April 5, 2008
A Symposium on Atomic Physics:
A Tribute to Walter Johnson
APPLICATIONS
ALL
APPLICATIONS OF
OF THE
THE ALLALLORDER
ALL--ORDER
METHOD:
METHOD: FROM
FROM PARITY
PARITY VIOLATION
VIOLATION
TO
TO ATOMIC
ATOMIC CLOCKS
CLOCKS
Marianna Safronova
UDRF
INTRODUCTION
INTRODUCTION OF
OF THE
THE
ALLALL
ALLALL--ORDER
ORDER METHOD
METHOD
HIGHHIGH-PRECISION CALCULATION OF THE ATOMIC PROPERTIES
MOTIVATION
MOTIVATION
Benchmark problems for understanding of atomic properties:
Tests of theory and experiment
MOTIVATION:
MOTIVATION: PNC
PNC
Parity violation studies with heavy atoms
http://CPEPweb.org, http://public.web.cern.ch/, Cs experiment, University of Colorado
WHY
ALL
WHY ALLALLORDER?
ALL--ORDER?
alkali--metal atoms
Correlation correction to ground state energies of alkali
Second order
4 terms
Third order
56 terms
RELATIVISTIC
ALL
RELATIVISTIC ALLALLORDER METHOD
METHOD
ALL--ORDER
Sum over infinite sets of many-body perturbation theory
(MBPT) terms.
Calculate the atomic wave functions and energies
Scheme:
Calculate various matrix elements
H Ψv = E Ψv
Calculate “derived” properties
useful for particular problems
RELATIVISTIC
ALL
RELATIVISTIC ALLALLORDER METHOD
METHOD
ALL--ORDER
Sum over infinite sets of many-body perturbation theory
(MBPT) terms.
Calculate the atomic wave functions and energies
Scheme:
Calculate various matrix elements
H Ψv = E Ψv
Calculate “derived” properties
such as PNC ampliudes
LOWESTLOWEST
LOWESTORDER ATOMIC
ATOMIC WAVE
WAVE FUNCTION
FUNCTION
LOWEST--ORDER
Cs: atom with single (valence)
electron outside of a closed core.
core
Lowest-order
wave function
1s2…5p6
Ψ
(0)
v
valence
electron
6s
†
= av
Creation operator
for state v
Ψ
core
Core
Core wave function
ALLALL
ALLORDER ATOMIC
ATOMIC WAVE
WAVE FUNCTION
FUNCTION (SD)
(SD)
ALL--ORDER
Lowest order
Core
core
valence electron
any excited orbital
Ψ v(0)
Single-particle
excitations
∑
ma
ρ ma am† aa Ψ v(0)
†
(0)
ρ
a
a
Ψ
∑ mv m v v
m≠v
Double-particle
excitations
1
2
∑
mnab
ρ mnab am† an† ab aa Ψ v(0)
∑
mna
ρ mnva am† an† aa av Ψ v(0)
Actual
Actual implementation:
implementation: Problem
Problem 11
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
ρ
:
a
a
a
a
:
:
a
a
a
a
a
a
:
Ψ
∑ 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
Actual
Actual implementation:
implementation: Problem
Problem 22
PROBLEM A WITH EQUATIONS:
There are some many of them !!!
ρmnab
Cs: a,b = 1s22s22p63s23p63d104s24p64d105s25p6
m,n : finite basis set = (35 × 13) × (35 × 13)
Total actually 15412 × 35 × 35 ~ 19 000 000 equations
to be solved iteratively!
Memory & storage of
ρmnab
: it is a really large file!
Actual
Actual implementation:
implementation: Problem
Problem 33
PROBLEM B WITH EQUATIONS:
These are really complicated equations !!!
• “Quadruple” term:
∑ gmnrs ρrsab
rs
Indices mnrs can be ANY orbitals
Basis set: nmax=35, lmax=6
17x17x(35x13)4=5 x 1012!
• Program has to be exceptionally efficient!
a,b core
(17 shells)
ALLALL
ALLORDER RESULTS
RESULTS
ALL--ORDER
alkali--metal atoms
Correlation correction to ground state energies of alkali
Second order
4 terms
All order
Third order
56 terms
M. S. Safronova, W. R. Johnson, and A. Derevianko, Phys. Rev. A 60, 4476 (1999)
RESULTS
ALKALI
RESULTS FOR
FOR ALKALIALKALIMETAL
ALKALI--METAL
ATOMS:
ATOMS: E1
E1 MATRIX
MATRIX ELEMENTS
ELEMENTS
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)
ATOMIC
ATOMIC PROPERTIES
PROPERTIES
Magic wavelength
BBR shifts
and
others
...
Parity
Fine-structure
nonconserving van der Waals
intervals
coefficients
amplitudes
Derived:
Electron
Weak charge QW,
electric-dipole
Hyperfine
Anapole moment
moment
constants
enhancement
Isotope
Energies
factors
shifts
Line strengths
Lifetimes
Transition
probabilities
Oscillator
strengths
ac and dc
Polarizabilities
Branching ratios
Wavelengths
Derived:
Fr nuclear magnetic
moment
Atom-wall
interaction
constants
OTHER
OTHER APPLICATIONS:
APPLICATIONS:
QAUNTUM
QAUNTUM INFORMATION
INFORMATION
Quantum communication, cryptography and quantum information processing
Need calculations of atomic properties
Optimizing the fast Rydberg quantum gate, M.S. Safronova, C. J. Williams, and C. W. Clark,
Phys. Rev. A 67, 040303 (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).
ATOMIC
ATOMIC CLOCKS
CLOCKS
Microwave
Transitions
High-accuracy calculation of black-body radiation shift in 133Cs primary
frequency standard", K. Beloy, U. I. Safronova and A. Derevianko,
Phys. Rev. Lett. 97, 040801 (2006).
Blackbody radiation shift in a 43Ca+ ion optical frequency standard,
Bindiya Arora, M.S. Safronova, and Charles W. Clark, Phys. Rev. A 76, 064501 (2007).
Optical
Transitions
EXTENSIONS
EXTENSIONS OF
OF THE
THE ALL
ALL ORDER
ORDER
METHOD
METHOD
Add more terms to the all order wave-function
Non-linear terms
Triple excitations
Non-linear terms:
R. Pal, M.S. Safronova, W.R. Johnson, A. Derevianko, S. G. Porsev,
Phys. Rev. A 75, 042515 (2007)
Triple excitations:
S. G. Porsev and A. Derevianko, Phys. Rev. A 73, 012501 (2006) (Na)
A. Derevianko and S. G. Porsev, Eur. Phys. J. A 32 (4), 517(2007) (Cs)
E. Iskrenova-Tchoukova and M.S. Safronova, in progress
Non-linear
Non-linear terms
terms
1 2
S2
2
Contract operators by Wick’s theorem
1 2 (0)
H S2 | Ψ v >→: ai+ a +j al ak : am+ an+ ar+ as+ ad ac ab aa av+ :| 0c >
2
800 terms!
Problem
Problem with
with all-order
all-order extensions:
extensions:
TOO
TOO MANY
MANY TERMS
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:
Solution: automated
automated code
code generation
generation !!
Automated
Automated code
code generation
generation
Codes
Codesthat
thatwrite
writeformulas
formulas
Codes
Codesthat
thatwrite
writecodes
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!
RELATIVISTIC
ALL-ORDER
METHOD
Singly-ionized
ions
FUTURE
FUTURE WORK:
WORK:
CONFIGURATION
CONFIGURATION INTERACTION
INTERACTION
++
ALLALL
ALLORDER METHOD
METHOD
ALL--ORDER
*under development
CONFIGURATION
CONFIGURATION INTERACTION
INTERACTION ++
ALLALL
ALLORDER METHOD
METHOD
ALL--ORDER
CI works for systems with many valence electrons
but can not accurately account for core-valence
and core-core correlations.
All-order method can not accurately describe
valence-valence correlation.
Therefore, two methods are combined to
acquire benefits from both approaches.
CI
ALL
CI ++ ALLALLORDER:
ALL--ORDER:
PRELIMINARY
PRELIMINARY RESULTS
RESULTS
Mg
Experiment
IP
182939
3s3p
3P
3s3p
3s4s
3s4s
3s3d
1P
3S
1S
1D
J=0
J=1
J=2
J=1
J=1
J=0
J=2
21850
21870
21911
35051
41197
43503
46403
Ionization potentials
Ca
CI
-4.1%
Ba
-6.4%
CI
DIF
CI+II
DIF
CI+ALL
DIF
179525
3414
182673
266
182848
91
20899
20919
20960
34486
40392
42664
45108
951
951
951
565
805
839
1295
21764
21785
21829
35048
41110
43428
46296
86
85
82
3
87
75
107
21824
21843
21888
35061
41151
43486
46367
CI+II
0.6%
1.7%
26
27
23
-10
46
17
36
CI+All-order
0.3%
0.5%
M.S. Safronova, M. Kozlov, and W.R. Johnson, in preparation
CONCLUSION
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 CHANDRA
JENNY TCHOUKOVA
DANSHA JIANG
COLLABORATORS:
COLLABORATORS:
Walter Johnson (University of Notre Dame)
Ulyana Safronova (University of Nevada-Reno)
Michael Kozlov (Petersburg Nuclear Physics Institute)
Andrei Derevianko (University of Nevada-Reno)
Victor Flambaum (UNSW)
Vladimir Dzuba (UNSW)
Sergey Porsev (PNPI, University of Nevada-Reno)
Luis Orozco (University of Maryland)
Carl Williams (NIST)
Charles Clarks (NIST)
TO WALTER:
THANK YOU!