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

A Bose-Einstein condensate was produced in a vapor of rubidium-87 atoms that was confined by magnetic fields and evaporatively cooled. The condensate fraction first appeared near a temperature of 170 nanokelvin and a number density of 2.5 x 10 12 per cubic centimeter and could be preserved for more than 15 seconds. Three primary signatures of Bose-Einstein condensation were seen. (i) On top of a broad thermal velocity distribution, a narrow peak appeared that was centered at zero velocity. (ii) The fraction of the atoms that were in this low-velocity peak increased abruptly as the sample temperature was lowered. (iii) The peak exhibited a nonthermal, anisotropic velocity distribution expected of the minimum-energy quantum state of the magnetic trap in contrast to the isotropic, thermal velocity distribution observed in the broad uncondensed fraction.

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Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor

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

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

Phase transitions to quantum condensed phases—such as Bose–Einstein condensation (BEC), superfluidity, and superconductivity—have long fascinated scientists, as they bring pure quantum effects to a macroscopic scale. BEC has, for example, famously been demonstrated in dilute atom gas of rubidium atoms at temperatures below 200 nanokelvin. Much effort has been devoted to finding a solid-state system in which BEC can take place. Promising candidate systems are semiconductor microcavities, in which photons are confined and strongly coupled to electronic excitations, leading to the creation of exciton polaritons. These bosonic quasi-particles are 109 times lighter than rubidium atoms, thus theoretically permitting BEC to occur at standard cryogenic temperatures. Here we detail a comprehensive set of experiments giving compelling evidence for BEC of polaritons. Above a critical density, we observe massive occupation of the ground state developing from a polariton gas at thermal equilibrium at 19 K, an increase of temporal coherence, and the build-up of long-range spatial coherence and linear polarization, all of which indicate the spontaneous onset of a macroscopic quantum phase. Bose–Einstein condensation (BEC), a form of matter first postulated in 1924, has famously been demonstrated in dilute atomic gases at ultra-low temperatures. Much effort is now being devoted to exploring solid-state systems in which BEC can occur. In theory semiconductor microcavities, where photons are confined and coupled to electronic excitations leading to the creation of polaritons, could allow BEC at standard cryogenic temperatures. Kasprzak et al. now present experiments in which polaritons are excited in such a microcavity. Above a critical polariton density, spontaneous onset of a macroscopic quantum phase occurs, indicating a solid-state BEC. BEC should also be possible at higher temperatures if coupling of light with solid excitations is sufficiently strong. Demokritov et al. have achieved just that, BEC at room temperature in a gas of magnons, which are a type of magnetic excitation. This paper presents a comprehensive set of experiments in which polaritons are excited in a semiconductor microcavity. Above a critical density of polaritons, massive occupation of the ground state at 19 K is observed and various pieces of experimental evidence point to a spontaneous onset of a macroscopic quantum phase.

Bose-Einstein condensation of exciton polaritons

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

We have demonstrated an output coupler for Bose condensed atoms in a magnetic trap. Short pulses of rf radiation were used to create Bose condensates in a superposition of trapped and untrapped hyperfine states. The fraction of out-coupled atoms was adjusted between 0% and 100% by varying the amplitude of the rf radiation. This configuration produces output pulses of coherent atoms and can be regarded as a pulsed ``atom laser.''

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Output Coupler for Bose-Einstein Condensed Atoms

Collective excitations of a dilute Bose condensate have been observed. These excitations are analogous to phonons in superfluid helium. Bose condensates were created by evaporatively cooling magnetically trapped sodium atoms. Excitations were induced by a modulation of the trapping potential, and detected as shape oscillations in the freely expanding condensates. The frequencies of the lowest modes agreed well with theoretical predictions based on mean-field theory. Before the onset of BoseEinstein condensation, we observed sound waves in a dense ultracold gas. [S0031-9007(96)00900-3] In 1941 Landau introduced the concept of elementary excitations to explain the properties of superfluid helium [1]. This phenomenological approach, based on quantum hydrodynamics, gave a quantitative description of the thermodynamic properties and transport processes in liquid helium. Landau rejected any relation to Bose-Einstein condensation (BEC). A microscopic derivation of the elementary excitation spectrum for a weakly interacting Bose gas was given by Bogoliubov in 1947 [2] and for He II by Feynman in 1955 [3], emphasizing the role of Bose statistics [3] and reconciling Landau’s approach with London’s explanation of superfluidity as being due to BEC [2,4]. The elementary excitations determine the spectrum of density fluctuations in a Bose liquid, and have been directly observed in He II by neutron scattering [5]. The low-frequency excitations are phonons, long-wavelength collective modes of the superfluid. So far, a satisfactory microscopic theory for an interacting bosonic system exists only for the dilute quantum gas. The recent realization of BEC in dilute atomic vapors [6 ‐ 8] has opened the door to test this theory experimentally. In this paper we report on the observation of shape oscillations of a trapped Bose condensate, modes analogous to phonons in homogeneous systems [9]. The experimental setup for creating Bose condensates was the same as in our previous work [10]. Briefly, sodium atoms were optically cooled and trapped, and transferred into a magnetic trap where they were further cooled by rf-induced evaporation [11,12]. Every 30 s, condensates containing 5 3 10 6 sodium atoms in the F › 1, mF › 21 ground state were produced. Evaporative cooling was extended well below the transition temperature to obtain a condensate without a discernible normal component. The condensate was confined in a cloverleaf magnetic trap which had cylindrical symmetry with trapping frequencies of 19 Hz axially and 250 Hz radially (see below). The trapping potential is determined by the axial curvature of the magnetic field B 00 › 125 Gc m 2 2 , the radial gradient B 0 › 150 Gc m 2 1 , and the bias field B0 › 1.2 G. The condensate was excited by a time-dependent modulation of the trapping potential. First, we used a sudden step in the gradient B 0 to identify several collective modes of the condensate and to find their approximate frequencies. B 0 was decreased by 15% for a duration of 5 ms with a transition time of about 1 ms, and then returned to its original value. A variable time delay was introduced between the excitation and the observation of the cloud. In this way, we strobed the free time evolution of the system after the excitation. The cloud was observed by absorption imaging after a sudden switch off of the magnetic trap and 40 ms of ballistic expansion. No trap loss was observed during the interval over which the delay was varied. The images were similar to the series shown in Fig. 1. Four modes were identified from the measured center-of-mass positions and the widths of the condensate. The radial and axial center-of-mass oscillations (dipole modes) were excited because a change in B 0 displaced the center of the trap slightly due to asymmetries in the field-producing coils. A fast shape oscillation predominantly showed up as a sinusoidal modulation of the radial width while a slow sinusoidal shape oscillation was observed in the axial width. When a strong parametric drive (see below) was used to excite the slow shape oscillation, a weak oscillation of the radial width was also detected. Note that the widths were observed after ballistic expansion and reflect a convolution of the initial spatial and velocity

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Collective Excitations of a Bose-Einstein Condensate in a Magnetic Trap.

Sound propagation has been studied in a magnetically trapped dilute Bose-Einstein condensate. Localized excitations were induced by suddenly modifying the trapping potential using the optical dipole force of a focused laser beam. The resulting propagation of sound was observed using a novel technique, rapid sequencing of nondestructive phase-contrast images. The speed of sound was determined as a function of density and found to be consistent with Bogoliubov theory. This method may generally be used to observe high-lying modes and perhaps second sound.

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Propagation of Sound in a Bose-Einstein Condensate

[S0031-9007(98)05702-0] We predicted that density perturbations should propagate along the axis of a cigar-shaped Bose-Einstein cond at a speedc ­ sn0Ũymd, wheren0 is the density at the center of the cloud, Ũ characterizes the repulsive interaction between atoms, and m is their mass. This prediction was based on an incorrect use of an energy functional. We have since corrected our theoretical approach to reducing the three-dimensional sound propagation to a dimensional equation. We considered the quantum Lagrangian for a Bose-Einstein condensate [1] confined cylindrial tube which is infinite in thez direction. We assumed a Thomas-Fermi solution in the radial direction b using a trial wave function of the form

C. G. Townsend

Papers

Observation of Interference Between Two Bose Condensates

Output Coupler for Bose-Einstein Condensed Atoms

Collective Excitations of a Bose-Einstein Condensate in a Magnetic Trap.

Propagation of Sound in a Bose-Einstein Condensate

Erratum: Propagation of Sound in a Bose-Einstein Condensate (Phys. Rev. Lett. 79, 553 (1997))