Determination of the superfluid gap in atomic Fermi gases by quasiparticle spectroscopy

Superfluid Signatures in a Dissipative Quantum Point Contact

We measure superfluid transport of strongly interacting fermionic lithium atoms through a quantum point contact with local, spin-dependent particle loss. We observe that the characteristic non-Ohmic superfluid transport enabled by high-order multiple Andreev reflections transitions into an excess Ohmic current as the dissipation strength exceeds the superfluid gap. We develop a model with mean-field reservoirs connected via tunneling to a dissipative site. Our calculations in the Keldysh formalism reproduce the observed nonequilibrium particle current, yet do not fully explain the observed loss rate or spin current.

Observing the Influence of Reduced Dimensionality on Fermionic Superfluids

Understanding the origins of unconventional superconductivity has been a major focus of condensed matter physics for many decades. While many questions remain unanswered, experiments have found the highest critical temperatures in layered two-dimensional materials. However, to what extent the remarkable stability of these strongly correlated 2D superfluids is affected by their reduced dimensionality is still an open question. Here, we use dilute gases of ultracold fermionic atoms as a model system to directly observe the influence of dimensionality on the stability of strongly interacting fermionic superfluids. We find that the superfluid gap follows the same universal function of the interaction strength regardless of dimensionality, which suggests that there is no inherent difference in the stability of two- and three-dimensional fermionic superfluids. Finally, we compare our data to results from solid state systems and find a similar relation between the interaction strength and the gap for a wide range of two- and three-dimensional superconductors.

Magic Gap Ratio for Optimally Robust Fermionic Condensation and Its Implications for High-T_{c} Superconductivity

Bardeen-Schrieffer-Cooper (BCS) and Bose-Einstein condensation (BEC) occur at opposite limits of a continuum of pairing interaction strength between fermions. A crossover between these limits is readily observed in a cold atomic Fermi gas. Whether it occurs in other systems such as the high temperature superconducting cuprates has remained an open question. We uncover here unambiguous evidence for a BCS-BEC crossover in the cuprates by identifying a universal magic gap ratio 2Δ/k_{B}T_{c}≈6.5 (where Δ is the pairing gap and T_{c} is the transition temperature) at which paired fermion condensates become optimally robust. At this gap ratio, corresponding to the unitary point in a cold atomic Fermi gas, the measured condensate fraction N_{0} and the height of the jump δγ(T_{c}) in the coefficient γ of the fermionic specific heat at T_{c} are strongly peaked. In the cuprates, δγ(T_{c}) is peaked at this gap ratio when Δ corresponds to the antinodal spectroscopic gap, thus reinforcing its interpretation as the pairing gap. We find the peak in δγ(T_{c}) also to coincide with a normal state maximum in γ, which is indicative of a pairing fluctuation pseudogap above T_{c}.

Excitation Spectrum and Superfluid Gap of an Ultracold Fermi Gas

Ultracold atomic gases are a powerful tool to experimentally study strongly correlated quantum many-body systems. In particular, ultracold Fermi gases with tunable interactions have allowed to realize the famous BEC-BCS crossover from a Bose-Einstein condensate (BEC) of molecules to a Bardeen-Cooper-Schrieffer (BCS) superfluid of weakly bound Cooper pairs. However, large parts of the excitation spectrum of fermionic superfluids in the BEC-BCS crossover are still unexplored. In this work, we use Bragg spectroscopy to measure the full momentum-resolved low-energy excitation spectrum of strongly interacting ultracold Fermi gases. This enables us to directly observe the smooth transformation from a bosonic to a fermionic superfluid that takes place in the BEC-BCS crossover. We also use our spectra to determine the evolution of the superfluid gap and find excellent agreement with previous experiments and self-consistent T-matrix calculations both in the BEC and crossover regime. However, toward the BCS regime a calculation that includes the effects of particle-hole correlations shows better agreement with our data.

Universal sound diffusion in a strongly interacting Fermi gas

Transport of strongly interacting fermions is crucial for the properties of modern materials, nuclear fission, the merging of neutron stars, and the expansion of the early Universe. Here, we observe a universal quantum limit of diffusivity in a homogeneous, strongly interacting atomic Fermi gas by studying sound propagation and its attenuation through the coupled transport of momentum and heat. In the normal state, the sound diffusivity D monotonically decreases upon lowering the temperature, in contrast to the diverging behavior of weakly interacting Fermi liquids. Below the superfluid transition temperature, D attains a universal value set by the ratio of Planck's constant and the particle mass. Our findings inform theories of fermion transport, with relevance for hydrodynamic flow of electrons, neutrons, and quarks.

Emergence of a Pseudogap in the BCS-BEC Crossover

Strongly correlated Fermi systems with pairing interactions become superfluid below a critical temperature T_{c}. The extent to which such pairing correlations alter the behavior of the liquid at temperatures T>T_{c} is a subtle issue that remains an area of debate, in particular regarding the appearance of the so-called pseudogap in the BCS-BEC crossover of unpolarized spin-1/2 nonrelativistic matter. To shed light on this, we extract several quantities of crucial importance at and around the unitary limit, namely, the odd-even staggering of the total energy, the spin susceptibility, the pairing correlation function, the condensate fraction, and the critical temperature T_{c}, using a nonperturbative, constrained-ensemble quantum Monte Carlo algorithm.

Linear response of a superfluid Fermi gas inside its pair-breaking continuum

We study the signatures of the collective modes of a superfluid Fermi gas in its linear response functions for the order-parameter and density fluctuations in the Random Phase Approximation (RPA). We show that a resonance associated to the Popov-Andrianov (or sometimes "Higgs") mode is visible inside the pair-breaking continuum at all values of the wavevector q, not only in the (order-parameter) modulus-modulus response function but also in the modulus-density and density-density responses. At nonzero temperature, the resonance survives in the presence of thermally broken pairs even until the vicinity of the critical temperature Tc, and coexists with both the Anderson-Bogoliubov modes at temperatures comparable to the gap Δ and with the low-velocity phononic mode predicted by RPA near Tc. The existence of a Popov-Andrianov-"Higgs" resonance is thus a robust, generic feature of the high-energy phenomenology of pair-condensed Fermi gases, and should be accessible to state-of-the-art cold atom experiments.

Pairing Correlations across the Superfluid Phase Transition in the Unitary Fermi Gas

In the two-component Fermi gas with a contact interaction, a pseudogap regime can exist at temperatures between the superfluid critical temperature T_{c} and a temperature T^{*}>T_{c}. This regime is characterized by pairing correlations without superfluidity. However, in the unitary limit of infinite scattering length, the existence of this regime is still debated. To help address this, we have applied finite-temperature auxiliary-field quantum Monte Carlo (AFMC) methods to study the thermodynamics of the superfluid phase transition and signatures of the pseudogap in the spin-balanced homogeneous unitary Fermi gas. We present results at finite filling factor ν≃0.06 for the condensate fraction, an energy-staggering pairing gap, the spin susceptibility, and the heat capacity, and compare them to experimental data when available. In contrast to previous AFMC simulations, our model space consists of the complete first Brillouin zone of the lattice, and our calculations are performed in the canonical ensemble of fixed particle number. The canonical ensemble AFMC framework enables the calculation of a model-independent gap, providing direct information on pairing correlations without the need for numerical analytic continuation. We use finite-size scaling to estimate T_{c} at the corresponding filling factor. We find that the energy-staggering pairing gap vanishes above T_{c}, showing no pseudogap effects, and that the spin susceptibility shows a substantially reduced signature of a spin gap compared to previously reported AFMC simulations.

Beyond Mean-Field Corrections to the Quasiparticle Spectrum of Superfluid Fermi Gases

We investigate the fermionic quasiparticle branch of superfluid Fermi gases in the Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensation (BEC) crossover and calculate the quasiparticle lifetime and energy shift due to its coupling with the collective mode. The only close-to-resonance process that low-energy quasiparticles can undergo at zero temperature is the emission of a bosonic excitation from the phononic branch. Close to the minimum of the branch we find that the quasiparticles remain undamped, allowing us to compute corrections to experimentally relevant quantities such as the energy gap, location of the minimum, effective mass, and Landau critical velocity.

Spectral Response and Contact of the Unitary Fermi Gas

We measure radio frequency (rf) spectra of the homogeneous unitary Fermi gas at temperatures ranging from the Boltzmann regime through quantum degeneracy and across the superfluid transition. For all temperatures, a single spectral peak is observed. Its position smoothly evolves from the bare atomic resonance in the Boltzmann regime to a frequency corresponding to nearly one Fermi energy at the lowest temperatures. At high temperatures, the peak width reflects the scattering rate of the atoms, while at low temperatures, the width is set by the size of fermion pairs. Above the superfluid transition, and approaching the quantum critical regime, the width increases linearly with temperature, indicating non-Fermi-liquid behavior. From the wings of the rf spectra, we obtain the contact, quantifying the strength of short-range pair correlations. We find that the contact rapidly increases as the gas is cooled below the superfluid transition.

High-Sensitivity rf Spectroscopy of a Strongly Interacting Fermi Gas

rf spectroscopy is one of the most powerful probing techniques in the field of ultracold gases. We report on a novel rf spectroscopy scheme with which we can detect very weak signals of only a few atoms. Using this method, we extended the experimentally accessible photon-energies range by an order of magnitude compared to previous studies. We directly verify a universal property of fermions with short-range interactions which is a power-law scaling of the rf spectrum tail all the way up to the interaction scale. We also determine, with high precision, the trap average contact parameter for different interaction strength. Finally, we employ our technique to precisely measure the binding energy of Feshbach molecules in an extended range of magnetic fields. These data are used to extract a new calibration of the Feshbach resonance between the two lowest energy levels of ^{40}K.

Landau Phonon-Roton Theory Revisited for Superfluid ^{4}He and Fermi Gases

Liquid helium and spin-1/2 cold-atom Fermi gases both exhibit in their superfluid phase two distinct types of excitations, gapless phonons and gapped rotons or fermionic pair-breaking excitations. In the long wavelength limit, revising and extending the theory of Landau and Khalatnikov initially developed for helium [Zh. Exp. Teor. Fiz. 19, 637 (1949)], we obtain universal expressions for three- and four-body couplings among these two types of excitations. We calculate the corresponding phonon damping rates at low temperature and compare them to those of a pure phononic origin in high-pressure liquid helium and in strongly interacting Fermi gases, paving the way to experimental observations.

Review of pseudogaps in strongly interacting Fermi gases

A central challenge in modern condensed matter physics is developing the tools for understanding nontrivial yet unordered states of matter. One important idea to emerge in this context is that of a 'pseudogap': the fact that under appropriate circumstances the normal state displays a suppression of the single particle spectral density near the Fermi level, reminiscent of the gaps seen in ordered states of matter. While these concepts arose in a solid state context, they are now being explored in cold gases. This article reviews the current experimental and theoretical understanding of the normal state of strongly interacting Fermi gases, with particular focus on the phenomonology which is traditionally associated with the pseudogap.

Two-terminal transport measurements with cold atoms

In recent years, the ability of cold atom experiments to explore condensed-matter-related questions has dramatically progressed. Transport experiments, in particular, have expanded to the point in which conductance and other transport coefficients can now be measured in a way that is directly analogous to solid-state physics, extending cold-atom-based quantum simulations into the domain of quantum electronic devices. In this topical review, we describe the transport experiments performed with cold gases in the two-terminal configuration, with an emphasis on the specific features of cold atomic gases compared to solid-state physics. We present the experimental techniques and the main experimental findings, focusing on-but not restricted to-the recent experiments performed by our group. We finally discuss the perspectives opened up by this approach, the main technical and conceptual challenges for future developments, and potential applications in quantum simulation for transport phenomena and mesoscopic physics problems.

Tuning across the BCS-BEC crossover in the multiband superconductor Fe1+y Se x Te1-x : An angle-resolved photoemission study

The crossover from Bardeen-Cooper-Schrieffer (BCS) superconductivity to Bose-Einstein condensation (BEC) is difficult to realize in quantum materials because, unlike in ultracold atoms, one cannot tune the pairing interaction. We realize the BCS-BEC crossover in a nearly compensated semimetal, Fe_1+y Se _x Te_1-x , by tuning the Fermi energy ε_F via chemical doping, which permits us to systematically change Δ/ε_F from 0.16 to 0.50, where Δ is the superconducting (SC) gap. We use angle-resolved photoemission spectroscopy to measure the Fermi energy, the SC gap, and characteristic changes in the SC state electronic dispersion as the system evolves from a BCS to a BEC regime. Our results raise important questions about the crossover in multiband superconductors, which go beyond those addressed in the context of cold atoms.

Homogeneous Atomic Fermi Gases

We report on the creation of homogeneous Fermi gases of ultracold atoms in a uniform potential. In the momentum distribution of a spin-polarized gas, we observe the emergence of the Fermi surface and the saturated occupation of one particle per momentum state: the striking consequence of Pauli blocking in momentum space for a degenerate gas. Cooling a spin-balanced Fermi gas at unitarity, we create homogeneous superfluids and observe spatially uniform pair condensates. For thermodynamic measurements, we introduce a hybrid potential that is harmonic in one dimension and uniform in the other two. The spatially resolved compressibility reveals the superfluid transition in a spin-balanced Fermi gas, saturation in a fully polarized Fermi gas, and strong attraction in the polaronic regime of a partially polarized Fermi gas.

Momentum-resolved spectroscopy of a Fermi liquid

We consider a recent momentum-resolved radio-frequency spectroscopy experiment, in which Fermi liquid properties of a strongly interacting atomic Fermi gas were studied. Here we show that by extending the Brueckner-Goldstone model, we can formulate a theory that goes beyond basic mean-field theories and that can be used for studying spectroscopies of dilute atomic gases in the strongly interacting regime. The model hosts well-defined quasiparticles and works across a wide range of temperatures and interaction strengths. The theory provides excellent qualitative agreement with the experiment. Comparing the predictions of the present theory with the mean-field Bardeen-Cooper-Schrieffer theory yields insights into the role of pair correlations, Tan's contact, and the Hartree mean-field energy shift.

Critical velocity in the BEC-BCS crossover

We map out the critical velocity in the crossover from Bose-Einstein condensation to Bardeen-Cooper-Schrieffer superfluidity with ultracold ^{6}Li gases. A small attractive potential is dragged along lines of constant column density. The rate of the induced heating increases steeply above a critical velocity v_{c}. In the same samples, we measure the speed of sound v_{s} by exciting density waves and compare the results to the measured values of v_{c}. We perform numerical simulations in the Bose-Einstein condensation regime and find very good agreement, validating the approach. In the strongly correlated regime our measurements of v_{c} provide a testing ground for theoretical approaches.

Local observation of pair condensation in a Fermi gas at unitarity

We present measurements of the local (homogeneous) density-density response function of a Fermi gas at unitarity using spatially resolved Bragg spectroscopy. By analyzing the Bragg response across one axis of the cloud, we extract the response function for a uniform gas which shows a clear signature of the Bose-Einstein condensation of pairs of fermions when the local temperature drops below the superfluid transition temperature. The method we use for local measurement generalizes a scheme for obtaining the local pressure in a harmonically trapped cloud from the line density and can be adapted to provide any homogeneous parameter satisfying the local density approximation.

Evolution of fermion pairing from three to two dimensions

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.

Onset of a pseudogap regime in ultracold Fermi gases

We show, using an ab initio approach based on Quantum Monte Carlo technique, that the pseudogap regime emerges in ultracold Fermi gases close to the unitary point. We locate the onset of this regime at a value of the interaction strength corresponding to (k(F)a)(-1)≈-0.05 (a-scattering length). We determine the evolution of the gap as a function of temperature and interaction strength in the Fermi gas around the unitary limit and show that our results exhibit a remarkable agreement with the recent wave-vector resolved radio frequency spectroscopy data. Our results indicate that a finite temperature structure of the Fermi gas around unitarity is more complicated and involves the presence of the phase with preformed Cooper pairs, which, however, do not contribute to the long range order.

Resonantly interacting fermions in a box

We use two fundamental theoretical frameworks to study the finite-size (shell) properties of the unitary gas in a periodic box: (1) an ab initio quantum Monte Carlo (QMC) calculation for boxes containing 4 to 130 particles provides a precise and complete characterization of the finite-size behavior, and (2) a new density functional theory (DFT) fully encapsulates these effects. The DFT predicts vanishing shell structure for systems comprising more than 50 particles, and allows us to extrapolate the QMC results to the thermodynamic limit, providing the tightest bound to date on the ground-state energy of the unitary gas: ξ(S)≤0.383(1). We also apply the new functional to few-particle harmonically trapped systems, comparing with previous calculations.

Probing non-abelian statistics of Majorana fermions in ultracold atomic superfluid

We propose an experiment to directly probe the non-abelian statistics of Majorana fermions by braiding them in an s-wave superfluid of ultracold atoms. We show that different orders of braiding operations give orthogonal output states that can be distinguished through Raman spectroscopy. Realization of Majorana states in an s-wave superfluid requires strong spin-orbital coupling and a controllable Zeeman field in the perpendicular direction. We present a simple laser configuration to generate the artificial spin-orbital coupling and the required Zeeman field in the dark-state subspace.

Finite-temperature pairing gap of a unitary fermi gas by quantum monte carlo calculations

We calculate the one-body temperature Green's (Matsubara) function of the unitary Fermi gas via quantum Monte Carlo, and extract the spectral weight function A(p,omega) using the methods of maximum entropy and singular value decomposition. From A(p,omega) we determine the quasiparticle spectrum, which can be accurately parametrized by three functions of temperature: an effective mass m{*}, a mean-field potential U, and a gap Delta. Below the critical temperature T{c}=0.15 epsilon{F} the results for m{*}, U, and Delta can be accurately reproduced using an independent quasiparticle model. We find evidence of a pseudogap in the fermionic excitation spectrum for temperatures up to T{*} approximately 0.20 epsilon{F}> T{c}.

Heavy-light fermion mixtures at unitarity

We investigate fermion pairing in the unitary regime for a mass ratio corresponding to a ;{6}Li-;{40}K mixture using quantum Monte Carlo methods. The ground-state energy and the average light- and heavy-particle excitation spectrum for the unpolarized superfluid state are nearly independent of the mass ratio. In the majority light system, the polarized superfluid is close to the energy of a phase separated mixture of nearly fully polarized normal and unpolarized superfluid. For a majority of heavy particles, we find an energy minimum for a normal state with a ratio of approximately 3ratio1 heavy to light particles. A slight increase in attraction to k_{F}a approximately 2.5 yields a ground state energy of nearly zero for this ratio. A cold unpolarized system in a harmonic trap at unitarity should phase separate into three regions, with a shell of unpolarized superfluid in the middle.

Observation of Fermi polarons in a tunable Fermi liquid of ultracold atoms

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.

Fermi condensates for dynamic imaging of electromagnetic fields

Ultracold gases provide micrometer size samples whose sensitivity to external fields may be exploited in sensor applications. Bose-Einstein condensates of atomic gases have been demonstrated to perform excellently as magnetic field sensors in atom chips. Here we propose that condensates of fermions can be used for noninvasive sensing of time-dependent and static magnetic and electric fields, by utilizing the tunable energy gap in the excitation spectrum as a frequency filter. Perturbations by the field create collective excitations and quasiparticles. The latter requires the frequency of the perturbation to exceed the gap. The frequencies of the field may be selectively monitored from the amount of quasiparticles which is measurable, e.g., by rf spectroscopy. We analyze the method by calculating the density-density susceptibility and discuss its sensitivity and spatial resolution.

Superfluidity in a gas of strongly interacting fermions

After an introduction into 100 years of research on superfluidity and the concept of the BCS-BEC crossover, we describe recent experimental studies of a spin-polarized Fermi gas with strong interactions. Tomographically resolving the spatial structure of an inhomogeneous trapped sample, we have mapped out the superfluid phases in the parameter space of temperature, spin polarization, and interaction strength. Phase separation between the superfluid and the normal component occurs at low temperatures, showing spatial discontinuities in the spin polarization. The critical polarization of the normal gas increases with stronger coupling. Beyond a critical interaction strength all minority atoms pair with majority atoms, and the system can be effectively described as a boson-fermion mixture. Pairing correlations have been studied by rf spectroscopy, determining the fermion pair size and the pairing gap energy in a resonantly interacting superfluid.

Bragg spectroscopy of a strongly interacting Fermi gas

We present a comprehensive study of the Bose-Einstein condensate to Bardeen-Cooper-Schrieffer (BEC-BCS) crossover in fermionic 6Li using Bragg spectroscopy. A smooth transition from molecular to atomic spectra is observed with a clear signature of pairing at and above unitarity. These spectra probe the dynamic and static structure factors of the gas and provide a direct link to two-body correlations. We have characterized these correlations and measured their density dependence across the broad Feshbach resonance at 834 G.