Atomic Clocks and the Search for Variation of Fundamental ... · Atomic Clocks and the Search for...
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Atomic Clocks and the Search for Variation of Fundamental Constants
MARIANNA SAFRONOVAMARIANNA SAFRONOVAMARIANNA SAFRONOVAMARIANNA SAFRONOVA
University of Maryland
January 22, 2015
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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
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Optical vs. microwave clocks
physics.aps.org
PTB Yb+
JILA Sr
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Motivation:
BBR shift gave the largest uncertainty for most accurate clocks
Very difficult to measure
Theoretical calculations and atomic clocksBlackbody radiation ( BBR ) shift:
Effect due to thermal radiation
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BLACKBODY RADIATION SHIFTSBLACKBODY RADIATION SHIFTSBLACKBODY RADIATION SHIFTSBLACKBODY RADIATION SHIFTS
T = 300 KT = 300 KT = 300 KT = 300 K
CLOCKCLOCKCLOCKCLOCK
TRANSITIONTRANSITIONTRANSITIONTRANSITION
LEVEL ALEVEL ALEVEL ALEVEL A
LEVEL BLEVEL BLEVEL BLEVEL B
∆BBRT = 0 KT = 0 KT = 0 KT = 0 K
Transition frequency should be corrected to account for the
effect of the black body radiation at T=300K.
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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)
4
2
BBR 0
1 ( )(0)(831.9 / ) (1+ )
2 300
T KV mν α η
∆ = −
Dynamic correction is generally small.
Multipolar corrections (M1 and E2) are suppressed by α2 [1].
Vector & tensor polarizability average out due
to the isotropic nature of field.
Dynamic correction
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(0)
v vΨ = Ω Ψ
Exact wave functionMany-body operator,
describes excitations from lowest-order
Dirac-Hartree-Fock
wave function (lowest order)
Cs: 55 electrons 55-fold excitations to get exact wave function
Even for 100 function basis set 10055
Approximate methods: perturbation theory does not
converge well, need to use all-order methods (for example
coupled-cluster method)
High-precision atomic calculationsWhy is it so difficult?
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Electron-electron correlation now separates into two problems
Problem 1: core-core and core-valence correlationsProblem 2: valence-valence correlations
1s2 ... 4d10 5s2 ground state
1s2...4d105s2
core
valenceelectrons
Example: Cd-like Nd12+
Two valence electrons outside of a closed core
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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.
Main idea: solve two problems by different methods
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Electron-electron correlation now separates into two problems
1s2...4d105s2
core
valenceelectrons
Cd-like Nd12+
core
valence
Use configurationinteraction (CI) method to
treat valence correlations
Use all-order
(coupled-cluster)method to treat core
and core-valence
correlations
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Linearized coupled-cluster method
Main idea: allow single and double excitationsof the initial wave functions to any orbital from finite basis set
1S2S
core excitation
valence
excitation
core excitations
core -
valence
excitations
Excitations are described by cluster excitation coefficients ρij, ρijkl.
valence excitation
5s 6s, 7s, ..35s, 5p, 6p, …35p, 5d, 6d, …
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• 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
Coupled-cluster method
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22
2
1S
2 (0)2 | | 0: :
1
2: cm n c adi j l k bv vr sa a aa a aa a a aaH aS a+ + + ++ ++ >→Ψ >
Contract operators by Wick’s theorem
800 TERMS!800 TERMS!800 TERMS!800 TERMS!
Too many terms beyond single and double excitations
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• 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 codes
Codes that write formulas
Codes that analyse results and estimate uncertainties
Coupled-cluster method
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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
• C3 and C6 coefficients
• 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
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
http://www.physics.udel.edu/~msafrono
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Configuration interaction method
i i
i
cΨ = Φ∑ Single-electron valencebasis states
( ) 0H E− Ψ =
1 1 1 2 2 1 2( ) ( ) ( , )
one bodypart
two bodypart
H h r h r h r r
− −
= + +
Example: two particle system: 1 2
1
−r r
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Configuration interaction +coupled-cluster method (CI+all-order)
H is modified using coupled-cluster method to calculate correction Σ
Advantages: most complete treatment of the correlations and applicable for many-valence electron systems
Note: this effectively accounts for up to dominant
quadrupole excitations
1 1 1
2 2 2
eff
eff
H H
H H
= + Σ
= + Σ
( ) 0effH E− Ψ =Run CI with effective Hamiltonian
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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
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Calculations should finish in “reasonable time”
Less then the lifetime of the Universe
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Before your current grant proposal
runs out
Calculations should finish in “reasonable time”
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Before your patience runs out
indieberries.blogspot.com
Calculations should finish in “reasonable time”
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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.
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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.
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Al+ energy levels, differences with experiment
Level CI CI+MBPT CI+All
3s2 1
S0 1.2% 0.043% 0.006%
3p2 1
D2 2.3% 0.07% -0.022%
3s4s3S1 1.4% 0.07% 0.015%
3p2 3
P0 1.6% 0.04% 0.008%
3p2 3
P1 1.6% 0.03% 0.004%
3p2 3
P2 1.6% 0.02% -0.004%
3s4s1S0 1.4% 0.05% 0.003%
3s3p3P0 3.1% 0.15% 0.007%
3s3p3P1 3.1% 0.14% 0.008%
3s3p3P2 3.1% 0.12% -0.017%
3s3p1P1 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, 143006 (2011) .
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Al+ polarizabilities (a.u.): 3 calculations
Accuracy of ∆α ∆α ∆α ∆α (3P0 - 1S0) ?
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%
CI CI+MBPT CI + All-order
α α α α (3s2 1S0) 24.405 24.030 24.408
α α α α (3s3p 3P0) 24.874 24.523 24.543
∆α∆α∆α∆α(3P0 - 1S0) 0.469 0.493 0.495
Precision Calculation of Blackbody Radiation Shifts for Optical Frequency Metrology , M. S. Safronova, M. G. Kozlov, and Charles W. Clark, Phys. Rev. Lett. 107, 143006 (2011) .
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Yb energy levels, differences with experiment
Level CI CI+MBPT CI+All
6s2 1S0 -7.4% 1.3% 0.7%
5d6s 3D1 4.1% 3.3% 2.5%
5d6s 1D2 -6.3% 4.2% 2.4%
6s7s 3S1 -9.4% 1.5% 1.2%
6s7s1S0 -8.7% 1.4% 1.2%
6s6p 3P0 -19% 5.6% 2.7%
6s6p 3P1 -18% 5.3% 2.5%
6s6p3P2 -18% 5.0% 2.7%
6s6p1P1 -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, 230802 (2012).
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Yb static polarizabilities (a.u.)
Method α(1S0) α(3P0) ∆α
CI 166 258 92
CI+MBPT 138 306 168
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 134-142 280-290
Expt. Sherman et al. (2012) 145.726(3)
Polarizability is calculated directly by solving of the inhomogeneous
differential equation in the valence sector – we do not use sum over states.
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Dynamic correction to the BBR shift in Yb
State α0 η1 η ∆νBBR(dyn)
T = 300 K
6s2 1S0 140.9 19261 0.00116 0.00116 −0.0014 Hz
6s6p 3P0 293.2 322862 0.00934 0.00963 -0.0243(8) Hz
3P0 -1S0 -0.0229(8) Hz
Expt. [1] -0.0226(6) Hz
[1] K. Beloy, J. A. Sherman, N. D. Lemke, N. Hinkley, C. W. Oates, and
A. D. Ludlow (2012).
Dynamic correction: 1.8% of the total BBRBBR uncertainty at T = 300K is reduced to 2×10-18
2
2
(0)g
gE
α∂
∂
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Sr DC polarizabilities: 3 calculations
E1 matrix elements
Energies αααα(1S0) αααα(3P0) ∆α∆α∆α∆α ∆α∆α∆α∆α
#1 CI+all CI+all 197.8 458.1 260.3 [1] 261(4)
Dominant terms
[1] S. Porsev and A. Derevianko, Phys. Rev. A 74, 020502R (2006)
[2] T. Middelmann, S. Falke, C. Lisdat, and U. Sterr, Phys. Rev. Lett. 109,
263004 (2012)
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Sr DC polarizabilities: 3 calculations
E1 matrix elements
Energies αααα(1S0) αααα(3P0) ∆α∆α∆α∆α ∆α∆α∆α∆α
#1 CI+all CI+all 197.8 458.1 260.3 [1] 261(4)
#2 CI+all Expt. 198.9 453.4 254.5
Dominant terms
-2.3%
[1] S. Porsev and A. Derevianko, Phys. Rev. A 74, 020502R (2006)
[2] T. Middelmann, S. Falke, C. Lisdat, and U. Sterr, Phys. Rev. Lett. 109,
263004 (2012)
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Sr DC polarizabilities: 3 calculations
E1 matrix elements
Energies αααα(1S0) αααα(3P0) ∆α∆α∆α∆α ∆α∆α∆α∆α
#1 CI+all CI+all 197.8 458.1 260.3 [1] 261(4)
#2 CI+all Expt. 198.9 453.4 254.5
#3CI+all + 4
correctionsExpt. 194.4 441.9 247.5
247.374(7)
Expt. [2]
Dominant terms
[1] S. Porsev and A. Derevianko, Phys. Rev. A 74, 020502R (2006)
[2] T. Middelmann, S. Falke, C. Lisdat, and U. Sterr, Phys. Rev. Lett. 109,
263004 (2012)
-2.3%
-2.8%
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Dynamic correction to the BBR shift
Theory [1] PTB [2]
-0.1492(16) Hz -0.1477(23) Hz
1. 5s5p 3P0 – 5s4d 3D1 transition contributes 98.2 % to
dynamic correction in Sr
2. Dynamic correction is 7% of the BBR shift in Sr!
Differences between Sr and Yb cases:
Dynamic correction ∆νBBR (dyn)
[1] Safronova et al., Phys. Rev. A 87, 012509 (2013).
[2] Middelmann et al., Phys. Rev. Lett. 109, 263004 (2012)
Our prediction of the 5s4d 3D1 lifetime: 2171(24) ns
Measurement of the 3D1 lifetime will yield dynamiccorrection to the BBR shift with the same accuracy!
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Dynamic correction to the BBR shift
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, 012509 (2013).
Our prediction of the 5s4d 3D1 lifetime: 2171(24) ns
Measurement of the 5s4d 3D1 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 1412.8621 (2015)
Sr JILA clock: BBR static shift uncertainty 3×10-19
BBR dynamic shift uncertainty 1.4×10-18
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Atom Clock transition ∆ν∆ν∆ν∆ν/νννν0 Uncertainty Reference
Rb 5s (F=2 - F=1) -1.25 ×××× 10-14 4 ×××× 10-17 Safronova et al. 2010
Cs 6s (F=4 - F=3) -1.7 ××××10-14 3 ×××× 10-17 Simon et al. 1998
Ca+ 4s - 3d5/2 9.2 ×××× 10-16 1 ×××× 10-17 Safronova et al. 2011
Sr+ 5s - 4d5/2 5.6 ×××× 10-16 2 ×××× 10-17 Jiang et al. 2009
Yb+ 6s - 5d 2D3/2 -5.3 ×××× 10-16 1 ×××× 10-16 Tamm et al. 2007
Yb+ 6s - 4f13 6s2 2F7/2 -5.7 ×××× 10-17 1 ×××× 10-17 Hosaka et al 2009
B+ 2s2 1S0 - 2s2p 3P0 1.42 ×××× 10-17 1 ×××× 10-18 Safronova et al. 2011
Al+ 3s2 1S0 - 3s3p 3P0 -3.8 ×××× 10-18 4 ×××× 10-19 Safronova et al. 2011
In+ 5s2 1S0 - 5s5p 3P0 -1.36 ×××× 10-17 1 ×××× 10-18 Safronova et al. 2011
Tl+ 6s2 1S0 - 6s6p 3P0 -1.06 ×××× 10-17 1 ×××× 10-18 Zuhrianda et al. 2012
Sr 5s2 1S0 - 5s5p 3P0 -5.5 ×××× 10-15 1.4 ×××× 10-18 Nicholson et al. (2015)
Yb 6s2 1S0 - 6s6p 3P0 -2.6 ×××× 10-15 2 ×××× 10-18
Sherman et al. 2012Safronova et al. 2012
Hg 6s2 1S0 - 6s6p 3P0 -1.6 ×××× 10-16 Hachisu et al. 2008
Summary of the fractional uncertainties ∆ν/ν0 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
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ARE ARE ARE ARE
FUNDAMENTAL FUNDAMENTAL FUNDAMENTAL FUNDAMENTAL
CONSTANTS CONSTANTS CONSTANTS CONSTANTS
CONSTANT???CONSTANT???CONSTANT???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)
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The New International System of Units based on Fundamental Constants
2
0
1
4
e
cα
πε=
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www.rikenresearch.riken.jp
The modern theories directed toward unifying gravitation with the
three other fundamental interactions suggest variation of the fundamental constants in an expanding universe.
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www.economist.com
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
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www.economist.com
Life needs very specific fundamental constants!
α
ħ~1/137
Nuclear reaction in stars are particularly sensitive to α.
If α were different by 4%: no carbon produced by stars. No life.
α~/
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www.economist.com
Life needs very specific fundamental constants!
α
ħ~1/137
No carbon produced by stars: No life in the Universe
α
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Search for the variation of the fine-structure
constant αααα
2
0
1
4
e
cα
πε=
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How to test if αααα changed with time?
Atomic transition energies depend on α2
Mg+ ion
Scientific American Time 21, 70 - 77 (2012)
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Julian Berengut, UNSW, 2010
2
0
0
1ZZ
E E qα
α
= + −
Laboratory frequencyObserved from quasarabsorption spectra
Astrophysical searches for variation of fine-structure constant αααα
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Conflicting results
Murphy et al., 2007
Keck telescope, 143 systems,
23 lines, 0.2<z<4.2
50.64(36) 10α α −∆ = − ×
50.06(0.06) 10α α −∆ = − ×
Srianand et al, 2004: VL
telescope, 23 systems, 12 lines,
Fe II, Mg I, Si II, Al II, 0.4<z<2.3
Molaro et al., 2007
Z=1.84
60.12(1.8) 10α α −∆ = − ×
65.7(2.7) 10α α −∆ = ×
Astrophysical searches for variation of fine-structure constant αααα
Julian Berengut, UNSW, 2010
Observed from quasarabsorption spectra
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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, 210802 (2014)
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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
17 11.6(2.3) 10 yα α − −= − ×
0( )x xν ν= + q ( )2
0 1x α α= −
Different optical atomic clocks use transitions that have
different contributions of the relativistic corrections to
frequencies.
Laboratory searches for αααα−−−−variation
Rosenband et al., Science 319, 1808 (2008)
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Need very precise
frequency standards
using systems with
very large q
0( )x xν ν= + q
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Highly-charged ions
for Atomic Clocks(1) Metastable level(2) Near optical transition
(3) Requirement for the αααα-variation searches:two clock levels can not belong to the same fine-structure of hyperfine-structure multiplet .
0( )x xν ν= + q
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HIGHLYHIGHLYHIGHLYHIGHLY----CHARGED IONS ???CHARGED IONS ???CHARGED IONS ???CHARGED IONS ???
100 200 300 400 500 600 Wavelength nm
3s2 1S0 – 3s3p 3P0 transition in Mg-like ions
Mg458 nm
Al+
267 nmSi2+
190 nm
Cl6+
102 nm
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Sn-like ions(present work)
[Kr] 4d105s2 core
Sn
5p6s 3P0
5p2 1S01D2
5p2 3P0,1,2
289 nm
Sn-like Pr9+
5p4f J=3
5p2 3P0
495(13) nm
Sn –like Ba6+
163 nm
5p2 1S0
1D23P1,2
5p2 3P0
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Clock proposals with highly-charged ions
1. Electron-hole transitions: Ir16+, Ir17+, W ions, …
2. Californium Cf16+, Cf17+ and similar ions
3. Nuclear-spin-zero f12 shells (clock only,
no α-variation enhancement)
4. Ag-like, Cd-like, In-like, Sn-like valence transitions
(present work)
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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 170-3000 nm.
(2) High sensitivity to α-variation.
(3) Stable isotopes.
(4) Relatively simple electronic structure:
one to four valence electrons above the closed core.
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Only ions in 4 isoelectronic sequnces satisfy the criteria:
Ag-like, Cd-likeIn-like, Sn-like ions
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Very difficult to accurately calculate energies!
Sn-like Pr9+
5p4f J=3
5p2 3P0
495(13) nm
- 2 497 720 cm -1
Two-electron energies
- 2 477 500 cm -1
20 220 ± 540 cm -1
Transition energy
Major corrections to the transition energy:
Higher-orders ( III+) : 2994 cm-1
Higher partial waves (l>6): -1078 cm-1
Breit interaction: -1750 cm-1
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M. S. Safronova et al., Phys. Rev. Lett. 113, 030801 (2014).
Ion Level Expt. Theory Diff. Diff. (%)
Nd13+ 5s 0 0 0
4f5/2 55870 55706 164 0.29%
4f7/2 60300 60134 166 0.28%
5p1/2 185028 185028 38 0.02%
5p3/2 234864 234887 -24 -0.01%
Sm15+ 4f5/2 0 0 0
4f7/2 6555 6444 111 1.69%
5s 60384 60517 -133 -0.22%
Ce9+ 5p1/2 0 0 0
5p3/2 33427 33450 -23 -0.07%
4f5/2 54947 54683 264 0.48%
Comparison of energy levels with experiment (cm-1)
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Nd13+: one valence electron (Ag-like)
5s
4f5/2
4f7/2
λ=179nm
λ=165nm
4400 cm-1
τ τ τ τ =15 days
τ =1 s
E3 q = 104 000 cm-1
0( )x xν ν= + q ( )2
0 1x α α= −
Quantity q describes sensitivity to α-variation
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Pr10+: three valence electrons (In-like)
5s25p1/2
5s24f5/2
5s24f7/2
λ=2700 nm
λ=1420 nm3330 cm-1
τ τ τ τ =1 day
τ τ τ τ =2.4 s
E2 q = 74 000 cm-1
5s25p3/2
λ=256 nm
τ = 0.002 s
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Pr9+: four valence electrons (Sn-like)
5s25p2 3P0
5s25p4f 3G3
λ=475 nm
λ=424 nm
τ τ τ τ = 20 000 000 years!
τ τ τ τ =58 s
M3
q = 43000 cm-1
λ=351 nm
τ = 0.003 s5s25p2 3P1
M1
5s25p4f 3F2
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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
10-19. Estimated sensitivity to the α-variation for transitions in
highly-charged ions approaches 10-20 per year [1, 2].
[1] A. Derevianko, V. A. Dzuba, and V. V. Flambaum, PRL 109, 180801 (2012).
[2] V. A. Dzuba, A. Derevianko, and V. V. Flambaum, PRA 86, 054502 (2012).
Summary
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[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, 030801 (2014).
[2] Ag-like and In-like ions: M. S. Safronova et al.,
Phys. Rev. A. 90, 042513 (2014)
[3] Cd-like and Sn-like ions, M. S. Safronova et al.,
Phys. Rev. A. 90, 052509 (2014)
Recommended ions: Nd13+, Sm15+, Ce9+, Pr10+,
Nd11+, Sm13+Nd12+, Sm14+, Pr9+, Nd10+
Summary: present work
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OTHER COLLABORATORSOTHER COLLABORATORSOTHER COLLABORATORSOTHER COLLABORATORS
Charles Clark, NISTAndrei Derevianko, University of Nevada-Reno Ephraim Eliav, Tel Aviv University, IsraelWalter Johnson, University of Notre Dame
Research scientist:Sergey Porsev
Graduate students:Z. Zhuriadna, D. Huang, A. Naing
HIGHLYHIGHLYHIGHLYHIGHLY----CHARGE ION COLLABORATIONCHARGE ION COLLABORATIONCHARGE ION COLLABORATIONCHARGE ION COLLABORATION
Michael Kozlov, PNPI, RussiaSergey Porsev, University of Delaware and PNPIUlyana Safronova, University of Nevada-RenoVladimir Dzuba, UNSW, AustraliaVictor Flambaum, UNSW, Australia