The quantitative description of atomic radiation reviewed in the previous chapter requires knowledge of energy levels and of level populations, which will be the principal subjects of chapters 6 and 7, and of oscillator strengths or of the closely related line strengths. Their calculation and measurement are the main topics of the present chapter. As to energy levels and wavelengths, a large body of high quality empirical data is available (Moore 1949-1958, Martin, Zalubas and Hagen 1978, Cowan 1981, Kelly 1987a and b, Bashkin and Stoner 1975, 1978, 1981), and atomic structure calculations (Cowan 1981, Sobel'man 1992) are usually of an accuracy that is sufficient for most plasma spectroscopy applications. Our knowledge of oscillator strengths, and therefore also of transition probabilities for spontaneous transitions and line strengths, has also greatly improved since the predecessor of this monograph (Griem 1964) was written. These advances are the result of improved experiments and computations and of critical evaluations of data (Wiese, Smith and Glennon 1966, Wiese, Smith and Miles 1969, Fuhr and Wiese 1995, Martin, Fuhr and Wiese 1988, Fuhr, Martin and Wiese 1988). Especially complete and accurate data are now available for atoms and ions of carbon, nitrogen and oxygen (Wiese, Fuhr and Deters 1995).

Although most of the electromagnetic radiation from many natural and laboratory plasmas is atomic in origin and therefore subject to quantum effects, it remains useful to introduce some of the basic radiative processes via classical theory. Other important foundations of plasma spectroscopy are atomic physics and plasma physics, especially the statistical mechanics of ionized gases. Generally, the basic theory is well established in these parent disciplines. However, the large variety of processes contributing to emission or absorption spectra often requires more or less drastic simplifications in their theoretical description or computer modeling. Critical experiments are playing an essential role in checking the reliability of various models and in delineating the region of their applicability.

Plasma spectroscopy, although being a highly specialized subfield, is at the same time a very interdisciplinary science. It not only owes its origins largely to astronomy, but also returns to astronomy and astrophysics methods of analysis of spectra and a multitude of basic data, which have both been subjected to experimental scrutiny. The state of stellar plasmas is significantly influenced by radiation, and the latter is more or less controlled by radiative energy transfer. Internally consistent treatments of the states of matter and radiation first developed by astronomers are now also becoming important for the description of plasma experiments.

In addition to the most general applications of spectroscopic methods to density and temperature measurements, which were discussed in the two preceding chapters, there are numerous special applications. Some of these will be discussed in this concluding chapter, without any prejudice against any well-established or recently developed methods which are omitted. If there is any common thread, it is in the strong role played by atomic collision theory and by the physics of atoms and ions in electric and magnetic fields. As in the other chapters, no attempt will be made to describe the often very sophisticated instrumentation or other experimental details, which the interested reader should be able to find with the help of the references in the original papers.

The first two special applications to be discussed, namely charge exchange recombination and beam emission spectroscopy, obviate the essential difficulty in emission or absorption spectroscopy in obtaining spatial resolution along the line of sight. This is accomplished by injecting heating or diagnostic neutral beams into magnetically confined plasmas either to preferentially populate some excited states of plasma ions by charge exchange recombination, as discussed in section 6.5, or by having the atoms in the beam ionized and excited by the plasma electrons, as discussed in sections 6.2 and 6.3, or even by protons and other plasma ions (Mandl et al. 1993), see section 6.5.

In this and the following chapters, various applications of plasma spectroscopy will be discussed. Their selection is necessarily somewhat arbitrary, but they will hopefully serve as useful demonstrations of the general methods and principles described in the preceding chapters. A very broad class of applications is concerned with the energy loss or gain of plasmas because of emission or absorption of electromagnetic radiation. As usual, the need for comprehensive calculations of these processes is shared with astrophysics. Here the requirement of energy conservation within a stellar atmosphere not subject to any significant nonradiative energy transport must be imposed by having zero divergence of the spectrally integrated radiative flux which, in turn, is obtained from the radiative energy transfer equations of the preceding chapter. In many laboratory plasmas such a general approach is not necessary, because most of the emission normally comes from optically thin layers and because radiative heating, except for radio-frequency (Golant and Fedorov 1989) and microwave heating (Bekefi 1966), is not involved.

Very notable exceptions to the last point are laser-produced plasmas, in which the absorption of the, typically, visible laser light is indeed essential (Kruer 1988). Other exceptions are x-ray heated plasmas produced, e.g., for the measurement of absorption coefficients of hot and dense low, medium and high Z materials (Davidson et al. 1988, Foster et al. 1991, Perry et al. 1991, Springer et al. 1992, 1994, Schwanda and Eidmann 1992, DaSilva et al. 1992, Eidmann et al. 1994, Winhart et al. 1995).

Next to the qualitative determination of the chemical composition of a plasma from emission and absorption line identifications, the measurement of electron and ion or atom temperatures is the oldest application of spectroscopic methods to plasma and gaseous electrical discharge physics, not to mention astronomy. It continues to play an important role, e.g., in fusion research (DeMichelis and Mattioli 1981, 1984 and Kauffman 1991). In the laboratory, independent methods based on laser light scattering and Langmuir probes are available, as already mentioned in the introduction to the preceding chapter. However, in astronomy spectroscopic methods normally must stand alone. Another important distinction is the usually dominant role of radiative transfer (see chapter 8) in astronomical applications, compared with the relatively small optical depth in some useful portion of the spectrum of most laboratory plasmas.

In many cases, it is necessary to distinguish between kinetic temperatures of electrons, ions and atoms, say, Te, Tz, and Ta. These temperatures may differ from each other even if the individual velocity distributions are close to Maxwellians, because, e.g., electron-electron energy transfer rates are much larger than electron-ion collision rates, as are ion-ion rates (Spitzer 1962). In most applications, at least the electrons do have a Maxwellian distribution, and we will assume this here.

This book was written for the benefit of young researchers in diverse disciplines ranging from experimental plasma physics to astrophysics, and graduate students wanting to enter the interdisciplinary area of research now generally called plasma spectroscopy. The author has attempted to develop the theoretical foundations of the numerous applications of plasma spectroscopy from first principles. However, some familiarity with atomic structure and collision calculations, with quantum-mechanical perturbation theory and with statistical mechanics of plasmas is assumed. The emphasis is on the quantitative mission spectroscopy of atoms and ions immersed in high-temperature plasmas and in weak radiation fields, where multi-photon processes are not important.

As in the author's previous books on plasma spectroscopy and spectral line broadening written, respectively, over three and two decades ago, various applications are discussed in considerable detail, as are the underlying critical experiments. Hopefully, the reader will find the numerous references useful and current up to the latter part of 1995. They provide advice concerning access to basic data, which are needed for the implementation of many of the experimental methods, and to descriptions of instrumentation.

The author has once more benefited from his experience in teaching special lecture courses at the University of Maryland and recently also at the Ruhr University in Bochum and some of its neighboring institutions.

Atoms and ions containing residual bound electrons do not quite resemble the simple harmonic oscillator model used so successfully in the classical theory of radiation. However, replacing the atoms or ions with sets of harmonic oscillators of a great number of discrete resonance frequencies and having various amplitudes, together with the results of classical radiation theory, go a long way toward a quantitative description of emission or absorption spectra. The set of resonance frequencies is obtained from measured or calculated energy levels using Ritz's combination principle. The amplitudes are associated with matrix elements of appropriate quantum mechanical operators between wave functions of the two energy eigenstates involved at a given frequency. In other words, quantities of the emitters, absorbers, or scatterers are described quantum-mechanically, whereas the electromagnetic field is treated classically.

Such semi-classical description of matter-electromagnetic field interactions became unnecessary very early in the development of quantum theory. It will therefore not be discussed in any detail. Instead, we will begin immediately with the combined theory of matter and radiation (Heitler 1954, Dirac 1958, Loudon 1983).

Quantum theory of particles and fields

There are various ways to also quantize the electromagnetic fields (Cohen-Tannoudji, DuPont-Roc and Grynberg 1989), of which that performed on the combined Hamiltonian equations of motion for the field-matter system is followed here.

Quantum coherence and correlations in atomic and radiation physics have led to many interesting and unexpected consequences. For example, an atomic ensemble prepared in a coherent superposition of states yields the Hanle effect, quantum beats, photon echo, self-induced transparency, and coherent Raman beats. In fact, in Section 1.4, we saw that the quantum beat effect provides one of the most compelling reasons for quantizing the radiation field.

A further interesting consequence of preparing an atomic system in a coherent superposition of states is that, under certain conditions, it is possible for atomic coherence to cancel absorption. Such atomic states are called trapping states†. The observation of nonabsorbing resonances via atomic coherence and interference impacts on the concepts of lasing without inversion (LWI),‡ enhancement of the index of refraction accompanied by vanishing absorption, and electromagnetically induced transparency.

In lasing without inversion, the essential idea is the absorption cancellation by atomic coherence and interference. This phenomenon is also the essence of electromagnetically induced transparency. Usually this is accomplished in three-level atomic systems in which there are two coherent routes for absorption that can destructively interfere, thus leading to the cancellation of absorption. A small population in the excited state can thus lead to net gain. A related phenomenon is that of resonantly enhanced refractive index without absorption in an ensemble of phase-coherent atoms (phaseonium). In a phaseonium gas with no population in the excited level, the absorption cancellation always coincides with vanishing refractivity.

Complementarity, e.g., the wave–particle duality of nature, lies at the heart of quantum mechanics. It distinguishes the world of quantum phenomena from the reality of classical physics. In the 1920s, quantum theory as we know it today was still new, and examples used to illustrate complementarity emphasized the position (particle-like) and momentum (wave-like) attributes of a quantum mechanical object, be it a photon or a massive particle. This is the historical reason why complementarity is often superficially identified with the so-called wave-particle duality of matter.

Complementarity, however, is a more general concept. We say that two observables are complementary if precise knowledge of one of them implies that all possible outcomes of measuring the other one are equally probable. We may illustrate this by two extreme examples. The first example consists of the position and momentum (along one direction) of a particle: if, say, the position is predetermined then the result of a momentum measurement cannot be predicted, all momentum values are equally probable (in a large range). The second extreme involves two orthogonal spin components of a spin- 1/2 particle: if, say, the vertical spin component has a definite value (up or down) then upon measuring a horizontal component both values (left or right, for instance) are found, each with a probability of 50%. Thus, in the microcosmos complete knowledge in the sense of classical physics is not available. The classic example of the merger of wave and particle behavior is provided by Young's double-slit experiment.

As discussed in the last three chapters, the fundamental source of noise in a laser is spontaneous emission. A simple pictorial model for the origin of the laser linewidth envisions it as being due to the random phase diffusion process arising from the addition of spontaneously emitted photons with random phases to the laser field. In this chapter we show that the quantum noise leading to the laser linewidth can be suppressed below the standard, i.e., Schawlow–Townes limit by preparing the atomic systems in a coherent superposition of states as in the Hanle effect and quantum beat experiments discussed in Chapter 7. In such coherently prepared atoms the spontaneous emission is said to be correlated. Lasers operating via such a phase coherent atomic ensemble are known as correlated spontaneous emission lasers (CEL). An interesting aspect of the CEL is that it is possible to eliminate the spontaneous emission quantum noise in the relative linewidths by correlating the two spontaneous emission noise events.

A number of schemes exist in which quantum noise quenching below the standard limit can be achieved. In two-mode schemes a correlation between the spontaneous emisson events in two different modes of the radiation field is established via atomic coherence so that the relative phase between them does not diffuse or fluctuate. In a Hanle laser and a quantum beat laser this is achieved by pumping the atoms coherently such that every spontaneously emitting atom contributes equally to the two modes of the radiation, leading to a reduction and even vanishing of the noise in the phase difference.

The phenomenon of resonance fluorescence provides an interesting manifestation of the quantum theory of light and is a “real world” application of the material of the Chapters 8 and 9. In this process, a two-level atom is typically driven by a resonant continuous-wave laser field and the spectral and quantum statistical properties of the fluorescent light emitted by the atom are measured. Experimentally this can be achieved by scattering a laser off a collimated atomic beam such that the directions of the laser beam, atomic beam, and detector axis are mutually perpendicular as shown in Fig. 10.1.

If the driving field is monochromatic, then at low excitation intensity the atom absorbs a photon at the excitation frequency and reemits it at the same frequency as a consequence of conservation of energy. The spectral width of the fluorescent light is therefore very narrow. The situation, however, is considerably more complicated when the excitation intensity increases and the Rabi frequency associated with the driving field becomes comparable to, or larger than, the atomic linewidth. At such intensity levels, the Rabi oscillations show up as a modulation of the quantum dipole moment and sidebands start emerging in the spectrum of the emitted radiation. This so-called dynamic Stark splitting is an interesting feature of the atom–field interaction. In addition to that, the fluorescent light exhibits certain nonclassical properties including photon antibunching and squeezing.

In this chapter, we develop a theory of resonance fluorescence to explain these phenomena.

Following the development of the quantum theory of radiation and with the advent of the laser, the states of the field that most nearly describe a classical electromagnetic field were widely studied. In order to realize such ‘classical’ states, we will consider the field generated by a classical monochromatic current, and find that the quantum state thus generated has many interesting properties and deserves to be called a coherent state. An important consequence of the quantization of the radiation field is the associated uncertainty relation for the conjugate field variables. It therefore appears reasonable to propose that the wave function which corresponds most closely to the classical field must have minimum uncertainty for all times subject to the appropriate simple harmonic potential.

In this chapter we show that a displaced simple harmonic oscillator ground state wave function satisfies this property and the wave packet oscillates sinusoidally in the oscillator potential without changing shape as shown in Fig. 2.1. This coherent wave packet always has minimum uncertainty, and resembles the classical field as nearly as quantum mechanics permits. The corresponding state vector is the coherent state |α〉, which is the eigenstate of the positive frequency part of the electric field operator, or, equivalently, the eigenstate of the destruction operator of the field.

Classically an electromagnetic field consists of waves with welldefined amplitude and phase. Such is not the case when we treat the field quantum mechanically.

Matter–wave interferometry dates from the inception of quantum mechanics, i.e., the early electron diffraction experiments. More recent neutron interferometry experiments have yielded new insights into many fundamental aspects of quantum mechanics. Presently, atom interferometry has been demonstrated and holds promise as a new field of optics – matter–wave optics. This field is particularly interesting since the potential sensitivity of matter–wave interferometers far exceeds that of their light-wave or ‘photon’ antecedents.

In this chapter we consider the physics of light-induced forces on the center-of-mass motion of atoms and their application to atom optics (Fig. 17.1). The most obvious being the recoil associated with the emission and absorption of light. This ‘radiation pressure’ is the basis for laser induced cooling.

Another very important mechanical effect is the gradient force due to, e.g., transverse variation in the laser beam. These, essentially semiclassical, forces are useful in guiding and trapping neutral atoms.

After considering the basic forces which allow us to cool, guide, and trap atoms, we turn to the optics of atomic center-of-mass de Broglie waves, i.e., atom optics. In keeping with the spirit of the present text, we will focus on the quantum limits to matter–wave interferometry. An analysis of a matter–wave gyro in an obvious extension of the laser gyro and the similarity and relative merits of the two will be compared and contrasted. Finally we derive the “recoil limit” to laser cooling; and show that it is possible to supersede this limit via atomic coherence effects.

Light occupies a special position in our attempts to understand nature both classically and quantum mechanically. We recall that Newton, who made so many fundamental contributions to optics, championed a particle description of light and was not favorably disposed to the wave picture of light. However, the beautiful unification of electricity and magnetism achieved by Maxwell clearly showed that light was properly understood as the wave-like undulations of electric and magnetic fields propagating through space.

The central role of light in marking the frontiers of physics continues on into the twentieth century with the ultraviolet catastrophe associated with black-body radiation on the one hand and the photoelectric effect on the other. Indeed, it was here that the era of quantum mechanics was initiated with Planck's introduction of the quantum of action that was necessary to explain the black-body radiation spectrum. The extension of these ideas led Einstein to explain the photoelectric effect, and to introduce the photon concept.

It was, however, left to Dirac to combine the wave-and particlelike aspects of light so that the radiation field is capable of explaining all interference phenomena and yet shows the excitation of a specific atom located along a wave front absorbing one photon of energy. In this chapter, following Dirac, we associate each mode of the radiation field with a quantized simple harmonic oscillator, this is the essence of the quantum theory of radiation.

In many problems in quantum optics, damping plays an important role. These include, for example, the decay of an atom in an excited state to a lower state and the decay of the radiation field inside a cavity with partially transparent mirrors. In general, damping of a system is described by its interaction with a reservoir with a large number of degrees of freedom. We are interested, however, in the evolution of the variables associated with the system only. This requires us to obtain the equations of motion for the system of interest only after tracing over the reservoir variables. There are several different approaches to deal with this problem.

In this chapter, we present a theory of damping based on the density operator in which the reservoir variables are eliminated by using the reduced density operator for the system in the Schrödinger (or interaction) picture. We also present a ‘quantum jump’ approach to damping. In the next chapter, the damping of the system will be considered using the noise operator method in the Heisenberg picture.

An insight into the damping mechanism is obtained by considering the decay of an atom in an excited state inside a cavity. The atom may be considered as a single system coupled to the radiation field inside the cavity. Even in the absence of photons in the cavity, there are quantum fluctuations associated with the vacuum state. As discussed in Chapter 1, the field may be visualized as a large number of harmonic oscillators, one for each mode of the cavity.

In the previous chapter, we developed the equation of motion for a system as it evolved under the influence of an unobserved (reservoir) system. We used the density matrix approach and worked in the interaction picture. In this chapter, we consider the same problem of the system-reservoir interaction using a quantum operator approach. We again eliminate the reservoir variables. The resulting equations for the system operators include, in addition to the damping terms, the noise operators which produce fluctuations. These equations have the form of classical Langevin equations, which describe, for example, the Brownian motion of a particle suspended in a liquid. The Heisenberg–Langevin approach discussed in this chapter is particularly suitable for the calculation of two-time correlation functions of the system operator as is, for example, required for the determination of the natural linewidth of a laser.

We first consider the damping of the harmonic oscillator by an interaction with a reservoir consisting of many other simple harmonic oscillators. This system describes, for example, the damping of a single-mode field inside a cavity with lossy mirrors. The reservoir, in this case, consists of a large number of phonon-like modes in the mirrors. We also consider the decay of the field due to its interaction with an atomic reservoir. An interesting application of the theory of the system–reservoir interaction is the evolution of an atom inside a damped cavity. It is shown that the spontaneous transition rate of the atom can be substantially enhanced if it is placed in a resonant cavity.

In the preceding chapters concerning the interaction of a radiation field with matter, we assumed the field to be classical. In many situations this assumption is valid. There are, however, many instances where a classical field fails to explain experimentally observed results and a quantized description of the field is required. This is, for example, true of spontaneous emission in an atomic system which was described phenomenologically in Chapter 5. For a rigorous treatment of the atomic level decay in free space, we need to consider the interaction of the atom with the vacuum modes of the universe. Even in the simplest system involving the interaction of a single-mode radiation field with a single two-level atom, the predictions for the dynamics of the atom are quite different in the semiclassical theory and the fully quantum theory. In the absence of the decay process, the semiclassical theory predicts Rabi oscillations for the atomic inversion whereas the quantum theory predicts certain collapse and revival phenomena due to the quantum aspects of the field. These interesting quantum field theoretical predictions have been experimentally verified.

In this chapter we discuss the interaction of the quantized radiation field with the two-level atomic system described by a Hamiltonian in the dipole and the rotating-wave approximations. For a single-mode field it reduces to a particularly simple form. This is a very interesting Hamiltonian in quantum optics for several reasons. First, it can be solved exactly for arbitrary coupling constants and exhibits some true quantum dynamical effects such as collapse followed by periodic revivals of the atomic inversion.