Conditions for Time-Independence of N-level Systems under the Rotating Wave Approximation (RWA) and Dipole Selection Rules

TL;DR

The paper classifies time-dependence in $N$-level systems under the Rotating Wave Approximation and dipole selection rules by rotating-frame analysis. It shows that $(N-1)+1$ topologies are unconditionally time-independent, while 2+2 and higher-level topologies require detuning conditions that depend on the system's topology (e.g., $ ext{Δ_D}$, $ ext{Δ_H}$, $ ext{Δ_T}$). The approach uses a diagonal rotating-frame unitary to count degrees of freedom against transitions, explaining when a TI form can be achieved and when detuning is necessary. The results have practical implications for constructing quantum gates and STIRAP-like protocols, and the authors discuss avenues for extending beyond-RWA effects via graph theory and Magnus expansions.

Abstract

We analyze the time-dependence of N-level systems under the Rotating Wave Approximation and dipole selection rules. Such systems can be solved straightforwardly if the Hamiltonian can be transformed into a time-independent form. The conditions under which a unitary transformation can be used to render time-dependent Hamiltonians into a time-independent form, thereby making the solution, are examined. After case-by-case analysis of different four and five-level systems, we conclude that systems having only one odd or even parity level achieve time-independence. In contrast, the others must satisfy a condition of laser detuning to achieve time-independence.

Figures (3)

• Figure 1: All possible diagrams of a 3+1 four-level systems. The names were chosen by the author according to symbol resemblance.
• Figure 2: This figure is the schematics of a one-level extension to the 3+1 $\lambda$ system.
• Figure 3: All possible diagrams of a 2+2 four-level systems.