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The Journal of Chemical Physics

Convergence of the Magnus expansion for time dependent two level systems

M. Matti Maricq

Published May 15, 1987• 51 citations
Publisher

Abstract

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.

DOI
10.1063/1.452541

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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.
The convergence of the Magnus expansion in the Schr\"odinger representation is investigated with the aid of two exactly solvable models. A baffling observation regarding the dynamics of a spin-(1/2 system driven by a superposition of a constant and a rotating magnetic field is elucidated. Perturbation theory is applied to the exponential solution to the Schr\"odinger equation for a time-dependent harmonic oscillator. The first terms of the perturbation expansion for the singularity of the exponent are exactly calculated. The scope and limitations of perturbation theory in obtaining the range of validity of the Magnus expansion are discussed.
We derive a Magnus expansion for a frequency chirped quantum two-level system. We obtain a time-independent effective Hamiltonian which generates a stroboscopic time evolution. At lowest order the according dynamics is identical to results from using a rotating wave approximation. We determine, furthermore, also the next higher-order corrections within our expansion scheme in correspondence to the Bloch-Siegert shifts for harmonically driven systems. Importantly, our scheme can be extended to more complicated systems, i.e., even many-body systems.
One of the most powerful and potentially useful methods of time-dependent matrix differential equations is the Magnus expansion. In this work, we applied the Magnus expansion for Schrödinger and master equations of a dissipative two-level system interacting with a time-dependent π pulse. Two different models of dissipative two-level systems are considered, namely (i) the system decays from two considered levels into some other levels with given decay rates; (ii) the system decays from its excited level to its ground level. The obtained approximate solutions of a time dynamics by the first-order Magnus expansion are well consistent with the results of numerical calculations obtained by the fourth-order Runge-Kutta method.
Using the Magnus expansion to the fourth order, we obtain analytic expressions for the atomic state of a two-level system driven by a laser pulse of arbitrary shape with small pulse area. We also determine the limitation of our obtained formulas due to limited range of convergence of the Magnus series. We compare our method to the recently developed method of Rostovtsev et al. (PRA 2009, 79, 063833) for several detunings. Our analysis shows that our technique based on the Magnus expansion can be used as a complementary method to the one in PRA 2009.