In all preceding chapters, we have tried to separate the effect of the light on the atoms from the reverse effect, the modification of the light by the atomic medium. In most cases this allows an explicit solution of the equations of motion and provides stable stationary states. In recent years, another approach to the interaction between light and matter has appeared, which emphasises strong coupling between the two. It uses the interactions as a tool to study general aspects of nonlinear dynamics. This chapter provides an introduction to this field, without trying to give a complete summary. Overview Resonant vapours as optically nonlinear media Mutual interactions So far, we have considered the effect of the laser on the atomic medium separately from the measurement of microscopic dynamics by the polarisation selective detection of transmitted light. In this approximation, the pump laser drives the atoms without suffering significant attenuation or polarisation changes. Conversely, the probe beam, which monitors the optical properties of the atomic medium, changes the microscopic state of the atoms only infinitesimally. This approach guarantees, e.g., that the response of the medium to the probe beam is linear. As stressed before, this assumption of one-sided interactions is always an approximation, since the conservation of energy and angular momentum make it impossible to change either of the two partial systems without compensating changes in the other part.
In all preceding chapters, we have tried to separate the effect of the light on the atoms from the reverse effect, the modification of the light by the atomic medium. In most cases this allows an explicit solution of the equations of motion and provides stable stationary states. In recent years, another approach to the interaction between light and matter has appeared, which emphasises strong coupling between the two. It uses the interactions as a tool to study general aspects of nonlinear dynamics. This chapter provides an introduction to this field, without trying to give a complete summary. Overview Resonant vapours as optically nonlinear media Mutual interactions So far, we have considered the effect of the laser on the atomic medium separately from the measurement of microscopic dynamics by the polarisation selective detection of transmitted light. In this approximation, the pump laser drives the atoms without suffering significant attenuation or polarisation changes. Conversely, the probe beam, which monitors the optical properties of the atomic medium, changes the microscopic state of the atoms only infinitesimally. This approach guarantees, e.g., that the response of the medium to the probe beam is linear. As stressed before, this assumption of one-sided interactions is always an approximation, since the conservation of energy and angular momentum make it impossible to change either of the two partial systems without compensating changes in the other part.
Nonlinear optics is the study of the response of dielectric media to strong optical fields. The fields are sufficiently strong that the response of the medium is, as its name implies, nonlinear. That is, the polarization, which is the dipole moment per unit volume in the medium, is not a linear function of the applied electric field. In the equation for the polarization, there is a linear term, but, in addition, there are terms containing higher powers of the electric field. This leads to significant new types of behavior, one of the most notable being that frequencies different from that of the incident electromagnetic wave, such as harmonics or subharmonics, can be generated. Linear media do not change the frequency of light incident upon them. The first observation of a nonlinear optical effect was, in fact, second-harmonic generation – a laser beam entering a nonlinear medium produced a second beam at twice the frequency of the original. Another type of behavior that becomes possible in nonlinear media is that the index of refraction, rather than being a constant, is a function of the intensity of the light. For a light beam with a nonuniform intensity profile, this can lead to self-focusing of the beam. Most nonlinear optical effects can be described using classical electromagnetic fields, and, in fact, the initial theory of nonlinear optics was formulated assuming the fields were classical. When the fields are quantized, however, a number of new effects emerge. Quantized fields are necessary if we want to describe fields that originate from spontaneous emission. For example, in a process known as spontaneous parametric down-conversion, a beam of light at one frequency, the pump, produces a beam at half the original frequency, the signal. This second beam is a result of spontaneous emission.
From a macroscopic viewpoint, atomic vapours appear homogeneous and isotropic. As we have seen during the discussion of the microscopic dynamics that light induces in atomic systems, however, the interaction with light as well as external fields can break the rotational and translational symmetries and make the system both inhomogeneous and anisotropic. This microscopic anisotropy appears directly in the optical properties of the system. This chapter discusses the propagation of light in such a medium and describes experiments that analyse the transmitted light to obtain information about the microscopic state of the medium. Isotropic atoms The Lorentz–Lorenz model Outline The interaction of electromagnetic radiation with matter, in particular light, has inspired philosophers of nature for many centuries and has led to heated debates like the one between Newton and Goethe. The original interest was with the difference in absorption of various materials as a function of the optical wavelength and with the refraction of light, i.e., its dispersion. The first theoretical analysis that connected these macroscopic effects to microscopic properties of the material was the Lorentz–Lorenz theory of dispersion (Lorentz 1880; Lorenz 1881), which was put forward shortly after the Maxwell equations and published in the year after Maxwell's death. (See Figure 6.1.) This theory models the material as a collection of dipoles, driven by the electromagnetic wave. We give here a brief summary of the theory, since its physical content is still the foundation of today's description although the mathematical formalism has changed significantly.
In the preceding chapters, we discussed the manipulation and observation of atomic variables associated with the internal degrees of freedom. A complete separation between internal and external degrees of freedom, like position and momentum, is not always possible, however. It therefore appears appropriate to include a discussion of the mechanical effects of light, which have received increasing attention in recent years. The transfer of momentum between photons and atoms, first observed more than sixty years ago, had to wait for the wide availability of lasers to become a useful experimental tool. In particular the cooling of atoms and atomic ions to very low temperatures has found numerous applications. Light-induced forces That light exerts mechanical forces on massive particles like atoms may appear surprising. To motivate the existence of such an effect two different approaches are possible. The first approach considers the light as a collection of photons that carry, apart from energy and angular momentum, linear momentum as well. Photons interacting with atoms can therefore change the momentum of the atoms. The second approach considers light as a wave, i.e., an inhomogeneous electromagnetic field interacting with the atomic dipole moment. Both approaches provide a possible description for the numerous phenomena that can occur in this context, but in many situations, one of them turns out to be more intuitive or more useful for calculations than the other.
Summary form only given. The dynamics of a medium consisting of a dense collection of two-level atoms interacting via the electromagnetic field has been derived from first principles. In dense media and condensed matter, the microscopic local field that couples with a single atom is related to the macroscopic field and volume propagation by the Lorentz-Lorenz relation written for an isotropic homogeneous medium. The dynamical equations for the medium, coupled self-consistently with the Maxwell inhomogeneous wave equation, are given in the rotating wave and dipole coupling approximations. The microscopic development is used within the framework of full quantization, and the explicit effects of spontaneous relaxation for the host as well as the atoms are treated, together with corresponding fluctuations. Particular effects upon intrinsic adiabatic inversion, self-induced transparency, and lasing without inversion, are presented and discussed. In particular, the equilibrium, stationary, and dynamical effects of the host medium, and medium and field fluctuations upon local field effects are discussed.
Two-dimensional spectroscopy is an extension of conventional spectroscopic methods that characterises resonant systems as a function of two frequency variables. We use this technique to improve on the characterisation of atomic media along the lines discussed in the preceding chapter. On a microscopic level, the most important physical process for two-dimensional spectroscopy is the transfer of coherence between different transitions. We introduce the basics of the technique and discuss a few specific examples to outline its potential. Fundamentals Motivation and principle Motivation The preceding chapter showed how light drives the internal dynamics of resonant atomic media and how the measurement of optical anisotropies allows us to monitor these dynamics. The experiments discussed in the preceding chapter, however, can provide only limited information about the system. Most physical systems have more degrees of freedom than we can observe by measurements on transmitted light. As another limitation, we have primarily considered atoms that evolve under their internal Hamiltonian, only weakly perturbed by the probe laser beam. The example of light-induced spin nutation showed that the dynamics of optically pumped atoms differ significantly from those of a free atom. Although it is possible to observe spin nutation for systems with more than two ground-state sublevels, such an experiment suffers from the damping that accompanies optical pumping. The damping drives the system rapidly to an equilibrium, too fast for detailed dynamical observations.
After the discussion of ionic solids we shift our attention to atomic vapours. Here, we discuss the dynamics of microscopic order within the angular momentum substates under the influence of light and magnetic fields. For this purpose, we combine optical pumping as outlined in Chapter 5 to prepare the ordered states, and polarisation-selective detection introduced in Chapter 6 for its observation. In contrast to the introductory sections, we no longer discuss these dynamics as mathematical models, but for specific physical systems, using experimental data to illustrate the theoretical description. The systems we use for this illustration are vapours of alkali metal atoms, in particular sodium. These one-electron systems allow the most direct application of the theoretical models developed above. In addition, we concentrate on electronic ground states, whose long lifetimes permit long observation times and correspondingly precise observations. Experimental arrangement General considerations Laser-induced dynamics This chapter surveys the coherent evolution of coherences between angular momentum sublevels. Optical pumping excites this microscopic order, and it evolves under the influence of external magnetic fields and the laser radiation. The primary goal of this section is to show how the mathematical models developed in the preceding sections apply to real physical systems. We discuss how the observed signals arise and by which parameters the experimenter can control the dynamics of these systems.