Ensembles of cold and ultracold atoms are natural objects for the development of quantum sensors. The study and application of such systems are of great interest both for the fundamental physics and numerous applications. The main aim of the development of sensors utilizing quantum states of systems (atoms, molecules, color centers, quantum dots, etc.) to measure physical fields is to increase the accuracy of measurements [1]. This is necessary not only for the solution of applied problems, e.g., navigation and mineral exploration but also for the solution of fundamental problems such as the detection of gravitational waves and dark matter and the study of drift of fundamental constants [2].

The use of quantum effects in sensors allows one to significantly increase the accuracy of measurements (e.g., when creating of atomic clocks or measuring the gravitational acceleration). The accuracy limit of a quantum sensor is determined by quantum projection noise (QPN). The accuracy of the sensor involving atoms is determined by the number of atoms N in the atomic ensemble of the sensor [3]:

$${\text{QPN}} \sim \frac{1}{{\sqrt N }}.$$
(1)

Significant efforts were focused on studies of experimental schemes to reach the maximum possible number of atoms in the sensor. Magneto-optical traps (MOTs) are used in the overwhelming majority of cases as sources of atoms for quantum sensors. A magneto-optical trap is necessary for the primary cooling and localization of atoms. In the initial stage of the preparation of the atomic ensemble in the MOT, its temperature corresponds to the Doppler cooling limit (145 μK for Rb atoms). In subsequent stages, the atomic ensemble is cooled below the Doppler limit due to sub-Doppler cooling mechanisms (about 10 μK) and then to a temperature of about 100 nK by means of the evaporative cooling method. Each of the listed stages, particularly, the evaporative cooling stage, results in the loss of atoms. For this reason, the maximum number of atoms in the MOT obtained in the stage of the preparation of the initial atomic ensemble primarily determines the accuracy of the quantum sensor.

In addition, ensembles of cold atoms formed in the MOT are used as sources of atoms for a large variety of experiments, e.g., experiments on the study of mechanisms of sub-Doppler cooling of atoms [4] in order to increase the number of atoms in a Bose–Einstein condensate [5]. The number of atoms in the MOT is important when studying the properties of an ultracold plasma [6] and atoms in Rydberg states [7].

Modern trends in the development of quantum sensors are aimed at their miniaturization and energy efficiency because it is necessary to place quantum sensors on mobile platforms both for fundamental studies and for applied problems. In particular, the use of quantum sensors in space is actively developed [8, 9]. The atom chip technology allows the complex solution of the problems of miniaturization and energy efficiency [10]. Indeed, the atom chip technology makes it possible to reduce the sizes of the quantum sensor and to increase its energy efficiency due to the reduction of the effective resistance of microwires used in the chip. Furthermore, high magnetic field gradients, which cannot be reached with quadrupole field coils, can be reached near microwires. This in turn allows one to rapidly and efficiently reach the Bose–Einstein condensation [11].

The nearly quadrupole magnetic field [12] necessary for the creation of the MOT can be formed near the surface of the atom chip. This allows one to cool and trap atoms near the atom chip without quadrupole field coils, which are larger and less energy efficient compared to the atom chip. The quadrupole magnetic field necessary for the formation of the primary ensemble of cold atoms on the chip is generated by the current flowing through the microwires of the atom chip placed in a uniform magnetic field [13]. This configuration ensures the required magnetic field distribution with zero magnetic field. At displacement from the central region of the trap, higher multipole components (hexagonal and octupole) of the magnetic field distort the quadrupole distribution. As a result, the effective volume of the MOT where cooling occurs decreases [14].

The number of atoms in the MOT is given by the expression [15]

$$N = R\tau ,$$
(2)

where R is the loading rate of atoms in the MOT and τ is the lifetime of atoms, which is determined by collisions of atoms with residual gases in the vacuum chamber. The loading rate of the MOT with the quadrupole magnetic field distribution is usually determined by the area of laser beams and the atomic flux. When the spatial distribution of the magnetic field deviates from quadrupole, the volume of the region of the effective trapping of atoms is determined not by the sizes of laser beams and by the effective size of the region where the deviation of the magnetic field from the “ideal” distribution is small. In this situation, R can be obviously increased by increasing the flux of atoms into the cooling region.

There are two approaches for the loading of atoms into the MOT: (i) from an atomic vapor, which can be created in the region of the MOT by means of a dispenser and (ii) from the atomic beam. Both approaches are studied in this work in order to de-termine the optimal parameters of increasing the loading rate of atoms into the MOT formed near the atom chip.

Methods for the localization of atoms near the atom chip and their detection were described in our previous work [16]. A mirror MOT, which is formed by two laser beams, is used to cool atoms near the atom chip. One of these beams propagates parallel to the plane of the atom chip, and the second beam is reflected from it at an angle of 45°. Further, these beam are reflected backward, passing twice through a quarter wave plate. In this configuration, the laser field near the atom chip is equivalent to the field used in the three-dimensional MOT.

The quadrupole magnetic field necessary for the MOT is generated by the current flowing through the U-shaped wire on the surface of the atom chip placed in the uniform magnetic field. The wire used in this work had a width of 2.9 mm, a length of 6.2 mm, and a thickness of 7 μm. This configuration allowed flowing an electric current of 8 A through the microwire without its significant heating, which enables -measurements with large loading times of atoms into the MOT.

The loading of atoms into the MOT from atomic vapors makes it possible to make the system compact. However, the achievement of a high loading rate R requires a significant increase in the pressure of Rb vapors in a vacuum chamber, which reduces the lifetime of atoms in the MOT. This approach was used in [16], and the number of localized atoms was small (about 105), which excludes the use of this method for quantum sensorics problems. In this work, we study the loading dynamics and determine the lifetime of atoms in the MOT loaded from vapors in order to compare with the loading from the atomic beam.

The experimental setup for the loading of the MOT from the atomic beam consisted of two vacuum chambers with differential pumping. One chamber with a vacuum relatively poor for modern cold atom systems was used for the atomic source. The atom chip forming the required magnetic field distribution was placed in the other ultrahigh vacuum chamber with a pressure of 10–10 Torr (Fig. 1). Atoms from the three-dimensional MOT created in the chamber at a pressure of about 10‒9–10–8 Torr formed a low-velocity atomic beam for the loading of the atom chip [17]. The average velocity of atoms in the direction of beam propagation was V = 14 m/s. To form the low-velocity atomic beam from the MOT, a 1-mm hole was made in a mirror served to create the required configuration of the laser fields of the three-dimensional MOT. This hole was also used for differential pumping. It formed a region with zero laser field in the reflected beam, which led to the imbalance of light pressure forces in the central region of the MOT: the net force acts on atoms in the direction to the hole. The atomic beam from the three-dimensional MOT propagated to the region with the atom chip, where further cooling and localization of atoms occurred.

Fig. 1.
figure 1

(Color online) Sketch of the experimental setup for the loading of atoms from the low-velocity atomic beam into the magneto-optical trap near the atom chip.

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A region of the interaction of atoms with transverse laser radiation and the two-dimensional quadrupole magnetic field was formed in our setup on the propagation path of the low-velocity atomic beam. This region can be used to focus the atom chip into the region of the atom chip [18, 19]. In this work, we used only laser radiation without the quadrupole magnetic field in the configuration of one-dimensional transverse laser cooling (one-dimensional optical molasses). This enables the spatial control of the position of the atomic beam in the plane parallel to the atom chip plane due to the creation of an additional force in the transverse direction. The force magnitude was controlled by the intensity of laser radiation. The optimal position of the atomic beam was reached at the intensity of transition saturation. The frequency of radiation was the same as that at the formation of the low-velocity atomic beam and cooling of atoms near the atom chip. The length of the interaction region was 15 mm.

We used 87Rb atoms. This determined the frequencies of laser fields used for cooling, control, and localization of atoms. For these aims, we used a frequency-stabilized laser with a detuning of \(\delta = - 2\Gamma \) (where \(\Gamma = 2\pi \times 6{\kern 1pt} \) MHz is the width of the atomic transition in Rb) with respect to the \(5{{S}_{{1/2}}}{\kern 1pt} F = 2 \to 5{{P}_{{3/2}}}{\kern 1pt} F' = \) 3 87Rb transition. The radiation of another frequency-stabilized laser near the \(5{{S}_{{1/2}}}{\kern 1pt} F = 1 \to 5{{P}_{{3/2}}}{\kern 1pt} F' = \) 2 87Rb transition was used for the optical pumping of Rb atoms. The radiation of the lasers was spatially matched and was guided to the experimental region through polarization-maintaining optical fibers.

A probe laser beam 1 mm in diameter was formed in the region of the atom chip. This beam propagated in the plane parallel to the atom chip perpendicular to the atomic beam. This beam allowed us to visualize the spatial position of the atomic beam in the region of the atom chip by resonant fluorescence of atoms, which was detected by a camera placed below the vacuum chamber.

The dynamics of the loading of atoms into the MOT is described by the equation

$$\frac{{dN(t)}}{{dt}} = R - \frac{{N(t)}}{\tau } - \beta {{N}^{2}}(t),$$
(3)

where \(N(t)\) is the number of atoms in the MOT at the time t measured from the beginning of loading, R is the loading rate, \(\tau \) is the lifetime of atoms in the MOT determined by collisions with the residual gas in the vacuum chamber, and \(\beta \) is the probability of loss of a cold atom colliding with another cold atom in the MOT (two-particle interaction) [20]. According to Eq. (3), the dynamics of the number of atoms in the MOT under the initial condition \(N(0) = 0\) is described by the expression

$$N(t) = \frac{{2R\tau \tanh \frac{{At}}{{2\tau }}}}{{A + \tanh \frac{{At}}{{2\tau }}}},$$
(4)

where

$$A = \sqrt {1 + 4R{{\tau }^{2}}\beta } .$$

When the atomic number density n = N/V, where V is the volume of the trap, in the MOT is low and two-particle interactions do not limit the loading of atoms, the last term in Eq. (3) can be neglected and Eq. (4) is reduced to the form

$$N(t) = R\tau (1 - {{e}^{{ - t/\tau }}}).$$
(5)

When the loading of atoms is turned off (the atomic beam is removed), the decay of the atomic ensemble begins in the MOT. The time dependence of the number of atoms in the MOT can be determined from Eq. (3) as

$$N(t) = \frac{{{{N}_{0}}{{e}^{{ - t/\tau }}}}}{{1 + {{N}_{0}}\tau \beta (1 - {{e}^{{ - t/\tau }}})}}.$$
(6)

When two-particle interactions do not affect the lifetime of atoms in the atomic ensemble, Eq. (6) takes the form

$$N(t) = {{N}_{0}}{{e}^{{ - t/\tau }}}.$$
(7)

Figure 2a presents dependences of the number of atoms in the MOT, which was formed near the atom chip, on the time measured from the time of the application of laser beams at various methods for the loading of the MOT. The number of atoms in the MOT was determined from the fluorescence signal. The number of atoms loaded into the MOT from vapors (Fig. 2a, black line) produced by the dispenser placed near the atom chip is minimal. It is seen that the number of atoms localized in the MOT increased dramatically when the atomic beam was used (Fig. 2a, green and blue line). Table 1 presents the found steady-state number of atoms in the trap, the loading rate, and the lifetime of atoms in the MOT.

Fig. 2.
figure 2

(Color online) (a) Measured time dependence of the number of atoms in the magneto-optical trap loaded with atoms (1) from thermal vapors (loading parameters \(R = 0.2 \times {{10}^{7}}{\kern 1pt} \) s–1 and t = 1.8 s), (2) from the low-velocity atomic beam whose position with respect to the center of the atom chip is shown in panel (c) (loading parameters R = \(0.7 \times {{10}^{7}}\) s–1 and \(t = 3.5{\kern 1pt} \) s), and (3) from the low-velocity atomic beam whose position was controlled by the one-dimensional optical molasses shown in panel (b) (loading parameters \(R = 1.1 \times {{10}^{7}}{\kern 1pt} \) s–1 and \(t = 4.7{\kern 1pt} \) s). (b, c) Photographs of the illuminated atom chip against the background of the atom chip. The atomic beam propagates from left to right and is illuminated by a narrow laser beam propagating from top to bottom. Intersection regions of the atomic and laser beams are shown in red. The microwire of the U-shaped trap through which the current flows to induce the magnetic field required for the magneto-optical trap is indicated in yellow.

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Table 1. Parameters of the magneto-optical trap (MOT) loaded by various methods
Full size table

The parameters summarized in Table 1 were estimated using Eq. (5) (simple exponential dependence). Two-particle collisions were disregarded when processing experimental data because the formation dynamics of the MOT for the low-velocity atomic beam affects the loading dynamics of atoms into the MOT near the atom chip. Due to features of our experimental setup, the formation of the MOT for the low-velocity atomic beam began at the time \({{t}_{0}} = 0\) simultaneously with the application of cooling laser radiation near the atom chip. As a result, the flow of atoms and thereby, the loading rate R, at the initial time will depend on the time. We took into account two-particle collisions and thus determined the parameter \(\beta \) from unloading curves (see below).

The loading of atoms from the low-velocity atomic beam is sensitive to its position with respect to the atom chip. This position can be controlled using the optical molasses. The green line in Fig. 2a corresponds to the loading of the MOT without the optical molasses. In this case, the position of the atomic beam in the region of the atom chip appeared to be not optimal (Fig. 2c). The position of the atomic beam passing through the central region of the atom chip is optimal. Figures 2b and 2c show images of the atomic beam (it propagates from left to right, the detecting laser beam propagates from top to bottom, and their intersection is marked in red) against the background of the atom chip, demonstrating its spatial position. The microwire of the U-shaped trap through which the current flows to form the magnetic field required for the MOT is indicated in yellow. It is seen in Fig. 2c that the atomic beam is displaced from the central region of the atom chip. This is due to the residual transverse velocity of atoms at the output of the diaphragm forming the atomic beam. Our estimates performed taking into account the geometry of our setup and the characteristic velocities of atoms in low-velocity atomic beams show that this velocity is about 0.3 m/s.

The blue line in Fig. 2a corresponds to the loading of the MOT with the use of the one-dimensional optical molasses when the position of the atomic beam with respect to the atom chip is optimal. The spatial scanning of the beam position allowed us to increase the loading rate of atoms by a factor of 1.5 and their number in the steady state by a factor of 2.

Our measurements showed that the lifetime τ of atoms in the MOT at loading from atomic vapors is small. This occurs because the operating dispenser increases the pressure of Rb vapors in the vacuum chamber. This factor is also responsible for a lower loading rate of atoms in the MOT R compared to loading from the atomic beam. Indeed, only atoms with velocities lower than the critical value \({{{v}}_{{\text{c}}}}\) ~ 35 m/s [15] are localized in the MOT. Atoms loaded from vapors have a Maxwellian velocity distribution and the number of atoms with the required velocities is small. The number of atoms with \({v}\) < \({{{v}}_{{\text{c}}}}\) loaded from the low-velocity atomic beam is significantly larger.

The position of the atomic beam relative to the atom chip should not change the pressure of vapors in the vacuum chamber. For this reason, the expected lifetime of atoms \(\tau \) should remain unchanged at the variation of the beam position. According to our data, this lifetime differs by 35%. This can be due to the difference of the loading dynamics from Eq. (3) at a large number of atoms in the MOT.

For this reason, we directly measured the lifetime of atoms in the MOT. This measurement implied the removal of the flow of atoms entering the cooling region after the achievement of the steady-state number of atoms in the MOT (“unloading” of atoms from the MOT). Since atomic vapors cannot be rapidly removed from the region of the atom chip at the use of the dispenser, such measurements were carried out only at the loading of the MOT from the atomic beam. To switch off the atomic beam, we removed the laser field of the three-dimensional MOT, which formed the low-velocity atomic beam.

Figure 3 shows the log–lin plots of the number of atoms in the MOT versus the time at switching-off of the atomic beam for the cases of loading with and without the correction of the beam position using the one-dimensional optical molasses. The number of atoms in the MOT in the steady state is larger in the former case. After the loading of atoms from the atomic beam without spatial correction, the unloading curve for atoms from the MOT is approximated by Eq. (6) with the parameters τ = 4.3 s and \(\beta = 3.3 \times {{10}^{{ - 9}}}{\kern 1pt} \) s–1 (coincide in order of magnitude with results obtained by other groups [21]).

Fig. 3.
figure 3

(Color online) Time dependence of the number of atoms in the magneto-optical trap at the switching-off of the low-velocity atomic beam (unloading of the magneto-optical trap) after loading (1) without the one-dimensional optical molasses (unloading parameters τ = 4.3 s and β = \(3.3 \times {{10}^{{ - 9}}}\) s–1) and (2) from the low-velocity atomic beam using the one-dimensional optical molasses (unloading parameters τ = 4.0 s and \(\beta = 3.4 \times {{10}^{{ - 9}}}{\kern 1pt} \) s–1). Red line 3 is the exponential approximation (with the parameter τfs = 3.6 s) of the time dependence of the number of atoms in the magneto-optical trap in the initial stage at a large number of atoms in the magneto-optical trap.

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When the number of atoms in the trap was increased by the spatial correction of the position of the atomic beam using the one-dimensional optical molasses at the loading of the MOT, the unloading curve in Fig. 3 is no longer approximated by Eq. (6). As shown in [21, 22], at a quite large number of atoms in the initial stage, the unloading curve is described by the exponential law given by Eq. (7) but with a different time constant τfs < τ. This effect is caused by the achievement of the limiting density of atoms in the MOT due to the effective repulsion between cold atoms. This region is shown in red in Fig. 3. In this case, the found time parameter in the exponent is τfs = 3.6 s. When the number of atoms is equal to or less than \(2.5 \times {{10}^{7}}\), the curve is approximated by Eq. (6) with the parameters τ = 4.0 s and \(\beta = 3.4 \times {{10}^{{ - 9}}}{\kern 1pt} \) s–1.

The data obtained indicate that the spatial correction of the atomic beam by means of the optical molasses makes it possible to achieve regimes where the number of atoms in the MOT begins to significantly affect the dynamics of their losses. The number of atoms in the MOT can be further increased by increasing the flow of atoms entering the effective cooling region. This can be achieved by focusing the atomic beam [18, 19].

Our studies have shown the advantage of the loading of the atom chip from the low-velocity atomic beam due to two factors. First, loading from the atomic beam allows one to reach larger lifetime of atoms in the MOT because the use of the atomic beam makes it possible to prevent the formation of Rb atomic vapors in the region of the atom chip and, therefore, to avoid additional collisions leading to losses of atoms in the MOT. The measurements have shown that the lifetime of atoms in the magneto-optical trap formed near the atom chip is about 4.1 s. Second, the use of the atomic beam allows the spatial control of the positions of atoms in the localization region, thus optimizing the loading rate of atoms. The large lifetime and the high loading rate of atoms enable an increase in the number of atoms localized near the atom chip to about \(4.9 \times {{10}^{7}}\). In particular, the number of atoms localized in the MOT in this work is larger than that in the experiment with the compensation of the hexagonal component of the magnetic field reported in [23]. The reached number of atoms is characteristic of modern atom chip systems [24].