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The Journal of Chemical Physics

Group theoretical approach to semiclassical dynamics: Single mode case

Robert Gilmore, Jianmin Yuan

Published Jan 1, 1987• 56 citations
Publisher

Abstract

The algebraic structure of a truncated semiclassical Hamiltonian describing the translational–vibrational interaction has been determined and exploited to compute the S matrix for the system. The equations of motion have been integrated in a 4×4 faithful nonunitary representation of the group, and the final results lifted to the infinite dimensional unitary representation of the group. Computations of transition probabilities are facilitated by using powerful disentangling theorems to write the S matrix in convenient operator-ordered form. These methods are in quantitative agreement with results of Gazdy and Micha, and generalize easily to the multimode case. The structure of the Ehrenfest equations of motion for the operators of the algebra is identical to the structure of the dynamical algebra itself. The S matrix is also reconstructed from these equations of motion. Finally, a simple geometric picture is presented to describe the two extreme limits of the parametric oscillator problem involving only single quantum transitions or double quantum transitions.

DOI
10.1063/1.452745

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The algebraic structure of a truncated semiclassical Hamiltonian describing the translational–vibrational interaction has been determined and exploited to compute the S matrix for the system. The equations of motion have been integrated in a 4×4 faithful nonunitary representation of the group, and the final results lifted to the infinite dimensional unitary representation of the group. Computations of transition probabilities are facilitated by using powerful disentangling theorems to write the S matrix in convenient operator-ordered form. These methods are in quantitative agreement with results of Gazdy and Micha, and generalize easily to the multimode case. The structure of the Ehrenfest equations of motion for the operators of the algebra is identical to the structure of the dynamical algebra itself. The S matrix is also reconstructed from these equations of motion. Finally, a simple geometric picture is presented to describe the two extreme limits of the parametric oscillator problem involving only single quantum transitions or double quantum transitions.
A signal-flow-graph representation of Schr\"odinger's equation appropriate to the analysis of multiphoton interactions in the semiclassical limit is described. Within the rotating-wave and quasistatic approximations the equations of motion for the density-matrix elements can be recast into a system of algebraic equations on the basis of which the signal flow graphs are constructed. By introducing selected graph manipulations, the graph can be simplified and direct extraction of information is facilitated by using standard graph algebra. Illustrations are given for typical cases involving three waves and three- or four-level systems for serial, parallel, and parametric interactions. Also, the construction of an ac Stark graph, which is specifically used for the determination of ac Stark shifts, is illustrated. We show that perturbation results for virtual transitions as well as for real transitions can be obtained directly from the graph.
The interaction of a quantum system with an oscillating field is studied in a formalism which replaces the semiclassical time-dependent Hamiltonian with a time-independent Hamiltonian represented by an infinite matrix. The formalism is developed as a mathematical equivalent to the semiclassical treatment, and interpreted as a classical approximation to the quantum treatment of the field. Combined with a perturbation theory for two nearly degenerate states, the formalism provides a convenient method for determining resonance transition probabilities including frequency shifts and multiple quantum transitions. The theory is illustrated by a detailed study of the simple case of a two-state system excited by a strong oscillating field.
A recently formulated continuum limit semiclassical initial value series representation (SCIVR) of the quantum dynamics of dissipative systems is applied to the study of vibrational relaxation of model harmonic and anharmonic oscillator systems. As is well known, the classical dynamics of dissipative systems may be described in terms of a generalized Langevin equation. The continuum limit SCIVR uses the Langevin trajectories as input, albeit with a quantum noise rather than a classical noise. Combining this development with the forward-backward form of the prefactor-free propagator leads to a tractable scheme for computing quantum thermal correlation functions. Here we present the first implementation of this continuum limit SCIVR series method to study two model problems of vibrational relaxation. Simulations of the dissipative harmonic oscillator system over a wide range of parameters demonstrate that at most only the first two terms in the SCIVR series are needed for convergence of the correlation function. The methodology is then applied to the vibrational relaxation of a dissipative Morse oscillator. Here, too, the SCIVR series converges rapidly as the first two terms are sufficient to provide the quantum mechanical relaxation with an estimated accuracy on the order of a few percent. The results in this case are compared with computations obtained using the classical Wigner approximation for the relaxation dynamics.
The structure of few-body Hamiltonian matrices is studied in the semiclassical regime. Given (A${^}_{1}$(q^,p^),A${^}_{2}$ (q^,p^)), a pair of operators, it can be shown that, under quite general conditions, A${^}_{2}$ takes the form of a banded matrix in the ordered eigenrepresentation of A${^}_{1}$. Moreover, the bandwidth depends only on \ensuremath{\Elzxh} and certain generalized microcanonical averages. In particular, if H=H0{H}_{0}H0​+\ensuremath{\epsilon}V, this implies that the perturbed Hamiltonian is banded in the appropriately ordered unperturbed basis.
Feynman’s disentangling theorem is applied to noncommuting operators in the problem of quantum parametric oscillator, which is mathematically equivalent to the problem of SU (1, 1) pseudospin rotation. The number states of the oscillator correspond to unitary irreducible representations of the SU (1, 1) group. Feynman disentangling is combined with group-theoretic arguments to obtain simple analytical formulas for the matrix elements and transition probabilities between the initial and final states of the oscillator. Feynman disentangling of time evolution operators is also discussed for an atom or ion interacting with a laser field and for a model Hamiltonian possessing the “ hidden” symmetry of the hydrogen atom.
Various equivalent representations of dynamical invariants (or constants of motion) are derived for N-level systems in mixed states with particular emphasis on so-called coherence vector invariants. They appear as certain homogeneous forms in the real solutions of the von Neumann equation with coefficients given by multilinear forms in the completely symmetric structure constants of the Lie-algebra of SU(N). The treatment is motivated by the close analogy between Lax pairs For classical dynamical systems and pairs given by density and Hamilton operators. In both cases the underlying mathematical structure is essentially determined by the properties of symmetric functions. However, in the quantum case more theorems of general validity can be derived due to the peculiar properties of density operators. In particular, the maximum number of functionally independent invariants in relation to the spectral properties is obtained, as well as bounds on their order of magnitude, the latter from an extremal property analysis. From group-theoretical methods a complete classification of all possibilities of completely incoherent state dynamics is deduced. As a by-product a simple algorithm for the explicit determination of all (N-1)-dimensional irreducible representation matrices of the symmetric group S N and an associated construction of hyperpolyhedra is worked out. Finally, the importance of invariants is stressed for the control of numerical accuracy in large-scale computations in very high dimensions which, after taking partial traces, can be used For the description of irreversible processes. In summary, the results will be of practical relevance for applications to problems of short-time dynamics in molecular laser spectroscopy, quantum optics and magnetic resonance.