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

Multiphoton single and double ionization of magnesium was studied by measuring electron energy spectra and ion mass spectra using 1-ps laser pulses in the 580-595-nm wavelength and 1012-1013-W/cm2 intensity range. In single ionization the (3p)2 1S doubly excited autoionizing state, resonant at the four-photon level, is found to play an important role. Single ionization leaving the Mg+ ion in the 3p excited state is strongly enhanced when resonant with the (3p)2 1S state. The amount of above threshold ionization observation also varies strongly in the wavelength range under study. At an intensity of 2 x 1013 W/cm2, sequential double ionization is found to occur dominantly via the excited ionic state Mg+(3p). The first ionization step populates this state less than the ion ground state Mg+(3s), but this is compensated by the lower order of the second ionization step.

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Multiphoton ionization of magnesium via an autoionizing state

In this review article, we describe the studies of 1D gases realised on atom-chip experiments.

Atom chips and one-dimensional Bose gases

We have measured the photon number probability distribution of a laser in which the inversion is not slaved to the field. For the experiments, we have used a Nd(3+):YVO(4) laser which has a sufficiently slow inversion to allow measurement of the photon fluctuations at a time scale much shorter than that of the relaxation oscillations. The photon distribution function becomes highly nonstandard (i.e., non-Poissonian) in such a laser; this is consistent with available theoretical work. We point out the relevance of our results for the case of the semiconductor microlaser.

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Photon statistics of a laser with slow inversion.

We describe the quantum-noise behavior of a laser that has nonorthogonal polarization modes. The nonorthogonality of the modes leads to excess quantum noise. Theoretically, we derive the excess noise dynamics of the laser, including the saturation of the gain medium. Experimentally, we have measured the noise dynamics in the polarization, intensity and phase degrees of freedom of a 3.51-µm HeXe laser and we obtain good agreement with theory. When the modes are made nonorthogonal we observe: (i) excess intensity and phase noise, (ii) spectral coloring of the excess intensity noise and (iii) correlations between the polarization-angle noise and the intensity noise. The excess phase noise is found to be partly suppressed by the polarization-anisotropic part of the saturation of the gain medium.

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Quantum noise in a laser with nonorthogonal polarization modes

Publisher Summary This chapter focuses on the quantification of the concept of “small”; this allows one to compare various lasers in a unified framework. This is in fact the realm of mesoscopic lasers—that is, lasers between macroscopic and microscopic. In macroscopic lasers the degrees of freedom of electromagnetic field and gain medium are effectively continuous variables, whereas in mesoscopic lasers they are discrete variables with the one-photon, one-atom laser as ultimate limit. The discreteness introduces quantum noise, the more so the smaller the number of degrees of freedom. Because the quantum noise depends on details of the system, the universality of the macroscopic laser gets more and more lost the deeper one penetrates into the mesoscopic domain. These generally allow a semiclassical description. A true quantum description becomes gradually essential, in particular at the end of the route where one reaches the cavity quantum electrodynamic (QED) limit in the form of a one-atom, one-photon device. The latter may come in many varieties; a good example would be a deterministic device that emits a single photon upon command. The transition between the semiclassical limit and the quantum limit is of course very vague; this is the realm of mesoscopic physics. The laser threshold transition of a macroscopic laser can be seen as a thermodynamic phase transition in a classical macroscopic system.

N.J. van Druten

Papers

Multiphoton ionization of magnesium via an autoionizing state

Atom chips and one-dimensional Bose gases

Photon statistics of a laser with slow inversion.

Quantum noise in a laser with nonorthogonal polarization modes

Quantum noise of small lasers