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the trapping potential does change between both sites. Hence, within the simulator region the physics is hardly influenced by the trap, cf. Fig. 1. Below we will show that this region is accurately described by the homoge- neous t - J model with controlled hole fraction x . The t - J model [19] (cf. [20] for other cold atom applications) de- scribes the doped fermionic Mott-insulator, with fermion hopping t and superexchange spin coupling J ( ≈ t/ 3 in cuprates [21]). Second-order superexchange is crucial for the physics of the doped fermionic Mott-insulator. We want to emphasize that our proposal, in which both spin coupling and hopping originate from superexchange, does not involve any process above second order. Our scheme is, in a sense, contrary to the slave-boson approach to the t - J model (where auxiliary bosons describing holes are introduced as a purely theoretical concept): We propose a physical system of real bosons and spin 1/2 fermions that behaves as a system of fermions alone. 

Before presenting the details of our proposal, we want to mention that recently another, interesting solution to problem (ii) of controlled hole doping [not to the issue of temperature, problem (i)] has been proposed [22]: The authors suggest realizing an attractive Hubbard model, which for a bipartite lattice can be mapped to a repulsive one (see, e.g., [23]). In particular, an imbalance of ↑ and ↓ fermions for attractive interaction, being relatively easy to control experimentally, corresponds to hole or particle doping in the repulsive model. Unfortunately, at the same time, the trap felt by the attractive fermions transforms to an inhomogeneous magnetic field, favoring a spatial separation of repulsive ↑ and ↓ fermions. 

Let us now describe in detail our approach to controlled hole doping. We consider a mixture of N$_{f}$ ≡ N$_{↑}$ + N$_{↓}$ fermionic and N$_{b}$ bosonic atoms ( N ≡ N$_{f}$ + N$_{b}$ ) in an op- 

tical lattice, with the fermions having equally populated “spin” states s = ↑ , ↓ . The system is described quantita- tively by the Hubbard model [2] 

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Here ˆ f † is and ˆ b † i denote creation operators for fermions and bosons at site i . We also define number opera- tors ˆ n$_{is}$ ≡ ˆ f † is ˆ f$_{is}$ , ˆ n$_{ib}$ ≡ ˆ b † $_{i}$ˆ b$_{i}$ , ˆ n$_{if}$ ≡ ˆ n$_{i}$$_{↑}$ + ˆ n$_{i}$$_{↓}$ , and ˆ n$_{i}$ ≡ ˆ n$_{if}$ + ˆ n$_{ib}$ . The first line of Eq. (1) comprises fermion and boson tunneling between neighboring sites, with pos- itive matrix elements t f ij and t b ij (depending exponentially on lattice depth [2]). The second line takes care of the repulsive on-site interaction between the different types of particles, with Hubbard energies U $^{ff}$, U $^{fb}$, and U bb (weakly depending on lattice depth and being propor- tional to the corresponding s -wave scattering lengths [2]). Finally, the third line includes co-centric trapping poten- tials V f i ≡ V$_{i}$ + δV$_{i}$ and V b i ≡ V$_{i}$ for fermions and bosons [ V f i ≡ V b i ≡ 0 in the center], with δV$_{i}$ to be tuned small. 

We are interested in the parameter regime giving rise to an extended mixed Mott region, with one particle (boson or fermion) per site, in the center of the trap. We assume a cubic lattice of spacing d with site loca- tions d r $_{i}$, as well as a parabolic confinement V$_{i}$ = 1 $_{2}$α r 2 $_{i}$. We also define both U$_{min}$ ≡ min( U $^{ff}$, U $^{fb}$, U $^{bb}$) and t$_{max}$ = max( { t f $_{ij}$} , { t b $_{ij}$} ). Temperatures are well lower than t$_{max}$ . Now, t$_{max}$ ≪ U$_{min}$ and µ < U$_{min}$ (while t$_{max}$ ≪ U$_{min}$ − µ ), with chemical potential µ , guaran- tees strong suppression of double occupancy. On the other hand, the trap V$_{i}$ prevents vacancies from enter- ing the central region: the occupied region of radius ρ ≈ (3 N/ 4 π ) 1 / 3 features only a thin shell (a few d wide) of reduced particle number, provided t$_{max}$ ≲ αρ ≡ Δ V$_{edge}$ , where Δ V$_{edge}$ is the energy difference between sites at radius ρ and ρ + 1. With the above requirements ful- filled, the chemical potential for bosons and fermions is basically given by the potential energy needed to place a particle at the edge of the occupied region, µ ≈ 1 $_{2}$αρ$^{2}$. One has Δ V$_{edge}$/µ = 2 ρ . The radii ρ might range from 10 to 30. All in all, t$_{max}$ ≲ Δ V$_{edge}$ ≪ µ < U$_{min}$ sum- marizes the parameter regime assumed here. In practice, t$_{max}$/U$_{min}$ can be adjusted via the lattice depths, while µ/U$_{min}$ can be tuned to a value of 1/2, say, by varying the trap depth α . 

In the bulk of the mixed Mott region, at each site i , strong repulsion in combination with the trapping con- finement V$_{i}$ gives rise to the constraint 

with the overall boson fraction x ≈ N$_{b}$/N . The system 

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