Development and Application of Magnus Expansion based Propagators for Problems in Spectroscopy and Quantum Dynamics

Author

Taner M. Ture, The Graduate Center, City University of New YorkFollow

Date of Degree

6-2024

Document Type

Dissertation

Degree Name

Ph.D.

Program

Chemistry

Advisor

Seogjoo J. Jang

Committee Members

Mark N. Kobrak

Vinod Menon

Jianbo Liu

Subject Categories

Computational Chemistry | Condensed Matter Physics | Optics | Quantum Physics

Keywords

Quantum Dynamics, Magnus Expansion, Open system quantum dynamics, Numerical Integrators

Abstract

Stable and accurate numerical propagators of time-evolution equations in quantum mechanics are required to capture correct dynamical behavior, especially in the long time limit. Magnus expansion (ME) provides a general way to expand the real time propagator of a time dependent Hamiltonian within the exponential such that the unitarity is satisfied at any order. Integrators are developed by truncating the ME and using explicit integration of Lagrange interpolation formulas for the time dependent Hamiltonian within each time interval. The derived approximations are studied in a numerical test and compared to other available expressions. The sixth order expression is applied to study the three-level $\lambda$ system under periodic driving. The dynamics are calculated with an exponential speedup by taking advantage of the periodicity of the time-dependent Hamiltonian. Numerical results are compared with steady-state analytical results. The rotating wave approximation is compared to the exact dynamics. One of the derived fourth order expression is used for the numerical propagation of several quantum master equations. The Arnoldi iteration is used to calculate the action of the Liouville superoperator onto the density matrix. Analytical expectation values are found as a basis for comparison, and the stability and accuracy of the time-evolution is assessed. Using the techniques in this work, the derived ME-based propagators can be used for stable and accurate time-evolution of time-local quantum master equations with arbitrary time dependent coefficients.

Recommended Citation

Ture, Taner M., "Development and Application of Magnus Expansion based Propagators for Problems in Spectroscopy and Quantum Dynamics" (2024). CUNY Academic Works.
https://academicworks.cuny.edu/gc_etds/5797