1. Blackbody radiation shifts and Theoretical contributions to atomic clock research EFTF - IEEE 2009 Besan ҫ on, France April 21, 2009 Marianna Safronova

2. Black-body radiation shifts Microwave vs. Optical transitions Methods of calculating BBR shifts in optical transitions Evaluation of uncertainties Monovalent systems: all-order method Other properties: quadrupole moments Other systems: CI + MBPT method Development of the CI + all-order method Future prospects Outline

3. Atomic clocks Microwave Transitions Optical Transitions

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. Frequency standard Transition frequency should be corrected to account for the effect of the black body radiation at T=300K. T = 0 K Clock transition Level A Level B

6. Frequency standard Transition frequency should be corrected to account for the effect of the black body radiation at T=300K. T = 300 K Clock transition Level A Level B  BBR

7. The temperature-dependent electric field created by the blackbody radiation is described by (in a.u.) : BBR shift of a level Dynamic polarizability Frequency shift caused by this electric field is:

8. 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

9. Effect on the frequency of clock transition is calculated as the difference between the BBR shifts of individual states. Example for optical transitions: Ca + BBR shift for a transition 4s 1/2 3d 5/2 729 nm

10. BBR shifts for microwave transitions Example: Cs primary frequency standard 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. Present estimated theory error in the BBR shift (0.35%) [1] implies 6x10 -17 fractional uncertainty in the clock frequency. The theoretical value of the BBR shift [1,2] is consistent with 0.2% measurement [3]. [1] K. Beloy, U. I. Safronova, and A. Derevianko, Phys. Rev.Lett. 97, 040801 (2006) [2] E. J. Angstmann, V. A. Dzuba, and V.V. Flambaum, Phys. Rev. Lett. 97, 040802 (2006). [3] E. Simon, P. Laurent, and A. Clairon, Phys. Rev. A 57, 436 (1998).

11. Blackbody radiation shifts in optical frequency standard with monovalent ions Main difficulty: accurately evaluating polarizability of the nd 5/2 state.

12. Examples: BBR shift in optical frequency standards with 43 C a + and 88 S r + ions Ca + : Bindiya Arora, M.S. Safronova, and Charles W. Clark, Phys. Rev. A 76, 064501 (2007) Sr + : Dansha Jiang, Bindiya Arora, M. S. Safronova, and Charles W. Clark, submitted to special issue of J. Phys. B (2009)

13. Motivation For Ca +, the contribution from Blackbody radiation gives the largest uncertainty to the frequency standard at T = 300K  BBR = 0.39(0.27) Hz [1] [1] C. Champenois et. al. Phys. Lett. A 331, 298 (2004)

14. 4s 1/2 4p 1/2 3d 3/2 397 nm 866 nm 729 nm 3d 5/2 854 nm 393 nm 4p 3/2 732 nm E2 Easily produced by non-bulky solid state or diode lasers The clock transition involved is 4s 1/2 F=4 M F =0 → 3d 5/2 F=6 M F =0 Level scheme Lifetime~1.2 s

15. BBR shifts & polarizabilities 4s 1/2 3d 5/2 729 nm We Need ground and excited state scalar static polarizabilities

16. Core term Valence term (dominant) Compensation term Scalar dipole polarizability Electric-dipole reduced matrix element Polarizability of a monovalent atom in a state v Sum over all possible excited states

17. Very precise calculation of atomic properties We also need to evaluate uncertainties of theoretical values! How to accurately calculate E1 matrix elements ?

18. Very precise calculation of atomic properties We also need to evaluate uncertainties of theoretical values! How to accurately calculate E1 matrix elements ? WANTED!

19. 43 Ca + ion 51 matrix elements are calculated accurately Relativistic all-order method

20. Theory: All-order method (relativistic linearized coupled- cluster approach)

21. Lowest order Core core valence electron any excited orbital Single-particle excitations Double-particle excitations All-order atomic wave function (SD)

22. Lowest order Core core valence electron any excited orbital Single-particle excitations Double-particle excitations All-order atomic wave function (SD)

23. We calculate the excitation coefficients  iteratively until valence energy converges. Each iteration picks up another order of many-body perturbation theory many-body perturbation theory (MBPT) terms. Therefore, this method includes dominant correlation contributions to all orders of MBPT. Relativistic all-order method Calculate various matrix elements that are expressed as functions of excitation coefficients . Review of all-order method and its applications: M. S. Safronova and W. R. Johnson, Advances in At. Mol. and Opt. Physics 55,191 (2007)

24. Features: simple input, essentially just type in a formula! Input: list of formulas to be programmed Output: final code (need to be put into a main shell) Codes that write codes Codes that write formulas Various extentions of the all-order method: Automated formula and code generation

25. Theory [ 1 ] Experiment* Excellent agreement with experiments ! Na K K Rb (3P 1/2 ) (3P 3/2 ) (4P 1/2 ) (4P 3/2 ) (5P 1/2 ) (5P 3/2 ) 359.9(4) -88.4(10) 616(6) -109(2) 807(14) 869(14) 361.6(4) -166(3) 359.2(6) -88.3 (4) 606.7(6) 614 (10) -107 (2) 810.6(6) 857 (10) 360.4(7) -163(3) 606(6) *Zhu et al. PRA 70 03733(2004) [1] Bindiya Arora, M.S. Safronova, and Charles W. Clark, Phys. Rev. A 76, 052509 (2007) Static polarizabilities of np states

26. Very brief summary of what we calculated with this approach Properties Energies Transition matrix elements (E1, E2, E3, M1) Static and dynamic polarizabilities & applications Dipole (scalar and tensor) Quadrupole, Octupole Light shifts Black-body radiation shifts Magic wavelengths Hyperfine constants C 3 and C 6 coefficients Parity-nonconserving amplitudes (derived weak charge and anapole moment) Isotope shifts (field shift and one-body part of specific mass shift) Atomic quadrupole moments Nuclear magnetic moment (Fr), from hyperfine data Systems Li, Na, Mg II, Al III, Si IV, P V, S VI, K, Ca II, In, In-like ions, Ga, Ga-like ions, Rb, Cs, Ba II, Tl, Fr, Th IV, U V, other Fr-like ions, Ra II

27. 5p 3/2 4p 3/2 0.010.01 6p 3/2 0.010.01 0.010.01 0.010.01 0.060.06 3.33.3 Core  tail 48.448.4 Total: Total: 76.1 ± 1.1 4s Contributions to the 4s 1/2 scalar polarizability ( ) 6p 1/2 5p 1/2 4p 1/2  43 Ca + 24.424.4

28. 2.42.4 5p 3/2 4p 3/2 0.010.01 np 3/2 tail 0.010.010.80.8 0.30.3 1.71.7 3.33.3 Core nf 7/2 22.822.8 Total: Total: 32.0 ± 1.1 3d 5/2 6f 7/2 5f 7/2 4f 7/2 nf 5/2 0.20.20.50.5 7-12f 7/2  43 Ca + Contributions to the 3d 5/2 scalar polarizability ( )

29. Comparison of black body radiation shift (Hz) for the 4s 1/2 - 3d 5/2 transition of 43 Ca + ion at T=300K (E=831.9 V/m). Black body radiation shift All-order[1]Champenois[2]Kajita [3]  (4s 1/2 → 3d 5/2 ) 0.38(1)0.39(27)0.4 [1] Bindiya Arora, M.S. Safronova, and Charles W. Clark, Phys. Rev. A 76, 064501 (2007) [2] C. Champenois et. al. Phys. Lett. A 331, 298 (2004) [3] Masatoshi Kajita et. al. Phys. Rev. A 72, 043404 (2005) An order of magnitude improvement is achieved with comparison to previous calculations

30. BBR shift in s r + PresentRef.[1]Ref. [2]  (5s 1/2 → 4d 5/2 ) 0.250(9)0.33(12)0.33(9) Present  0 (5s 1/2 ) 91.3(9)  0 (4d 5/2 ) 62.0(5) Present: Dansha Jiang, Bindiya Arora, M. S. Safronova, and Charles W. Clark, submitted to special issue of J. Phys. B (2009). [1] A. A. Madej, J. E. Bernard, P. Dube, and L. Marmet, Phys. Rev. A 70, 012507 (2004). [2] H. S. Margolis, G. Barwood, G. Huang, H. A. Klein, S. N. Lea, K. Szymaniec, and P. Gill, Science 306, 19 (2004). nf tail contribution issue has been resolved Need precise lifetime measurements 1% Dynamic correction, E2 and M1 corrections negligible

31. Quadrupole moments Electric quadrupole moments of metastable states of Ca+, Sr+, and Ba+, Dansha Jiang and Bindiya Arora and M. S. Safronova, Phys. Rev. A 78, 022514 (2008)

32. Quadrupole moments

33. how to evaluate uncertainty of theoretical calculations?

34. Theory: evaluation of the uncertainty HOW TO ESTIMATE WHAT YOU DO NOT KNOW? I. Ab initio calculations in different approximations: (a) Evaluation of the size of the correlation corrections (b) Importance of the high-order contributions (c) Distribution of the correlation correction II. Semi-empirical scaling: estimate missing terms

35. Example: quadrupole moment of 3d 5/2 state in Ca +

36. Lowest order 2.451 3D 5/2 quadrupole moment in Ca +

37. Third order 1.610 Lowest order 2.451 3D 5/2 quadrupole moment in Ca +

38. All order (SD) 1.785 Third order 1.610 Lowest order 2.451 3D 5/2 quadrupole moment in Ca +

39. All order (SDpT) 1.837 All order (SD) 1.785 Third order 1.610 Lowest order 2.451 3D 5/2 quadrupole moment in Ca +

40. Coupled-cluster SD (CCSD) 1.822 All order (SDpT) 1.837 All order (SD) 1.785 Third order 1.610 Lowest order 2.451 3D 5/2 quadrupole moment in Ca +

41. Coupled-cluster SD (CCSD) 1.822 All order (SDpT) 1.837 All order (SD) 1.785 Third order 1.610 Lowest order 2.451 3D 5/2 quadrupole moment in Ca + Estimate omitted corrections

42. All order (SD), scaled 1.849 All-order (CCSD), scaled 1.851 All order (SDpT) 1.837 All order (SDpT), scaled 1.836 Third order 1.610 Final results: 3d 5/2 quadrupole moment Lowest order 2.454 1.849 (17)

43. All order (SD), scaled 1.849 All-order (CCSD), scaled 1.851 All order (SDpT) 1.837 All order (SDpT), scaled 1.836 Third order 1.610 Final results: 3d 5/2 quadrupole moment Lowest order 2.454 1.849 (17) Experiment 1.83(1) Experiment: C. F. Roos, M. Chwalla, K. Kim, M. Riebe, and R. Blatt, Nature 443, 316 (2006).

44. relativisticAll-ordermethod Singly-ionized ions

45. More complicated systems Mg, Ca, Zn, Cd, Sr, Al +, In + ( ns 2 1 S 0 – nsnp 3 P) Yb, Hg ( ns 2 1 S 0 – nsnp 3 P) Hg + (5d 10 6s – 5d 9 6s 2 ) Main difficulty: accurate treatment of the correlation corrections.

46. Configuration interaction method ( CI ) Single-electron valence basis states Example: two particle system:

47. Configuration interaction + many-body perturbation theory CI works for systems with many valence electrons but can not accurately account for core-valence and core-core correlations. MBPT 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.

48. Configuration interaction method + MBPT H eff is modified using perturbation theory expressions are obtained using perturbation theory V. A. Dzuba, V. V. Flambaum, and M. G. Kozlov, Phys. Rev. A 54, 3948 (1996) V. A. Dzuba and W. R. Johnson, Phys. Rev. A 57, 2459 (1998) V. A. Dzuba, V. V. Flambaum, and J. S. Ginges, Phys. Rev. A 61, 062509 (2000) S. G. Porsev, M. G. Kozlov, Yu. G. Rakhlina, and A. Derevianko, Phys. Rev. A 64, 012508 (2001) M. G. Kozlov, S. G. Porsev, and W. R. Johnson, Phys. Rev. A 64, 052107 (2001) I. M. Savukov and W. R. Johnson, Phys. Rev. A 65, 042503 (2002) Sergey G. Porsev, Andrei Derevianko, and E. N. Fortson, Phys. Rev. A 69, 021403 (2004) V. A. Dzuba and J. S. Ginges, Phys. Rev. A 73, 032503 (2006) V. A. Dzuba and V. V. Flambaum, Phys. Rev. A 75, 052504 (2007)

49. Sergey Porsev and Andrei Derevianko, Physical Review A 74, 020502R (2006)

50. Configuration interaction method + MBPT H eff is modified using perturbation theory expressions are obtained using perturbation theory Problem: (1) Accuracy deteriorates for heavier systems owing to larger correlation corrections. (2) Accuracy will not be ultimately sufficient.

51. Configuration interaction + all-order method H eff is modified using all-order excitation coefficients Advantages: most complete treatment of the correlations and applicable for many-valence electron systems

52. CI + ALL-ORDER: PRELIMINARY RESULTS for mg Experiment CI Dif(%)CI+MBPTDif(%) CI+all- orderDif(%) 3s 21S01S0 1829381795361.91827180.121828800.03 3s4s 3S13S1 41197403991.9411210.19411620.08 3s4s 1S01S0 43503426621.9434350.16434780.06 3s3d 1D21D2 46403451172.8463190.18463730.06 3s3d 3D13D1 47957469672.1478920.13479440.03 3s3d 3D23D2 47957469672.1478920.13479410.03 3s3d 3D33D3 47957469672.1478930.13479360.04 3s3p 3P03P0 21850 20905 4.3217820.31218370.06 3s3p 3P13P1 21870209264.3218040.30218560.06 3s3p 3P23P2 21911209664.3218470.29219010.04 3s3p 1P11P1 35051344881.6350530.0035068-0.05 3s4p 3P03P0 47841 46914 1.9477660.16478130.06 3s4p 3P13P1 47844469171.9477690.16478160.06 3s4p 3P23P2 47851469241.9477760.16478230.06 3s4p 1P11P1 49347484871.7492900.12493290.04

53. CI + ALL-ORDER: PRELIMINARY RESULTS CI CI + MBPT CI + All-order Mg 1.9% 0.12%0.03% Ca 4.1% 0.6%0.3% Cd9.6% 1.0%0.02% Sr5.2% 0.9%0.3% Zn8.0% 0.9%0.4 % Ba6.4% 1.7%0.5% Ionization potentials, differences with experiment

54. Expt.DIF(%) State J CICI+MBPTCI+All-order 5s 2 1S0208915100.02 5s5p3P°03011419-3.2-0.53 13065619-3.1-0.40 23182719-3.1-0.46 5s5p1P°14369211-0.09 5s6s3S15148414-1.6-0.49 5s6s1S05331013-1.4-0.35 5s5d1D25922014-1.5-0.24 5s5d3D15948614-1.4-0.22 25949814-1.4-0.22 35951614-1.4-0.22 C d energies, differences with experiment

55. C d, Z n, and S r Polarizabilities, preliminary results (a.u.) ZnCICI+MBPTCI + All-order 4s 2 1 S 0 44.1337.2237.02 4s4p 3 P 0 75.9466.2064.97 CdCICI+MBPTCI+All-order 5s 2 1 S 0 52.6641.5042.11 5s5p 3 P 0 86.9470.72 SrCI+ All-orderRecomm.* 5s 2 1 S 0 197.4197.2 *From expt. matrix elements, S. G. Porsev and A. Derevianko, PRA 74, 020502R (2006).

56. Conclusion I. BBR shift results for optical frequency standards with Ca+ and Sr+ ions are presented. Dynamic and multipolar corrections to BBR shift are evaluated for Sr+ ion. The issue of large uncertainty in tail contributions to nd polarizabilities is resolved. An order of magnitude improvement in accuracy is achieved. 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 are presented.

57. collaborations: Michael Kozlov (Petersburg Nuclear Physics Institute) Walter Johnson (University of Notre Dame) Charles Clark (NIST) Ulyana Safronova (University of Nevada-Reno) Graduate students: Bindiya Arora (graduated August 2008) Rupsi Pal (graduated January 2009) Jenny Tchoukova (graduated August 2008) Dansha Jiang