1. Test of Variation in m p /m e using 40 CaH + Molecular Ions in a String Crystal NICT Masatoshi Kajita TMU Minori Abe 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 ) 40 CaH + (in a string crystal in a linear trap) X 1  (v,N) = (0,0)→(v u,0) (v u = 1,2,3,4) The transition is observed by Raman transition (one photon forbidden).

2. Test of the variation in a constant X Measurement of the ratio of transition frequencies (f a,b ), that have different dependence on X f ∝ X λ The variation in f a /f b :  (f a /f b ) should be measured.  (f a /f b )/(f a /f b )=( a - b )(  X/X) >  (f a /f b )/(f a /f b ) (  f/f): frequency uncertainty The variation in  from the variation in the frequency ratio between different atomic transitions ～ 10 -17 /yr (T. Rosenband et al., Science 319, 1808 (2008)) 199 Hg + : = -3.2 27 Al +  = 0.008  f(Hg + )/f(Al + )]/  f(Hg + )/f(Al + )] = 5 x 10 -17

3. Test of the variance in m p /m e f ∝ (m p /m e ) atomic transitions : | |< 10 -4 (difficult to test using only atomic trans.) molecular pure harmonic vibrational transition : ～ -0.5 molecular pure rotational transition :  ～ -1 Measurement of the ratio of molecular transition freq. atomic transition freq. 1 S 0 – 3 P 0 transition of 87 Sr(optical lattice) or 27 Al + ～ 0 for all constants (m p /m e,  etc.) frequency uncertainty ～ 10 -17 From astronomical estimation variance in m p /m e ～ 10 -16 /yr hopeful condition for molecular transition |[  f /f]/  < 10 -16 low frequency uncertainty ? high sensitivity on m p /m e ?

4. Trapped XH + molecular ion (X: 40 Ca, 88 Sr, 174 Yb etc.) XH + molecular ion can be produced via collision between X + and H 2 X + + H 2  XH + + H XH + molecular ion trapped in a cryogenic chamber is localized in the J = 0 F =1/2 state (no hyperfine splitting) XH + molecular ion can be sympathetically cooled with laser cooled Y + ion

5. Prepare in the J =0, F = 1/2 state Equilibrium distribution in each quantum state is determined by Boltzmann distribution in balance between transition induced by blackbody radiation + spontaneous emission We need a cryogenic chamber with temperature lower than 4 K (distribution in the J = 0 state higher than 90 % because of large rotational const.) XH + molecular ion will be pumped to the J = 0 state within 1 s (there is no hyperfine splitting in the J = 0 state)

6. XH + 1  state v = v u v = 0 magnetic field f+f+ f-f- M = 1/2 M = -1/2 | f + - f - |/ f + < 4.4×10 -16 /mT N = J = 0 F = 1/2 No Electric quadrupole shift We can observe with tow photon absorption with a stable infrared laser N = 0 -> 1 order of 1 kHz/G N = 1 -> 1 order of 10 Hz/G

7. 40 CaH + transition freq, Nat. linewidth Allan variance (×(τe/τN) 1/2 ) v = 0 -> 1 43.4 THz 2.5 Hz 3.7 x 10 -14 Stark shift with two photon absorption 1.5 x 10 -14 (2.4 W/cm 2 ) v = 0 -> 2 85.6 THz 5.5 Hz 4.1 x 10 -14 Stark shift with two photon absorption -8.2 x 10 -14 (5.5 W/cm 2 ) v = 0 -> 3 126.5THz 10.0 Hz 5.0 x 10 -14 v = 0 -> 4 166.1 THz 15.9 Hz 6.1 x 10 -14 Stark shift can be eliminated by (1) measuring with different laser intensities (2) measurement with hyper Ramsey π/2 → (free) →-π→π/2 v = 0 → 1 transition is most advantageous for precise measurement

8. Raman vibrational transition of 40 CaH + molecular ion B 1  cm   B 1  cm   A 1  cm   X 1  v,J) = (0,0) X 1  v,J) = (v u,0) v u = 1,2,3,4 f(vu)f(vu) Laser 0 frequency: f L0 Power density: I 0 Laser 1 frequency: f L1 (= f L0 – f v ) Power density: I 1

10. Magic Raman frequencies in the quasi-resonant area (I 0 = I 1 : saturation power)

11. Magic Raman frequencies in the quasi-resonant area (I 0 = I 1 ) f L （ THz ） I S (W/cm 2 ) d [  f R /f]/df L (/cm-1) |  f R0 /f| v = 0->1 735.3 + 691.9 0.25 1.7 x 10 -16 2.8 x 10 -14 v = 0->2 737.5 + 651.9 0.40 2.5 x 10 -16 2.5 x 10 -14 v = 0->3 786.6 + 660.0 0.25 1.4 x 10 -16 4.0 x 10 -14 v = 0->4 760.1 + 594.0 0.40 2.2 x 10 -16 2.8 x 10 -14 f L : laser frequencies I S : saturation power density Stark shift by two lasers  f R =  f R0 +  f R1 = 0

12. Other frequency shifts Zeeman shift (magnetic field < 0.3 G) < 10 -17 (not necessary to choose M = 1/2 or -1/2) Electric quadrupole shift is zero because of F = ½ Stark shift by trap electric field ( < 0.1 V/cm) < 10 -16 Quadratic Doppler shift (< 1 mK) < 10 -17 Blackbody radiation shift < 10 -20

14. Conclusion Precise measurement of the XH + （ 1 , v = 0, J =0, F = 1/2, M = ±1/2 ） → （ 1 , v = v u, J =0, F = 1/2, M = ±1/2 ） v u = 1,2,3,4,,, transition frequency is proposed. Uncertainty < 10 -16 (detailed analysis only for 40 CaH + ) XH + molecular ions are (1)Easy to be produced by the X + + H 2  XH + + H reaction (2)Easy to be localized in the J = 0, F = 1/2 state without hyperfine splitting in a cryogenic chamber (3)Easy to be sympathetically cooled with atomic ion Y + (4) The frequency uncertainty is dominated by the Stark shift by the probe laser. (5) The v = 0 -> 1 transition is much more advantageous than overtone transitions to measure with Raman transition using far- off resonant frequencies. (6)Measuring with Raman transition, it is hopeful to measure two lasers with same intensities. (7)Using Raman lasers in the quasi-resonant area, the Stark shift can be nulled.

15. With room temperature, the measurement cycle is perturbed by the blackbody radiation rate of unhopeful transition (v, N) = (0,0)→(1,1) order of 1 /s Thank you for attention!!