BCS Crossover Research Articles

Pseudogap Effects in the Strongly Correlated Regime of the Two-Dimensional Fermi Gas.

The two-species Fermi gas with attractive short-range interactions in two spatial dimensions provides a paradigmatic system for the understanding of strongly correlated Fermi superfluids in two dimensions. It is known to exhibit a BEC to BCS crossover as a function of ln(k_{F}a), where a is the scattering length, and to undergo a Berezinskii-Kosterlitz-Thouless superfluid transition below a critical temperature T_{c}. However, the extent of a pseudogap regime in the strongly correlated regime of ln(k_{F}a)∼1, in which pairing correlations persist above T_{c}, remains largely unexplored with controlled theoretical methods. Here, we use finite-temperature auxiliary-field quantum MonteCarlo methods on discrete lattices in the canonical ensemble formalism to calculate thermodynamical observables in the strongly correlated regime. We extrapolate to continuous time and the continuum limit to eliminate systematic errors and present results for particle numbers ranging from N=42 to N=162. We estimate T_{c} by a finite-size scaling analysis, and observe clear pseudogap signatures above T_{c} and below a temperature T^{*} in both the spin susceptibility and free-energy gap. We also present results for the contact, a fundamental thermodynamic property of quantum many-body systems with short-range interactions.

A quantum engine in the BEC–BCS crossover

Heat engines convert thermal energy into mechanical work both in the classical and quantum regimes1. However, quantum theory offers genuine non-classical forms of energy, different from heat, which so far have not been exploited in cyclic engines. Here we experimentally realize a quantum many-body engine fuelled by the energy difference between fermionic and bosonic ensembles of ultracold particles that follows from the Pauli exclusion principle2. We employ a harmonically trapped superfluid gas of 6Li atoms close to a magnetic Feshbach resonance3 that allows us to effectively change the quantum statistics from Bose–Einstein to Fermi–Dirac, by tuning the gas between a Bose–Einstein condensate of bosonic molecules and a unitary Fermi gas (and back) through a magnetic field4–10. The quantum nature of such a Pauli engine is revealed by contrasting it with an engine in the classical thermal regime and with a purely interaction-driven device. We obtain a work output of several 106 vibrational quanta per cycle with an efficiency of up to 25%. Our findings establish quantum statistics as a useful thermodynamic resource for work production.

Solution of the BEC to BCS Quench in One Dimension.

A gas of interacting fermions confined in a quasi one-dimensional geometry shows a BEC to BCS crossover upon slowly driving its coupling constant through a confinement-induced resonance. On one side of the crossover the fermions form tightly bound bosonic molecules behaving as a repulsive Bose gas, while on the other they form Cooper pairs, whose size is much larger than the average interparticle distance. Here we consider the situation arising when the coupling constant is varied suddenly from the BEC to the BCS value. Namely, we study a BEC-to-BCS quench. By exploiting a suitable continuum limit of recently discovered solvable quenches in the Hubbard model, we show that the local stationary state reached at large times after the quench can be determined exactly by means of the quench action approach. We provide an experimentally accessible characterization of the stationary state by computing local pair correlation function as well as the quasiparticle distribution functions. We find that the steady state is increasingly dominated by two-particle spin singlet bound states for stronger interaction strength, but that bound state formation is inhibited at larger BEC density. The bound state rapidity distribution displays quartic power-law decay suggesting a violation of Tan's contact relations.

A generalized molecule approach capturing the Feshbach-induced pairing physics in the BEC–BCS crossover

By including the effect of a trap with characteristic energy given by the Fermi temperature T F in a two-body two-channel model for Feshbach resonances, we reproduce the measured binding energy of ultracold molecules in a 40K atomic Fermi gas. We also reproduce the experimental closed-channel fraction Z across the BEC–BCS crossover and into the BCS regime of a 6Li atomic Fermi gas. We obtain the expected behavior at unitarity, together with the recently measured proportionality constant. Our results are also in agreement with recent measurements of the Z dependency on T F on the BCS side, where a significant quantitative discrepancy between experimental data and theory’s predictions has been repeatedly reported. In order to contrast with future experiments we report the proportionality constant at unitarity between Z and predicted by our model for a 40K atomic Fermi gas.

Dynamical structure factors of a two-dimensional Fermi superfluid within random phase approximation

Based on random phase approximation, we numerically calculate dynamical structure factors of a balanced two-dimensional (2D) Fermi superfluid, and discuss their energy, momentum and interaction strength dependence in the 2D BEC–BCS crossover. At a small transferred momentum, a stable Higgs mode is observed in the unitary 2D Fermi superfluid gas where the particle–hole symmetry is not satisfied. Stronger interaction strength will make the visibility of the dispersion of Higgs mode harder to be observed. We also discuss the dimension effect and find that the signal of the Higgs mode in two dimension is more obvious than that in 3D case. At a large transferred momentum regime, stronger interaction strength will induce the weight of the molecules excitation increasing, while in verse the atomic one decreasing, which shows the pairing information of Fermi superfluid. The theoretical results qualitatively agree with the corresponding quantum Monte Carlo data.

Order parameter fluctuations in the holographic superconductor

We investigate the effect of order parameter fluctuations in the holographic superconductor. In particular, following an introduction to the concept of intrinsic dynamics and its implementation within holographic models, we compute the intrinsic spectral functions of the order parameter in both the normal and the superconducting phase, using a fully backreacted bulk geometry. We also present a vector-like large-N version of the Ginzburg–Landau model that accurately describes our long-wavelength results in both phases. Our results indicate that the holographic superconductor describes a relativistic multi-component superfluid in the universal regime of the BEC–BCS crossover.

Anomalous dispersion and band gap reduction in UO2+x and its possible coupling to the coherent polaronic quantum state

Hypervalent UO2, UO2(+x) formed by both addition of excess O and photoexcitation, exhibits a number of unusual or often unique properties that point to it hosting a polaronic Bose–Einstein(-Mott) condensate. A more thorough analysis of the O X-ray absorption spectra of UO2, U4O9, and U3O7 shows that the anomalous increase in the width of the spectral features assigned to predominantly U 5f and 6d final states that points to increased dispersion of these bands occurs on the low energy side corresponding to the upper edge of the gap bordered by the conduction or upper Hubbard band. The closing of the gap by 1.5eV is more than twice as much as predicted by calculations, consistent with the dynamical polaron found by structural measurements. In addition to fostering the excitation that is the proposed mechanism for the coherence, the likely mirroring of this effect on the occupied, valence side of the gap below the Fermi level points to increased complexity of the electronic structure that could be associated with the Fermi topology of BEC–BCS crossover and two band superconductivity.

Fluctuation effects on the transport properties of unitary Fermi gases

In this Rapid Communication we investigate the fluctuation effects on the transport properties of unitary Fermi gases in the vicinity of the superfluid transition temperature ${T}_{\mathrm{c}}$. Based on the time-dependent Ginzburg-Landau formalism of the Bose-Einstein condensate--BCS crossover, we investigate both the residual resistivity below ${T}_{\mathrm{c}}$ induced by phase slips and the paraconductivity above ${T}_{\mathrm{c}}$ due to pair fluctuations. These two effects have been well studied in the weak-coupling BCS superconductor and here we generalize them to the unitary regime of ultracold Fermi gases. We find that while the residual resistivity below ${T}_{\mathrm{c}}$ increases as one approaches the unitary limit, consistent with recent experiments, the paraconductivity exhibits nonmonotonic behavior. Our results can be verified with the recently developed transport apparatus using mesoscopic channels.

Extension of the Ginzburg–Landau approach for ultracold Fermi gases below a critical temperature

In the context of superfluid Fermi gases, the Ginzburg–Landau (GL) formalism for the macroscopic wave function has been successfully extended to the whole temperature range where the superfluid state exists. After reviewing the formalism, we first investigate the temperature-dependent correction to the standard GL expansion (which is valid close to Tc). Deviations from the standard GL formalism are particularly important for the kinetic energy contribution to the GL energy functional, which in turn influences the healing length of the macroscopic wave function. We apply the formalism to variationally describe vortices in a strong-coupling Fermi gas in the BEC–BCS crossover regime, in a two-band system. The healing lengths, derived as variational parameters in the vortex wave function, are shown to exhibit hidden criticality well below Tc.

BCS–BEC CROSSOVER IN RELATIVISTIC FERMI SYSTEMS

We review the BCS–BEC crossover in relativistic Fermi systems, including the QCD matter at finite density. In the first part, we study the BCS–BEC crossover in a relativistic four-fermion interaction model and show how the relativistic effect affects the BCS–BEC crossover. In the second part, we investigate both two-color QCD at finite baryon density and pion superfluid at finite isospin density, by using an effective Nambu–Jona–Lasinio model. We will show how the model describes the weakly interacting diquark and pion condensates at low density and the BEC–BCS crossover at high density.

Snake instability of dark solitons in fermionic superfluids

We present numerical calculations of the snake instability in a Fermi superfluid within the Bogoliubov-de Gennes theory of the BEC to BCS crossover using the random phase approximation complemented by time-dependent simulations. We examine the snaking behaviour across the crossover and quantify the timescale and lengthscale of the instability. While the dynamic shows extensive snaking before eventually producing vortices and sound on the BEC side of the crossover, the snaking dynamics is preempted by decay into sound due to pair breaking in the deep BCS regime. At the unitarity limit, hydrodynamic arguments allow us to link the rate of snaking to the experimentally observable ratio of inertial to physical mass of the soliton. In this limit we witness an unresolved discrepancy between our numerical estimates for the critical wavenumber of suppression of the snake instability and recent experimental observations with an ultra-cold Fermi gas.

Competition between BCS-pairing and “moth-eaten effect” in BEC–BCS crossover

We study the change in condensation energy from a single pair of fermionic atoms to a large number of pairs interacting via the reduced BCS potential. We find that the energy-saving due to correlations decreases when the pair number increases because the number of empty states available for pairing gets smaller (“moth-eaten effect”). However, this decrease dominates the 3D kinetic energy increase of the same amount of noninteracting atoms only when the pair number is a sizable fraction of the number of states available for pairing. As a result, in BEC–BCS crossover of 3D systems, the condensation energy per pair first increases and then decreases with pair number while in 2D, it always is controlled by the “moth-eaten effect” and thus simply decreases.

BEC–BCS crossover in a ( [formula omitted])-wave pairing Hamiltonian coupled to bosonic molecular pairs

We analyse a ( p + i p )-wave pairing BCS Hamiltonian, coupled to a single bosonic degree of freedom representing a molecular condensate, and investigate the nature of the BEC–BCS crossover for this system. For a suitable restriction on the coupling parameters, we show that the model is integrable and we derive the exact solution by the algebraic Bethe ansatz. In this manner we also obtain explicit formulae for correlation functions and compute these for several cases. We find that the crossover between the BEC state and the strong pairing p + i p phase is smooth for this model, with no intermediate quantum phase transition.

Resonant superfluidity in an optical lattice

We have studied a system of ultracold fermionic potassium (40K) atoms in a three-dimensional (3D) optical lattice in the vicinity of an s-wave Feshbach resonance. Close to resonance, the system is described by a multi-band Bose–Fermi Hubbard Hamiltonian. We derive an effective lowest-band Hamiltonian in which the effect of higher bands is incorporated by a self-consistent mean-field approximation. The resulting model is solved by means of generalized dynamical mean-field theory. In addition to the BEC–BCS crossover, we find a phase transition to a fermionic Mott insulator at half-filling, induced by the repulsive fermionic background scattering length. We also calculate the critical temperature of the BEC/BCS state and find it to be minimal at resonance.

Superfluid Fermi Gas in Optical Lattices: Self-Trapping, Stable, Moving Solitons and Breathers

We predict the existence of self-trapping, stable, moving solitons and breathers of Fermi wave packets along the Bose-Einstein condensation (BEC)-BCS crossover in one dimension (1D), 2D, and 3D optical lattices. The dynamical phase diagrams for self-trapping, solitons, and breathers of the Fermi matter waves along the BEC-BCS crossover are presented analytically and verified numerically by directly solving a discrete nonlinear Schrödinger equation. We find that the phase diagrams vary greatly along the BEC-BCS crossover; the dynamics of Fermi wave packet are different from that of Bose wave packet.

The axial anomaly and the phases of dense QCD

The QCD axial anomaly, by coupling the chiral condensate and BCS pairing fields of quarks in dense matter, leads to a new critical point in the QCD phase diagram [1, 2], which at sufficiently low temperature may terminate the line of phase transitions between chirally broken hadronic matter and color superconducting quark matter. The critical point indicates that matter at low temperature should cross over smoothly from the hadronic to the quark phase, as suggested earlier on the basis of symmetry. We review here the arguments, based on a general Ginzburg–Landau effective Lagrangian, for the existence of the new critical point as well as discuss possible connections between the QCD phase structure and the BEC–BCS crossover in ultracold trapped atomic fermion systems at unitarity and implications for the presence of quark matter in neutron stars.

Fluctuations of the number of particles within a given volume in cold quantum gases

In ultracold gases many experiments use atom imaging as a basic observable. The resulting image is averaged over a number of realizations and mostly only this average is used. Only recently the noise has been measured to extract physical information. In the present paper we investigate the quantum noise arising in these gases at zero temperature. We restrict ourselves to the homogeneous situation and study the fluctuations in particle number found within a given volume in the gas, and more specifically inside a sphere of radius $R$. We show that zero-temperature fluctuations are not extensive and the leading term scales with sphere radius $R$ as ${R}^{2}\text{ }\text{ln}\text{ }R$ (or $\text{ln}\text{ }R$) in three- (or one-) dimensional systems. We calculate systematically the next term beyond this leading order. We consider first the generic case of a compressible superfluid. Then we investigate the whole Bose-Einstein-condensation (BEC) --BCS crossover, and in particular the limiting cases of the weakly interacting Bose gas and of the free Fermi gas.

Optical spectra and excitonic BEC–BCS crossover in coherent electron–hole pair condensation

We investigate coherent pair condensation in three-dimensional electron–hole (e–h) systems using the two-band Hubbard model with both repulsion and attraction, especially from the viewpoint of optical properties. Applying the self-consistent t-matrix and local approximations, the crossover behavior between the e–h BCS-like state and the excitonic Bose–Einstein condensation (BEC) at finite temperatures is described as a function of the e–h attraction strength. The changes of absorption and luminescence spectra are also discussed along the excitonic BEC–BCS crossover.

Renormalization flow and universality for ultracold fermionic atoms

A functional renormalization-group study for the Bose-Einstein condensate (BEC)-BCS crossover for ultracold gases of fermionic atoms is presented. We discuss the fixed point which is at the origin of universality for broad Feshbach resonances. All macroscopic quantities depend only on one relevant parameter, the concentration ak{sub F}, besides their dependence on the temperature in units of the Fermi energy. In particular, we compute the universal ratio between molecular and atomic scattering length in vacuum. We also present an estimate to which level of accuracy universality holds for gases of Li and K atoms.

Dynamic structure factor of a Fermi superfluid in the BEC-BCS crossover

We consider cigar-shaped Fermi superfluid in the Bose-Einstein condensation (BEC)-BCS crossover. Using the polytropic form of equation of state, we derive low energy multibranch bosonic excitations and the corresponding density fluctuations in three different regimes along the crossover, namely weak-coupling BCS, unitarity, and molecular BEC regimes. Bragg spectroscopy can be used to probe the multibranch nature of the low-energy bosonic excitations by measuring the dynamic structure factor. Therefore we calculate the dynamic structure factor in those three different regimes. In Bragg spectroscopy, an actual observable is momentum imparted to the superfluid due to the Bragg potential. We also present results of the momentum imparted to the superfluid due to the Bragg pulses.