Measuring the electric dipole moment of the electron in BaF

The NL-eEDM collaboration: Parul Aggarwal1, Hendrick L. Bethlem2, Anastasia Borschevsky1, Malika Denis1,Kevin Esajas1, Pi A. B. Haase1, Yongliang Hao1, Steven Hoekstra1 a, Klaus Jungmann1, Thomas B. Meijknecht1,Maarten C. Mooij2, Rob G. E. Timmermans1, Wim Ubachs2, Lorenz Willmann1, Artem Zapara1

1 Van Swinderen Institute for Particle Physics and Gravity (VSI), University of Groningen, Groningen, The Netherlands, andNikhef, National Institute for Subatomic Physics, Science Park 105, 1098 XG Amsterdam, The Netherlands

2 LaserLaB, Department of Physics and Astronomy, Vrije Universiteit, De Boelelaan 1081, 1081 HV Amsterdam, The Nether-lands

April 27, 2018

Abstract We investigate the merits of a measurement of the permanent electric dipole moment of theelectron (eEDM) with barium monofluoride molecules, thereby searching for phenomena of CP violationbeyond those incorporated in the Standard Model of particle physics. Although the BaF molecule has asmaller enhancement factor in terms of the effective electric field than other molecules used in currentstudies (YbF, ThO and ThF+), we show that a competitive measurement is possible by combining Stark-deceleration, laser-cooling and an intense primary cold source of BaF molecules. With the long coherentinteraction times obtainable in a cold beam of BaF, a sensitivity of 5×10−30 e·cm for an eEDM is feasible.We describe the rationale, the challenges and the experimental methods envisioned to achieve this target.

PACS. XX.XX.XX No PACS code given

1 Introduction

The discovery of the Higgs boson [1, 2] at CERN’s LargeHadron Collider (LHC) started a new era in particle physics.The LHC will continue to test the Standard Model (SM)by mapping out the properties of the Higgs boson withincreasing precision. Moreover, now that we know thatthe Higgs boson exists, unanswered questions in particlephysics and cosmology come into focus. The Higgs fieldis responsible for the electroweak symmetry breaking inthe very early Universe. The most natural explanation ofwhy this breaking occurs at a scale of 100 GeV predictsnew particles with masses in the TeV range, some of whichcould make up the dark matter in the Universe [3]. Lep-tons and quarks acquire mass via the Higgs field, such thatparticles and antiparticles have slightly different weak in-teractions. This symmetry breaking between matter andantimatter is called CP violation, where C denotes chargeconjugation and P parity. CP violation within the SM,however, is insufficient, by orders of magnitude, to explainthe matter-antimatter asymmetry in the Universe. Thisfailure of the SM provides a major motivation to searchfor new sources of CP violation at the TeV scale. This isthe mission of, for example, the LHCb experiment [4].

Experiments that search for permanent electric dipolemoments (EDMs) are complementary to, and competewith, searches for CP violation with antiparticles at high-energy colliders [5]. A nonzero electron EDM (eEDM) im-

a corresponding author: s.hoekstra@rug.nl

plies the electron effectively has an aspherical charge dis-tribution along its spin axis, which is forbidden by time-reversal (T) invariance. According to the CPT theoremof quantum field theory, T violation is equivalent to CPviolation. In the SM the eEDM is zero up to the three-loop level. Its predicted value [5] is de = O(10−38)e·cm,which is experimentally far out of reach. However, exten-sions of the SM invariably predict much larger values, andthus provide a window of opportunity for experimentalsearches. The reason is that in such theories new heavyparticles are introduced that have interactions with CP-violating quantum-mechanical phases, giving rise to aneEDM at the one- or two-loop level, or even already attree level [6]. For instance, supersymmetry (SUSY), a con-jectured symmetry between bosons and fermions, predictsan eEDM due to interactions with virtual selectrons andphotinos, the SUSY partners of the electron and photon,at the one-loop level. If such new particles with TeV-scalemasses exist, the eEDM will have a measurable value inthe ongoing and upcoming experiments.

The relation between the masses of the yet to be dis-covered particles and the eEDM value in different typesof models is illustrated in Fig. 1. The blue and green linesshow the limits on the eEDM set by experiments withmolecular beams of YbF [7] and ThO [8], respectively.The purple line shows the limit aimed at in our exper-iment with BaF. It is obvious from Fig. 1 that existingeEDM experiments put strong constraints on the valid-ity of the SUSY models and other speculative SM exten-

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2 Aggarwal et al.: Measuring the electric dipole moment of the electron in BaF

YbF limit

ThO limit

BaF limit (this proposal)

10-30 limit

10−29

10−28

10−27

10−26

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ron-

EDM

size

(e.cm

)Probed energy scale (TeV)

Multi-Higgs Supersymmetry Leptoquarks

# loops needed to generate the eEDM

10−31

10−30

012

Figure 1. The experimental upper limits on the value of the electron EDM translated into a limit on the probed energy scalethrough de = eαn sinφ (me/Λ

2). Here α is the fine-structure constant, n is the number of loops needed to generate the eEDM.φ is a typical CP-violating phase, which is set to the maximum value of π/2, and Λ is the energy scale that is being probed. Theregions above the dotted lines are the energy ranges that are excluded for the indicated classes of models. The models differ inthe number of loops required to generate an eEDM. The eEDM value predicted by the Standard Model, in which the eEDMonly appears at 4-loop level, is of the order of 10−38 e·cm.

sions. Improved eEDM experiments will probe energiesapproaching the PeV scale and will thereby contribute tothe roadmap of particle physics beyond the LHC.

In this article, we outline our plans to construct andperform an experiment to search for an eEDM with an in-tense cold beam of barium monofluoride (BaF) molecules.Our program exploits novel techniques to manipulate andcontrol the quantum states of molecules with electric, mag-netic and light fields. Such methods have in recent yearspromoted diatomic molecules to ultrasensitive probes ofparticle physics [9, 10]. We show that this experiment willbe sensitive to an eEDM value of de = 5×(10−30) e·cm,which is an improvement by more than one order of mag-nitude compared to the current upper limit.

2 The eEDM in atoms and molecules

Searches for permanent dipole moments of fundamen-tal particles started already before the discovery of P andCP violation, with the investigation of the neutron [11].Ref. [12] contains an early proposal to measure the EDMof the proton. In later years, atomic species such as mer-cury [13], thallium [14], and xenon [15] were employed indedicated searches for a permanent EDM and the accom-panying CP violation, and the neutron also remained atthe focus of attention [16]. For a historical overview of thisfield of research we refer to Ref. [17].

The EDM of composite systems, such as atoms ormolecules, can originate in nuclear or electronic contri-butions (or both), depending on the electronic structure.In addition, in such systems, large enhancement factorscan occur for the particle EDMs due to the strong inter-nal electric field near a heavy nucleus. This enhancementis denoted the effective electric field (Eeff). In some atomsthese enhancements can reach values of 103-104, while inmolecules they can go up to 106 [18]. Such systems there-fore form sensitive testing grounds for probing new sourcesof CP violation. In paramagnetic systems, CP violationgives rise not only to an eEDM, which is usually the focusof the investigations, but also to semileptonic CP-violatinginteractions between the electrons and the quarks in theatomic nuclei. Theoretical tools are needed to disentan-gle the different contributions and to relate these to CP-violating observables measured at colliders [19, 20].

In recent years, experiments with neutral diatomicshave set new stringent experimental eEDM limits. TheHinds group at Imperial College, London, uses a super-sonic beam of YbF molecules [7]; YbF is predicted tohave Eeff of 23 GV/cm [21]. The ACME collaboration atYale/Harvard uses a buffer-gas beam of metastable ThOmolecules [8], exploiting the large predicted Eeff in therange of 75-84 GV/cm [22, 23, 24, 25, 26, 27]. The lat-ter experiment was sensitive to frequency shifts of < 6mHz or an energy shift of < 3× 10−18 eV (3 aeV), whichcan be translated, knowing the Eeff , to a limit on theelectron-EDM of |de| < 8.7 × 10−29 e·cm. This is cur-

Aggarwal et al.: Measuring the electric dipole moment of the electron in BaF 3

rently the most stringent constraint on the electron EDM.Other neutral species have also been considered for elec-tron EDM searches, such as the WC [28] and the PbO[29] molecules. There is also an endeavor to use trappedmolecular ions [30] and a study based on the 180Hf19F+

ion has led to a constraint of |de| < 1.3× 10−28 e·cm [31].We aim to use the neutral BaF molecule for an elec-

tron EDM measurement. In heavy paramagnetic diatomicmolecules such as BaF the valence electron is exposed toa huge internal electric field, parametrized by Eeff , whichenhances the effect of the eEDM and results in a linearStark shift. This shift is the experimental signal that weaim to measure [32]. The magnitude of Eeff cannot bemeasured directly and has to be calculated using electronicstructure methods. BaF has a somewhat smaller predictedEeff of 6-8.5 GV/cm compared to other molecules usedin such experiments, depending on the computational ap-proach [33, 34, 22, 35, 36, 37]. We are currently performingbenchmark quality calculations of the Eeff of BaF, whichis needed for the interpretation of the measurements andextraction of the limit on the electron EDM. The rela-tivistic coupled cluster approach (CCSD(T)), consideredto be one of the most powerful computational methods, isused in these calculations, and we are also investigatingthe influence of various computational parameters on thevalue of obtained Eeff [38]. The modest value of Eeff ishowever compensated by the fact that BaF is lighter thanYbF and ThO, which allows it to be efficiently deceler-ated in a molecular beam machine of realistic size. Fur-thermore, unlike ThO, the experiment will be performedin the electronic ground state that has a suitable structurefor efficient laser cooling and detection.

3 Scientific challenges

3.1 Boosting the sensitivity of an eEDM measurement

The experimental strategy to search for a finite EDM ofthe electron is illustrated in Fig. 2. A superposition of hy-perfine substates is created, which builds up a phase dif-ference during the interaction with external electric andmagnetic fields. The EDM signal is detectable through adifference in the total accumulated phase for the paralleland the anti-parallel orientation of the magnetic and elec-tric fields. The phase difference φEDM due to the EDM ismeasured by subtracting the average fluorescence countrates for the relative orientations, and is directly propor-tional to the eEDM

φEDM = de|P |Eeffτ/h , (1)

where P is the polarisation factor of the molecule and τis the coherent interaction time in the interaction zone.The experimental resolution that can be reached dependson the fringe contrast, i.e. on the statistics, on the stabil-ity and homogeneity of the applied electric and magneticfields, as well as on the velocity and flux stability of themolecular beam.

Ba

F

Ba

F

E B

F=0

F=1

mF=-1 mF=0 mF=1

rf ~70 MHz

X2Σ+(N=0)

2deεe

Figure 2. The ingredients of an eEDM measurement in BaFmolecules. A superposition of hyperfine substates is created,which builds up a phase difference during the interaction withthe electric and magnetic fields. This phase difference, propor-tional to the product of the eEDM magnitude and the effectiveelectric field inside the molecule, is read out when the super-position is projected back onto the N = 0, F = 0 state.

The statistical uncertainty σd of a molecular eEDMexperiment is determined by the interplay of four crucialparameters: the rate of detected molecules N = dN/dt,the coherent interaction time of the molecules with theelectric field τ , the measurement time T , and |P |Eeff , andis given by

σd =h

e

1

2|P |Eeffτ√NT

. (2)

It is attractive to use a long coherent interaction time τ ,as the sensitivity improves linearly with this parameter.Until recently, increasing the interaction time was accom-panied by a significant decrease in the counting rate N .However, recent advances in decelerating molecular beams[39, 40, 41, 42] combined with spectacular progress inmolecular laser cooling [43] and the demonstration of in-tense cryogenic molecular beam sources [44] have openeda route to circumvent this limitation and make long in-teraction times possible. A similar approach has recentlybeen suggested for the YbF molecule [45].

Our experimental strategy is summarized in Fig. 3.The use of advanced slowing and cooling techniques cre-ates a slow and well-collimated molecular beam, whichleads to a coherent interaction time one order of magni-tude longer than that of competing experiments, withoutsacrificing the counting rate. Slow BaF molecules can bedetected efficiently by exploiting the closed cycling tran-sition, which further boosts the sensitivity.

We made a conservative estimate of the statistical sen-sitivity that we can obtain, resulting in 5 × 10−30 e·cm.This is based on the detection of 7× 105 molecules/shot,with the experiment running at 10 Hz for a period of 24hours. All essential numbers leading to this estimate aresummarized in Table 1. In section 4 the four main compo-nents of the proposed experiment are described in detail,and more information is given to support the numbers inthe table.

4 Aggarwal et al.: Measuring the electric dipole moment of the electron in BaF

cryogenic source decelerator state preparation interaction optical detectionlaser coolingguide

Figure 3. Schematic overview of the proposed experimental approach. The molecular beam of BaF molecules travels fromthe left to the right. A beam with a velocity of ∼ 180 m/s is created in the cryogenic source, and subsequently decelerated to∼ 30 m/s in the Stark decelerator. A lasercooling section reduces the transverse velocity spread of the beam, preventing it fromspreading out during the ∼ 15 ms it takes the molecules to travel through the 0.5 m long interaction zone. In the magneticallyshielded interaction zone the eEDM is probed using Ramsey interferometry in an electric field. Readout of the signal is doneby fluorescence detection.

Table 1. The estimate of the number of molecules that can be detected per repetition of the experiment. We aim to run theexperiment at 10 Hz.

Item Number Units Resulting # mol./shot

Source 1013 Molecules/shot0.005 Extraction efficiency from buffer gas cell0.24 Fraction in v = 0, N = 2 5×1010 from source; 4×109 in desired state,0.3 Fraction in low-field seeking states vlong=(180 ± 50) m/s, vtrans=±30 m/s.

Decelerator 0.002 Fraction in velocity acceptance0.3 Fraction in spatial acceptance0.7 Efficiency of deceleration relative to guiding 2 × 106, vlong=(30 ± 6) m/s, vtrans=±5 m/s.

Laser cooling 0.8 Laser cooling efficiency0.7 State transfer efficiency 9×105, vlong=(30±6) m/s, vtrans=±0.2 m/s.

Interaction zone 0.8 Transmission and state transfer efficiency1.0 Detection efficiency 7 × 105

3.2 Systematic effects

The achievable limit on the EDM of the electron dependsnot only on the statistical uncertainty (see eq. 2) but alsoon the control and understanding of experimental proce-dures, i.e, on systematic uncertainties. The measurementprinciple and the interaction zone are similar to that ofthe ongoing molecular EDM experiments such as ACMEat Harvard [46] and the YbF experiment at Imperial Col-lege [47]. The systematic effects are thus comparable; how-ever, they need to be evaluated explicitly in the contextof the current BaF experiment. We expect to control thecombined systematic effects at a level corresponding to aneEDM sensitivity of 10−30 e·cm.

In order to suppress systematics all experimental pa-rameters, such as magnetic and electric fields as well asrf and light polarizations, will be switched (reversed) [48].Such reversals can in practice be achieved with finite accu-racy. For example, reversing the electric field may result ina slightly different field strength, which can cause an asym-metry in the experimental conditions. As imperfections inthese fields can mimic EDM signals, their mapping andin situ monitoring will be implemented by electro-opticalfield sensors. Systematic effects that cannot be canceledby parameter reversal need to be carefully addressed in-dividually. These effects will be studied making use of the

well controlled molecular beam and, where possible, takeninto account by their deliberate exaggeration.

A major distinction of the current approach comparedto the previous eEDM experiments is the long interactiontime, which results in increased demands on the magneticfield control. The low value of the magnetic holding field(∼ 600 pT) requires careful shielding of both static anddynamic stray magnetic fields. For this the backgroundmagnetic field will be compensated (to about 5%) by 3sets of large electric current driven field coils around theinteraction zone setup. The interaction zone itself will beembedded in a cylindrical multilayer µ-metal shield forprimarily static field compensation. A coaxial, few mmthick Al shield will be used for AC environmental fieldsuppression, with particular emphasis on the AC field fre-quencies with periods comparable to the transit time ofthe molecules.

4 Experimental set-up

4.1 Cryogenic source

Until now all Stark deceleration experiments (includingthe experiments with SrF at the Van Swinderen Institute(VSI) at the University of Groningen [42, 49]) have usedpulsed supersonic beams as the source. However, over the

Aggarwal et al.: Measuring the electric dipole moment of the electron in BaF 5

Figure 4. Panoramic view of the Stark decelerator, operational at VSI in Groningen. This device, with a length of 4.5 meter,was built with the purpose of studying fundamental symmetry violations with slows beams of heavy diatomic molecules.

last decade several groups have demonstrated a promisingnew type of molecular-beam source using the cryogenicbuffer gas method [44, 50, 51, 52, 53]. With this method,cold beams of refractory species are generated by ablatinga solid precursor inside a cold cell. This cold cell, withtypical dimensions of a few centimeters, is filled with he-lium or neon atoms cooled to 2 − 20 K by a closed-cyclecryocooler. The buffer gas in the cell is kept at a specif-ically tuned atom number density, which is low enoughto prevent simple three-body collision cluster formationyet high enough to provide enough collisions for thermal-ization before the molecules touch the walls of the coldcell. A beam of cold molecules can be formed when thebuffer gas and target molecules escape the cell through afew-millimeter sized opening into a high vacuum region.When the helium or neon flow through the cell is small, aneffusive beam is formed with a relatively low forward ve-locity, which scales as the square root of the temperatureover the mass of the molecule. At higher flow rates thebeam velocity increases, eventually becoming equal to thesquare root of the temperature over the mass of the atomused as buffer gas, while the extraction efficiency increasesas well and the beam becomes more collimated. Beams ofslow (< 200 m/s) heavy molecules, such as SrF, BaF, YbFand ThO, with a brightness of above 1011 molecules persteradian per pulse have been reported [54, 55].

We will use a cryogenic buffer gas source with a de-sign similar to that demonstrated by Hutzler et al. [55]and Bulleid et al. [51], to create pulses of BaF moleculeswith a forward velocity of around 180 m/s. At these veloc-ities, the cryogenic source will generate an intense beamof forward-directed molecules that can be effectively cap-tured by our traveling wave decelerator. As a conservativeestimate, 1013 molecules are created per ablation pulse, of

which a fraction of 5× 10−3 is extracted into the molecu-lar beam. We therefore expect to be able to create a beamwith 5×1010 molecules in a pulse of 5−10 ms with a rota-tional temperature of 2 K, a longitudinal velocity spread of100 m/s, and a transverse velocity spread of 60 m/s. Thisintensity is similar to that demonstrated for BaF by Zhou[56], and about a factor 10 smaller than demonstrated forThO [50].

In order to prevent the buffer gas from entering thedecelerator beamline, we will add a series of quadrupolelenses of design similar to those used in the molecularfountain and molecular synchrotron experiments at theVrije Universiteit, Amsterdam [57, 58], which will guidethe BaF molecules from the exit of the cryogenic sourceto the entrance of the decelerator over a distance of 0.5to 1 meter. By adding some length, the coupling to thelongitudinal phase-space matching of the decelerator canbe improved [59]. The guide has a much larger transverseacceptance than the decelerator, so molecules that are losttransversally in the guide would not have been deceleratedanyway. The last section of the guide will be used to matchthe transverse phase-space distribution of the beam to theacceptance of the decelerator [60].

4.2 Stark deceleration

To reduce the forward velocity of the molecular beam wewill use a traveling-wave decelerator. This decelerator (de-picted in Fig. 4) for heavy diatomic molecules was re-cently built and taken into operation at VSI. In recentyears we have performed a number of experiments to pre-pare samples of molecules for precision measurements us-ing traveling-wave decelerators [61, 62, 63, 42, 41, 49].

6 Aggarwal et al.: Measuring the electric dipole moment of the electron in BaF

12 13 14 15 16 17 18 19

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Figure 5. (a) Time of flight profile showing the arrival time of SrF molecules captured from a supersonic expansion when awaveform is applied to guide (red) or decelerate (blue and green) the molecules [42, 49]. These measurements are performed witha 4 meter long decelerator. The central peak in these plots corresponds to a packet of molecules kept together by the electricfields throughout the deceleration process. (b) A numerical simulation of the time of flight profile illustrates the deceleration ofBaF molecules from a cryogenic source, from 180 to 30 m/s, in a 4.5 meter long decelerator. Molecules from the long pulse arecaptured into neighboring traps of the traveling-wave decelerator, resulting in multiple peaks in the time of flight profile. Alldecelerated molecules can be used for the EDM measurement because they will all propagate, after deceleration, through theinteraction zone at (30 ± 6) m/s.

The traveling-wave decelerator consists of many ring-shaped electrodes that effectively form a 4 mm diametertube through which the molecular beam travels. Recentresults, shown in Fig. 5(a), illustrate the efficient decel-eration of SrF molecules from a supersonic source withthis device. From these results, combined with numericaltrajectory simulations, we can deduce the deceleration ef-ficiency for BaF molecules, illustrated in Fig. 5(b). Dueto its length, the VSI decelerator can operate at modestdeceleration strengths, which results in the decelerationof the heavy BaF molecules with relatively high efficiency.In this process the molecules are kept together in closelyspaced traps of 3 by 6 mm at all times. As the cryogenicbeam emits a rather long pulse, multiple co-moving trapsinside the decelerator will be filled. The molecules in thesetraps will all be decelerated to the same final speed andwill all contribute to the eEDM measurement.

The number of molecules accepted by the decelera-tor is determined by the required deceleration strength,the applied voltage, and the Stark shift of the molecu-lar state of interest. The Stark shift of the lowest threerotational levels in the electronic ground state of BaF isshown in Fig. 6. The low-field seeking states of BaF have aturning point in their Stark shift that limits the acceptedtransverse and longitudinal velocities, which makes decel-eration of molecules in the first excited rotational state(N = 1) optimal at 5 kV, and deceleration of molecules inthe N = 2 state optimal at 10 kV. At 5 kV, BaF moleculesin the N = 1 state are accepted if they enter the deceler-ator with a longitudinal velocity within ±5 m/s of the setvelocity and a transverse velocity up to ±3 m/s. At an in-creased voltage amplitude of 10 kV, BaF molecules in theN = 2 state are accepted if they enter the decelerator with

−0,5

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1,5 N=2

0 10 20 30 40 50 60

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gy re

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-1)

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Figure 6. The lowest three rotational levels within the elec-tronic ground state of the BaF molecule, as a function of elec-tric field strength. The N = 0 level is used for the eEDMmeasurement; deceleration is most efficient in the N = 2 level,while lasercooling and detection is best performed in the N = 1level.

a longitudinal velocity within ±8.5 m/s of the set velocityand a transverse velocity up to 5 m/s. In order to deceler-ate a beam of BaF in the N = 2 state from 180 m/s to 30m/s in 4.5 meters, the traveling well will be decelerated at3.5 km/s2, which reduces the longitudinal acceptance ofthe decelerator to 70% compared to the situation wherethe well is traveling at a constant speed. Assuming a cryo-genic beam of BaF with a density and velocity spread as

Aggarwal et al.: Measuring the electric dipole moment of the electron in BaF 7

discussed in the previous section, we find that a fraction of4×10−4 of the cryogenic beam can be decelerated. Conse-quently, the decelerated beam exiting the traveling-wavedecelerator will contain 2× 106 molecules per pulse, witha longitudinal velocity spread of ±6 m/s and a transversevelocity spread of ±5 m/s.

4.3 Laser cooling

The slow molecular beam exiting the decelerator divergesdue to the transverse velocity spread. To allow the molec-ular beam to pass through the 50 cm long EDM measure-ment zone, laser cooling will be employed to reduce thetransverse velocity component. Since laser cooling requiresthe molecules to be in the N = 1 level, the molecules willbe transferred from the low-field seeking F = 2 and F = 3states of the N = 2 level to the F = 1 and F = 2 statesof the N = 1 using microwave radiation at 25.2 GHz, atthe beginning of the laser cooling section.

The powerful technique of laser cooling has only re-cently been extended from atoms to molecules [43, 64,65, 66, 67], where the leaky optical cooling cycle leads tochallenges comparable to the laser cooling of atoms likeBa [68]. Some molecules though, like BaF, have however anumber of properties that make them well-suited for lasercooling, that is i) a low probability for vibrational excita-tions upon electronic excitation, ii) a short excited-statelifetime [69, 70], which is important for a fast cooling cycle,and iii) convenient transition wavelengths [71, 72, 73, 74],where diode lasers with sufficient intensity are available.Recently, a number of investigations have highlighted thepossibility of lasercooling of BaF molecules [75, 76]. Wehave carried out relativistic coupled cluster calculationsof the branching ratios and relativistic multireference con-figuration interaction (MRCI) calculations of the transi-tion dipole moments of the relevant transitions [77]. Theobtained Franck-Condon factors are indicated along withthe experimentally known energy level structure of BaF inFig. 7. One main cooling transition and repumping fromthe vibrational states v′′ = 1 and v′′ = 2 is sufficient forthe transverse cooling of the BaF beam. The branchingto the Delta state A′2Δ is strongly suppressed due to thesmall energy separation [78, 79]. Frequency sidebands willbe created on both the main cooling transition and therepump transition to match the hyperfine structure in theground state [80].

The efficient transverse laser cooling is possible be-cause of the advantageous properties of the slowed BaFbeam at the exit of the decelerator. A single photon at860 nm results in a recoil of BaF of about 3 mm/s. In2D optical molasses, scattering of 2000 photons suffices todissipate the 5 m/s transverse velocity at the exit of thedecelerator, taking less than 2 ms at saturation intensity.During the transverse cooling the molecules travel ∼ 6 cmand the beam diameter increases to ∼ 1 cm. Saturation in-tensity in the area of the transverse cooling requires on theorder of 100 mW of laser power. Conveniently, the requiredlaser wavelengths and intensities are all available as diode

X2Σ

A2Π1/2

v''=0v''=1

v''=2

v'=0v'=1

860 nm

96% 4%

0.1% 4%

88%

8%

896 nm

829 nm 862 nm

A’ 2Δ<10-5

31 μm

Figure 7. The lowest vibrational levels within the electronicground state and electronically excited A2

Π1/2 state of theBaF molecule. The wavelengths of the main transitions aretaken from experiment [74]; the branching ratios from the ex-cited state are from calculations performed in our group [77].Given the low number of photons required for the cooling, twoadditional lasers repumping molecules from the v′′ = 1 andv′′ = 2 are sufficient.

lasers with tapered amplifiers. In order to cover the hyper-fine structure splitting of about 25 MHz, sidebands to thelaser frequency will be generated by electro-optical andacousto-optical modulation or by frequency offset lockingof individual diode lasers. The absolute wavelength of thelight will be stabilized against the optical frequency combat the VSI for stable long-term operation. From simula-tions using the calculated branching ratios, we find thatless than 20% of the molecules are lost to states other thanthe cooling state.

After being laser-cooled, the molecules will be opti-cally pumped into a single hyperfine state of the N = 1level, and subsequently transferred to the F = 0 hyperfinelevel of the N = 0 rotational ground state by resonant mi-crowave radiation at 12.6 GHz. The combined efficiency ofthe microwave transfers and optical pumping to the EDMstate is estimated to be better than 0.7. Hence we expectthe molecular beam emerging from the laser cooling sec-tion to have 9 · 105 molecules/shot, with the transversevelocity reduced to ±0.2 m/s.

4.4 Interaction zone and measurement

The interaction zone with a length of 0.5 m will be mag-netically shielded using multiple cylindrical layers of µ-metal [81] with end-caps inside a set of magnetic coilsfor environmental static field compensation. The externalstatic and dynamic magnetic fields Bext will be cancelledto |Bext| < 50 pT. The magnetic field will be monitoredwith sensitivity at the several 10 fT level [82] with flux-gate (and if needed SERF) magnetometers at fixed loca-tions outside and inside the shielded volume, based on theresults of the magnetic field survey at the experiment site.Of particular importance are dynamic field changes at fre-quencies originating from laboratory technological equip-ment (such as, e.g., 50 Hz), the period of which is com-

8 Aggarwal et al.: Measuring the electric dipole moment of the electron in BaF

parable to the travel time of the molecular beam throughthe interaction volume. During the measurements we willuse a set of additional coils external to the apparatus inthe interaction zone as a feedback system for dynamicfield compensation to provide sufficient control of environ-ment induced magnetic field changes on fast (sub-second)timescales. In order to avoid phase locking of the data ac-quisition to the frequency of potential perturbations wewill adjust the repetition rate for the measurements to beasynchronous with technical frequencies. These measuresalso prepare the interaction zone for future experimentswith slower beams.

A homogeneous vertical magnetic holding field of 600 pTcorresponds to a spin precession angle of π/4 when themolecules pass through the interaction zone inside thepassive magnetic shield similar to that employed in the129Xe EDM experiment [83]. A static field homogeneityof < 10−3 can be reached with an additional coil systeminside the µ-metal shield for higher magnetic multipolecompensation. The magnetic field polarity will be regu-larly reversed throughout the measurements in order tocontrol and check for systematics.

Two parallel plane electrodes, made from a thin lowconductance material deposited on glass, provide an elec-tric field of order 10 kV/cm parallel to the magnetic hold-ing field. For control of systematics the electric field strengthwill be monitored by a recently developed ultra-low fre-quency electro-optic sensor [84]. The molecular state prepa-ration and state detection regions are located near bothends of and inside the electric field volume. This yieldsquantitative state identification in BaF via fluorescencedetection. For the EDM measurements we will repeatedlyreverse the electric field on timescales that are long com-pared to the coherence time. This procedure will includesettling times for field stabilization.

After the molecules pass the electric field region themolecular superposition state is projected back onto thefirst excited rotational level, from where ∼ 103 photonscan be scattered per molecule. These photons are im-aged with 1% efficiency onto a position-sensitive low-noisesingle-photon detector. A cooled EMCCD camera, with aquantum efficiency of 50% at 860 nm, is capable of quanti-tatively detecting, without pileup, the high instantaneousphoton rates of above 107 photons per shot, which we ex-pect at the detector. Thus, 5 photons on average will bedetected for each molecule, which results in a detectionefficiency of close to unity.

5 Future perspectives

The proposed use of a slow, intense, and cold molecularbeam is, with the current level of technology, a promis-ing approach to measure the eEDM and will allow for ameasurement that is more than one order of magnitudebelow the current limit. The great sensitivity of the pro-posed scheme (mainly) stems from the interaction timethat is over ten times longer than used in ongoing ex-periments. We believe that in future experiments the in-teraction times can be increased even further. The inter-

action time in a horizontal beam machine is limited toabout 15 ms, as with longer times the beam will fall un-der gravity and will miss the detection zone. An elegantway to circumvent this is by using a vertical beam machinein fountain geometry. We have recently demonstrated thefirst-ever fountain for molecules, which enables the studyof ammonia molecules in free fall for up to 266 millisec-onds [57]. A crucial part of this machine is the travelling-wave decelerator that enables us to manipulate the mo-tion of polar molecules virtually without losses. Using acombination of quadrupole lenses and bunching elements,the slow ammonia beam exiting the decelerator is shapedsuch that it has a large position spread and a small veloc-ity spread when the molecules are in free fall, while beingstrongly focused at the detection region. Using a similarlens system to focus the laser cooled BaF beam, it shouldbe possible to create a (quasi-cw) fountain offering inter-action times of up to a second without significant loss influx. Ultimately, the most sensitive experiment could bebased on optically trapped molecules. However, significanthurdles still have to be overcome to realize this: a meansto accumulate a sufficiently large number of molecules hasto be demonstrated as well as cooling to significantly lowertemperatures than can currently be obtained.

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