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Since the experimental realization of atomic Bose-Einstein condensates (BECs) [1], this field has experienced an enormous increase of interest. Indeed, the perspectives are very promising and range from coherent matter-wave optics to precision measurements and quantum information processing. Nowadays there exists an unprecedented control over the external magnetic, electric, or optical fields that are used to confine or alter the properties of the BECs, while matter-wave devices, such as the atom chips [2], have been developed as well. These advances, along with the formal similarities between nonlinear and matter-wave optics [3] indicate that coherent matter-waves, such as matter-wave solitons and vortices, may in principle be controlled similarly to their optical siblings in optical fibers, waveguides, photonic crystals, and so on [4]. [ABSTRACT FROM AUTHOR]

The phenomenon of Bose-Einstein condensation, initially predicted by Bose [1] and Einstein [2, 3] in 1924, refers to systems of particles obeying the Bose statistics. In particular, when a gas of bosonic particles is cooled below a critical transition temperature Tc, the particles merge into the Bose-Einstein condensate (BEC), in which a macroscopic number of particles (typically 103 to 106) share the same quantum state. Bose-Einstein condensation is in fact a quantum phase transition, which is connected to the manifestation of fundamental physical phenomena, such as superfluidity in liquid helium and superconductivity in metals (see, e.g., [4] for a relevant discussion and references). Dilute weakly-interacting BECs were first realized experimentally in 1995 in atomic gases, and specifically in vapors of rubidium [5] and sodium [6]. In the same year, first signatures of Bose-Einstein condensation in vapors of lithium were also reported [7] and were later more systematically confirmed [8]. The significance and importance of the emergence of BECs has been recognized through the 2001 Nobel prize in Physics [9, 10]. During the last years there has been an explosion of interest in the physics of BECs. Today, over fifty experimental groups around the world can routinely produce BECs, while an enormous amount of theoretical work has ensued. [ABSTRACT FROM AUTHOR]

Vortices are pervasive in nature, representing the breakdown of laminar fluid flow and hence playing a key role in turbulence. The fluid rotation associated with a vortex can be parameterized by the circulation Γ = ∮ dr · v(r) about the vortex, where v(r) is the fluid velocity field. While classical vortices can take any value of circulation, superfluids are irrotational, and any rotation or angular momentum is constrained to occur through vortices with quantized circulation. Quantized vortices also play a key role in the dissipation of transport in superfluids. In BECs quantized vortices have been observed in several forms, including single vortices [1, 2], vortex lattices [3-6] (see also Part VII), and vortex pairs and rings [7-9]. The recent observation of quantized vortices in a fermionic gas was taken as a clear signature of the underlying condensation and superfluidity of fermion pairs [10]. In addition to BECs, quantized vortices also occur in superfluid helium [11, 12], nonlinear optics, and type-II superconductors [13]. [ABSTRACT FROM AUTHOR]

The one-dimensional solitons described in Parts II and III of this book can be extended into two and three dimensions. Such extensions are generally unstable [1]. However, in the tightly confined geometries associated with trapped Bose-Einstein condensates (BECs) both bright and dark solitons extended into two and three dimensions can be stabilized for times longer than the lifetime of experiments [2-4]. BECs offer the opportunity to tune a matter-wave gradually from one to two and to three dimensions [5, 6]. In the crossover regimes, new nonlinear objects can appear, such as the svortex, a solitary wave which is a soliton-vortex hybrid [7, 8]. The general question of crossover dimensions is an intriguing one in physics. [ABSTRACT FROM AUTHOR]

Various analytical techniques are reviewed in the context of asymptotic reductions of the Gross—Pitaevskii (GP) equation, which is the nonlinear Schrödinger (NLS) equation with an external potential. When the external potential is periodic, the GP equation can be reduced to the coupled-mode (Dirac) system, the continuous NLS equation and the discrete NLS equation by using formal multi-scale expansion methods and their rigorous mathematical analogues. When the external potential is decaying at infinity, finite-dimensional reductions of the GP equation can be derived for modeling of dynamics of localized modes. When the external potential is confining, the GP equation can be recovered from the multi-particle linear Schrödinger equation. [ABSTRACT FROM AUTHOR]

A large number of effects related to the phenomenon of Bose-Einstein Condensation (BEC) can be understood in terms of lowest order mean-field theory, whereby the entire system is assumed to be condensed, with thermal and quantum fluctuations completely ignored [1]. Such a treatment leads to the Gross-Pitaevskii Equation (GPE) [2] used extensively throughout this book. Although this theory works remarkably well for a broad range of experimental parameters, a more complete treatment is required for understanding various experiments, including experiments with solitons and vortices. Such treatments should include the dynamical coupling of the condensate to the thermal cloud, the effect of dimensionality, the role of quantum fluctuations, and should also describe the critical regime, including the process of condensate formation. [ABSTRACT FROM AUTHOR]

This chapter presents an overview of models providing for theoretical description of basic one- and two-dimensional nonlinear patterns and their stability in mixtures of degenerate Bose and/or Fermi gases, in the mean-field approximation. The models are based on coupled Gross-Pitaevskii equations for bosonic gases, and mean-field-hydrodynamic equations for gases of fermions. [ABSTRACT FROM AUTHOR]

The dawning recognition of the link between Bose-Einstein condensation (BEC) and superfluidity in 4He led to the first theoretical contemplations of multi-component condensates in the mid-late 1950s [1]. The first candidate mixtures were of 4He and 6He, but the 0.8 s lifetime of the latter prevented any serious approach towards a true superfluid mixture, as had been recognized in experiments performed earlier that decade [2]. However, theoretical treatment of this mixture laid important groundwork for the later study of multi-component dilute-gas condensates [3]. The discovery of superfluidity in 3He in the early 1970s presented a new possibility, in that the vector order parameter of the 3He superfluid allows for internal degrees of freedom that manifest themselves in behavior such as domain-walls and spin textures [4]. Internal degrees of freedom were subsequently considered theoretically in a dilute Bose-condensed gas inhabiting two different hyperfine levels of spinpolarized hydrogen [5]. [ABSTRACT FROM AUTHOR]

In the early days of laser cooling, which together with magnetic trapping techniques led to the first observation of BEC in 1995, it was realized that the optical interference between the cooling beams could lead to a spatially periodic density modulation of the trapped atoms. This notion of a "three-dimensional egg-carton" for cold atoms quickly led to a number of experiments confirming the existence of such a light-bound crystal and exploring its properties [1-3]. It was found that, indeed, cold atoms could be trapped in such periodic structures, and it was possible to measure, for example, the quantized motion of the atoms inside the potential wells. While early experiments were carried out in the dissipative regime using near-resonant lattices in which the atoms were continuously cooled through the Sisyphus mechanism involving light scattering [4], more sophisticated experimental techniques later enabled studies on far-detuned lattices in which atoms evolved coherently. [ABSTRACT FROM AUTHOR]

This chapter presents an overview of the properties of a Bose-Einstein condensate (BEC) trapped in a periodic potential. This system has attracted a wide interest in the last years, and a few excellent reviews of the field have already appeared in the literature (see, for instance, [1-3] and references therein). For this reason, and because of the huge amount of published results, we do not pretend here to be comprehensive, but we will be content to provide a flavor of the richness of this subject, together with some useful references. On the other hand, there are good reasons for our effort. Probably, the most significant is that BEC in periodic potentials is a truly interdisciplinary problem, with obvious connections with electrons in crystal lattices, polarons and photons in optical fibers. Moreover, the BEC experimentalists have reached such a high level of accuracy to create in the lab, so to speak, paradigmatic Hamiltonians, which were first introduced as idealized theoretical models to study, among others, dynamical instabilities or quantum phase transitions. [ABSTRACT FROM AUTHOR]

This chapter examines quantized vortex lattices from an experimental perspective. While vortices in dilute Bose gases have been the subject of a number of theoretical review papers [1-3] (see also Chaps. 9 and 11), here we wish to present a pedagogical overview from the experimental side. We will emphasize manipulating and observing the lattices, and what can be learned about their structure and dynamics1. This chapter is not a comprehensive review of all experiments to date; rather, it is focused on the basic techniques and experiments. The reader will be familiar with some experimental methods from reading the chapter on individual vortex nucleation and formation. For completeness, we briefly review the method of creation and detection before discussing the specifics pertaining to the lattice structures. [ABSTRACT FROM AUTHOR]

Some of the basic principles and recent developments of vortex lattices in a dilute Bose-Einstein condensate are reviewed. The topics covered include nucleation dynamics of scalar and spinor vortices, collective modes, and fast rotation. [ABSTRACT FROM AUTHOR]

As discussed in the previous section, one of the most spectacular manifestation of superfluidity in Bose-Einstein condensates is the possibility of nucleating quantized vortices when the system is set into rotation. This property, first predicted by Feynman and Onsager [1, 2], has since been verified experimentally in superfluid 4He and gaseous Bose-Einstein condensates. The observation of vortices has also been used to probe superfluidity in fermionic systems, where quantum ordering arises from Cooper pairing of two opposite spin particles [3]. Vortices and vortex lattices have been observed both in ultra-cold Fermi gases [4] and in liquid 3He [5]. In this latter case, the coupling between the orbital and spin degrees of freedom leads to a new class of vortex-like excitations called skyrmions [6]. Finally, the formal identity between Coriolis and Lorentz forces permits to draw an interesting parallel between rotating systems and charged particles in magnetic fields where the flux quantization observed in superconductors is the complete analog of the circulation's quantization in superfluids. [ABSTRACT FROM AUTHOR]