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Physical Review Letters

Time-Dependent Perturbation Theory with a Classical Limit

Shamik Banerjee, Jayanta K. Bhattacharjee

Published Sep 14, 2004• 9 citations
Publisher

Abstract

We construct a quantum mechanical perturbation theory which uses the multiple time scale technique. Working with the time translation operator, we use a variant on the method of Bender and Bettencourt. Our perturbation theory smoothly crosses over to the classical result as Planck's -->0. It is seen that this technique has a nonperturbative element built into it.

DOI
10.1103/physrevlett.93.120403

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We construct a quantum mechanical perturbation theory which uses the multiple time scale technique. Working with the time translation operator, we use a variant on the method of Bender and Bettencourt. Our perturbation theory smoothly crosses over to the classical result as Planck's -->0. It is seen that this technique has a nonperturbative element built into it.
We consider the classical limit of quantum mechanics from the viewpoint of perturbation theory. The main focus is time dependent perturbation theory, in particular, the time evolution of a harmonic oscillator coherent state in an anharmonic potential. We explore in detail a perturbation method introduced by Bhaumik and Dutta-Roy [J. Math. Phys. 16, 1131 (1975)] and resolve several complications that arise when this method is extended to second order. A classical limit for coherent states used by the above authors is then applied to the quantum perturbation expansions and, to second order, the classical Poincaré–Lindstedt series is retrieved. We conclude with an investigation of the connection between the classical limits of time dependent and time independent perturbation theories, respectively.
Abstract The time-dependent quantum perturbation theory developed by Born, Heisenberg and Jordan in 1926 is revisited. We show that it not only reproduces the standard theory formulated in the interaction picture, but also allows one to construct more accurate approximations if time averaging techniques are employed. The theory can be rendered unitary even if the expansion is truncated by using a transformation previously suggested by Heisenberg. We illustrate the main features of the procedure on a simple example which clearly shows its advantages in comparison with the standard perturbation theory.
The multiple-linear-time-scale method is used to construct a perturbation theory for NNN-level quantum systems subjected to time-dependent perturbations. The secular and small-denominator terms which plague conventional time-dependent perturbation theory are avoided; consequently, the theory is useful for treating long-term behavior and resonant interactions. The introduction of a self-energy operator allows the shifts in energy levels to be displayed explicitly. When the perturbation is the dipole interaction with an electromagnetic field, the successive approximations yield the well-known rotating-wave approximation, corrections due to counter rotating terms, and the Bloch-Siegert frequency shift. The formalism is applicable to arbitrary pulse shapes, provided that $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}\ensuremath{\ll}\ensuremath{\hbar}\overline{\ensuremath{\omega}}$, where $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}$ represents the inter-action strength and $\overline{\ensuremath{\omega}}$ is a characteristic frequency of the field. Rabi oscillations induced by multiphoton resonances are automatically included, and this effect is demonstrated in the case of a square pulse with frequency approximately one half the resonance frequency for a transition between two vibrational levels of a molecule. These calculations are compared to an exact numerical solution in order to find the limits of validity of the approximation. Even at high intensities ($I\ensuremath{\sim}{10}^{14}$ W/cm2{\mathrm{cm}}^{2}cm2) the shifted resonance frequency is quite accurate; however, the approximate and exact solutions are slightly out of phase, so that the approximate solution can only be trusted for a few Rabi cycles. For much lower intensities ($I\ensuremath{\sim}{10}^{10}$ W/cm2{\mathrm{cm}}^{2}cm2), the approximate solution is valid for many thousands of Rabi cycles.
We set up a semiclassical approximation which helps us clarify by means of several simple examples the rich variety of time scale in the quantum domain. The underlying structure of quantum and classical mechanics is so completly different that it is naive to expect to reach a classical regime by counting powers of the quantum scale variant Planck's over 2pi. We show although it is possible to define a time scale for nonclassical phenomena, but it is impossible to characterize quantum dynamics through a unique time scale, such as Ehrenfest's time. We use simple systems to critically discuss and illustrate these features of the quantum-classical limit.