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ICCMSE 2009
RHODES, GREECE
October 30, 2009
Atomic Calculations for
Future Technology and
Study of Fundamental
Problems
Marianna Safronova
Outline
• Selected applications of atomic calculations
• Study of fundamental symmetries: parity violation
• Quantum information
• Atomic clocks
• Methods for high-precision atomic calculations
• Overview
• Computational challenges
• Evaluation of the uncertainty
• Development of the LCCSD + CI method
• Future prospects
study of Fundamental
symmetries
Parity violation studies with heavy atoms & search for
Electron electric-dipole moment
r ─r
Parity Violation
Parity-transformed world:
Turn the mirror image upside down.
The parity-transformed world is not identical
with the real world.
Parity is not
conserved.
Searches for New Physics
Beyond the Standard Model
High energies
(1) Search for new processes
or particles directly
(2) Study (very precisely!)
quantities which Standard Model
predicts and compare the result
with its prediction
Weak charge QW
Low energies
http://public.web.cern.ch/, Cs experiment, University of Colorado
The most precise measurement
of PNC amplitude (in cesium)
C.S. Wood et al. Science 275, 1759 (1997)
F=4
7s
F=3
2
1
6s
F=4
F=3
Im  EPNC 
0.3% accuracy
b
 1.6349(80) mV cm

 1.5576(77) mV cm
Stark interference scheme to measure ratio of the
PNC amplitude and the Stark-induced amplitude b
1
2
Parity violation
Nuclear
spin-independent
PNC:
Searches for new
physics
beyond the
Standard Model
e
q
e
Z0
q
Weak Charge QW
Nuclear
spin-dependent
PNC:
Study of PNC
In the nucleus
a
Nuclear anapole
moment
Analysis of CS PNC experiment
Nuclear
spin-independent
PNC
7s
Nuclear
spin-dependent
PNC
F=4
7s
F=3
2
1
6s
F=3
Difference of 1 & 2
Average of 1 & 2
Im  E
si
PNC
b
  1.5935(56)
F=4
6s
mV
Weak Charge QW

cm

sd

  Im EPNC
b


3443
 0.077(11) mV cm
Nuclear anapole moment
Spin-dependent parity violation:
Nuclear anapole moment
a
Parity-violating nuclear moment
F=4
7s
F=3
2
H
(a)
PNC

GF
2
 a α  I v ( r )
1
6s
F=4
F=3
Valence
nucleon
density
Anapole moment
Nuclear anapole moment is parity-odd, time-reversal-even
E1 moment of the electromagnetic current operator.
Constraints on nuclear weak
coupling contants
W. C. Haxton and C. E. Wieman, Ann. Rev. Nucl. Part. Sci. 51, 261 (2001)
Nuclear anapole moment:
test of hadronic weak interations
The constraints obtained from the Cs experiment
were found to be inconsistent with constraints
from other nuclear PNC measurements, which
favor a smaller value of the133Cs anapole moment.
All-order (LCCSD) calculation of spin-dependent PNC amplitude:
k = 0.107(16)* [ 1% theory accuracy ]
No significant difference with previous value k = 0.112(16) is found.
NEED NEW EXPERIMENTS!!!
*M.S. Safronova, Rupsi Pal, Dansha Jiang, M.G. Kozlov,
W.R. Johnson, and U.I. Safronova, Nuclear Physics A 827 (2009) 411c
Quantum 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).
Quantum Computer
(Innsbruck)
P1/2
D5/2
„quantum
bit“
S1/2
Quantum communication
Need to interconnect
flying and
stationary qubits
~100 µm
Optical dipole traps
(ac Stark shift)
Thin ion trap inside a cavity (Monroe/Chapman, Blatt)
problem
Atom in state A
sees potential UA
Atom in state B
sees potential UB
What is magic wavelength?
Atom in state A
sees potential UA
Atom in state B
sees potential UB
Magic wavelength magic is the wavelength for
which the optical potential U experienced
by an atom is independent on its state
U   ( )
Atomic polarizability
Locating magic wavelength
 magic
α 
S State
P State
wavelength
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 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, to appear
in Special Issue of IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control (2009).
motiation: next generation
Atomic clocks
Next - generation
ultra precise atomic clock
http://CPEPweb.org
Atoms trapped by laser light
The ability to develop more precise optical frequency
standards will open ways to improve global positioning
system (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.
applications
Parity Violation
P1/2
Atomic
Clocks
NEED
ATOMIC
PROPERTIES
D5/2
„quantum
bit“
S1/2
Quantum information
How to accurately calculate
atomic properties?
Very precise calculation of atomic properties
We also need to evaluate uncertainties of
theoretical values!
Atomic Properties
Magic wavelength
BBR shifts
Parity
nonconserving van der Waals
coefficients
amplitudes
Derived:
Weak charge QW,
Hyperfine
Anapole moment
constants
Lifetimes
Isotope
shifts
Line strengths
Oscillator
strengths
ac and dc
Polarizabilities
Energies
Quadrupole
moments
Transition
probabilities
and
others
...
Fine-structure
intervals
Electron
electric-dipole
moment
enhancement
factors
Branching ratios
BBR
shifts
Atom-wall
interaction
Wavelengths constants
Theory:
All-order method
(relativistic linearized coupledcluster approach)
Perturbation theory:
Correlation correction to ground state
energies of alkali-metal atoms
Linearized coupledcluster method
The linearized coupled-cluster method sums infinite
sets of many-body perturbation theory terms. The
wave function of the valence electron v is represented
as an expansion that includes all possible single,
double, and partial triple excitations.
Cs: atom with single (valence) electron
outside of a closed core.
1s22s22p63s23p63d104s24p64d105s25p66s
core
1s2…5p6
6s
valence
electron
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)
Triples excitations, non-linear
terms, extra perturbation
theory terms, …
Need for symbolic computing
1 2 (0)
H S2 | v : ai aj al ak : am an ar as ad ac ab aa av :| 0c 
2
800 terms!
The code was developed to implement Wick’s
theorem and simplify the resulting expressions.
Symbolic program for coupled-cluster
method & perturbation theory
Input: expression of the type
gijkl mnwa : ai a j al ak :: am an aa aw :
in ASCII format.
Output: simplified resulting formula in the LaTex format, ASCII
output is also generated.
Program features
1) The code is set to work with two or three normal products (all possible
cases) with large number of operators.
2) The code differentiates between different types of indices, i.e.
core (a,b,c,…), valence (v,w,x,y,...), excited orbitals (m,n,r,s,…), and
general case (i, j, k, l,…).
3) The operators are ordered as required in the same order for all terms.
gmnba rswc am an ar as aa ab ac aw
4) The expression is simplified to account for the identical
terms and symmetry rules,
gijkl  g jilk .
5) The direct and exchange terms are joined together,
gijkl  gijkl  gijlk .
Symbolic computing for program
generation
The resulting expressions that need to be evaluated
numerically contain very large number of terms, resulting in
tedious coding and debugging.
The symbolic program generator was developed for this
purpose to automatically generate efficient numerical codes for
coupled-cluster or perturbation theory terms.

 (1)
abcd k1k2 k3
J  k1  k2  k3  jc  jd  jw  jw '
J

k2

jc
jw '
jd   J

jv '  k3
ja
jw
jb   J

jv  k1
jc
jb
X k1 (cdab) X k2 (v ' w ' cd ) Z k3 (vwab)
 c   d   v '   w '   a   b   v '   w' 
jd 

ja 
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!
All-order method:
Correlation correction to ground state
energies of alkali-metal atoms
Results for alkali-metal
atoms: E1 matrix elements (a.u.)
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:
Theory
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)
M.S. Safronova, W.R. Johnson, and A. Derevianko,
Phys. Rev. A 60, 4476 (1999)
Static polarizabilities of np states
Theory [ 1 ]
Experiment*
Na
 0 (3P1/2)
 0 (3P3/2)
 2 (3P3/2)
K
 0 (4P1/2)
 0 (4P3/2)
 2 (4P3/2)
606(6)
616(6)
-109(2)
606.7(6)
614 (10)
-107 (2)
Rb
 0 (5P1/2)
 0 (5P3/2)
 2 (5P3/2)
807(14)
869(14)
-166(3)
810.6(6)
857 (10)
-163(3)
359.9(4)
361.6(4)
-88.4(10)
Excellent agreement with experiments !
359.2(6)
360.4(7)
-88.3 (4)
*Zhu et al. PRA
70 03733(2004)
[1] Bindiya Arora, M.S. Safronova, and Charles W. Clark, Phys. Rev. A 76, 052509 (2007)
Very brief summary of what we
calculated with this approach
Properties
Systems
• Energies
• Transition matrix elements (E1, E2, E3, M1)
Li, Na, Mg II, Al III,
• Static and dynamic polarizabilities & applications Si IV, P V, S VI, K,
Dipole (scalar and tensor)
Ca II, In, In-like ions,
Quadrupole, Octupole
Ga, Ga-like ions, Rb,
Light shifts
Cs, Ba II, Tl, Fr, Th IV,
Black-body radiation shifts
U V, other Fr-like ions,
Magic wavelengths
Ra II
• 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
http://www.physics.udel.edu/~msafrono
how to evaluate
uncertainty of
theoretical
calculations?
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
Example:
quadrupole moment of
3d5/2 state 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)
3D5/2 quadrupole moment in Ca+
Lowest order
2.451
3D5/2 quadrupole moment in Ca+
Lowest order
2.451
Third order
1.610
3D5/2 quadrupole moment in Ca+
Lowest order
2.451
Third order
1.610
All order (SD)
1.785
3D5/2 quadrupole moment in Ca+
Lowest order
2.451
Third order
1.610
All order (SD)
1.785
All order (SDpT)
1.837
3D5/2 quadrupole moment in Ca+
Lowest order
2.451
Third order
1.610
All order (SD)
1.785
All order (SDpT)
1.837
Coupled-cluster SD (CCSD)
1.822
3D5/2 quadrupole moment in Ca+
Lowest order
2.451
Third order
1.610
All order (SD)
1.785
All order (SDpT)
1.837
Coupled-cluster SD (CCSD)
1.822
Estimate
omitted
corrections
Final results: 3d5/2 quadrupole moment
Lowest order
2.454
Third order
1.849 (13)
1.610
All order (SD), scaled
All-order (CCSD), scaled
All order (SDpT)
All order (SDpT), scaled
1.849
1.851
1.837
1.836
Final results: 3d5/2 quadrupole moment
Lowest order
2.454
Third order
1.849 (13)
1.610
All order (SD), scaled
All-order (CCSD), scaled
All order (SDpT)
All order (SDpT), scaled
1.849
1.851
1.837
1.836
Experiment
1.83(1)
Experiment: C. F. Roos, M. Chwalla, K. Kim, M. Riebe, and R. Blatt, Nature 443, 316 (2006).
More complicated systems
relativistic
All-order
method
Singly-ionized
ions
Summary of theory methods for
PNC studies
• Configuration interaction (CI)
• Many-body perturbation theory
• Relativistic all-order method (coupled-cluster)
• Perturbation theory in the screened Coulomb
interaction (PTSCI), all-order approach
• Configuration interaction + second-order MBPT
• Configuration interaction + all-order method*
*under development
Configuration interaction
method
   ci i
H
Single-electron valence
basis states
i
eff
 E  0
1
r1  r2
Example: two particle system:
H
eff
 h1 (r1 )  h1 (r2 )  h2 (r1 , r2 )
onebody
part
twobody
part
Configuration interaction +
many-body perturbation
theory
CI works for systems with many valence electrons
but can not accurately account for core-valence
and core-core correlations.
MBPT 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
acquire benefits from both approaches.
Configuration interaction
method + MBPT
Heff is modified using perturbation theory expressions
h1  h1  1
h2  h2   2
1 ,  2
H
eff
 E  0
are obtained using perturbation theory
Problem:
(1) Accuracy deteriorates for heavier systems
owing to larger correlation corrections.
(2) Accuracy will not be ultimately sufficient.
Configuration interaction
+ all-order method
Heff is modified using all-order excitation coefficients
~
1 mn   n   m  mn
L
L
~
~
 2 mnkl   k   l   m   n  mnkl
Advantages: most complete treatment of the
correlations and applicable for many-valence
electron systems
CI + ALL-ORDER RESULTS
Two-electron binding energies, differences with experiment
Atom
Mg
Ca
Cd
Sr
Zn
Ba
Hg
CI
1.9%
4.1%
9.6%
5.2%
8.0%
6.4%
11.8% 2.4%
CI + MBPT
CI + All-order
0.12%
0.6%
1.0%
0.9%
0.9%
1.7%
0.03%
0.3%
0.02%
0.3%
0.4 %
0.5%
0.5%
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).
CI + ALL-ORDER: Mg energies
Experiment
CI
Dif(%)
CI+MBPT
Dif(%)
CI+allorder
Dif(%)
3s2
1S
0
182938
179536
1.9
182718
0.12
182880
0.03
3s4s
3S
1
41197
40399
1.9
41121
0.19
41162
0.08
3s4s
1S
0
43503
42662
1.9
43435
0.16
43478
0.06
3s3d
1D
2
46403
45117
2.8
46319
0.18
46373
0.06
3s3d
3D
1
47957
46967
2.1
47892
0.13
47944
0.03
3s3d
3D
2
47957
46967
2.1
47892
0.13
47941
0.03
3s3d
3D
3
47957
46967
2.1
47893
0.13
47936
0.04
3s3p
3P
0
21850
20905
4.3
21782
0.31
21837
0.06
3s3p
3P
1
21870
20926
4.3
21804
0.30
21856
0.06
3s3p
3P
2
21911
20966
4.3
21847
0.29
21901
0.04
3s3p
1P
1
35051
34488
1.6
35053
0.00
35068
-0.05
3s4p
3P
0
47841
46914
1.9
47766
0.16
47813
0.06
3s4p
3P
1
47844
46917
1.9
47769
0.16
47816
0.06
3s4p
3P
2
47851
46924
1.9
47776
0.16
47823
0.06
3s4p
1P
1
49347
48487
1.7
49290
0.12
49329
0.04
Cd energies, differences with experiment
Expt.
State
J
DIF(%)
DIF(%)
DIF(%)
CI
CI+MBPT
CI+All-order
5s2
1S
0
208915
10
-1.0
0.02
5s5p
3P°
0
30114
19
-3.2
-0.53
1
30656
19
-3.1
-0.40
2
31827
19
-3.1
-0.46
5s5p
1P°
1
43692
11
-1.0
-0.09
5s6s
3S
1
51484
14
-1.6
-0.49
5s6s
1S
0
53310
13
-1.4
-0.35
5s5d
1D
2
59220
14
-1.5
-0.24
5s5d
3D
1
59486
14
-1.4
-0.22
2
59498
14
-1.4
-0.22
3
59516
14
-1.4
-0.22
Cd, Zn, and Sr Polarizabilities,
preliminary results (a.u.)
Zn
CI
CI+MBPT
CI+All-order
4s2 1S0
44.13
37.22
37.02
4s4p 3P0
75.94
66.20
64.97
CI
CI+MBPT
CI+All-order
5s2 1S0
52.66
41.50
42.11
5s5p 3P0
86.94
70.72
70.72
CI+ All-order
Recomm.*
197.4
197.2
Cd
Sr
5s2 1S0
*From expt. matrix elements, S. G. Porsev and A. Derevianko, PRA 74, 020502R (2006).
Conclusion
Parity Violation
Atomic
Clocks
P1/2
Future:
New Systems
New Methods,
New Problems
D5/2
„quantum
bit“
S1/2
Quantum information
Graduate
students:
Rupsi Pal
Dansha Jiang
Bindiya Arora
Jenny Tchoukova
Other collaborations:
Michael Kozlov (PNPI, Russia)
(Visiting research scholar at the University of Delaware)
Walter Johnson (University of Notre Dame), Charles Clark (NIST)
Ulyana Safronova (University of Nevada-Reno)