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

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

QUANTUM gravitational effects are usually ignored in calculations of the formation and evolution of black holes. The justification for this is that the radius of curvature of space-time outside the event horizon is very large compared to the Planck length (Għ/c3)1/2 ≈ 10−33 cm, the length scale on which quantum fluctuations of the metric are expected to be of order unity. This means that the energy density of particles created by the gravitational field is small compared to the space-time curvature. Even though quantum effects may be small locally, they may still, however, add up to produce a significant effect over the lifetime of the Universe ≈ 1017 s which is very long compared to the Planck time ≈ 10−43 s. The purpose of this letter is to show that this indeed may be the case: it seems that any black hole will create and emit particles such as neutrinos or photons at just the rate that one would expect if the black hole was a body with a temperature of (κ/2π) (ħ/2k) ≈ 10−6 (M/M)K where κ is the surface gravity of the black hole1. As a black hole emits this thermal radiation one would expect it to lose mass. This in turn would increase the surface gravity and so increase the rate of emission. The black hole would therefore have a finite life of the order of 1071 (M/M)−3 s. For a black hole of solar mass this is much longer than the age of the Universe. There might, however, be much smaller black holes which were formed by fluctuations in the early Universe2. Any such black hole of mass less than 1015 g would have evaporated by now. Near the end of its life the rate of emission would be very high and about 1030 erg would be released in the last 0.1 s. This is a fairly small explosion by astronomical standards but it is equivalent to about 1 million 1 Mton hydrogen bombs. It is often said that nothing can escape from a black hole. But in 1974, Stephen Hawking realized that, owing to quantum effects, black holes should emit particles with a thermal distribution of energies — as if the black hole had a temperature inversely proportional to its mass. In addition to putting black-hole thermodynamics on a firmer footing, this discovery led Hawking to postulate 'black hole explosions', as primordial black holes end their lives in an accelerating release of energy.

Black Hole Explosions

We study the coherent atomic tunneling between two zero-temperature Bose-Einstein condensates (BEC) confined in a double-well magnetic trap. Two Gross-Pitaevskii equations for the self-interacting BEC amplitudes, coupled by a transfer matrix element, describe the dynamics in terms of the interwell phase difference and population imbalance. In addition to the anharmonic generalization of the familiar ac Josephson effect and plasma oscillations occurring in superconductor junctions, the nonlinear BEC tunneling dynamics sustains a self-maintained population imbalance: a novel ``macroscopic quantum self-trapping'' effect.

Quantum Coherent Atomic Tunneling between Two Trapped Bose-Einstein Condensates

The standard quantum limit defines the performance of the best available measurement devices for quantities such as time or position. Many of these sensors are interferometers in which the standard quantum limit can be overcome by using quantum-entangled states (in particular, spin squeezed states) at the two input ports. A team from the University of Heidelberg's Kirchhoff Institute for Physics now demonstrates spin squeezed states suitable for atomic interferometry by splitting a Bose–Einstein condensate into a few parts using an optical lattice potential. The measurements imply entanglement between the particles, a resource that would allow a precision gain of 3.8 dB over the standard quantum limit for interferometric measurements. Entanglement, a key feature of quantum mechanics, is a resource that allows the improvement of precision measurements beyond the conventional bound attainable by classical means1. This results in the standard quantum limit, which is reached in today’s best available sensors of various quantities such as time2 and position3,4. Many of these sensors are interferometers in which the standard quantum limit can be overcome by using quantum-entangled states (in particular spin squeezed states5,6) at the two input ports. Bose–Einstein condensates of ultracold atoms are considered good candidates to provide such states involving a large number of particles. Here we demonstrate spin squeezed states suitable for atomic interferometry by splitting a condensate into a few parts using a lattice potential. Site-resolved detection of the atoms allows the measurement of the atom number difference and relative phase, which are conjugate variables. The observed fluctuations imply entanglement between the particles7,8,9, a resource that would allow a precision gain of 3.8 dB over the standard quantum limit for interferometric measurements.

Squeezing and entanglement in a Bose-Einstein condensate

We have investigated the expansion of a Bose-Einstein condensate of strongly magnetic chromium atoms. The long-range and anisotropic magnetic dipole-dipole interaction leads to an anisotropic deformation of the expanding chromium condensate which depends on the orientation of the atomic dipole moments. Our measurements are consistent with the theory of dipolar quantum gases and show that a chromium condensate is an excellent model system to study dipolar interactions in such gases.

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Observation of Dipole-Dipole Interaction in a Degenerate Quantum Gas

A paper in this issue addresses a hot topic in the physics of ultracold atoms — quantum ferrofluids, which are superfluid quantum gases that consist of polarized dipoles, either electric or magnetic. The symmetry-breaking interaction between such dipoles is expected to lead to new physical phenomena. Now a group from Stuttgart University has succeeded in producing a chromium Bose–Einstein condensate with strong anisotropic magnetic dipole–dipole interaction between the atoms, which induces a pronounced change of the aspect ratio of the cloud. This much-sought form of matter should enable fundamental studies of physical systems in previously unexplored regimes. This paper reports the realization of a chromium Bose-Einstein condensate (BEC) with strong anisotropic magnetic dipole–dipole interaction between the atoms, which induces a pronounced change of the aspect ratio of the cloud. The experiment opens the way for exploration of the unique properties of quantum ferrofluids. Symmetry-breaking interactions have a crucial role in many areas of physics, ranging from classical ferrofluids to superfluid 3He and d-wave superconductivity. For superfluid quantum gases, a variety of new physical phenomena arising from the symmetry-breaking interaction between electric or magnetic dipoles are expected1. Novel quantum phases in optical lattices, such as chequerboard or supersolid phases, are predicted for dipolar bosons2,3. Dipolar interactions can also enrich considerably the physics of quantum gases with internal degrees of freedom4,5,6. Arrays of dipolar particles could be used for efficient quantum information processing7. Here we report the realization of a chromium Bose–Einstein condensate with strong dipolar interactions. By using a Feshbach resonance, we reduce the usual isotropic contact interaction, such that the anisotropic magnetic dipole–dipole interaction between 52Cr atoms becomes comparable in strength. This induces a change of the aspect ratio of the atom cloud; for strong dipolar interactions, the inversion of ellipticity during expansion (the usual ‘smoking gun’ evidence for a Bose–Einstein condensate) can be suppressed. These effects are accounted for by taking into account the dipolar interaction in the superfluid hydrodynamic equations governing the dynamics of the gas, in the same way as classical ferrofluids can be described by including dipolar terms in the classical hydrodynamic equations. Our results are a first step in the exploration of the unique properties of quantum ferrofluids.

Strong dipolar effects in a quantum ferrofluid

We report on the realization of a Chromium Bose-Einstein condensate (BEC) with strong dipolar interaction. By using a Feshbach resonance, we reduce the usual isotropic contact interaction, such that the anisotropic magnetic dipole-dipole interaction between 52Cr atoms becomes comparable in strength. This induces a change of the aspect ratio of the cloud, and, for strong dipolar interaction, the inversion of ellipticity during expansion - the usual "smoking gun" evidence for BEC - can even be suppressed. These effects are accounted for by taking into account the dipolar interaction in the superfluid hydrodynamic equations governing the dynamics of the gas, in the same way as classical ferrofluids can be described by including dipolar terms in the classical hydrodynamic equations. Our results are a first step in the exploration of the unique properties of quantum ferrofluids.

S. Giovanazzi

Papers

Quantum Coherent Atomic Tunneling between Two Trapped Bose-Einstein Condensates

Squeezing and entanglement in a Bose-Einstein condensate

Observation of Dipole-Dipole Interaction in a Degenerate Quantum Gas

Strong dipolar effects in a quantum ferrofluid

A quantum ferrofluid