Pairing in fermionic systems: A quantum-information perspective

The notion of “paired” fermions is central to important condensed-matter phenomena such as superconductivity and superfluidity. While the concept is widely used and its physical meaning is clear, there exists no systematic and mathematical theory of pairing that would allow us to unambiguously characterize and systematically detect paired states. We propose a definition of pairing and develop methods for its detection and quantification applicable to current experimental setups. Pairing is shown to be a quantum correlation different from entanglement, giving further understanding in the structure of highly correlated quantum systems. In addition, we will show the resource character of paired states for precision metrology, proving that the BCS states allow phase measurements at the Heisenberg limit.

Figure 1

(Color online) Bloch sphere representation of the expectation values of $j_k^{\left(x\right)}$, $j_k^{\left(y\right)}$, and $j_k^{\left(z\right)}$ for a variational BCS state. All pure states lie on the surface of the sphere. Unpaired states lie on the $z$ axis, while the maximally paired states lie on the equator.

Figure 2

(Color online) Expectation values of the vector Eq. (25). For all number-conserving states these lie within the convex set $C_{{\vec{O}}_3}^{\text{all}}$ indicated by the dashed back lines. The extreme points of the polytope are given by (0, 0, 0), (2, 0, 0), (4, 2, 0), and $(2,1,\pm 1)$. Unpaired states have expectation values in the smaller convex set $C_{{\vec{O}}_3}^{\text{unpaired}}$ (solid red), which has extreme points (0, 0, 0), (2, 0, 0), (4, 2, 0), and $(2,1∕2,\pm 1∕2)$.

Figure 3

Scheme of the Ramsey interferometer setup. The incoming wave function $\mid {\Psi }_{\mathrm{in}}⟩$ enters a beam splitter (BS). Then particles in the modes $a_{\pm l}^{†}$ evolve under the Hamiltonian $\phi H$. At the end, a particle number measurement is performed on all particles.

Figure 4

Setup that allows interferometry with paired states at the Heisenberg limit. Particles in modes $a_k^{†}$ and $b_k^{†}$ evolve under the complex coupling Hamiltonian $H$ (for the detailed form of $H$, refer to the text). In the end, particle numbers are measured.