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Physical review. B, Condensed matter

Infinite-order semiclassical expansion for a two-level system coupled quadratically to an oscillator

J. Florencio

Published Apr 1, 1992• 0 citations
Publisher

Abstract

We use a functional-integral technique to evaluate all the terms of the semiclassical expansion series for the density matrix of a mean-field two-level system coupled to an oscillator with ${\mathit{q}}^{2}$ interaction. Our calculation reproduces the exact result found by direct diagonalization. We also show that our theory yields a lower free energy than that obtained by using the adiabatic approximation.

DOI
10.1103/physrevb.45.7165

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We present a functional-integral method that is suitable for obtaining nonperturbative solutions to the partition function of fermion systems. The density matrix is cast as an expansion about the classical path contributions, and the quantum corrections can be calculated systematically by using retarded propagators in the space holomorphic functions. It is shown that the semiclassical expansion amounts to a high-temperature expansion, where the first nonvanishing quantum correction is of O(${\mathrm{\ensuremath{\beta}}}^{2}$). Each term of the semiclassical expansion may contain the coupling parameter in all orders of perturbation theory. The method can also be extended to boson systems.
This paper shows how to calculate the terms of a semiclassical (WKB) expansion of the quantum-mechanical propagator corresponding to the quartic anharmonic-oscillator potential, V=mω2q2/2 +λq4/4. This nonperturbative treatment expresses each term in the series as a path integral, which is then evaluated in the framework of a formalism, introduced by C. DeWitt-Morette, which does not entail the usual time-slicing operation followed by a limiting procedure. The Gaussian measure used absorbs all the quadratic terms in the expansion of the action functional about a classical path. The covariance of this Gaussian measure is Feynman’s Green function for the small-disturbance operator of the system. This function can be obtained by varying the constants of integration in the classical solution, and therefore the coefficients of the expansion depend only on this classical solution. If the latter is chosen to be the one which tends to its harmonic counterpart when λ→0, then it is seen that the propagator also tends to its harmonic counterpart when λ→0.
We present a numerical study of a two-level system coupled quadratically to a harmonic oscillator. The possibility of an instability as a function of the coupling strength is studied. The model might be relevant for the behavior of quadratically coupled polarons in solids. In the adiabatic limit the thermodynamical properties of the system can be calculated exactly over the whole coupling range. In addition, we discuss computations based on a discrete path-integral representation for the partition function. In the adiabatic limit there is a transition as a function of the coupling strength from the weak-coupling regime to a state where the oscillator mode is soft.