题目：Quantum fluctuations in the BCS-BEC crossover
The experimental realization of ultracold atomic Fermi gases with tunable interatomic interactions has opened a new era for the study of some longstanding theoretical proposals in many-fermion systems. One interesting proposal is the smooth crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid with largely overlapping Cooper pairs to a Bose-Einstein condensate (BEC) of tightly bound bosonic molecules. While the BCS-BEC crossover at T=0 can be qualitatively described by the BCS-Leggett mean-field theory, it has been shown by experimental measurements and quantum Monte Carlo calculations that the quantum fluctuations are rather important. In two dimensions (2D), the pressure in the mean-field theory is predicted to be equal to that of a noninteracting Fermi gas in the entire BCS-BEC crossover, which is not consistent with the features of a weakly interacting Bose condensate in the BEC limit and a weakly interacting Fermi liquid in the BCS limit. The inadequacy of the 2D mean-field theory indicates that the quantum fluctuations are much more pronounced than those in 3D. In this talk, I show that the inclusion of the Gaussian quantum fluctuations naturally recovers the above features in both the BEC and the BCS limits. In the BEC limit, the missing logarithmic dependence on the boson chemical potential is recovered by the quantum fluctuations. Near the quantum phase transition from the vacuum to the BEC phase, we compare our equation of state with the known grand canonical equation of state of 2D Bose gases and determine the ratio of the composite boson scattering length aB to the fermion scattering length a2D. We find aB ? 0.56a2D, in good agreement with the exact four-body calculation. We compare our equation of state in the BCS-BEC crossover with recent results from the quantum Monte Carlo simulations and the experimental measurements and find good agreements.