Loading and spatially-resolved characterization of a cold atomic ensemble inside a hollow-core fiber
aa r X i v : . [ phy s i c s . a t o m - ph ] J a n Loading and spatially-resolved characterization of a cold atomic ensemble inside ahollow-core fiber
Thorsten Peters, Leonid P. Yatsenko, and Thomas Halfmann Institute of Applied Physics, Technical University of Darmstadt, Hochschulstrasse 6, 64289 Darmstadt, Germany Institute of Physics, National Academy of Science of Ukraine, Nauky Ave. 46, Kyiv 03028, Ukraine (Dated: January 14, 2021)We experimentally study the loading of laser-cooled atoms from a magneto-optical trap into anoptical dipole trap located inside a hollow-core photonic bandgap fiber, followed by propagation ofthe atoms therein. Although only limited access in 1D is available to probe atoms inside such afiber, we demonstrate that a detailed spatially-resolved characterization of the loading and trappingprocess along the fiber axis is possible by appropriate modification of probing techniques combinedwith theoretical analysis. Specifically, we determine the ensemble temperature, spatial density(profile), loss rates, axial velocity and acceleration. The spatial resolution along the fiber axisreaches a few millimeters, which is much smaller than the typical fiber length in experiments. Wecompare our results to other fiber-based as well as free-space optical dipole traps. Moreover, weidentify limits to the loading efficiency and potential for further improvements.
I. INTRODUCTION
Since its proposal by Ol’shanii et al. [1] laser guid-ing of atoms into or through hollow-core optical fibers[2] has received increasing experimental interest overthe past years [3–30] This is due to their potential forcontrolled guiding of atoms over macroscopic distancesfor atom interferometry [13, 22] and quantum optics[10, 31]. The latter application requires long coherencetime [23] and strong light-matter coupling, i.e., trans-versely tightly confined atomic ensembles of large opticaldepth d opt = σ nL [15], where σ is the atomic absorp-tion cross-section, n is the atomic number density, and L is the length of the ensemble. Therefore, in recent ef-forts have been devoted to load laser-cooled atoms froma magneto-optical trap (MOT) located at some distancefrom the hollow-core fiber into the core [9–30] by use ofa standing- or traveling-wave optical dipole trap [32].After pioneering work on laser-guiding of atomsthrough hollow fibers which had to rely on lossy glasscapillaries of large core diameters [3–8], later experimentsemployed single-mode, low-loss photonic cyrstal fibers[33] due to their more advantageous properties. Nowa-days, basically two types of fibers are applied. Kagom´e-structured fibers [34] and hollow-core photonic bandgapfibers (HCPBGFs) [35]. The first type of fiber pro-vides single-mode guidance of light over a large wave-length range, low birefringence, and low numerical aper-ture for core diameters of several ten micrometers. Thelarge bandwidth and low birefringence make Kagom´efibers ideally suited for far-off-resonance guiding with on-resonance probing and controlled manipulation of dif-ferent atomic species. Their low numerical aperture,and thus slowly diverging laser beams, allow for di-rect transport from the distant MOT into the fiber core[16, 18–20, 22, 27–30]. Experimental and theoreticalloading studies using Kagom´e fibers were reported re-cently [18, 19, 28]. A drawback of these type of fibers istheir relatively large core diameter, resulting in weaker light-matter coupling strengths, compared to HCPBGFs.HCPBGFs, on the other hand, are available with core di-ameters below 10 µ m to enable stronger light-matter cou-pling and suffer less from micro-lensing [21, 36], but pro-vide smaller guiding bandgaps and larger, uncontrolledbirefringence [37]. Nonetheless, efficient loading of atomsinto such fibers [9, 10, 15] and quantum optics exper-iments [10, 38], even down to the quantum level [39],were demonstrated recently. The larger numerical aper-ture of HCPBGFs usually prohibits a direct transfer ofatoms from the MOT into the fiber, as the trapping po-tential (proportional to the laser intensity) quickly de-creases outside of the fiber. Thus, a different approachto loading has to be chosen. So far, loading via free-fall,magnetic guiding, and hollow-beam guiding was investi-gated experimentally by Bajcsy et al. [11]. The moreefficient loading technique, using a dark funnel to guidethe atoms into the fiber [15, 38, 39], has however not yetbeen investigated thoroughly.We report in the following on a detailed experimen-tal study of loading atoms from a MOT into a far-off-resonance trap (FORT) [40, 41] located inside a HCP-BGF by using a dark funnel guide. Although FORTsin free space have been studied for years (see, e.g.,[32, 41, 42]) and a variety of probing techniques are avail-able, there are some interesting differences compared toa FORT located inside a hollow-core fiber. Loading andprobing is only possible along the direction of the fiberaxis and the axial extension of the FORT can be four or-ders of magnitude larger than the radial extension. Theatoms are therefore basically untrapped along the fiberaxis and propagation of the ensemble becomes relevant.We will present experimental data along with compari-son to theoretical models to characterize the system–evenspatially-resolved along the fiber axis. Specifically, weinvestigate the loading efficiency into the fiber, and de-termine loss rates, temperature, velocity as well as thespatial density distribution in the fiber. This allows usto identify limits to the loading efficiency and possibleimprovements as well as to compare our loading schemeto others.This paper is organized as follows: In Sec. II we givean overview of the experimental system and describethe probing techniques applied in our work. Then, weexperimentally characterize the fiber-guided FORT inSec. III A. In Sec. III B we study the atomic ensembleinside the fiber with a focus on spatially-resolved mea-surements in both radial and axial dimensions. Next, westudy the number and propagation of atoms loaded fromthe MOT into the FORT for various parameters with afocus on efficient loading in Sec. III C. Finally, we presenta summary and outlook in Sec. IV. II. EXPERIMENTAL DETAILS
Details on the experimental setup and sequence can befound in our previous work on related subjects [15, 38,39]. Thus, we give a brief summary of the setup now.
A. Experimental setup and sequence
We first load laser-cooled Rb atoms into a cigar-shaped vapor cell magneto-optical trap (MOT) locatedabout 5 . FIG. 1. (a) Schematic experimental setup. DM: dichroicmirror; aHWP: achromatic half wave plate; PBS: polarizingbeam splitter: BS: beam splitter; APD: avalanche photodi-ode; SPCM: single-photon counting module. (b) Couplingscheme. (c) Scanning electron micrograph of the HCPBGFand false color intensity profile of the FORT.
HCPBGF (
NKT Photonics
HC800-02) (see Fig. 1). TheHCPBGF has a length of 14 cm and a core diameter of ∼ µ m. The 1 /e intensity mode field diameter insidethe fiber is around 5 . µ m. After the MOT is loadedfor about 1 s yielding typically around 10 atoms, weshift the cloud down towards the HCPBGF tip duringthe next 40 ms by applying a magnetic offset field in thevertical direction. Simultaneously, we transfer the atomsfrom the bright MOT into a dark funnel guide [15] byswitching off the MOT repumper and ramping up thequadrupole gradient of the MOT coils which compressesthe atom cloud towards the HCPBGF. Also, the trap-ping detuning is increased to −
30 MHz for optimizedcooling during the transfer. The dark funnel guide con-sists of two nearly orthogonally aligned repumper beamswith a funnel-shaped shadow in their center, termed darkfunnel repumper (DFR) in the following. This dark fun-nel is aligned from the side onto the HCPBGF. In orderto improve the density above the HCPBGF and confinethe atoms in F = 1 we use a depumper beam travel-ing upwards through the HCPBGF [43]. The depumperis locked to the cross-over transition 5 S / F = 2 → P / F ′ = 2 ,
3. As the effective transition light-shiftby the FORT is estimated to be near +130 MHz for ourtypical FORT depth, the depumper is almost resonantwithin and slightly above the HCPBGF, where loadingand guiding is relevant. All trapping, MOT repumper,and DFR beams have diameters of nearly 20 mm thuscovering the whole region of MOT and HCPBGF tipwhere trapping and guiding is relevant. After the cloudhas reached the fiber tip we hold it for 10 ms in place.Here, the atoms are transferred into a FORT of depth . . µ m of thefiber tip for the FORT being strong enough to guide theatoms into the core. Thus, cooling and compression ofthe atom cloud above the HCPBGF is crucial for obtain-ing a good loading efficiency. Once the atoms are insidethe fiber they can be probed as described in the next sec-tion. In order to prevent detection of atoms that mightstill be left above the HCPBGF during probing, we usea strong repumper beam above the fiber to purge thevolume through which the probe beam propagates. Thisconfines the population in F = 2 above the HCPBGF,whereas we always probe the atoms in F = 1 inside thefiber. FIG. 2. (a) Level scheme for TROP on the D line. (b)Transmission for TROP with N a = 2 . × atoms ( (cid:3) )and without atoms ( ◦ ) inside the HCPBGF. The vertical axisis calibrated to show the actual power inside the HCPBGF. B. Probing techniques
1. Time-resolved optical pumping
The number of atoms inside the HCPBGF N a canbe inferred, e.g., by nonlinear saturation measurements[11] or spectroscopic studies followed by comparison to asimulation [19]. A simpler method infers N a from time-resolved optical pumping (TROP) [15].Here, population is first prepared in the lower groundstate F = 1. Then, the optical trap is quickly switchedoff and a strong pump laser, tuned, e.g., to the transi-tion F = 1 → F ′ = 2 [see Fig. 2(a)], is sent through thefiber and detected by an avalanche photodiode (APD).This beam optically pumps the population into F = 2where it does not interact with the pump anymore. Themedium thus becomes transparent. If the APD is cali-brated and the transmission from the fiber to the APDis known, the difference of the temporally-resolved trans-missions without and with atoms is thus a measure forthe energy absorbed by the atoms until they are in theuncoupled level F = 2. With the branching ratio from F ′ = 2 into F = 1 ,
2, here 0.5, the number of photons anatom absorbs on the average before it ends up in F = 2is 2. From the known energy of a photon, hc/λ , onetherefore gets the number of atoms. The result of suchmeasurement is shown in Fig. 2(b). We use this techniqueto study the loading efficiency for various parameters aswe will discuss in the following sections.
2. Electromagnetically induced transparency
The other technique suitable to extract informationfrom within the HCPBGF is electromagnetically inducedtransparency (EIT) [44, 45]. Here, a weak probe and astrong control field couple two long-lived states | , i viaa decaying intermediate state | i , e.g., in a Λ − scheme.The strong control field renders the previously opaque medium transparent for the weak probe within a fre-quency window around two-photon resonance betweenthe two long-lived states. The width of the EIT window∆ ω EIT = Ω c / (Γ p d opt ) depends on the control Rabi fre-quency Ω c , the intermediate state decay rate Γ and theoptical depth d opt of the atomic ensemble. The depthof the EIT window depends on the ground state de-phasing rate γ between states | , i relative to ∆ ω EIT .This enables determination of γ from EIT spectra [46].For Λ − type coupling of nearly energetically degenerateground states and collinear probe and control beamswith wavevectors k p,c the two-photon resonance is notvery susceptible to atomic motion, as the correspond-ing Doppler shift ( k p − k c ) · v a is very small, where v a is the atomic velocity along the laser beam direction.This allows one to determine the effective dephasing rate γ comprised of transit-time broadening, two-photonlinewidth, and inhomogeneous transition shifts due to theradially varying control field intensity [38]. For simplicitywe do not discriminate here between reversible dephas-ing processes, e.g., due to a spatial magnetic field gradi-ent and irreversible decoherence, e.g., due to transit-timebroadening. For anti-collinear probe and control beams,however, the two-photon resonance is very susceptible toatomic motion as the two-photon Doppler shifts is now( k p − k c ) · v a ≈ · k p,c · v a . This can be used in ultracoldmedia to obtain information on the atomic velocity (dis-tribution) along the laser beam direction [47]. Thus, EITcan be used to obtain information on the magnetic field(gradient) and atomic velocities inside the HCPBGF, de-pending on the configuration used. III. EXPERIMENTAL STUDIES
The following experimental studies allow us to gaininformation on the atoms as they propagate throughthe HCPBGF. Unlike in free-space setups, optical accessfrom the side is not possible for a HCPBGF. Thereforeinformation on the trap depth, the number, position andvelocity of the atoms, temperature, magnetic field (gra-dients) and radial distribution have to be obtained frommeasurements performed along the fiber axis only.
A. Characterization of the FORT
As the atoms are guided into the HCPBGF by a deepFORT, which affects the atomic temperature, peak num-ber density and its distribution, resonance shifts, scat-tering rate, and collisions [32], precise knowledge of theFORT potential is desirable. For far-detuned traps,where the FORT detuning is much larger than the hy-perfine and fine structure splitting, the following radialpotential is applicable [32, 48] U F ( r ) = − πc Γ2 ω " (cid:0) g F m F √ − ǫ (cid:1) ( ω D − ω )+ (cid:0) − g F m F √ − ǫ (cid:1) ( ω D − ω ) I ( r ) . (1)Here Γ is the excited state decay rate and ω is the cor-responding transition frequency of either the D or D line, g F is the Land´e factor, m F is the magnetic quantumnumber, ω is the frequency of the FORT field, and I ( r ) isits radial intensity profile. The ellipticity ǫ of the polar-ization is defined as ǫ = ( P max − P min ) / ( P max + P min ),where P max,min are the maximum/minimum transmittedpowers through a rotated linear polarizer. In the follow-ing we present measurements that yield information onthe fiber-guided FORT potential.
1. Light-shift spectroscopy
The FORT potential inside the HCPBGF, given byEq. (1), can easily be calculated from the measured powerand beam profile. In our case, the FORT intensity profileis nearly Gaussian with a 1 /e half width of w = 2 . µ m.For a wavelength λ F ORT ∼
820 nm and a power insidethe fiber of up to 115 mW we can thus calculate the peakFORT depth as T F = − U F (0) /k B ∼ . k B is the Boltzmann constant.The FORT potential directly determines the atomiclevel shifts via ∆ ls ( r ) = U F ( r ) / ~ , with the reducedPlanck constant ~ , and also the thermalized atomic num-ber density distribution [32] (see also Sec. III B 1) n ( r ) = n exp (cid:18) − U F ( r ) − U F (0) k B T (cid:19) , (2)where n is the peak density and T < T F is the ensem-ble temperature inside the deep FORT. Therefore, spec-troscopy of the atoms while the FORT is on will yield in-formation on the FORT depth, density distribution andtemperature [14, 27].We recorded absorption spectra with the FORTswitched on during the measurement [see Fig. 3(a)]. Theeffective transition shifts (including ground and excitedstate shifts, which are not equal) are near 130 MHz forour typical FORT parameters. As this is larger than thescanning range of our probe frequency, we performed thespectroscopy at reduced FORT power. Clearly, the reso-nance is blue-shifted and asymmetric. The red line showsthe result of a numerical simulation [38] where we consid-ered a radial atomic density profile based on Eq. (1)-(2).This leads to a radially-dependent absorption and shiftof the levels and we require integration over the radial co-ordinate in a similar fashion as in [27] for obtaining thetransmission through the fiber. We obtain good agree-ment with the experimental data for an atomic temper-ature of T = 250 µ K for a calculated T F = 900 µ K. FIG. 3. (a) Absorption spectrum of the transition 5 S / F =1 → P / F ′ = 1 when the FORT is on ( ◦ ). The mea-surement was recorded 15 ms after the loading was finishedand the FORT power was P F = 18 . λ F ORT = 813 nm for this measurement, resultingin T F = 900 µ K. The red solid line shows the result of a nu-merical simulation where only the number and temperatureof the atoms were free parameters. We obtain best agree-ment for T = 250 µ K. (b) Remaining fractional atom numberinside the FORT after a sinusoidal modulation duration of10 ms and a modulation depth of 0.5 %. The dashed verticallines mark the positions of the calculated ω r , ω r . The arrowmarks an additional resonance at nearly 4 ω r . These results show that the atomic number densitydistribution is also in our HCPBGF well-represented byEq. (2), i.e., the density is highest on the fiber axis asobserved in [27], in contrast to other predictions [14, 49],which favored ring-shaped distributions.
2. Determination of the FORT trap frequency
The trap frequencies ω r,z are important parameters ofa cylindrically symmetric FORT [32]. Already small dis-turbances of the trapping potential occurring at frequen-cies ω mod = 2 ω r,z /n with n = 1 , , ... will lead to para-metric heating and losses from the trap [50, 51]. This iscrucial in our experiment, as we intentionally modulatethe FORT on/off for probing the atoms without inhomo-geneous broadening induced by the FORT [11, 38, 39].This modulation frequency should thus be chosen suchthat parametric heating and loss does not occur. Whenthe FORT is inside a HCPBGF, the axial trap frequencycannot be defined due to the constant potential alongthe fiber axis leaving only the radial frequency. By ap-proximating the potential U F ( r ) by a cylindrically sym-metric harmonic potential the corresponding radial trapfrequency is ω r = (4 | U minF | /mw ) / , where w is the1 /e width of the radial Gaussian intensity profile.To investigate the FORT trap frequency we followedthe method of Friebel et al. [51] and modulated the trap-ping potential sinusoidally with different modulation fre-quencies ω mod and then measured the remaining numberof atoms left in the FORT. Figure 3(b) shows the resultsfor the fractional N a left after a 10 ms modulation du-ration vs. ω mod / π . Although it would be preferablehaving the same number of modulation cycles for eachmodulation frequency ω mod , we had to limit the mea-surement to a fixed duration of 10 ms where the atomsare all loaded and N a had not yet decayed significantly(see Sec. III C 4). For our measurements the number ofcycles therefore varies between 100 ≤ N cyc ≤ P F = 108 mW anda wavelength of λ F ORT = 820 nm we expect to have aradial trap frequency of ω r = 2 π ×
51 kHz. The tworesonances of lowest frequency agree well with the calcu-lated radial resonance frequencies at ω mod = ( ω r , ω r ).Interestingly, a third resonance (marked by an arrow)near 4 ω r is clearly visible in the data. In addition, theresonances appear to be slightly red-shifted. Such addi-tional resonance including a red-shift has been observedin optical lattices where the trapping potential along theaxial direction is anharmonic due to the sinusoidal in-terference of the lattice beams [52, 53]. In our case, us-ing a single-beam FORT of near-Gaussian profile insidethe HCPBGF, however, such resonance due to the radialdirection comes unexpected. As the temperature of theatoms inside the FORT was measured to be near 1 mK forthe data shown in Fig. 3(b) (see Sec. III B 1), i.e., a signif-icant fraction of the FORT potential with T F = 3 . ω mod = 2 π ×
250 kHz, the typical FORT modulation frequency we usefor performing quantum optics experiments [39].
3. Determination of the FORT scattering rate
Photon scattering potentially results in heating andloss as well as unwanted population dynamics. As toavoid hyperfine-changing collisions [54] that could leadto additional losses from the FORT inside the HCP-BGF ( h · .
835 GHz /k b = 328 mK ≫ T F ) we keep adepumper beam, tuned near the transition 5 S / F =2 → P / F ′ = 2, on whenever the FORT is on. Thisconfines the population in F = 1 inside and outside ofthe HCPBGF. Population redistribution between the twohyperfine ground states should therefore only be due tothe FORT and depumper.If the FORT intensity spectrum I ( ω ) is known, the FIG. 4. (a) Measured normalized power spectrum of theFORT laser system. (b) Temporal evolution of the popu-lation in F = 1 ( ◦ ) after the depumper beam is switched offand only the FORT is present. The red line is a fit to the ex-perimental data of the form N a ( t ) = N + N exp( − γt ) with γ = 690(280) s − , N = 0 . , N = 0 . FORT scattering rate can be calculated according toΓ F = πc Γ ~ ω Z ω D − ω ) + 1( ω D − ω ) ! I ′ ( ω ) dω, (3)where I ′ ( ω ) = dI ′ ( ω ) /dω is the intensity per frequencyinterval dω , we assumed linear polarization and that thehyperfine structure is not resolved. We thus measuredthe power spectrum of the FORT with an optical spec-trum analyzer as shown in Fig. 4(a). Frequency-resolvedanalysis of the scattering rate showed that even thoughthe resonant fractional power is around 50 dB smallerthan at the FORT peak wavelength, resonant scatteringstill dominates. We therefore employed a notch filter withan optical depth around 6 near the rubidium resonancesto suppress resonant scattering to a negligible level. Withthe filter in place, we obtain from the measured spec-trum and Eq. (3) a spontaneous FORT scattering rate ofΓ F ∼
186 s − (for 101 mW of FORT power).In order to compare this value with the experiment, wemeasured the population in F = 1 as a function of timeafter the depumper had been switched off while only theFORT was on. As the FORT couples both ground statesoff-resonantly to the excited states, optical pumping oc-curs with an effective relaxation rate γ , leading to totaldepolarization of the population. A fit of type N a ( t ) = N + N · exp( − γt ) to the measured data in Fig. 4(b)yields γ = 690(280) s − , N = 0 . , N = 0 . et al. , the relaxation rate can be orders of magni-tude smaller than the total spontaneous scattering rate,due to an interference between the D , lines suppressingRaman scattering [55]. When we consider this interfer-ence effect, we obtain a relaxation rate of γ rel = 0 . F between F = 1 ,
2, i.e., about fifty times smaller than mea-sured. To solve this discrepancy, we note that our FORTsystem is comprised of two independent and unstabilizedlaser diodes. As a result, the FORT field intensity showsfluctuations, whose spectral components can drive Ra-man transitions between F = 1 , B. Spatially-resolved characterization of theatomic ensemble
Typically, HCPBGFs are loaded with the aim to max-imize the number of atoms inside the fiber as to createthe highest possible d opt or to guide atoms with lossesas small as possible from the FORT. This is achievedwhen the fiber can be filled over a length as large aspossible. However, experimental studies of the confinedatomic ensemble thereafter only yield an average over theensemble length of several centimeters (up to the lengthof the HCPBGF; 14 cm in our case). This impedes gain-ing information on, e.g., possible magnetic field gradientsalong the fiber, the axial atomic density profile etc. Alsothe radial distribution of atoms is of interest. It relatesto the temperature of the atoms inside the FORT andaffects inhomogeneous broadening as the atomic densityprofile width is typically comparable to the mode fielddiameter of the laser beams. We thus describe in the fol-lowing methods to extract such information. To this end,we load the HCPBGF only with a short pulse of atomsand interrogate them as a function of time t if they havepropagated inside the fiber. We do this by switching onthe FORT for only a short period of 1-3 ms for loading.This creates an atomic pulse of 1-3 mm length, as can beestimated from the initial velocity of ∼
1. Determination of the temperature inside the FORT
We start by determining the temperature of the atomicensemble inside the HCPBGF which affects the radialdensity profile according to Eq. (2). This can be done byemploying a time-of-flight (TOF) measurement, e.g., viaa release-and-recapture technique as shown by Bajcsy etal. [11]. Here, we simply measure the atom number byoptical pumping as a function of time after the FORT hasbeen switched off. If the pumping time is much shorterthan the time the atoms reach the fiber wall, we veri-fied that this simple method yields the same tempera-ture as the release-and-recapture technique. We assumethat the atoms are not detectable once they reached thefiber wall [6, 56]. The result of such measurement, takenat t = 1050 ms, i.e., 5 ms after loading has ended, isshown in Fig. 5(a). For this measurement the fiber wasloaded to its maximum by keeping the FORT on dur- FIG. 5. (a,c,d) Time-of-flight measurements to determine theensemble temperature inside the HCPBGF. The normalizedatom number ( ◦ ) is plotted vs. the time after the FORT hasbeen switched off. The red solid lines show the results of acalculation based on a Gaussian atomic density distribution[see Eq. (4)]. For maximum loading of the fiber (a) we obtaina temperature of T = 1000(200) µ K and a radial ensemble1 /e half width of σ a = 1 . µ m for T F = 3 . ∼ µ s period in the calculation. For a pulsed loadingof 3 ms only, the results are shown for times t if = 1 . t if = 21 . T = 2700(200) µ K (c) and T = 1000(400) µ K(d). In (b) we plotted the obtained temperatures ( ◦ ) vs. thetime t if . The temperature from (a) is shown for reference ( (cid:3) and gray-shaded area). ing the transfer process from the MOT. The atoms havetherefore spent 5 ms ≤ t if ≤
35 ms inside the fiber atthe time of the measurement (see Fig. 10). Thus, andbecause we do not observe an increase of the tempera-ture for t > n ( r )can be written as a Gaussian of 1 /e -width σ a [36]. Afterrelease from the FORT the density evolves as n ( r, t ) = N πσ t exp (cid:18) − r σ t (cid:19) , (4)where σ t = σ a + v a t and v = p k B T /m . The depen-dence of n on the axial coordinate z is irrelevant for thismeasurement and can therefore be neglected. Integra-tion of Eq. (4) over 0 ≤ r ≤ r f , where r f is the fiber coreradius and normalization leads to the solid red curve inFig. 5(a). We obtain a temperature of T = 1000(200) µ Kfor a FORT depth of T F = 3 . T /T F agrees reasonably well with the ratio from Fig. 3(a) fora much shallower FORT, but is larger than the ratio in[11]. We note that the temperature is the only free pa-rameter in our model after the FORT potential has beencharacterized as shown in Sec.III A.Next, we performed TOF measurements with a vari-able delay after the atoms are loaded for a short period of3 ms only. This enabled us to measure the temperatureas a function of time t if the atoms have spent inside thefiber. In Fig. 5(d) we show the results for t if ∼ . T = 1000(400) µ K. The temperature agrees well with theone obtained for maximum loading, where the atoms havespent > t if ∼ . T = 2700(200) µ K. Figure 5(b) depicts the results forthe temperatures at different times t if ( ◦ ). The tem-perature is initially much larger than the one obtainedfor maximum loading ( (cid:3) ) from Fig. 5(a), but decreaseswith time inside the fiber. We believe this decrease oftemperature during the course of about 20-30 ms to bethe result of free evaporative cooling inside the FORT,as for instance observed in [57]. As the atoms enter theFORT of finite depth T F . T ∼ T F . The trap is there-fore overheated and a significant fraction of atoms hasa potential energy close to zero. Thus, any heating pro-cess present leads to immediate loss of high-energy atomsfrom the trap and therefore cooling. The rate of evapo-ration due to elastic collisions can be estimated [58] forour estimated in-fiber atomic density of 10 cm − andtrap depth as Γ ev ∼ − , which is too small to explainthe observed decrease in temperature. However, as wewill discuss in Sec. III C 4, losses from the FORT are cur-rently dominated by heating due to an ellipticity of theFORT polarization [48] leading to a loss rate of 40 s − .This matches the quite well to the here observed decreaseof temperature. We note that the observation of evapo-ration on such a short timescale shortly after loading theFORT is facilitated by the ability to selectively probe theatoms inside the HCPBGF. In free-space FORTs, prob-ing is restricted to timescales where the untrapped atomshave fallen out of the interaction volume [57].
2. Axially resolved measurements: velocity and acceleration
To gain information on the time-dependent position z ( t ) inside the HCPBGF we recorded EIT spectra withanti-collinear probe and control beams for pulsed loadingof the fiber. Here, the spectra are maximally susceptibleto the atomic motion along the fiber axis as the two-photon resonance is Doppler-shifted by ∼ k p,c v a . Thisallows to determine the average axial velocity v a ( t ) = v + a eff t as a function of time, and the effective accel- FIG. 6. (a) EIT spectra with anti-collinear probe and con-trol fields taken at t if = 35 ms ( ◦ ) and t if = 65 ms ( (cid:3) )after the atoms entered the HCPBGF. The solid lines showthe results from a numerical simulation with d opt = (2 .
2; 1 . c = 1 . D , v a = (0 .
86; 1 .
16) m/s (black;red), γ = (0 .
16; 0 . D (black; red), T = 1000 µ K. (b)Measured velocities of the atoms (positive when parallel tothe direction of gravity) vs. the time t if they have spent in-side the HCPBGF. The estimated uncertainty for determiningthe velocities is ± .
05 m/s. For regular loading conditions ( ◦ ),we obtain v = 0 . a eff = 6 . from alinear least-squares fit. When the depumper is turned on foronly 200 µ s just before the measurement, the effective accel-eration is larger ( (cid:3) ). The black solid (red dashed) line showsthe result of the rate equation model (see App. A 1) when thedepumper is on (off) during loading. Here, we used the mea-sured depumper power, Γ ( R ) F = 410 s − , and v = 0 .
74 m/swhich are both within the experimentally determined uncer-tainties. If the acceleration were solely determined by grav-ity, the gray dotted line would show the corresponding de-pendence. When the notch filter used to suppress resonantscattering by the FORT is removed and the depumper is con-tinuously on during the loading process ( ⋄ ), the atoms areeffectively accelerated upwards. The blue dashed-dotted lineshows the least-squares linear fit yielding v = 0 . a eff = − . eration a eff . The position inside the HCPBGF is thensimply given by z ( t ) = v t + a eff t .For an atomic cloud entering the HCPBGF at a down-ward velocity v > t if inside the fiber for adownward propagating probe beam.To illustrate this we first show two EIT spectra takenat t if = 35 ms ( ◦ ) and t if = 65 ms ( (cid:3) ) in Fig. 6(a). Thesolid lines show calculated spectra based on the theoryin [38]. As expected, both spectra are increasingly red-shifted for larger t if . We then extracted the velocity asa function of t if from a series of measurements which isshown in Fig. 6(b) by black circles. A linear fit to thedata yields v = 0 . T F = 3 . a eff = 6 . is smaller than expectedfrom gravity only (gray dotted line). Our explanation forthis is radiation pressure from the FORT and depumperbeams, which are both aligned anti-parallel to gravityand are typically on during the loading and guiding pe-riod. We therefore recorded spectra when the depumperwas switched on for only 200 µ s just before the measure-ment to avoid any influence on the atomic motion. Theextracted velocity is shown by the red square in Fig. 6(b).It is consistent with the one expected for a purely grav-itational acceleration within the error bars. To furtherinvestigate the acceleration, we employed a populationrate equation model (see App. A), which then allowsfor calculation of the light pressure force via the photonmomentum, excited state scattering rate and population[59]. In our model, we consider photon scattering bythe FORT, optical pumping by the depumper via the D line and pumping between F = 1 , µ s). We here used the totalFORT scattering rate of Γ F = 186 s − (see Sec.III A 3),depumping rate according to the measured depumperpower, an initial velocity of v = 0 .
74 m/s, and a FORTRaman pumping rate of Γ ( R ) F = 410 s − [ ≡ γ in Fig. 4(b)within the measurement uncertainty]. The total FORTscattering rate Γ F has a direct influence on the change ofatomic momentum. The Raman pumping rate Γ ( R ) F onlyaffects the momentum if the depumper is on, as the netmomentum transfer of ± ~ · π/c · . a eff = 5 . (black solid line) with and a eff = 8 . (red dashed line) without the depumperbeam on during loading. The former one agrees withthe experimental data quite well while the latter one isslightly smaller than measured. We suspect that the dis-crepancy between the measured and simulated velocitywithout the depumper ( (cid:3) , red dashed line) is due to aslightly higher initial velocity when the depumper is notpresent, as it will be near-resonant with the cycling tran-sition F = 2 → F ′ = 3 just outside of the HCPBGFwhere the light-shift by the FORT is smaller and there-fore slow the atoms down when it is present. Notes: (i) The previous measurements in this sec-tion were done with the notch filter in place to sup-press resonant scattering by the FORT. This filter wasinstalled only after acquiring the data shown in Fig. 6(b)by blue diamonds. A linear fit of the data is shownby the blue dashed-dotted line with v = 0 . a eff = − . Again, the initial velocitymatches quite well to the FORT depth of T F = 4 .
3. Axially resolved measurements: dephasing
Whereas EIT with anti-collinear probe/control beamsis maximally sensitive to atomic motion, it is minimallysensitive for collinear alignment. EIT spectra taken withcollinear beams at different times t if therefore poten-tially allow for spatially resolved determination of theeffective ground state dephasing rate γ . This rateis, e.g., affected by transit-time-broadening, two-photonlinewidth of the probe/control beams, and an inhomo-geneous broadening due to the similar transverse dimen-sions of the control and the atomic density distribution[38]. In addition, it could depend on the spatial positiondue to a possible magnetic field (gradient) which changesacross the fiber length of 14 cm, leading to a spatiallyvarying ground state splitting. A more detailed investiga-tion of the ground state dephasing rate is of importance,as in our first work on EIT in a HCPBGF using the D line, we found γ = 0 . D [38], whereas in ourfollowing work at the D line, we observed a much larger γ = 0 . D [39]. This came as a surprise, as γ in[38] was dominated by inhomogeneous broadening due tothe transverse beam and atomic density profiles and weexpected this to be significantly reduced at the D linedue to its larger hyperfine splitting. An EIT spectrumfor collinear probe/control beams taken at t if = 15(1)is shown in Fig. 7(a). As before, we adjusted the pa-rameters of a numerical simulation (solid line) until wereached best agreement with the experimental data ( ◦ )[38]. Such spectra allow us to extract γ as a function of t if and therefore position inside the HCPBGF as shownin Fig. 7(b). Notably, for early as well as later times t if , γ = 0 . D is roughly the same. For 20 ms ≤ t if ≤
35 ms, however, the dephasing rate increases upto γ = 0 . D . We confirmed this behavior dur-ing several measurement runs on different days and thedata shown here are averages of these multiple measure-ments. This only temporary increase of the dephasingrate is again quite surprising as we would have expected asteady increase for a spatially varying magnetic field (gra-dient). Using the results from Sec. III B 2, we estimatethe atoms to be around 1.8 cm from the fiber tip dur-ing the time period of larger γ . This position matchesquite well with the start of the µ -metal shielding of theHCPBGF, which is wrapped around the fiber except forthe first/last 2 cm. This might however just be a coin- FIG. 7. (a) EIT spectra with collinear probe and control fieldstaken at t if = 15(1) ms ( ◦ ) after the atoms entered the HCP-BGF. The solid line shows the result of a numerical simulationwith d opt = 2 . c = 1 . D , T = 1 mK, v = 1 . γ = 0 . D . (b) Averaged decoherence rates γ vs. t if as extracted from multiple measurements such as in (a). Theerror bars represent an estimated uncertainty of 0 . D . cidence, as we cannot see any reason why the dephasingrate should be similar inside and outside of the shieldedregion, except for the transition. Nonetheless, the mea-sured spatially-resolved maximum dephasing rate γ isof the same order of magnitude as in [39], but its originis yet undetermined.The data and simulations presented above show thatit is possible to gain information on the radial as wellas axial position and distribution of an atomic ensembleloaded into a HCPBGF. For the spatially-resolved mea-surements to succeed with an acceptable signal-to-noiseratio, we required to load around 5000-6000 atoms for apulsed loading of 2-3 ms duration. C. Study of the loading process
Usually, the goal is to achieve the highest possible load-ing efficiency from the MOT into the fiber-guided FORT.We thus now investigate the transfer from the MOT tothe HCPBGF and the maximum number of atoms N a loaded into the HCPBGF. It is obvious that N a is propor-tional to the number of atoms N ( MOT ) a inside the MOTand increases with decreasing atomic temperature abovethe fiber due to the finite trapping potential. Therefore,we performed the following studies at an empirically op-timized laser and magnetic field timing sequence whichproduced the largest loading efficiency [39].
1. MOT to HCPBGF atom transfer
In contrast to experiments loading laser-cooled atomsinto Kagom´e-structured hollow fibers of low numericalaperture, we first have to transfer the atoms from theMOT near the HCPBGF tip before the FORT poten-tial is deep enough to guide atoms into the fiber. Wedo this by employing a DFR to maximize the atomic
FIG. 8. Measured number of atoms N ( MOT ) a above the HCP-BGF vs. time t ′ after start of the transfer from the MOT tothe HCPBGF. The atom number above the HCPBGF is mea-sured by fluorescence imaging. The error bars correspond tothe standard deviation of 20 successive measurements. TheMOT repumper is switched off near t ′ = 0 ms. The atomcloud reaches the HCPBGF tip around t ′ = 15 ms. TheMOT quadrupole field is switched off at t ′ = 45 ms. Theinsets show false color images of the atom cloud above theHCPBGF tip (yellow structure at lower end) at two differenttimings. density above the HCPBGF (see Sec. II A). We there-fore first investigate the transfer of atoms from the MOT(centered ∼ . ≤ t ≤ × in the MOT we ob-tained N a = 1 . × inside the fiber. The loadingefficiency from MOT to HCPBGF-based FORT is there-fore roughly η = 2 . η is basedon the shot-to-shot fluctuations and does not accountfor an incorrect calibration of the fluorescence imagingsetup. Therefore we expect the uncertainty to be actuallylarger than 0 . t ′ . ≤ t ′ ≤
15 ms). Once the lower end of the cloud hasreached the HCPBGF tip at t ′ ∼
15 ms, however, thenumber of atoms decreases quickly due to the losses in theregions where the HCPBGF mount blocks the trappinglight. At t ′ = 45 ms, where the MOT quadrupole fieldconfining the atoms is switched off, basically no atomsare left above the HCPBGF.0 FIG. 9. (a) Measured N a vs. relative FORT power P F /P maxF with ( ◦ ) and without ( (cid:3) ) sub-Doppler cooling above the HCP-BGF. The solid lines are linear fits to the corresponding ex-perimental data for P F /P maxF ≤ .
7. (b) Number of atoms N a loaded into the HCPBGF vs. the power of the dark funnelrepumper (of a single beam) for different depumper powers.The error bars in the vertical direction correspond to an esti-mated 8% uncertainty. The depumper power P dep was 2.3 nW( ◦ ), 9.7 nW ( (cid:3) ), and 69 nW ( ⋄ ) corresponding to peak Rabifrequencies inside the HCPBGF of around 1 . D , 2 . D ,and 7 . D , respectively.
2. Dependence on the FORT power
The trap depth T F , is proportional to the FORT power P F . Thus we expect N a to increase with P F when thetemperature above the HCPBGF is kept constant.We measured N a for different P F during the load-ing phase for our standard sequence which produces thelargest N a . In Fig. 9(a) we show the results for thecase where the trapping detuning is kept constant at − . D (red squares) and when the detuning is rampeddown to − D during the loading process (black cir-cles) to optimize sub-Doppler cooling above the HCP-BGF. In both cases the atom number initially increasesnearly linearly with the FORT power as expected. For P F /P maxF > .
7, however, the atom number saturatesaround N a ∼ . × in the case where sub-Dopplercooling is performed. The solid lines are linear fits tothe corresponding experimental data for P F /P maxF ≤ . P F /P maxF = 1. When no sub-Doppler cooling is performed, the linear region extendsfurther, but also here N a saturates near N a ∼ . × .We interpret these results as follows: As the maximumatom number inside the HCPBGF seems to saturate atdifferent values for the two conditions, and this maximumvalue can change from day to day (going sometimes upto N a ≥ × ), we believe that this limit is due to thedensity and temperature of the atoms above the fiber.Loading more atoms inside the MOT, better compres-sion and/or cooling during the transfer should thereforelead to a better loading efficiency and is only a techni-cal issue. We note that a similar saturation behavior hasalso been observed for a different loading technique usinga Kagom´e fiber [19].
3. Dependence on the repumper power
The DFR [15] is at the heart of our HCPBGF loadingmethod as it serves to enhance the density of atoms asthey approach the fiber tip. Therefore, we now study theloading efficiency dependence on the repumper configu-ration and power. Most repumper light available for theDFR is blocked by the funnel mask in the center of thebeam and thus the repumper light available in the MOTplus the number of atoms is limited. We therefore employan additional Gaussian-shaped MOT repumper which co-propagates with the trapping beams and is switched bya mechanical shutter. If only this MOT repumper ispresent (with the DFR blocked) during the MOT phase( t ≤ µ W fora maximum N a while the DFR is still blocked, we load N a = 6 . × atoms. This corresponds to a com-pression and cooling of the cloud during the transferas it is shifted towards the fiber. Density-limiting col-lisions are, however, not suppressed. If we switch off theMOT repumper at the beginning of the HCPBGF load-ing phase ( t = 1000 ms) while keeping the DFR on at apower of 25(1) µ W per beam, N a increases by a factor of ∼ .
7. This demonstrates the effectiveness of our guid-ing method based on the DFR. It also nicely matches theresults by Kuppens et al. who studied the loading of afree-space dipole trap and reported a factor of two im-provement in the loading efficiency when using a shadowin the repumper [42].In the following we study the dependence of N a onthe DFR power which is only relevant during the HCP-BGF loading phase when the stronger MOT repumper isoff. The results are shown in Fig. 9(b) for three differentdepumper powers. The depumper serves to confine theatoms in F = 1 inside and outside of the HCPBGF. Forall depumper powers N a first increases with DFR power P DF R , reaches a maximum and then decreases again forhigher DFR powers. The maximum N a is reached for P DF R ∼ µ W and P dep ∼
10 nW. An exact interpreta-tion of this behavior is difficult, as the depumper beamserves to pump the atoms both inside and outside of theHCPBGF into F = 1 where the intensities are very dif-ferent due to the diverging beams above the fiber. Also,the loading process temporally coincides with propaga-tion of atoms already inside the fiber and thus a tempo-ral modulation of the depumper power for the spatiallyseparated regions is not possible. Therefore we will tryto qualitatively understand the data in Fig. 9(b). Anoptimum value of the DFR power is to be expected asfor larger power scattered light (e.g. from the vacuumcell) can inhibit the dark funnel. To some degree thiscan be compensated by a so-called forced dark SPOT1where an additional depumper beam confines the pop-ulation in the non-cycled ground state [43]. For thisto work we require the depumper to be (i) sufficientlystrong to perform optical pumping ( R Ω dep / Γ D dt > dep = 1 . D inside the HCPBGF (spot size 2 . µ m)where the Rabi frequency at the MOT position is onlya fraction of 0 . dep (spot size ∼ µ m), opticalpumping is in principle efficient over the whole MOT-HCPBGF distance due to the long pumping times. Thelatter requirement can be estimated as follows. We as-sume that repumper stray light is mainly coming fromreflections at the vacuum cell windows, which are onlycoated at the outside. With an estimated maximum DFRRabi frequency of Ω DF R ∼ . D this shows that evenfor the lowest depumper power in Fig. 9(b) the depumperRabi frequency is at least comparable to stray light bythe DFR. For regions close to the fiber tip, we thereforeexpect the depumper to effectively confine the populationin the ground state F = 1.Interestingly, the datasets for Ω dep ∼ . D ( ◦ ) andΩ dep ∼ . D ( (cid:3) ) agree reasonably well with each otherfor DFR powers below 10 µ W whereas the data forΩ dep ∼ . D ( ◦ ) and Ω dep ∼ . D ( ⋄ ) do for DFRpowers larger 30 µ W. We believe this dependence and ob-servation of an optimum depumper Rabi frequency mightbe due to a light-shift induced by the depumper above thefiber where the depumper should be slightly off-resonantdue to the weaker FORT (note that the depumper fre-quency is adjusted such that it is nearly resonant insidethe fiber considering the light-shift by the FORT). Amore detailed modeling of the loading would be requiredto understand the data, which is, however, beyond thescope of this work.Nonetheless, our results clearly show the benefit of us-ing a DFR instead of mere cooling to guide the atomsmore efficiently into the HCPBGF.
4. Temporal evolution of the loading process
So far, the measurements in Sec. III C were all doneat the optimum time where N a is largest. In order tounderstand what possibly limits this optimum we noware interested in how N a evolves in time after the transferfrom the MOT has started. This allows us to determinethe initial loading as well as loss rates inside the FORT.To study the loading process in time we modified thesequence described above by adjusting the switch-off timeof the quadrupole field and the timings of all laser beamscorresponding to the time of each measurement shown inFig. 10 ( ◦ ). Here, a time of t ′ = 0 ms corresponds to t =1000 ms after the MOT loading sequence has started, i.e.,loading of the FORT begins. At around t ′ ∼
15 ms the number of atoms inside the HCPBGF rises rapidly witha rate of roughly 4000 ms − . According to images takenwith a CCD camera (see Fig. 8 inset) this time coincideswith the time the lower part of the atom cloud reachesthe fiber tip. During the next ∼
20 ms the rate decreaseswhile N a reaches its maximum value at t ′ ∼
50 ms. Thisagrees well with the data in Fig. 8 showing the evolutionof atoms above the fiber. The maximum N a is reachedaround 5 ms after the quadrupole field has been switchedoff. For times t ′ >
50 ms N a decreases during the courseof around 80 ms.To understand this behavior we first need to deter-mine the HCPBGF loading rate. This rate obviouslydepends on the evolution of the atomic density abovethe fiber where the FORT potential is deep enough toguide atoms into the HCPBGF. We estimate the regionfrom which atoms are guided into the HCPBGF to bein the range of 200 µ m (axially) and 50 µ m (radially),respectively. As these values are much smaller than the(initial) dimensions of the atomic cloud above the HCP-BGF ( l z , l r ) ∼ (3000 µ m × µ m) the measured atomnumber inside the HCPBGF is proportional to the localdensity ρ tip of the atom cloud at the fiber tip. In our case, FIG. 10. Measured N a inside the HCPBGF ( ◦ ) vs. time t ′ after the start of the transfer from the MOT. The MOTquadrupole field strength is schematically shown by the graydotted line. The error bars correspond to a 10% measure-ment uncertainty of N a . Numerical results of the evolu-tion based on Eqns. (6)-(7) for a eff = g = +9 .
81 m/s ,Γ L = β L = 0 are shown by the gray dashed-dotted line andfor a eff = +6 . , v = 0 . L = 40(5) s − and β L = 1(1) × − cm s − by the blue solid line. The load-ing rate R ( t ) is taken from the data shown in the inset withan amplitude scaling factor of S f = 2 to account for lossesduring the initial 5 ms inside the fiber. The red dashed lineshows the results when an acceleration of a eff = −
23 m/s inside the HCPBGF is assumed (see text). Inset: N a vs. t ′ for a loading time of 1 ms, i.e., the FORT is only on for abrief period just before the measurement. The error bars cor-respond to an uncertainty in the measurement of 1300 atoms.The red solid line shows the idealized evolution of N a usedfor the simulations. ∼ .
33 m/s. Thus, by loading the HCPBGF for only aperiod of 1-3 ms (by pulsing the FORT, see above) theatom cloud moves much less than its length of ∼ N a as a function of time t ′ therefore yieldsa value proportional to the local atomic density ρ tip ( t ′ )above the HCPBGF. The result of such measurement isshown in the inset of Fig. 10 ( ◦ ). N a ( t ′ ) increases fromaround 10 ms to 25 ms as the cloud approaches the fiber.Then, the atom number stays roughly constant (neglect-ing the data point at t ′ = 34 ms) for the next ∼
25 msas the cloud is held near the fiber tip. After the MOTquadrupole field is switched off at 45 ms N a ( t ′ ) decreasesquickly for t ′ >
48 ms. We thus estimate the peak load-ing rate as ∼ − which is in reasonable agreementwith the initial loading rate obtained from Fig. 10.The data shown in the inset of Fig. 10 can now beused to simulate how N a is expected to evolve in time.To this end we used an idealized atom number/densityevolution shown by the red solid line. In regular focused-beam FORTs of large N a the evolution of trapped atomscould now be modeled according to dN a dt = R ′ ( t ) − Γ ′ L N a − β ′ L N a , (5)where R ′ ( t ) is the time-dependent loading rate, Γ ′ L de-scribes losses due to collisions with background gas aswell as heating and β ′ L describes two-body collisions, re-spectively [42]. In the case of our extended fiber-basedFORT, however, propagation of the atoms through thefiber as well as their acceleration (see Sec. III B 2) has tobe taken into account. We therefore employ the follow-ing continuity equation with loss terms and consider achanging atomic velocity due to an effective acceleration a eff ∂∂t n + ∂∂z ( n · v a ) = − Γ L n − β L n , (6) ∂∂t v a = a eff . (7)Here, the unprimed rates Γ L , β L refer to the loss of den-sity instead of atom number. Using the loading rate R ′ ( t ) as shown in the inset of Fig. 10 by the red lineas initial condition at the fiber input (after conversioninto a density rate r ( t ) using the known FORT poten-tial and temperature of the atoms), as well as an initialvelocity of v = +0 . n ( z, t ) = e − Γ L T ( z ) β L R ( t − T ( z )) v Γ L (1 − e − Γ L T ( z ) ) r ( t − T ( z )) v , (8)where the time T ( z ) is defined in the appendix. Finally,the density can be converted into an atom number N a by integration over the axial and radial dimensions. We notethat since the data in the inset of Fig. 10 were obtained5 ms after the atoms entered the fiber, losses are presentduring this period that have to be accounted for by ascaling factor S f in the initial loading rate used for thesimulations.The gray dashed-dotted line in Fig. 10 shows the resultfor ideal conditions with an acceleration of +9.81 m/s due to gravity and vanishing loss rates Γ L , β L . Initially,the agreement with the measured data is reasonable. Forlonger times, however, a huge discrepancy in terms ofmaximum N a and decay is obvious that illustrates theimportance of losses and heating [42, 48]. The resultwhich matches our experimental data ( ◦ ) best is shownin Fig. 10 by the blue solid line with Γ L = 40(5) s − and β L = 1(1) × − cm s − , and S f = 2. An animationof the results can be found here. The factor S f deter-mines mainly the peak atom number that can be loaded.The same factor was also used for the simulation withvanishing loss rates (gray dashed-dotted line). The rateΓ L clearly dominates the losses, which is about 14 and5 times larger than reported by Okaba et al. [16] andHilton et al. [19], respectively, for Kagom´e-structuredfibers. It is also significantly larger than reported byKuppens et al. for a free-space trap, where they foundlosses being dominated by β L ∼ × − cm s − whileΓ L was found to be near zero [42]. Due to the large Γ L in our case, we can only determine the upper limit of as β L < × − cm s − . This value is roughly in agree-ment with the lower limits reported in [19] and [42] andcould currently be limited by light-assisted collisions dueto the depumper.The loss rate Γ L is determined by collisions of trappedatoms with background gas inside the fiber as well asheating. In both studies using Kagom´e-structured fibers[16, 19], collisional losses due to residual gases inside thefiber were assumed to dominate Γ L while heating wasneglected. Heating by photon scattering of the FORTcan be estimated for our trap as ˙ T ∼ µ K/s − [32]and is therefore negligible on the considered timescale.However, heating induced losses due to a possible ellip-tical polarization of the FORT, as discussed by Corwin et al. [48], resulting in a hopping between Zeeman levelsof different potential depths, can be estimated as around57(23) s − . This fits quite well to the loss rate determinedabove from Fig. 10. Here we assumed an ellipticity ofthe FORT inside the HCPBGF of 0.9, a hopping rate of690(280) s − according to the results in Sec. III A 3, po-tentials for states F = 1 , m F = +1 and F = 2 , m F = +2according to Eq. (1), and initial and final temperaturesof T i = 1 mK and T f = 4 . L ,can be explained well by heating and the rate β L is of sim-ilar magnitude as in other FORTs. It also explains wellthe time required to reach the steady-state temperatureas shown in Fig. 5(b) (iii) As was shown in Sec. III B 2,the atoms experience an acceleration of a = −
23 m/sinside the fiber and reverse their direction when reso-nant scattering by the FORT is not suppressed. Thiswas the case, e.g., in our previous work [39]. Suppressionof resonant scattering did however not affect the load-ing efficiency. It might be surprising why this effectiveupward acceleration resulting in the atoms being pushedout at the top of the fiber does not affect the loadingefficiency. We therefore tried to simulate these condi-tions. However, the theoretical model used to simulatethe temporal and spatial density evolution can only beused when the atoms are traveling into a single direction.To model the case of direction reversal we had to employa cruder model which divides the initial loading rate intoslices, each assigned with a distinct velocity that wouldchange in each time step according to the acceleration.The results are shown in Fig. 10 by the red dashed line.Surprisingly, the results are quite similar to the case ofa falling cloud without reversal of direction. This is dueto the fact that the atoms are pushed out of the fiberonly at late times where losses, dominated by heatingand collisions, have already significantly reduced N a asillustrated in an animation of the results. (iv) Currently,Γ L limits our loading efficiency. In-fiber cooling (duringthe loading and guiding process) might therefore lead toa larger loading efficiency. (v) Optical pumping betweenthe hyperfine ground states via a Raman coupling by thebroadband FORT significantly affects the acceleration forour loading scheme. Employing a narrowband FORT istherefore crucial when using such system for interferom-etry [22].We conclude that the loading efficiency is limited bythe number of atoms and density above the HCPBGF aswell as the loss rates. Especially the loss rate Γ L seemsto be dominated by heating inside the slightly ellipticalFORT. The simulation shows that for loss rates such asin [54], the loading efficiency can be easily increased bya factor beyond 1.8. IV. SUMMARY AND OUTLOOK
We presented a series of measurements along with the-oretical calculations to carefully characterize the loadingof laser-cooled atoms from a MOT into a FORT inside ahollow-core fiber. Although optical access in such a fiberis restricted to the input and output ports, we demon-strated spatially-resolved probing of the atoms inside thefiber with a resolution of a few millimeters using EITand a time-of-flight technique. Besides the FORT po-tential parameters (trap depth and frequency, scatteringrates), we determined the atomic ensemble temperatureas about 1/4 of the trap depth, an in-fiber acceleration of the atoms of ∼ . ∼ cm − and loss rates of 40(5) s − , dominated by heating, and1(1) × − cm due to two-body collisions. We foundgood agreement between the experimental data and thesimple theoretical models used. Although our results forour fiber-guided FORT basically confirm the expecta-tions for free-space FORTs, there are also some signifi-cant differences. Most notably, propagation of the atomsinside the extended fiber has to be taken into account,which is not required for standard focused-beam FORTs.We also found indications that free evaporative coolingtakes place in the FORT after the atoms entered the fiber.Analysis of the velocity of the atoms inside the fiber pro-vided an initially surprising acceleration which preventeda complete filling of the fiber in our previous work. Af-ter identifying resonant scattering by the FORT as thesource of this acceleration, we could then resolve this is-sue by application of a suitable notch filter resulting asignificant reduction of acceleration by the laser beams.Our results have direct implications for applications us-ing such atom-filled hollow-core fibers. First of all, the re-sults on the loss rates show that there is significant roomfor improvement on our record d opt = 1000 achieved sofar [15]. An even larger d opt would be required for exper-iments on many-body physics with light [31, 61], whichseems to be feasible. Secondly, as the loading efficiency isin part limited by density-dependent losses, in-fiber cool-ing [62] will have to be applied with care. On the onehand, cooling will lead to a reduction of inhomogenousbroadening and compensation of heating. On the otherhand, the atomic number density will also increase if theFORT potential is kept constant, leading to increasedlosses. Thus, careful modeling and characterization ofthe loading will be required for achieving optimum con-ditions, as we have done in our present work. Finally,our results on the atomic velocity inside the fiber showsthat a careful analysis is required when such a platformis used for atom interferometry [22]. ACKNOWLEDGMENTS
We thank M. Schlosser for helpful discussions on theFORT bandwidth and comments on the manuscript, C.-Y. Hsu and M. Coelle for assistance with time-of-flightmeasurements, and the group of T. Walther for pro-viding us with a home-made ultra-low noise laser diodedriver with high modulation bandwidth. This projecthas received funding from the European Union’s Hori-zon 2020 research and innovation programme under theMarie Sklodowska Curie grant agreement No. 765075.4
Appendix A: Appendix: Simulation of opticalpumping and acceleration by the FORT anddepumper beams
Both FORT and depumper are basically alwaysswitched on during loading of the HCPBGF. Therefore,optical pumping by both fields has to be considered, asit affects the population redistribution between the twoground states F = 1 , a = F/m . Here, F = ~ k Γ ρ ee is the force onan atom induced by absorption of photons followed byspontaneous emission at rate Γ and ρ ee is the populationin the excited state [59].
1. Population rate equation model
In order to calculate the relevant state-resolved popu-lations in our multi-level system, we numerically solved the following population rate equations, where the in-dex i corresponds to the following hyperfine states in as-cending energetic order (ground states F = 1 ,
2, D line F ′ = 1 , , F = 1 , ( R ) F = 690(280) s − (see Sec. III A 3) whilespontaneous scattering of the FORT at rate Γ F = 186 s − leads predominantly to Rayleigh scattering due to an in-terference effect between the D , lines [55] and there-fore does not affect the populations. We thus only needto consider the ground state populations as well as theexcited states coupled by the depumper at the D line.Ω d ( δ i ) are the depumper Rabi frequencies depending onthe detuning from levels F ′ = 1 , ,
3. The branching ra-tios are taken from [63].˙ N = Γ D (cid:18) N + 12 N (cid:19) −
58 Γ ( R ) F N + 38 Γ ( R ) F N , (A1)˙ N = Γ D (cid:18) N + 12 N + 1 N (cid:19) + 58 Γ ( R ) F N −
38 Γ ( R ) F N , (A2) − Ω d ( δ ) ( N − N ) − Ω d ( δ ) ( N − N ) − Ω d ( δ ) ( N − N ) , ˙ N = − Γ D N + Ω d ( δ ) ( N − N ) , (A3)˙ N = − Γ D N + Ω d ( δ ) ( N − N ) , (A4)˙ N = − Γ D N + Ω d ( δ ) ( N − N ) . (A5)Furthermore, we assumed an unpolarized FORT anddepumper, i.e., we neglected the Zeeman substructureand only considered stimulated emission by the near-resonant depumper.
2. Acceleration by the FORT and depumper
The rate equations (A1)-(A5) can be either solved nu-merically, where the effective acceleration a eff = g − F/m of the atoms is then given by a eff = g − ~ km Γ F + Γ D X i =3 N i ! , (A6)or analytically by using the stationary solutions. Then,the acceleration can be determined as a eff = g − ~ km (cid:18) Γ F + 58 Γ ( R ) F p (cid:19) , (A7)where the first term is due to downward acceleration bygravity and the other terms due to the upward directedlight pressure by the FORT and depumper beams. The parameter p is between 1.5 and 2 and depends on thedetuning of the depumper from the transition F = 1 → F ′ = 2. For the previously used FORT scattering rateswe thus obtain a eff ∼ , close to the measuredvalue. Appendix B: Analytical derivation of the densityevolution inside the fiber
We consider the propagation of an ensemble inside afiber where atoms are moving with positive accelerationinto a single direction, i.e., there is no change of direction.The atom density n ( z, t ) inside the fiber at time t inposition z is determined by the atom density at beginning n ( z = 0 , t ) in the previous time t − T ( z ), where T ( z ) isthe time to travel from z = 0 to z with the acceleration a eff . This time is a solution of the equation z = v T + 12 a eff T , (B1)5where v is the initial velocity. The solution of this equa-tion reads T ( z ) = zv p a eff z/v + 1 . (B2)If the fiber is loaded with the density rate r ( t ), then thedensity n ( z = 0 , t ) is given by n ( z = 0 , t ) = r ( t ) /v . (B3)Thus, the density n ( z, t ) in time t at the position z isequal to the density n ( z = 0 , t − T ( z )) but subject tocollisional losses during the time T ( z ). 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