Atomic Clocks and the Search for Variation of Fundamental Constants

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1 January 22, 2015 Atomic Clocks and the Search for Variation of Fundamental Constants MARIANNA SAFRONOVA University of Maryland

2 Outline Blackbody radiation shifts in atomic clocks: Al +, Yb, Sr Theoretical Method Variation of fundamental constants: an introduction How to search for the variation of the fine-structure constant? Highly-charged ions for clocks and search for α-variation

3 Optical vs. microwave clocks PTB Yb + JILA Sr physics.aps.org

4 Theoretical calculations and atomic clocks Blackbody radiation ( BBR ) shift: Effect due to thermal radiation Motivation: BBR shift gave the largest uncertainty for most accurate clocks Very difficult to measure

5 BLACKBODY RADIATION SHIFTS CLOCK TRANSITION LEVEL B LEVEL A T = 0 K T = 300 K BBR Transition frequency should be corrected to account for the effect of the black body radiation at T=300K.

6 BBR shift and polarizability BBR shift of atomic level can be expressed in terms of a scalar static polarizability to a good approximation [1]: T( K) BBR 0(0)(831.9 V / m) (1+ ) ν = α η Dynamic correction Dynamic correction is generally small. Multipolar corrections (M1 and E2) are suppressed by α 2 [1]. Vector & tensor polarizability average out due to the isotropic nature of field. [1] Sergey Porsev and Andrei Derevianko, Physical Review A 74, R (2006)

7 High-precision atomic calculations Exact wave function Cs: 55 electrons Why is it so difficult? Ψ v = Ω Ψ (0) v Dirac-Hartree-Fock wave function (lowest order) Many-body operator, describes excitations from lowest-order 55-fold excitations to get exact wave function Even for 100 function basis set Approximate methods: perturbation theory does not converge well, need to use all-order methods (for example coupled-cluster method)

8 Electron-electron correlation now separates into two problems Example: Cd-like Nd 12+ Two valence electrons outside of a closed core 1s d 10 5s 2 ground state core 1s 2...4d 10 5s 2 valence electrons Problem 1: core-core and core-valence correlations Problem 2: valence-valence correlations

9 Main idea: solve two problems by different methods Configuration interaction works well for systems with a few valence electrons but can not accurately account for core-valence and core-core correlations. Coupled-cluster method accounts well for core-core and core-valence correlations (as demonstrated by work on alkali-metal atoms). Therefore, two methods are combined to acquire benefits from both approaches.

10 Electron-electron correlation now separates into two problems Cd-like Nd 12+ core 1s 2...4d 10 5s 2 valence electrons Use all-order (coupled-cluster) method to treat core and core-valence correlations core valence Use configuration interaction (CI) method to treat valence correlations

11 Linearized coupled-cluster method Main idea: allow single and double excitations of the initial wave functions to any orbital from finite basis set core excitation S1 S2 valence excitation core excitations core - valence excitations Excitations are described by cluster excitation coefficients ρ ij, ρ ijkl. 5s 6s, 7s,..35s, 5p, 6p, 35p, 5d, 6d, valence excitation

12 Coupled-cluster method Main feature: includes correlations to all orders of perturbation theory Implementation has to be very efficient Both formula derivations and required coding are very extensive

13 Too many terms beyond single and double excitations 1 S Contract operators by Wick s theorem H (0) v i j l k d b S2 Ψ > : a a a a : aman ar as a aca aaav 0 : c > 800 TERMS!

14 Coupled-cluster method Main feature: includes correlations to all orders of perturbation theory Implementation has to be very efficient Both formula derivations and required coding are very extensive Codes that write formulas Codes that write codes Codes that analyse results and estimate uncertainties

15 Monovalent systems: very brief summary of what we calculated with all-order method 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 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 Parity-nonconserving amplitudes (derived weak charge and anapole moment) EDM enhancement factors Isotope shifts (field shift and one-body part of specific mass shift) Atomic quadrupole moments Nuclear magnetic moment (Fr), from hyperfine data

16 Configuration interaction method Ψ = i c i Φ i Single-electron valence basis states ( H E) Ψ = 0 Example: two particle system: 1 2 H = h ( r ) + h ( r ) + h ( r, r ) one body two body part part r 1 r

17 Configuration interaction + coupled-cluster method (CI+all-order) H is modified using coupled-cluster method to calculate correction Σ H H eff = H + Σ eff = H + Σ Run CI with effective Hamiltonian ( H E) Ψ = 0 Advantages: most complete treatment of the correlations and applicable for many-valence electron systems Note: this effectively accounts for up to dominant quadrupole excitations eff

18 Computational and other challenges 1. Computer calculations should finish within reasonable time. 2. Evaluation and reduction of numerical uncertainties 3. Estimation of missing physics uncertainties

19 Calculations should finish in reasonable time Less then the lifetime of the Universe

20 Calculations should finish in reasonable time Before your current grant proposal runs out

21 Calculations should finish in reasonable time Before your patience runs out indieberries.blogspot.com

22 How to estimate missing physics uncertainties? 1. Calculate properties of similar reference systems where experimental data exist. 2. Use several different methods of increasing precision and compare results. 3. Calculate all major corrections separately, check for possible cancelations use to estimate uncertainty. 4. Test the methods of evaluating uncertainties on reference systems.

23 How to estimate what we do not know? 1. Calculate properties of similar reference systems where experimental data exist. 2. Use several different methods of increasing precision and compare results. 3. Calculate all major corrections separately, check for possible cancelations use to estimate uncertainty. 4. Test the methods of evaluating uncertainties on reference systems.

24 Al + energy levels, differences with experiment Level CI CI+MBPT CI+All 3s 2 1 S 0 1.2% 0.043% 0.006% 3p 2 1 D 2 2.3% 0.07% % 3s4s 3 S 1 1.4% 0.07% 0.015% 3p 2 3 P 0 1.6% 0.04% 0.008% 3p 2 3 P 1 1.6% 0.03% 0.004% 3p 2 3 P 2 1.6% 0.02% % 3s4s 3s3p 3s3p 3s3p 3s3p 1 S 0 1.4% 0.05% 0.003% 3 P 0 3.1% 0.15% 0.007% 3 P 1 3.1% 0.14% 0.008% 3 P 2 3.1% 0.12% % 1 P 1 0.4% -0.17% -0.14% Precision Calculation of Blackbody Radiation Shifts for Optical Frequency Metrology, M. S. Safronova, M. G. Kozlov, and Charles W. Clark, Phys. Rev. Lett. 107, (2011).

25 Al + polarizabilities (a.u.): 3 calculations CI CI+MBPT CI + All-order α (3s 2 1 S 0 ) α (3s3p 3 P 0 ) α( 3 P 0-1 S 0 ) Accuracy of α ( 3 P 0-1 S 0 )? Difference (CI+MBPT CI + all-order) = 0.4% Difference (CI CI + all-order) = 2.6 % Other uncertainties: 1.4% (Breit) and 2% (core) Estimate: 10% Precision Calculation of Blackbody Radiation Shifts for Optical Frequency Metrology, M. S. Safronova, M. G. Kozlov, and Charles W. Clark, Phys. Rev. Lett. 107, (2011).

26 Yb energy levels, differences with experiment Level CI CI+MBPT CI+All 6s 2 1 S 0-7.4% 1.3% 0.7% 5d6s 5d6s 6s7s 6s7s 6s6p 6s6p 6s6p 6s6p 3 D 1 4.1% 3.3% 2.5% 1 D 2-6.3% 4.2% 2.4% 3 S 1-9.4% 1.5% 1.2% 1 S 0-8.7% 1.4% 1.2% 3 P 0-19% 5.6% 2.7% 3 P 1-18% 5.3% 2.5% 3 P 2-18% 5.0% 2.7% 1 P 1-4.7% 5.6% 3.6% Ytterbium in quantum gases and atomic clocks: van der Waals interactions and blackbody shifts, M. S. Safronova, S. G. Porsev, and Charles W. Clark, Phys. Rev. Lett. 109, (2012).

27 Yb static polarizabilities (a.u.) Method α( 1 S 0 ) α( 3 P 0 ) α CI CI+MBPT Final CI+all-order (ab initio) 141(2) 293(10) 152 Porsev & Derevianko (2006) 111.3(5) 266(15) 155 Zhang & Dalgarno (2007) 143 Dzuba & Derevianko (2010) 141(6) 302(14) 161 Beloy (2012) from expt. data Expt. Sherman et al. (2012) (3) Polarizability is calculated directly by solving of the inhomogeneous differential equation in the valence sector we do not use sum over states.

28 Dynamic correction to the BBR shift in Yb 2 State α 0 αg (0) η 1 η ν BBR (dyn) 2 T = 300 K E g 6s 2 1 S Hz 6s6p 3 P (8) Hz 3 P 0-1 S 0 Expt. [1] (8) Hz (6) Hz Dynamic correction: 1.8% of the total BBR BBR uncertainty at T = 300K is reduced to [1] K. Beloy, J. A. Sherman, N. D. Lemke, N. Hinkley, C. W. Oates, and A. D. Ludlow (2012).

29 Sr DC polarizabilities: 3 calculations Dominant terms E1 matrix elements Energies α( 1 S 0 ) α( 3 P 0 ) α α #1 CI+all CI+all [1] 261(4) [1] S. Porsev and A. Derevianko, Phys. Rev. A 74, R (2006) [2] T. Middelmann, S. Falke, C. Lisdat, and U. Sterr, Phys. Rev. Lett. 109, (2012)

30 Sr DC polarizabilities: 3 calculations Dominant terms E1 matrix elements Energies α( 1 S 0 ) α( 3 P 0 ) α α #1 CI+all CI+all [1] 261(4) #2 CI+all Expt % [1] S. Porsev and A. Derevianko, Phys. Rev. A 74, R (2006) [2] T. Middelmann, S. Falke, C. Lisdat, and U. Sterr, Phys. Rev. Lett. 109, (2012)

31 Sr DC polarizabilities: 3 calculations Dominant terms E1 matrix elements Energies α( 1 S 0 ) α( 3 P 0 ) α α #1 CI+all CI+all [1] 261(4) #2 CI+all Expt #3 CI+all + 4 corrections Expt (7) Expt. [2] -2.3% -2.8% [1] S. Porsev and A. Derevianko, Phys. Rev. A 74, R (2006) [2] T. Middelmann, S. Falke, C. Lisdat, and U. Sterr, Phys. Rev. Lett. 109, (2012)

32 Dynamic correction to the BBR shift Differences between Sr and Yb cases: 1. 5s5p 3 P 0 5s4d 3 D 1 transition contributes 98.2 % to dynamic correction in Sr 2. Dynamic correction is 7% of the BBR shift in Sr! Dynamic correction ν BBR (dyn) Theory [1] PTB [2] (16) Hz (23) Hz Measurement of the 3 D 1 lifetime will yield dynamic correction to the BBR shift with the same accuracy! Our prediction of the 5s4d 3 D 1 lifetime: 2171(24) ns [1] Safronova et al., Phys. Rev. A 87, (2013). [2] Middelmann et al., Phys. Rev. Lett. 109, (2012)

33 Dynamic correction to the BBR shift Our prediction of the 5s4d 3 D 1 lifetime: 2171(24) ns Blackbody radiation shift in the Sr optical atomic clock, M. S. Safronova, S. G. Porsev, U. I. Safronova, M. G. Kozlov, and Charles W. Clark, Phys. Rev. A 87, (2013). Measurement of the 5s4d 3 D 1 lifetime: 2180(10) ns total uncertainty in an atomic clock, T.L. Nicholson, S.L. Campbell, R.B. Hutson, G.E. Marti, B.J. Bloom, R.L. McNally, W. Zhang, M.D. Barrett, M.S. Safronova, G.F. Strouse, W.L. Tew, and J. Ye, submitted to Nature Physics, arxiv (2015) Sr JILA clock: BBR static shift uncertainty BBR dynamic shift uncertainty

34 Summary of the fractional uncertainties ν/ν 0 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 Atom Clock transition ν/ν 0 Uncertainty Reference Rb 5s (F=2 - F=1) Safronova et al Cs 6s (F=4 - F=3) Simon et al Ca + 4s - 3d 5/ Safronova et al Sr + 5s - 4d 5/ Jiang et al Yb + 6s - 5d 2 D 3/ Tamm et al Yb + 6s - 4f 13 6s 2 2 F 7/ Hosaka et al 2009 B + 2s 2 1 S 0-2s2p 3 P Safronova et al Al + 3s 2 1 S 0-3s3p 3 P Safronova et al In + 5s 2 1 S 0-5s5p 3 P Safronova et al Tl + 6s 2 1 S 0-6s6p 3 P Zuhrianda et al Sr 5s 2 1 S 0-5s5p 3 P Nicholson et al. (2015) Yb 6s 2 1 S 0-6s6p 3 P Safronova et al Sherman et al Hg 6s 2 1 S 0-6s6p 3 P Hachisu et al. 2008

35 ARE FUNDAMENTAL CONSTANTS CONSTANT??? Being able to compare and reproduce experiments is at the foundation of the scientific approach, which makes sense only if the laws of nature do not depend on time and space. J.-P. Uzan, Rev. Mod. Phys. 75, 403 (2003)

36 The New International System of Units based on Fundamental Constants α = 4 1 πε 0 e 2 ħc

37 The modern theories directed toward unifying gravitation with the three other fundamental interactions suggest variation of the fundamental constants in an expanding universe.

com If α is too big small nuclei can not exist Electric

38 Life needs very specific fundamental constants! α ħ ~1/137 If α is too big small nuclei can not exist Electric repulsion of the protons > strong nuclear binding force α~1/137 α~1/10 will blow carbon apart

39 Life needs very specific fundamental constants! α ħ ~1/137 α~ / Nuclear reaction in stars are particularly sensitive to α. If α were different by 4%: no carbon produced by stars. No life.

40 Life needs very specific fundamental constants! α ħ ~1/137 α No carbon produced by stars: No life in the Universe

41 Search for the variation of the fine-structure constant α α = 4 1 πε 0 e 2 ħc

42 How to test if α changed with time? Atomic transition energies depend on α 2 Mg + ion Scientific American Time 21, (2012)

43 Astrophysical searches for variation of fine-structure constant α Observed from quasar absorption spectra Laboratory frequency 2 α Z EZ = E0 + q 1 α 0 Julian Berengut, UNSW, 2010

Astrophysical searches for variation of fine-structure constant α Observed from quasar absorption spectra Conflicting results α α = 0.64(36) 10 5 Murphy et al.

44 Astrophysical searches for variation of fine-structure constant α Observed from quasar absorption spectra Conflicting results α α = 0.64(36) 10 5 Murphy et al., 2007 Keck telescope, 143 systems, 23 lines, 0.2<z<4.2 α α = 0.06(0.06) 10 5 Srianand et al, 2004: VL telescope, 23 systems, 12 lines, Fe II, Mg I, Si II, Al II, 0.4<z<2.3 Julian Berengut, UNSW, 2010 Molaro et al., 2007 α α = 0.12(1.8) 10 Z=1.84 α α = 5.7(2.7)

45 Laboratory searches for variation of fundamental constants Ratio of two clock frequencies N. Huntemann, B. Lipphardt, Chr. Tamm, V. Gerginov, S. Weyers, E. Peik, Phys. Rev. Lett. 113, (2014)

46 ν Laboratory searches for α variation Different optical atomic clocks use transitions that have different contributions of the relativistic corrections to frequencies. ( x) ν = + q x ( ) 2 0 x = α α 1 0 Therefore, comparison of different clocks can be used to search for α variation. Optical only clock test only α-variation Example: Al + / Hg + atomic clocks α α = 1.6(2.3) 10 y 17 1 Rosenband et al., Science 319, 1808 (2008)

47 Need very precise frequency standards using systems with very large q ν ( x) ν = + qx 0

48 Highly-charged ions for Atomic Clocks (1) Metastable level (2) Near optical transition ν ( x) = ν + qx 0 (3) Requirement for the α-variation searches: two clock levels can not belong to the same fine-structure of hyperfine-structure multiplet.

49 HIGHLY-CHARGED IONS??? Wavelength nm Cl nm Si nm Al nm Mg 458 nm 3s 2 1 S 0 3s3p 3 P 0 transition in Mg-like ions

50 Sn-like ions (present work) [Kr] 4d 10 5s 2 core Sn 5p6s 3 P 0 Sn like Ba 6+ 5p 2 1 S 0 Sn-like Pr 9+ 5p4f J=3 5p 2 1 S 0 1 D nm 1 D 2 3 P 1,2 163 nm 495(13) nm 5p 2 3 P 0,1,2 5p 2 3 P 0 5p 2 3 P 0

51 Clock proposals with highly-charged ions 1. Electron-hole transitions: Ir 16+, Ir 17+, W ions, 2. Californium C f16+, Cf 17+ and similar ions 3. Nuclear-spin-zero f 12 shells (clock only, no α-variation enhancement) 4. Ag-like, Cd-like, In-like, Sn-like valence transitions (present work)

52 This work: Exhaustive search of transitions in highly-charged ions that are particularly well suited for the current experimental explorations. Our criteria: (1) Metastable states with transition frequencies to the ground state ranging between nm. (2) High sensitivity to α-variation. (3) Stable isotopes. (4) Relatively simple electronic structure: one to four valence electrons above the closed core.

53 Only ions in 4 isoelectronic sequnces satisfy the criteria: Ag-like, Cd-like In-like, Sn-like ions

54 Very difficult to accurately calculate energies! Sn-like Pr 9+ 5p4f J=3 Two-electron energies Transition energy cm ± 540 cm (13) nm 5p 2 3 P cm -1 Major corrections to the transition energy: Higher-orders ( III+) : 2994 cm -1 Higher partial waves (l>6): cm -1 Breit interaction: cm -1

55 Comparison of energy levels with experiment (cm -1 ) Ion Level Expt. Theory Diff. Diff. (%) Nd 13+ 5s f 5/ % 4f 7/ % 5p 1/ % 5p 3/ % Sm 15+ 4f 5/ f 7/ % 5s % Ce 9+ 5p 1/ p 3/ % 4f 5/ % M. S. Safronova et al., Phys. Rev. Lett. 113, (2014).

56 Nd 13+ : one valence electron (Ag-like) λ=165nm 5s 4f 7/ cm -1 τ =15 days 4f 5/2 τ =1 s E3 q = cm -1 λ=179nm Quantity q describes sensitivity to α-variation ν ( x) ν = + q x ( ) 2 0 x = α α 1 0

57 Pr 10+ : three valence electrons (In-like) 5s 2 5p 3/2 τ = s λ=256 nm λ=1420 nm 5s 2 4f 7/2 τ =2.4 s 3330 cm -1 τ =1 day λ=2700 nm 5s 2 4f 5/2 E2 q = cm -1 5s 2 5p 1/2

58 Pr 9+ : four valence electrons (Sn-like) 5s 2 5p 2 3 P 1 τ = s λ=351 nm λ=424 nm 5s 2 5p4f 3 F 2 τ =58 s M1 5s 2 5p4f 3 G 3 M3 τ = years! q = cm λ=475 nm -1 5s 2 5p 2 3 P 0

59 Summary Selected highly-charged ions have several metastable states representing a level structure and other properties that are not present in any neutral and low-ionization state ions and may be advantageous for the development of atomic clocks as well as provide new possibilities for quantum information storage and processing. Estimated fractional accuracy of the transition frequency in the clocks based on highly-charged ions can be smaller than Estimated sensitivity to the α-variation for transitions in highly-charged ions approaches per year [1, 2]. [1] A. Derevianko, V. A. Dzuba, and V. V. Flambaum, PRL 109, (2012). [2] V. A. Dzuba, A. Derevianko, and V. V. Flambaum, PRA 86, (2012).

60 Summary: present work Recommended ions: Nd 13+, Sm 15+, Ce 9+, Pr 10+, Nd 11+, Sm 13+ Nd 12+, Sm 14+, Pr 9+, Nd 10+ [1] Highly Charged Ions for Atomic Clocks, Quantum Information, and Search for α-variation, M. S. Safronova, V. A. Dzuba, V. V. Flambaum, U. I. Safronova, S. G. Porsev, and M. G. Kozlov, Phys. Rev. Lett. 113, (2014). [2] Ag-like and In-like ions: M. S. Safronova et al., Phys. Rev. A. 90, (2014) [3] Cd-like and Sn-like ions, M. S. Safronova et al., Phys. Rev. A. 90, (2014)

61 HIGHLY-CHARGE ION COLLABORATION Michael Kozlov, PNPI, Russia Sergey Porsev, University of Delaware and PNPI Ulyana Safronova, University of Nevada-Reno Vladimir Dzuba, UNSW, Australia Victor Flambaum, UNSW, Australia OTHER COLLABORATORS Research scientist: Sergey Porsev Graduate students: Z. Zhuriadna, D. Huang, A. Naing Charles Clark, NIST Andrei Derevianko, University of Nevada-Reno Ephraim Eliav, Tel Aviv University, Israel Walter Johnson, University of Notre Dame

INTERACTION PLUS ALL-ORDER METHOD FOR ATOMIC CALCULATIONS

DAMOP 2010 May 29, 2010 DEVELOPMENT OF A CONFIGURATION-INTERACTION INTERACTION PLUS ALL-ORDER METHOD FOR ATOMIC CALCULATIONS MARIANNA SAFRONOVA MIKHAIL KOZLOV PNPI, RUSSIA DANSHA JIANG UNIVERSITY OF DELAWARE