Scandium Sc Atomic Number: 21 Atomic Mass: 44.955910 Atomic Number: 21 Atomic Mass: 44.955910.
Polarizabilities, Atomic Clocks, and Magic Wavelengths DAMOP 2008 focus session: Atomic polarization...
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![Page 1: Polarizabilities, Atomic Clocks, and Magic Wavelengths DAMOP 2008 focus session: Atomic polarization and dispersion May 29, 2008 Marianna Safronova Bindiya.](https://reader035.fdocuments.us/reader035/viewer/2022062714/56649d585503460f94a371f0/html5/thumbnails/1.jpg)
Polarizabilities, Atomic Clocks, and Magic Wavelengths
Polarizabilities, Atomic Clocks, and Magic Wavelengths
DAMOP 2008 focus session:DAMOP 2008 focus session:Atomic polarization and dispersionAtomic polarization and dispersion
May 29, 2008 Marianna SafronovaMarianna SafronovaBindiya aroraBindiya arora
Charles W. clarkCharles W. clarkNIST, GaithersburgNIST, Gaithersburg
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• Motivation
• Method
• Applications
• Frequency-dependent polarizabilities of alkali atomsand magic frequencies
• Atomic clocks: blackbody radiation shifts
• Future studies
OutlineOutline
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State-insensitive cooling and trapping for quantum
information processing
Motivation: 1Optically trapped atoms
Motivation: 1Optically trapped atoms
Atom in state A sees potential UA
Atom in state B sees potential UB
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Atomic clocks: Next Generation
Motivation: 2Motivation: 2
MicrowaveTransitions
OpticalTransitions
http://tf.nist.gov/cesium/fountain.htm,NIST Yb atomic clock
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Parity violation studies with heavy atoms & search for Electron electric-dipole moment
Motivation: 3Motivation: 3
http://CPEPweb.org, http://public.web.cern.ch/, Cs experiment, University of Colorado
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MotivationMotivation
• Development of the high-precision methodologies• Benchmark tests of theory and experiment• Cross-checks of various experiments• Data for astrophysics • Long-range interactions • Determination of nuclear magnetic and anapole
moments• Variation of fundamental constants with time
![Page 7: Polarizabilities, Atomic Clocks, and Magic Wavelengths DAMOP 2008 focus session: Atomic polarization and dispersion May 29, 2008 Marianna Safronova Bindiya.](https://reader035.fdocuments.us/reader035/viewer/2022062714/56649d585503460f94a371f0/html5/thumbnails/7.jpg)
Atomic polarizabilitiesAtomic polarizabilities
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c vc v
Core term
Valence term (dominant)
Compensation term
2
02 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
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Very precise calculation of atomic properties
We also need to evaluate uncertainties of theoretical values!
How to accurately calculate various matrix elements ?
How to accurately calculate various matrix elements ?
![Page 10: Polarizabilities, Atomic Clocks, and Magic Wavelengths DAMOP 2008 focus session: Atomic polarization and dispersion May 29, 2008 Marianna Safronova Bindiya.](https://reader035.fdocuments.us/reader035/viewer/2022062714/56649d585503460f94a371f0/html5/thumbnails/10.jpg)
Very precise calculation of atomic properties
WANTED!
We also need to evaluate uncertainties of theoretical values!
How to accurately calculate various matrix elements ?
How to accurately calculate various matrix elements ?
![Page 11: Polarizabilities, Atomic Clocks, and Magic Wavelengths DAMOP 2008 focus session: Atomic polarization and dispersion May 29, 2008 Marianna Safronova Bindiya.](https://reader035.fdocuments.us/reader035/viewer/2022062714/56649d585503460f94a371f0/html5/thumbnails/11.jpg)
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)
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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 nmnm
ab b vn
aab
aa aa
All-order atomic wave function (SD)All-order atomic wave function (SD)
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The derivation gets really complicated if you add triples!
Solution: develop analytical codes that do all the work for you!
Input: ASCII input of terms of the type
(0† † † )† †: : ::ijkl l kijkl
m n ri jm vaab vnrmnr
bab
vg a a a a a aa aa a
Output: final simplified formula in LATEX to be used in the all-order equation
Actual implementation: codes that write formulasActual implementation:
codes that write formulas
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Problem with all-order extensions: TOO MANY TERMS
Problem with all-order extensions: TOO MANY TERMS
The complexity of the equations increases.Same issue with third-order MBPT for two-particle systems (hundreds of terms) .What to do with large number of terms?
Solution: automated code generation !Solution: automated code generation !
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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
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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 atomsResults for alkali-metal atoms
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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
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Polarizabilities: Applications
Polarizabilities: Applications
• Optimizing the Rydberg gate
• Identification of wavelengths at which two different alkali atoms have the same oscillation frequency for simultaneous optical trapping of two different alkali species.
• Detection of inconsistencies in Cs lifetime and Stark shift experiments
• Benchmark determination of some K and Rb properties
• Calculation of “magic frequencies” for state-insensitive cooling and trapping
• Atomic clocks: problem of the BBR shift• …
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Polarizabilities: Applications
Polarizabilities: Applications
• Optimizing the Rydberg gate
• Identification of wavelengths at which two different alkali atoms have the same oscillation frequency for simultaneous optical trapping of two different alkali species.
• Detection of inconsistencies in Cs lifetime and Stark shift experiments
• Benchmark determination of some K and Rb properties
• Calculation of “magic frequencies” for state-insensitive cooling and trapping
• Atomic clocks: problem of the BBR shift• …
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ApplicationsFrequency-dependent polarizabilities of alkali
atoms from ultraviolet through infrared spectral regions
ApplicationsFrequency-dependent polarizabilities of alkali
atoms from ultraviolet through infrared spectral regions
Goal:
First-principles calculations of the frequency-dependent polarizabilities of ground and excited states of alkali-metal atoms
Determination of magic wavelengths
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Excited states: determination of magic frequencies in alkali-metal atoms for state-insensitive cooling and trapping, i.e.
When does the ground state and excited np state has the same ac Stark shift?
Magic wavelengthsMagic wavelengths
Bindiya Arora, M.S. Safronova, and Charles W. Clark,Phys. Rev. A 76, 052509 (2007)Na, K, Rb, and Cs
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( )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?
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magic
wavelength
α
S State
P State
Locating magic wavelengthLocating magic wavelength
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What do we need?What do we need?
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What do we need?What do we need?
Lots and lots of Lots and lots of matrix elements!matrix elements!
![Page 26: Polarizabilities, Atomic Clocks, and Magic Wavelengths DAMOP 2008 focus session: Atomic polarization and dispersion May 29, 2008 Marianna Safronova Bindiya.](https://reader035.fdocuments.us/reader035/viewer/2022062714/56649d585503460f94a371f0/html5/thumbnails/26.jpg)
What do we need?What do we need?
Lots and lots of matrix elements!Cs
1/ 2 1/ 2 1/ 2 1/ 2
3 / 2 3 / 2 3 / 2 3 / 2
6 7 8 9
6 7 8 9
6,7,8,9
P D nS P D nS P D nS P D nS
P D nS P D nS P D nS P D nS
n
1/ 2 3/ 2 1/ 2 3/ 2 1/ 2 3/ 2 1/ 2 3/ 2
3/ 2 3/ 2 3/ 2 3/ 2 3/ 2 3/ 2 3/ 2 3/ 2
3/ 2 5/ 2 3/ 2 5/ 2 3/ 2 5/ 2 3/ 2 5/ 2
6 7 8 8
6 7 8 8
6 7 8 8
5,6,7
P D nD P D nD P D nD P D nD
P D nD P D nD P D nD P D nD
P D nD P D nD P D nD P D nD
n
56 matrix elements in main
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What do we need?What do we need?
Lots and lots of matrix elements!
All-order “database”: over 700 matrix elements for alkali-
metal atoms and other monovalent systems
All-order “database”: over 700 matrix elements for alkali-
metal atoms and other monovalent systems
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Theory (This work)
Theory (This work) Experiment*Experiment*
0
=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)
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Frequency-dependent polarizabilities of Na atom in the ground and 3p3/2 states.
The arrows show the magic wavelengths
0 2v MJ = ±3/2
MJ = ±1/20 2v
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Magic wavelengths for the 3p1/2 - 3s and 3p3/2 - 3s transition of Na.
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Magic wavelengths for the 5p3/2 - 5s transition of Rb.
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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.
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(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
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ac Stark shifts for the transition from 6p3/2F′=5 M′
sublevels to 6s FM sublevels in Cs.The electric field intensity is taken to be 1 MW/cm2.
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Applications:atomic clocksApplications:atomic clocks
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atomic clocksblack-body radiation ( BBR ) shift
atomic clocksblack-body radiation ( BBR ) shift
Motivation:
BBR shift gives the larges uncertainties for some of the optical atomic clock schemes, such as withCa+
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Blackbody radiation shift in optical frequency standard with 43Ca+ ion
Blackbody radiation shift in optical frequency standard with 43Ca+ ion
Bindiya Arora, M.S. Safronova, and Charles W. Clark, Phys. Rev. A 76, 064501 (2007)
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MotivationMotivation
For Ca+, the contribution from Blackbody radiation gives
the largest uncertainty
to the frequency standard at T = 300K
BBR = 0.39(0.27) Hz [1]
[1] C. Champenois et. al. Phys. Lett. A 331, 298 (2004)
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Frequency standardFrequency standard
Transition frequency should be corrected to
account for the effect of the black body radiation at
T=300K.
T = 0 K
Clocktransition
Level A
Level B
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Frequency standardFrequency standard
Transition frequency should be corrected to
account for the effect of the black body radiation at
T=300K.
T = 300 K
Clocktransition
Level A
Level B
BBR
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4s1/2
4p1/2
3d3/2397 nm
866 nm
729 nm
3d5/2
854 nm
393 nm
4p3/2
732 nmE2
Easily produced by non-bulky solid state or diode lasers
The clock transition involved is 4s1/2F=4 MF=0 → 3d5/2 F=6 MF=0
Why Ca+ ion?Why Ca+ ion?
Lifetime~1.2 s
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• The temperature-dependent electric field created by the blackbody radiation is described by (in a.u.) :
BBR shift of a levelBBR shift of a level
32 8( )
exp( / ) 1
dE
kT
2BBR ( ) ( ) v A E d
Dynamic polarizability
• Frequency shift caused by this electric field is:
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42
BBR 0
1 ( )(0)(831.9 / )
2 300
T KV m
BBR shift can be expressed in terms of a scalar static polarizability:
BBR shift and polarizabilityBBR shift and polarizability
Vector & tensor polarizability average out due to the isotropic nature of field.
Dynamic correction
Dynamic correction ~10-3 Hz.
At the present level of accuracy the dynamic
correction can be neglected.
(1 )
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Effect on the frequency of clock transition
is calculated as the difference between the
BBR shifts of individual states.
BBR shift for a transitionBBR shift for a transition
BBR 1/ 2 5/ 2 BBR 5/ 2 BBR 1/ 2(4 3 ) (3 ) (4 )v s d v d v s
4s1/2
3d5/2
729 nmBBR 0 (0)v
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Need ground and excited state scalar static polarizability
NOTE: Tensor polarizability calculated in this work is also of experimental interest.
Need BBR shifts
2
0 1
3(2 1)vnv n v
n D v
j E E
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5p3/2
4p3/2
0.010.010.010.01
6p3/2
0.010.010.010.010.010.010.010.01
0.010.010.010.01
0.060.060.060.06
3.33.33.33.3
CoreCore
tailtail
48.448.448.448.4
Total: Total: 76.1 ± 1.1Total: Total: 76.1 ± 1.1 4s
Contributions to the 4s1/2 scalar polarizability ( )
Contributions to the 4s1/2 scalar polarizability ( )
30a
6p1/2
5p1/2
4p1/2
43Ca+ (= 0)43Ca+ (= 0)
24.424.424.424.4
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2.42.42.42.4
5p3/2
4p3/2
0.010.010.010.01
np3/2 tail
0.010.010.010.01 0.80.80.80.8
0.30.30.30.3
1.71.71.71.7
3.33.33.33.3
CoreCore
nf7/2nf7/2
22.822.822.822.8
Total: Total: 32.0 ± 1.1Total: Total: 32.0 ± 1.13d5/2
6f7/2
5f7/2
4f7/2
nf5/2
0.20.20.20.2 0.50.50.50.5
7-12f7/2
43Ca+ 43Ca+
Contributions to the 3d5/2 scalar polarizability ( )
Contributions to the 3d5/2 scalar polarizability ( )
30a
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Present Ref. [1] Ref. [2] Ref. [3]
0(4s1/2) 76.1(1.1) 76 73 70.89(15)
0(3d5/2) 32.0(1.1) 31 23
Comparison of our results for scalar static polarizabilities for the 4s1/2 and
3d5/2 states of 43Ca+ ion with other
available results
Comparison of our results for scalar static polarizabilities for the 4s1/2 and
3d5/2 states of 43Ca+ ion with other
available results
[1] C. Champenois et. al. Phys. Lett. A 331, 298 (2004)[2] Masatoshi Kajit et. al. Phys. Rev. A 72, 043404, (2005)[3] C.E. Theodosiou et. al. Phys. Rev. A 52, 3677 (1995)
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Comparison of black body radiation shift (Hz) for the 4s1/2- 3d5/2 transition of 43Ca+ ion at T=300K (E=831.9 V/m).
Black body radiation shiftBlack body radiation shift
Present Champenois[1]
Kajita [2]
(4s1/2 → 3d5/2) 0.38(1) 0.39(27) 0.4
[1] C. Champenois et. al. Phys. Lett. A 331, 298 (2004)[2] Masatoshi Kajit et. al. Phys. Rev. A 72, 043404, (2005)
An order of magnitude improvement is achieved with comparison to
previous calculations
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Comparison of black body radiation shift (Hz) for the 4s1/2- 3d5/2 transition of 43Ca+ ion at T=300K (E=831.9 V/m).
Black body radiation shiftBlack body radiation shift
Present Champenois[1]
Kajita [2]
(4s1/2 → 3d5/2) 0.38(1) 0.39(27) 0.4
[1] C. Champenois et. al. Phys. Lett. A 331, 298 (2004)[2] Masatoshi Kajit et. al. Phys. Rev. A 72, 043404, (2005)
Sufficient accuracy to establishThe uncertainty limits for the
Ca+ scheme
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relativisticrelativisticAll-orderAll-ordermethodmethod
Singly-ionized ions
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Future studies: more complicated system development of the
CI + all-order approach*
Future studies: more complicated system development of the
CI + all-order approach*
M.S. Safronova, M. Kozlov, and W.R. Johnson, in preparation
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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 method can account for core-core and core-valence correlation can not accuratelydescribe valence-valence correlation.
Therefore, two methods are combined to Therefore, two methods are combined to acquire benefits from both approaches. acquire benefits from both approaches.
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CI + ALL-ORDER: PRELIMINARY RESULTS
CI + ALL-ORDER: PRELIMINARY RESULTS
CI CI + MBPT CI + All-order
Mg 1.9% 0.12% 0.03%Ca 4.1% 0.6% 0.3%Sr 5.2% 0.9% 0.3%Ba 6.4% 1.7% 0.5%
Ionization potentials, differences with experiment
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ConclusionConclusion
• Benchmark calculation of various polarizabilities and tests of theory and experiment
• Determination of magic wavelengths for state-insensitive optical cooling and trapping
• Accurate calculations of the BBR shifts
Future studies: Development of generally applicable CI+ all-order method for more complicated systems
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ConclusionConclusion
AtomicClocks
S1/2
P1/2D5/2
„quantumbit“
Parity Violation
Quantum information
Future:Future:New SystemsNew SystemsNew Methods,New Methods,New ProblemsNew Problems
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P3.8 Jenny Tchoukova and M.S. SafronovaTheoretical study of the K, Rb, and Fr lifetimes
Q5.9 Dansha Jiang, Rupsi Pal, and M.S. SafronovaThird-order relativistic many-body calculation of transition probabilities for the beryllium and magnesium isoelectronic sequences
U4.8 Binidiya Arora, M.S. Safronova, and Charles W. ClarkState-insensitive two-color optical trapping
Graduate students: Bindiya AroraRupsi palJenny TchoukovaDansha Jiang