Research On Correlation Of Two Anyons In One-dimensional Optical Lattices

In recent years in the field of condensed matter physics of cold atoms, quantum correlation in one-dimensional many-body quantum systems receives widespread attention. So, considering a vivid simulation of the solid lattice environment by optical lattice, people construct various one-dimensional optical lattice models formed by bosons and fermions based on the Boson-Hubbard model. The implementation of the quantum correlation dynamics manipulation through the study of multiparticle quantum walks, to a great extent, promotes the developments of quantum algorithms and quantum computations, which are based on the theory of quantum walks.Beginning with continuous-time quantum walks theories, via the numerical method, this paper firstly discusses the quantum walks of two identical particles in one-dimensional lattices with periodic boundary conditions, and focuses on the effects of quantum statistics、initial quantum state、interaction strength and interaction pattern on the correlation of two quantum walkers. Secondly, we analyses the bloch oscillations of entangled state induced by linear potential, which indirectly show the periodic evolution of the dynamics of quantum walks, and the nonlinear terms breaks the oscillation periods. And then, a brief discussion about Anderson Localization of quantum correlation in a disordered lattice is given.On the base of the above works, we mainly investigate the impact of the fractional statistical parameter on second-order correlations of two interacting anyonic quantum walkers as well as hard-core anyonic quantum walkers based on the Anyon-Boson mapping and Anyon-Fermion mapping, respectivly. The results show that the two-body correlations in position space are symmetric about the initial sites of two quantum walkers in the Bose limit and Fermi limit, while in momentum space this happens only in the Bose limit. An interesting asymmetry arises in the correlation once the statistical parameter deviates from the two limits. It turns out that the origin of this asymmetry comes from the fractional statistics that anyons obey. On the other hand, the two-body correlations of the anyons defined by Girardeau, i.e. the hard-core anyons, show uniform behaviors from antibunching to cowalking regardless of the statistical parameter with increasing nearest-neighbor interaction. The momentum correlations in the case of strong interaction undergo a smooth process of two stripes smoothly merging into a single one with increasing fractional statistical parameter, which is representative of the evolution of fermions into hard-core bosons. Eventually, two mappings do not realize a perfect transition between bosons and fermions, any one only converges to one limit. Thus anyons are not simply intermediate particles between bosons and fermions.