Continuous matrix product states & cavity QED

Abstract

Existing studies [1] have proposed similarities between the continuous Matrix Product States (cMPS) which describes low energy states of quantum field theories, and the input-output formalism which describes the quantum state of electromagnetic field leaking out of an optical cavity. This research investigates the above relationship in greater detail, to understand and develop new tools to simulate exotic quantum field systems, focusing on bosonic gases in one dimension. Using the formalism developed by [2, 1] we analysed the Lieb-Liniger model which describes a bosonic one-dimensional gas subject to a delta potential interaction. We produced a simpler algorithm to find the ground state energy of the Lieb-Liniger model. We demonstrated that its expectation value in terms of matrix products states can be rewritten as the expectation value of an effective closed Jaynes-Cummings Hamiltonian when we use a cavity-atom system as a simulator. The Hamiltonian of this simulator corresponds to a laser-driven atom in the Jaynes-Cummings model with a lossy cavity. This analytical approach makes two contributions to existing knowledge. First, it corrects an intrinsic error in the numerical simulation of [3] due to the normal order of the annihilation operators. Second, it gives us a better understanding of why the simulator used in [3], is so effective in reproducing the correlation properties of the Lieb-Liniger model. Furthermore, we also find that depending on the Hamiltonian used as a simulator, the form of the effective Hamiltonian that would describe the Lieb-Liniger model will reflect the dynamical properties of said simulator. Finally, we derived the transformation rules that relate the Hamiltonian of the simulator with the analytical form of the effective Hamiltonian, concluding with the analysis for the case of a mixture of gases. Thus, our findings could help in the selection of appropriate simulators for any given field theory.