Triplet q-cat states of the Biedenharn-Macfarlane q-oscillator with q > 1

The normalized triplet q-cat states associated with three orthogonal subspaces of the Fock space representation of the Biedenharn-Macfarlane q-oscillator are constructed as eigenstates of the cubic power of the q-boson annihilation operator. Their corresponding resolution of the identity conditions by three appropriate positive definite measures in the q-integral on the whole complex plane is realized. We show that the antibunching effect and sub-Poissonian statistics as two nonclassical behaviors of the light field are demonstrated by three and two the measurable triplet q-cat states, respectively. The bipartite entanglements via the parity symmetry are another nonclassical behavior of the triplet q-cat states that are examined in this work. One remarkable property is that, contrary to the simple harmonic oscillator, the amount of entanglement in the bipartite entangled triplet q-cat states is an oscillatory function of coherency parameters and oscillation pattern depends on the deformation parameter q. In general, the entanglement becomes weaker as the parameter q increases. Clearly, the triplet q-cat states and their measures as well as the three nonclassical behaviors mentioned above are converted to their corresponding counterparts in the simple harmonic oscillator in the limit q -> 1.