1. Polarizabilities, Atomic Clocks, and Magic Wavelengths DAMOP 2008 focus session: Atomic polarization and dispersion May 29, 2008 Marianna Safronova Bindiya arora Charles W. clark NIST, Gaithersburg

2. Motivation Method Applications Frequency-dependent polarizabilities of alkali atoms and magic frequencies Atomic clocks: blackbody radiation shifts Future studies Outline

3. State-insensitive cooling and trapping for quantum information processing Motivation: 1 Optically trapped atoms Atom in state A sees potential U A Atom in state B sees potential U B

4. Atomic clocks: Next Generation Motivation: 2 Microwave Transitions Optical Transitions http://tf.nist.gov/cesium/fountain.htmhttp://tf.nist.gov/cesium/fountain.htm, NIST Yb atomic clock

5. Parity violation studies with heavy atoms & search for Electron electric-dipole moment Motivation: 3 http://CPEPweb.orghttp://CPEPweb.org, http://public.web.cern.ch/, Cs experiment, University of Coloradohttp://public.web.cern.ch/

6. Motivation Development of the high-precision methodologies Benchmark tests of theory and experiment Cross-checks of various experiments Data for astrophysics Long-range interactions Determination of nuclear magnetic and anapole moments Variation of fundamental constants with time

7. Atomic polarizabilities

8. Core term Valence term (dominant) Compensation term Example: Scalar dipole polarizability Electric-dipole reduced matrix element Polarizability of an alkali atom in a state v

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

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

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

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

13. The derivation gets really complicated if you add triples! Solution: develop analytical codes that do all the work for you! Input: ASCII input of terms of the type Output: final simplified formula in LATEX to be used in the all-order equation Actual implementation: codes that write formulas

14. Problem with all-order extensions: TOO MANY TERMS The complexity of the equations increases. Same issue with third-order MBPT for two-particle systems (hundreds of terms). What to do with large number of terms? Solution: automated code generation !

15. 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 Automated code generation

16. Experiment Na,K,Rb:U. Volz and H. Schmoranzer, Phys. Scr. T65, 48 (1996), Cs:R.J. Rafac et al., Phys. Rev. A 60, 3648 (1999), Fr:J.E. Simsarian et al., Phys. Rev. A 57, 2448 (1998) Theory M.S. Safronova, W.R. Johnson, and A. Derevianko, Phys. Rev. A 60, 4476 (1999) Results for alkali-metal atoms

17. 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

18. Polarizabilities: Applications Optimizing the Rydberg gate Identification of wavelengths at which two different alkali atoms have the same oscillation frequency for simultaneous optical trapping of two different alkali species. Detection of inconsistencies in Cs lifetime and Stark shift experiments Benchmark determination of some K and Rb properties Calculation of “magic frequencies” for state-insensitive cooling and trapping Atomic clocks: problem of the BBR shift …

19. Polarizabilities: Applications Optimizing the Rydberg gate Identification of wavelengths at which two different alkali atoms have the same oscillation frequency for simultaneous optical trapping of two different alkali species. Detection of inconsistencies in Cs lifetime and Stark shift experiments Benchmark determination of some K and Rb properties Calculation of “magic frequencies” for state-insensitive cooling and trapping Atomic clocks: problem of the BBR shift …

20. Applications Frequency-dependent polarizabilities of alkali atoms from ultraviolet through infrared spectral regions Goal: First-principles calculations of the frequency-dependent polarizabilities of ground and excited states of alkali-metal atoms Determination of magic wavelengths

21. Excited states: determination of magic frequencies in alkali-metal atoms for state-insensitive cooling and trapping, i.e. When does the ground state and excited np state has the same ac Stark shift? Magic wavelengths Bindiya Arora, M.S. Safronova, and Charles W. Clark, Phys. Rev. A 76, 052509 (2007) Na, K, Rb, and Cs

22. 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 What is magic wavelength?

23. wavelength α  S State P State Locating magic wavelength

24. What do we need?

25. Lots and lots of matrix elements!

26. What do we need? Lots and lots of matrix elements! Cs 56 matrix elements in

27. What do we need? Lots and lots of matrix elements! All-order “database”: over 700 matrix elements for alkali-metal atoms and other monovalent systems

28. Theory (This work) Experiment*  =0 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)

29. Frequency-dependent polarizabilities of N a atom in the ground and 3p 3/2 states. The arrows show the magic wavelengths M J = ±3/2 M J = ±1/2

30. Magic wavelengths for the 3p 1/2 - 3s and 3p 3/2 - 3s transition of N a.

31. Magic wavelengths for the 5p 3/2 - 5s transition of R b.

32. ac Stark shifts for the transition from 5p 3/2 F ′ =3 M ′ sublevels to 5s FM sublevels in R b. The electric field intensity is taken to be 1 MW/ cm 2.

33.  0 -  2  0 +  2 magic M J = ±3/2 M J = ±1/2 Other* magic around 935nm * Kimble et al. PRL 90(13), 133602(2003) Magic wavelength for Cs

34. ac Stark shifts for the transition from 6p 3/2 F ′ =5 M ′ sublevels to 6s FM sublevels in C s. The electric field intensity is taken to be 1 MW/ cm 2.

35. Applications: atomic clocks

36. atomic clocks black-body radiation ( BBR ) shift Motivation: BBR shift gives the larges uncertainties for some of the optical atomic clock schemes, such as with Ca+

37. Blackbody radiation shift in optical frequency standard with 43 Ca + ion Bindiya Arora, M.S. Safronova, and Charles W. Clark, Phys. Rev. A 76, 064501 (2007)

38. 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)

39. 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

40. 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

41. 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 Why Ca + ion? Lifetime~1.2 s

42. 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:

43. BBR shift can be expressed in terms of a scalar static polarizability: BBR shift and polarizability Vector & tensor polarizability average out due to the isotropic nature of field. Dynamic correction Dynamic correction ~10 -3 Hz. At the present level of accuracy the dynamic correction can be neglected.

44. Effect on the frequency of clock transition is calculated as the difference between the BBR shifts of individual states. BBR shift for a transition 4s 1/2 3d 5/2 729 nm

45. Need ground and excited state scalar static polarizability NOTE: Tensor polarizability calculated in this work is also of experimental interest. Need BBR shifts

46. 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 + (  = 0) 24.424.4

47. 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 ( )

48. PresentRef. [1]Ref. [2]Ref. [3]  0 (4s 1/2 ) 76.1(1.1)767370.89(15)  0 (3d 5/2 ) 32.0(1.1)3123 Comparison of our results for scalar static polarizabilities for the 4s 1/2 and 3d 5/2 states of 43 Ca + ion with other available results [1] C. Champenois et. al. Phys. Lett. A 331, 298 (2004) [2] Masatoshi Kajit et. al. Phys. Rev. A 72, 043404, (2005) [3] C.E. Theodosiou et. al. Phys. Rev. A 52, 3677 (1995)

49. 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 PresentChampenois [1] Kajita [2]  (4s 1/2 → 3d 5/2 ) 0.38(1)0.39(27)0.4 [1] C. Champenois et. al. Phys. Lett. A 331, 298 (2004) [2] Masatoshi Kajit et. al. Phys. Rev. A 72, 043404, (2005) An order of magnitude improvement is achieved with comparison to previous calculations

50. 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 PresentChampenois [1] Kajita [2]  (4s 1/2 → 3d 5/2 ) 0.38(1)0.39(27)0.4 [1] C. Champenois et. al. Phys. Lett. A 331, 298 (2004) [2] Masatoshi Kajit et. al. Phys. Rev. A 72, 043404, (2005) Sufficient accuracy to establish The uncertainty limits for the Ca + scheme

51. relativisticAll-ordermethod Singly-ionized ions

52. Future studies: more complicated system development of the CI + all-order approach* M.S. Safronova, M. Kozlov, and W.R. Johnson, in preparation

53. 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 method can account for core-core and core-valence correlation can not accurately describe valence-valence correlation. Therefore, two methods are combined to acquire benefits from both approaches.

54. 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% Sr5.2% 0.9%0.3% Ba6.4% 1.7%0.5% Ionization potentials, differences with experiment

55. Conclusion Benchmark calculation of various polarizabilities and tests of theory and experiment Determination of magic wavelengths for state- insensitive optical cooling and trapping Accurate calculations of the BBR shifts Future studies: Development of generally applicable CI+ all-order method for more complicated systems

56. Conclusion Atomic Clocks S 1/2 P 1/2 D 5/2 „quantum bit“ Parity Violation Quantum information Future: New Systems New Methods, New Problems

57. P3.8 Jenny Tchoukova and M.S. Safronova Theoretical study of the K, Rb, and Fr lifetimes Q5.9 Dansha Jiang, Rupsi Pal, and M.S. Safronova Third-order relativistic many-body calculation of transition probabilities for the beryllium and magnesium isoelectronic sequences U4.8 Binidiya Arora, M.S. Safronova, and Charles W. Clark State-insensitive two-color optical trapping Graduate students: Bindiya Arora Rupsi pal Jenny Tchoukova Dansha Jiang