Peter van der Straten

Cooling and trapping of neutral atoms

Abstract As early as 1917, Einstein had predicted that momentum is transferred in the absorption and emission of light, but it was not until the 1980s that such optical momentum transfer was used to cool and trap neutral atoms. By properly tuning laser light close to atomic transitions, atomic samples can be cooled to extremely low temperatures, the brightness of atomic beams can be enhanced to unprecedented values, and atoms can be manipulated with extraordinary precision. In this review several of the techniques for laser cooling and trapping of neutral atoms are described.

Optical forces and laser cooling

The usual form of electromagnetic forces is given by but for neutral atoms q = 0. The next order of force is the dipole term, but this also vanishes because neutral atoms have no inherent dipole moment. However, a dipole moment can be induced by a field, and this is most efficient if the field is alternating near the atomic resonance frequency. Since these frequencies are typically in the optical range, dipole moments are efficiently induced by shining nearly resonant light on the atoms. Laser cooling and trapping rely on the interaction between laser light and atoms to exert a controllable force on the atoms, and many sophisticated schemes have been developed using the special properties of the interaction. This chapter begins with the simplest schemes for exerting optical forces on atoms, namely a single-frequency light field interacting with a two-level atom. Although this is the simplest possible scheme, it is pedagogically valuable because it shows many of the features that will be encountered further on. The ultimate temperature using such a scheme is limited, but lower temperatures can be achieved with multilevel atoms. This topic is addressed later in the chapter. Two kinds of optical forces When an atom absorbs nearly resonant light, it makes a transition to the excited state, and its subsequent return to the ground state can be either by spontaneous or by stimulated emission. The nature of the optical force that arises from these two different processes is quite different, and will be described separately [1]. The spontaneous emission case is different from the familiar quantum mechanical calculations using state vectors to describe the state of the system, because spontaneous emission causes the state of the system to evolve from a pure state into a mixed state (see Exercise 6.9). Spontaneous emission is an essential ingredient for the dissipative nature of the optical forces in monochromatic light, and the density matrix is needed to describe it (see Chap. 6). When an atom absorbs light, its energy excites the atom and its angular momentum changes the electrons orbit. Both of these are internal atomic properties. However, the linear momentum of the light, can change only the external, translational motion of the atom.

Interaction of two-level atoms and light

The use of light has been a very powerful tool to study the internal structure of atoms. Since absorption of light takes place only when the frequency of the light is nearly resonant with an atomic transition, and since the frequency of laser light can be controlled to a high degree, detailed information can be obtained this way. However, the careful analysis of the interaction between atoms and light is complicated and merits a study of its own. This chapter starts with the analysis of the simplest case, namely when only two atomic levels are important. The study of the internal structure of more complicated atoms is the subject of the second part of this book.

Ultra-cold chemistry

Introduction Interactions between separated atoms are complicated by the large number of degrees of freedom of a many-atom system, and are very difficult to describe quantitatively. Although the Hamiltonian for such systems can be written down easily, finding the eigenfunctions requires a sophisticated effort and large computer resources. This area of quantum chemistry, with all its intricacies, is a complex subfield of science that cannot be adequately described in a single book chapter. Therefore the student is referred to specialized books in this field for further reading [165, 170]. In the 1990s a new field of chemistry emerged that considered such interactions between atoms at ultra-cold temperatures. The field focuses strongly on atomic interactions at large internuclear distances where the atoms are well separated and their electronic clouds do not strongly overlap. In such circumstances the interactions are dominated by the properties of the states of the atoms and they can be understood from an atomic point of view. In this chapter the basic features of this field will be described. The interaction energy (electronic potential energy) between two colliding partners separated by a large distance is called the dissociation limit. Light interactions with the particles can occur with the total energy above or below this limit. The relative kinetic energy does not contribute because it is so very small at such low temperatures. In the presence of a light field tuned below this limit for a pair that has one of the atoms in an excited state, two unbound atoms can be associated into a bound molecular state (for example, an S–P collision of two alkali atoms). Such photo-association is more likely at a large interatomic separation because there is good radial overlap between the incoming, low-energy scattering state and the outer turning point of the bound excited state, so it usually produces a molecule in a high vibrational state. Studying these states provides important information about interatomic interactions near the dissociation limit. The long-range part of the potential is dominated by atomic parameters and in Sec. 18.2 it is shown how these potentials can be constructed. The LeRoy–Bernstein method, which was developed long before the advent of cold-atom photo-association, is described in Sec. 18.3 and can be used to extract these parameters from measurements of the bound states.

The hydrogen atom

Introduction In 1885 Balmer discovered a simple arithmetic relation among the wavelengths of the spectral lines of hydrogen that led to the Rydberg formula. Balmers 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 Schrodinger 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.

Angular momentum in quantum mechanics

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General Properties Concerning Laser Cooling

This chapter presents some of the general ideas regarding laser cooling. One of the characteristics of optical control of atomic motion is that the speed of atoms can be reduced by a considerable amount. Since the spread of velocities of a sample of atoms is directly related to its temperature, the field has been dubbed laser cooling, and this name has persisted throughout the years. Laser cooling has much in common with the field of optics. In laser cooling, light is used to manipulate atoms, whereas in optics matter is used to manipulate light. The more proper identification for the field would therefore be “atom optics” or “optical control of atomic motion”. The similarities between atom optics and electromagnetic optics will be pointed out.

Review of Quantum Mechanics

This chapter presents a brief review of those aspects of quantum mechanics that are important for understanding some of the material to be found elsewhere in this book. Its purpose is not to teach the subject, because that is so very well done in numerous other textbooks [1–9]. Rather, the intent is to bring together certain things that are sometimes scattered throughout such texts, to establish notation and conventions, and to provide a reference point for many important and useful formulas.

Deceleration of an Atomic Beam

The origin of optical forces on atoms has been discussed in Chapter 3, and here a specific application is introduced. The use of electromagnetic forces to influence the motion of neutral atoms has been a subject of interest for some years, and several review articles and books on the subject are listed in Appendix B. The force caused by radiation, particularly by light at or near the resonance frequencies of atomic transitions, originates from the momentum associated with light. In addition to energy E = ℏω, each photon carries momentum ℏk and angular momentum ℏ. When an atom absorbs light, it stores the energy by going into an excited state; it stores the momentum by recoiling from the light source with a momentum ℏk; and it stores the angular momentum in the form of internal motion of its electrons. The converse applies for emission, whether it is stimulated or spontaneous. It is the velocity change of the atoms, vr = ℏk/M ≃ few cm/s, that is of special interest here, and although it is very small compared with thermal velocity, multiple absorptions can be used to produce a large total velocity change. Proper control of this velocity change constitutes a radiative force that can be used to decelerate and/or to cool free atoms.

Optical Traps for Neutral Atoms

The force on atoms confined in the magnetic traps described in Chapter 10 arises from the permanent magnetic dipole moments of the atoms in the inhomogeneous field of the trap. By contrast, the inversion symmetry of atomic wave functions prevents them from having permanent electric dipole moments, so optical trapping of neutral atoms by electrical interaction must proceed by inducing a dipole moment. This can be accomplished either by electrostatic fields or by nearly resonant optical frequency fields. Inducing appropriate dipole moments with dc fields can be accomplished in atoms that have a sufficiently close-lying energy states of opposite parity (this excludes most atomic ground states but favors Rydberg states). By contrast, there are several types of optical traps that employ various configurations of laser beams [40, 41]. These produce not only the mixing of atomic states of opposite parity needed to provide dipole moments for interaction with the field, but also the strong field gradients appropriately arranged for such trapping.