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arXiv:2101.11323

Universal relations between atomic dipolar relaxation and van der Waals interaction

Yuangang Deng, Yi-Quan Zou, Gao‐Ren Wang, Qi Liu, Su Yi, Meng Khoon Tey, Li You

Published Jan 27, 2021• 1 citation
arXivHTMLPDF

Abstract

Dipolar relaxation happens when one or both colliding atoms flip their spins exothermically inside a magnetic ($B$) field. This work reports precise measurements of dipolar relaxation in a Bose-Einstein condensate of ground state $^{87}$Rb atoms together with in-depth theoretical investigations. Previous perturbative treatments fail to explain our observations except at very small $B$-fields. By employing quantum defect theory based on analytic solutions of asymptotic van der Waals interaction $-C_6/R^6$ ($R$ being interatomic spacing), we significantly expand the applicable range of perturbative treatment. We find the $B$-dependent dipolar relaxation lineshapes are largely universal, determined by the coefficient $C_6$ and the associated $s$-wave scattering lengths $a_{\rm sc}$ of the states before and after spin flips. This universality, which applies generally to other atomic species as well, implicates potential controls of dipolar relaxation and related cold chemical reactions by tuning $a_{\rm sc}$.

DOI
10.48550/arxiv.2101.11323

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Dipolar relaxation happens when one or both colliding atoms flip their spins exothermically inside a magnetic (BBB) field. This work reports precise measurements of dipolar relaxation in a Bose-Einstein condensate of ground state 87^{87}87Rb atoms together with in-depth theoretical investigations. Previous perturbative treatments fail to explain our observations except at very small BBB-fields. By employing quantum defect theory based on analytic solutions of asymptotic van der Waals interaction −C6/R6-C_6/R^6−C6​/R6 (RRR being interatomic spacing), we significantly expand the applicable range of perturbative treatment. We find the BBB-dependent dipolar relaxation lineshapes are largely universal, determined by the coefficient C6C_6C6​ and the associated sss-wave scattering lengths asca_{\rm sc}asc​ of the states before and after spin flips. This universality, which applies generally to other atomic species as well, implicates potential controls of dipolar relaxation and related cold chemical reactions by tuning asca_{\rm sc}asc​.
Magnetically tunable scattering resonances have been used with great success for precise control of s-wave scattering lengths in ultracold atomic collisions. We describe relatively simple yet quite powerful analytic treatments of such resonances based on the analytic properties of the van der Waals long range potential. This theory can be used to characterize a number of properties of specific resonances that have been used successfully in various experiments with 87^{87}87Rb, 85^{85}85Rb, 40^{40}40K, and 6^{6}6Li. Optical Feshbach resonances are also possible and may be practical with narrow intercombination line photoassociative transitions in species like Sr and Yb.
In this dissertation, I present ab initio theoretical results on cold and ultra-cold collision of atomic and molecular systems with magnetic and electric dipole moments. The collisions of different isotopes of maximally spin-polarized chromium atoms (52Cr and 53Cr), alkali-metal atom (Rb) - chromium atom, and magnetically trapped hydroxide (OH) with cold supersonic helium beam are considered in this thesis. Spin-changing cross sections and rate constants in Cr-Cr isotopic collision in the presence of a magnetic field are calculated by propagating a set of multichannel equations in the field. The coupling between dipole-dipole allowed and hyperfine channels are included. The short range electrostatic interactions between atoms and atom-molecule are represented with highly-accurate Born-Oppenheimer (BO) potential energy curves. Due to the large spin multiplicity in Cr-Cr and Rb-Cr interactions, more than the usual singlet and triplet channels are obtained. The long-range van der Waals (vdW) interactions between atoms are calculated, for Cr, by including accurate transition frequencies, discrete dipole matrix elements, and photoionization continuum oscillator strengths, and for Rb, using published dynamic polarizabilities. The vdW coefficients for Cr-Cr and Rb-Cr are obtained. In the cold collision regime, I investigate the influence of angular momentum shape resonances on the elastic and inelastic (loss channel) rate coefficients, and when the Zeeman relaxation, resulting in the loss of atoms from the trap, dominates elastic collision. Dipolar collisions are crucial to achieving quantum degeneracy. Magnetically tuned Fano-Feshbach closed-channel resonances are used to obtain zero-energy scattering lengths in comparison with observation of trap loss. In bosonic chromium collisions, resonances appear due only to dipolar interaction. The interplay between hyperfine and dipolar resonances in mixed-species Cr-Cr and Rb-Cr collisions is studied. In Rb-Cr, it is found that the molecule has broken symmetry and therefore contains a large permanent electric dipole moment in the ground state. A recent collisional experiment at JILA of a supersonic helium beam with a magnetically trapped OH radical found the onset of a quantum threshold in the total OH loss rate. Using ab initio BO energy surfaces, I calculated the spin-changing collisions which removed OH from the trap. Only after including the effect of the trapping potential on the collision dynamics, some agreement with observation was found. ^
We study dipolar relaxation in both ultracold thermal and Bose-condensed Cr atom gases. We show three different ways to control dipolar relaxation, making use of either a static magnetic field, an oscillatory magnetic field, or an optical lattice to reduce the dimensionality of the gas from three-dimensional (3D) to two-dimensional (2D). Although dipolar relaxation generally increases as a function of a static magnetic-field intensity, we find a range of nonzero magnetic-field intensities where dipolar relaxation is strongly reduced. We use this resonant reduction to accurately determine the S=6S=6S=6 scattering length of Cr atoms: ${a}_{6}=103\ifmmode\pm\else\textpm\fi{}4{a}_{0}$. We compare this new measurement to another new determination of a6{a}_{6}a6​, which we perform by analyzing the precise spectroscopy of a Feshbach resonance in ddd-wave collisions, yielding ${a}_{6}=102.5\ifmmode\pm\else\textpm\fi{}0.4{a}_{0}$. These two measurements provide, by far, the most precise determination of a6{a}_{6}a6​ to date. We then show that, although dipolar interactions are long-range interactions, dipolar relaxation only involves the incoming partial wave l=0l=0l=0 for large enough magnetic-field intensities, which has interesting consequences on the stability of dipolar Fermi gases. We then study ultracold Cr gases in a one-dimensional (1D) optical lattice resulting in a collection of independent 2D gases. We show that dipolar relaxation is modified when the atoms collide in reduced dimensionality at low magnetic-field intensities, and that the corresponding dipolar relaxation rate parameter is reduced by a factor up to 777 compared to the 3D case. Finally, we study dipolar relaxation in the presence of rf oscillating magnetic fields, and we show that both the output channel energy and the transition amplitude can be controlled by means of the rf frequency and Rabi frequency.
The van der Waals coefficient ${C}_{6}(\ensuremath{\theta};nlJM)$ of two like Rydberg atoms in their identical Rydberg states $|nlJM\ensuremath{\rangle}$ is resolved into four irreducible components called scalar Rss{R}_{ss}Rss​, axial (vector) Raa{R}_{aa}Raa​, scalar-tensor RsT=RTs{R}_{sT}={R}_{Ts}RsT​=RTs​, and tensor-tensor RTT{R}_{TT}RTT​ parts in analogy with the components of dipole polarizabilities. The irreducible components determine the dependence of ${C}_{6}(\ensuremath{\theta};nlJM)$ on the angle $\ensuremath{\theta}$ between the interatomic and the quantization axes of atoms. The spectral resolution for the biatomic Green's function with account of the most contributing terms is used for evaluating the components ${R}_{\ensuremath{\alpha}\ensuremath{\beta}}$ of atoms in their Rydberg series of doublet states of the low angular momenta (2S,2P,2D,2F^{2}S, ^{2}P, ^{2}D, ^{2}F2S,2P,2D,2F). The polynomial presentations in powers of the Rydberg-state principal quantum number nnn taking into account the asymptotic dependence ${C}_{6}(\ensuremath{\theta};nlJM)\ensuremath{\propto}{n}^{11}$ are derived for simplified evaluations of irreducible components. Numerical values of the polynomial coefficients are determined for Rb atoms in their n2S1/2,n2P1/2,3/2,n2D3/2,5/2n{\phantom{\rule{0.222222em}{0ex}}}^{2}{S}_{1/2}, n{\phantom{\rule{0.222222em}{0ex}}}^{2}{P}_{1/2,3/2}, n{\phantom{\rule{0.222222em}{0ex}}}^{2}{D}_{3/2,5/2}n2S1/2​,n2P1/2,3/2​,n2D3/2,5/2​, and n2F5/2,7/2n{\phantom{\rule{0.222222em}{0ex}}}^{2}{F}_{5/2,7/2}n2F5/2,7/2​ Rydberg states of arbitrary high nnn. The transformation of the van der Waals interaction law $\ensuremath{-}{C}_{6}/{R}^{6}$ into the dipole-dipole law C3/R3{C}_{3}/{R}^{3}C3​/R3 in the case of close dipole-connected two-atomic states (the F\"orster resonance) is considered and the dependencies on the magnetic quantum numbers MMM and on the angle $\ensuremath{\theta}$ of the constant ${C}_{3}(\ensuremath{\theta};nlJM)$ are determined together with the ranges of interatomic distances RRR, where the transformation appears.
Progress in ultracold experiments with polar molecules requires a clear understanding of their interactions and reactivity at ultralow collisional energies. Two important theoretical steps in this process are the characterization of interaction potentials between molecules and the modeling of the reactive scattering mechanism. Here, we report on the ab initio calculation of isotropic and anisotropic van der Waals interaction potentials for polar KRb and RbCs colliding with each other or with ultracold atoms. Based on these potentials and two short-range scattering parameters, we then develop a single-channel scattering model with flexible boundary conditions. Our calculations show that, at low temperatures (and in the absence of an external electric field), the reaction rates between molecules or between molecules and atoms have a resonant character as a function of the short-range parameters. We also find that both the isotropic and anisotropic van der Waals coefficients have significant contributions from dipole coupling to excited electronic states. Their values can differ dramatically from those solely obtained from the permanent dipole moment. A comparison with recently obtained reaction rates of fermionic 40 K 87 Rb shows that the experimental data cannot be explained by a model where the short-range scattering parameters are independent of the relative orbital angular momentum or partial wave.
We investigate the scattering dynamics governed by the long-range van der Waals plus dipole-dipole interaction potential, $\ensuremath{-}{C}_{6}/{R}^{6}\ensuremath{-}{C}_{3}/{R}^{3}$, which describes the long-range interaction between two polar molecules in an electric field. In the spirit of quantum defect theory, a set of parameters which are nearly constants in the threshold regime is defined to characterize the scattering process. Using appropriate boundary conditions for the scattering wave functions and relevant parameters, we explore the quantum reflection by and quantum tunneling through the long-range potential. As a sample application, the reactive collision rates of 40{}^{40}40K87{}^{87}87Rb +++ 40{}^{40}40K87{}^{87}87Rb are calculated.