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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
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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
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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.
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