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The Journal of Chemical Physics

Application of a folding transformation to the Magnus solution for the evolution of periodically time dependent systems

M. Matti Maricq

Published Nov 1, 1986• 31 citations
Publisher

Abstract

Two difficulties arise in the Magnus expansion for the evolution operator of a periodic time dependent Hamiltonian when the eigenvalues of the Hamiltonian are separated by more than 2π/τ, where τ is the period: (1) The Magnus expansion generally does not converge, and (2) the equilibrium properties of the system cannot be determined by statistical mechanical techniques analogous to those for time independent systems. A transformation is introduced that folds the eigenvalue spectrum into the interval −π/τ<λi<π/τ, yet retains the periodicity of the Hamiltonian, thereby alleviating the above difficulties. The method is compared to the approach taken by Shirley [Phys. Rev. B 138, 979 (1965)] for the Floquet solution to the evolution operator. It is applied to a dipole coupled spin system in an oscillating transverse field and the implications of the folding transformation on the spin thermodynamics of this system and the ‘‘Magnus paradox’’ are discussed.

DOI
10.1063/1.451710

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Two difficulties arise in the Magnus expansion for the evolution operator of a periodic time dependent Hamiltonian when the eigenvalues of the Hamiltonian are separated by more than 2π/τ, where τ is the period: (1) The Magnus expansion generally does not converge, and (2) the equilibrium properties of the system cannot be determined by statistical mechanical techniques analogous to those for time independent systems. A transformation is introduced that folds the eigenvalue spectrum into the interval −π/τ<λi<π/τ, yet retains the periodicity of the Hamiltonian, thereby alleviating the above difficulties. The method is compared to the approach taken by Shirley [Phys. Rev. B 138, 979 (1965)] for the Floquet solution to the evolution operator. It is applied to a dipole coupled spin system in an oscillating transverse field and the implications of the folding transformation on the spin thermodynamics of this system and the ‘‘Magnus paradox’’ are discussed.
The effective, or Floquet, Hamiltonian for a periodically time-dependent many-body system plays a role analogous to the energy of a conservative system and can therefore be used to describe the equilibrium state. Restrictions on this analogy and the necessity of a series approximation of the effective Hamiltonian complicate the interpretation of the long-time evolution. Analysis of the effective Hamiltonian for the examples of a dipolar coupled spin system in a weak nonresonant field and for the pulsed spin-locking problem shows it to be characteristically the sum of two (or more) commuting observables plus small off-diagonal terms. This implies that the system evolves to a quasistationary state comprised of two or more independent thermodynamic baths. The properties of this state and its subsequent decay to equilibrium are discussed in detail and compared to the results of magnetic resonance experiments. This analysis overcomes the difficulties, noted in the past as the ``Magnus paradox,'' with the use of time-independent model Hamiltonians to describe approximately the true evolution of the system.
The Floquet-Magnus expansion is a widely used tool to derive effective descriptions of time-periodic quantum systems by approximating their dynamics with a time-independent Hamiltonian. However, its standard formulation is, strictly speaking, restricted to bounded Hamiltonians. In this work, we extend its definition and analysis to a broad class of time-periodic unbounded Hamiltonians. Our approach is based on an a priori distinct nonperturbative framework for the construction of effective Hamiltonians, which we show to reproduce the Floquet-Magnus expansion. A particular strength of our framework is that it allows us to prove that the resulting effective dynamics approximates the original time evolution propagators to arbitrary order in the high-frequency limit without requiring convergence of the Floquet-Magnus expansion, a condition that is already highly restrictive even in the bounded setting. We illustrate the scope of the method on representative models: the quantum Rabi Hamiltonian in the interaction picture, and the periodically driven quantum harmonic oscillator.
We introduce a variational framework for approximating the time-evolution operator U^(t)\hat{U}(t)U^(t) within a physically motivated operator manifold, reformulating quantum dynamics as a tractable problem in operator space using stationary action principle. For periodically driven systems, the resulting approximate evolution operator directly yields an effective Floquet Hamiltonian, offering a non-perturbative alternative to conventional expansion-based methods. The framework is systematically improvable by enlarging the operator pool and naturally incorporates symmetries and physical constraints. When the operator manifold is chosen from the terms of a truncated Magnus expansion, the variational procedure effectively resums the Magnus series within the restricted space, significantly enhancing accuracy. We benchmark the approach on the driven Rabi model, the driven Lipkin-Meshkov-Glick model, and the one-dimensional driven Ising chain, yielding effective Floquet Hamiltonians that are systematically more accurate than low-order Magnus expansions, particularly in regimes where the latter converge poorly, and illustrating applicability to systems with exponentially large Hilbert spaces. Although we focus here on Floquet systems, the formalism applies equally to generic time-dependent Hamiltonians, providing a versatile tool for non-equilibrium quantum dynamics.
This paper derives criteria for the convergence of the Magnus expansion, applied to an arbitrary time dependent two level system, in terms of the eigenvalues of the Hamiltonian for that system. When a weak to moderate intensity field perturbs a two level system the resonance frequency of the system must lie below the frequency of the applied field for the series to converge. An alternative solution for the exponential form of the evolution operator is developed that converges for all values of the expansion parameter and clearly shows the singularities that lead to the failure of the Magnus solution. Finally, a simple unitary transformation is presented that extends considerably the radius convergence of the Magnus series, as illustrated by a spin 1/2 in a linearly polarized field.
We investigate Krylov spread complexity for states evolving under time-dependent Hamiltonians. For periodically driven systems, we formulate the problem within Floquet theory and show how the Magnus expansion provides a systematic approximation when the Floquet Hamiltonian is not available in closed form. We then extend this framework beyond periodic driving and demonstrate that, in addition to the globally truncated Magnus expansion, a piecewise Magnus expansion provides a reliable method when the global expansion loses convergence or accuracy. Our results provide practical tools for analyzing complexity growth in a broad class of time-dependent quantum systems.
A discussion is presented on the use of statistical thermodynamics to describe the long-time behavior of a many-body Hamiltonian that also depends periodically on time. Floquet's theorem is applied to the dynamics of the system in order to exploit the time symmetry. This approach produces an effective Hamiltonian that propagates the system over one time cycle. This is taken as the fundamental constant of the motion for the system and forms the basis for the thermodynamic description. Under suitable conditions, the resulting formulas for the equilibrium values of observables have a strong analogy to the usual thermodynamic expressions for time independent systems. These ideas are applied to describe the equilibrium magnetization for multiply pulsed spin systems, in particular to the Ostroff-Waugh and Waugh-Huber-Haeberlen pulse sequences. For these systems a quasistationary state develops after a few times T2{T}_{2}T2​. This is followed by a slow decay to an equilibrium state. The dynamics of the decay is discussed in terms of an application of the Provotorov theory of saturation to the effective Hamiltonian.