There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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

We have observed Bose-Einstein condensation of sodium atoms. The atoms were trapped in a novel trap that employed both magnetic and optical forces. Evaporative cooling increased the phase-space density by 6 orders of magnitude within seven seconds. Condensates contained up to 5\ifmmode\times\else\texttimes\fi{}${10}^{5}$ atoms at densities exceeding ${10}^{14}$ ${\mathrm{cm}}^{\ensuremath{-}3}$. The striking signature of Bose condensation was the sudden appearance of a bimodal velocity distribution below the critical temperature of \ensuremath{\sim}2\ensuremath{\mu}K. The distribution consisted of an isotropic thermal distribution and an elliptical core attributed to the expansion of a dense condensate.

Bose-Einstein condensation in a gas of sodium atoms.

Publisher Summary This chapter focuses on the concept of evaporative cooling of trapped neutral atoms. The recent observations of Bose–Einstein condensation have shown dramatically the potential of evaporative cooling. Through evaporative cooling, phase-space density could be increased by six orders of magnitude in these experiments. In such experiments, evaporative cooling was used to reach temperature and densities that are unprecedented for trapped atoms and greatly exceeded what had been reached before by laser cooling. Laser cooling has recently broken the recoil limit in three dimensions (3D) and reached extremely cold temperatures of 3 nK in 1D. However, none of these optical sub-recoil techniques has been realized so far at high densities. The current density limitations are caused by the absorption of light, radiation trapping, and excited state collisions. An often-mentioned disadvantage of evaporative cooling is the loss of atoms. However, as discussed in the chapter, the efficiency of evaporative cooling is quite high.

Evaporative Cooling of Trapped 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.

Bose-Einstein condensation (BEC) of an ideal gas is investigated for a finite number of particles. In three dimensions, we find a transition temperature which is lower than in the thermodynamic limit. Lowering the dimension increases the transition temperature and is therefore favorable for BEC. This is in contrast to the standard result obtained in the thermodynamic limit which states that BEC is not possible in, e.g., a one-dimensional (1D) harmonic potential. As a result, 1D atom traps, such as radially tightly confining magnetic traps or optical dipole traps, are promising for studying BEC. \textcopyright{} 1996 The American Physical Society.

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N.J. van Druten

Papers

Bose-Einstein condensation in a gas of sodium atoms.

Bose-Einstein condensation in a gas of sodium atoms

Evaporative Cooling of Trapped Atoms

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

Bose-Einstein condensation of a finite number of particles trapped in one or three dimensions