Local Spin Dynamics with the Electron Electric Dipole Moment

Transcription

1 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, (2016). 1

2 Time evolution based on QED Heisenberg operators QED Hamiltonian Spacetime-resolved physical quantity Wave function QED is able to predict which position of the screen electrons will reach moment by moment, while in quantum mechanics, it is a mystery. Electron gun Electron Double-slit Observing screen 2

3 Local picture of spin based on QED Equation of motion of electronic spin Spin angular momentum density Spin torque density Zeta force density A. Tachibana, J. Mol. Model, 11, 301 (2005). A. Tachibana, J. Mol. Struct. (THEOCHEM) 943, 138 (2010). M. Fukuda, K. Soga, M. Senami, and A. Tachibana, Int. J. Quantum Chem., in press (2016). The equation of motion of electronic spin based on QED does not lose the information of local contribution, and it gives a new perspective even for the spin stationary state. 3

4 Electron EDM and T violation Electric dipole moment (EDM) of electron is a significant key to reveal a violation of the time-reversal symmetry. Time reversal Time : t t Spin : s s EDM : d d With the CPT invariance, T violation means CP violation. CP violation may be a hint of the mystery of the dominance of matter over antimatter in our universe. 4

5 Search for physics beyond the standard model Values of electron EDM d e Standard model of particle physics Too small to be observed by present experiments : Supersymmetric model Much larger : d e e cm d e Present experimental bound : e cm d e < e cm for ThO molecule d e < e cm for YbF molecule Experiments in the near future will find or rule out extension 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, (2012). 5

6 Spin precession with the electron EDM Nucleus System energy Expansion of electron field Expectation value of physical quantities Time evolution of O x depends on ω + ω = 2E EDM /ħ. 6

7 Effective electric field for the electron EDM Since the electron EDM d e cannot be derived only by experiments, we must evaluate the EDM effective electric field. Electron EDM Proof of the violation of T invariance What we want to know in particle physics Interaction energy of d e and internal electric field Experimentally determined EDM effective electric field Derived by relativistic calculation Heavy polar diatomic molecules are used in the present experiments. Relativistic effects and correlation effects are essentially important for the computation of heavy atoms. 7

8 Calculation method for the EDM effective electric field Using no approximation for molecular internal electric field needs much time and computational costs due to the product of four creation and annihilation operators in the electric field of electrons. Contribution from electric field of nuclei Contribution from electric field of electrons 8

9 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

10 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 μ K and electrons. γ : Space components of gamma matrices I : Nuclear spin quantum number Molecular electric dipole moment (DM) Ω : Total electronic angular momentum projection onto the internuclear axis 10

11 Local spin torque dynamics In QED, time evolution of spin angular momentum density is governed by spin torque density and zeta force density. Equation of motion of electronic spin Spin angular momentum density Spin angular momentum density Spin torque density Zeta force density Spin torque density Zeta force density 11

12 Relation between QED and quantum mechanics (QM) Equation of motion of electronic spin based on QED Heisenberg equation of spin in relativistic QM Π e t : Kinetic momentum In QM, the local contribution of the zeta force density is lost because the zeta force density is the gradient of the zeta potential φ 5 x. 12

13 Local spin torque dynamics with EDM The EDM Lagrangian density L EDM = d e i 2 ψσμν F μν ψ gives the additional local spin torque density t EDM 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

14 Computational details 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 QEDynamics, M. Senami, K. Ichikawa, A. Tachibana, Method : Dirac-Hartree-Fock, Method : Restricted Active Space Configuration Interaction Method : (All single and double excitations are included.) Target : YbF 2 Σ 1/2, ThO 3 Δ 1, BaF 2 Σ 1/2, HF + 2 Π 1/2 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. 14

15 Calculation results of EDM effective electric field for YbF 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, (2014). 15

16 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). 16

17 Calculation results of EDM effective electric field for ThO a P. Hess, Ph.D. thesis, Harvard University, 2014; see b T. Fleig, M. K. Nayak, J. Mol. Spectroscopy, 300, 16 (2014). c L. V. Skripnikov and A. V. Titov, J. Chem. Phys. 142, (2015). 17

18 Calculation results of EDM effective electric field for HF + Summary of EDM effective electric field for each molecule 18

19 Electron density and spin angular momentum density YbF BaF ThO HF + Distributions of (a) the electron density (b) the norm of the spin angular momentum density A remarkable feature of the spin angular momentum density in YbF, BaF, and ThO is that its distribution is not symmetric for both sides of internuclear axis around nuclei and is concentrated at a little distance away from nuclei, while it is not seen in HF +. 19

20 Small component of spin angular momentum density YbF BaF HF + ThO Distributions of the norm of the small component of spin angular momentum density Small component of the spin angular momentum density distribution in YbF, BaF, and ThO is also asymmetric though its magnitude of symmetry breaking is smaller than the spin angular momentum density itself. 20

21 Spin angular momentum density and its small component Yb F Yb F F H F H The value of the EDM effective electric field depends on the scalar product of ψ S ħ 2 σ ψ S and E nuc. As shown in figures, the distribution pattern of ψ S ħ 2 σ ψ S in HF + is nearly antisymmetric to a plane which intersects orthogonally with the internuclear axis on the F nucleus. On the other hand, the distribution pattern in YbF is asymmetric. 21

22 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 EDM effective electric field if the small component of the spin angular momentum density has an asymmetric distribution pattern. 22

23 Local spin torque density induced by external fields Yb F 23

24 Conclusion Local 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 demonstrated that the local pictures of the spin enables us to understand some of the physical 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. 24