The coupling of the spin of electrons to their motional state lies at the heart of recently discovered topological phases of matter. Here we create and detect spin-orbit coupling in an atomic Fermi gas, a highly controllable form of quantum degenerate matter. We directly reveal the spin-orbit gap via spin-injection spectroscopy, which characterizes the energy-momentum dispersion and spin composition of the quantum states. For energies within the spin-orbit gap, the system acts as a spin diode. We also create a spin-orbit coupled lattice and probe its spinful band structure, which features additional spin gaps and a fully gapped spectrum. In the presence of s-wave interactions, such systems should display induced p-wave pairing, topological superfluidity, and Majorana edge states.

We have observed the superfluid phase transition in a strongly interacting Fermi gas via highprecision measurements of the local compressibility, density and pressure down to near-zero entropy. Our data completely determine the universal thermodynamics of strongly interacting fermions without any fit or external thermometer. The onset of superfluidity is observed in the compressibility, the chemical potential, the entropy, and the heat capacity. In particular, the heat capacity displays a characteristic lambda-like feature at the critical temperature of Tc/TF = 0.167(13). This is the first clear thermodynamic signature of the superfluid transition in a spin-balanced atomic Fermi gas. Our measurements provide a benchmark for many-body theories on strongly interacting fermions, relevant for problems ranging from high-temperature superconductivity to the equation of state of neutron stars.Phase transitions are ubiquitous in Nature: water freezes into ice, electron spins suddenly align as materials turn into magnets, and metals become superconducting. A pervasive feature of continuous phase transitions is their critical behavior, namely singularities in thermodynamic quantities: the magnetic susceptibility diverges at a ferromagnetic transition, the specific heat shows a jump at superconducting transitions [1] as well as at the superfluid transition of 3 He [2]. In 4 He, at the famous λ-transition into the the superfluid state, the jump is even resolved, in zero gravity, to be a near-diverging, singular peak [3]. A novel form of superfluidity has been realized in trapped, ultracold atomic gases of strongly interacting fermions, particles with half-integer spin [4][5][6]. Thanks to an exquisite control over relevant system parameters, these gases have recently emerged as a versatile system well suited to solve open problems in many-body physics [6]. However, while superfluidity has been established via the observation of vortex lattices in rotating gases [7], no clear thermodynamic signature of the superfluid transition has previously been observed.Initial measurements on the thermodynamics of strongly interacting Fermi gases have focused on trap averaged quantities [8][9][10] in which the superfluid transition is inherently difficult to observe. It is also challenging to reveal the critical behavior through the study of local thermodynamic quantities. The emergence of the condensate of fermion pairs in a spin-balanced Fermi gas is accompanied by only minute changes in the density [4]. Therefore, quantities that involve integration of the density over the local potential, such as the energy E [11] and the pressure P [12], are only weakly sensitive to the sudden variations in the thermodynamics of the gas that one expects near the superfluid phase transition [13].A thermodynamic quantity involving the second derivative of the pressure P is expected to become singular at the second order phase transition into the superfluid state. An example is the (isothermal) compressibility κ = 1 n ∂n ∂P | T , the relative change of...

We have observed Fermi polarons, dressed spin down impurities in a spin up Fermi sea of ultracold atoms. The polaron manifests itself as a narrow peak in the impurities' rf spectrum that emerges from a broad incoherent background. We determine the polaron energy and the quasiparticle residue for various interaction strengths around a Feshbach resonance. At a critical interaction, we observe the transition from polaronic to molecular binding. Here, the imbalanced Fermi liquid undergoes a phase transition into a Bose liquid coexisting with a Fermi sea.

Transport of fermions is central in many fields of physics. Electron transport runs modern technology, defining states of matter such as superconductors and insulators, and electron spin, rather than charge, is being explored as a new carrier of information [1]. Neutrino transport energizes supernova explosions following the collapse of a dying star [2], and hydrodynamic transport of the quark-gluon plasma governed the expansion of the early Universe [3]. However, our understanding of non-equilibrium dynamics in such strongly interacting fermionic matter is still limited. Ultracold gases of fermionic atoms realize a pristine model for such systems and can be studied in real time with the precision of atomic physics [4,5]. It has been established that even above the superfluid transition such gases flow as an almost perfect fluid with very low viscosity [3,6] when interactions are tuned to a scattering resonance. However, here we show that spin currents, as opposed to mass currents, are maximally damped, and that interactions can be strong enough to reverse spin currents, with opposite spin components reflecting off each other. We determine the spin drag coefficient, the spin diffusivity, and the spin susceptibility, as a function of temperature on resonance and show that they obey universal laws at high temperatures. At low temperatures, the spin diffusivity approaches a minimum value set by /m, the quantum limit of diffusion, where is the reduced Planck's constant and m the atomic mass. For repulsive interactions, our measurements appear to exclude a metastable ferromagnetic state [7][8][9].Understanding the transport of spin, as opposed to the transport of charge, is of high interest for the novel field of spintronics [1]. While charge currents are unaffected by electron-electron scattering due to momentum conservation, spin currents will intrinsically damp due to collisions between opposite spin electrons, as their relative momentum is not conserved. This phenomenon is known as spin drag [10,11]. It is expected to contribute significantly to the damping of spin currents in doped semiconductors [12]. The random collision events also lead to spin diffusion, the tendency for spin currents to flow such as to even out spatial gradients in the spin density, which has been studied in high-temperature superconductors [13] and in liquid 3 He-4 He solutions [14,15]. Creating spin currents poses a major challenge in electronic systems where mobile spins are scattered by their environment and by each other. However, in ultracold atoms we have the freedom to first prepare an essentially non-interacting spin mixture, separate atoms spatially via magnetic field gradients, and only then induce strong interactions. Past observations of spin currents in ultracold Fermi gases [16,17] were made in the weaklyinteracting regime. Here we access the regime near a Feshbach resonance [5], where interactions are as strong as allowed by quantum mechanics (the unitarity limit). We measure spin transport properties, the spin drag coeffici...

From studies of exotic quantum many-body phenomena to applications in spintronics and quantum information processing, topological materials are poised to revolutionize the condensed-matter frontier and the landscape of modern materials science. Accordingly, there is a broad effort to realize topologically nontrivial electronic and photonic materials for fundamental science as well as practical applications. In this work, we demonstrate the first simultaneous site-and time-resolved measurements of a time-reversalinvariant topological band structure, which we realize in a radio-frequency photonic circuit. We control band-structure topology via local permutation of a traveling-wave capacitor-inductor network, increasing robustness by going beyond the tight-binding limit. We observe a gapped density of states consistent with a modified Hofstadter spectrum at a flux per plaquette of ϕ ¼ π=2. In situ probes of the band gaps reveal spatially localized bulk states and delocalized edge states. Time-resolved measurements reveal dynamical separation of localized edge excitations into spin-polarized currents. The radio-frequency circuit paradigm is naturally compatible with nonlocal coupling schemes, allowing us to implement a Möbius strip topology inaccessible in conventional systems. This room-temperature experiment illuminates the origins of topology in band structure, and when combined with circuit quantum electrodynamics techniques, it provides a direct path to topologically ordered quantum matter. [7], and 2DEGs [8,9]. In a condensed-matter context, such "topologically protected" properties include single-particle features of the band structure and many-particle ground-state degeneracies, with the latter typically emerging from the former in conjunction with strong interactions. To explore the nature of topologically derived material properties, it is desirable to develop materials that not only support conserved topological quantities but that may be precisely produced, manipulated, and probed. The aim, then, is to realize material test beds that marry favorable coherence properties, strong interactions, and topologically nontrivial single-particle dynamics.Metamaterials, where interaction strengths and length scales can be engineered, are a promising avenue for studying topological physics. Efforts are ongoing to produce the requisite topological single-particle dynamics in ultracold atomic gases [10][11][12][13][14][15][16], gyrotropic metamaterials [17,18], and photonic systems [17,[19][20][21][22][23][24][25][26].In cold atomic gases, gauge fields are generated either through spatially dependent Raman coupling of internal atomic states [10,14], or time-and space-periodic modulation of lattice tunneling rates [15,27,28]. In the optical domain, synthetic magnetic fields were realized via strain of a honeycomb lattice [29]. A Floquet topological insulator [30,31] was realized under a space-to-time mapping of an array of tunnel-coupled waveguides modulated along their propagation direction [21]. A photonic topologic...

Solitons-solitary waves that maintain their shape as they propagate-occur as water waves in narrow canals, as light pulses in optical fibres and as quantum mechanical matter waves in superfluids and superconductors. Their highly nonlinear and localized nature makes them very sensitive probes of the medium in which they propagate. Here we create long-lived solitons in a strongly interacting superfluid of fermionic atoms and directly observe their motion. As the interactions are tuned from the regime of Bose-Einstein condensation of tightly bound molecules towards the Bardeen-Cooper-Schrieffer limit of long-range Cooper pairs, the solitons' effective mass increases markedly, to more than 200 times their bare mass, signalling strong quantum fluctuations. This mass enhancement is more than 50 times larger than the theoretically predicted value. Our work provides a benchmark for theories of non-equilibrium dynamics of strongly interacting fermions.

Precise understanding of strongly interacting fermions, from electrons in modern materials to nuclear matter, presents a major goal in modern physics. However, the theoretical description of interacting Fermi systems is usually plagued by the intricate quantum statistics at play. Here we present a cross-validation between a new theoretical approach, Bold Diagrammatic Monte Carlo (BDMC), and precision experiments on ultra-cold atoms. Specifically, we compute and measure with unprecedented accuracy the normal-state equation of state of the unitary gas, a prototypical example of a strongly correlated fermionic system. Excellent agreement demonstrates that a series of Feynman diagrams can be controllably resummed in a non-perturbative regime using BDMC. This opens the door to the solution of some of the most challenging problems across many areas of physics.

We follow the evolution of fermion pairing in the dimensional crossover from three-dimensional to two-dimensional as a strongly interacting Fermi gas of ^{6}Li atoms becomes confined to a stack of two-dimensional layers formed by a one-dimensional optical lattice. Decreasing the dimensionality leads to the opening of a gap in radio-frequency spectra, even on the Bardeen-Cooper-Schrieffer side of a Feshbach resonance. The measured binding energy of fermion pairs closely follows the theoretical two-body binding energy and, in the two-dimensional limit, the zero-temperature mean-field Bose-Einstein-condensation to Bardeen-Cooper-Schrieffer crossover theory.