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In this work, multi-particle quantum optimal control problems are studied in the framework of time-dependent density functional theory (TDDFT).
Quantum control problems are of great importance in both fundamental research and application of atomic and molecular systems. Typical applications are laser induced chemical reactions, nuclear magnetic resonance experiments, and quantum computing.
Theoretically, the problem of how to describe a non-relativistic system of multiple particles is solved by the Schrödinger equation (SE). However, due to the exponential increase in numerical complexity with the number of particles, it is impossible to directly solve the Schrödinger equation for large systems of interest. An efficient and successful approach to overcome this difficulty is the framework of TDDFT and the use of the time-dependent Kohn-Sham (TDKS) equations therein.
This is done by replacing the multi-particle SE with a set of nonlinear single-particle Schrödinger equations that are coupled through an additional potential.
Despite the fact that TDDFT is widely used for physical and quantum chemical calculation and software packages for its use are readily available, its mathematical foundation is still under active development and even fundamental issues remain unproven today.
The main purpose of this thesis is to provide a consistent and rigorous setting for the TDKS equations and of the related optimal control problems.
In the first part of the thesis, the framework of density functional theory (DFT) and TDDFT are introduced. This includes a detailed presentation of the different functional sets forming DFT. Furthermore, the known equivalence of the TDKS system to the original SE problem is further discussed.
To implement the TDDFT framework for multi-particle computations, the TDKS equations provide one of the most successful approaches nowadays. However, only few mathematical results concerning these equations are available and these results do not cover all issues that arise in the formulation of optimal control problems governed by the TDKS model.
It is the purpose of the second part of this thesis to address these issues such as higher regularity of TDKS solutions and the case of weaker requirements on external (control) potentials that are instrumental for the formulation of well-posed TDKS control problems. For this purpose, in this work, existence and uniqueness of TDKS solutions are investigated in the Galerkin framework and using energy estimates for the nonlinear TDKS equations.
In the third part of this thesis, optimal control problems governed by the TDKS model are formulated and investigated. For this purpose, relevant cost functionals that model the purpose of the control are discussed.
Henceforth, TDKS control problems result from the requirement of optimising the given cost functionals subject to the differential constraint given by the TDKS equations. The analysis of these problems is novel and represents one of the main contributions of the present thesis.
In particular, existence of minimizers is proved and their characterization by TDKS optimality systems is discussed in detail.
To this end, Fréchet differentiability of the TDKS model and of the cost functionals is addressed considering \(H^1\) cost of the control.
This part is concluded by deriving the reduced gradient in the \(L^2\) and \(H^1\) inner product.
While the \(L^2\) optimization is widespread in the literature, the choice of the \(H^1\) gradient is motivated in this work by theoretical consideration and by resulting numerical advantages.
The last part of the thesis is devoted to the numerical approximation of the TDKS optimality systems and to their solution by gradient-based optimization techniques.
For the former purpose, Strang time-splitting pseudo-spectral schemes are discussed including a review of some recent theoretical estimates for these schemes and a numerical validation of these estimates.
For the latter purpose, nonlinear (projected) conjugate gradient methods are implemented and are used to validate the theoretical analysis of this thesis with results of numerical experiments with different cost functional settings.
The topic of this PhD thesis is the combination of topologically non-trivial phases with correlation effects stemming from Coulomb interaction between the electrons in a condensed matter system. Emphasis is put on both emerging benefits as well as hindrances, e.g. concerning the topological protection in the presence of strong interactions.
The physics related to topological effects is established in Sec. 2. Based on the topological band theory, we introduce topological materials including Chern insulators, topological insulators in two and three dimensions as well as Weyl semimetals. Formalisms for a controlled treatment of Coulomb correlations are presented in Sec. 3, starting with the topological field theory. The Random Phase Approximation is introduced as a perturbative approach, while in the strongly interacting limit the theory of quantum Hall ferromagnetism applies. Interactions in one dimension are special, and are treated through the Luttinger liquid description. The section ends with an overview of the expected benefits offered by the combination of topology and interactions, see Sec. 3.3.
These ideas are then elaborated in the research part. In Chap. II, we consider weakly interacting 2D topological insulators, described by the Bernevig-Hughes-Zhang model. This is applicable, e.g., to quantum well structures made of HgTe/CdTe or InAs/GaSb. The bulk band structure is here a mixture stemming from linear Dirac and quadratic Schrödinger fermions. We study the low-energy excitations in Random Phase Approximation, where a new interband plasmon emerges due to the combined Dirac and Schrödinger physics, which is absent in the separate limits. Already present in the undoped limit, one finds it also at finite doping, where it competes with the usual intraband plasmon. The broken particle-hole symmetry in HgTe quantum wells allows for an effective separation of the two in the excitation spectrum for experimentally accessible parameters, in the right range for Raman or electron loss spectroscopy. The interacting bulk excitation spectrum shows here clear differences between the topologically trivial and topologically non-trivial regime. An even stronger signal in experiments is expected from the optical conductivity of the system. It thus offers a quantitative way to identify the topological phase of 2D topological insulators from a bulk measurement.
In Chap. III, we study a strongly interacting system, forming an ordered, quantum Hall ferromagnetic state. The latter can arise also in weakly interacting materials with an applied strong magnetic field. Here, electrons form flat Landau levels, quenching the kinetic energy such that Coulomb interaction can be dominant. These systems define the class of quantum Hall topological insulators: topologically non-trivial states at finite magnetic field, where the counter-propagating edge states are protected by a symmetry (spatial or spin) other than time-reversal. Possible material realizations are 2D topological insulators like HgTe heterostructures and graphene. In our analysis, we focus on the vicinity of the topological phase transition, where the system is in a strongly interacting quantum Hall ferromagnetic state. The bulk and edge physics can be described by a nonlinear \sigma-model for the collective order parameter of the ordered state. We find that an emerging, continuous U(1) symmetry offers topological protection. If this U(1) symmetry is preserved, the topologically non-trivial phase persists in the presence of interactions, and we find a helical Luttinger liquid at the edge. The latter is highly tunable by the magnetic field, where the effective interaction strength varies from weakly interacting at zero field, K \approx 1, to diverging interaction strength at the phase transition, K -> 0.
In the last Chap. IV, we investigate whether a Weyl semimetal and a 3D topological insulator phase can exist together at the same time, with a combined, hybrid surface state at the joint boundaries. An overlap between the two can be realized by Coulomb interaction or a spatial band overlap of the two systems. A tunnel coupling approach allows us to derive the hybrid surface state Hamiltonian analytically, enabling a detailed study of its dispersion relation. For spin-symmetric coupling, new Dirac nodes emerge out of the combination of a single Dirac node and a Fermi arc. Breaking the spin symmetry through the coupling, the dispersion relation is gapped and the former Dirac node gets spin-polarized. We propose experimental realizations of the hybrid physics, including compressively strained HgTe as well as heterostructures of topological insulator and Weyl semimetal materials, connected to each other, e.g., by Coulomb interaction.
Optical antennas work similar to antennas for the radio-frequency regime and convert electromagnetic radiation into oscillating electrical currents. Charge density accumulations form at the antenna surface leading to strong and localized near-fields. Since most optical antennas have dimensions of a few hundred nanometers, their near-fields allow the focusing of electromagnetic fields to volumes much smaller than the diffraction limit, with intensities several orders of magnitude larger than achievable with classical diffractive and refractive optical elements. The task to maximize the emission of a quantum emitter, a point-like entity capable of reception and emission of single photons, is identical to the task to maximize the field intensity at the position of the quantum emitter. Therefore it is desirable to optimize the capabilities of focusing optical antennas.
Radio-frequency-antenna designs scaled to optical dimensions of several hundred nanometers show already a decent performance. However, optical frequencies lie near the plasma frequency of the metals used for optical antennas and the mass of electrons cannot be neglected anymore. This leads to new physical phenomena. Light can couple to charge density oscillations, yielding a so-called Plasmon. Effects emerge which have no equivalent in the very advanced field of radio-frequency-technology, e.g.~volume currents and shortened effective wavelengths. Additionally the conductivity is not infinite anymore, leading to thermal losses. Therefore, the question for the optimal geometry of a focusing optical antenna is not easy to answer. However, up to now there was no evidence that there exist better alternatives for optical antennas than down-scaled radio-frequency designs.
In this work the optimization of focusing optical antennas is based on an approach, which often proved successful for radio-frequency-antennas in complex applications (e.g.~broadband and isotropic reception): evolutionary algorithms. The first implementation introduced here allows a large freedom regarding particle shape and count, as it arranges cubic voxels on a planar, square grid. The geometries are encoded in a binary matrix, which works as a genome and enables the methods of mutation and crossing as mechanism of improvement. Antenna geometries optimized in this way surpass a comparable dipolar geometry by a factor of 2. Moreover, a new working principle can be deduced from the optimized antennas: a magnetic split-ring resonance can be coupled conductively to dipolar antennas, to form novel and more effective split-ring-antennas, as their currents add up constructively near the focal point.
In a next step, the evolutionary algorithm is adapted so that the binary matrices describe geometries with realistic fabrication constraints. In addition a 'printer driver' is developed which converts the binary matrices into commands for focused ion-beam milling in mono-crystalline gold flakes. It is shown by means of confocal two-photon photo-luminescence microscopy that antennas with differing efficiency can be fabricated reliably directly from the evolutionary algorithm. Besides, the concept of the split-ring antenna is further improved by adding this time two split-rings to the dipole-like resonance.
The best geometry from the second evolutionary algorithm inspires a fundamentally new formalism to determine the power transfer between an antenna and a point dipole, best termed 'three-dimensional mode-matching'. Therewith, for the first time intuitive design rules for the geometry of an focusing optical antenna can be deduced. The validity of the theory is proven analytically at the case of a point dipole in from of a metallic nano sphere.
The full problem of focusing light by means of an optical antenna can, thus, be reduced to two simultaneous mode-matching conditions -- on the one hand with the fields of a point dipole, on the other hand with a plane wave. Therefore, two types of ideal focusing optical antenna mode patterns are identified, being fundamentally different from the established dipolar antenna mode. This allows not only to explain the functionality of the evolutionary antennas and the split-ring antenna, but also helps to design novel plamonic cavity antennas, which lead to an enhanced focusing of light. This is proven numerically in direct comparison to a classical dipole antenna design.
We develop the formalism of holographic renormalization to compute two-point functions in a holographic Kondo model. The model describes a (0 + 1)-dimensional impurity spin of a gauged SU(N ) interacting with a (1 + 1)-dimensional, large-N , strongly-coupled Conformal Field Theory (CFT). We describe the impurity using Abrikosov pseudo-fermions, and define an SU(N )-invariant scalar operator O built from a pseudo-fermion and a CFT fermion. At large N the Kondo interaction is of the form O\(^{†}\)O, which is marginally relevant, and generates a Renormalization Group (RG) flow at the impurity. A second-order mean-field phase transition occurs in which O condenses below a critical temperature, leading to the Kondo effect, including screening of the impurity. Via holography, the phase transition is dual to holographic superconductivity in (1 + 1)-dimensional Anti-de Sitter space. At all temperatures, spectral functions of O exhibit a Fano resonance, characteristic of a continuum of states interacting with an isolated resonance. In contrast to Fano resonances observed for example in quantum dots, our continuum and resonance arise from a (0 + 1)-dimensional UV fixed point and RG flow, respectively. In the low-temperature phase, the resonance comes from a pole in the Green’s function of the form −i〈O〉\(^{2}\), which is characteristic of a Kondo resonance.
We present the computer code RECOLA2 along with the first NLO electroweak corrections to Higgs production in vector-boson fusion and updated results for Higgs strahlung in the Two-Higgs-Doublet Model and Higgs-Singlet extension of the Standard Model. A fully automated procedure for the generation of tree-level and one-loop matrix elements in general models, including renormalization, is presented. We discuss the application of the Background-Field Method to the extended models. Numerical results for NLO electroweak cross sections are presented for different renormalization schemes in the Two-Higgs-Doublet Model and the Higgs-Singlet extension of the Standard Model. Finally, we present distributions for the production of a heavy Higgs boson.
Higgs production in association with off-shell top-antitop pairs at NLO EW and QCD at the LHC
(2017)
We present NLO electroweak corrections to Higgs production in association with off-shell top-antitop quark pairs. The full process pp → e +νeµ −ν¯µbb¯H is considered, and hence all interference, off-shell, and non-resonant contributions are taken into account.
The electroweak corrections turn out to be below one per cent for the integrated cross section but can exceed 10% in certain phase-space regions. In addition to its phenomenological relevance, the computation constitutes a major technical achievement as the full NLO virtual corrections involving up to 9-point functions have been computed exactly. The results of the full computation are supported by two calculations in the double-pole approximation. These also allow to infer the effect of off-shell contributions and emphasise their importance especially for the run II of the LHC. Finally, we present combined predictions featuring both NLO electroweak and QCD corrections in a common set-up that will help the experimental collaborations in their quest of precisely measuring the aforementioned process.
We calculate the next-to-leading order electroweak corrections to the production of a photon pair in association with zero, one and two jets at the LHC. We use GoSam and Sherpa to obtain the results in a fully automated way. For a typical set of fiducial cuts the electroweak corrections lead to a modification of the total cross section of up to 3%, depending on the jet multiplicity. We find substantial contributions in differential distributions, leading to tens of per cent corrections for phase space regions within the reach of the LHC. Furthermore we investigate the importance of photon induced processes as well as subleading contributions. Photon induced processes are found to be negligible, subleading contributions can have a sizeable impact however they can be removed by appropriate phase space cuts.
Complete NLO corrections to W\(^{+}\)W\(^{+}\) scattering and its irreducible background at the LHC
(2017)
The process pp → μ\(^{+}\)ν\(_{μ}\)e\(^{+}\)ν\(_{e}\)jj receives several contributions of different orders in the strong and electroweak coupling constants. Using appropriate event selections, this process is dominated by vector-boson scattering (VBS) and has recently been measured at the LHC. It is thus of prime importance to estimate precisely each contribution. In this article we compute for the first time the full NLO QCD and electroweak corrections to VBS and its irreducible background processes with realistic experimental cuts. We do not rely on approximations but use complete amplitudes involving two different orders at tree level and three different orders at one-loop level. Since we take into account all interferences, at NLO level the corrections to the VBS process and to the QCD-induced irreducible background process contribute at the same orders. Hence the two processes cannot be unambiguously distinguished, and all contributions to the μ\(^{+}\)ν\(_{μ}\)e\(^{+}\)ν\(_{e}\)jj final state should be preferably measured together.
The production of a neutral and a charged vector boson with subsequent decays into three charged leptons and a neutrino is a very important process for precision tests of the Standard Model of elementary particles and in searches for anomalous triple-gauge-boson couplings. In this article, the first computation of next-to-leading-order electroweak corrections to the production of the four-lepton final states μ\(^{+}\)μ\(^{−}\)e\(^{+}\)ν\(_{e}\), μ\(^{+}\)μ\(^{−}\)e\(^{−}\)ν\(_{e}\), μ\(^{+}\)μ\(^{−}\)μ\(^{+}\)ν\(_{μ}\), and μ\(^{+}\)μ\(^{−}\)μ\(^{−}\)ν\(_{μ}\) at the Large Hadron Collider is presented. We use the complete matrix elements at leading and next-to-leading order, including all off-shell effects of intermediate massive vector bosons and virtual photons. The relative electroweak corrections to the fiducial cross sections from quark-induced partonic processes vary between −3% and −6%, depending significantly on the event selection. At the level of differential distributions, we observe large negative corrections of up to −30% in the high-energy tails of distributions originating from electroweak Sudakov logarithms. Photon-induced contributions at next-to-leading order raise the leading-order fiducial cross section by +2%. Interference effects in final states with equal-flavour leptons are at the permille level for the fiducial cross section, but can lead to sizeable effects in off-shell sensitive phase-space regions.
Next-to-leading-order electroweak corrections to the production of four charged leptons at the LHC
(2017)
We present a state-of-the-art calculation of the next-to leading-order electroweak corrections to ZZ production, including the leptonic decays of the Z bosons into μ\(^+\)μ\(^ −\)e\(^+\)e\(^−\) or μ\(^+\)μ\(^−\)μ\(^+\)μ\(^−\) final states. We use complete leading-order and next-to-leading-order matrix elements for four-lepton production, including contributions of virtual photons and all off-shell effects of Z bosons, where the finite Z-boson width is taken into account using the complex-mass scheme. The matrix elements are implemented into Monte Carlo programs allowing for the evaluation of arbitrary differential distributions. We present integrated and differential cross sections for the LHC at 13 TeV both for an inclusive setup where only lepton identification cuts are applied, and for a setup motivated by Higgs-boson analyses in the four-lepton decay channel. The electroweak corrections are divided into photonic and purely weak contributions. The former show the well-known pronounced tails near kinematical thresholds and resonances; the latter are generically at the level of ∼ −5% and reach several −10% in the high-energy tails of distributions. Comparing the results for μ\(^+\)μ\(^−\)e\(^+\)e\(^−\) and μ\(^+\)μ\(^−\)μ\(^+\)μ\(^−\) final states, we find significant differences mainly in distributions that are sensitive to the μ\(^+\)μ\(^−\) pairing in the μ\(^+\)μ\(^−\)μ\(^+\)μ\(^−\) final state. Differences between μ\(^+\)μ\(^−\)e\(^+\)e\(^−\) and μ\(^+\)μ\(^−\)μ\(^+\)μ\(^−\) channels due to interferences of equal-flavour leptons in the final state can reach up to 10% in off-shell-sensitive regions. Contributions induced by incoming photons, i.e. photon-photon and quark-photon channels, are included, but turn out to be phenomenologically unimportant.