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Physical Review Letters

Theory of Atomic Motion in a Resonant Electromagnetic Wave

R. J. Cook

Published Dec 25, 1978• 45 citations
Publisher

Abstract

A new theory of atomic motion in a resonant standing or traveling electromagnetic wave is presented. It is shown that, when effects of spontaneous emission are negligible, the motion of a two-level atom in the resonant radiation is determined by two noninterfering wave functions, each of which satisfied a time-dependent Schr\"odinger equation with time-independent potential energy. An experiment is proposed to test the theory.

DOI
10.1103/physrevlett.41.1788

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A new theory of atomic motion in a resonant standing or traveling electromagnetic wave is presented. It is shown that, when effects of spontaneous emission are negligible, the motion of a two-level atom in the resonant radiation is determined by two noninterfering wave functions, each of which satisfied a time-dependent Schr"odinger equation with time-independent potential energy. An experiment is proposed to test the theory.
A general theory of the motion of a two-level atom in a resonant or near-resonant electromagnetic wave of arbitrary amplitude and phase, including effects of radiative relaxation due to interaction with the quantized vacuum field, is developed from first principles. Particular emphasis is placed on the effects of quantum-mechanical fluctuations of the radiation force and on the associated diffusion of atomic momentum due to spontaneous and induced absorption and emission processes. Analytic results and numerical examples are presented for (1) the lower bound on the temperature achievable by radiation cooling in a standing wave tuned below resonance, (2) the heating rate in a strong resonant standing wave, (3) the maximum confinement time for an atom in a Gaussian radiation trap, (4) the deflection and spreading of an atomic beam transversely illuminated by a strong resonant running wave, and (5) the transverse cooling of an atomic beam by a strong running wave tuned below resonance.
The effect of field fluctuations on the motion of a two-level atom in resonant laser radiation is investigated theoretically. The internal motion of the atom is treated quantum mechanically (optical Bloch equations); the translational motion, which is coupled to the internal motion, is treated classically (Ehrenfest's theorem); and the fluctuating radiation is treated as a classical electromagnetic wave with stochastic phase (phase-diffusion model). An expression for the radiation force and equations of motion for the ensemble-averaged Bloch vector are derived and used to calculate (i) the radiation force in a plane running wave, (ii) the steady-state radiation force in a general standing wave, and (iii) the dissipative force in a plane standing wave in the limit of strong phase fluctuations. In case (iii) it is found that the atomic velocity is damped when the radiation is tuned above resonance, which is opposite to the case of a weak coherent standing wave, in which damping occurs for radiation tuned below resonance. It is shown that fluctuations have a substantial effect on the radiation force only when the spectral width of the radiation approaches or exceeds the spontaneous emission rate.
In this chapter a quantum mechanical treatment of atomic response to the fields of an electromagnetic wave is presented. Absorption and emission of energy by atoms involves transition between their quantized energy levels driven by an interaction Hamiltonian. Schrodinger equation is used to derive the equations of motion for the density matrix of the atom, which is better suited to include the effects of mechanisms other than the interaction Hamiltonian – collisions, level decays, level shifts, etc. that also affect the state of the atom. Using the density matrix equation of motion and focusing on two atomic levels, induced dipole moment for a single two-level atom and polarization and susceptibility for a collection of two-level atoms, including the effects of level decays, and collisions, are calculated. Equations of motion for level populations are also derived. These expressions naturally predict stimulated emission as well as absorption. Expressed in terms of spontaneous emission rate they confirm the validity of equations derived in Chapter 5 using the dipole oscillator model of atomic response. The limit in which quantum mechanical equations for atomic response reduce to the classical dipole oscillator equations is also discussed.
In Chap. 1 we introduced the Schrodinger equation and gave solutions for a number of simple cases. The general solution was then obtained by a linear superposition of these solutions. In our work, we are particularly concerned with the way in which atomic systems interact with electromagnetic radiation. These systems are still described by wave functions given by superpositions of simple solutions, but the expansion coefficients Cn (probability amplitudes) become functions of time. In this chapter we derive equations of motion for the expansion coefficients using Schrddinger’s equation and introduce the electric-dipole interaction energy. Then, specializing to a two-level system and a monochromatic electromagnetic field, we solve the equations of motion both by time-dependent perturbation theory and by the more exact Rabi method.
In this paper, we study the behavior of a two-level atom in an electromagnetic wave of circular polarization. A general solution to the Schr\"odinger equation is obtained, and the action of the electromagnetic wave on the atom is analyzed in detail. It is shown that the levels of the atom would be split, with the energy difference between the split levels the same for the two states of the atom. The atom would acquire an average momentum, with its direction the same as the propagating direction of the electromagnetic wave, showing the expelling action of the electromagnetic wave on the atom; or with its direction opposite to the propagating direction of the electromagnetic wave, showing the trapping action of the electromagnetic wave on the atom. The properties of the action are dependent on the initial state of the atom. The average expelling and trapping forces exerted on the atom by the electromagnetic wave are determined. Two cases of ``resonance'' and ``strict detuning'' are discussed particularly. Under the resonance case, the transition probability reaches its maximum for a given time. In the strict detuning case, the atom is not influenced by the electromagnetic wave.