We analyze the momentum entanglement of homogeneous Bardeen--Cooper--Schrieffer (BCS) states between spin-up and spin-down particles at finite temperature. This is achieved by studying the structure of the partial transposition of the two-particle density matrix (denoted by ${\ensuremath{\rho}}_{2}^{{T}_{A}}$) in momentum space, as well as the corresponding negativity. We show that ${\ensuremath{\rho}}_{2}^{{T}_{A}}$ consists of infinitely many decoupled $2\ifmmode\times\else\texttimes\fi{}2$ submatrices, and momentum entanglement is a necessary and sufficient condition for a nonzero order parameter $\ensuremath{\Delta}$. From the distribution of eigenvalues of ${\ensuremath{\rho}}_{2}^{{T}_{A}}$, we discover that momentum entanglement is most significantly contributed by pairs with momenta slightly above a surface related to the Fermi energy. In addition, we propose an entanglement witness operator for detecting momentum entanglement in BCS states.
An investigation was conducted within the limits of the Bardeen-Cooper- Schrieffer (BCS) theory of the properties of the l = 1, 2, 3,... states of condensed fermions. The study is confined to states only of zero-total momentum. The results indicated that the most unusual properties were connected with the angular momentum. Less spectacular were the superfluidity or superconductivity properties. Unlike the case for l = 0, the states for all higher l's are sensitive to impurity scattering effects. (B.O.G.)
We determine the exact dynamics of an initial Bardeen-Cooper-Schrieffer (BCS) state of ultracold atoms in a deep hexagonal optical lattice. The dynamical evolution is triggered by a quench of the lattice potential such that the interaction strength
Uf is much larger than the hopping amplitude
Jf. The quench initiates collective oscillations with frequency $|{U}_{f}|/2\ensuremath{\pi}$ in the momentum occupation numbers and imprints an oscillating phase with the same frequency on the BCS order parameter $\mathrm{\ensuremath{\Delta}}$. The oscillation frequency of $\mathrm{\ensuremath{\Delta}}$ is not reproduced by treating the time evolution in mean-field theory. In our theory, the momentum noise (i.e., density-density) correlation functions oscillate at frequency $|{U}_{f}|/2\ensuremath{\pi}$ as well as at its second harmonic. For a very deep lattice, with zero tunneling energy, the oscillations of momentum occupation numbers are undamped. Nonzero tunneling after the quench leads to dephasing of the different momentum modes and a subsequent damping of the oscillations. The damping occurs even for a finite-temperature initial BCS state, but not for a noninteracting Fermi gas. Furthermore, damping is stronger for larger order parameter and may therefore be used as a signature of the BCS state. Finally, our theory shows that the noise correlation functions in a honeycomb lattice will develop strong anticorrelations near the Dirac point.