Kerr magnon assisted asymptotic stationary photon-phonon squeezing

Bosonic two-mode squeezed states are paradigmatic entangled states in continuous variable systems, which have broad applications in quantum information processing. In this work, we propose a photon-phonon squeezing protocol assisted by a Kerr magnon within a hybrid cavity magnomechanical system. We construct an effective Hamiltonian that accounts for photon-phonon squeezing through strong photon-magnon interaction and precise modulation over the driving frequency on the photon mode. The effective Hamiltonian can be confirmed by a fascinating method about the diagonalization of the system's Liouvilian superoperator. This method can address the level attractions rather than avoided level crossings in the energy diagram of the whole system. With the effective Hamiltonian and quantum Langevin equation, we provide a rigorous theoretical solution for the dynamical process of squeezing generation. Our finding indicates that asymptotic stationary squeezing can be obtained by optimizing the squeezing quadrature operator, even when the covariance matrix of the system still varies with time. This squeezing level can exceed the maximum value under stable conditions. Moreover, our analysis also reveals that the Kerr nonlinearity of the magnon can further promote the squeezing generation. Our work provides an extendable framework for generating squeezed states that entangle two Gaussian modes with indirect coupling.