BEC-BCS crossover in an optical lattice

Experimental Realization of the BCS-BEC Crossover with a Fermi Gas of Atoms

This article reviews recent progresses in ultracold quantum gases, and it includes three subjects which are the Fermi gases across a Feshbach resonance, quantum gases in the optical lattices and the fast rotating quantum gases. In this article, we discuss many basic physics pictures and concepts in quantum gases, for examples, the resonant interaction, universality and condensation in the lowest Landau level; we introduce fundamental theoretical tools for studying these systems, such as mean-field theory for BEC-BCS crossover and for the boson Hubbard model; also, we emphasize the important unsolved problems in the forefront of this field, for instance, the temperature effect in optical lattices.

Atomic Fermi gases have been an ideal platform for simulating conventional and engineering exotic physical systems owing to their multiple tunable control parameters. Here we investigate the effects of mixed dimensionality on the superfluid and pairing phenomena of a two-component ultracold atomic Fermi gas with a short-range pairing interaction, while one component is confined on a one-dimensional (1D) optical lattice whereas the other is in a homogeneous 3D continuum. We study the phase diagram and the pseudogap phenomena throughout the entire BCS-BEC crossover, using a pairing fluctuation theory. We find that the effective dimensionality of the non-interacting lattice component can evolve from quasi-3D to quasi-1D, leading to strong Fermi surface mismatch. Upon pairing, the system becomes effectively quasi-two dimensional in the BEC regime. The behavior of Tc bears similarity to that of a regular 3D population imbalanced Fermi gas, but with a more drastic departure from the regular 3D balanced case, featuring both intermediate temperature superfluidity and possible pair density wave ground state. Unlike a simple 1D optical lattice case, Tc in the mixed dimensions has a constant BEC asymptote.

We calculate the momentum distribution and the second-order correlation function in momentum space, $g^{(2)}({\bf p},{\bf p}',t)$ for molecular dimers that are coherently formed from an ultracold atomic gas by photoassociation or a Feshbach resonance. We investigate using perturbation theory how the quantum statistics of the molecules depend on the initial state of the atoms by considering three different initial states: a Bose-Einstein condensate (BEC), a normal Fermi gas of ultra-cold atoms, and a BCS-type superfluid Fermi gas. The cases of strong and weak coupling to the molecular field are discussed. It is found that BEC and BCS states give rise to an essentially coherent molecular field with a momentum distribution determined by the zero-point motion in the confining potential. On the other hand, a normal Fermi gas and the unpaired atoms in the BCS state give rise to a molecular field with a broad momentum distribution and thermal number statistics. It is shown that the first-order correlations of the molecules can be used to measure second-order correlations of the initial atomic state.

Imagine you could produce an artificial crystal for quantum matter, defect free and with complete control over the periodic crystal potential. The shape of the periodic potential, its depth and the interactions between the underlying particles could be changed at will and the particles could be moved around in a highly controlled way, at essentially zero temperature. This sounds almost too good to be true, but it is in fact what optical lattices have made possible for cold and ultracold atoms.

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.

We propose a scheme for making a Bose-Einstein condensate (BEC) of molecules from a BEC of atoms in a strongly confining two-dimensional optical lattice and a weak one-dimensional optical lattice in the third dimension. The stable solutions obtained for the order parameters take the form of a different type of gap soliton, with both atomic and molecular BECs, and also standard gap solitons with only a molecular BEC. The strongly confining dimensions of the lattice stabilize the BEC against inelastic energy transfer in atom-molecule collisions. The solitons with atoms and molecules may be obtained by starting with an atomic BEC, and gradually tuning the resonance by changing the external magnetic-field strength until the desired atom-molecule soliton is obtained. A gap soliton of a BEC of only molecules may be obtained nonadiabatically by starting from an atom-only gap soliton, far from a Feshbach resonance and adjusting the magnetic field to near Feshbach resonance. After a period of time in which the dimer field grows, change the magnetic field such that the detuning is large and negative and Feshbach effects wash out, turn off the optical lattice in phase with the atomic BEC, and turn on an optical lattice in phase with the molecules. The atoms disperse, leaving a gap soliton composed of a molecular BEC. Regarding instabilities in the dimension of the weak optical lattice, the solitons which are comprised of both atoms and molecules are sometimes stable and sometimes unstable---we present numerically obtained results. Gap solitons comprised of only molecules have the same stability properties as the standard gap solitons: stable from frequencies slightly below the middle of the band gap to the top, and unstable below that point. Instabilities are only weakly affected by the soliton velocities, and all instabilities are oscillatory.

We theoretically investigate normal-state properties of an ultracold Fermi gas with an orbital Feshbach resonance (OFR). Recently, OFR has attracted much attention as a promising pairing mechanism to realize a superfluid 173Yb Fermi gas. Including pairing fluctuations within a T-matrix approximation, and removing effects of an experimentally inaccessible deep bound state, we evaluate strong-coupling corrections to single-particle excitations. With increasing the strength of an OFR-induced tunable pairing interaction, the open channel is shown to exhibit the pseudogap phenomenon in the BCS-BEC crossover region, as in the case of a broad magnetic Feshbach resonance (MFR) in 6Li and 40K Fermi gases. We also show that the strong pairing interaction affects the closed channel, leading to the coexistence of particle and hole branches in the single-particle spectral weight. Since the latter phenomenon cannot be observed in the conventional MFR case, it may be viewed as a characteristic strong-coupling phenomenon peculiar to the OFR case.

Liquid helium 4 had been the only bosonic superfluid available in experiments for a long time. This situation was changed in 1995, when a new superfluid was born with the realization of the Bose–Einstein condensation in ultracold atomic gases. The liquid helium 4 is strongly interacting and has no spin; there is almost no way to change its parameters, such as interaction strength and density. The new superfluid, Bose–Einstein condensate (BEC), offers various advantages over liquid helium. On the one hand, BEC is weakly interacting and has spin degrees of freedom. On the other hand, it is convenient to tune almost all the parameters of a BEC, for example, the kinetic energy by spin–orbit coupling, the density by the external potential, and the interaction by Feshbach resonance. Great efforts have been devoted to studying these new aspects, and the results have greatly enriched our understanding of superfluidity. Here we review these developments by focusing on the stability and critical velocity of various superfluids. The BEC systems considered include a uniform superfluid in free space, a superfluid with its density periodically modulated, a superfluid with artificially engineered spin–orbit coupling, and a superfluid of pure spin current. Due to the weak interaction, these BEC systems can be well described by the mean-field Gross–Pitaevskii theory and their superfluidity, in particular critical velocities, can be examined with the aid of Bogoliubov excitations. Experimental proposals to observe these new aspects of superfluidity are discussed.

The dynamics of an ultracold dilute gas of bosonic atoms in an optical lattice can be described by a Bose-Hubbard model where the system parameters are controlled by laser light. We study the continuous (zero temperature) quantum phase transition from the superfluid to the Mott insulator phase induced by varying the depth of the optical potential, where the Mott insulator phase corresponds to a commensurate filling of the lattice (``optical crystal''). Examples for formation of Mott structures in optical lattices with a superimposed harmonic trap, and in optical superlattices are presented.

We derive pseudo-potentials that describe the scattering between two\nparticles in two spatial dimensions for any partial wave m, whose scattering\nstrength is parameterized in terms of the m-dependent phase shift. Using our\nm=0 pseudo-potential, we develop a coupled channel model with 2D zero-range\ninteractions, which describes the two-body physics across a Feshbach resonance.\nOur model predicts the scattering length, the binding energy and the "closed\nchannel molecular fraction" of two particles; these observables can be measured\nin experiments on ultracold quasi-2D atomic Bose and Fermi gases with\npresent-day technology.\n

In ultracold atomic gases, optical lattices and spin-orbit coupling are two important and useful linear means to study and control the nonlinear and quantum dynamics of matter waves, and both have been well realized in experiments. Here, nonlinear localization of spin-orbit-coupled Bose-Einstein condensates in moiré optical lattices is revealed theoretically, stressing the existence and stability of localized modes of diverse types including fundamental, half-vortex matter-wave gap solitons, and full-vortex ones (of both components). Notably, to our knowledge, a new type of vector vortex gap solitons, which is composed of a gap vortex soliton (singular mode) and a rhombic-array gap vortex (quadruple mode), is predicted in our two-component theoretical framework. It is also found that the model exhibits unique linear Bloch band structures including band splitting and extremely-flat bands, because of which all the predicted localized modes tend to be robustly stable, backed by our direct perturbed simulations. Our results demonstrate that the moiré pattern (and its joint cooperation with spin-orbit coupling) provides great possibilities to explore flat-band and moiré physics in ultracold atoms, and to further investigate the nonlinear dynamics.

We studied fermionic pairing in an ultracold two-component gas of 6Li atoms by observing an energy gap in the radio-frequency excitation spectra. With control of the two-body interactions through a Feshbach resonance, we demonstrated the dependence of the pairing gap on coupling strength, temperature, and Fermi energy. The appearance of an energy gap with moderate evaporative cooling suggests that our full evaporation brought the strongly interacting system deep into a superfluid state.

Recent success in manipulating ultracold atomic systems allows one to probe different strongly correlated regimes in one dimension. Regimes such as the (spin-coherent) Luttinger liquid and the spin-incoherent Luttinger liquid can be realized by tuning the interatomic interaction strength and trap parameters. We identify the noise correlations of density fluctuations as a robust observable (uniquely suitable in the context of trapped atomic gases) to discriminate between these two regimes. Finally, we address the prospects of realizing and probing these phenomena experimentally using optical lattices.

We present the Bogoliubov theory for the Bose-Einstein condensation of Feshbach molecules in a balanced Fermi mixture. Because the Bogoliubov theory includes (Gaussian) fluctuations, we can in this manner accurately incorporate both the two-body and many-body aspects of the BEC-BCS crossover that occurs near a Feshbach resonance. We apply the theory in particular to the very broad Feshbach resonance in atomic Li-6 at a magnetic field of B_0 = 834 G and find good agreement with experiments in that case. The BEC-BCS crossover for more narrow Feshbach resonances is also discussed.