2025 Theses Doctoral

Exact and Approximate Numerical Methods for Simulating Open Quantum Systems

Nguyen, Hai Mi

This thesis presents progress towards formulating and applying exact and approximate methods for simulating various open quantum systems.

Chapter 1 presents the formulation of the influence functional of the spin-boson model using a form of tensor networks called the Matrix Product States (MPS) to calculate correlation functions using distinct time contours. In particular, we demonstrate the calculation via two distinct approaches: the construction of the equilibrium influence functional from imaginary and real-time propagation and the construction of the steady state by a uniform MPS approach under the assumption of ergodicity. We find that the uniform MPS method provides the best efficiency and simplicity in implementation, enabling the calculation of the correlation functions for long times.

Chapter 2 explores another system-environment model, the Holstein model, with semiclassical methods. Due to the nature of the Holstein bath as well as the number of discrete sites, the exact methods of Chapter 1 cannot be directly applied. However, the bath can be directly sampled classically and propagated efficiently. We utilize mean-field Ehrenfest and the Modified Approach to Surface Hopping (MASH) methods to compute spectral functions for the Holstein model in various parameter regimes. We find that both give reasonable and encouraging results. This is especially true for the Ehrenfest approach, which can resolve peak splittings at the band edges in small lattices. We apply the mean-field Ehrenfest method to an ab initio model of a hole polaron in LiF, which yields results consistent with a previous result from the cumulant expansion.

Finally, in Chapter 3, we study the open Dicke model, particularly the fate of putative discrete time crystalline behavior under the modulation of the coupling strength for a finite number of two-level atoms. We assess the performance of the second-order cumulant expansion and truncated Wigner approximation methods, and show that the truncated Wigner approximation gives superior performance for the dynamics of the model. By applying the exact Monte Carlo wavefunction method at small system sizes and the approximate truncated Wigner approximation at larger sizes, we demonstrate that discrete time crystalline behavior becomes more robust with increasing system size, as expected, although the trends do not monotonically follow intuitive expectations with an increasing number of two-level atoms.