Krylov Complexity of Time-Dependent Optical Hamiltonians

Abstract

State Krylov complexity becomes experimentally concrete in driven optics because its chain can be interrogated through excitation counting, parity, return probability, and phase. Its meaning, however, is fixed jointly by the control history, the prepared state, and the reference used to follow explicit time dependence. We disentangle these ingredients for oscillator, $SU(2)$, $SU(1,1)$, and three-level Hamiltonians and determine when preparation dependence preserves exact solvability. A fixed group displacement of an extremal state can be absorbed into transformed controls, giving an undo-and-count readout for arbitrary drives. A polynomial preparation instead Christoffel-transforms the spectral measure of a fixed operator; an affine moving frame then transports its complete seed-adapted chain even when laboratory Hamiltonians at different times do not commute. We also resolve seed dependence in stroboscopic dynamics and driven three-level geometry. A periodically forced mode started from any finite Fock state has cyclic dimension equal to the denominator of a nonzero-displacement rational rotation and infinite dimension for an irrational rotation. At a half turn its complexity becomes a Laguerre survival deficit and a magnitude-sensitive Wigner readout. In a driven $V$ atom, bright-state geometry interferes with excited-manifold dynamics and controls bright-dark decoupling. Preparation therefore changes not only the initial state but also the chain and the optical observable that realizes its complexity.