Preparation of individual magnetic sub-levels of $^4$He($2^3$S$_1$) in a supersonic beam using laser optical pumping and magnetic hexapole focusing

We compare two different experimental techniques for the magnetic-sub-level preparation of metastable $^4$He in the $2^3$S$_1$ level in a supersonic beam, namely magnetic hexapole focusing and optical pumping by laser radiation. At a beam velocity of $v = 830\,$m/s, we deduce from a comparison with a particle trajectory simulation that up to $99\,$\% of the metastable atoms are in the $M_{J^{''}} = +1$ sub-level after magnetic hexapole focusing. Using laser optical pumping via the $2^3$P$_2-2^3$S$_1$ transition, we achieve a maximum efficiency of $94\pm3\,$\% for the population of the $M_{J^{''}} = +1$ sub-level. For the first time, we show that laser optical pumping via the $2^3$P$_1-2^3$S$_1$ transition can be used to selectively populate each of the three $M_{J^{''}}$ sub-levels ($M_{J^{''}} = $ -1, 0, +1). We also find that laser optical pumping leads to higher absolute atom numbers in specific $M_{J^{''}}$ sub-levels than magnetic hexapole focusing.


I. INTRODUCTION
In a He gas discharge, two long-lived, excited ("metastable") atomic levels are formed by electron-impact excitation from the 1 1 S 0 electronic ground level: the 2 3 S 1 level (electronic energy E = 19.8 eV [1], natural lifetime τ = 7870 s [2]) and the 2 1 S 0 level (E = 20.6 eV [1], τ = 19.7 ms [3]). In the following, the metastable He atoms are referred to as spinpolarized, when only a single magnetic sub-level of He(2 3 S 1 ) is populated. Such spinpolarized metastable He (He SP ) is used for a wide range of applications. Special interest is currently devoted to He magnetometry for the quantum sensing of very small magnetic fields, e.g. see Refs. [4,5]. In metastable atom electron spectroscopy [6], also referred to as metastable de-excitation spectroscopy, He SP has, for example, been used for probing surface magnetism [7]. In atom optics, He SP has found applications in nanolithography, as well as in atomic waveguides and beamsplitters for atom interferometry [8,9]. He SP also serves as a source of polarized electrons [10,11] and ions [12], e.g., for atomic and high-energy nuclear scattering experiments. Besides that, spin-polarized samples of 3 He(2 3 S 1 ) are used for biomedical imaging, e.g., to visualize the human lung [13][14][15].
Supersonic beams of He SP are typically produced by optical pumping [16], as well as by magnetic (de-)focusing and magnetic deflection. Optical pumping of 4 He(2 3 S 1 ) via the 2 3 P J − 2 3 S 1 transition (where J = 0, 1, 2) at a wavelength of λ = 1083 nm has first been achieved by Franken, Colegrove and Schaerer using a helium lamp [17][18][19]. A few years later, also the optical pumping of the 2 3 S 1 level of the 3 He isotope has been demonstrated using a similar setup [20,21]. The more recent use of narrowband laser radiation has proven to be particularly efficient for the optical pumping of He SP [22][23][24][25].
A comparison between the different techniques for He SP preparation in a supersonic beam is of paramount importance for experimental design considerations. In this article, we describe the results of a comparative study aimed at the laser optical pumping of 4 He(2 3 S 1 ) into a single M J sub-level (where M J = −1, 0, +1) and at the magnetic hexapole focusing (defocusing) of the M J = +1 (M J = −1) sub-level of 4 He(2 3 S 1 ) using an array of two magnetic hexapoles. We have determined the efficiency for M J -sub-level selection using low-cost fluorescence and surface-ionization detectors, respectively, which can easily be implemented in other experiments.

II. EXPERIMENTS
Major parts of the experimental setup have already been described elsewhere [41,42]. Briefly, a pulsed 4 He beam is produced by a supersonic expansion of 4 He gas from a high-pressure reservoir (30 − 40 bar) into the vacuum using a home-built CRUCS valve [43] (30 µs pulse duration). An electron-seeded plate discharge (attached to the front plate of the valve) is used to excite an ≈ 4 · 10 −5 fraction of He atoms from the 1 1 S 0 electronic ground level to the two metastable levels, 2 1 S 0 and 2 3 S 1 , referred to as He * hereafter [41]. After passing through an 1 mm-diameter skimmer at a distance of 130 mm from the valve exit, the supersonic beam enters a second vacuum chamber, in which a specific magnetic sub-level of the 2 3 S 1 level is prepared using laser optical pumping or selected using magnetic hexapole focusing (see below). The distance between the skimmer tip and the center of the optical pumping region (hexapole magnets) is 228 mm (331 mm). The He * flux and the He * beam velocity are determined using Faraday cup detection at well-known positions along the supersonic beam axis y.
For the experiments on optical pumping, the pulsed valve is operated at room temperature resulting in a supersonic beam of He * with a mean longitudinal velocity of v = (1844±6) m/s (250 m/s full width at half maximum, FWHM). For the experiments on magnetic hexapole focusing, the pulsed valve is cooled by a cryocooler (CTI Cryogenics, 350CP), and the valve temperature is actively stabilized to 42 K using PID-controlled resistive heating. This results in a supersonic beam of He * with a mean longitudinal velocity of v = (830 ± 17) m/s (≈ 130 m/s FWHM).

A. Laser optical pumping
The energy-level schemes and the experimental setup used for laser optical pumping are shown in Figs. 1 (a) and (b), respectively. Optical pumping is achieved by laser excitation via the 2 3 P 1 − 2 3 S 1 transition or via the 2 3 P 2 − 2 3 S 1 transition at λ = 1083 nm, respectively.
The laser light for optical pumping is generated by a combination of a fiber laser and a fiber amplifier (NKT Photonics, Koheras BOOSTIK Y10 PM FM, 2.2 W output power, 10 kHz line width). The laser frequency is stabilized using frequency-modulated, saturated absorption spectroscopy in a He gas discharge cell. Since the frequency difference between the 2 3 P 1 and 2 3 P 2 spin-orbit levels is only ∆f ≈ 2 GHz [1], the laser frequency can be changed in between the different transitions without effort.
The laser light is guided to the vacuum chamber using a polarization-maintaining singlemode fiber, where it is collimated to a beam diameter of 2w 0 ≈ 14 mm (w 0 is the Gaussian beam waist). Before the laser beam enters the vacuum chamber, it passes a polarizing beam splitter for polarization clean-up, and a quarter wave plate for the production of circularly polarized light. Inside the vacuum chamber, the laser beam crosses the supersonic beam at right angles and parallel to the direction of the magnetic field produced by two coils in near-Helmholtz configuration (radius R = 55 cm). The thus produced homogeneous magnetic field of B z ≤ 4.5 G provides a uniform quantization axis for all He atoms in the supersonic beam.
The level-preparation efficiency is determined by measuring the laser induced-fluorescence (LIF) of the He atoms in the direction perpendicular to the supersonic beam and the laser beam. The fluorescence light is collected and focused onto an InGaAs photodiode (Hamamatsu, 1 mm active area diameter, photosensivity of R PD = 0.63 A/W at λ = 1080 nm) using two anti-reflection-coated, aspheric lenses (Thorlabs, 75 mm diameter, 60 mm focal length). Due to the symmetric lens configuration (as shown in Fig. 1 (b)), the fluorescence collection region in the yz plane is expected to be of the same size as the detection region, which is given by the active area of the photodiode. The output of the photodiode is amplified using a home-built transimpedance amplifier with a gain of G PD ≈ 5 · 10 5 V/A. A rotatable linear polarizer (Thorlabs, extinction ratio > 400 : 1 at λ = 1083 nm) is mounted in between the lenses in order to analyze the polarization of the fluorescence light. All the optical components of the fluorescence detector are placed into a single lens tube system to ensure the correct alignment of the optical parts. The entire detector assembly is positioned on a translational stage outside the vacuum chamber which can be moved along the y axis.
Under normal operating conditions, the number of He atoms in the 2 3 S 1 level is ≈ 10 9 /pulse, as inferred from the signal on the Faraday-cup detector [41]. For excitation via the 2 3 P 2 − 2 3 S 1 transition, the time-dependent signal of the photodiode has a peak voltage of U PD,max ≈ 41 mV and an FWHM of 27 µs. The peak flux of detected photons is then inferred from U PD,max usinġ where h is Planck's constant and ν is the corresponding transition frequency. From these measurements, we infer a root-mean-square noise amplitude of U noise = 6.4 mV for a single measurement which improves to U noise = 0.4 mV by averaging over 300 gas pulses. This results in a signal-to-noise ratio of At SN R = 0 dB, we thus infer a detection limit ofṄ ph,lim ≈ 1 · 10 11 photons/s (Ṅ ph,lim ≈ 7 · 10 9 photons/s) for a single measurement (300 averages).

B. Magnetic hexapole focusing
For the magnetic focusing of He(2 3 S 1 , M J = +1), we use a set of two Halbach arrays [44,45] in hexapole configuration, sketched in Fig. 1 (c), whose design has already been described previously [46,47]. Each hexapole array consists of 12 magnetized segments (Arnold Magnetic Technologies, NdFeB, N42SH, remanence of B 0 = 1.3 T) which are glued into an aluminium housing and placed on a position-adjustable rail at a center-to-center distance of 14.6 mm.
To determine the focusing properties of the magnet assembly, a thin stainless-steel wire (diameter d wire = 0.2 mm, labelled as "W" in Fig. 1 (d)) is used as a position-sensitive Faraday-cup-type detector. Its position along the y and x axes can be varied by a maximum of 180 mm and 50 mm, respectively, using a set of two precision linear translators. A second Faraday-cup detector (labelled as "FC" in Fig. 1 (d)), i.e., a stainless-steel plate of 30 mm diameter, is placed behind the wire detector to determine the He * beam velocity. In order to compare the sub-level preparation efficiencies, we define for producing a specific magnetic sub-level population ρ( . For the 2 3 P 2 − 2 3 S 1 transition, the efficiency for optical pumping into the where I F (P B z ) and I F (P ⊥ B z ) are the fluorescence intensities for emission at polarizer axes P B z and P ⊥ B z , respectively. efficiency for pumping into the 2 3 S 1 , M J = +1 (−1) sub-level is determined using where I F (σ +(−) ) and I F (σ + + σ − ) are the fluorescence intensities for excitation with pure σ +(−) polarization and with a mixture of σ + and σ − polarization, respectively. sub-level preparation efficiency is thus obtained using where I F (π B x ) and I F (π ⊥ B x ) are the fluorescence intensities for excitation using πpolarized light in a direction parallel and perpendicular to the magnetic field component B x , respectively.

Optimization of the sub-level preparation efficiency
We have identified several parameters which strongly affect the sub-level preparation efficiency: the interaction time between the excitation laser light and the sample, the laser intensity, the magnetic field strength and the purity of the input polarization.
During the excitation process, an atom typically scatters several photons before it is We have studied the influence of the interaction time on the sub-level preparation efficiency η i by monitoring the fluorescence intensity at different fluorescence detector positions along the y axis. As can be seen from the colored markers in Fig. 5, the efficiency η +1 for σ + excitation of the 2 3 P 2 − 2 3 S 1 and 2 3 P 1 − 2 3 S 1 transitions, respectively, increases to a nearly constant value as the detector is moved towards the midpoint of the excitation laser beam. This confirms that, in our experiment, the interaction time does not limit the sub-level preparation efficiency.
We have simulated the population transfer process using rate-equation calculations. A detailed description of the rate-equation model can be found in App. A. The best fit to our experimental data for excitation via the 2 3 P 2 − 2 3 S 1 transition and via the 2 3 P 1 − 2 3 S 1 transition, respectively, is found by assuming that the excitation light is a mixture of 95 % σ + -and 5 % σ − -polarized light. The admixture of wrongly polarized light also explains why the observed sub-level preparation efficiency is below 100 %. In addition to that, as can be seen from Fig. 1 (a), the relative transition strengths for optical pumping with wrongly polarized light is 1/6 for the 2 3 P 1 − 2 3 S 1 transition, while it is only 1/30 for the 2 3 P 2 − 2 3 S 1 transition. Thus, optical pumping via the 2 3 P 1 −2 3 S 1 transition is more sensitive to wrongly polarized excitation light which explains the observed difference in the sub-level preparation efficiency. In our setup, such an admixture of wrong input polarization might be caused by imperfections of the quarter wave plate or by the birefringence of the vacuum window. Secondly, the laser intensity has to be high enough so that the laser-induced power broadening compensates for a detuning of the laser frequency from the atomic resonance. This detuning is caused by the Doppler broadening due to the transverse velocity of the atoms (∆ Doppler ≈ 12 MHz FWHM) and by the Zeeman shift of the atomic levels (∆ Zeeman < 14 MHz). The FWHM of the power broadening can be expressed as where I is the intensity of the laser light and I sat ≈ 0.16 mW/cm 2 (assuming a two-level system) is the saturation intensity of the transition. Therefore, in order to compensate for the Doppler broadening and for the Zeeman shift, the laser intensity has to be I ≥ 12 mW cm 2 , corresponding to a laser power of ≥ 9 mW for our experiments. From Fig. 6, we can see that the sub-level preparation efficiency for σ + excitation of the 2 3 P 2 − 2 3 S 1 transition is constant for laser powers P > 50 mW. Unfortunately, measurements of the sub-level preparation efficiency at lower laser powers suffer from low signal intensities and are thus less representative. For σ + excitation of the 2 3 P 1 −2 3 S 1 transition, we observe that more than 300 mW of laser power are required to reach a constant sub-level preparation efficiency. This power difference might be attributed to a weaker power broadening of the 2 3 P 1 − 2 3 S 1 line compared to the 2 3 P 2 −2 3 S 1 line as a result of a higher saturation intensity for this transition.
As both transitions have the same line width, the same initial level and approximately the same transition frequency, we can see from Eq. (A4) that the squared dipole matrix elements |µ J | 2 are proportional to the degeneracy factors 2J + 1. As I sat ∝ 1/ |µ J | 2 , it follows that Thirdly, the magnetic bias field has to be large enough to ensure a uniform quantization axis within the optical pumping region so that the contributions of stray fields along other spatial directions is small.
In Fig. 7, a scan of the sub-level preparation efficiency η +1 for σ + excitation of the 2 3 P 2 − 2 3 S 1 and 2 3 P 1 − 2 3 S 1 transitions, respectively, is shown as a function of the magnetic field component B z . The highest efficiency is achieved at field strengths between 2 G ≤ B z ≤ 3 G for both transitions. This magnetic field range is in line with previous observations reported in the literature [24,49,50]. At magnetic field strengths B z > 3 G, the sub-level preparation efficiency for excitation via the 2 3 P 1 − 2 3 S 1 (2 3 P 2 − 2 3 S 1 ) transition is decreased (remains constant) compared to the optimum B z field range. This is consistent with a decreased scattering rate at higher magnetic fields caused by the increased Zeeman detuning. We have also analyzed the influence of stray magnetic fields along the x and y directions.
Using a high-accuracy, three-axis Gauss probe (Stefan Mayer Instruments, ≤ 1 G, 0.05 mG resolution), we obtain B x ≈ 0.2 G and B y ≈ 0.1 G. At B z = 3 G, this results in an angle of θ = B 2 x + B 2 y /B z ≈ 80 mrad between the magnetic field and the z axis (cf. Gillot et al. [49]). We have observed that a further compensation of the magnetic stray fields using additional coils along the x axis (resulting in θ < 40 mrad) does not result in an improved sub-level preparation efficiency. In addition to that, a non-perfect alignment of the laser propagation direction parallel to the quantization axis can induce a similar limit to the achievable sub-level preparation efficiency as the presence of magnetic stray fields.
Furthermore, small magnetic-field inhomogeneities within the interaction region, resulting from e.g. a not perfect Helmholtz coil arrangement or electronic devices in the laboratory, may also limit the sub-level-preparation efficiency.
In summary, we conclude that the imperfect polarization of the laser light (see discussion above) is the main limiting factor for the sub-level preparation efficiency.

Comparison with literature values
In Tab. I, we present a summary of the maximum sub-level preparation efficiencies η i,max obtained from our experimental data, and a comparison with literature values. As can be seen from the table, our η i,max values are in good agreement with previous results for the laser optical pumping of He(2 3 S 1 ). To the best of our knowledge, we are the first to obtain a maximum efficiency > 90 % for optical pumping into M J = 0. The only previous attempt to selectively populate M J = 0 has been by Giberson et al. [22] using linearly polarized light resonant with the 2 3 P 0 − 2 3 S 1 transition and propagating along the quantization axis.
For optical pumping into the spin-stretched sub-levels (M J = ±1), we report a somewhat lower maximum efficiency than previous groups which we attribute to the aforementioned imperfect laser polarization in our experiments. In addition, we see a deviation of η i,max for optical pumping with σ + and σ − -polarized light especially while exciting via the 2 3 P 1 − 2 3 S 1 transition. This might be induced by a systematic asymmetry in our setup resulting from e.g. small magnetic-field inhomogeneities as discussed above.

B. Magnetic hexapole focusing
The red circles in Fig. 8 show the results of a series of measurements which were obtained using the setup for the magnetic hexapole focusing of He(2 3 S 1 , M J = +1) (cf. Figs. 1 (c) and (d)). In order to interpret these results, we did numerical three-dimensional particle trajectory simulations in MATLAB. For these simulations, we use random number distributions for the particle positions and velocities (deduced from the experimental data obtained at the Faraday-cup detector) and a velocity-Verlet algorithm. An intial number of 5 · 10 6 particles in each Zeeman sub-level of He(2 3 S 1 ) and He(2 1 S 0 ) are propagated at a time. The magnetic field by the two Halbach arrays is implemented using the analytical expressions given in Ref. [47]. Particles are removed from the simulation if their transverse position inside a Halbach array exceeds the 3.0 mm inner radius of the assembly (cf. Fig. 1 (c)).
In each xy detection plane, the output of the trajectory simulation (black lines in Fig.   8) is analyzed over a certain interval of x positions corresponding to the diameter of the wire detector. The experimental results are matched to the simulated data by comparing  The given uncertainties (two standard deviations) of our experimental results are statistical only.  sub-levels. This is also consistent with previous observations [34]. At time t 0 = 0, we assume a He(2 1 S 0 )/He(2 3 S 1 ) ratio of 66 % which is in line with the results of previous measurements in our laboratory [42]. He * signal intensities obtained from a numerical particle trajectory simulation.

IV. CONCLUSION
We conclude that both laser optical pumping and magnetic hexapole focusing are very efficient methods for the selective preparation of magnetic sub-levels of He(2 3 S 1 ) in a supersonic beam. We find that optical pumping into the spin-stretched sub-levels of He(2 3 S 1 ) via the 2 3 P 2 − 2 3 S 1 transition is more efficient than excitation via the 2 3 P 1 − 2 3 S 1 transition. The best performance is achieved for σ +(−) excitation via the 2 3 P 2 − 2 3 S 1 transition yielding a maximum efficiency of 94 ± 3 % (90 ± 3 %) for optical pumping into M J = +1 (M J = −1).
Magnetic hexapole focusing is observed to be highly sub-level selective at low forward velocities of the supersonic beam. At v = 830 m/s and at the focal point of the hexapole lens system, we infer that up to 99 % of the metastable atoms are in the M J = +1 sub-level, if an 0.2 mm-diameter region around the center of the supersonic beam axis is considered.
The magnetic-hexapole-sub-level-selection technique is attractive, because it allows for the quantum-state manipulation of all atomic and molecular species with non-zero spin. Compared to optical pumping, the mechanical setup for magnetic focusing is rather simple, especially when commercial magnets are used [53].
However, optical pumping has several advantages compared to magnetic hexapole focusing. While magnetic focusing is limited to the preparation of sub-level-selected samples in low-field-seeking sub-levels only, optical pumping allows for the selective population of all M J sub-levels, as shown here for the 2 3 P 1 − 2 3 S 1 transition in He. For optical excitation with π-polarized light, we obtain an efficiency of 93 ± 4 % for population transfer into M J = 0. The creation of a pure M J = 0 sub-level might be possible by using magnetic focusing as well but would require a strong overfocusing of the low-field seeking quantum states. In our experiments, this may be realized by further reducing the forward velocity of the He* atoms or by using a longer hexapole magnet array. However, we observe that the number of metastable helium atoms decreases by a factor of ≈ 2 when the valve temperature is decreased from 300 K to 50 K. At the same time, the peak He * flux within the gas pulse decreases by a factor of ≈ 50, as the longer flight time to the detection region leads to a larger longitudinal spreading of the beam. Optical pumping can be applied independently of the velocity of the atoms as long as the discussed requirements for reaching the equilibrium sublevel efficiency are fulfilled. Thus, this technique results in a greater flexibility in choosing the valve temperature and, as mentioned above, running the valve at higher temperatures leads to much higher peak fluxes of M J -sub-level-selected atoms. These high peak fluxes are particularly important for applications which benefit from high local densities, such as collision experiments. Besides that, optical pumping relies on a transfer of population from a statistical mixture of M J sub-levels into a single sub-level, whereas magnetic hexapole focusing relies on the spatial focusing (defocusing) of the desired (unwanted) M J -sub-level population. Further transmission losses are due to an aperture which has to be inserted into the beam path behind the magnet assembly in order to eliminate contributions by the 2 3 S 1 , M J = 0 and 2 1 S 0 , M J = 0 sub-levels, whose motion is not influenced by a magnetic field.
In the future, we will use the presented sources of M J -sub-level-selected He(2 3 S 1 ) for quantum-state-controlled Penning-ionization studies [41]. Furthermore, magnetic-sub-levelselected beams of He(2 3 S 1 ) are useful as a starting point for the generation of coherent superposition states. The coherent control of Penning and associative ionization cross sections with such superposition states, for instance, involving the M J = 0 sub-level of He(2 3 S 1 ), has been predicted [55]. In addition to that, helium is of particular interest for high-precision tests of few-electron quantum electrodynamics theory, as it is the simplest two-electron atom [56,57]. Accurate transition frequency measurements have been performed on ultracold trapped samples [58][59][60] as well as on atomic beams [28,61,62]  The equations used for the characterization of the optical pumping process are of the forṁ where N i and N j denote the populations in the i-th and j-th magnetic sub-levels of He(2 3 S J =1 ) and He(2 3 P J ), respectively, and Γ = 1/τ is the spontaneous decay rate of the excited sub-levels according to their natural lifetime τ . The matrix elements for the excitation rate and for the branching ratio between the i-th and j-th magnetic sub-level are denoted as W ij and ξ ij , respectively. The former are expressed as where L = 0 and L = 1 are the quantum numbers for the orbital angular momenta of the lower and the upper level, S = 1 is the quantum number for the total spin and q = M J ,i − M J ,j = 0, ±1 denote π and σ ∓ polarization, respectively. The spontaneous decay rate Γ is used to calculate the squared reduced dipole matrix element |µ L | 2 : Here, ω 0 is the zero-field transition frequency.
In addition to that, we consider a Gaussian distribution of the laser intensity along the y axis, where w 0 is the beam radius and I 0 = f · 2P laser /(πw 2 0 ) is the peak intensity calculated from the laser power P laser . The factor f = 0.1341 is used to correct for the limited spatial overlap between the laser beam and the supersonic beam. We use the mean forward velocity of the He * beam in order to transform from the time frame of the rate equations to the position frame of the intensity distribution and to the detector position along the y axis.
The matrix elements for the branching ratio are calculated using the 3-j symbol (A7)

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.