  We propose a reliable scheme to simulate tunable and ultrastrong mixed
(first-order and quadratic optomechanical couplings coexisting) optomechanical
interactions in a coupled two-mode bosonic system, in which the two modes are
coupled by a cross-Kerr interaction and one of the two modes is driven through
both the single- and two-excitation processes. We show that the
mixed-optomechanical interactions can enter the single-photon strong-coupling
and even ultrastrong-coupling regimes. The strengths of both the first-order
and quadratic optomechanical couplings can be controlled on demand, and hence
first-order, quadratic, and mixed optomechanical models can be realized. In
particular, the thermal noise of the driven mode can be suppressed totally by
introducing a proper squeezed vacuum bath. We also study how to generate the
superposition of coherent squeezed state and vacuum state based on the
simulated interactions. The quantum coherence effect in the generated states is
characterized by calculating the Wigner function in both the closed- and
open-system cases. This work will pave the way to the observation and
application of ultrastrong optomechanical effects in quantum simulators.

Quantum Physics