aa r X i v : . [ phy s i c s . a t o m - ph ] M a y AC Stark effect in ThO H ∆ for the electron EDM search A.N. Petrov
1, 2, ∗ National Research Centre “Kurchatov Institute” B.P. Konstantinov PetersburgNuclear Physics Institute, Gatchina, Leningrad district 188300, Russia Division of Quantum Mechanics, St.Petersburg State University, 198504, Russia
A method and code for calculations of diatomic molecules in the external variable electromagneticfield have been developed. Code applied for calculation of systematics in the electron’s electric dipolemoment search experiment on ThO H ∆ state related to geometric phases, including dependenceon Ω-doublet, rotational level, and external static electric field. It is found that systematics decreasecubically with respect to the frequency of the rotating transverse component of the electric field.Calculation confirms that experiment on ThO H ∆ state is very robust against systematic errorsrelated to geometric phases. The experimental measurement of a non-zero electronelectric dipole moment ( e EDM, d e ) would be a clear sig-nature of physics beyond the Standard model [1–4]. Thecurrent limit for e EDM, | d e | < × − e · cm (90%confidence), was set with a buffer-gas cooled molecularbeam[5–7] of thorium monoxide (ThO) molecules in themetastable electronic H ∆ state. It was shown thatdue to existence of closely-spaced levels of opposite par-ity of Ω-doublet the experiment on ThO is very robustagainst a number of systematic effects related to mag-netic fields[8, 9] or geometric phases[10]. However, theupper and lower Ω-doublet states have slightly differ-ent properties and systematic effects related to magneticfield imperfections and geometric phases can still mani-fest themselves as a false e EDM. The dependence of g -factors of the ThO H ∆ state on Ω-doublets and exter-nal electric field was considered in Ref. [9]. The aim ofthe present work is to consider geometric phase shifts.Following the computational scheme of [9, 11], the en-ergies of the rotational levels in the H ∆ electronic stateof the Th O molecule in external static electric ~ E = E ˆ z and magnetic ~ B = B ˆ z fields are obtained by numeri-cal diagonalization of the molecular Hamiltonian ( ˆH mol )over the basis set of the electronic-rotational wavefunc-tions Ψ Ω θ JM, Ω ( α, β ). Here Ψ Ω is the electronic wave-function, θ JM, Ω ( α, β ) = p (2 J + 1) / πD JM, Ω ( α, β, γ = 0)is the rotational wavefunction, α, β, γ are Euler angles,and M (Ω) is the projection of the molecule angularmomentum J on the lab ˆ z (internuclear ˆ n ) axis. De-tailed feature of the Hamiltonian is described in [9]. Inthe paper the M = ± e EDM search experiment are considered. For elec-tric field E = 20 −
200 V / cm, used in the experiment,lower rotational levels with M = 0 can be labeled by | J, M, Ω > quantum numbers. States | J, M =1 , Ω=1 > , | J, M = − , Ω= − > correspond to the upper and | J, M = − , Ω=1 > , | J, M =1 , Ω= − > to the lowerΩ-doublet levels. External magnetic field removes thedegeneracy between Ω-doublet components: ∆ E u = E ( | J, M =1 , Ω=1 > ) − E ( | J, M = − , Ω= − > ), ∆ E l = E ( | J, M =1 , Ω= − > ) − E ( | J, M = − , Ω=1 > ). The rel- evant energy levels can be seen in Figure 2 of Ref. [12] orFigure 3 of Ref. [13]. Provided g-factors for upper andlower Ω-doublet levels are close enough ∆ E u and ∆ E l remain equal unless both parity and time reversal sym-metries are violated. The difference in splitting gives thevalue for e EDM d e = | ∆ E l − ∆ E u | E eff , here E eff = 81 . / cm[12, 14] is the effective internal electric field. However,there are systematic effects which can give additional en-ergy shifts δ ∆ E l and δ ∆ E u for ∆ E l and ∆ E u whichmanifest as a false e EDM. This leads to a systematic er-ror δd e ( sys ) = δ ∆ E l − δ ∆ E u E eff . It is also useful to considersystematic effects ˜ δd e ( sys ) = δ ∆ E l ( u ) E eff related to one ofthe Ω-doublet component . One of the effect is the in-teraction with transverse component of the electric field ~ E ( t ) = E ⊥ (ˆ xcos ( ω ⊥ t ) + ˆ ysin ( ω ⊥ t )) which appears due toa spatial inhomogeneties in the applied electric field [13].Let us consider this effect.The corresponding part of the Hamiltonian is ˆH tf = − ~d · ~ E ( t ) = −E ⊥ / d + e − iω ⊥ t + d − e iω ⊥ t ) , (1)where d ± = d x ± id y . It is more convenient to describethe interaction of the molecule with the quantized elec-tromagnetic fields. The corresponding Hamiltonian is ˆH int = ¯ hω ⊥ a + a − r π ¯ hω ⊥ V ( d + a + + d − a ) , (2)where a + and a are creation and annihilation operators, V is a volume of the system. To work with Hamiltonian(2) one need to add the quantum number | n > , where n = V E ⊥ hω ⊥ is number of photons. The approach is similarto the formalism outlined in Ref. [10]. For this paperwe consider the case E ⊥ << E , such that the additionalenergy shifts can be calculated in the framework of thesecond order perturbation theory. δ ∆ E u = E ⊥ / × ( X J ′ , Ω ′ | < J, M =1 , Ω=1 | d + | J ′ , M =0 , Ω ′ > | E ( J, M =1 , Ω=1) − E ( J ′ , M =0 , Ω ′ ) + ¯ hω ⊥ + X J ′ , Ω ′ | < J, M =1 , Ω=1 | d − | J ′ , M =2 , Ω ′ > | E ( J, M =1 , Ω=1) − E ( J ′ , M =2 , Ω ′ ) − ¯ hω ⊥ − X J ′ , Ω ′ | < J, M = − , Ω= − | d + | J ′ , M = − , Ω ′ > | E ( J, M = − , Ω= − − E ( J ′ , M = − , Ω ′ ) + ¯ hω ⊥ −− X J ′ , Ω ′ | < J, M = − , Ω= − | d − | J ′ , M =0 , Ω ′ > | E ( J, M = − , Ω= − − E ( J ′ , M =0 , Ω ′ ) − ¯ hω ⊥ ) , (3) δ ∆ E l = E ⊥ / × ( X J ′ , Ω ′ | < J, M =1 , Ω= − | d + | J ′ , M = 0 , Ω ′ > | E ( J, M =1 , Ω= − − E ( J ′ , M =0 , Ω ′ ) + ¯ hω ⊥ + X J ′ , Ω ′ | < J, M =1 , Ω= − | d − | J ′ , M =2 , Ω ′ > | E ( J, M =1 , Ω= − − E ( J ′ , M =2 , Ω ′ ) − ¯ hω ⊥ − X J ′ , Ω ′ | < J, M = − , Ω=1 | d + | J ′ , M = − , Ω ′ > | E ( J, M = − , Ω=1) − E ( J ′ , M = − , Ω ′ ) + ¯ hω ⊥ − X J ′ , Ω ′ | < J, M = − , Ω=1 | d − | J ′ , M =0 , Ω ′ > | E ( J, M = − , Ω=1) − E ( J ′ , M =0 , Ω ′ ) − ¯ hω ⊥ ) . (4)Major contribution to δ ∆ E u ( l ) comes from coupling ofstates with the same J . The most simple is the pic-ture for J =1 state. | J =1 , M =1 , Ω , n > interact with | J =1 , M =0 , Ω , n +1 > and | J = 1 , M = − , Ω , n > with | J, M =0 , Ω , n − > . Note, that ˆH mol can only couple thestates with the same n , whereas ˆH int couples the stateswith ∆ n = ±
1. Energies of states | J, M =0 , Ω , n +1 > and | J, M =0 , Ω , n − > are different by 2¯ hω ⊥ , andstates | J =1 , M =1 , Ω , n > and | J =1 , M = − , − Ω , n > by 2 µ B . This leads to different energy denominatorsin eqs. (3,4) and results in different energy shift for | J =1 , M =1 , Ω , n > and | J =1 , M = − , − Ω , n > . How-ever, for J =1 level, it was shown in Ref. [13] that (consid-ering the interaction with | J =1 , M =0 , Ω > states only)provided tensor Stark (∆ E ST = E(J,M= ± δ ∆ E u and δ ∆ E l will also be equal. This allowsone to reject systematic errors due to geometric phasesby performing measurements in both Ω − doublet states.However the tensor Stark splittings do not coincide ex-actly. Also including interaction with other states lift thedegeneracy. It is particularly important to include the in-teraction with the neighbor rotational levels. The latterinteraction increases value for δd e ( sys ) on several orders of magnitude whereas ˜ δd e ( sys ) is almost unaffected bythis interaction. See also influence of perturbation by J = 2 level on J = 1 g-factors in Refs. [9, 11, 15, 16].Tables I and II list the calculated ˜ δd e ( sys ) and δd e ( sys ) as a functions of ω ⊥ and E for J = 1 and J = 2,correspondingly. Though for smaller E the E ⊥ value willbe smaller as well, for the calculation I take the same E ⊥ = 10mV / cm given in Ref. [13] for all E . Using thefact that ˜ δd e ( sys ) and δd e ( sys ) are quadratic functionsof E ⊥ the results can be easily recalculated for any E ⊥ .For static magnetic field the value B = 40mG used in theexperiment [5] is used. One can see that δd e ( sys ) is twoorders of magnitude larger for J =2 than for J =1 thoughmuch smaller than the current limit on d e .Calculation for ω ⊥ / π less than 250 kHz is not per-formed due to the limited computational accuracy. Forsmaller ω ⊥ one can expect further decreasing of ˜ δd e ( sys )and δd e ( sys ). Each term in Eqs. (3,4) has form b u ( l ) a u ( l ) +¯ hω ⊥ − b u ( l ) a u ( l ) − ¯ hω ⊥ . Retaining terms up to the thirdorder in ω ⊥ we have δ ∆ E u ( l ) ≈ − B u ( l ) A u ( l ) ¯ hω ⊥ A u ( l ) + ¯ h ω ⊥ A u ( l ) ! . (5)Formulae for B u ( l ) and A u ( l ) for J =1 are given below. Eq.(5) explains the fact that δ ∆ E u ( l ) decreases linerly withsmall ω ⊥ listed in Tables I and II. Similarly to δ ∆ E u ( l ) the major contribution to difference δ ∆ E l − δ ∆ E u comesfrom coupling of states with the same J (terms with J ′ = J in Eqs. (3,4)). However, important role playsthe perturbation by the closest rotational levels whichmakes matrix elements and denominators for upper andlower components of Ω-doublet slightly different. Let usconsider this effect for the simplest case the J =1 level.Without perturbation by the J = 2 level the parameters A = ∆ E ST = −E < J =1 , M =1 , Ω | d z | J =1 , M = 1 , Ω > = −E dM Ω /J ( J +1)(6)and B = −E ⊥ / < J =1 , M =1 , Ω | d + | J =1 , M ′ =0 , Ω > = − E ⊥ d Ω2 p ( J − M +1)( J + M ) J ( J +1) (7)up to the sign are the same for upper and lower Ω-doubletlevels. ∆ E ST is positive for upper and negative for lowerlevels. Note that dipole moment d <
0. Eqs. (5,6) ex-plain the fact that δ ∆ E u ( l ) decreases quadratically with E . Perturbation by J = 2 changes the parameters: A u ( l ) = ∆ E ST + δ ∆ E u ( l ) ST + δ ∆ E u ( l ) ST , (8) B u ( l ) = B + δ B u ( l ) + δ B u ( l ) . (9) TABLE I. The ˜ δd e ( sys ) (in units 10 − e · cm ) and δd e ( sys )(in units 10 − e · cm ) calculated for the J = 1 H ∆ state in Th O. E ω ⊥ / π = 4MHz ω ⊥ / π = 1MHz ω ⊥ / π = 250kHz(V/cm) ˜ δd e ( sys ) δd e ( sys ) ˜ δd e ( sys ) δd e ( sys ) ˜ δd e ( sys ) δd e ( sys )20. -1299. -2144. -313. -31. -78. -0.4830. -565. -608. -139. -9.1 -35. -0.1440. -315. -253. -78. -3.8 -19. -0.05950. -201. -128. -50. -1.9 -12. -0.03060. -139. -74. -35. -1.1 -8.6 -0.01870. -102. -47. -25. -0.70 -6.3 -0.01180. -78. -31. -19. -0.47 -4.8 -0.007390. -62. -22. -15. -0.34 -3.8 -0.0053100. -50. -16. -12. -0.24 -3.1 -0.0038110. -41. -12. -10. -0.18 -2.6 -0.0028120. -35. -9.2 -8.6 -0.14 -2.2 -0.0022130. -30. -7.3 -7.3 -0.11 -1.8 -0.0017140. -26. -5.8 -6.3 -0.088 -1.6 -0.0014150. -22. -4.7 -5.5 -0.071 -1.4 -0.0011160. -20. -3.9 -4.9 -0.059 -1.2 -0.00093170. -17. -3.2 -4.3 -0.050 -1.1 -0.00078180. -15. -2.7 -3.8 -0.043 -0.96 -0.00067190. -14. -2.3 -3.4 -0.034 -0.86 -0.00053200. -12. -2.0 -3.1 -0.030 -0.77 -0.00047 δ ∆ E ST is the correction to ∆ E ST due to shift-ing down | J =1 , M = ± , Ω > ( | J =1 , M =0 , Ω > ) lev-els when interacting with J =2. δ ∆ E u ( l ) ST is negative.It decrease (increase absolute value) ∆ E ST for upper(lower) Ω-doublet levels. In turn δ ∆ E u ( l ) ST is posi-tive. It increases (decreases absolute value) ∆ E ST forupper (lower) Ω-doublet levels. δ B is the correc-tion to B due to the perturbation of the wavefunction | J =1 , M = ± , Ω > ( | J =1 , M =0 , Ω > ) by | J =2 , M = ± , Ω > ( | J =2 , M =0 , Ω > ) one. It is shown in AP-PENDIX that for J =1 level the corrections δ ∆ E u ( l ) ST , δ ∆ E u ( l ) ST , δ B u ( l ) , and δ B u ( l ) are correlated in such away that B u ( l ) A u ( l ) = B A . (10)Eq. (10) is correct up to the second order in small param-eter ∆ E ST / ∆ E rot , where ∆ E rot = E ( J =1) − E ( J =2) isenergy difference between the first and second rotationallevels. Due to Eq. (10) the linear term in the difference δd e ( sys ) = δ ∆ E l − δ ∆ E u E eff is canceled and δd e ( sys ), in theleading order, is a cubic function of ω ⊥ for J =1. Thisbehavior can be seen from the data in Table I. Depen-dence of the δd e ( sys ) for J =2 level on ω ⊥ has also nearlythe cubic character.The calculations confirm that the experiment on ThO H ∆ state is very robust against systematic errors re-lated to geometric phases. Developed code can be appliedfor calculation of molecules in an ion trap at presence ofrotating field [17, 18]. TABLE II. The ˜ δd e ( sys ) (in units 10 − e · cm ) and δd e ( sys )(in units 10 − e · cm ) calculated for the J = 2 H ∆ state in Th O. E ω ⊥ / π = 4MHz ω ⊥ / π = 1MHz ω ⊥ / π = 250kHz(V/cm) ˜ δd e ( sys ) δd e ( sys ) ˜ δd e ( sys ) δd e ( sys ) ˜ δd e ( sys ) δd e ( sys )20. -1996. -213498. -320. -1663. -78. -33.30. -658. -42436. -140. -495. -35. -9.940. -342. -15657. -79. -211. -20. -4.250. -212. -7546. -50. -107. -13. -2.160. -144. -4229. -35. -61. -8.7 -1.270. -105. -2613. -26. -39. -6.4 -0.7880. -80. -1728. -20. -26. -4.9 -0.5290. -63. -1204. -15. -18. -3.9 -0.37100. -51. -872. -13. -14. -3.1 -0.27110. -42. -653. -10. -10. -2.6 -0.20120. -35. -501. -8.7 -7.7 -2.2 -0.15130. -30. -393. -7.4 -6.1 -1.9 -0.12140. -26. -314. -6.4 -4.9 -1.6 -0.098150. -22. -255. -5.6 -4.0 -1.4 -0.080160. -20. -210. -4.9 -3.1 -1.2 -0.063170. -17. -175. -4.3 -2.7 -1.1 -0.055180. -15. -147. -3.9 -2.2 -0.97 -0.044190. -14. -125. -3.5 -1.9 -0.87 -0.038200. -13. -107. -3.1 -1.6 -0.78 -0.032 The author is grateful to A. V. Titov for useful dis-cussions. Codes for calculations of diatomic moleculeswere developed with the support of the Russian ScienceFoundation grant (project No. 14-31-00022). Calcula-tion of the ThO molecule were performed with the sup-port of Saint Petersburg State University, research grant0.38.652.2013 and RFBR Grant No. 13-02-0140.
APPENDIX
In the first order in the small parameter∆ E ST / ∆ E rot ∼ E d/ ∆ E rot for δ ∆ E u ( l ) ST , δ ∆ E u ( l ) ST , δ B u ( l ) , and δ B u ( l ) we have δ ∆ E u ( l ) ST = E / ∆ E rot | < J =1 , M =1 , Ω | d z | J ′ =2 , M = 1 , Ω > | = E d ∆ E rot (( J +1) − M )(( J +1) − Ω )(2 J +1)(2 J +3)( J +1) ,δ ∆ E u ( l ) ST = −E / ∆ E rot | < J =1 , M =0 , Ω | d z | J ′ =2 , M = 0 , Ω > | = − E d ∆ E rot ( J +1) (( J +1) − Ω )(2 J +1)(2 J +3)( J +1) ,δ B u ( l ) = E ⊥ / < J ′ =2 , M =1 , Ω | d + | J =1 , M ′ =0 , Ω > ×E / ∆ E rot < J =1 , M = 1 , Ω | d z | J ′ =2 , M = 1 , Ω > = − E ⊥ E d E rot s ( J + M )( J + M +1)(( J +1) − Ω )(2 J +1)(2 J +3)( J +1) × s (( J +1) − M )(( J +1) − Ω )(2 J +1)(2 J +3)( J +1) , δ B u ( l ) = E ⊥ / < J =1 , M =1 , Ω | d + | J ′ =2 , M ′ =0 , Ω > ×E / ∆ E rot < J =1 , M ′ =0 , Ω | d z | J ′ =2 , M ′ =0 > = E ⊥ E d E rot s ( J − M +2)( J − M +1)(( J +1) − Ω )(2 J +1)(2 J +3)( J +1) × s ( J +1) (( J +1) − Ω )(2 J +1)(2 J +3)( J +1) . ∆ E rot is negative, therefore δ (1) ∆ E u ( l ) ST < δ (2) ∆ E u ( l ) ST > E ST / ∆ E rot we have B u ( l ) A u ( l ) = ( B + δ B u ( l ) + δ B u ( l ) ) (∆ E ST + δ ∆ E u ( l ) ST + δ ∆ E u ( l ) ST ) ≈ B + 2 Bδ B u ( l ) + 2 Bδ B u ( l ) ∆ E ST + 2∆ E ST δ ∆ E u ( l ) ST + 2∆ E ST δ ∆ E u ( l ) ST = E ⊥ d Ω J − M +1)( J + M )( J ( J +1)) E d M Ω ( J ( J +1)) × (cid:16) h ( J + M +1) − ( J +1) q J − M +2 J + M i K ( J ) / Ω (cid:17) (1 + M K ( J ) / Ω) = B (cid:16) h ( J + M +1) − ( J +1) q J − M +2 J + M i K ( J ) / Ω (cid:17) A (1 + M K ( J ) / Ω) , (11)where K ( J ) = 2 E d ∆ E rot J ( J +1)(( J +1) − Ω )(2 J +1)(2 J +3)( J +1) . Substituting M =1, J =1 to Eq. (11) we have got Eq.(10). Eq. (11) is obtained for M =+1 level. The re-sult is the same for M = −
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