# The Fulde-Ferrell-Larkin-Ovchinnikov state for ultracold fermions in lattice and harmonic potentials: a review

## TL;DR

This review discusses the theoretical understanding and experimental prospects of the FFLO state in ultracold fermionic gases, emphasizing systems like optical lattices and spin-orbit coupled setups, and addressing key open questions.

## Contribution

It provides a comprehensive overview of the theoretical basis, potential experimental realizations, and open questions regarding the FFLO state in ultracold quantum gases.

## Key findings

- Identifies promising systems for FFLO observation, such as optical lattices and spin-orbit coupled gases.
- Analyzes theoretical bounds and experimental limitations affecting FFLO detection.
- Summarizes current ideas and open questions about FFLO stability and properties.

## Abstract

We review the concepts and the present state of theoretical studies of spin-imbalanced superfluidity, in particular the elusive Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, in the context of ultracold quantum gases. The comprehensive presentation of the theoretical basis for the FFLO state that we provide is useful also for research on the interplay between magnetism and superconductivity in other physical systems. We focus on settings that have been predicted to be favourable for the FFLO state, such as optical lattices in various dimensions and spin-orbit coupled systems. These are also the most likely systems for near-future experimental observation of the FFLO state. Theoretical bounds, such as Bloch's and Luttinger's theorems, and experimentally important limitations, such as finite-size effects and trapping potentials, are considered. In addition, we provide a comprehensive review of the various ideas presented for the observation of the FFLO state. We conclude our review with an analysis of the open questions related to the FFLO state, such as its stability, superfluid density, collective modes and extending the FFLO superfluid concept to new types of lattice systems.