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The analytical expressions of eigenstates and eigenenergies are demonstrated using a parameter ? without the assumption of Bethe anstz for two kinds of N-mode mixing models. The parameter is shown to be determined by the roots of a simple polynomial and is solvable analytically or numerically. In addition, the exact analytical expressions of eigenstates and eigenenergies are illustrated without any unknown parameter.

We present the explicit analytical results of all the eigenstates and eigenvalues using a parameter without the assumption of Bethe ansatz for three different kinds of six bosonic modes models. The parameter is determined by the roots of a polynomial and is solvable analytically or numerically.

We discuss the generation of certain kinds of multi-component entangled states for three-level atomic system with hyperfine structure. The method proposed here is based on the interactions of dispersive cavity with only one atom driven by a strong classical field. It is shown that, with a judicious choice of the cavity detuning and applied coherent field detuning, the atom can interact dispersively with the quantized field but the classical driving field gives rise to the creation or destruction of photons conditional on the state of the system. In comparison with previous schemes, our method is likely to be extremely easy to realize in practice.

We discuss and analyze the absorption-amplification properties of a weak probe field in a typical four-level atomic system in the presence of an additional coherence term, the spontaneously generated coherence term. The influences of the spontaneously generated coherence and a coherent pump field on the probe absorption (amplification) are investigated in detail. We show that the absorption of such a weak probe field can be dramatically enhanced due to the presence of the spontaneously generated coherence. At the same time, the probe-absorption profile exhibits the double-peak structure and the probe-absorption peak gradually decreases as the pump intensity increases. On the contrary, the amplification of such a weak probe field near the line center of the probe transition can be achieved by adjusting the coherent pump field intensity in the absence of the spontaneously generated coherence.

We present an efficient approach to studying the spectra and eigenstates for the model describing interactions among five bosonic modes without using the assumption of the Bethe ansatz. The exact analytical results of all the eigenstates and eigenvalues are in terms of a parameter for a class of models describing five-mode multiphoton process. The parameter is determined by the roots of a polynomial and is solvable analytically or numerically.