Quantum Fisher Information For Su(2) Atomic Coherent States And Su(1,1) Coherent States

As an extension of Fisher information, Quantum Fisher information plays a critical role in quantum estimation theory, quantum information theory and quantum metrology. The parameter to be estimated is bounded from below by Cramer -Rao inequality.On the other hand, the most fundamental parameter estimation task is utilizing the unitary dynamics U(?)= e-i?H, which is common in many experimental setups such as Mach-Zehnder interferometers and the four-wave mixer interferometers. In this paper, We estimate the parameter of the unitary dynamics U(?)=e-i?H via making measurements on su(2) atomic coherent states and su(1,1) coherent states respectively.The su(2) atomic coherent states can be obtained by rotating the Dick states through the angle (?, ?) in angular momentum space. They are also named Bloch states in view of their resembalance to spin states common in nuclear-induction problems. The su(1,1) coherent stat-es to be employed in this paper are those of Perelomov, which are generated via a displace-ment type operator, rather those of Barut and Girardello, which are eigenstates of the lowering operator. Here we consider the parametrized su(2) atomic coherent states and su(1,1) coherent states. Intriguingly, our analytical results demonstrate that the QFI has a rich and subtle physical structure:(i) F? remains constant for the su(2) atomic coherent states and F? remains constant for su(1,1) coherent states, when we determine N (the number of atoms) and k (the Bargmann index); (ii) For su(2) atomic coherent states, F? is govern by ? and F?, for su(1,1) coherent states, F? is govern by ? and F?. More importantly, F? possesses the symmetry with respect to ?=?/2 in the su(2) atomic coherent states, and for su(1,1) coherent states, the symmetry of F? is respect to ?= 0.In this paper, we also study Quantum Fisher information in the Yang-Baxter system. We find that F? remains constant and the F?, is affected by parameter ?. When ?=?/2, F? is a maximum. In addition, the symmetry of F? is respect to ?= ?/2.