The possibility of using an atom chip and a diffraction grating to form a compact magneto-optical trap for ytterbium atoms, which can be used to develop compact atomic interferometers and optical clocks based on ultracold atoms, has been studied. An experiment on the laser cooling of the ^{171}Yb and ^{174}Yb isotopes in a first‑stage magneto-optical trap has been carried out to determine initial requirements for the mentioned elements. The design of the atom chip forming a magnetic field gradient up to 60 G/cm has been calculated. The optimal configurations of the diffraction grating that allow forming both the first- and second-stage magneto-optical traps have been evaluated.

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## 1 INTRODUCTION

Ultracold atoms are actively studied in many laboratories worldwide because they can be potentially applied in quantum computing [1], quantum metrology [2], fundamental physics [3], and other fields [4–6]. Ultracold atoms are used in atomic interferometers in order to reach the extreme sensitivity. Gravimeters [7], gradiometers [8], gyroscopes [9], and optical clocks [4] based on ultracold atoms have been already demonstrated. The accuracy and sensitivity of such devices are higher than the respective parameters of “classical” analogs [4, 10]. This circumstance makes it possible to use them not only for applied problems but also for some fundamental problems, e.g., for the detection of gravitational waves [11].

Atomic rubidium, which is an alkali metal, is usually used to develop atomic interferometers. This approach is characterized by a relative simplicity of obtaining ultracold atoms and their subsequent utilization. Atoms of other groups of the periodic system, primarily Sr and Yb atoms, are usually used as ultracold atoms in order to develop optical clocks. These atoms are used due to their metrological characteristics such as the existence of transitions with narrow spectral lines (also called “clock”) in the optical range and their insensitivity to thermal radiation. However, the use of such atoms complicates experimental setups. For this reason, the development of atomic interferometers with Sr and Yb atoms is complicated. First experimental studies on the development of such atomic interferometers were considered only recently [12, 13] and were carried out in the case of Sr atoms in [14].

Some characteristics of alkaline-earth metal atoms make them promising for application both in optical clocks and in atom interferometry. First, due to their zero orbital angular momentum in the ground state \(^{1}{{S}_{0}}\), these atoms are less sensitive, compared to alkali metal atoms, to perturbations caused by fluctuations of magnetic fields. Second, the presence of transitions in ultraviolet and near-ultraviolet spectral ranges (461 nm for Sr and 399 nm for Yb) will allow increasing the momentum transferred to an atom in an atomic interferometer, which is necessary for increasing the sensitivity. Third, the clock transition makes it possible to develop atomic interferometers with an extremely low level of phase noise [15] for precise measurements of gravity.

The development of atomic interferometers and optical clocks based on ultracold atoms requires magneto-optical traps (MOTs), where atoms are cooled to several microkelvins. To construct optical clocks, such atoms are trapped in optical lattices for precise spectroscopy [16, 17]. In the atomic interferometer, atoms leaving the MOT interact with different laser pulses. The gravitational acceleration of freely falling atoms in the gravitational potential can be determined from, e.g., the interference pattern. The accuracy of measurements will be determined by the standard quantum limit [18]. For this reason, the MOT should form an atomic ensemble with a large number of atoms at the minimum temperature. Furthermore, the dimensions of the MOT should be minimal in order to construct compact devices, which can be placed on mobile platforms [19].

The MOT consists of two fundamental elements: the magnetic and laser fields (see Fig. 1). Atoms are cooled in the laser field. Moreover, cooled atoms are subjected to the force depending on the coordinate. This allows one to concentrate atoms at a certain spatial point using a nonuniform magnetic field.

The magnetic field gradient in classical three-dimensional MOTs (see Figs. 1a and 1c) is produced by two coils in an anti-Helmholtz configuration (see Fig. 1a). This approach is inconvenient because of large dimensions and high energy consumption, which are necessary for the production of magnetic field gradients of 10–50 G/cm, depending on atoms that should be cooled in the MOT. Laser cooling is implemented by using six beams (see Fig. 1c); consequently, six windows for optical access are necessary in the vacuum chamber. In addition, the intensity and polarization of each beam should be controlled to reach the optimal temperature and the number of atoms in the MOT.

To develop transportable optical clocks and atomic interferometers, every elements of the MOT can be optimized in dimensions using planar technologies. An atom chip can be used to produce the necessary configuration of magnetic fields (see Fig. 1b) [20]. The application of atom chips is based on high magnetic field gradients, which occur near current-conducting microwires. At the same time, this approach does not require a high power consumption [21], which is an advantage for the development of compact devices. In turn, to form the configuration of laser fields necessary to cool atoms, one can use conical hollow mirrors [22], inclined prisms, and diffraction gratings [24]. In the last case, the planar technology is used (see Fig. 1d), which ensures not only smaller dimensions but also the possibility of the combination with the atom chip to produce the united system of forming the MOT. This was demonstrated on rubidium atoms in [25].

In this work, using experimental data, we consider for the first time the possibility of utilizing diffraction gratings and atom chips to develop a MOT for ytterbium atoms. The compact MOT will make it possible to form transportable atomic interferometers and optical clocks with the advantages of ytterbium including the ultranarrow clock transition, low sensitivity to perturbations by magnetic fields, increased momentum transferred to atoms in the atomic interferometer.

## 2 EXPERIMENTAL DETERMINATION OF THE REQUIRED PARAMETERS OF THE MAGNETIC FIELD AND LASER RADIATION FOR Yb ATOMS IN THE MAGNETO-OPTICAL TRAP

Ytterbium atoms can be cooled and localized in two stages [26]. First, cooling is performed in the “first stage” MOT formed by the magnetic field gradient and laser radiation at the wavelength λ_{B} = 399 nm corresponding to the \(^{1}{{S}_{0}}{\kern 1pt} {{ - }^{1}}{\kern 1pt} {{P}_{1}}\) transition. This transition has a relatively large linewidth (\({{\Gamma }_{s}}{\text{/}}2\pi \) \( \simeq 28\) MHz), which determines the Doppler cooling limit *T*_{D} ≈ 1 mK. To reach lower temperatures, it is necessary to use the intercombination \(^{1}{{S}_{0}}\)–\(^{3}{{P}_{1}}\) transition with a linewidth of \({{\Gamma }_{s}}{\text{/}}2\pi \simeq 182\) kHz at a wavelength of \({{\lambda }_{G}}\) = 556 nm for making the “second-stage” MOT. The Doppler limit for this transition is *T*_{D} ≈ 1 μK. The necessity of use of two-frequency laser radiation imposes additional requirements on the plane diffraction grating and atom chip. Furthermore, high magnetic field gradients up to 50 G/cm usually used to form the MOT for Yb atoms impose requirements on the atom chip.

To determine the parameters of the magnetic and laser fields necessary for the calculation of the atom chip and plane diffraction grating, we experimentally studied the three-dimensional MOT. Key parameters of the MOT are the magnetic field gradient, total intensity of laser beams, and the relations between the intensities of these beams. The first parameter is key for developing the atom chip and the other, for the plane diffraction grating. The criteria of the optimal values of these parameters are the maximum number of atoms in the MOT and their lowest temperature. The method for the calculation of the number and temperature of atoms, which is based, in particular, on work [27], as well as details of the experimental setup are presented in the supplementary material.

The optimal parameters of the magnetic field were determined in the first stage MOT for the ^{174}Yb and ^{171}Yb isotopes. Figure 2а shows the dependence of the number of atoms on the magnetic field gradient *A*. It is seen that the magnetic field gradient for the ^{174}Yb and ^{171}Yb atoms should be larger than 25 and 26 G/cm, respectively.

The temperature of atoms in the MOT was measured at the total intensity of the laser beams about 2*I*_{s} (*I*_{s} is the saturation intensity; for the \(^{1}{{S}_{0}}\)–\(^{1}{{P}_{1}}\) transition, *I*_{s} = 63 mW/cm^{2}). The results presented in Fig. 2b show that ^{171}Yb atoms were cooled to a lower temperature than ^{174}Yb under the same conditions. It was po-ssible to cool ^{171}Yb atoms to a temperature of (12 ± 3) mK at the magnetic field gradient *A* = 27.4 G/cm, whereas ^{174}Yb atoms were cooled to *T* = (22 ± 5) mK at *A* = 35.1 G/cm. In addition, it is seen in Fig. 2b that an increase in the magnetic field gradient reduces the temperature to which ^{174}Yb atoms can be cooled. The lower temperature obtained for ^{171}Yb atoms is due to the structure of magnetic sublevels of odd isotopes [28] and is characteristic of such systems.

Тhree-dimensional MOT formed by two coils with currents in an anti-Helmholtz configuration has a separate axis along the straight line connecting the centers of the coils. The magnetic field gradient in this direction is higher than that in the perpendicular directions. For this reason, the optimal intensity of laser radiation propagating along the central axis will differ from the optimal intensity of radiation propagating along other two axes. To determine the optimal intensity parameters that ensure the maximum number of localized atoms at the minimum temperature, we examined both the total intensity of the MOT beams *I*_{sum} and the relation between the intensities of the central and two side beams. The latter parameter is equivalent to the ratio *P*_{side}/*P*_{center} of the power of an side beam in the MOT *P*_{side} to the power of the central beam *P*_{center}. The number of atoms in the MOT was measured for these parameters from a certain region.

Figure 3а presents the (*I*_{sum}/*I*_{s}, *P*_{side}/*P*_{center}) map of the number *N*_{at} of ^{171}Yb atoms in the first stage MOT, where *I*_{sum} is the total intensity of cooling beams, *I*_{s} is the saturation intensity, and *P*_{side} and *P*_{center} are the powers of the side and central beams, respectively. The measurements were carried out at a magnetic field gradient of 27.4 G/cm, which was previously determined as the optimal value. It is seen that the maximum number of atoms reached in the experiment was \({{N}_{{{\text{at}}}}} = 3 \times {{10}^{6}}\) at the total intensity in the cooling region *I*_{sum} = 1.6*I*_{s}. The center of the region with the maximum number of trapped atoms corresponds to the ratio *P*_{side}/*P*_{center} ≈ 1.5. The solid and dashed lines mark the optimal region of the parameters at which the MOT is formed at magnetic field gradients of 27.4 and 30.0 G/cm, respectively. It is seen that the increase in the magnetic field gradient is accompanied by the expansion of the optimal region; its center is shifted to *P*_{side}/*P*_{center} ≈ 2 because of the difference between magnetic field gradients in these directions.

To determine the temperature of the atomic ensemble, it was measured at the total intensity of the laser beams *I*_{sum} = 1.4*I*_{s} and a magnetic field gradient of 27.4 G/cm. With these fixed parameters, the ratio *P*_{side}/*P*_{center} was scanned and the temperature was measured. The results are presented in Fig. 3b. It is seen that the minimum temperature is reached at the ratio *P*_{side}/*P*_{center} ≈ 1 because of the equality of the frictional or damping forces and, therefore, of momentum diffusion coefficients along all axes of cooling of atoms.

It is seen that the achievement of a lower temperature of the atomic ensemble requires the equality of the frictional forces caused by laser radiation. At the same time, to reach the optimal number of atoms in the MOT, the balance of forces ensuring the localization of atoms in the MOT is required. For this, it is necessary to ensure the isotropy of the magnetic field gradient with the equality of the intensities of laser beams. Unfortunately, this is impossible in the three-dimensional MOT with anti-Helmholtz coils.

According to the conducted measurements, the diffraction grating and atom chip should satisfy the following requirements:

—the atom chip should provide an isotropic magnetic field gradient of no less than 27.4 and 40–50 G/cm for ^{171}Yb and ^{174}Yb atoms, respectively, by a preliminary estimate;

—the intensities of the beam incident on the diffraction grating and the beams reflected from it should be identical to reach the minimum temperature of atoms in the MOT.

## 3 ATOM CHIP FOR THE MAGNETO-OPTICAL TRAP OF YTTERBIUM ATOMS

The experimental data show that the first stage MOT features a relatively high magnetic field gradient *A* \( > \) 30 G/cm, which is typical of ytterbium atoms [29]. Currents *I* ≈ 50 A passing through anti-Helmholtz magnetic coils were used to produce such a magnetic field gradient. This circumstance limits the development of energy efficient compact devices because high-power sources are required to generate such currents in magnetic coils.

The required quadrupole magnetic field can be produced by an atom chip. The simplest model of the atom chip is an infinitely long thin wire with the current in combination with an uniform external magnetic field. In this model, the coordinate *z*_{0} of the localization point of atoms (point of the minimum magnetic field gradient) and the magnetic field gradient in it are given by the expressions [20]

where \({{\mu }_{0}}\) is the Bohr magneton, *I* is the current flowing through the microwire, and \({{\tilde {B}}_{{{\text{bias}}}}}\) is the uniform external magnetic field. In this model, the magnetic field gradient \(B_{z}^{'}({{z}_{0}})\) = 50 G/cm at a reasonable distance to the atom chip of *z*_{0} = 2.5 mm (certainly larger than the radius of the atomic cloud) can be reached at \({{\tilde {B}}_{{{\text{bias}}}}} = 12.5\) G and *I* = 15 A. The generation of such electric current through the microwire of the atom chip does not require high-power sources because the resistance of the microwire is low (about *R* ≈ 1 Ω) compared to the resistance of magnetic coils of the three-dimensional MOT. It is worth noting that coils with current in an Helmholtz configuration are necessary for the generation of the external magnetic field. However, these coils are more compact than anti-Helmholtz coils used to form the classical MOT and require a lower power.

To generate the quadrupole magnetic field near the atom chip, it is necessary to use a U-shaped microwire [20]. However, the deviation of the real distribution of the magnetic field from quadrupole restricts the region of effective localization of atoms [30]. Deviation can be partially compensated by using the wide U-shaped microwire [20, 30] and an additional vertical magnetic field. The wide microwire also allows one to reduce the thermal loading of the atom chip due to a decrease in the resistance.

Figure 4 shows the proposed model of the atom chip. To form the so-called mirror MOT near the atom chip, the geometry of laser beams is such that laser radiation is reflected at an angle of 45° from the surface [31]. For this reason, the atom chip is completely coated with a metal layer. The proposed design of the atom chip consists of two 10-μm-thick wide U-shaped microwires (their boundaries are indicated in red in Fig. 4), one of which is reserved. These microwires are formed due to 100-μm-wide gaps in the metallic layer. The analysis with the variation of the length of the U‑shaped strip the range of 5–11 mm showed that the optimal is a length of 6 mm. In this case, the magnetic field gradient along the *Y* axis is 40 G/cm. The analysis shows that the width of the wire should be about the distance from the center of the formed MOT to the atom chip. This distance should be at least *z*_{0} = 2.5 mm (certainly larger than the radius of the atomic cloud); for this reason, calculations were performed for the 3‑mm-wide microwire of the atom chip. The microwires are broadened closer to the edge of the atom chip in order to reduce the heat release due to a decrease in the resistance and to form contact areas.

Figure 5 presents the distribution of the magnetic field near the proposed atom chip with the optimized uniform external magnetic field. The magnetic field gradient is 60 G/cm in the direction of laser beams reflected from the surface of the atom chip and is 40 G/cm along the wire (along the *Y* axis). Such magnetic field gradients are enough to form the MOT for ytterbium atoms (see Fig. 2a). The presented distribution of the magnetic field was calculated with the electric current through the microwire *I* = 15 A and the uniform external magnetic field *B*_{bias} = (12.6, 0, 6.3) G.

The electric current flowing through the microwire of the atom chip heats it. This is one of the significant problems of atom chips. First, insufficient heat dissipation leads to the overheating and destruction of the microwire. Second, even if the temperature is below the melting temperature and the microwire is not destroyed, additional heating can locally degrade the vacuum near the atom chip. As a result, the lifetime of atoms in the MOT and, thereby, the total number of localized atoms will decrease. The last circumstance will affect the stability of the atomic clock.

Figure 4 presents the temperature distribution calculated with a current of 15 A flowing through one of the microwires and with other parameters similar to those used in [32]. It is seen that the maximum temperature is about 50°C, which is not critical for the operation of the atom chip and is much lower than 100°C. For this reason, we do not expect a significant change in the pressure of residual gases near the atom chip. A temperature of 100°C was chosen as the threshold allowed value for the operation of the atom chip for the following reasons. First, the active evaporation of water and other impurities from the surface of the atom chip begins above this temperature. This degrades the local vacuum near the atom chip and, as a result, reduces the lifetime of atoms in the MOT. Second, the temperature dependence of the resistance becomes strong above 100°C. As a result, the temperature of the surface becomes a nonlinear function of the flowing current. Consequently, a small increase in the current can lead to a sharp increase in the temperature and to the failure of the microwires.

The proposed layer structure of the atom chip, which can be formed using the approach considered in [32], is described in the supplementary material.

Such an atom chip should be loaded by the standard method from an atomic beam. In this case, to increase the atomic beam density, the atomic beam can be focused to the cooling region near the atom chip [33, 34].

## 4 DIFFRACTION GRATING FOR THE FORMATION OF LASER BEAMS OF THE MAGNETO-OPTICAL TRAP FOR YTTERBIUM ATOMS

To construct an optical system for the development of the compact MOT, one can use plane diffraction gratings [24]. This approach is based on the generation of a cooling laser field acting in all spatial directions due to the diffraction of a single laser beam on a planar structure (see Fig. 1d). This allows the development of compact MOTs, which require only one laser beam rather than six beams as in the “classical” three-dimensional variant. Various configurations of planar structures can form the laser field configuration required for cooling [35]. In the case of the cooling of ytterbium atoms, both diffraction gratings for the formation of only the first stage MOT [36] and gratings that make it possible to form the cooling field at two wavelengths simultaneously [37] were considered. Planar gratings for two wavelengths were experimentally demonstrated for Sr atoms [38]. We considered the formation of the cooling region due to the overlapping of three mutually perpendicular beams for both the first- and second-stage MOTs.

Laser beams necessary for the generation of the optical field in the MOT should form a maximally possible overlapping volume in the atom trapping region [39] at the equal power and the conservation of the circular polarization in the beams. It is necessary to take into account that diffraction gratings have dispersion, and a single diffraction grating cannot form the same field distribution at different wavelengths referring to two stages of cooling of atoms in the MOT. Let us consider three configurations shown in Fig. 6. For a convenient design, we calculate the parameters of the plane diffraction grating for the diffraction angle \(\varphi _{m}^{{{\text{ref}}}} = 45^\circ \). Then, the period should be *d*_{B} = 564 nm and *d*_{G} = 786 nm for radiation at the wavelengths λ_{B} = 399 nm and λ_{G} = 556 nm, respectively. The angle of diffraction of radiation with the wavelength λ_{B} = 399 nm on the diffraction grating with the period *d*_{G} = 786 nm will be \(\varphi _{m}^{{{\text{ref}}}} \approx 30.5^\circ \), which is also acceptable for the balance of light pressure forces [37].

Table 1 summarizes the geometrical parameters of the considered diffraction gratings. The relief profile height \(h\) was determined from the solution of the problem of diffraction on the periodic structure. Since the circular polarization should be kept in diffraction orders, it was necessary to minimize the difference between the diffraction efficiencies for mutually perpendicular TE and TM polarizations, where the electric field vector oscillates along and across the grooves of the diffraction grating, respectively.

The volume of overlapping of diffraction orders is an important parameter determining the region of efficient cooling of atoms. We consider the diffraction grating with the working region 25 mm in diameter. In the considered diffraction geometry, it is necessary to analyze the dependence of the diffraction efficiency of the +1 diffraction maximum on the relief profile height. Since parasitic diffraction to the other maxima cannot be minimized in the case of the symmetric binary relief, we consider the diffraction efficiency at the +1 and –1 maxima. Figure 7 shows the diffraction efficiency for three types of the considered diffraction gratings determined by the coupled wave method [40].

The dependence of the diffraction efficiency on the filling factor was not studied. For each case, we considered three types of metallization of diffraction gratings with aluminum, silver, and gold. The vertical red line indicates the relief profiled heights that satisfy the condition of the minimum difference between the diffraction efficiencies for TE and TM polarizations. The minimum possible relief height was chosen for the subsequent fabrication in order to increase the accuracy of the fabrication.

Aluminum is an optimal material for the metallization of the three-section diffraction grating of the first type (see Fig. 7a). The optimal diffraction efficiencies are \(\eta _{{ \pm 1}}^{{{\text{TM}}}} = 32.7{\kern 1pt} \% \) and \(\eta _{{ \pm 1}}^{{{\text{TE}}}} = 45.2{\kern 1pt} \% \) (\(\eta _{{ \pm 1}}^{{{\text{TM}}}} = 47.0{\kern 1pt} \% \) and \(\eta _{{ \pm 1}}^{{{\text{TE}}}} = 32.7{\kern 1pt} \% \)) for λ_{B} = 399 nm (λ_{G} = 556 nm). These values are reached at the groove profile height \({{h}_{1}}\) = 116 nm. The overlapping region of laser beams (indicated in violet in Fig. 6b) has a volume of ~511 mm^{3}. It is seen in Fig. 7a that it is impossible to completely match the diffraction efficiency for the TE and TM polarizations at both wavelengths. For this reason, diffracted radiation will have an elliptic polarization.

Silver is the best material for the fabrication of square elements of the second-type diffraction grating with a square packing (see Fig. 7b). In this case, the diffraction efficiency will be the same for the TE and TM polarizations due to symmetry. The optimal diffraction efficiency reached at the groove height \({{h}_{2}} = 113\) nm is \(\eta _{{ \pm 1}}^{{{\text{TM}}}} = \eta _{{ \pm 1}}^{{{\text{TE}}}} = 14.0{\kern 1pt} \% \) for both λ_{B} = 399 nm and λ_{G} = 556 nm. The overlapping region of laser beams (see Fig. 6d) has a volume of \( \sim {\kern 1pt} 2045\) mm^{3}, which is the maximum value among all three types of diffraction gratings.

Aluminum columns with a height of \({{h}_{3}} = 97\) nm are optimal for the diffraction grating of the third type with a triangular packing of circular elements (see Fig. 7с). In this case, the diffraction efficiency for the TE and TM polarizations will be the same \(\eta _{{ \pm 1}}^{{{\text{TM}}}} = \) \(\eta _{{ \pm 1}}^{{{\text{TE}}}}\) = 10.0% for λ_{B} = 399 nm and λ_{G} = 556 nm due to symmetry. The overlapping region of laser beams (see Fig. 6d) has a volume of \( \sim {\kern 1pt} 885\) mm^{3}.

## 5 CONCLUSIONS

To summarize, our calculations have shown that the atom chip can provide the quadrupole field with the parameters required to localize ytterbium atoms, which have been determined in our experiment. Magnetic field gradients near the surface at currents through the microwire of about 15 A reach 50 G/cm, which are enough to localize both the ^{171}Yb and ^{174}Yb isotopes. In this case, laser radiation can propagate in the mirror magneto-optical trap.

According to the calculation of diffraction gratings, a single optimal solution can barely ensure the balance of three factors: high diffraction efficiency, conservation of the polarization state after diffraction, and a large spatial volume of the overlapping region of laser beams. Consequently, all proposed implementations should be experimentally verified in order to determine the number of atoms trapped with all types of diffraction gratings.

After the independent experimental study of the proposed approaches, we are going to analyze the possibility of jointly using the atom chip and the diffraction grating in a single compact optical frequency standard device based on cold ytterbium atoms.

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## Funding

The development of the atom chip in this work was supported by the Ministry of Science and Higher Education of the Russian Federation (state assignment no. FFUU-2021-0003) and by the Russian Metrological Institute of Technical Physics and Radioengineering (ongoing institutional funding). The other studies were supported by the Russian Metrological Institute of Technical Physics and Radioengineering (ongoing institutional funding).

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Vyalykh, A.P., Skakunenko, P.I., Shishova, M.V. *et al.* Atom Chip and Diffraction Grating for the Laser Cooling of Ytterbium Atoms.
*Jetp Lett.* **119**, 285–293 (2024). https://doi.org/10.1134/S0021364023604189

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DOI: https://doi.org/10.1134/S0021364023604189