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Atomic Calculations for Future Technology and Study of Fundamental Problems
Atomic Calculations for Future Technology and Study of Fundamental Problems
ICCMSE 2009ICCMSE 2009RHODES, GREECERHODES, GREECE
October 30, 2009
Marianna SafronovaMarianna Safronova
• Selected applications of atomic calculations
• Study of fundamental symmetries: parity violation
• Quantum information
• Atomic clocks
• Methods for high-precision atomic calculations
• Overview
• Computational challenges
• Evaluation of the uncertainty
• Development of the LCCSD + CI method
• Future prospects
OutlineOutline
Parity violation studies with heavy atoms & search for Electron electric-dipole moment
study of Fundamental symmetries
study of Fundamental symmetries
r ─ r
Parity-transformed world:
Turn the mirror image upside down.
The parity-transformed world is not identical with the real world.
Parity is not conserved.
Parity Violation Parity Violation
(1) Search for new processes or particles directly
(2) Study (very precisely!) quantities which Standard Model
predicts and compare the result with its prediction
High energies
http://public.web.cern.ch/, Cs experiment, University of Colorado
Searches for New Physics Beyond the Standard Model
Searches for New Physics Beyond the Standard Model
Weak charge QWLow energies
C.S. Wood et al. Science 275, 1759 (1997)
0.3% accuracy0.3% accuracy
The most precise measurement of PNC amplitude (in cesium)
The most precise measurement of PNC amplitude (in cesium)
6s
7s F=4
F=3
F=4
F=3
12
Stark interference scheme to measure ratio of the PNC amplitude and the Stark-induced amplitude
mVcmPNC
mVcm
1.6349(80)Im
1.5576(77)
E
1
2
Parity violation Parity violation
Nuclear spin-independent
PNC: Searches for new
physics beyond the
Standard Model
Weak Charge QWWeak Charge QW
Nuclear spin-dependent
PNC:Study of PNC
In the nucleus
e
e
q
qZ0
a
Nuclear anapole moment
Nuclear anapole moment
Analysis of CS PNC experimentAnalysis of CS PNC experiment
Nuclear spin-independent
PNC
Weak Charge QWWeak Charge QW
Nuclear spin-dependent
PNC
Nuclear anapole momentNuclear anapole moment
6s
7s F=4
F=3
F=4
F=3
12
6s
7s
Average of 1 & 2 Difference of 1 & 2
PNC
sd
mVcm
34 43Im
0.077(11)
E
siPNC
mVcm
Im1.5935(56)
E
(a)PNC 2
( )vaFG
rH α I
Valence nucleon density
Parity-violating nuclear moment
Anapole moment
Spin-dependent parity violation: Nuclear anapole moment
Spin-dependent parity violation: Nuclear anapole moment
Nuclear anapole moment is parity-odd, time-reversal-even E1 moment of the electromagnetic current operator.
6s
7s F=4
F=3
F=4
F=3
12
a
Constraints on nuclear weak coupling contants
Constraints on nuclear weak coupling contants
W. C. Haxton and C. E. Wieman, Ann. Rev. Nucl. Part. Sci. 51, 261 (2001)
Nuclear anapole moment:test of hadronic weak interations
Nuclear anapole moment:test of hadronic weak interations
*M.S. Safronova, Rupsi Pal, Dansha Jiang, M.G. Kozlov, W.R. Johnson, and U.I. Safronova, Nuclear Physics A 827 (2009) 411c
NEED NEW EXPERIMENTS!!!
The constraints obtained from the Cs experiment were found to be inconsistentinconsistent with constraints from other nuclear PNC measurements, which
favor a smaller value of the133Cs anapole moment.
All-order (LCCSD) calculation of spin-dependent PNC amplitude:
k = 0.107(16)* [ 1% theory accuracy ]
No significant difference with previous value k = 0.112(16) is found.
Quantum communication, cryptography and quantum information processing
Quantum informationQuantum information
Need calculations of atomic properties
Optimizing the fast Rydberg quantum gate, M.S. Safronova, C. J. Williams, and C. W. Clark, Phys. Rev. A 67, 040303 (2003) .
Magic wavelengths for the ns-np transitions in alkali-metal atoms, Bindiya Arora, M.S. Safronova, and C. W. Clark, Phys. Rev. A 76, 052509 (2007).
S1/2
P1/2
D5/2
„quantumbit“
Quantum Computer(Innsbruck)
Quantum communicationQuantum communication
Need to interconnect flying and stationary qubits
~100 µm
Thin ion trap inside a cavity (Monroe/Chapman, Blatt)
Optical dipole traps(ac Stark shift)
Atom in state A sees potential UA
Atom in state B sees potential UB
problemproblem
( )U
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 UA
Atom in state B sees potential UB
What is magic wavelength?What is magic wavelength?
Atomic polarizability
magic
wavelength
α
S State
P State
Locating magic wavelengthLocating magic wavelength
Magic wavelengths for the ns-np transitions in alkali-metal atoms, Bindiya Arora, M.S. Safronova, and C. W. Clark, Phys. Rev. A 76, 052509 (2007).
Atomic clocksAtomic clocks
MicrowaveTransitions
OpticalTransitions
Blackbody Radiation Shifts and Theoretical Contributions to Atomic Clock Research, M. S. Safronova, Dansha Jiang, Bindiya Arora, Charles W. Clark, M. G. Kozlov, U. I. Safronova, and W. R. Johnson, to appear in Special Issue of IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control (2009).
motiation: next generation Atomic clocks
motiation: next generation Atomic clocks
Next - generation ultra precise atomic clock
Atoms trapped by laser light
http://CPEPweb.org
The ability to develop more precise optical frequency standards will open ways to improve global positioningsystem (GPS) measurements and tracking of deep-spaceprobes, perform more accurate measurements of the physical constants and tests of fundamental physics such as searches for gravitational waves, etc.
applicationsapplications
AtomicClocks
S1/2
P1/2D5/2
„quantumbit“
Parity Violation
Quantum information
NEEDNEEDATOMIC ATOMIC PROPERTIESPROPERTIES
Very precise calculation of atomic properties
WANTE
D!
We also need to evaluate uncertainties of theoretical values!
How to accurately calculate atomic properties?
How to accurately calculate atomic properties?
Electronelectric-dipole
momentenhancement
factors
Lifetimes
Energies
Wavelengthsac and dc
Polarizabilities
Fine-structure intervals
Hyperfineconstants
Line strengths Branching ratios
Isotope shifts
van der Waals coefficients
Oscillator strengths Atom-wall
interactionconstants
Transitionprobabilities
and others
...
Derived:Weak charge QW,Anapole moment
Paritynonconserving
amplitudes
Atomic PropertiesAtomic Properties
Magic wavelength BBR shifts
Quadrupole moments BBR
shifts
Theory:All-order method
(relativistic linearized coupled-cluster approach)
Theory:All-order method
(relativistic linearized coupled-cluster approach)
Perturbation theory:Correlation correction to ground state
energies of alkali-metal atoms
Perturbation theory:Correlation correction to ground state
energies of alkali-metal atoms
Linearized coupled-cluster method
Linearized coupled-cluster method
The linearized coupled-cluster method sums infinite sets of many-body perturbation theory terms. The wave function of the valence electron v is represented as an expansion that includes all possible single, double, and partial triple excitations.
1s22s22p63s23p63d104s24p64d105s25p66s
1s1s22…5p…5p66 6s6scorevalenceelectron
Cs: atom with single (valence) electron outside of a closed core.
Lowest order Core
core valence electron
any excited orbital
Single-particle excitations
Double-particle excitations
All-order atomic wave function (SD)All-order atomic wave function (SD)
Lowest order Core
core
valence electron any excited orbital
Single-particle excitations
Double-particle excitations
(0)v
(0)† †mn m n
ma av v
navaa a a
† (0)a a
am m
mva a
† (0)v v v
vm m
ma a
† † (0)12 m nmn
mab b v
na
abaa aa
All-order atomic wave function (SD)All-order atomic wave function (SD)
2 (0)2 | | 0: :
12
: cm n c adi j l k bv vr sa a aa a aa a a aaH aS a
800 terms!800 terms!
Triples excitations, non-linear terms, extra perturbation
theory terms, …
Triples excitations, non-linear terms, extra perturbation
theory terms, …
Need for symbolic computingNeed for symbolic computing
The code was developed to implement Wick’s theorem and simplify the resulting expressions.
Symbolic program for coupled-cluster method & perturbation theory
Symbolic program for coupled-cluster method & perturbation theory
:: : :ijkl mnwa i j l m wak ng a a aaa aaa Input: expression of the type
in ASCII format.
Output: simplified resulting formula in the LaTex format, ASCII output is also generated.
Program features Program features
1) The code is set to work with two or three normal products (all possible cases) with large number of operators.
2) The code differentiates between different types of indices, i.e.
core (a,b,c,…), valence (v,w,x,y,...), excited orbitals (m,n,r,s,…), and general case (i, j, k, l,…).
3) The operators are ordered as required in the same order for all terms.
4) The expression is simplified to account for the identical
terms and symmetry rules, .
5) The direct and exchange terms are joined together,
.
ijkl jilkgg
ijkl ijkl ijlkg gg
mnba rswc m n r s a c wbg a a a a a a a a
Symbolic computing for program generation
Symbolic computing for program generation
The resulting expressions that need to be evaluated numerically contain very large number of terms, resulting in tedious coding and debugging.
The symbolic program generator was developed for this purpose to automatically generate efficient numerical codes for coupled-cluster or perturbation theory terms.
1 2 3 '
2 ' ' 3 11 2 3
1 2 3
' '
( 1)
( ) ( ' ' ) ( )
c d a b c dJ k k k j j j jc d w w
abcd k k k w v w v b a
k k k
c d v w a b
J j j J j j J j j
k j j k j j k j j
X cdab X v w cd Z vwab
' 'v w
Features: simple input, essentially just type in a formula!
Input: list of formulas to be programmedOutput: final code (need to be put into a main shell)
Codes that write codesCodes that write codes
Codes that write formulasCodes that write formulas
Automated code generation Automated code generation
All-order method:Correlation correction to ground state
energies of alkali-metal atoms
All-order method:Correlation correction to ground state
energies of alkali-metal atoms
Na3p1/2-3s
K4p1/2-4s
Rb5p1/2-5s
Cs6p1/2-6s
Fr7p1/2-7s
All-order 3.531 4.098 4.221 4.478 4.256Experiment 3.5246(23) 4.102(5) 4.231(3) 4.489(6) 4.277(8)Difference 0.18% 0.1% 0.24% 0.24% 0.5%
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: E1 matrix elements (a.u.)
Results for alkali-metal atoms: E1 matrix elements (a.u.)
Theory [ 1 ]Theory [ 1 ] Experiment*Experiment*
0
Excellent agreement with experiments !Excellent agreement with experiments !
Na Na
K K
Rb Rb
(3P1/2)0(3P3/2)(3P3/2)2(4P1/2)0(4P3/2)0(4P3/2)2(5P1/2)0(5P3/2)0(5P3/2)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 Static polarizabilities of np states
Very brief summary of what wecalculated with this approach
Very brief summary of what wecalculated with this approach
Properties• Energies• Transition matrix elements (E1, E2, E3, M1) • Static and dynamic polarizabilities & applications
Dipole (scalar and tensor) Quadrupole, OctupoleLight shiftsBlack-body radiation shiftsMagic wavelengths
• Hyperfine constants• C3 and C6 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
http://www.physics.udel.edu/~msafrono
how to evaluateuncertainty of
theoretical calculations?
how to evaluateuncertainty of
theoretical calculations?
Theory: evaluation of the uncertainty
Theory: evaluation of the uncertainty
HOW TO ESTIMATE WHAT YOU DO NOT KNOW?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
Example: quadrupole moment of 3d5/2 state in Ca+
Example: quadrupole moment of 3d5/2 state in Ca+
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)
Lowest order2.451
3D5/2 quadrupole moment in Ca+3D5/2 quadrupole moment in Ca+
Third order1.610
Lowest order2.451
3D5/2 quadrupole moment in Ca+3D5/2 quadrupole moment in Ca+
All order (SD)1.785
Third order1.610
Lowest order2.451
3D5/2 quadrupole moment in Ca+3D5/2 quadrupole moment in Ca+
All order (SDpT) 1.837
All order (SD)1.785
Third order1.610
Lowest order2.451
3D5/2 quadrupole moment in Ca+3D5/2 quadrupole moment in Ca+
Coupled-cluster SD (CCSD)1.822
All order (SDpT) 1.837
All order (SD)1.785
Third order1.610
Lowest order2.451
3D5/2 quadrupole moment in Ca+3D5/2 quadrupole moment in Ca+
Coupled-cluster SD (CCSD)1.822
All order (SDpT) 1.837
All order (SD)1.785
Third order1.610
Lowest order2.451
3D5/2 quadrupole moment in Ca+3D5/2 quadrupole moment in Ca+
Estimateomittedcorrections
All order (SD), scaled 1.849All-order (CCSD), scaled 1.851 All order (SDpT) 1.837All order (SDpT), scaled 1.836
Third order1.610
Final results: 3d5/2 quadrupole momentFinal results: 3d5/2 quadrupole moment
Lowest order2.454
1.849 (13)1.849 (13)
All order (SD), scaled 1.849All-order (CCSD), scaled 1.851 All order (SDpT) 1.837All order (SDpT), scaled 1.836
Third order1.610
Final results: 3d5/2 quadrupole momentFinal results: 3d5/2 quadrupole moment
Lowest order2.454
1.849 (13)1.849 (13)
Experiment1.83(1)
Experiment1.83(1)
Experiment: C. F. Roos, M. Chwalla, K. Kim, M. Riebe, and R. Blatt, Nature 443, 316 (2006).
More complicated systemsMore complicated systems
relativisticrelativisticAll-orderAll-ordermethodmethod
Singly-ionized ions
Summary of theory methods for PNC studies
Summary of theory methods for PNC studies
• Configuration interaction (CI)• Many-body perturbation theory• Relativistic all-order method (coupled-cluster)• Perturbation theory in the screened Coulomb interaction (PTSCI), all-order approach
• Configuration interaction + second-order MBPT• Configuration interaction + all-order method*
*under development
Configuration interaction method
Configuration interaction method
i ii
c Single-electron valencebasis states
0effH E
1 1 1 2 2 1 2( ) ( ) ( , )eff
one bodypart
two bodypart
H h r h r h r r
Example: two particle system: 1 2
1
r r
Configuration interaction +many-body perturbation
theory
Configuration interaction +many-body perturbation
theory
CI works for systems with many valence electrons but can not accurately account for core-valenceand core-core correlations.
MBPT can not accurately describe valence-valencecorrelation for large systems but accounts well for core-core and core-valence correlations.
Therefore, two methods are combined to Therefore, two methods are combined to acquire benefits from both approaches. acquire benefits from both approaches.
Configuration interaction method + MBPT
Configuration interaction method + MBPT
1 1 1
2 2 2
1 2,
h h
h h
Heff is modified using perturbation theory expressions
are obtained using perturbation theory
Problem:
(1) Accuracy deteriorates for heavier systemsowing to larger correlation corrections.
(2) Accuracy will not be ultimately sufficient.
0effH E
Configuration interaction + all-order method
Configuration interaction + all-order method
Heff is modified using all-order excitation coefficients
Advantages: most complete treatment of the correlations and applicable for many-valenceelectron systems
L
mnklnmlkL
mnkl
mnmnmn
~~
~
2
1
CI + ALL-ORDER RESULTSCI + ALL-ORDER RESULTS
Atom CI CI + MBPT CI + All-order
Mg 1.9% 0.12% 0.03%Ca 4.1% 0.6% 0.3%Cd 9.6% 1.0% 0.02%Sr 5.2% 0.9% 0.3%Zn 8.0% 0.9% 0.4 %Ba 6.4% 1.7% 0.5%Hg 11.8% 2.4% 0.5%
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).
CI + ALL-ORDER: Mg energiesCI + ALL-ORDER: Mg energies
Experiment CI
Dif(%) CI+MBPT Dif(%)
CI+all- order Dif(%)
3s2 1S0 182938 179536 1.9 182718 0.12 182880 0.03
3s4s 3S1 41197 40399 1.9 41121 0.19 41162 0.08
3s4s 1S0 43503 42662 1.9 43435 0.16 43478 0.06
3s3d 1D2 46403 45117 2.8 46319 0.18 46373 0.06
3s3d 3D1 47957 46967 2.1 47892 0.13 47944 0.03
3s3d 3D2 47957 46967 2.1 47892 0.13 47941 0.03
3s3d 3D3 47957 46967 2.1 47893 0.13 47936 0.04
3s3p 3P0 21850 20905 4.3 21782 0.31 21837 0.06
3s3p 3P1 21870 20926 4.3 21804 0.30 21856 0.06
3s3p 3P2 21911 20966 4.3 21847 0.29 21901 0.04
3s3p 1P1 35051 34488 1.6 35053 0.00 35068 -0.05
3s4p 3P0 47841 46914 1.9 47766 0.16 47813 0.06
3s4p 3P1 47844 46917 1.9 47769 0.16 47816 0.06
3s4p 3P2 47851 46924 1.9 47776 0.16 47823 0.06
3s4p 1P1 49347 48487 1.7 49290 0.12 49329 0.04
Expt. DIF(%) DIF(%) DIF(%)
State J CI CI+MBPT CI+All-order
5s2 1S 0 208915 10 -1.0 0.02
5s5p 3P° 0 30114 19 -3.2 -0.53
1 30656 19 -3.1 -0.40
2 31827 19 -3.1 -0.46
5s5p 1P° 1 43692 11 -1.0 -0.09
5s6s 3S 1 51484 14 -1.6 -0.49
5s6s 1S 0 53310 13 -1.4 -0.35
5s5d 1D 2 59220 14 -1.5 -0.24
5s5d 3D 1 59486 14 -1.4 -0.22
2 59498 14 -1.4 -0.22
3 59516 14 -1.4 -0.22
Cd energies, differences with experiment
Cd, Zn, and Sr Polarizabilities, preliminary results (a.u.)
Zn CI CI+MBPT CI+All-order
4s2 1S0 44.13 37.22 37.02
4s4p 3P0 75.94 66.20 64.97
Cd CI CI+MBPT CI+All-order
5s2 1S0 52.66 41.50 42.11
5s5p 3P0 86.94 70.72 70.72
Sr CI+ All-order Recomm.*
5s2 1S0 197.4 197.2
*From expt. matrix elements, S. G. Porsev and A. Derevianko, PRA 74, 020502R (2006).
ConclusionConclusion
AtomicClocks
S1/2
P1/2D5/2
„quantumbit“
Parity Violation
Quantum information
Future:Future:New SystemsNew SystemsNew Methods,New Methods,New ProblemsNew Problems
Other collaborations: Michael Kozlov (PNPI, Russia)(Visiting research scholar at the University of Delaware)Walter Johnson (University of Notre Dame), Charles Clark (NIST)Ulyana Safronova (University of Nevada-Reno)
Graduatestudents:
Rupsi PalDansha JiangBindiya AroraJenny Tchoukova