The basic building blocks of atoms, molecules and solids are positively charged nuclei and negatively charged electrons. Their mutual interactions determine most physical and chemical properties of matter, such as the electrical conductivity or the absorption of light. The laws that guide this delicate interplay between electrons and nuclei are those of quantum electrodynamics (QED), in which particles interact via the exchange of photons, which are the quanta of light. However, the equations of QED are so complex that in practice we have to simplify them to be able to make any prediction for real materials. A very common simplification in quantum chemistry and solid-state physics is to neglect the quantum nature of light. Although this assumption works well for many applications, recent experiments have uncovered situations where the photons can dramatically change the material properties and give rise to new collective behavior and phenomena.

### Quantum-Electrodynamical Density-Functional Theory

In order to simulate such situations on a computer we are developing a novel theoretical approach that also includes the interaction with photons. The main idea is to employ a recent density-functional reformulation of QED in terms of coupled quantum fluids, which we termed Quantum-Electrodynamical Density-Functional Theory (QEDFT). This reformulation allows to represent the coupled matter-photon system by two uncoupled, yet non-linear equations. The resulting Maxwell-Kohn-Sham equations employ effective fields that fully take into account the particle-particle (Coulomb) and the particle-photon interactions. This allows simulating situations where the photon-matter coupling becomes important and can change the properties of materials dramatically.

In the theory department we advance the foundations of QEDFT, construct the necessary approximations to the effective Kohn-Sham fields and develop an accurate as well as efficient numerical implementation of the resulting Maxwell-Kohn-Sham equations.

### Many-Body Perturbation Theory of Photon-Matter Systems

Many-body perturbation theory (MBPT) opens the possibility to construct approximations to every desired order in the interaction of photon-matter systems. A significant advantage is incorporated in the possibility to explicitly construct conserving approximations via partial summation of the infinite series of Feynman diagrams, leading to accurate results depending on the relation between system and summation scheme. The drawback is an in general non-local interaction in space and time, and it is therefore a demanding task to apply it to realistic systems in its accurate non-local form. In combination with QEDFT, the MBPT allows a systematic construction of Kohn-Sham potentials that describe the coupling to photons. We therefore investigate the connection between MBPT and density-functional approaches to coupled matter-photon systems.

### Control Theory

The density-functional reformulation of QED also provides a novel quantum-control approach that allows to temporally and spatially control particles coupled to photons. Hence, a further research activity of the theory department is to design control fields that bring complex quantum systems into new states of matter by hybridization with the photon field. The signatures of the quantum effects of light in ultrafast optical measurements have great potential for use in spectroscopic applications to resolve individual pathways of matter and predict novel quantum effects. The quantum nature of light is essential for achieving this degree of control.

### Correlated Spectroscopy

Systems composed of strongly interacting matter-photon degrees of freedom offer a rich setting for the emergence of novel correlated multi-mode properties wherein suitable quantum control via individual constituent modes can be exercised. In this ongoing research effort we focus on the spectroscopic investigation of such systems.

We aim to develop an ab-initio framework, e.g. QEDFT, lattice QED to describe extended, correlated photon-matter multi-mode dynamics on which quantum-field spectroscopy can be performed systematically. The eventual goal lies in performing few photon quantum optics with molecules, probing of topology of molecular potential energy surfaces, and active control of nonlinear responses.

### High-Harmonic Generation

Under ultrashort and very intense laser pulses, matter reacts in a nonlinear way. In this context, the most investigated phenomenon is high-harmonic generation (HHG), where the system reacts by emitting harmonics of the incoming laser. This phenomenon has been studied in atoms and molecules for decades and it has been recently observed in solids, opening the door to numerous applications such as a tabletop X-ray source or synchrotron. In spite of the impressive number of studies already published in a short time, using mostly simplified empirical few-band models and non-interacting electrons, HHG in solids is still not fully understood, and the contributions of major physical effects have not yet been explored. Time-dependent density functional theory (TDDFT) is a framework of choice in this context, especially to understand the microscopic mechanism responsible for HHG in solids.

In the Theory Department, we develop the ab-initio quantum mechanical methods and numerical tools to obtain fundamental insights concerning the leading mechanisms responsible for HHG in solids.

### Nanoplasmonics and Surfaces

The coupling of light with the free carriers of a metal or a doped semiconductor is used to built waveguides which confine and manipulate photonic signals. The incident photons can excite other quasiparticles, such as excitons or phonons, and all excitations can strongly couple to the light and form polaritons. In order to achieve the desired functionality, the material of choice needs to be structured on the nanometer scale, where quantum confinement effects become very important. Also low-dimensional materials such as graphene or nanoparticles are used.

Yet, the modeling of these nanoplasmonic systems is very challenging because of the need to bridge several time and length scales. Time-dependent density functional theory can not only be used to describe plasmon excitations and other light-matter interactions but also allows for the calculation of nonlinear phenomena. The accuracy, however, depends on the chosen exchange-correlation kernel. Many-body perturbation theory, on the other hand, is able to predict absorption peaks with high accuracy and can describe excitons with the help of the Bethe-Salpeter equation but is practically often limited to linear response and computationally demanding. Thus, to gain deeper insights and achieve a complete description of nanoplasmonic systems, a combination of these approaches is desirable. Research along these lines is carried out in the Theory Department.

Further research in this direction concerns the interaction between different quasiparticle excitations, such as the interaction of excitons with surface plasmons and surface plasmon polaritons, or the periodic driving of the quasiparticle system in order to form Floquet states.

#### Further research lines pursued in the group

- Lattice QED and QEDFT
- Exact solutions and Tensor-Network approaches of coupled matter-photon systems

#### Software development

- Extension of the real-space real-time TDDFT code OCTOPUS to QEDFT
- Development of a general purpose nuclear-electron-photon code

#### Papers on Photon-Matter Interaction

- Time-dependent Kohn-Sham approach to quantum electrodynamics; Phys. Rev. A
**84**, 042107 (2011) - Time-Dependent Density Functional Theory for Many-Electron Systems Interacting with Cavity Photons; Phys. Rev. Lett.
**110**, 233001 (2013) - Quantum-electrodynamical density-functional theory: Bridging quantum optics and electronic-structure theory; Phys. Rev. A
**90**, 012508 (2014) - Quantum electrodynamical time-dependent density-functional theory for many-electron systems on a lattice; Phys. Rev. B
**90**, 195149 (2014) - Optimized Effective Potential for Quantum Electrodynamical Time-Dependent Density Functional Theory; Phys. Rev. Lett.
**115**, 093001 (2015) - Kohn–Sham approach to quantum electrodynamical density-functional theory: Exact time-dependent effective potentials in real space; PNAS, doi:10.1073/pnas.1516362112 (2015)