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