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Abstract
We demonstrate, for the first time, a hollow-conical atomic beam from a standard low-velocity intense source. Experimental results and numerical simulations indicate that the hollow-conical feature is caused by the converging-diverging extraction process. The degree of hollowness can be reduced by using a weaker push beam and extending the length of transverse cooling. Analytical models are proposed to quantitatively describe the hollowness of the atomic beam. This study can find applications where a compact and solid atomic beam is needed, such as coupling cold atoms into matter waveguides or continuous cold atomic beam interferometers.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Cold atomic beams have been widely used in atomic clocks [1,2], atom interferometry [3–6], atomic imaging [7], atom optics [8] and atom lithography [9–11]. Beams of slow atoms can be created by cooling atoms from a hot oven [12–14] or extracting cold atoms from a magneto-optical trap (MOT), such as two-dimensional MOT (2D-MOT) [15–17] or the low-velocity intense source (LVIS) system [18]. The LVIS apparatus, which is a modification of a vapor-cell MOT (VCMOT), creates a narrow dark column at the center of one of the six MOT laser beams. This modification allows for the production of an atomic beam with a controllable low velocity and a much narrower longitudinal velocity distribution compared to that of 2D-MOT [19]. These advantages have contributed to its extensive application in loading optical molasses in atomic clocks and atom interferometers [1,2]. The main features of the LVIS beam have been studied in detail, including the flux, divergence, longitudinal velocity distribution, and transverse density profile [10,19–21].
The transverse density and velocity distributions of atomic beams are particularly important in some applications. When loading cold atoms into matter waveguides, the number of atoms that can be captured by the waveguide is determined by the overlap between the transverse beam profile and the waveguide entrance [22–24]. In atom interferometers that use continuous atomic beams, the fringe contrast is reduced due to the Doppler effect and thermal expansion of the atomic beam, caused by the transverse velocity spread [6]. The transverse density profile of an atomic beam is typically measured by recording the fluorescence intensity from a thin probe laser sheet that scans across the atomic beam transversely [11,25]. While another method records the fluorescence image induced by the atomic beam, using a thin probe sheet which is aligned perpendicularly to the atomic beam, and its fluorescence emission is imaged by a scientific charge-coupled device (CCD) camera that is perpendicular to both the atomic beam and the probe sheet [26]. In previous studies, the two-dimensional transverse density distributions are integrated into a one-dimensional distribution during the process of one-dimensional scanning or imaging, resulting in the loss of information in the other dimension. The measured transverse density is typically consistent with a Gaussian distribution.
In this study, we demonstrate for the first time that the transverse density profile of a standard LVIS beam has a minimum in the central area, resulting in a hollow-conical shape of the atomic beam. This hollow-conical feature is experimentally observed by directly imaging the transverse density profile and is replicated through a Monte Carlo simulation. We investigate the mechanism that causes this feature and discuss key parameters that affect the degree of hollowness.
2. Experimental setup
The experimental setup is sketched in Fig. 1. Rubidium 87 atoms are laser-cooled in the source chamber with dimensions of $80 ~\textrm {mm} \times 80~\textrm {mm} \times 110~\textrm {mm}$. The chamber has five glass windows for the entrance of the cooling laser beams and for observation. The cooling laser beams, each with a power of 15 mW, are derived from a diode laser with frequency detuned −1.5 $\Gamma$ from the $|{5S_{1/2},F=2}\rangle$ to $|{5P_{3/2},F=3}\rangle$ transition of $^{87}\textrm {Rb}$. The longitudinal (Z-direction) cooling beam is also referred to as the push beam in this paper. The repump beam is supplied by another diode laser with frequency locked to the $|{5S_{1/2},F=1}\rangle$ to $|{5P_{3/2},F=2}\rangle$ transition. The repump beam has a power of 5 mW and is overlapped with the push beam. These laser beams, with a diameter ($1/e^{2}$) of $15 \textrm {mm}$, are retro-reflected by quarter-wave retarders coated with high-reflection dielectric layers. A pair of anti-Helmholtz coils along with the push beam generate the quadrupole magnetic field with the gradient of 8 G/cm for the trap. Three pairs of orthogonally placed Helmholtz coils are used to finely adjust the position of the atomic cloud in order to maximize the flux of the atomic beam. A quarter-wave plate with a hole in the center acts as a mirror and creates an extraction column for the atomic beam. The science chamber is made of glass with anti-reflection coatings on the outer surface. A 1 mm $\times$ 15 mm thin laser beam (referred to as probe beam) shines on the atomic beam perpendicularly at a distance of 350 mm downstream from the QP. The probe beam is resonant on transition $|{5S_{1/2},F=2}\rangle$ to $|{5P_{3/2},F=3}\rangle$ and it is retro-reflected to prevent deflection of the atomic beam and preserve its transverse profile. The fluorescence emitted by the atoms passing through the probe beam is captured by an electron-multiplying CCD (EMCCD) at a 45-degree angle, revealing the transverse density distribution of the atomic beam. A $0.175\times$ objective lens is used to ensure a sufficiently large field of view and depth of field. The flux of the atomic beam is measured to be approximately $2\times 10^{9}$ atoms per second by detecting the fluorescence using a photodiode.
Fig. 1. The top view of the experimental setup. The source chamber (left) and the science chamber (right) are separated by a quarter-wave plate with a hole of 1 mm in diameter drilled through its center. The QP serves as both an optical reflector and a differential pumping separator. A 40 l/s ion pump is used to maintain the pressure in the chambers between $1 \times 10^{-7}$ Pa to $5 \times 10^{-8}$ Pa. The two chambers are connected to the same ion pump by two tubes separately.
3. Two-dimensional transverse profile of the atomic beam
A typical image of the transverse profile is shown in Fig. 2(a). The atomic density is proportional to the brightness of the gray-level map. The image reveals a hollow-core density distribution of the atomic beam. Extending this feature to the Z-dimension reveals a hollow-conical shape of the atomic beam. The degree of hollowness of the beam is quantified by extracting the gray level of the vertical pixels in the center of the atomic beam, as shown in Fig. 2(b). The density in the valley is only about half of the density at the peak. We introduce two parameters to characterize the hollow-conical feature: the beam width and the relative depth. The beam width is defined as the distance between the two peaks, while the relative depth is the normalized depth of the valley.
Fig. 2. (a) The image shows the transverse profile of the LVIS beam. The raw image is horizontally stretched to compensate for the distortion caused by the viewing angle. The displayed image is smoothed by averaging 20 consecutive images, and the background is removed by subtracting the image without atoms. The vertical dark edges on both sides results from the limited width of the probe laser beam. The faint straight line in the horizontal direction is caused by the probe beam coming out of the 1 mm hole. (b) The vertical density distribution near the center of the atomic beam. The abscissa is calibrated based on the magnification of the objective lens and the pixel size on the CCD. To reduce fluctuation, each data point represents the average brightness of ten adjacent pixels in the same row, as indicated by the white box in (a). (c) The simulated transverse profile of the LVIS beam. (d) The simulated vertical density distribution near the center of the atomic beam.
4. Numerical simulation
The characteristics of the hollow-conical atomic beam are quantitatively studied using Monte Carlo simulations. A simple semi-classical model is developed to simulate the cooling and accumulation of the atoms and subsequent generation of the LVIS beam. The atoms are treated as two-level atoms, and the spontaneous scattering force from each of the cooling laser beams is calculated as follows [27]:
The Monte Carlo simulation starts with $5\times 10^{4}$ atoms randomly distributed in the vacuum chamber, following a Maxwell-Boltzmann velocity distribution. The initial temperature of the vapor is set to several Kelvin degrees to accelerate the simulation. The final results are not sensitive to the initial temperature because the atoms are considered independent of each other. We assume that atoms collide elastically with the inner wall of the vacuum chamber. Once an atom reaches the probe plane, it is eliminated and replaced by a new atom randomly placed in the source chamber. The motion equation for each atom is parallel solved using the Runge-Kutta method with a time step of 50 $\mu$s.
The transverse profile of the simulated atomic beam is obtained by recording the positions of the atoms that are pushed out of the hole and reach the probing beam. A typical simulated transverse profile is shown in Fig. 2(c), which presents a similar hollow-conical feature as the experiment. Figure 2(d) is obtained using the same method as in Fig. 2(b). The beam width and relative depth of the simulated profile are approximately the same as those of the experimental results. However, the simulated results have steeper peaks, especially at the extremities, than the experimental results because they do not take into account the collisions and radiation trapping force between the atoms [27]. These effects broaden the velocity distribution of the atoms in the transverse direction, resulting in a wider distribution of transverse positions.
5. Explanation for the hollow-conical atomic beam
The mechanism causing the hollow-conical feature of the atomic beam is revealed by observing the extraction process of the LVIS. The extraction process is illustrated in Fig. 3. We consider a typical case in which a pre-cooled atom diffuses towards the extraction volume with a non-zero transverse velocity $v_\textrm {0}$. The initial longitudinal velocity is ignored because it is insignificant compared with the huge acceleration during the extracting process. For the $^{87}\textrm {Rb}$ atom, the maximum acceleration under saturated intensity can be $10^{5}\space \textrm {m/s}^2$. Once the atom enters the extraction column, it is swiftly propelled out of the cloud region by the push beam before it reaches the center of the cloud. This feature is clearly demonstrated by the gray-level images shown in Fig. 3 where a smoke-ring-like LVIS is observed. During the extraction process, the atoms are decelerated by the transverse cooling beams, and the final transverse velocity depends on the cooling time which is proportional to the transverse cooling length and is inverse to the longitudinal velocity. The trajectory of those atoms with symmetric initial positions and velocities intersect on the symmetry axis, forming a focus point of the atomic beam. After the focus point, the atoms move out of the push beam and move under gravity until it reach the plane of probe beam. This process of converging and diverging leads to a transverse velocity distribution with a non-zero most probable velocity. This distribution results in a similar transverse density distribution, which explains the hollow-conical shape of the atomic beam.
Fig. 3. The process of extracting atoms from the MOT into the atomic beam. HOP: height of peaks, DOV: depth of the valley. Blue dashed lines represent the trajectory of the extracted atoms. Blue solid line: the transverse density distribution of the atomic beam. It is a superposition of two Gaussian functions, depicted by the gray dashed lines. The inset shows a typical LVIS image captured by placing the camera at the window of the source chamber with a slight tilted angle. The smoke-ring-like shape becomes visible by using a small aperture and reducing the exposure time of the camera. The 1 mm hole can be seen on the right side of the image.
The converging-diverging extraction process of the atomic beam is directly observed through simulation and experimentation. To facilitate the observation of the extraction process, we initialized the positions and velocities of the atoms in the X direction to zero. As a result, the atoms primarily move in the Y-Z plane (defined in Fig. 1). A simulated stationary atomic beam is shown in Fig. 4(a), and the zoomed figure in Fig. 4(b) presents the extraction process in the MOT area. The atoms rarely reach the center of the cloud, resulting in a cloud of atoms with a hollow core. This extraction process is experimentally demonstrated by observing a transversely compressed LVIS, as shown in Fig. 4(c). The LVIS is compressed in the X direction by increasing the intensity of the cooling beam in that direction, making the extraction process observable. The image of the compressed LVIS shares similar features with the simulated image, showing a clear extraction tunnel of the atoms. The LVIS becomes faint on the right side of the image due to the reduced atomic density and the Doppler effect. With these tools at our disposal, we can quantitatively stud y the key factors that influence the width and hollowness of the atomic beam.
Fig. 4. (a) The simulated LVIS and atomic beam. (b) The zoomed image of the simulated LVIS. The gathering, extraction and converging of the atoms are illustrated. (c) The compressed LVIS in the source chamber. The upper part has more fluorescence because the unbalanced transversal cooling beam and the defect at the edge of the 1 mm hole in the QP which is indicated by the white box in (b) and (c).
6. Width of the hollow-conical atomic beam
The width of atomic beams is typically determined by the full width at half maximum (FWHM) when the transverse density distribution closely approximates a Gaussian function. For the hollow-conical atomic beam, the beam width is defined as the distance between the peaks (see Fig. 3). The beam width can be calculated using the velocity distributions. We ignore the finite size of the focal point and assume that the trajectories of all the extracted atoms intersect at the focal point. We consider a narrow ring in the probe plane with an inner radius of $r$ and an outer radius of $r+\Delta r$ (see Fig. 3). Only the atoms with transverse velocities between $r/L \cdot \bar {v}_\textrm {z}$ and $(r+\Delta r)/L \cdot \bar {v}_\textrm {z}$ can reach this ring. Here, $\bar {v}_\textrm {z}$ represents the average longitudinal velocity, and $L$ denotes the distance between the focus and the probe plane. The number of atoms in the ring is given by $N(v_\textrm {r}) \cdot \Delta v_\textrm {r}$ with $N(v_\textrm {r})$ the transverse density of the velocity distribution and $\Delta v_\textrm {r} = \Delta r / L \cdot \bar {v}_\textrm {z}$. The transverse density of the position distribution, $\rho (r)$, is given by
The peak appears at the position where $\rho (r)$ reaches its maximum. If we denote the corresponding transverse velocity by $v_\textrm {rm}$, the beam width is then given by
which shows that the beam width is proportional to the ratio of $v_\textrm {rm}$ to $\bar {v}_\textrm {z}$. By recording the velocities of the atoms that reach the probe plane in the simulations, beam widths can be calculated by using Eq. (3).To validate the numerical simulation and theoretical model, we scan the intensity of the push beam to change the ratio of $v_\textrm {rm}$ to $\bar {v}_\textrm {z}$. The numerical simulation and the theoretical model are in good agreement with the experimental results, as shown in Fig. 5(a). The beam width increases approximately linearly with the intensity of the push beam, indicating that the push beam has a stronger effect on $v_\textrm {rm}$ than $\bar {v}_\textrm {z}$. The normalized velocity distributions of atoms reaching the probe plane at different intensity of push beams are shown in Fig. 6. Both the transverse and longitudinal mean velocities increase with a stronger push beam. A push beam with higher intensity implies a larger longitudinal acceleration and hence a shorter transverse cooling time, which results in higher $\bar {v}_\textrm {z}$ and $v_\textrm {rm}$. Because the radial velocity $v_\textrm {rm}$ increases more rapidly than the longitudinal velocity $\bar {v}_\textrm {z}$, the beam width, which is proportional to the ratio of $v_\textrm {rm}$ to $\bar {v}_\textrm {z}$, is larger in stronger push beams. This statement is clarified with the black arrows in Fig. 6.
Fig. 5. (a) The relationship between the beam width and the intensity of the pushing beam. The intensity of the transverse cooling beams is the same for all data. (b) The relationship between the beam width and the transverse cooling length. (c) The relationship between the relative depth (RD) of the hollow core and the intensity of the push beam. (d) The relationship between the RD and the transverse cooling length. The error bars indicate the level of uncertainty. $I_\textrm {z}$: intensity of the push beam. In the calculation of the theoretical model, we set $r_{0} = 1 ~\textrm {mm}$.
Fig. 6. The Monte Carlo simulation yielded normalized density distributions of transverse and longitudinal velocity at different push beam intensities. The orange and blue curves represent the distributions of transverse and longitudinal velocity, respectively. The horizontal components of the black vectors are represented by $v_\textrm {rm}$, while the vertical components are represented by the average mean longitudinal velocities $\bar {v}_\textrm {z}$.
Another key factor influencing the beam width is the transverse cooling length, which determines the distribution of transverse velocities. The transverse cooling length is adjusted by moving the atomic cloud back and forth along the push beam. This is achieved by altering the electric currents of the offset coils in the Z direction. The transverse cooling length is scanned from 7 mm to 11.5 mm, as shown in Fig. 5(b). The beam width decreases with a longer transverse cooling length because a longer cooling length results in a lower transverse velocity. Therefore, the beam width can be reduced by moving the atomic cloud away from the reflecting mirror and using a larger transverse cooling beam. It should be noted that the intensity of the push beam and the transverse cooling length in our parameter ranges have only a minor influence on the flux.
7. Hollowness of the hollow-conical atomic beam
The hollowness of the atomic beam is evaluated by the RD defined as the ratio of DOV to HOP (see Fig. 3). RD reflects the solidness of the atomic beam. Assuming that atoms that reach the central region within a radius of $r_{0}$ contribute to DOV, RD is then given by
Based on these results, we conclude that both the beam width and RD decrease with a weaker push beam and a longer transverse cooling length. An extreme example is the 2D-MOT, which has a long transverse cooling length but uses a weak push beam. It is reasonable to deduce that an atomic beam from a 2D-MOT would have a solid core, while the transverse profiles of other beams based on hollow-core mirror systems, such as the conical or pyramidal traps [21,26,29–31], should be investigated.
8. Conclusion
In this article, we reported a hollow-conical atomic beam from a standard LVIS setup. This hollow-conical feature is observed by directly imaging the transverse profile of the atomic beam and is replicated through a Monte Carlo numerical simulation. By studying the extraction process through experiments and simulations, we hypothesize that the hollow-conical feature is caused by the converging-diverging process that occurs during the extraction of the atomic beam. Analytical models are proposed to quantitatively describe the beam width and the RD of the hollow-conical atomic beam. The experimental results and the analytical models indicate that the level of hollowness can be decreased by employing a weaker push beam and increasing the length of transverse cooling. On the bright side, the hollow-conical feature of the atomic beam is advantageous for applications such as continuously loading of cold atoms into hollow-core fibers. The loading efficiency can be significantly increased by placing the fiber tip at the focal point of the atomic beam, thereby enlarging the overlapping volume between the atomic beam and the fibers. By loading pre-cooled atomic beams from LVIS into the fiber and applying the in-fiber cooling technique [32], it is possible to create a magneto-optical trap of atoms inside hollow-core fibers. This technology has various applications, including fiber-guided atom gravimeters, gradiometers and accelerometers [33].
Funding
China National Postdoctoral Program for Innovative Talents (BX2021028).
Acknowledgments
We gratefully acknowledge Dr. D. Hu, Dr. X. X. Lu, and Dr. J. X. Liu for fruitful discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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