State Data

PHYSICAL REVIEW A **, ****** (****)

Magnetic and electric dipole moments of the H state in ThO

1

A. C. Vutha,1,* B. Spaun,2 Y. V. Gurevich,2 N. R. Hutzler,2 E. Kirilov,1 J. M. Doyle,2 G. Gabrielse,2 and D. DeMille1

1

Department of Physics, Yale University, New Haven, Connecticut 06520, USA

2

Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

(Received 13 July 2011; published 29 September 2011)

The metastable H 3 1 state in the thorium monoxide (ThO) molecule is highly sensitive to the presence of a

CP-violating permanent electric dipole moment of the electron (eEDM) [E. R. Meyer and J. L. Bohn, Phys. Rev. A

78, 010502 (2008)]. The magnetic dipole moment H and the molecule- xed electric dipole moment DH of this

state are measured in preparation for a search for the eEDM. The small magnetic moment H = 8.5(5) 10 3 B

displays the predicted cancellation of spin and orbital contributions in a 3 1 paramagnetic molecular state,

providing a signi cant advantage for the suppression of magnetic eld noise and related systematic effects in

the eEDM search. In addition, the induced electric dipole moment is shown to be fully saturated in very modest

electric elds (

in the ThO eEDM search experiment.

DOI: 10.1103/PhysRevA.84.034502 PACS number(s): 31.30.jp, 11.30.Er, 33.15.Kr

Measurable CP violation is predicted in many proposed was subsequently probed a few millimeters downstream by

extensions to the standard model, and could provide a clue exciting laser-induced uorescence (LIF). Both the 943 nm

light to drive the X A pump transition and the 908 nm light

to the observed dominance of matter over antimatter in the

for the H E probe transition were derived from external

universe [1]. The permanent electric dipole moment of the

electron (eEDM) is a sensitive probe for avor-diagonal CP cavity diode lasers. Fluorescence from E X at 613 nm

violation in the lepton sector [2]. A number of experimental was collected with an f /1.0 lens, channeled through a quartz

efforts are currently focused on searching for this elusive lightpipe and a bandpass interference lter and monitored

quantity [3]. Many of these experiments take advantage of with a photomultiplier tube. Detection of uorescence at a

the large internal electric eld Emol experienced by valence wavelength signi cantly to the blue of the excitation laser

electrons in a polar molecule [4]. Following the suggestion of suppresses background due to scattered laser light. The pump

Meyer et al. [5], states with a 3 1 character in heavy molecules and probe lasers were perpendicular to the molecular beam and

the transverse Doppler width on the H E probe transition

are being used in several new eEDM experiments [6 8].

was 5 MHz. With the pump laser locked to resonance, the

In addition to a large intrinsic eEDM sensitivity, there are

two key attractive features of this kind of molecular state probe laser s frequency was tuned. The frequency steps were

for eEDM searches: closely spaced opposite parity doublets calibrated by monitoring the laser s transmission through a

( -doublets) [5,9] and small magnetic moments [5,10]. The scanning confocal interferometer, which was actively stabi-

-doublets enable the molecule to be completely polarized lized to a 1064 nm Nd:YAG laser (in turn locked to an iodine

in small electric elds. The fully polarized molecule accesses cell) [13]. The free spectral range of the interferometer was

the full eEDM sensitivity of the molecule, while suppressing independently determined from its length and from off-line

E - eld-induced systematic errors such as those due to leakage measurements of spectra with RF sidebands added to the laser.

currents and geometric phases [7,11,12]. The extremely small

magnetic moment of the H 3 1 state makes the molecule

less sensitive to effects arising from uctuating B elds

II. MAGNETIC DIPOLE MOMENT

and motional (v E /c2 ) magnetic elds [7]. Here we report

3

A 1 molecular state has two units of orbital angular

measurements of both of these key parameters in the H state

momentum ( = 2) and one unit of spin angular momentum

of ThO.

( = 1) projected onto the internuclear axis. The contribu-

tions of these to the magnetic moment cancel out to a large

I. EXPERIMENTAL SETUP

extent, since the orbital g factor (gL = 1) is very nearly half as

The measurements were carried out using a molecular beam large as the spin g factor (gS = 2.002); hence, the effective g

of ThO, produced in an apparatus similar to one described factor is geff = gL + gS 0 [5,10]. The magnetic moment

elsewhere [7]. The apparatus uses helium buffer gas at 4 K of such a pure molecular state is nonzero only because of

to cool a pulse of ThO molecules (produced by pulsed laser small effects such as the nonzero value of gS 2. Ab initio

ablation of ThO2 ), which are extracted into a molecular beam calculations indicate that the H state in ThO has 99% 3 1

and probed 30 cm downstream. The lowest rovibrational level content, with small admixtures of other Hund s case (a)

(v = 0,J = 1) in the H state was populated by optical pump- states (1 1, 3 1 ) due to off-diagonal spin-orbit mixings [14].

ing from the ground electronic X 1 + (v = 0,J = 1) state via These spin-orbit admixtures are expected to be the dominant

the higher-lying, short-lived A 3 0+ (v = 0,J = 0) state. It contribution to the nonzero magnetic moment of the H state,

at the level of geff 0.01. Other effects such as the magnetic

moment due to the rotation of this polar molecule are expected

to be much smaller [15].

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1050-2947/2011/84(3)/034502(4) 2011 American Physical Society

BRIEF REPORTS PHYSICAL REVIEW A 84, 034502 (2011)

FIG. 2. (Color online) Spectra of LIF from the H,J = 1 state in

a magnetic eld B = 1.9(1) kG, acquired with 16 averages per data

point. The x -error bars account for the standard error in the laser s

frequency offset (derived from the rms frequency excursion of the

lock s error signal). The y -error bars indicate the quadrature sum of

the standard error of the LIF signal due to shot-to-shot uctuations

in the yield of molecules in a pulse, and the y -error derived from

the x -error bars using the numerically calculated slope of the data.

(a) The probe laser s polarization B ; in this con guration the

mJ = 0 sublevel is probed. (b) With the laser polarization B, the

FIG. 1. (Color online) (a) The magnet assembly used in the

mJ = 1 sublevels are probed. The t to the spectrum in (b) yields

measurement of H . Arrows on the NdFeB magnets indicate the

= 22.66(41) MHz for the Zeeman splitting between mJ = 1.

direction of magnetization. (b) The z component of the B eld, Bz,

measured along the x axis. The shaded area indicates the calculated

acceptance of the LIF detection optics. (c) Measured (red squares)

have the same value for both data sets. We included minority

and calculated (solid green line) values of Bz along the y axis. The

peaks corresponding to the orthogonal polarization (i.e., the

shaded area indicates the LIF excitation region, de ned by the probe

mJ = 0 peak in the B t and the mJ = 1 peaks in the

laser s intensity pro le.

B t) to model the effect of residual circular polarization

(due to an imperfect waveplate and birefringence in vacuum

In order to split the mJ sublevels in the H state by a windows). The measured amount of circular polarization

frequency greater than the Doppler width in our molecular ( 15%) was in fair agreement with that deduced from the

beam, it was found necessary to apply a large magnetic eld relative size of the minority peaks ( 20%). We veri ed that

B 1 kG. To do this, we constructed a compact permanent changes in the size of the minority peaks did not affect the Zee-

magnet assembly using NdFeB magnets (see Fig. 1). The man shift extracted from the t within its uncertainty. Varying

separation and alignment of the magnets was adjusted to the width of the minority peaks shifted the t value of by

obtain a uniform magnetic eld over the region probed by 0.33 MHz; this effect is included as a systematic contribution

the laser. The magnet was oriented so B v, the velocity to the t uncertainty. The t frequency separation between the

mJ = 1 peaks in Fig. 1(b) is = 22.66(41) MHz.

of the molecular beam, in order to avoid spurious effects

due to motional electric elds (Emot = v B ) polarizing the is related to the intrinsic magnetic moment of the H state, H,

by the formula h = J2( H B [16].

molecular state. The probe laser was collimated (intensity J +1)

full width at half maximum 0.6 mm) to spatially select a The magnetic eld sampled by molecules in the experiment

well-de ned region near the center of the magnet assembly; was characterized as follows. The magnetic eld pro les in

its k vector was aligned along x in Fig. 1(a). The polarization Figs. 1(b) and 1(c) were measured with a Hall probe whose

active area (0.127 mm 0.127 mm) is small compared to the

of the probe laser was adjusted to be parallel (perpendicular)

to the B eld in order to probe the unshifted mJ = 0 (Zeeman- area illuminated by the laser beam. Spatial selectivity along

shifted mJ = 1) states. The LIF collection lens and optics the x axis was provided by the LIF detection optics, as shown

were positioned along y . in Fig. 1(b). The acceptance of the optical system was modeled

The spectra of LIF collected from the H,J = 1 state in the with ray-tracing software and used to extract the central value

magnetic eld region are shown in Fig. 2. Spectra with B of the B eld along this dimension. As the separation between

and B were simultaneously t to a sum of 3 Gaussian the magnets was too small to allow a direct measurement of

Bz along z, we accounted for the spatial dependence in the yz

line shapes, with the line centers and linewidths constrained to

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BRIEF REPORTS PHYSICAL REVIEW A 84, 034502 (2011)

plane in the following way. The value of the B eld measured system in an E eld is described by the Hamiltonian

at the origin in the Bz vs. y pro le was used to calibrate the pole

DJ E

0 /2

strength in a 2D numerical calculation in the yz plane based on H=,

DJ E

the (measured) magnet geometry. When weighted over an area 0 /2

corresponding to the Hall probe, the calculation reproduced the

where DJ is the electric dipole matrix element connecting the

measured B - eld pro le, as shown in Fig. 1(c). The calculated

two basis states. The energy spacing between the eigenstates

B - eld pattern was weighted by the Gaussian intensity pro le

is = 2 0 /4 + DJ E . In the limit where DJ E

2 22

0, the

of the probe laser beam in the yz plane and averaged. (The

parity eigenstates are completely mixed and the energy spacing

LIF yield on the H E detection transition was linear in

between the polarized eigenstates is 2 DJ E .

the probe laser s intensity.) We calculated the average B eld,

In our experiment, the probe laser couples the E - eld-

B, in this way for a range of displacements of the laser beam

polarized states in H (v = 0) to the E (v = 0,J p = 0+ ) state

pro le ( 0.5 mm) from the exact center of the magnets in the

for LIF. The excited E state does not have -doublets, and

yz plane to account for the experimental uncertainty in the

in an E eld the predominant mixing of the E,J p = 0+ state

laser s position and pointing. This leads to the estimate that

is with the neighboring E,J p = 1 rotational state. Since the

B = 1.9(1) kG in the probed volume. As a further check on

rotational spacing in the E state ( 20 GHz) is much larger

systematic errors arising from the probe laser s alignment, we

than the zero- eld -doublet spacing ( 0 400 kHz) in

repeated the measurement with a complete realignment of the

the H state [17], whereas the dipole matrix elements are of

laser s path through the magnet and obtained a value for

comparable size, there is a range of electric elds (1 V/cm

that was within the estimated range of possible changes in B

E 1 kV/cm), where the H,J p = 1 -doublets are fully

due to misalignment (5%).

mixed while the E,J p = 0+ state remains a parity eigenstate

We combine the t uncertainty in quadrature with uncer-

to a good approximation. In this regime, therefore, LIF signals

tainties in the probe laser s frequency calibration (1%) and the

from both the polarized eigenstates in H should be visible

central value of the B eld in the probed volume (5%) to obtain

with equal intensity. Since the laser polarization E, only

H = 0.0085(5) B for the magnetic moment of the H state.

mJ = 1 states are excited by the laser.

The sample LIF spectrum in Fig. 3(a) shows peaks from

the E - eld-polarized eigenstates that are separated by a

III. MOLECULE-FIXED ELECTRIC DIPOLE MOMENT frequency . They are also well separated from the residual

E - eld-free signal, which was due to molecules excited outside

The presence of -doublets, levels of opposite parity

spaced much closer than the rotational splittings, in a 3 1

state leads to a polarizability that is typically 109 atomic

units. This means that the molecule can be fully polarized

(i.e., the levels of opposite parity completely mixed) even in

a static electric eld as small as a few V/cm. This feature

enables the suppression of a number of systematic errors in the

measurement of the eEDM (see Ref. [7] for more details). We

measured the molecule- xed dipole moment DH of the H state

by spectroscopically resolving the Stark shift in an applied

electric eld and veri ed that the molecule was completely

polarized in electric elds as low as 10 V/cm.

For these measurements, a pair of glass plates, coated with

transparent conducting indium tin oxide (ITO) on one side

and broadband antire ection coating on the other, were used

to make a capacitor with 25 mm 30 mm plates separated by

a 3.00(5) mm vacuum gap. The molecular beam was passed

between the plates and a linearly polarized probe laser was sent

at right angles to the molecular beam, normal to and through

the transparent plates. LIF was collected with the same optical

arrangement used for the magnetic moment measurement, at

right angles to both the molecular beam velocity and the probe

laser s k vector.

In the H (v = 0) state, we focus our analysis on the J p = FIG. 3. (Color online) (a) LIF spectrum from the H,J = 1 state

1, mJ = 1 -doublet states (p denotes the parity of the in an electric eld E = 20 V/cm, acquired with 16 averages per data

state). In the absence of an electric eld, these states are parity point. The error bars are assigned in the same way as in Fig. 2. The

eigenstates and are separated by an energy 0 . (The Zeeman two E - eld-polarized states are separated by a frequency . The LIF

sublevels with mJ = 0 do not mix in an electric eld as a result signal near zero offset is due to molecules outside the electric eld

plates (see text). In (b), is plotted as a function of the E eld across

of angular momentum selection rules, and we ignore them in

the plates. The solid line is a t to the function = DJ E . The slope

the rest of this analysis.) In the two-state space spanned by the

basis states J = 1,mJ = +1 (or identically, the space with of the plot yields DJ =1 = h 2.13(2) MHz/(V/cm) for the dipole

mJ = 1, since mJ = 1 are degenerate in an E eld), the matrix element between the -doublets in H,v = 0,J = 1 state.

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BRIEF REPORTS PHYSICAL REVIEW A 84, 034502 (2011)

the capacitor plates. The narrowness of the E - eld-polarized IV. SUMMARY

spectral peaks was due to additional collimation of the

We have measured the magnetic moment H and molecule-

molecular beam by the capacitor plates. A pair of identical

xed electric dipole moment DH of the metastable H state in

Gaussian line shapes were t to the E - eld-polarized spectral

the ThO molecule. The suppression of H predicted for an

peaks and their separation was extracted. The E - eld-free

eEDM-sensitive 3 1 molecular state has been experimentally

signal was ignored for the purpose of tting and extracting

veri ed. The H state in ThO can be polarized with very small

from the data (we veri ed that the residual slope due to this

electric elds due to the presence of -doublets. This com-

signal did not affect the t value of within its uncertainty).

bination of a small magnetic moment and large polarizability

In Fig. 3(b), the frequency separations extracted from a set

in the H state enables the strong suppression of systematic

of LIF spectra are shown plotted against the E eld across

effects in our ongoing experiment to search for the eEDM

as a function of E

the plates. The linear dependence of

with ThO.

indicates that the H state was fully polarized over the range

of E elds applied during the measurements. The t yields

the value DJ =1 = h 2.13(2) MHz/(V/cm) and constrains

ACKNOWLEDGMENTS

2 MHz in agreement with the result of Ref. [17]. The

0

relation between DJ and the molecule- xed dipole moment A.V. acknowledges helpful discussions with Alexei

in the H state, DH, is DJ = J (D+1) [16]. We nd DH = Buchachenko. We thank Elizabeth Petrik and Paul Hess for

H

J

technical assistance and Wes Campbell for comments on the

1.67(4) ea0, where the reported error is a quadrature sum of

manuscript. This work was supported by the National Science

the t uncertainty (1%), and systematic uncertainties due to

Foundation.

laser frequency calibration (1%) and eld plate spacing (2%).

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