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arXiv:2409.04156

Krylov Complexity of Time-Dependent Optical Hamiltonians

Abhishek Chowdhury, Aryabrat Mahapatra, Mahapatra, Ajit Prasad

Published Sep 6, 2024• 0 citations
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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.

DOI
10.48550/arxiv.2409.04156

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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(2)SU(2), SU(1,1)SU(1,1)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 VVV 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.
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This study formulates observer-dependent Krylov spreading for non-inertial quantum systems driven by linear Bogoliubov transformations. Starting with the closed single Rindler-pair SU(1,1)SU(1,1)SU(1,1) sector, we show that its Lanczos basis is identical to the Rindler pair-number basis. As a result, the Krylov spread complexity reduces exactly to the mean number of correlated Rindler pairs, Ck=∣βk∣2C_k=\vertβ_k\vert^2Ck​=∣βk​∣2. Within this framework, we demonstrate that Krylov spreading dynamics are governed by the competition between the detuning parameter and the coupling constant, naturally dividing the dynamics into three distinct regimes. Notably, Krylov complexity becomes localized in the detuning-dominated regime. By extending this to a multimode, strictly quadratic Bogoliubov Hamiltonian, we find that inequivalent Rindler wave-packet pairs violate the Ck=∣βk∣2C_k=\vertβ_k\vert^2Ck​=∣βk​∣2 correspondence, thereby highlighting the single-pair SU(1,1)SU(1,1)SU(1,1) model as an exactly solvable, observer-adapted benchmark. In such multimode scenarios, the mean pair-number eigenstates no longer dictate Krylov complexity. Overall, our work provides a new perspective for analyzing Krylov complexity in non-inertial quantum systems.
Krylov complexity is a powerful diagnostic of quantum dynamics, with clear connections to other measures of quantum chaos and operator growth. One such measure is the Loschmidt amplitude, defined as the overlap of initially identical states evolved under two slightly different Hamiltonians. Its decay in certain systems is controlled by the classical Lyapunov exponent. Using the algebraic properties of the Krylov complexity operator, we express Krylov complexity as the derivative of a Loschmidt amplitude whose perturbation is parameterized by an angular variable φφφ. This formulation allows us to define a spectral propagator that encodes the entire complexity distribution, which we characterize for specific types of systems. We study the two-dimensional quantum geometry spanned by time and φφφ where the original and deformed trajectories reside, demonstrating that Krylov complexity is upper-bounded by its volume. We also express the time derivative of Krylov complexity in terms of a distinct Loschmidt amplitude. Depending on the growth of the Lanczos coefficients, the perturbation term in this amplitude can be truncated. We propose that the strength of this perturbation provides a classification scheme for Krylov complexity dynamics and relate it to the φφφ-derivative of the spectral propagator. Using this analytical framework, we derive general relations between the time-dependence of the survival amplitude and Krylov space measures.
We investigate Krylov state complexity in a tilted extended Bose-Hubbard chain in which both the on-site interaction UUU and nearest-neighbor interaction VVV are present. The tilted extended Bose-Hubbard model can be realized with Rydberg-dressed interactions in optical lattices. Using exact diagonalization and Lanczos recursion from three physically motivated Fock states, we compute the time-dependent complexity, its long-time saturation value, and the fluctuations of the Lanczos coefficients for an open chain at unit filling. Crucially, we identify a pronounced quasi-chaotic diagonal regime near U≃VU\simeq VU≃V in which the Krylov complexity saturation exhibits a sharp ridge while the spectral level-spacing statistics remain partially chaotic. This dissociation arises from the competition between the on-site and nearest-neighbor interaction, which generates a dense manifold of near-resonant many-body configurations that is efficiently coupled by the state-dependent dynamics but retains residual spectral structure. Our results show that Krylov state complexity provides complementary, state-sensitive information about many-body chaos that is not captured by the spectral diagnostics alone, establishing Krylov complexity as a useful diagnostic for studying thermalization and information scrambling in experimentally accessible quantum simulators.
At fixed Hamiltonian HHH, changing the initial state changes the cyclic pair and hence generally the Lanczos basis and spread. For every normalizable polynomial preparation ∣ψQ⟩=NQ−1/2Q(H)∣K0⟩\lvertψ_Q\rangle=N_Q^{-1/2}Q(H)\lvert K_0\rangle∣ψQ​⟩=NQ−1/2​Q(H)∣K0​⟩, 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 ∣Q∣2\lvert Q\rvert^2∣Q∣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 Kr(t)≥K0(t)K_r(t)\ge K_0(t)Kr​(t)≥K0​(t), with strict inequality for r≥1r\ge1r≥1 away from revivals. At every fixed finite rrr, the Hermite endpoint has exact all-level amplitudes, whereas the continuous-qqq-Hermite chord chain of double-scaled SYK has an all-level root-free re-Lanczos reconstruction. Writing DrHD_r^{\rm H}DrH​ and Drch(t)D_r^{\mathrm{ch}}(t)Drch​(t) for the mean absolute Fock and chord displacements, respectively, the bounds are Kr≥DrHK_r\ge D_r^{\rm H}Kr​≥DrH​ and Kr(t)≥Drch(t)≥∣n‾r(t)−r∣K_r(t)\ge D_r^{\mathrm{ch}}(t)\ge\lvert\overline n_r(t)-r\rvertKr​(t)≥Drch​(t)≥∣nr​(t)−r∣. Thus rebuilt spread bounds physical displacement and signed chord drift. At fixed 0≤q<10\le q<10≤q<1 and time, Kr(t)−Drch(t)→0K_r(t)-D_r^{\mathrm{ch}}(t)\to0Kr​(t)−Drch​(t)→0 as r→∞r\to\inftyr→∞, 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.
Krylov complexity is an important dynamical quantity with relevance to the study of operator growth and quantum chaos and has recently been much studied for various time-independent systems. We initiate the study of K complexity in time-dependent (driven) quantum systems. For periodic time-dependent (Floquet) systems, we develop a natural method for doing the Krylov construction and then define (state and operator) K complexity for such systems. Focusing on kicked systems, in particular the quantum kicked rotor on a torus, we provide a detailed numerical study of the time dependence of Arnoldi coefficients as well as of the K complexity with the system coupling constant interpolating between the weak and strong coupling regimes. We also study the growth of the Krylov subspace dimension as a function of the system coupling constant.