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From strongly interacting atomic systems to optical lattices W. Ketterle In my comment, I discussed the frontiers of cold atom science. Ultracold atoms are the building blocks to realize novel Hamiltonians, or in other words, to explore Hilbert space. Hilbert space is vast, and so far we have explored only a small section of it. Using quantum control and cooling we will advance further. For more than a decade, cold atoms represented weakly interacting systems. Laser cooling achieved temperatures typically around 100 µK in a classical gas, far away from quantum degeneracy. Evaporative cooling led to Bose-Einstein condensation at temperatures typically around 100 nK. Bose-Einstein condensates are weakly interacting many-body systems described by mean field physics through the Gross-Pitaevskii equation. The next milestone was the cooling of fermions to quantum degeneracy. Strong interactions led to pairing and fermionic superfluidity. The exploration of the BEC-BCS crossover was a major achievement. These systems show strong pair correlations, but the pairs can still be described by a mean field approximation. Correlations become stronger when the kinetic energy is small compared to the interaction energy. Strongly correlated matter has been created in optical lattices which reduces the kinetic energy, and trough Feshbach resonances which enhance the interaction energy. Strong correlations in optical lattices lead to Mott insulator physics. All these developments focused on the external degree of freedom, motion. Superfluidity is coherent motion, Mott insulator physics is the suppression of motion by interactions. To go beyond motion is one of the frontiers of cold atom science, and this involves spin ordering in the form of quantum magnetism. One of the major goals is the realization of the low-temperature phases of the fermionic Hubbard model. Many people regard this model to be the minimum model for high Tc superconductors. So the goal is to observe d-wave superfluidity by hole doping the well known antiferromagnetic phase of the Fermi-Hubbard model at half filling. A major challenge is the temperature requirement. The Hubbard model is usually parameterized by two parameters: the tunneling or hopping 1 matrix element t, and U, the on-site interaction between two atoms occupying the same lattice site. So far, the physics which has been explored involves particle hole excitations at energy U , and direct first order tunneling at energy scale t. However, antiferromagnetic ordering is caused by exchange interactions, or second order tunneling which has a rate t2 /U . This is usually on the order of 100 picokelvin. Such temperatures have been achieved in my group in proof of principle experiments through adiabatic cooling, but not in a situation where spin ordering could have occurred. In the Fermi-Hubbard model, dwave superfluidity occurs at even lower temperatures than the Neel temperature at half doping. Therefore, an intermediate goal for the realization of d-wave superfluidity is antiferromagnetic ordering at half filling. Besides pursuing novel cooling schemes to reach lower temperatures, there are at least three possibilities to raise the phase transition temperature. (1) Use light atoms (lithium) which tunnel faster due to their smaller mass. (2) Use stronger coupling than second-order tunneling in the form of electrostatic interactions. This can be realized with Rydberg admixtures, or polar molecules which interact via strong electric dipole moments. (3) A third possibility is to realize magnetic ordering not with spins, but in the density sector. This was recently accomplished at Harvard, where an Ising model was realized where spin up and spin down correspond to different occupation numbers (zero or two) on each lattice site. Let me briefly mention two other frontiers of cold atom science. One is precision many body physics. Usually, the Hamiltonian for a cold atoms system is exactly known since the interactions in cold atomic gases are short range. Therefore, precision calculations of transition temperatures and other thermodynamic quantities are possible, and can be directly compared to experiments. It is unprecedented to have calculations for a superfluid of strongly interacting fermions at the level of a few %, but this has become possible now due to advances in quantum Monte Carlo simulations. These methods have been validated through experiments which have reached a similar level of precision. Finally, another frontier is quantum dynamics. I mentioned above that the slow speed of tunneling can be challenge (since phase transitions have a very low temperature), but it is also a blessing, because it means 2 that systems don’t equilibrate quickly. Therefore, it is possible to prepare cold atoms systems far away from equilibrium and study such states and their dynamics. 3