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