Most of this review has focused on collisions of cold, trapped atoms in a light field. Understanding such collisions is clearly a significant issue for atoms trapped by optical methods, and historically this subject has received much attention by the laser cooling community. However, there is also great interest in ground-state collisions of cold neutral atoms in the absence of light. Most of the early interest in this area was in the context of the cryogenic hydrogen maser or the attempt to achieve Bose–Einstein condensation (BEC) of trapped doubly spin-polarized hydrogen. More recently the interest has turned to new areas such as pressure shifts in atomic clocks or the achievement of BEC in alkali systems. The actual realization of BEC in 87Rb [15], 23Na [103], 7Li [56, 57], 4He* [310, 330] and H [138] has given a tremendous impetus to the study of collisions in the ultracold regime. Collisions are important to all aspects of condensates and condensate dynamics. The process of evaporative cooling which leads to condensate formation relies on elastic collisions to thermalize the atoms. The highly successful mean field theory of condensates depends on the sign and magnitude of the s-wave scattering length to parameterize the atom interaction energy that determines the mean field wavefunction. The success of evaporative cooling, and having a reasonably long lifetime of the condensate, depend on having sufficiently small inelastic collision rates that remove trapped atoms through destructive processes.

Introduction

If, while approaching on an unbound ground-state potential, two atoms absorb a photon and couple to an excited bound molecular state, they are said to undergo photoassociation. Figure 5.1 illustrates the process. At long range electrostatic dispersion forces give rise to the ground-state molecular potential varying as C6/R6. If the two atoms are homonuclear, then a resonant dipole–dipole interaction sets up ±C3/R3 excited-state repulsive and attractive potentials. Figure 5.2 shows the actual long-range excited potential curves for the sodium dimer, originating from the 2S½ + 2P3/2 and 2S½ + 2P½ separated atom states. For cold and ultracold photoassociation processes the long-range attractive potentials play the key role; the repulsive potentials figure importantly in optical shielding and suppression, the subject of Chapter 6. In the presence of a photon with frequency ωp the colliding pair with kinetic energy kBT couples from the ground-state to the attractive molecular state in a free–bound transition near the Condon point RC, the point at which the difference potential just matches ћωp.

Scanning the probe laser ωp excites population of vibration–rotation states in the excited bound potential and generates a free–bound spectrum. This general class of measurements is called photoassociative spectroscopy (PAS) and can be observed in several different ways. The observation may consist of bound-state decay by spontaneous emission, most probably as the nuclei move slowly around the outer turning point, to some distribution of continuum states on the ground potential controlled by bound–free nuclear Franck–Condon overlap factors.

In the 1980s the first successful experiments [312] and theory [98], demonstrating that light could be used to cool and confine atoms to submillikelvin temperatures, opened several exciting new chapters in atomic, molecular, and optical (AMO) physics. Atom interferometry [6, 8], matter–wave holography [294], optical lattices [192], and Bose–Einstein condensation in dilute gases [18, 95] all exemplified significant new physics where collisions between atoms cooled with light play a pivotal role. The nature of these collisions has become the subject of intensive study not only because of their importance to these new areas of AMO physics but also because their investigation has led to new insights into how cold collision spectroscopy can lead to precision measurements of atomic and molecular parameters and how radiation fields can manipulate the outcome of a collision itself. As a general orientation Fig. 1.1 shows how a typical atomic de Broglie wavelength varies with temperature and where various physical phenomena situate along the scale. With de Broglie wavelengths on the order of a few thousandths of a nanometer, conventional gas-phase chemistry can usually be interpreted as the interaction of classical nuclear point particles moving along potential surfaces defined by their associated electronic charge distribution. At one time liquid helium was thought to define a regime of cryogenic physics, but it is clear from Fig. 1.1 that optical and evaporative cooling have created “cryogenic” environments below liquid helium by many orders of magnitude.

Cold and ultracold collisions occupy a strategic position at the intersection of several powerful themes of current research in chemical physics, in atomic, molecular and optical physics, and even in condensed matter. The nature of these collisions has important consequences for optical manipulation of inelastic and reactive processes, precision measurement of molecular and atomic properties, matter–wave coherence and quantum-statistical condensates of dilute, weakly interacting atoms. This crucial position explains the wide interest and explosive growth of the field since its inception in 1987. Obviously due to continuing rapid developments the very latest new results cannot appear in book form, but the field is sufficiently mature that a fairly comprehensive account of the principal research themes can now be undertaken. The hope is that this account will prove useful to newcomers seeking a point of entry and as a reference for those already initiated.

After a general introduction and a brief review of the elements of scattering theory in Chapters 1 and 2, the next four chapters treat collisions taking place in the presence of one or more light fields. The reason for this is simply historical. After the development of the physics of optical cooling and trapping from the early to mid 1980s, the first generation of collisions experiments applied this light-force physics to cool and confine atoms in traps and beams.

Introduction

Photoassociation uses optical fields to produce bound molecules from free atoms. Optical fields can also prevent atoms from closely approaching, thereby shielding them from shortrange inelastic or reactive interactions and suppressing the rates of these processes. Recently several groups have demonstrated shielding and suppression by shining an optical field on a cold atom sample. Figure 6.1(a) shows how a simple semiclassical picture can be used to interpret the shielding effect as the rerouting of a ground-state entrance channel scattering flux to an excited repulsive curve at an internuclear distance localized around a Condon point. An optical field, blue detuned with respect to the asymptotic atomic transition, resonantly couples the ground and excited states. In the cold and ultracold regime particles approach on the ground state with very little kinetic energy. Excitation to the repulsive state effectively halts their approach in the immediate vicinity of the Condon point, and the scattering flux then exits either on the repulsive excited state or on the ground state. Figure 6.1(b) shows how this picture can be represented as a Landau–Zener (LZ) avoided crossing of field-dressed potentials. As the blue-detuned suppressor laser intensity increases, the avoided crossing gap around the Condon point widens, and the semiclassical particle moves through the optical coupling region adiabatically. The flux effectively enters and exits on the ground state, and the collision becomes elastic.

Predicting the distribution of measured fluctuations in intensity is one of the great challenges in describing electromagnetic propagation through random media. The amplitude variance addressed in Chapter 3 describes the width of this distribution but tells one nothing about the likelihood that very large or very small fluctuations will occur. By contrast, the probability density function for intensity variations provides a complete portrait of the signal's behavior. This wider perspective is important for engineering applications in which one must predict the complete range of signal values. The same description provides an important insight into the physics of scattering of waves by turbulent irregularities.

The Rytov approximation predicts that the distribution is log-normal. That forecast is confirmed by measurements made over a wide range of conditions on terrestrial links. It is also confirmed by astronomical observations. The agreement is independent of the model of turbulent irregularities employed. It therefore provides a test for the basic theoretical approach to electromagnetic propagation. Second-order refinements to the log-normal distribution predict a skewed log-normal distribution and agree with numerical simulations. This success represents a significant achievement for the Rytov approach.

The distribution provided by Rytov theory needs to be enlarged in some situations. It describes the short-term fluctuations observed over periods of a few minutes to a few hours. One must also consider variations of signal strength measured over weeks, months or years. Long-term statistics are important for communication services employing terrestrial or satellite relays.

All of the development so far has concentrated on the variances of phase and amplitude – or their correlations with respect to separation, time delay and frequency separation. We turn now to moments of the field strength itself. These are the quantities that are often measured in astronomical observations and terrestrial experiments. They are usually calculated with theories that characterize strong scintillation. It is significant that the Rytov approximation gives identical results for many of these quantities. This approach requires only the tools we have already developed.

The average field strength and mean irradiance set important reference levels for measurement programs. They are also needed in order to describe other features of propagation in random media. Accurate estimates for these quantities require both first- and second-order solutions in the Rytov expansion. These depend on the double-scattering expressions which were established for plane, spherical and beam waves in Chapter 8. Field-strength moments are more difficult to estimate because they contain the Born terms in exponential form. We shall learn that the mean field is attenuated very rapidly with distance. By contrast, the mean irradiance for plane and spherical waves is everywhere equal to its free-space value.

These calculations set the stage for analyzing two important features of the electromagnetic field. With the second Rytov approximation one can demonstrate that energy is conserved in a nonabsorbing medium. The mutual coherence function is calculated in the same way and provides an expression identical to that predicted both by geometrical optics and by strong-scattering theories.