: even isotopes of group II atoms)

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<ul><li><p>PHYSICAL REVIEW A 84, 045401 (2011)</p><p>Magnetic-field effects in transitions of X Li molecules (X: even isotopes of group II atoms)</p><p>Geetha Gopakumar,* Minori Abe, and Masahiko HadaDepartment of Chemistry, Tokyo Metropolitan University, Minami-Osawa, Hachioji, Tokyo 192-0397, Japan</p><p>Masatoshi Kajita</p><p>National Institute of Information and Communications Technology, Koganei, Tokyo 184-8795, Japan(Received 21 July 2011; published 4 October 2011)</p><p>We analyze the Zeeman shift in the (v,N ) = (0,0) (1,0) transition frequency of X Li molecules (X: evenisotopes of group II atoms), which is of interest in metrology. The Zeeman shift in the transition frequencybetween stretching states is found to be less than 1 mHz with a magnetic field of 1 G. X 6Li molecules are moreadvantageous than X 7Li molecules for measuring the transition frequency without the Zeeman shift because ofthe smaller g factor of the Li nuclear spin.</p><p>DOI: 10.1103/PhysRevA.84.045401 PACS number(s): 32.60.+i, 06.30.Ft, 33.20.t</p><p>I. INTRODUCTION</p><p>Using high-precision laser spectroscopy to measure thetransition frequencies of ultracold molecules is a novelapproach for testing the variance in the proton-to-electronmass ratio. One of the important effects responsible for thefrequency uncertainty in measuring transition frequency isthe Zeeman shift. We discuss the possibility of eliminatingthe Zeeman shift of the transition frequency to reduce themeasured frequency uncertainty. Bakalov et al. analyzed theZeeman shift of the HD+ rovibrational transition frequencies[1], which in turn could be used to determine the most suitabletransitions for a metrology experiment.</p><p>We analyzed the Zeeman shift of the (v,N ) = (0,0) (1,0) transition frequencies of X Li (X = 24Mg, 40Ca,88Sr, 138Ba, and 174Yb) molecules, where v and N are thequantum numbers of the molecular vibrational and rotationalstates, respectively. The electronic ground state is 2 withthe fine structure quantum number J = 1/2 when N = 0.The X nuclear spins are zero, the 6Li (7Li) nuclear spin is1 (3/2), and the possible hyperfine quantum numbers F are3/2 and 1/2 (2 and 1). The vibrational transition frequenciesof X Li molecules (produced using laser-cooled X and Liatoms) are advantageous for precise measurement because Liis the lightest atom that can be laser cooled in the groundstate, and the vibrational transition frequencies are highest</p><p>*geetha@tmu.ac.jpkajita@nict.go.jp</p><p>among the ultracold diatomic molecules. The vibrationaltransition frequency of optically trapped 174Yb6Li molecules(presumably also with some other X 6Li molecules) can bemeasured by eliminating the Stark shift induced by the trapand probe lasers [2]. Therefore, it is useful to estimate theZeeman shift of the transitions between different hyperfinesubstates.</p><p>The present discussion is based on the molecular constantsobtained using ab initio calculations [3] at the coupled clustersingles and doubles with partial triples [CCSD(T)] level ofcorrelation using the MOLCAS [4] software. We used relativisticcorrelation-consistent atomic natural orbital (ANO-RCC) [5]basis sets for both X and Li atoms.</p><p>II. ZEEMAN ENERGY SHIFTS AT EACH HYPERFINESUBSTATE</p><p>A. Eigenvalues of v(F,M) states</p><p>The 6Li and 7Li nuclear spins are 1 and 3/2, respectively,and the energy eigenvalues of the v(F,M)(|X2,v,N =0,J = 1/2,F,M, where M is the component of F parallelto the magnetic field) states with magnetic-field strength B inthe format [v(F,M) eigenvalues are given as follows.</p><p>For X 6Li,</p><p>v</p><p>(3</p><p>2, 3</p><p>2</p><p>): BB</p><p>[1</p><p>2ge + gI(6Li) (v)</p><p>]+ hhf (v)</p><p>2, (1)</p><p>v</p><p>(3</p><p>2, 1</p><p>2</p><p>): larger eigenvalue of matrix M1/2 =</p><p>[hhf (v)</p><p>2 13BB[</p><p>12ge + gI (Li)(v)</p><p>] 23 BB[ge gI (6Li)(v)]</p><p>2</p><p>3 BB[ge gI (6Li)(v)] hhf (v)</p><p>2 13BB[</p><p>12ge gI (6Li)(v)</p><p>]],</p><p>(2)</p><p>v</p><p>(1</p><p>2, 1</p><p>2</p><p>): smaller eigenvalue of matrix M1/2 , (3)</p><p>045401-11050-2947/2011/84(4)/045401(4) 2011 American Physical Society</p><p>http://dx.doi.org/10.1103/PhysRevA.84.045401</p></li><li><p>BRIEF REPORTS PHYSICAL REVIEW A 84, 045401 (2011)</p><p>and for X 7Li</p><p>v(2, 2) : BB[</p><p>1</p><p>2ge + 3</p><p>2gI (7Li)(v)</p><p>]+ hhf (v)</p><p>2, (4)</p><p>v (2, 1) : larger eigenvalue of matrix</p><p>M1 =[</p><p>hhf (v)2 12BB[ 12ge + 32gI (7Li)(v)] </p><p>3</p><p>4 BB[ge gI (7Li)(v)</p><p>]</p><p>3</p><p>4 BB[ge gI (Li)(v)] hhf (v)</p><p>2 12BB[</p><p>12ge 52gI (Li)(v)</p><p>]], (5)</p><p>v(2,0) : larger eigenvalue of matrix M0 =[</p><p>hhf (v)2</p><p>12BB[ge gI (7Li)(v)]</p><p>12BB[ge gI (7Li)(v)] hhf (v)2</p><p>], (6)</p><p>v(1, 1) : smaller eigenvalue of matrix M1, (7)</p><p>v(1,0) : smaller eigenvalue of matrix M0, (8)</p><p>respectively, where B is the Bohr magneton (B/h =1.4 107 Hz/mT), hf (v) is the F = 3/2 1/2 hyperfinetransition frequency, and ge (= 2.003) and gI (Li)(v)(gI (6Li) =4.4 104,gI (7Li) = 1.15 103) [6] are the g factors ofelectron and Li nuclear spins, respectively. Note that hf (v)and gI (Li)(v) have dependence on the vibrational state (seeSec. II B). Table I lists the values of hf for X 6Li andX 7Li molecules in the 0 and 1 states computed using theGAUSSIAN03 [7] program. The gauge-including atomic orbital(GIAO) [8] method is used with correlation consistent gaussianbasis sets with valence quintuple-zeta quality (cc-pV5Z)(Li)and Stuttgart/Dresden relativistic effective core potential(SDD) (Mg, Ca, Sr, Ba, and Yb) basis set for the computationof magnetic shielding tensors. In our calculations, magneticshielding tensors are divided into spin-dependent and spin-freeterms. For the former (spin-dependent term given in Table I),we used only the Fermi-contact term, which depends on thespin density of the nucleus. We used unrestricted Becke three-parameter Lee-Yang-Parr hybrid functionals (B3LYP) with apoint nucleus model for calculating spin density. From ourcalculations, we found that the hf values are larger for X 7Lithan for X 6Li, with the factor of gI (7Li)/gI (6Li) as expected.In Fig. 1, we show as an example of the energy levels of thehyperfine substates of (a) 24Mg6Li and (b) 24Mg7Li moleculesin the 0(F,M) state with a magnetic field in the range of 0</p></li></ul>