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

Kaushik Mitra, C. J. Williams and C. A. R. Sá de Melo  Joint Quantum
Institute, University of Maryland, College Park, Maryland 20742,
and National Institute of Standards and Technology, Gaithersburg, Maryland
20899

###### 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 Lm

Fermi 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 4He
(boson) and in 3He (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 4He density and increasing amounts of 3He, 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 4He and 3He.

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 40K and 87Rb. 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 6Li and 7Li, 39K and 40K, or 172Yb and 173Yb;
or systems where the masses are quite different like 6Li and 40K, 7Li and 39K,
6Li and 23Na, 6Li and 87Rb, 40K and 23Na, or 40K and 87Rb. 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

$\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle K_{B}+K_{F}-\sum_{\bf
r}\mu_{B}({\bf r}){\hat{n}}_{B}({\bf r})-\sum_{\bf r}\mu_{F}({\bf
r}){\hat{n}}_{F}({\bf r})$ $\displaystyle+$
$\displaystyle\frac{U_{BB}}{2}\sum_{\bf r}{\hat{n}}_{B}({\bf
r})\left[{\hat{n}}_{B}({\bf r})-1\right]+U_{BF}\sum_{\bf r}{\hat{n}}_{B}({\bf
r}){\hat{n}}_{F}({\bf r}),$

where $K_{B}=-t_{B}\sum_{\langle{\bf r},{\bf r}^{\prime}\rangle}b_{\bf
r}^{\dagger}b_{{\bf r}^{\prime}}$ and $K_{F}=-t_{F}\sum_{\langle{\bf r},{\bf
r}^{\prime}\rangle}f_{\bf r}^{\dagger}f_{{\bf r}^{\prime}}$ are the kinetic
energies of boson and fermions with nearest-neighbor hoppings $t_{B}$ and
$t_{F}$, and $b^{\dagger}_{\bf r}$ and $f^{\dagger}_{\bf r}$ are the bosonic
and fermionic creation operators at site ${\bf r}$. 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 ${\hat{n}}_{B}({\bf r})=b_{\bf
r}^{\dagger}b_{\bf r}$ and $\hat{n}_{F}({\bf r})=f_{\bf r}^{\dagger}f_{\bf
r}$, and the corresponding local chemical potentials are $\mu_{F}({\bf
r})=\mu_{F}-V_{F}({\bf r})$ and $\mu_{B}({\bf r})=\mu_{B}-V_{B}({\bf r})$,
where $V_{F}({\bf r})=\Omega_{F}(r/a)^{2}/2$ and $V_{B}({\bf
r})=\Omega_{B}(r/a)^{2}/2$ are the harmonically confining potentials and
$\mu_{F}$ and $\mu_{B}$ are the chemical potentials for fermions and bosons.
The origin of the lattice with spacing $a$ is chosen to be at the minimum of
the harmonically confining potential. The terms containing $U_{BB}$ $(U_{BF})$
represent the boson-boson (boson-fermion) interaction.

When $t_{B}=t_{F}=0$, the Hamiltonian ${\hat{H}}={\hat{H}}_{0}$ is a sum of
single-site contributions, and the eigenstates are tensor products of number
states with state vectors
$|\psi\rangle=|n_{B,0},n_{B,1},\cdots\rangle|n_{F,0},n_{F,1},\cdots\rangle$,
with $n_{B,{\bf r}}=0,1,2,...$ and $n_{F,{\bf r}}=0,1$ representing the
occupation number of bosons and fermions at site ${\bf r}$, respectively. At
site ${\bf r}$ the local energy is $E_{n_{B},n_{F}}({\bf
r})=U_{BB}n_{B}(n_{B}-1)/2+U_{BF}n_{B}n_{F}-\mu_{B}({\bf r})n_{B}-\mu_{F}({\bf
r})n_{F}$. For the ground state wavefunction the number of bosons at site
${\bf r}$ is determined by $max(0,\lfloor(U_{BB}+\mu_{B}({\bf
r}))/2U_{BB}\rfloor)$ if $E_{n_{B},0}({\bf r})<E_{n_{B},1}({\bf r})$ and
$max(0,\lfloor(U_{BB}+\mu_{B}({\bf r})-U_{BF})/2U_{BB}\rfloor)$ otherwise.
Similarly, the number of fermions per site is zero if $E_{n_{B},0}({\bf
r})<E_{n_{B},1}({\bf r})$ and one otherwise. The symbol $\lfloor\cdot\rfloor$
is the floor function. In the ground state solution shells $(n_{B},n_{F})$
with $n_{B}$ bosons and $n_{F}$ fermions are formed by those lattice sites
${\bf r}$ for which the local energy is the same. For our harmonic traps these
shells are nearly spherically symmetric. The boundary between shells
$(n_{B},n_{F})$ and $(n_{B}\\!+\\!1,n_{F})$ is determined by
$E_{n_{B},n_{F}}({\bf r})=E_{n_{B}\\!+\\!1,n_{F}}({\bf r})$, leading to the
radius $R_{B,n_{B},n_{F}}=a\sqrt{\Omega_{n_{B},n_{F}}/\Omega_{B}}$, where
$\Omega_{n_{B},n_{F}}=2(\mu_{B}-n_{B}U_{BB}-n_{F}U_{BF})$. Similarly, the
boundary between shells with occupation numbers $(n_{B},0)$ and $(n_{B},1)$ is
determined by equating the local energies $E_{n_{B},n_{F}}({\bf r})$ and
$E_{n_{B},n_{F}+1}({\bf r})$, leading to the radius
$R_{F,n_{B}}=a\sqrt{2(\mu_{F}-n_{B}U_{BF})/\Omega_{F}}$. We consider the
number of particles to be sufficiently large such that the radii of the
boundaries are much larger than $a$.

Next, we begin our discussion of finite hoppings by taking first $t_{B}\neq
0$, with $t_{F}=0$. The Bose superfluid region emerges due to kinetic
fluctuations at the boundaries between the $(n_{B},n_{F})$ and
$(n_{B}+1,n_{F})$ shells. At this boundary the local energy
$E_{n_{B}+1,n_{F}}({\bf r})$ is degenerate with $E_{n_{B},n_{F}}({\bf r})$. To
describe the emergence of superfluid regions, we introduce the order parameter
for superfluidity $\psi_{B,j}$ via the transformation
$b_{i}^{\dagger}b_{j}\to\psi_{B,i}^{*}b_{j}+b_{i}^{\dagger}\psi_{B,j}-{\psi_{B,i}^{*}\psi_{B,j}}$,
and then for analytical convenience make the continuum approximation
$\psi({\bf r}+{\bf\it a})=\psi({\bf r})+a_{i}\partial_{i}\psi({\bf
r})+(1/2)a_{i}a_{j}\partial_{i}\partial_{j}\psi({\bf r})$.

In the limit of $U_{BB}\gg t_{B}$, we can restrict our Hilbert space to the
number basis states $|n_{B},n_{F}\rangle$ and $|n_{B}+1,n_{F}\rangle$, as any
contribution from other basis states to the local energy is of order
$t_{B}^{2}/U_{BB}$. The hopping term $t_{B}$ affects the energies
$E_{n_{B}+1,n_{F}}({\bf r})$ and $E_{n_{B},n_{F}}({\bf r})$ by removing their
degeneracy, thus creating finite-width superfluid regions between shells
$(n_{B}+1,n_{F})$ and $(n_{B},n_{F})$. The effective local Hamiltonian then
becomes,

$H_{\mathbf{r}}^{\textrm{eff}}=\left(\begin{array}[]{cc}E_{n_{B},n_{F}}({\bf
r})+\Lambda({\bf r})&-\sqrt{n_{B}+1}\Delta({\bf r})\\\
-\sqrt{n_{B}+1}\Delta^{*}({\bf r})&E_{n_{B}+1,n_{F}}({\bf r})+\Lambda({\bf
r})\end{array}\right),$ (1)

where $\Lambda({\bf r})=\frac{1}{2}(\Delta({\bf r})\psi^{*}({\bf r})+cc)$ and
$\Delta({\bf r})=t_{B}(z\psi({\bf r})+{\it a}^{2}\nabla^{2}\psi({\bf r}))$.
Here, $z$ is the coordination number which depends on the lattice dimension
$d$.

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

$E_{\pm}(\mathbf{r})=E_{s}(\mathbf{r})\pm\sqrt{\left[E_{d}(\mathbf{r})\right]^{2}+(n_{B}+1)\left|\Delta(\mathbf{r})\right|^{2}},$

where $E_{s}(\mathbf{r})=\left[E_{n_{B}+1,n_{F})}({\bf
r})+E_{n_{B},n_{F}}({\bf r})\right]/2+\Lambda({\bf r})$ is proportional to the
sum of the diagonal terms, and $E_{d}(\mathbf{r})=\left[E_{n_{B}+1,n_{F}}({\bf
r})-E_{n_{B},n_{F}}({\bf r})\right]/2=(n_{B}U_{BB}+n_{F}U_{BF}-\mu_{B}({\bf
r})/2$ is proportional to their difference. Notice that $E_{-}({\bf r})$ is
the lowest local energy leading to the total ground state energy ${\bf
E}=\frac{1}{L^{d}}\int d{\bf r}E_{-}({\bf r}).$

The order parameter equation (OPE) is determined by minimization of ${\bf E}$
with respect to $\psi^{*}(\mathbf{r})$ leading to

$\Delta(\mathbf{r})-\frac{(n_{B}+1)t_{B}(z+a^{2}\nabla^{2})\Delta(\mathbf{r})}{2\sqrt{\left|E_{d}(\mathbf{r})\right|^{2}+(n_{B}+1)\left|\Delta(\mathbf{r})\right|^{2}}}=0.$
(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 $\psi$ in Eq. 2 leads to the
spatially dependent order parameter

$\left|\psi(\mathbf{r})\right|^{2}=\frac{n_{B}+1}{4}-\frac{\left(n_{B}U_{BB}+n_{F}U_{BF}-\mu_{B}({\bf
r})\right)^{2}}{4z^{2}t_{B}^{2}(n_{B}+1)}.$ (3)

Since $\left|\psi(\mathbf{r})\right|^{2}\geq 0$, hence
$|n_{B}U_{BB}+n_{F}U_{BF}-\mu_{B,\bf{r}}|\leq(n_{B}+1)zt_{B}$, and the inner
$R_{n_{B},n_{F},-}$ and outer $R_{n_{B},n_{F},+}$ radii for the superfluid
shell between the $(n_{B},n_{F})$ and $(n_{B}+1,n_{F})$ Mott regions are
obtained by setting $\left|\psi(\mathbf{r})\right|^{2}=0$ leading to

$R_{n_{B},n_{F},\pm}=R_{B,n_{B},n_{F}}\sqrt{1\pm\frac{2zt_{B}(n_{B}+1)}{\Omega_{B}}\frac{a^{2}}{R_{B,n_{B},n_{F}}^{2}}}.$

This relation shows explicitly that $t_{B}$ splits the spatial degeneracy of
the $(n_{B},n_{F})$ and $(n_{B}+1,n_{F})$ insulating shells at
$r=R_{c,n_{B},n_{F}}$ or $\mu_{B}({\bf r})=n_{B}U_{B}B+n_{F}U_{BF}$ by
introducing a superfluid region of width $\Delta
R_{n_{B},n_{F}}=R_{n_{B},n_{F},+}-R_{n_{B},n_{F},-}$. (See Fig. 1 for
characteristic widths).

In addition, the local bosonic filling fraction

$n_{B}(\mathbf{r})=-\frac{\partial
E_{-}(\mathbf{r})}{\partial\mu_{B}}=n_{B}+\frac{1}{2}-\frac{n_{B}U_{BB}+n_{F}U_{BF}-\mu_{B}({\bf{r}})}{2zt_{B}(n_{B}+1)}$

in the same region interpolates between $n_{B}+1$ for $r\lesssim
R_{n_{B},n_{F},-}$ and $n$ for $r\gtrsim R_{n_{B},n_{F},+}$, while the
chemical potential $\mu_{B}$ is fixed by the total number of bosons
$N_{B}=\int d{\bf r}n({\bf r})$. The local bosonic compressibility
$\kappa_{B}({\bf r})=\partial n_{B}({\bf
r})/\partial\mu_{B}=1/2zt_{B}(n_{B}+1)$ of the superfluid shells is non-zero,
in contrast to the incompressible ($\kappa_{B}=0$) $(n_{B},n_{F})$ and
$(n_{B}+1,n_{F})$ insulating shells for $r<R_{n_{B},n_{F},-}$ and
$r>R_{n_{B},n_{F},+}$, respectively.

Figure 1:  (color online) (a) Shell structure of Bose-Fermi mixtures in
harmonically confined optical lattices showing a coexisting Bose-Mott and
Fermi-band insulator region at the center $(n_{B}=1,n_{F}=1)$, a coexisting
Bose-Mott insulator and Fermi liquid region (in blue), a Bose-Mott insulator
region $(n_{B}=1,n_{F}=0)$, and Bose-Superfluid region at the edge (in red).
(b) filling factors for fermions shown as the solid (dark blue) curve and for
bosons shown at the dashed (red) curve. The solid parabolic curve (light blue)
shows the order parameter in the superfluid region. The parameters used are
$t_{F}=t_{B}=0.0325U_{BB}$, $U_{BF}=0.1U_{BB}$, $\mu_{B}=0.8U_{BB}$,
$\mu_{F}=0.4U_{BB}$ and $\Omega_{F}=\Omega_{B}=8\times 10^{-6}U_{BB}$, which
are representative of Bose-Fermi mixtures with nearly the same mass such as
6Li and 7Li, 39K and 40K, or 172Yb and 173Yb. For the parameters chosen, the
widths of the superfluid and FL shells are several times larger than the
lattice spacing $a$.

Now, we consider finite $t_{F}$. 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 $n_{B}$ 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

$n_{F}({\bf r})=\int_{\epsilon_{\rm min}}^{\epsilon_{\rm max}}d\epsilon{\cal
D}(\epsilon)f[\epsilon-\mu^{\rm eff}_{F}({\bf r})]$ (4)

where $f[x]$ is the Fermi function at temperature $T$, and ${\cal
D}(\epsilon)=\sum_{{\bf k}}\delta(\epsilon-\epsilon_{{\bf k}})$ is the density
of fermion states with energy dispersion $\epsilon_{{\bf
k}}=-2t_{F}\sum_{\ell}^{d}\cos(k_{\ell}a)$. The effective chemical potential
$\mu^{\rm eff}_{F}({\bf r})=\mu_{F}({\bf r})-n_{B}U_{BF}$ accounts for the
effect of the bosons. The band minimum and maximum of $\epsilon_{{\bf k}}$ are
$\epsilon_{\rm min}=-2dt_{F}$ and $\epsilon_{\rm max}=2dt_{F}$, respectively.
Thus, the Fermi liquid region is limited by the boundaries $\epsilon_{\rm
min}\leq\mu_{F}({\bf r})\leq\epsilon_{\rm max}$, leading to
$R_{F,\pm}=R_{F,n_{B}}\sqrt{1\pm 2zt_{F}a^{2}/\Omega_{F}R_{F,n_{B}}^{2}}$ for
the inner $R_{F,-}$ and outer $R_{F,+}$ radius of the FL shell. The width of
the FL region is $\Delta R_{F,n_{B}}=R_{F,+}-R_{F,-}$. (See Fig. 1 for
characteristic widths). The isothermal compressibility of the FL region is
$\kappa_{F}({\bf r})=\partial n_{F}({\bf r})/\partial\mu_{F}$, which leads at
zero temperature to $\kappa_{F}({\bf r})=0$ outside the FL shell, indicating
the presence of insulating regions and $\kappa_{F}({\bf r})={\cal
D}[\mu_{F}({\bf r})]$ inside the FL shell, indicating the presence of
conducting regions. The superfluid and FL shells for finite $t_{B}$ and
$t_{F}$, 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.

Figure 2:  (Color online) Schematic plot for the detection of outer (red)
superfluid and inner (blue) Fermi liquid shells using Bragg spectroscopy. The
angles $\theta_{1}$ and $\theta_{2}$ indicate the locations of strongest
momentum transfer from the Bragg beams (large green arrows) to the rotating
superfluid and Fermi liquid shells of radii $R_{1}$ and $R_{2}$. The gray
arrows indicate the sense of rotation of the conducting shells.

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 $\hbar(k+k^{\prime}){\bf x}$ to
the atoms of mass $m$ which satisfy the energy conservation condition

$\hbar(\omega-\omega^{\prime})=\epsilon_{f}-\epsilon_{i}-v_{x}\hbar(k+k^{\prime})+\frac{\hbar^{2}(k+k^{\prime})^{2}}{2m},$
(5)

where $v_{x}$ is the component of the velocity ${\bf v}({\bf r})={\bf p}({\bf
r})/m$ along the x direction and $\epsilon_{i}$ and $\epsilon_{f}$ 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 ${\bf
v}_{i}=\hbar\hat{\theta}/mr$. Therefore, within a conducting shell with radius
$r=R$ atoms get a linear momentum kick of $\hbar(k+k^{\prime})\hat{\bf x}$
when the velocity $v_{x}=\hbar\sin\theta/mR$ satisfies the condition given in
Eq. 5. This leads to two Bragg angles $\theta=-\sin^{-1}(mRv_{x}/\hbar)$, and
$\pi-\theta$ for each conducting shell. As can be seen in Fig. 2, the Bragg
angles are $\theta_{1}$ and $\pi-\theta_{1}$ for the outer superfluid shell
labelled by $R_{1}$, and are $\theta_{2}$ and $\pi-\theta_{2}$ for the fermi
liquid shell labelled by $R_{2}$. 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 $c({\mathbf{r}})=\sqrt{\rho_{s}({\bf
r})/\kappa}$, where $\rho_{s}({\bf r})=2t_{B}a^{2}|\psi({\bf r})|^{2}$ is the
local superfluid density, and $\kappa$ is the compressibility. Thus,
$c({\mathbf{r}})=2\sqrt{(n_{B}+1)z}ta|\psi(\mathbf{r})|$ vanishes at the
insulator boundaries where $|\psi({\bf r})|=0$, and only close to the edge of
the superfluid regions the local rotational speed $v({\bf r})=\hbar/mr$
exceeds $c({\bf r})$, 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

$\Sigma_{F}(k)=U_{BF}^{2}T^{2}\sum_{q_{1},q_{2}}G_{B}(q_{1})G_{B}(q_{2})G_{F}(k+q_{1}-q_{2}),$

where $k=({\bf k},i\omega)$ and $q_{i}=({\bf q}_{i},i\nu_{i})$, with $\omega$
$(\nu_{i})$ are fermionic (bosonic) Matsubara frequencies and $T$ is
temperature. The bare inverse bosonic propagator in the Bose-Mott phase is

$G_{B}^{-1}(q,{\bf r})=\epsilon_{{\bf q}}\left[1+\epsilon_{\bf
q}\left(\frac{n_{B}+1}{i\hbar\omega-E_{1}({\bf r})}-\frac{n_{B}}{i\hbar\omega-
E_{2}({\bf r})}\right)\right],$

where $E_{1}({\bf r})=(n_{B}-1)U_{BB}-\mu_{B}({\bf r})$, $E_{2}({\bf
r})=(n_{B}-2)U_{BB}-\mu_{B}({\bf r})$ and $\epsilon_{\bf
q}=-2t_{B}\sum_{\ell}^{d}\cos(k_{\ell}a)$. The bare inverse fermionic
propagator in the Fermi liquid phase is $G_{F}^{-1}(k,{\bf
r})=i\hbar\omega-\epsilon_{F}({\bf k},{\bf r}),$ where $\epsilon_{F}({\bf
k},{\bf r})=\epsilon_{F}({\bf k})-\mu_{F}({\bf r})$. For $n_{B}=1$ and $T=0$,
the imaginary part of the fermionic self-energy is

${\rm Im}\Sigma_{F}(k,{\bf
r})=-\pi{U^{2}_{BF}}\left[F(\hbar\omega)+F(-\hbar\omega)\right],$ (6)

where $F(\hbar\omega)=\Theta(\hbar\omega)\Theta(-\hbar\omega+\mu_{B}({\bf
r})){\cal D}(\hbar\omega+\mu_{F}({\bf r}))$, $\Theta$ is the Heaviside step
function, and ${\cal D}(\epsilon)$ is the Fermion density of states. For a
two-dimensional Fermi liquid shell, there is a Van Hove singularity in ${\cal
D}(\epsilon)$ 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 $\mu_{B}({\bf r})$ and $\mu_{F}({\bf r})$, leading to a
characteristic decay time $\tau_{\bf r}=-h/{\rm Im}\Sigma(k,{\bf r})$. For the
parameters used in Fig. 1, (with $U_{BF}/h=1~{}{\rm kHz}$) the time scale for
the persistence of the flow near the edges (away from the Van Hove
singularity) is $\tau_{\bf r}\approx 13~{}{\rm ms}$. However, near the center
of the Fermi liquid region (close to the Van Hove singularity) $\tau_{\bf r}$
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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