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Journal of the Physical Society of Japan

Precise Measurement of Transition Frequencies of Optically Trapped 40Ca19F Molecules

Masatoshi Kajita

Published Sep 18, 2018• 9 citations
Publisher

Abstract

The laser cooling and trapping of molecules have recently been achieved. In this paper, the possibility of precisely measuring the vibrational transition frequencies of optically trapped 40Ca19F molecules is discussed. The quadratic Stark shift induced by the trap laser is eliminated using the trap laser frequencies where the Stark energy shifts at upper and lower states are equal (magic frequencies). At one value of the magic frequency, molecules are trapped at the positions where the power density is minimum, therefore, the measurement is not sensitive to the detuning of the trap laser frequency from the magic frequency. Measurement with an uncertainty of 10−17 appears to be possible using the \((v,N,J,F,M) = (0,0,1/2,1, \pm 1) \to (1,0,1/2,1, \pm 1)\) transition frequency.

DOI
10.7566/jpsj.87.104301

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The laser cooling and trapping of molecules have recently been achieved. In this paper, the possibility of precisely measuring the vibrational transition frequencies of optically trapped 40Ca19F molecules is discussed. The quadratic Stark shift induced by the trap laser is eliminated using the trap laser frequencies where the Stark energy shifts at upper and lower states are equal (magic frequencies). At one value of the magic frequency, molecules are trapped at the positions where the power density is minimum, therefore, the measurement is not sensitive to the detuning of the trap laser frequency from the magic frequency. Measurement with an uncertainty of 10−17 appears to be possible using the ((v,N,J,F,M) = (0,0,1/2,1, \pm 1) \to (1,0,1/2,1, \pm 1)) transition frequency.
Transition frequencies of cold molecules must be accurately evaluated to test the variance in the proton-to-electron mass ratio. Measuring the $X{\phantom{\rule{0.16em}{0ex}}}^{2}\ensuremath{\Sigma}(v,N)=(0,0)\ensuremath{\rightarrow}(1,0)$ transition frequency of optically trapped 174{}^{174}174Yb6{}^{6}6Li molecules is a promising method for achieving this goal. The Stark shifts induced by trap and probe (for the Raman transition) lasers are eliminated by choosing appropriate frequencies (magic frequencies) during the construction of the optical lattice. In the far-off resonance region, the Stark shift is found to be less than 10${}^{\ensuremath{-}16}$ even when the laser frequencies are detuned from the magic frequencies by $\ensuremath{\sim}$1 MHz.
The attainable accuracy of the $X\phantom{\rule{0.16em}{0ex}}{}^{2}\phantom{\rule{-0.16em}{0ex}}\ensuremath{\Sigma}(v,N)=(0,0)\ensuremath{\rightarrow}({v}_{u},0)$ transition frequencies of optically trapped 174{}^{174}174Yb6{}^{6}6Li molecules are analyzed (vu=1,2,3,4)({v}_{u}=1,2,3,4)(vu​=1,2,3,4) to test the variation in the proton-to-electron mass ratio. We eliminate the Stark shifts induced by the trap and Raman lasers by choosing appropriate frequencies (magic frequencies). For vu=1{v}_{u}=1vu​=1 and 2, we obtain more than one experimentally useful magic frequency, thereby leading to more choice in the trap laser. For vu=3{v}_{u}=3vu​=3 and 4, the choice in the trap laser is limited. The systematic frequency uncertainty is given by the trap laser frequency detuning from the magic frequency and the fluctuation of the power ratio of the two Raman lasers and can be lower than 10${}^{\ensuremath{-}16}$ for all these transition frequencies. To obtain lower statistical uncertainty, measurement with vu=4{v}_{u}=4vu​=4 is more advantageous than that with ${v}_{u}=1\ensuremath{-}3$.
Precision measurements of the X 2 Σ( v , N ) = (0, 0) → ( v u , 0) ( v u = 1, 2, 3, 4) transition frequencies from optically trapped 88 Sr 6 Li and 40 Ca 6 Li molecules are proposed, with the intention to test the variation in the proton-to-electron mass ratio m p / m e . The Stark shifts induced by the trap and Raman lasers are eliminated by choosing appropriate frequencies (magic frequencies). These transitions are measurable with an uncertainty lower than 10 −16 and can be used to test the variation in m p / m e , as already shown with 174 Yb 6 Li molecular transitions.
We study the magic wavelength for two-photon 1S–nS transitions in a hydrogen and deuterium atom, as well as 2S–nS transitions, where the lower level is the metastable 2S state. At the magic wavelength, the dynamic Stark shifts of the ground and the excited state of the transition coincide, so that the transition frequency is independent of the intensity of the trapping laser field. Experimentally feasible magic wavelengths of transitions with small slopes in the atomic polarizabilities are determined; these are the most stable magic wavelengths against variations of the laser frequency. We provide data for the magic wavelengths for the 1S–nS and 2S–nS transitions in hydrogen and deuterium, with n=2,⋯,8. We also analyze the stability of the elimination of the ac Stark shift at the magic wavelength against tiny variations of the trapping laser frequency from the magic value.
We demonstrate experimentally the existence of magic wavelengths and determine the ratio of oscillator strengths for a single trapped ion. For the first time, two magic wavelengths near 396 nm for the ^{40}Ca^{+} clock transition are measured simultaneously with high precision. By tuning the applied laser to an intermediate wavelength between transitions 4s_{1/2}→4p_{1/2} and 4s_{1/2}→4p_{3/2}, the sensitivity of the clock transition Stark shift to the oscillator strengths is greatly enhanced. Furthermore, with the measured magic wavelengths, we determine the ratio of the oscillator strengths with a deviation of less than 0.5%. Our experimental method may be applied to measure magic wavelengths for other ion clock transitions. Promisingly, the measurement of these magic wavelengths paves the way to building all-optical trapped ion clocks.