We estimate the Stark and Zeeman shifts in the transition frequencies of the ${{}^{16}\mathrm{O}_2}^{+}$ molecular ion as a step in the search for the variation in the proton-to-electron mass ratio $\mu$. The $X^2\mathrm{\Pi }v=21-a^4\mathrm{\Pi }v=0$ or the $X^2\mathrm{\Pi }v=21-a^4\mathrm{\Pi }v=1$ transition frequencies (THz region) of the ${{}^{16}\mathrm{O}_2}^{+}$ molecular ion have particularly high sensitivity to the variation in $\mu$. Note also that the Stark shift in the ${{}^{16}\mathrm{O}_2}^{+}$ transition frequencies is expected to be much smaller than that for heteronuclear diatomic molecules. However, the actual systematic uncertainties for the ${{}^{16}\mathrm{O}_2}^{+}$ transition frequencies have never been estimated. We estimated the Stark and Zeeman shifts in the different ${{}^{16}\mathrm{O}_2}^{+}$ transition frequencies. When the molecular ions in a string crystal formed in a linear trap (trap electric field $<0.1$ V/cm, and Stark shift $<{10}^{-20}$) are used, the $X^2{\mathrm{\Pi }}_{1/2}(v,J)=(0,1/2)-(v^{\prime },1/2)(v^{\prime }\ge 1)$ transition frequencies are most advantageous for the search for the variation in $\mu \left(\mathrm{\Delta }\mu /\mu <{10}^{-17}\right)$ because the Zeeman shift is easily suppressed to lower than ${10}^{-18}$ and the electric quadrupole shift is zero. On the other hand, the $X^2{\mathrm{\Pi }}_{1/2}\left(v,J\right)=\left(21,1/2\right)-a^4{\mathrm{\Pi }}_{1/2}\left(v,J\right)=\left(0,1/2\right)$ transition frequency has another merit in that the positive Stark shift induced by the trap electric field can be canceled by the quadratic Doppler shift. Therefore, the measurement using molecular ions in a Coulomb crystal broadened in the radial direction is also possible, when the Zeeman shift is effectively eliminated.

• Received 14 November 2016

DOI:https://doi.org/10.1103/PhysRevA.95.023418