ISSN: 0256-307X 中国物理快报 Chinese Physics Letters Volume 28 Number 7 July 2011 A Series Journal of the Chinese Physical Society Distributed by IOP Publishing Online: http://iopscience.iop.org/cpl http://cpl.iphy.ac.cn C HINESE P HYSICAL S OCIETY Institute of Physics PUBLISHING CHIN. PHYS. LETT. Vol. 28, No. 7 (2011) 073201 Magic Wavelength of an Optical Clock Transition of Barium * YU Geng-Hua({庚u)1;2;3, ZHONG Jia-Qi(®Wj)1;2;3, LI Run-Bing(odW)1;2, WANG Jin(王>)1;2, ZHAN Ming-Sheng(詹²))1;2** 1 State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071 2 Center for Cold Atom Physics, Chinese Academy of Sciences, Wuhan 430071 3 Graduate University, Chinese Academy of Sciences, Beijing 100049 (Received 30 March 2011) Similar to most of the other alkaline earth elements, barium atoms can be candidates for optical clocks, thus the magic wavelength for an optical lattice is important for the clock transition. We calculate the magic wavelength 2 1 3 of a possible clock transition between 6s S0 and 6s5d D2 states of barium atoms. Our theoretical result shows that there are three magic wavelengths 615.9 nm, 641.2 nm and 678.8 nm for a linearly polarized optical lattice laser for barium. PACS: 32.60.+i, 37.10.Jk, 32.70.Jz DOI:10.1088/0256-307X/28/7/073201 Optical clocks based on neutral atoms have at- us to build an accurate optical clock based on barium tracted great attention due to their variable ap- atoms trapped in an optical lattice. In this Letter, we 2 1 3 plications. Possible applications include next- propose to use the Ba 6s S0–6s5d D2 transition as generation frequency standards, new tests of funda- a clock transition and present a calculation of magic mental physics, more accurate measurements of funda- wavelength for this clock. mental constants and their time dependence, and fur- ther improvement in global positioning systems.[1−4] Optical lattice traps formed by a far off-resonant laser are very useful for confining cold atoms. Collision shift and broadening of atoms confined in optical lattices can be effectively eliminated. In addition, when atoms are confined to a Lamb–Dicke region, the first-order Doppler shift will vanish.[5;6] For high precision frequency standards, light shift Clock transition of the clock transitions caused by a trapping laser needs to be avoided. Thus the wavelength of the trapping light should be tuned to a region where the Fig. 1. Diagram of the energy levels for barium atoms and some possible laser couplings. light shifts of two states of the clock transition can- cel each other. This special wavelength is called the Calculation of energy shift caused by the trap- magic wavelength. Katori et al.[7−9] have proposed to ping laser has been reported before.[16−19] Accord- utilize optical lattice traps formed with a magic wave- ing to second-order perturbation theory, when atoms length trapping laser, and this ingenious technique can are in state jii with energy Ei and Zeeman sublevel greatly improve the established high-accuracy optical mi, emerged in a trapping laser field with frequency frequency standard based on neutral atoms. Opti- v = !=2휋, polarization p, and intensity I, the energy cal lattice clocks of alkaline-earth atom Sr and rare- shift of the atomic state, Ui(!; p; mi), can be expressed earth atom Yb have been widely studied and have as [10−13] 훼i(!; p; mi)I achieved great success in recent years. Alkaline- Ui(!; p; mi) = − ; (1) earth metal Ba has a very sharp optical transition 2"0c 2 1 3 [14] 6s S0–6s5d D2 with a lifetime of about 69 s of where 훼i(!; p; mi) is the induced polarizability, which the upper level. The natural linewidth of this clock can be calculated by summation over the contribu- transition is about 2 mHz, thus Ba can also be a good tions coming from all dipole transitions from the de- candidate for an optical frequency standard. The cool- sired level i to all other levels k with the respective ing and trapping of barium atoms has been realized Einstein coefficients Aki (spontaneous emission rate 0 and reported in Ref. [15], which makes it possible for for Ek > Ei), Zeeman sublevels m and transition fre- *Supported by the National Natural Science Foundation of China under Grant Nos 11004227 and 11074281, the National Basic Research Program of China under Grant No 2010CB832805, and the Chinese Academy of Sciences. **To whom correspondence should be addressed. Email: [email protected] © 2011 Chinese Physical Society and IOP Publishing Ltd 073201-1 CHIN. PHYS. LETT. Vol. 28, No. 7 (2011) 073201 1 quencies vik = !ik=2휋, plified, as shown in Fig. 1. Here, S0 is the ground 3 state and D2 is the metastable triplet state with a !2 lifetime of about 69 s. X Aki(2Jk + 1) Ji 1 Jk 훼 = 6휋c3" : Table1 shows the wave numbers (WN) and prob- i 0 !2 (!2 − !2) m p −m0 k;m0 ik ik i abilities (Einstein coefficientski A ) of transitions orig- 2 1 3 (2) inating from the 6s S0 and 6s5d D2 states of bar- The expression in large parentheses denotes a 3J sym- ium. We choose the available experimental values as bol. If we know !ik and Aki in Eq. (2), we can obtain much as possible from Refs. [20,21], then we take part the polarizability 훼i. of the theoretical data from Ref. [22], and for the rest For barium atoms, the energy diagram can be sim- we mainly use the values in Ref. [23]. Table 1. Atom transition data of barium. −1 6 −1 Lower state jii Upper state jki WN (cm ) Aki (10 s ) References 1 6s7p P1 32547.033 0.38 [22] 3 6s7p P2 30987.240 2.1 [22] 3 6s7p P1 30815.512 8.6 [22] 1 5d6p P1 28554.221 0.090 [21] 1 5d6p F3 26 816.266 0.34 [22] 3 5d6p P2 25956.519 16.2 [21] 3 5d6p P1 25704.110 56.0 [20] 3 −1 3 6s5d D2 (9215.501 cm ) 5d6p D3 24979.834 11.6 [21] 3 5d6p D2 24531.513 33.0 [20] 3 5d6p D1 24192.033 18.9 [20] 1 5d6p D2 23074.387 0.065 [21] 3 5d6p F3 22947.423 32.0 [20] 3 5d6p F2 22064.645 7.6 [20] 1 6s6p P1 18060.261 0.11 [20] 3 6s6p P2 13514.745 0.1 [22] 3 6s6p P1 12636.623 0.32 [23] 1 6s18p P1 41494.39 0.086 [20] 1 6s17p P1 41411.04 0.15 [20] 1 6s16p P1 41307.88 0.23 [20] 1 6s15p P1 41183.60 0.56 [20] 3 5d8p P1 41097.20 0.72 [20] 1 6s14p P1 40991.23 0.14 [20] 3 5d8p D1 40893.76 0.45 [20] 1 6s13p P1 40765.23 0.081 [20] 1 6s12p P1 40428.68 0.46 [20] 1 6s11p P1 39982.14 1.5 [20] 1 2 1 6s10p P1 39311.95 0.041 [20] 6s S0 (ground state) 1 5d7p P1 38499.86 [20] 1 6s9p P1 37775.28 1.1 [20] 3 5d7p P1 36989.98 [20] 3 5d7p D1 36495.732 [20] 1 6s8p P1 35892.465 [20] 1 6s7p P1 32547.033 42.0 [20] 3 6s7p P1 30815.512 0.013 [20,22] 1 5d6p P1 28554.221 35.0 [20] 3 5d6p P1 25704.110 1.1 [20] 3 5d6p D1 24192.033 1.5 [20] 1 6s6p P1 18060.261 119 [20] 3 6s6p P1 12636.623 0.299 [23] We now calculate the light shift for m = 0 lev- considered in this work, the magic wavelength value 2 1 els of the optical clock transition from 6s S0 to corresponds to the crossing of the light shift curves for 3 2 1 3 6s5d D2 of a boson barium isotope with nuclear spin the 6s S0 and 6s5d D2 states. of I = 0. The magic wavelength is the wavelength Figure2 displays the dependence of energy level of the trapping laser where the light shifts of the two shift difference for barium on wavelength of a linearly states for a corresponding transition are the same. For polarized trapping laser. The light shift is defined as 2 1 3 the 6s S0–6s5d D2 clock transition of barium atoms U=h (U is the energy shift of a state), and the inten- 073201-2 CHIN. PHYS. LETT. Vol. 28, No. 7 (2011) 073201 sity of the trapping laser is 1 × 104 W/cm2. As shown Boltzmann’s constant). The slope of light shift dif- d by the arrows in Fig. 2, at 615.9 nm, 641.2 nm, and ference at a magic wavelength defined as k = 푑휆 (∆U) 1 3 3 1 678.8 nm the states S0 and D2 have the same light (∆U = U D2 − U S0 ) shows how fast the difference shifts. Thus these wavelengths are magic wavelengths changes with the laser wavelength. These are listed in for the Ba clock. Table2. We also consider the magic wavelength caused by Table 2. Trap depth and light shift slope at different magic a circularly polarized laser. The light shifts caused wavelengths. by linearly polarized and circularly polarized lasers 휆 (nm) µ K with wavelengths around 615 nm, 640 nm and 680 nm magic Trap depth ( K) (MHz/nm) 615.9 29.1 0.11 are presented in Figs.3(a),3(b) and3(c), respectively.