The Q-deformed Barut-Girardello SU(1,1) Coherent States And Schr(?)dinger Cat States And Their Nonclassical Properties

The Yang-Baxter Equation is the solid basics of the quantum statistical model and quantum many-body system etc., which promotes the investigation greatly and obtains some outstanding results in these fields. The quantum Group Theory, established by V. G. Drinfel'd in 1985, offer a perfect mathematical method for the investigation about the quantum statistical model, quantum many-body system and so on in physics.The quantum Group is not a real group, it is a Hopf algebra, and also described as a q-deformation of a Lie algebra. Since the quantum Group was first introduced by V. G.Drinfel'd, some researchers begin to study the quantum Group and their applications in different branches of physics and some outstanding results have been worked out in the several decades. In the Lie algebra, Some researchers have found the q-deformed Bose algebra, SU_q(n) Lie algebra, SU_q(1,1) Lie algebra and so on. In the quantum optics, since L. C. Biedenharn was devoted to extend the concept of coherent states to q-deformed coherent states, various paper about the study of the q-deformed coherent states and their superposition states were published in the several decades. In this paper, we will obtain q-deformed Barut-Girardello SU(1,1) coherent states by the lowering operator of SU_q(1,1) Lie algebra and define the SU(1,1) Schr(?)dinger cat states as the superpositions of these coherent states. Then we discussed the statistical properties of photons and the quadrature squeezing for the q-deformed SU(1,1) Schr(?)dinger cat states.In this paper, we construct the generators of q-deformed SU(1,1) Lie algebra by the q-deformed bosonic operator, and we also obtain the q-deformed Barut-Girardello SU(1,1) coherent states by the lowering operator of SU_q(1,1) Lie algebra. Then we find that the two quadratures X and Y are identical to each other, which satisfies the minimum uncertainty relation of Heisenberg uncertainty relation, and we also get that the statistical properties of photons are always sub-Poissonian for q-deformed Barut-Girardello SU(1,1) coherent states.We define the q-deformed SU(1,1) Schr(?)dinger cat states as the superpositions of q-deformed Barut-Girardello SU(1,1) coherent states with an adjustable angle ? in this paper, and then discuss the quadrature squeezing and the statistical properties of photons for the q-deformed SU(1,1) Schr(?)dinger cat states. We will focus on the cases of odd Schr(?)dinger cat states and even Schr(?)dinger cat states, and then discuss the quadrature squeezing for these Schr(?)dinger cat states. We also discuss the statistical properties of photons for three limiting cases of ?=0, ?/2, ? in the q-deformed SU(1,1) Schr(?)dinger cat states. We find that:For the quadrature squeezing properties of q-deformed Schr(?)dinger cat states, we find that unlike the coherent states, uncertainties in two quadratures are not equal to each other. The quadrature Y is squeezed below the right-hand side of the uncertainty relation, whereas the quadrature X is expanded correspondingly such that the uncertainty relation holds, but the quadratures X and Y always submit to the Heisenberg uncertainty relation. We also plot the variance of the distribution about X and Y with the parameters |z| and q. Notice that one can adjust the values of |z|,q,k and (p in such a way that one can get the minimum uncertainty states.For a measurement of the number of photons in the cases of ?=0, ?=?/2 and ?-? in the q-deformed SU(1,1) Schr(?)dinger cat states, we find only states of even n contribute for ?=0, and for ?=?, only those states for odd n; For ?=?/2, the q-deformed SU(1,1) Schr(?)dinger cat states have the same photon statistics with that of the q-deformed Barut-Girardello SU(1,1) coherent states by comparing the photon number distribution function. We also obtain the statistical properties of photons are always sub-Poissonian when ?=?/2, and the other cases are hard to determine which are dependent on the parameter q and k. Moreover, we find some amusing properties about the q-deformed SU(1,1) Schr(?)dinger cat states in the limit |z|?0, the Mandel's parameter Q_q approaches-1 for ?-? and approaches 0 for ?=?/2, which are not all dependent on the parameter q and k. While, for the case of ?=0 and |z|?0, the Mandel's parameter Q_q does depend on the parameter q and then we can easily change the statistical properties of photons by adjusting the parameter q.