Population trapping in three-state quantum loops revealed by Householder reflections

TL;DR

This paper reveals how Householder reflections can break quantum loops into chains, uncover hidden population trapping states, and simplify complex N-state systems for quantum state engineering.

Contribution

It introduces a method using Householder reflections to transform loop linkages into chain structures, revealing hidden states and constants of motion in multi-state quantum systems.

Findings

Householder reflections break three-state loops into chains.

Spectator states generalize dark states with hidden population trapping.

Any N-dimensional Hamiltonian can be reduced to a chain form.

Abstract

Population trapping occurs when a particular quantum-state superposition is immune to action by a specific interaction, such as the well-known dark state in a three-state lambda system. We here show that in a three-state loop linkage, a Hilbert-space Householder reflection breaks the loop and presents the linkage as a single chain. With certain conditions on the interaction parameters, this chain can break into a simple two-state system and an additional spectator state. Alternatively, a two-photon resonance condition in this Householder-basis chain can be enforced, which heralds the existence of another spectator state. These spectator states generalize the usual dark state to include contributions from all three bare basis states and disclose hidden population trapping effects, and hence hidden constants of motion. Insofar as a spectator state simplifies the overall dynamics, its existence facilitates the derivation of analytic solutions and the design of recipes for quantum state engineering in the loop system. Moreover, it is shown that a suitable sequence of Householder transformations can cast an arbitrary N-dimensional hermitian Hamiltonian into a tridiagonal form. The implication is that a general N-state system, with arbitrary linkage patterns where each state connects to any other state, can be reduced to an equivalent chainwise-connected system, with nearest-neighbor interactions only, with ensuing possibilities for discovering hidden multidimensional spectator states and constants of motion.