additional shifts, due predominantly to Berry’s phase , are discussed in Sec. IV but for now we
note only that in the absence of new physics (such as a nonzero eEDM) the energy levels of a molecule in time-
varying electromagnetic fields obey time-reversal symmetry. Reversing the direction of the electric field rotation while
-3/2 -1/2 +1/2 +3/2
F = 1/2
-3/2 -1/2 +1/2 +3/2
|Ω = -1〉
|Ω = +1〉
Σ = -1
|Ω = -1〉
|Ω = +1〉
-3/2 -1/2 +1/2 +3/2
Σ = +1
Σ = +1
Σ = -1
F = 3/2
FIG. 1: Energy levels of HfF+in the3∆1, J = 1 state including hyperfine structure associated with the fluorine I = 1/2 nucleus.
Λ and Σ are defined as the projection along the molecular axis of the electronic orbital angular momentum, and spin, respectively.
Ω = Λ + Σ. (a) In zero electric field, the eigenstates of the system are states of good parity, |e? = (|Ω = +1? − |Ω = −1?)/√2
and |f? = (|Ω = +1? + |Ω = −1?)/√2, separated by a small Λ-doublet splitting. (b) An electric field, Erot, mixes the parity
eigenstates yielding states with well defined Ω. (c) A small magnetic field lifts the degeneracy between states with the same
value of mFΩ. A permanent electron electric dipole moment further breaks this degeneracy, but with opposite sign for the
upper (solid arrow) and lower (dotted arrow) transition. Energy splittings not to scale.
chopping the sign of the magnetic field amounts to cleanly reversing the direction of time, and will leave certain
transition energies rigorously unchanged if de= 0. These are our “science transitions”, which we will measure with
our highest precision.
E.Electronic Levels, Spin Preparation, and Spin Readout
The density of trapped molecular ions will be too low to permit direct detection of the radio frequency or microwave
science transitions. (A possible exception could involve the use of a superconducting microwave cavity, but this would
add considerable experimental complexity.) We will of necessity rely on electronic transitions to prepare the initial
electron spin state, and on a double resonance method to detect the spin flips. The details of these steps will depend
on the specific molecule we use. For a qualitative illustration, we present a schematic of the calculated low-lying
electronic potential curves of HfF+(Fig. 2). We note that HfH+and ThF+have similar level structures [30, 101].
The molecules will be formed by laser ablation and cooled by supersonic expansion such that a large portion of the
molecular population will be in1Σ0ground state with a few rotational levels occupied (Sec. IIIA). Spin-orbit mixing
between states of identical |Ω| are enhanced by relativistic effects in the high-Z Hf atom. The b(1) and c(1) states
are well-mixed combinations of1Π1,3Π1, and3Σ−
1states, allowing for electric dipole transitions to and from these
states that do not respect spin selection rules. The1Σ0state, on the other hand, has no nearby |Ω| = 0 state with
which to mix, and thus Σ and Λ are good quantum numbers. Similarly, the3∆1state has so little contamination of
1Π1in it that a rough calculation indicates that it is metastable against spontaneous decay, with a lifetime of order
300 ms [30, 101].
The Ramsey resonance experiment will begin with a two-photon, stimulated Raman pulse, off-resonant from the
FIG. 2: Potential energy curves for select states of HfF+. The b(1) and c(1) states are well-mixed combinations of1Π1,3Π1,
intermediate1,3Π1 states, which will coherently transfer population from the1Σ0, J = 0 ground state to the two
|mF| = 3/2 magnetic sublevels of the3∆1, J = 1 level. The relative phase between the two magnetic levels evolves
at a rate given by the energy difference. After a variable dwell time, a second Raman pulse is applied, which will
coherently transfer a fraction of the population back down to the1Σ0 state, with probability determined by the
accumulated relative phase. By varying the dwell time between Raman pulses, the population in the1Σ0state will
oscillate at a frequency given by the energy difference between the two spin states in the3∆1manifold.
The final step in the resonance experiment is to measure the number of molecules remaining in the3∆1state. This
we propose to do with state-selective photodissociation. Molecules in the3∆1state will be dissociated via a two-color
pulse, back up through the3Π1state to a repulsive curve, generating a Hf+atomic ion and a neutral fluorine atom.
Molecules in the1Σ0state will not be affected by the two-color laser pulse and will remain as HfF+molecular ions.
The Paul trap parameters will be adjusted to confine only ions with the Hf+atomic mass, and not the HfF+molecular
mass with mass difference ∆M = 19 amu. Finally, the potential on an endcap electrode will be lowered, and the
remaining ions in the trap will be dumped onto a ion-counting device.
Details of this procedure will depend on the molecule ultimately selected for this experiment. We are also inves-
tigating alternative modes of spin state readout, including large-solid-angle collection of laser-induced fluorescence,
and high finesse optical cavities .
We are interested in studying molecular radicals and therefore must create the molecules in situ. As described in
Sec. IIB, we have a small collection of molecules that satisfy our selection criteria and our final choice of molecule
has not been made. However, for clarity this section will describe the production, detection, and characterization of
a beam containing neutral HfF molecules and HfF+molecular ions.
The molecules are made in a pulsed supersonic expansion (Fig. 3). A pulse valve isolates ∼ 7 atmospheres of argon
that is seeded with 1% sulfur hexafluoride (SF6) gas from the vacuum chamber. The pulse valve opens for ∼ 200 µs
allowing the Ar + 1% SF6mixture to expand into the vacuum chamber. This creates a gas pulse moving at 550 m/s in
the laboratory frame, but in the co-moving frame the expansion cools the translational temperature of the Ar atoms
to a few Kelvin.
Immediately after entering the vacuum chamber, the gas pulse passes over a Hf metal surface. Neutral Hf atoms
and Hf+ions are ablated from this surface with a 50 mJ, 10 ns, 1064 nm Nd:YAG laser pulse. The ablation plume is
entrained in the Ar + 1% SF6gas pulse and the following exothermic chemical reactions occur:
Hf + SF6 −→ HfF + SF5,
Hf++ SF6 −→ HfF++ SF5,
In the co-moving frame, the resulting neutral HfF molecules and HfF+molecular ions are cooled through collisions
with the Ar gas to rotational, vibrational, and translational temperatures of order a few Kelvin. The molecular
beam then passes through a skimmer, first entering a region where laser induced fluorescence (LIF) spectroscopy is
performed and finally arriving at an rf (Paul) trap where the ions are stopped and confined.
rf (Paul) trap & quadrupole mass flter
Ar + 1% SF6
FIG. 3: Experimental setup. Laser ablation of a metal Hf target creates neutral Hf atoms and Hf+ions that react with SF6 to
produce neutral HfF molecules and HfF+molecular ions, respectively (Eqs. 1 and 2). The molecules (both neutral and ionic)
are cooled in a supersonic expansion with a He buffer gas. The molecular beam is illuminated with a pulse dye laser beam
and the resulting fluorescence is collected with a photomultiplier tube (PMT) yielding laser induced fluorescence (LIF) spectra
(Fig. 4). At the end of the beamline, the ions are loaded into an rf (Paul) trap where the electron spin resonance experiment
is performed. The Paul trap also acts as a quadrupole mass filter and ions of a particular mass/charge ratio are detected with
a microchannel plate (MCP) (Fig. 5). Additionally, the spatial resolution of the MCP allows for the temperature of the ion
cloud to be determined from the detected cloud size.
fluorescence signal [arb. units]
photon wavenumber [cm-1]
FIG. 4: Laser induced fluorescence (LIF) spectroscopy. The top trace is experimental data for a newly detected neutral HfF
transition: [14.2] |Ω| = 3/2 |v′= v′′,J′? ← X2∆3/2|v′′,J′′?. The transition highlighted with a vertical arrow originates from
the rotational ground state. The bottom trace is a theoretical prediction assuming a rotational temperature of 5 K. The traces
are offset vertically for clarity.
LIF spectroscopy is performed by transversely illuminating the molecular beam with a ∼ 500 µJ, 10 ns, ∼ 700 nm
dye laser pulse. The linewidth of the dye laser is specified to be less than 0.1 cm−1. Fluorescence photons are collected
and imaged onto a photomultiplier tube (PMT).
Using this technique we have found previously unobserved neutral HfF molecular transitions, one of which is shown
in Fig. 4 (for previous neutral HfF spectroscopy see Ref. ). The data shows that entrained neutral HfF molecules
are cooled to rotational temperatures of order 5 K, with a large fraction of the population in the rotational ground
state. We expect that entrained HfF+molecular ions should be similarly cooled.
To detect the presence of HfF+molecular ions in the beam the rf (Paul) trap is operated as a quadrupole mass
Hf+ HfF+ HfF+ HfF+
ion signal [arb. units]
FIG. 5: Mass spectrometry. Operating the rf (Paul) trap as a quadrupole mass filter gives mass-dependent trapping potentials
such that Hf+(M = 180 amu), HfF+(M = 199 amu), HfF+
trapped and detected. The ion detector signal is a non-linear function of ion number, but a level of 0.4 corresponds to ∼ 100,000
2(M = 218 amu), and HfF+
3(M = 237 amu) can be separately
filter. All of the ions in the beam are stopped and loaded into the trap. The voltages applied to the trap electrodes
are then adjusted only to confine ions of a particular mass/charge ratio. Finally, the ions remaining in the trap are
released onto the ion detector and counted. A typical mass spectrum is shown in Fig. 5, which clearly resolves the
HfF+molecular ions from the other atomic and molecular ions in the trap.
Our experimental count rate will be limited by space charge effects of the trapped ions. Therefore, any ions trapped
that are not used in measuring the eEDM limit the statistical sensitivity of our measurement. In order to maximize
our count rate, we wish to create and trap only HfF+ions of a single Hf isotope and in a single internal quantum
state. One scheme is to filter out all of the ions created from laser ablation and use photoionization techniques to
ionize neutral HfF in as state-selective a way as possible. Using two color, two photon excitation, we excite to a high
lying Rydberg state, in an excited vibrational level, that then undergoes vibrational autoionization . The ion
core of these Rydberg state molecules will occupy a single rotational level and consist of a single Hf isotope. The
autoionization process is seen, in our preliminary (unpublished) data, to leave the ion core rotational level largely
unperturbed. It should be possible to excite a Rydberg level that corresponds to an excited3∆1ion core with v = 1,
J = 1 (where v is the vibrational quantum number). The Rydberg state might then vibrationally autoionize to the
v = 0, J = 13∆1level that will be used to measure the eEDM.
B.Radio Frequency (Paul) Trap
For our preliminary studies of ion production, the ions are confined by a linear rf (Paul) trap shown schematically in
Fig. 6. The ideal hyperbolic electrodes are replaced by cylinders of radius a ≈ 1.15ρ0, where ρ0is the minimum radial
separation between the trap center and the surface of the electrodes. This choice produces the best approximation to
a perfect radial two-dimensional electric quadrupole field .
For HfF+(M = 199 amu), an example set of operating parameters for the ion trap would be ρ0 = 25 mm,
Vrf = 550 mV, and ωrf = 2π × 15 kHz. This produces a ponderomotive potential that is well within the harmonic
pseudo-potential approximation given by Urf(ρ) = Mω2
ωsec= qωrf/√8 with q = 4eVrf/Mρ2
rf. For the above parameters, q = 0.2, ωsec= 2π × 1 kHz, and Urf(ρ0) = 300 K.
Under these conditions, an ion cloud at a temperature of 15 K would have an rms radius of 5 mm. The trap can also
be operated in mass filter mode .
In addition to supplying the oscillating electric quadrupole field for radial confinement, the cylindrical electrodes
can also be driven with voltages to produce the rotating electric bias field, Erot, needed to polarize the molecular ions
secρ2/2, where the radial secular frequency is approximately
Vrf sin(ωrft) + Vrot sin(ωrott - π/2)
Vrf sin(ωrft - π) + Vrot sin(ωrott - π)
Vrf sin(ωrft) + Vrot sin(ωrott - 3π/2)
A < 2π
A > 2π
A = 2π
Vrf sin(ωrft - π) + Vrot sin(ωrott)
FIG. 6: Linear rf (Paul) trap. Neighboring cylindrical electrodes are driven with rf voltages 180◦out of phase. Axial confinement
is provided by d.c. voltages applied to the end cap electrodes. The cylindrical electrode rods have radius a and the radial distance
from the trap center to the nearest electrode surface is ρ0. See Ref.  for further details of rf (Paul) trap operation. In
addition to the voltages oscillating at ωrf, there is also a component of the voltages oscillating at ωrot. Over a period of time
2π/ωrot, the electric field at the axial center (z=0) of the trap will trace out a trajectory which subtends a solid angle A of
exactly 2π. Ions to the left (right) of trap center will experience an electric field whose trajectory subtends slightly less (greater
than) 2π. Consequences of this time variation are explored in discussed in Sec. IVC and IVD. Not to scale.
(Fig. 6). In order to generate Erotneighboring electrodes will be driven 90◦out of phase at a frequency ωrot. The net
voltage applied to each electrode is the sum of the voltages Vrf+ Vrot.
At present we are designing a second-generation ion trap with geometry designed for optimal precision eEDM
spectroscopy, rather than for mass selection. The perfect ion trap would have very large optical access for collection
of laser-induced fluorescence, and idealized electric and magnetic fields as follows
?E = Erotˆ ρ′+ E′
rf(xˆ x − yˆ y)cos(ωrft) + E′
z(−zˆ z + yˆ y/2 + xˆ x/2)(3)
?B = Brotˆ ρ′
where ˆ ρ′= cos(ωrott)ˆ x + sin(ωrott)ˆ y and E′
If we assume ωrot ≫ ωrf, that ωrot/ωrf is not a rational fraction, and that ω2
separate out the ion motion into three components: rf micromotion, circular micromotion, and secular motion.
rf micromotion involves a rapid oscillation at ωrf whose amplitude grows as the ion’s secular trajectory takes it
away from trap center. The kinetic energy of this motion, averaged over an rf cycle, is given by
rf/M, then we can cleanly
Erf= (x2+ y2)e2E′
where x and y in this case refer to the displacement of the ion’s secular motion.
The displacement of the ion’s circular micromotion is given by
? rrot= −e?Erot
The kinetic energy of the circular motion, averaged over a rotation cycle, is given by
The time-averaged kinetic energies of the two micromotions act as ponderomotive potentials that contribute to the
potential that determines the relatively slowly varying secular motion:
Usec= Erot(x,y,z) + Erf(x,y,z) + eE′
z(2z2− y2− x2)/4.(8)
In the idealized case, the secular motion corresponds to 3-d harmonic confinement with secular or “confining” fre-
for i = x,y,z. In the idealized case, confinement is cylindrically symmetric, ωx= ωy, and Erotis spatially uniform, so
the circular micromotion does not contribute to the confining frequencies.
The density of ions will be low enough that there will be few momentum-changing collisions during a single mea-
surement. Thus, any given ion’s trajectory will be well approximated by the simple sum of three contributions:
(i) a 3-d sinusoidal secular motion, specified by a magnitude and initial phase for each of the ˆ x, ˆ y, and ˆ z directions.
In a thermal ensemble of ions, the distribution of initial phases will be random and the magnitudes, Maxwell-
Boltzmannian. For typical experimental parameters (see the Appendix) the secular frequencies ωiwill each be about
2π × 1 kHz and the typical magnitude of motions, r, will be about 0.5 cm.
(ii) the more rapid, smaller amplitude rf micromotion, of characteristic frequency about 2π×15 kHz and radius per-
haps 0.05 cm. This rf micromotion, purely in the x-y plane, is strongly modulated by the instantaneous displacement
of the secular motion in the x-y plane, and vanishes at secular displacement x=y=0.
(iii) The still more rapid rotational micromotion, purely circular motion in the x-y plane, at frequency ωrotabout
2π × 100 kHz and of radius comparable to the rf motion, around 0.05 cm. In the idealized case, the rotational
micromotion (in contrast to the rf micromotion) is not modulated by the secular motion.
As described in Secs. IVE and V below, for spectroscopic reasons we must operate with trapping parameters such
∼30kBT. Under that condition, relatively small imperfections in Erot, say a spatial variation of 1.5%, can
give rise to contributions to Usecof the same scale as the ions’ thermal energy, and thus significantly distort the shape
of the trapped ion cloud or even deconfine the ions.
For improved optical access we had to shrink the radius of the linear electrodes a with respect to their spacing ρ0
The spectroscopic requirement for highly uniform Erotthen forced the redesign of the second-generation ion trap to
be based on six near-linear elements arranged on a hexagon, rather the four electrodes arranged on square shown in
Fig. 6. The trap will be discussed in more detail in a future publication, but simulations project spatial uniformity of
Erotbetter than 0.5% with good optical access. The design led to significant compromises in the spatial uniformity of
Erf, so in future operation, mass selectivity in ion detection will come not from a quadrupole mass filter, but rather
from pulsing Erot to a very high value for a small fraction of a rotation cycle and then doing time-of-flight mass
discrimination on the ions thus ejected. Brotwill be imposed by means of time-varying currents flowing lengthwise
along the same electrodes that generate Erot.
IV.SPECTROSCOPY IN ROTATING AND TRAPPING FIELDS
On the face of it, an ion trap, with its inhomogeneous and rapidly time-varying electric fields, is not necessarily
a promising environment in which to perform sub-Hertz spectroscopic measurements on a polar molecule. In this
section we will explore in more detail the effects of the various components of the electric and magnetic fields on
the transition energies relevant to our science goals. The theoretical determination of the energy levels of heavy
diatomic molecules in the presence of time-varying electric and magnetic fields is a tremendously involved problem
in relativistic few-body quantum mechanics. State-of-the-art ab initio molecular structure calculations are limited to
an energy accuracy of perhaps 1013Hz, a quantity which could be compared with the size of a hypothetical “science
signal”, which could be on the order of 10−3Hz or smaller.
Fortunately, we can take advantage of the fact that at the energy scales of molecular physics, time-reversal invariance
is an exact symmetry except to the extent that there is a time-violating moment associated with the electron (or
nuclear) spin. In this section, except in those terms explicitly involving de, we will assume that time-reversal invariance
is a perfect symmetry in order to analyze how various laboratory effects can cause decoherence or systematic shifts
in the relevant resonance measurements. The results can be compared to the size of the line shift that would arise
from a given value of the electron EDM, which is treated theoretically as a very small first-order perturbation on the
otherwise T-symmetric system.
In the subsections below, we bring in sequentially more realistic features of the trapping fields.
A. Basic Molecular Structure
We begin by considering in detail the relevant molecular structure in zero electric and magnetic fields, thus quan-
tifying the qualitative discussion of the experiment given in Sec. II. Although the molecular structure cannot be
calculated in detail from ab initio structure calculations, nevertheless its analytic structure is well known. Because
the measurements will take place in nominally a single electronic, vibrational, and rotational state, we will employ an
effective Hamiltonian within this state, as elaborated by Brown and Carrington . This approach will specify a
few undetermined numerical coefficients, whose values can be approximated from perturbation theory, but which will
ultimately be measured.
Brown [118–120] and co-workers have done thorough work on deriving an effective Hamiltonian for3∆ molecules.
The complete Hamiltonian in the absence of deis given by
Hstruct= Helec+ Hvib+ HSO+ Htum+ HSS+ HSR+ HHFS+ HLD,(10)
listed in rough order of decreasing magnitude. Since we are concerned only with terms acting within the subspace of
the3∆ manifold, other electronic and vibrational states will enter only as perturbations that help to determine the
effective Hamiltonian. Thus we consider eigenstates of Helecand Hvib.
The remaining terms in Eq. (10) are corrections to the Born-Oppenheimer curves. They describe couplings between
various angular momenta (HSR, HHFS), parity splittings (HLD, HHFS), and spin-dipolar interactions (HSS, HHFS). In
typical Hund’s case (a) molecules these interactions are small compared to the rotational energy governed by Htum.
The relevant interactions that act within the |Ω| = 1 manifold of states take the explicit form
HSO = AΛΣ
Htum = Be(J − S)2− D(J − S)4
HSR = γSR(J − S) · S
HHFS = aIzLz+ bFI · S +c
2(o∆+ 3p∆+ 6q∆)(S2
3(3IzSz− I · S) +1
The constants in the first four terms are as follows: A is the molecular spin-orbit constant, Bethe rotational constant
for the electronic level of interest, D the effect of centrifugal distortion on rotation (typically D ∼ Be(me/mmol)2,
with me the electron mass and mmol the reduced mass of the molecule), λ governs the strength of the spin-spin
dipolar interaction, and γSRdetermines the strength of the interaction of the spin with the end-over-end rotation of
the molecule. These four terms primarily describe an overall shift of the3∆1J-level, and can be ignored in evaluating
energy differences in the states we care about. They can, however, contribute small perturbations to these basic levels,
as we will describe below.
Within the3∆1, J = 1 manifold of interest, the energy levels are distinguished by the hyperfine and Λ-doubling
terms. The hyperfine Hamiltonian HHFSincludes the familiar contact (bF), nuclear-spin-orbit (a) and spin-nuclear
spin terms (c). By estimating the parameters in perturbation theory, it is expected that the resulting hyperfine
splitting is on the order of 2π ×50 MHz . The hyperfine interaction also contains a previously unreported term,
with constant denoted e∆, that is connected to the Λ-doubling. This term is expected to be even smaller than the
already small Λ-doublet splitting itself , however, and will be ignored.
The Λ-doubling Hamiltonian arises from Coriolis-type mixing of states with differing signs of Λ due to end-over-end
rotation of the molecule. For a3∆ state this interaction is characterized by three constants, of which the parameter
o∆ is the dominant one. These terms describe how the3∆ state is perturbed by electronic states with2S+1Π and
2S+1Σ symmetry. Since we are primarily concerned with terms in the Hamiltonian that affect the ground rotational
state of the3∆1 electronic level, we only need to keep the term which connects Ω = 1 to Ω = −1. This term has
the general form, with numerical prefactors CΠ,Σ,Π′ that depend on Clebsch-Gordon coefficients and wavefunction
|o∆+ 3p∆+ 6q∆| = ˜ o∆≈
(E∆− EΠ)(E∆− EΣ)(E∆− EΠ′),(17)
where the sum is over all intermediate Σ and Π states of singlet and triplet spin symmetries. For HfF+this perturbation
leads to a Λ-doublet splitting on the order of 2π × 10 kHz. This estimate was carried out assuming a σδ molecular
orbital configuration, where the δ orbital has total angular momentum L = 2 in the pure precession approximation.
The ground X1Σ is a σ2molecular orbital but has some admixture of atomic d0orbitals. We therefore expand the
molecular wavefunction into atomic orbitals and reduce the amount of admixture by the factor ǫdthat describes the
d0character. From here on, we shall express the energy difference in parity levels for the J = 1 as ωef= 4˜ o∆, rather
than ˜ o∆itself.
Thus the basic molecular structure of interest to the3∆1, J = 1 state is governed by two constants: the hyperfine
splitting Ehf(given by 3A||/4 for J = 1,I = 1/2) and the Λ-doublet splitting ωef. These constants give the structure
depicted in Fig. 1(a). These basic levels may be perturbed by couplings to other levels, especially rotational or
electronic excited states. However, for the J = 1 state of interest, some simplifications are possible, namely: (1)
Off-diagonal couplings in Ω are zero since J · S preserves the value of J (there is no level with J = 1 and Ω = 2);
(2) Off-diagonal contributions that mix J = 2 into the J = 1 manifold thus depend solely on the applied fields and
the hyperfine interactions. Since the value of the spin-orbit constant is expected to be far larger than the rotational
constant and we are concerned with a J = 1 state, the operators that connect Ω to Ω ± 1 will be ignored. The
contributions to the ground state characteristics by terms off diagonal in Ω are smaller by a factor of the hyperfine
interaction energy to the spin-orbit separation energy, hence a factor of 10−6. This is the value which appears in front
of any term connecting Ω to Ω ± 1 in the ground J = 1 state.
B. Effect of Non-rotating Electric and Magnetic Fields
The influence of external fields presents new terms in the Hamiltonian of the form
HStark = −?dmf·?E
HZeeman = −? µ ·?B.
Here?E and?B are the electric and magnetic fields, assumed for the moment to be collinear so that they define the
axis along which mF is a good quantum number; while?dmf and ? µ are the electric and magnetic dipole moments of
The electric dipole moment arises from the body-fixed molecular dipole moment, at fields sufficiently small not
to disturb the electronic structure. We assume that the field is sufficiently large to completely polarize this dipole
moment, i.e., dmfE ≫ ωef, in which case the Stark energies are given by
where γF is a geometric factor, analogous to a Land´ e g-factor, which accounts for the Stark effect in the total angular
momentum basis F. In the limit where the electric field is weak compared to rotational splittings, it is given by
γF =J(J + 1) + F(F + 1) − I(I + 1)
2F(F + 1)J(J + 1)
Its numerical values in the J = 1 state are therefore γF=3/2= 1/3 and γF=1/2= 2/3. The electric field therefore
raises the energy of the states with mFΩ < 0 (denoted “upper” states with superscript u), and lowers the energy of
states with mFΩ > 0 (“lower” states with superscript ℓ). This shift in energy levels is shown in Fig. 1(b), where |a?
and |b? are upper and |c? and |d? are lower states.
The form of the Zeeman interaction is somewhat more elaborate, as the magnetic moment of the molecule can arise
from any of the angular momenta L, S, J, and I. Quite generally, however, in the weak-field limit where µBB ≪ Ehf,
the Zeeman energies are given by mFgu/ℓ
and lower states. In general, gu
F, and this difference can depend on electric field, a possible source of systematic
error. We will discuss this in Sec. IVG below.
The leading order terms in the Zeeman energy are those that preserve the signed value of Ω. They are given by
FµBB, where µBis the Bohr magneton and gu/ℓ
are g-factors for the upper
HZeeman= (γF[((gL+ gr)Λ + (gS+ gr)Σ)Ω − grJ(J + 1)] − gIκF)mFµBB,
where κF = (F(F + 1) + I(I + 1) − J(J + 1))/2F(F + 1) is another Land´ e-type g-factor, but for nuclear spin. The
orbital and spin g-factors are gLand gS, while the rotation and nuclear spin g-factors are grand gI. Both grand gI
are small, being on the order of the electron-to-molecular mass ratio ∼ me/mmol∼ 10−3. Thus for an idealized3∆1
molecule where gL= 1, Λ = ±2, gS= 2, Σ = ∓1, we would expect molecular g-factors on the order of 10−3. More
realistically, gsdiffers from 2 by a number on the order of α, the fine structure constant, and a g-factor ∼ 10−2might
be expected. In heavy-atom molecules such as ours for which spin-orbit effects mix Λ, we may expect instead the
difference 2gL− gSto be as large as ∼ 0.1 in magnitude. If we assume the dominant contribution comes from these
spin-orbit type effects, we can define the g-factor for the J = 1 state as
gF=3/2= γF=3/2(gLΛ + gSΣ)Ω<
Finally, the effect of the EDM itself introduces a small energy shift
HEDM= −?de·?Eeff= deEeff? σ1· ˆ n,(25)
where ? σ1is the spin of the s-electron contributing to the EDM signal; and ˆ n denotes the intermolecular axis, with ˆ n
pointing from the more negative atom to the more positive one; in our case from the fluorine or hydrogen to thorium,
platinum, or hafnium. Also in this convention we take Eeff as positive if it is anti-parallel to ˆ n. The energy shift
arising from this Hamiltonian depends only on the relative direction of the electron spin and the internuclear axis,
and is given by
Polarizing the molecule in the external field selects a definite value of Ω, hence a definite energy shift, positive or
negative, due to the EDM. This additional shift is illustrated in Fig. 1(c).
For a range of field strengths and parameters, the energies of the sublevels within the J = 1 manifold are well
approximated by a linear expansion in the electric and magnetic fields. We define
?B = B||
Taking ωef≪ dmfE ≪ Ehfand dmfE ≫ gFµBB||, and setting B⊥= 0, we get for the non-rotating energies,
3(F(F + 1) −11
4)Ehf− mFΩγFdmfE + mFgu/ℓ
FµBB − (deEeff/2|Ω|)Ω,(28)
where Ω is either 1 or -1, and the prefactor in front of Ehfis such that for the J = 1 level, E(F = 3/2) - E(F = 1/2)
= 3A||/4 = Ehf. F and Ω are good quantum numbers only to the extent that the electric field is neither too large
nor too small, but we will use F and Ω as labels for levels even as these approximations begin to break down.
For notational compactness, we introduce special labels for particular states as follows (see Fig. 1(b)):
|a? = |F = 3/2,m = 3/2,Ω = −1?
|b? = |F = 3/2,m = −3/2,Ω = 1?
|c? = |F = 3/2,m = 3/2,Ω = 1?
|d? = |F = 3/2,m = −3/2,Ω = −1?
with corresponding energies, Ea, Eb, Ec, and Ed, and identify the energies of two particularly interesting transitions,
Wu= Ea− Eb, and Wℓ= Ec− Edsuch that
FµBB + deEeff
FµBB − deEeff.
Taking this analysis a step farther, it is possible that the electric field energy dmfE is not small compared to the
hyperfine splitting Ehf. In this case the electric field mixes the different total-F states and perturbs the above energies.
Ignoring the magnetic field and EDM energies, the energy levels take the form
Enr(˜F ∼ 3/2,mFΩ = +1/2) = −1
Enr(˜F ∼ 1/2,mFΩ = +1/2) = −1