# Superfluid and Fermi liquid phases of Bose-Fermi mixtures in optical lattices

###### Abstract

We describe interacting mixtures of ultracold bosonic and fermionic atoms in harmonically confined optical lattices. For a suitable choice of parameters we study the emergence of superfluid and Fermi liquid (non-insulating) regions out of Bose-Mott and Fermi-band insulators, due to finite Boson and Fermion hopping. We obtain the shell structure for the system and show that angular momentum can be transferred to the non-insulating regions from Laguerre-Gaussian beams, which combined with Bragg spectroscopy can reveal all superfluid and Fermi liquid shells.

###### pacs:

03.75.Hh, 03.75.Kk, 03.75 LmFermi and Bose degenerate quantum gases and liquids are amazing systems, which have revealed individually several macroscopic quantum phenomena. For instance, superfluidity is known to exist in neutral liquids such as He (boson) and in He (fermion), as well as in a variety of electronic materials studied in standard condensed matter physics. The role of quantum statistics and interactions is of fundamental importance to understand the phases emerging from purely bosonic or purely fermionic systems, and a substantial amount of understanding of these individual Bose or Fermi systems can be found in the atomic and condensed matter physics literature. However, new frontiers can be explored when mixtures of bosons and fermions are produced in harmonic traps or optical lattices. An important example of the richness of quantum degenerate Bose-Fermi mixtures was revealed in standard condensed matter systems, where for fixed He density and increasing amounts of He, the critical temperature for superfluidity is reduced, and below a tricritical point phase separation appears reppy-1967 (). In this Bose-Fermi mixture of standard condensed matter physics, essentially the only control parameter is the ratio between the densities of He and He.

In atomic physics, a spectacular degree of control has been achieved in Bose-Fermi mixtures, where not only the ratio between densities of bosons and fermions can be adjusted, but also the interactions between fermions and bosons can be controlled through the use of Feshbach resonances bongs-2006 (), as demonstrated in mixtures of harmonically trapped Bose-Fermi polarized Fermion mixtures of K and Rb. Furthermore, these same atoms have been succesfully loaded into optical lattices esslinger-2006 () and have produced a system that has no counterpart in standard condensed matter systems. By controlling the depths of optical lattices we can change not only the interactions between boson and fermions, but also their hopping from site to site, thus allowing the exploration of a very rich phase space, where supersolid and phase separated states have been suggested blatter-2003 (). The list of Bose-Fermi mixtures in atomic physics is growing, and include systems where the masses are close like Li and Li, K and K, or Yb and Yb; or systems where the masses are quite different like Li and K, Li and K, Li and Na, Li and Rb, K and Na, or K and Rb. This suggests a wide possibility of regimes that can be reached by tuning interactions, density and geometry, which is not is possible in ordinary condensed matter physics.

A few studies of quantum phases of Bose-Fermi mixtures have focused on homogeneous three dimensional systems with lewenstein-2004 (); batrouni-2007 (), and without optical lattices viverit-2000 (). However, only one effort focused on harmonically confined optical lattices illuminati-2004 (). Most of the descriptions of Bose-Fermi mixtures in optical lattices have relied on numerical methods using either Gutzwiller projection lewenstein-2004 (); illuminati-2004 () or quantum Monte Carlo batrouni-2007 () techniques. In this paper, we present a fully analytical theory of boson and spin polarized fermion mixtures in harmonically confined optical lattices by using degenerate perturbation theory for finite hopping in conjunction with the local density approximation. This work provides insight into the phase diagram of Bose-Fermi mixtures, and into the detection of superfluid and Fermi liquid shells at low temperatures.

This paper we analyses in detail the regime where the hopping parameters of bosons and fermions are comparable and the repulsion between bosons and fermions is a substantial fraction of the boson-boson repulsion. In this case, the system presents regions of (I) coexisting Bose-Mott and Fermi-band insulator, (II) coexisting Bose-Mott insulator and Fermi liquid, (III) Bose-Mott insulator, and (IV) Bose superfluid, as shown in Fig. 1. We compute analytically the boundaries between various phases, and obtain the spatially dependent boson and fermions filling fractions in each region. Although one can envisage other situations where, for example, one can have coexistence of superfluid and Fermi band insulator, we confine our discussion to the situation above for the sake of simplicity. Finally, we propose a detection method of the shell strucuture of Bose-Fermi mixtures by using Laguerre-Gaussian beams helmerson-2007 () and Bragg spectroscopy raman-2006 (), where angular momentum is transferred only to regions with extended states such as the superfluid and Fermi-liquid shells.

To describe Bose-Fermi mixtures in harmonically confined square (2D) or cubic (3D) optical lattices we start with the Hamiltonian

where and are the kinetic energies of boson and fermions with nearest-neighbor hoppings and , and and are the bosonic and fermionic creation operators at site . Here, the lattice sites for bosons and fermions are assumed to be the same, but the hopping parameters can be different. The number operators are and , and the corresponding local chemical potentials are and , where and are the harmonically confining potentials and and are the chemical potentials for fermions and bosons. The origin of the lattice with spacing is chosen to be at the minimum of the harmonically confining potential. The terms containing represent the boson-boson (boson-fermion) interaction.

When , the Hamiltonian is a sum of single-site contributions, and the eigenstates are tensor products of number states with state vectors , with and representing the occupation number of bosons and fermions at site , respectively. At site the local energy is . For the ground state wavefunction the number of bosons at site is determined by if and otherwise. Similarly, the number of fermions per site is zero if and one otherwise. The symbol is the floor function. In the ground state solution shells with bosons and fermions are formed by those lattice sites for which the local energy is the same. For our harmonic traps these shells are nearly spherically symmetric. The boundary between shells and is determined by , leading to the radius , where . Similarly, the boundary between shells with occupation numbers and is determined by equating the local energies and , leading to the radius . We consider the number of particles to be sufficiently large such that the radii of the boundaries are much larger than .

Next, we begin our discussion of finite hoppings by taking first , with . The Bose superfluid region emerges due to kinetic fluctuations at the boundaries between the and shells. At this boundary the local energy is degenerate with . To describe the emergence of superfluid regions, we introduce the order parameter for superfluidity via the transformation , and then for analytical convenience make the continuum approximation .

In the limit of , we can restrict our Hilbert space to the number basis states and , as any contribution from other basis states to the local energy is of order . The hopping term affects the energies and by removing their degeneracy, thus creating finite-width superfluid regions between shells and . The effective local Hamiltonian then becomes,

(1) |

where and . Here, is the coordination number which depends on the lattice dimension .

The eigenvalues of Eq. (1) are given by,

where is proportional to the sum of the diagonal terms, and is proportional to their difference. Notice that is the lowest local energy leading to the total ground state energy

The order parameter equation (OPE) is determined by minimization of with respect to leading to

(2) |

Notice that the OPE is not of the Gross-Pitaeviskii (GP) type, since the superfluid regions emerge from local fluctuations between neighboring Mott shells. Ignoring the spatial derivatives of in Eq. 2 leads to the spatially dependent order parameter

(3) |

Since , hence , and the inner and outer radii for the superfluid shell between the and Mott regions are obtained by setting leading to

This relation shows explicitly that splits the spatial degeneracy of the and insulating shells at or by introducing a superfluid region of width . (See Fig. 1 for characteristic widths).

In addition, the local bosonic filling fraction

in the same region interpolates between for and for , while the chemical potential is fixed by the total number of bosons . The local bosonic compressibility of the superfluid shells is non-zero, in contrast to the incompressible () and insulating shells for and , respectively.

Now, we consider finite . In order to have a tractable theory we assume that the shell boundaries of the bosons and fermions are well separated. This allows us to investigate the Fermi liquid near the shell boundary of the fermions in the presence of a Bose-Mott insulator with bosons per site. Furthermore, if we assume that the local density of the Fermi gas is smoothly varying then the local number of fermions is

(4) |

where is the Fermi function at temperature , and is the density of fermion states with energy dispersion . The effective chemical potential accounts for the effect of the bosons. The band minimum and maximum of are and , respectively. Thus, the Fermi liquid region is limited by the boundaries , leading to for the inner and outer radius of the FL shell. The width of the FL region is . (See Fig. 1 for characteristic widths). The isothermal compressibility of the FL region is , which leads at zero temperature to outside the FL shell, indicating the presence of insulating regions and inside the FL shell, indicating the presence of conducting regions. The superfluid and FL shells for finite and , and their density profiles are shown in Fig. 1 for the two-dimensional case.

Next, we propose an experiment to detect superfluid and Fermi liquid shells in Bose-Fermi mixtures using a combination of Gaussian and Laguerre-Gaussian beams followed by Bragg spectroscopy. To illustrate the idea, we discuss the simpler case of a nearly two-dimensional configuration, where the harmonic trap is very tight along the z-direction, loose along the x- and y- directions. Upon application of Gaussian and Laguerre-Gaussian beams along the z-direction, only angular momentum is transferred to the atoms in the conducting phases (superfluid or Fermi liquid), imposing a rotating current with a well defined velocity profile, while the insulating regions do not absorb angular momentum due to their large gap in the excitation spectrum.

To probe the rotating superfluid and Fermi liquid phases we propose the use of two counter-propagating Bragg beams applied in the xy plane along the x direction, as indicated in Fig. 2.

The Bragg beams transfer a net linear momentum to the atoms of mass which satisfy the energy conservation condition

(5) |

where is the component of the velocity along the x direction and and are the experimentally accessible energies of the initial and final internal states atom. For an atom carrying one unit of angular momentum, the velocity is . Therefore, within a conducting shell with radius atoms get a linear momentum kick of when the velocity satisfies the condition given in Eq. 5. This leads to two Bragg angles , and for each conducting shell. As can be seen in Fig. 2, the Bragg angles are and for the outer superfluid shell labelled by , and are and for the fermi liquid shell labelled by . Once these atoms are kicked out of the conducting shells, they form two small expanding clouds, which can be detected by direct absorption imaging.

Next, we discuss the time scales over which the rotation in the conducting regions persist and can be detected experimentally. In the case of the superfluid region we use the Landau criterion to show that the velocity imposed to the superfluid through the angular momentum transfer is much smaller than the local sound velocity , where is the local superfluid density, and is the compressibility. Thus, vanishes at the insulator boundaries where , and only close to the edge of the superfluid regions the local rotational speed exceeds , which means that essentially all the superfluid region can be detected and the angular momentum transferred does not decay over time scales of at least seconds, limited by the lifetime of the trapped system.

In the case of the Fermi liquid region, the time scale over which the flow of the fermions persist in presence of the Bose-Mott insulator background can be calculated from the imaginary part of the fermionic self-energy

where and , with are fermionic (bosonic) Matsubara frequencies and is temperature. The bare inverse bosonic propagator in the Bose-Mott phase is

where , and . The bare inverse fermionic propagator in the Fermi liquid phase is where . For and , the imaginary part of the fermionic self-energy is

(6) |

where , is the Heaviside step function, and is the Fermion density of states. For a two-dimensional Fermi liquid shell, there is a Van Hove singularity in at half filling. The expression in Eq. (6) is independent of momentum, since the dominant excitations in the Bose-Mott region are number-conserving and low-momentum particle-hole excitations, but strongly dependent on position through and , leading to a characteristic decay time . For the parameters used in Fig. 1, (with ) the time scale for the persistence of the flow near the edges (away from the Van Hove singularity) is . However, near the center of the Fermi liquid region (close to the Van Hove singularity) is extremely short, indicating that it is much easier to detect fermions at the edge than at the center of Fermi liquid shells.

We have discussed the phase diagram of Bose-Fermi mixtures in harmonically confined optical lattices in the regime where the hopping parameters of bosons and fermions are comparable and the repulsion between bosons and fermions is a substantial fraction of the boson-boson repulsion. We showed that the system exhibits regions of (I) coexisting Bose-Mott and Fermi-band insulator, (II) coexisting Bose-Mott insulator and Fermi liquid, (III) Bose-Mott insulator, and (IV) Bose-superfluid. We have calculated analytically the boundaries between these phases and obtained the spatially dependent filling fraction for each region. Finally, we proposed a detection method of the superfluid and Fermi liquid shells of Bose-Fermi mixtures by using Gaussian and Laguerre-Gaussian beams followed by Bragg spectroscopy.

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