  A suitable scheme to continuously create inversion on an optical clock
transition with negligible perturbation is a key missing ingredient required to
build an active optical atomic clock. Re- pumping of the atoms on the narrow
transition typically needs several pump lasers in a multi step process
involving several auxiliary levels. In general this creates large effective
level shifts and a line broadening, strongly limiting clock accuracy. Here we
present an extensive theoretical study for a realistic multi-level
implementation in search of parameter regimes where a sufficient inversion can
be achieved with minimal perturbations. Fortunately we are able to identify a
useful operating regime, where the frequency shifts remain small and
controllable, only weakly perturbing the clock transition for useful pumping
rates. For practical estimates of the corresponding clock performance we
introduce a straightforward mapping of the multilevel pump scheme to an
effective energy shift and broadening parameters for the reduced two-level
laser model system. This allows to evaluate the resulting laser power and
spectrum using well-known methods.
