The metastable $H \ {}^3\Delta_1$ 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). The magnetic dipole moment
$\mu_H$ and the molecule-fixed electric dipole moment $D_H$ of this state are
measured in preparation for a search for the eEDM. The small magnetic moment
$\mu_H = 8.5(5) \times 10^{-3} \ \mu_B$ displays the predicted cancellation of
spin and orbital contributions in a ${}^3 \Delta_1$ paramagnetic molecular
state, providing a significant advantage for the suppression of magnetic field
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 fields ($<$ 10 V/cm). This feature is favorable for the suppression of
many other potential systematic errors in the ThO eEDM search experiment.Comment: 4 pages, 3 figure

A. C. Vutha

B. L. Roberts

B. Spaun

D. DeMille

E. Kirilov

G. Gabrielse

I. B. Khriplovich

J. M. Brown

J. M. Doyle

N. R. Hutzler

Y. V. Gurevich

Physical Review A

English

arXiv

The metastable $H\ ^3\Delta_1$ 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). The magnetic dipole moment $\mu_H$ and the molecule-fixed electric dipole moment $D_H$ of this state are measured in preparation for a search for the eEDM. The small magnetic moment $\mu_H = 8.5(5) \times 10^{-3}\ \mu_B$ displays the predicted cancellation of spin and orbital contributions in a $^3\Delta_1$ paramagnetic molecular state, providing a significant advantage for the suppression of magnetic field 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 fields ($<$ 10 V/cm). This feature is favorable for the suppression of many other potential systematic errors in the ThO eEDM search experiment.PhysicsAccepted Manuscrip

Vutha, Amar

Spaun, Ben

Gurevich, Yulia Vsevolodovna

Hutzler, Nicholas

Kirilov, E

Doyle, John

Gabrielse, Gerald

DeMille, David

Harvard University - DASH

 Magnetic and Electric Dipole Moments of the $H\ ^3\Delta_1$ Statein ThO  (Article begins on next page)The Harvard community has made this article openly available.Please share how this access benefits you. Your story matters.Citation Vutha, Amar, Benjamin Spaun, Yulia Gurevich, Nicholas Hutzler,Emil Kirilov, John Doyle, Gerald Gabrielse, and David DeMille.2011. Magnetic and electric dipole moments of the $H\^3\Delta_1$ state in ThO. Physical Review A 84:034502.Published Version doi:10.1103/PhysRevA.84.034502Accessed February 19, 2015 9:13:26 AM ESTCitable Link http://nrs.harvard.edu/urn-3:HUL.InstRepos:10184313Terms of Use This article was downloaded from Harvard University's DASHrepository, and is made available under the terms and conditionsapplicable to Open Access Policy Articles, as set forth athttp://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#OAPMagnetic and electric dipole moments of the H 31 state in ThOA. C. Vutha1, B. Spaun2, Y. V. Gurevich2, N.R. Hutzler2, E. Kirilov1, J. M. Doyle2, G. Gabrielse2 and D. DeMille11 Department of Physics, Yale University, New Haven, CT 06520, USA and2 Department of Physics, Harvard University, Cambridge, MA 02138, USAThe metastable H31 state in the thorium monoxide (ThO) molecule is highly sensitive to thepresence of a CP-violating permanent electric dipole moment of the electron (eEDM) [1]. Themagnetic dipole moment H and the molecule-xed electric dipole moment DH of this state aremeasured 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 a31 paramagneticmolecular state, providing a signicant advantage for the suppression of magnetic eld noise andrelated systematic eects in the eEDM search. In addition, the induced electric dipole moment isshown to be fully saturated in very modest electric elds (< 10 V/cm). This feature is favorable forthe suppression of many other potential systematic errors in the ThO eEDM search experiment.PACS numbers: 31.30.jp, 11.30.Er, 33.15.Kr,Measurable CP-violation in physics beyond the Stan-dard Model is predicted in many proposed extensionsto the Standard Model, and could provide a clue tothe observed dominance of matter over antimatter inthe universe [2]. The permanent electric dipole mo-ment of the electron (eEDM) is a sensitive probe foravor-diagonal CP-violation in the lepton sector [3]. Anumber of experimental eorts are currently focused onsearching for this elusive quantity [4]. Many of these ex-periments take advantage of the large internal electriceld Emol experienced by valence electrons in a polarmolecule [5]. Following the suggestion of Meyer et al.[6], states with a 31 character in heavy molecules arebeing used in several new eEDM experiments [7{9]. Inaddition to a large intrinsic eEDM sensitivity, there aretwo key attractive features of this kind of molecular statefor eEDM searches: closely spaced opposite parity dou-blets (-doublets) [6, 10] and small magnetic moments[6, 11]. The -doublets allow complete polarization ofthe molecule in small electric elds, taking full advantageof the eEDM sensitivity of the molecule while suppress-ing E-eld induced systematic errors, such as those dueto leakage currents and geometric phases [8, 12, 13]. Theextremely small magnetic moment of the H 31 statemakes the molecule less sensitive to eects arising fromuctuating B-elds and motional (~ v  ~ E=c2) magneticelds [8]. Here we report measurements of both of thesekey parameters in the H state of ThO.I. EXPERIMENTAL SETUPThe measurements were carried out using a molecu-lar beam of ThO, produced in an apparatus similar toone described elsewhere [8]. The apparatus uses heliumbuer gas at 4 K to cool a pulse of ThO molecules (pro-duced by pulsed laser ablation of ThO2), which are ex-tracted into a molecular beam and probed 30 cm down-stream. The lowest rovibrational level (v = 0;J = 1)in the H state was populated by optical pumping fromthe ground electronic X 1+(v = 0;J = 1) state via thehigher-lying, short-lived A 30+(v = 0;J = 0) state. Itwas subsequently probed a few mm downstream by ex-citing laser-induced uorescence (LIF). Both the 943 nmlight to drive the X ! A pump transition and the 908 nmlight for the H ! E probe transition were derived fromexternal cavity diode lasers. Fluorescence from E   Xat 613 nm was collected with an f/1.0 lens, channeledthrough a quartz lightpipe and a bandpass interferencelter and monitored with a photomultiplier tube. De-tection of uorescence at a wavelength signicantly tothe blue of the excitation laser suppresses backgrounddue to scattered laser light. The pump and probe laserswere perpendicular to the molecular beam and the trans-verse Doppler width on the H ! E probe transition was5 MHz. With the pump laser locked to resonance, theprobe laser's frequency was tuned. The frequency stepswere calibrated by monitoring the laser's transmissionthrough a scanning confocal interferometer, which wasactively stabilized to a 1064 nm YAG laser (in turn lockedto an iodine cell) [14]. The free spectral range of the inter-ferometer was independently determined from its length,and from o-line measurements of spectra with RF side-bands added to the laser.II. MAGNETIC DIPOLE MOMENTA 31 molecular state has 2 units of orbital angularmomentum ( = 2) and one unit of spin angular momen-tum ( =  1) projected onto the inter-nuclear axis. Thecontributions of these to the magnetic moment cancel outto a large extent, since the orbital g-factor (gL = 1) isvery nearly half as large as the spin g-factor (gS = 2:002);hence the eective g-factor is ge = gL+gS  0 [6, 11].The magnetic moment of such a pure molecular state isnonzero only because of small eects such as the nonzerovalue of gS   2. Ab initio calculations indicate that theH state in ThO has 99% 31 content, with small admix-tures of other Hund's case (a) states (11, 31) due too-diagonal spin-orbit mixings [15]. These spin-orbit ad-mixtures are expected to be the dominant contributionarXiv:1107.2287v1  [physics.atom-ph]  12 Jul 201123 mm25.4 mm2.8 mmFIG. 1: a) The magnet assembly used in the measurementof H. Arrows on the NdFeB magnets indicate the directionof magnetization. b) The ^ z component of the B-eld, Bz,measured along the x-axis. The shaded area indicates thecalculated acceptance of the LIF detection optics. c) Mea-sured (red squares) and calculated (solid green line) values ofBz along the y-axis. The shaded area indicates the LIF exci-tation region, dened by the probe laser's intensity prole.to the non-zero magnetic moment of the H state, at thelevel of ge  0:01. Other eects such as the magneticmoment due to the rotation of this polar molecule areexpected to be much smaller [16].In order to split the mJ azimuthal sublevels in theH state by a frequency greater than the Doppler widthin our molecular beam, it was found necessary to applya large magnetic eld B  1 kG. To do this, we con-structed a compact permanent magnet assembly usingNdFeB magnets (see Fig.1). The separation and align-ment of the magnets was adjusted to obtain a uniformmagnetic eld over the region probed by the laser. Themagnet was oriented so that ~ B k ~ v, the velocity of themolecular beam, in order to avoid spurious eects dueto motional electric elds (~ Emot = ~ v  ~ B) polarizing themolecular state. The probe laser was collimated (inten-sity FWHM ' 0.6 mm) to spatially select a well-denedregion near the center of the magnet assembly; its k-vector was aligned along ^ x in Fig.1a. The polarization^  of the probe laser was adjusted to be parallel (perpen-m=-1m=-1 0 +10 +1FIG. 2: 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'sfrequency oset (derived from the rms frequency excursion ofthe lock's error signal) and y-error bars indicate the standarderror of the LIF signal due to shot-to-shot uctuations in theyield of molecules in a pulse. a) The probe laser's polarization^  k ~ B; in this conguration the mJ = 0 sublevel is probed.b) With the laser polarization ^  ? ~ B, the mJ = 1 sublevelsare probed. The t to the spectrum in (b) yields  =22.66(41) MHz for the Zeeman splitting between mJ = 1.dicular) to the B-eld in order to probe the unshiftedmJ = 0 (Zeeman-shifted mJ = 1) states. The LIFcollection lens and optics were positioned along ^ y.The spectra of LIF collected from the H;J = 1 statein the magnetic eld region are shown in Fig. 2. Spec-tra with ^  k ~ B and ^  ? ~ B were simultaneously t to asum of 3 gaussian lineshapes, with the line centers andlinewidths constrained to have the same value for bothdata sets. We included `minority' peaks correspondingto the orthogonal polarization (i.e. the mJ = 0 peak inthe ^  ? ~ B t, and the mJ = 1 peaks in the ^  k ~ B t)to model the eect of residual circular polarization (dueto an imperfect waveplate and birefringence in vacuumwindows). The measured amount of circular polarization( 15%) was in fair agreement with that deduced fromthe relative size of the minority peaks ( 20%). We ver-ied that changes in the size of the minority peaks didnot aect the Zeeman shift extracted from the t withinits uncertainty. Varying the width of the minority peaksshifted the t value of  by 0.33 MHz; this eect is in-cluded as a systematic contribution to the t uncertainty.The t frequency separation between the mJ = 1 peaksin Fig.1b is  = 22.66(41) MHz.  is related to theintrinsic magnetic moment of the H state, H, by the for-mula h =2HBJ(J+1) [17].The magnetic eld sampled by molecules in the ex-3periment was characterized as follows. The magneticeld proles in Fig.1b,c were measured with a Hall probewhose active area (0.127 mm  0.127 mm) is small com-pared to the area illuminated by the laser beam. Spatialselectivity along the x-axis was provided by the LIF de-tection optics, as shown in Fig.1b. The acceptance ofthe optical system was modeled with ray-tracing soft-ware and used to extract the central value of the B-eldalong this dimension. As the separation between themagnets was too small to allow a direct measurementof Bz along ^ z, we accounted for the spatial dependencein the yz plane in the following way. The value of theB-eld measured at the origin in the Bz vs. y prole wasused to calibrate the pole strength in a 2D numerical cal-culation in the yz plane based on the (measured) magnetgeometry. When weighted over an area corresponding tothe Hall probe, the calculation reproduced the measuredB-eld prole, as shown in Fig.1c. The calculated B-eldpattern was weighted by the Gaussian intensity prole ofthe probe laser beam in the yz plane and averaged. (TheLIF yield on the H ! E detection transition was linearin the probe laser's intensity). We calculated the averageB-eld, hBi, in this way for a range of displacements ofthe laser beam prole ( 0.5 mm) from the exact centerof the magnets in the yz plane to account for the exper-imental uncertainty in the laser's position and pointing.This leads to the estimate that hBi = 1.9(1) kG in theprobed volume. As a further check on systematic errorsarising from the probe laser's alignment, we repeated themeasurement with a complete realignment of the laser'spath through the magnet and obtained a value for that was within the estimated range of possible changesin hBi due to misalignment (5%).We combine the t uncertainty in quadrature with un-certainties in the probe laser's frequency calibration (1%)and the central value of the B-eld in the probed volume(5%) to obtain jHj = 0.0085(5) B for the magneticmoment of the H state.III. MOLECULE-FIXED ELECTRIC DIPOLEMOMENTThe presence of -doublets, levels of opposite parityspaced much closer than the rotational splittings, in a31 state leads to a polarizability that is typically  109atomic units. This means that the molecule can be fullypolarized (i.e. the levels of opposite parity completelymixed) even in a static electric eld as small as a fewV/cm. This feature enables the suppression of a numberof systematic errors in the measurement of the eEDM (see[8] for more details). We measured the molecule-xeddipole moment DH of the H state by spectroscopicallyresolving the Stark shift in an applied electric eld, andveried that the molecule was completely polarized inelectric elds as low as 10 V/cm.For these measurements, a pair of glass plates, coatedwith transparent conducting indium tin oxide (ITO) onm=-1 0 +1 m=-1 0 +1FIG. 3: a) LIF spectrum from the H;J = 1 state in an electriceld E =20 V/cm, acquired with 16 averages per data point.The error bars are assigned in the same way as in Figure 2.The two E-eld polarized states are separated by a frequency. The LIF signal near zero oset is due to molecules outsidethe electric eld plates (see text). In b),  is plotted asa function of the E-eld across the plates. The solid lineis a t to the function  = jDJEj. The slope of the plotyields DJ=1 = h2:13(2) MHz/(V/cm) for the dipole matrixelement between the -doublets in H;v = 0;J = 1 state.one side and broadband anti-reection coating on theother, were used to make a capacitor with 25 mm x 30mm plates separated by a 3.00(5) mm vacuum gap. Themolecular beam was passed between the plates and a lin-early polarized probe laser was sent at right angles to themolecular beam, normal to and through the transparentplates. LIF was collected with the same optical arrange-ment used for the magnetic moment measurement, atright angles to both the molecular beam velocity and theprobe laser's k-vector.In the H(v = 0) state, we focus our analysis on theJp = 1;jmJj = 1 -doublet states (p denotes the parityof the state). In the absence of an electric eld, thesestates are parity eigenstates and are separated by anenergy 0. (The Zeeman sublevels with mJ = 0 donot mix in an electric eld as a result of angular mo-mentum selection rules, and we ignore them in the restof this analysis.) In the two-state space spanned bythe basis states J = 1;mJ = +1 (or identically, thespace with mJ =  1, since mJ = 1 are degeneratein an E-eld), the system in an E-eld is described bythe Hamiltonian H =  0=2  DJE DJE 0=2, where DJ isthe electric dipole matrix element connecting the twobasis states. The energy spacing between the eigen-4states is  = 2p20=4 + D2JE2. In the limit wherejDJEj  0, the parity eigenstates are completely mixedand the energy spacing between the polarized eigenstatesis   2jDJEj.In our experiment, the probe laser couples the E-eld-polarized states in H(v = 0) to the E(v = 0;Jp = 0+)state for LIF. The excited E state does not have -doublets, and in an E-eld the predominant mixing of theE;Jp = 0+ state is with the neighboring E;Jp = 1  ro-tational state. Since the rotational spacing in the E state( 20 GHz) is much larger than the zero-eld -doubletspacing (0  400 kHz) in the H state [18], whereasthe dipole matrix elements are of comparable size, thereis a range of electric elds (1 V/cm . jEj . 1 kV/cm)where the H;Jp = 1 -doublets are fully mixed whilethe E;Jp = 0+ state remains a parity eigenstate to agood approximation. In this regime therefore, LIF sig-nals from both the polarized eigenstates in H should bevisible with equal intensity. Since the laser polarization^  ? ~ E, only jmJj = 1 states are excited by the laser.The sample LIF spectrum in Fig.3a shows peaks fromthe E-eld polarized eigenstates that are separated by afrequency . They are also well separated from the resid-ual E-eld-free signal, which was due to molecules excitedoutside the capacitor plates. The narrowness of the E-eld polarized spectral peaks was due to additional col-limation of the molecular beam by the capacitor plates.A pair of identical Gaussian lineshapes were t to the E-eld polarized spectral peaks and their separation  wasextracted. The E-eld-free signal was ignored for thepurpose of tting and extracting  from the data (weveried that the residual slope due to this signal did notaect the t value of  within its uncertainty). In Fig.3b,the frequency separations  extracted from a set of LIFspectra are shown plotted against the E-eld across theplates. The linear dependence of  as a function of jEjindicates that the H state was fully polarized over therange of E-elds applied during the measurements. Thet yields the value DJ=1 = h  2.13(2) MHz/(V/cm),and constrains 0  2 MHz in agreement with the resultof [18]. The relation between DJ and the molecule-xeddipole moment in the H state, DH, is DJ = DHJ(J+1) [17].We nd jDHj = 1.67(4) ea0, where the reported error isa quadrature sum of the t uncertainty (1%), and sys-tematic uncertainties due to laser frequency calibration(1%) and eld plate spacing (2%).IV. SUMMARYWe have measured the magnetic moment H andmolecule-xed electric dipole moment DH of themetastable H state in the ThO molecule. The suppres-sion of H predicted for an eEDM-sensitive 31 molecu-lar state has been experimentally veried. The H statein ThO can be polarized with very small electric eldsdue to the presence of -doublets. This combination of asmall magnetic moment and large polarizability in the Hstate enables the strong suppression of systematic eectsin our ongoing experiment to search for the eEDM withThO.A.V. acknowledges helpful discussions with AlexeiBuchachenko. We thank Elizabeth Petrik and Paul Hessfor technical assistance and Wes Campbell for commentson the manuscript. This work was supported by the Na-tional Science Foundation.[1] E. R. Meyer and J. L. Bohn, Phys. Rev. A 78, 010502(2008).[2] N. Fortson, P. Sandars, and S. Barr, Phys. Today 56, 33(2003).[3] I. B. Khriplovich and S. K. Lamoreaux, CP-violationwithout strangeness : electric dipole moments of parti-cles, atoms, and molecules (Springer-Verlag, 1997).[4] B. L. Roberts and W. J. Marciano, eds., Lepton DipoleMoments (World Scientic, 2009).[5] P. G. H. Sandars, Phys. Rev. Lett. 19, 1396 (1967).[6] E. R. Meyer, J. L. Bohn, and M. P. Deskevich, Phys.Rev. A 73, 062108 (2006).[7] J. Lee, E. R. Meyer, R. Paudel, J. L. Bohn, and A. E.Leanhardt, J. Mod. Opt. 56, 2005 (2009).[8] A. C. Vutha, W. C. Campbell, Y. V. Gurevich, N. R.Hutzler, M. Parsons, D. Patterson, E. Petrik, B. Spaun,J. M. Doyle, G. Gabrielse, et al., J. Phys. B 43, 074007(2010).[9] A. E. Leanhardt, J. L. Bohn, H. Loh, P. Maletinsky, E. R.Meyer, L. C. Sinclair, R. P. Stutz, and E. A. Cornell,arXiv:1008.2997v2 [physics.atom-ph] (2010).[10] D. DeMille, F. Bay, S. Bickman, D. Kawall, andL. Hunter, Art and Symmetry in Experimental Physics:Festschrift for Eugene D. Commins (AIP Conf. Proc.)pp. 72{83 (2001).[11] F. H. Crawford, Rev. Mod. Phys. 6, 90 (1934).[12] D. Kawall, F. Bay, S. Bickman, Y. Jiang, and D. DeMille,Phys. Rev. Lett. 92, 133007 (2004).[13] S. Bickman, P. Hamilton, Y. Jiang, and D. DeMille,Phys. Rev. A 80, 023418 (2009).[14] D. M. Farkas, Ph.D. thesis, Harvard University (2007).[15] J. Paulovic, T. Nakajima, K. Hirao, R. Lindh, and P. A.Malmqvist, J. Chem. Phys. 119, 798 (2003).[16] S. P. A. Sauer, Chem. Phys. Lett. 297, 475 (1998).[17] J. M. Brown and A. Carrington, Rotational spectroscopyof diatomic molecules (Cambridge University Press,2003).[18] G. Edvinsson and A. Lagerqvist, Phys. Scr. 30, 309(1984).

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Magnetic and electric dipole moments of the $H\ {}^3\Delta_1$ state in
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