Generating overlap between compass states and squeezed, displaced, or Fock states

We investigate a broad class of nonclassical states, composed of superposed squeezed and displaced number states. The phase-space structure is analyzed, keeping in mind the Heisenberg limited sensitivity in parameter estimation. Appropriate squeezing and displacement parameters are identified, wherein the state fidelity in comparison to the metrologically sensitive compass state is more than $99%$. Also, the equal variance in small shift measurements, for the proposed and compass state, shows identical behavior of the average photon number. In metrological application, the number variance of the compass state, which is small for low coherent amplitude, suggests its potential to estimate damping parameters. Finally, we present the theoretical models corresponding to their preparation.