Atomic calculations: recent advances and modern applications

Atomic calculations: recent advances and modern applications

JQI seminarJQI seminarJuly 8, 2010

Marianna SafronovaMarianna Safronova

• Selected applications of atomic calculations

• Study of fundamental symmetries

• Atomic clocks

• Optical cooling and trapping and quantum information

• Examples: magic wavelengths

• Present status of theory

• How accurate are theory values?

• Present challenges

• Development of CI+all-order method for group II atoms

• Future prospects

OutlineOutline

Atomic calculations for study of fundamental symmetries

Atomic calculations for study of fundamental symmetries

Transformations and symmetries

Transformations and symmetries

Translation Momentum conservation

Translation in time Energy conservation

Rotation Conservation of angular momentum

[C] Charge conjugation C-invariance

[P] Spatial inversion Parity conservation (P-invariance)

[T] Time reversal T-invariance

[CP]

[CPT]

Transformations and symmetries

Transformations and symmetries

Translation Momentum conservation

Translation in time Energy conservation

Rotation Conservation of angular momentum

[C] Charge conjugation C-invariance

[P] Spatial inversion Parity conservation (P-invariance)

[T] Time reversal T-invariance

[CP]

[CPT]

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 Violationr ─ r

Parity violation in atomsParity violation in atoms

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

Standard ModelStandard Model

(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 in atomsParity violation in atoms

Nuclear spin-dependent

PNC:Study of PNC

In the nucleus

a

Nuclear anapole moment

Nuclear anapole moment

Present status:agreement with the

Standard Model

Nuclear spin-independent

PNC: Searches for new

physics beyond the

Standard Model

(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.

Fr, Yb, Ra+

Why do we need Atomic calculations to study parity violation?

Why do we need Atomic calculations to study parity violation?

(1)Present experiments can not be analyzed without theoretical value of PNC amplitude in terms of Qw.

(2)Theoretical calculation of spin-dependent PNC amplitude is needed to determine anapole moment from experiment.

(3) Other parity conserving quantities are needed.

Note: need to know theoretical uncertainties!

Transformations and symmetries

Transformations and symmetries

Translation Momentum conservation

Translation in time Energy conservation

Rotation Conservation of angular momentum

[C] Charge conjugation C-invariance

[P] Spatial inversion Parity conservation (P-invariance)

[T] Time reversal T-invariance

[CP]

[CPT]

Permanent electric-dipole moment ( EDM )

Permanent electric-dipole moment ( EDM )

Time-reversal invariance must be violated for an elementary particle or atom to possess a permanent EDM.

S

d

S

dt t

S S

d d

0

dS

d

Sd

EDM and New physicsEDM and New physics

Many theories beyond the Standard Model predict EDM within or just beyond the present experimental capabilities.

10-40

10-39

10-34

10-33

10-32

10-31

10-30

10-29

10-28

10-27

10-26

10-25

de (ecm)

Naïve SUSY

Multi-Higgs

Heavy sFermions

Standard Model

Lef t-Right Symmetric

Accidental Cancellations

ExtendedTechnicolor

Yale I(projected)

Yale I I(projected)

~

Berkeley(2002)

ExactUniversality

Split SUSY

Seesaw Neutrino Yukawa Couplings

Lepton Flavor-Changing

Approx. Universality

Approx. CP

SO(10) GUT

Alignment

10-40

10-39

10-34

10-33

10-32

10-31

10-30

10-29

10-28

10-27

10-26

10-25

de (ecm)

Naïve SUSY

Multi-Higgs

Heavy sFermions

Standard Model

Lef t-Right Symmetric

Accidental Cancellations

ExtendedTechnicolor

Yale I(projected)

Yale I I(projected)

~

Berkeley(2002)

ExactUniversality

Split SUSY

Seesaw Neutrino Yukawa Couplings

Lepton Flavor-Changing

Approx. Universality

Approx. CP

SO(10) GUT

Alignment

David DeMille, YalePANIC 2005

Atomic calculations and search for EDM Atomic calculations and search for EDM

EDM effects are enhanced in some heavy atoms and molecules.

Theory is needed to calculate enhancement factors and search for new systems for EDM detection.

Recent new limit on the EDM of 199Hg

Phys. Rev. Lett. 102, 101601 (2009)

| d(199Hg) | < 3.1 x 10-29 e cm

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, Special Issue of IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 57, 94 (2010).

(1)Prediction of atomic properties required for new clock proposals

New clock proposals require both estimation of the atomic properties for details of the proposals (transition rates, lifetimes, branching rations, magic wavelength, scattering rates, etc.) and evaluation of the systematic shifts (Zeeman shift, electric quadruple shift, blackbody radiation shift, ac Stark shifts due to various lasers fields, etc.).

(2)Determination of the quantities contributing to the uncertainty budget of the existing schemes.

In the case of the well-developed proposals, one of the main current uncertainty issues is the blackbody radiation shift.

Atomic calculations & more precise clocksAtomic calculations & more precise clocks

Blackbody radiation shiftBlackbody radiation shift

T = 300 K

Clocktransition

Level A

Level B

BBRT = 0 K

Transition frequency should be corrected to account for the effect of the black body radiation at T=300K.

BBR shift and polarizabilityBBR 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)

42

BBR 0

1 ( )(0)(831.9 / ) (1+ )

2 300

T KV m

Dynamic correction is generally small. Multipolar corrections (M1 and E2) are suppressed by 2 [1].

Vector & tensor polarizability average out due to the isotropic nature of field.

Dynamic correctionDynamic correction

microWave transitionsmicroWave transitions optical transitionsoptical transitions

4d5/2

Sr+

Lowest-order polarizability

2

0 1

3(2 1)vnv n v

n D v

j E E

5s1/2

Cs

6s F=3

6s F=4

In lowest (second) order the polarizabilities of ground hyperfine 6s1/2 F=4 and F=3 states are the same.

Therefore, the third-order F-dependent polarizability F (0) has to be calculated.

(1) (1) (1) 2, ,DDT DT D T D terms

2D term

Blackbody radiation shifts in optical frequency standards:

(1) monovalent systems(2) divalent systems(3) other, more complicated systems

Blackbody radiation shifts in optical frequency standards:

(1) monovalent systems(2) divalent systems(3) other, more complicated systems

+1/2 5/2

+1/2 5/2

+1/2 5/2

+1/2 5/2

C (4 3 )

S (5 4 )

B (6 5 )

R (7 6 )

a s d

r s d

a s d

a s d

Mg, Ca, Zn, Cd, Sr, Al+, In+, Yb, Hg ( ns2 1S0

– nsnp 3P)

Hg+ (5d 106s – 5d 96s2)Yb+ (4f 146s – 4f 136s2)

Example: BBR shift in sr+Example: BBR shift in sr+

Present Ref.[1] Ref. [2]

(5s1/2 → 4d5/2) 0.250(9) 0.33(12) 0.33(9)

Present

0(5s1/2) 91.3(9)

0(4d5/2) 62.0(5)

Sr+: Dansha Jiang, Bindiya Arora, M. S. Safronova, and Charles W. Clark, J. Phys. B 42 154020 (2010).

Ca+: Bindiya Arora, M.S. Safronova, and Charles W. Clark, Phys. Rev. A 76, 064501 (2007)

nf tail contribution issue has been resolved

Need precise lifetime measurements

1% Dynamic correction, E2 and M1 corrections negligible

[1] A. A. Madej et al., PRA 70, 012507 (2004); [2] H. S. Margolis et al., Science 306, 19 (2004).

Summary of the fractional uncertainties 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.

M. S. Safronova et al., IEEE - TUFFC 57, 94 (2010).

510-17Present

Atoms in optical latticesAtoms in optical lattices

• Cancellations of ac Start shifts: state-insensitive optical cooling and trapping• State-insensitive bichromatic optical trapping schemes• Simultaneous optical trapping of two different alkali-metal species• Determinations of wavelength where atoms will not be trapped• Calculations of relevant atomic properties: dipole matrix elements, atomic polarizabilities, magic wavelengths, scattering rates, lifetimes, etc.

Optimizing the fast Rydberg quantum gate, M.S. Safronova, C. J. Williams, and C. W. Clark, Phys. Rev. A 67, 040303 (2003) .

Frequency-dependent polarizabilities of alkali atoms from ultraviolet through infraredspectral regions, M.S. Safronova, Bindiya Arora, and Charles W. Clark, Phys. Rev. A 73, 022505 (2006)

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).

Theory and applications of atomic and ionic polarizabilities (review paper), J. Mitroy, M.S. Safronova, and Charles W. Clark, submitted to J. Phys. B (2010)

State-insensitive bichromatic optical trapping, Bindiya Arora, M.S. Safronova, and C. W. Clark, submitted to Phys. Rev. A (2010)

( )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

magic wavelengthmagic wavelength

magic

wavelength

α

S State

P State

Locating magic wavelengthLocating magic wavelength

c vc v

Core term

Valence term (dominant)

Compensation term

2

2 2

1

3(2 1)

n v

vnv n v

E E n D v

j E E

Example:Scalar dipole polarizability

Electric-dipole reduced matrix

element

Polarizability of an alkali atom in a state vPolarizability of an alkali atom in a state v

Sum over infinite sets of many-body perturbation theorymany-body perturbation theory (MBPT) terms.

Calculate the atomic wave functions and energies

Calculate various matrix elements

Calculate “derived” propertiesuseful for particular problems

Scheme:

Relativistic all-order methodRelativistic all-order method

Ψ Ψv vH = E

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 transition matrix elements in Results for alkali-metal atoms:

E1 transition matrix elements in

Example: “Best set” Rb matrix elements

Magic wavelengths for the 5p3/2 - 5s transition of Rb.

ac Stark shifts for the transition from 5p3/2F′=3 M′

sublevels to 5s FM sublevels in Rb.The electric field intensity is taken to be 1 MW/cm2.

(nm)

925 930 935 940 945 950 955

(a.

u.)

0

2000

4000

6000

8000

10000

6S1/ 26P3/ 2

932 nm938 nm

0- 2

0+ 2

magic

0 2v MJ = ±3/2

MJ = ±1/20 2v

Other*Other*

magic around 935nm

* Kimble et al. PRL 90(13), 133602(2003)

Magic wavelength for CsMagic wavelength for Cs

A combination of trapping and control lasers is used to minimize the variance of the potential experienced by the atom in ground and excited states.

Trap laser Control laser

atom

bichromatic optical trappingbichromatic optical trapping

Surface plot for the 5s and 5p3/2 |m| = 1/2 state polarizabilities as a function of laser wavelengths 1

and 2 for equal intensities of both lasers.

Magic wavelengths for the the 5s and 5p3/2 | m| = 1/2 states for 1 =800-810nm and 2=2 1 for various intensities of both lasers.

The intensity ratio (1/2)2 ranges from 1 to 2.

Other applicationsOther applications

• Variation of fundamental constants• Long-range interaction coefficients• Data for astrophysics• Actinide ion studies for chemistry models• Benchmark tests of theory and experiment• Cross-checks of various experiments• Determination of nuclear magnetic moment in Fr• Calculation of isotope shifts• …

MONovalent systems: Very brief summary of what we

calculated with all-order method

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, 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).

Development of high-precision methods

Present status of theory and need for further development

Development of high-precision methods

Present status of theory and need for further development

All-orderCorrelation potential

CI+MBPT

• Atomic clocks • Study of parity violation (Yb)• Search for EDM (Ra)• Degenerate quantum gases, alkali-group II mixtures• Quantum information• Variation of fundamental constants

Motivation: study of group II – type systems

Motivation: study of group II – type systems

Divalent ions: Al+, In+, etc.Divalent ions: Al+, In+, etc.

Mg

Ca SrBaRa

ZnCdHgYb

Mg

Ca SrBaRa

ZnCdHgYb

Summary of theory methods for atomic structure

Summary of theory methods for atomic structure

• Configuration interaction (CI)• Many-body perturbation theory• Relativistic all-order method (coupled-cluster)• Correlation - potential method

• Configuration interaction + second-order MBPT• Configuration interaction + all-order method*

*under development

GOAL of the present project:

calculate properties of group II

atoms with precision comparable

to alkali-metal atoms

GOAL of the present project:

calculate properties of group II

atoms with precision comparable

to alkali-metal atoms

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 +all-order method

Configuration interaction +all-order method

CI works for systems with many valence electrons but can not accurately account for core-valenceand core-core correlations.

All-order (coupled-cluster) method can not accurately describe valence-valence correlation 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 + all-order

Configuration interaction method + all-order

1 1 1

2 2 2

1 2,

h h

h h

Heff is modified using all-order calculation

are obtained using all-order method used for alkali-metal atoms with appropriatemodification

0effH E

In the all-order method, dominant correlation corrections are summed to all orders of perturbation theory.

CI + ALL-ORDER RESULTSCI + ALL-ORDER RESULTS

Atom CI CI + MBPT CI + All-order

Mg 1.9% 0.11% 0.03%Ca 4.1% 0.7% 0.3%Zn 8.0% 0.7% 0.4 %Sr 5.2% 1.0% 0.4%Cd 9.6% 1.4% 0.2%Ba 6.4% 1.9% 0.6%Hg 11.8% 2.5% 0.5%Ra 7.3% 2.3% 0.67%

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).

Cd, Zn, and Sr Polarizabilities, preliminary results (a.u.)

Zn CI CI+MBPT CI + All-order

4s2 1S0 46.2 39.45 39.28

4s4p 3P0 77.9 69.18 67.97

Cd CI CI+MBPT CI+All-order

5s2 1S0 59.2 45.82 46.55

5s5p 3P0 91.2 76.75 76.54

*From expt. matrix elements, S. G. Porsev and A. Derevianko, PRA 74, 020502R (2006).

Sr CI +MBPT CI+all-order Recomm.*

5s2 1S0 195.6 198.0 197.2(2)

5s5p 3P0 483.6 459.4 458.3(3.6)

Cd, Zn, Sr, and Hg magic wavelengths, preliminary results (nm)

Sr Present Expt. [1]813.45 813.42735(40)

PresentCd 423(4)Zn 414(5)

Present Theory [2]Hg 365(5) 360

[1] A. D. Ludlow et al., Science 319, 1805 (2008)[2] H. Hachisu et al., Phys. Rev. Lett. 100, 053001 (2008)

ConclusionConclusion

AtomicClocks

S1/2

P1/2D5/2

Qubit

Parity Violation

Quantum information

Future:Future:New SystemsNew SystemsNew Methods,New Methods,New ProblemsNew Problems