Low-temperature physics

Paired in one dimension

The trend towards using ultracold atomic gases to explore emergent phenomena in many-body systems continues to gain momentum. This time around, they have been used to explore novel pairing mechanisms in one dimension. See Letter p.567

Atomic gases cooled down to nanokelvin temperatures and confined in optical or magnetic traps have helped to realize and investigate fundamental many-body quantum phases of matter1,2. An investigation by Liao et al.3 on page 567 of this issue now shows how such ultracold systems are also moving to centre stage in the quest for an exotic form of superconductivity — the elusive FFLO superconducting state of matter that was proposed more than 40 years ago by Fulde and Ferrell4 and Larkin and Ovchinnikov5.

In condensed-matter physics, an arbitrarily small attraction between fermions (particles with half-integer spin, such as electrons) of identical but opposing spin and momentum can lead to the formation of bound pairs that have bosonic character (bosons being particles with whole-integer spin). Under specific conditions, such pairs can undergo the phenomenon of Bose–Einstein condensation (BEC), transforming the many-body system into a 'giant matter wave' with spectacular frictionless-flow properties — a superconductor or superfluid is born. This remarkable outcome of pairing, first proposed by Bardeen, Cooper and Schrieffer (BCS), is considered to be the conventional way in which superconductivity emerges in a wide range of materials. In the world of atomic physics, the same pairing mechanism has been studied thoroughly in three dimensions with equal two-component gas mixtures of fermionic neutral atoms1,2, each component comprising atoms with one of two spin states (up or down). But what happens to such a BCS superfluid state if the two fermionic spin states are not present in equal numbers in the system?

In a solid-state material, such a spin-imbalance condition can be created by applying a magnetic field to the system. In ultracold atomic gases, a simple initial difference in the number of spin-up and spin-down atoms will do the job. Intuitively, one might think that an increasing mismatch in the number of spin-up and spin-down particles would make it harder for the opposing spins to meet each other and pair up, thus hindering superconductivity. And this is indeed what happens in experiments. Put in more technical terms, the Fermi surfaces of the two system components will have different sizes, and this difference will hamper the formation of the pairs and the ensuing BCS superfluid state (the Fermi surface is the boundary in momentum space that separates unoccupied states from occupied ones).

Fulde and Ferrell4, as well as Larkin and Ovchinnikov5, proposed a clever solution that would still allow a superfluid state to exist under spin-imbalanced conditions. They suggested a paired state in which the pairs are not at rest but instead have a net momentum. This FFLO state can be viewed as a kind of microscale phase separation, containing alternating superfluid regions and normal, non-superfluid regions, in which the extra atoms of the spin species that are in excess squeeze in. Although searches for such an exotically paired FFLO state have been carried out exhaustively in condensed-matter systems, and more recently in ultracold atomic gases, unambiguous experimental evidence has remained elusive. In their study, Liao et al.3 take a major step towards creating an FFLO state using ultracold fermionic atoms.

In three dimensions (3D), the FFLO state is believed to occupy only a tiny portion of the phase diagram of possible states of matter, making an observation of this state almost impossible. In one dimension (1D), however, this prospect seems more promising: a 'nesting' effect that allows the edges of the Fermi surfaces of the two system components to be connected with one another makes the FFLO state a much more robust phase in 1D6, one that occupies large parts of the phase diagram7. Liao and colleagues3 therefore confined a fermionic two-spin gas mixture of atoms in one-dimensional tubes (Fig. 1). To do this, they overlapped two perpendicular standing light waves: the interference between the two waves produces an array of one-dimensional tubes in which the atoms are trapped. In each tube, the atoms are free to move along the longitudinal direction of the tube but have their motion completely restricted in the tube's radial direction. Next, the researchers measured the radial profiles of the number-density difference between the two spin-state components, as well as the number density of the minority spin state, for a range of overall spin imbalances (polarizations) of the gas mixture.

Figure 1: Trapping and pairing atoms in one dimension.

Liao et al.3 trapped two-component spin mixtures of ultracold fermionic atoms in an array of one-dimensional tubes, which are produced by the interference between two optical standing waves (not shown). Each component consists of atoms that are in one of two spin states (up or down), and the system has an overall spin imbalance caused by a difference in number between the spin-up and spin-down atoms. For a wide range of spin imbalances, the authors find that the system has a partially polarized core and fully paired wings — a result that is in good agreement with theory but in striking contrast to results obtained for systems trapped in three dimensions.

Their data revealed a striking result. In contrast to the 3D case, for which previous observations invariably displayed a fully paired gas core, the authors3 find that the opposite can occur in 1D: for a large range of polarizations, the wings of their gas mixture are fully paired and the core exhibits partial polarization, as indicated by an excess of one spin-state component (Fig. 1). This behaviour — and the general variation in radius of the density of both the minority spin-state component and the difference between the two spin states as a function of polarization — are in excellent agreement with theoretical calculations of a one-dimensional system with FFLO characteristics. But Liao and colleagues' work is also remarkable for another reason. Simple extensions of their experiment should allow the crossover from 1D to 3D to be investigated. In 3D, the nature of the paired states can be markedly different from that in 1D.

What remains to be demonstrated, however, is whether the partially polarized core observed by the authors3 is indeed a superfluid FFLO state. So far, they have detected neither signs of superfluidity nor the 'smoking gun' signature of the FFLO state: the two momentum components of the superfluid that are caused by the microscale phase separation. In an extension of Liao and colleagues' experiment, this characteristic momentum distribution should be measurable in a single tube. According to theory, success in reaching the FFLO phase may also require lowering the temperature of the experiments further. More than 40 years after the original proposal, the race for the unambiguous observation of the FFLO state is therefore still on, but Liao and colleagues have opened a path towards making it a reality.


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Bloch, I. Paired in one dimension. Nature 467, 535–536 (2010). https://doi.org/10.1038/467535a

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