1. Precise Measurement of Vibrational Transition Frequency of Optically Trapped molecules NICT Masatoshi Kajita TMU G. Gopakumar, M. Abe, M. Hada We propose to test the variation in the proton-to-electron mass ratio, via the precise measurement of the vibrational transition freq. of ultra-cold molecules. (  f/f < 10 -16 ) X 6 Li or X 23 Na molecules X: 174 Yb, 88 Sr, 40 Ca (in an optical lattice) X 2  (v,N) = (0,0)->(v u,0) (v u = 1,2,3,4) The transition is observed by Raman transition (one photon forbidden).

2. Ultra-cold XLi or XNa molecules Production from atoms in a optical lattice -> Feshbach resonance (very challenging for X 6 Li) photo-association ->localization in the vibrational-rotational ground state K. Aikawa et al. Phys. Rev. Lett. 105, 203001(2010) -> trap by laser light -> measurement of X 2  (v, N ) = (0, 0) -> (v u, 0) transition freq. v u = 1,2,3,4, -> detection of molecules in the v = v u state by selective photo-ionization (afterwards also v = 0)

3. Transition Frequency(THz) λ f ∝ (m p /m e )- 174 Yb 6 Li v = 0 ->1 4.17 0.47 v = 0 ->4 15.37 0.43 88 Sr 6 Li v = 0 ->1 5.06 0.48 v = 0 ->4 19.05 0.45 40 Ca 6 Li v = 0 ->1 5.77 0.47 v = 0 ->4 21.46 0.43 88 Sr 23 Na v = 0 ->1 2.29 0.56 v = 0 ->4 8.65 0.50 40 Ca 23 Na v = 0 ->1 2.57 0.48 v = 0 ->4 9.72 0.45

4. Measurement with molecules in a optical lattice Problem Stark shift induced by the trap laser light (this depends on the laser frequency) Merits (1)long interaction time (narrow homogeneous broadening) (2) trap inside the Lamb-Dicke region -> no Doppler broadening (3) measurement with many molecules -> high S/N ratio (low statistic uncertainty) (4) supression of the collision shift

5. Stark shift in the transition frequency Upper state Lower state without light with light without light transition frequency transition frequency EuEu ElEl Stark shift (  E u –  E l )/h No Stark shift (  E u =  E l ) In general With a proper laser freq. (magic frequency)

6. Are there magic frequency with molecular transitions? There exist magic frequency in the electronic quasi- resonant region （ already proposed withSr 2 ) Upper state Lower state frequency but sensitive to the slight detuning from the magic frequency resonance freq.

7. Stark shift in 174 Yb 6 Li transitions by trap laser 23 kW/cm 2

8. Magic frequencies (power density to get the potential depth of 10  K ） Transition Freq. (THz) power density (kW/cm 2 ) Slope (/MHz) 174 Yb 6 Li v = 0 -> 1 361.4 17 5.8 x 10 -17 v = 0 -> 4 374.2 12 3.2 x 10 -17 88 Sr 6 Li v = 0 -> 1 313.5 11 1.1 x 10 -15 v = 0 -> 2 271.3 11 1.8 x 10 -16 v = 0 -> 4 325.2 8 1.3 x 10 -15 40 Ca 6 Li v = 0 -> 1 268.9 11 1.0 x 10 -16 v = 0 -> 2 258.3 12 1.2 x 10 -16 v = 0 -> 3 261.6 12 1.8 x 10 -16 88 Sr 23 Na v = 0 -> 1 287.8 8 1.2 x 10 -15 v = 0 -> 4 318.4 15 5.7 x 10 -16 40 Ca 6 Li v = 0 -> 1 335.7 25 1.2 x 10 -15 v = 0 -> 3 340.0 21 2.6 x 10 -16 v = 0 -> 4 344.1 20 1.5 x 10 -16

9. (v, N) = (0,0)→(v u,0) transition is induced by Raman transition X 2  (0,0) X 2  (v u,0) Electronically excited states We have a choice with the Raman laser frequency (difference must be resonant)

10. Stark shift in the vibrational transition frequency magic frequency 0 laser frequency Raman frequencies transition frequency cancel each other

11. Raman laser frequencies when intensities of tow Raman lasers are equal ( intensity to get Rabi frequency of 1/6 Hz) Transition Freq. (THz) power density (mW/cm 2 ) One laser shift 174 Yb 6 Li v = 0 -> 1 364.3 + 360.1 100 8.0 x 10 -16 v = 0 -> 2 277.9 + 269.8 43 1.5 x 10 -15 v = 0 -> 4 385.8 + 370.4 112 9.5 x 10 -16 88 Sr 6 Li v = 0 -> 1 313.7 + 308.6 17 3.0 x 10 -16 v = 0 -> 2 313.5 + 303.6 50 1.7 x 10 -15 v = 0 -> 4 298.6 + 279.5 6.7 1.7 x 10 -15 40 Ca 6 Li v = 0 -> 1 261.8 + 256.0 75 1.3 x 10 -15 v = 0 -> 2 291.1 + 279.9 6.7 1.7 x 10 -15 v = 0 -> 3 298.6 + 282.1 6.7 1.3 x 10 -15 88 Sr 23 Na v = 0 -> 1 316.1 + 313.8 29 9.6 x 10 -16 v = 0 -> 3 315.1 + 308.5 160 2.0 x 10 -15 40 Ca 23 Na v = 0 -> 1 334.1 + 331.6 27 3.0 x 10 -16 v = 0 -> 2 340.1 + 335.1 59 1.5 x 10 -15 v = 0 -> 4 309.7 + 300.0 5.8 5.7 x 10 -15

12. Other frequency shifts DC Stark shift v = 0 ->1 174 Yb 6 Li -2 x 10 -15 /(V/cm) 2 88 Sr 6 Li 4 x 10 -14 /(V/cm) 2 40 Ca 6 Li 1 x 10 -13 /(V/cm) 2 88 Sr 23 Na 7 x 10 -13 /(V/cm) 2 40 Ca 23 Na 9 x 10 -13 /(V/cm) 2 Stark shift by the blackbody radiation ( ∝ T 4 ) (300K) order of X 6 Li 1 x 10 -14 X 23 Na 2 x 10 -14 Zeeman shift F = 3/2 M = 3/2 -> 3/2 174 Yb 6 Li -5 x 10 -17 /G 88 Sr 6 Li -3 x 10 -17 /G 40 Ca 6 Li -2 x 10 -17 /G F = 2 M = 2 -> 2 88 Sr 23 Na -6 x 10 -16 /G 40 Ca 23 Na -6 x 10 -16 /G 2 nd order Doppler shift <10 -19

13. Which molecule is advantageous? 174 Yb 6 Li is most advantageous (but difficult to be produced) 40 Ca 6 Li is first candidate to be realized (Fourier spec. is observed) Large DC Zeeman and Stark shift with X 23 Na This method is not applicable for 88 Sr 87 Rb (no magic frequencies)

14. Application to 40 CaH molecule? Measurement with 40 CaH molecule seems to be advantageous Magic frequency: 378.43 THz (792 nm) 50 kW/cm 2 ( potential 10  K) Slope 1.1 x 10 -17 /MHz DC Stark shift 1 x 10 -14 /(V/cm) Zeeman shift 2 x 10 -18 /G Blackbody radiation with 300 K 1.3 x 10 -16 But it is a subject to get ultra-low temperature buffer-gas cooling -> laser cooling (Doppler limit 60  K) for further cooling (1) PGC -> optical trap (2) optical trap with intense laser -> sideband Raman cooling

15. A 2  1/2 v = 0 N = 1 J = 1/2 A 2  1/2 v = 1 N = 1 J = 1/2 X 2  v = 0 N = 0,F = 1 N = 0,F = 0 N = 2,J = 3/2,F=1 N = 2,J = 3/2,F=2 Laser C0-F1 Laser C0-F0 Laser C2 X 2  v = 1 X 2  v = 2 N = 0 N = 2 Laser R0-1 Laser R2-1 Laser R0-2 Laser R2-2 Laser transition Spontaneous emission 1.4 x 10 7 /s 2.4 x 10 5 /s 2.2 x 10 3 /s 1.4 x 10 7 /s

16. Conclusion (v, N) = (0,0)-> (v u,0) (v u = 1,2,3,4) transition frequency can be measured with the systematic uncertainty of 10 -16 for XLi or XNa molecules The Stark shift induced by trap laser Raman lasers can be eliminated Detuning of trap laser frequency with the order of 0.1 MHz does not give significant effect Advantageous to get low systematic uncertainty 40 CaH > 174 Yb 6 Li > 88 Sr 6 Li > 40 Ca 6 Li > 88 Sr 23 Na, 40 Ca 23 Na This method is not applicable for 88 Sr 87 Rb The linewidth of Raman lasers are hope to be less than 0.1 Hz. This frequency measurement is useful to test the variance in m p /m e

17. Thank you for attention! Danke fuer Aufmerksamkeit! Merci pour attention! Tante Grazie!