Polarizabilities, Atomic Clocks, and Magic Wavelengths DAMOP 2008 focus session: Atomic polarization and dispersion May 29, 2008 Marianna Safronova Bindiya.

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

• Motivation

• Method

• Applications

• Frequency-dependent polarizabilities of alkali atomsand magic frequencies

• Atomic clocks: blackbody radiation shifts

• Future studies

OutlineOutline

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

Atomic clocks: Next Generation

Motivation: 2Motivation: 2

MicrowaveTransitions

OpticalTransitions

http://tf.nist.gov/cesium/fountain.htm,NIST Yb atomic clock

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

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

Atomic polarizabilitiesAtomic polarizabilities

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

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 ?

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 ?

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 nmnm

ab b vn

aab

aa aa

All-order atomic wave function (SD)All-order atomic wave function (SD)

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

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 !

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

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

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

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• …

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• …

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

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

( )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?

magic

wavelength

α

S State

P State

Locating magic wavelengthLocating magic wavelength

What do we need?What do we need?

What do we need?What do we need?

Lots and lots of Lots and lots of matrix elements!matrix elements!

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

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

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)

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

Magic wavelengths for the 3p1/2 - 3s and 3p3/2 - 3s transition of Na.

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

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.

Applications:atomic clocksApplications:atomic clocks

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+

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)

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)

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

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

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

• 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:

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 )

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

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

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

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

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)

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

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

relativisticrelativisticAll-orderAll-ordermethodmethod

Singly-ionized ions

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

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.

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

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

ConclusionConclusion

AtomicClocks

S1/2

P1/2D5/2

„quantumbit“

Parity Violation

Quantum information

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

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