$CPT$ tests with the antihydrogen molecular ion

High-precision radio-frequency, microwave, and infrared spectroscopic measurements of the antihydrogen molecular ion ${\overline{\text{H}}}_2^{-}$ ($\overline{p}\overline{p}{e}^{+}$) compared with its normal matter counterpart provide direct tests of the $CPT$ theorem. The sensitivity to a difference between the positron-antiproton and electron-proton mass ratios and to a difference between the positron-antiproton and electron-proton hyperfine interactions can exceed that obtained by comparing antihydrogen with hydrogen by several orders of magnitude. Practical schemes are outlined for measurements on a single ${\overline{\text{H}}}_2^{-}$ ion in a cryogenic Penning trap that use nondestructive state identification by measuring the cyclotron frequency and bound-positron spin-flip frequency and also for creating an ${\overline{\text{H}}}_2^{-}$ ion and initializing its quantum state.

Figure 1

Examples of positron spin-flip (Zeeman) transitions, 1, 3, and a rotational Zeeman-hyperfine transition, 2, in ${\overline{\text{H}}}_2^{-}$, for the case of $N=2$ in a magnetic field of 5 T (not to scale). Using the Breit-Rabi formula [60] and Zeeman and hyperfine coefficients from Refs. [57, 69], for $v=0$, transitions 1, 2, and 3 have calculated frequencies of 140 082.6, 56.150, and 140 040.4 MHz, respectively; and for $v=1$, 140 085.2, 54.499, and 140 045.6 MHz. This illustrates how $M_N$ and $v$ (and also by extension $M_I$ and $N$) can be identified by measuring the positron spin-flip frequencies, with additional information given by measuring Zeman-hyperfine transition frequencies. For ${\text{H}}_2^{+}$, the level structure is the same, but with the signs of $M_S$ and $M_N$ reversed.