Dynamics In Correlated Quantum Systems

The interactions between the particles that are responsible of novel phenomena in the context of condensed matter physics, such as high temperature superconductivity,fractional quantum Hall effect, quantum phase transitions, etc. And the correlations among the particles can also induce a range of exotic dynamics of matter wave, including fractional Bloch oscillation, many body coherent destruction of tunneling, and BlochZener oscillation. With the latest advances in the coherent manipulation of quantum systems such as atoms in optical lattices, trapped ions, nuclear spins, superconducting circuits, or waveguide array, simulations of the conventional many body correlated systems of condensed matter physics are accessible. Meanwhile, preparing, manipulating and detecting quantum states can also be realized in these system. In addition, people find that the non-Hermitian Hamiltonian with complex interaction constant can still have full real spectrum. And the evolution of the quantum state exhibits unusual properties that have no counterpart in Hermitian system. Furthermore, the development of the non-Hermitian theory and implementation of optical emulation make the creating new optical and even quantum devices possible. Therefore, the investigation of matter wave dynamics of various correlated systems not only enrich the method of manipulation of quantum state but also connect the research to the basic concepts of quantum physics.In this thesis, we focus on three main topics.Firstly, we systematically investigate the influence of Hubbard interaction on the dynamics of pair state. On the one hand, we study the scattering problem of the two boson or fermion in one dimensional lattice. The scattering matrix of the two particles collision reveals dynamical properties that can have also implications to many-particle systems. Based on this, we propose a scheme to generate the distant entangled states through the collision between particles. On the other hand, we study the dynamics of bound pairs and find that the interactions between the particles can lead to the incomplete wave vectors of the energy band. In the presence of the external field, the imperfect energy band can induce a sudden death of Bloch oscillation, which is associated with disappearance of the correlations between two particles.Secondly, we investigate the influence of the complex external field and coupling constant on the dynamics as well as ground state properties of the the quantum spin model. On the one hand, we study the perfect state transfer in parity- and time-reversal(PT)-symmetric non-Hermitian spin network, which can be ascribed to the evolution operator at a certain period is equivalent to the PT operator. On the other hand, we extend the research scope of the pseudo-Hermitian theory and propose the rotation-timereversal(RT)-symmetric rather than PT-symmetric spin model, which can have full real spectrum. The corresponding Hermitian counterpart possess the local interaction and can be accessed in experiment. Based on the analytical solution, we connect the divergence of the geometric phase to the boundary of the non-Hermitian phase transition and give the corresponding scaling behavior. Our results indicate that the spectrum,dynamics and critical behaviour of the non-Hermitian system are analogous to that of the Hermitian system.Finally, we investigate the unidirectional propagation of the non-Hermitian system.We find that the unidirectional propagation can be achieved when the parameter of the system reaches to the exceptional point or spectral singularity. Using it as a base, we explore the exotic phenomenon of dynamics at exceptional point(spectral singularity),including reflectionless absorption, self-sustain emission and momentum-independent reflectionless transmission. On the other hand, we find that the existence of the exceptional point of the finite system is not a necessary condition for the appearance of the unidirectional plane wave. The results also show that the non-Hermitian system can be used to describe the counterintuitive physical phenomenon, which can’t readily explained in the context of Hermitian system.