We show that spin squeezing implies entanglement for 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 starts 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, specially when the interaction occurs off-resonantly without and with a nonlinear medium contained in the cavity field subsystem. We illustrate that, in 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 non-linear medium. In this case, we illustrate that the insight into entanglement dynamics becomes more clearly in 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.
The interaction between two identical two-level atoms and single-mode binomial light field is studied in anti-Tavis-Cummings model,and the atoms stay in entangled state initially.When the classical field is strong enough,the interaction between the atoms and the light field can turn into an effective anti-Tavis-Cummings model and Tavis-Cummings model under the condition of large detuning.In this paper,we mainly probe into the squeezing property of the binomial state in anti-Tavis-Cummings model,and the influence of initial field parameters on the squeezing property of the field are discussed.The results show that the degree and period of squeezing depend strongly on the initial parameters,and energy nonconservation item has a great impact on squeezing property.