Density-matrix approach to dynamics of multilevel atoms in laser fields

Abstract

The theoretical foundations of atom dynamics in laser fields are reviewed in relation with applications to laser spectroscopy, control of atomic motion, atom traps and frequency standards. We present an ab initio approach to the description of internal and translational dynamics of multilevel atoms in laser fields based on the equations for the atomic density matrix. Semiclassical density matrix equations are reviewed and applied to the description of properties of atomic populations and coherences for a classically moving atom. Quantum-kinetic equations for the atomic density matrix are reviewed for the multilevel interaction schemes. The procedure of reduction of the quantum-kinetic equations to the Fokker–Planck quasiclassical kinetic equation for the atomic distribution function is described. Quasiclassical kinetic equations are applied to the multilevel atomic schemes to describe the translational atomic dynamics. Basic types of the dipole radiation forces on atoms are considered for realistic cases of multilevel dipole interaction schemes. The applications of the theory of atomic dynamics in laser fields to the laser cooling, magneto-optical and optical dipole traps, and optical lattices are discussed.

Introduction

The purpose of this review is to describe the density matrix approach to atomic motion in laser fields, present theoretical fundamentals of translational dynamics of atoms in laser fields, and outline the applications of theoretical approaches to laser control of atomic motion, including laser cooling of atoms and atom traps. For the past decades both internal and translational dynamics of atoms in laser fields have been investigated for many specific dipole interaction schemes and under different conditions. Extensive theoretical and experimental studies of atomic dynamics resulted in the development of the effective techniques to control both the internal and translational atomic states. Among such techniques one can mention optical pumping, velocity-selective excitation of atoms, coherent population trapping and methods of cooling and trapping atoms, deflection, reflection and splitting atomic beams, and guiding atoms in laser fields. It can nowadays be said that the development of the above methods resulted in the creation of foundations of atom manipulation with laser light and atom optics (Letokhov and Chebotayev, 1977; Demtröder, 1996; Minogin and Letokhov, 1987; Kazantsev et al., 1990; Arimondo et al., 1992; Berman, 1997; Grimm et al., 1999; Metcalf and van der Straten, 1999; Balykin et al., 2000).

From a general physical point of view both the internal and translational dynamics of an atom in a laser field can be attributed to one of the two basic types according to the relation between the contributions of the induced and spontaneous transitions. At short interaction time τ_int compared with the spontaneous decay times, τ_int⪡τ_sp, spontaneous transitions cannot play a noticeable role in atomic dynamics. In this relatively simple pure quantum-mechanical case the atomic dynamics is mostly a coherent one, well defined by the time evolution of the initial atom state and initial shape of the atom wavepacket. This case is of basic importance for the coherent atom control by pulsed laser fields and for atom optics. Quite a different and most complicated case occurs when the interaction time is of the order of or exceeds the characteristic relaxation times defined by the spontaneous decays, τ_int≳τ_sp. In this most frequently investigated case atomic transitions induced by a laser field are interrupted by a stochastic process of spontaneous photon emission. As a result, spontaneous decays lead to a relaxation of the internal atom states to the quasi-stationary states while the quantum-statistical fluctuations in atomic momentum cause the atomic wave packet to perform a stochastic motion and drift in the momentum space. This latter case of quantum-statistical atom dynamics is of importance for applications related with spectroscopic studies of atoms and control of atomic motion by continuous laser fields.

In this paper we concentrate on the quantum-statistical atom dynamics paying basic attention to the excitation processes and dynamics for multilevel interaction schemes. While internal and translational dynamics of a two-level atom is relatively simple, the dynamics of multilevel atoms exhibits many new features specific of multilevel interaction schemes.

Among studies of the problems of translational dynamics of multilevel atoms in the laser fields the most important is the quantum-statistical approach based on the quantum-kinetic equations for the atomic density matrix. This general approach can be applied to any specific dipole interaction scheme between an atom and the laser field. Depending on the level of simplification this approach can give a simple description of the internal atomic states in the framework of the semiclassical approach, or describe the time evolution of the internal and translational atomic state in terms of quasiclassical approach, and finally give the most complete description of atomic dynamics in terms of a fully quantum-kinetic approach.

This paper aims at the review of basic physical principles of atomic dynamics with applications to basic schemes of laser cooling and trapping of multilevel atoms. We discuss atomic dynamics for practically important laser field configurations and the multilevel dipole interaction schemes relevant to the experiments in the field. The review considers three basic levels of the theoretical description of atomic dynamics in laser fields. First, a relatively simple semiclassical approach is used for describing the internal atomic dynamics and dipole radiation forces on atoms. This approach treats atoms as classically moving systems possessing quantized internal states (2 Semiclassical atomic density matrix, 3 Dipole radiation forces). The most general quantum-statistical description in terms of the quantum-kinetic equations for the atomic density matrix is given in Section 4. The quasiclassical level of description is discussed first in general in Section 5 and is applied later in 6 Laser cooling of atoms, 7 Magneto-optical trap, 8 Optical dipole traps for description of laser cooling of atoms and atomic motion in the atom traps.