This paper reviews recent experimental and theoretical progress concerning many-body phenomena in dilute, ultracold gases. It focuses on effects beyond standard weak-coupling descriptions, such as the Mott-Hubbard transition in optical lattices, strongly interacting gases in one and two dimensions, or lowest-Landau-level physics in quasi-two-dimensional gases in fast rotation. Strong correlations in fermionic gases are discussed in optical lattices or near-Feshbach resonances in the BCS-BEC crossover.

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Many-Body Physics with Ultracold Gases

The phenomenon of Bose-Einstein condensation of dilute gases in traps is reviewed from a theoretical perspective. Mean-field theory provides a framework to understand the main features of the condensation and the role of interactions between particles. Various properties of these systems are discussed, including the density profiles and the energy of the ground-state configurations, the collective oscillations and the dynamics of the expansion, the condensate fraction and the thermodynamic functions. The thermodynamic limit exhibits a scaling behavior in the relevant length and energy scales. Despite the dilute nature of the gases, interactions profoundly modify the static as well as the dynamic properties of the system; the predictions of mean-field theory are in excellent agreement with available experimental results. Effects of superfluidity including the existence of quantized vortices and the reduction of the moment of inertia are discussed, as well as the consequences of coherence such as the Josephson effect and interference phenomena. The review also assesses the accuracy and limitations of the mean-field approach.

https://arxiv.org/pdf/cond-mat/9806038.pdf

Theory of Bose-Einstein condensation in trapped gases

For a system at a temperature of absolute zero, all thermal fluctuations are frozen out, while quantum fluctuations prevail. These microscopic quantum fluctuations can induce a macroscopic phase transition in the ground state of a many-body system when the relative strength of two competing energy terms is varied across a critical value. Here we observe such a quantum phase transition in a Bose-Einstein condensate with repulsive interactions, held in a three-dimensional optical lattice potential. As the potential depth of the lattice is increased, a transition is observed from a superfluid to a Mott insulator phase. In the superfluid phase, each atom is spread out over the entire lattice, with long-range phase coherence. But in the insulating phase, exact numbers of atoms are localized at individual lattice sites, with no phase coherence across the lattice; this phase is characterized by a gap in the excitation spectrum. We can induce reversible changes between the two ground states of the system.

Quantum Phase Transition From a Superfluid to a Mott Insulator in a Gas of Ultracold Atoms

Bose-Einstein condensation (BEC) has been observed in a dilute gas of sodium atoms. A Bose-Einstein condensate consists of a macroscopic population of the ground state of the system, and is a coherent state of matter. In an ideal gas, this phase transition is purely quantum-statistical. The study of BEC in weakly interacting systems which can be controlled and observed with precision holds the promise of revealing new macroscopic quantum phenomena that can be understood from first principles.

Bose-Einstein condensation in a gas of sodium atoms

Feshbach resonances are the essential tool to control the interaction between atoms in ultracold quantum gases. They have found numerous experimental applications, opening up the way to important breakthroughs. This review broadly covers the phenomenon of Feshbach resonances in ultracold gases and their main applications. This includes the theoretical background and models for the description of Feshbach resonances, the experimental methods to find and characterize the resonances, a discussion of the main properties of resonances in various atomic species and mixed atomic species systems, and an overview of key experiments with atomic Bose-Einstein condensates, degenerate Fermi gases, and ultracold molecules.

Feshbach resonances in ultracold gases

It has long been predicted that the scattering of ultracold atoms can be altered significantly through a so-called ‘Feshbach resonance’. Two such resonances have now been observed in optically trapped Bose–Einstein condensates of sodium atoms by varying an external magnetic field. They gave rise to enhanced inelastic processes and a dispersive variation of the scattering length by a factor of over ten. These resonances open new possibilities for the study and manipulation of Bose–Einstein condensates.

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Observation of Feshbach resonances in a Bose–Einstein condensate

Interference between two freely expanding Bose-Einstein condensates has been observed. Two condensates separated by ∼40 micrometers were created by evaporatively cooling sodium atoms in a double-well potential formed by magnetic and optical forces. High-contrast matter-wave interference fringes with a period of ∼15 micrometers were observed after switching off the potential and letting the condensates expand for 40 milliseconds and overlap. This demonstrates that Bose condensed atoms are “laser-like”; that is, they are coherent and show long-range correlations. These results have direct implications for the atom laser and the Josephson effect for atoms.

https://science.sciencemag.org/content/sci/275/5300/637.full.pdf

Observation of Interference Between Two Bose Condensates

Bose-Einstein condensates of sodium atoms have been confined in an optical dipole trap using a single focused infrared laser beam. This eliminates the restrictions of magnetic traps for further studies of atom lasers and Bose-Einstein condensates. More than $5\ifmmode\times\else\texttimes\fi{}{10}^{6}$ condensed atoms were transferred into the optical trap. Densities of up to $3\ifmmode\times\else\texttimes\fi{}{10}^{15}{\mathrm{cm}}^{\ensuremath{-}3}$ of Bose condensed atoms were obtained, allowing for a measurement of the three-body loss rate constant for sodium condensates as ${K}_{3}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.1(3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}30}{\mathrm{cm}}^{6}{\mathrm{s}}^{\ensuremath{-}1}$. At lower densities, the observed $1/e$ lifetime was longer than 10 s. Simultaneous confinement of Bose-Einstein condensates in several hyperfine states was demonstrated.

Optical confinement of a bose-einstein condensate

Bose–Einstein condensates — a low-temperature form of matter in which a macroscopic population of bosons occupies the quantum-mechanical ground state — have been demonstrated for weakly interacting, dilute gases of alkali-metal1,2,3 and hydrogen25 atoms. Magnetic traps are usually employed to confine the condensates, but have the drawback that spin flips in the atoms lead to untrapped states. For this reason, the spin orientation of the trapped alkali atoms cannot be regarded as a degree of freedom. Such condensates are therefore described by a scalar order parameter, like the spinless superfluid 4He. In contrast, a recently realized optical trap4 for sodium condensates confines atoms independently of their spin orientations. This offers the possibility of studying ‘spinor’ condensates in which spin comprises a degree of freedom, so that the order parameter is a vector rather than scalar quantity. Here we report the observation of equilibrium states of sodium spinor condensates in an optical trap. The freedom of spin orientation leads to the formation of spin domains in an external magnetic field, which can be either miscible or immiscible with one another.

Spin domains in ground-state Bose-Einstein condensates

Bose-Einstein condensates of dilute atomic gases, characterized by a macroscopic population of the quantum mechanical ground state, are a new, weakly interacting quantum fluid. In most experiments condensates in a single weak field seeking state are magnetically trapped. These condensates can be described by a scalar order parameter similar to the spinless superfluid 4He. Even though alkali atoms have angular momentum, the spin orientation is not a degree of freedom because spin flips lead to untrapped states and are therefore a loss process. In contrast, the recently realized optical trap for sodium condensates confines atoms independently of their spin orientation. This opens the possibility to study spinor condensates which represent a system with a vector order parameter instead of a scalar. Here we report a study of the equilibrium state of spinor condensates in an optical trap. The freedom of spin orientation leads to the formation of spin domains in an external magnetic field. The structure of these domains are illustrated in spin domain diagrams. Combinations of both miscible and immiscible spin components were realized.

H.-J. Miesner

Papers

Observation of Feshbach resonances in a Bose–Einstein condensate

Observation of Interference Between Two Bose Condensates

Optical confinement of a bose-einstein condensate

Spin domains in ground-state Bose-Einstein condensates

Spin domains in ground state spinor Bose-Einstein condensates