We discuss various aspects of coherence of a Bose-Einstein condensate, and summarize the experimental evidence obtained thus far. It is shown that the mean-field energy of a condensate is a measure of second-order coherence. ``Release-energy'' measurements therefore provide direct evidence of the suppression of density fluctuations in a condensate compared to a thermal cloud.
We discuss various aspects of coherence of a Bose-Einstein condensate, and summarize the experimental evidence obtained thus far. It is shown that the mean-field energy of a condensate is a measure of second-order coherence. ``Release-energy'' measurements therefore provide direct evidence of the suppression of density fluctuations in a condensate compared to a thermal cloud.
This thesis features fundamental experiments on the coherence properties of Bose-Einstein condensates (BEC) and atom lasers, made possible by a newly developed apparatus combining a Bose-Einstein condensate with an ultrahigh nesse optical cavity.Using the cavity as a single atom detector, atoms extracted from a trapped cloud are detected with unprecedented sensitivity, and the rst time-resolved counting of single atoms from a quantum degenerate source was demonstrated.O-diagonal long-range order is a dening property of a BEC and equivalent to longrange phase coherence.Its formation is a complex process accompanied by spontaneous symmetry breaking.We witnessed this formation in an ultracold cloud of 87 Rb atoms crossing the phase transition of BEC.Two overlapping atomic beams originating from dierent positions inside the condensate propagated downwards into the cavity.They showed a matter wave interference pattern with a visibility determined by the coherence of the probed regions.The modulation of the ux of atoms, recorded with single-atom resolution, yields the visibility, whereas its mean value gives information about the density of the sample.By shock cooling, a cloud of atoms was prepared in a highly nonequilibrium state slightly above the phase transition.While the ensemble subsequently relaxed, it crossed the phase transition into an equilibrium with a small condensate fraction.Employing our minimally invasive measurement method, the growth of both the rst-order coherence and the density of the sample was tracked in real time while the system crossed the phase transition.The growth of the coherence was found to start later, proceed faster and nish earlier than the increase in density.The coherent region increased at one fth of the speed of sound.Employing time-resolved single atom counting with the cavity detector, the full counting statistics of an atom laser beam was recorded.In a Hanbury Brown and Twiss type experiment, the second-order correlation function was found to be constant g (2) (τ ) = 1.00±0.01,proving the atom laser's second-order coherence and the absence of excess intensity uctuations that were observed in a pseudothermal beam.An analysis of the time interval distributions of atoms in the beam conrmed these ndings.The iii
For a dilute, interacting Bose gas of magnetically-trapped atoms at temperatures below the critical temperature T0 for Bose-Einstein condensation, we determine the second-order coherence function g(2)(r1, r2) within the framework of a finite-temperature quantum field theory. We show that, because of the different spatial distributions of condensate and thermal atoms in the trap, g(2)(r1, r2) does not depend on |r1 - r2| alone. This means that the experimental determinations of g(2) reported to date give only its spatial average. Such an average may underestimate the degree of coherence attainable in an atom laser by judicious engineering of the output coupler.
For a dilute, interacting Bose gas of magnetically-trapped atoms at temperatures below the critical temperature T0 for Bose-Einstein condensation, we determine the second-order coherence function g (2) (r1 ; r2) within the framework of a finite-temperature quantum field theory. We show that, because of the different spatial distributions of condensate and thermal atoms in the trap, g (2) (r1 ; r2) does not depend on jr1 r2j alone. This means that the experimental determinations of g (2) reported to date give only its spatial average. Such an average may underestimate the degree of coherence attainable in an atom laser by judicious engineering of the output coupler.
We investigate the properties of a coherent array containing about 200 Bose-Einstein condensates produced in a far detuned 1D optical lattice. The density profile of the gas, imaged after releasing the trap, provides information about the coherence of the ground-state wave function. The measured atomic distribution is characterized by interference peaks. The time evolution of the peaks, their relative population, as well as the radial size of the expanding cloud are in good agreement with the predictions of theory. The 2D nature of the trapped condensates and the conditions required to observe the effects of coherence are also discussed.
By extending Einstein's separation of wave and particle parts of the second order thermal fluctuation to encompass "generalized fluctuations" in any Bose field, P. E. Gordon has proposed alternative definitions for nth order coherence and nth order coherent states. The main point of this paper is to explore some of the physical insights to be gained by extending dualism to higher orders. Recent experiments have examined aspects of the coherence of Bose-Einstein condensates. It has been argued that the condensate state is coherent to (at least) second or third order, but the coherence properties of Bose-Einstein condensates remain somewhat controversial. Using probability distributions developed by M. O. Scully and V. V. Kocharovsky et. al., we apply Gordon's dualistic expression of the coherence conditions to investigate coherence properties in Bose-Einstein condensation. Via numerical calculations, we present a graphical survey of wave-like and particle-like fluctuations in condensed and uncondensed fractions. Near the critical point, we find a very marked peak in the ratio of nth order wave to nth order particle fluctuations in the condensate. Not surprisingly, n-point correlations between the positions of condensate atoms also peak near the critical temperature, and this apparently mirrors, to higher orders, the well-known relation between the integral of the 2-point correlation function over a certain volume and the rms fluctuation in the number of particles in that volume.
We investigate the coherence properties of an atomic beam evaporatively cooled in a magnetic guide, assuming thermal equilibrium in the quantum degenerate regime. The gas experiences two-dimensional, transverse Bose-Einstein condensation rather than a full three-dimensional condensation because of the very elongated geometry of the magnetic guide. First order and second order correlation functions of the atomic field are used to characterize the coherence properties of the gas along the axis of the guide. The coherence length of the gas is found to be much larger than the thermal de Broglie wavelength in the strongly quantum degenerate regime. Large intensity fluctuations present in the ideal Bose gas model are found to be strongly reduced by repulsive atomic interactions; this conclusion is obtained with a one-dimensional classical field approximation valid when the temperature of the gas is much higher than its chemical potential, kBT ≫ |µ|. Pacs: 03.75.Fi, 42.50.-p The first experimental achievements of Bose-Einstein condensates in atomic vapors [1, 2, 3, 4] have opened promising perspectives for atom optics: condensates constitute indeed atomic waves sources of much better coherence properties than the usual ‘thermal ’ sources like the standard magneto-optical trap. These