Abstract:
we describe recent efforts to study cooper pairs in atomic nuclei. we consider a self-consistent hartree fock mean field for the even sm isotopes and compare results based on three treatments of pairing correlations: a bcs treatment, a number-projected bcs treatment and an exact treatment using the richardson ansatz. significant differences are seen in the pairing correlation energies. furthermore, because it does not average over the properties of the fermion pairs, the richardson solution permits a more meaningful definition of the cooper wave function and of the fraction of pairs that are collective. our results confirm that only a few pairs near the fermi surface in realistic atomic nuclei are collective.

Abstract:
We consider the development of Cooper pairs in a self-consistent Hartree Fock mean field for the even Sm isotopes. Results are presented at the level of a BCS treatment, a number-projected BCS treatment and an exact treatment using the Richardson ansatz. While projected BCS captures much of the pairing correlation energy that is absent from BCS, it still misses a sizable correlation energy, typically of order $1 MeV$. Furthermore, because it does not average over the properties of the fermion pairs, the exact Richardson solution permits a more meaningful definition of the Cooper wave function and of the fraction of pairs that are collective.

Abstract:
Describimos los recientes esfuerzos en el estudio de pares de Cooper en el núcleo atómico. Consideramos un campo promedio autoconsistente Hartree Fock para los isótopos pares del Sm y comparamos los resultados de tres tratamientos de correlaciones de pairing: un tratamiento BCS, un tratamiento BCS con proyección de número, y un tratamiento exacto usando el ansatz de Richardson. Se encuentran diferencias significativas en las energías de correlación de pairing. Además, ya que la solución de Richardson no promedia sobre las propiedades de los pares de fermiones, permite una definición más significativa de la función de onda de Cooper y de la fracción de pares que son colectivos. Nuestros resultados confirman que solo unos pocos pares cerca de la superficie de Fermi en núcleos atómicos reales son colectivos.

Abstract:
A setup based on the Franson optical interferometer is analyzed, which allows us to detect the coherence properties of Cooper pairs emerging via tunneling from a superconductor in contact with two one-dimensional channels. By tuning the system parameters we show that both the internal coherence of the emitted Cooper pairs, which is proportional to Pippard's length, and the de Broglie wavelength of their center-of-mass motion can be measured via current-current correlation measurements.

Abstract:
In an ultracold mixture of two different Fermi species of atoms, Cooper pairs can be formed between two different atoms. The masses of one atom and its partner in this kind of Cooper pairs may differ by order of magnitude. In this system, each species of atoms are in the same atomic spin state and two species have the same atomic densities. The pairing gap diminishes if two species have different densities and vanishes when the density imbalance reaches a critical value.

Abstract:
We consider a gas of trapped Cooper-paired fermionic atoms which are manipulated by laser light. The laser induces a transition from an internal state with large negative scattering length (superfluid) to one with weaker interactions (normal gas). We show that the process can be used to detect the presence of the superconducting order parameter. Also, we propose a direct way of measuring the size of the gap in the trap. The efficiency and feasibility of this probing method is investigated in detail in different physical situations.

Abstract:
We examine the basic mode structure of atomic Cooper pairs in an inhomogeneous Fermi gas. Based on the properties of Bogoliubov quasi-particle vacuum, the single particle density matrix and the anomalous density matrix share the same set of eigenfunctions. These eigenfunctions correspond to natural pairing orbits associated with the BCS ground state. We investigate these orbits for a Fermi gas in a spherical harmonic trap, and construct the wave function of a Cooper pair in the form of Schmidt decomposition. The issue of spatial quantum entanglement between constituent atoms in a pair is addressed.

Abstract:
We study the two-particle wave function of paired atoms in a Fermi gas with tunable interaction strengths controlled by Feshbach resonance. The Cooper pair wave function is examined for its bosonic characters, which is quantified by the correction of Bose enhancement factor associated with the creation and annihilation composite particle operators. An example is given for a three-dimensional uniform gas. Two definitions of Cooper pair wave function are examined. One of which is chosen to reflect the off-diagonal long range order (ODLRO). Another one corresponds to a pair projection of a BCS state. On the side with negative scattering length, we found that paired atoms described by ODLRO are more bosonic than the pair projected definition. It is also found that at $(k_F a)^{-1} \ge 1$, both definitions give similar results, where more than 90% of the atoms occupy the corresponding molecular condensates.

Abstract:
A superconductor is a natural source of spin-entangled spatially separated electron pairs. Although the first Cooper-pair splitter devices have been realized recently, an experimental confirmation of the spin state and the entanglement of the emitted electron pairs is lacking up to now. In this paper a method is proposed to confirm the spin-singlet character of individual split Cooper pairs. Two quantum dots (QDs), each of them holding one spin-prepared electron, serve as the detector of the spin state of a single Cooper pair that is forced to split when it tunnels out from the superconductor to the QDs. The number of charges on the QDs, measured at the end of the procedure, carries information on the spin state of the extracted Cooper pair. The method relies on the experimentally established toolkit of QD-based spin qubits: resonant spin manipulation, Pauli blockade, and charge measurement.

Abstract:
An atomic grating generated by a pulsed standing wave laser field is proposed to manipulate the superfluid state in a quantum degenerate gas of fermionic atoms. We show that in the presence of atomic Cooper pairs, the density oscillations of the gas caused by the atomic grating exhibit a much longer coherence time than that in the normal Fermi gas. Our result indicates that the technique of a pulsed atomic grating can be a potential candidate to detect the atomic superfluid state in a quantum degenerate Fermi gas.