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Physical Review A

Exact solutions for the resonant response of multiplets undergoing stochastic modulation

Ernst E. van Faassen

Published Oct 1, 1991• 3 citations
Publisher

Abstract

Line-shape analysis of transitions in a quantum system is a method widely applied to study relaxation processes on a microscopic scale. In many practical cases, this relaxation arises from stochastic modulation of the Hamiltonian due to interactions with a surrounding heat bath. Using a separable class of stochastic operators, the line-shape problem for a Hamiltonian with a single transition has been solved in closed form. This method is not appropriate for more complicated Hamiltonians with closely spaced multiplets, as the modulation may lead to significant coupling between the individual transitions in the multiplet. In this work, therefore, an investigation of the response function of a modulated multiplet is undertaken. The dynamical equation is derived from the stochastic Liouville equation for the density matrix. It is solved rigorously for two practical examples drawn from magnetic-resonance spectroscopy. These exact results are used to investigate the accuracy of approximations found in the literature.

DOI
10.1103/physreva.44.4430

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Line-shape analysis of transitions in a quantum system is a method widely applied to study relaxation processes on a microscopic scale. In many practical cases, this relaxation arises from stochastic modulation of the Hamiltonian due to interactions with a surrounding heat bath. Using a separable class of stochastic operators, the line-shape problem for a Hamiltonian with a single transition has been solved in closed form. This method is not appropriate for more complicated Hamiltonians with closely spaced multiplets, as the modulation may lead to significant coupling between the individual transitions in the multiplet. In this work, therefore, an investigation of the response function of a modulated multiplet is undertaken. The dynamical equation is derived from the stochastic Liouville equation for the density matrix. It is solved rigorously for two practical examples drawn from magnetic-resonance spectroscopy. These exact results are used to investigate the accuracy of approximations found in the literature.
A short-time approximation for the evolution of quantum systems governed by a Hamiltonian with stochastic time dependence is derived. The evolution operator of the system is replaced by an ensemble of evolution operators with time-independent Hamiltonians, weighted by a distribution function related to the ‘‘line shape function’’ of a randomly modulated harmonic oscillator. The approximation conserves the trace of the density matrix and converges to the exact solution in the case of very slow and in the case of very rapid stochastic modulation. The approximation is applied to hyperfine-induced singlet–triplet transitions of a biradical-like system with fluctuating exchange interaction.
Half century has past since the pioneering works of Anderson and Kubo on the stochastic theory of spectral line shape were published in J. Phys. Soc. Jpn. 9 (1954) 316 and 935, respectively. In this review, we give an overview and extension of the stochastic Liouville equation focusing on its theoretical background and applications to help further the development of their works. With the aid of path integral formalism, we derive the stochastic Liouville equation for density matrices of a system. We then cast the equation into the hierarchy of equations which can be solved analytically or computationally in a nonperturbative manner including the effect of a colored noise. We elucidate the applications of the stochastic theory from the unified theoretical basis to analyze the dynamics of a system as probed by experiments. We illustrate this as a review of several experimental examples including NMR, dielectric relaxation, Mössbauer spectroscopy, neutron scattering, and linear and nonlinear laser spectroscopies. Following the summary of the advantage and limitation of the stochastic theory, we then derive a quantum Fokker–Planck equation and a quantum master equation from a system–bath Hamiltonian with a suitable spectral distribution producing a nearly Markovian random perturbation. By introducing auxiliary parameters that play a role as stochastic variables in an expression for reduced density matrix, we obtain the stochastic Liouville equation including temperature correction terms. The auxiliary parameters may also be interpreted as a random noise that allows us to derive a quantum Langevin equation for non-Markovian noise at any temperature. The results afford a basis for clarifying the relationship between the stochastic and dynamical approaches. Analytical as well as numerical calculations are given as examples and discussed.
Relaxation of a motionless three-level atom with a non-equidistant spectrum in a photon heat bath and an external stochastic field is investigated. In the Markovian approximation of the relaxation processes a master equation for the reduced density matrix is derived by making use of the averaging over realizations of the stochastic field. The master equation is solved exactly. The correlation functions, the shapes of the radiation lines, the intensity correlation and photon number spectrums are calculated for the V, Ξ and Λ atoms and realizations of the stochastic field by a delta-correlated and a Kubo-Anderson processes.
When a dynamical system has a perturbation which is considered as a stochastic process, the Liouville equation for the system in the phase space or the space of quantum-mechanical density operators is a sort of stochastic equation. The ensemble average of its formal integral defines the relaxation operator Φ(t) of the system. By the definition Φ(t) = exp K(t), the cumulant function K(t) may be introduced. Some general properties are first discussed for a simple example of an oscillator with random frequency modulation and then, concepts of slow and fast modulation are considered. These concepts can be generalized to more general types of stochastic Liouville equations. It is shown that by various possibilities of defining generalized exponential functions, this approach may be useful to understand some essential features of the problem from an unified point of view.
A theory is formulated for time dependent fluctuations of the spectrum of a single molecule in a dynamic environment. In particular, we investigate the photon counting statistics of a single molecule undergoing a spectral diffusion process. Based on the stochastic optical Bloch equation, fluctuations are characterized by Mandel's QQQ parameter yielding the variance of number of emitted photons and the second order intensity correlation function, g(2)(t)g^{(2)}(t)g(2)(t). Using a semi-classical approach and linear response theory, we show that the QQQ parameter can be described by a three-time dipole correlation function. This approach gene ralizes the Wiener-Khintchine formula that gives the average number of fluorescent photons in terms of a one-time dipole correlation function. We classify the time ordering properties of the three-time dipole correlation function, and show that it can be represented by three different pulse shape functions similar to those used in the context of nonlinear spectroscopy. An exact solution is found for a single molecule whose absorption frequency undergoes a two state random telegraph process (i.e., the Kubo-Anderson sudden jump process.) Simple expressions are obtained from the exact solution in the slow and fast modulation regimes based on appropriate approximations for each case. In the slow modulation regime QQQ can be large even in the long time limit, while in the fast modulation regime it becomes small.
Absorption spectra are calculated for four model systems using two different methods for the calculation: The first is the standard linear response technique, while the second involves direct calculation of the energy dissipated in the sample interacting with the applied field, thus corresponding exactly to what is in fact measured experimentally. This latter technique, which we call the direct method, agrees with a linear response (as it must) for weakly applied fields for which the first-order perturbation analysis underlying linear response is in fact valid. For cases in which the system/field interaction is stronger, the linear approximation becomes poor, and the direct method, which includes the field-induced dynamics and in itself contains no approximations, is far better. The four systems considered consist of either single or coupled pairs of harmonic oscillators or two-level systems (spins) each in a thermal bath and coupled to the field by a displacement coupling. For both linear-response and direct calculations the system/field dynamics is solved exactly and the system/bath dynamics treated using a semigroup technique. For the harmonic oscillators, for which the population in any single excited state can never become large, the nonlinear effects (those present in the direct but not in the linear response) are quite minor. For the two-level/spin/systems, however, substantial corrections to the linear-response result are found in the direct calculation. These include power (or saturation) broadening and the appearance, when the two spins are coupled strongly to each other and one is coupled strongly to the field, of a new line at the averaged frequency of the two spins. For even stronger coupling, the line shape becomes very complex (corresponding to the full 15D algebra of the two spins plus the field), exhibiting several peaks and Fano antiresonances. The direct method, which is based on work of Lebowitz, seems to offer substantial advantages for the analysis of systems in which the applied field interactions are strong: This situation, common in microwave and magnetic resonance spectroscopy, is also relevant to high-energy laser studies.