A Viewpoint on : Evolution of Fermion Pairing from Three to Two Dimensions

Superconductivity and superfluidity in two dimensions (2D) has long been a subject of great interest. Materials with the highest known transition temperature Tc, the cuprates, are quasi-2D systems with superconductivity originating in the 2D copper-oxide planes. A deeper understanding of strongly interacting 2D superconductors and their normal states could potentially give insights into the unsolved problem of high-Tc superconductivity. Further, 2D is known to be the lower critical, or marginal, dimension both for pairing of fermions and for superfluidity. Thus 2D represents a borderline dimensionality, above and below which the system behaves qualitatively differently from a classical and a quantum perspective. Classically, thermal fluctuations of the order parameter destroy long-range order in 2D and lead to algebraic order at finite temperatures. In quantum mechanics, 2D is the marginal dimension for bound-state formation, which has important implications for the pairing of fermions that is essential for superfluidity. Finally, the effects of strong interactions in 2D Fermi systems present a formidable theoretical challenge. A paper [1] appearing in Physical Review Letters reports experiments with ultracold Fermi gas of lithium-6 atoms that give new insight into how dimensionality and interactions affect the binding of fermions to form a superfluid. Ariel Sommer and his colleagues at the Massachusetts Institute of Technology, Cambridge, use two independent knobs: a magnetic field to tune interatomic interactions and an optical lattice to tune the dimensionality between 3D and 2D. They then use radio-frequency (rf) spectroscopy to probe pairing in the many-particle system and relate it to bound-state formation in the twobody problem in the 2D limit. Although theorists have predicted how this pairing energy evolves with interaction strength and dimensionality, until now there were no real systems on which to test these ideas. In recent years there has been tremendous progress in controlling the interaction between atoms in ultracold gases with the Feshbach resonance technique, in which a magnetic field is used to tune the attractive s-wave interaction between Fermi atoms in two different hyperfine states [2–4]. This is the analog of tuning the attraction between spin-up and spin-down electrons in a metal, which is, of course, impossible in solid-state materials. An important outcome of this progress has been the ability to explore the crossover from a condensate of Cooper pairs, described by Bardeen-Cooper-Schrieffer (BCS) theory, to a Bose-Einstein condensate (BEC) of tightly bound diatomic molecules in 3D ultracold gases. The most significant new insights have come from studies of the very strongly interacting states at resonance [2, 3], which has scale-invariant properties [4] analogous to problems in nuclear physics and string theory. The experiment of Sommer et al.[1] builds on all of this progress with an optical lattice, which is a periodic potential that arises from interfering laser beams. By tuning the strength of this potential, V0, one can go from an anisotropic 3D system, for weak V0, to an essentially decoupled stack of 2D layers, in the limit of strong V0, which is shown schematically in Fig. 1. The MIT team probes the pairing between atoms with rf spectroscopy [2], in which photons transfer Fermi atoms from one hyperfine state to another. The resulting absorption intensity as a function of photon energy has a characteristic asymmetric line shape with a threshold for bound-to-free transitions that contains information about the spectrum of fermionic excitations. The meanfield theory for a BCS-BEC crossover predicts that the rf threshold is given by [2] h̄ωth = √ μ2 + ∆2 − μ, where μ is the chemical potential and ∆ the BCS gap parameter. It is important to emphasize that the rf threshold is not simply ∆, in contrast to spectroscopic probes of electronic excitations in superconductors like tunneling. The reason is that rf photons excite Fermi atoms in all k states and the threshold is determined by k = 0 fermions. Sommer et al. interpret the rf threshold h̄ωth as the “pair binding energy” in the many-body system. In 2D, the zero temperature mean-field equations for