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Population trapping occurs when a particular quantum-state superposition isimmune to action by a specific interaction, such as the well-known dark statein a three-state lambda system. We here show that in a three-state looplinkage, a Hilbert-space Householder reflection breaks the loop and presentsthe linkage as a single chain. With certain conditions on the interactionparameters, this chain can break into a simple two-state system and anadditional spectator state. Alternatively, a two-photon resonance condition inthis Householder-basis chain can be enforced, which heralds the existence ofanother spectator state. These spectator states generalize the usual dark stateto include contributions from all three bare basis states and disclose hiddenpopulation trapping effects, and hence hidden constants of motion. Insofar as aspectator state simplifies the overall dynamics, its existence facilitates thederivation of analytic solutions and the design of recipes for quantum stateengineering in the loop system. Moreover, it is shown that a suitable sequenceof Householder transformations can cast an arbitrary N-dimensional hermitianHamiltonian into a tridiagonal form. The implication is that a general N-statesystem, with arbitrary linkage patterns where each state connects to any otherstate, can be reduced to an equivalent chainwise-connected system, withnearest-neighbor interactions only, with ensuing possibilities for discoveringhidden multidimensional spectator states and constants of motion.

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