Stephan Schiller

Sympathetic cooling of trapped ions has been established as a powerful technique for manipulation of non-laser-coolable ions (Raizen1992,Waki1992,Bowe1999,Barrett2003). For molecular ions, it promises vastly enhanced spectroscopic resolution and accuracy. However, this potential remains untapped so far, with the best resolution achieved being not better than $5\times10^{-8}$ fractionally, due to residual Doppler broadening being present in ion clusters even at the lowest achievable translational temperatures (Bressel2012). Here we introduce a general and accessible approach that enables Doppler-free rotational spectroscopy. It makes use of the strong radial spatial confinement of molecular ions when trapped and crystallized in a linear quadrupole trap, providing the Lamb-Dicke regime for rotational transitions. We achieve a line width of $1\times10^{-9}$ fractionally and $1.3~\textrm{kHz}$ absolute, an improvement by $50$ and nearly $3\times10^{3}$, respectively, over other methods. The systematic uncertainty is $2.5\times10^{-10}$. As an application, we demonstrate the most precise test of $\textit{ab initio}$ molecular theory and the most precise ($1.3~\textrm{PPB}$) spectroscopic determination of the proton mass. The results represent the long overdue extension of Doppler-free microwave spectroscopy of laser-cooled atomic ion clusters (Berkeland1998) to higher spectroscopy frequencies and to molecules. This approach enables a vast range of high-precision measurements on molecules, both on rotational and, as we project, vibrational transitions.

We demonstrate a simple method for the determination of the magnetic field in an ion trap using laser-cooled Be+ ions. The method is not based on magnetic resonance and thus does not require delivering radiofrequency (RF) radiation to the trap. Instead, stimulated Raman spectroscopy is used, and only an easily generated optical sideband of the laser cooling wave is required. The d.c. magnetic vector, averaged over the Be+ ion ensemble, is determined. Furthermore, the field strength can be minimized and an upper limit for the field gradient can be determined. The resolution of the method is 0.04 G at present. The relevance for precision rovibrational spectroscopy of molecular hydrogen ions is briefly discussed.

We calculate the DC Stark effect for three molecular hydrogen ions in the non-relativistic approximation. The effect is calculated both in dependence on the rovibrational state and in dependence on the hyperfine state. We discuss special cases and approximations. We also calculate the AC polarisabilities for several rovibrational levels, and therefrom evaluate accurately the black-body radiation shift, including the effects of excited electronic states. The results enable the detailed evaluation of certain systematic shifts of the transitions frequencies for the purpose of ultra-high-precision optical, microwave or radio-frequency spectroscopy in ion traps.

The systematic shifts of the transition frequencies in the molecular hydrogen ions are of relevance to ultra-high-resolution radio-frequency, microwave and optical spectroscopy of these systems, performed in ion traps. We develop the ab-initio description of the interaction of the electric quadrupole moment of this class of molecules with the static electric field gradients present in ion traps. In good approximation, it is described in terms of an effective perturbation hamiltonian. An approximate treatment is then performed in the Born-Oppenheimer approximation. We give an expression of the electric quadrupole coupling parameter valid for all hydrogen molecular ion species and evaluate it for a large number of states of H2+, HD+, and D2+. The systematic shifts can be evaluated as simple expectation values of the perturbation hamiltonian. Results on radio-frequency (M1), one-photon electric dipole (E1) and two-photon E1 transitions between hyperfine states in HD+ are reported. For two-photon E1 transitions between rotationless states the shifts vanish. For a subset of rovibrational one-photon transitions the quadrupole shifts range from 0.2 to 10 Hz for an electric field gradient of 0.1 GV/m2. We point out an experimental procedure for determining the quadrupole shift which will allow reducing its contribution to the uncertainty of unperturbed rovibrational transition frequencies to the 1.10^(-15) relative level and, for selected transitions, even below it. The combined contributions of black-body radiation, Zeeman, Stark and quadrupole effects are considered for a large set of transitions and it is estimated that the transition frequency uncertainty of selected transitions can be reduced below the 1.10^(-15) level.

We review the current status of the field of sympathetically cooled molecular ions, i.e. the method used for cooling the translational degrees of freedom, the techniques applied for detection and analysis of the cold trapped ion ensembles produced as well as first applications. We summarize a few salient results and point out various possibilities for future developments.