Coherent States - Coherent States of Bose–Einstein Condensates

Coherent States of Bose–Einstein Condensates

• A Bose–Einstein condensate (BEC) is a collection of boson atoms that are all in the same quantum state. In a thermodynamic system, the ground state becomes macroscopically occupied below a critical temperature — roughly when the thermal de Broglie wavelength is longer than the interatomic spacing. Superfluidity in liquid Helium-4 is believed to be associated with the Bose–Einstein condensation in an ideal gas. But 4He has strong interactions, and the liquid structure factor (a 2nd-order statistic) plays an important role. The use of a coherent state to represent the superfluid component of 4He provided a good estimate of the condensate / non-condensate fractions in superfluidity,consistent with results of slow neutron scattering. Most of the special superfluid properties follow directly from the use of a coherent state to represent the superfluid component — that acts as a macroscopically occupied single-body state with well-defined amplitude and phase over the entire volume. (The superfluid component of 4He goes from zero at the transition temperature to 100% at absolute zero. But the condensate fraction is about 6% at absolute zero temperature, T=0° K.)
• Early in the study of superfluidity, Penrose and Onsager proposed a metric ("order parameter") for superfluidity. It was represented by a macroscopic factored component (a macroscopic eigenvalue) in the first-order reduced density matrix. Later, C. N. Yang proposed a more generalized measure of macroscopic quantum coherence, called "Off-Diagonal Long-Range Order" (ODLRO), that included fermion as well as boson systems. ODLRO exists whenever there is a macroscopically large factored component (eigenvalue) in a reduced density matrix of any order. Superfluidity corresponds to a large factored component in the first-order reduced density matrix. (And, all higher order reduced density matrices behaved similarly.) Superconductivity involves a large factored component in the 2nd-order ("Cooper electron-pair") reduced density matrix.
• The orders of reduced density matrices used to describe macroscopic quantum coherence in superfluids are formally the same as the correlation functions used to describe orders of coherence in radiation. Both are examples of macroscopic quantum coherence. The macroscopically large coherent component, plus noise, in the electromagnetic field, as given by Glauber's description of signal-plus-noise, is formally the same as the macroscopically large superfluid component plus normal fluid component in the two-fluid model of superfluidity.
• Every-day electromagnetic radiation, such as radio and TV waves, is also an example of near coherent states (macroscopic quantum coherence). That should "give one pause" regarding the conventional demarcation between quantum and classical.