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

The term photonic crystals appears because of the analogy between electron waves in crystals and the light waves in artificial periodic dielectric structures. During the recent years the investigation of one-, two-and three-dimensional periodic structures has attracted a widespread attention of the world optics community because of great potentiality of such structures in advanced applied optical fields. The interest in periodic structures has been stimulated by the fast development of semiconductor technology that now allows the fabrication of artificial structures, whose period is comparable with the wavelength of light in the visible and infrared ranges.

http://ieeexplore.ieee.org/iel5/6354/16969/00781867.pdf

Photonic crystals

Single trapped ions represent elementary quantum systems that are well isolated from the environment. They can be brought nearly to rest by laser cooling, and both their internal electronic states and external motion can be coupled to and manipulated by light fields. This makes them ideally suited for quantum-optical and quantum-dynamical studies under well-controlled conditions. Theoretical and experimental work on these topics is reviewed in the paper, with a focus on ions trapped in radio-frequency (Paul) traps.

/pdf/quantum-dynamics-of-single-trapped-ions-5b12pz9r16.pdf

Quantum dynamics of single trapped ions

(b) Write out the equations for the time dependence of ρ11, ρ22, ρ12 and ρ21 assuming that a light field interaction V = -μ12Ee iωt + c.c. couples only levels |1> and |2>, and that the excited levels exhibit spontaneous decay. (8 marks) (c) Under steady-state conditions, find the ratio of populations in states |2> and |3>. (3 marks) (d) Find the slowly varying amplitude ̃ ρ 12 of the polarization ρ12 = ̃ ρ 12e iωt . (6 marks) (e) In the limiting case that no decay is possible from intermediate level |3>, what is the ground state population ρ11(∞)? (2 marks) 2. (15 marks total) In a 2-level atom system subjected to a strong field, dressed states are created in the form |D1(n)> = sin θ |1,n> + cos θ |2,n-1> |D2(n)> = cos θ |1,n> sin θ |2,n-1>

The Quantum Theory of Light

The change of dynamics in a quantum system under frequent or continuous observation, known as the quantum Zeno effect, is generally derived from the projection or reduction of the wave-packet hypothesis that is the central postulate in the theory of quantum measurements The only experiment in which the Zeno effect has yet been clearly demonstrated, though, allows no conclusion on the necessity or validity of the projection postulate This is shown by calculating, in detail, the outcome of the experiment on the basis of the standard three-level Bloch equations These equations follow from the quantum theory of irreversible processes with no additional assumptions necessary, such as which part of the system serves as measuring apparatus or how efficient the measurement would be

Quantum Zeno effect without collapse of the wave packet

The macroscopic interference of two Bose condensates released from a double minimum potential has been demonstrated recently [M. R. Andrews et al., Science 275, 637 (1997)]. In this Letter we show the excellent agreement between those experiments and theoretical predictions based on the nonlinear Schr\"odinger equation. In addition, the transition from interference of coupled condensates, comparable with the Josephson effect in superconductors, to the interference of independent Bose condensates is studied.

Transition from phase locking to the interference of independent bose condensates : theory versus experiment

We study theoretically the macroscopic interference of two independent Bose condensates released from a double potential trap. The observation of fringes could serve as a test for the paradigm of broken gauge symmetry. By numerical solution of the nonlinear Schr\"odinger equation in three dimensions, the consecutive stages of expansion, overlap, and interference are investigated in order to facilitate the design of future experiments. The phase-space dynamics of the condensates is analyzed by means of the Wigner function. It turns out that the distance of interference fringes grows linearly in time with a velocity inversely proportional to the initial distance of the two condensates. The collisional reduction of the fringe visibility is estimated.

Phase-space dynamics of bose condensates: interference versus interaction

In this paper we study the quantum Zeno effect in real space due to a position measurement. The motion of a particle is decelerated or comes to a complete stop when its position is observed. Instead of using the von Neumann collapse hypothesis, we treat a real measurement process. The measurement consists of coupling a two-level atom in a double-well potential to a resonant laser beam. Subsequent resonance fluorescence can be used to determine the atom's position within the double well, provided the laser wavelength is short enough to ensure a resolvable scattering pattern of the fluorescence photons. Treating this process in the framework of dissipative quantum mechanics, we derive a master equation which describes the measurement process in all relevant details. Solving the master equation analytically as well as numerically we study the conditions for the decay of the nondiagonal elements. This leads directly to the inhibition of the center-of-mass motion, i.e., to a quantum Zeno effect. Because we treat the measurement process in detail we are able to investigate the conditions for a complete measurement. In particular, we study the role of the intensity and the wavelength of the probing laser field.

Quantum Zeno effect in a double-well potential: A model of a physical measurement

We study an output coupler for matter waves that could be used to develop a cw atom-laser, based on current Bose-Einstein-condensation experiments with alkali-metal atoms. Assuming a cw pump mechanism for the condensate and taking into account losses due to outgoing atoms allows a description of the output by a coherently driven and damped Schr\"odinger equation. By contrast to a pulsed atom-laser output the stationary outgoing matter wave is extremely monoenergetic. Phase fluctuations of the condensate and technical noise limit the coherence length of the output beam. Using a Monte Carlo simulation we obtain the spatial correlation function of the output wave function, also including the influence of gravity.

Axel Schenzle

Papers

Quantum Zeno effect without collapse of the wave packet

Transition from phase locking to the interference of independent bose condensates : theory versus experiment

Phase-space dynamics of bose condensates: interference versus interaction

Quantum Zeno effect in a double-well potential: A model of a physical measurement

Phase diffusion and the output properties of a cw atom-laser