Stark deceleration of CaF molecules in strong- and weak-field seeking states
T.E. Wall, J.F. Kanem, J. M. Dyne, J.J. Hudson, B.E. Sauer, E.A. Hinds, M.R. Tarbutt
SStark deceleration of CaF molecules in strong- and weak-field seeking states
T.E. Wall, J.F. Kanem, J. M. Dyne, J.J. Hudson, B.E. Sauer, E.A. Hinds and M.R. Tarbutt
Centre for Cold Matter, Blackett Laboratory, Imperial College London, Prince Consort Road, London, SW7 2AZ, UK.
We report the Stark deceleration of CaF molecules in the strong-field seeking ground state andin a weak-field seeking component of a rotationally-excited state. We use two types of decelerator,a conventional Stark decelerator for the weak-field seekers, and an alternating gradient deceleratorfor the strong-field seekers, and we compare their relative merits. We also consider the applicationof laser cooling to increase the phase-space density of decelerated molecules.
PACS numbers: 37.10.Mn, 37.20.+j, 29.20.Ba
I. INTRODUCTION
A Stark decelerator is a device for slowing down pulsesof cold molecules [1]. The force that acts on the moleculesis the spatial gradient of their Stark shift arising from theinhomogeneous electric field of the decelerator. The abil-ity of a Stark decelerator to deliver cold polar moleculeswith a centre-of-mass velocity at or near zero has provenexceedingly useful for a wide variety of experiments in-cluding high precision spectroscopy [2], measurements ofthe lifetimes of long-lived molecular states [3, 4], low tem-perature collision studies [5, 6], and tests of fundamentalphysics [7, 8].To date, only relatively light polar molecules in weak-field seeking states have been decelerated to low speed bythe Stark deceleration method [1, 9–14]. For many ap-plications it would be useful to extend the range of slowmolecules available by applying the method to heaviermolecules. This is particularly relevant for molecule-based tests of fundamental physics, such as the mea-surement of the electron’s electric dipole moment [15]and tests of parity violation in nuclei [16] and chiralmolecules [17], all of which require heavy molecules toreach high precision and could benefit from the use ofslower molecules [18].The kinetic energy to be removed by the decelerator isproportional to the molecular mass, so heavier moleculesrequire more deceleration stages. Fortunately, the decel-erator focusses the bunches of molecules both longitudi-nally [19] and transversally [20], so the bunches do notspread out as more stages are added. This important fo-cussing property applies only to molecules in weak-fieldseeking states, since they are naturally attracted to theaxis of the machine where the field is lowest. Focussing ofstrong-field seekers in all three directions is not possiblebecause of the impossibility of creating an electric fieldmaximum in free space [21]. Unfortunately, the low-lyingstates of heavy molecules are all strong-field seeking inthe large electric fields needed for a decelerator. This is amajor obstacle to extending Stark deceleration to heav-ier molecules. There are two possible ways to circumventthis difficulty. One option is to prepare the molecules in ahigher-lying rotational state which does have a weak-fieldseeking component at the relevant fields, and then decel-erate them in the usual way. The heavier the molecule the greater the rotational excitation will need to be in or-der to find a suitable weak-field seeking state. The secondoption is to an use an alternating gradient (AG) decel-erator [22–24] which works with molecules in strong-fieldseeking states. In this type of decelerator each decelera-tion stage focusses the molecules in one transverse direc-tion and defocusses them in the other, the focussing anddefocussing directions alternating from one stage to thenext. This series of focussing and defocussing lenses can,when properly arranged, provide net focussing in bothtransverse directions. An AG decelerator can be used formolecules in any state, including the ground state whichalways has the largest Stark shift, and its application toheavy molecules has been demonstrated [23, 25].In this paper we describe the deceleration of CaFmolecules in both weak-field seeking and strong-fieldseeking states. For the former case, we use CaF moleculesin a weak-field seeking component of the fourth rotation-ally excited state ( N = 4) so as to maximize the Starkshift, and we use a decelerator that follows a conventionaldesign - we call it a WF-decelerator. For deceleratingstrong-field seeking CaF we use molecules in the groundrotational state and an AG decelerator. We comparethese two strategies for decelerating heavy molecules. Wealso consider the prospects of laser cooling CaF so as toincrease the number and phase-space density of deceler-ated molecules. II. EXPERIMENTAL SET-UP
A pulsed supersonic beam of CaF is produced bylaser ablation of Ca into a pulsed supersonic carriergas (Ar, Kr or Xe) containing a small fraction of SF [26]. The translational and rotational temperatures ofthe molecules are both approximately 3 K. Their speedis usually a little higher than the terminal supersonicspeed of the carrier gas. The molecules pass through askimmer, then through the decelerator, and are finallydetected 805 mm from the source by time-resolved cwlaser-induced fluorescence on a chosen rotational compo-nent of the A Π / − X Σ + (0 −
0) transition at 606 nm.The experiment runs at a repetition rate of 10 Hz. Fur-ther details of the methods we use to produce and detectcold CaF molecules are given in [27]. a r X i v : . [ phy s i c s . a t o m - ph ] A p r zxy zx y (a)(b) FIG. 1: Photographs of the two decelerators. (a) Three stagesof the AG decelerator. (b) The first few stages of the WFdecelerator.
The AG decelerator, shown in Fig. 1(a), consists of 21deceleration stages, each of which is also a lens that fo-cusses in one transverse direction and defocusses in theother. Each of these lenses is formed by a pair of stain-less steel cylindrical electrodes, 14 mm long and 6 mm indiameter, with hemispherically-rounded ends of radius3 mm, their axes parallel to the z -axis, and their surfacesseparated by 2 mm. The centre-to-centre spacing of thelenses (along z ) is 24 mm, and successive lenses are ro-tated through 90 ◦ about the z -axis. When the electrodepair lies in the xz -plane the lens focusses strong-fieldseeking molecules along y and defocusses them along x .The following lens has its electrodes in the yz -plane anddoes the opposite. Each electrode is attached by a pairof dowels to one of four 16 mm-diameter stainless steelsupport rods, 504 mm in length. Each support rod isattached by insulating macor stand-offs to two stainlesssteel support rings, one at either end of the decelerator.The WF decelerator, shown in Fig. 1(b), consists of100 deceleration stages. Each stage is formed by twoparallel stainless steel cylindrical electrodes of diameter3 mm, with their axes parallel to either x or y , and theirsurfaces separated by 2 mm. The centre-to-centre spac-ing of the stages is 6 mm along z , and successive stagesare rotated through 90 ◦ about the z -axis. When the elec- trodes have their axes along x they focus weak-field seek-ing molecules along y , and do nothing along x . For theelectrodes aligned along y the opposite is true. Each elec-trode is pushed into one of four 16 mm-diameter stainlesssteel support rods, 594 mm in length, and these supportrods are attached by insulating alumina stand-offs to twostainless steel support rings.For both decelerators, four independent 20 kV switchesare used to switch the high voltages applied to each of thesupport rods between two values, ± V HI and ± V LO , withrise and fall times of approximately 500 ns. The supportrings are grounded. For the AG decelerator, ± V LO iszero. To avoid nonadiabatic transitions in the WF decel-erator [28], V LO is not zero but is still far smaller than V HI . After high voltage conditioning, both deceleratorsare able to support ±
20 kV across the 2 mm gaps betweenelectrode surfaces.
III. PROPERTIES OF THE TWODECELERATORS
As is usual, we introduce a reduced position θ = πz/L (modulo 2 π ) where L is the distance between decelerationstages. The decelerators can be in one of four high volt-age configurations: (i) even stages at ± V HI , odd stages at ± V LO , (ii) odd stages at ± V HI , even stages at ± V LO , (iii)all stages at ± V LO , and (iv) all stages at ± V HI . The WFdecelerator is switched back and forth between configu-rations (i) and (ii) in a sequence generated such that amolecule with a chosen initial speed v i reaches the samepoint relative to the periodic array every time the decel-erator is switched. The value of θ at this point is termedthe synchronous phase angle, φ [19]. The switching se-quence for the AG decelerator is slightly more compli-cated - the decelerator is switched into state (i) when thesynchronous molecule reaches the position z on inside aneven stage, to state (iii) when it reaches the position z off ,to state (ii) when at z on in an odd stage, and back tostate (iii) when it reaches z off again.In Fig. 2 we compare the basic properties of the twodecelerators. Here, we take the electrodes to be chargedto ±
20 kV so that the field on the beamline is approx-imately 200 kV/cm, and we take the CaF molecules tobe in the (
N, M ) = (0 ,
0) state for the AG deceleratorand in the (4 ,
0) state for the WF decelerator. Figure2(a) shows the Stark potential as a function of position( z ) along the beamline for configurations (i) and (ii). Al-though the electric field on the beamline is approximatelythe same in the two decelerators, the depth of the Starkpotential is about 4 times higher in the AG than in theWF because of the larger Stark shift of the ground state.However, the stages are packed 4 times closer together inthe WF and so the deceleration per unit length of decel-erator is approximately the same in the two cases. Figure2(b) shows the longitudinal acceptance areas of the twodecelerators, calculated analytically in both cases usingthe relevant effective potentials [19, 30]. In both cases, -20 -10 0 10 20-200-150-100-50050-15-10-5051015 -4 -2 0 2 4 6-1.0 -0.5 0.0 0.5 1.0-4-2024 z (mm) S t a r k s h i f t ( GH z ) R e l a ti v e s p ee d ( m / s ) Relative position (mm)Relative position (mm) R e l a ti v e s p ee d ( m / s ) (a)(b)(c) z on z off FIG. 2: Key information for deceleration of CaF in the twodecelerators: WF (solid red lines) and AG (dashed blue lines).Both decelerators are charged to ±
20 kV. The molecules arein the (
N, M ) = (4 ,
0) rotational state for the WF decelera-tor and in the (0 ,
0) state for the AG decelerator. (a) Starkshift versus longitudinal position with even stages chargedand odd stages grounded (bold) and vice versa (thin). (b)Longitudinal acceptance when the energy loss per stage ishalf the maximum possible value. (c) Transverse acceptancewhen the energy loss per stage is half the maximum possiblevalue. For the AG decelerator we show the acceptance areaat the entrance of a focussing lens, we took an effective lenslength of 5.5 mm, an effective drift length of 18.5 mm and aforward speed of 430 m/s. the energy loss per stage is chosen to be half the maxi-mum possible value. The longitudinal acceptance of theAG is larger than the WF because the stages are longerand the the Stark shift is larger. Figure 2(c) shows theacceptance areas of the two decelerators in one of the twotransverse directions. For the WF decelerator this is ob-tained by calculating the mean transverse spring constant [20], while for the AG decelerator it is found by using theCourant-Snyder formalism for AG focussing [24, 29, 30].We find that the transverse acceptance areas are approx-imately the same for the two decelerators. In the AG de-celerator each lens focusses/defocusses strongly, becauseof the large Stark shift. The net effect of the alternatinglenses is to focus the molecules, and this can be describedas an effective focussing lens whose power is consider-ably smaller than that of the individual lenses [29]. Inthe WF decelerator the molecules are focussed through-out, but the focussing is relatively weak because of thesmaller Stark shift and because, whenever the moleculesapproach a region of strong field where the focussing isstrongest the decelerator is switched to make the fieldsmall again.We have used approximate methods to calculate thephase-space acceptance areas shown in Fig. 2. In partic-ular we have used an effective potential to describe thelongitudinal motion of molecules relative to that of thesynchronous molecule, we have assumed that the trans-verse forces are harmonic, and that the longitudinal andtransverse motions are uncoupled. This last approxima-tion is particularly poor, the motions in the longitudinaland transverse directions being quite strongly coupled inboth the WF decelerator [20] and the AG decelerator [24].The true acceptances will be smaller as a consequence,but still of the same order of magnitude as obtained fromthese approximate methods (see [18] for example.)
IV. AG DECELERATION RESULTS
In this section we discuss the deceleration of groundstate CaF from an initial speed of v i = 433 m/s. TheAG decelerator is used with V HI = 20 kV, V LO = 0, z on = 0 mm and z off = 6 mm, as indicated in Fig. 2(a).The deceleration switching sequence is applied on everyeven-numbered pulse of the experiment, while on everyodd-numbered pulse the decelerator is turned off. Figure3(a) shows how the time-of-flight profile of the moleculeschanges as the number of deceleration stages used in theexperiment, n , is increased. It is the last n stages thatare used, the decelerator being off until the synchronousmolecule is n stages from the decelerator exit. To elimi-nate the effects of drifts in the source flux, each profile hasbeen normalized to the amplitude of the correspondingdecelerator-off profile obtained (almost) simultaneously.The zero of time is the arrival time of the synchronousmolecule (with speed v i ) when the decelerator is off. Af-ter a few stages the phase-stable molecules have beenbunched about the synchronous molecule, resulting in anarrow peak in the measured time-of-flight profiles. As n is increased the narrow peak moves to later arrival times,showing that the speed of this bunch has been reduced.Figure 3(b) shows the time-of-flight profiles obtainedby simulating the motion of molecules through the decel-erator, using a field map obtained from a finite elementmodel and removing any molecule that crashes into an Arrival time ( µ s) T i m e - o f-f li gh t s i gn a l (a) (b)off39513211917 -200 -100 0 100 -200 -100 0 100 200 FIG. 3: (a) Points: Measured time-of-flight profiles of ground-state CaF with the last n stages of the AG decelerator beingused. The parameters are V HI = 20 kV, V LO = 0, z on = 0 mm, z off = 6 mm, v i = 433 m/s. The bottom profile was taken withthe decelerator turned off. Lines: Triple Gaussian fits to thedata. (b) Points: Simulated time-of-flight profiles to matchthe experiments. Lines: Triple Gaussian fits. electrode. The simulation results agree fairly well withthe experimental data. To make a quantitative com-parison, we fit each of the experimental and simulateddatasets to a model which is a sum of three Gaussians.In this model, one Gaussian ( G ) represents the broadbackground of undecelerated molecules, a second ( G )represents the decelerated bunch, and a third ( G ), withnegative amplitude, represents the hole in the undeceler-ated distribution where the decelerated bunch would oth-erwise have been. These fits are shown as lines in figure3. We see that this model fits well to the data, except forsome modulations in the undecelerated background thatthe model does not capture. These modulations, whichare due to bunching of molecules in the neighbouring de-celeration stages, are most clearly seen in the simulateddata for small n , but are also present to a lesser extentin the experimental data, and in the data for higher n .As one would expect, the central arrival times of G and G have no significant dependence on n , while the arrivaltime of G increases with n . This is shown in Fig. 4(a)where we plot, as a function of n , the delay in arrivaltime between the decelerated bunch and the arrival timeof the synchronous molecule when the decelerator is off.Note that for the relatively small reductions in speed ob-tained here, the narrow bunch of molecules representedby G is actually a mixture of phase-stable moleculesand some molecules that are not phase-stable but havenot yet separated from the phase-stable bunch. The ex-perimental delays are similar to the predictions of thesimulations but deviate for n = 19 and 21 where the sim-ulations predict a larger delay than is measured. We do Number of stages D e l a y ( µ s ) R e l a ti v e m o l ec u l e nu m b e r Number of stages(a)(b)
FIG. 4: Comparison of experimental (filled circles, red) andsimulated (open circles, blue) results for the deceleratedbunch, as determined by the parameters of a three-Gaussianfit. The error bars are the uncertainties on the fitted param-eters; when not visible, they are smaller than the size of thepoints. (a) Delay in arrival time as a function of n . (b) Num-ber of molecules in the decelerated bunch as a function of n ,normalized to the total number of molecules measured whenthe decelerator is off. not know the reason for this discrepancy. In the exper-iments for n = 21 the phase stable bunch is deceleratedfrom 433 m/s to 399 m/s, corresponding to a removal of2.1 THz of kinetic energy, or 15% of the initial kineticenergy.Figure 4(b) shows how the area of the deceleratedbunch (i.e. of G ) depends on n for both the experimen-tal and simulated data. We have normalized these areasto the total area of the signal obtained when the decel-erator is off. The simulations predict an increase in thesignal ratio up to n = 13 and then a decrease at higher n . When n is small, the distance from the source tothe effective entrance of the decelerator is large, and therange of transverse speeds that can enter the deceleratoris smaller than the range the decelerator can accept. Thedecelerator’s acceptance is thus not filled. As n increasesthe effective entrance moves closer to the source, the ac-ceptance is more completely filled and so the number ofmolecules in the decelerated bunch increases. At around n = 13 the acceptance is completely filled and furtherincreasing n now reduces the signal ratio because thereis a loss mechanism at work in the decelerator, as we dis-cuss below. The measured signal ratio for the deceleratedbunch is always smaller than the simulations predict, re-maining constant up to n = 15, and then dropping. Thisis the behaviour we would expect if the continuous losseswithin the decelerator are larger than expected. We notethat there is a similar reduction with increasing n in thetransmission of the undecelerated molecules.The transverse forces at the ends of each lens are re-sponsible for these continuous losses. In an ideal AGdecelerator, where the transverse forces are linear in theoff-axis displacement with equal and opposite force con-stants, the transverse acceptance does not change as morelenses are added, at least not once the decelerator lengthis longer than the wavelength of the macromotion. Ashas been discussed previously [24], the ends of the AGdecelerator electrodes are detrimental to proper AG fo-cussing because the magnitude of the defocussing forceconstant k y = − ∂F y /∂y greatly exceeds the magnitudeof the focussing force constant k x = − ∂F x /∂x near theseends, F x,y being the components of the force in the fo-cussing ( x ) and defocussing ( y ) directions. This is shownin Fig. 5(a), where we plot the quantity − ( k x + k y ) as afunction of position along the axis, z . For reference, theelectric field along z is shown on the same plot. Whilethe focussing and defocussing force constants are well-balanced inside the lens, the defocussing is far strongerthan the focussing in the region near the ends of thelens. Figure 5(b) demonstrates the impact of these endeffects. Here, we have simulated AG deceleration with n = 17, V HI = 20 kV, V LO = 0, z on = 0 mm, z off = 6 mmand v i = 436 m/s. The speed distribution entering thedecelerator is a Gaussian centred at 436 m/s. From thesimulation we can select out those molecules that aretransmitted by the decelerator and then look at theirdistribution of initial speeds. It is this distribution thatis plotted in Fig. 5(b). We see that the transmission de-pends very strongly on the speed. We draw attention tomolecules with three specific velocities, 420 m/s, 430 m/sand 440 m/s, indicated by the vertical lines, where thetransmission is bad, good and bad, respectively. Eachof these molecules passes through a 6 mm-wide region ofthe first lens during the time when it is switched on. Foreach molecule, the centre of this region is indicated by avertical line in Fig. 5(a). The molecules that are poorlytransmitted are the ones that see the two bad regions ofthis first lens. The simulations show that these moleculesrepeatedly experience the bad regions as they move fromone lens to the next, and that is why they tend to be lost.The 430 m/s molecule sees only the good part of the firstlens, and to a large extent avoids the bad regions of theother lenses too, and so is transmitted well by the decel-erator. Unfortunately, the bad region at the exit of eachlens is also the region where the molecules are deceler-ated, and so the decelerated molecules necessarily haveto move through this bad region and tend to be thrownout of the decelerator for this reason.Our simulations use an accurate map of the field, cal-culated numerically using a finite element model, but theexperimental data shows more molecule loss than in the Longitudinal speed (m/s) N u m b e r o f m o l ec u l e s
350 400 450 500 550 (b)(a) -20 -10 0 10 20
Position (mm)
E-( k x + k y ) -( k x + k y ) ( c m - / mm )
420 430 440
FIG. 5: (a) The quantity − ( k x + k y ) plotted as a function ofdistance z along the AG decelerator, indicating the regionsof the lens where the defocussing is far stronger than the fo-cussing. For reference, the electric field as a function of z isalso plotted (on a different scale). (b) Simulation result show-ing the distribution of initial velocities that are successfullytransmitted through the decelerator. Molecules with threevelocities are indicated by vertical lines and are discussed inthe text. Their central positions inside the first lens, duringthe time that it is turned on, are shown by the equivalentvertical lines in (a). simulations. We do not know the reason for this. OtherAG deceleration and focussing experiments also reportgreater losses than expected, the suspected cause beingelectrode misalignments [22, 25, 31, 32]. V. WF DECELERATION RESULTS
In this section we discuss the WF deceleration of CaFmolecules that emerge from the source in the (
N, M ) =(4 ,
0) state and with an initial speed of 340 m/s. Forthese experiments, it is important to suppress nonadi-abatic transitions to other (4 , M ) states which may bedriven by the rotation of the electric field when the de-celerator switches, especially in regions where the electricfield is small [28]. These unwanted transitions can besuppressed by ensuring that the Stark splitting betweenthe various M states is much larger than the angularfrequency at which the field rotates. For this reason, in-stead of switching between high voltage and ground, weswitch between V HI and a bias voltage of V LO (cid:39)
300 500 700 -400 0 400-400 0 400 S i gn a l (i) 334m/s(ii) 322 m/s(iii) 305 m/s(iv) 279 m/s(v) 260 m/s(vi) 224 m/s (a) (b) FIG. 6: (a) Measured time-of-flight profiles of CaF in the N = 4 rotational state following deceleration in the WF de-celerator. The zero of time is the arrival time of the syn-chronous molecule when the decelerator is off. For all thedata, v i = 340 m/s, V LO = 1 . V HI )and synchronous phase angles ( φ ) for the individual tracesare (i) 18 kV, 15 ◦ , (ii) 18 kV, 30 ◦ , (iii) 18 kV, 45 ◦ , (iv) 18 kV,60 ◦ , (v) 20 kV, 60 ◦ , (vi) 20 kV, 75 ◦ . The inset shows the de-celerated bunch for trace (vi) in more detail. (b) Numericalsimulations of these same experiments but with V LO = 0 andnon-adiabatic transitions neglected. the energy loss per stage.Figure 6 shows the results of these experiments, alongwith the associated numerical simulations. The laser-induced fluorescence signal from molecules in N = 4 isplotted versus arrival time for several different values ofthe synchronous phase angle and the high voltage V HI .The zero of time is the arrival time of the synchronousmolecule when the decelerator is turned off, or when itis operated at φ = 0. As φ increases the deceler-ated bunch moves to later arrival times, correspondingto slower speeds, but also reduces in amplitude becausethe longitudinal phase space acceptance of the deceler-ator decreases as φ increases. In the uppermost trace,we have set φ = 75 ◦ and V HI = 20 kV. In this casethe molecules are decelerated to a final speed of 224 m/s,corresponding to an energy reduction of 4.9 THz or a re-moval of 57% of the initial kinetic energy. The simulatedtime-of flight profiles shown in Fig. 6(b) agree well withthe experimental data. In particular, the measured ar- Synchronous phase angle (degrees) E n e r gy l o ss ( T H z ) Synchronous phase angle (degrees) R e l a ti v e nu m b e r o f d ece l e r a t e d m o l ec u l e s (a)(b) FIG. 7: Results from the WF decelerator operated at V HI =18 kV. (a) Points: Measured kinetic energy loss for the de-celerated bunch versus synchronous phase angle. Line: Ex-pected energy loss calculated from the Stark potential. (b)Points: Measured number of decelerated molecules versussynchronous phase angle, normalized to the number at φ = 0.Line: Longitudinal acceptance versus synchronous phase an-gle. rival time and size of the decelerated bunch are the sameas predicted by the simulations.Figure 7 (a) shows how the measured energy losschanges with φ when V HI = 18 kV. Here, we have deter-mined the energy loss from the measured arrival time ofthe decelerated bunch and the measured distances fromsource to decelerator and decelerator to detector, by as-suming that the decelerator applies a constant force tothe molecules over its entire length. Since there are manydeceleration stages, each applying the same net deceler-ation, this is a good approximation. The energy lossfollows the prediction obtained from the Stark potentialplotted in Fig. 2(a). In Fig. 7(b) we show the number ofdecelerated molecules as a function of φ , normalized tothe number at φ = 0 and compare this to the (similarlynormalized) longitudinal acceptance. As φ increases thenumber of decelerated molecules decreases because thelongitudinal acceptance decreases. While the experimen-tal results follow the trend of the longitudinal acceptance,there is some deviation from this simple model. This isbecause the transverse acceptance also has some depen-dence on φ , through the coupling of the transverse andlongitudinal motions. This coupling tends to reduce thenumber of decelerated molecules at low phase angles andto increase it at high phase angles [20], as we indeed ob-serve in Fig. 7(b). VI. PROSPECTS FOR COMBINING STARKDECELERATION AND LASER COOLING
Laser cooling of a diatomic molecule has recently beendemonstrated for the first time [33]. Laser cooling is fea-sible if the molecule has a short-lived state that can beexcited with a convenient laser and that decays (exclu-sively) to the ground state with a Franck-Condon factorclose to unity. Then, only a few excitation lasers areneeded to address all relevant vibrational components ofthe electronic transition [34]. The A Π / − X Σ + (0 − v (cid:48) = 0) − X( v (cid:48)(cid:48) = 0),and the other at 628 nm resonant with the P(1) line ofA( v (cid:48) = 0) − X( v (cid:48)(cid:48) = 1). Sidebands are applied to ad-dress the 4 hyperfine components of each of these tran-sitions, so that there are 8 discrete frequencies in total.A small magnetic field is used to prevent optical pump-ing into a dark state. The only remaining dark statesare the vibrational states with v (cid:48)(cid:48) >
1, and the branch-ing ratio for the v (cid:48) = 0 state to decay into these darkstates is assumed to be approximately 10 − . Neglectingcoherence, the internal dynamics of the molecule inter-acting with this set of laser beams is calculated using arate model that includes all the Zeeman sub-levels (24in the ground state, 4 in the upper state). For the de-cay rate and the 0 − / (2 π ) = 8 . Z = 0 . Longitudinal position (mm)Transverse position (mm) L ong it ud i n a l s p ee d ( m / s ) T r a n s v e r s e s p ee d ( m / s ) (a)(b)
100 150 200 250300320340360380 -4 -2 0 2 4-15-10-5051015
FIG. 8: Simulations of laser cooling CaF prior to Stark de-celeration in a travelling-trap decelerator. The points showthe phase-space distributions in (a) the longitudinal directionand (b) the transverse direction. Green points: molecules en-tering the decelerator at z = 103 mm with no laser coolingapplied. Red points: molecules entering the decelerator at z = 170 mm with laser cooling applied. Line: Phase-spaceacceptance of the decelerator for CaF( N = 1) decelerated ata rate of 2 × m/s . From these calculations we find that when the intensityat each of the 8 frequencies is I , and the frequencies havea common detuning, the photon scattering rate is likethat of a two-level system with an effective decay rateΓ eff / (2 π ) = 2 . s eff = I/I s,eff , where I s,eff (cid:39)
50 mW/cm . We use thiseffective scattering rate to calculate the scattering forceson the molecule.We take a supersonic beam of CaF with a mean speedof 340 m/s and a translational temperature of 3 K. Thespatial distribution of the source is Gaussian in all di-rections with a standard deviation of 5 mm in the longi-tudinal direction and 2.5 mm in the transverse direction.The beam passes through a 3 mm diameter skimmer at z = 68 mm and then into the decelerator. We consider us-ing the travelling-trap decelerator demonstrated in Berlin[35] to decelerate CaF molecules in the N = 1 state, thesebeing the ones that are subjected to laser cooling. In thetravelling-trap decelerator the molecules are confined ina three-dimensional potential well throughout the decel-eration process and the instabilities that arise in the WFdecelerator [36] are absent. We take the geometry tobe exactly that of [35] and the peak-to-peak amplitudeof the applied waveform to be 16 kV. In the case wherethere is no laser cooling applied we place the deceleratorat z = 103 mm, close to the exit of the skimmer. In thelaser cooling case the entrance of the decelerator is moveddownstream to z = 253 mm. Cooling is applied in bothtransverse directions in the 150 mm gap between the exitof the skimmer and the entrance of the decelerator usingretro-reflected laser beams with a common red-detuningof 7.5 MHz and with s eff = 0 .
5. Longitudinal cooling isapplied using a laser beam that propagates along − z , has s eff = 0 .
5, and is pulsed on for 550 µ s starting 200 µ s af-ter the molecules are produced, so that they interact withthe laser in the region between the skimmer entrance andthe decelerator entrance. When this light turns on, thecommon detuning is -578 MHz, and then the frequencyis chirped linearly at a rate of 0.048 MHz/ µ s so that thetotal frequency change is 26 MHz. The decelerator trapshave an acceleration of − × m/s . Their initial speedis chosen as 340 m/s for the no-cooling case and 333 m/sfor the cooling case.The results of these simulations are shown in Fig. 8,where we plot the phase-space distributions of moleculesentering the decelerator in the two cases, and comparethese to the phase-space acceptance of the deceleratorin both the longitudinal and transverse directions. Inthe longitudinal direction the laser slows down moleculeswith speeds in the 335–345 m/s range, substantially com-pressing the speed distribution into a narrow bunch cen-tred near 335 m/s. The laser cooling does nothing to com-press the spatial distribution which continues to expandso that at the decelerator entrance the laser-cooled bunchis longer than the uncooled bunch. The over-all effect ofthe longitudinal laser cooling is to increase the number ofmolecules within the decelerator acceptance by a factorof 3.1. The picture is similar for transverse cooling: thecooling is very effective at reducing the transverse veloc-ity spread, but the spatial spread increases because of theextra distance the molecules have to travel. The numberof molecules overlapping with the transverse acceptanceincreases by a factor of 1.4. With the laser cooling ap-plied in all three dimensions the number of deceleratedmolecules is predicted to increase by a factor of 6. The de-celerated molecules occupy a far smaller volume of phasespace when the cooling is applied, because in the trans-verse direction the velocity spread is compressed to anarea far smaller than the acceptance area. The simula-tions show that the phase-space density of the moleculesexiting the decelerator increases by a factor of about 2000when the laser cooling is applied. So we see that althoughthe laser cooling results in only a modest increase in thenumber of decelerated molecules, it greatly increases thephase-space density of slow molecules. A further increasein phase-space density could be obtained by using a sec-ond short pulse of longitudinal cooling as the moleculesexit the decelerator. VII. CONCLUSIONS
We have reported the Stark deceleration of CaFmolecules for the first time and have compared theperformance of two types of decelerator for slowingthese molecules. The AG decelerator is used to slowdown molecules in the strong-field seeking ground state,whereas the WF decelerator works only for molecules inweak-field seeking states and has been applied here toCaF in the (
N, M ) = (4 ,
0) state.Using a 100-stage WF decelerator at 20 kV and at aphase angle of 75 ◦ , the molecules have been deceleratedfrom 340 m/s to 224 m/s. A 200-stage decelerator of thesame design would be able to bring these molecules torest. For this case, the relatively high phase angle re-sults in a small acceptance. This could be increased byusing a longer decelerator at a lower phase angle. Atravelling-trap decelerator of the type recently demon-strated in Berlin [35] is also suitable for deceleratingCaF and should increase the number of slow moleculesobtained. The WF or travelling trap decelerator isalso suitable for slowing down molecules of considerablyhigher mass than CaF, although as the mass increaseshigher-lying rotationally-excited states are needed for ef-ficient deceleration, and the required decelerator lengthincreases. The required length could be reduced by theuse of cryogenically-cooled sources [37] to provide slower,more intense beams. Cryogenic sources to date produceeither continuous beams or very long pulses that are notwell matched to the small size of the potential wells in atypical decelerator. However, there does not seem to beany barrier to making far shorter pulses.Using a 21-stage AG decelerator at 20 kV, groundstate molecules have been decelerated from 433 m/s to399 m/s. We observe a smaller decelerated signal than ex-pected from numerical simulations, and in the case whereall 21 stages are used the decelerated bunch is moving alittle faster than expected. The advantage of the AG de-celerator is its ability to focus molecules in strong-fieldseeking states. For large molecules, these are the onlystates available [25]. There are several difficulties in de-celerating large molecules to low velocity. Firstly, themolecules must stay inside the stability region for AGfocussing and that requires either the lens length or thefocussing power to change as the molecules slow down.One approach is to use long lenses at the start of the de-celerator and short ones at the end, with the long lensessegmented into a series of shorter ones so as to maintainthe same deceleration per unit length. This arrangementhas been discussed in detail in the context of deceleratingground state YbF [18]. An alternative is to build eachlens from four electrodes instead of two [24] in which casethe focussing (and defocussing) direction of each lens isset by the polarities of the electrodes. This provides amore versatile structure because the exact arrangementof lenses is no longer fixed by the layout of the machine.Even with these more sophisticated lens arrangementsit becomes very difficult to maintain stability of AG fo-cussing at the lowest speeds [18]. A further difficulty forstrong-field seeking molecules is that AG deceleration isalways accompanied by excess defocussing, as discussedabove. This can be minimized by tapering the ends ofthe electrodes more gradually, for example by replacingthe hemispherical ends of the electrodes in the presentdesign with ellipsoidal ends. Finally, experimental workon AG deceleration and focussing consistently reportsmore loss than expected from detailed numerical sim-ulations, and these additional losses are thought to bedue to mechanical misalignments. Exceptionally goodalignment is needed to make the decelerator perform asexpected. A cryogenically-cooled source operated in theeffusive regime offers an alternative, and simpler, methodto obtain slow-moving beams of large, heavy molecules.For some molecules, CaF being a good example, lasercooling promises to increase the phase-space densityenormously. Laser cooling can both cool and deceleratethe molecules. For the latter, a large number of photonshave to be scattered, requiring a particularly ‘closed’ vi- brational structure. An alternative, requiring far fewerspontaneous emissions, is to combine laser cooling withStark deceleration. We have simulated a possible exper-iment where transverse and longitudinal laser cooling isapplied to CaF prior to Stark deceleration. The coolingis very effective in compressing the velocity distribution,but does not compress the spatial distribution. Instead,the size of the beam at the decelerator entrance increasesbecause the decelerator needs to be moved downstream toallow space for laser cooling. In our simulations, the lasercooling increases the number of decelerated molecules bya factor of about 6, but increases the phase-space densityof slow molecules by about 2000. Acknowledgments
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