Ajit Prasad Mahapatra

arXiv2026

Krylov Complexity of Time-Dependent Optical Hamiltonians

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.

quant-ph↗

arXiv2026

Polynomial Initial-State Jumps and Christoffel Transforms in Krylov Complexity

At fixed Hamiltonian $H$, changing the initial state changes the cyclic pair and hence generally the Lanczos basis and spread. For every normalizable polynomial preparation $\lvertψ_Q\rangle=N_Q^{-1/2}Q(H)\lvert K_0\rangle$, we reconstruct its state-Krylov problem exactly from reference cyclic data. The transfer assumes no integrability and applies on finite or infinite cyclic support. Reweighting the reference spectral measure by $\lvert Q\rvert^2$ gives a finite-band connector for every prepared amplitude and a finite-rank Christoffel-Darboux projection for cumulative probabilities and spread, without rerunning ambient-space Lanczos. Fixed-degree seeds preserve the limiting Jacobi coefficients of asymptotically constant chains. Every Charlier number-state jump obeys $K_r(t)\ge K_0(t)$, with strict inequality for $r\ge1$ away from revivals. At every fixed finite $r$, the Hermite endpoint has exact all-level amplitudes, whereas the continuous-$q$-Hermite chord chain of double-scaled SYK has an all-level root-free re-Lanczos reconstruction. Writing $D_r^{\rm H}$ and $D_r^{\mathrm{ch}}(t)$ for the mean absolute Fock and chord displacements, respectively, the bounds are $K_r\ge D_r^{\rm H}$ and $K_r(t)\ge D_r^{\mathrm{ch}}(t)\ge\lvert\overline n_r(t)-r\rvert$. Thus rebuilt spread bounds physical displacement and signed chord drift. At fixed $0\le q<1$ and time, $K_r(t)-D_r^{\mathrm{ch}}(t)\to0$ as $r\to\infty$, with both approaching the same folded-Bessel limit. In Liouville space, the exact unnormalized gap measure family on an open inverse-temperature interval determines the positive transition-resolved measure, assuming the thermal kernel is known and strictly positive and the requisite exponential moments exist. One cyclic solution can therefore be reused across polynomial seeds while separating physical propagation, basis response, and rebuilt spread.

hep-th↗