Application of matrix product states for few photon dynamics and quantum walks in reduced dimensions

Abstract

Numerical simulations of low-dimensional quantum many-body systems have been a very active field in recent years. New techniques have enabled the experimental realization of these systems and have shed light on both theoretical and technological developments. However, the numerical simulations of these systems have been challenging due to the exponential growth of the Hilbert space with the system size. In addition, quantum correlations such as entanglement play an important role in many-body systems. Therefore approximate methods have been developed. One of the methods to simulate such quantum systems in one dimension is the Matrix Product States (MPS) Formalism. In this thesis, we concentrate on the application of MPS to quantum optical systems and quantum walks. For this purpose, we have developed a pedagogical numerical library that consists of functions responsible for the efficient representation of the wave function and its time evolution. We have tested the efficiency of these functions for different parameters. The quantum optical system we consider is a one-dimensional coupled cavity array interacting with a two-level system. One of the techniques to simulate the long-time dynamics of a quantum many-body system in a computationally manageable grid is to impose absorbing boundary conditions. We have applied absorbing boundary conditions in the form of an imaginary potential and determined the optimum parameter intervals for efficient simulation. Another objective of this thesis is to examine the photon dynamics and the decay of the two-level system from its excited state for different interaction strengths. We have shown that in the strong interaction regime where rotating wave approximation (RWA) is applicable, the results obtained from exact diagonalization and MPS simulations are in perfect agreement. For higher interaction strengths we have used polaron transformation to lower the effective interaction and applied RWA afterward. We have discussed the differences between the results in terms of photon numbers and the excited-state population of the two-level system. As part of this thesis, we have studied two types of discrete-time quantum walks. Firstly, we have considered a quantum walk with a single-phase impurity and investigated the effects of the bound states on its spatial localization and non-Markovianity properties. In Markovian systems, there is an irreversible flow of information from the system under consideration to its environment, whereas in non-Markovian systems some of this information flows back to the system. Our findings show that there is a strong relation between localization and non-Markovianity in this model. Secondly, we turned our attention to a quantum walk coupled with a spin chain environment where there is a dynamic spin attached to each site. Using our MPS algorithm, we have studied the relationship between the quasi-energy spectrum obtained from the exact diagonalization of finite systems, dynamical localization, entanglement entropy, and spin dynamics of this walk. We have observed that due to the extensive number of conserved quantities it possesses, this model is similar to the disorder-free localization models found in literature, where disorder is induced due to the interaction between the constituents of the system.