… in which we take a wild left turn into the mysterious world of quantum simulations!

This article is a lengthy review article on a topic that I have no clue about, so I’m going to do my best to digest it bit by bit. The topic is on using ultracold quantum gases to precisely simulate an assembly of quantum particles. The reason why there is interest in performing quantum simulations, rather than quantum computations, for these particles is because the cost of describing quantum particles increases exponentially with additional particles. Therefore, quantum physics that is primarily interesting on the large scale, such as high temperature superconductors, are very difficult to do a full computation of. It is sort of similar to doing a numerical integration, where rather than trying to get the full analytical result, you compute small steps of the function and add them together.

This paper discusses the ability for quantum gases to be finely controlled to simulate complex quantum states. There are three basic requirements for quantum simulations to work properly:

- The quantum simulator should map to the proper Hamiltonian of the gas in question
- The simulator should be able to be set into an well-identified state
- The simulator needs to be able to carry out measurements with high precision

If all three of these requirements are met, then quantum simulations can be valid. After laying out the requirements, the paper begins to analyze why quantum gases meet these requirements. They analyze the use of quantum gases under three different regimes: Feshbach resonances, control of energy landscape, and control of the topology in which the quantum fluid evolves. The first of these corresponds to ultracold Fermi gases, the second corresponds to optical lattices, and the last corresponds to artificial gauge fields. I realize that I understand none of these, so let’s try to at least grasp the first of these topics, especially since it will be relevant to my research this summer!

The Feshbach resonances are linked to ultracold Fermi gases, or near 0K clouds of fermions. Fermions are particles with spin 1/2 that obey Fermi-Dirac statistics, as compared to integer spin Bosons that obey Bose-Einstein statistics. Some familiar principles that apply to fermions include Pauli’s exclusion principle, which prevents identical fermions from occupying the same quantum state, which leads to degeneracy pressure in the middle of neutron stars. Ultracold fermi gases cannot form Bose-Einstein Condensates, because BECs require bosons that can collapse to the same quantum state, while fermions cannot.

The paper discusses the use of Feshbach resonances to “tune the strength of interactions between atoms over several orders of magnitude by means of an external magnetic field.” I’m not entirely sure how this mechanism is able to work, but it appears that the ultracold Fermi gases can be characterized with just a few parameters, one of which is their scattering length. When the strength of interactions is adjusted through the Feshbach resonance, this parameter is able to be adjusted.

For the weakly attractive interactions, the gas is able to be understood through the Bardeen-Cooper-Schrieffer theory, or BCS. BCS is applied because the particles are able to weakly pair into Cooper pairs, a quantum phenomenon that is important for understanding superconductivity. Cooper pairs have opposite spin and velocity, but the actual gas particles are not constrained to be close together. Instead, they can have some distance between them and are bound weakly. On the other hand, for very strongly attractive interactions, the Fermi particles are able to pair together to form a pseudo-bosonic particle which then acts as a BEC. Since there are different behaviors at either extreme, there is also a crossover region in the middle, called the BEC-BCS crossover. In this region, it is very difficult to get analytical results. Therefore, it is easier to create an ultracold Fermi gas, tune the magnetic field to affect the interaction strength through the Feshbach resonance, and then create a state that can then be measured.

One such measurement that can be made is the proportionality constant that relates the chemical potential with the Fermi energy. Because this ratio is used in many other contexts beyond any specific Fermi gas, it is important to determine. However, it is rather difficult to compute analytically. Therefore, it is better to just measure the constant within the gas, instead of predicting it from something else. In regions especially close to the unitary limit, where the BEC-BCS crossover occurs, it can be much better to use a quantum simulator than using a quantum computer.

That concludes this portion on ultracold Fermi gases, and I don’t think I can really move on to the optical lattices and gauge fields! I might return to this review article on another date, and get more into it :)

Reference: Bloch, I. *et al. *Quantum Simulations with Ultracold Quantum Gases *Nature Phys. ***8 **267-276 (2012)