Effective Evolution of Open Quantum Systems

科学家用数学来描述各种现象。天气预报、政治民调、金融和消费市场的演变、交通流量，都是由特定的数学方程建模的。解这些方程式往往等同于预见未来，并让我们洞察一个人将如何影响未来。当然，预测和操纵未来并不是那么简单。一方面，我们只能建立现实的近似模型，无法捕捉到所有方面。一般来说，更复杂的模型能更好地描述自然，但由此产生的方程变得(无可救药地)难以求解。然后，人们被引导得出有效的方程，这些方程只关注系统的一小部分，而不必描述它所接触的复杂环境的每一个微小细节。我的研究重点是遵守量子理论薛定谔方程的系统，例如基本粒子之间的相互作用(电子、原子核、原子、分子)和物质与辐射的相互作用(电磁场、光)。长期以来，量子理论一直在数学、物理、化学以及最近的生物学中扮演着重要角色。为了降低复杂性，人们通常只描述系统的一部分，比如分子中的特定原子或固定在衬底上的几个量子比特(量子计算机的基本构件)。这一部分(原子、量子比特)被称为开放系统，因为它与称为环境(分子、底物)的复杂的外部系统相互作用。我的目标是(1)推导出开放量子系统的有效方程，(2)分析这些方程的解。在过去的几年里，我与其他科学家合作，一直在开发量子动力学共振理论，以实现这些目标。除了被不断进步的基础研究所激励外，我还希望能够找到在应用中有用的新技术。例如，我们最近推导并求解了一个描述藻类光合作用中发生的化学过程的有效量子方程。我们发现了过程速度与某些参数(温度、光强……)的精确关系。因此，现在可以尝试优化速度来加速反应，这与生物燃料生产实验有关。虽然我与理论物理学家、化学家和生物学家合作，但我的方法是基于严格的数学方法。它涉及数学分析的工具，如(无限维非正规)算符的谱理论，C*-和冯·诺伊曼代数理论，量子场论和概率论。在接下来的五年里，我将进一步推动动力共振理论。我将采取的方向是受到数学创新、创造新技术和物理相关性的推动，解决与相关学科的科学家合作确定的关键问题。

Scientists use mathematics to characterize all kinds of phenomena. Weather forecasting, political polling, the evolution of financial and consumer markets, traffic flow, are all modeled by specific mathematical equations. Solving the equations is often tantamount to seeing the future and gives insight into how one could influence it. Of course, predicting and manipulating the future is not quite so simple. For one thing, we can only build approximate models of reality, unable to capture all aspects. Generally, more complex models describe nature better, but the resulting equations become (hopelessly) hard to solve. One is then led to come up with effective equations, which focus on a small part of a system only, without having to describe every little detail of the complex surroundings with which it is in contact. My research focuses on systems obeying the Schrödinger equation of quantum theory, such as elementary particles interacting with each other (electrons, nuclei, atoms, molecules) and matter interacting with radiation (electromagnetic field, light). Quantum theory has long played an important role in mathematics, physics, chemistry and more recently also in biology. To reduce complexity, one often describes part of a system only, say a specific atom in a molecule or a few qubits pinned on a substrate (basic building blocks of a quantum computer). That part (atom, qubits) is called an open system, as it interacts with a complex `outside' system, called an environment (molecule, substrate). My continuing goal is to (1) derive effective equations for open quantum systems and (2) analyze the solutions of those equations. In collaboration with other scientists, I have been developing the quantum dynamical resonance theory to reach these goals over the past few years. Other than being motivated by progressing fundamental research, I hope to be able to find new techniques useful in applications. As an example, we recently derived and solved an effective quantum equation describing chemical processes happening in photosynthesis in algae. We found how exactly the process speed depends on certain parameters (temperature, light intensity...). So now one might try to optimize the speed to accelerate the reaction, which is relevant in biofuel production experiments. While I collaborate with theoretical physicists, chemists and biologists, my approach is based on mathematically rigorous methods. It involves tools from mathematical analysis, like the spectral theory of (infinite-dimensional, non-normal) operators, C*- and von Neumann algebra theory, quantum field theory and probability theory. Over the next five years, I will push the dynamical resonance theory further. The direction I will take is motivated by mathematical innovation, creating new techniques, and by physical relevance, addressing crucial issues identified in alliance with scientists of related disciplines.