Local Spin Dynamics with the Electron Electric Dipole Moment

March 7-9, 2016

New Generation Quantum Theory

-Particle Physics, Cosmology, and Chemistry-

○ Kota Soga, Masahiro Fukuda,Masato Senami, and Akitomo Tachibana

(Kyoto University)

M. Fukuda, K. Soga, M. Senami, and A. Tachibana, Phys. Rev. A 93, 012518 (2016). 1

Time evolution based on QED

Electron gun

Double-slit

Observing screen

Electron

Heisenberg operators

Wave function

Spacetime-resolvedphysical quantity

QED Hamiltonian

QED is able to predict whichposition of the screen electronswill reach moment by moment,while in quantum mechanics,it is a mystery.

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Local picture of spin based on QED

M. Fukuda, K. Soga, M. Senami, and A. Tachibana,Int. J. Quantum Chem., in press (2016).

Spin torquedensity

Zeta forcedensity

Spin angularmomentum density

Equation of motion of electronic spin

A. Tachibana, J. Mol. Model, 11, 301 (2005).A. Tachibana, J. Mol. Struct. (THEOCHEM) 943, 138 (2010).

The equation of motion of electronic spin based on QED does notlose the information of local contribution, and it gives a newperspective even for the spin stationary state.

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Time reversal

Electron EDM and T violation

Time : 𝒕 → −𝒕

Spin : 𝒔 → −𝒔

EDM : 𝒅 → 𝒅

With the CPT invariance,T violation

meansCP violation.

Electric dipole moment (EDM) of electron is a significant key to reveal a violation of the time-reversal symmetry.

CP violation may be a hint of the mystery of thedominance of matter over antimatter in ouruniverse.

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Search for physics beyond the standard modelValues of electron EDM 𝒅𝒆

Standard model of particle physicsToo small to be observed by present experiments :

𝒅𝒆 ∼ 𝟏𝟎−𝟒𝟎 𝐞 𝐜𝐦

Supersymmetric modelMuch larger :

𝒅𝒆 ∼ 𝟏𝟎−𝟐𝟕 − 𝟏𝟎−𝟐𝟗 𝐞 𝐜𝐦

Present experimental bound :

𝒅𝒆 < 𝟖. 𝟕 × 𝟏𝟎−𝟐𝟗 𝐞 𝐜𝐦 for ThO molecule†

𝒅𝒆 < 𝟏. 𝟎𝟓 × 𝟏𝟎−𝟐𝟕 𝐞 𝐜𝐦 for YbF molecule††

Experiments in the near future will find or rule outextension models of the standard model.†† The ACME Collaboration et al., Science 343, 269 (2014).†† J. J. Hudson et al., Nature (London) 473, 493 (2011); D. M. Kara et al., New J. Phys. 14, 103051 (2012).

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Spin precession with the electron EDM

Expectation value of physical quantities

Expansion of electron fieldSystem energy

Time evolution of 𝑂 𝑥 depends on 𝝎+ − 𝝎− = 𝟐𝑬𝐄𝐃𝐌/ℏ.

Nucleus

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Effective electric field for the electron EDMSince the electron EDM 𝑑𝑒 cannot be derived only by experiments,

we must evaluate the EDM effective electric field.

Electron EDMProof of the violation of 𝑇 invariance

What we want to know in particle physics

EDM effective electric fieldDerived by relativistic calculation

Interaction energy of 𝒅𝒆 and internal electric field

Experimentally determined

Heavy polar diatomic molecules are used in the present experiments.

Relativistic effects and correlation effects are essentially important for the computation of heavy atoms.

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Calculation method for the EDM effective electric fieldUsing no approximation for molecular internal electric field needsmuch time and computational costs due to the product of fourcreation and annihilation operators in the electric field of electrons.

Contribution fromelectric field of electrons

Contribution fromelectric field of nuclei

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Calculation method for the EDM effective electric field

To avoid time consuming calculations,some approximation methods are widely used. Approximation method “nuc” uses only the electric field of nuclei.

The deviation by this approximation is reported to be within 3% in YbF†.

Approximation method “ob” uses the effective EDM one-body operator.Electric field of electrons is included in the computation.

† H. M. Quiney et al, J. Phys. B. 31, L85 (1998). 9

Parallel magnetic hyperfine interaction constant and molecular electric dipole moment

Both quantities are useful to estimate the accuracy of wave functions around the vicinity of nuclei. Parallel magnetic hyperfine structure constant 𝑨∥

𝐴∥ characterizes the strength of the electromagnetic interaction between the nuclear magnetic dipole moment 𝜇𝐾 and electrons.

Molecular electric dipole moment (DM)

𝜸 : Space components of gamma matrices

𝑰 : Nuclear spin quantum number

𝛀 :Total electronic angular momentum projection onto the internuclear axis

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Local spin torque dynamicsIn QED, time evolution of spin angular momentumdensity is governed by spin torque density and zetaforce density.

Zeta force density

Spin torquedensity

Zeta forcedensity

Spin angularmomentum density

Equation of motion of electronic spin

Spin torque density

Spin angular momentum density

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Relation between QED andquantum mechanics (QM)

Equation of motion of electronic spin based on QED

Heisenberg equation of spin in relativistic QM

In QM, the local contribution of the zeta force density is lost because

the zeta force density is the gradient of the zeta potential 𝜙5 𝑥 .

Π𝑒 𝑡 : Kinetic momentum

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Local spin torque dynamics with EDMThe EDM Lagrangian density 𝐿EDM = −𝑑𝑒

𝑖

2 𝜓𝜎𝜇𝜈 𝐹𝜇𝜈

𝜓 gives

the additional local spin torque density 𝑡EDM 𝑥 .

Equation of motion of electronic spin with the electron EDM

EDM torque density

This state is chosen for calculating both EDM effective electric field and spin torque. 13

Relativistic four-component wave function is used as an approximation of those in QED.

Program package : — Electronic structure calculation : DIRAC13— Calculation for physical quantities : QEDynamics

Method : Dirac-Hartree-Fock, Method : Restricted Active Space Configuration Interaction

Method : (All single and double excitations are included.)

Target : YbF 2Σ1/2 , ThO 3Δ1 , BaF 2Σ1/2 , HF+ 2Π1/2

Basis set : Uncontracted Dyall’s four-component Basis set : double zeta (DZ), triple zeta (TZ), quadruple zeta (QZ) Basis set : including correlating function for all shells

All the values without specifying unit are written in atomic units.

QEDynamics, M. Senami, K. Ichikawa, A. Tachibana,http://www.tachibana.kues.kyoto-u.ac.jp/qed/index.html.

Computational details

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a R. J. VanZee, M. L. Seely, T. C. DeVore, and W. Weltner, Jr., J.Phys.Chem., 82, 1192(1978).

b B. E. Sauer, J. Wang, and E. A. Hinds, J. Chem. Phys., 105, 7412 (1996).

c N. S. Mosyagin, M. G. Kozlov and A. V. Titov, J. Phys. B: At. Mol. Opt. Phys. 31 L763 (1998).

d M. K. Nayak and R. K. Chaudhuri, Pramana 73, 581 (2009).

e M. Abe, G. Gopakumar, M. Hada, B. P. Das, H. Tatewaki, D. Mukherjee, Phys. Rev. A 90, 022501 (2014).

Calculation results of EDM effective electric field for YbF

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Calculation results of EDM effective electric field for BaF

a W. E. Ernst, J. Kändler, and T. Törring, J. Chem. Phys., 80, 2283 (1984).

b L. B. Knight Jr., W. C. Easley, W. Weltner Jr., and M. Wilson, J. Chem. Phys., 41, 2836 (1964).

c M. G. Kozlov, A. V. Titov, N. S. Mosyagin, and P. V. Souchko, Phys. Rev. A 56, R3326(R) (1997).

d M. K. Nayak and R. K. Chaudhuri, J. Phys. B: At. Mol. Opt. Phys. 39, 1231 (2006).

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a P. Hess, Ph.D. thesis, Harvard University, 2014; see http://laserstorm.harvard.edu/edm//publications.html.

b T. Fleig, M. K. Nayak, J. Mol. Spectroscopy, 300, 16 (2014).

c L. V. Skripnikov and A. V. Titov, J. Chem. Phys. 142, 024301 (2015).

Calculation results of EDM effective electric field for ThO

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Calculation results of EDM effective electric field for HF+

Summary of EDM effective electric field for each molecule

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Electron density and spin angular momentum density

Distributions of (a) the electron density(b) the norm of the spin angular momentum density

YbF

HF+

BaF ThO

A remarkable feature of the spin angular momentumdensity in YbF, BaF, and ThO is that its distribution is notsymmetric for both sides of internuclear axis aroundnuclei and is concentrated at a little distance away fromnuclei, while it is not seen in HF+.

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Small component of spin angular momentum density

Distributions of the norm of the small component of spin angularmomentum density

YbF

HF+

BaF

ThO

Small component of the spin angular momentumdensity distribution in YbF, BaF, and ThO is alsoasymmetric though its magnitude of symmetrybreaking is smaller than the spin angularmomentum density itself.

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Spin angular momentum density and its small component

YbF

F

H

YbF

F

H

The value of the EDM effective electric field depends on the scalar product of 𝜓𝑆† ℏ

2 𝜎 𝜓𝑆 and

𝐸nuc.

As shown in figures, the distribution pattern of 𝜓𝑆† ℏ

2 𝜎 𝜓𝑆 in HF+ is nearly antisymmetric to a plane

which intersects orthogonally with the internuclear axis on the F nucleus. On the other hand, thedistribution pattern in YbF is asymmetric.

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Scalar product of small component of spin angular momentum density and nuclear electric field

Asymmetric distribution Not canceling out Large EDM effective

electric field

Antisymmetric distribution Canceling out Small EDM effective

electric field

It can be predicted that even light atomic molecules could have the large EDMeffective electric field if the small component of the spin angular momentumdensity has an asymmetric distribution pattern.

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Local spin torque density induced by external fields

Yb

F

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ConclusionLocal picture of electron spin based on QED is studied,in relation to the electron EDM. We have calculated and have clarified that

an asymmetric distribution pattern of the small component of the spin angular momentum density yields large .

We have shown the local spin torque density and have demonstratedthat the local pictures of the spin enables us to understand some of thephysical origin of spin phenomena.

Future work Clarify the mechanism that the small component of the spin angular

momentum density has an asymmetric distribution pattern. Calculate molecular internal electric field accurately and evaluate the

EDM torque and without approximation. Explore new prediction methods of the spin precession.

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