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Physical Review A

Magic wavelengths for optical cooling and trapping of potassium

Marianna S. Safronova, Marianna S. Safronova, Charles W. Clark

Published May 6, 2013• 29 citations
Publisher

Abstract

We carry out a systematic study of the static and dynamic polarizabilities of the potassium atom using a first-principles high-precision relativistic all-order method in which all single, double, and partial triple excitations of the Dirac-Fock wave functions are included to all orders of perturbation theory. Recommended values and uncertainties are provided for the relevant electric-dipole matrix elements. Polarizabilities of the $4s$, $4{p}_{j}$, $5s$, $5{p}_{j}$, and $3{d}_{j}$ states are compared with other theoretical and experimental values when possible. We identify magic wavelengths for the $4s\ensuremath{-}np$ transitions for $n=4,5$, i.e., those wavelengths for which the two levels have the same ac Stark shifts. The magic wavelengths for the $4s\ensuremath{-}5p$ transitions are of particular interest for attaining a quantum gas of potassium at high phase-space density. We find 20 such wavelengths in the technically interesting region of $1050--1130$ nm.

DOI
10.1103/physreva.87.052504

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We carry out a systematic study of the static and dynamic polarizabilities of the potassium atom using a first-principles high-precision relativistic all-order method in which all single, double, and partial triple excitations of the Dirac-Fock wave functions are included to all orders of perturbation theory. Recommended values and uncertainties are provided for the relevant electric-dipole matrix elements. Polarizabilities of the 4s4s4s, 4pj4{p}_{j}4pj​, 5s5s5s, 5pj5{p}_{j}5pj​, and 3dj3{d}_{j}3dj​ states are compared with other theoretical and experimental values when possible. We identify magic wavelengths for the 4s\ensuremath−np4s\ensuremath{-}np4s\ensuremath−np transitions for n=4,5n=4,5n=4,5, i.e., those wavelengths for which the two levels have the same ac Stark shifts. The magic wavelengths for the 4s\ensuremath−5p4s\ensuremath{-}5p4s\ensuremath−5p transitions are of particular interest for attaining a quantum gas of potassium at high phase-space density. We find 20 such wavelengths in the technically interesting region of 1050−−11301050--11301050−−1130 nm.
We carry out a systematic study of the static and dynamic polarizabilities of the potassium atom using a first-principles high-precision relativistic all-order method in which all single, double, and partial triple excitations of the Dirac-Fock wave functions are included to all orders of perturbation theory. Recommended values are provided for a large number of electric-dipole matrix elements. Static polarizabilities of the 4s, 4p_j, 5s, 5p_j, and 3d_j states are compared with other theory and experiment where available. We use the results of the polarizability calculations to identify magic wavelengths for the 4s-np transitions for n=4,5n = 4, 5n=4,5, i.e. those wavelengths for which the two levels have the same ac Stark shifts. These facilitate state-insensitive optical cooling and trapping. The magic wavelengths for the 4s−5p4s-5p4s−5p transitions are of particular interest for attaining a quantum gas of potassium at high phase-space density. We find 20 such wavelengths in the technically interest region of 1050-1130 nm. Uncertainties of all recommended values are estimated.
Using first-principles calculations, we identify magic wavelengths ${\ensuremath{\lambda}}_{\mathrm{magic}}$ for the 2s2s2s-2p2p2p and 2s2s2s-3p3p3p transitions in lithium. The nsnsns and npnpnp atomic levels have the same ac Stark shifts at the corresponding magic wavelength, which facilitates state-insensitive optical cooling and trapping. Tune-out wavelengths for which the ground-state frequency-dependent polarizability vanishes are also calculated. Differences of these wavelengths between 6{}^{6}6Li and 7{}^{7}7Li are reported. Our approach uses high-precision, relativistic all-order methods in which all single, double, and partial triple excitations of the Dirac-Fock wave functions are included to all orders of perturbation theory. Recommended values are provided for a large number of Li electric-dipole matrix elements. Static polarizabilities for the 2s2s2s, 2p2p2p, 3s3s3s, 3p3p3p, and 3d3d3d levels are compared with other theory and experiment where available. Uncertainties of all recommended values are estimated. The magic wavelengths for the uv 2s2s2s-3p3p3p transition are of particular interest for the production of a quantum gas of lithium [Duarte et al., Phys. Rev. A 84, 061406R (2011)].
Extensive calculations of the electric-dipole matrix elements in alkali-metal atoms are conducted using the relativistic all-order method. This approach is a linearized version of the coupled-cluster method, which sums infinite sets of many-body perturbation theory terms. All allowed transitions between the lowest nsnsns, np1∕2n{p}_{1∕2}np1∕2​, np3∕2n{p}_{3∕2}np3∕2​ states and a large number of excited states are considered in these calculations and their accuracy is evaluated. The resulting electric-dipole matrix elements are used for the high-precision calculation of frequency-dependent polarizabilities of the excited states of alkali-metal atoms. We find ``magic'' wavelengths in alkali-metal atoms for which the nsnsns and np1∕2n{p}_{1∕2}np1∕2​ and np3∕2n{p}_{3∕2}np3∕2​ atomic levels have the same ac Stark shifts, which facilitates state-insensitive optical cooling and trapping.
We report here oscillator strengths, transition rates, branching ratios, and lifetimes due to the allowed transitions in the potassium (K) atom. We evaluate electric dipole (E1E1E1) amplitudes using an all-order relativistic many-body perturbation method. The obtained results are compared with previously available experimental and theoretical studies. Using the E1E1E1 matrix elements mentioned above and estimated from the lifetimes of the 4p2P1/2,3/24p{\phantom{\rule{4pt}{0ex}}}^{2}\phantom{\rule{-0.16em}{0ex}}{P}_{1/2,3/2}4p2P1/2,3/2​ states, we determine precise values of static and dynamic polarizabilities for the first five low-lying states in the considered atom. The static polarizabilities of the ground and 4p2P1/2,3/24p{\phantom{\rule{4pt}{0ex}}}^{2}\phantom{\rule{-0.16em}{0ex}}{P}_{1/2,3/2}4p2P1/2,3/2​ states in the present work are more precise than the available measurements in these states. Only the present work employs relativistic theory to evaluate the polarizabilities in the 3d2D3/2,5/23d{\phantom{\rule{4pt}{0ex}}}^{2}\phantom{\rule{-0.16em}{0ex}}{D}_{3/2,5/2}3d2D3/2,5/2​ states for which no experimental results are known with which to compare. We also reexamine ``magic wavelengths'' for the $4p{\phantom{\rule{4pt}{0ex}}}^{2}\phantom{\rule{-0.16em}{0ex}}{P}_{1/2}\ensuremath{\rightarrow}4s{\phantom{\rule{4pt}{0ex}}}^{2}\phantom{\rule{-0.16em}{0ex}}{S}_{1/2}$ and $4p{\phantom{\rule{4pt}{0ex}}}^{2}\phantom{\rule{-0.16em}{0ex}}{P}_{3/2}\ensuremath{\rightarrow}4s{\phantom{\rule{4pt}{0ex}}}^{2}\phantom{\rule{-0.16em}{0ex}}{S}_{1/2}$ transitions due to the linearly polarized light, which are useful to perform the state-insensitive trapping of K atoms.
We demonstrate magic wavelengths for the 6s 2S1/2−6p 2P3/2,1/26s ~ {^2}S_{1/2}-6p ~ {^2}P_{3/2,1/2}6s 2S1/2​−6p 2P3/2,1/2​ transitions in the Cs atom due to the circularly polarized light. In addition, we also obtain magic wavelengths using the linearly polarized light in order to verify the previously reported values and make a comparative study with the values obtained for the circularly polarized light. Most of these wavelengths are found to be in the optical region and could be of immense interest to the experimentalists for carrying out high precision measurements. To obtain these wavelengths, we have calculated dynamic dipole polarizabilities of the ground, 6p 2P1/26p ~{^2}P_{1/2}6p 2P1/2​ and 6p 2P3/26p ~{^2}P_{3/2}6p 2P3/2​ states. We use the available precise values of the electric dipole (E1) matrix elements for the most dominantly contributing transitions from the lifetime measurements of the above excited states and obtain other relevant important E1 matrix elements by employing a relativistic coupled-cluster method approximating at the singles and doubles level. Accuracies in these quantities are substantiated by comparing our evaluated scalar polarizability values with the corresponding experimental results.
Using first-principles calculations, we identify ``tune-out'' optical wavelengths, ${\ensuremath{\lambda}}_{\mathrm{zero}}$, for which the ground-state frequency-dependent polarizabilities of alkali-metal atoms vanish. Our approach uses high-precision, relativistic all-order method in which all single, double, and partial triple excitations of the Dirac-Fock wave functions are included to all orders of perturbation theory. We discuss the use of tune-out wavelengths for sympathetic cooling in two-species mixtures of alkali metals with group II and other elements of interest. Special cases in which these wavelengths coincide with strong resonance transitions in a target system are identified.