Energies of Two-State Systems

TL;DR

This paper introduces a phase-based semi-classical approach to analyze energies in two-state quantum systems, revealing stationary states, quasi-energies, and dynamic precession modes through theoretical and simulation methods.

Contribution

It presents a novel phase-emphasized framework for studying energies in two-state systems, including driven cases and the existence of quasi-energies, supported by simulations.

Findings

Identification of stationary states and energies in non-driven TSSs

Discovery of two quasi-energies in driven TSSs

Visualization of a breathing-spiral precession mode

Abstract

Energies of quantum states are given by the arguments of phase-evolution exponentials. It follows then that an analysis of the energies of a two-state system (TSS) can revolve around phase-emphasized description of states' probability amplitudes in the Schrodinger picture. Here, studying energies of TSSs semi-classically, we suggest an energy-revealing format in which the time-dependence of the probability amplitudes is expressed by phase-evolution factors only. With this fresh energy-studying approach, we first revisit non-driven TSSs, write the conditions for setting a system (in general) in a stationary state, and identify the associated (single) definite energy. Then, more importantly, we revisit driven-TSSs, identify the two stationary states and prove the existence of two quasi-energies associated with each stationary state. Resulting from our phase-keeping framework, we display a "breathing-spiral" type precession-mode of a strongly-driven spin one-half TSS. Further, we exemplify our findings through a set of two coupling-probe computer simulations. For specific examples, we compute and list the energies of several concrete TSSs under typical operating conditions.