Nonclassical properties and quantum resources of hierarchical photonic superposition states

We motivate and introduce a class of "hierarchical" quantum superposition states of $N$ coupled quantum oscillators. Unlike other well-known multimode photonic Schr\"{o}dinger cat states such as entangled coherent states, the hierarchical superposition states are characterized as two-branch superpositions of tensor products of single-mode Schr\"{o}dinger cat states. In addition to analyzing the photon statistics and quasiprobability distributions of prominent examples of these nonclassical states, we consider their usefulness for high-precision quantum metrology of nonlinear optical Hamiltonians and quantify their mode entanglement. We propose two methods for generating hierarchical superpositions in $N=2$ coupled microwave cavities which exploit currently existing quantum optical technology for generating entanglement between spatially separated electromagnetic field modes.