A robust, high-flux source of laser-cooled ytterbium atoms
E. Wodey, R. J. Rengelink, C. Meiners, E. M. Rasel, D. Schlippert
AA robust, high-flux source of laser-cooled ytterbium atoms
E. Wodey, R. J. Rengelink, C. Meiners, E. M. Rasel, and D. Schlippert
Leibniz Universit¨at Hannover, Institut f¨ur Quantenoptik, Welfengarten 1, 30167 Hannover, Germany (Dated: August 18, 2020)We present a high-flux source of cold ytterbium atoms that is robust, lightweight and low-maintenance. Our apparatus delivers 1 × atoms/s into a 3D magneto-optical trap withoutrequiring water-cooling or high current power supplies. We achieve this by employing a Zeemanslower and a 2D magneto-optical trap fully based on permanent magnets in Halbach configurations.This strategy minimizes mechanical complexity, stray magnetic fields, and heat production whilerequiring little to no maintenance, making it applicable to both embedded systems that seek tominimize electrical power consumption, and large scale experiments to reduce the complexity oftheir subsystems. I. INTRODUCTION
Neutral cold atoms are a versatile resource for manydomains of modern physics. They lie at the core ofmodern quantum engineering, realizing quantum simula-tors [1] and proposals for quantum computers [2]. Theyalso constitute a key element of state-of-the-art quan-tum metrology, producing the most stable frequency ref-erences [3], contributing to the determination of funda-mental constants [4, 5], searching for new physics [6],and enabling inertial sensing with unprecented stability[7, 8]. Furthermore, owing to increased robustness andcompactness of cold-atoms technology [9–12], even verylarge scale experiments [13–15] and eminently challeng-ing environments are accessible, including zero-g aircrafts[16], sounding rockets [17], and orbiting spacecrafts [18].These developments are paving the way for long-term op-eration in space [19] and a new era for quantum metrol-ogy [20–23].Light-pulse atom interferometry [24] is instrumental inmany of these applications. It relies on the precise controlover the phase of matter waves and is able to exploit thesimultaneous interrogation of a large number of atoms todeliver accurate and stable measurements. Since eachmeasurement destroys the atomic sample, minimizingaliasing and dead-time effects while preserving low noisecharacteristics for applications in metrology [25] and iner-tial navigation [26, 27] requires reliable high-flux sourcesof cold atoms. In addition, setups of all extents, butespecially large-scale and transportable devices benefittremendously from low maintenance, robust, low powerconsumption strategies for the initial atomic cooling stepsin order to increase up-time and overall reliability.Alkali atoms, mainly rubidium and cesium, are usedto achieve the current state of the art in atom interfer-ometry. Recently, however, ytterbium and other alkaline-earth-like atoms gained interest from the community dueto their low magnetic susceptibility in the ground stateand richer electronic structure. In particular, the avail-ability of long-lived metastable states allows for novelcoherent matterwave manipulation schemes [28–30] andhas implications for gravitational wave detection [31]. Inaddition, these elements constitute interesting choices for D - M O T D - M O T Zeeman slowerOven c m Slowingbeam35 cm
FIG. 1. Overview of our cold ytterbium apparatus. An atomicbeam is produced from metallic ytterbium chunks heated inan oven with a microchannel nozzle. The combination of aZeeman slower and 2D-MOT slows down, redirects, and rec-ollimates the atomic beam for delivery in a 3D-MOT. tests of the universality of free fall [32].Alkaline-earth-like atoms have far lower vapor pres-sures compared to alkali atoms. This requires hot ovensto obtain sufficient content in the gaseous phase to ini-tiate laser cooling. The correspondingly large atomicvelocities at the exit of the oven prompted the use ofZeeman slowers [33] or, more recently, 2D magneto-optical traps [34] to enable efficient loading into a three-dimensional trap. In their conventional implementation,however, these techniques are power consuming and usu-ally lead to bulky and heavy setups as they feature high-power electrical circuits which require water cooling andassociated maintenance.We present a robust, lightweight and low-maintenancesource of slow ytterbium atoms delivering 1 × at/sinitial loading rate into a three-dimensional magneto-optical trap (3D-MOT), comparable to other state-of-the-art strontium [35] and ytterbium [36] systems. Adistinctive feature of our setup is the use of permanentmagnets to generate all the required magnetic fields, thusreducing the weight and complexity of the setup whilealso requiring less electrical power and no water-coolingfor operation.Our apparatus is depicted in figure 1. We describeand characterize its four distinct elements in the follow- a r X i v : . [ phy s i c s . a t o m - ph ] A ug ing sections. First, a hot atomic beam is produced frommetallic ytterbium chunks in an oven terminated by a mi-crochannel nozzle (sec. II). The mean atomic velocity inthe beam is then reduced from around 300 m/s to 20 m/sin a Zeeman slower made of permanent magnets in a Hal-bach configuration (sec. III). A 2D-MOT (sec. IV) servesas a deflection and recollimation stage, preventing theresidual fast atoms from entering the main experimentalchamber without introducing in-vacuum moveable parts,thus limiting failure points. Finally, the slowed atomsare captured in a 3D-MOT, demonstrating the usabilityand performance of our apparatus (sec. V). II. OVEN WITH MICROCHANNEL NOZZLEA. Design
Figure 2 shows the design of our ytterbium oven.About 5 g of solid metallic chunks (natural isotopic abun-dance, approximate chunk volume 10 mm ) are vapor-ized in a cylindrical vacuum chamber with 19 mm in-ner diameter. This crucible is connected to the oven’snozzle piece via CF (ConFlat) flanges. The vacuum an-nealed oxygen-free copper gasket sealing the connectionis silver-plated to avoid corrosion by the hot ytterbiumvapor. The nozzle is made of N tubes = 104 microchan-nels assembled in a quasi-hexagonal lattice constrained ina triangular holder (inset of fig. 2). The microchannelsare AISI 316L stainless-steel capillaries of inner diame-ter 2 a = 280 µ m and outer diameter 320 µ m cut to length L = 12 mm. The triangular prism holder allows for an al-most defect-free lattice, minimizing apertures larger thanthe microchannel diameter 2 a [37]. The microchannelsare stacked inside the holder and clamped from one faceof the prism. However, machining the prism’s edge op-posing the clamping face with conventional milling tech-niques leads to a non-negligible finite radius of curvature.To properly constrain the microchannel array, we inserta (cid:31) . . ◦ C and the nozzle piecearound 40 ◦ C higher to avoid clogging of the microchan-nels. A major loss channel is thermal conduction throughthe vacuum chamber. We measured temperatures up to100 ◦ C around 100 mm downstream of the nozzle part.The atomic beam extracted from the vapor throughthe nozzle travels towards the Zeeman slower in a vacuumof approximately 1 × − mbar maintained by a 30 L/sion pump. Before entering the Zeeman slower, the beamtraverses an area offering a 38 mm-diameter optical ac- CrucibleT oven ∼ ° C mm NozzleT nozzle ∼ ° C
75 mm34 mm (cid:31)
16 mm Laser beam λ =
399 nmIon pump, photodiode θ max = . ° to Zeeman slower FIG. 2. Sketch of the ytterbium beam apparatus. Solid yt-terbium chunks are heated to T oven = 490 ◦ C to form partialytterbium vapor at 2 × − mbar using circular band heaters(depicted in red). A directed beam is extracted by a noz-zle made of 104 large length-to-diameter ratio tubes (inset:photograph of the microchannel stackup with homogeneousbacklight). The properties of the atomic beam (flux, diver-gence) are measured around 100 mm downstream the nozzleby means of laser absorption spectroscopy near the S → P resonance at 399 nm. cess on an axis perpendicular to the atomic beam. Thisenables the use of laser absorption spectroscopy to char-acterize the flux and divergence of the atomic beam. B. Ytterbium atomic beam
We determine the divergence and flux of the atomicbeam shaped by the microchannel nozzle using laser ab-sorption spectroscopy around the S → P resonanceat 399 nm (figure 2, inset figure 3). We use a low sat-uration ( s = I / I sat = 0 .
3) laser probe of / e diameter2 r b = 7 . Yb,
Yb,
Yb,
Yb) and fermionic (
Yb ( F (cid:48) = / , F (cid:48) = / ), Yb ( F (cid:48) = / , F (cid:48) = / , F (cid:48) = / )) ytterbium [38]. Thestable boson Yb cannot be resolved due its low naturalabundance (0.3 %) and signal-to-noise ratio limitations inthis setup.To obtain quantitative information about the atomicbeam, we adjust the sum of nine Lorentzian profiles offull width at half maximum Γ + 2˜ σ to the spectrum offigure 3. Γ = (2 π ) ·
29 MHz is the natural width ofthe S → P electronic transition and ˜ σ quantifies thespectral broadening that results from residual velocitycomponents along the probe laser beam. We found aLorentzian shape to fit the data better than a Gaus-sian or Voigt profile. In the adjustment, the spectralpositions of the features as well as their relative ampli-tudes are fixed, respectively set from tabulated values[39], and calculated from the isotopic natural abundancesand standard L - S coupling theory [40]. This leaves onlyfive parameters free for the adjustment: the Dopplerbroadening ˜ σ , a global amplitude quantifying the at-tenuation of the laser probe by the atomic beam, andthree technical parameters that are not physically rele-vant (global offsets in signal and frequency ranges andfrequency ramp calibration). For the spectrum in fig-ure 3, we find ˜ σ = (2 π ) ·
53 MHz which corresponds to atransverse velocity ˜ v t = 21 m / s or a half-opening angle˜ θ / = 69 mrad.We estimate the atomic flux for all isotopes by inte-grating absorption spectra d ( ν ) like the one in figure 3.Normalizing by the spectrally-integrated scattering cross-section gives the linear density of atoms along the ab-sorption column. Multiplying the column density by theabsorption surface πr b leads to the instantaneous num-ber of atoms N atoms in the overlap volume of the laserand atomic beams: N obs = (cid:82) ∞−∞ d ( ν ) d ν (cid:82) ∞−∞ σ (2 πν ) / Γ ] d ν × πr b (1)where r b is the probe beam’s / e radius and σ = λ / (2 π ) ≈ . µ m is the on-resonance scattering crosssection [41]. For a probe beam perpendicular to theatomic beam, the transit time of atoms with longitudinalvelocity ¯ v is on the order of r b / ¯ v ≈ µ s. If the probebeam is large enough to intersect the entire atomic beam,the total flux of atoms emerging from the oven is˙ N = N obs × ¯ v r b (2)where we estimate ¯ v by averaging over a Maxwell-Boltzmann distribution at the crucible temperature T oven .Figure 4 shows the variation of flux with the crucibletemperature T oven . At our typical operation tempera-ture of 490 ◦ C, the flux of ytterbium atoms passing thespectroscopy zone exceeds 2 × at/s. C. Discussion
The fit to the spectrum in figure 3 implies that theatomic beam diverges with a half-angle ˜ θ / ≈
70 mrad,much larger than the angle predicted from the collision-less theory for the microchannel nozzle [42, 43] θ / = . a / L ≈
20 mrad. This suggests a non-negligible influ-ence of interatomic collisions in the nozzle, as evidencedby the mean free path calculation below.We evaluate the vapor pressure of ytterbium at T oven =490 ◦ C to 2 × − mbar using a Clausius-Clapeyron-typelaw and tabulated values for the sublimation latent heat[44]. Assuming an ideal gas, the corresponding den-sity is n ≈ . × at / m and the mean velocity¯ v ≈
300 m / s. The molecular-flow conductance of thenozzle piece is only 0.02 L/s. This supports five orders ofmagnitude pressure difference between the entrance andthe exit of the microchannels. We assume a constant − , −
500 0 500 1 ,
000 1 ,
500 2 , . . . . Yb Yb (5 / Yb Yb Yb (3 / Yb (7 / Yb (3 / Yb (1 / Yb Yb Frequency ν (MHz) O p t i c a l d e n s i t y d MeasuredFit
FIG. 3. Laser absorption spectrum recorded around the S → P resonance at 399 nm in atomic ytterbium. Thelaser probe beam is perpendicular to the atomic beam (seefigure 2), such that the spectral width results from the natu-ral (2 π · Γ) and transverse Doppler (2 π · σ ) widths. We identifysix main features, corresponding to nine electronic transitions.For fermions, the number in parentheses indicates the totalatomic angular momentum F in the excited state. The boson Yb is not resolved due to its low natural abundance.
Inset :simplified energy level diagram for bosonic atomic ytterbium.
400 430 460 490 52010 × at/s at 490 ◦ CCrucible temperature T oven ( ◦ C) T o t a l flu x ˙ N ( a t / s ) Measured Transparent Opaque . . · ¯Λ < L ¯Λ > L p ˙ N (cid:16)p at / s (cid:17) ˜ θ / ( m r a d )
450 470 490 510 T oven ( ◦ C) FIG. 4. Flux of ytterbium atoms ˙ N emerging from the mi-crochannel nozzle versus crucible temperature T oven . Exper-imental points are derived from absorption spectra applyingequation (2). The solid line corresponds to equation (4),truncated at θ max (see figure 2). The dotted line continuesthat model for temperatures above 485 ◦ C where the averagemean free path ¯Λ (equation 3) is smaller than the microchan-nel length L . The dashed curve corresponds to the opaquesource model. Inset : onset of the interacting regime. When¯Λ (cid:46) L , the spectral half-width ˜ θ / scales linearly with thesquare-root of the measured flux [42]. gradient density profile n ( z ) = n (1 − z / L ) along the mi-crochannels’ length 0 ≤ z ≤ L [42]. Let ¯ σ ≈
400 pm theeffective atomic diameter of ytterbium. Then, the aver-age mean free path for atoms inside the nozzle capillariesreads:¯Λ = L (cid:90) L d z Λ( z ) with Λ( z ) = (cid:16) π √ n ( z )¯ σ (cid:17) − . (3)We find ¯Λ = 11 mm at T oven = 490 ◦ C, which is indeed onthe order of the microchannels’ length L = 12 mm. How-ever, equation (3) gives ¯Λ = 36 mm for T oven = 450 ◦ Cand ¯Λ = 5 mm at T oven = 520 ◦ C, indicating a changeof regime across the temperature range in figure 4, goingfrom transparent (¯Λ (cid:29) L ) to opaque (¯Λ (cid:28) L ) channels.Giordmaine and Wang [42] derive atomic beam angulardistributions J ( θ ) for both regimes. The total flux followsby integration over the forward half solid angle. However,according to figure 2, the vacuum tubing between thenozzle piece and the spectroscopy region truncates theatomic beam’s solid angle at the spectroscopy zone toa polar angle θ max = 6 . ° . The prediction for the fluxmeasured by absorption spectroscopy therefore reads:˙ N = 2 π (cid:90) θ max J ( θ ) sin θ d θ (4)The solid and dotted lines in figure 4 correspond to thismodel. At the highest temperature probed, the averagemean free path is less than half the length of the mi-crochannels. Nevertheless, the collisionless model (dot-ted line) still qualitatively reproduces the data betterthan the opaque source model (dashed line). The trun-cation represents a loss of flux of around an order ofmagnitude but the geometric acceptance angle of the fullZeeman slower, around 20 mrad, is far smaller than θ max and therefore constitutes the major constraint on useableflux.Despite the qualitative agreement for the total flux,the theoretical prediction doesn’t match the observedhalf-width ˜ θ / quantitatively. The half angle divergencein the collisionless theory is around 20 mrad and above100 mrad in the opaque theory, compared to measuredvalues between 60 mrad and 80 mrad. This mismatchis likely due to the mixed regime in which we operatethe nozzle. The inset in figure 4 shows some evidencefor the opaque regime at higher temperatures where thebeam half-width scales with the square root of the totalflux, while it is predicted to be independent of flux in thetransparent regime. Nevertheless, the truncation of theatomic beam by the vacuum chamber prevents a faith-ful observation of the angular distribution, although notlimited by a thin aperture as in other work [45]. A studyof the onset of the opaque regime is of high interest forfuture applications of high flux ovens, for example in viewof gravitational wave detection with atomic test masses [31]. Cascaded designs [46] potentially enable higher col-limation and better recycling of atoms with highly diverg-ing trajectories. Also, in highly opaque configurations,even moderate optical recollimation can be effective atthe exit of the oven [35]. III. PERMANENT MAGNET ZEEMANSLOWER IN HALBACH CONFIGURATION At T oven = 490 ◦ C, the oven and its microchannel noz-zle produce a beam of around 2.1 × atoms/s travel-ling at a mean forward velocity ¯ v ≈
300 m / s. We employZeeman slowing to reduce the forward velocity to fit thefew tens of m/s capture velocity of our 2D-MOT oper-ating around the S → P transition. We focus on themost abundant isotope, Yb.We set up rare-earth magnets in a Halbach config-uration to generate the magnetic field profile requiredfor Zeeman deceleration. This approach is straightfor-ward to assemble and requires neither high-power electri-cal supplies nor water cooling, thus increasing reliability[9, 47]. Also, the Halbach configuration provides efficientsuppression of stray fields.In this section, we briefly review elements of Zeemanslowing theory and motivate our design parameters (sec-tion III A). We then describe our permanent magnet lay-out in depth and characterize its magnetic properties(section III B). Finally, we evaluate the performance ofthe slower on the atomic beam (section III C).
A. Zeeman slower design
Zeeman slowing [48] exploits the fact that atoms canbe decelerated when scattering photons from a counter-propagating laser beam. However, the change in velocityleads to a varying Doppler shift that needs to be ac-counted for to maintain the resonance condition for thescattering process. This is achieved using the shift ofelectronic energy levels arising from the coupling of anexternal magnetic field to the electronic angular momen-tum, the so-called “Zeeman effect”.The theory of Zeeman slowing is extensively coveredin the literature [41]. We review the essential aspects tofix the notation. We consider atomic two-level systemswith an zero-field energy splitting (cid:126) ω travelling at ve-locity v ( z ) along an axis labelled by coordinate z . Thedifferential linear Zeeman shift between the energy levelswhen the atom is at position z is ˜ µB ( z ) where | ˜ µ | equals1.035 Bohr magnetons for bosonic ytterbium atoms onthe S → P transition [49]. In frequency units, thisamounts to ±
14 MHz/mT. A laser beam with frequency ω / (2 π ) is directed against the atomic beam. At position z , the detuning δ to the atomic resonance reads: [50] δ ( z ) = ∆ + kv ( z ) − ˜ µB ( z ) (cid:126) (5)where ∆ = ω − ω is the laser detuning to the resonancefor atoms at rest, k = π / λ , the light’s wavenumber, and (cid:126) the reduced Planck constant.Setting δ ( z ) = 0 for the whole length of the Zeemanslower, the equation of motion of atoms in this regionfollows from standard scattering theory:d v d t = − s s (cid:124) (cid:123)(cid:122) (cid:125) η × (cid:126) k Γ2 m (cid:124)(cid:123)(cid:122)(cid:125) a max (6)with s the saturation parameter, m the atomic mass andΓ the natural width of the transition. The parameter η < − a max required for the atoms tostay on resonance and continue on the slowing trajectory.A solution of equation (6) reads v ( z ) = v c · (cid:114) − zL stop (7)where v c = v ( z = 0) is the capture velocity (the largestvelocity class addressed by the slowing process), and L stop = v c / ηa max is the distance required to bring theatoms to rest with the constant deceleration − ηa max .It is however necessary for the atoms to be extractedfrom the slower that they have a small exit velocity v e . The slowing length required to reach v e is L = L stop − v e / ηa max < L stop .Replacing equation (7) in equation (5) for δ ( z ) = 0 andsolving for the magnetic field profile B ( z ), one finds: B ( z ) = B + B L · (cid:114) − zL stop < z < L B = (cid:126) ∆ / ˜ µ , and the amplitude B L = (cid:126) kv c / ˜ µ . The field profileis truncated at z = L < L stop to be able to extract theslow atoms. The field therefore varies from B + B L at z = 0 to B + (cid:126) kv e / ˜ µ at z = L . The exit velocity v e can be experimentally tuned either by adjusting themagnitude of the field near the end of the slower, or byvarying the detuning ∆. We design the slower for L stop ,not forgetting that slight adjustments in detuning andfield magnitude near the end will be necessary.Table I summarizes the design parameters for our Zee-man slower. We set the capture velocity v c = 390 m / sthat corresponds to around 75 % of the atoms in thebeam with a forward velocity below the slowing thresh-old. The minimal slower length is 14 cm ( η = 1). Wechoose L stop = 30 cm, i.e. a deceleration margin param-eter η ≈ .
5. The detuning ∆ determines the offset B and hence the magnitude of the maximum field to gen-erate. Minimizing this quantity would lead to the choice B = − B L / , that is ∆ = − π ·
490 MHz. However, for ge-ometrical reasons, the Zeeman slowing laser beam passes
Parameter Symbol ValueAngular momen-tum change ˜ µ − . · µ B Maximumdeceleration − a max = − (cid:126) k Γ2 m − × m/s Detuning ∆ − π ·
700 MHzCapture velocity v c
390 m/sLength L stop
30 cmDecelerationmargin η = v c L stop a max s = η − η B L = (cid:126) kv c ˜ µ −
67 mTField offset B = (cid:126) ∆˜ µ
48 mT
TABLE I. Zeeman slower design parameters through, or very close to, the center of the subsequent2D-MOT which needs not to be unbalanced or lifetimelimited through excessive losses to the D states (see insetfigure 3). We choose ∆ = − π ·
700 MHz, which corre-sponds to a loss time constant above 200 s [51]. Also, theinfluence of the Zeeman slowing beam on the scatteringrate is more than 1000 times smaller than that of the2D-MOT beams. This results in a maximum field ampli-tude of 40 mT. The magnetic field profile correspondingto equation 8 with the parameters in table I is shown asthe solid green line in the middle panel in figure 5.
B. Zeeman slowing field from permanent magnetsin Halbach array configuration
We use permanent magnets in a Halbach configura-tion to produce the magnetic field described by equa-tion (8). Halbach arrays are particular magnetic config-urations that produce magnetic multipole fields in a wellconstrained domain in space, while suppressing the fieldoutside this region [52]. We consider a magnetic cylin-der with large length-to-diameter ratio and prescribe thatthe material’s magnetization M rotates as follows in theplane (ˆ e ρ , ˆ e φ ) transverse to the cylinder’s axis: M | M | ≡ ˆ e M = cos (2 φ ) ˆ e ρ + sin (2 φ ) ˆ e φ . (9)The corresponding magnetic field is transverse and homo-geneous along +ˆ e ρ and zero outside the cylinder. If the ° ° mm
30 15 B z xy − . . . . . − Longitudinal distance z (m) T r a n s v e r s e fi e l d ( m T ) − R e s i du a l s ( % ) ° ° mm
30 15 B z xy − . . . . . − Longitudinal distance z (m) T r a n s v e r s e fi e l d ( m T ) − R e s i du a l s ( % ) FIG. 5. Construction and magnetic field of the permanent magnets-based Zeeman slower.
Top panel : mechanical arrangementof the two main magnets and four trimmer magnets in one of the eight blades forming the magnet assembly. The inset showsthe magnetization direction for the entrance trimmer magnet on all eight blades, effectively forming an eight-fold discreteHalbach configuration.
Middle panel : ideal ( ), calculated ( ), and measured ( , ) strong transverse component of themagnetic field along the slower’s magnet. represents the externally trimmed configuration used for loading the 2D-MOTof section IV.
Lower panel : residuals of the measurements ( ) to the corresponding calculated configuration ( ). Most pointslie within 10 % of the numerical model. magnetic object is not a cylinder but a cone, the mag-nitude of the transverse field decreases with increasingcone radius, however also introducing a longitudinal fieldcomponent except on the cone’s axis. We assert numeri-cally, as already shown in previous work [53, 54], that aHalbach cone produces a percent-level approximation ofthe ideal Zeeman slowing field from equation (8).Since a continuous magnetic material with the magne-tization rotation described by equation (9) is not prac-tical, and similarly to the work in references [53, 54],we discretize the array using an eight-fold symmetry.However, contrary to previous work, our design exhibitsa zero crossing and a higher overall average gradient(200 mT/m versus 120 mT/m for Ref. [53], and 33 mT/mfor Ref. [54]).The top panel in figure 5 shows the construction ofour magnet. The main field is generated by two setsof eight NdFeB rectangular cuboids [55] with length128 mm, 6 mm × B r = 1 .
08 T, magnetized along one of the transverse directions. These long cuboids are arrangedin two Halbach cones with slopes +9 . ° and − ° , gen-erating the main part of the downwards and upwardspointing transverse field (figure 5, middle panel). Weadditionaly use 8 × × × B r = 1 .
17 T [56]) as adjust-ment variables to better fit the starting field, the zero-field transition regime in the middle of the slower, andhelp decreasing the field faster at the exit of the slower.We numerically optimize the positions and orientationsof all the permanent magnets, while enforcing the Hal-bach symmetry. We use the analytic expression de-rived in Ref. [54] to calculate the field from each mag-net. The optimized configuration leads to the field profilereprensented by the solid black curve in the middle panelin figure 5.The full magnet consists of eight sets of six magnetsclamped between aluminium blades. The positions ofthe magnets on the plates is engraved (e.g. 3 mm deepper plate for the long magnets) during the manufacturingprocess. The assembly of the magnet takes less than twohours, dominated by the identification of the magnetiza-tion direction for the 48 permanent magnets. We kept themanufacturing tolerances below 0.5 mm to constrain theassembly sufficiently while taking size variations of theindividual magnets into account. The plates are pressedagainst each other using five M3 bolts and mounted inan octagonal holder (figure 5, inset) to obtain the desiredconfiguration. The total weight of this assembly is 10 kg.We measure the magnetic field along the magnet’s axisusing a 3-axes teslameter [57]. As shown in figure 5 (mid-dle panel: blue triangles), we reproduce the calculatedfield (solid black line) to around 10 % (lower panel). Inthe transverse direction, the field decays to backgroundvalues within 10 cm outside the magnet assembly.
C. Slowing performance
Our design achieves slowing by scattering photons onthe S → P ( m J = −
1) transition, which requires σ − polarized light to account for the change in electronic an-gular momentum. However, since the generated field istransverse to the atomic propagation direction, the con-figuration maximizing the σ − polarization content cor-responds to light polarized linearly along ˆ e x , perpendic-ular to the magnetic field direction ˆ e y (see inset of fig-ure 5). This however implies that only half of the opticalpower has σ − polarization and contributes to the dom-inant slowing effect. The σ + polarization component isresonant with the atoms at some positions in the slowerbut this doesn’t affect overall performance. An influencevia (coherent) population dynamics is ruled out by thenon-magnetic S ground state. The slowing beam has a / e diameter of 4 mm and total power of 80 mW result-ing in a peak intensity of 3.2 × mW/cm ( ≈ . × I sat with σ − polarization).Coupling the slowing light beam inside the vacuumchamber requires special care. Since the slowing beam isanti-parallel to the atomic beam, an optical surface mustbe facing the atomic beam directly. A common solutionto reduce metallic deposition is to use an uncoated, z-cutsapphire viewport heated to several hundreds of degreescelsius [58]. We found this inconvenient, namely becauseof the high maintenance cost and added heat source nearthe main experimental chamber. An alternative solutionis an in-vacuum protected aluminum mirror, as demon-strated in reference [59] (see inset in figure 1). Whilethe chemical mechanism for maintaining high reflectivityunder metallic deposition is not well-known, operation ofour atomic beam over several months evaporated morethan 100 mg of ytterbium from the oven with neitherperformance loss nor visible degradation of the mirrorsurface.We characterize the slowed atomic beam using absorp-tion spectroscopy under a 30 ° angle at the exit of theslower. Figure 6 shows the longitudinal velocity profilefor Yb atoms. The capture velocity around 390 m/s is clearly visible, as well as the collection of low velocity( <
50 m / s) atoms. We maximize the flux of slow atomsby adjusting the laser detuning to the S → P tran-sition and tuning the magnetic field with two additionaltrimming bar magnets placed symmetrically at the exitof the magnet assembly. The resulting field is plottedusing pink crosses in the middle panel in figure 5. Themain effect of these extra trimmers is to avoid signifi-cantly overshooting the ideal magnetic field profile, thuskeeping the local deceleration parameter η close to itsdesign value and therefore avoid loosing atoms from theslowing profile due to missing light intensity.For the data presented in figure 6, the detuning is∆ = − (2 π ) ·
580 MHz, 4Γ away from the design value − (2 π ) ·
700 MHz. This is caused by the decrease in fieldmagnitude due to the external trimmers, and the needfor a finite exit velocity. The most probable exit veloc-ity in figure 6 is v e = 15 m / s. We measured a trimmedfield maximum of 35 mT, which matches reasonably theprediction from equation (8) (cid:126) ∆ / ˜ µ + (cid:126) kv e / ˜ µ = 37 mT.Ideally, all atoms below the capture velocity should beslowed down. This is evidently not the case in figure 6,mainly due to the divergence of the atomic beam insidethe slower. Atoms travelling too far off-axis are not in-teracting with the slowing beam and therefore not decel-erated further. This effect is amplified with decreasingvelocity since the longitudinal velocity becomes compa-rable to the transverse one. Increasing the slowing beamdiameter or decreasing the atomic beam divergence withradiation pressure prior to the Zeeman slower [35] caneffectively mitigate this issue, at the cost of increasedoptical power requirements. IV. 2D-MOT RECOLLIMATION/DEFLECTIONSTAGE
The divergence of the atomic beam during Zeemanslowing is a major limitation for the flux of slow atomscapturable by a 3D-MOT. Moreover, constraints on thetarget apparatus for this atomic source dictate a largesize of the main experimental vacuum chamber, leadingto a minimum distance of more than 30 cm between theexit of the slower and the center of the 3D trap. Wetherefore have to keep the atoms’ longitudinal velocitylarge compared to their transverse velocity in order toenable efficient loading of the 3D-MOT.In order to mitigate the beam divergence issue, we im-plement a 2D-MOT between the Zeeman slower and the3D-MOT. This presents the following advantages. First,it efficiently recollimates the atomic beam, preventingsignificant losses even when passing through a 4 mm di-ameter differential pumping aperture between the 2D-MOT and 3D-MOT chambers. Second, since we loadit under a 30 ° angle, it acts as a dump for fast, nonslowed atoms, preventing them from entering the mainchamber. Finally, the Zeeman slowing beam can be cou-pled through the 2D-MOT chamber, thus reducing the v c Atomic longitudinal velocity (m/s) O p t i c a l d e n s i t y ( · − ) ThermalSlowed
FIG. 6. Absorption spectroscopy of the atomic beam undera 30 ° angle at the exit of the Zeeman slower. The result-ing spectrum shows the longitudinal velocity distribution andthe effect of the slowing laser beam. Atoms exiting the ovenwith a velocity below v c ≈
390 m / s are slowed down to below50 m/s. The frequency axis is calibrated using a simultaneousspectroscopy measurement under an axis perpendicular to theatomic beam. complexity around the 3D-MOT chamber and avoidingperturbing the 3D-MOT with the slowing light (see fig-ure 1). A. Design
We follow the same design guidelines for the mag-netic configuration of the 2D-MOT as for the Zee-man slower and use four permanent magnet bars(80 mm × × B r = 1 .
17 T [60])) in a four-poleHalbach configuration [61] to produce around 5 . / cmgradients. Figure 7 shows the magnetic configuration, aswell as simulated and measured magnetic field profiles.Our design includes electromagnetic coils for fine tun-ing the position of the quadrupole zero but we do notuse them for the following results. Therefore, like forthe Zeeman slower, the magnetic field for the 2D-MOTis generated permanently and passively, increasing therobustness and reliability of the system.We generate the optical radiation pressure using tworetro-reflected 1 cm × ≈ . × I sat ) tuned 16 MHz (0 . S → P resonance. Beam shaping is achieved usingpairs of cylindrical lenses. B. Characterization
We characterize the atomic beam produced by the2D-MOT with retro-reflected spectroscopy in the mainchamber. Since the 2D-MOT mostly affects velocity com-ponents perpendicular to its axis, the longitudinal ve-locity of atoms exiting the 2D-MOT corresponds to theprojection of the slower’s exit velocity on the 2D-MOTaxis. We find a flux optimum for a 2D-MOT exit veloc-ity ¯ v l = 20 m / s, characterizing the trade-off between lowoutput velocity and excessive losses due to beam diver-gence in the Zeeman slower.Since the transverse velocity is not resolved in ourretro-reflected spectroscopy setup, we deduce it from thesize of the atomic beam in the center of the main cham-ber. Varying the probe beam diameter, we estimate anatomic beam diameter of 1 cm at a distance of 23 cm fromthe differential pumping pinhole. This corresponds to amaximum transverse velocity ( /
23 cm ) · ¯ v l < / s orfive times the Doppler limit (0.18 m/s) and represents afactor 20 improvement over the atomic beam divergenceat the exit of the oven.The trade-off between divergence and low exit veloc-ity is also relevant at the exit of the 2D-MOT. Indeed,the longitudinal velocity must be small enough to en-able capture by the 3D-MOT, while maintaining decentbeam collimation despite the 23 cm travel distance be-tween the differential pumping pinhole and the center ofthe 3D-MOT. Since the transverse velocity is on the or-der of 1 m/s, matching the few meters per second capturevelocity of a typical MOT operating on the intercombina-tion transition S → P (Γ = 2 π ·
180 kHz) would resultin an atomic beam diameter of around ten centimeters,which is impractical. With 20 m/s however, the atomicbeam size is restricted to single digit centimeters whilestill capturable by a MOT operated on the S → P (Γ = 2 π ·
29 MHz) transition.
V. FULL SYSTEM CHARACTERIZATION
We characterize the overall performance of the sys-tem consisting of the oven, the Zeeman slower, and the2D-MOT by capturing the ytterbium atoms in a largevolume 3D-MOT and measuring the trap’s loading rate.This provides a realistic estimate of the useable flux forfurther cooling steps but also removes the need for geo-metrical assumptions associated with spectroscopic fluxmeasurements.
A. 3D-MOT
We operate the 3D MOT on the S → P transition.The laser beams have a / e diameter around 2 cm. Themagnetic field gradient is produced by a pair of coils inanti-Helmholtz configuration. − − −
20 0 20 40 60 − − − P o s i t i o n ( mm ) − − − −
20 0 20 40 60 − M ag n e t i c fi e l d ( m T ) M ag n e t i c fi e l d ( m T ) FIG. 7. Permanent magnet configuration and transverse fieldprofile for the 2D-MOT generated with permanent magnets.Top: simulation and measurement showing a gradient of(-)5.4 mT/cm. The shaded area indicates the interior of thevacuum chamber. Bottom: simulation of the transverse cross-section of the field profile. The red/black rectangles indicatethe positions and orientations of the four permanent bar mag-nets with red(black) indicating their north(south) pole. Theblack circle represents the inner wall of the vacuum chamber. . . · Loading time (s) A t o m nu m b e r FIG. 8. Loading curve of the 3D MOT. The solid line is afit of eq. 10, with best fit parameters γ = 1 × at / s and R = 2 / s. The errorbars indicate the standard deviation ofthe measured datapoints. We determine the number of trapped atoms by ab-sorption imaging and collection of the fluorescence fromtrapped atoms. While both methods are in qualitativeagreement, we exclusively use absorption imaging for thequantitative results below, as it provides the more con-servative values and requires less assumptions.We optimize the trap for loading rate and maximumatom number. We find the optimal loading performancefor a detuning ∆ = −
32 MHz ( − . S → P resonance, a magnetic field gradient δB = 210 mT / m,and P MOT = 30 mW per beam (0 . × I sat per beam).Figure 8 shows an average loading curve. Our trap sat-urates at a steady-state atom number of 5 × atomswithin about 2 s. We extract the loading rate using aone-body loss rate model: N ( t ) = γR (1 − exp( − Rt )) , (10)where γ is the loading rate. The one-body loss rate R contains losses due to background collisions but is dom-inated by radiative loss via the D to the P states(inset figure 3), making it highly dependent on beam in-tensity and detuning [62]. Despite the relatively highdensity ( ≈ × at / cm ) no significant two-body losseswere observed compared to the strong one-body loss rateand are therefore neglected.Adjusting the model of equation (10) to the data infigure 8, we find γ = 1 × at / s and R = 2 / s. Athigher powers the maximum loading rate remains thesame but relaxes the requirements on detuning and gra-dient, allowing the same loading rate in a larger part ofthe parameter space. This behaviour suggests that inthese configurations the loading rate is limited only bythe flux into the main chamber and would allow for largersteady-state numbers at larger detunings and gradientsas long as sufficient laser power can be provided. B. System performance evaluation
Starting from a flux of 2 × at/s at the exit ofthe oven, our system loads 5 × Yb atoms in 2 s(1 × at/s initial loading rate) in a 3D-MOT on the S → P transition. Accounting for the natural abun-dance of Yb (32 %), this represents a loss of five ordersof magnitude in atom numbers during the slowing, redi-rection, and recollimation processes.Table II shows a summary of diagnostic flux measure-ments performed along the beamline. The major losscontributor is the Zeeman slower. The loss of total flux,cumulating all velocity classes at the exit of the Zeemanslower’s vacuum pipe, is accounted for by the slower’slength (60 cm including all connection pieces and iso-lation valves) and the divergence of the atomic beamexiting the oven (70 mrad, see figure 4). This gives amaximum throughput of 1.3 %, i.e. a maximum fluxof 8 × at/s, in good agreement with the measured0value of 5 × at/s. The loss of flux during the slow-ing process is split between the slower’s velocity accep-tance (25 % of the atoms are above the capture veloc-ity v c = 390 m / s) and atomic beam divergence insidethe slower (see discussion in section III), with the latterdominating largely.Reducing the divergence of the atomic beam at theexit of the oven is therefore a good way of improving theslowing efficiency and preserving flux along the beam-line. As an example, reducing the divergence half-angleby a factor 3 increases the geometric throughput by al-most a factor 8 and we expect this to also improve onthe slowing losses. Since the oven is operated near theopaque channel regime to achieve high flux, the colli-mation needs to be performed optically. With our ap-paratus, the above recollimation would require around150 mW of laser power which are currently not at ourdisposal. Improvements by simple optical collimationsafter the oven have been observed by experiments, forexample in reference [35]. Nevertheless, when the ovennozzle is in the opaque channel regime, the divergence in-creases with flux, correspondingly constraining the lasersystem or the geometry of the optical collimation cham-ber. Alternatively, the slower’s length could be reduced,limiting divergence effects but decreasing it’s safety mar-gin η , or the slowing beam’s diameter can be increased,at the expense of a considerable rise in laser power con-sumption.As discussed in section IV, the current apparatus isunsuitable for direct loading of a MOT operating on the S → P transition [63]. This is mostly due to thesize of the 3D trap chamber, constrained by other de-sign requirements for this apparatus. In order to keepa reasonably sized MOT, the transverse velocity around1 m/s of the atoms exiting the 2D-MOT require the for-ward velocity to stay above 20 m/s while not significantlysacrificing flux. In order to benefit from narrow line cool-ing, further optimization of the 2D-MOT stage up to theDoppler limit is a possibility. However, it is more versa-tile to add an extra slowing stage in the main chamber.The most straightforward would be sequential loadingvia a singlet MOT but other schemes such as two-stagecooling [64] or a core-shell MOT [36] might also be con-sidered. VI. CONCLUSION
We have built and characterized a source of laser-cooled ytterbium atoms delivering 1 × atoms/s in a3D magneto-optical trap. All necessary magnetic fieldson the atomic beamline are produced by permanent mag-nets in Halbach configurations. This provides easily re-producible designs with low stray magnetic fields, as wellas robustness since no maintenance on e.g. water cool-ing circuits is required. Also, the resulting assembly islightweight and can be disassembled for investigation,vacuum bakeout, or transportation. Apart from the laser Position along beamline Flux in atoms/sOven 6 × Zeeman slower total 5 × Zeeman slower ( <
50 m / s) 6 × Captured by 3D-MOT 1 × TABLE II. Flux of
Yb atoms at different positions alongthe beamline. The oven flux is given in figure 4 weighted bythe natural abundance of
Yb. For the Zeeman slower, wedistinguish between the total flux of atoms, cumulating allvelocity classes, and the slowed flux, accounting only atomswith less than 50 m/s forward velocity. Finally, the capturedflux is the loading rate into the 3D-MOT according to equa-tion (10). systems, the only electrical power consumption stemsfrom the oven ( <
50 W) and could be further reducedusing in-vacuum heating [45]. In particular, we foundthe use of an in-vacuum mirror for coupling the Zeemanslowing laser beam in the chamber to be an efficient al-ternative to the commonly used heated viewports.Besides a Zeeman slower, our apparatus features a 2D-MOT operated as a deflection and recollimation stage.While adding little complexity, this component is crucialto keeping a high fraction of the flux exiting the Zeemanslower capturable by the subsequent 3D trap. This is inparticular relevant in setups, like ours, where the mainexperimental chamber needs to have a large physical vol-ume due to other design constraints, at the expense oflittle loss of flux.Our study involves a detailed characterization of theatomic beam emerging from a microchannel nozzle. Weconfirm that systems aiming at very high fluxes operateat the onset of the opaque channel regime, where inter-atomic interactions inside the nozzle cannot be neglectedand the atomic beam divergence increases with increas-ing flux. This has crucial implications for the design offuture devices aiming at even higher flux, for example forthe detection of gravitational waves with atoms [31].Overall, our apparatus operates reliably at its maxi-mum performance level with no other maintenance thanthat associated with laser systems at a level comparableto that achieved in other ytterbium [36] or strontium [35]setups. This constitutes a solid starting point for com-plex cold atoms experiments such as, but not limited to,high-performance atom interferometers.
ACKNOWLEDGEMENTS
This work is part of the Hannover very long base-line atom interferometry facility, a major researchequipment funded by the German Research Founda-tion (Deutsche Forschungsgemeinschaft, DFG). We ac-knowledge support by the Collaborative Research Cen-ters 1128 “geo-Q” and 1227 “DQ-mat”, and Germany’s1Excellence Strategy within EXC-2123 “QuantumFron-tiers” (project No. 390837967). D. S. acknowl-edges funding from the German Federal Ministry ofEducation and Research (BMBF) through the fund-ing program Photonics Research Germany (contractNo. 13N14875). E. W. acknowledges support from“Nieders¨achsisches Vorab” through the “Quantum- and Nano-Metrology (QUANOMET)” initiative (project No.QT3). We thank M. Robert-de-Saint-Vincent for in-sightful discussions on the microchannel nozzle and in-vacuum mirrors, and D. Tell for careful proof-reading ofthe manuscript. We are grateful to C. Schubert, D. Tell,and K. Zipfel for their contributions and W. 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