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).

From prehistoric times has come recognition that sunlight and firelight provide warmth, and that such illumination casts shadows. Expressed in more contemporary terms one would say that light travels in rays, and that this radiation has the potential to provide heat energy to absorbing material. From the time of Newton it has been known that light sources emit radiation comprising a distribution of colors. During the nineteenth century it was recognized that radiation had characteristics of transverse waves (with wavelength associated to color) but until the late twentieth century, when lasers became laboratory tools, it was hardly necessary to delve into the equations of electromagnetic theory to treat such experiments as were then possible; interest lay primarily with thermodynamic considerations of energy flow or with measurements of the dark or bright lines seen in the distribution of light that, after passing through a slit, was dispersed by a prism or grating into constituent colors. Although laser light sources are essential for the types of atomic excitation considered in this monograph, the legacy from thermal radiation still influences many interpretations of the interaction between radiation and matter, and it is therefore useful to summarize some of those concepts.

The mathematics needed for describing laser radiation, or polarized light in general, has much in common with the mathematics of quantum theory discussed starting in Sec. 3.5 and specialized to two-level atoms in Sec. 5.6. In both cases one deals with two complex-valued functions – independent electric field amplitudes or probability amplitudes – whose absolute squares are measurable.

The quantum world within an atom or molecule that once attracted explorations only by academic physicists now provides fertile sustenance for chemists seeking control of chemical reactions and for engineers developing ever smaller electronic devices or tools for processing information with greater security. Whereas the first pioneers could only discover the most elementary properties – the discrete energy levels that characterize the internal structures of atoms and the radiative transitions that link these structures to our external world – it is now possible to alter that structure at will, albeit briefly.

Objective

This monograph presents the physical principles that describe such deliberately crafted changes, namely how single atoms or molecules (or other simple quantum systems) are affected by coherent interactions, primarily laser light – a subject that has been regarded first as a part of quantum electronics and then quantum optics but is most generally described as coherent atomic excitation [Sho90].

This physics has relevance to such basic concerns as the detection and quantitative analysis of trace amounts of chemicals, the catalysis or control of chemical reactions, the alignment of molecules, and the processing of quantum information. The physics necessarily involves elementary quantum mechanics, but it has many associations to the classical dynamics that governs macroscopic objects – waves and particles. The mathematics that quantifies the changes is that of differential equations, specifically coupled ordinary differential equations (ODEs), whose parameters incorporate the controls of experimenters and whose solutions, appropriately interpreted, quantify the resulting changes.

The philosophers who first hypothesized the existence of “atoms” had in mind the smallest particles of matter that could preserve identifiable chemical properties – building blocks that could be assembled into familiar substances. Little more than this definition – tiny masses that carry kinetic energy and undergo collisions – led to the fruitful quantitative explanation of vapor properties in the kinetic theory of gases, and to such devices as mass spectrometers and ion accelerators.

Atoms and molecules. As became clear during the early twentieth century, these “chemical atoms’ from which materials are constructed have internal structure that endows them with their chemical attributes: one or more positively charged nuclei, each a few femtometers (1 fm = 10−13 cm = 10−15 m) in diameter, surrounded by one or more much lighter negatively charged electrons whose motion fills a volume of at least a few cubic angstroms (1 Å= 10−8 cm = 10 nm) in diameter. Nowadays we distinguish between particles having multiple nuclei (molecules) and those with a single nucleus (atoms); when the positive and negative charges are unbalanced these are ions (positive or negative). The simplest atom, hydrogen, has a single electron; the most complex atoms have more than a hundred electrons. Although I will often refer to “atoms”, usually the discussion applies equally well to molecules or to any other structure whose constituents exhibit distinct quantum properties, as manifested by discrete energies.

Nuclei. The nuclei of atoms are, in turn, composed of protons and neutrons.