Grant: Discovery Grants Program - Individual

In quantum theory, physical systems exposed to external noise or forces are said to be open. These systems are highly sensitive to the interactions with their surroundings, which can lead to significant changes in their properties. Indeed, noise has the potential to disrupt useful quantum features such as entanglement and coherence, while also enhancing desired effects, such as speedup and efficiency in quantum chemical processes. Prominent examples of open systems include qubits (quantum bits) in contact with substrates and radiation, as well as light harvesting complexes within molecular environments found in plants. The dynamic behaviour of open systems differs fundamentally from their closed counterparts (without external noises). In terms of the mathematical equations predicting the behaviour of the system, while closed systems obey the famous Schrödinger equation of quantum theory, the appropriate evolution equation of open systems is the so-called master equation. Deriving master equations is a challenging task, too hard to carry out unless one makes certain simplifications or approximations, such as the commonly encountered Markovian approximation. The problem is that often, the approximations are plausible heuristically, but making (mathematically rigorously) sure that the predictions obtained using the approximate equations are a good fit for the true behaviour is very difficult.  The research proposed here aims to overcome this difficulty by developing a variety of mathematical tools that allow us to measure precisely the deviation from the true dynamics resulting from the approximations. It builds upon the progress achieved during the cycle of my previous NSERC Discovery Grant. My work advances fundamental knowledge in basic science. In the longer term, it will be useful in applications to quantum computing, quantum physics and quantum chemistry. Indeed, one of my recent achievements is the rigorous derivation of the Markovian master equation for a wide class of systems used in physics and chemistry. Unlike heuristic derivations containing uncontrollable error terms, this work provides a rigorous control of approximation errors for all times, even in the asymptotic long-time regime. I will extend this work in three directions: (1) I aim to develop a comprehensive theory for correlation effects. I want to investigate the relation between correlation (entanglement) and memory, and  to see if correlations can be used to protect quantum properties useful for quantum computation. (2) I plan on establishing a formalism in situations when the Markovian approximation fails. The exploration of this challenging realm will shed light on the rich behaviour that arises when system-reservoir correlations have long-lasting effects. (3) I will combine tools from statistical physics, linear algebra, probability theory, perturbation theory and spectral analysis to create a new approach to advance the rigorous analysis of open system dynamics.