Engineering entangled Schrödinger cat states of separated cavity modes in cavity QED

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 Schr"odinger 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.