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.
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.
Quantum correlations in fermionic system are central to many fascinating effects in condensed matter physics. In this Thesis we investigate these correlations from the perspective of quantum information theory. Motivated by recent experiments on ultra cold gases, we develop a “pairing theory” for fermionic systems. Next, we investigate possibilities to gain insight into the correlations of these systems using simulation techniques. On the one hand, we study the power of quantum systems for the simulation of fermionic, bosonic and spin systems. On the other hand, we develop numerical methods applicable to interacting fermionic systems. First, we construct a class of variational states well suited for the approximation of ground and thermal states of systems with short-range interaction. Then, we develop a method that allows the numerical approximation of observables in dynamical processes as well as in equilibrium.
Fermion pairing underlies collective quantum phenomena across widely different forms of matter. In extended systems such as ultracold Fermi gases, pairing is commonly understood through the BCS--BEC crossover, where the pair size evolves from large, overlapping Cooper pairs to tightly bound dimers. In finite systems such as atomic nuclei, superconducting grains and quantum dots, however, the same pairing tendency competes with confinement, shell filling and spatial inhomogeneity, making the microscopic structure of pairs much harder to access. Here, we image pair correlations in a finite, tunable system of few fermionic atoms with single-particle resolution and full counting statistics. We observe that confinement and shell structure re-organize pairing in real space: In the weakly interacting, confinement-dominated regime, closed-shell configurations suppress correlations in the high-density trap center. Pairing is mainly observed toward the low-density surface. Open-shell systems, however, support substantially stronger central pairing. Already for surprisingly small systems, increasing either interaction strength or particle number restores a locally bulk-like Cooper-pair profile in the trap center, whereas the edge retains dimer-like correlations. By resolving where pairs form and how their character changes from localized dimers to overlapping Cooper pairs, our measurements provide a microscopic view of pairing in finite fermionic matter and connect the physics of mesoscopic cold atoms to pairing phenomena in nuclei and superconducting nanostructures.
Superfluidity is one of the most striking manifestations of quantum mechanics at macroscopic scales, characterized by frictionless flow and the emergence of collective quantum phenomena. The theoretical foundation of superfluidity in fermionic systems was revolutionized by the Bardeen-Cooper-Schrieffer (BCS) theory, developed in 1957 to explain superconductivity in metals. This groundbreaking work provided a microscopic description of how attractive interactions between electrons near the Fermi surface can lead to the formation of Cooper pairs-correlated pairs of fermions that condense into a coherent quantum state. The BCS theory not only explained superconductivity but also laid the groundwork for understanding superfluidity in systems like liquid helium-3 and ultracold fermionic gases, becoming a cornerstone of condensed matter physics.
The study of entanglement in systems composed of identical particles raises interesting challenges with far-reaching implications in both, our fundamental understanding of the physics of composite quantum systems, and our capability of exploiting quantum indistinguishability as a resource in quantum information theory. Impressive theoretical and experimental advances have been made in the last decades that bring us closer to a deeper comprehension and to a better control of entanglement. Yet, when it involves composites of indistinguishable quantum systems, the very meaning of entanglement, and hence its characterization, still finds controversy and lacks a widely accepted definition. The aim of the present paper is to introduce, within an accessible and self-contained exposition, the basic ideas behind one of the approaches advanced towards the construction of a coherent definition of entanglement in systems of indistinguishable particles, with focus on fermionic systems. We also inquire whether the corresponding tools developed for studying entanglement in identical-fermion systems can be exploited when analysing correlations in distinguishable-party systems, in which the complete information of the individual parts is not available. Further, we open the discussion on the broader problem of constructing a suitable framework that accommodates entanglement in the presence of generalized statistics. This article is part of the theme issue 'Identity, individuality and indistinguishability in physics and mathematics'.
The two-particle spectrum in a dense Fermion system is treated using a thermodynamic Green function approach. A self-consistent description of possible bound states and a superfluid condensate in a correlated medium is given. A detailed understanding of superfluidity and superconductivity in correlated Fermion systems, especially the transition from the Cooper-paired state (weak-coupling limit) to the Bose condensed state of tightly bound pairs of Fermions (strong coupling limit) [1], is of great interest for very different physical systems. The problem of a unified treatment of Bose–Einstein condensation (BEC) and the Bardeen–Cooper–Schrieffer (BCS) phase arises not only in describing the electron structure of strongly correlated electron superfluids such as superconductors [2], the electronhole system in semiconductors [3], spin-polarized hydrogen and liquid He [4], but also in the theory of nuclear matter [5] and quark–gluon systems [6]. Recently, there have been several new approaches to this stimulating problem. A Monte-Carlo simulation of a finite He system at zero temperature has been performed in Ref. [7]. The microscopic theory of strongly coupled quantum fluids has been treated within the Jastrow approximation (cf. Ref. [8]) to obtain the ground state and low-lying excited states of strongly correlated boson quantum fluids. The crossover from weak to strong coupling superconductivity has been considered using a functional integral representation [2] (see also Ref. [7]).
Quantum correlations are crucial features in both quantum information theory and many-body physics. Characterization and quantification of quantum correlations have delivered a rich body of work and helped to understand some quantum phenomena. Entanglement is a unique quantum correlation for which it is a resource in many quantum information tasks. Developed for quantification of entanglement, entanglement witness formalism is a remarkable tool in the quantum information toolkit. It can be deployed beyond the entanglement theory to quantify quantum correlations in many-body systems. For many-body systems, the particle statistics and indistinguishability are essential. The relation between indistinguishability and entanglement has to be studied carefully. In this thesis, first, a class of witness operators for pairing correlations in both fermionic and bosonic systems have been proposed and the necessary and sufficient conditions for being a witness operator are found. Then, a Clauser-Horne-Shimony-Holt type test and its bounds for fermions have been demonstrated.