This chapter introduces the order parameter and the Gross--Pitaevskii equation. Although a Bose-Einstein C condensate (BEC) is a quantum many-body system which that can be described fully only using the many-body wavefunction, many of its simplest spatial dynamics can be understood using the order parameter and its dynamics. We first describe how the order parameter can be formally defined, and how this can be considered an effective spatial wavefunction of the BEC. We then derive the time-dependent evolution of the order parameter via the Gross--Pitaevskii equation, then and study various solutions of it. This includes plane-wave solutions, an infinite potential well solutions, and excited state solutions such as vortices and solitons. After introducing key concepts such as the Thomas--Fermi approximation and the healing length, we also discuss how the Gross--Pitaevskii equation can be cast in the form of hydrodynamic equations.

This chapter introduces the topic of quantum simulation and the various approaches that are being pursued in cold atom systems. We first start by describing what the aims of quantum simulation are, and why this is considered a difficult yet important problem. The example of the transverse field Ising model is discussed to illustrate the type of phenomenology, more commonly studied in condensed matter physics, that is of interest in quantum simulation. We then discuss two main approaches to quantum simulation. The first is digital quantum simulation, where a quantum computer is used to simulate the time evolution of a system, and methods to obtain static quantities is are discussed. The second is analogue quantum simulation, where experimental methods are used to physically create a tailored system in the laboratory. The toolbox of methods that is available to the cold atom physicist is explained, such aswhich includes optical lattices, Feshbach resonances, artificial gauge fields, spin-orbit coupling, time-of-flight measurements, and the quantum gas microscope are explained. We then consider a specific case study of one of the earliest quantum simulation experiments, where a Bose--Hubbard model was realized to observe a superfluid to Mott insulator transition in cold atoms.

This chapter describes the use of squeezed states to improve the sensitivity in the estimation of the relative phase accumulated in a Bose--Einstein condensate (BEC) beyond the standard quantum limit. We begin by examining the specific example of a two-component BEC, where atoms in two different hyperfine levels interact with each other. By evolving an initial superposition of states under an effective Hamiltonian, a squeezed state is realized. Next, the squeezing is visualized using the Q-function, giving the mean spin direction and shape of the BEC distribution. We discuss the basic operation of Ramsey interferometry, which is a classic example of two-slit atom diffraction. We then calculate the error in estimating the relative phase shift using the error propagation formula that allows us to define the squeezing parameter. This measures the amount of metrological gain using squeezed states over unsqueezed states. We give an example of the metrological gain in the estimation of the relative phase shift using a squeezed state of the two-component BEC as an input state to an atom Mach--Zehnder interferometer. The Fisher information is used to find which rotation axis gives the optimal information from a given initial state. Finally, we parameterize the nonlinear phase per atom responsible for squeezing and discuss how this parameter is controlled in different experiments.

In this final chapter, we focus more on quantum information and quantum computing applications of atomic ensembles. We first examine ways of implementing continuous variables in quantum information processing using atomic ensembles, based on the Holstein--Primakoff approximation. Methods to perform quantum teleportation using this method, and some seminal experiments using this approach are introduced. We then introduce other approaches not based on the Holstein--Primakoff approximation to represent quantum information, namely the spinor quantum computing scheme. After showing a simple example of how such a scheme works with Deutsch's algorithm, we describe how adiabatic quantum computing can be performed, which displays the key feature of quantum error suppression.

This chapter discusses the basic physics of atom diffraction. Starting with a short review on the diffraction of light, we describe the absorption of photons by an atom and its subsequent emission, where the atom to changes its momentum and internal state. Assuming a two-level atom, we describe the coherent interactions of atoms with light, showing its effect on the ground state of the atoms. We further illustrate how the light impacts forces on the atom that can be used for trapping it in the light field. Next, we discuss Bragg diffraction of atoms, where the internal state of an atom is unchanged after diffraction. Here the external motional state of the atom is put in a linear superposition of its movement. Finally, we describe the diffraction of atoms by Raman pulses, where the internal state of the atom is changed after diffraction, while achieving directed motion.

This chapter introduces the notion of entanglement, methods of detection, and various types of two atomic ensemble entangled states. Bipartite The bipartite system constitutes is the simplest and most straightforward way of understanding entanglement, where there is a well-developed theory of detecting and quantifying it. We discuss how von Neumann entropy and negativity are two simple and powerful quantifiers of entanglement in pure and mixed systems, respectively. In the absence of a way to topographically reconstruct the density matrix, other correlation-based entanglement criteria are important practically, since they only involve the measurement of some key correlations. Various approaches to this are introduced, such as the Duan--Giedke--Cirac--Zoller criterion, Hillery--Zubairy criteria, entanglement witness approaches, and covariance matrices. Finally, two types of squeezed states for two atomic ensembles are introduced, : namely the one-axis two-spin and two-axis two-spin squeezed states are introduced.

This chapter introduces the basic physics of Bose--Einstein condensation. We first start with why there is a difference between distinguishable and indistinguishable particles in terms of the number of states that are available in a composite system. Then Bose and Einstein's argument of why one expects a high probability of occupation of the ground state is discussed. This is derived then more rigorously for the grand canonical ensemble, showing that at some critical temperature there should be a macroscopic occupation of the ground state. Next, the low-lying energy spectrum of an interacting Bose--Einstein condensate is derived, leading to the Bogoliubov dispersion. The significance of the Bogoliubov dispersion as the origin of superfluidity is then discussed, in terms of superfluditysuperfluidity. Laudau'sLandau's criterion for superfluidity is derived, by general principles of Galilean transformations of Schrodinger's equation.

This chapter discusses spinor Bose--Einstein condensates and the various common states that are encountered with such systems. Many analogous concepts tooptical systems are discussed, such as spin coherent states, spin squeezed states, uncertainty relations, and quasiprobability distributions such as the Q- and Wigner functions. A gallery of different spin states is shown for both the Q- and Wigner functions is shown, including that for highly non-classical states such as the Schrodinger cat state. Due to the different operators involved for spin systems, we describe the similarities and differences to optical squeezing, and introduce the one-axis and two-axis countertwisting spin squeezed states. The notion of entanglement in such systems, along with and ways of detecting this is are discussed. Several mappings, such as the Holstein--Primakoff transformation between spins and bosonic operators, as well asand the equivalence between condensed and uncondensed systems, is are discussed. We also introduce some key mathematical results involving important states, such as formulas to perform basis transformations between Fock states.

Long time tails in Green-Kubo formulas for transport coefficients indicate that long range correlations in non-equilibrium fluids cause divergent transport coefficients in the Navier-Stokes equations for two dimensional fluids, and divergent higher order gradient corrections to these equations for three dimensional fluids. Possible resolutions of these difficulties are considered, for transport of momentum or energy in fluids maintained in non-equilibrium stationary states. The resolutions of the divergence difficulties depend on the particular flow under consideration. For stationary Couette flow in a gas, the divergences are resolved by including non-linear terms in the kinetic equations, leading to logarithmic terms in the velocity gradients for the equations of fluid flow for two dimensional gases, and fractional powers for three dimensional flows. Methods used for Couette flow do not resolve the divergence problems for stationary heat flow. Instead, the difficulties are resolved by taking the finite size of the system into account.

Kinetic theory is defined as a branch of statistical mechanics that attempts to describethe non-equilibrium properties of macroscopic systems in terms of microscopic propertiesof the constituent particles or quantum excitations. The history of kinetic theory is summarizedfrom the first understandings of the connections of temperature and pressure ofperfect gases with their average kinetic energy and with the average momentum transferto the walls by particle-wall collisions. The history continues with a discussion of the contributionsof Maxwell and Boltzmann, and the development of the Boltzmann transportequation. Modern developments include extending the Boltzmann equation to moderatelydense gases, formulation of kinetic theory for hard sphere systems, discovery of long timetail contributions to the Green-Kubo expressions for transport coefficients, applications ofkinetic theory to fluctuations in gases, to quantum gases and to granular particles. Thecontents of each chapter are then summarized.

Symmetric functions of phase variables of N particles, can be expanded, formally, in terms of a series of Ursell cluster functions. They depend successively on one, two, ..., particle variables. For equilibrium systems, cluster expansions are used to obtain virial expansions of the thermodynamic functions, The cluster expansion method can be applied to N-particle time displacement operators and to the initial distribution function for a non-equilibrium system. Assuming a factorization property of the initial distribution function, one obtains expansions of the time dependent two and higher particle distributions in terms of successively higher products of one particle functions. This expansion of the pair distribution function, together with the first hierarchy equation generalizes the Boltzmann equation to higher densities in terms of the dynamics of successively higher number of particles considered in isolation. Contributions from correlated binary collision sequences appear and for hard spheres the Enskog equation is an approximation.

The Lorentz model consists of non-interacting, point particles moving among a collection of fixed scatterers of radius a, placed at random, with or without overlapping, at density n 6 in space. This model was designed to be, and serves as, a model for the motion of electrons in solids. The kinetic equation for the moving particles must be linear, and for low scatterers density, nad >> 1, it is the Lorentz-Boltzmann equation. If external fields are absent, the Chapman-Enskog method leads to the diffusion equation. For three dimensional systems with hard sphere scatterers, the Lorentz-Boltzmann equation can be solved exactly, and the range of validity of the Chapman-Enskog solution can be examined. Electrical conduction and magneto-transport can be studied for charged, moving particles. In both cases there are unexpected results. The Lorentz model with hard sphere scatterers is a chaotic system, and one can calculate Lyapunov exponents and related dynamical quantities.