4 smaller. Such low energy scales are challenging experi- mentally. However, they are not very specific to our pro- posal, but rather are a generic consequence of the fact that the t - J physics is inevitably the physics of superex- change processes. Therefore, the temperatures needed here are only moderately lower than those that would be needed in an experimental setup where—somehow— a Mott insulator of fermionic atoms can be doped with real holes in a controlled fashion. While in the latter case U $^{ff}$/tf ≫ 1 would be required, in our proposal one has U $^{ff}$/tf ≫ 2( t/J ) ≈ 6 (combine τu$_{f}$ ≃ 2 t/J and U fb ≫ t $^{b}$), reducing J by a factor of 6 only. that the presence of a trap can fundamentally change the nature of the transition between an insulating and a superfluid quantum phase. The reason is not that the trapped system is too small to sharply distinguish be- tween the two phases. It is more subtle: Because of the inhomogeneity, the transition occurs locally by displac- ing the interface separating spatial regions of different bulk phases. However, this spatial interface might be a smooth crossover rather than a sharp transition. A direct consequence can be that the adiabatic passage between two quantum phases is greatly facilitated by the trap. A simple lattice model that would allow to study such phenomena is given by spinless hard-core bosons on a square lattice with nearest-neighbor repulsion U$_{nn}$ and hopping t . It features a transition from a checkerboard insulator at half filling and small t/U$_{nn}$ to a superfluid phase [30]. The model (being equivalent to an XXZ spin 1/2 model) can be realized experimentally by creating a mixed Mott insulator having one boson of either species s = 1 or 2 on each site [17]. Mapping ˆ b † i ≡ ˆ b † i $_{1}$ˆ b$_{i}$$_{2}$ , the sys- tem is described by a single type of hard-core boson only, with U$_{nn}$ = ( t 2 1 + t 2 $_{2}$) /U$_{12}$ − 4 t 2 $_{2}$/U$_{22}$ , t = 2 t$_{1}$t$_{2}$/U$_{12}$ and residual trapping potential W$_{i}$ = V$_{i}$$_{1}$ − V$_{i}$$_{2}$ . Here U$_{ss}$ ′ , t$_{s}$ , and V$_{is}$ denote on-site interaction, tunneling, and traps for the species s = 1 and 2. Now W$_{i}$ can be tuned con- stant without making particles leak out. Thus, a ramp of t/U$_{nn}$ at different rates can now be performed both with and without parabolic potential, and the degree of adiabaticity in passing the transition can be compared. At hole doping x ≲ O (0 . 01) and temperatures k$_{B}$T ≲ J , the homogeneous t - J model (9) is known to give rise to (quasi-)long-range antiferromagnetic N´eel order. At larger doping x ∼ O (0 . 1) and even lower temperatures, the model is conjectured to possess yet another ordered phase, namely the d$_{x}$ $^{2}$− y 2 -wave superconducting one ob- served also in the cuprates [9, 10]. Superfluidity of ˆ c - fermions (that is in the t - J model) is connected to super- fluidity of both ˆ f -fermions and ˆ b -bosons [28]; both have to be probed. The bosonic superfluidity is related to a (quasi)condensate at quasimomentum k = 0, visible in time-of-flight absorption images. $^{2}$. In order to verify d -wave superfluidity of the fermions one can resort to different schemes for measuring quasiparticle excitations (as well as their interference) that have been proposed for cold atom systems for this purpose [11, 29]. We have proposed a robust implementation of a quan- tum simulator for the homogeneous t - J model with well controlled hole doping, using a sample of ultra- cold bosonic and fermionic atoms in an optical lattice. We believe that—once the necessary temperatures are realized—our scheme can serve to gain crucial insight into the physics of strongly correlated quantum matter. Moreover, realizing a homogeneous bosonic system allows one to investigate experimentally the role of inhomogene- ity when passing from one quantum phase to another. In the cuprates, at optimal doping, superconducting behavior appears at temperatures that are at least one order of magnitude smaller than J [10, 21]. It is challeng- ing to achieve these temperatures with cold atoms, and novel cooling techniques, as those proposed in Refs. [14], have to be implemented. The system has to be divided into a low-entropy part (lacking low-lying excitations) and a part carrying most of the entropy (a metallic shell or a boson-gas reservoir). The latter is removed before the system is adiabatically transferred into the desired state. In our fermion-boson system, a suitable gapped low-entropy phase would be an insulator of two fermions and an integer number of bosons at each site, to be cre- ated by making the trap steep and the lattice deep. We thank P. Massignan, A. Muramatsu, V. Ahufinger, and A. Sanpera for discussion. Support by the Spanish MICINN (FIS2008-00784, FIS2007-29996-E), the A. v. Humboldt Foundation, ERC Grant QUAGATUA, and EU STREP NAMEQUAM is acknowledged. Finally, we would like to mention that neutralizing the trapping potential the way described here is also possible and interesting for bosons. In Refs. [16] it has been shown 2 Bosons and ↑ and ↓ fermions can be imaged separately by se- lective absorption or Stern-Gerlach separation. Fisher, Phys. Rev. B 40 , 546 (1989). [5] M. Greiner, O. Mandel, T. Esslinger, T.W. H¨ ansch, and I. Bloch, Nature 415 , 39 (2002). [6] S. Trotzky, L. Pollet, F. Gerbier, U. Schnorrberger, I. Bloch, N. Prokof’ev, B. Svistunov, and M. Troyer, arXiv:0905.4882 (2009). [7] C. Lhuillier, arXiv:cond-mat/0502464 (2005); F. Alet, A.M. Walczak, and M.P.A. Fisher, Physica A 369 , 122 [1] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen(de), and U. Sen, Adv. Phys. 56 , 243 (2007); I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80 , 885 (2008). [2] D. Jaksch, C. Bruder, J.I. Cirac, C.W. Gardiner, and P. Zoller, Phys. Rev. Lett. 81 , 3108 (1998). [3] R. Feynman, Int. J. Theor. Phys. 21 ,467 (1982). [4] M.P.A. Fisher, P.B. Weichman, G. Grinstein, and D.S.