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arXiv:1008.2997v3 [physics.atomph] 18 Jul 2011
Highresolution 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 803090440, USA
2Department of Physics, University of Michigan, Ann Arbor, Michigan 481091040, USA
3Physics Department, Harvard University, Cambridge, Massachusetts, 02138, USA
(Dated: July 19, 2011)
Highresolution 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 highresolution 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 timevarying 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: Highresolution spectroscopy, radiofrequency, fundamental symmetries, Stark and Zeeman inter
actions, molecular ions
I. INTRODUCTION
A. HighResolution 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 [1], which have spectroscopic signatures that are both theoretically
calculable and experimentally detectable. Parityviolating 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 [4] with sufficient precision to test electroweak theory at the ∼ 1% level. In addition, highresolution
molecular spectroscopy experiments are underway to probe parity violation in chiral polyatomic molecules [5–8] and
to probe nuclear spindependent parity violation in diatomic molecules [9, 10]. Looking outside of the Standard
Model, precision molecular spectroscopy experiments have been designed to search for timevariation of fundamental
constants, such as the electrontoproton mass ratio [11–15] and the fine structure constant [11, 16], as well as to
search for simultaneous parity and timereversal symmetry violation in the form of permanent electric dipole moments
(EDMs) [17–30].
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 highflux 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 highresolution
∗Electronic address: ecornell@jilau1.colorado.edu; URL: http://jilawww.colorado.edu/bec/CornellGroup/
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TABLE I: Theoretical predictions of the electron electric dipole moment, de. Current listings are taken from Ref. [61], which
extracted the numbers from Refs. [57, 58].
CP Violating Model
Standard Model
Supersymmetric models
Leftright symmetric models
Higgs 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 degreesoffreedom 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 Λdoublet 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 nextgeneration 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
highresolution 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 [33] agrees with
the predictions [34] of quantum electrodynamics out to fourloop corrections. Compared to the electron work, muonic
g2 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 newphysics signal out of the muon g2 measurement is made difficult
by uncertainty in the hadronic contributions to the Standard Model prediction [55]. 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 [56]. 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 [60]. 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 factoroften 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.
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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 Λdoubled 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 longestablished
technology.
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 labframe 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 timevarying 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 lowenergy [62] 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 [68], 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 [69];
there are ongoing neutron EDM searches [69–71]. Beamline measurements on bare nucleons are envisioned [72].
The current best atomic dipole measurement is an experiment is in the diamagnetic species, Hg, by the Washington
group [73]. Many other groups are looking for EDMs in diamagnetic (that is, net electron spin S = 0), groundstate
electronic levels in Hg [73], Xe [74–77], Rn [78], Yb [79] 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. Higherorder contributions from eEDM contribute to the atomic EDM
of S = 0 atoms [83], 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 atomicbeam 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 figureofmerit 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 spinflips
that can be counted in some reasonable experimental integration time, for instance one week. The statisticslimited
sensitivity to the eEDM is just the inverse of our figureofmerit. We will discuss the three terms in order.
The conceptually simplest version of an eEDM experiment would simply be to measure the spinflip frequency of
a free electron in an electric field Elab, ωd= deElab, where deis the electric dipole moment of the electron [85]. Alas,
a free electron in a large electric field would not stay still long enough for one to make a careful measurement of its
spinflip 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 spinflip frequency
ωdis enhanced by relativistic effects occurring near the highZ 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 [88] and +114 [89], respectively. Practical DC electric fields in
a laboratory vacuum are limited by electric breakdown to about 105V/cm. The Commins experiment used a very
highZ atom, thallium, and achieved an Eeff of about 7 × 107V/cm [60]. 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 [93]
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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 highZ 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 [22] 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 [24] has proposed to work with
PbF, which has a limiting value of Eeff≃ 29 GV/cm [25, 26]. The ACME collaboration [27] will use ThO, with Eeff
≃ 100 GV/cm [28]. The Michigan group is working with WC, with Eeff≃ 54 GV/cm [29]. 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 [24], or with a decelerated beam [102]. Groups
working in lasercooled cesium anticipate coherence times of around 1 s, using either a fountain [90] 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 excitedstate lifetime to 2 ms [27]. A
now discontinued experiment at Amherst [93] 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 figureofmerit 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. Vaporcell 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 figureofmerit. 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 solidstate 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 figureofmerit.
Finally, atoms with diamagnetic ground states may have S ?= 0 metastable states amenable to an eEDM search
[107]. Closely spaced opposite parity states in Ra can give rise to an Eeff on the electron spin larger [108] than in Tl
or Cs, but very short coherence times [108] may make complicate efforts [67] to measure the eEDM in Ra.
√N, which, assuming good initial polarization, good final
E.Outline
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 timevarying
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 ionion 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.
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TABLE II: Figureofmerit comparison between several recently completed and ongoing eEDM experiments. For ongoing
experiments these numbers are subject to change and are often orderofmagnitude estimates. For the JILA entry M is Hf, Th,
or Pt and x is H or F.
Group Refs.Species Elab [V/cm] Eeff [V/cm]
Berkeley[60] Tl1.23 × 105
Amherst[93] Cs4 × 103
LBNL[90] Cs105
Texas[92] Cs105
Penn State[91] Cs105
Yale[17–21]PbO 10
Imperial[22, 23] YbF8.3 × 103
Oklahoma[24–26]PbF7 × 104
ACME[27, 28] ThO102
Michigan [29] WC
JILA This workMx+
5
τ [s]
˙N [s−1]
109
7 × 107
4.6 × 105
107
107
107
2.5 × 1010
1.3 × 1010
2.9 × 1010
1011
5.4 × 1010
3 − 9 × 1010
2.4 × 10−3
1.5 × 10−2
1
1
1
8 × 10−5
10−3
109
2 × 10−3
10−3
0.2 − 1
105
∼ 10
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 Λdoublet 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 Λdoublet 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. [30] 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. [101], 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].
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C.Ω = 1 vs. Ω = 3
We mention one final valuable feature we look for in a candidate molecule: a small magnetic gfactor, 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 [24]. This is because Σ = −Λ/2, and because the spin gfactor
is ∼2 times the orbital gfactor. 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 spontaneousdecay
lifetime), the comparison with the3∆1result would allow one to reject many systematic errors.
D.Ω = 1, J = 1 Λdoublet
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 Λdoublet, 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 Λdoubling 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 Λdoublet 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 Λdoublet 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 highZ 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 gfactor. Singly charged molecules with spin triplet states will necessarily have an oddZ 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 nearidentical pairs of mFlevels with opposite parity. As seen in Fig. 1(b), an external electric
field, Erot, mixes these opposite parity states to yield pairs of mFlevels with opposite sign of Eeff [19] 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 timereversal 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 gfactors are identical for the states
involved [109]. With nonzero eEDM the energy difference Wu− Wlalso gives 2deEeff.
Potential additional shifts, due predominantly to Berry’s phase [110], 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 timereversal symmetry. Reversing the direction of the electric field rotation while
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3/2 1/2 +1/2 +3/2
1/2 +1/2
e〉
F = 1/2
(a)(b)
f〉
e〉
f〉
EHF
3/2 1/2 +1/2 +3/2
1/2 +1/2
Hf++
F 
^n
Ω = 1〉
F 
Ω = +1〉
Σ = 1
Ω = 1〉
Ω = +1〉
3/2 1/2 +1/2 +3/2
mF
1/2 +1/2
(c)
mF
^n
Σ = +1
^n
Σ = +1
^n
Σ = 1
?rot
?rot
?rot?
?rot?
mF
3gFµB?+de?eff
3gFµB?−de?eff
mF
mF
mF
3γF?rotdmf
b〉
a〉
c〉
d〉
Wu
Wl
?ef
?ef
ωef
Hf++
F 
F 
Hf++
Hf++
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 Λdoublet 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 lowlying
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). Spinorbit mixing
between states of identical Ω are enhanced by relativistic effects in the highZ Hf atom. The b(1) and c(1) states
are wellmixed 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 twophoton, stimulated Raman pulse, offresonant from the
Page 8
8
b(1)
3??
3??
3??
3??
c(1)
3??
1??
FIG. 2: Potential energy curves for select states of HfF+[30]. The b(1) and c(1) states are wellmixed combinations of1Π1,3Π1,
and3Σ−
1states.
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 stateselective photodissociation. Molecules in the3∆1state will be dissociated via a twocolor
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 twocolor 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 ioncounting 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 largesolidangle collection of laserinduced fluorescence,
and high finesse optical cavities [111].
III.EXPERIMENTAL APPARATUS
A.Molecular Beamline
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 comoving 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,
(1)
(2)
In the comoving 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.
Page 9
9
+
+
skimmerskimmer
rf (Paul) trap & quadrupole mass flter
+
~100 pisg
Ar + 1% SF6
pulse valve
Hf
target
+
+
+
+
+
+
+
+
+
+
+
microchannel plate
Nd:YAG
1064 nm
"click"
"click"
photomultiplier tube
dye laser
700 nm
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]
14228 14224 14220
photon wavenumber [cm1]
14216
Q
J'=J''
}
P
J'=J''1
R
J'=J''+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∆3/2v′′,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. [112]). 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
Page 10
10
Hf+ HfF+ HfF+ HfF+
2 3
0.5
0.4
0.3
0.2
0.1
0.0
ion signal [arb. units]
240220 200180160
mass [amu]
FIG. 5: Mass spectrometry. Operating the rf (Paul) trap as a quadrupole mass filter gives massdependent trapping potentials
such that Hf+(M = 180 amu), HfF+(M = 199 amu), HfF+
trapped and detected. The ion detector signal is a nonlinear function of ion number, but a level of 0.4 corresponds to ∼ 100,000
ions.
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 stateselective 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 [113]. 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 twodimensional electric quadrupole field [115].
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
pseudopotential 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 [116].
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
0ω2
Page 11
11
+VDC
Vrf sin(ωrft) + Vrot sin(ωrott  π/2)
Vrf sin(ωrft  π) + Vrot sin(ωrott  π)
Vrf sin(ωrft) + Vrot sin(ωrott  3π/2)
Ez
+VDC
+VDC
+VDC
A < 2π
A > 2π
A = 2π
Vrf sin(ωrft  π) + Vrot sin(ωrott)
Ez
2a
2ρ0
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. [114] 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 secondgeneration 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 laserinduced 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′
(4)
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≈ −2Vrf/ρ2
0.
rf≫ eE′
rf/M, then we can cleanly
Erf= (x2+ y2)e2E′
rf
2
4Mω2
rf
(5)
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
Mω2
rot
. (6)
The kinetic energy of the circular motion, averaged over a rotation cycle, is given by
Erot=
e2E2
2Mω2
rot
rot
.(7)
Page 12
12
The timeaveraged 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 3d harmonic confinement with secular or “confining” fre
quencies
ωi=
?1
M
∂2Usec
∂i2
?1/2
,(9)
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 momentumchanging 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 3d 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 xy plane, is strongly modulated by the instantaneous displacement
of the secular motion in the xy plane, and vanishes at secular displacement x=y=0.
(iii) The still more rapid rotational micromotion, purely circular motion in the xy 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
that Erot>
∼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 secondgeneration ion trap to
be based on six nearlinear 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 timeofflight mass
discrimination on the ions thus ejected. Brotwill be imposed by means of timevarying 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 timevarying electric fields, is not necessarily
a promising environment in which to perform subHertz 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 timevarying electric and magnetic fields is a tremendously involved problem
in relativistic fewbody quantum mechanics. Stateoftheart 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, timereversal invariance
is an exact symmetry except to the extent that there is a timeviolating moment associated with the electron (or
nuclear) spin. In this section, except in those terms explicitly involving de, we will assume that timereversal 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 firstorder perturbation on the
otherwise Tsymmetric system.
In the subsections below, we bring in sequentially more realistic features of the trapping fields.
Page 13
13
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 [117]. 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 coworkers 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 BornOppenheimer curves. They describe couplings between
various angular momenta (HSR, HHFS), parity splittings (HLD, HHFS), and spindipolar 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
HSS =
3λ(3Σ2− S2)
HSR = γSR(J − S) · S
HHFS = aIzLz+ bFI · S +c
1
2(o∆+ 3p∆+ 6q∆)(S2
(11)
(12)
2
(13)
(14)
3(3IzSz− I · S) +1
2e∆(J+I+S2
++ J−I−S2
−)(15)
HLD =
+J2
++ S2
−J2
−).(16)
The constants in the first four terms are as follows: A is the molecular spinorbit 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 spinspin
dipolar interaction, and γSRdetermines the strength of the interaction of the spin with the endoverend rotation of
the molecule. These four terms primarily describe an overall shift of the3∆1Jlevel, 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 Λdoubling
terms. The hyperfine Hamiltonian HHFSincludes the familiar contact (bF), nuclearspinorbit (a) and spinnuclear
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 [101]. The hyperfine interaction also contains a previously unreported term,
with constant denoted e∆, that is connected to the Λdoubling. This term is expected to be even smaller than the
already small Λdoublet splitting itself [121], however, and will be ignored.
The Λdoubling Hamiltonian arises from Coriolistype mixing of states with differing signs of Λ due to endoverend
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 ClebschGordon coefficients and wavefunction
overlap, [118]
o∆+ 3p∆+ 6q∆ = ˜ o∆≈
?
Π,Σ,Π′
CΠ,Σ,Π′
A2B2
e
(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 Λdoublet splitting on the order of 2π × 10 kHz. This estimate was carried out assuming a σδ molecular
Page 14
14
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 Λdoublet 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)
Offdiagonal couplings in Ω are zero since J · S preserves the value of J (there is no level with J = 1 and Ω = 2);
(2) Offdiagonal 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 spinorbit 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 spinorbit 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 Nonrotating Electric and Magnetic Fields
The influence of external fields presents new terms in the Hamiltonian of the form
HStark = −?dmf·?E
HZeeman = −? µ ·?B.
(18)
(19)
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 molecule.
The electric dipole moment arises from the bodyfixed 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
EStark= −mFΩγFdmfE,
where γF is a geometric factor, analogous to a Land´ e gfactor, 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
(20)
γF =J(J + 1) + F(F + 1) − I(I + 1)
2F(F + 1)J(J + 1)
.(21)
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 weakfield 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
F
are gfactors for the upper
F?= gℓ
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´ etype gfactor, but for nuclear spin. The
orbital and spin gfactors are gLand gS, while the rotation and nuclear spin gfactors are grand gI. Both grand gI
are small, being on the order of the electrontomolecular 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 gfactors 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 gfactor ∼ 10−2might
be expected. In heavyatom molecules such as ours for which spinorbit effects mix Λ, we may expect instead the
(22)
Page 15
15
difference 2gL− gSto be as large as ∼ 0.1 in magnitude. If we assume the dominant contribution comes from these
spinorbit type effects, we can define the gfactor for the J = 1 state as
gF=3/2= γF=3/2(gLΛ + gSΣ)Ω<
∼0.03, (23)
while
gF=1/2= 2gF=3/2. (24)
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 selectron 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 antiparallel 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
EEDM= −deEeff
2ΩΩ. (26)
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
?E
E+?B⊥
(27)
Taking ωef≪ dmfE ≪ Ehfand dmfE ≫ gFµBB, and setting B⊥= 0, we get for the nonrotating energies,
nr(F,mF,Ω;E,B) ≈1
Eu/ℓ
3(F(F + 1) −11
4)Ehf− mFΩγFdmfE + mFgu/ℓ
FµBB − (deEeff/2Ω)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?
(29)
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
Wu= 3gu
FµBB + deEeff
Wℓ= 3gℓ
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 totalF states and perturbs the above energies.
Ignoring the magnetic field and EDM energies, the energy levels take the form
(30)
Enr(˜F ∼ 3/2,mFΩ = +1/2) = −1
2
?dmfE
?dmfE
2
+Ehf
3
?
?
+1
2
??dmfE
??dmfE
6
− Ehf
?2
?2
+ 2
?dmfE
?dmfE
3
?2
?2
−
3ω2
4dmfE
3ω2
8dmfE
ef
(31)
Enr(˜F ∼ 1/2,mFΩ = +1/2) = −1
22
+Ehf
3
−1
26
− Ehf
+ 2
3
−
ef
Page 16
16
Enr(˜F ∼ 3/2,mFΩ = −1/2) =
1
2
?dmfE
?dmfE
2
−Ehf
3
?
?
+1
2
??dmfE
??dmfE
6
− Ehf
?2
?2
+ 2
?dmfE
?dmfE
3
?2
?2
+
3ω2
4dmfE
3ω2
8dmfE,
ef
(32)
Enr(˜F ∼ 1/2,mFΩ = −1/2) =
1
22
−Ehf
3
−1
26
− Ehf
+ 2
3
+
ef
The equations of this section have so far been to one degree or another approximate results. But in the absence of
exotic particle physics we can invoke timereversal symmetry and write exact relations:
Enr(F,mF,Ω;E,B) − Enr(F,−mF,−Ω;E,B) = Enr(F,−mF,−Ω;E,−B)− Enr(F,mF,Ω;E,−B)
which, for B = 0, becomes
Enr(F,mF,Ω;E) = Enr(F,−mF,−Ω;E).
This exact degeneracy is, in fact, an example of the Kramers degeneracy that follows from timereversal invari
ance [122]. For our purposes, the key result here is that, in the limit of nonrotating fields, zero applied magnetic field,
and an electron EDM, the energy of the science transitions mF,Ω? ↔  − mF,−Ω? (and in particular, Wuand Wℓ)
are independent of the magnitude of the electric field. This is an important property because we are using spatially
inhomogeneous electric fields to confine the ions in the trap, and we want to minimize the resulting decoherence.
This degeneracy in turn means that the energy differences Wuand Wldepend only on the magnetic field and, of
course, the EDM term as shown in Eq. (30). The magnetic contribution reverses sign upon reversing the direction of
B with respect to the electric field direction (which also sets the quantization axis, since dmfE ≫ µBB). Therefore
the science measurement is given by the combinations
(33)
(34)
Wu(E,B) + Wu(E,−B) = 2deEeff
Wl(E,B) + Wl(E,−B) = −2deEeff, (35)
where a + sign on B denotes that it points in the same direction as E.
C.Rotating Fields, SmallAngle Limit
Many EDM experiments over the years have been complicated by the problem of “Berry’s phase”, the term in this
context used as a catchall to describe a variety of effects related to the motion of the particles in inhomogeneous
fields.
The sketch in Fig. 7(a) illustrates the classic Berry’s phase result: if the field that defines the quantization axis, as
experienced locally by a particle (or atom, or molecule), precesses about the laboratory axis at some angle, θ, then,
in the limit of slow precession, with each cycle of the precession the wavefunction Ψ picks up a phase given by mFA,
where mF is the instantaneous projection of the particle’s total angular momentum on the quantization axis, and A
is the solid angle subtended by the cone. If the precession is periodic with period τ, one can (with provisos, as we will
discuss) think of this phaseshift as being associated with a frequency, or indeed energy, mFA/τ . In a spectroscopic
measurement of the energy difference between two states whose mF values differ by δmF, there will be a contribution
to the transition angular frequency AδmF/τ.
In neutron EDM experiments, motional magnetic fields, in combination with uncharacterized fixed gradients from
magnetic impurities, Berry’s phase can be a dangerous systematic whose dependence on applied fields can mimic an
EDM signal [123]. In Sec. IVL we will see that the effects of motional fields in our experiment are negligible.
Neutral atoms or molecules may be confined in traps consisting of static configurations of electric or magnetic
fields. These traps are based on the interaction between the trapped species’ magnetic or electric dipoles and the
inhomogeneous magnetic or electric fields, respectively, of the trap. Especially in cases where the traps are axially
symmetric, so that the singleparticle trajectory of an atom can orbit many times one way or the other about the axis
of the trap, the coherence time of an ensemble of atoms with a thermal distribution of trajectories can be severely
restricted [124]. Our system is quite different, because in an ion trap the forces arise from the interaction between
the trapping fields and the monopole moment of our trapped ion. Assuming the temperature, size of bias field, and
radius of confinement are the same, the trapping fields for an ion are spatially much more homogeneous than would
be those for a neutral molecule or atom.
That said, the fact that we can speak of a “bias” electric field at all in an ion trap comes at the cost of having the
applied electric field constantly rotating.
Page 17
17
????????
A?~ ???? ??

(b)
?
A?~ ??
?
(a)
?
F
F
?
FIG. 7: (a) Smallangle limit. When the quantization axis F follows a slow periodic perturbation characterized by tilt angle
θ, angular frequency ω and enclosed solid angle A, two states whose instantaneous projection of angular momentum along F
differs by δm will have their effective relative energy displaced by a “Berry’s energ” ωAδm/2π. (b) Largeangle limit. When
instead the quantization axis sweeps out a full 2π steradians per cycle (α=0), the differential phase shift between the two levels
is indistinguishable from zero, and in the most natural conceptual framework, the Berry’s energy vanishes.
D. Rotating Fields, LargeAngle Limit (Dressed States)
The basic dressedstate idea is an extension of the more common idea of an energy eigenstate: a system governed
by a timeinvariant Hamiltonian H will have solutions Ψ such that Ψ(t + T) = e−iωTΨ(t) for all T and t; such a
solution Ψ is called an energy eigenstate, with ω being then the corresponding energy. Similarly, a system governed by
a periodic Hamiltonian with period τ such that H(t+τ) = H(t) for all values of t, will have socalled “dressedstate”
solutions Ψ such that Ψ(t + nτ) = e−inφΨ(t) for all t and all integer values of n. It is tempting to call φ/τ the
“energy” of the dressed state, but there will be an ambiguity in that energy because we can always replace φ with
φ + 2π.
Operationally, the dressed state energies are derived from the eigenvalues of a formally timeindependent Hamilto
nian. If H0denotes the Hamiltonian in the absence of the field, then the appropriate rotationdressed Hamiltonian is
given by
Hdressed= H0−?dmf·?Erot+ Hrot, (36)
Hrotis defined as [125]
Hrot= −ωrot(cos(θ)Fz− sin(θ)Fx)(37)
where Fzand Fxare the projections of the total angular momentum?F into a set of axes where z coincides with the
instantaneous direction of the electric field. We now make explicit the rotating electric field with?Erot. The cos(θ)
Page 18
18
term thus provides an energy which, when multiplied by the rotational period τ = 2π/ωrot, gives the ordinary Berry
phase,
− 2πcos(θ)mF → 2π(1 − cos(θ))mF
(38)
where we have taken the liberty of adding an arbitrary phase 2πmF to reveal explicitly the solid angle 2π(1 −cosθ).
In the experiment, the applied electric field should lie very nearly in the plane orthogonal to the rotation axis, i.e.,
θ ≈ π/2. It is therefore useful to consider the small angular deviation from this plane, α = π/2 − θ (Fig. 7). Then
the apparent energy shift arising from the geometric phase is
Egeo= −mFωrotsin(α) ≈ −mFωrotα.(39)
Now consider two states which are, in the absence of rotation, degenerate, say the states a?, with m = 3/2,Ω = −1,
and the state b?, with m = −3/2,Ω = 1, indicated in Fig 1(b). Rotation breaks this degeneracy, by adding the
energies ∼ ±(3/2)ωrotα, as shown by the dashed lines in Fig. 8. These levels cross at α = 0, leading to their apparent
degeneracy when the electric field lies in the horizontal plane.
In addition, the rotation of the field also incurs coupling between states with different mF values, arising from the
sin(θ) term in Eq. (37). This perturbation, treated at thirdorder in perturbation theory, connects the two levels and
turns the crossing into an avoided one, as shown by the solid lines in Fig. 8. Since the energy contribution due to the
rotating field is small compared to Stark energy splittings, we can use ideas similar to the derivation of Λdoubling,
i.e., we take a sum of the perturbing components and take them to the appropriate power. We look for terms in this
expansion that can connect the state a? = mFΩ? to b? =  −mF−Ω?. Therefore, the power of perturbation theory
needed is 2mF+ 1, where the 2mF takes mF → −mF and the extra power takes Ω → −Ω. The two terms in the
Hamiltonian that can do this are the Λdoubling term and the mFchanging terms of the rotating electric field. Our
expansion is, schematically, the following
Hcoup=(HLD+ Hrot)2mF+1
(∆EmF)2mF
. (40)
The (∆EmF)2mFare the energy level differences between states with different mF values, thus are related to the Stark
splittings. This tells us that
∆ ∼ ωef
?
ωrot
dmfErot
?2mF
, (41)
where ∆ is the energy splitting at the level crossing between otherwise degenerate states with mF > 0 and mF < 0.
The numerical prefactor in this expression has a rather complicated form within perturbation theory. However, its
value can be computed by numerically diagonalizing the relevant hyperfineplusrotation dressed Hamiltonian [121].
The result, for the mF= ±3/2 states in Fig. 1(b), is
∆u/ℓ≈ 170ωef
?
ωrot
dmfErot
?3
, (42)
where the superscript u refers to mixing between the a? and b? states, and the superscript ℓ to mixing between c?
and d? states. In the absence of the hyperfine interaction, the average value of the numerical prefactor is 170 and
the upper and lower states have the same avoided crossing. However, small fractional differences between ∆uand ∆ℓ
turn out to be significant, and are discussed further below.
The presence of the electric field causes the states with F = 1/2,mF= ±1/2? and F = 3/2,mF= ±1/2? to mix.
Including the hyperfine interaction into the numerical diagonalization yields
∆ =1
2(∆u+ ∆ℓ) ≈ 170ωef
?
?
ωrot
dmfErot
ωrot
dmfErot
?3
?3?dmfErot
,(43)
δ∆ =1
2(∆u− ∆ℓ) ≈ 127ωef
Ehf
?2
.(44)
It is evident that the average shift is the same, but now the upper and lower levels acquire a different splitting due
to the rotationinduced mixing within the sublevels. The difference is suppressed relative to the average value of the
splitting by a factor of (dmfE/Ehf)2, reflecting the fact that higher orders of perturbation theory are needed to include
the effects of the hyperfine interaction. For ωef= 2π × 10 kHz, ωrot= 2π × 100 kHz, dmfErot= 2π × 10 MHz, Ehf=
2π × 45 MHz, then ∆ = 2π × 2 Hz and δ∆ = 2π × 0.06 Hz.
Page 19
19
E/?
E/?
?mFrot
? ? ? /
(a)
(b)
FIG. 8: The apparent energy shifts between mF = +3/2 and mF = −3/2 states in upper (a,b) and lower (c,d) Λdoublet levels
versus α, the angle of the electric field to the plane orthogonal the rotation axis of Erot (α is shown in Fig. 7(b)). (a) At α = 0,
there is an avoided crossing that mixes mF = ±3/2 states, with an energy splitting at the crossing of ∆u/ℓ. (b) Since α = 0 at
the axial trap center, and since we need mF to be a signed quantity in order to measure de, we will bias away from the avoided
crossing using a magnetic field Brot. δmFgFµBBrot > ∆u/ℓis required for mF to be a quantity of definite sign. This picture
is intuitively correct in the limit that ∆u/ℓ> ωmax (see Sec. IVE). The experiment will be performed in the opposite limit.
However, solving the time dependent Schr¨ odinger equation (Eq. 55) gives the same requirement of δmFgFµBBrot > ∆u/ℓin
both limits.
The magnitude of the rotationinduced mixing within any of the four pairs of otherwise degenerate m = ±1/2 states
is much larger than the mixing within either pair of m = ±3/2 states, ∆uor ∆ℓ. For this reason, the m = ±1/2 levels
are probably not great candidates for precision metrology in rotating fields.
An ion in a trap will feel an axial force pushing it towards the axial position where the axial electric field vanishes,
that is, the location at which α is identically zero. This poses a problem, because at α = 0, each dressed state is an
equal mixture of states with Ω = 1 and with Ω = −1. In other words, the dressed states right at the avoided crossing
will have vanishing eEDM signal. The solution is to bias the avoided crossing away from α = 0 by adding to the
trapping fields a uniform, rotating magnetic field which is instantaneously always parallel or antiparallel to vector
Erot,
?Brot=Brot
Erot
In our convention,?Erotdefines the quantization axis, so that the number Erotwill always be taken to be positive. The
sign of Brotthen determines whether the corotating magnetic field is parallel (Brot> 0) or antiparallel (Brot< 0).
The energy levels are now as shown in Fig. 8(b). As derived below in Sec. IVE, in the limit BrotgFµB≫ ∆u/ℓ, the
dressed states near α = 0 are once again states of good mF and Ω. The energy splitting between the two states, as
altered by the rotation of the field, are given approximately by
?Erot. (45)
Wu/ℓ(Erot,Brot) = Ea/c− Eb/d= −3αωrot+ 3gu/ℓ
FµBBrot+
(∆u/ℓ)2
6gu/ℓ
FµBBrot− 6αωrot
± deEeff, (46)
where the + sign corresponds to u states, and the − sign to ℓ states.
Page 20
20
Over the course of one axial oscillation of the ion in the trap, α which is approximately proportional to the axial
electric field, will average to zero. Unfortunately, the contributions to δW from Erotand from Brotare larger than
that from the scale of the physics we most care about, deEeff, and the spatial and temporal variation in Erotand in
Brotwill reduce the coherence time of the spectroscopy, as discussed in Sec. IVH  IVJ below. But to the extent that
one is able quite precisely to chop Brotto −Broton alternate measurements, the science signal still arises from the
same combination as in Eq. 35:
?Wu/ℓ(Erot,Brot)? + ?Wu/ℓ(Erot,−Brot)? ≈ ±2deEeff,(47)
where the +/ corresponds to the u/ℓ superscripts respectively, and the brackets denote averaging over the excursions
of α, which is assumed to vary symmetrically about zero.
The equation above relies on several approximations. One needs in particular that dmfErot≫ ωef, 3gFµBBrot≫
170ωef(ωrot/dmfErot)3, and dmfErot≫ ωrot, α ≪ 1, and dmfErot< Ehf. These are all good approximations, but they
are not perfect. For example, using values from the Appendix, ωrot/(dmfErot) ≈ 0.01, a small number, but not zero.
To what extent will imperfections in these approximations mimic an eEDM signal?
The driving principle of our experimental design is to measure dewith as close to a null background as possible.
We are not especially concerned if the right hand side of Eq. (47) is 1.9 deEeffrather than 2.0 deEeff. More important
to us is that, if de= 0, the righthand side of Eq. (47) be as close to zero as possible. As we shall see, as long as
we preserve certain symmetries of the system we are guaranteed a very high quality null. A preliminary remark is
that the “energy” of a dressed state, or more precisely the phase shift per period τ, is unaffected by an offset in how
the zero of time is defined. A second observation is that, in the absence of exotic particle physics (such as nonzero
eEDM), the energy levels of a diatomic molecule in external electromagnetic fields are not affected by a global parity
inversion.
Under the action of this inversion, all the fields and interactions in the Hamiltonian transform according to their
classical prescriptions, whereas quantum states are transformed into their parityrelated partners. In a parityinvariant
system, parity thus changes quantum numbers, but leaves energies of the eigenstates unchanged. This is true for the
dressed states as well, since their eigenenergies emerge formally from a timeindependent Hamiltonian.
To formulate the effect of inversion symmetry we write the electric and magnetic fields as
?E = Erotˆ ρ′+ Ezˆ z
?B = Brotˆ ρ′
(48)
(49)
where ˆ ρ′= cos(ωrott)ˆ x + sin(ωrott)ˆ y and α = tan−1(Ez/Erot). The dressed states defined by the rotating field are
characterized by the projection mF of total angular momentum on the axis defined by the rotating electric field,
?Erot/Erot. Because the magnetic field is not strictly collinear with the electric field, and because of the field rotation,
mF is only approximately a good quantum number. Nevertheless, considering the effect of parity on all the mF’s
simultaneously, we can still map each dressed eigenstate into its parityreversed partner.
Assuming the ions are “nailed down” in their axial oscillation, at a particular value of Ezand thus α, our various
spectroscopic measurements would give dressed energy differences E(Erot,Brot,α,mF,Ω) − E(Erot,Brot,α,−mF,−Ω).
Now we invoke the following symmetry argument: if we take the entire system, electric fields, magnetic field, and
molecule, and apply a parity inversion, that will leave the energy of the corresponding levels unchanged. If further we
then shift the zero of time by π/ωrot, in effect letting the system advance through half a cycle of the field rotation,
that also will not change the corresponding energy levels of the dressed state, which are after all defined over an entire
period of the rotation. This transformation effectively connects measurements made for α > 0, above the midplane,
to those with α < 0, below the midplane. The combined transform acts as follows:
F → F
?Brot → −?Brot
?Erot →?Erot
Ez → −Ez
α → −α
? ωrot → ? ωrot
mF → −mF
?S ·?B →?S ·?B
ˆd ·?E →ˆd ·?E
Ω → −Ω
(50)
Page 21
21
The last of these is equivalent to ˆ n·? σ1, i.e., our symmetry operation would change the sign of the EDM energy shift.
However, in the absence of this shift we can expect the following exact relations between the dressed state energies:
(51)
E(Erot,Brot,α,mF,Ω) − E(Erot,−Brot,−α,−mF,−Ω) = 0
E(Erot,−Brot,α,mF,Ω) − E(Erot,Brot,−α,−mF,−Ω) = 0
E(Erot,Brot,−α,mF,Ω) − E(Erot,−Brot,α,−mF,−Ω) = 0
E(Erot,−Brot,−α,mF,Ω) − E(Erot,Brot,α,−mF,−Ω) = 0.
Summing four equations and rearranging terms, we get that
Wu/ℓ(Erot,Brot,α) + Wu/ℓ(Erot,−Brot,α) + Wu/ℓ(Erot,−Brot,−α) + Wu/ℓ(Erot,Brot,−α) = 0.
If we assume that the axial confinement is symmetric (not necessarily harmonic), and that our spectroscopy averages
over an ensemble of ions oscillating in the axial motion with no preferred initial phase of the axial motion (we will
later explore the consequences of relaxing this assumption) then the ions will spend the same amount of time on
average at any given positive value of α as they do at the corresponding negative value of α, and thus the averaged
results yield:
(52)
< Wu/ℓ(Erot,Brot) > + < Wu/ℓ(Erot,−Brot) >= 0 (53)
The combined result, in the absence of exotic particle physics, is zero by symmetry. We did not need to invoke the
various approximations that went into Eq. 47. In particular, this null result is, unlike the traditional Berry’s phase
result, not based on the assumption of very small (ωrot/dmfErot). Also, for conceptual simplicity we have discussed
the result as being based on an average over quasistatic values of α, but the symmetry argument does not hinge on
the axial frequency being infinitely slow compared to ωrot.
E. Frequency or PhaseModulation of Axial Oscillation
The trapped ions will oscillate in the axial direction at a frequency ωz, confined by an approximately harmonic
axial trapping potential Uz = (1/2)Mω2
ence an oscillating axial electric field Ez(t) = −Mω2
−mFωrotαmaxcos(ωzt), where αmax= Ez,max/Erotis the maximum excursion of the tilt angle. Because the product
ωrotαmax is again an energy, it is convenient to redefine the geometric energy contribution in terms of a frequency
ωmax,
zz2. Upon moving away from the midplane z = 0, the ions will experi
zz(t)/e. The geometric phase correction to the energy is then
Egeo= ωmaxcos(ωzt), (54)
with ωmax= −δmFωrotαmax.
For ωz= 2π×1 kHz, an ion cloud temperature of 15 K, an ion whose axial energy Ezis twice the thermal value, for
ωrotand Erotas shown in the Appendix, then a transition such as Wu/ℓwith δm = 3 will have a maximum frequency
modulation ωmax= 2π × 400 Hz.
Thus the electric field at the ion’s location undergoes two motions, the comparatively fast radial rotation, and the
comparatively slow axial wobble. We exploit the different time scales to create, for each instantaneous value of α, the
rotationdressed states worked out in the previous section. The effect of the axial wobble is then described by the
time variation of the amplitudes in these dressed states. The timedependent Schr¨ odinger equation of motion for this
is
i∂
∂t
?
a
b
?
=
?
3
2gFµBBrot+ωmax
2
cos(ωzt)
∆
2
∆
2
−3
2gFµBBrot−ωmax
2
cos(ωzt)
??
a
b
?
,(55)
where a and b are the probability amplitudes for being in the a? and b? states, respectively. For typical experimental
values, ωzis about 2π ×1 kHz, ωmaxwill range as high as 2π ×1 kHz, and ∆ (given by Eq. 42) is perhaps 2π ×2 Hz,
and 3µBgFBrotis about 2π × 8 Hz.
Eq. 55 describes a system again governed by a periodic Hamiltonian, and we will therefore follow a similar course
to Sec. IVD and search for dressedstate solutions Ψ such that Ψ(t + nτ) = e−inφΨ(t). Of course, this will only be
Page 22
22
???
mB
BFFroteff
g/
eff
E/?eff
FIG. 9: The apparent energy shifts between mF = +3/2 and mF = −3/2 states in upper (a,b) and lower (c,d) Λdoublet levels
versus Brot,“dressed” first by the electric field rotation (ωrot) and then by the ion’s axial trap oscillation (ωz). At Brot = 0,
there is an avoided crossing that mixes mF = ±3/2 states, with an energy splitting at the crossing of ∆u/ℓ
δmFgFµBBrot ≫ ∆eff, the dressed states are of good mF with an energy splitting slightly modified by ∆eff.
eff. In the limit
valid in the limit that ωrot≫ ωz, a necessary condition to write the timedependent Hamiltonian in Eq. 55. First, we
get rid of fast timedependence by guessing solutions:
a(t) = A(t)
∞
?
n=−∞
∞
?
Jn
?ωmax
?−ωmax
2ωz
?
?
e−iωznt
(56)
b(t) = B(t)
n=−∞
Jn
2ωz
e−iωznt,
where Jnare Bessel’s functions of the first kind and A(t) and B(t) are slowly varying functions. We then substitute
our trial solutions into Eq. 55 and use the recurrence relation (2n/x)Jn(x) = Jn−1(x)+Jn+1(x). We multiply through
∞
?
period, 2π/ωz, and make the approximation that A(t) and B(t) are unchanged over this small time interval. This
approximation should be good as long as ωz≫ ∆ and ωz≫ gFµBBrot. The integration then yields,
?
B
2
by
n′=−∞
Jn′
?ωmax
2ωz
?
eiωzn′tor
∞
?
n′=−∞
Jn′
?−ωmax
2ωz
?
eiωzn′tas appropriate. We then integrate over an axial time
i∂
∂t
A
?
=
?
3
2gFµBBrot
∆eff
∆eff
2
−3
2gFµBBrot
??
A
B
?
(57)
with
∆eff=
∞
?
n=−∞
Jn
?ωmax
2ωz
?
Jn
?−ωmax
2ωz
?
∆ = J0
?ωmax
ωz
?
∆.(58)
This results in dressedstate energies, now as a function of Brot, and not α, as seen in Fig. 9. This clearly shows
the requirement of 3gFµBBrot> ∆eff in order to keep a? and b? as the dressed states. This is true despite the fact
that an ion will sample the avoided crossing in Fig. 8 during its axial oscillation in the trap, as ωmax≫ 3gFµBBrot
in our experiment. ∆eff will have a maximum value of ∆ at ωmax/ωz= 0 and will oscillate about zero according to
J0(ωmax/ωz).
For finite ωmax/ωz, the dressed states from Eq. 57 only appear stationary if measured at integer multiples of the
axial trapping period, 2π/ωz. Consider states +? and −?, symmetric and antisymmetric combinations of states a?
and b?, respectively. In the limit that δmFgFµBBrot≫ ∆eff, an ion initially in state +? will oscillate between +?
and −? at the precession frequency ω0= ((3gu/ℓ
axial trapping frequency. However, if our EDM measurement is made after a noninteger number of axial oscillations,
or if the ions have different axial frequencies in the trap, the +? to −? oscillation will be frequency modulated at
ωmax. For the example parameters, the frequencymodulation index ωmax/ωz is less than 1, and thus the spectral
FµBBrot+deEeff)2+∆2
eff)
1
2, when measured at integer multiples of the
Page 23
23
power of transition is overwhelmingly at ω0, the quantity which symmetry arguments above show is unaffected by
Berry’s phase. In an ensemble of ions which have a random distribution of initial axial motions, the sidebands on the
transition average to zero, and won’t pull the frequency of the measured central transition. If instead the process of
loading ions into the trap has left the ions with an initial nonzero axial velocity or axial offset from trap center, the
measured frequency can be systematically pulled from ω0.
We note that increasing Erotor decreasing ωrotreduces the value of ωmaxand thus the frequency modulation index.
On the other hand, these changes also would have the effect of increasing the energy Erot of the micromotion of
the ions in the rotating fields. For harmonic axial confinement, we find the frequency modulation for a δm = 3
transition obeys the following relation ωmax/ωz = 3(Ez/Erot)1/2. Thus to keep the modulation safely under unity
for a comfortable majority of an ensemble of ions with an average Ez given by Tz, one needs to choose operating
parameters such that Erot> 30kBTz. This inequality in turn places stringent requirements on the spatial uniformity
of Erot. On a timescale slow compared to 1/ωrot, Erotacts like a sort of ponderomotive potential analogous to the
effective confining potential in a Paul trap. If Erot = 30kBTz, then a spatial inhomogeneity in Erot of only 1.5%
already gives rise to structure in the ponderomotive potential comparable to Tz.
To summarize the effect of axial motion: in the limit 3gFµBBrot> ∆, ions prepared, for instance by optical pumping,
in state a? (or b?) will remain in a? (or b?). The energy difference between dressed states which are predominantly
either a? or b? will be slightly modified by the avoided crossing. But the important combined measurement described
by Eq. (47) will continue to yield zero for de = 0, and the sensitivity of that combined measurement to a nonzero
EDM will not be much affected as long as ωmax/ωz<
∼1.
F. Structure of the Measurements. What Quantities Matter
In the remainder of this section, we will look at the possible effects of various experimental imperfections on our
measurement.
The symmetry argument in Sec. IVB presupposes the ability to impose a perfect “Bchop”, i.e., to collect data
with alternating measurements changing quite precisely only the sign of Brot. If not only the sign but the magnitude
of the rotating magnetic field alternates, the situation is more complicated. There will likely be contributions to the
rotating magnetic field that are not perfectly reversed in our Bchop, including displacement currents associated with
sinusoidally charging the electrodes that create the rotating electric field. These effects can be quantified with a value
Bstray
Wu/ℓ(?E,?Brot+?Bstray
This offset is very nearly the same for the upper and lower states, to the extent that gu
that δgF ≡ 1/2(gu
and lower states, in the form of a “fourway chop”:
rot , and to lowest order they would appear as a frequency offset in the chopped measurement:
rot ) + Wu/ℓ(?E,−?Brot+?Bstray
rot ) = 6gu/ℓ
FµBBstray
rot
± 2deEeff.
F≈ gℓ
(59)
F, i.e., to the extent
F− gℓ
F) ≪ gF. The effect of the stray field is reduced by combining measurements from the upper
?
−
= 12δgFµBBstray
Wu(?Erot,?Brot+?Bstray
?
rot ) + Wu(?Erot,−?Brot+?Bstray
Wℓ(?Erot,?Brot+?Bstray
+ 4deEeff.
rot )
?
rot ) + Wℓ(?Erot,−?Brot+?Bstray
rot )
?
rot
(60)
It may prove to be advantageous to shim the Bchop by deliberately adding a nonchopped rotating magnetic field,
Bshim
Wu(Brot+ Bstray
Then, a measurement in the lower Λdoublet state gives
rot, and adjusting its value until experimentally we measure
rot
+ Bshim
rot) + Wu(−Brot+ Bstray
rot
+ Bshim
rot) = 0.(61)
Wℓ(Brot+ Bstray
rot
+ Bshim
rot) + Wℓ(−Brot+ Bstray
rot
+ Bshim
rot) = −2(1 +gℓ
F
gu
F
)deEeff,(62)
yielding a still more accurate value for 4deEeff.
What we care about most then are: (1) Things that perturb Wuand Wℓdifferently, in particular the quantity δgF,
but also the quantity δη, to be defined and estimated in Sec. IVH, and (2) to a lesser extent, we care about effects
which affect Wu(Brot) + Wu(−Brot) the same way as they affect Wℓ(Brot) + Wℓ(−Brot), because, to the extent that
they lead to a measurement
Wu(Brot) + Wu(−Brot) = +2deEeff+ δsyst,(63)
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24
we can mistake a nonzero value for δsyst as an indicator for a nonzero value of Bstray
procedure discussed above to remove Bstray
(4δgF/gF)δsyst. This is down by a relative factor of (δgF/gF) compared to the effects that differentially perturb
Wuversus Wℓ, but they could still be troublesome. And (3) to a still lesser extent, we care about imperfections
that perturb individual measurements such as Wu(Brot), even if they do not perturb the Bchop measurement,
Wu(Brot)−Wu(−Brot), because, to the extent that they vary over time, or depend on the trajectory of an individual
ion in the trap, they can reduce coherence times. This leads not to systematic errors, but to a reduction in the overall
precision.
In addition to the B chop, state chop, and fourway chop discussed above, we can perform a rotation chop, by
changing the sign of ωrot. Our hope is to keep experimental imperfections to a level where the fourway chop is by
itself already good enough to suppress systematic error below the desired level. Then repeating the entire series of
measurements with the opposite sign of ωrot(rotating the field CW instead of CCW) will to the extent it yields the
same final value of 4Eeffdeprovide a useful redundant check.
rot . In that case, the shimming
rot
would lead to a combined result from the fourway chop of 4deEeff+
G. An Estimate of δgF=3/2
There are two leading contributions to δgF=3/2in our molecule. In the regime in which we will operate (a regime
wherein Ω is a signed quantity) they are to a good approximation independent of each other. These two contributions
are the zerofield difference and the induced difference caused by the applied electric field. In the zerofield limit, the
former is dominant. However, in the limit in which we are working, the latter dominates.
The zero electric field contribution arises due to centrifugal distortion effects in the molecular Hamiltonian. In
Sec. IVB, we wrote the Zeeman Hamiltonian in Eq. 22. We omitted two terms which connect states of Ω → −Ω.
Therefore, these states will give rise to a parity dependent gfactor for each Jlevel. The Hamiltonians which govern
this interaction can be found with the use of perturbation theory in a manner similar to the approach used to find
the Λdoubling parameters. Brown et al. [118] and Nelis et al. [120] have written these terms as
HZeemanDist = −1
2grSµB(B+J−S+S−+ B−J+S−S+)
HZeemanDoub =
1
2g′
rSµB(B+J+S2
++ B−J−S2
−), (64)
where HZeemanDistis the centrifugal distortion induced by the magnetic field and HZeemanDoubis the Zeeman induced
Λdoubling. HZeemanDist is parity independent while HZeemanDoub is parity dependent. Due to the nature of the
perturbation approach, we can estimate the size of g′
rSin terms of the Λdoubling J = 1 energy splitting ωef
g′
rS ≈
????
ωef
2Be
????,(65)
In addition, if the3∆1state of interest is composed of a (s)σ(d)δ molecular orbital (where (s) and (d) refer to atomic
orbitals with l = 0,2), then grS = g′
value in Eq. (65). It is evident that this effect is quite small, of the order 10−6for HfF+.
The electric field dependent gfactor arises due to the mixing of rotational levels J in the molecule. The levels with
J = 2, while far away in energy compared to the Stark energy dmfErot, are perturbers. In the signed Ω basis, the mF
sublevels in the J = 2 level have a smaller γF value than do the mF sublevels in the J = 1 level. Therefore, the
states which go up (down) in energy in the J = 1 level “gain” (“run”) on (from) the J = 2 level. When one includes
the effects of Hyperfine interactions, there are multiple connections to each sublevel. In the J = 1, mF= ±3/2 levels
that we are interested in, we can write an analytic expression for the electric field dependent δgF factor
rSis expected. The difference in zerofield gfactors is then given by twice the
δgF(Erot) =
?
J′,F′
dmfErot
Be(J + 1)
gF
γFΩ[F,F′,J,J′]2
?
F1 F′
−mF 0 mF
?2?
J1 J′
−Ω 0 Ω
?2?
F′J′I
JF 1
?2
,(66)
where [J,J′,...] =
is a 6Jsymbol. The sum runs on all states connected to J,F? by the electric field. In the case of HfF+with a
J = 1,F = 3/2 ground state, the sum contains the J′= 2 and F′= 3/2,5/2 states. Since the rotation constant
Beis far larger than either dmfErotor Ehf, only Beis included in the perturbative expression for δgF(Erot). For the
parameters here, this contribution is
?(2J + 1)(2J′+ 1).... The terms in parentheses are 3Jsymbols while the term in curly brackets
δgF=3/2
gF=3/2
(Erot) =9dmfErot
40Be
, (67)
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25
which means that the fractional shift δgF=3/2/gF=3/2is a few 10−4. The same approach gives that the electric field
“g” factor, γF, will shift in the same manner such that δγF/γF≈ 10−4.
For rotating fields, another contribution to δgF arises from nonvanishing value of ωrot/(dmfErot). The states
with Ω = 1 and Ω = −1 are equally affected by the rotating field since they have an equal Stark shift in the
absence of hyperfine interactions. However, because the levels with F = 3/2,mF= ±1/2? are repelled by the lower
F = 1/2,mF= ±1/2? states, the effective Stark difference between mF levels with Ω = −1 (upper levels) is smaller
than the same mF levels with Ω = +1 (lower levels). The scale at which this difference will appear is then determined
by how much the lower hyperfine state pushes on the upper due to the coupling induced by the electric field.
√6
γ2
F=3/2
δgF=3/2
gF=3/2
=
ω2
rot
dmfErotEhf. (68)
This fractional shift is of the order a few 10−4and is therefore about the same magnitude as the electric field induced
mixing of higher rotational levels.
H. Dependencies on Erot
Proximity to the avoided crossing shown in Fig. 8(b) means that the transitions Wuand Wℓwill have residual
dependencies on Erot, which in turn may lead to decoherence or systematic errors. We characterize the sensitivity of
Wu/ℓto small changes in Erotwith the following expansion
Wu/ℓ(E0
with
rot+ δErot,Brot) = Wu/ℓ(E0
rot,Brot) + ηu/ℓδErot
(69)
ηu/ℓ≡∂Wu/ℓ
∂Erot
E0
rot,Brot=
(∆u/ℓ)2
gu/ℓ
FµBBrotErot
, (70)
using the expressions in Eqs. (42) and (46). Any spatial inhomogeneity in Erotthat does not average away with ion
motion will lead to a decoherence rate given approximately by ηδErot.
In terms of systematic errors, if chopping the sign of Brot gives rise to an unintended systematic change in the
magnitude of Erot(call it δEchop), for instance due to motional fields discussed later, or due to ohmic voltages generated
by the eddy currents, then there will be a frequency shift in a Bchop combination, 2ηu/ℓδEchop. To the extent that
δη =1
is likely from δ∆, rather than from δgF. Assuming this limit, the systematic error surviving is
2(ηu− ηℓ) is nonzero, some of this shift will survive a fourway chop as well. The dominant contribution to δη
8δ∆
∆ηδEchop≈ 6
?dmfErot
Ehf
?2
ηδEchop. (71)
For a large but not inconceivable value for δEchopof 100 µV/cm, and for other values as in the Appendix, this works
out to comfortably less than 100 µHz, and is therefore not a problem. But this error would scale as E−5
could cause trouble if for other reasons we chose to decrease Erot. The science signal is roughly independent of Erot,
which should allow for the source of error to be readily identified.
rot, and thus
I.Perpendicular BFields
The quantization axis is essentially defined by Erot. The shift of the various levels a?, b?, c?, d? due to a component
of the magnetic field perpendicular to Erotis on the order of
±3
4γFdmfErot
for the upper/lower states. In the absence of rotation, the lowestorder correction to Wu/ℓ(Brot) goes as
−3
2
(gFµBB⊥)2
(72)
g3
(γFdmfErot)2.
Fµ3
BB2
⊥Brot
(73)
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26
For reasonable experimental parameters, this will be a negligible number. The lowestorder correction to the statechop
combination, Wu(Brot) − Wℓ(Brot) is smaller still and goes as
3
2
g3
γFdmfErotEhf.
Fµ3
BB2
⊥Brot
(74)
It is similar in form to the difference in gfactors caused by the rotation of the field.
When we turn on rotation, there is an additional larger contribution to Wu/ℓ(Brot). If we assume (as a worst case)
that B⊥ is purely axial, not azimuthal, then the lowestorder effect of B⊥ is to tilt the quantization axis by angle
given by
±gFµBB⊥
γFdmfErot± µBgFBrot, (75)
with the +() in the numerator corresponding to the upper(lower) states and the +() in the denominator corresponding
to the Ω = 1(+1) states. This has the leading order effect on Wu/ℓof
3ωrotg2
(γFdmfErot)2
Fµ2
BB⊥Brot
(76)
even a rudimentary nulling of the Earth’s magnetic field, say to below 25 mG, will leave this term negligible, for
parameters in the Appendix. Its contribution to the state chop, Wu(Brot) − Wℓ(Brot), is still smaller by dmfErot/Ehf
ωrotg2
BB⊥Brot
γFdmfErotEhf
Fµ2
(77)
J.Stray Contributions to B: Uniform or TimeVarying B Fields
In the previous section we have seen that the effects of B⊥are small. Spatial or shottoshot variation in B, on the
other hand, can limit coherence time through its contribution to Wu/ℓ. The biggest contribution to Bis of course
the intentionally applied rotating field Brot. Let’s examine the various other contributions to B.
Static, uniform fields: B fields of this nature are relatively harmless. Bis defined relative to the quantization
axisˆErot. The timeaverage of Bis ?B? = ??B ·ˆErot?. SinceˆErotsweeps out a circle with angular velocity ωrot, the
contribution to the timeaveraged Bfrom a uniform, static magnetic field averages nearly to zero in a single rotation
of the bias electric field, and still more accurately after a few cycles of axial and radial motion in the trap. The average
electric field in the ion trap must be very close to zero, or the ions would not remain trapped. In the case of certain
anharmonicities in the trapping potential, however, one can find that the average value ofˆErotis nonzero, even if the
average value of?Erotis zero. For instance, an electrostatic potential term proportional to z3, along with a uniform
axial magnetic field Bz, will for an ion with nonzero axial secular motion, yield a nonzero ?B?. In addition, nonzero
Bz will interact with the tilt of?Erot oscillating with an ion’s axial motion at ωz to cause a frequency modulation
similar to the one discussed in Sec. IVE. A uniform magnetic field in the xy plane will cause a frequency modulation
at ωrot. If the modulation index for either of these modulations approaches one, the modulation will begin to suppress
the contrast of spectroscopy performed at the carrier frequency. For uniform magnetic fields with amplitude less
than 10 mG (achievable for instance by roughly nulling the earth’s field with Helmholtz coils), frequency modulation
indices will be small, and, barring pathologically large z3electrostatic terms, the mean shifts from uniform, static B
fields will be less than 1 Hz and can be can be dealt with by means of an applied Bshim
Timevarying magnetic fields with frequency near ωrotcan cause more trouble. If the time between the two Ramsey
pulses used to interrogate the frequency is tRamsey, then the dangerous bandwidth is 1/tRamsey, centered on ωrot. We
discuss in order (i) thermally generated fields from the electrodes, (ii) ambient magnetic field noise in laboratory, (iii)
magnetic fields associated with the application of Erot, oscillating coherently with Erot, (iv) shot toshot variation in
magnitude of applied Brot, and (v) spatial inhomogeneities in Brot.
(i) Proposed EDM experiments on trapped atomic species such as Cesium are vulnerable to magnetic field noise
generated by thermally excited currents in conductors located close to the trapped species [126]. In our case, the
effect is less worrisome because, visavis the trapped atom experiments, our bandwidth of vulnerability is centered
at much higher frequency fields, because our molecules are trapped considerably further from the nearest conductors,
and because the sensitivity of our measurement of de to magnetic field noise, which goes as gFµB/Eeff is down by
a factor of 104. The spectral density of thermal magnetic field noise (which is calculated in reference [127] in the
rot
as discussed in Sec. IVF.
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27
simplified geometry of a semiinfinite planar conductor) will surely be less than 1 pG/Hz1/2in our bandwidth of
vulnerability. This effect is negligible.
(ii) Like thermal magnetic noise, technical magnetic noise in our lab arising for instance from various nearby
equipment will not so much decohere an individual measurement as generate shottoshot irreproducibility between
measurements. What level of noise are we sensitive to? As we discuss in Sec. VI below, the precision of a single trap
load is unlikely to be better than 300 mHz, meaning magnetic field noise less than 0.2 µG/Hz1/2won’t hurt us, for a
1 s interrogation time. Measurements made in our lab show that there are a number of magnetic field “tones” of very
narrow bandwidth, associated with harmonics of 60 Hz power and various power supplies. As long as we choose ωrot
to not coincide with one of these frequencies, in the range of 50 kHz to 300 kHz ambient magnetic frequency noise
in our lab has spectral density typically less than 0.02 µG/Hz1/2. For this reason, at least for the first generation
experiment, there will be no explicit effort to shield ambient magnetic field other than to use Helmholtz coils to
roughly null the earth’s dc field. The steel vacuum chamber will in addition provide some shielding at 100 kHz.
(iii) In traditional eEDM experiments, one of the most difficult unwanted effects to characterize and bring under
control is magnetic fields generated by leakage currents associated with the high voltages on the electrodes that
generate the principal electric field. In our case the bulk of the electric field Eeffis generated inside the molecule. The
laboratory electric fields are measured in V/cm, not kV/cm, and leakage currents as traditionally conceived will not
be a problem for us. On the other hand, the electric field does rotate rapidly, and thus the electrode potentials must
constantly oscillate. Displacement currents in the trapping volume between the electrodes, and real currents in the
electrodes themselves and in the wire leads leading to them, will generate magnetic fields with spatial gradients and
strengths that oscillate coherently with Erotat the frequency ωrot.
The spatial structure of the oscillating magnetic fields will depend on the geometry of the electrodes and in particular
on the layout of the wire leads that provide the current to charge them. In principle, shim coils can be constructed just
outside the trap electrodes and driven with various phases and amplitude of current oscillating at ωrot, all in order to
further control the shape of the magnetic field. The one immutable fact is the Maxwell equation, ∇×?B = c−2∂?E/∂t.
The dominant time dependence of the electric field is from the spatially uniform rotating field, and thus for a
circular field trajectory, the dominant contribution to the magnetic field structure goes as
∇ ×?B = kˆ y′
(78)
with
k = c−2Erotωrot= 350 nG/cm ×
?
ωrot
2π × 100 kHz
??
Erot
5 V/cm
?
,(79)
where ˆ y′is the direction in the xy plane orthogonal to the instantaneous electric field.
The curl determines only the spatial derivatives of B; B itself only depends on the boundary conditions. An idealized
arrangement of current carrying leads and shim coils could in principle force the B field to be
?Bideal= kx′ˆ z.
where k is given by Eq. 79 and x′is displacement in the xy plane along the direction of the instantaneous rotating
electric field. These fields would be perpendicular to the quantization axis provided by the electric field, and would
have negligible effect on the transitions of interest.
While realizing such an idealized displacement field would be very difficult, there are relatively simple steps to
take to minimize the displacement fields. For instance, each rodlike electrode can be charged up by two leads, one
connected to each end of the rod, with the leads running along respective paths symmetric in reflection in the z=0
plane to a common oscillating voltage source outside of the vacuum can, at z=0. It is worth considering a maximally
bad electrode layout, to put a limit on worstcase performance. Our electrodes will be spaced by about 10 cm and
mounted in such a way that their capacitance to each other or to ground will be at worst 5 pF. If the charging current
is provided entirely by a single lead connected to one end of the rod, the peak current running down the rod near its
center will be 80 µA, leading to a worstcase field magnitude at the trap center of about 20 µG, and a contribution
to Wu/ℓof perhaps 2.5 Hz. Spatial gradients of this effect, and shottoshot irreproducibility of this effect will not
contribute to decoherence at the 0.1 Hz level. As for its contribution to systematic error, this shift will survive the
Bchop, but will be suppressed in the fourway chop by the factor (δgF/gF), perhaps a factor of a thousand. For still
better accuracy the shift should be nulled out of the B chop by adjusting Bshim
(iv)Given that the main effect of Brotis to apply an offset frequency, 3gFµBBrotof perhaps 8 Hz, and given that
(see Sec. VI) the singleshot precision is unlikely to be any better than 300 mHz, the shottoshot reproducibility of
Brotneed be no better than a part in 30, a very modest requirement on stability. Decoherence then is not a problem,
but a potential source of systematic error arises if the the B chop is not “clean” that is if Brotbefore the chop is not
(80)
rot, as discussed in Sec. IVF.
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exactly equal to −Brotafter the chop. This sort of error could arise for instance from certain offset errors in opamps
generating the oscillating current. Experimentally, one adjusts Bshim
of that procedure, the fourway chop cleans up these sorts of errors. For a rather egregious fractional deviation from
Bchop cleanliness of, for instance, 1%, and for (δgF/gF) < 0.001, the systematic error remaining after the fourway
chop is 10−5of the offset frequency of perhaps 8 Hz. In HfF+this is a systematic error on deof 10−29e cm. For
ThF+the error as referred to de is smaller still, and of course if we avail ourselves of Bshim
Bchop signal to <100 mHz, the systematic error on dewill be less than 10−29e cm for either species.
(v) The largest single contribution to decoherence (with the exception of spontaneous decay of the3∆1line to a
lower electronic state) will likely be due to spatial inhomogeneity in the applied rotating bias field Brot. That is to say,
spatial inhomogeneities in?B that rotate in the xy plane at frequency ωrot. Firstorder spatial gradients in Brotare
not important, because ion secular motion in the trap will average away the effects of these gradients leaving only the
value of Brotat the center of the trap. Secondorder spatial gradients on the other hand will lead to nonzero average
frequency shifts whose value will vary from ion to ion in a thermal sample of ions, depending on conserved quantities of
individual ion motion like the axial secular energy Ezor radial secular energy Eρ, quantities with thermally averaged
values of kTzand kTρ, respectively, and with iontoion variation comparable to their mean values. The Brotwill be
generated by currentcarrying rods which are of necessity within the vacuum chamber because of the screening effects
of a metal vacuum chamber. Unless particular care is taken in the design of these rods, the secondorder spatial
gradients in Brot will scale as 1/X2, where X is the characteristic size (and spacing) of the current carrying rods.
The contribution to the inhomogeneity of the timeaveraged value of Brot experienced by a thermal sample of ions
orbiting in a cloud with r.m.s size r is then of order (r2/X2)Brot, leading to an iontoion frequency variability of order
(r2/X2)3gFµBBrotFor planned parameters of the experiment, (r2/X2) is of order 0.01. We have seen from Sec. IVE
above that the quantity 3gFµBBrotmust be at least about five times larger than ∆ in order to make the eigenstates
in the rotating fields be states of good mF. Thus in the absence of explicit apparatus design to null the secondorder
spatial gradient in Brot(The rodlike electrodes that bear the charge that generates Erotare in the secondgeneration
trap the same objects that carry the current that generates Brotand thus their shape is already subject to multiple
design constraints) we may have to live with a decoherence rate from this effect on the order of 0.05∆, perhaps 0.5
s−1, for the experimental values given in the Appendix.
The inhomogeneity in Brotshould reverse quite cleanly with the B chop, and residual imperfections there will be
cleaned up with the fourway chop, and thus the effects of the secondorder gradients in Brot are expected to be
predominantly a source of decoherence, rather than systematic error on measured de.
rot
to cancel these offsets, but even in the absence
rot
so as to null the post
K.Stray Contributions to B: Static BField Gradients
We now return to discussing static magnetic fields, now including the effects of spatial gradients. With the charac
teristic size of the ion cloud r being smaller than the characteristic distance X from cloud center to source of magnetic
field by a ratio of 0.1 or smaller, it makes sense to expand the field about the uniform value at the trap center. The
most general firstorder correction to a static magnetic field in the absence of local sources can be characterized by
five linearly independent components as follows:
?B = B′
axgrad(zˆ z −x
+B′
+B′
+B′
+B′
2ˆ x −y
2ˆ y)
trans(xˆ x − yˆ y)
1(yˆ x + xˆ y)
2(zˆ x + xˆ z)
3(yˆ z + zˆ y) (81)
By far the most important effect of these terms is the “micromotionaxial gradient interaction.” As discussed in
Sec. IIIB above, the displacement of an ion’s circular micromotion ? rrotis exactly out of phase with the rotation of its
quantization axisˆE, see Eq. 6. Averaged over a cycle of ωrot, this will give rise to a nonzero average contribution to
Band cause a shift in Wu/ℓgiven by 3gFµBB′
of stray B′
Hz, and this shift would survive the B chop. As with the effect of displacement currents, one expects the systematic
effect of the shift to be reduced after the fourway chop by (δgF/gF), but for maximum accuracy the effect should
be shimmed out of the B chop, either by adjusting the value of Bshim
external to the vacuum chamber) a compensating value of B′
A smaller effect arises from the interaction of the magnetic field gradient with the component of the electric fields
responsible for providing ion confinement, which after averaging over cycles of ωrot and ωrf, always point inward,
axgradrrot= 3gFµBB′
axgradeErot/(Mω2
rot). A guess for a possible value
axgradis 2 mG/cm, which for anticipated experimental parameters would lead to a shift in Wu/ℓof order 4
rot, or by applying (say with antiHelmholtz coils
axgrad.
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29
giving rise to a net inwardpointing time average ofˆE. If we look at only the component of the firstorder magnetic
field gradient that points towards or away from the trap center
?Bcentral= (B′
trans− B′
axgrad/2)xˆ x + (−B′
trans− B′
axgrad/2)yˆ y + B′
axgradzˆ z. (82)
The net contribution to Bcomes from integrating, along the rf and rotation micromotion trajectories, over first a
rotational cycle, and then an rf cycle, and then a secular cycle in a given direction. We assume that the trap is suffi
ciently harmonic that there is no crossdimensional mixing of secular energy, that ωx, ωy, and ωzare incommensurate
and with principle axes as defined in Eq. 81, and that hardmomentumchanging collisions are rare enough so that,
during the duration of a spectroscopic measurement, there is no change in Ei, the sum of the kinetic and potential
energy associated with an individual ion’s secular motion in the ith direction. The contribution to Bis then,
δB= −((B′
The contribution to Baveraged over a thermal sample of ions is given by the above expression with Ei replaced
by Ti. Note that for Tx = Ty = Tz, several terms cancel and the thermally averaged contribution to Bis just
B′
general differ from one another for a given ion, and between different ions. For B′
mG/cm, ion temperatures about 15 K, the mean shift in Wu/ℓfor typical experimental parameters given in the
Appendix might be 30 mHz, with a comparable contribution to dephasing.
The three remaining terms in the firstorder gradient, B′
combined with other (usually small) trap imperfections, for instance the plane of rotation of Erot being tilted with
respect to the principal axes of the confining potential. The net effects will be correspondingly smaller than those
from B′
Just as with the second spatial derivative of Brot, the spatial derivative of B′
the size of ion orbits, can give rise to decoherence. Of course, Baxgradis defined already as a first spatial derivative
of a magnetic field, thus the dephasing arises from a third derivative of the field, and its rate should be down from
the mean size of the shift (roughly estimated above at 4 Hz) by a factor of order (r/X)2, or a factor of one hundred.
Even spatially uniform B′
change in rrotis the same as the fractional change in Erot. As discussed in Sec. IIIB, this should be smaller than 0.5%
over the typical size of the ion sample.
As a coda to this subsection, it is worth considering that applying a very spatially uniform Brot may be very
challenging because of difficulttomodel eddy currents induced in electrodes and lightgathering mirrors. On the other
hand a purposely applied static B′
materials, which can be minimized and modeled. One way or another we will need to bias away from the avoided
crossing discussed in Sec. IVD, but it may turn out that this can be accomplished with greater spatial uniformity
and thus with a lower total decoherence rate by omitting the applied Brotaltogether, and providing the bias with a
deliberately applied B′
invariance argument of Sec. IVD above can readily be modified to describe a chop of B′
Brot.
To sum up subsections IV.J and IV.K, we have looked at a range of ways in which various contributions to Bcan
shift Wuand Wℓ. Decoherence due to shottoshot fluctuations or spatial inhomogeneity should not be a problem
out to beyond 1 s coherence times. Various effects can shift Wuand Wℓby as much as a few Hz, and this shift can
survive a B chop. With δgF/gF on order of 10−4, and Eeff estimated at 90 GV/cm in ThF+, after a fourway chop
the remaining systematic error will be a few 10−29e cm, but this can be dramatically reduced by tuning away the
postBchop signal with Bshim. The most dangerous systematic error would be if Brotwere systematically different
between measurements on the upper and on the lower states. Chopping between upper and lower states will be
determined by variations in optical pumping, which should be well decoupled from the mechanisms that generate
Brot.
trans/2 − B′
axgrad/4)Ex+ (−B′
trans/2 − B′
axgrad/4)Ey+ B′
axgradEz)/(eErot). (83)
axgradEz/(2eErot). The decohering effect is comparable because within a thermal sample, Ex, Ey, and Ez will in
axgradand B′
transeach about 2
1, B′
2and B′
3, will contribute to a shift in Bonly when
trans.
axgrad, coupled to a thermal spread in
axgradcould give rise to decoherence if there is a spatial dependence in rrot. The fractional
axgradwould be perturbed only by the magnetic permeability of trap construction
axgradfield. The B chop could be accomplished by chopping the sign of B′
axgrad. The parity
axgradrather than a chop in
L. Relativistic (IonMotionInduced) Fields
The largest component of the velocity on the ions is that of the micromotion induced by Erot; for reasonable
experimental parameters it will be less than 1000 m/s. In typical labframe magnetic fields of a few mG, the motion
will give rise, through relativistic transformation, to electric fields of order of a few µV/cm, which are irrelevant
to our measurement. Conversely, motion at 1000 m/s in typical labframe electric fields of 10 V/cm generates a
magnetic field of 0.1 µG. This field will be rigorously perpendicular to the electric field, the quantization axis, and
thus represents only a negligible modification to the generally unimportant B⊥.
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30
(a)(b)
?x
?x
?z
?y
?rot
?rf
?z
FIG. 10: Over one rotation of Erot, both Erf and Ez are quasistatic. The total electric field is the sum of all three and its
trajectory over one cycle of Erot is plotted as the dotted line projected onto (a) the xy and (b) the xz planes. The electric
field trajectory is a circle of radius Erot, parallel to and displaced from xy plane, a circle whose center is offset from the zaxis
by Erf. In the limit  Erf  ≪  Erot , the solid angle subtended from the origin by this circle differs only slighlty from that
subtended by a circle with vanishing Erf. The magnitudes of both Erf and Ez relative to Erot are very much exaggerated for
clarity.
M.Effect of RF Fields
The effects of the rf electric fields providing Paul trap confinement are best understood by putting them in the
context of a threetier hierarchy of electric field magnitudes and frequencies.
(i) Erot, the nominally uniform, rotating electric field, with field magnitude of perhaps 5 V/cm and frequency
ωrot= 2π × 100 kHz.
(ii) Erf, the Paultrap fields, are highly inhomogeneous, but at a typical displacement in the xy plane of perhaps 0.5
cm, the field strength might be 75 mV/cm, or two orders of magnitude less than that of Erot, oscillating at a frequency,
ωrf = 2π × 15 kHz which is one order of magnitude less than ωrot. At a fixed point in space, the rf fields average
rigorously to zero over time, but averaged instead along an ion’s rf micromotion trajectory, the rf fields contribute to
(iii) the inwardpointing trapping electric field, again very inhomogeneous but with typical strength down from peak
rffield values by factor of (ωrf/ωi), another order of magnitude, to perhaps 5 mV/cm. From the ion’s perspective,
the direction of the trapping fields oscillate with the ion’s secular motions in the trap, at frequencies ωiof perhaps
2π × 1 kHz, the slowest time scale by an order of magnitude.
The effects of the strong, fast Erothave been discussed extensively throughout Sec. IV, and those of the weak, slow
trapping fields were covered in Sec. IVK above. In this subsection we argue that the rf electric fields, intermediate in
both frequency and strength, are the least significant of the three categories.
The effects of the rf fields averaged over the rf micromotion trajectory are discussed in Sec. IVK. The remaining
part averages to zero in one rf cycle, but is roughly frozen at a single value over the duration of one cycle of ωrot. The
dominant source of the rf fields’ timeaveraged contribution to transitions Wu/lis in very small corrections to Berry’s
phase energy associated with the rotation of Erot. See Fig. 10. The correction to the solid angle arising from Erfgoes as
(Ez/Erot)(Erf/Erot)2. If we include a factor of ωrotto get a Berry’s energy shift and evaluate for typical experimental
parameters, the magnitude of the resulting frequency shift will be about 20 mHz, and will oscillate in sign with the
axial secular motion. The magnitude of radial rf fields scales linearly with the radial secular displacement. If secular
freqencies were commensurate, in particular if ωz = 2ωr, then this 20 mHz shift could contribute to a decoherence
rate at the negligible level of a few tens of mHz. For incommensurate ratios of ωz/ωxor ωz/ωy, the rf fields will be
still less important.
N. Systematic Errors Associated with Trap Asymmetries
The symmetry argument of Sec. IVD was based on parity invariance. This argument is only as good as reflection
symmetry of the electric and magnetic fields in the region of the trapped ions. In this section we look, as an example,
at the consequences of a symmetry imperfection.
The electrodes used to generate Erothave been numerically designed to make Erotas spatially uniform as possible,
but imperfections in design and construction of the trap and imperfect drive electronics will lead to some residual field
nonuniformity. Suppose that the magnitude of the Erotwas consistently larger in the region of the trap for which z>0,
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31
so that the value of Erotover the z>0 half of an axial secular oscillation is about 0.3% larger than that experienced
over the z<0 half. Thus the frequency modulation of perhaps ± 500 Hz, discussed in Sec. IVE will no longer average
to precisely zero over an axial cycle but instead a net contribution of about 1.5 Hz to Wu. Such a frequency shift
would survive a B chop, and, following the protocol discussed in section IV.F, we could very likely incorrectly identify
this shift as arising from the presence of a Bstray
a complete fourway chop, we would be left with a systematic error on the order of (δgF/gF)×1.5 Hz, or about 0.4
mHz.
For the value of Eeff estimated for HfF+, a 0.4 mHz error corresponds to a systematic error on de of the order
of a few 10−29e cm. For ThF+, the error on dewould be about three times smaller. We continue a more general
discussion on systematic errors in Sec. VID below.
rot , and apply a value of Bshimto largely null the 1.5 Hz shift. After
V.COLLISIONS
The overarching strategy of the trappedion approach to precision spectroscopy is to accept low count rates in
exchange for very long coherence times. In some previous precision measurement experiments with trapped ions, the
very best results have come from taking this to the extreme limit of working with only one ion [128–135], or in some
cases a pair of ions [136], in the trap at any given time. More often however, optimal precision is achieved working
with a small cloud of trapped ions. In this section we evaluate various detrimental effects of ionion interactions.
A. MeanField
With no electrons present to neutralize overall charge, even a relatively low density cloud of ions can have a
significant meanfield potential. A spherically symmetric sample of Nionions confined within a sphere of radius r will
give rise to a meanfield potential
Umean−field
kB
≈ 3 K ×
?Nion
1000
??
r
0.5 cm
?−1
. (84)
At values of the meanfield interaction energy comparable to or larger than kBT, there is a risk of instabilities, viscous
heating, and other undesirable effects; even in their absence, systematic errors are more difficult to analyze in the
strong meanfield limit. Iontrap experiments have been performed at much higher meanfield strengths, and indeed
there have been precision spectroscopy experiments done in systems for which the interaction potential even between
an individual pair of nearestneighbor ions is much larger than kBT. However, these systems exhibit a high degree of
spontaneous symmetry breaking including crystallization [137].
For the purpose of this paper, we assume the experiments will be done in the low meanfield limit, say
Umean−field<
∼
1
3kBT. (85)
In this limit, meanfield effects are relatively benign, and can be modeled as a modest decrease in the trap confining
frequencies, ωi, plus the addition of some anharmonic terms to the potential. Crucially for the arguments presented
in Sec. IVD, these additional modifications do not break any of the reflection or rotationbased symmetries of the
trapping fields. We note that Eqs. 84 and 85 combine to set limits on various combinations of the ion number, Nion,
ion temperature, T, cloud radii, ri ∝
compromises in selecting operating parameters.
In Sec. IVE, we saw that the axial component of the electric field at the ion’s location, Ez, tilts the rotating electric
field and gives rise to an apparent shift of the energy of our spectroscopic transition, linear in Ez. This energy shift
integrated over time in turn gives rise to an oscillatory phase shift, ∆φ =ωrot
the effects of longrange, grazingangle ionion collisions may be thought of as simply a fluctuating component to the
local electric field, and the integrated effect of those fluctuations will make a random contribution to the phase shift.
We present a simple argument to show that the resulting rms spread in phase does not continue to increase with
time but reaches a steadystate asymptote. This is because Ez not only shifts the transition energy, it also causes
an axial force and corresponding acceleration, which, like the shift in transition energy, is linear in Ez. Integrated
over time, ∆pz= e?Ezdt, this fluctuating force results in a fluctuating momentum. But we know that the combined
without bound but rather to be loosely bounded by a characteristic thermal value,
nature of the thermal equilibration process – once an ion has developed a superthermal momentum, further collisions
?
kBT
Mω2
i, and mean ion density, n ∝
Nion
rxryrz. This necessitates making various
Erot
?Ezdt. In a onecomponent ion cloud,
effect of a trapping field and a large number of random collisions will not cause the rms momentum to randomly walk
??p2
z? ≈√MkBTz. This is the
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32
εrot
εrot
(a)(b)
εion
θ
FIG. 11: Geometric phases accumulated during an ionion collision. (a) A typical ionion collision trajectory (red), resultant
Rutherford scattering angle, θ, and ionion interaction electric field, Eion, are shown in the collision plane (blue). For clarity, the
collision plane has been taken perpendicular to the instantaneous direction of Erot. (b) During an ionion collision the molecular
axis adiabatically follows the net electric field vector,?Erot+?Eion, and traces out the contour (black) on the unit sphere (yellow).
The solid angle, ∆A(θ), subtended by this contour gives rise to a geometric phase accumulated by the eigenstates during the
collision. This leads to decoherence of the spectroscopic transition, see text.
are biased to reduce the momentum. Since both the phase excursion and the momentum excursion are linear in the
??∆φ2? ≈ωrot
∼30kBTz, the phase fluctuations for each ion’s spectroscopic transition will be bounded by a value less than
one radian, so that there will be no loss in spectroscopic contrast in a Ramseytype experiment.
The argument in the paragraph above hinges on the assumption that the electric field arising from the ion cloud’s
meanfield distribution and from grazingangle collisions is small in magnitude compared to Erot, so that the shift in
Berry energy is linear in the axial component of the electric field. For higher values of the ion temperature or lower
values of Erot, a pair of colliding ions can get so close to each other that the electric field is, transiently, comparable
to or larger than Erot. We discuss the consequences in the next subsection.
timeintegrated axial electric field, we can estimate
if Erot>
Erot
??p2
z? ≈
?
kBTz
2Erot. Again, as discussed in Sec. IVE,
B. Geometric Phases Accumulated During an IonIon Collision
As discussed in Sec. IVD, when a spin adiabatically follows a timevarying quantization axis it acquires a geometric
(Berry’s) phase. For the eigenstates in Fig. 1(b), the geometric phase factor can be written as exp(±imFA), where
A is the solid angle subtended by the contour on the unit sphere traced out by the timevarying quantization axis.
Thus, the relative phase generated between the F = 3/2,mF = ±3/2? states used for spectroscopy is φ = 3A. The
concern of this subsection is how ionion collisions cause uncontrolled excursions of the quantization axis leading to
random geometric phase shifts and decoherence between spin states. These uncontrolled phase shifts will be written
as ∆φ = 3∆A to distinguish them from the calibrated geometric phases in the experiment.
The instantaneous quantization axis for the molecular ion eigensates is defined by the net electric field vector at
the location of the ion. During a collision, this axis is defined by the vector sum of the rotating electric field,?Erot, and
the ionion interaction electric field,?Eion. Both of these are timevarying vectors, however typical ionion collisions
have a duration short compared to the rotation period of Erotso for the purpose of this discussion Erotwill be taken
as stationary. Thus, the problem is reduced to calculating the excursion of the quantization axis under the time
variation of Eion. A typical ionion collision is shown in Fig. 11(a) and the effect of this collision on the quantization
axis is shown in Fig. 11(b).
At the temperatures of our trapped ion samples, no two ions are ever close enough for the details of the intermolecular
potential to matter. Only monopolemonopole and monopoledipole interactions matter. Further, the translational
degree of freedom may be treated as purely classical motion in a 1/r ionion potential, with the initial condition of
a given collisional event characterized by an impact parameter and relative velocity. The outcome of the collision
depends not only on the magnitudes of the impact parameter and of the velocity, but also on their angles with respect
to the ambient electric bias field, Erot. Each initial condition contributes a particular amount to the variance in
the phase between the relevant internal states. These contributions can be converted to partial contributions to a
decoherence rate, and a numerical integral over a thermal distribution of collisional initial conditions can yield the
total decoherence rate. We have pursued this program to a greater or lesser extent with the decoherence mechanisms
discussed in this subsection and the one immediately following, but the results are not especially illuminating and we
have used them primarily to confirm that the powerlaw expressions discussed below represent only overestimates of
the decoherence rate, and that for experimental parameters of interest, the decoherence rate will be conservatively
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33
less than 1 s−1.
The main question is whether T is high enough to include significant phase space for collision trajectories for which
the peak value of Eion > Erot (which is to say, large enough to transiently tip the direction of the total field by
more than a radian). If so, then a single collision can cause decoherence and one can get a simple estimate of the
crosssection for decoherence simply from the size of the impact parameter that leads to those events. There is a
significant probability for collisions with Eion>
∼Erotwhen
T ≫ 18 K
?
Erot
5 V/cm
?1/2
,(86)
which leads to a decoherence rate
τ−1≈ 0.47 ×
?
n
1000 cm−3
??
T
15 K
?1/2?
Erot
5 V/cm
?−1
s−1.(87)
If T is instead so low that the Coulomb barrier suppresses collisions that could lead to a sufficiently large value of
A and cause decoherence with a single collision, then decoherence will arise only from the combined effects of many
collisions each causing small phase shifts that eventually random walk the science transition into decoherence. In this
regime, the decoherence rate falls off very fast at low temperatures. For
T ≪ 18 K
?
Erot
5 V/cm
?1/2
,(88)
typical collisions have Eion<
∼Erotand the decoherence rate is
τ−1≈ 0.13 ×
?
n
1000 cm−3
??
T
15 K
?13/2?
Erot
5 V/cm
?−4
s−1.(89)
Both Eqs. 87 and 89 represent conservative estimates of the decoherence rate, and for an intermediate range of
temperature, the decoherence rate will be less than whichever estimate gives the smaller value (Fig. 12).
C.mLevel Changing Collisions
A second source of decoherence can arise from ionion collisions that induce transitions between internal levels of a
molecule. The dominant inelastic channel will be transitions between mFlevels induced by a sufficiently sudden tilt in
the quantization axis defined by the instantaneous local electric field. There are two conditions for such a transition
to occur: (i) the direction of the total field must change by nearly a radian or more, so that there is significant
amplitude for, e.g., an mF = +3/2 level in the unperturbed electric field to suddenly have nonnegligible projection
on an mF = +1/2 level in the collisionperturbed field, and (ii) the time rate of change of the electric field direction
must be comparable to or larger than the energy splitting between an mF= 3/2 level and its nearest mF= 1/2 level
in the field Erot.
Note that the first requirement is the same as the requirement for picking up an appreciable single collision Berry’s
phase. However, not all collisions that satisfy the first requirement will satisfy the second requirement. In particular,
if the relative velocity in a collision is too low, then the time rate of change of the electric field direction will be too
slow to satisfy the second requirement. Thus, given that the first requirement is satisfied, then the second requirement
will not be satisfied when
T < 5 K ×
Erot
5 V/cm,(90)
In this limit, the second requirement is more stringent than the first requirement, which means that the rate of
mlevel changing collisions will be smaller than the rate of singlecollision Berry’s phaseinduced decoherence. In the
opposite limit, we expect the second requirement will be met whenever the first requirement is met, and thus we would
expect that the two channels of decoherence, mlevel changing and singlecollision Berry’s phase, will be comparable
in magnitude.
Looking at particular collision trajectories in more detail, we see that there are trajectories that can cause an mlevel
change but for which there is no contribution to Berry’s phase because the electric field traces out a trajectory with no
solid angle (for instance, if the classical impact parameter?b is parallel to?Erot). We also note that our formulation of the
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34
102
100
102
104
106
108
τ1 [s1]
(a)
T [K]
1 10
100
2
20 505
εrot = 1 V/cm
εrot = 10 V/cm
εrot = 100 V/cm
102
100
102
104
106
108
τ1 [s1]
(b)
εrot [V/cm]
0.20.5125102050 100 200
T = 2 K
T = 10 K
T = 50 K
FIG. 12: Inverse coherence times, τ−1, due to geometric phases accumulated during ionion collisions as a function of (a)
collision energy in temperature units and (b) Erot. Dotted lines are approximations given in Eqs. 87 and 89. Solid lines
are more involved estimates based on integrals over collision parameters, but are still based on approximations so as to be
conservative. The ion density was taken to be n = 1000 cm−3.
requirement of sweep rate for mlevel changing collisions neglects the fact that Eionwill not only change the direction
of the total electric field (?Eion+?Erot) but also in general will change its magnitude. For most impact parameters,
the magnitude of the total electric field will increase, thus suppressing nonadiabatic effects. However, a narrow range
of impact parameters exists where the magnitude of the total electric field decreases, thus enhancing nonadiabatic
effects. However, the above scaling laws account for the majority of collisions.
In the end, we are less interested in the actual rates than we are in putting conservative limits on decoherence rates.
For instance, in calculating the curves in Fig. 12, we pessimistically took a worstcase geometry, Erot⊥ Eion, which
gives an upper limit on the size of the effect. Thus we estimate that:
• For T < 5 K ×
effects, will be less than or equal to the value given by solid curves in Fig. 12, while
Erot
5 V/cmthe total collisional decoherence, including both mlevelchanging and Berry’sinducing
• For T > 5 K×
by those curves.
Erot
5 V/cm, the total collisional decoherence will be no greater than twice as large as the rate indicated
VI.CONCLUSIONS: PRECISION AND ACCURACY
Recall from Sec. I the three components to the sensitivity figureofmerit:
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35
A.Coherence Time
Conclusions of Sec. IV and V: Taking into account only collisional decoherence, and all the questions associated
with being in rotating fields and in trapping fields, we would anticipate a coherence time longer than one second.
Blackbody thermal excitation of the J=1 rotational level will also be well over one second. Vibrational blackbody
excitation for the v=0 state is estimated at 6 s for HfF+in a 300 K environment. Thus the dominant limitation to
coherence will likely be the radiative lifetime of the3∆1state, estimated [138] at 390 ms for HfF+, and still longer
for ThF+, for which the3∆1 state is predicted to be still lower in energy. The largest uncertainty in the lifetime
calculation is the uncertainty in the3∆1→1Σ decay energy, calculated to be 1600 cm−1in HfF+.
B.Eeff
Eeffin HfF+is calculated by Meyer and coworkers to be 30 GV/cm [28], and by Titov et al. to be 24 GV/cm [101].
For ThF+, Meyer calculates 90 GV/cm [28]. The uncertainties in these numbers are hard to assess, but they are
very likely accurate to better than a factor of two and, if ongoing spectroscopic studies provide experimental values
of hyperfine and fine structure that confirm the ab initio values predicted by the St. Petersburg group, our confidence
in the precision of calculated Eeffwill be much higher.
C.Count Rate and Summary of Expected Precision
We are producing HfF+ions by photoionization in a relatively narrow range of quantum states, and can estimate
yield per quantum level within the desired trapping volume at perhaps 100 ions per shot, but we have just begun to
characterize the efficiency of the process and very little optimization has been done. Our design efficiency for reading
out spin states of trapped ions via laserinduced fluorescence (LIF) is 4%, but that has not been verified yet. With
a large uncertainty, then, we may detect about one ion per shot with four shots per second. Overall, precision in
one hour could be about 10 mHz. For ten hours of data, we anticipate (very roughly) a raw precision at 5 × 10−28
e cm in HfF+, and 1.5×10−28e cm in ThF+. We are investigating several more efficient alternatives to LIF for spin
readout, including in particular resonantly enhanced photodissociation or second photoionization. Even if we detect
as many as four ions in a shot, single shot precision will be no better than 300 mHz, which sets a relaxed requirement
for suppressing experimental shottoshot noise.
D.Accuracy, Systematic Error
We have not completed a systematic study of the consequences of all possible violations of reflection symmetry in
the trapping fields, but work in this direction is ongoing.
For now, we make the following three observations:
i) For the field asymmetries we have analyzed to date, realistic estimates for the magnitude in asconstructed field
imperfections lead to systematic errors on the order of a few 10−29e cm or less. While this is not yet as accurate as
our ultimate ambitions, it would represent roughly a factor of thirty improvement on the existing best experimental
limit.
ii) Asymmetries analyzed to date lead to systematic errors whose signs reverse when the direction of rotation ωrot
reverses. If we combine measurements made with clockwise and counterclockwise field rotation, the errors vanish.
Ideally, we’d like to design sufficient accuracy into the experiment so that the chop in field rotation is not needed to
achieve desired accuracy, but as a practical matter we will of course run the experiment both ways, averaging the
results to get ultimate accuracy, and differencing them to diagnose experimental flaws.
iii) Auxiliary measurements are envisioned to characterize and shim out flaws in the asconstructed trap. For
instance, we plan to be able to shim the equilibrium position of the ion cloud up and down along the trap axis, and at
each location measure the energy difference EbEd. Unlike EaEb, EbEdis highly electricfield sensitive. The result
will be a precise measurement of any spatial gradient in Erot.
iv) All systematic errors we have analyzed to date have strong dependencies on quantities such as ωrot, Brot, Erot,
and on the ioncloud temperature and density, and the trap confining frequencies. A true signal from a nonzero value
of dewill be largely independent of all those quantities. We anticipate making a number of auxiliary measurements
with the experimental parameters tuned far away from their optimal values to deliberately exaggerate the size of
systematic errors and allow us thus to characterize their dependencies in less integration time than that required for
ultimate sensitivity. Even so, and as is often the case in precision measurement experiments, sensitivity and accuracy
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36
are coupled. To the extent we can measure deto high precision at many combinations of experimental parameters,
we will better be able to detect and reject false signals.
We believe the experiment as we have described it should have the capability to improve the limit on the electron’s
electric dipole moment to 10−29e cm. As of this writing, the largest contribution to the uncertainty in our ultimate
capability has to do with unknown efficiencies of state preparation and read out. More specialized publications from
our group addressing progress in these areas are forthcoming.
E. Summary
Until now, molecular ions have not been viable candidates for symmetry violation searches largely due to the fact
that applying electromagnetic fields to manipulate the internal states of the molecule would also violently perturb
the translational motion of the ions. In this work, we have proposed a technique to overcome this obstacle – namely
applying an electric field that rotates at radio frequencies. The specifics of performing highresolution electron spin
resonance spectroscopy under these conditions were analyzed. In particular, we have shown that a significant advance
towards detecting the permanent electric dipole moment of the electron can be made by probing the valence electrons
in a ground or metastable3∆1level of an ensemble of trapped diatomic molecular ions.
Note added in proof: Since the submission of this work, a new experimental limit on the electric dipole moment of
the electron has been achieved using YbF molecules: de < 10.5 × 10−28e cm [139].
Appendix: Typical Experimental Parameter Values
deEeff= 2π ×0.36 mHz, transition energy between mF= +3/2 and mF= −3/2 states in ThF+if de= 1.7 x 10−29e
cm.
dmf= +1.50 a.u. ≈ 2π × 2 MHz/(V/cm), electric dipole moment of HfF+in the molecular rest frame.
Erot= 5 V/cm, rotating electric field.
ωrot= 2π × 100 kHz, frequency of rotating electric field.
Erot= 1800 K, typical kinetic energy in rotational micromotion.
rrot= 0.6 mm, radius of circular micromotion.
dmfErot= 2π × 10 MHz.
(3/2)γF=3/2dmfErot= 2π × 5 MHz, Stark shift of mF = ±3/2 states of3∆1levels in rotating electric field.
ωef= 2π × 10 kHz, Λdoublet splitting between opposite parity3∆1J=1 states.
gF=3/2= 0.03, magnetic gfactor in3∆1mF= ±3/2 states.
Brot= 70 µG, rotating magnetic field.
3gFµBBrot= 2π × 8 Hz, Zeeman splitting between mF= +3/2 and mF= −3/2 states due to Brot.
δgF=3/2/gF=3/2≈ 3 × 10−4, fractional difference of magnetic gfactor for upper and lower levels, for parameters
shown in the Appendix.
∆ ≈ 2π × 2 Hz, splitting at the avoided crossing between mF = +3/2 and mF = 3/2 levels, for parameters shown
in the Appendix.
B⊥= 25 mG, anticipated scale of transverse magnetic field.
r = 0.5 cm, characteristic rms radius of trapped ion cloud.
T = 15 K, characteristic temperature of trapped cloud.
ωi= 2π × 1 kHz, typical trap confining frequency.
Ez= 5 mV/cm, typical axial electric field applied for confinement.
Erf= 75 mV/cm, typical Paul trap electric field strength, at typical cloud radius.
< Erf> = 5 mV/cm, typical radial confining electric field, averaged over one Paul cycle.
ωrf= 2π × 15 kHz, typical “rf freq” for Paul trap.
Erf= 15 K, typical kinetic energy in Paul micromotion.
Ehf= 2π × 45 MHz, hyperfine splitting between F = 1/2 and F = 3/2 states of3∆1J=1 level.
Acknowledgments
We gratefully acknowledge many useful discussions. On the topic of molecular and atomic spectroscopy, we ben
efited from discussions with Peter Bernath, Carl Wieman, Tom Gallagher, Ulrich Hechtfische, and Jim Lawler. On
molecular structure, Andrei Derevianko, Svetlana Kotochigova, Richard Saykally, Laura Gagliardi, and especially the
St. Petersburg group, Anatoly Titov, Mikhail Kozlov, and Alexander Petrov. On molecular dynamics, we had useful
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discussions with Carl Lineberger, David Nesbitt, and especially Bob Field. On EDM measurements, Pat Sandars,
Dave DeMille, and Neil ShaferRay. We thank Gianfraco DiLonardo for the loan of a hollow cathode lamp and Tobin
Munsat for the loan of a highcurrent power supply. We acknowledge useful contributions in our lab from Herbert
Looser, Tyler Yahn, Tyler Coffey, Matt Grau, and Will Ames. We thank Jun Ye and members of his group for sharing
their innovative ideas in combbased spectroscopy, and Konrad Lehnert for enlightening us about sensitive microwave
detection. This work was supported by NIST, NSF, and funds from a Marsico Chair of Excellence.
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