We show that spin-squeezing implies entanglement for the quantum tripartite state, where the subsystem of the bipartite state is identical. We study the relation between spin-squeezing parameters and entanglement through the quantum entropy of a system, starting initially in a pure state when the cavity is binomial. We show that spin-squeezing can be a convenient tool to give some insight into the subsystems entanglement dynamics when the bipartite subsystem interacts simultaneously with the cavity field subsystem, especially when the interaction occurs off-resonantly without and with a nonlinear medium contained in the cavity field subsystem. We illustrate that, in the case of large off-resonance interaction, spin-squeezing clarifies the properties of entanglement almost with full success. However, it is not a general rule when the cavity is assumed to be filled with a nonlinear medium. In this case, we illustrate that the insight into entanglement dynamics becomes more clear in the case of a weak nonlinear medium than in strong nonlinear medium. In parallel, the role of the phase-space distribution in quantifying entanglement is also studied. The numerical results of Husimi Q-function show that the integer strength of the nonlinear medium produces Schrödinger cat states, which is necessary for quantum entanglement.
Cat states in a driven superfluid: role of signal shape and switching protocol
