Download Polarizabilities, Atomic Clocks, and Magic Wavelengths

DAMOP 2008 focus session:
Atomic polarization and dispersion
Polarizabilities, Atomic
Clocks, and Magic
Wavelengths
May 29, 2008
Marianna Safronova
Bindiya arora
Charles W. clark
NIST, Gaithersburg
Outline
• Motivation
• Method
• Applications
• Frequency-dependent polarizabilities of alkali atoms
and magic frequencies
• Atomic clocks: blackbody radiation shifts
• Future studies
Motivation: 1
Optically trapped atoms
Atom in state A
sees potential UA
Atom in state B
sees potential UB
State-insensitive cooling
and trapping for
quantum information
processing
Motivation: 2
Atomic clocks: Next Generation
Microwave
Transitions
http://tf.nist.gov/cesium/fountain.htm,
NIST Yb atomic clock
Optical
Transitions
Motivation: 3
Parity violation studies with heavy atoms & search for
Electron electric-dipole moment
http://CPEPweb.org, http://public.web.cern.ch/, Cs experiment, University of Colorado
Motivation
• 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 polarizabilities
Polarizability of an alkali atom in a state
   c   vc   v
Core term
Example:
Scalar dipole polarizability
v
Valence term
(dominant)
Compensation term
Electric-dipole
reduced matrix
element
E n  Ev  n D v

1
 v0    

3(2 jv  1) n  En  Ev  2   2
2
How to accurately
calculate various matrix
elements ?
Very precise calculation of atomic properties
We also need to evaluate uncertainties of
theoretical values!
How to accurately calculate
various matrix elements ?
Very precise calculation of atomic properties
We also need to evaluate uncertainties of
theoretical values!
All-order atomic wave function (SD)
Lowest order
Single-particle
excitations
Double-particle
excitations
Core
core
valence electron
any excited orbital
All-order atomic wave function (SD)
Lowest order
Core
core
valence electron
any excited orbital
 v(0)
Single-particle
excitations

ma
 ma am† aa  v(0)
†
(0)

a
a

 mv m v v
mv
Double-particle
excitations
1
2

mnab
 mnab am† an† ab aa  v(0)

mna
 mnva am† an† aa av  v(0)
Actual implementation:
codes that write formulas
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

mnrab
† †
† † †
(0 )
g

:
a
a
a
a
:

:
a
a
a
a
a
a
:

 ijkl mnrvab i j l k m n r b a v v
ijkl
Output: final simplified formula in LATEX to be used
in the all-order equation
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 !
Automated code generation
Codes that write formulas
Codes that write codes
Input:
Output:
list of formulas to be programmed
final code (need to be put into a main shell)
Features: simple input, essentially just type in a formula!
Results for alkali-metal atoms
Na
3p1/2-3s
K
4p1/2-4s
Rb
5p1/2-5s
Cs
Fr
6p1/2-6s 7p1/2-7s
All-order 3.531
4.098
4.221
4.478
4.256
Experiment 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),
Theory
Cs:
R.J. Rafac et al., Phys. Rev. A 60, 3648 (1999),
Fr:
J.E. Simsarian et al., Phys. Rev. A 57, 2448 (1998)
M.S. Safronova, W.R. Johnson, and A. Derevianko,
Phys. Rev. A 60, 4476 (1999)
Theory: evaluation of the
uncertainty
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
• 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
• 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
• …
Applications
Frequency-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
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?
Bindiya Arora, M.S. Safronova, and Charles W. Clark,
Phys. Rev. A 76, 052509 (2007)
Na, K, Rb, and Cs
What is magic wavelength?
Atom in state B
sees potential UB
Atom in state A
sees potential UA
Magic wavelength magic is the wavelength for
which the optical potential U experienced
by an atom is independent on its state
U   ( )
Locating magic wavelength
α 
S State
P State
wavelength
 magic
What do we need?
What do we need?
Lots and lots of
matrix elements!
What do we need?
Cs
Lots and lots of matrix elements!
56 matrix elements in
 main
6 P1/ 2 D nS 7 P1/ 2 D nS 8 P1/ 2 D nS 9 P1/ 2 D nS
6 P3 / 2 D nS 7 P3 / 2 D nS 8 P3 / 2 D nS 9 P3 / 2 D nS
n  6, 7, 8, 9
6 P1/ 2 D nD3 / 2 7 P1/ 2 D nD3 / 2 8 P1/ 2 D nD3 / 2 8 P1/ 2 D nD3 / 2
6 P3 / 2 D nD3 / 2 7 P3 / 2 D nD3 / 2 8 P3 / 2 D nD3 / 2 8 P3 / 2 D nD3 / 2
6 P3 / 2 D nD5 / 2 7 P3 / 2 D nD5 / 2 8 P3 / 2 D nD5 / 2 8 P3 / 2 D nD5 / 2
n  5, 6, 7
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
Theory
=0
Na
K
Rb
 0 (3P1/2)
 0 (3P3/2)
 2 (3P3/2)
(This work)
Experiment*
359.9(4)
359.2(6)
361.6(4)
-88.4(10)
360.4(7)
-88.3 (4)
 0 (4P1/2)
 0 (4P3/2)
 2 (4P3/2)
606(6)
616(6)
-109(2)
614 (10)
-107 (2)
 0 (5P1/2)
 0 (5P3/2)
 2 (5P3/2)
807(14)
869(14)
-166(3)
810.6(6)
857 (10)
-163(3)
Excellent agreement with experiments !
606.7(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
v  0   2
MJ = ±3/2
v  0  2
MJ = ±1/2
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.
Magic wavelength for Cs
v  0   2
MJ = ±3/2
v  0  2
MJ = ±1/2
10000
6S1/2
6P3/2
 (a.u.)
8000
6000
magic
932 nm
Other*
938 nm
0+ 2
4000
2000
0
925
magic around 935nm
0- 2
930
935
940
(nm)
945
950
955
* Kimble et al. PRL 90(13), 133602(2003)
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 clocks
atomic clocks
black-body radiation ( BBR ) shift
Motivation:
BBR shift gives the larges uncertainties
for some of the optical atomic clock
schemes, such as with
Ca+
Blackbody radiation
shift in optical
frequency standard
43
+
with Ca ion
Bindiya Arora, M.S. Safronova, and Charles W. Clark,
Phys. Rev. A 76, 064501 (2007)
Motivation
For Ca+, the contribution from
Blackbody radiation gives
the largest uncertainty
to the frequency standard at T = 300K
DBBR = 0.39(0.27) Hz [1]
[1] C. Champenois et. al. Phys. Lett. A 331, 298 (2004)
Frequency standard
Level B
Clock
transition
Level A
T=0K
Transition frequency should be corrected to
account for the effect of the black body radiation at
T=300K.
Frequency standard
Level B
Clock
transition
DBBR
Level A
T = 300 K
Transition frequency should be corrected to
account for the effect of the black body radiation at
T=300K.
Why Ca+ ion?
The clock transition involved is
4s1/2F=4 MF=0 → 3d5/2 F=6 MF=0
Easily produced by
non-bulky solid state
o r 854
d i o dnm
e lasers
4p3/2
4p1/2
393 nm
866 nm
3d5/2
397 nm
3d3/2
732 nm
E2
Lifetime~1.2 s
729 nm
4s1/2
BBR shift of a level
• The temperature-dependent electric field
created by the blackbody radiation is described
by (in a.u.) :
3
8


d
2
E ( ) 
 exp( / kT )  1
• Frequency shift caused by this electric field is:
DvBBR   A    ( ) E 2 ( ) d
Dynamic polarizability
BBR shift and polarizability
BBR shift can be expressed in terms of a scalar
static polarizability:
4
1
 T ( K )  (1   )
DBBR    0 (0)(831.9V / m) 2 

2
300


Dynamic correction
Dynamic correction ~10-3 Hz.
At the present level of accuracy the dynamic
correction can be neglected.
Vector & tensor polarizability average
out due to the isotropic nature of field.
BBR shift for a transition
Effect on the frequency of clock transition
is calculated as the difference between the
BBR shifts of individual states.
DvBBR (4s1/ 2  3d5/ 2 )  DvBBR (3d5/ 2 )  DvBBR (4s1/ 2 )
3d5/2
729 nm
4s1/2
DvBBR   0 (0)
Need BBR shifts
Need ground and excited state
scalar static polarizability
2
n Dv
1
   

3(2 jv  1) n En  Ev
0
v
NOTE: Tensor polarizability calculated in
this work is also of experimental interest.
Contributions to the 4s1/2 scalar
3
polarizability ( a 0 )
43Ca+
(= 0)
Stail
6p3/2
6p1/2
0.01
0.01
5p3/2
0.06
5p1/2
0.01
0.01
4p1/2
4p3/2
24.4
48.4
3.3
Core
4s
Total: 76.1 ± 1.1
Contributions to the 3d5/2 scalar
3
polarizability ( a 0 )
43Ca+
nf7/2
nf5/2
np3/2 tail
5p3/2
4p3/2
0.2
1.7
7-12f7/2
6f7/2
0.5
0.3
0.01
0.8
0.01
Core
4f7/2
2.4
22.8
3.3
5f7/2
3d5/2
Total: 32.0 ± 1.1
Comparison of our results for
scalar static polarizabilities for
the 4s1/2 and 3d5/2 states of 43Ca+ ion
with other available results
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
[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)
Black body radiation shift
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).
D(4s1/2 → 3d5/2)
Present
Champenois
[1]
Kajita [2]
0.38(1)
0.39(27)
0.4
An order of magnitude
improvement is achieved with
comparison to previous
calculations
[1] C. Champenois et. al. Phys. Lett. A 331, 298 (2004)
[2] Masatoshi Kajit et. al. Phys. Rev. A 72, 043404, (2005)
Black body radiation shift
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).
D(4s1/2 → 3d5/2)
Present
Champenois
[1]
Kajita [2]
0.38(1)
0.39(27)
0.4
Sufficient accuracy to establish
The uncertainty limits for the
Ca+ scheme
[1] C. Champenois et. al. Phys. Lett. A 331, 298 (2004)
[2] Masatoshi Kajit et. al. Phys. Rev. A 72, 043404, (2005)
relativistic
All-order
method
Singly-ionized
ions
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
CI works for systems with many valence electrons
but can not accurately account for core-valence
and core-core correlations.
All-order method can account for core-core and
core-valence correlation can not accurately
describe valence-valence correlation.
Therefore, two methods are combined to
acquire benefits from both approaches.
CI + ALL-ORDER:
PRELIMINARY RESULTS
Ionization potentials, differences with experiment
CI
Mg
Ca
Sr
Ba
1.9%
4.1%
5.2%
6.4%
CI + MBPT
0.12%
0.6%
0.9%
1.7%
CI + All-order
0.03%
0.3%
0.3%
0.5%
Conclusion
•
•
•
Benchmark calculation of various polarizabilities
and tests of theory and experiment
Determination of magic wavelengths for stateinsensitive optical cooling and trapping
Accurate calculations of the BBR shifts
Future studies: Development of generally applicable
CI+ all-order method for more complicated systems
Conclusion
Parity Violation
Atomic
Clocks
P1/2
Future:
New Systems
New Methods,
New Problems
D5/2
„quantum
bit“
S1/2
Quantum information
Graduate students:
Bindiya Arora
Rupsi pal
Jenny Tchoukova
Dansha Jiang
P3.8 Jenny Tchoukova and M.S. Safronova
Theoretical study of the K, Rb, and Fr lifetimes
Q5.9 Dansha Jiang, Rupsi Pal, and M.S. Safronova
Third-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. Clark
State-insensitive two-color optical trapping