### Abstract

The idea that light carries a mechanical momentum and can modify the trajectories of massive objects can be traced back to Kepler, who offered it as an explanation for the direction of the tail of comets away from the Sun. More rigorously, the force exerted by light on atoms is implicit in Maxwell's equations. For example, it is readily derived from the classical Lorentz model of atom-radiation interaction, where the force of light on atoms is found to be {A formula is presented} Here r is the center-of-mass coordinate of the atom, x is the position of the electron relative to the nucleus, {A formula is presented} is the dipole potential due to the light, E ( r, t ) is the electric field at the center-of-mass location of the atom and q = - e is the electron charge. The force F ( r, t ) is often called the dipole force, or the gradient force. It indicates that it is possible to use light to manipulate atomic trajectories, even when considered at the classical level. An additional key element in understanding the motion of atoms in light fields derives from basic quantum mechanics: since the 1923 work of Louis de Broglie we know that any massive particle of mass M possesses wave-like properties, characterized by a de Broglie wavelength {A formula is presented} where h is Planck's constant and v the particle velocity. Combining de Broglie's matter wave hypothesis with the idea that light can exert a mechanical action on atoms, it is easy to see that in addition to conventional optics, where the trajectory of light is modified by material elements such as lenses, prisms, mirrors, and diffraction gratings, it is also possible to manipulate matter waves with light, resulting in atom optics. Indeed, atom optics (Meystre [2001]) often (but not always) proceeds by reversing the roles of light and matter, so that light provides the "optical" elements for matter waves. Very much like conventional optics can be organized into ray, wave, nonlinear, and quantum optics, matter-wave optics has recently witnessed parallel developments. Ray atom optics is concerned with those aspects of atom optics where the wave nature of the atoms does not play a central role, and the atoms can be treated as point particles. Wave atom optics deals with topics such as matter-wave diffraction and interference. Nonlinear atom optics considers the mixing of matter-wave fields, such as in atomic four-wave mixing, and the photoassociation of ultracold atoms - a matter-wave analog of second-harmonic generation. In such cases, the nonlinear medium appears to be the atoms themselves, but in a more proper treatment it turns out to be the electromagnetic vacuum, as we discuss in some detail later on. Finally, quantum atom optics deals with topics where the quantum statistics of the matter-wave field are of central interest. Examples include the generation of entangled and squeezed matter waves. In contrast to photons, which obey bosonic statistics, atoms can be either composite bosons or fermions. Hence, in addition to the atom optics of bosonic matter waves, which finds much inspiration in its electromagnetic counterpart, the atom optics of fermionic matter waves is now actively studied by a number of groups. This emerging line of investigations is likely to lead to the discovery of novel phenomena completely absent from bosonic atom optics. This chapter reviews some of the key recent developments in nonlinear and quantum atom optics that result from the availability of Bose-Einstein condensates and quantum-degenerate Fermi systems. After an elementary review of the formalism of second quantization, which describes atoms as a quantum field and leads to a simple understanding of much of atom optics in direct analogy to the optical case, we recall some important features of Bose-Einstein condensation and of quantum-degenerate Fermi systems. One important distinction between optical and matter-wave fields is that the latter ones are self-interacting, a result of atomic collisions. As it turns out, collisions play for atoms a role analogous to that of a nonlinear medium for light; hence it is important to introduce their main characteristics in the context of ultracold atoms. We show that attractive two-body interactions are the de Broglie waves analog of a self-focusing medium in optics, while repulsive interactions correspond to defocusing. We also discuss at some length the physics of Feshbach resonances, which provide us with an exquisite tool to change two-body collisions from being attractive to repulsive, with important implications in nonlinear atom optics. Indeed, much of the recent work in that field relies heavily on these resonances, as we shall see. After having understood the source of nonlinearities in de Broglie optics in this way, we turn our attention to the mean-field description of bosonic matter-wave fields, the analog of the semiclassical approximation in optics. We introduce the Gross-Pitaevskii equation, and study departures from its predictions in a linearized approach that introduces the concept of quasiparticles. We also introduce particle-hole operators that are of particular use in the description of fermionic fields. This formalism being established, we turn to nonlinear atom optics per se. Concentrating first on bosonic atoms, we discuss the focusing and defocusing of coherent atomic matter waves and the generation of dark and bright solitons. For lack of space, we omit several important topics including much of the fascinating work on optical lattices and the generation of vortices and vortex lattices. We also omit the nonlinear mixing of optical and matter waves, where the progress has been somewhat slower in the last three years than in the topics that we cover. The reader is referred to Chapter 13 of the monograph by Meystre [2001] for a discussion of this topic that is still reasonably current. We then turn to four-wave mixing, starting with bosonic atoms, which was one of the first nonlinear atom-optical effects demonstrated experimentally. We then extend our considerations to the four-wave mixing and phase conjugation of fermionic matter waves, drawing an analogy between this process and Dicke superradiance. Further extending the analogy with optics, the following section discusses three-wave mixing. We first return to quasiparticles to interpret Baliev and Landau damping in terms of nonlinear wave mixing, and then proceed with discussion of the mixing between atomic and molecular (dimer) matter-wave fields. This allows us to make some comments of a general nature on the so-called BEC-BCS cross-over and the potential use of Feshbach resonances to achieve resonant superfluidity in ultracold bosonic atomic samples. We conclude this section with the discussion of a molecular analog of the cavity QED micromaser.

Original language | English (US) |
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Title of host publication | Progress in Optics |

Publisher | Elsevier |

Pages | 139-214 |

Number of pages | 76 |

ISBN (Print) | 0444515984, 9780444515988 |

DOIs | |

State | Published - Jan 1 2005 |

### Publication series

Name | Progress in Optics |
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Volume | 47 |

ISSN (Print) | 0079-6638 |

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### ASJC Scopus subject areas

- Electronic, Optical and Magnetic Materials
- Surfaces and Interfaces

### Cite this

*Progress in Optics*(pp. 139-214). (Progress in Optics; Vol. 47). Elsevier. https://doi.org/10.1016/S0079-6638(05)47003-1