Realization of BEC in a Dilute 87 Rb Vapor

1. Realization of BEC in a Dilute 87Rb Vapor In principle, the lighter the bosons, the greater TC. For example, the BE condensation of excitons (light-induced electron-hole pairs) in semiconductors has been observed before the BE condensation in dilute gases (electron is a fermion, but an electron-hole pair has an integer spin). The first observation of the BEC with weakly-interacting gases was observed with relatively heavy atoms of 87Rb. 10,000 rubidium-87 atoms were confined within a “box” with dimensions ~ 10 m (the density ~ 1019 m-3). The spacing between the energy levels: The transition was observed at ~ 0.1 K. This is in line with the estimate: again, it is worth emphasizing that the BEC occurs at kT ≫  :

2. The system of bosons at a very low temperature The system of bosons at a relatively low temperature Bose-Einstein condensation is a pure quantum effect; it is due to the symmetrization requirement for the wave function. It does not occur for fermions, or if each particle in the box is distinguishable from every other particle. “Distinguishable” should here be taken to mean that there are no antisymmetrization requirements, as there are not if each particle in the system is a different type of particle from every other particle. Given a microscopic system of distinguishable particles with half in the single-particle ground state, if you hold the temperature constant while increasing the system size, the size of the cloud of occupied states in wave number space remains about the same. However, the bigger macroscopic system has much more energy states, spaced much closer together in wave number space. Distinguishable particles spread out over these additional states, leaving only a vanishingly small fraction in the lowest energy state. This does not happen if you scale up a Bose-Einstein condensate; here the fraction of bosons in the lowest energy state stays finite regardless of system size.

3. The atoms are not very close to each other in the classic sense - in fact, the average density of this condensate is very low—one billionth the density of normal solids or liquids. But at this temperature, the quantum volume becomes comparable to the average volume per atom: At T=0.9TC, the number of atoms in the ground state is degeneracy of the 1st excited state In the first excited state: The ratio N0/N1, which is ~ 5 for N = 104, rapidly increases with N at a fixed T/TC (it becomes ~ 25 for N = 106). Velocity-distribution data of a gas of rubidium atoms, confirming the discovery of BEC. Left: just before the appearance of a BEC. Center: just after the appearance of the condensate. Right: after further evaporation, leaving a sample of nearly pure condensate.

4. By integrating the heat capacity at constant volume, we can get the entropy: The Helmholtz free energy ( = 0): The pressure exerted by a degenerate Bose gas: does not depend on volume! Pressure of the ideal Bose gas vs the specific volume v = V/N for two temperatures T1 > T2. This is due to the fact that, when compressing a degenerate Bose gas, we just force more particles to occupy the ground state. The particles in the ground state do not contribute to pressure – except of the zero-motion oscillations, they are at rest.

5. The pictures show schematically the evolution of the ground state from the BCS limit with large, spatially overlapping Cooper pairs to the BEC limit with tightly bound molecules. The ground state at unitarity (1/(kFas) = 0) has strongly interacting pairs with size comparable to 1/kF. As a function of increasing attraction, the pair-formation crossover scale T* diverges away from Tc below which a condensate exists. Most Fermi superfluids and superconductors are close to the BCS limit where these two temperatures coincide.

6. Blackbody Radiation Ultraviolet Catastrophe The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was a prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium will emit radiation with infinite power. The ultraviolet catastrophe results from the equipartition theorem of classical statistical mechanics which states that all harmonic oscillator modes (degrees of freedom) of a system at equilibrium have an average energy of kT. Normal modes (standing waves) of EM wave in a box: Each mode has degeneracy 2 degrees of freedom (polarization). From equipartition theorem: Total energy of the EM standing waves: Max Planck solved the problem by postulating that electromagnetic energy did not follow the classical description, but could only be emitted in discrete packets of energy proportional to the frequency, as given by Planck's law. This has the effect of reducing the number of possible excited modes with a given frequency in the cavity, and thus the average energy at those frequencies.

7. Solution: Light consist of quantized energy units, called photons with energy  and momentum p: The partition function of photon gas in a box: Average photon energy on ith level: Average number of photons with energy (Planck distribution) Bose-Einstein distribution: For photons: Photons are bosons with =0 !!!

8. Number of photons is not conserved because they can be emitted or absorbed by matter. Therefore, the free energy cannot depends on total number (Nph) of photons, i.e. The total energy of photon gas at temperature T: … accounting for two polarization states The energy density per unit photon energy : (Planck spectrum)