Link between Alhassid-Levine and Hioe-Eberly formalisms of $SU(N)$ equation of motion

TL;DR

This paper establishes a connection between two Lie algebra-based formalisms for quantum system dynamics, demonstrating their equivalence and analyzing their implications through a two-level system example.

Contribution

It links the Alhassid-Levine and Hioe-Eberly formalisms using $su(n)$ Lie algebra, providing new insights into quantum dynamics and constants of motion.

Findings

Established a formal link between two Lie algebra approaches.

Identified two constants of motion under time-dependent detuning.

Showed the existence of disjoint subspaces in quantum evolution.

Abstract

Geometric representations of solutions provides intuitive physical insights. To which end studying dynamics of Quantum systems via $su (n)$ Lie algebra proves to be convenient way of obtaining geometric solution. In this paper link is established between two formalisms that made use of Lie algebra to describe equation of motion for quantum system. In both approaches the Hamiltonian and the density matrix are expressed as a linear combination of the Lie group. To exemplify the approach we consider a very well studied two level system coupled by a laser pulse. Beyond establishing link between these two formalism we obtained two constants of motion by assuming time dependent detuning whose time profile is assumed to be same as the laser pulse. Consequently we have shown how one can have two disjoint subspaces whose evolution vector is independent of each other.