Atomic Polarizabilitiespolarizabilities

ICCMSEICCMSE 2010,2010, KOS,KOS, GREECEGREECE

ATOMICATOMIC POLARIZABILITIESPOLARIZABILITIES

MariannaMarianna SafronovaSafronova

October 4, 2010 OUTLINEOUTLINE

• Atomic Dipole Polarizability • Applications

• Atomic clocks • Cooling and trapping of atoms • Other applications • Methods for calculation of atomic polarizabilities • Summary of high-precision results • How to determine theoretical uncertainties • Development of combined CI + RLCCSD(T) method ATOMICATOMIC DIPOLEDIPOLE POLARIZABILITYPOLARIZABILITY

The advent of cold-atom physics owes its existence to the ability to manipulate groups of atoms with electromagnetic fields. Many topics in the area of field-atom interactions have recently been the subject of considerable interest and heightened importance. U ∝ α λ( ) Electric-dipole polarizability governs the first-order response of an atom to an applied electric field. ATOMICATOMIC DIPOLEDIPOLE POLARIZABILITYPOLARIZABILITY

The interaction of field E directed along z-axis with the atom is described by the Hamiltonian E +++ Hext= −=−− e∑∑∑ zaa ijij ij First-order correction to the wave function satisfies

(1) (((HVE0 + −))) Ψ =− H ext Ψ

where (((H0 + V))) Ψ= E Ψ . 1 E(2)=Ψ H Ψ (1) =− e 22E ααα ext 2 SUMSUM---OVER----OVER-OVEROVER-OVEROVER----STATES---STATESSTATESSTATES METHODMETHOD

Example: scalar static electric-dipole polarizability

Absorption oscillator strength

fgn ααα0 === ∑∑∑ 2 n ((()(En−−− E g )))

Mixed approach:

(1) Get polarizability by direct solution method (2) Extract the most important terms using the sum over states (3) Replace these terms using the most accurate available data APPLICATIONSAPPLICATIONS OFOF ATOMICATOMIC POLARIZABILITIESPOLARIZABILITIES (1)(1) AtomicAtomic clocksclocks (2)(2) CoolingCooling && trappingtrapping ofof atomsatoms (3)(3) OtherOther applicationsapplications ATOMICATOMIC CLOCKSCLOCKS

Optical Microwave Transitions Transitions

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). MOTIATION:MOTIATION: NEXTNEXT GENERATIONGENERATION ATOMICATOMIC CLOCKSCLOCKS

Next - generation http://CPEPweb.org ultra precise atomic clock Atoms trapped by laser light

The ability to develop more precise optical frequency standards will open ways to improve global positioning system (GPS) measurements and tracking of deep-space probes, perform more accurate measurements of the physical constants and tests of fundamental physics such as searches for gravitational waves, etc. ATOMICATOMIC CLOCKSCLOCKS ANDAND VARIATIONVARIATION OFOF FUNDAMENTALFUNDAMENTAL CONSTANTSCONSTANTS

(1) Astrophysical constraints on variation of α: 4σ4σ4σ ! Study of quasar absorption spectra Changes in isotopic abundances mimic shift of α

(2) Laboratory atomic clock experiments : Compare rates of different clocks over long period of time to study time variation of fundamental constants Need: ultra precise clocks! ATOMICATOMIC CLOCKSCLOCKS ANDAND POLARIZABILITYPOLARIZABILITY

(1) Magic Wavelengths

(2) Blackbody Radiation Shift MAGICMAGIC WAVELENGTHWAVELENGTH

Atom in state A Atom in state B sees potential U sees potential U A B

MagicMagic wavelengthwavelength λλmagicmagic isis thethe wavelengthwavelength forfor whichwhich thethe opticaloptical potentialpotential UU experiencedexperienced byby anan atomatom isis independentindependent onon itsits statestate U ∝ α λ( ) LOCATINGLOCATING MAGICMAGIC WAVELENGTHWAVELENGTH

State B λλλ m agic (λ) (λ) (λ) (λ) α α α α State A

wavelength BLACKBODYBLACKBODY RADIATIONRADIATION SHIFTSHIFT

LEVEL B CLOCK TRANSITION 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. BBRBBR SHIFTSHIFT OFOF AA LEVELLEVEL

• The temperature-dependent electric field created by the blackbody radiation is described by (in a.u.) : 8 ωα ω 3d E 2 (ω ) = ωπ exp( /kT )− 1

• Frequency shift caused by this electric field is:

2 ∆vBBR =−× A∫α ω ω( ω ) Ed ( )

Dynamic polarizability BBRBBR SHIFTSHIFT ANDAND POLARIZABILITYPOLARIZABILITY

BBR shift of atomic level can be expressed in terms of a scalar static polarizability to a good approximation [1]:

1T ( K )  4 η α ∆ν = − (0)(831.9V / m )2   (1+ ) BBR2 0  300 

DynamicDynamic correctioncorrection Dynamic correction is generally small. Multipolar corrections (M1 and E2) are suppressed by a2 [1]. VECTOR & TENSOR POLARIZABILITY AVERAGE OUT DUE TO THE ISOTROPIC NATURE OF FIELD.

[1] Sergey Porsev and Andrei Derevianko, Physical Review A 74, 020502R (2006) BLACKBODYBLACKBODY RADIATIONRADIATION SHIFTSSHIFTS ININ OPTICALOPTICAL FREQUENCYFREQUENCY STANDARDS:STANDARDS:

(1)(1) MONOVALENTMONOVALENT SYSTEMS SYSTEMS (2)(2) DIVALENTDIVALENT SYSTEMSSYSTEMS (3)(3) OTHER,OTHER, MOREMORE COMPLICATEDCOMPLICATED SYSTEMSSYSTEMS

+ + + Ca (4 s1/2→ 3 d 5/2 ) Mg, Ca, Zn, Cd, Sr, Al , In , Yb , Hg + ( ns 2 1 S – nsnp 3P) Sr (5 s1/2→ 4 d 5/2 ) 0 Ba+ (6 s→ 5 d ) 1/2 5/2 Hg + (5d 10 6s – 5d 96s 2) Ra+ (7 s→ 6 d ) 1/2 5/2 Yb + (4f 14 6s – 4f 13 6s 2) Quantum Computer (Innsbruck)

P1/2 D5/2

Quantum bit

S1/2 COOLINGCOOLING ANDAND TRAPPINGTRAPPING OFOF NEUTRALNEUTRAL ATOMSATOMS

Atom in state A Atom in state B sees potential U sees potential U A B

StateState----insensitiveinsensitive cooling and trapping for quantum information processing MAGICMAGIC WAVELENGTHWAVELENGTH FORFOR CSCS

α α αv =0 + 2 MJ = ±3/2

α α αv =0 − 2 MJ = ±1/2 10000 6S 1/2 8000 6P 3/2

λλλmagic 6000 932 nm 938 nm OtherOther** (a.u.) a + a α α α α 4000 0 2 λλλmagic around 935nm a a 2000 0- 2

0 925 930 935 940 945 950 955 λ (nm) * Kimble et al. PRL 90(13), 133602(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). OTHEROTHER APPLICATIONSAPPLICATIONS

• Quantum computing with Rydberg atoms • Cold degenerate gases • Study of fundamental symmetries • Thermometry and other macroscopic standards • Benchmark tests of theory and experiment • Atomic transition rate determinations CURRENTCURRENT STATUSSTATUS OFOF THEORYTHEORY ANDAND EXPERIMENTEXPERIMENT

ATOMICATOMIC DIPOLEDIPOLE POLARIZABILITIESPOLARIZABILITIES

JPB TOPICAL REVIEW (2010): Theory and applications of atomic and ionic polarizabilities , J. Mitroy, M.S. Safronova, and Charles W. Clark, in press SUMMARYSUMMARY OFOF THEORYTHEORY METHODSMETHODS FORFOR ATOMICATOMIC POLARIZABILITIESPOLARIZABILITIES

• Configuration interaction (CI) • CI calculations with a semi-empirical core potential (CICP) • Density functional theory • Correlated basis functions (Hyl.,ECG) • Many-body perturbation theory (MBPT) • Coupled-cluster methods (CCSDT) • Correlation - potential method

• Configuration interaction + second-order MBPT (CI+MBPT) • Configuration interaction + coupled-cluster method*

*under development

ATOMICATOMIC POLARIZABILITIES:POLARIZABILITIES:

HOWHOW ACCURATEACCURATE AREARE THEORYTHEORY VALUESVALUES ?? POLARIZABILITYPOLARIZABILITY OFOF ANAN ALKALIALKALI ATOMATOM ININ AA STATESTATE vv Sum-over-statesSum-over-states approachapproach

α α α α=c + vc + v Valence term (dominant)

Core term Compensation term

Example: Electric-dipole reduced Scalar dipole polarizability matrix element

2 n D v 0 1 αααv === ∑∑∑ 3(2jv+ 1) n E n − E v

HOWHOW TOTO ESTIMATEESTIMATE UNCERTAINTYUNCERTAINTY OFOF AA MATRIXMATRIX ELEMENTELEMENT ?? THEORY:THEORY: EVALUATIONEVALUATION OFOF THETHE UNCERTAINTYUNCERTAINTY

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:EXAMPLE: QUADRUPOLEQUADRUPOLE MOMENT MOMENT OFOF 3D3D3D STATE IN C ++++++ 3D3D3D 5/25/25/25/25/25/2 STATE IN C aa

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) 3D3D3D QUADRUPOLE MOMENT IN CA ++++++ 3D3D3D 5/25/25/25/25/25/2 QUADRUPOLE MOMENT IN CA

Lowest order 2.451 3D3D3D QUADRUPOLE MOMENT IN CA ++++++ 3D3D3D 5/25/25/25/25/25/2 QUADRUPOLE MOMENT IN CA

Lowest order 2.451 Third order 1.610 3D3D3D QUADRUPOLE MOMENT IN CA ++++++ 3D3D3D 5/25/25/25/25/25/2 QUADRUPOLE MOMENT IN CA

Lowest order 2.451 Third order 1.610 All order (SD) 1.785 3D3D3D QUADRUPOLE MOMENT IN CA ++++++ 3D3D3D 5/25/25/25/25/25/2 QUADRUPOLE MOMENT IN CA

Lowest order 2.451 Third order 1.610 All order (SD) 1.785

All order (SDpT) 1.837 3D3D3D QUADRUPOLE MOMENT IN CA ++++++ 3D3D3D 5/25/25/25/25/25/2 QUADRUPOLE MOMENT IN CA

Lowest order 2.451 Third order 1.610 All order (SD) 1.785

All order (SDpT) 1.837 Coupled-cluster SD (CCSD) 1.822 3D3D3D QUADRUPOLE MOMENT IN CA ++++++ 3D3D3D 5/25/25/25/25/25/2 QUADRUPOLE MOMENT IN CA

Lowest order 2.451 Third order 1.610 All order (SD) 1.785

All order (SDpT) 1.837 Estimate Coupled-cluster SD (CCSD) omitted 1.822 corrections Final results: 3d quadrupole moment Final results: 3d 5/25/2 quadrupole moment

Lowest order 2.454 1.849 (13) Third order 1.849 (13) 1.610 All order (SD), scaled 1.849 All-order (CCSD), scaled 1.851 All order (SDpT) 1.837 All order (SDpT), scaled 1.836 Final results: 3d quadrupole moment Final results: 3d 5/25/2 quadrupole moment

Lowest order 2.454 1.849 (13) Third order 1.849 (13) 1.610 All order (SD), scaled 1.849 All-order (CCSD), scaled 1.851 ExperimentExperiment All order (SDpT) 1.837 All order (SDpT), scaled 1.836 1.83(1)1.83(1)

Experiment: C. F. Roos, M. Chwalla, K. Kim, M. Riebe, and R. Blatt, Nature 443, 316 (2006). DEVELOPMENTDEVELOPMENT OFOF HIGHHIGH-HIGHHIGH----PRECISION---PRECISIONPRECISIONPRECISION METHODSMETHODS

PRESENTPRESENT STATUSSTATUS OFOF THEORYTHEORY ANDAND NEEDNEED FORFOR FURTHERFURTHER DEVELOPMENTDEVELOPMENT Polarizabilities of excited states

Coupled-cluster Correlation potential

CI+MBPT MOTIVATION:MOTIVATION: MgMg STUDYSTUDY OFOF GROUPGROUP IIII –––––– TYPE TYPE SYSTEMSSYSTEMS CaCa SrSrSrSrSrSr BaBaBaBaBaBa • Atomic clocks RaRaRaRaRaRa • Study of parity violation (Yb) • Search for EDM (Ra) ZnZnZnZnZnZn • Degenerate quantum gases, CdCdCdCdCdCd alkali-group II mixtures HgHgHgHgHgHg • Quantum information YbYbYbYbYbYb • Variation of fundamental constants

Divalent ions: ++++++ ++++++ Divalent ions: AlAlAlAlAlAl ,, In, InIn, In ,, etc.etc. SUMMARYSUMMARY OFOF THEORYTHEORY METHODSMETHODS FORFOR ATOMICATOMIC POLARIZABILITIESPOLARIZABILITIES

• Configuration interaction + second-order MBPT (CI+MBPT) • Configuration interaction + all-order (RLCCSD(T) coupled- cluster) method* *under development CONFIGURATIONCONFIGURATION INTERACTIONINTERACTION + + ALLALL-ALLALL----ORDER---ORDERORDERORDER METHODMETHOD

CI works for systems with many valence electrons but can not accurately account for core-valence and 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,Therefore, twotwo methodsmethods areare combinedcombined toto acquireacquire benefitsbenefits fromfrom bothboth approaches.approaches. CONFIGURATIONCONFIGURATION INTERACTIONINTERACTION METHODMETHOD

Single-electron valence Ψ =∑ci Φ i basis states i (Heff − E ) Ψ = 0 1

Example: two particle system: r1− r 2 Heff = hr() + hr () + hrr (,) 11 12 212 one− body two− body part part CONFIGURATIONCONFIGURATION INTERACTIONINTERACTION METHODMETHOD ++ ALLALL-ALLALL----ORDER---ORDERORDERORDER

Heff is modified using all-order calculation h→ h + Σ 1 1 1 (Heff − E ) Ψ = 0 h2→ h 2 + Σ 2

Σ1, Σ 2 are obtained using all-order, RLCCSD(T), method used for alkali-metal atoms with appropriate modifications

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

core (0) valence electron Lowest order Core Ψv any excited orbital

Single-particle excitations † (0) † (0) ∑ ρma a m a a Ψv ∑ ρmv a m a vΨ v ma m≠v

Double-particle excitations 1 † † (0) † † (0) 2 ∑ ρmn ab ama n a b aa Ψv ∑ ρmnva a m a n a a a v Ψv mnab mna MONOVALENTMONOVALENT SYSTEMS: SYSTEMS: VERYVERY BRIEFBRIEF SUMMARYSUMMARY OFOF WHATWHAT WEWE CALCULATEDCALCULATED WITHWITH ALLALL-ALLALL----ORDER---ORDERORDERORDER METHODMETHOD Properties • Energies Systems • Transition matrix elements (E1, E2, E3, M1) Li, Na, Mg II, Al III, • Static and dynamic polarizabilities & applications Si IV, P V, S VI, K, Dipole (scalar and tensor) Ca II, In, In-like ions, Quadrupole, Octupole Ga, Ga-like ions, Rb, Light shifts Cs, Ba II, Tl, Fr, Th IV, Black-body radiation shifts U V, other Fr-like ions, Magic wavelengths Ra II • Hyperfine constants

•C3 and C 6 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

http://www.physics.udel.edu/~msafrono CONFIGURATIONCONFIGURATION INTERACTIONINTERACTION ++ ALLALL-ALLALL----ORDER---ORDERORDERORDER METHOD METHOD

Heff is modified using all-order excitation coefficients ~ (Σ1 )mn = (ε n −ε m )ρmn L ~ ~ L ()()Σ2 mnkl = ε k + ε l −ε m −ε n ρmnkl

Advantages: most complete treatment of the correlations and applicable for many-valence electron systems CICI ++ ALLALL-ALLALL----ORDER---ORDERORDERORDER RESULTSRESULTS

TwoTwo----electronelectron binding energies, differences with experimexperimentent

Atom CI CICICI + MBPT CI + AllAll----orderorder

Mg 1.9% 0.11% 0.03% Ca 4.1% 0.7% 0.3% ZnZnZn8.0% 0.7% 0.4 % SrSrSr5.2% 1.0% 0.4% CdCdCd9.6% 1.4% 0.2% BaBaBa6.4% 1.9% 0.6% HgHgHg11.8% 2.5% 0.5% RaRaRa7.3% 2.3% 0.67%

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). SSSr POLARIZABILITIES PRELIMINARY RESULTS (a.u.)

Sr CI +MBPT CI + all-order Recomm.* 2 1 5s S0 195.6 198.0 197.2(2) 3 5s5p P0 483.6 459.4 458.3(3.6)

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

Polarizabilities are of significant importance for various applications ranging from study of fundamental symmetries to development of more precise clocks.

Significant improvement in accuracy is needed for polarizabilities of systems with more than one valence electrons.

Development of new method for calculating atomic properties of divalent systems is reported. • Improvement over best present approaches is demonstrated. • Results for group II atoms from Mg to Ra are presented. GRADUATE STUDENTS:

Rupsi Pal* Dansha Jiang* Bindiya Arora* Jenny Tchoukova* Z. Zhuriadna Matt Simmons OTHER COLLABORATIONS: Michael Kozlov (PNPI, Russia) (Visiting research scholar at the University of Delaware) Walter Johnson (University of Notre Dame), Charles Clark (NIST) Jim Mitroy (Darwin), Ulyana Safronova (University of Nevada-Reno)