Browsing by master's degree program "Teoreettisten ja laskennallisten menetelmien maisteriohjelma (Theoretical Calculation Methods)"

Topological defects are some of the more common phenomena of many extensions of the standard model of particle physics. In some sense, defects are a consequence of an unresolvable misalignment between different regions of the system, much like cracks in ice or kinks in an antiquated telephone cord. In our context, they present themselves as localised inhomogeneities of the fundamental fields, emerging at the boundaries of the misaligned regions at the cost of, potentially massive, trapped energy. Should the cosmological variety exist in nature, they are hypothesised to emerge from some currently unknown cosmological phase transition, leaving their characteristic mark on the evolution of the nascent universe. As of date, so called cosmic strings are perhaps the most promising type of cosmic defect, at least with respect to their observational prospects. Cosmic strings, as the name suggest, are linelike topological defects; exceedingly thin, yet highly energetic. Given the advent of gravitational wave astronomy, a substantial amount of research is devoted to detailed and expensive real-time computer simulations of various cosmic string models in hopes of extracting their effects on the gravitational wave background. In this thesis we discuss the Abelian-Higgs model, a toy model of a gauge theory of a complex scalar field and a real vector field. Through a choice of a symmetry-breaking scalar potential, this model permits line defects, so called local strings. We discuss some generalities of classical field theory as well as those of the interesting mathematical theory of topological defects. We apply these to our model and present the necessary numerical methods for writing our own cosmic string simulation. We use the newly written simulation to reproduce a number of contemporary results on the scaling properties of the string networks and present some preliminary results from a less investigated region of the model parameter space, attempting to compare the effects of different types of string-string interactions. Furthermore, preliminary results are presented on the thermodynamic evolution of the system and the effects a common computational trick, comoving string width, are discussed with respect to the evolution of the equation of state.

In quantum field theory the objects of interest are the n-point vacuum expectations which can be calculated from the path integral. The path integral usually used in physics is not well-defined and the main motivation for this thesis is to give axioms that a well-defined path integral candidate has to at least satisfy for it to be physically relevant - we want the path integral to have properties which allow us to reconstruct the physically interesting objects from it. The axioms given in this thesis are called the Osterwalder-Schrader axioms and the reconstruction of the physical objects from the path integral satisfying the axioms is called the Osterwalder-Schrader reconstruction. The Osterwalder-Schrader axioms are special in the sense that they are stated in terms of the Euclidean spacetime instead of the physically relevant Minkowski spacetime. As the physical objects live in Minkowski spacetime this means that when reconstructing the physically relevant objects we have to go back to Minkowski spacetime at some point. This thesis has three parts (and an introduction). In the first part we give a brief introduction to parts of functional analysis which we will need later - theory about distributions and about generators of families of operators. The second part is about the Osterwalder-Schrader axioms and about the reconstruction of the physically relevant objects from the path integral. In the last part we check that the path integral for the free field of mass m satisfies the Osterwalder-Schrader axioms.

Numerical techniques have become powerful tools for studying quantum systems. Eventually, quantum computers may enable novel ways to perform numerical simulations and conquer problems that arise in classical simulations of highly entangled matter. Simple one dimensional systems of low entanglement are efficiently simulatable on a classical computer using tensor networks. This kind of toy simulations also give us the opportunity to study the methods of quantum simulations, such as different transformation techniques and optimization algorithms that could be beneficial for the near term quantum technologies. In this thesis, we study a theoretical framework for a fermionic quantum simulation and simulate the real-time evolution of particles governed by the Gross-Neveu model in one-dimension. To simulate the Gross-Neveu model classically, we use the Matrix Product State (MPS) method. Starting from the continuum case, we discretise the model by putting it on a lattice and encode the time evolution operator with the help of fermion-to-qubit transformations, Jordan-Wigner and Bravyi-Kitaev. The simulation results are visualised as plots of probability density. The results indicate the expected flavour and spatial symmetry of the system. The comparison of the two transformations show better performance of the Jordan-Wigner transformation before and after the gate reduction.

Quantum computers are one of the most prominent emerging technologies of the 21st century. While several practical implementations of the qubit—the elemental unit of information in quantum computers—exist, the family of superconducting qubits remains one of the most promising platforms for scaled-up quantum computers. Lately, as the limiting factor of non-error-corrected quantum computers has began to shift from the number of qubits to gate fidelity, efficient control and readout parameter optimization has become a field of significant scientific interest. Since these procedures are multibranched and difficult to automate, a great deal of effort has gone into developing associated software, and even technologies such as machine learning are making an appearance in modern programs. In this thesis, we offer an extensive theoretical backround on superconducting transmon qubits, starting from the classical models of electronic circuits, and moving towards circuit quantum electrodynamics. We consider how the qubit is controlled, how its state is read out, and how the information contained in it can become corrupted by noise. We review theoretical models for characteristic parameters such as decoherence times, and see how control pulse parameters such as amplitude and rise time affect gate fidelity. We also discuss the procedure for experimentally obtaining characteristic qubit parameters, and the optimized randomized benchmarking for immediate tune-up (ORBIT) protocol for control pulse optimization, both in theory and alongside novel experimental results. The experiments are carried out with refactored characterization software and novel ORBIT software, using the premises and resources of the Quantum Computing and Devices (QCD) group at Aalto University. The refactoring project, together with the software used for the ORBIT protocol, aims to provide the QCD group with efficient and streamlined methods for finding characteristic qubit parameters and high-fidelity control pulses. In the last parts of the thesis, we evaluate the success and shortcomings of the introduced projects, and discuss future perspectives for the software.

Low-frequency $1/$ noise is ubiquitous, found in all electronic devices and other diverse areas such as as music, economics and biological systems. Despite valiant efforts, the source of $1/f$ noise remains one of the oldest unsolved mysteries in modern physics after nearly 100 years since its initial discovery in 1925. In metallic conductors resistance $1/f$ noise is commonly attributed to diffusion of mobile defects that alter the scattering cross section experienced by the charge carriers. Models based on two-level tunneling systems (TLTS) are typically employed. However, a model based on the dynamics of mobile defects forming temporary clusters would naturally offer long-term correlations required by $1/f$ noise via the nearly limitless number of configurations among a group of defects. Resistance $1/f$ noise due to such motion of mobile defects was studied via Monte Carlo simulations of a simple resistor network resembling an atomic lattice. The defects migrate through the lattice via thermally activated hopping motion, causing fluctuations in the resistance due to varying scattering cross section. The power spectral density (PSD) $S(f)$ of the simulated resistance noise was then calculated and first compared to $S(f)=C/f^\alpha$ noise, where $C$ is a constant and $\alpha$ is ideally close to unity. The value of $\alpha$ was estimated via a linear fit of the noise PSD on a log-log scale. The resistor network was simulated with varying values of temperature, system size and the concentration of defects. The noise produced by the simulations did not yield pure $1/f^\alpha$ noise, instead the lowest frequencies displayed a white noise tail, changing to $1/f^\alpha$ noise between $10^{-4}$ to $10^{-2}$~Hz. In this way the spectrum of the simulated noise resembles a Lorentzian. The value of $\alpha$ was found to be the most sensitive to temperature $T$, which directly affects the motion of the defects. At high $T$ the value of $\alpha$ was closer to 1, whereas at low $T$ it was closer to $1,5$. Varying the size of the system was found to have little impact on $\alpha$ when the temperature and concentration of defects were kept fixed. Increasing the number of defects was found to have slightly more effect on $\alpha$ when the temperature and system size were kept fixed. The value of $\alpha$ was closer to unity when the concentration of defects was higher, but the effect was not nearly as pronounced compared to varying the temperature. In addition, the simulated noise was compared to a PSD of the form $S(f)\propto e^{-\sqrt{N}/T}1/f$, where $N$ is the size of the system, according to recent theoretical proceedings. The $1/f^\alpha$ part of the simulated noise was found to roughly follow the above equation, but the results remain inconclusive. Although the simple toy model did not produce pure $1/f^\alpha$ noise, the dynamics of the mobile defects do seem to have an effect on the noise PSD, yielding noise closer to $1/f$ when there are more interactions between the defects due to either higher mobility or higher concentration of defects. However, this is disregarding the white noise tail. Recent experimental research on high quality graphene employing more rigorous kinetic Monte Carlo simulations have displayed more promising results. This indicates that the dynamics of temporary cluster formation of mobile defects is relevant to understand $1/f$ noise in metallic conductors, offering an objective for future work.

Despite the continuous efforts to unveil the true nature of Dark Matter (DM), it still remains as a mystery. In this thesis we propose one model that can produce the correct relic abundance of DM in the current Universe, while fitting into the existing experimentally obtained constraints. In this model we add a singlet fermion, which is a not completely sterile right-handed neutrino, and a heavy real scalar singlet into the Standard Model of Particle Physics (SM) and carry out the relic density calculations. The DM candidate here is the singlet fermion, which acts as a thermally decoupled Weakly Interacting Massive Particle. Theoretical framework is laid out in detail. Special attention is given to obtaining the definition for relic abundance from Lee-Weinberg equation in terms of yield, and the decoupling temperature. It is found, that the usual way of handling the thermally averaged cross section appearing in these definitions is not suitable in this case. In fact, the usual approximations can only be done when thermally averaged cross section is almost linear in $s$, and this is a demand that very few models can satisfy. The correct way to treat the cross section by taking the expansion in terms of the relative velocity is presented with careful attention to detail. This methodology is then applied to the extension of the SM we introduced. Only tree-level processes are being considered. Cross sections are calculated for each possible process to obtain the total cross section needed for the DM relic density calculations. We present how the different free parameters in the theory affect the relic abundance and what masses are allowed for the right-handed neutrino to obtain. It is found out that the parameters in this model are heavily constrained. Yet the model is able to fit into the constraints obtained from branching ratio and direct detection (DD) experiments, while producing the correct relic density. This is true when the mixing angle $\theta$ is of the order $1 \times 10^{-4}$, and right-handed neutrino has the mass of exactly half of the mass of the heavy scalar or higher than the mass of the heavy scalar. It is proposed that allowing lepton mixing and adding a separate mass term for fermion in the model could make the model less restricted. Investigating this would be interesting thing to do in the future. However, the proposed DM candidate remains viable and the upcoming DD experiments will relatively soon reveal if the singlet fermion is the DM particle we are seeking.

One of the main ways of physically realizing quantum bits for the purposes of quantum technology is to manufacture them as superconducting circuits. These qubits are artificially built two-level systems that act as carriers of quantum information. They come in a variety of types but one of the most common in use is the transmon qubit. The transmon is a more stable, improved version of the earlier types of superconducting qubits with longer coherence times. The qubit cannot function properly on its own, as it needs other circuit elements around it for control and readout of its state. Thus the qubit is only a small part of a larger superconducting circuit interacting with the qubit. Understanding this interaction, where it comes from and how it can be modified to our liking, allows researchers to design better quantum circuits and to improve the existing ones. Understanding how the noise, travelling through the qubit drive lines to the chip, affects the time evolution of the qubit is especially important. Reducing the amount of noise leads to longer coherence times but it is also possible to engineer the noise to our advantage to uncover novel ways of quantum control. In this thesis the effects of a variable temperature noise source on the qubit drive line is studied. A theoretical model describing the time evolution of the quantum state is built. The model starts from the basic elements of the quantum circuit and leads to a master equation describing the qubit dynamics. This allows us to understand how the different choices made in the manufacturing process of the quantum circuit affect the time evolution. As a proof of concept, the model is solved numerically using QuTiP in the specific case of a fixed-frequency, dispersive transmon qubit. The solution shows a decohering qubit with no dissipation. The model is also solved in a temperature range 0K < T ≤ 1K to show how the decoherence times behave with respect to the temperature of the noise source.

A quantum computer is a new kind of computer which utilizes quantum phenomena in computing. This machine has the potential to solve specific tasks faster than the most powerful supercomputers and therefore has potential real-life applications across various sectors of society. One promising approach to realize a quantum computer is to store information in superconducting qubits, which are artificial two-level quantum systems made from superconducting electrical circuits. Extremely precise control of these qubits is essential but also challenging due to the excitations out of the two lowest energy states of the quantum system that constitute the computational subspace. In this thesis, we propose a new way to control a superconducting multimode qubit using the unimon qubit as an example. By coupling differently to the different modes of the multimode qubit circuit, we cancel the transition from the first excited state to the second excited state, which is typically the main transition causing a leakage out of the computational subspace. We present a theoretical description of this model by utilizing methods of circuit quantum electrodynamics to compute the energy spectrum and the transition matrix elements of the qubit. By using these results, we simulate the dynamics of the driven unimon qubit undergoing as a single-qubit gate. The result of the simulation shows that this method decreases the leakage relative to the conventional method of driving a qubit, where only one external drive is applied. However, by improving the conventional method with a more advanced pulse optimization method, the leakage becomes smaller than in the standard case of two drive fields. In addition, we find that the practical implementation of our method may be sensitive to variations in the qubit parameters. Therefore, the practical implementation of the method needs further research in the future. By cancelling one energy-level transition of the qubit, we find that other transitions in modes of similar frequency were strongly suppressed. Therefore, this method might be potentially utilized in other qubit operations than the quantum gates, such as in the qubit resetting process, where driving to higher frequency modes of the unimon is preferred.

Fysikaalisten kenttien rakenteista solmittuja linkkejä ja solmuja on osoitettu esiintyvän useissa eri systeemeissä, optisista ja akustisista kentistä aina supranesteisiin. Tällaisten solmujen evoluutio ideaaleissa systeemeissä rajoittuu perinteisten solmujen, kuten kengännauhoista löytyvien, tyyppiseksi. Fysikaalisissa systeemeissä, joissa energiahäviöt ovat mahdollisia, mahdolliset evoluutiot ovat eksoottisempia; kentässä olevan solmun säikeet --- kentän topologisten pyörteiden ytimet --- voivat mahdollisesti kokea solmun topologiaa muuttavia evoluutioita ja purkautua erillisiksi solmuttomiksi silmukoiksi. Hyödyntämällä väritettyä linkkidiagrammiesitystä osoitamme yllä mainittuja evoluutioita välttävien topologisesti suojattujen pyörresolmujen olemassaolon spin-2 Bosen--Einsteinin kondensaatin syklisessä faasissa. Nämä ovat ensimmäiset suojatut pyörresolmut, jotka on löydetty fysikaalisesti toteutettavasta systeemistä. Todistuksessa käytetty väritysformalismi on yleinen viitekehys. Sitä voidaan soveltaa useisiin muihin systeemeihin, kunhan näiden järjestysparametriavaruuksien toinen homotopiaryhmä on triviaali. Syklisen faasin pyörresolmut ja -linkit mahdollistavat myös osittaisen luokittelun: jokainen solmu on ekvivalentti joukkoon irtonaisia apilasolmuja ja/tai kahdeksikkosolmuja, tai on triviaali. Kokonaislukuaskelin kvantittuneista pyörteistä muodostuvat linkit ovat ekvivalentteja joko Borromeon renkaisiin, suljettuun kolmen silmukan ketjuun tai triviaaliin silmukkaan.

Particle dark matter (DM) as a solution to the missing mass problem in astronomy has been examined widely and with different models. Among the most studied are weakly interacting massive particles, WIMPs, for short. As dark matter constitutes roughly a quarter of the energy budget of the universe, and due to its vital role in galaxy structures through gravitational interaction, the motivation to uncover the nature and properties of it is strong. In this master’s thesis, a specific particle dark matter model is examined. The model consists of a hidden dark sector added to the Standard Model of Particle Physics (SM). The dark sector introduces a new SU(2) gauge field that acts as a vector dark matter candidate, as well as a complex SU(2) scalar field and interactions between the two. Due to spontaneous symmetry breaking, the dark vector gains a non-zero mass. This relocation of degrees of freedom allows us to write the dark scalar field as having only one real degree of freedom. The dark scalar field also experiences mass mixing with the SM Higgs field, leaving the two propagating scalar mass eigenstates as superpositions of the dark scalar field and the Higgs field. One of these is then identified as the observed Higgs field with a mass of 125 GeV. The four free parameters of the model can be chosen as the masses of the dark matter candidate and the propagating dark scalar field, the angle of the rotation between mass and gauge eigenbasis in the scalar sector and the dark gauge coupling constant. To produce the observed relic density of dark matter, the DM particles need to pair-annihilate with a cross section of order 1.64 × 10^(−9) GeV^(−2). Further constraints are given by collider and direct detection experiments, leaving the parameter space of the model rather constrained. Depending on the values of the other free parameters, a viable mass range of around 100-200 GeV is found for the vector dark matter. The possibility of probing the properties of dark matter through experiments and observations exists. The existence and properties of the dark scalar field could be examined in the Large Hadron Collider. Possible phenomena in the scalar sector of the model, such as phase transitions, could be studied with upcoming gravitational wave detectors, namely the Laser Interferometer Space Antenna. Direct detection experiments provide a way of seeking the dark matter particle itself. With all these possibilities, the future seems interesting.