  We present theoretical and experimental progress towards a new approach for
the precision spectroscopy, coherent manipulation and state-to-state chemistry
of single isolated molecular ions in the gas phase. Our method consists of a
molecular beam for creating packets of rotationally cold neutrals from which a
single molecule is state-selectively ionized and trapped inside a
radiofrequency ion trap. In addition to the molecular ion, a single co-trapped
atomic ion is used to cool the molecular external degrees of freedom to the
ground state of the trap and to detect the molecular state using
state-selective coherent motional excitation from a modulated optical-dipole
force acting on the molecule. We present a detailed discussion and theoretical
characterization of the present approach. We simulate the molecular signal
experimentally using a single atomic ion indicating that different rovibronic
molecular states can be resolved and individually detected with our method. The
present approach for the coherent control and non-destructive detection of the
quantum state of a single molecular ion opens up new perspectives for precision
spectroscopies relevant for, e.g., tests of fundamental physical theories and
the development of new types of clocks based on molecular vibrational
transitions. It will also enable the observation and control of chemical
reactions of single particles on the quantum level. While focusing on N$_2^+$
as a prototypical example in the present work, our method is applicable to a
wide range of diatomic and polyatomic molecules.
