Density-potential mapping in the standard and quantum electrodynamical time-dependent density functional theory

This thesis is devoted to the formulation and implications of the time-dependent density functional theory (TDDFT). The work is divided into two main parts. In the first part we develop rigorous theorems for the density-potential mapping in quantum many-body systems on a lattice. We prove the uniqueness of the TDDFT map and demonstrate that a given density is v-representable if both the initial many-body state and the density satisfy certain well-defined conditions. In particular, we show that for a system evolving from its ground state, any density with a continuous second time derivative is v-representable . Then the lattice TDDFT formulation is extended to cover a system of interacting lattice electrons strongly coupled to cavity photons. We prove that under some mathematical conditions the electron-photon wave function is a unique functional of the electronic density and the expectation value of the photonic coordinate. Then we further generalize the ground state v-representability theorem to include the ground state of a general lattice electron-photon Hamiltonian. The second part of this thesis is focused on the applications of the analytic densitypotential maps in TDDFT and the current-potential map in the time-dependent current density functional theory (TDCDFT). We use the analytic lattice map to analyze and quantify the role of non-adiabaticity (”memory effects”) in the exchange-correlation functional for describing non-linear dynamics of many-body systems. Studying time-dependent resonant processes using the available functional in TDDFT is a big challenge and that is due to their strong nonlinear and non-adiabatic character. Here we study the Rabi oscillations within the solvable 2-site Hubbard model as an example for the resonant processes. We construct the exact adiabatic exchange-correlation functional and show that it cannot reproduce resonant Rabi dynamics correctly. It turns out the non-adiabatic contribution to the exact exchange-correlation potential is significant throughout the dynamics and it is small only when the ground state population is significant. Afterwards we reconstruct the exact time-dependent exchange-correlation functional within the two-level approximation. This fully non-adiabatic and explicit density functional captures Rabi dynamics both for resonant and detuned oscillations. Finally we apply the analytic current-potential maps of TDCDFT in a completely different context. We use them to inverse engineer analytically solvable time-dependent quantum problems. In this approach the driving potential (the control signal) and the corresponding solution of the Schrödinger equation are analytically parametrized in terms of the basic TD(C)DFT observables. We describe the general reconstruction strategy and illustrate it with few explicit examples ranging from real space one-particle dynamics to controlling quantum dynamics in a discrete space. In particular we construct a time-dependent potential which generates prescribed dynamics on a tight-binding chain. We also apply our method to the dynamics of spin-1/2 driven by a time-dependent magnetic field and construct an analytic control pulse for the quantum NOT gate.