Grassmann Phase Space Theory for the BEC/BCS Crossover in Cold Fermionic Atomic Gases

TL;DR

This paper introduces Grassmann Phase Space Theory (GSPT) to analyze the BEC/BCS crossover in cold fermionic gases, focusing on the evolution of quantum correlation functions related to Cooper pair formation and interactions.

Contribution

It develops a novel GSPT framework to compute the evolution of quantum correlation functions during the BEC/BCS crossover, including analytical expressions for small time or temperature steps.

Findings

Derived linear relations between stochastic averages of Grassmann momentum fields over time.

Provided analytical formulas for matrix elements for numerical evolution studies.

Analyzed initial conditions for non-interacting and high-temperature fermionic gases.

Abstract

Grassmann Phase Space Theory (GSPT) is applied to the BEC/BCS crossover in cold fermionic atomic gases and used to determine the evolution (over either time or temperature) of the Quantum Correlation Functions (QCF) that specify: (a) the positions of the spin up and spin down fermionic atoms in a single Cooper pair and (b) the positions of the two spin up and two spin down fermionic atoms in two Cooper pairs The first of these QCF is relevant to describing the change in size of a Cooper pair, as the fermion-fermion coupling constant is changed via Feshbach resonance methods through the crossover from a small Cooper pair on the BEC side to a large Cooper pair on the BCS side. The second of these QCF is important for describing the correlations between the positions of the fermionic atoms in two Cooper pairs, which is expected to be small at the BEC or BCS sides of the crossover, but is expected to be significant in the strong interaction unitary regime, where the size of a Cooper pair is comparable to the separation between Cooper pairs. In GPST the QCF are ultimately given via the stochastic average of products of Grassmann stochastic momentum fields, and GPST shows that the stochastic average of the products of Grassmann stochastic momentum fields at a later time (or lower temperature) is related linearly to the stochastic average of the products of Grassmann stochastic momentum fields at an earlier time (or higher temperature), and that the matrix elements involved in the linear relations are all c-numbers. Expressions for these matrix elements corresponding to a small time or temperature increment have been obtained analytically, providing the formulae needed for numerical studies of the evolution that are planned for a future publication. Various initial conditions are considered, including those for a non-interacting fermionic gas at zero temperature and a high temperature gas.