  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.
