1. Blackbody radiation shifts and magic wavelengths for atomic clock research IEEE-IFCS 2010, Newport Beach, CA June 2, 2010 Marianna Safronova 1, M.G. Kozlov 1,2, Dansha Jiang 1, and U.I. Safronova 3 1 University of Delaware, USA 2 PNPI, Gatchina, Russia 3 University of Nevada, Reno, USA

2. Black-body radiation shifts Microwave vs. Optical transitions BBR shift in Rb frequency standard How to calculate its uncertainty? Development of new methodology for precision calculations of Group II-type system properties Polarizabilities Magic wavelengths Outline

3. Blackbody radiation shifts T = 300 K Clock transition Level A Level B  BBR T = 0 K Transition frequency should be corrected to account for the effect of the black body radiation at T=300K.

4. atomic clocks black-body radiation ( BBR ) shift Motivation: BBR shift gives large contribution into uncertainty budget for some of the atomic clock schemes. Accurate calculations are needed to achieve ultimate precision goals.

5. BBR shift and polarizability BBR shift of atomic level can be expressed in terms of a scalar static polarizability to a good approximation [1]: [1] Sergey Porsev and Andrei Derevianko, Physical Review A 74, 020502R (2006) Dynamic correction is generally small. Multipolar corrections (M1 and E2) are suppressed by  2 [1]. Vector & tensor polarizability average out due to the isotropic nature of field. Dynamic correction

6. microWave transitions optical transitions 4d 5/2 Sr + Lowest-order polarizability 5s 1/2 Cs 6s F=3 6s F=4 In lowest (second) order the polarizabilities of ground hyperfine 6s 1/2 F=4 and F=3 states are the same. Therefore, the third-order F-dependent polarizability  F (0) has to be calculated. terms term

7. BBR shifts for microwave transitions Atom Transition Method Ref.  7 Li 2s (F=2 – F=1) LCCSD[pT][1] -0.5017  10 -14 23 Na 3s (F=2 – F=1) LCCSD[pT] [7] -0.5019  10 -14 39 K 4s (F=2 – F=1) LCCSD[pT][2] -1.118  10 -14 87 Rb 5s (F=2 – F=1) CP [3] -1.26(1)  10 -14 133 Cs 6s (F=3 – F=4) LCCSD[pT] [4] -1.710(6)  10 -14 CP [3] -1.70(2)  10 -14 Experiment [5]-1.710(3)  10 -14 137 Ba + 6s (F=2 – F=1) CP [3] -0.245(2)  10 -14 171 Yb + 6s (F=1 – F=0) CP [3] -0.0983  10 -14 MBPT3 [6] -0.094(5)  10 -14 137 Hg + 6s (F=1 – F=0) CP [3] -0.0102(5)  10 -14 [1] W.R. Johnson, U.I. Safronova, A. Derevianko, and M.S. Safronova, PRA 77, 022510 (2008) [2] U.I. Safronova and M.S. Safronova, PRA 78, 052504 (2008) [3] E. J. Angstmann, V.A. Dzuba, and V.V. Flambaum, PRA 74, 023405 (2006) [4] K. Beloy, U.I. Safronova, and A. Derevianko, PRL 97, 040801 (2006) [5] E. Simon, P. Laurent, and A. Clairon, PRA 57, 426 (1998) [6] U.I. Safronova and M.S. Safronova, PRA 79, 022510 (2009) [7] M. S. Safronova et al., IEEE - TUFFC 57, 94 (2010).

8. BBR shifts for microwave transitions Atom Transition Method Ref.  7 Li 2s (F=2 – F=1) LCCSD[pT][1] -0.5017  10 -14 23 Na 3s (F=2 – F=1) LCCSD[pT] [7] -0.5019  10 -14 39 K 4s (F=2 – F=1) LCCSD[pT][2] -1.118  10 -14 87 Rb 5s (F=2 – F=1) CP [3] -1.26(1)  10 -14 LCCSD[pT] Present -1.255(4)  10 -14 133 Cs 6s (F=3 – F=4) LCCSD[pT] [4] -1.710(6)  10 -14 CP [3] -1.70(2)  10 -14 Experiment [5]-1.710(3)  10 -14 137 Ba + 6s (F=2 – F=1) CP [3] -0.245(2)  10 -14 171 Yb + 6s (F=1 – F=0) CP [3] -0.0983  10 -14 MBPT3 [6] -0.094(5)  10 -14 137 Hg + 6s (F=1 – F=0) CP [3] -0.0102(5)  10 -14 [1] W.R. Johnson, U.I. Safronova, A. Derevianko, and M.S. Safronova, PRA 77, 022510 (2008) [2] U.I. Safronova and M.S. Safronova, PRA 78, 052504 (2008) [3] E. J. Angstmann, V.A. Dzuba, and V.V. Flambaum, PRA 74, 023405 (2006) [4] K. Beloy, U.I. Safronova, and A. Derevianko, PRL 97, 040801 (2006) [5] E. Simon, P. Laurent, and A. Clairon, PRA 57, 426 (1998) [6] U.I. Safronova and M.S. Safronova, PRA 79, 022510 (2009) [7] M. S. Safronova et al., IEEE - TUFFC 57, 94 (2010).

9. BBR shift in R b  = -1.255(4)  10 -14 Uncertainty estimate How to determine theoretical uncertainty?

10. BBR shift in R b  = -1.255(4)  10 -14 Scalar Stark shift coefficient Uncertainty estimate How to determine theoretical uncertainty?

11. The third-order static scalar electric-dipole polarizability of the hyperfine level F can be written as: CoefficientEach term involves sums with two electric-dipole and one hyperfine matrix element. The summations in these terms range over core, valence bound and continuum states. Third-order polarizability calcualtion Electric-dipole matrix elementsHyperfine matrix elements

12. Sources of uncertainties Strategy: dominant terms (m, n=5-12) are calculated with ``best set’’ matrix elements and experimental energies. The remaining terms are calculated in Dirac-Hartree-Fock approximation. Uncertainty calculation: (1) Uncertainty of each of the 157 matrix elements contributing to dominant terms is estimated. (2) Uncertainties in all remainders are evaluated.

13. 157 “Best-set” matrix elements Relativistic all-order matrix elements or experimental data TransitionValueTransitionValueTransitionValue 5s – 5p 1/2 4.231(3)5s – 6p 1/2 0.325(9)5s – 7p 1/2 0.115(3) 6s – 5p 1/2 4.146(27)6s – 6p 1/2 9.75(6)6s – 7p 1/2 0.993(7) 7s – 5p 1/2 0.953(2)7s – 6p 1/2 9.21(2)7s – 7p 1/2 16.93(9) 8s – 5p 1/2 0.502(2)8s – 6p 1/2 1.862(8)8s – 7p 1/2 16.00(2) 9s – 5p 1/2 0.331(1)9s – 6p 1/2 0.936(5)9s – 7p 1/2 3.00(2)

14. Uncertainty of the remainders: Term T fast convergence slow convergence 15% of the term T DHF approximation is determined to be accurate to 4% by comparing accurate results for main terms with DHF values. Therefore, we adjust the DHF tail by 4%. Entire adjustment (4%) is taken to be uncertainty in the tail.

15. Blackbody radiation shifts in optical frequency standards: (1) monovalent systems (2) divalent systems (3) other, more complicated systems Mg, Ca, Zn, Cd, Sr, Al +, In +, Yb, Hg ( ns 2 1 S 0 – nsnp 3 P) Hg + (5d 10 6s – 5d 9 6s 2 ) Yb + (4f 14 6s – 4f 13 6s 2 )

16. GOAL of the present project: calculate properties of group II atoms with precision comparable to alkali-metal atoms

17. Configuration interaction + all-order method CI works for systems with many valence electrons but can not accurately account for core-valence and core-core correlations. All-order (coupled-cluster) method can not accurately describe valence-valence correlation for large systems but accounts well for core-core and core-valence correlations. Therefore, two methods are combined to acquire benefits from both approaches.

18. CI + ALL-ORDER RESULTS Atom CI CI + MBPT CI + All-order Mg 1.9% 0.11%0.03% Ca 4.1% 0.7%0.3% Zn 8.0% 0.7%0.4 % Sr 5.2% 1.0%0.4% Cd 9.6% 1.4%0.2% Ba 6.4% 1.9%0.6% Hg 11.8% 2.5%0.5% Ra 7.3% 2.3%0.67% Two-electron binding energies, differences with experiment Development of a configuration-interaction plus all-order method for atomic calculations, M.S. Safronova, M. G. Kozlov, W.R. Johnson, Dansha Jiang, Phys. Rev. A 80, 012516 (2009).

19. C d, Z n, and S r Polarizabilities, preliminary results (a.u.) ZnCICI+MBPTCI + All-order 4s 2 1 S 0 46.239.4539.28 4s4p 3 P 0 77.969.1867.97 CdCICI+MBPTCI+All-order 5s 2 1 S 0 59.245.8246.55 5s5p 3 P 0 91.276.7576.54 *From expt. matrix elements, S. G. Porsev and A. Derevianko, PRA 74, 020502R (2006). SrCI +MBPTCI+all-orderRecomm.* 5s 2 1 S 0 195.6198.0197.2(2) 5s5p 3 P 0 483.6459.4458.3(3.6)

20. Magic wavelength magic is the wavelength for which the optical potential U experienced by an atom is independent on its state Magic wavelength magic is the wavelength for which the optical potential U experienced by an atom is independent on its state Atom in state A sees potential U A Atom in state B sees potential U B magic wavelength

21. C d, Z n, S r, and Hg magic wavelengths, preliminary results (nm) [1] A. D. Ludlow et al., Science 319, 1805 (2008) [2] H. Hachisu et al., Phys. Rev. Lett. 100, 053001 (2008)

22. Summary of the fractional uncertainties   due to BBR shift and the fractional error in the absolute transition frequency induced by the BBR shift uncertainty at T = 300 K in various frequency standards. M. S. Safronova et al., IEEE - TUFFC 57, 94 (2010). 5  10 -17 Present

23. Conclusion I.New BBR shift result for Rb frequency standard is presented. The new value is accurate to 0.3%. II. Development of new method for calculating atomic properties of divalent and more complicated systems is reported (work in progress). Improvement over best present approaches is demonstrated. Preliminary results for Mg, Zn, Cd, and Sr polarizabilities are presented. Preliminary results for magic wavelengths in Cd, Zn, and Hg are presented.