The major portion of this monograph discusses how given pulses of laser radiation affect individual atoms. This chapter inverts that relationship, describing how the matter alters the fields. Those field changes provide quantitative measures of the quantum-state changes produced in the atoms. Their description therefore is an adjunct to Chap. 19.

Basically the incident radiation produces excitation which, in turn, alters the various multipole moments of the atoms. When viewed as a macroscopic sample of matter, such changes alter the electric polarization field P and the magnetization field M of the matter through which the radiation must pass. The Maxwell equations, see App. C.1, provide the needed description of how the P and M fields alter the electric and magnetic fields E and B. The combination of the Maxwell equations for the fields and the Schrödinger, Bloch, or Liouville equations for the atoms provide the tools needed to construct a self-consistent description of radiation passing through matter – atoms responding to a pulsed field and traveling waves being modified by the resulting atomic changes [All87; Ale92; Muk99; Die06; Vit01a]. The present chapter, drawing on [Sho90, Chap. 12] and App. C, discusses this theory.

Incoherent radiation passing through matter typically undergoes exponential attenuation in accord with eqn. (6.9). A measure of the incremental change of intensity in distance L by absorption coefficient κ is the optical depth κL. This parameter appears in rateequation treatments of incoherent light attenuation.

The nature of measurements, and their place within quantum theory, has engaged physicists and philosophers for generations [Bra92; Sch03]. Much of that interest centered on variables such as position and momentum of free particles, whose values form a continuum. The present monograph deals with discrete quantum states; the measurements are those required to specify as completely as possible a particular discrete quantum state Ψ or, more generally, a density matrix ρ defined within a finite-dimensional Hilbert space.

General remarks

General system. At the outset we assume that the possible quantum states are a small number – the N essential states used in formulating the time-dependent Schrödinger equation or specifying the dimensions of the density matrix. To completely characterize a density matrix for such a system we require the N2 elements. Of these the N diagonal elements are real valued, while the off-diagonal elements of the upper right side are complex conjugates of those on the lower left. Thus with allowance for the requirement of unit trace a total of N2 – 1 real numbers suffice to completely specify the density matrix. These values must be consistent with the constraints discussed in Sec. 16.6.3.

Pure state. If it is known that the system is in a pure state, we require the magnitude and phase of N probability amplitudes. These are constrained by normalization, and so only 2N –1 real numbers are needed. Out of these 2N – 1 parameters the overall phase of the statevector is usually not of interest (but see Chap. 20).

As light passes an atom, it exerts forces on the charges, electrons, and nuclei that alter the atomic structure. These may be slight distortions (perturbations) of the electron cloud or they may be more severe, as described in subsequent sections of this monograph. We wish to determine those changes, given the radiation field, or to devise a radiation field that will produce specified changes.

The changes to the atomic structure also affect the radiation that subsequently passes the atom. To describe those effects we must consider wave equations for radiation in the presence of altered atomic structure. More generally we must find self-consistent equations for the atoms and the field together, as discussed in Chaps. 21 and 22. Here we consider mathematical descriptions of the influence of coherent radiation on individual atoms, molecules, or other single quantum systems.

Individual atoms

Traditional sources of emission and absorption spectra, though revealing energy states of the constituent atoms and molecules, are macroscopic samples. One observes averaged characteristics of many individual particles, see Chap. 16. Quantum theory offers the basic formalism for dealing with individual atoms exposed to controlled radiation fields. Several experimental techniques provide acceptable approximations to this ideal. The following paragraphs note some of these examples.

Vapors

Neutral atoms or molecules in a vapor move freely along straight-line paths, interrupted by brief collisions that redirect the two collision partners. When the kinetic energies of the two partners are small, there can be no transfer of kinetic energy into internal energy of either particle – the collision is elastic.

Quantum changes of three-state systems have some similarities with those of two-state systems. When subject to steady illumination the populations may undergo oscillations similar to the Rabi cycling of two-state systems, and various forms of adiabatic following are possible. Analytic solutions to the relevant TDSE exist [Sho90, Chap. 23]. The additional degree of freedom, typically allowing controllable parameters of a second laser pulse, allows a wider variety of controlled excitation. The resulting differences and similarities to two-state systems have been discussed at length [Whi76; Sho77; Rad82b; Yoo85; Car87].

Three-state linkages

Two-field linkages. The simplest extension of two-state excitation allows two laser fields, here identified by letters P (for pump) and S (for Stokes), as befits the stimulated Raman process discussed in Chap. 14. The carrier frequencies of the two fields, ωP and ωS, are each assumed to be close to resonance with one, and only one, Bohr frequency, so that each field can be uniquely identified with a particular transition (failure of this restriction, and the resulting linkage ambiguity, is discussed in [Una00]). I will assume that the P field is (near) resonance only with the 1–2 transition, while the S field is (near) resonant only with the 2–3 transition; these interactions thereby form a two-step linkage chain. This system has three possible linkage patterns, shown in Fig. 13.1.

The linkage patterns (sometimes called configurations) differ by the ordering of the energies of the linked states.With the assumption that population initially occupies state 1, as in Fig. 13.1, the definitions are:

Ladder: The ladder system has the energy ordering E1 < E2 < E3.

Manipulation of the internal structure of atoms and molecules – altering the quantum states of submicroscopic systems – makes an increasingly significant contribution to contemporary technology, as electronic circuits continue to shrink in size and new opportunities appear for applying abstract quantum theory to the creation of practical devices. The structural changes range from simple perturbative distortions of the electronic charge distribution to complete transformation into an excited energy state or the creation of superposition states whose properties cannot be fully described without quantum theory.

This monograph discusses ways of inducing such changes, primarily (but not only) with pulses of laser light, and ways of picturing the changes with the aid of suitable mathematical tools. Aiming at a level suitable for advanced undergraduates or researchers it explains the basic principles that underly the quantum engineering of devices used for such applications as coherent atomic excitation and quantum information processing.

Presupposing some familiarity with quantum mechanics, it first introduces notions of atoms (or other localized quantum systems) and quantum states, and of radiation (specifically laser pulses), defining thereby the essential observable quantities with which theory must deal. It presents the constructs – probabilities, probability amplitudes, wavefunctions, and statevectors – that serve as variables for the mathematics. It then discusses the differential equations that describe laser-induced changes to atomic structure. It contrasts the pre-laser incoherent absorption of energy, governed by rate equations, with the coherent regime of laser-induced changes, governed by the time-dependent Schrödinger equation that is the foundation for all descriptions of quantum-mechanical changes.

Between every pair of quantum states for which a nonzero electric or magnetic multipole moment exists there can take place a radiative transition. When the quantum states are both discrete, as they are for pairs of bound states, the radiation is discrete – a spectral line. The frequency of that spectral line is set by the difference of the two energies, and can occur in any region of the electromagnetic spectrum. If there is available a source of coherent radiation at that frequency, then coherent quantum-state manipulation is possible.

For many years spectroscopists routinely assembled collections of wavelengths and line strengths (or transition probabilities) for various elements and molecules. The National Bureau of Standards (NBS), now the National Institute for Standards and Technology (NIST), collected, organized, and published much of this data. Their website, www.nist.gov/pml/data/handbook/index.cfm, provides ready access to this information for all the elements and many molecules. Much of this data has appeared in the Journal of Physical and Chemical Reference Data, published by the American Institute of Physics (AIP). Diagrams showing the relative positions of energies and the connecting transitions, often called Grotrian diagrams [Bas75; Moo68; Lan99], are helpful for presenting the excitation linkages.

Spectroscopic parameters

From traditional spectroscopic studies come several parameters with which to describe the resonant interaction between light and an atom – one that exists, ideally, in free space and which therefore has degenerate energy levels.

A simple redefinition of the quantization axis can change the linkage pattern of a linearly polarized field from one in which only pairs of state are linked (e.g. quantization axis along the linear polarization direction) to one in which several states are linked, with consequences such as those discussed in Sec. 12.4. The change of linkage pattern accompanies a redefinition of the basis states. It is natural to wonder whether a suitable change of basis can reduce other linkage patterns into ones in which only pairs of states are linked. If so, then one might make use of the very simple analytic solutions for two-state systems in treatments of more elaborate systems. Such simplification is possible under some conditions [Sho08]. The procedure involves a change of basis states known as a Morris–Shore (MS) transformation [Mor83; Vit00a; Vit03; Kis04; Iva06; Ran06].

The Morris–Shore transformation

An interaction pattern comprising multiple linkages can, when the pattern satisfies appropriate conditions, be replaced by a set of independent two-state interactions. The necessary conditions are [Mor83]

1. I. Two sets of states:

2. Set A (initial, ground), with NA elements.

3. Set B (excited), with NB elements.

1. II. There are no couplings within the A set or the B set, only couplings between A and B. (The graph corresponding to this linkage pattern is therefore bipartite.)

2. III. The two sets share common diagonal elements. In the RWA these are the two detunings ΔA and ΔB.

The examples of quantum-state manipulation and coherent excitation discussed in this monograph present idealizations of actual quantum systems, simplifications that allow straightforward theoretical description. As one moves beyond the models of isolated atoms, few essential states, and transform-limited pulses to deal with more realistic models that can describe experimental reality, the basic tools described hitherto require elaboration and extension. Theoretical treatments of large molecules and chemical reactions involving laser-induced changes rely upon numerical simulation more than on analytic solutions. This final chapter discusses two of the themes applicable to that work: control theory and optimization.

Control theory

Classical control theory, as followed by mathematicians and engineers, deals with procedures for manipulating the input (the “controls”) of a dynamically changing system to obtain a desired output of the system. In a closed-loop control system some device measures the output and, using a feedback loop, alters the input (via a control element) to bring the output closer to conformity with a goal. These techniques typically find application in control of experiments but they also work for theoretical modeling. An open-loop control system is one without such feedback; the controls are adjusted in accord with some established plan. Design of a suitable control mechanism (a control function ratioing output to input) must ensure that the system is stable (i.e. a finite input signal produces a finite output signal) and controllable (i.e. it is possible to obtain the desired output).

The traditional Raman process alluded to in Chap. 13 is a three-state sequence of transitions in which radiative excitation (induced by a pump field) is followed by spontaneous emission that produces a final state differing from the initial state [Her50b] [Sho90, § 17.5]. When the final state of the sequence is more energetic than the initial state the resulting emission line (to the red of the pump wavelength) is known as a Stokes spectral line. The difference between the pump frequency and the Stokes frequency, the Raman frequency, defines the excitation energy of the final state relative to the initial state. When, instead, the final state has lower energy than the initial state, as can occur when the initial quantum state is already excited, then the emission is an anti-Stokes line, at a bluer wavelength than the pump field. The overall Raman scattering is a two-photon process.

Typically Raman spectroscopy deals with molecules; the two-photon transitions are then between vibrational-rotational states, through electronically excited intermediate states, that are characterized in part by vibrational quantum number v and rotational angular momentum quantum numbers J, M. From any given excited electronic state there are many fluorescing transitions, corresponding to various vibrational and rotational quantum numbers of the final state. The wavelengths of the various Stokes and anti-Stokes lines (i.e. the Raman frequencies) characterize the particular molecular species, and so they have provided a valuable diagnostic tool for spectroscopists.

Ensembles and expectation values

Equations of motion provide predictions of time-evolving variables of a dynamical system, and thereby enable us to predict the changing behavior. These equations are characteristic of a particular type of system, say a classical harmonic oscillator or a two-state quantum system. To complete the description of behavior one must specify initial conditions that select, from the many possible solutions to the equations of motion, a particular realization.

In the example of a classical point particle the equation is second order in time, and so we need two initial conditions, of position and velocity, to specify a particular solution. These two numbers, taken with the classical equation of motion, provide a complete description of the behavior of a specific harmonic oscillator. Similarly the two-state quantum system requires two initial values to complete its definition.

The initial conditions are never defined with infinite precision, even for classical mechanics. More generally we deal with an ensemble, that is, a collection of similar dynamical systems (e.g. oscillators) that differ in their preparation. The individual cases share some common attributes, but not necessarily all. Thus we have only incomplete information about any single system. We might, for example, consider planetary motion in which we know only the mean energy and the orbital plane. The loci of positions of the planets then form ellipses that make up the ensemble.

Although it is possible to prepare a single two-state quantum system in which the initial state is known (apart from an overall phase), often one deals with ensembles of such systems, each of which may differ in some uncontrollable property.

The discrete energies En of bound states are not the only indicators of quantum-mechanical properties and quantization. Rotational motion of atoms or molecules, associated with angular momentum, is also quantized, both in magnitude and in direction; only discrete orientations are allowed with respect to any selected (but arbitrary) axis of quantization [Sho90, §18.1]. In the absence of external fields, the energy of a free atom or molecule does not depend on this orientation, and the energy states are degenerate. This chapter discusses that degeneracy, and the theory of coherent excitation of such degenerate quantum states.

Angular momentum degeneracy. The theoretical building blocks for describing rotational motion are angular momentum states |J,M,〉 discussed in App.A, associated with a dimensionless vector operator Ĵ and its component Ĵz along a quantization axis, taken as defining the z axis. The label J, the angular momentum quantum number, derives from the eigenvalue J(J + 1) of the operator Ĵ2 and therefore quantifies the magnitude of the angular momentum. It may be an integer or half integer. The label M, the magnetic quantum number, is the eigenvalue of Ĵz; it quantifies the projection of angular momentum along a reference axis. The values of M differ by integers and range from −J to +J in integer steps. The total number of such values, 2J +1, is an integer, the degeneracy of the quantum state |J,M〉.

Angular momentum of an isolated quantum system, such as an atom or molecule, refers always to the center of mass (which may be moving).