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Bhanu Pratap Das
Theoretical Physics and Astrophysics GroupIndian Institute of Astrophysics
Bangalore
Collaborators:H. S. Nataraj, B. K. Sahoo, D. Mukherjee, M. Nayak,M. Kallay, M. Abe, G. Gopakumar and M. Hada
PCPV 2013, Mahabaleshwar22 Feb, 2013
Theory of Electric Dipole Moments of Atoms and
Molecules
Outline of the talk
➢ General features of EDMs and relationship to the Standard Model
➢ Relationship between the electron EDM and atomic and molecular EDMs.
➢ Need for a relativistic many-body theory of atomic and molecular EDMs
➢ Future improvements in atomic and molecular theory of EDMs
Permanent EDM of a particle VIOLATES both P - & T – invariance.
T-violation implies CP-violation via CPT theorem.
⟨ ∣ D ∣ ⟩ = c ⟨ ∣ J ∣ ⟩
⇒ D = 0
EDM and Degeneracy :
D = ⟨∣e r∣ ⟩ ≠ 0
Consider the degeneracy of opposite parity states in a physical system
∣ ⟩ = a ∣e⟩ b ∣
o⟩
EDM can be nonzero for degenerate states.
P and T violations in non-degenerate systems implies nonzero EDM.
Sources of Atomic EDM
Elementary Coupling
Particles Nucleon Nucleus constant Atomic e (de) de Da (open shell)
Cs Da (open shell)
e-q e-n e-N
CT Da (closed shell)
q (dq) dn dN Q Da (closed shell)
q-q dn, n-n dN Q Da (closed shell)
Standard Model < 10-38
Super-symmetric Model 10-24 – 10-28
Left-Right Symmetric Model 10-25 – 10-30
Multi-Higgs Model 10-25 - 10-29
Particle Physics Model Electron EDM (e-cm)
ATOMIC EDM DUE TO THE ELECTRON EDM
( NON-RELATIVISTIC CASE )
The interaction between the electron spin and internal electric field exerted by the nucleus and the other electrons gives,
The total atomic Hamiltonian is then,
H = ∑i
{p
i2
2m−
Z e
ri
} ∑i j
e2
rij
− de∑
i
i⋅E
iI
−de⋅E I
E I = −∇ {∑iV
Nr
i ∑i j
VCr
ij}where,
HO= ∑
i
{p
i2
2m−
Z e
ri
} ∑i j
e2
rij
; H / =−de ∑
i
i⋅E
iI
; HO∣
O ⟩ = EO∣
O ⟩
H = HO
H /Using perturbation theory :
When there is an external electric field, induced electric dipole moment arises.
The induced electric dipole moment of an atom is given by
The atomic EDM is
Using perturbation theory
As de is small, determ can be neglected.
er
Da = ∑i{d e i e r i}
⟨ Da⟩ = ⟨ ∣ Da ∣ ⟩
∣ ⟩ =∣O⟩ d e∣
1⟩ d e
2∣
2⟩ ⋯
D1DO
Assume, the applied field is in the z direction
Is even under parity and is odd under parity
⟨ Da⟩ = d
e⟨
O∣∑i
z i
∣ O⟩ d
ee{⟨
O ∣∑iz i
∣ 1 ⟩ ⟨
1 ∣∑iz i
∣ 0 ⟩ }
∣ 1
⟩ = ∑I≠
∣ IO⟩
⟨IO ∣ H / ∣
O ⟩
EO−E
IO
H /=−d
e⋅E
i
From the Time-independent Non-degenerate perturbation theory, we have,
and are of opposite parity, then the non-
vanishing terms of the EDM are:
∣ O⟩ ∣
1⟩
⟨D1⟩ =−d
e⟨
O∣∑i
z
i
∣ O⟩⟨DO
⟩ = de⟨
O∣∑i
z
i
∣ O⟩
⟨ Da⟩ = ⟨ DO
⟩ ⟨ D1⟩
Hence, in the non-relativistic scenario, even though the electron is assumed to have a EDM, when all the interactions in the atom are considered, the total atomic EDM becomes zero.
⟨ Da⟩ = 0 ( Sandars 1968 )
D1DO
⟨ Da⟩ = d
e⟨
O∣∑i
i
zi
∣ O⟩ d
ee {⟨
O∣∑i
zi∣
1 ⟩ ⟨1∣∑i
zi∣
0 ⟩ }
H = ∑i
{ci⋅pi i m c2 −Z er i
}∑i j
e2
r ij
− de∑i
i i⋅EiI
The total atomic Hamiltonian, including intrinsic electron EDM is,
The expectation value of atomic EDM in the presence of
applied electric field is given by,
ATOMIC EDM DUE TO THE ELECTRON EDM
( RELATIVISTIC CASE )
H /H0
⟨Da⟩ ≠ 0
⟨ Da ⟩ =2 c de
ℏ ∑I≠
[⟨
O ∣z ∣ IO ⟩ ⟨ I
O ∣ i 5 p2 ∣ O ⟩
EO
− E IO h.c.]
Finally, the expression for Atomic EDM reduces to,
Sandars (1968) and Das (1988)
R =< Da> / de : is the enhancement factor
Effective H EDM=2icde
ℏβγ5 p2
: Relativistic
E=−⟨D a . E ext ⟩=−R E extd eEnergy Shift
Effective field seen by an electron in an atom = R Eext
Effective field in certain molecules can be several orders of magnitude larger than in an atom
Ha=∑
i
{c i⋅p
i
im c2
VNr
i} ∑
i j
e2
rij
The relativistic atomic Hamiltonian is,
Theory of Atomic EDMs
Treating HEDM as a first-order perturbation, the atomic wave function is given by
∣Ψ ⟩ = ∣Ψ(0)
⟩ + de ∣Ψ(1)
⟩
The atomic EDM is given by Da =⟨Ψ∣ D ∣Ψ ⟩
⟨Ψ∣Ψ⟩
R=Da
d e
=⟨Ψ
(0)∣ D ∣Ψ
(1)⟩ + ⟨Ψ
(1)∣ D ∣Ψ
(0)⟩
⟨Ψ(0)
∣Ψ(0)
⟩This ratio, known as the enhancement factor, is calculated by relativistic many-body theory.
Unique many-body problem involving the interplay of the long range Coulomb interaction and short range P- and T-violating interactions.
Accuracy depends on precision to which ∣ 0
⟩ ∣ 1
⟩ are calculated.and
Relativistic Wavefunctions of Atoms
Atoms of interest for EDM studies are relativistic many-body systems;
Wavefunctions of these atoms can be written in the mean field approximation
∣0⟩ = Det {
1
2⋯
N} (Relativistic Dirac-Fock
wavefunction)
∣0⟩ T
1∣
0⟩ T
2∣
0⟩
T1= ∑
a , p
tap
a †pa
aT 2 = ∑
a ,b , p ,qtabpq a †p a †q ab aa T = T
1 T
2 ⋯
∣ 0 ⟩ = exp T ∣0⟩Relativistic Coupled-cluster (CC)
wavefunction;
H0− E
0∣ 1 ⟩ = − H
PTV∣ 0 ⟩First-order EDM Perturbed RCC wfn.
satisfies :
CC wfn. has electron correlation to all-orders of perturbation theory for any level of excitation.
∣ ⟩ = ∣ 0 ⟩ d e∣1 ⟩ = exp {T de T 1}∣0 ⟩In presence of EDM,
EDM enhancement factor in the RCC method
∣Ψ(0)
⟩=eT( 0 )
{1+ Sv(0)
}∣Φ⟩
R =Da
d e
=⟨
0∣D∣
1⟩⟨
1∣D∣
0⟩
⟨ 0 ∣ 0 ⟩
=⟨∣DSv
1DT 1 DT 1 Sv0 +Sv
0 † DSv 1+Sv
0 † DT 1 +Sv0 † DT 1 Sv
0 ∣⟩+h .c .
⟨0
∣0
⟩
Unperturbed RCC wave function:
EDM enhancement factor:
∣ ⟩=eT 0 deT
1
{1+S v 0
de Sv 1}∣ ⟩
Perturbed RCC wave function:
D =eT (0 ) †
DeT (0)
where
∣Ψ ⟩=∣Ψ(0 )
+ de∣Ψ(1)
⟩
Da=⟨Ψ∣D∣Ψ ⟩
⟨Ψ∣Ψ ⟩
DT 1
D Sv1
Sv0 † DT 1
DT 1Sv0
Sv0 † D Sv
1
Sv0 † DT 1Sv
0
5.18
122.21
94.19DF
0.53
-0.01
-7.34
-0.05
Total 120.53 124*
RCCSD(T) term
Cs EDM enhanc- ment factor
Tl EDM enhanc-ment factor
-422.02
-333.33
-101.07
-24.82
-7.12
-4.26
-0.56
-466.31 -582* -585** -573***
Cs: Nataraj et al., Phys. Rev. Lett. (2008) *Dzuba and Flambaum PRA (2009)Tl: Nataraj et al,. Phys. Rev. Lett. (2011) **Liu and Kelly PRA(1992)
***Porsev et al PRL(2012)
T and S are core and valence excitation operators.
The measured value of Da in combination with the calculated value of Da/d
e will give d
e .
From Tl EDM experiment (Regan et al, PRL 2002) and theory (Nataraj et al, PRL 2011) :
de< 2.0 X 10-27 e-cm (90% confidence limit)
This is a new upper limit for the electron EDM
Most recent new limit from YbF: de< 1.0 X 10-27 e-cm (90% confidence limit) Hudson et al, Nature, 2011
New Electron EDM limit
Ongoing EDM Experiments and Theory Using Paramagnetic Atoms
Improved accuracies in experiments and relativistic many-body theory for de might be possible in the future.
Rb: Weiss, Penn State
Cs: Gould, LBNL ; Heinzen, UT, Austin; Weiss, Penn State
Fr: Sakemi, Tohoku
Ra*: Jungmann, KVI, Netherlands
Theory :Theory : Flambaum, UNSW, Sydney ; Porsev and Kozlov, St. Petersburg, State Univ.; Safronova, U of Delaware; Sahoo, PRL, Ahmedabad; Nataraj, IIT Roorkee, Das, IIA, Bangalore
Experiments :Experiments :
Molecular EDMs
The shift in energy is given by:
The effective electric field in certain molecules interacting with the electron EDM can be several orders of magnitude larger than those in atoms. It can be expressed as
Some of the current molecular EDM experiments that are underway are :
YbF : Hinds, Imperial College, London
PbO * and ThO : DeMille, Yale, Doyle and Gabrielse, Harvard
HfF + : Cornell, JILA, Colorado
The sensitivities of these experiments could be 2-3 orders of magnitude better than that of the best electron EDM limit from atomic Tl.
Calculations of the effective fields in molecules are currently in their infancy.
H = Hm − de∑i
i i⋅EiI
Δ E=− ⟨Ψm∣d e∑i
βiσ i⋅EiI∣Ψm⟩
Δ Ede
= − ⟨Ψm∣∑i
βi σ i⋅EiI∣Ψm ⟩= −2ic ⟨Ψm∣βγ5 p2
∣Ψm ⟩
Calculations of effective fields in molecules using Coupled Cluster Theory
∣Ψ ⟩=eS∣Φ0 ⟩ where S=S1+ S2+ ... S1=∑
a , p
Sapa p
† aa; S2= ∑ab , pq
Sabpq a p
† aq† ab aaand
⟨ Ψ∣=⟨Φ0∣S e−S where S=1+ S1+ S2+ ... and S1=∑a , p
Sapaa
†a p; S2= ∑ab , pq
Sabpq aa
† ab† aqa p
S , S amplitudes are solved using suitable equations :
⟨ A ⟩=⟨ Ψ∣A∣Ψ ⟩
⟨ Ψ∣Ψ ⟩=⟨ Ψ∣A∣Ψ ⟩=⟨Φ0∣S e−S A eS
∣Φ0 ⟩
The effective field can be expressed as an expectation value as mentioned in the previous slide.
Expectation Values in CC Theory
H∣Ψ ⟩=E∣Ψ ⟩
⟨ Ψ∣H=⟨ Ψ∣E
For molecular EDMs, A = 2icdeβγ5p2
Future : Extended Coupled Cluster Method
⟨Ψ∣=⟨Φ0∣eS †
Normal Coupled Cluster Method
Cs, Fr, YbF, HfF+, ThO
2010
Limits on de : Past, Present and Future
Conclusions
Atomic and molecular EDMs arising the electron EDM could serve as excellent probes of physics beyond the standard model and shed lighton CP violation.
Relativistic many-body theory plays a crucial role in determining an upper limit for the electron EDM
The current best electron EDM limits come from Tl and YbF
Several Atomic ( Rb, Cs, Fr, etc. ) and Molecular ( YbF, HfF+, ThO, etc ) EDM experiments are underway. Results of some of these experiments could in combination with relativistic many-body calculations improve the limit for the electron EDM.
Aside
. . . Dirac - Fock Theory
For a relativistic N-particle system, we have a Dirac-Fock equation given by,
H0 =∑I
{c I⋅p
I
I−1 m c2
VNr
I} ∑
I J
e2
rIJ
The single particle wave functions ’s expressed in Dirac form as,
0=
1
N ! ∣
1x
1
1x
2
1 x
3 ⋯
1x
N
2x
1
2x
2
2x
3 ⋯
2x
N
⋯ ⋯ ⋯ ⋯ ⋯
Nx
1
Nx
2
Nx
3 ⋯
Nx
N∣
We represent the ground state wave function as an N×N Slater determinant,
a=
1
r Par
a, m
a
iQar
−a,m
a
METHOD OF CALCULATION
∣ 0 ⟩ = eT 0
∣ 0 ⟩
∣v⟩ = eT 0
{1S0}∣v⟩
. . . Coupled Cluster Theory
The coupled cluster wave function for a closed shell atom is given by,
Since the system considered here has only one valence electron, it reduces to
T 0= T 1
0 T 2
0 ⋯ S0
= S10
S20
⋯Where, and
The RCC operator amplitudes can be solved in two steps; first we solve for
closed shell amplitudes using the following equations:
H 0 = e−T 0
H 0 eT 0
Where,
⟨0∣ H
0∣
0⟩ = E
g⟨0
∗ ∣ H0 ∣ 0 ⟩ = 0and
The total atomic Hamiltonian in the presence of EDM as a perturbation is given by,
∣v⟩ = e
T 0 d
eT 1
{1 S0 d
eS1}∣
v⟩
The effective ( one-body ) perturbed EDM operator is given by,
⟨v∣ H
op{1S
v0}∣
v⟩ = − E
v
H = H 0 H EDM
Thus, the modified atomic wave function is given by,
H EDMeff = 2 i c d e 5 p2
⟨v∗ ∣ H
op{1S
v0}∣
v⟩ = − E
v⟨
v∗ ∣ {S
v0 }∣
v⟩
The open shell operators can be obtained by solving the following two equations :
Where, is the negative of the ionization potential of the valence electron v.
Ev
⟨0∗ ∣ H
N0 T 1 H
EDMeff ∣
0⟩ = 0
⟨v∗ ∣ H
N0 − E
v S
v1 H
N0 T 1 H
EDMeff {1 S
v0 }∣
v⟩ = 0
The perturbed cluster amplitudes can be obtained by solving the following
equations self consistently :
⟨Da⟩ =
⟨v∣ D
a∣
v⟩
⟨v∣
v⟩
The atomic EDM is given by,
HN
= H0− ⟨
0∣ H
0∣
0⟩Where,
EXPERIMENTS ON ATOMIC EDM
. . . Principle of Measurement
If the atomic EDM Da ~ 10-26 e-cm and E = 105 V/cm; ∆ ~ 10-5 Hz
Major source of error:
HI= − D
a⋅E − ⋅B
2
EB1 =
2⋅B 2 Da⋅E
ℏ
2 =2⋅B − 2 D
a⋅E
ℏ
= 1 − 2 =4 D
a⋅E
ℏ1
EB
Bm
=v×E
c2