DJ Strouse

Project Title: The Nature of Equilibration in the Quantum World
Advisor: Professor Paolo Zanardi (USC & Institute for Scientific Interchange, Torino, Italy)
Collaborator(s): Lorenzo Campos Venuti (ISI Quantum Information Group)
Sponsorship: USC Provost Summer Research Fellowship

Timeline: Summer 2009

For Grandma*:
    Thermalization is the property of a large, dynamic system to still have certain, static values. For instance, the air particles in a jar might be constantly moving and bumping into one another, but we can still speak of a constant temperature of the air in this jar because the average kinetic energy for these particles is well-defined. We say that the air in the jar has reached thermalization. Yet, as mentioned in the description of my project at IQC, tiny systems dominated by the strange effects of quantum mechanics tend to act very differently than the everyday systems that we are used to. For example, perfect thermalization is impossible in quantum systems. So for tiny quantum systems, we need a new definition of thermalization. This is important because many techniques in physics are developed for studying systems that have thermalized, so to better analyze tiny quantum systems, we need to understand their thermalization behavior. I spent the summer of 2009 working in Torino, Italy on how to define a quantum version of thermalization.

For Einstein:
    The unitary dynamics of quantum mechanics do not allow convergence in a strong sense of quantum states to equilibrium states, since a closed quantum system cannot “relax” into thermal equilibrium. How then should we definite quantum equilibration? A good place to begin is to borrow from probability theory other definitions of convergence, such as convergence in probability, and study the equilibration properties of quantum systems under these alternative definitions.
    Throughout the summer of 2009, I worked with USC Professor Paolo Zanardi and his colleague Lorenzo Campos Venuti in Torino, Italy to define and justify an appropriate definition of thermalization in quantum mechanics. We focused our studies on quantum spin systems such as the XXY and Ising models. Drs. Zanardi and Venuti had already studied the Ising model and my role was to explore the robustness of their results and to extend their analysis to the XXY model.
    Just to give you a taste of the strange things that can happen in quantum systems, here’s an interesting result that emerged from our investigation. In the example described above “For Grandma”, the thermalization is with respect to temperature, but we can also speak of thermalization with respect to other variables. In our case, we were studying the thermalization of the Loschmidt echo for various perturbations on various spin models. The Loschmidt echo acts essentially as a measure of how close the perturbed quantum state is to the original quantum state. Dr. Zanardi and his colleague Lorenzo Campos Venuti had analyzed the Loschmidt echo for a model called the Ising model in transverse field and found a strange result (Venuti & Zanardi 2009). Instead of the Loschmidt echo of the system having a well-defined average, the distribution for the Loschmidt echo had two peaks, forming a shape that resembles a Batman-hood. So instead of the system thermalizing to one well-defined average value for its Loschmidt echo, it seemed to be jumping back and forth between two different values. This is as if the box of air I mentioned in the earlier example kept hopping between 20 degrees Celius and 30 degrees Celsius without us repeatedly heating it up or cooling it down!

*”You do not really understand something unless you can explain it to your grandmother.” – Albert Einstein