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The Journal of Chemical Physics

The adiabatic and related approximations for quantum particles in harmonic systems

John M. Vail

Published Feb 15, 1987• 2 citations
Publisher

Abstract

The adiabatic approximation is set up to estimate variationally the energy of a quantum particle interacting with a quantum harmonic oscillator. This example is seen as a prototype for a wide range of problems, including molecular clusters, free or in condensed matter. A heirarchy of approximations is introduced, labelled simplified adiabatic, average field, and classical field. The quasiadiabatic limit is then introduced, in which the oscillator takes account, as much as possible, of the instantaneous particle position, and an interpolation procedure between the average field adiabatic and quasiadiabatic limits is derived. Questions of implementation are discussed. In particular, an approximation is introduced, in which the quasiadiabatic generalization of the average field case is found to be energetically favored, and which reduces to a problem similar to the static-field case, for which many-electron computational methods are well developed. Now, however, oscillator zero-point effects are incorporated in a renormalized oscillator self-energy and a modified effective particle-oscillator interaction.

DOI
10.1063/1.452117

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The adiabatic approximation is set up to estimate variationally the energy of a quantum particle interacting with a quantum harmonic oscillator. This example is seen as a prototype for a wide range of problems, including molecular clusters, free or in condensed matter. A heirarchy of approximations is introduced, labelled simplified adiabatic, average field, and classical field. The quasiadiabatic limit is then introduced, in which the oscillator takes account, as much as possible, of the instantaneous particle position, and an interpolation procedure between the average field adiabatic and quasiadiabatic limits is derived. Questions of implementation are discussed. In particular, an approximation is introduced, in which the quasiadiabatic generalization of the average field case is found to be energetically favored, and which reduces to a problem similar to the static-field case, for which many-electron computational methods are well developed. Now, however, oscillator zero-point effects are incorporated in a renormalized oscillator self-energy and a modified effective particle-oscillator interaction.
Quantum control could be implemented by varying the system Hamiltonian. According to adiabatic theorem, a slowly changing Hamiltonian can approximately keep the system at the ground state during the evolution if the initial state is a ground state. In this paper we consider this process as an interpolation between the initial and final Hamiltonians. We use the mean value of a single operator to measure the distance between the final state and the ideal ground state. This measure resembles the excitation energy or excess work performed in thermodynamics, which can be taken as the error of adiabatic approximation. We prove that under certain conditions, this error can be estimated for an arbitrarily given interpolating function. This error estimation could be used as guideline to induce adiabatic evolution. According to our calculation, the adiabatic approximation error is not linearly proportional to the average speed of the variation of the system Hamiltonian and the inverse of the energy gaps in many cases. In particular, we apply this analysis to an example in which the applicability of the adiabatic theorem is questionable.
The von Neumann trace form of quantum statistical mechanics is transformed to an integral over classical phase space. Formally exact expressions for the resultant position-momentum commutation function are given. A loop expansion for wave function symmetrization is also given. The method is tested for quantum harmonic oscillators. For both the boson and fermion cases, the grand potential and the average energy obtained by numerical quadrature over classical phase space are shown to agree with the known analytic results. A mean field approximation is given which is suitable for condensed matter, and which allows the quantum statistical mechanics of interacting particles to be obtained in classical phase space.
In dense media, the Lorentz local-field condition results in a generalization of the usual semiclassical Maxwell-Bloch formulation for two-level atoms in which the optical Bloch equations are recast in terms of macroscopic spatially averaged atomic variables. We derive an adiabaticlike, or stationary, approximation for the generalized Bloch equations in the ultratransient limit and establish validity criteria. The significance of this quasiadiabatic approximation is that a point response solution of the generalized Bloch equations is obtained in terms of the field strength, thereby allowing the atomic response to be expressed as a nonlinear susceptibility that can be used to predict and analyze propagation effects in dense media. The quasiadiabatic approximation is quite general, allowing for a combination of detuning, chirping, and a variety of local-field and mean-field effects. It spans the range from cases for which local-field effects can be neglected, as in the usual adiabatic approximation, to cases in which local-field effects cause a large inversion-dependent frequency renormalization. The approximation is used to interpret previous numerical results for the resonant interaction of ultrashort pulses with dense media and it is shown that dense media can exhibit behavior analogous to adiabatic following and adiabatic inversion. The nonlinear index of refraction in the quasiadiabatic limit is purely real for large detunings. However, at and near resonance, the nonlinear index is purely imaginary. The imaginary index relates to coherent reflectivity arising from the reaction field of the cooperating atoms, rather than absorption, and transmission and reflection coefficients for thin films are derived. In the intermediate range of detunings, the intensity-dependent index of refraction can undergo a phase transition due solely to the variation of the field intensity during excitation by an ultrashort pulse. Propagation effects are investigated using a finite-difference time-domain method to integrate the generalized Bloch-Maxwell equations and the results of the numerical simulations are analyzed in the context of the nonlinear index of refraction obtained using the quasiadiabatic approximation.
The nature of the adiabatic approximation is investigated. This approximation is at the center of the Born-Oppenheimer approximation, commonly employed in the construction of molecular potentials. First, general behaviors of the approximation and some implications of these results are discussed. We show that the ground-state energy, calculated using the adiabatic approximation, will be lower than the calculated energy when no approximations are made. Next, a numerical and analytical application of the adiabatic approximation to systems comprising two coupled harmonic oscillators is described. These systems are used to demonstrate the results derived in the first part of the paper and to investigate how the accuracy of the approximation depends on the frequencies of the oscillators and the coupling strength.
Based on a functional-integral formalism, a generalization of the self-energy-functional theory (SFT) is proposed which is applicable to systems of interacting electrons with disorder. Similar to the pure case without disorder, a variational principle is set up which gives the physical (disorder) self-energy as a stationary point of the (averaged) grand potential. Although the resulting self-energy functional turns out to be more complicated, the formal structure of the theory can be retained since the unknown part of the functional is universal. This allows one to construct nonperturbative and thermodynamically consistent approximations via searching for a stationary point on a restricted domain of the functional. The theory and the possible approximations are worked out for models with local interactions and local disorder. This results in a derivation of different mean-field approaches and various cluster extensions, including well-known concepts such as the statistical dynamical mean-field theory, the molecular coherent-potential approximation, and the dynamical cluster approximation. Due to the common formal framework provided by the SFT, one achieves a general systematization of dynamical approaches, i.e., approaches based on the spectrum of one-particle excitations. Different mean-field and different cluster schemes naturally appear in this framework and complement the existing ones. Their prospects for future applications are discussed.