  We provide a scheme by utilizing a two-cavity setup to generate useful
quantum mechanically entangled states of two cavity fields, which themselves
are prepared in Schrodinger cat states. The underlying atom-field interaction
is considered off-resonant and three atoms are successively sent through the
cavities, initially fed with coherent fields. Analytical solution of the
protocol, followed by conditional measurements on the atoms, produce a family
of eight such entangled states. Entanglement properties of the obtained states
are characterized by the Von Neumann entropy. We reveal the parameter domain
for tuning the entanglement, the prime tuning parameters being the atom-field
interaction time and the field amplitudes. The parameter domains for both
quasi-Bell and non quasi-Bell states are discussed. We also present a Wigner
phase space representation of the reduced state of the cavity, showing negative
values and interference patterns similar to those of a compass state, used in
quantum precision measurements, and despite its large entropy.
