Principles and design of a Zeeman-Sisyphus decelerator for molecular beams

We explore a technique for decelerating molecules using a static magnetic field and optical pumping. Molecules travel through a spatially varying magnetic field and are repeatedly pumped into a weak-field seeking state as they move towards each strong field region, and into a strong-field seeking state as they move towards weak field. The method is time-independent and so is suitable for decelerating both pulsed and continuous molecular beams. By using guiding magnets at each weak field region, the beam can be simultaneously guided and decelerated. By tapering the magnetic field strength in the strong field regions, and exploiting the Doppler shift, the velocity distribution can be compressed during deceleration. We develop the principles of this deceleration technique, provide a realistic design, use numerical simulations to evaluate its performance for a beam of CaF, and compare this performance to other deceleration methods.


Introduction
There has been much recent progress in the formation and control of cold molecules, motivated by numerous potential applications, [1,2] including quantum information processing, [3] tests of fundamental physics, [4][5][6][7][8][9][10] and understanding chemistry and intermolecularc ollisions at the quantum level. [11][12][13][14] With many experimentsb eginning with ar elativelyf ast molecular beam, deceleration techniques such as Stark deceleration, [15] Zeemand eceleration, [16] opticalS tark deceleration [17] andc entrifuge deceleration [18] have been at the forefront of cold molecule research, being used to provide velocity-controlled beams and to load traps that can store cold molecules for many seconds. [19] Once trapped, ac ommon goal is to cool the molecules to lower temperatures by sympathetic, [20,21] Sisyphus, [22][23][24][25] adiabatic [26] or evaporative [27] cooling. Direct laser slowing and cooling is another viable option for certain species, [28][29][30][31] being capable of both decelerating and subsequently trapping and cooling the molecules under study.I ndeed, such an approach has recently led to the demonstration [32] and optimization [33,34] of the first molecular magneto-opticalt rap (MOT).I nt his experiment, ac ryogenic buffer-gas source produces intense molecular pulses, typically1 -10 ms in duration with speeds in the range 50-200 ms À1 ,d epending on the source geometry and gas flow rate. [35,36] Then, by scattering about 10 4 photonsf rom ac ounter-propagating laser beam, [37] the molecules are decelerated to the capture velocityo ft he MOT,w hich is about 10 ms À1 . [38] So far,o nly af ew thousand molecules have been captured, mainly because the slowing method is inefficient. There are several reasons for this:1 )the stopping distance is large compared to the capturea rea of the trap, and so the solid-angle that can be captured is small; 2) the molecular beam is slowed longitudinally, but is not cooled transversely, and so the beam divergence grows as the moleculesa re slowed; 3) the photons cattering that slows down the beam also causes transverse heating, which increases the divergence even further; 4) molecules are lost if they decay out of the cooling cycle, and addressing those decaysi ncreases the experimental complexity.
Ac urrent focus of research in this area is to increase the number of molecules loaded into MOTsb yi mproving the efficiencyo ft he deceleration process. Such progress is important both for current experiments and to extend laser slowing and cooling to diatomic and polyatomic [39] speciesw ith less favorable vibrational branching ratios. Decelerators that use time-dependentf ields, such as Stark decelerators, are not wells uited to this application because they slow af ew slices of the molecular beam that are only af ew mm in length,ahundred times shorter than the beams emitted by at ypicalb uffer-gas source. At raveling-wave decelerator [40] or centrifuge decelerator [18] can handle long pulses, but these methods have not yet been widely adopted. DeMille et al. [41] explore methods to confine amolecular beam transversely as it is slowedb yradiation pressure, and conclude that guiding using microwave fields is ag ood option.
Here, we explore at echnique that we call Zeeman-Sisyphus deceleration. Molecules in ab eam travel through an array of permanent magnets that produces aspatially varying magnetic field, and are optically pumpedi nto aw eak-field seekings tate as they move towards regionso fs trong field, and into We explore at echnique for decelerating molecules using as tatic magnetic field and opticalp umping. Molecules travel through as patially varying magnetic field and are repeatedly pumped into aw eak-field seekings tate as they move towards each strong field region,a nd into as trong-field seeking state as they move towards weak field. The methodi st ime-independenta nd so is suitable for decelerating both pulsed and continuous molecular beams. By using guiding magnets at each weakf ield region,t he beam can be simultaneously guided and decelerated. By tapering the magnetic field strength in the strong field regions, and exploitingt he Doppler shift, the velocity distribution can be compressed during deceleration. Wed evelop the principles of this deceleration technique, providearealistic design, use numerical simulations to evaluate its performance for ab eam of CaF,a nd comparet his performance to other deceleration methods.
as trong-field seeking state as they move towards regions of weak field. In this way,t here is always af orce opposing their forwardm otion.T hisg eneral idea has al ong history.I n1 981, Breedena nd Metcalf suggested as imilar methodf or decelerating atoms in Rydberg states. [42] More recently,t he use of Sisyphus-type forces due to the Stark effect has been proposed as ad ecelerationt echnique [43] and demonstrated with great success as ac ooling method for electrostatically trapped molecules. [23,24] The magnetic-field analogs, relying on the Zeeman effect, have also been discussed [25] and already form the basis of an established trap-loading technique. [44,45] Here, we analyze in detail the prospects for extending these techniques to molecular beam deceleration, finding ad esign that provides both longitudinal slowing and net transverse guiding, as required of av iable decelerationm ethod. The approach is capable of bringingatypical buffer-gas-cooled molecular beam to rest by scattering only af ew hundred photons, far less than the typical % 10 4 scattered photonsr equiredf or direct laser slowing, and so could be applied to molecular species with only quasiclosed optical cycling transitions withoutn eedingn umerous repump lasers.

Principles of Zeeman-Sisyphus Deceleration
The general idea of Zeeman-Sisyphus deceleration is illustrated in Figure 1. Here, am olecule (black dot) with al ower state L and upper state U propagates through two regionso fs trong magnetic field separated by ar egion of weak field. L is degenerate in zero field with two substatesw hich shift oppositely in the appliedf ield:aweak-field seeking( wfs) state whose energy increases with fields trength, and as trong-field seeking (sfs) state whose energy decreases with field strength. The upper state has no Zeemans hift, and it can decay to either of the two lower states,b ut not to any other state of the molecule. Molecules that are amenable to laser cooling approach this ideal, while for others there may be transitions to other rotationalo rv ibrational states whichw ould need to be addressed. The two lower states are coupledt ot he upper state by two pump lasers, L w!s and L s!w . L w!s has an egative detuning of -D w!s relative to the zero-field resonance frequency, while L s!w has ap ositive detuning of D s!w .T he magnetic field values at which the molecules come into resonance with one of the lasers are called the resonance fields,a nd the locations in space where this occursa re called the resonance points. Both D w!s and D s!w are positive quantities, and they are arranged with D w!s > D s!w .W ith this configuration, wfs (sfs) molecules come into resonance with L w!s (L s!w )i nr egions of strong (weak) magnetic fielda nd are then optically pumped to the other state by absorption and subsequent spontaneous decay,a si ndicated by the solid and dashedv ertical arrows,r espectively.T he distance movedb yt he molecules during the opticalp umping process is negligible. The moleculesd ecelerate because they move into each strong field region in aw eak-field seekings tate, and out of those regions in as trong-field seeking state. This process is repeated until the molecules reach the desired final velocity.
For the arrangement shown in Figure 1, the averaged eceleration force is given by [Eq. (1)]: where 2 L is the spatialp eriodicity and the laser detunings are given in Hz. For the largest force we should set hD w!s = U max , where U max is the maximum Zeeman shift, and D s!w = 0. However,t oe nsure that L s!w only pumps molecules out of strongfield seeking states, D s!w should not be too close to zero. For af ixed decelerator length, the change in speed due to the average constant force of Equation (1) is inverselyp roportional to the mean speed, ands od eceleration increases the spread of velocities in the beam. In Section6 we show how the Doppler shift can be used to counter this effect under certain conditions. Because the deceleration methodi st ime-independent, it is applicable to long-pulse or even continuous molecular beams. It works for molecules of all longitudinal positions and speeds, and so itsl ongitudinal phase-spacea cceptance is unbounded. We would also like to arrange al arget ransverse phase-space acceptance,m eaning that molecules should be guided as they are decelerated. Because the strongest fields are at the magnet surfaces, molecules will tend to be anti-guided while in the strong-field seeking state and guidedw hile in the weak-field seekings tate. Witht he arrangemento fd etuningsi llustrated in Figure 1, the molecules spend more of their time in the weakfield seekings tate, and so net guidings eems possible. Moreover,w en ote that the molecules are in the weak-field seeking Figure 1. Illustration of the Zeeman-Sisyphus decelerationt echnique. A ground-state molecule (black dot) propagatest hrough two regionso fl arge magnetic field (two hills/valleys in energy) and is periodically optically pumpedb etween weak-field and strong-field seeking states such that it is perpetually decelerated. state as they pass through the field minimum, and that guiding magnets naturally have zero magnetic field on the axis. This presentsa no pportunity to interleave decelerating magnets, where the field is strong and uniform,w ith guidingm agnets, where the field is weak and increases with transverse displacementfrom the axis. As uitable arrangementofp ermanent magnets that achievest his is presentedi nS ection 3.
The deceleration methodr elies on efficient opticalp umping of molecules between the weak-and strong-field seeking ground states. Molecules that are not optically pumped will not be decelerated as efficiently,a nd if they remaini n as trong-field seekings tate fort oo long their trajectories are likely to become transversely unstable. To understand how the opticalp umpinge fficiency depends on various experimental parameters, we introduce as imple analytical model of the optical pumping process. Let p be the probability that the molecule switches from one ground state to the other after scattering as ingle photon, and let R(t)b et he scatteringr ate at time t.T he meann umber of photonss cattered by am olecule that is not optically pumped as it passes through the resonance point is " n ¼ R RðtÞdt,w here the integral is taken over the period of time where R(t)i sa ppreciable. In terms of p and n, the optical pumping probability c is given by [Eq. (2)]: For at wo-level system,w hich is ar easonable approximation for our opticalp umping arrangement, the steady-state scattering rate is given by [Eq. (3)]: [46] R ¼ G 2 where G is the excited state decay rate, s = I/I sat is the saturation parameter of the pump laser,a nd d is the laser detuning [Eq. (4)]: Here, l is the transition wavelength, v z is the forward velocity of the molecule, ÀUi st he Zeemans hift of the transition energy in am agnetic fieldB,a nd D 0 = f laser Àf 0 is the detuning of the laser from the transition frequency for astationary molecule in zero field. Over the small region of space aroundt he resonance point where Ri sl arge, d changesa pproximatelyl inearly with z and hence with t,s ow et ake d = bt.T his gives [Eq. (5)]: Assuming am agnetic momento fm B ,a nd neglecting the small change in speed as the molecule passes through the resonancepoint, we have [Eq. (6) where @ z B is the longitudinal component of the gradient of the magnetic field magnitude at the resonance point. Thus, [Eq. (7)]: To gether,E quations (2) and (7) determine the average optical pumping probability as af unction of the relevant experimentalp arameters. This probability needs to be high enough to ensure that molecules pass through most of the guiding magnetsi nt he weak-field seeking state, setting ar equirement on n.R earranging Equation (7) then gives am aximum allowable value for the field gradienta tt he resonancep oints. This maximum scales inversely with v z ,a nd scales linearly with s when s ! 1b ut only as ffiffi s p when s @ 1. The value of p depends on the molecular transition and particular Zeeman sublevel, the magnitude ofB at the resonance point,a nd the polarization of the optical pumping light relative toB.W ei nvestigate these detailsi nS ections 4a nd 5, and find that ac onstant p % 1/2 is agood approximation.
In this paper,w ec onsider the prototypical case of decelerating CaF molecules emitted from ac ryogenic buffer-gas source. The molecules are optically pumped on the A 2 P 1/2 ÀX 2 S + transition which has l = 606 nm, G = 2p 8.3 MHz, and I sat = 5mWcm À2 .Atypical initial speed is v z = 150 ms À1 ,a nd the corresponding kinetic energy is h 1660GHz. The molecules move through an array of permanent magnets that produce ap eak field of ' 1T .T he magnetic dipole moments of the ground states are AE m B ,a nd so the maximum energy that can be removedp er strong-fieldr egion,r eferred to as ad eceleration stage,i sh 28 GHz. The minimum number of stages neededt ob ring the moleculet or est is 60. Using p = 1/2, the average number of photonss cattered by decelerated molecules is 240, about 40 times smaller than using radiation pressure alone. Choosing n = 5g ives an opticalp umping probability of approximately 97 %. Ar easonable laser intensity is 250 mW cm À2 ,corresponding to s = 50, which gives amaximum allowable field gradienta tt he resonancep oints of about 2Tcm À1 .T his sets an approximate scale of about2cm for the periodicity of the decelerator,g iving an overall decelerator length of 1.2 m.

Decelerator Design
Figure2ai llustrates our decelerator design,w hich follows the designp rinciples outlined above.I tc onsistso fa na rray of cylindricalp ermanent magnets whose axes are concentric with the molecular beam axis (z). The magnets alternate between two types of approximate Halbachc ylinders, [47] which are discussed in detail below.T he two types are denoted K = 2a nd K = 6, with angle labels representing an absolute rotationr elative to the global coordinate axes. Each cylinder is 8mmt hick longitudinally with an outer diametero f4 0mma nd an inner diameter of 5mmt hrough which the molecules propagate. A 2mmg ap between the cylinders allows background gas to escape from the inner bore, and only slightly weakens the longitudinal magnetic-field gradient. Constructingt his geometry out of N52 NdFeB wedgem agnets with ar emanent magnetization of 1.44 Tr esults in the magnetic field shown in Figures 2b,c, calculated using finite-elementm ethods. Figure2b shows the magnetic field over as lice through the xz-plane. The contours are lines of equal magnetic field magnitude and the white arrows show the fieldd irection. We see that the K = 2c ylinders produce stronga nd fairly uniform magnetic fields while the K = 6c ylinders provide regions of low magnetic field and transverse guiding of wfs molecules. Note that every other K = 2c ylinder is rotated 1808 to produce strong-field directions that alternate between AEx and fringe fields that cancela tt he longitudinal centers of the K = 6s tages. Without this rotation the fringe fields from the strong-field stages produce an undesirable non-zero field offset in the guiding regions. The orientation of the K = 6s tages is chosen to give approximately equal field gradients in the two transverse directions.F igure2c shows the magnetic field magnitude (solid line), and its gradient (dashedl ine), along the z-axis. The on-axis field magnitude spatially oscillates between 0a nd 1T .T he peak gradient is about 150 Tm À1 ,a nd so the condition on the maximum field gradientd iscussed in Section2 is satisfied everywhere.T his meanst hat the resonance points can be chosen freely.
As mentioned above, the individual cylindrical decelerator stagesc onsist of two types of approximate Halbachc ylinders. In an ideal case, the local magnetization is given by [Eq. (8)]: where M r is the remanentm agnetization amplitude,Î andĴ form al ocal Cartesian basis perpendicular to the cylinder axis, f is the polar angle, and K is the number of rotations made by the local magnetization around ac losed path that encompasses the inner aperture. 1 In general,c hoice of K yields a" 2(KÀ1)"pole field, where the field magnitude in the bore depends on the radius as j B j % r KÀ2 .W eu se K = 2t op roduce ar egion of strong uniform magnetic field, and K = 6t og uide molecules in weak-field seekings tates. For molecules to be pumped from the strong-to the weak-fields eeking state, they must pass through regions of sufficiently small magnetic field that they can come into resonance with L s!w .I ft hey repeatedly fail to do that, they will be lost from the decelerator.T herefore, we would like the guiding magnets to have al arge area where the field is low,a nd steep potential walls that do the guiding. The K = 6c ylinder has this property,w hich is why we choose it. We have found that these large weak-field regionsa re essential for efficient deceleration, as discussed further in Section 5.
In practice, it is difficult to manufacture strong permanent magnetsw ith locally varying magnetization. Instead, we approximate each of the Halbach cylinders using 12 wedges, as showni nF igures 3a,b. The magnetization of each wedge relative to the coordinate axes can be expressed by Equation (8) with the substitution 0 ! 2p W w À 1 2

ÀÁ
,w here there are W discrete wedges labeled by w2{1,…,W}. Choosing W = 12 andr ecognizing the symmetry of the wedgem agnet array,o ne finds that only six unique magnets are required to construct either the K = 2o rK = 6H albach cylinders. These are denoted A-F in Figures3a,b, where as uperscript *i ndicates aw edge has been flipped into the page. The required magnetization directions relative to the radius vector that bisects the wedgea re 158,458,758,1058,1358,a nd 1658 for A-F,r espectively.F igures 3c-f shows the resulting magnetic fields as calculated by Figure 2. Zeeman-Sisyphusd eceleratordesign. a) The magnet geometry consists of as tack of two typesofa pproximate Halbach cylinders, denoted K = 2a nd K = 6, which produce regionso fs trong and weak magnetic field, respectively.b)Aslice of the magnetic field magnitude as calculated by finite-element analysis methods. c) The on-axis field magnitude (solid line) and its gradient (dashed line).
finite-elementm ethods, with the left (right) column showing results for the K = 2( K = 6) cases. The geometry is identical to that of the final decelerator design,d escribed above.A s shown, the K = 2a nd K = 6c ylinders produce the desired field characteristics for strong-field and guiding-field cases, respectively.

Application to CaF
In the rest of this paper we explore the dynamics of molecules traveling through the Zeeman-Sisyphus decelerator,u sing calcium monofluoride (CaF) as ap rototypical molecule. CaF is amenable to laser cooling, [30] with at least two opticalc ycling transitions known, being [38] In strong magnetic fields, both the X and B states have large Zeeman shifts approximately equal to that of af ree electron. Conversely,t he A state has as mallm agnetic momentb ecause the spin and orbital magneticm oments are almost exactly equal and opposite.
In the X and B states the electron spin is uncoupled from all other angular momenta in the large magneticf ields of the decelerator, and the Zeemans ub-levels are characterizedb yM s , the projectiono ft he spin onto the field axis. Since M s cannot change in an electric dipole transition, the optical pumping between strong-and weak-field seeking states cannot be achieved on the B-X transition. Because the spin-orbit interaction of the A state is vastly larger than the Zeeman shift at all relevant fields, the Zeemans ub-levels are characterizedb yM J ,t he projection of the total electronic angular momentum onto the magnetic field axis. These levels are of mixed M s character,a nd so the opticalp umping works well. Because of these features of the A-X transition, the simplified scheme illustrated in Figure 1i sagood representation of deceleratoro peration for this molecule and transition. Figure 4s hows the Zeeman shifts of the relevant states of CaF. [48] The behavioro ft he ground (excited) state is showni n the lower (upper)p lots, with the low-(high-) field regime showno nt he left (right). Figure 4a showst he Zeemans hifts in the A state at low field. At zero field there are two hyperfine levels whose splitting is known to be smaller than 10 MHz. Fol-lowingR ef. [38],w eh ave set this splitting to 4.8 MHz, though the exact value is too small to be of any relevance. There are four magnetic sub-levels labeled by (F,M F )i nw eak fields. In strong fields, they are labeled by (M J ,M I )a nd form aw fs and sfs manifold as shown in Figure 4b.The individual components of each manifold have equal gradients with magnetic field, and they are spacedb ya bout 2MHz. Figure4cs hows the shifts of the X state in lowf ields. This state consists of four hyperfine components labeled by their total angular momentum as F = {1 À ,0,1 + ,2} in ascending energy,w here the AE superscripts act only to distinguish between the two F = 1l evels. These hyperfine levels split into 12 magnetic sublevels, six weak-field seekinga nd six strong-field seeking, each labeled by (F,M F ). These two manifolds play the part of the singlew fs and sfs ground states in the simplified picture of Figure 1. Figure 4d shows how these states shift at high magnetic field. The six levels of each manifold have an early uniform spacing of about 20 MHz, and they have equal gradients with magnetic field which is about 50 times larger than that of the A levels. In this high-field regime, the states are properly labeled by (M S ,M I ,M N ). However,w ec hoose to label each level at all fields according to the (F,M F )s tate it becomes as the field is adiabatically reduced to zero.
We see from Figure 4t hat the pump lasers must address transitions between multiple levels. Since the Zeeman shift is far larger than the splitting between the levels of both the wfs and sfs manifolds, the longitudinally varying magnetic field will bring the varioust ransitions into resonance at slightly different longitudinal positions. This means that, despite the multiple levels, only one laser frequency is neededt oo ptically pump molecules in one direction. However,t he presence of multiple levels is expected to make the opticalp umpingm ore likely to fail. Consider, for example, ag round state molecule in the (2,À1) state enteringar egiono fl arge magnetic field. As the lowest level in the wfsm anifold, this molecule will come into resonance with the pump laser at the most advanced position. The excited state may decay to ad ifferent sublevel of the wfs manifold,and the difference in energy betweenthese two sublevels might be large enough that the molecule is now too far out of resonance with the pump laser to be excited as econd time. This effect can be worsened by the non-zeroZ eeman shift of the excited state sublevels. Specifically,t he new ground sublevel mayp redominantly couple to the opposite Zeemanm anifold in the upper state compared to the initial ground level, taking the molecule even furtherfrom resonance. Because of these multi-level effects, am olecule may pass through the resonance point withoutb eing optically pumped and will continue through as tage of the decelerator in the wrong state. All is not lost, however,b ecause am olecule that fails to be pumped has as econd chance on the opposite side of the potential energy hill. These effects are not captured by the simple modelp resented in Section 2, but are included in the simulations discussed in Section 5. The effects can be mitigated by reducing the magnetic field gradient, increasing the laser power, or addings idebands to the laser to increase its frequency spread.
As econd potentialp roblem for the optical pumping is level crossings with other quantum states not yet considered. For example, the sfs manifold of the N = 1g round state crosses the wfs manifold of the N = 0s tate at af ield of % 0.75 T. Am olecule transferred to N = 0a tt his crossing will be lost from the decelerator since the lasers are tuned to drive the cycling transition from N = 1a nd do not address the N = 0l evels.F ortu-nately,t here is no coupling between these two states because they are of opposite parity and the magnetic field can only couple states of the same parity. An electric field turns the crossingi nto an avoided crossing and so must be kept sufficiently small. The electric field arising from the motion of the molecules through the magneticf ield is too small to be of concern. The situationi ss imilar near 1.5 T, where the wfs N = 1m anifold crosses the sfs N = 2m anifold. The first problematic crossingi sb etween N = 1a nd N = 3, since they have the same parity,b ut this occurs near 2.5 T, which is well above the fields present in the decelerator.
At hird concern for the reliability of the opticalp umping is that other transitions from the N = 1state might come into resonancew ith the laser light andt ransfer molecules out of the cycling transition. In this case, the only such transition is the Q(1) transition, whichi sa pproximately 30 GHz higher in frequencyt han the P(1) cycling transition at zero field. For at ypical choice of detuning, the Q(1) transition comes into resonance with L s!w when the fieldi sa bout 1.75 T. Fortunately, this is highert han the largest field present in the decelerator. We see that, at least for CaF,n oo ther states or transitions play any role in the decelerator and our analysisc an focus solely on the 12 ground states and 4e xcited states shown in Figure 4. This good fortune does not necessarily carry over to other moleculeso fi nterest;as imilar analysiss hould be completed for each case. To understand the optical pumping of the multi-level CaF system in the decelerator,w eh ave calculated the relative transition strengthsb etween each of the ground and excited sublevels for various magnetic field strengths and laser polarizations. Figure 5s hows the transition intensities for excitation out of the ground states (top row) and the branching ratios for the decay of the excited states (bottom row), for magnetic fields of 0( left column), 0.2 (middle column), and 1T (right column). The last two fieldv alues are typical values where the two optical pumpingp rocesseso ccur.I nc alculating the transition intensities we have taken light linearly polarized perpendicular to the strong magnetic-field direction, which is the configuration used in the decelerator.W es ee that the transition intensities and branching ratios change significantly between the zero-and nonzero-field cases,b ut change very little between the two non-zerof ield values. In fact, we find that all branching ratios change by less than 2% in absolutev alue as the field increases beyond0 .03 T. This makes sense in the context of Figures 4a,c where we can see (by extrapolation) that the levels are already grouped into well-spaced wfs and sfs manifolds once the field reaches this value. Since the optical pumpingo ccurs at fields much highert han this, we take the branching ratios and transition intensities to be constants in the numerical simulations presented in Section5.T hischoice is discussed further in the Section 7.
Let us consideri nm ore detail ap articularo pticalp umping event. As our example, we consider am olecule in the sfs state (1 À ,À1) entering ar egion of weak magnetic field and coming into resonancew ith the L s!w laser.T he laser drives, almost exclusively,t he transition to the (1,0)* excited state (see Figure 5b,3 rd column). This state can decay to the wfs states (2,0) or (2,1), each with 33 %p robability (see Figure 5e,2 nd row) ands ot here is a6 6% probability that the molecule switchesb etween the sfs and wfs manifolds after scattering as ingle photon. The excited state can also decay to the sfs states (1 À ,1) or (1 À ,À1), each with 17 %p robability.B oth states remain near resonance with the pump laser,a nd so the mole-cule is likely to be re-excited, again to the (1,0)* state, giving it as econd 66 %c hance of switching between sfs and wfs manifolds. In the notation of Equation (2), p = 2/3, andavalue of n = 4i ss ufficient to ensure c > 0.98. After successful optical pumping, the molecule has a5 0% chance of being in either of the two participating wfs ground states.
As an example of as tate that does not optically pump as efficiently,c onsider am olecule in the sfs (1 À ,0) state under the same conditions. Again, the pumping laser almoste xclusively drives as ingle transition, in this case to (1,1)*. The subsequent spontaneous decay takes the molecule to aw fs state [either (2,0) or (2,2)] only 33 %o ft he time, giving p = 1/3. Them olecule is returned to an sfs state [either (1 À ,1) or the original (1 À ,0) state] with 66 %p robability.T he decay to (1 À ,1) is particularly troublesome, because this state couples only to (1,0)* in the wfs upper manifold, whereas the original( 1 À ,0) state couples only to (1,1)* in the sfs upper manifold.T hus, the resonance condition may be lost due to the Zeeman shift of the excited state. This is ag reater concern for pumping from the wfs to the sfs ground-statem anifolds,s ince that process occursa tl arger fields where the upper-state manifolds are further separated.
Repeating the optical-pumping analysis for each of the twelveg round states reveals that eight of the states have p = 2/3, while the remaining four have p = 1/3. These four all exhibit the behavior described above where af ailure to optically pump may take the molecule to as tate where the optical pumping transition is further from resonance due to the excited-state Zeeman splitting. None of the eight states that pump with high efficiency exhibit this behavior.
Assuming that the optical pumping proceeds with unit probability despite the aforementioned difficulties,the molecular ensemblec ontinually exchanges population between the wfs and sfs manifolds at each resonance point. Details of the population transfer betweent he two ground-state manifolds is summarized in Figure 6, which can be derived by continually propagating the set of ground states through the excitation and decay processes presented in Figure 5. Here, populations pump from initial states indicated by column to final states indicatedb yr ow.T he eights tates that pump efficiently transfer strongly to only two states in the opposite manifold,w hile the four that pump less efficiently are transferred to three states in the opposite manifold.

Trajectory Simulations
We now study the dynamics of CaF molecules in the decelerator in more detail by using trajectory simulations.Asimulation takes as its inputa ni nitial phase-spaced istribution, am ap of the magnetic field calculated using af inite element model, and at able of transition strengths and branching ratios between the ground and excited states,w hich we take to be independento fB as discussed in Section 4. The direction of the magnetic field changes little over the set of positions where the opticalp umping occurs, being purely AEx to ag ood approximation. For all the simulations presented here the pump lasers are linearly polarized alongŷ and the transition strengthsa re independent of whetherB is parallel or anti-parallel tox.I na ll our simulations, the laser intensity profile is assumed to be Gaussianw ith af ull width at half maximum (FWHM) of 5mm, the same as the inner diameter of the decelerator.Weuse linear Zeemanshifts and the hyperfine splittings shown in Figure 4b,d. This assumesthat molecules never experience magnetic fields below about 0.01 T, which is ag ood assumption for nearly all trajectories. The effects of variations in both magnetic field magnitude and direction at the opticalpumpingl ocations are discussed furtheri nS ection 7. The cycling transition for this system,c onsisting of the 12 ground states and 4e xciteds tates, is considered to be closed;t he excited states always decay to one of the 12 ground states. In reality,s omer epumping of population that leaks into v = 1m ay be required.
The simulationp ropagates each molecule through the deceleratoru nder the action of the forceF ¼ ÀrUB ,a nd keeps track of its state as it is optically pumped. During at ime step Dt,t he probability of am olecule initially in ground state i scattering ap hoton via excitation to state j is calculated as [Eq. (9)]: Here, R is given by Equation (3) and T ij is the pre-calculated transition intensity between states i and j.T he total probability of scattering ap hoton duringt his time step is P ¼ P j P j .W e choose the time step so that P ! 1, typically Dt = 10 ns. A random number, r,i ss elected from au niform distribution between 0a nd 1. If r > P,n ot ransition occurs. If r < P,atransition occurs andt he excited state is selected at random according to the relative probabilities P j .T he molecule then decays, with the final ground state selected randomly according to the precalculated branching ratios.T he photon is emitted in ar andom direction chosen from an isotropic distribution. The new position and speed at the end of the time step are then calculated, including the small changes in momentum duet o the absorbed and spontaneously emitted photons. The simulation then proceeds to the next time step.
The initial phase-space distribution used for the simulations starts all molecules at t = 0, z = 0, but with ar ange of initial forward speeds.F or the transverse degrees of freedom, we typically use ad istribution that is uniform in the range from AE 2.5 mm and AE 7.5 ms À1 for both transverse dimensions.T his range is larger than the decelerator can accept, and so most molecules are lost via collisions with the inner magnet surfaces in the first % 25 cm of the decelerator.B yo verfilling the transverse phase space in this way,w ee nsure that the molecular distributionsa tt he exit of the deceleratora re indicative of the deceleration dynamics and not the particular choice of initial conditions. Combined, the initial transverse and longitudinal phase-space extents of the molecular distribution do an acceptable job of simulating molecules with not only differing forwards peeds but also differently directed initial velocityv ectors.

Guiding Performance
We first turn off the opticalp umping light and study the performance of the magnet array as ag uide for molecules in wfs states. This is useful in identifying dynamical instabilities that arise from the coupling of longitudinal andt ransverse motions, and helps to elucidate why molecules are lost as they are decelerated. [49] Figure 7s hows the relative number of molecules reaching the end of the magnet array as af unction of their initial forward speed. Molecules with v z 14 ms À1 have insufficient kinetic energy to climb over as ingle potentialenergy hill, so this sets al ower limit to the speed am olecule can have in order to reach the exit. Above 100 ms À1 ,t he number guided does not depend strongly on the speed, but below 100 ms À1 the number that reach the end falls off rapidly with decreasing speed. This is because there are stronger guiding forces in the guidingm agnets than in the strong-fieldm agnets. The slow molecules are guided too strongly by the guiding magnets and are then lost in the strong-field magnets where the guiding is weak. Moreover,t he modulation of the transverse guiding can couple energy from the longitudinalm otion into the transverse motion,c ausing further loss. These effects set in once v z 4 L/T,w here L = 2cmi st he spatialp eriodicity of the magneta rray and T is the transverse oscillation period.T he guide is not harmonic, so there is ar ange of oscillation periods, but T % 1msi st ypical. Thus,w ee xpect the losses to set in when v z % 80 ms À1 ,w hichi sr oughlyw hat we observe in the simulations. Because of this loss mechanism,iti sadvantageous to decelerate the molecules as rapidly as possible once they reach low speed. Fortuitously,t he highest optical-pumping efficiency,a nd therefore the highest deceleration, is naturally realized by the slowestm olecules. We note that the low-speed stabilityc an be improved by reducing the spatial periodicity (L)n ear the end of the decelerator,o ri ncreasing the bore size of the magnetsn ear the end so that the oscillationp eriod (T) increases.
The inset to Figure 7s hows the transverse phase-space distribution of molecules that exit the guide. The spatiale xtent of AE 2.5 mm is set by the bore diameter of the magnets, and the velocitys pread of AE 6ms À1 is set by the energetic depth of the guide.   Figure 8f ollowss ome molecules,a ll with initial velocities around1 50 ms À1 ,a st hey propagate through the decelerator. Here, the pump laser powers are 200 mW with detuningso f D w!s = 13.75 GHz and D s!w = 2.5GHz. In Figure 8a,q uantumstate trackingf or three molecules is shown for the first few cm of the decelerator,w ith the solid vertical lines representing strong-field regions near which molecules should optically pump from wfs states (red) to sfs states (blue). Transitions from sfs to wfs states shouldo ccur in the weak-field regions between the vertical solid lines. In most cases,t he optical pumping is successful. At z ' 28 mm there is an example of afailure to switch. The molecule in the wfs state (2,1) is excited at this positionb ut decays to the (2,0) state which is another wfs state. The molecule is not excited as econd time, and so travels through as tage of the decelerator in the wrong state. Another interesting example occurs near z = 50 mm. The molecule in the wfs state (2,À1) first switches to the wfs state (1 + ,1), then back to (2,À1) before finally being pumped to the sfs state (1 + ,À1). Figure 8b shows the molecules approximately following the vÀz curvese xpected for ac onstant deceleration. Occasional failures to optically pump can be seen as horizontal propagationw ith no net change in forward velocity. The inset follows as ingle molecule over as hort region of the decelerator,s howingf ive switches between wfs ands fs states and the associated deceleration. Figure8cs hows the distribution of forwardv elocities at variousl ongitudinal positions with the initial distribution divided down for easier comparison. The velocityd istribution spreadsa sm olecules are decelerated, in accordance with the dynamics of constantd eceleration from variousi nitial velocities, as discussed in Section 2. Approximately 10 %o ft he population present at 50 cm, with am ean speed of 130 ms À1 ,s uccessfully propagates to 150 cm where the mean speed has been reduced to 55 ms À1 .F igure 8d shows some molecular trajectories in the xz-plane. Wes ee that some molecules transversely oscillatet hrough the decelerator on stable trajectories, showing the effect of the transverse guiding.S ome hit the walls at x = AE 2.5 mm before reaching the end of the decelerator, while others come to rest before they get to the decelerator'se xit. Both cases are indicated by at rajectory abruptly ending. From the simulation, we find that the average number of photons scattered by molecules that reach z = 150 cm is 225, corresponding to 1.55 photons per resonance point.T his is near the expected value for p = 2/3, indicatingt hat the few states which pump less efficiently do not play al arge role in the deceleration dynamics.

Deceleration Performance
As discussed in Section 2, setting D s!w close to zero gives the maximum deceleration butr educes the transverse stability because molecules in sfs states may never reach low enough fields to come into resonance with L s!w .F igure 9e xplores this effect. Here, we use the same simulations ettings as before, except that the decelerator length is fixed at 1m and D s!w is varied. Figure9as hows how D s!w influences the final velocity distribution. As expected, bringing L s!w closer to zero reduces the final averages peed, but also reduces the number of molecules at the exit of the decelerator.F igure 9b shows the final phase-space distribution, in one transverse dimension, of those molecules that successfully exit the decelerator,f or two different values of D s!w .I nd eceleratort erminology, the set of stable molecule positions and velocities defines the transverse phase-space acceptance, though in this case the concept is less well defined because the stochastic nature of the optical pumping meanst hat am olecule can be lost even if it appears to be well inside the acceptance region.
To understand in more detail why the acceptance decreases with decreasing D s!w ,c onsider the magnetic fields experienced by molecules at various distances from the decelerator axis. The on-axis field varies between 0a nd 1T ,b ut further away from the axis the field does not reach low values. There will be some radius where the minimum field is above that required to bring molecules into resonance with L s!w at all longitudinalp ositions. Beyond this radius, opticalp umping out of the sfs states fails, and the molecules stuck in sfs states are anti-guided andl ost. For our magnet geometry and D s!w = 3.5 GHz, optical pumping should cease beyondaradius of 1mm, consistent with the observed region of transverse stability shown. Figure 9c plots the deceleration and the phasespace acceptance in one transverse direction, both as functions of D s!w .T he deceleration is determined using Equation (1). The acceptance is determined in an approximate way by calculating the area of an ellipse that encloses9 0% of the transverse phase-space distribution of molecules exiting the decelerator.W es ee that there is am odest reduction in deceleration as D s!w is increased from 0.5 to 5GHz, but av ery large increasei nt he transverse phase-space acceptance.U nless there is as trong penalty for having al onger decelerator, it is best to keep D s!w relatively large to give the largestd ecelerated flux.

Prospects for MOT Loadingand Comparisonw ith Other Deceleration Methods
An attractive application of Zeeman-Sisyphus decelerationi s the productiono fs low molecules for loading into am agnetoopticaltrap (MOT). We can use our trajectory simulations to estimate how many molecules mightb el oaded using this technique, and comparet he result to other deceleration methods. There are many available deceleration techniques, and they depend strongly on the choice of molecule and molecule source.W el imit our discussiont od eceleration of CaF molecules in the N = 1s tate, emitted by ac ryogenic buffer-gas source.U nless otherwise noted, we assume aC aF beam with 10 11 molecules per steradian per shot in N = 1, and ad istribution of forward speeds approximatedb yaG aussian distribution with a150 ms À1 mean and aFWHM of 93 ms À1 . [31] The distance from source to detector will be fixed at 1.3 m. In all comparisons, we assume a1 0cmf ree-flight distance between the source and the location where deceleration begins, as is typically neededd ue to geometric or pressure constraints. The detector area is taken to be ac ircle of diameter 5mm, whichi s equal to the diameter of the decelerator aperture described in this paper.
With the Zeeman-Sisyphus decelerator design presented above,m olecules with speeds below 14 ms À1 cannot get over the final potential hill and are either lost transversely or trapped by the magnetic field. This could be mitigated by using aw eaker magnetic field near the deceleratore xit. Instead, we take as our figure of merit the number of molecules exiting the decelerator below 50 ms À1 ,s ince this is approaching the MOT capturevelocity andslow enough that ashort distance of direct laser slowing could then be used to load the MOT. Figure 10 compares the simulated velocity distribution exiting a1 .2 ml ong Zeeman-Sisyphus decelerator, to the simulated distribution detected without the decelerator present. The decelerator parameters are identicalt ot hose described in Section 5.2 except the optical pumpingl aser detunings have been set to {D w!s , D s!w } = {À14.25, 3} GHz. This choice ensures molecules are optically pumped right at the potential-energy hill-top for wfs states. As expected, the initially widei nput velocity distribution from the buffer-gas source is both shiftedd own in velocitya nd broadened by the presence of the decelerator. The effects of net guiding are also apparent;d espite being slowed down, more molecules reach the detector when the beam is decelerated. Also as expected, the distribution approaches zero at the velocity equivalentt ot he potentialenergy hill height ( % 14 ms À1 ), thought here are al arge number of molecules with velocities just above this limit. In free flight, approximately 10 6 moleculesper pulse pass through the detection area. When the beam is decelerated, this number increases by af actor of 2.1, and 15 %o ft hese have forwards peeds below 50 ms À1 .T hus, the deceleratorp roduces about 3 10 5 molecules in our chosen velocity range.
Let us compare this to the direct laser slowing results presentedi nR ef. [31],w here the beam source andd istance to the detectora re the same. In this comparison the free-flight curves presented in Ref. [31] are identical to Figure 10, and represent the same absolute number of detected molecules. In these experiments, about 2 10 5 molecules are slowed to speeds below 50 ms À1 .T his is similar to the result above,i ndicating that Zeeman-Sisyphus deceleration is competitive with stateof-the-art direct laser slowing techniques. The number of photons that have to be scattered is about % 10 4 for directl aser slowing but only about 300 for Zeeman-Sisyphusd eceleration. This makes the decelerator ap articularly attractive option for decelerating molecules where direct laser slowing may be impractical because the branching ratios are less favorable than for CaF.
Both the traditional [15] and traveling-wave [50] Stark deceleration methods are also capable of slowingm olecules into the velocityr ange of interest when startingw ith our beam parameters and using the same deceleration distance. These methods are typically not well suited for deceleration of buffer-gascooled molecularb eams due to the typically long (1-10 ms) molecular pulses.O ur source is unusual because it produces ap articularly short pulse, approximately2 50 msF WHM, making these time-dependent deceleration methods feasible.T oe stimate the number of slow molecules that could be produced, we determine how many molecules from the initial distribution are within the longitudinal phase-space acceptance of the deceleratorw hen it is turned on. For the N = 1s tate of CaF,t he maximum electric field that can be appliedi sa pproximately 30 kV cm À1 ,w hich is al imitation in the traditional decelerator geometry.I ti sn ot al imitation for the traveling-wave decelerator,w hich by design uses smaller peak electric fields. An acceleration of À8.3 10 3 ms À2 is sufficient to decelerate molecules from 150 ms À1 to 50 ms À1 in 1.2 m, corresponding to as ynchronous molecule phase angle of 24.58 for the Stark decelerator.W ec alculate longitudinalp hase-spacea cceptances of 65 and 16 mm ms À1 for the traditional and traveling-wave decelerator,r espectively.T he % 3t imes larger solid angle subtended by the traveling-wave decelerator makes up most of the difference, ands ob oth methods yield roughly the same number of slow molecules, approximately 3 10 5 ,spread over 10 potential wells. We note that this simple one-dimensional estimate is optimisticf or the traditional Starkd ecelerator, as it neglects coupling between longitudinal and transverse motions and other loss mechanisms at slow forwards peeds, [49] but it should be relativelya ccurate for the traveling-wave case. Intriguingly,t he results are comparablew ith both Zeeman-Sisyphus deceleration and direct laser cooling, thoughw es tress again that the short pulse produced by our source is crucial for obtaining such high numbersaccepted into the Stark decelerator(s).
Finally,w ec onsider direct laser slowing from a" two-stage" buffer-gas cell. [51] Relative to the molecular beam from as ingle-stage source, two-stage sourcesp roduce slower beams at the expense of molecular flux. In Ref. [52] the authors report ab eam of 10 9 molecules/steradian/shot, some two orders of magnitude lower than as ingle-stage beam, but with the mean velocityr educed to about60ms À1 .Using direct laser slowing over a5 0cml ength, the authors show that about 20 %o ft he beam can be slowed below 50 ms À1 ,c orresponding to about 1.5 10 4 slow molecules passing through a5mm diameter detector located 50 cm from the source.T his number is less than the estimates above, but improvements to the molecular beam source,t he effectiveness of the laser cooling, or as hortening of the source-to-detector distance, could produce significantly more slow molecules.
To summarize,w es ee that an umber of techniques can slow CaF moleculest ol ow velocities, and that they can have similar efficiencies. Other options not evaluated here, but certainly worthy of consideration, are the Zeeman [16] and centrifuge [18] deceleration methods. The Zeeman-Sisyphus decelerator is competitive with other methodsi nt erms of efficiency,d oes not require the exceptional branching ratios neededf or direct laser slowing, andw orks with long, or even continuous molecular pulses that are not well suited to time-dependent deceleration methods such as used in Stark or Zeemand ecelerators.

Simultaneous Slowing and Cooling
As described in Section2,t he spread of longitudinal velocities increases as the molecules are decelerated. This is an atural consequence of ac onstant deceleration over af ixed decelerator length. The simulations reveal that the spread of velocities actually increasesm ore rapidly than expected from this simple picture,e specially for low laser powers. This is because the optical pumping efficiency is greater for the slower molecules, which spend more time near the resonance points [n in Eq. (7) scales as 1/v z ], ands ot he mean deceleration is larger fors low molecules than for fast ones. In addition, because of the Doppler shift of the counter-propagating light, slower molecules must climb further up the potentiale nergy hills to come into resonance. Again, this resultsi ns lower molecules experiencing more average deceleration.I nt his section, we consider some alterations to the design of the decelerator that can minimize, or even reverse, the spread of velocities. In this way,w ea im to cool and decelerate the molecules simultaneously.
We first consider how to use the Doppler shift to introduce an on-monotonic velocity-dependentc omponent to the force. To achieve this, we detune L w!s above the potential-energy hilltop sot hat the fast molecules are Doppler-shifted into resonance at the hilltop and are optically pumped with high probability,b ut the slower ones are not. We also find it necessary to reduce the L w!s power to just af ew mW,s ot hat the velocity-dependent effect is not washedo ut by powerb roadening. Fortunately,t hose molecules that fail to optically pump are left in weak-field seeking states and so are still guidedt hrough the decelerator.T his means that the transverse stability is not adversely affected, thought he decelerator does need to be made longerb ecause of the frequent opticalp umping failures.
We introduce the quantity D top ,t he detuning of the light from resonance for as tationary molecule at the top of the potential hill. Figure 11 shows the transition probability, c,a s af unction of v z for 1mWo fp ower and two choices of detuning, D top = 0a nd À350 MHz. The value of c is calculated by integrating the scattering rate as am olecule climbso ver the top of the hill. The dashed line shows the monotonic velocity-dependence of c,a nd therefore also the force, in the case where L w!s is tuned to the potentiale nergy hilltop, that is, D top = 0. By contrast, when D top = À350 MHz (solidl ine), the optical pumping efficiency and associated force is largerf or faster molecules, as desired. At very low velocities, the transition probability againi ncreasesa sm olecules spend al ong time near the top of the potential energy hill. Figure 11. Detuning the L w!s pumpl aser abovet he potential-energy hilltop (at 14 GHz)and lowering the pump laser power( to 1mWi nthis case)creates an on-monotonic velocity-selective transition probability and corresponding deceleration. We would like to compensatet he changing Doppler shift as the molecules slow down, by changing D top .W ew ish to maintain the time-independence of the deceleration process, so instead of chirping the laser frequency in time, we introduce as caling of the magnetic fields ot hat the magnitude at the hilltops increases with z.I nt his way, the solid curve shown in Figure 11 will be swept inwards towards lower velocities, bunching molecules in velocitya st hey proceed through the decelerator.T he faster molecules are Doppler-shifted into resonance with the light throughout the decelerator,w hile the slower ones join the deceleration processl ater on. This is similar to the traditional Zeemanslower for atoms. [53] This mechanism of velocity compression is inhibited by any effect that makes D top inhomogeneous. Thisi ncludes the hyperfine structure, the different Zeeman shifts of molecules at differentradial positions at the hilltop, and the Zeeman shift of the excited state. The last of these is relevant because some of the ground-states ublevels couple only to the wfs manifold of the excited state, while others couple only to the sfs manifold. We find that for CaF,t he upper state Zeeman splitting is the biggest concern, being % 600 MHz at the 1T hilltops, which is 3.6 times larger than the Doppler shift of a1 00 ms À1 molecule. Ap ossible solution to this problem is to couple togethert he two excited state manifoldsu sing an rf magnetic field tuned to the Zeemans plitting of the excited state at the hilltop. In this way,a ll the lower levels can couple to the lowest energy manifold of the excited state and the problem of the excited state Zeeman splitting is eliminated. The field uniformity at the hilltop is also ac oncern, thought his could easily be improved with somem inor adjustments to the wedge magneta rray. [54] To investigate the basic mechanism of the velocityc ompression without these complications, we set the upper state Zeeman shift to zero and limit the initial transverse distribution to be 1mm( FWHM)a nd 1ms À1 (FWHM). Figure 12 shows how molecules with ab road range of initial speeds propagate through the refined decelerator for four different conditions. In Figure12a, L w!s is detuned to bring wfs molecules into resonance right at the potential energyh illtop (D top = 0). The velocitys pread increases enormously as the molecules propagate through the decelerator.T his is the same effect seen in Figure 8b but amplified by the lower L w!s power,which is only 3mW, and the broader initial velocity distribution for these simulations. In Figure 12 b, L w!s is detuned above the hilltop (D top = À300 MHz). In this case, the transition probability resembles the solid curve in Figure 11 and the molecules are not decelerated efficiently.F igure 12 ci s identicale xcept that the magnetic field amplitude is multiplied by the scalingf actor 1 + 0.001z 2 .T his brings the fastestm olecules into the slowing cycle before the slower ones. We see that this strategy counteracts the increase in the velocity spread,e ven slightly reducing it. Figure12d shows this strategy again, but with the molecules restricted to the decelerator axis. With all molecules experiencing the same magnetic field at the hilltop, the effects are much clearer.F ast molecules decelerate more efficiently,w hile slow ones do not decelerate until the magnetic field scaling brings them into resonance with the pump light. The result is as ubstantial compression of the longitudinal velocity distribution during the deceleration process. Similar results should be attainable without the restriction to the decelerator axis, by improving the field uniformity in the strong-field regions.

Justification of Simplifying Assumptions Made in the Trajectory Simulations
Our trajectory simulations assume that the transition strengths and branchingr atios between the various sublevels are constant throughout the decelerator.T ot est this assumption, we calculated the state couplings over aw ide range of magnetic field magnitudes and found that they are nearly constant for all fields above about 30 mT.W hile the branching ratios depend only on the field magnitude, the excitations trengths also dependo nt he direction of the magnetic field relative to the laser polarization.T he opticalp umping from wfs to sfs states occurs in the strong field region where the fieldd irection is uniform. However,p umping from sfs to wfs states occurs in relatively low fields ( % 0.2 T) where the magnetic field direction is more variable. To explore this, we used the trajectory simulations to record the magnitude and direction of the local magnetic field each time am olecule scattered ap hoton. We found that the fieldd irection is fairly uniform even for pumping from sfs to wfs states. Specifically,t he magnetic field at the resonant points is restricted to the xy-plane, and is centered on the AEx axis (the strong-field directions) with an angular extent of AE 458.C alculating the transition intensities over this range of magnetic field directions, with the laser polarization fixed alongŷ,r esults in variations of only af ew percent, justifying our approximation of constant couplings. The numerical values fort he transition intensitiesa nd branching ratios used in the simulations appear in Tables 1  and 2, respectively.
In addition to assuming constant transitions trengths and branching ratios everywhere in the decelerator, the simulations also assume linear Zeemans hifts for all molecular states of interest. This approximation holds as long as molecules do not experience spatialr egions where the magnetic field strength is below 100 mT or so, as showni nF igure 4. According to the finite-elementm odel of the magnetic fields presenti nt he decelerator,t he field amplitude only drops below this value within % 10 mmo ft he (on-axis) K = 6g uiding-stage centers. These regions constitute only af ew parts per billion of the internald ecelerator volumew here the molecules propagate and thus this assumption holds for nearly all molecular trajectories.

Conclusions
We have discussed in detail the principles and design of aZ eeman-Sisyphus decelerator and presenteds everala dvantages over other methods. Because it is time-independent, it is applicable to continuous beams or long molecular pulses such as those typically emitted by cryogenic buffer gas sources. A molecule such as CaF,e mittedf rom such as ource,c an be broughtt or est after scattering just af ew hundred photons. It follows thatm olecules whose vibrational branching ratios are not so favorable for direct laser cooling couldstill be decelerated using this technique,w ithout needing too many repump lasers.W ith our magnetic field design, molecules are simultaneously guided and decelerated. This is an advantage over direct laser slowing where many molecules are lost due to the ever-increasing divergence of the slowed beam. Ours imulations suggest that, for CaF,t he efficiency of Zeeman-Sisyphus deceleration is comparable to direct laser slowing. For heavier molecules, or those where the photon scattering rate is lower, the slowing requires al ongerd istance and beam divergence can be particularly problematic. In thesec ases, the decelerator may provide ab etter way to load molecular MOTs. The decelerator uses only static magnetic fields and should be relatively straightforward to construct using readily availablep ermanent magnets.
Our simulations with CaF use the real level structure, Zeeman shifts and transition intensities in the molecule, the full 3D magnetic field map of arealistic magnet array,and ar ealistic laser intensity distribution. These details introduces ome subtle andi mportant effects, but the deceleration dynamics remains imilar to those expected from the very simple model presented in Figure 1. While we have analyzed only the case of Table 1. Transition intensities used in the simulations, calculated in a1Tm agnetic field with linearly polarized light polarized 908 with respect to the strong-field direction.V alues less than 1 10 À4 are shown as zero.
( CaF in detail, it seems likely that the methodw ill be applicable to aw ide range of molecules. In Section4,w ei dentified some potentialp itfallst hat are not problematic for CaF but might be for other species, and we recommend an analysiso ft he particular level structure and state couplings involved in the optical pumpingt ransitionfor each case of interest. Heavier diatomics, for example, usually have smaller rotational splittings. This can be ap articular concern if the g-factor in the excited state is not small;o ther transitions out of X(N = 1) coming into resonance in the magnetic fields presenti nt he decelerator may violate the cycling transition requirement and necessitate repump lasers to re-introducel eaked molecules back into the opticalpumping cycle.
We have shownt hat, with some refinements, the Zeeman-Sisyphus decelerator could compress the velocity distribution of the molecular beam during deceleration.T hatw ould make it an especially powerful new tool for producing cooledm olecular beams at low speed.
Data underlying this article can be accessed from Ref. [55].