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The Journal of Chemical Physics

First-Order Dispersion Forces

M. J. Stephen

Published Feb 1, 1964• 362 citations
Publisher

Abstract

The interaction energy between two atoms, one of which is excited, is investigated. The form of the interaction depends on the relation between the distance R separating the atoms and the wavelength k1—1 of the transition from the ground state to the excited state. If k1R<1 the photon may be trapped between the two atoms leading to the well-known dipole—dipole interaction. If k1R>1 the interaction is small, the state of the interacting atoms is not a stationary state and the photon is emitted after a time determined by the spontaneous emission probability.

DOI
10.1063/1.1725188

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The interaction energy between two atoms, one of which is excited, is investigated. The form of the interaction depends on the relation between the distance R separating the atoms and the wavelength k1—1 of the transition from the ground state to the excited state. If k1R<1 the photon may be trapped between the two atoms leading to the well-known dipole—dipole interaction. If k1R>1 the interaction is small, the state of the interacting atoms is not a stationary state and the photon is emitted after a time determined by the spontaneous emission probability.
The usual form of electromagnetic forces is given by but for neutral atoms q = 0. The next order of force is the dipole term, but this also vanishes because neutral atoms have no inherent dipole moment. However, a dipole moment can be induced by a field, and this is most efficient if the field is alternating near the atomic resonance frequency. Since these frequencies are typically in the optical range, dipole moments are efficiently induced by shining nearly resonant light on the atoms. Laser cooling and trapping rely on the interaction between laser light and atoms to exert a controllable force on the atoms, and many sophisticated schemes have been developed using the special properties of the interaction. This chapter begins with the simplest schemes for exerting optical forces on atoms, namely a single-frequency light field interacting with a two-level atom. Although this is the simplest possible scheme, it is pedagogically valuable because it shows many of the features that will be encountered further on. The ultimate temperature using such a scheme is limited, but lower temperatures can be achieved with multilevel atoms. This topic is addressed later in the chapter. Two kinds of optical forces When an atom absorbs nearly resonant light, it makes a transition to the excited state, and its subsequent return to the ground state can be either by spontaneous or by stimulated emission. The nature of the optical force that arises from these two different processes is quite different, and will be described separately [1]. The spontaneous emission case is different from the familiar quantum mechanical calculations using state vectors to describe the state of the system, because spontaneous emission causes the state of the system to evolve from a pure state into a mixed state (see Exercise 6.9). Spontaneous emission is an essential ingredient for the dissipative nature of the optical forces in monochromatic light, and the density matrix is needed to describe it (see Chap. 6). When an atom absorbs light, its energy excites the atom and its angular momentum changes the electron's orbit. Both of these are internal atomic properties. However, the linear momentum of the light, can change only the external, translational motion of the atom.
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The excitation spectrum arising from the interaction between two identical atoms (molecules), one of which is excited in the presence of a resonant strong electromagnetic field (pump field), is investigated. The excitation spectrum is found to consist of those describing the symmetric and antisymmetric modes, respectively. The form of both spectra depends on the relation between the distance RRR separating the atoms and the wavelength $\ensuremath{\lambda}$ of the transition from the ground state to the excited state. For $R<\ensuremath{\lambda}$, and when certain conditions prevail, the spectral function for the symmetric modes consists of three Lorentzian lines describing the central peak and the two sidebands whose radiative widths are equal to ${\ensuremath{\gamma}}_{0}$, $\frac{3{\ensuremath{\gamma}}_{0}}{2}$, and $\frac{3{\ensuremath{\gamma}}_{0}}{2}$, which are twice the corresponding ones arising from an isolated single atom interacting with the pump field. For the antisymmetric modes the central peak has a $\ensuremath{\delta}$-function distribution indicating the stability of the mode in question, while, when certain conditions are satisfied, the two sidebands are described by Lorentzian lines each having a radiative width of the order of $\frac{{\ensuremath{\gamma}}_{0}}{2}$, which is equal to the natural linewidth for a photon spontaneously emitted from an isolated atom. For both the symmetric and antisymmetric modes, the dipole-dipole interaction between the atoms brings about small energy shifts. For $R>\ensuremath{\lambda}$, apart from the small energy shifts caused by the dipole-dipole interaction, the spectral functions for symmetric and antisymmetric modes are similar to that for the single atom interacting with the pump field.
The spontaneous emission of a set of NNN identical two-level atoms interacting with a quantized electromagnetic field is studied. The atoms are assumed to be close together compared to the mean wavelength of the emitted radiation, and their initial state is either a Dicke state or a superposition of the ground and excited state for each atom. Approximate expressions are obtained for transition probabilities, expectation values, and correlations as functions of time. The spontaneous emission for a very large separation of the atoms is also studied. In this case the directional properties of the emitted radiation, as well as the time within which the atoms decay to their ground state, depend on the position of each atom.