Theory of Electric Dipole Moments of Atoms and Molecules Bhanu Pratap Das

Transcription

1 Theory of Electric Dipole Moments of Atoms and Molecules Bhanu Pratap Das Theoretical Physics and Astrophysics Group Indian 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, Mahabaleshwar 22 Feb, 2013

2 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

3 D = c J D = 0 Permanent EDM of a particle VIOLATES both P - & T invariance. T-violation implies CP-violation via CPT theorem.

4 EDM and Degeneracy : Consider the degeneracy of opposite parity states in a physical system = a e b o D = e r 0 EDM can be nonzero for degenerate states. P and T violations in non-degenerate systems implies nonzero EDM.

5 Sources of Atomic EDM Elementary Coupling Particles Nucleon Nucleus constant Atomic e (d e ) d e D a (open shell) C s D a (open shell) e-q e-n e-n C T D a (closed shell) q (d q) d n d N Q D a (closed shell) q-q d n, n-n d N Q D a (closed shell)

6 Particle Physics Model Electron EDM (e-cm) Standard Model < Super-symmetric Model Left-Right Symmetric Model Multi-Higgs Model

7 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, d e E I where, E I = { i V N r i i j V C r ij } The total atomic Hamiltonian is then, H = i 2 { p i 2m Z e r i } i j e 2 r ij d e i i E i I Using perturbation theory : H = H O H / H O = i 2 { p i 2m Z e r i } i j e 2 r ij ; H / = d e i i E i I ; H O O = E O O

8 When there is an external electric field, induced electric dipole moment arises. The induced electric dipole moment of an atom is given by e r The atomic EDM is D a = i {d e i e r i } Using perturbation theory D a = D a As d e is small, d e term can be neglected. = O d e 1 d e 2 2

9 Assume, the applied field is in the z direction Is even under parity and is odd under parity and O 1 are of opposite parity, then the nonvanishing terms of the EDM are: D a = d e O i zi O d e e{ O i z i 1 1 i z i 0 } D O D 1 From the Time-independent Non-degenerate perturbation theory, we have, 1 = O I O H / O I I E O O E I H / = d e E i

10 D a = D O D 1 D O = d e O i z i O D 1 = d e O i zi O D a = 0 ( Sandars 1968 ) 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.

11 ATOMIC EDM DUE TO THE ELECTRON EDM ( RELATIVISTIC CASE ) The total atomic Hamiltonian, including intrinsic electron EDM is, H = i {c i p i i m c 2 Z e H 0 r i } i j e 2 r ij d e i i i E i I The expectation value of atomic EDM in the presence of applied electric field is given by, H / D a = d e O i i zi O d e e{ O i z i 1 1 i z i 0 } D O D 1

12 Finally, the expression for Atomic EDM reduces to, D a = 2 c d e ħ I [ O z O I O I i 5 p 2 O h.c.] E O O E I D a 0 : Relativistic Sandars (1968) and Das (1988) Effective H EDM = 2icd e ħ β γ 5 p 2 R =< D a > / d e : is the enhancement factor Energy Shift E= D a. E ext = R E ext d e Effective field seen by an electron in an atom = R E ext Effective field in certain molecules can be several orders of magnitude larger than in an atom

13 H a = i Theory of Atomic EDMs The relativistic atomic Hamiltonian is, {c i p i i m c 2 V N r i } i j Treating H EDM as a first-order perturbation, the atomic wave function is given by Ψ = Ψ (0) + d e Ψ (1) The atomic EDM is given by D a = Ψ D Ψ Ψ Ψ R= D a 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. and Accuracy depends on precision to which 0 e 2 r ij 1 are calculated.

14 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 T 1 = a, p t ap a p a a T 2 = a,b, p,q t abpq a p a q a b a a T = T 1 T 2 Relativistic Coupled-cluster (CC) wavefunction; 0 = exp T 0 CC wfn. has electron correlation to all-orders of perturbation theory for any level of excitation. In presence of EDM, = 0 d e 1 = exp {T d e T 1 } 0 First-order EDM Perturbed RCC wfn. satisfies : H 0 E 0 1 = H PTV 0

15 EDM enhancement factor in the RCC method Unperturbed RCC wave function: Ψ ( 0) =e T ( 0 ) {1+ S v (0) } Φ Perturbed RCC wave function: EDM enhancement factor: D a = Ψ D Ψ Ψ Ψ =e T 0 d e T 1 0 { 1+S 1 v d e S v } Ψ = Ψ (0 ) + d e Ψ (1) R = D a d e = 0 D 1 1 D = D S v 1 D T 1 D T 1 S v 0 +S v 0 D S v 1 +S v 0 DT 1 +S v 0 D T 1 S v 0 +h. c. 0 0 where D =e T (0 ) De T ( 0)

16 RCCSD(T) term DF D T 1 D S v 1 S v 0 D T 1 Cs EDM enhancment factor Tl EDM enhancment factor D T 1 S v 0 S v 0 D S v T and S are core and valence excitation operators. S v 0 D T 1 S v Total * * -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)

17 New Electron EDM limit The measured value of D a in combination with the calculated value of D a /d e will give d e. From Tl EDM experiment (Regan et al, PRL 2002) and theory (Nataraj et al, PRL 2011) : d e < 2.0 X e-cm (90% confidence limit) This is a new upper limit for the electron EDM Most recent new limit from YbF: d e < 1.0 X e-cm (90% confidence limit) Hudson et al, Nature, 2011

18 Experiments : Ongoing EDM Experiments and Theory Using Paramagnetic Atoms Rb: Cs: Fr: Ra*: Weiss, Penn State Gould, LBNL ; Heinzen, UT, Austin; Weiss, Penn State Sakemi, Tohoku Jungmann, KVI, Netherlands Theory : Flambaum, UNSW, Sydney ; Porsev and Kozlov, St. Petersburg, State Univ.; Safronova, U of Delaware; Sahoo, PRL, Ahmedabad; Nataraj, IIT Roorkee, Das, IIA, Bangalore Improved accuracies in experiments and relativistic many-body theory for d e might be possible in the future.

19 The shift in energy is given by: Molecular EDMs 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 Δ E d e = Ψ m i 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 H = H m d e i i i E i I Δ E= Ψ m d e β i σ i E I i Ψ m 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. i β i σ i E i I Ψ m = 2ic Ψ m β γ 5 p 2 Ψ m

20 Calculations of effective fields in molecules using Coupled Cluster Theory The effective field can be expressed as an expectation value as mentioned in the previous slide. Expectation Values in CC Theory Normal Coupled Cluster Method H Ψ = E Ψ Ψ H = Ψ E Ψ =e S Φ 0 where S=S 1 + S and S 1 = Ψ = Φ 0 e S Ψ = Φ 0 S e S where S=1+ S 1 + S and S 1 = a, p S, S amplitudes are solved using suitable equations : A = Ψ A Ψ Ψ Ψ = Ψ A Ψ = Φ 0 S e S A e S Φ 0 For molecular EDMs, A = 2icd e βγ 5 p 2 Future : Extended Coupled Cluster Method a, p S a p a p a a ; S 2 = ab, pq S p a a a a p ; S 2 = ab, pq S pq ab a p a q a b a a S pq ab a a a b a q a p

21 Limits on d e : Past, Present and Future Cs, Fr, YbF, HfF +, ThO 2010

22 Conclusions Atomic and molecular EDMs arising the electron EDM could serve as excellent probes of physics beyond the standard model and shed light on 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.

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26 METHOD OF CALCULATION... Dirac - Fock Theory For a relativistic N-particle system, we have a Dirac-Fock equation given by, H 0 = I {c I p I I 1 m c 2 V N r I } I J We represent the ground state wave function as an N N Slater determinant, x1 1 x2 1 x3 1 x N 0 = 1 2 x 1 2 x 2 2 x 3 2 x N N! N x 1 N x 2 N x 3 N x N 1 The single particle wave functions s expressed in Dirac form as, a = 1 r P a r a, m a iq a r a,m a e 2 r IJ

27 ... Coupled Cluster Theory The coupled cluster wave function for a closed shell atom is given by, 0 = e T 0 0 Since the system considered here has only one valence electron, it reduces to v = e T 0 {1 S 0 } v Where, T 0 = T 1 0 T 2 0 S 0 = S 1 0 S 2 0 and The RCC operator amplitudes can be solved in two steps; first we solve for closed shell amplitudes using the following equations: 0 H 0 0 = E and g 0 H 0 0 = 0 Where, H 0 0 T = e H 0 e T 0

28 The open shell operators can be obtained by solving the following two equations : eff H EDM = 2 i c d e 5 p 2 v H op {1 S v 0 } v = E v v {S v 0 } v E v Where, electron v. is the negative of the ionization potential of the valence The total atomic Hamiltonian in the presence of EDM as a perturbation is given by, v = e T 0 d e T 1 {1 S 0 d e S 1 } v The effective ( one-body ) perturbed EDM operator is given by, v H op {1 S v 0 } v = E v Thus, the modified atomic wave function is given by, H = H 0 H EDM

29 The perturbed cluster amplitudes can be obtained by solving the following equations self consistently : 0 H 0 N T 1 eff H EDM 0 = 0 v H 0 N E v S 1 v H 0 N T 1 eff H EDM {1 S v 0 } v = 0 Where, H N = H 0 0 H 0 0 The atomic EDM is given by, D a = v D a v v v

30 EXPERIMENTS ON ATOMIC EDM... Principle of Measurement H I = D a E B B E 1 B E 2 1 = 2 B 2 D a E ħ 2 = 2 B 2 D a E ħ = 1 2 = 4 D a E ħ If the atomic EDM D a ~ e-cm and E = 10 5 V/cm; ~ 10-5 Hz B m = v E c 2