Soon Hoe Lim
Applied Mathematics GIDP
On Site Research Dates
June 1 to June 30, 2017
Title of Research Project:
"Multiscale Dynamics for a Class of Quantum Brownian Motions"
During the month of June in 2017, I visited the Institute of Photonic Sciences in the seaside town of Castelldefels in the metropolitan Barcelona area. The institute is also known, in abbreviation, as ICFO (Institut de Ciencies Fotoniques) in Spanish and was established in 2002 by the Government of Catalonia and the Technical University of Catalonia – Barcelona to perform both fundamental and applied research to advance photonics, i.e. the science and technology of harnessing light. The institute was not new to me as I had previously visited there two years ago. The first visit ignited a research collaboration that kick-started most of my PhD projects. I am grateful to be able to visit ICFO this summer again to resume the collaboration, thanks in part to the Raphael and Jolene Gruener Research Travel Award.
During my visit there, I collaborated with the members of the Quantum Optics Theory group led by Dr. Maciej Lewenstein, a long time collaborator of my PhD advisor, Dr. Jan Wehr. We worked on several projects that aim to investigate the dynamics of a class of open quantum systems known as quantum Brownian motion (QBM). QBM describes a Brownian particle (the central system of interest) interacting with a heat bath (the environment) in thermal equilibrium. Such systems are quantum generalizations of the ones discovered by Robert Brown and studied by Albert Einstein, Marian Smoluchowski, among others. In particular, we are interested in more realistic models where the heat bath is inhomogeneous. A throughout understanding of such systems is a stepping stone to understand many interesting, experimentally realizable quantum systems, for instance a quantum impurity interacting with a Bose-Einstein condensate.
I spent most of my time at ICFO to study the problem of Smoluchowski-Kramers limit (or the small mass limit) of various variants of the QBM models. I also presented a seminar talk on the problem to the ICFO community. The problem can be roughly described as follows. It is often desirable to derive effective and tractable mathematical models that reduce the number of degrees of freedom of the original model, while still capturing its complex nature. Such mathematical models are studied by using stochastic differential equations (SDEs), which reduce the degrees of freedom of the environment to a single noise term. This motivates our plan to study various effective dynamics of the QBM model – one of which is to derive a SDE that describes the limiting dynamics when the mass of the Brownian particle becomes small. Solving such problem not only requires technical tools from mathematics but also physical insights from physicists. Working with Dr. Lewenstein and his group members has helped me to gain valuable physical insights into the QBM models. The collaboration was a successful one and culminated in a research preprint, which is available at arXiv – “On the Small Mass Limit of Quantum Brownian Motion with Inhomogeneous Damping and Diffusion” (arXiv:1708.03685).
The results that we obtained in the research preprint are significant as they extend the existing results for classical systems and exhibit the important roles played by the quantum nature of the noise. These results are also important from my academic career perspective, as they will constitute the main parts of my PhD dissertation. However, as Einstein famously said “The more I learn, the more I realize how much I don't know”, more interesting research questions followed up upon my visit there. It is my greatest hope to continue this fruitful collaboration and push the boundaries of knowledge about the QBM models. Overall my visit to ICFO was productive and intellectually stimulating. As ICFO is an amazing place that provides me the ideal setup to conduct top class research in quantum sciences, I very much hope to return there as a researcher in the near future.