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ATOMIC POLARIZABILITIESATOMIC POLARIZABILITIESATOMIC POLARIZABILITIESATOMIC POLARIZABILITIESATOMIC POLARIZABILITIESATOMIC POLARIZABILITIESATOMIC POLARIZABILITIESATOMIC POLARIZABILITIES
ICCMSE 2010, KOS, GREECEICCMSE 2010, KOS, GREECEICCMSE 2010, KOS, GREECEICCMSE 2010, KOS, GREECEICCMSE 2010, KOS, GREECEICCMSE 2010, KOS, GREECEICCMSE 2010, KOS, GREECEICCMSE 2010, KOS, GREECE
October 4, 2010
Marianna Marianna Marianna Marianna Marianna Marianna Marianna Marianna SafronovaSafronovaSafronovaSafronovaSafronovaSafronovaSafronovaSafronova
• 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
OUTLINEOUTLINEOUTLINEOUTLINEOUTLINEOUTLINEOUTLINEOUTLINE
ATOMIC DIPOLE POLARIZABILITYATOMIC DIPOLE POLARIZABILITYATOMIC DIPOLE POLARIZABILITYATOMIC DIPOLE POLARIZABILITYATOMIC DIPOLE POLARIZABILITYATOMIC DIPOLE POLARIZABILITYATOMIC DIPOLE POLARIZABILITYATOMIC DIPOLE POLARIZABILITY
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.
Electric-dipole polarizability governs the first-order response of an atom to an applied electric field.
( )U α λ∝
ATOMIC DIPOLE POLARIZABILITYATOMIC DIPOLE POLARIZABILITYATOMIC DIPOLE POLARIZABILITYATOMIC DIPOLE POLARIZABILITYATOMIC DIPOLE POLARIZABILITYATOMIC DIPOLE POLARIZABILITYATOMIC DIPOLE POLARIZABILITYATOMIC DIPOLE POLARIZABILITY
The interaction of field E directed along z-axis with the atom
is described by the Hamiltonian
ext ij i j
ij
H e z a a++++= −= −= −= − ∑∑∑∑E
First-order correction to the wave function satisfies
(((( )))) (1)
0 extH V E H+ − Ψ = − Ψ+ − Ψ = − Ψ+ − Ψ = − Ψ+ − Ψ = − Ψ
where . (((( ))))0H V E+ Ψ = Ψ+ Ψ = Ψ+ Ψ = Ψ+ Ψ = Ψ
(2) (1) 2 21
2ext
E H e αααα= Ψ Ψ = −= Ψ Ψ = −= Ψ Ψ = −= Ψ Ψ = − E
SUMSUMSUMSUM----OVEROVEROVEROVER----STATES METHODSTATES METHODSTATES METHODSTATES METHODSUMSUMSUMSUM----OVEROVEROVEROVER----STATES METHODSTATES METHODSTATES METHODSTATES METHOD
(((( ))))0 2
gn
nn g
f
E Eαααα ====
−−−−∑∑∑∑
Example: scalar static electric-dipole polarizability
Absorption oscillator strength
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
APPLICATIONS OF APPLICATIONS OF APPLICATIONS OF APPLICATIONS OF
ATOMIC POLARIZABILITIESATOMIC POLARIZABILITIESATOMIC POLARIZABILITIESATOMIC POLARIZABILITIES
(1) Atomic clocks
(2) Cooling & trapping of atoms
(3) Other applications
APPLICATIONS OF APPLICATIONS OF APPLICATIONS OF APPLICATIONS OF
ATOMIC POLARIZABILITIESATOMIC POLARIZABILITIESATOMIC POLARIZABILITIESATOMIC POLARIZABILITIES
(1) Atomic clocks
(2) Cooling & trapping of atoms
(3) Other applications
ATOMIC CLOCKSATOMIC CLOCKSATOMIC CLOCKSATOMIC CLOCKSATOMIC CLOCKSATOMIC CLOCKSATOMIC CLOCKSATOMIC CLOCKS
Microwave
Transitions
Optical
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: NEXT MOTIATION: NEXT MOTIATION: NEXT MOTIATION: NEXT
GENERATION ATOMIC CLOCKSGENERATION ATOMIC CLOCKSGENERATION ATOMIC CLOCKSGENERATION ATOMIC CLOCKS
MOTIATION: NEXT MOTIATION: NEXT MOTIATION: NEXT MOTIATION: NEXT
GENERATION ATOMIC CLOCKSGENERATION ATOMIC CLOCKSGENERATION ATOMIC CLOCKSGENERATION 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-space
probes, perform more accurate measurements of the physical constants and tests of fundamental physics such as
searches for gravitational waves, etc.
ATOMIC CLOCKS AND VARIATION OF ATOMIC CLOCKS AND VARIATION OF ATOMIC CLOCKS AND VARIATION OF ATOMIC CLOCKS AND VARIATION OF
FUNDAMENTAL CONSTANTS FUNDAMENTAL CONSTANTS FUNDAMENTAL CONSTANTS FUNDAMENTAL CONSTANTS
ATOMIC CLOCKS AND VARIATION OF ATOMIC CLOCKS AND VARIATION OF ATOMIC CLOCKS AND VARIATION OF ATOMIC CLOCKS AND VARIATION OF
FUNDAMENTAL CONSTANTS FUNDAMENTAL CONSTANTS FUNDAMENTAL CONSTANTS FUNDAMENTAL CONSTANTS
(1) Astrophysical constraints on variation of α: 4σ4σ4σ4σ!
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!
ATOMIC CLOCKS AND POLARIZABILITYATOMIC CLOCKS AND POLARIZABILITYATOMIC CLOCKS AND POLARIZABILITYATOMIC CLOCKS AND POLARIZABILITYATOMIC CLOCKS AND POLARIZABILITYATOMIC CLOCKS AND POLARIZABILITYATOMIC CLOCKS AND POLARIZABILITYATOMIC CLOCKS AND POLARIZABILITY
(1) Magic Wavelengths
(2) Blackbody Radiation Shift
MAGIC WAVELENGTHMAGIC WAVELENGTHMAGIC WAVELENGTHMAGIC WAVELENGTHMAGIC WAVELENGTHMAGIC WAVELENGTHMAGIC WAVELENGTHMAGIC WAVELENGTH
Atom in state A sees potential UA
Atom in state Bsees potential UB
( )U α λ∝
Magic wavelength λmagic is the wavelength for which the optical potential U experienced
by an atom is independent on its state
Magic wavelength λmagic is the wavelength for which the optical potential U experienced
by an atom is independent on its state
m agicλλλλ
wavelength
αα αα(λ
)(λ
)(λ
)(λ
)
State B
State A
LOCATING MAGIC WAVELENGTHLOCATING MAGIC WAVELENGTHLOCATING MAGIC WAVELENGTHLOCATING MAGIC WAVELENGTHLOCATING MAGIC WAVELENGTHLOCATING MAGIC WAVELENGTHLOCATING MAGIC WAVELENGTHLOCATING MAGIC WAVELENGTH
BLACKBODY RADIATION SHIFTBLACKBODY RADIATION SHIFTBLACKBODY RADIATION SHIFTBLACKBODY RADIATION SHIFTBLACKBODY RADIATION SHIFTBLACKBODY RADIATION SHIFTBLACKBODY RADIATION SHIFTBLACKBODY RADIATION SHIFT
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.
• The temperature-dependent electric field
created by the blackbody radiation is described
by (in a.u.) :
BBR SHIFT OF A LEVELBBR SHIFT OF A LEVELBBR SHIFT OF A LEVELBBR SHIFT OF A LEVELBBR SHIFT OF A LEVELBBR SHIFT OF A LEVELBBR SHIFT OF A LEVELBBR SHIFT OF A LEVEL
32 8( )
exp( / ) 1
dE
kT
α ω ωω
π ω=
−
2
BBR ( ) ( ) v A E dα ω ω ω∆ = − × ∫
Dynamic polarizability
• Frequency shift caused by this electric field is:
BBR SHIFT AND POLARIZABILITYBBR SHIFT AND POLARIZABILITYBBR SHIFT AND POLARIZABILITYBBR SHIFT AND POLARIZABILITYBBR SHIFT AND POLARIZABILITYBBR SHIFT AND POLARIZABILITYBBR 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)
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 a2 [1].
VECTOR & TENSOR POLARIZABILITY AVERAGE
OUT DUE TO THE ISOTROPIC NATURE OF FIELD.
Dynamic correctionDynamic correction
BLACKBODY RADIATION SHIFTS IN BLACKBODY RADIATION SHIFTS IN BLACKBODY RADIATION SHIFTS IN BLACKBODY RADIATION SHIFTS IN
OPTICAL FREQUENCY STANDARDS:OPTICAL FREQUENCY STANDARDS:OPTICAL FREQUENCY STANDARDS:OPTICAL FREQUENCY STANDARDS:
(1) (1) (1) (1) MONOVALENTMONOVALENTMONOVALENTMONOVALENT SYSTEMSSYSTEMSSYSTEMSSYSTEMS
(2) DIVALENT SYSTEMS(2) DIVALENT SYSTEMS(2) DIVALENT SYSTEMS(2) DIVALENT SYSTEMS
(3) OTHER, MORE COMPLICATED SYSTEMS(3) OTHER, MORE COMPLICATED SYSTEMS(3) OTHER, MORE COMPLICATED SYSTEMS(3) OTHER, MORE COMPLICATED SYSTEMS
BLACKBODY RADIATION SHIFTS IN BLACKBODY RADIATION SHIFTS IN BLACKBODY RADIATION SHIFTS IN BLACKBODY RADIATION SHIFTS IN
OPTICAL FREQUENCY STANDARDS:OPTICAL FREQUENCY STANDARDS:OPTICAL FREQUENCY STANDARDS:OPTICAL FREQUENCY STANDARDS:
(1) (1) (1) (1) MONOVALENTMONOVALENTMONOVALENTMONOVALENT SYSTEMSSYSTEMSSYSTEMSSYSTEMS
(2) DIVALENT SYSTEMS(2) DIVALENT SYSTEMS(2) DIVALENT SYSTEMS(2) DIVALENT SYSTEMS
(3) OTHER, MORE COMPLICATED SYSTEMS(3) OTHER, MORE COMPLICATED SYSTEMS(3) OTHER, MORE COMPLICATED 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)
StateStateStateState----insensitive cooling insensitive cooling insensitive cooling insensitive cooling
and trapping for and trapping for and trapping for and trapping for
quantum information quantum information quantum information quantum information
processingprocessingprocessingprocessing
COOLING AND TRAPPING OF COOLING AND TRAPPING OF COOLING AND TRAPPING OF COOLING AND TRAPPING OF
NEUTRAL ATOMSNEUTRAL ATOMSNEUTRAL ATOMSNEUTRAL ATOMS
COOLING AND TRAPPING OF COOLING AND TRAPPING OF COOLING AND TRAPPING OF COOLING AND TRAPPING OF
NEUTRAL ATOMSNEUTRAL ATOMSNEUTRAL ATOMSNEUTRAL ATOMS
Atom in state A sees potential UA
Atom in state Bsees potential UB
λ λ λ λ (nm)
925 930 935 940 945 950 955
αα αα (a.u.)
0
2000
4000
6000
8000
10000
6S1/26P3/2
932 nm
938 nm
a0- a2
a0+ a2
λλλλmagic
0 2vα α αα α αα α αα α α= += += += + MJ = ±3/2
MJ = ±1/20 2vα α αα α αα α αα α α= −= −= −= −
Other*Other*
λλλλmagic around 935nm
* Kimble et al. PRL 90(13), 133602(2003)
MAGIC WAVELENGTH FOR CSMAGIC WAVELENGTH FOR CSMAGIC WAVELENGTH FOR CSMAGIC WAVELENGTH FOR CSMAGIC WAVELENGTH FOR CSMAGIC WAVELENGTH FOR CSMAGIC WAVELENGTH FOR CSMAGIC WAVELENGTH FOR CS
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).
OTHER APPLICATIONSOTHER APPLICATIONSOTHER APPLICATIONSOTHER APPLICATIONSOTHER APPLICATIONSOTHER APPLICATIONSOTHER APPLICATIONSOTHER APPLICATIONS
• 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
CURRENT STATUS OF CURRENT STATUS OF CURRENT STATUS OF CURRENT STATUS OF
THEORY AND EXPERIMENTTHEORY AND EXPERIMENTTHEORY AND EXPERIMENTTHEORY AND EXPERIMENT
ATOMIC DIPOLE POLARIZABILITIESATOMIC DIPOLE POLARIZABILITIESATOMIC DIPOLE POLARIZABILITIESATOMIC DIPOLE POLARIZABILITIES
CURRENT STATUS OF CURRENT STATUS OF CURRENT STATUS OF CURRENT STATUS OF
THEORY AND EXPERIMENTTHEORY AND EXPERIMENTTHEORY AND EXPERIMENTTHEORY AND EXPERIMENT
ATOMIC DIPOLE POLARIZABILITIESATOMIC DIPOLE POLARIZABILITIESATOMIC DIPOLE POLARIZABILITIESATOMIC DIPOLE POLARIZABILITIES
JPB TOPICAL REVIEW (2010):
Theory and applications of atomic and ionic polarizabilities,
J. Mitroy, M.S. Safronova, and Charles W. Clark, in press
SUMMARY OF THEORY METHODS FOR SUMMARY OF THEORY METHODS FOR SUMMARY OF THEORY METHODS FOR SUMMARY OF THEORY METHODS FOR
ATOMIC POLARIZABILITIESATOMIC POLARIZABILITIESATOMIC POLARIZABILITIESATOMIC POLARIZABILITIES
SUMMARY OF THEORY METHODS FOR SUMMARY OF THEORY METHODS FOR SUMMARY OF THEORY METHODS FOR SUMMARY OF THEORY METHODS FOR
ATOMIC POLARIZABILITIESATOMIC POLARIZABILITIESATOMIC POLARIZABILITIESATOMIC POLARIZABILITIES
• 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
ATOMIC POLARIZABILITIES:ATOMIC POLARIZABILITIES:ATOMIC POLARIZABILITIES:ATOMIC POLARIZABILITIES:ATOMIC POLARIZABILITIES:ATOMIC POLARIZABILITIES:ATOMIC POLARIZABILITIES:ATOMIC POLARIZABILITIES:
HOW ACCURATE ARE THEORY VALUES HOW ACCURATE ARE THEORY VALUES HOW ACCURATE ARE THEORY VALUES HOW ACCURATE ARE THEORY VALUES HOW ACCURATE ARE THEORY VALUES HOW ACCURATE ARE THEORY VALUES HOW ACCURATE ARE THEORY VALUES HOW ACCURATE ARE THEORY VALUES ??
c vc vα α α αα α α αα α α αα α α α= + += + += + += + +
Core term
Valence term
(dominant)
Compensation term
2
0 1
3(2 1)v
nv n v
n D v
j E Eαααα ====
+ −+ −+ −+ −∑∑∑∑
Example:Scalar dipole polarizability
Electric-dipole reduced matrix element
POLARIZABILITY OF AN ALKALI ATOM IN A STATEPOLARIZABILITY OF AN ALKALI ATOM IN A STATEPOLARIZABILITY OF AN ALKALI ATOM IN A STATEPOLARIZABILITY OF AN ALKALI ATOM IN A STATE vSum-over-states approach
POLARIZABILITY OF AN ALKALI ATOM IN A STATEPOLARIZABILITY OF AN ALKALI ATOM IN A STATEPOLARIZABILITY OF AN ALKALI ATOM IN A STATEPOLARIZABILITY OF AN ALKALI ATOM IN A STATE vSum-over-states approach
HOW TO ESTIMATE UNCERTAINTY OF HOW TO ESTIMATE UNCERTAINTY OF HOW TO ESTIMATE UNCERTAINTY OF HOW TO ESTIMATE UNCERTAINTY OF HOW TO ESTIMATE UNCERTAINTY OF HOW TO ESTIMATE UNCERTAINTY OF HOW TO ESTIMATE UNCERTAINTY OF HOW TO ESTIMATE UNCERTAINTY OF
A MATRIX ELEMENT A MATRIX ELEMENT A MATRIX ELEMENT A MATRIX ELEMENT A MATRIX ELEMENT A MATRIX ELEMENT A MATRIX ELEMENT A MATRIX ELEMENT ??
THEORY: EVALUATION OF THE THEORY: EVALUATION OF THE THEORY: EVALUATION OF THE THEORY: EVALUATION OF THE
UNCERTAINTYUNCERTAINTYUNCERTAINTYUNCERTAINTY
THEORY: EVALUATION OF THE THEORY: EVALUATION OF THE THEORY: EVALUATION OF THE THEORY: EVALUATION OF THE
UNCERTAINTYUNCERTAINTYUNCERTAINTYUNCERTAINTY
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:EXAMPLE:EXAMPLE:EXAMPLE:
QUADRUPOLEQUADRUPOLEQUADRUPOLEQUADRUPOLE MOMENT OF MOMENT OF MOMENT OF MOMENT OF
3D3D3D3D5/25/25/25/2 STATE IN CSTATE IN CSTATE IN CSTATE IN Ca++++
EXAMPLE:EXAMPLE:EXAMPLE:EXAMPLE:
QUADRUPOLEQUADRUPOLEQUADRUPOLEQUADRUPOLE MOMENT OF MOMENT OF MOMENT OF MOMENT OF
3D3D3D3D5/25/25/25/2 STATE IN CSTATE IN CSTATE IN CSTATE 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 order
2.451
3D3D3D3D5/25/25/25/2 QUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CA++++3D3D3D3D5/25/25/25/2 QUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CA++++
Third order
1.610
Lowest order
2.451
3D3D3D3D5/25/25/25/2 QUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CA++++3D3D3D3D5/25/25/25/2 QUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CA++++
All order (SD)
1.785
Third order
1.610
Lowest order
2.451
3D3D3D3D5/25/25/25/2 QUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CA++++3D3D3D3D5/25/25/25/2 QUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CA++++
All order (SDpT)
1.837
All order (SD)
1.785
Third order
1.610
Lowest order
2.451
3D3D3D3D5/25/25/25/2 QUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CA++++3D3D3D3D5/25/25/25/2 QUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CA++++
Coupled-cluster SD (CCSD)
1.822
All order (SDpT)
1.837
All order (SD)
1.785
Third order
1.610
Lowest order
2.451
3D3D3D3D5/25/25/25/2 QUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CA++++3D3D3D3D5/25/25/25/2 QUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CA++++
Coupled-cluster SD (CCSD)
1.822
All order (SDpT)
1.837
All order (SD)
1.785
Third order
1.610
Lowest order
2.451
3D3D3D3D5/25/25/25/2 QUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CA++++3D3D3D3D5/25/25/25/2 QUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CAQUADRUPOLE MOMENT IN CA++++
Estimateomittedcorrections
All order (SD), scaled 1.849All-order (CCSD), scaled 1.851All order (SDpT) 1.837All order (SDpT), scaled 1.836
Third order
1.610
Final results: 3d5/2 quadrupole momentFinal results: 3d5/2 quadrupole moment
Lowest order
2.454
1.849 (13)1.849 (13)
All order (SD), scaled 1.849All-order (CCSD), scaled 1.851All order (SDpT) 1.837All order (SDpT), scaled 1.836
Third order
1.610
Final results: 3d5/2 quadrupole momentFinal results: 3d5/2 quadrupole moment
Lowest order
2.454
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 HIGHDEVELOPMENT OF HIGHDEVELOPMENT OF HIGHDEVELOPMENT OF HIGH----PRECISION PRECISION PRECISION PRECISION
METHODSMETHODSMETHODSMETHODS
PRESENT STATUS OF THEORY ANDPRESENT STATUS OF THEORY ANDPRESENT STATUS OF THEORY ANDPRESENT STATUS OF THEORY AND
NEED FOR FURTHER DEVELOPMENTNEED FOR FURTHER DEVELOPMENTNEED FOR FURTHER DEVELOPMENTNEED FOR FURTHER DEVELOPMENT
DEVELOPMENT OF HIGHDEVELOPMENT OF HIGHDEVELOPMENT OF HIGHDEVELOPMENT OF HIGH----PRECISION PRECISION PRECISION PRECISION
METHODSMETHODSMETHODSMETHODS
PRESENT STATUS OF THEORY ANDPRESENT STATUS OF THEORY ANDPRESENT STATUS OF THEORY ANDPRESENT STATUS OF THEORY AND
NEED FOR FURTHER DEVELOPMENTNEED FOR FURTHER DEVELOPMENTNEED FOR FURTHER DEVELOPMENTNEED FOR FURTHER DEVELOPMENT
• 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: MOTIVATION: MOTIVATION: MOTIVATION:
STUDY OF GROUP II STUDY OF GROUP II STUDY OF GROUP II STUDY OF GROUP II –––– TYPE SYSTEMSTYPE SYSTEMSTYPE SYSTEMSTYPE SYSTEMS
MOTIVATION: MOTIVATION: MOTIVATION: MOTIVATION:
STUDY OF GROUP II STUDY OF GROUP II STUDY OF GROUP II STUDY OF GROUP II –––– TYPE SYSTEMSTYPE SYSTEMSTYPE SYSTEMSTYPE SYSTEMS
Divalent ions:Divalent ions:Divalent ions:Divalent ions:AlAlAlAl++++, In, In, In, In++++, etc., etc., etc., etc.Divalent ions:Divalent ions:Divalent ions:Divalent ions:AlAlAlAl++++, In, In, In, In++++, etc., etc., etc., etc.
Mg Mg Mg Mg Ca Ca Ca Ca SrSrSrSrBaBaBaBaRaRaRaRaZnZnZnZnCdCdCdCdHgHgHgHgYbYbYbYb
Mg Mg Mg Mg Ca Ca Ca Ca SrSrSrSrBaBaBaBaRaRaRaRaZnZnZnZnCdCdCdCdHgHgHgHgYbYbYbYb
SUMMARY OF THEORY METHODS FOR SUMMARY OF THEORY METHODS FOR SUMMARY OF THEORY METHODS FOR SUMMARY OF THEORY METHODS FOR
ATOMIC POLARIZABILITIESATOMIC POLARIZABILITIESATOMIC POLARIZABILITIESATOMIC POLARIZABILITIES
SUMMARY OF THEORY METHODS FOR SUMMARY OF THEORY METHODS FOR SUMMARY OF THEORY METHODS FOR SUMMARY OF THEORY METHODS FOR
ATOMIC POLARIZABILITIESATOMIC POLARIZABILITIESATOMIC POLARIZABILITIESATOMIC POLARIZABILITIES
• 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 + all-order (RLCCSD(T) coupled-cluster) method*
*under development
CONFIGURATION INTERACTION +CONFIGURATION INTERACTION +CONFIGURATION INTERACTION +CONFIGURATION INTERACTION +
ALLALLALLALL----ORDER METHOD ORDER METHOD ORDER METHOD ORDER METHOD
CONFIGURATION INTERACTION +CONFIGURATION INTERACTION +CONFIGURATION INTERACTION +CONFIGURATION INTERACTION +
ALLALLALLALL----ORDER METHOD ORDER METHOD ORDER METHOD ORDER METHOD
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, two methods are combined to Therefore, two methods are combined to
acquire benefits from both approaches. acquire benefits from both approaches.
CONFIGURATION INTERACTION CONFIGURATION INTERACTION CONFIGURATION INTERACTION CONFIGURATION INTERACTION
METHODMETHODMETHODMETHOD
CONFIGURATION INTERACTION CONFIGURATION INTERACTION CONFIGURATION INTERACTION CONFIGURATION INTERACTION
METHODMETHODMETHODMETHOD
i i
i
cΨ = Φ∑ Single-electron valence
basis states
( ) 0effH E− Ψ =
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 CONFIGURATION INTERACTION CONFIGURATION INTERACTION CONFIGURATION INTERACTION
METHOD + ALLMETHOD + ALLMETHOD + ALLMETHOD + ALL----ORDERORDERORDERORDER
CONFIGURATION INTERACTION CONFIGURATION INTERACTION CONFIGURATION INTERACTION CONFIGURATION INTERACTION
METHOD + ALLMETHOD + ALLMETHOD + ALLMETHOD + ALL----ORDERORDERORDERORDER
1 1 1
2 2 2
1 2,
h h
h h
→ + Σ
→ + Σ
Σ Σ
Heff is modified using all-order calculation
are obtained using all-order, RLCCSD(T),
method used for alkali-metal atoms with
appropriate modifications
( ) 0effH E− Ψ =
In the all-order method, dominant correlation corrections are summed to all orders of perturbation theory.
Lowest order Corecorevalence electron any excited orbital
Single-particle excitations
Double-particle excitations
(0)
vΨ
(0)† †
mn m nm
a av vna
vaa a aρ Ψ∑
† (0)
a aa
m mm
va aρ Ψ∑ † (0)
v v vv
m mm
a aρ≠
Ψ∑
† † (0)12 m nmn
mab b v
na
ab
aa aaρ Ψ∑
RLCCSDRLCCSDRLCCSDRLCCSD ATOMIC WAVE FUNCTION ATOMIC WAVE FUNCTION ATOMIC WAVE FUNCTION ATOMIC WAVE FUNCTION RLCCSDRLCCSDRLCCSDRLCCSD ATOMIC WAVE FUNCTION ATOMIC WAVE FUNCTION ATOMIC WAVE FUNCTION ATOMIC WAVE FUNCTION
MONOVALENTMONOVALENTMONOVALENTMONOVALENT SYSTEMS: SYSTEMS: SYSTEMS: SYSTEMS:
VERY BRIEF SUMMARY OF WHAT WEVERY BRIEF SUMMARY OF WHAT WEVERY BRIEF SUMMARY OF WHAT WEVERY BRIEF SUMMARY OF WHAT WE
CALCULATED WITH ALLCALCULATED WITH ALLCALCULATED WITH ALLCALCULATED WITH ALL----ORDER METHODORDER METHODORDER METHODORDER METHOD
MONOVALENTMONOVALENTMONOVALENTMONOVALENT SYSTEMS: SYSTEMS: SYSTEMS: SYSTEMS:
VERY BRIEF SUMMARY OF WHAT WEVERY BRIEF SUMMARY OF WHAT WEVERY BRIEF SUMMARY OF WHAT WEVERY BRIEF SUMMARY OF WHAT WE
CALCULATED WITH ALLCALCULATED WITH ALLCALCULATED WITH ALLCALCULATED WITH ALL----ORDER METHODORDER METHODORDER METHODORDER 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
CONFIGURATION INTERACTION CONFIGURATION INTERACTION CONFIGURATION INTERACTION CONFIGURATION INTERACTION
+ + + + ALLALLALLALL----ORDERORDERORDERORDER METHODMETHODMETHODMETHOD
CONFIGURATION INTERACTION CONFIGURATION INTERACTION CONFIGURATION INTERACTION CONFIGURATION INTERACTION
+ + + + ALLALLALLALL----ORDERORDERORDERORDER METHODMETHODMETHODMETHOD
Heff is modified using all-order excitation coefficients
Advantages: most complete treatment of the
correlations and applicable for many-valenceelectron systems
( ) ( )
( ) ( ) L
mnklnmlk
L
mnkl
mnmnmn
ρεεεε
ρεε
−−+=Σ
−=Σ
~~
~
2
1
CI + ALLCI + ALLCI + ALLCI + ALL----ORDER RESULTSORDER RESULTSORDER RESULTSORDER RESULTSCI + ALLCI + ALLCI + ALLCI + ALL----ORDER RESULTSORDER RESULTSORDER RESULTSORDER RESULTS
AtomAtomAtomAtom CI CI CI CI CICICICI + MBPT+ MBPT+ MBPT+ MBPT CI + AllCI + AllCI + AllCI + All----orderorderorderorderMg 1.9%Mg 1.9%Mg 1.9%Mg 1.9% 0.11%0.11%0.11%0.11% 0.03%0.03%0.03%0.03%Ca Ca Ca Ca 4.1%4.1%4.1%4.1% 0.7%0.7%0.7%0.7% 0.3%0.3%0.3%0.3%ZnZnZnZn 8.0%8.0%8.0%8.0% 0.7%0.7%0.7%0.7% 0.4 %0.4 %0.4 %0.4 %SrSrSrSr 5.2%5.2%5.2%5.2% 1.0%1.0%1.0%1.0% 0.4%0.4%0.4%0.4%CdCdCdCd 9.6% 9.6% 9.6% 9.6% 1.4%1.4%1.4%1.4% 0.2%0.2%0.2%0.2%BaBaBaBa 6.4% 6.4% 6.4% 6.4% 1.9%1.9%1.9%1.9% 0.6%0.6%0.6%0.6%HgHgHgHg 11.8%11.8%11.8%11.8% 2.5%2.5%2.5%2.5% 0.5%0.5%0.5%0.5%RaRaRaRa 7.3%7.3%7.3%7.3% 2.3%2.3%2.3%2.3% 0.67%0.67%0.67%0.67%
TwoTwoTwoTwo----electron binding energies, differences with experimentelectron binding energies, differences with experimentelectron binding energies, differences with experimentelectron 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).
SSSSr POLARIZABILITIESPOLARIZABILITIESPOLARIZABILITIESPOLARIZABILITIES
PRELIMINARY RESULTS PRELIMINARY RESULTS PRELIMINARY RESULTS PRELIMINARY RESULTS (a.u.)
* From expt. matrix elements, S. G. Porsev and A. Derevianko, PRA 74, 020502R (2006).
458.3(3.6)459.4483.65s5p 3P0
197.2(2)198.0195.65s2 1S0
Recomm.*CI + all-orderCI +MBPTSr
CONCLUSIONCONCLUSIONCONCLUSIONCONCLUSIONCONCLUSIONCONCLUSIONCONCLUSIONCONCLUSION
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.
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.
OTHER COLLABORATIONS:OTHER COLLABORATIONS:OTHER COLLABORATIONS: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)
GRADUATEGRADUATEGRADUATEGRADUATE
STUDENTS: STUDENTS: STUDENTS: STUDENTS:
Rupsi Pal*Dansha Jiang*Bindiya Arora*Jenny Tchoukova*Z. ZhuriadnaMatt Simmons