Quantum memories for squeezed and coherent superpositions in a driven-dissipative nonlinear oscillator

Quantum oscillators with nonlinear driving and dissipative terms have gained significant attention due to their ability to stabilize cat states for universal quantum computation. Recently, superconducting circuits have been employed to realize such long-lived qubits stored in coherent states. We present a generalization of these oscillators, which are not limited to coherent states. The key ingredient lies in the presence of different nonlinearities in driving and dissipation, beyond the quadratic one. Through an extensive analysis of the asymptotic dynamical features for different nonlinearities, we identify the conditions for the storage and retrieval of quantum states, such as squeezed states, in both coherent and incoherent superpositions. We explore their applications in quantum computing, where squeezing prolongs the lifetime of memory storage for qubits encoded in the superposition of two symmetric squeezed states, and in quantum associative memory, which has so far been limited to the storage of classical patterns.