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American Journal of Physics

Quantum mechanics of periodic systems

J. C. Garrison

Published Mar 1, 1999• 9 citations
Publisher

Abstract

The quantum description of a periodic system can be naturally embedded in an auxiliary problem with an extended configuration space. Standard methods applied to the solution of the extended Schrödinger equation yield the results usually obtained by the use of group theory or extensions of Floquet’s theorem. This method is used to prove Bloch’s theorem and to construct a multimode Floquet method for the semiclassical theory of lasers interacting with atoms.

DOI
10.1119/1.19226

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The quantum description of a periodic system can be naturally embedded in an auxiliary problem with an extended configuration space. Standard methods applied to the solution of the extended Schrödinger equation yield the results usually obtained by the use of group theory or extensions of Floquet’s theorem. This method is used to prove Bloch’s theorem and to construct a multimode Floquet method for the semiclassical theory of lasers interacting with atoms.
In this chapter,we shall analyze the particular case of an atom interacting with a laser pulse whose duration is sufficiently long, so that the evolution of the atom in the laser field is adiabatic. When this condition is fulfilled, the atom can be considered to interact with a monochromatic laser field. As a consequence, the Hamiltonian of the system is periodic in time, and the Floquet theory [1] can be used to solve the time-dependent Schrödinger equation (TDSE) non-perturbatively. We begin in Section 4.1 by considering the Hermitian Floquet theory. We first derive the Floquet theorem for a monochromatic, spatially homogeneous laser field and show that the solutions of the TDSE correspond to dressed states having real quasi-energies , which can be obtained by solving an infinite system of time-independent coupled equations. We then generalize the Floquet theory to multicolor laser fields and to “non-dipole” laser fields which are not spatially homogeneous. In Section 4.2, the Floquet theory is applied to study the dynamics of a model atom having M discrete levels interacting with a monochromatic laser field. In this case, the coupling between the bound and continuum atomic states is neglected.We analyze the relationship between the Floquet theory and the rotating wave approximation, and examine the perturbative limit of the Floquet theory. We also consider the population transfer between Floquet dressed states.
A classical theorem of Stone and von Neumann says that the Schrödinger representation is, up to unitary equivalences, the only irreducible representation of the Heisenberg group on the Hilbert space of square-integrable functions on configuration space. Using the Wigner-Moyal transform we construct an irreducible representation of the Heisenberg group on a certain Hilbert space of square-integrable functions defined on phase space. This allows us to extend the usual Weyl calculus into a phase-space calculus and leads us to a quantum mechanics in phase space, equivalent to standard quantum mechanics. We also briefly discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase space Schrödinger equation
The semiclassical interaction of a two-level system with a multimode electromagnetic field is discussed in detail. The equations of motion for the macroscopic density matrix are transformed into a single integral equation for the population density, with a complex kernel which can take into account both running- and standing-wave configurations. The iterative solution to this integral equation, which corresponds to an expansion in terms of powers of the electromagnetic field amplitudes, turns out to be divergent when the field amplitudes are greater than some critical value. In the case in which the kernel is a periodic or quasiperiodic function of time, a general solution to the integral equation is found, and this solution turns out to be unique and convergent for every value of the electromagnetic (e.m.) field amplitude. This solution is expressed through a continued-fraction expansion, whose terms are matrices, which can be readily obtained from the kernel. This solution is a generalization of the continued-fraction expansion usually met in the strong-signal semiclassical theory of lasers. Some particular cases are then treated in detail; among them, the solution to the multimode operation in laser devices is given in which the excited level lifetime is long compared with the period of the beating term between two adjacent modes. Although the theory presented is directed towards the steady-state regime in multimode lasers, it may also treat the transient regime of laser operations, or it may be extended to include phenomena where a strong e.m. field, made up of several modes equally spaced in frequency, interacts with a two-level system.
This chapter introduces the basic theoretical tools for handling many-body quantum systems. Starting from second quantized operators, we discuss how it is possible to describe the composite wavefunction of multi-particle systems, and discuss representations in various bases. The algebra of Fock states is described for single and multi-mode systems, and how they relate to the eigenstates of the Schrodinger equation. Finally, we describe how interactions between particles can be introduced in a general way, and then describe the most common type of interaction in cold atom systems, the s-wave interaction