† Corresponding author. E-mail:
Project supported by the National Key Research and Development Program of China (Grant No. 2017YFA0304203), the Chang Jiang Scholars and Innovative Research Team in the University of the Ministry of Education of China (Grant No. IRT13076), the National Natural Science Foundation of China (Grant Nos. 61722507, 61675121, 61705123, and 11434007), the Fund for Shanxi 1331 Project Key Subjects Construction, China, and the Applied Basic Research Project of Shanxi Province, China (Grant No. 201701D221002).
We study the influence of external magnetic field on the shift of the resonant frequency in the photoassociation of ultracold Cs atoms, which are captured in a magnetically levitated optical crossed dipole trap. With the increase of the photoassociation laser intensity, the linear variation of the frequency shift is measured by recording the photoassociation spectra of the long-range
The formation and manipulation of ultracold molecules have been attracting a large amount of research interest due to their potential applications.[1] Ultracold molecules as the basic units of chemical matter at ultralow temperatures provide an efficient way to coherently control the chemical reaction at vanishing entropy.[2,3] Their rich internal structure is available to obtain higher accuracy in the precise measurement of fundamental physical constants.[4–8] The strong dipolar interaction of ultracold heteronuclear molecules induced by an external electric field can be used for the quantum simulation of the strongly interacting regime and quantum computation.[9,10] In this context, the photoassociation (PA) of ultracold atoms is a common but vital tool for the formation of ultracold molecules, which has been extensively investigated to facilitate the promising applications of ultracold molecules.[11–16]
In PA spectra, an interesting phenomenon is the shift of the resonant transition frequency from ground atoms to excited molecules, which has to be considered for the efficient formation of ultracold molecules.[17] The shift is mainly attributed to the coupling between the scattering atomic state and excited molecular state. The shift has a strongly linear dependence on the intensity of the PA laser.[18,19] A large number of atomic species have been experimentally studied for exhibiting the frequency shift of PA laser-induced transition.[20–25] In these previous studies, the slope of the shift to PA laser intensity has been illustrated as an efficient way to measure the s-wave scattering length of colliding atoms.[26] It is also closely related to both the optical Feshbach resonance of alkali-metal atoms[27] and the formation of ultracold molecules.[17] However, there are very few studies on the effect of the external field on the slope and the techniques that are likely to alter the slope and manipulate the frequency shift in PA spectra.
Although Feshbach resonance of 7Li atoms has been experimentally confirmed to have a great influence on the shift of the transition frequency in the PA,[28] the result is only qualitatively explained and quantitative investigations are still lacking. In this paper, we present the variation of the shift of the resonant transition frequency in the PA of ultracold Cs atoms with the intensity of the PA laser at different magnetic fields. We develop a square well model with a tunable depth, where the magnetic field can be used to alter the atomic collision during the PA of ultracold atoms. The quantitative analysis agrees with the observed dependence of the slope of the shift on the external magnetic field.
The Cs atoms are collected and prepared in a standard vapor-loaded magneto-optical trap (MOT). Successively compressed MOT and optical molasses are implemented to increase the atomic density and lower the temperature of the atomic sample by increasing the magnetic field gradient, reducing the intensity of the repumping laser, and enlarging the detuning of the trapping laser. Furthermore, the pre-cooled atoms are efficiently loaded into a three-dimensional (3D) optical lattice, which consists of four linearly polarized far-off-resonance laser beams with an optically optimized frequency detuning.[29] The degenerated Raman sideband cooling (DRSC) is then performed by pumping the degenerated adjacent vibrational levels with the different Zeeman sublevels to a dark state with the lowest vibrational number of v = 0 at the magnetic field of ∼ 200 mG. Through 10 ms of 3D DRSC, the atoms are cooled to the temperature of ∼ 1.7 μK and prepared in the F = 3, mF = 3 state.
After the 3D DRSC, the 3D optical lattice is adiabatically released with a 1/e time of 100 μs. Meanwhile, a crossed optical dipole trap is used to collect the cooled atoms, where two 1064 nm, high-power of ∼7 W laser beams are focused on the center of the atomic sample with the 1/e beam radius of ∼ 240 μm at the crossing angle of 90°, as shown in Fig.
PA is performed by illuminating a narrow linewidth Ti:sapphire laser with the widely tunable output frequency on the atomic sample in the optical trap for 100 ms. The PA laser is focused on the center of the atomic cloud with a beam waist of 150 μm. When the frequency of the PA laser is resonantly tuned to the transition frequency from the scattering atom state to the excited molecular state, the number of remaining atoms in the optical trap reduces because the formed weakly excited molecules spontaneously decay either into pairs of hot atoms that escape the trap or into the ground state molecules. Therefore PA spectroscopy can be obtained by recording the number of atoms in the dipole trap as a function of the PA laser frequency, which is monitored by a high precision wavemeter (HighFinese WSU). The frequency of the PA laser is tuned to ∼ 11155.059 cm−1 near the resonant transition from the scattering state F = 3, mF = 3 of Cs atoms to the rovibrational level of vD − v = 190, J = 0 of the excited
The shift of the resonant transition frequency in the PA of ultracold colliding atoms is induced by the change of the PA laser intensity IPA. As the intensity of the PA laser varies in each experimental cycle, the position at which the number of optically trapped atoms exhibits the maximum loss shifts in the PA spectrum. Figure
According to the simple and approximate expression of the frequency shift in Refs. [27] and [32], the magnitude of the shift of the transition frequency in PA is given as βΩ2/δεv, where
For the variation of the slope of the frequency shift in PA with the magnetic field, there is only a qualitative explanation for the previously experimental observations by considering the direct effect of the external magnetic fields on the coupling strength between the atomic and excited molecular states.[28] In the low temperature limit, the magnetic field alters the atomic s-wave scattering length and then leads to the variation of the scattering phase shift, which is a significant parameter in the atomic wavefunction. Nevertheless, the atomic scattering wavefunction in a long range is insufficient to capture the behavior of the colliding atoms in the effective range in which PA occurs.
An analytically theoretical model in Ref. [33] was used to calculate the atomic scattering length near a Feshbach resonance, and this model was used to explain the enhanced PA by magnetic field.[34] Here, a single channel square-well potential with a tunable depth is considered to generate a versatile wavefunction of Cs atoms in the effective PA range in Fig.
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The radius of the model square-well potential is chosen to be the mean scattering length R0 = ā = 4πΓ(1/4)−2RvdW,[35] where RvdW = (2mrC6/ℏ)1/4/2 is the van der Waals radius, mr is the reduced mass, and C6 is the van der Waals coefficient at R−6 in the realistic long-range potential. The mean scattering length of 95.7a0 securely covers the range of the upper bound state wavefunction of the excited Cs molecule. The atomic scattering wavefunction in the range of r > R0, containing the incident plane wave and a scattered wave, approaches
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![](cpb_28_1_013702/cpb_28_1_013702_ieqn6.gif)
For 133 Cs atoms in the hyperfine state F = 3, mF = 3, the s-wave Feshbach resonance of B = −11.74 G leads to the smooth variation of the scattering length a in the low-field region from 0 to 150 G and the dependence of the scattering length on B can be well approximated as[37,38]
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Compared to the radius of the square-well potential R0, the small effective range of the excited molecule state wavefunction ψv,J(r) determines that only the atomic wavefunction ϕk,r<R0(r) in the region of r < R0 dedicates to the effective overlap between the atomic and molecular wavefunctions. To explain the observed variation of the slope with the magnetic field, we construct the tunable depth of the square-well potential in Fig.
We have presented the linear dependence of the frequency shift on the intensity of the PA laser at three different magnetic fields. The derived slope of the shift is strongly dependent on the magnetic field, which has potential applications on the formation of cold molecules, precision measurement, and coherent manipulation of atoms and molecules. The observed variation of the slope with magnetic field is reproduced with a reasonable quality by using the model of a single-channel square-well potential with the tunable depth. Through this model, we can relate the magnetic field to the atomic wavefunction in the effective PA region. In the previous experiments, the influence of magnetic field on the shift in PA was explained by directly considering that the magnetic field controlled scattering length alters the atomic scattering wavefunction outside of the effective PA range. In contrast, our result shows the clear physical mechanism for the effect of external magnetic field on the slope of the frequency shift in PA.
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