\chapter{Conclusion and outlook}

This thesis explores the bridge between the theories and experiments of three-dimensional optical lattices, through large-scale computational numerics in the quantitative aspect.
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\noindent Experiments with bosons at equilibrium have been largely successful in developing state-of-art techniques to probe for time-of-flight images \cite{trotsky}, as well as in-situ density profiles up to the precision of single-site resolution \cite{bloch1}.
On the other hand, the theoretical picture of describing by the boson Hubbard model has been quantitatively confirmed with numerical-exact large scale Quantum Monte Carlo simulations implemented in the directed worm algorithm \cite{prokofev1, pollet1}, at least for the time-of-flight images \cite{trotsky}. 
New theoretical proposals are currently made feasible due to the high precision in experiments, in particularly the fluctuation-dissipation thermometry scheme \cite{zhou1, thermometry}.
In this thesis, we perfected this concept proposed by Zhou and Ho \cite{zhou1} for bosonic optical lattices in reality by reducing the undesirable noise through window sizing technique.
This modified scheme has been numerically tested to give realiable estimates of the thermodynamic temperature even in the presence of various experimental imperfections, therefore proving itself to be a feasible quantitative thermometer for optical lattices.  
Highlighted in this thesis is also the wing thermometry scheme which is numerically tested to work for large enough normal-wings of the bosonic cloud.
As the bosonic cloud increases in size for future experiments, the normal-wings get spread out due to waist effects, therefore enhancing the feasibility of the wing thermometry scheme in the soon coming future.
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\noindent Future experiments with bosons will involve a much bigger bosonic cloud with different lattice geometries at lower temperatures, thereby challenging the theoretical limits of numerical computation for direct quantitative validations.
Facing this challenge, we need therefore an ultra-robust implementation of the QMC directed worm algorithm \cite{pollet1} that can handle simulations of that monstrous scale.
For the benefit of the scientific community, we decide to publish our code\footnote{
The DWA code is published in reference \cite{alps_webpage} as a part of the ALPS project. 
It is written in C++ for maximal performance, and interfaced in Python for easy data management and analysis.
It has been designed for easy automation of large-scale simulation chains.
With ALPS, the DWA code can now handle robustly large datasets, as well as lattice geometries of all kinds.
}
which has been designed for such purposes.
Its performance is ultra-robust, such that it could handle simulations of up to some 10 million lattice sites numerical-exactly, and some of which manage to converge in a couple of hours.
In fact, this is definitely one of the fastest, if not the fastest, in the world till date.
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\noindent Luck, on the other hand, is not (yet) with the fermions.
Current experiments with fermions are still relatively hot to realize the conjectured antiferromagnetic or superconducting phases predicted from theory.
While the experimentalists are tediously cooling the fermions, the theoreticians are also questing to solve the Hubbard model which is believed to be the theory behind high-Tc superconductors as well as other strongly-correlated materials. 
Handicapped by the notorious negative-sign problem for fermions, Quantum Monte Carlo methods prove to be pretty undesirable.
Motivated by the success of density functional theory for conventional materials, we apply it for the first time to ultracold fermions in a shallow optical lattice.
We find that Stoner ferromagnetism, which was recently discovered to exist in ultracold fermionic gases \cite{mit}, gets stablized by the optical lattice due to band structure effects. 
Besides, the antiferromagnetic phase is recovered qualitatively in the Hubbard model limit at half-filling by density functional theory, in which we find the existence of a spin-density-wave gap that could be used as an indirect probe for antiferromagentism in future experiments.
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\noindent Perhaps not in the distant future would optical lattices be eventually realized as quantum emulators, thereby solving the long theoretical puzzle of strongly-correlated physics.
Only by then could new materials with surprisingly exotic properties be predicted and engineered, which is in fact the ultimate aim of the optical lattice emulator (OLE) project.
Today, optical lattices provide a vast unprecented opportunity to explore also the non-equilibrium dynamics of quantum many-body systems, and it is the view of the author that we will embrace a new era of quantum engineering for our human civilization in the coming decades.

