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American Journal of Physics

A modified time-dependent perturbation approach to quantum dynamics and its application to relaxation phenomena in magnetic molecules

César Augusto Dartora, G. G. Cabrera, Kleber Zuza Nobrega

Published Jul 13, 2009• 1 citation
Publisher

Abstract

We introduce a modified version of the interaction picture applicable to situations in which the nonperturbed Hamiltonian is time-dependent but can be diagonalized by the same basis of states at all times. We probe the method using a simple model Hamiltonian to study the spin dynamics of molecular magnets with time-dependent fields. Fully quantum mechanical calculations show the superiority of our method compared to the usual time-dependent perturbation theory. Our approach takes into account time-dependent fields and converges rapidly to the exact solution.

DOI
10.1119/1.3133088

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We introduce a modified version of the interaction picture applicable to situations in which the nonperturbed Hamiltonian is time-dependent but can be diagonalized by the same basis of states at all times. We probe the method using a simple model Hamiltonian to study the spin dynamics of molecular magnets with time-dependent fields. Fully quantum mechanical calculations show the superiority of our method compared to the usual time-dependent perturbation theory. Our approach takes into account time-dependent fields and converges rapidly to the exact solution.
We propose a method to describe the evolution of two spins coupled by hyperfine i nteraction in an external time- dependent magnetic field. We apply the approach to the case of hyperfine interaction with axial symmetry, which can be solved exactly in a constant, appropriately oriented magnetic field. In order to t reat t he n onstationary d ynamical p roblem, we modify the time-dependent Schrödinger equation through a change of representation that, by exploiting an instantaneous (adiabatic) basis makes the time-dependent Hamiltonian diagonal at any time instant. The solution of the transformed time-dependent Schrödinger FRVBUJPO in the form of chronologically ordered exponents with transparent pre-exponential coefficients is reported. This solution is highly simplified when an adiabatically varying magnetic field perturbs the system. The approach here proposed may be used for the perturbative treatment of other dynamical problems with no exact solution.
A perturbation series is derived for the Schr\"odinger equation where the perturbation is permitted to go to \ensuremath{\infty}. Such an asymptotic series can be obtained in a representation where the time evolution of the states is due to the unperturbed part of the Hamiltonian of the problem and the time evolution of the observables is determined by the perturbation, contrary to the interaction picture. The equivalence between the series given by such a picture, and an asymptotic perturbation scheme is given. The method is applied to a spin-1/2 particle in a two-component magnetic field, one component of which, considered as a perturbation, can be time varying. Choosing a constant perturbation we show the equivalence between the exact solution and the approximate one when a component of the magnetic field is much larger than the other. In this case the limit t\ensuremath{\rightarrow}\ensuremath{\infty} cannot be taken, as mixed-secular terms appear in the asymptotic series. Taking a linear time-varying dependence for a component of the magnetic field, we get a nonanalytic asymptotic series.
We study the dynamics of molecular nanomagnets through a fully quantum mechanical model describing high-spin and high-anisotropy magnetic molecules subjected to a time-dependent magnetic field along the quantization axis, which continuously inverts the population of spin states. Crystals of molecular nanomagnets placed inside a resonant cavity interact with a quantized electromagnetic field. Relaxation of excited states takes place by means of spin-photon interaction, allowing stimulated emission of radiation and creating a maser effect.
We apply our recently developed resonance perturbation theory to describe the dynamics of magnetization in paramagnetic spin systems interacting simultaneously with local and collective bosonic environments. We derive explicit expressions for the evolution of the reduced density matrix elements. This allows us to calculate explicitly the dynamics of the macroscopic magnetization, including characteristic relaxation and dephasing timescales. We demonstrate that collective effects (i) do not influence the character of the relaxation processes but merely renormalize the relaxation times and (ii) significantly modify the dephasing times, leading in some cases to a complicated (time inhomogeneous) dynamics of the transverse magnetization, governed by an effective time-dependent magnetic field.
In 1937, Rabi treated the problem of a magnetic moment in an applied time-dependent magnetic field. This became the well-known magnetic resonance situation. The Hamiltonian is often taken to be [Formula: see text] = – µ · [[Formula: see text]]. In this paper, the Rabi oscillations formula, describing the spin flipping, is derived in an unusual way. The method uses a modification of a method due to Heitler. In the Heitler method, one uses the Interaction Picture of quantum mechanics. Due to the time-dependence in the problem, the usual Heitler method fails. However, the solution is found after quantizing the electromagnetic field. To better understand the origin of the spin flipping, the analogous time-independent problem is also solved. It is made clear that the origin of the Rabi oscillations is not due to the time-dependent magnetic field. The spin flipping is essentially due to the fact that the spin system, when initially prepared, is not in an eigenstate of the Hamiltonian. Thus, as times progresses, the system naturally evolves through the noneigenstate basis states.PACS Nos.: 03.65.–w, 76.20.+q
We develop a systematic method of operator perturbation theory in the Heisenberg picture, which is the formal analog of time-independent stationary-state perturbation theory in the Schr\"odinger picture. However, we use the eigenoperators and operator eigenfrequencies to calculate the time evolution of the dynamical variables of interest. The spectrum of the unperturbed Hamiltonian is assumed to be discrete. The method is especially well-suited for treating systems where the unperturbed Hamiltonian represents noninteracting fermions and/or bosons, and therefore, we illustrate the method using the exactly solvable model of Jaynes and Cummings: a single two-level atom interacting with a single quantized field mode. The time evolution of these operator solutions is shown to be unitary order by order in powers of the interaction strength. We find that the time evolution of exact operator solutions is well approximated for significantly longer times by the time evolution of this method's operator solutions than by the time evolution of the same operator solutions calculated using the Dyson expansion. We also demonstrate that time-dependent operator solutions are very convenient for computing quantities such as correlation functions.