Finite dimensional ordered vector spaces with Riesz interpolation and Effros-Shen's unimodularity conjecture

Peer reviewed

Memory effects and geometrical considerations in open quantum systems

This thesis discusses memory effects in open quantum systems with an emphasis on the Breuer, Laine, Piilo (BLP) measure of non-Markovianity. It is shown how the calculation of the measure can be simplifed and how quantum information protocols can bene t from memory e ects. The superdense coding protocol is used as an example of this. The quantum Zeno effect will also be studied from the point of view of memory e ects. Finally the geometric ideas used in simplifying the calculation of the BLP measure are applied in studying the amount of resources needed for detecting bipartite quantum correlations. It is shown that to decide without prior information if an unknown quantum state is entangled or not, an informationally complete measurement is required. The first part of the thesis contains an introduction to the theoretical ideas such as quantum states, closed and open quantum systems and necessary mathematical tools. The theory is then applied in the second part of the thesis as the results obtained in the original publications I-VI are presented and discussed.

Geometrical measurements in three-dimensional quantum gravity

A set of observables is described for the topological quantum field theory which describes quantum gravity in three space-time dimensions with positive signature and positive cosmological constant. The simplest examples measure the distances between points, giving spectra and probabilities which have a geometrical interpretation. The observables are related to the evaluation of relativistic spin networks by a Fourier transform.

Degenerate mixtures of rubidium and ytterbium for engineering open quantum systems

In the last two decades, experimental progress in controlling cold atoms and ions now allows us to manipulate fragile quantum systems with an unprecedented degree of precision. This has been made possible by the ability to isolate small ensembles of atoms and ions from noisy environments, creating truly closed quantum systems which decouple from dissipative channels. However in recent years, several proposals have considered the possibility of harnessing dissipation in open systems, not only to cool degenerate gases to currently unattainable temperatures, but also to engineer a variety of interesting many-body states. This thesis will describe progress made towards building a degenerate gas apparatus that will soon be capable of realizing these proposals. An ultracold gas of ytterbium atoms, trapped by a species-selective lattice will be immersed into a Bose-Einstein condensate (BEC) of rubidium atoms which will act as a bath. Here we describe the challenges encountered in making a degenerate mixture of rubidium and ytterbium atoms and present two experiments performed on the path to creating a controllable open quantum system. The first experiment will describe the measurement of a tune-out wavelength where the light shift of $\Rb{87}$ vanishes. This wavelength was used to create a species-selective trap for ytterbium atoms. Furthermore, the measurement of this wavelength allowed us to extract the dipole matrix element of the $5s \rightarrow 6p$ transition in $\Rb{87}$ with an extraordinary degree of precision. Our method to extract matrix elements has found use in atomic clocks where precise knowledge of transition strengths is necessary to account for minute blackbody radiation shifts. The second experiment will present the first realization of a degenerate Bose-Fermi mixture of rubidium and ytterbium atoms. Using a three-color optical dipole trap (ODT), we were able to create a highly-tunable, species-selective potential for rubidium and ytterbium atoms which allowed us to use $\Rb{87}$ to sympathetically cool $\Yb{171}$ to degeneracy with minimal loss. This mixture is the first milestone creating the lattice-bath system and will soon be used to implement novel cooling schemes and explore the rich physics of dissipation.

Phase Transitions in Engineered Ultracold Quantum Systems

The study of quantum degenerate gases has many applications in topics such as condensed matter dynamics, precision measurements and quantum phase transitions. We built an apparatus to create 87Rb Bose-Einstein condensates (BECs) and generated, via optical and magnetic interactions, novel quantum systems in which we studied the contained phase transitions. For our first experiment we quenched multi-spin component BECs from a miscible to dynamically unstable immiscible state. The transition rapidly drives any spin fluctuations with a coherent growth process driving the formation of numerous spin polarized domains. At much longer times these domains coarsen as the system approaches equilibrium. For our second experiment we explored the magnetic phases present in a spin-1 spin-orbit coupled BEC and the contained quantum phase transitions. We observed ferromagnetic and unpolarized phases which are stabilized by the spin-orbit coupling’s explicit locking between spin and motion. These two phases are separated by a critical curve containing both first-order and second-order transitions joined at a critical point. The narrow first-order transition gives rise to long-lived metastable states. For our third experiment we prepared independent BECs in a double-well potential, with an artificial magnetic field between the BECs. We transitioned to a single BEC by lowering the barrier while expanding the region of artificial field to cover the resulting single BEC. We compared the vortex distribution nucleated via conventional dynamics to those produced by our procedure, showing our dynamical process populates vortices much more rapidly and in larger number than conventional nucleation.

The solutions and tunneling properties for a linear potential and an inverted harmonic oscillator in an infinite well

In this work we look at two different 1-dimensional quantum systems. The potentials for these systems are a linear potential in an infinite well and an inverted harmonic oscillator in an infinite well. We will solve the Schrödinger equation for both of these systems and get the energy eigenvalues and eigenfunctions. The solutions are obtained by using the boundary conditions and numerical methods. The motivation for our study comes from experimental background. For the linear potential we have two different boundary conditions. The first one is the so called normal boundary condition in which the wave function goes to zero on the edge of the well. The second condition is called derivative boundary condition in which the derivative of the wave function goes to zero on the edge of the well. The actual solutions are Airy functions. In the case of the inverted oscillator the solutions are parabolic cylinder functions and they are solved only using the normal boundary condition. Both of the potentials are compared with the particle in a box solutions. We will also present figures and tables from which we can see how the solutions look like. The similarities and differences with the particle in a box solution are also shown visually. The figures and calculations are done using mathematical software. We will also compare the linear potential to a case where the infinite wall is only on the left side. For this case we will also show graphical information of the different properties. With the inverted harmonic oscillator we will take a closer look at the quantum mechanical tunneling. We present some of the history of the quantum tunneling theory, its developers and finally we show the Feynman path integral theory. This theory enables us to get the instanton solutions. The instanton solutions are a way to look at the tunneling properties of the quantum system. The results are compared with the solutions of the double-well potential which is very similar to our case as a quantum system. The solutions are obtained using the same methods which makes the comparison relatively easy. All in all we consider and go through some of the stages of the quantum theory. We also look at the different ways to interpret the theory. We also present the special functions that are needed in our solutions, and look at the properties and different relations to other special functions. It is essential to notice that it is possible to use different mathematical formalisms to get the desired result. The quantum theory has been built for over one hundred years and it has different approaches. Different aspects make it possible to look at different things.

The solutions and tunneling properties for a linear potential and an inverted harmonic oscillator in an infinite well

In this work we look at two different 1-dimensional quantum systems. The potentials for these systems are a linear potential in an infinite well and an inverted harmonic oscillator in an infinite well. We will solve the Schrödinger equation for both of these systems and get the energy eigenvalues and eigenfunctions. The solutions are obtained by using the boundary conditions and numerical methods. The motivation for our study comes from experimental background. For the linear potential we have two different boundary conditions. The first one is the so called normal boundary condition in which the wave function goes to zero on the edge of the well. The second condition is called derivative boundary condition in which the derivative of the wave function goes to zero on the edge of the well. The actual solutions are Airy functions. In the case of the inverted oscillator the solutions are parabolic cylinder functions and they are solved only using the normal boundary condition. Both of the potentials are compared with the particle in a box solutions. We will also present figures and tables from which we can see how the solutions look like. The similarities and differences with the particle in a box solution are also shown visually. The figures and calculations are done using mathematical software. We will also compare the linear potential to a case where the infinite wall is only on the left side. For this case we will also show graphical information of the different properties. With the inverted harmonic oscillator we will take a closer look at the quantum mechanical tunneling. We present some of the history of the quantum tunneling theory, its developers and finally we show the Feynman path integral theory. This theory enables us to get the instanton solutions. The instanton solutions are a way to look at the tunneling properties of the quantum system. The results are compared with the solutions of the double-well potential which is very similar to our case as a quantum system. The solutions are obtained using the same methods which makes the comparison relatively easy. All in all we consider and go through some of the stages of the quantum theory. We also look at the different ways to interpret the theory. We also present the special functions that are needed in our solutions, and look at the properties and different relations to other special functions. It is essential to notice that it is possible to use different mathematical formalisms to get the desired result. The quantum theory has been built for over one hundred years and it has different approaches. Different aspects make it possible to look at different things.

Entanglement and boundary critical phenomena

We investigate boundary critical phenomena from a quantum-information perspective. Bipartite entanglement in the ground state of one-dimensional quantum systems is quantified using the Renyi entropy S-alpha, which includes the von Neumann entropy (alpha -> 1) and the single-copy entanglement (alpha ->infinity) as special cases. We identify the contribution of the boundaries to the Renyi entropy, and show that there is an entanglement loss along boundary renormalization group (RG) flows. This property, which is intimately related to the Affleck-Ludwig g theorem, is a consequence of majorization relations between the spectra of the reduced density matrix along the boundary RG flows. We also point out that the bulk contribution to the single-copy entanglement is half of that to the von Neumann entropy, whereas the boundary contribution is the same.

Reduction of infinite dimensional systems to finite dimensions: Compact convergence approach

We consider parameter dependent semilinear evolution problems for which, at the limit value of the parameter, the problem is finite dimensional. We introduce an abstract functional analytic framework that applies to many problems in the existing literature for which the study of asymptotic dynamics can be reduced to finite dimensions via the invariant manifolds technique. Some practical models are considered to show wide applicability of the theory. © 2013 Society for Industrial and Applied Mathematics.

Magnetoresistance oscillations in multilayer systems: Triple quantum wells

Magnetoresistance of two-dimensional electron systems with several occupied subbands oscillates owing to periodic modulation of the probability of intersubband transitions by the quantizing magnetic field. In addition to previous investigations of these magnetointersubband (MIS) oscillations in two-subband systems, we report on both experimental and theoretical studies of such a phenomenon in three-subband systems realized in triple quantum wells. We show that the presence of more than two subbands leads to a qualitatively different MIS oscillation picture, described as a superposition of several oscillating contributions. Under a continuous microwave irradiation, the magnetoresistance of triple-well systems exhibits an interference of MIS oscillations and microwave-induced resistance oscillations. The theory explaining these phenomena is presented in the general form, valid for an arbitrary number of subbands. A comparison of theory and experiment allows us to extract temperature dependence of quantum lifetime of electrons and to confirm the applicability of the inelastic mechanism of microwave photoresistance for the description of magnetotransport in multilayer systems.

Emergent and reentrant fractional quantum Hall effect in trilayer systems in a tilted magnetic field

Magnetotransport measurements in triple-layer electron systems with high carrier density reveal fractional quantum Hall effect at total filling factors nu>2. With an in-plane magnetic field we are able to control the suppression of interlayer tunneling which causes a collapse of the integer quantum Hall plateaus at nu=2 and nu=4, and an emergence of fractional quantum Hall states with increasing tilt angles. The nu=4 state is replaced by three fractional quantum Hall states with denominator 3. The state nu=7/3 demonstrates reentrant behavior and the emergent state at nu=12/5 has a nonmonotonic behavior with increasing in-plane field. We attribute the observed fractional quantum Hall plateaus to correlated states in a trilayer system.

Anomalous dephasing scattering rate of two-dimensional electrons in double quantum well structures

The results on the measurement of electrical conductivity and magnetoconductivity of a GaAs double quantum well between 0.5 and 1.1 K are reported. The zero magnetic-field conductivity is well described from the point of view of contributions made by both the weak localization and electron-electron interaction. At low field and low temperature, the magnetoconductivity is dominated by the weak localization effect only. Using the weak localization method, we have determined the electron dephasing times tau(phi) and tunneling times tau(t). Concerning tunneling, we concluded that tau(t) presents a minimum around the balance point; concerning dephasing, we observed an anomalous dependence on temperature and conductivity (or elastic mean free path) of tau(phi). This anomalous behavior cannot be explained in terms of the prevailing concepts for the electron-electron interaction in high-mobility two-dimensional electron systems.

Finite element modelling of three-dimensional convection problems in fluid-saturated porous media heated from below

We use the finite element method to model three-dimensional convective pore-fluid flow in fluid-saturated porous media when they are heated from below. In particular, we employ the particle-tracking technique to mimic the trajectories of particles in three-dimensional fluid flow problems. The related numerical results demonstrated that: (1) The progressive asymptotic approach procedure, which was previously developed for the finite element modelling of two-dimensional convective pore-fluid flow problems, is equally applicable to the finite element modelling of three-dimensional convective pore-fluid flow in fluid-saturated porous media heated from below. (2) The perturbation of gravity at different planes has a significant effect on the pattern of three-dimensional convective pore-fluid flow and therefore, may influence the pattern of orebody formation and mineralization in three-dimensional hydrothermal systems. Copyright (C) 2001 John Wiley & Sons, Ltd.

Quantum-Critical Spin Dynamics in Quasi-One-Dimensional Antiferromagnets

By means of nuclear spin-lattice relaxation rate T-1(-1), we follow the spin dynamics as a function of the applied magnetic field in two gapped quasi-one-dimensional quantum antiferromagnets: the anisotropic spin-chain system NiCl2-4SC(NH2)(2) and the spin-ladder system (C5H12N)(2)CuBr4. In both systems, spin excitations are confirmed to evolve from magnons in the gapped state to spinons in the gapless Tomonaga-Luttinger-liquid state. In between, T-1(-1) exhibits a pronounced, continuous variation, which is shown to scale in accordance with quantum criticality. We extract the critical exponent for T-1(-1), compare it to the theory, and show that this behavior is identical in both studied systems, thus demonstrating the universality of quantum-critical behavior.