Bose–Einstein statistics

Abstract: Bose–Einstein condensation has been observed in several physical systems, but is not predicted to occur for blackbody radiation such as photons. However, it becomes theoretically possible in the presence of thermalization processes that conserve photon number. Martin Weitz and colleagues have now realized such conditions experimentally, observing Bose–Einstein condensation of photons in a dye-filled optical microcavity. The effect is of interest for fundamental studies and may lead to new coherent ultraviolet sources. Bose–Einstein condensation has been observed in several physical systems, but is not predicted to occur for blackbody radiation such as photons. However, it becomes theoretically possible in the presence of thermalization processes that conserve photon number. These authors experimentally realise such conditions, observing Bose–Einstein condensation of photons in a dye-filled optical microcavity. The effect is of interest for fundamental studies and may lead to new coherent ultraviolet sources. Bose–Einstein condensation (BEC)—the macroscopic ground-state accumulation of particles with integer spin (bosons) at low temperature and high density—has been observed in several physical systems1,2,3,4,5,6,7,8,9, including cold atomic gases and solid-state quasiparticles. However, the most omnipresent Bose gas, blackbody radiation (radiation in thermal equilibrium with the cavity walls) does not show this phase transition. In such systems photons have a vanishing chemical potential, meaning that their number is not conserved when the temperature of the photon gas is varied10; at low temperatures, photons disappear in the cavity walls instead of occupying the cavity ground state. Theoretical works have considered thermalization processes that conserve photon number (a prerequisite for BEC), involving Compton scattering with a gas of thermal electrons11 or photon–photon scattering in a nonlinear resonator configuration12,13. Number-conserving thermalization was experimentally observed14 for a two-dimensional photon gas in a dye-filled optical microcavity, which acts as a ‘white-wall’ box. Here we report the observation of a Bose–Einstein condensate of photons in this system. The cavity mirrors provide both a confining potential and a non-vanishing effective photon mass, making the system formally equivalent to a two-dimensional gas of trapped, massive bosons. The photons thermalize to the temperature of the dye solution (room temperature) by multiple scattering with the dye molecules. Upon increasing the photon density, we observe the following BEC signatures: the photon energies have a Bose–Einstein distribution with a massively populated ground-state mode on top of a broad thermal wing; the phase transition occurs at the expected photon density and exhibits the predicted dependence on cavity geometry; and the ground-state mode emerges even for a spatially displaced pump spot. The prospects of the observed effects include studies of extremely weakly interacting low-dimensional Bose gases9 and new coherent ultraviolet sources15.

Abstract: Recent observations of angular distributions of $\ensuremath{\pi}$ mesons in $\overline{p}\ensuremath{-}p$ annihilation indicate a deviation from the predictions of the usual Fermi statistical model. In order to shed light on these phenomena, a modification of the statistical model is studied. We retain the assumption that the transition rate into a given final state is proportional to the probability of finding $N$ free $\ensuremath{\pi}$ mesons in the reaction volume, but express this probability in terms of wave functions symmetrized with respect to particles of like charge. The justification of this assumption is discussed. The model reproduces the experimental results qualitatively, provided the radius of the interaction volume is between one-half and three-fourths of the pion Compton wavelength; the dependence of angular correlation effects on the value of the radius is rather sensitive. Quantitatively, there seems to remain some discrepancy, but we cannot say whether this is due to experimental uncertainties or to some other dynamic effects. In the absence of information on $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ interactions and of a fully satisfactory explanation of the mean pion multiplicity for annihilation, we wish to emphasize the preliminary nature of our results. We consider them, however, as an indication that the symmetrization effects discussed here may well play a major role in the analysis of angular distributions. It is pointed out that in this respect the energy dependence of the angular correlations may provide valuable clues for the validity of our model.

Abstract: The Dilute Bose Gas in 3D- The Dilute Bose Gas in 2D- Generalized Poincare Inequalities- Bose-Einstein Condensation and Superfluidity for Homogeneous Gases- Gross-Pitaevskii Equation for Trapped Bosons- Bose-Einstein Condensation and Superfluidity for Dilute Trapped Gases- One-Dimensional Behavior of Dilute Bose Gases in Traps- Two-Dimensional Behavior in Disc-Shaped Traps- The Charged Bose Gas, the One- and Two-Component Cases- Bose-Einstein Quantum Phase Transition in an Optical Lattice Model

Abstract: Observations of Bose–Einstein condensates—macroscopic populations of ultracold atoms occupying a single quantum state—in dilute alkali-metal and hydrogen gases have stimulated a great deal of research into the statistical physics of weakly interacting quantum degenerate systems1,2 Recent experiments offer a means of exploring fundamental low-temperature physics in a controllable manner A current experimental goal in the study of trapped Bose gases is the observation of superfluid-like behaviour, analogous to the persistent currents seen in superfluid liquid helium which flow without observable viscosity The ‘super’ properties of Bose-condensed systems occur because the macroscopic occupation of a quantized mode provides a stabilizing mechanism that inhibits decay through thermal relaxation3 Here we show how to selectively generate superfluid vortex modes with different angular momenta in a Bose–Einstein condensate Our approach involves solving the time-dependent equation of motion of a two-component condensate with strongly coupled internal atomic states, as recently investigated experimentally4,5 The generation of vortices relies on the coupling between the states (achieved by applying an electromagnetic field), combined with mechanical rotation of the trapping potentials which confine the condensate