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Z. Phys. B 91, 235-244 (1993) ZEITSCHRIFT FOR PHYSIK B ?9 Springer-Verlag 1993 Quantum system driven by periodic external field M. Ichiyanagi* Theoretische Festk~Srperphysik, Technische Hochschule Darmstadt, W-6100 Darmstadt, Germany Received: 15 October 1992 / Revised version: 15 December 1992 Abstract. A quantum Floquet Theory for a periodically driven system is studied. For this purpose the periodic external fields are changed to have an increasing ampli- tude with exp(r/t), where r/ is an infinitesimal positive number to be taken to be zero at the end. By using the expansions in terms of inverse powers of the driving fre- quency, periodic factors of the time evolution operator are factorized successively. Each step corresponds to a periodically driven system with different strength of ex- ternal field. This approach produces a time-independent effective Hamiltonian. The effectiveness of the method is examined by applying it to simple models; 1) a forced harmonic oscillator, 2) a particle in the double-well po- tential, and 3) a hydrogen atom in an electric potential. 1. Introduction Over the past ten years, the theory of quantum system perturbed by a rapidly varying intense external field has advanced considerably. In one of the earliest discussions of the interaction of atoms with periodic fields, Bloch and Siegert [1] considered the interaction of stationary spin-half atom in a static magnetic field with an applied rf field. They found that increasing rf field strength causes a shift in the resonance frequency. This problem is ap- proached by the Magnus expansion [2] or by the Floquet theory [3]. In classical mechanics a common way of describing the motion of a Hamiltonian system disturbed by a rap- idly varying external periodic field is to try to separate the mean motion from the superimposed oscillations caused by the external field. This procedure results in a time-independent effective Hamiltonian, in terms of which a stationary state of the driven system can be described. * Present address: Nagasaki Institute of Applied Science, Nagasaki 851-01, Japan Since such an effective Hamiltonian can describe the average effect of the external field over one cycle of its oscillation, such an effective Hamiltonian often is termed as an average Hamiltonian. Resting on this principle, the average Hamiltonian theory has developed as a powerful method of analysis of the motion of quantum system interacting periodic fields [4]. The existence of a time- independent effective Hamiltonian in the case where a time-dependent Hamiltonian is periodic is a direct con- sequence of the Floquet theorem. A source of difficulties arises in calculating it under conditions which allow res- onances. In recent papers, Sauermann and Zhang [5] have pro- posed a formal prescription how to construct a time- independent effective Hamiltonian and have found in- teresting applications of it [6]. Their theory involves the adoption of a factor exp (r/t) on the driving amplitude. The limit r/~0 will be taken after the thermodynamic limti has been performed. Then, they have obtained the decomposition of Floquet type for the evolution operator which is essentially identical to the one obtained for the case r/= 0 (see, Appendix A). This attractive idea has not been completed yet. The aim of this paper is to supplement their theory by developing a new method which leads to a factori- zation of the time evolution operator into periodic and exponential aperiodic components. In order to be able to attack such a problem it is not sufficient to have only the so-called Floquet operator at one's disposal, but it turns out to be necessary to have an expression for the periodic part of the evolution operator. In this paper, we establish the relation between a time-independent effective Ham- iltonian and the unperturbed Hamiltonian of the driven system. The present paper is of a rather physical than mathematical nature in that it aims mainly at the analysis of the naive treatment of perturbative method, not clar- ifying the limit of applicability. We believe that the for- malism developed here in its generality can probably be justified. In Sect. 2, in order to have a basis for our subsequent discussions we recall first some general properties of the