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.

Introduction

Up to here this book has considered in detail only neutral atoms, namely a single nucleus surrounded by its electrons. This chapter will consider molecules, systems with at least two nuclei bound together by their electrons. This change increases the number of possibilities considerably, since all atoms can be combined to form molecules in principle, and the number of atoms in a molecule is unlimited. The study and engineering of molecules forms the heart of chemistry, but this chapter does not intend to give a proper account of this field of science.

Moreover, the discussion here will be restricted to molecules having only two nuclei. The reason is that such diatomic molecules can be described with the techniques that have already been discussed earlier in this book, and therefore they are the atomic physicists’ favorite kind of molecule. Furthermore, the atmosphere of the earth consists of more than 99% diatomic molecules, namely N2 and O2, and the most abundant molecule in the universe is H2. Nevertheless, it is important to remember that polyatomic molecules give us life and reasons to live, even though they will not be discussed here.

The paired nuclei of diatomic molecules provide important additional degrees of freedom. In an atom, the center-of-mass and the nucleus nearly coincide, and in the center-of-mass frame the nucleus is nearly at rest. However, in a molecule the kinetic energy of the nuclei is not negligible since the center of mass is generally well separated from each of them, and the energy of their motion in the center-of-mass frame adds significantly to the total energy of the system.

Since molecules consist of at least three particles, the equations of motion cannot be solved exactly, neither in classical mechanics nor in quantum mechanics. In order to find solutions, one has to resort to approximations. One of the most important of these approximations is enabled by the large difference in mass between the nuclei and the electrons. This results in the nuclei moving much more slowly than the electrons in the center-of-mass frame. It therefore seems natural to treat the nuclei as fixed at a given separation R, calculate the energy of the system, and then repeat the calculations at a different value of R. The result provides a potential V(R) created by the electrons in which the nuclei move.

Introduction

Helium is the natural connection between the hydrogen-like atoms (one electron outside a spherical core, such as the alkali-metal atoms of Chap. 10) and all the others in the periodic table. For these one-electron atoms the field-free description of the electron motion in terms of three quantum numbers n, ℓ, and m suffices. The other electrons are considered to form the “core” and its effect on the energy levels is reduced to the quantum defects δℓ, as discussed in Chap. 10. The case of helium is very different since the two ground-state electrons are in the same electronic state and differ only in the orientation of their spins. Thus spin is of paramount importance for the description of all but the hydrogenic atoms. Since helium has only two spins to deal with, their roles can be written explicitly.

Because the two electrons are identical, quantum mechanics requires that the wavefunction undergoes nothing more than an unobservable phase shift when they are exchanged. This consequence is called the Pauli symmetrization principle, and is sometimes referred to as “the source of our individuality”. Moreover, it has a profound influence on the wavefunction and thus on the allowed energy states. It will be shown that this symmetry requirement dictates that two electrons cannot be in states with identical quantum numbers, and this topic will be discussed in detail in this chapter.

Symmetry

The profound difference between the hydrogen and helium wavefunctions arises because the two electrons are indistinguishable, and this results in a fundamental change of the energy-level structure. The concept of “spin”, introduced as a shortcut for the intrinsic magnetic moment of the electron in Chap. 7, now becomes a dominating feature of the description. Before discussing the helium energy levels, the effects of this symmetry with respect to exchange of identical particles is presented.

The exchange operator

In quantum mechanics, identical particles are inherently indistinguishable, so the exchange of the two electrons cannot change any observable and there should be no measurable effect on the overall wavefunction. However, there can be an overall phase change whose nature can easily be found by defining an operator P12 that exchanges the electrons and applying it to the wavefunction. The exchange applies to all coordinates of the electron including its spin coordinate, but for simplicity only the spatial coordinate will be indicated here.

Atomic physics has played a vital role in defining the progress of physics since the late nineteenth century. One could say that it began with the observation of discrete spectral lines and Balmer's categorization of their wavelengths in the middle 1880s that culminated in the success of the Bohr model 30 years later. Precise spectral measurements led to the discovery of the Zeeman effect, atomic fine structure, the Lamb shift, the anomalous electron magnetic moment, and a host of other phenomena that made great impacts on the physical description of nature.

Apart from precision measurements, atomic and molecular physics experiments have led to a very long list of important discoveries that have had major impacts on fundamental physics. These include, but are not limited to, the discovery of deuterium and thereby the existence of isotopes, violation of the Bell inequalities, the development of the maser and laser, magnetic resonance imaging, and Bose– Einstein condensation along with its impact on condensed matter physics. Modern commercial aviation, the Internet, and the GPS would not exist without the atomic clocks that synchronize them.

Arguably, the origin and development of quantum mechanics was driven by the inconsistencies between laboratory observations and the classical physics of Newton and Maxwell. Most of what was known about nuclear physics in the 1950s was based on atomic hyperfine studies, and there is currently much information on hadron structure derived from atomic physics [322]. Modern field theories derive from quantum electrodynamics that was founded and established by studies of atomic structure.

More recently there there have been attempts to detect an intrinsic electric dipole moment (EDM) of either the electron or the neutron. In the case of the electron, a non-zero EDM would have implications for supersymmetry theories of physics beyond the standard model. Such theories have predictions in the 10-26 to 10-29 e·cm range (1 e·cm = 1.602 178 ×10-21 C · m), and current limits in the few ×10-28 e·cm range are already casting doubt on some of these. By contrast, the standard model without such additions predicts 10-38 e·cm, which is well beyond any current capabilities.

Introduction

Atomic spectroscopy provided a wealth of information about nuclear physics before the advent of huge accelerator facilities. The multiple, closely spaced atomic spectral lines could be ascribed to various nuclear properties, including the existence of isotopes, the magnitude of the nuclear magnetic moments, and even quadrupole and higher-order non-spherical aspects of nuclear structure. Some aspects of these effects are discussed below.

Motion, size, and shape of the nucleus

In Chap. 7 it is assumed that the nucleus of the atom is a point particle located at rest in the center of the coordinate system. Since the nucleus is much heavier than the electrons and the size of the nucleus is much smaller than the radius of the electron's orbit, this is a good approximation. In this section the effects of the nuclear motion, size, and shape are discussed.

Nuclear motion

The kinetic energy in the Hamiltonian of the Schrödinger equation is represented by the operator p2/2m, whose constituent terms are the momentum operators px = −iħd/dx, and similarly for py and pz. But these operate only on the electron coordinates, and thus provide only its kinetic energy, and not that of the moving nucleus. As suggested in Chap. 7 the nuclear kinetic energy is small compared with that of the electron.

Consider that the motion of this two-body system may be divided into the overall center-of-mass motion and the motion of the constituent particles with respect to the center of mass. Then, in the center-of-mass rest frame, the total momentum of the constituents is zero so, for the two-body system of the hydrogen atom, MV + mv = 0, where upper (lower) case refers to the nucleus (electron). Then so the kinetic energy of the nucleus is smaller than that of the electron by the ratio of the masses, one part in 2,000 for hydrogen, and still smaller for heavier atoms. For this two-body system, the kinetic energy of the nucleus with respect to the center of mass can be included by making a simple correction by replacing the mass of the electron with the reduced mass μ = mM/(m + M) as in Eq. (7.29). For an atom, M ≫ m so μ ⋍ m (see Tab. 7.5).