Bose-Einstein Condensation Regime Research Articles

BCS-BEC Crossover in a Two-Dimensional Fermi Gas

We investigate the crossover from Bardeen-Cooper-Schrieffer (BCS) superfluidity to Bose-Einstein condensation (BEC) in a two-dimensional Fermi gas at T=0 using the fixed-node diffusion MonteCarlo method. We calculate the equation of state and the gap parameter as a function of the interaction strength, observing large deviations compared to mean-field predictions. In the BEC regime our results show the important role of dimer-dimer and atom-dimer interaction effects that are completely neglected in the mean-field picture. Results on Tan's contact parameter associated with short-range physics are also reported along the BCS-BEC crossover.

Cooper pair insulators and theory of correlated superconductors

The pseudogap state of cuprate high-temperature superconductors has been often viewed as either a yet unknown competing order or a precursor state to superconductivity. While awaiting the resolution of the pseudogap problem in cuprates, we demonstrate that local pairing fluctuations, vortex liquid dynamics, and other precursor phenomena can emerge quite generally whenever fermionic excitations remain gapped across the superconducting transition, regardless of the gap origin. Our choice of a tractable model is a lattice band insulator with short-range attractive interactions between fermions in the $s$-wave channel. An effective crossover between Bardeen-Cooper-Schrieffer (BCS) and Bose-Einstein condensate (BEC) regimes can be identified in any band insulator above two dimensions, while in two dimensions only the BEC regime exists. The superconducting transition is ``unconventional'' (non-pair-breaking) in the BEC regime, identified by either the bosonic mean field or $\mathit{XY}$ universality class. The insulator adjacent to the superconductor in the BEC regime is a bosonic Mott insulator of Cooper pairs, which may be susceptible to charge density wave ordering. We construct a function of the many-body excitation spectrum whose nonanalytic changes define a sharp distinction between band and Mott insulators. The corresponding ``second-order transition'' can be observed out of equilibrium by driving a Cooper pair laser in the Mott insulator. We explicitly show that the gap for charged bosonic excitations lies below the threshold for Cooper pair breakup in any BEC regime, despite quantum fluctuations. Our discussion ends with a view of possible consequences for cuprates, where antinodal pair dynamics has certain features in common with our simple $s$-wave picture.

Dc Josephson effect with Fermi gases in the Bose-Einstein regime

We show that the dc Josephson effect with ultracold fermionic gases in the Bose-Einstein-condensate (BEC) regime of composite molecules can be described by a nonlinear Schr\odinger equation (NLSE). By comparing our results with Bogoliubov--de Gennes calculations [Phys. Rev. Lett. 99, 040401 (2007)] we find that our superfluid NLSE, which generalizes the Gross-Pitaevskii equation taking into account the correct equation of state, is reliable in the BEC regime of the BCS-BEC crossover up to the limit of very large (positive) scattering length. We also predict that the Josephson current displays relevant beyond-mean-field effects.

BCS-BEC crossover and phase structure of relativistic systems: A variational approach

We investigate here the BCS-BEC crossover in relativistic systems using a variational construct for the ground state and the minimization of the thermodynamic potential. This is first studied in a four-fermion point interaction model and with a BCS type ansatz for the ground state with fermion pairs. It is shown that the antiparticle degrees of freedom play an important role in the BCS-BEC crossover physics, even when the ratio of Fermi momentum to the mass of the fermion is small. We also consider the phase structure for the case of fermion pairing with imbalanced populations. Within the ansatz, thermodynamically stable gapless modes for both fermions and antifermions are seen for strong coupling in the Bose-Einstein condensation (BEC) regime. We further investigate the effect of fluctuations of the condensate field by treating it as a dynamical field and generalize the BCS ansatz to include quanta of the condensate field also in a boson-fermion model with quartic self-interaction of the condensate field. It is seen that the critical temperature decreases with inclusion of fluctuations.

Superfluid equation of state of cold fermionic gases in the Bose-Einstein regime

We present an exact many-body theory of ultracold fermionic gases for the Bose-Einstein condensation (BEC) regime of the BEC-BCS crossover. This is a purely fermionic approach which treats explicitly and systematically the dimers formed in the BEC regime as made of two fermions. We consider specifically the zero temperature case and calculate the first terms of the expansion of the chemical potential in powers of the density $n$. We derive first the mean-field contribution, which has the expected standard expression when it is written in terms of the dimer-dimer scattering length ${a}_{M}$. We go next in the expansion to the Lee-Huang-Yang order, proportional to ${n}^{3∕2}$. We find the far less obvious result that it retains also the same expression in terms of ${a}_{M}$ as for elementary bosons. The composite nature of the dimers appears only in the next term proportional to ${n}^{2}$.

Role of the Majorana Fermion and the Edge Mode in Chiral Superfluidity near ap-Wave Feshbach Resonance

The visualization of chiral p-wave superfluidity in Fermi gases near p-wave Feshbach resonances is theoretically examined. It is proposed that the superfluidity becomes detectable in the entire BCS-BEC regimes through (i) vortex visualization by the density depletion inside the vortex core and (ii) intrinsic angular momentum in vortex-free states. It is revealed that both (i) and (ii) are closely connected with the Majorana zero energy mode of the vortex core and the edge mode, which survive until the strong coupling BCS regime is approached from the weak coupling limit and vanish in the Bose-Einstein condensation regime.

BCS-BEC crossover in a relativistic boson-fermion model beyond mean field approximation

We investigate the fluctuation effect of the di-fermion field in the crossover from Bardeen-Cooper-Schrieffer (BCS) pairing to a Bose-Einstein condensate (BEC) in a relativistic superfluid. We work within the boson-fermion model obeying a global U(1) symmetry. To go beyond the mean field approximation we use Cornwall-Jackiw-Tomboulis formalism to include higher-order contributions. The quantum fluctuations of the pairing condensate is provided by bosons in nonzero modes, whose interaction with fermions gives the two-particle-irreducible effective potential. It changes the crossover property in the BEC regime. With the fluctuations the superfluidity phase transition becomes the first order in a grand canonical ensemble. We calculate the condensate, the critical temperature ${T}_{c}$ and particle abundances as functions of a crossover parameter of the boson mass.

BCS-BEC crossover of a quasi-two-dimensional Fermi gas: The significance of dressed molecules

We study the crossover of a quasi-two-dimensional Fermi gas trapped in the radial plane from the Bardeen-Cooper-Schrieffer (BCS) regime to the Bose-Einstein condensation (BEC) regime by crossing a wide Feshbach resonance. We consider two effective two-dimensional Hamiltonians within the mean-field level, and calculate the zero-temperature cloud size and number density distribution. For a model 1 Hamiltonian with renormalized atom-atom interaction, we observe a constant cloud size for arbitrary detunings. For a model 2 Hamiltonian with renormalized interactions between atoms and dressed molecules, the cloud size deceases from the BCS to BEC side, which is consistent with the picture of BCS-BEC crossover. This qualitative discrepancy between the two models indicates that the inclusion of dressed molecules is essential for a mean-field description of quasi-two-dimensional Fermi systems, especially on the BEC side of the Feshbach resonance.

Asymmetric Fermi superfluid with different atomic species in a harmonic trap

We study the dilute fermion gas with pairing between two species and unequal concentrations in a harmonic trap using the mean-field theory and the local density approximation. We found that the system can exhibit a superfluid shell structure sandwiched by the normal fermions. This superfluid shell structure occurs if the mass ratio is larger than a certain critical value, which increases from the weak-coupling BCS region to the strong-coupling Bose-Einstein condensation (BEC) side. In the strong-coupling BEC regime, the radii of the superfluid phase are less sensitive to the mass ratios and are similar to the case of pairing with equal masses. However, the lighter leftover fermions are easier to mix with the superfluid core than the heavier ones. A partially polarized superfluid can be found if the majority fermions are lighter, whereas phase separation is still found if they are heavier.

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.

Mixtures of ultracold fermions with unequal masses

We analyze the phase diagram of superfluidity for two-species fermion mixtures from the Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensation (BEC) limit as a function of scattering parameter, population imbalance, and mass anisotropy. We identify regions corresponding to normal, or uniform or nonuniform superfluid phases, and discuss topological quantum phase transitions in the BCS, unitarity, and BEC limits. We derive the Ginzburg-Landau equation near the critical temperature, and show that it describes a dilute mixture of paired and unpaired fermions in the BEC limit. We also obtain the zero temperature low frequency and long wavelength collective excitation spectrum, and recover the Bogoliubov relation for weakly interacting dilute bosons in the BEC limit. Lastly, we discuss the effects of harmonic traps and the resulting density profiles in the BEC regime.

Derivation of Hydrodynamics for the Gapless Mode in the BEC-BCS Crossover from the Exact One-Loop Effective Action

We derive generalized two-superfluid continuity equations for the BEC-BCS crossover in the presence of a Feshbach resonance at T=0. In addition, we calculate the velocity of sound throughout both BCS and Bose-Einstein condensation (BEC) regimes.

BCS–BEC crossover in one-dimensional integrable model

Abstract The crossover from Bardeen–Cooper–Schrieffer (BCS) superfluid with singlet pairs to Bose–Einstein condensation (BEC) of molecules is studied in one dimension. By use of the nested Bethe ansatz method, the ground state properties of spin-1/2 fermions interacting through attractive δ-function are analyzed explicitly for strong and weak couplings. Based on those results, we confirm a crossover picture, that is, in the BEC regime (strong couplings) the system is described by molecules with weak repulsion while in the BCS regime (weak couplings) it behaves as the weakly attractive fermions.

SUPERFLUID REGIMES IN DEGENERATE ATOMIC FERMI GASES

We give a brief overview of recent studies of quantum degenerate regimes in ultracold Fermi gases. The attention is focused on the regime of Bose-Einstein condensation of weakly bound molecules of fermionic atoms, formed at a large positive scattering length for the interspecies atom-atom interaction. We analyze remarkable collisional stability of these molecules and draw prospects for future studies.

Magnetic-modulation spectroscopy of an atomic Fermi gas in the BCS-BEC crossover: Dissociation spectra in the Bose-Einstein condensate regime

The effect of magnetic-field modulation on a Fermi gas of atoms in the BCS-BEC crossover is studied analytically. Recent experimental findings on the system response to a sinusoidal variation of the field are explained. Specifically, the dissociation processes induced by the modulation in the Bose-Einstein condensate regime are described. The role played by the frequency, amplitude, and application time of the perturbation in the emergence of the observed behavior is clarified. The results uncover also the relevance of the detuning from the Feshbach resonance to the appearance of particular spectral features. The applicability of the field modulation as a spectroscopic tool for probing the crossover is discussed.

Fluctuations between the BCS and BEC Limits in the System of Ultracold Alkali Atoms

We investigate the fluctuations induced by time-dependent interchange between the Bardeen–Cooper–Schrieffer and Bose–Einstein condensation regimes in the ultracold gas of fermion atoms. Such crossover can be triggered by varying the external magnetic field across the Feshbach resonance. Experimental realization is usually done via very fast switching which leads to the nonequilibrium effects. In this paper we focus on the ground state properties. In particular, we analyze time dependence of the wave function and consider fluctuations of the order parameters.

Cluster formation in the system of interacting Bose particles

Based on statistical approach we described possible formation of spatially inhomogeneous distribution in the system of interacting Bose particles. The condition of cluster formation in both gas and condensed phases was obtained in this system. We studied the dynamics of cluster formation in the limit case of high temperatures. We compared the cluster-formation processes in the attractive system (with short-range interaction) and in the gravitational system at the low temperatures of Bose-Einstein condensate regime.

Particle density distributions in Fermi gas superfluids: Differences between one- and two-channel models in the Bose-Einstein-condensation limit

We show how to describe the $T \neq 0$ behavior associated with the usual BCS- Bose Einstein condensation (BEC) crossover ground state. We confine our attention here to the BEC and near-BEC regime where analytical calculations are possible. At finite $T$, non-condensed fermion pairs must be included, although they have been generally ignored in the literature. Within this BEC regime we compute the equations of state for the one and two channel models; these two cases correspond to whether Feshbach resonance effects are omitted or included. Differences between these two cases can be traced to differences between the nature of a Cooper pair and bosonic condensate. Our results are also compared with the Gross Pitaevskii equations of state for true bosons. Differences found here are associated with the underlying fermionic character of the system. Finally, the particle density distribution functions for a trap containing superfluid fermionic atoms are computed using a Thomas-Fermi approach. The one and two channel behavior is found to be very different; we find a narrowing of the density profile as a result of Feshbach resonance effects. Importantly, we infer that the ratio between bosonic and fermionic scattering lengths depends on the magnetic detuning and is generally smaller than 2. Future experiments will be required to determine to what extent this ratio varies with magnetic fields.

Density Profiles of Strongly Interacting Trapped Fermi Gases

We study density profiles in trapped fermionic gases, near Feshbach resonances, at all T< or =Tc and in the near Bose-Einstein condensation and unitary regimes. For the latter, we characterize and quantify the generally neglected contribution from noncondensed Cooper pairs. As a consequence of these pairs, our profiles are rather well fit to a Thomas-Fermi (TF) functional form, and equally well fit to experimental data. Our work lends support to the notion that TF fits can be used in an experimental context to obtain information about the temperature.

Doping-dependent superconducting properties of two-dimensional metals with different types of interparticle coupling (Review)

The superconducting properties of a two-dimensional metallic system with arbitrary carrier density and both local and various types of attractive indirect boson-exchange interaction in the cases of s- and d-wave pairing are analyzed and reviewed at T=0. In particular, the possibility of a crossover from the Bose–Einstein condensation regime to Bardeen–Cooper–Schrieffer-like superconductivity with growing carrier density and changing coupling in the case of different pairing channels is discussed. Gaussian fluctuations of the order parameter are taken into account, and the carrier density dependence of the gap magnitude is studied. The role of the form of the interparticle attractive interaction in the physical behavior of the system is also considered.