Introduction

Bose–Einstein condensation (BEC) is one of the most intriguing phenomena in physics. The basic idea goes back to the 1920s, when Einstein made predictions for the behavior of atoms at sufficiently low temperatures and high densities. However, it took seven decades for experimental physicists to realize this phase under the low-density conditions envisioned by Einstein. In the intermediate period there were many experiments in which aspects of BEC played important roles. Therefore, phenomena such as superfluidity and superconductivity were connected to BEC long before its 1995 realization in ultra-cold, low-density atomic vapors. Since BEC can be discussed with very many different backgrounds and applications in mind, the subject can sometimes be overwhelming for interested students.

There is a very good reason why a chapter on BEC does not belong in this book. In all the other chapters here, the processes described involve one or at most two atoms, and the interactions that play a role are between the particles that constitute the atom, namely between the nucleus (or nuclei) and their electrons. The interactions are described quantum mechanically by their Hamiltonian that can always be written exactly in non-relativistic terms, and in some cases can even be solved exactly. This makes atoms and molecules a branch of physics where comparisons between theory and experiments become very meaningful since theoretical predictions are often confirmed to very high accuracy.

In BEC, it is the interactions among many atoms that are the key for understanding the phenomenon. Although in principle one could write a Hamiltonian for a Bose condensed gas, its large number of particles makes this not very practical. The starting point for the description of a BEC is in the area of many-body physics that belongs to the expertise of condensed-matter physicists. The interactions between ~1023 electrons in a solid state with a lattice formed by ~1023 ions resemble the kind of interactions in a BEC much more closely than the simple fewbody interactions in an atom or molecule. Thus all the theoretical models that have been described up to now are not applicable for the contents of this chapter.

Introduction

In 1885 Balmer discovered a simple arithmetic relation among the wavelengths of the spectral lines of hydrogen that led to the Rydberg formula. Balmer's formula was later to be found consistent with the Bohr model of the hydrogen atom, and for this reason hydrogen has served as the paradigm for the study of all atoms. The Bohr model serves to describe the energy of electrons in an atom, and also for the study of atoms in external fields, both optical and dc.

It can easily be shown that the behavior of electrons in the atom should be described quantum mechanically, and thus that the Bohr model using semi-classical arguments cannot be definitive. One of the triumphs of the establishment of quantum mechanics in the 1920s was that the Schrödinger equation for hydrogen could be solved exactly. Since the outcome of the theory corresponded completely with the experimental results known at that time, the quantum mechanical description of the hydrogen atom served as one of the first proofs of the validity of quantum mechanics.

The quantum mechanical model of the hydrogen atom is discussed in many textbooks about both quantum mechanics and atomic physics. It is discussed in this book because it serves as the basis for much of the remainder of the book, both in techniques and in notation. Furthermore, hydrogen is the only element (apart from its isotopes and single-electron ions) that can be solved exactly. More important, its solution will be discussed more in terms that are relevant for atomic physics, and not so much in terms of quantum mechanical aspects that are more mathematically oriented.

The system of units used in this book is the internationally accepted SI system. Atomic physics calculations are sometimes very cumbersome in this system, particularly for the internal structure, but for consistency SI units are used in this part as well. This usage is especially important for experimentalists so that formulas can be evaluated in terms of laboratory quantities. The atomic units can be found in App. B.1 at the end of this book.

The Schrödinger equation is exactly solvable for hydrogenic atoms and provides excellent results for the energies and wavefunctions. For all the other atoms, the Schrödinger equation can be written down and the Hamiltonian is known exactly, but the resulting equation cannot be solved analytically. The main problem for these systems is that the electron–electron interaction does not admit the central field solutions of the hydrogen atom. The solution is intractable, and one has to resort to approximations to describe these atoms.

A special case is the alkali-metal atoms, where all but one of the electrons are in a spherically symmetric core consisting of closed shells. This one remaining valence electron determines most of the atomic properties. Since the size of the core is much smaller than the orbital radius of the valence electron, the wavefunction is concentrated outside the core, where the field is purely Coulombic and the hydrogen solutions are appropriate. Only electrons that penetrate the core are strongly affected by it, and these electrons have low orbital angular momentum ℓ.

In the nineteenth century it was discovered that the energy of the states of the alkali-metal atoms can be accurately described by the Rydberg formula when the quantum number n is replaced by an effective quantum number n∗ that differs from n by an ℓ-dependent constant denoted δℓ. The shift was named the quantum defect by Schrödinger before the advent of quantum mechanics. It is remarkable that δℓ depends only on ℓ and is nearly independent of n.

In this chapter many different techniques that are more or less standard in quantum mechanics will be used to describe the alkali-metal atoms. For each of these there is sufficient background provided to be able to understand the line of reasoning even without a formal introduction. For further details one should consult the standard textbooks on quantum mechanics. For example, a model will be introduced that can explain the physical background of the quantum defect and show how to calculate the various δℓ values from first principles. The defects will subsequently be used to calculate the energy levels and the transition rates between several states.

In order to confine any object, it is necessary to exchange kinetic for potential energy in the trapping field, and in neutral atom traps, the potential energy must be stored as internal atomic energy. Thus practical traps for ground-state neutral atoms are necessarily very shallow compared with thermal energy because the energy level shifts that result from conveniently sized fields are typically considerably smaller than kBT, even for T = 1 K. Neutral atom trapping therefore depends on substantial cooling of a thermal atomic sample, and is often connected with the cooling process. In most practical cases, atoms are loaded from magneto-optical traps, where they have been efficiently accumulated and cooled to mK temperatures (see Sec. 20.3), or from optical molasses, where they have been optically cooled to μK temperatures (see Sec. 19.4).

The small depth of typical neutral atom traps dictates stringent vacuum requirements, because an atom cannot remain trapped after a collision with a thermal energy background gas molecule. Since these atoms are vulnerable targets for thermal energy background gas, the mean free time between collisions must exceed the desired trapping time. The cross-section for destructive collisions is quite large because even a gentle collision (i.e. large impact parameter) can impart enough energy to eject an atom from a trap. At pressure P sufficiently low to be of practical interest, the trapping time is ~(10-8/P) s, where P is in torr.

This chapter begins with a discussion of ordinary, single-center traps, but also considers arrays of traps. Such “optical lattices” are playing an increasingly important role in research in atomic physics, so discussion of their properties is an appropriate part of the education of a student of the subject.

Dipole force optical traps

Single-beam optical traps for two-level atoms

The simplest imaginable optical trap consists of a single, strongly focused Gaussian laser beam (see Fig. 20.1) [214, 215] whose intensity at the focus varies transversely with r as where is the beam waist size. Such a trap has a well-studied and important macroscopic classical analog in a phenomenon called optical tweezers (see App. 1.C) [216–218].

Introduction

The preceding chapters have focused on atomic transitions induced by applied radiation. That is, there is an optical field illuminating the atoms, and their response is calculated. It is implicitly assumed that an excited atom left alone will spontaneously return to the ground state, and conserve energy by emitting light that satisfies E2 − E1 = ħω21 in accordance with the Planck and Bohr pictures. But this spontaneous emission process is not addressed quantitatively in these earlier chapters.

Spontaneous emission is one of the most pervasive phenomena of atomic physics, and yet its origins remain among the least well-understood. The number of misconceptions is enormous, and they are to be found in textbooks, journal articles, and lecture notes. Because the natural decay of an excited atom adds energy to what is usually an empty mode of the electromagnetic field, the process is fundamentally quantum mechanical. Classical or even semi-classical descriptions cannot be assured to give correct answers. This is not to say that such discussion is to be avoided, but only that caution is needed in interpreting the results.

Toward the end of the nineteenth century, many experimental results made it clear that classical mechanics needed to be supplemented by new theories. The first venture into the new physics was Planck's hypothesis that the energy of classical radiating oscillators like those discussed in Chap. 1 would have to be quantized into integer multiples of ħω. This notion led to his famous formula for the spectrum of thermal radiation from a non-reflective object (black body), and this formula describes a radiation field in thermodynamic equilibrium with its environment. The result of Planck's hypothesis in the context of thermal equilibrium plays a vital role in the discussion of spontaneous emission.

Einstein A- and B-coefficients

Soon after Bohr proposed a model of atomic structure consisting of discrete energy levels, connected by radiation whose energy satisfied ΔE = ħω when atoms made transition between them, Einstein published the first serious attempt to connect the absorption and emission processes. Absorption, he reasoned, could occur only in the presence of applied radiation, whereas spontaneous decay occurred without the application of any field.

Introduction

While it might appear that the nineteenth-century physics presented in this chapter has no place in a topic as quantum mechanically oriented as atomic physics, this is simply not the case. The purpose of this first chapter is to try to convince the readers that the material learned in their early years of studying physics is not disjoint from modern topics, but in fact can provide a foundation for their further understanding. Other examples will occur later.

The reader may be a bit surprised to see that the results of the classical description turn out to be so close to the quantum mechanical results. Where there are significant disagreements between classical and quantum descriptions, it is necessary to explain why these occur. The correspondence principle requires that in the limit of large quantum numbers the complete quantum mechanical description must agree with the classical one. It is always better to need fewer of these explanations and to be able to treat physics as a coherent whole instead of a collection of topics.

The consequences of classical physics are intuitively familiar, and this is often most helpful in gaining deeper understanding. In fact, the approach taken in very many of the chapters in this book is to start with a purely classical description of a topic, and then transport it directly into the Schrödinger equation. Radiation from an accelerated charge, as treated here in the harmonic case, provides an ideal setting for such pedagogy. Even though classical mechanics is used in place of quantum mechanics, for the case of harmonic motion, the results are quite compatible.

The early history of atomic physics contains many topics that are well worth reading about, especially the experimental facilities. For example, there were no electric motors, wire was not readily available, and making even a crude vacuum was an experimental tour de force. Very many textbooks have discussions of these topics, and so they will not be presented here. But the reader is warned that certain oft-repeated modern descriptions are simply wrong.

Perhaps the most important notion to be gained from this chapter is that the classical description of a harmonically bound electron provides quite accurate results about the interaction between atoms and light. Of course, the utility of this description is limited because the classical harmonic oscillator has no internal structure, is infinitely deep, and has a continuous energy spectrum.

Slowing, cooling, and trapping molecules

One of the major achievements in physics during the last two decades of the twentieth century was the ability to cool and trap atoms [1]. Laser cooling can produce previously inconceivable atomic temperatures of less than 1 μK. Atoms with sufficiently low kinetic energy can be trapped in magnetic and optical traps and thus studied for a prolonged time, thereby improving the resolution of spectroscopy by many orders of magnitude as well as reducing Doppler broadening. Furthermore, besides the external degrees of freedom, the internal degrees of freedom of the atoms can also be manipulated, leading to full control of the atoms. One of the holy grails of physics, the achievement of Bose–Einstein condensation in a dilute gas, was achieved mainly because of laser cooling and trapping of atoms.

One of the natural extensions of laser cooling and trapping of atoms is the application of the techniques to molecules. Molecules offer a much richer internal structure than atoms, and thus provide extended possibilities to experimentalists. However, atomic laser cooling is based on the repetitive scattering of light and requires a closed, or at least nearly closed, two-level system. Because of the complicated internal structure of molecules, a closed two-level system cannot usually be found because they have many different electronic, vibrational, and rotational states and in general can decay to many different states after absorption of light (see Chap. 16). This requires a complete rethinking of the schemes that are used to cool atoms.

Still, because of the extended possibilities cold molecules can offer with respect to cold atoms, there have been a large number of techniques developed to slow, cool, and trap them. These techniques can be divided into two classes, namely direct and indirect cooling.

In the first case, molecules at cryogenic or room temperature are directly slowed and cooled. Although many different slowing techniques are used, this chapter focuses on Stark slowing of molecules since it was the first technique developed. Apart from the Stark shift of the molecules, the Zeeman and light shift of molecules can also be used to slow them. The idea behind Stark slowing is to subject the moving molecules to a continuously increasing electric field so that their kinetic energy is converted into potential energy.

By any measure, atomic physics is among the fastest-growing, most dynamic, and best-recognized areas of physics. The student attendance at conferences devoted to this subject area has burgeoned, and the number of new university tenure track positions in atomic physics is disproportionately high. Recently there have been several documents from national and international science-oriented agencies extolling the growth and importance of atomic physics, with statements such as “Light influences our lives today in ways we could never have imagined a few decades ago,” and “AMO science (not only) provides the basis for new technology, it is also a source of the intellectual capital on which science and technology depend for growth and development.” The General Assembly of the United Nations and UNESCO declared 2015 as the “International Year of Light”, further underscoring its importance to the world. Applications are not restricted to further exploration of our specialized field, but have expanded to include substantial impact on other areas of physics such as condensed matter, quantum information, thermodynamics, and fluid mechanics. For these and other reasons, we have decided to provide an introductory text appropriate to the emerging laser era in atomic, molecular, and optical science.

This book is intended for multiple purposes. First and foremost, we are experimentalists, so the material is presented in an intuitive and very physical tone. Many ideas are developed from the classical physics perspective rather than from mathematical formalism. We try to connect the concepts with measurements where it makes sense to do so, and to motivate each topic by the observations that produced the information about it.

Our intent is to use it as a text for a course in atomic physics. It requires a knowledge of quantum mechanics at the level of the well-known textbooks by Griffiths or Liboff, and of elementary electricity and magnetism. Thus it is suitable for an advanced undergraduate course or a beginning graduate course (certainly for a course that serves both populations together). In addition, we have tried to write in a sufficiently familiar style that a student can read and understand the material even without the benefit of a course. That is, the material is presented in a descriptive mode to maximize understanding. Some applications and detailed calculations relevant to a particular chapter are provided as appendices to the specific chapter.