Consistent Mori-Projector Theory in Two Dimensions

Quantum many-body systems are traditionally studied in thermal equilibrium where the methodology of statistical mechanics has proven very successful in describing them. In recent years, an increasing number of experiments in diverse technologies including superconducting circuits, semiconductors and atomic Rydberg media are however giving rise to a new, largely unexplored class of quantum many-body systems with interacting particles in driven and dissipative, non-equilibrium regimes. These pose a new scientific challenge and thus generate a need to develop novel techniques that are capable of modelling them efficiently and accurately. I recently started to develop a new technique called consistent Mori Projectors (c-MoP) which derives coupled but non-linear equations of motion for the reduced density matrices of the subsystems of a many-body system and thus leads to an efficient description of local quantities. Starting from my promising initial results, I will in this project derive a powerful theory for the efficient description of two-dimensional systems in non-equilibrium scenarios that is based on the c-MoP concept. I expect this achievement to open up possibilities for modelling two-dimensional quantum many-body systems that hitherto defy a tractable but accurate description. As first applications that exploit these new possibilities I will employ the developed techniques for calculating stationary states and their phase diagram for driven-dissipative Bose-Hubbard and Jaynes-Cummings-Hubbard models, which play a central role among driven-dissipative quantum many-body systems. The results of the project will provide a basis for my future research plans and and will lay foundations for future technologies that exploit quantum effects.

This project will develop a versatile and powerful theory framework for the description of quantum many-body systems in driven and dissipative non-equilibrium scenarios and explore first applications of these methods to models of central interest. The impact of the project will be most direct in related academic communities where its results can be applied or may enable further progress. l It will also accelerate the development of the research group I am currently building up at Heriot-Watt university. I here summarise the project's impact, grouped according to the categories "science" and "people", where direct impact can be expected, as well as the category "technology and society", where the impact will be of a more indirect nature.

Science: The development of methods that are capable of modelling large quantum many-body systems is an important research area with the most prominent efforts focusing on numerical methods such as the Density Matrix Renormalisation Group (DMRG) and Tensor Networks or on quantum simulation, i.e. modelling one quantum system experimentally with another one that offers better control and measurement access. The theory I will develop in this project uses an alternative strategy and its successes will thus trigger "out of the box" thinking in these mainstream research communities. In particular my plans to develop hybrid approaches that exploit synergies between the c-MoP approach of this project and well established techniques, such as DMRG, will have significant impact on efforts for developing numerical tools that can successfully simulate quantum many-body systems on two-dimensional lattices.
Moreover, the c-MoP theory I will develop may find applications in the description of elementary biological systems such as energy transfer complexes in photosynthesis and my work will thus have impact on the emerging field of exploring possible quantum effects in biological systems.

People: The grant will also create immediate benefits for people who are involved with it or closely interact with me. The RA to be hired with the project funds will be supervised and trained by myself and receive training in complimentary approaches during visits to collaborators. Moreover, the PhD students in my group will receive training that ideally prepares them for an academic career. The PhD students will also receive training in many skills that are highly transferable, e.g. into the area of modelling of complex systems. I personally will benefit from the grant as it will contribute to building up a research group at Heriot-Watt university.

Technology and Society: Modern technological devices are characterised by a growing degree of complexity in the sense that ever more units are integrated into single devices. As this project will derive efficient methods for predicting properties of interest for complex systems with exceedingly many degrees of freedom, its techniques may thus find applications or at least influence the development of related approaches in the engineering and modelling of complex systems in technology. Further areas where the project may influence the development of simulation algorithms comprise climate models or social and economic networks.