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ABENDA, SIMONETTA. "Analysis of Singularity Structures for Quasi-Integrable Hamiltonian Systems". Doctoral thesis, SISSA, 1994. http://hdl.handle.net/20.500.11767/4499.
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Nagaj, Daniel. "Local Hamiltonians in quantum computation". Thesis, Massachusetts Institute of Technology, 2008. http://hdl.handle.net/1721.1/45162.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2008.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Includes bibliographical references (p. 169-176).
In this thesis, I investigate aspects of local Hamiltonians in quantum computing. First, I focus on the Adiabatic Quantum Computing model, based on evolution with a time- dependent Hamiltonian. I show that to succeed using AQC, the Hamiltonian involved must have local structure, which leads to a result about eigenvalue gaps from information theory. I also improve results about simulating quantum circuits with AQC. Second, I look at classically simulating time evolution with local Hamiltonians and finding their ground state properties. I give a numerical method for finding the ground state of translationally invariant Hamiltonians on an infinite tree. This method is based on imaginary time evolution within the Matrix Product State ansatz, and uses a new method for bringing the state back to the ansatz after each imaginary time step. I then use it to investigate the phase transition in the transverse field Ising model on the Bethe lattice. Third, I focus on locally constrained quantum problems Local Hamiltonian and Quantum Satisfiability and prove several new results about their complexity. Finally, I define a Hamiltonian Quantum Cellular Automaton, a continuous-time model of computation which doesn't require control during the computation process, only preparation of product initial states. I construct two of these, showing that time evolution with a simple, local, translationally invariant and time-independent Hamiltonian can be used to simulate quantum circuits.
by Daniel Nagaj.
Ph.D.
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Assis, Paulo. "Non-Hermitian Hamiltonians in field theory". Thesis, City University London, 2009. http://openaccess.city.ac.uk/2118/.
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This thesis is centred around the role of non-Hermitian Hamiltonians in Physics both at the quantum and classical levels. In our investigations of two-level models we demonstrate [1] the phenomenon of fast transitions developed in the PT -symmetric quantum brachistochrone problem may in fact be attributed to the non-Hermiticity of evolution operator used, rather than to its invariance under PT operation. Transition probabilities are calculated for Hamiltonians which explicitly violate PT -symmetry. When it comes to Hilbert spaces of infinite dimension, starting with non-Hermitian Hamiltonians expressed as linear and quadratic combinations of the generators of the su(1; 1) Lie algebra, we construct [2] Hermitian partners in the same similarity class. Alongside, metrics with respect to which the original Hamiltonians are Hermitian are also constructed, allowing to assign meaning to a large class of non-Hermitian Hamiltonians possessing real spectra. The finding of exact results to establish the physical acceptability of other non-Hermitian models may be pursued by other means, especially if the system of interest cannot be expressed in terms of Lie algebraic elements. We also employ [3] a representation of the canonical commutation relations for position and momentum operators in terms of real-valued functions and a noncommutative product rule of differential form. Besides exact solutions, we also compute in a perturbative fashion metrics and isospectral partners for systems of physical interest. Classically, our efforts were concentrated on integrable models presenting PT - symmetry. Because the latter can also establish the reality of energies in classical systems described by Hamiltonian functions, we search for new families of nonlinear differential equations for which the presence of hidden symmetries allows one to assemble exact solutions. We use [4] the Painleve test to check whether deformations of integrable systems preserve integrability. Moreover we compare [5] integrable deformed models, which are thus likely to possess soliton solutions, to a broader class of systems presenting compacton solutions. Finally we study [6] the pole structure of certain real valued nonlinear integrable systems and establish that they behave as interacting particles whose motion can be extended to the complex plane in a PT -symmetric way.
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Ramaswami, Geetha Pillaiyarkulam. "Numerical solution of special separable Hamiltonians". Thesis, University of Cambridge, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.627541.
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Moore, David Jeffrey. "Non-adiabatic Berry phases for periodic Hamiltonians". Thesis, University of Canterbury. Physics, 1991. http://hdl.handle.net/10092/8072.
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A method for the calculation of Berry phases for periodic, but not necessarily adiabatic, Hamiltonians is reported. This method is based on a novel factorisation of the evolution operator and is in the spirit of the theory of systems of linear differential equations with periodic coefficients.
The use of this approach in practical situations is greatly facilitated by exploiting the Fourier decomposition of the Hamiltonian. This converts the problem into an equivalent time-independent form. The solution to the problem is then expressible in terms of the eigenvectors and eigenvalues of a certain self-adjoint operator called the Floquet Hamiltonian. This operator can be calculated from the Fourier decomposition of the original Hamiltonian.
Our formalism has several calculational advantages over the other methods used in the literature. These advantages are best seen by considering standard quantum optical systems such as the semi-classical model of a two-level atom strongly irradiated by a near resonant laser beam. The utility of our formalism is not confined to systems of this type however. For example it can be used to great advantage in the study of systems with time-odd electron-phonon coupling.
Apart from its calculational utility, our formalism also has important theoretical applications. Here it is used to clarify the relationship between Berry phases and the time dependence of the Hamiltonian.
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Yildirim, Yolcu Selma. "Eigenvalue inequalities for relativistic Hamiltonians and fractional Laplacian". Diss., Atlanta, Ga. : Georgia Institute of Technology, 2009. http://hdl.handle.net/1853/31649.
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Thesis (Ph.D)--Mathematics, Georgia Institute of Technology, 2010.Committee Chair: Harrell, Evans; Committee Member: Chow, Shui-Nee; Committee Member: Geronimo, Jeffrey; Committee Member: Kennedy, Brian; Committee Member: Loss, Michael. Part of the SMARTech Electronic Thesis and Dissertation Collection.
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Bartlett, Bruce. "Flow equations for hamiltonians from continuous unitary transformations". Thesis, Stellenbosch : Stellenbosch University, 2003. http://hdl.handle.net/10019.1/53428.
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Thesis (MSc)--Stellenbosch University, 2003.ENGLISH ABSTRACT: This thesis presents an overview of the flow equations recently introduced by Wegner. The little known mathematical framework is established in the initial chapter and used as a background for the entire presentation. The application of flow equations to the Foldy-Wouthuysen transformation and to the elimination of the electron-phonon coupling in a solid is reviewed. Recent flow equations approaches to the Lipkin model are examined thoroughly, paying special attention to their utility near the phase change boundary. We present more robust schemes by requiring that expectation values be flow dependent; either through a variational or self-consistent calculation. The similarity renormalization group equations recently developed by Glazek and Wilson are also reviewed. Their relationship to Wegner's flow equations is investigated through the aid of an instructive model.
AFRIKAANSE OPSOMMING: Hierdie tesis bied 'n oorsig van die vloeivergelykings soos dit onlangs deur Wegner voorgestel is. Die betreklik onbekende wiskundige raamwerk word in die eerste hoofstuk geskets en deurgans as agtergrond gebruik. 'n Oorsig word gegee van die aanwending van die vloeivergelyking vir die Foldy-Wouthuysen transformasie en die eliminering van die elektron-fonon wisselwerking in 'n vastestof. Onlangse benaderings tot die Lipkin model, deur middel van vloeivergelykings, word ook deeglik ondersoek. Besondere aandag word gegee aan hul aanwending naby fasegrense. 'n Meer stewige skema word voorgestel deur te vereis dat verwagtingswaardes vloei-afhanklik is; óf deur gevarieerde óf self-konsistente berekenings. 'n Inleiding tot die gelyksoortigheids renormerings groep vergelykings, soos onlangs ontwikkel deur Glazek en Wilson, word ook aangebied. Hulle verwantskap met die Wegner vloeivergelykings word bespreek aan die hand van 'n instruktiewe voorbeeld.
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Duffus, Stephen N. A. "Open quantum systems, effective Hamiltonians and device characterisation". Thesis, Loughborough University, 2018. https://dspace.lboro.ac.uk/2134/33672.
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We investigate the some of the many subtleties in taking a microscopic approach to modelling the decoherence of an Open Quantum System. We use the RF-SQUID, which will be referred to as a simply a SQUID throughout this paper, as a non-linear example and consider different levels of approximation, with varied coupling, to show the potential consequences that may arise when characterising devices such as superconducting qubits in this manner. We first consider a SQUID inductively coupled to an Ohmic bath and derive a Lindblad master equation, to first and second order in the Baker-Campbell-Hausdorff expansion of the correlation-time-dependent flux operator. We then consider a SQUID both inductively and capacitively coupled to an Ohmic bath and derive a Lindblad master equation to better understand the effect of parasitic capacitance whilst shedding more light on the additions, cancellations and renormalisations that are attributed to a microscopic approach.
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Hyder, Asif M. "Green's operator for Hamiltonians with Coulomb plus polynomial potentials". California State University, Long Beach, 2013.
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Engeler, Marco Bruno Raphael. "New model Hamiltonians for improved orbital basis set convergence". Thesis, Cardiff University, 2006. http://orca.cf.ac.uk/54563/.
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The standard approach in quantum chemistry is to expand the eigenfunctions of the non relativistic Born Oppenheimer Hamiltonian in terms of Slater determinants. The quality improvements of such wavefunctions in terms of the underlying one electron basis is frustratingly slow. The error in the correlation energy decreases only with L 3 where L is the maximum angular momentum present in the basis. The integral evaluation effort that grows with 0(N4) prevents the use of ever larger bases for obtaining more accurate results. Most of the developments are therefore focused on wavefunction models with explicit correlation to get faster convergence. Although highly successful these approaches are computationally very demanding. A different solution might be provided by constructing new operators which take care of the information loss introduced by truncating the basis. In this thesis different routes towards such new operators are investigated.
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Kaasalainen, Mikko K. J. "On the construction of invariant tori and integrable Hamiltonians". Thesis, University of Oxford, 1994. http://ora.ox.ac.uk/objects/uuid:399aa26d-4f86-4100-81e2-ba34b6def947.
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The main principle of this thesis is to employ the geometric representation of Hamiltonian dynamics: in a broad sense, we study how to construct, in phase space, geometric structures that are related to a dynamical system. More specifically, we study the problem of constructing phase-space tori that are approximate invariant tori of a given Hamiltonian; also, using the constructed tori, we define an integrable Hamiltonian closely approximating the original one. The methods are generally applicable; as examples, we use gravitational potentials that are of interest in stellar dynamics. First, we construct tori for box and loop orbits in planar, barred potentials, thus demonstrating the applicability of the scheme to potentials that have more than one major orbit family. Also, we show that, in general, the construction scheme needs two types of canonical transformations together: point transformations as well as those expressed by generating functions. To complete the construction scheme, we show how to furnish the tori with consistent coordinate systems, i.e., how to recover the angle variables of a torus labelled by its actions. Next, the developed methods are employed in creating invariant phase-space tori in nonintegrable potentials supporting minor-orbit families. These tori are used to define an integrable Hamiltonian H0, and a modified form of the standard Hamiltonian perturbation theory is then used to demonstrate that a minor-orbit family can be treated as one made up of orbits trapped by a resonance of H0. Finally, we generalize the scheme further by constructing tori in time-reversal asymmetric Hamiltonians (by considering the motion in a rotating frame of reference), and study the transition from locally contained stochasticity to global chaos. Using both near-integrable 'laboratory' Hamiltonians and those for which we construct tori, we investigate the transition in the light of the resonance overlap criterion.
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Wessels, Gert Jermia Cornelus. "A numerical and analytical investigation into non-Hermitian Hamiltonians". Thesis, Stellenbosch : University of Stellenbosch, 2009. http://hdl.handle.net/10019.1/2894.
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Thesis (MSc (Physical and Mathematical Analysis))--University of Stellenbosch, 2009.In this thesis we aim to show that the Schr odinger equation, which is a boundary eigenvalue problem, can have a discrete and real energy spectrum (eigenvalues) even when the Hamiltonian is non-Hermitian. After a brief introduction into non-Hermiticity, we will focus on solving the Schr odinger equation with a special class of non-Hermitian Hamiltonians, namely PT - symmetric Hamiltonians. PT -symmetric Hamiltonians have been discussed by various authors [1, 2, 3, 4, 5] with some of them focusing speci cally on obtaining the real and discrete energy spectrum. Various methods for solving this problematic Schr odinger equation will be considered. After starting with perturbation theory, we will move on to numerical methods. Three di erent categories of methods will be discussed. First there is the shooting method based on a Runge-Kutta solver. Next, we investigate various implementations of the spectral method. Finally, we will look at the Riccati-Pad e method, which is a numerical implemented analytical method. PT -symmetric potentials need to be solved along a contour in the complex plane. We will propose modi cations to the numerical methods to handle this. After solving the widely documented PT -symmetric Hamiltonian H = p2 (ix)N with these methods, we give a discussion and comparison of the obtained results. Finally, we solve another PT -symmetric potential, illustrating the use of paths in the complex plane to obtain a real and discrete spectrum and their in uence on the results.
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Musumbu, Dibwe Pierrot. "The metric for non-Hermitian Hamiltonians : a case study". Thesis, Stellenbosch : Stellenbosch University, 2006. http://hdl.handle.net/10019.1/17403.
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Thesis (MSc)--University of Stellenbosch, 2006.ENGLISH ABSTRACT: We are studying a possible implementation of an appropriate framework for a proper non- Hermitian quantum theory. We present the case where for a non-Hermitian Hamiltonian with real eigenvalues, we define a new inner product on the Hilbert space with respect to which the non-Hermitian Hamiltonian is Quasi-Hermitian. The Quasi-hermiticity of the Hamiltonian introduces the bi-orthogonality between the left-hand eigenstates and the right-hand eigenstates, in which case the metric becomes a basis transformation. We use the non-Hermitian quadratic Hamiltonian to show that such a metric is not unique but can be uniquely defined by requiring to hermitize all elements of one of the irreducible sets defined on the set of all observables. We compare the constructed metric with specific known examples in the literature in which cases a unique choice is made.
AFRIKAANSE OPSOMMING: Ons ondersoek die implementering van n gepaste raamwerk virn nie-Hermitiese kwantumteorie. Ons beskoun nie-Hermitiese Hamilton-operator met reele eiewaardes en definieer in gepaste binneproduk ten opsigtewaarvan die operator kwasi-Hermitiese is. Die kwasi- Hermities aard van die Hamilton operator lei dan tot n stel bi-ortogonale toestande. Ons konstrueer n basistransformasie wat die linker en regter eietoestande van hierdie stel koppel. Hierdie transformasie word dan gebruik omn nuwe binneproduk op die Hilbert-ruimte te definieer. Die oorspronklike nie-HermitieseHamilton-operator is danHermitiesmet betrekking tot hierdie nuwe binneproduk. Ons gebruik die nie-Hermitiese kwadratieseHamilton-operator omte toon dat hierdie metriek nie uniek is nie, maar wel uniek bepaal kan word deur verder te vereis dat dit al die elemente van n onherleibare versameling operatoreHermitiseer. Ons vergelyk hierdie konstruksiemet die bekende voorbeelde in die literatuur en toon dat diemetriek in beide gevalle uniek bepaal kan word.
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Santos, Filipe André Paulino. "Bochi-Mañé dichotomy for 2n-Hamiltonians, Random Perturbation Techniques". Doctoral thesis, Instituto Superior de Economia e Gestão, 2020. http://hdl.handle.net/10400.5/20575.
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Doctor in Applied Mathematics for Economics and ManagementWe prove the high dimensional version of the Bochi-Mañé dichotomy for Hamiltonian systems, achieving for the fi rst time in a continuous setting such a general result. That is, we nd the existence of a C2-residual set of Hamiltonians for which there is an open mod 0 dense set of regular energy surfaces each being either Anosov or for almost every x, either LE(x) = 0 or there is a partially hyperbolic splitting . A generalization of Bochi's random perturbative technique is developed and used in the Hamiltonian framework, to show the extended reach of probabilistic methods and their importance on the nesting and iterative process of perturbations. The main technique consists in letting the original dynamics component act on the invariant subspaces while the random component acts on the direction we are iteratively perturbing. We also connect reachability properties of dynamically guided stochastic processes with the existence or lack of domination on orbits
info:eu-repo/semantics/publishedVersion
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Gutiérrez, i. Serrés Pere. "Estabilitat efectiva i tors invariants de sistemes hamiltonians quasi-integrables". Doctoral thesis, Universitat de Barcelona, 1995. http://hdl.handle.net/10803/2108.
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La memòria recull contribucions a diversos aspectes del problema de l'estabilitat en sistemes hamiltonians quasi-integrables. Aquests aspectes inclouen resultats d'estabilitat efectiva, que comporten el confinament de trajectòries durant un interval de temps molt gran, i també resultats que estableixen l'existència de tors invariants, entre els quals distingim els tors KAM i tors de dimensió inferior.Considerem un sistema hamiltonià quasi-integrable, amb n graus de llibertat, en el qual la mida de la pertorbació és "Epsilon". Malgrat la possibilitat de difusió en aquest tipus de sistemes, els teoremes de Nekhoroshev i KAM (Kolmogorov-Arnol'd-Moser) són resultats molt valuosos que asseguren certs tipus d'estabilitat. Amb tot, les proves habituals d'aquests teoremes no posen en relleu la profunda relació que existeix entre els diferents tipus d'estabilitat a què donen lloc. Gran part de la memòria és dedicada doncs a donar un enfocament unificat per als dos teoremes.
Després d'un capítol d'introducció, al capítol 2 descrivim el mètode seguit per a la prova d'ambdós teoremes, consistent a construir iterativament una transformació canònica que porti el hamiltonià de partida a una forma normal que depengui de menys angles. Per a l'obtenció de la forma normal fem ús del formalisme de les sèries de Lie, que descrivim a la secció 2.1. Aquest és un procediment molt apropiat per a aplicacions pràctiques, perquè permet dur a terme càlculs explícits en exemples concrets, i pot ésser directament implementat en ordinadors. Per tal d'evitar l'efecte causat pels petits divisors, prop de la ressonància associada a un mòdul fixat acceptem que la forma normal pugui dependre de certes combinacions d'angles. De fet només cal considerar ressonàncies fins a un ordre finit apropiat, ja que l'efecte de les ressonàncies d'ordre més alt és exponencialment petit. Basant-nos en el mètode de les sèries de Lie, construïm el procés iteratiu, el qual és finit en la prova del teorema de Nekhoroshev i infinit per al teorema KAM (en aquest darrer cas, sempre prenem el mòdul nul). De fet, descrivim un algorisme lineal i un de quadràtic. Tot i que l'algorisme lineal és d'aparença més senzilla, mostrem que el càlcul explícit de la forma normal podria ésser una mica més ràpid usant l'algorisme quadràtic.
A les seccions 2.3 i 2.4 obtenim les versions lineal i quadràtica del lema iteratiu, que ens donen les fites per a un pas concret del procés iteratiu en cadascun dels dos algorismes. Utilitzem una norma per a camps vectorials hamiltonians (introduïda a la secció 2.2), la qual ens permet d'optimitzar les fites respecte les d'altres autors. Duent a terme un nombre adequat de passos, i aplicant reiteradament el lema iteratiu (en qualsevol de les seves dues versions), obtenim a la secció 2.5 el teorema de la forma normal, en el qual la fita de la resta és exponencialment petita. La prova d'aquest resultat esdevé molt simple degut al fet que el lema iteratiu ha estat optimitzat.
Al capítol 3 obtenim, a partir del teorema de la forma normal, la prova del teorema de Nekhoroshev en el cas quasiconvex. En primer lloc, donem a les seccions 3.1 i 3.2 fites d'estabilitat vàlides sobre regions no ressonants i regions ressonants, respectivament (per al cas ressonant imposem la condició de quasiconvexitat). A la secció 3.3 recobrim tot l'espai de fases amb una família de conjunts, que reben el nom de blocs, associats a diferents mòduls de ressonàncies. Així obtenim a la secció 3.4 un temps d'estabilitat exponencialment gran en 1/Epsilon. per a totes les trajectòries, completant la prova del teorema de Nekhoroshev amb l'exponent òptim 1/2n.
Obtenim també al capítol 3 altres resultats sobre estabilitat efectiva. Hem considerat a la secció 3.1 una pertorbació d'un sistema de n oscil·ladors harmònics amb freqüències satisfent una condició diofàntica. En aquest cas l'exponent de les fites és 1/(Tau + 1), essent Tau l'exponent de la condició diofàntica. A la secció 3.5 veiem que podem millorar les fites de Nekhoroshev si ens restringim a un entorn de la ressonància associada a un mòdul fixat, i obtenim uns exponents d'estabilitat particulars, que depenen de la dimensió del mòdul. A més, apliquem aquestes fites al conegut exemple d'Arnol'd.
Al capítol 4 provem la versió isoenergèica de teorema KAM de manera directa sense usar aplicació de Poincaré) i introduïm la noció de tor quasi-invariant. Comencem veient a la secció 4.1 les dificultats que sorgeixen en el cas isoenergètic, i les resolem amb els lemes tècnics que donem a la secció 4.2. El mètode iteratiu que usem per a provar el teorema KAM isoenergètic és paral·lel, en línies generals, al que usa Arnol'd en el cas ordinari. A la secció 4.3 donem fites per a un pas concret del procés a partir del lema iteratiu. A la secció 4.4 completem la prova del teorema KAM isoenergètic, veient que les restes tendeixen ràpidament cap a zero i obtenint tors invariants n-dimensionals (tors KAM), però només sobre un conjunt cantorià que ve donat per freqüències diofàntiques.
A més, obtenim a la secció 4.5 un resultat d'estabilitat que constitueix un pont entre els teoremes KAM i de Nekhoroshev. Cal considerar les freqüències que satisfan aproximadament una condició diofàntica, fins una precisió donada r. Aquestes freqüències donen lloc a tors quasi-invariants, noció que expressa que les trajectòries que parteixen d'un d'aquests tors hi romanen a prop durant un temps exponencialment gran en 1/r. Així, la precisió r passa a constituir el paràmetre de pertorbació (per a r = 0 tenim els tors KAM). Obtenim aquest resultat dins del mateix esquema iteratiu usat per al teorema KAM però aturant-lo en el moment adequat, en comptes de dur-lo fins al límit. El resultat és molt proper, des del punt de vista quantitatiu, al teorema KAM. Qualitativament, sacrifiquem l'estabilitat perpètua dels tors KAM però, en canvi, tenim un resultat més significatiu des del punt de vista pràctic, ja que per tal d'associar un tor quasi-invariant a una freqüència donada només cal comprovar la condició diofàntica aproximadament. Aquest resultat és lleugerament diferent dels d'altres autors, que estableixen que els tors KAM són "enganxosos" (prenent com a paràmetre la distància a un tor KAM fixat). El nostre resultat és més útil a la pràctica, car no requerim l'existència prèvia d'un tor KAM.
Estudiem a la secció 4.6 l'existència de tors invariants per a un hamiltonià a l'entorn d'un punt fix el·líptic. Sota les condicions adequades, el teorema KAM ens diu que en un entorn de radi r existeix un gran nombre de tors invariants. Fins i tot, si les freqüències del punt el·líptic satisfan una condició diofàntica, llavors la mesura del complementari dels tors invariants és exponencialment petita en 1/r.
Al capítol 5 estudiem els tors invariants de dimensió inferior prop de la ressonància associada a un mòdul de dimensió d < n. La localització d'aquests tors, especialment els tors hiperbòlics, és important com a primer pas per a establir l'existència de difusió d'Arnol'd al llarg d'una cadena de transició. En primer lloc, posem el hamiltonià en forma normal respecte el mòdul fixat i la resta és petita. Fent un canvi canònic lineal (secció 5.2), podem suposar que la part en forma normal només depèn de d angles. Menyspreant la resta, fem un estudi de la forma normal, la qual constitueix un sistema intermedi entre el hamiltonià no pertorbat i el hamiltonià pertorbat. A la secció 5.1 donem condicions per tal que la forma normal tingui tors invariants de dimensio n-d, els quals poden ésser el·líptics, hiperbòlics i d'altres categories. Considerem a la secció 5.3 el cas d'una ressonància simple (d=1), en el qual la forma normal és integrable i per tant podem dur a terme un estudi complet de les varietats invariants dels tors hiperbòlics i les connexions homoclíniques que tenen lloc. Remarquem que, si bé l'existència dels tors hiperbòlics per al sistema original ha estat establerta per altres autors, cal esperar que aquests tors es trobin molt a prop dels de la forma normal si aquesta ha estat obtinguda fins un ordre prou alt. Llavors podem obtenir més informació sobre les varietats invariants.
The main results concerning stability in nearly-integrable Hamiltonian systems are revisited: Nekhoroshev theorem (effective stability) and KAM theorem (existence of invariant tori). We prove both theorems using a common method, which allows to stress the close relationship between them.
The method consists of bringing our Hamiltonian to normal form using an iterative procedure based on Lie series. We describe two algorithms (linear and quadratic) which can both be directly implemented in computers. To give estimates for the remainder of the normal form along the iterative process, we use a vectorfield norm which allows to optimize the estimates.
Iterating these estimates an appropiate (finite) number of steps, we get an exponentially small remainder. Assuming quasiconvexity, we get Nekhoroshev theorem (with the optimal exponent). Further results on effective stability are also obtained.
We prove the isoenergetic version of KAM theorem in a direct way (without using a Poincaré map). In this case, in order to make the remainder tend to zero, we consider an infinite iterative process. In this way the majority of trajectories lie in invariant tori, but these tori fill a Cantorian set given by Diophantine frequencies. Moreover, we introduce the notion of nearly-invariant torus by stopping the process at an appropiate step. We associate a nearly-invariant torus to the frequencies satisfying, up to a given precision, a Diophantine condition (the precision becomes the parameter of perturbation). We also prove the existence of a large number of invariant tori near an elliptic fixed point with Diophantine frequencies: we give for the complement of the invariant tori an exponentially small estimate.
Finally, we study low dimensional tori near resonances and the invariant manifolds of hyperbolic tori near simple resonances. This constitutes a first step towards finding Arnol'd diffusion in nearly-integrable Hamiltonian systems.
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Krüger, Andreas. "Effective hamiltonians from field theory for one and two nucleons". [S.l. : s.n.], 2000. http://deposit.ddb.de/cgi-bin/dokserv?idn=961473347.
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Bookatz, Adam Darryl. "Control, gates, and error suppression with Hamiltonians in quantum computation". Thesis, Massachusetts Institute of Technology, 2016. http://hdl.handle.net/1721.1/104502.
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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Physics, 2016.Cataloged from PDF version of thesis.
Includes bibliographical references.
In this thesis we are primarily interested in studying how to suppress errors, perform simulation, and implement logic gates in quantum computation within the context of using Hamiltonian controls. We also study the complexity class QMA-complete. We first investigate a method (introduced by Jordan, Farhi, and Shor) for suppressing environmentally induced errors in Hamiltonian-based quantum computation, involving encoding the system with a quantum error-detecting code and enforcing energy penalties against leaving the codespace. We prove that this method does work in principle: in the limit of infinitely large penalties, local errors are completely suppressed. We further derive bounds for the finite-penalty case and present numerical simulations suggesting that the method achieves even greater protection than these bounds indicate. We next consider the task of Hamiltonian simulation, i.e. effectively changing a system Hamiltonian to some other desired Hamiltonian by applying external time-dependent controls. We propose protocols for this task that rely solely on realistic bounded-strength control Hamiltonians. For systems coupled to an uncontrollable environment, our approach may be used to perform simulation while simultaneously suppressing unwanted decoherence. We also consider the scenario of removing unwanted couplings in many-body quantum systems obeying local system Hamiltonians and local environmental interactions. We present protocols for efficiently switching off the Hamiltonian of a system, i.e. simulating the zero Hamiltonian, using bounded-strength controls. To this end, we introduce the combinatorial concept of balanced-cycle orthogonal arrays, show how to construct them from classical error-correcting codes, and show how to use them to decouple n-qudit l-local Hamiltonians using protocols of length at most O(l-1 log n). We then present a scheme for implementing high-fidelity quantum gates using a few interacting bosons obeying a Bose-Hubbard Hamiltonian on a line. We find high-fidelity logic operations for a gate set (including the CNOT gate) that is universal for quantum information processing. Lastly, we discuss the quantum complexity class QMA-complete, surveying all known such problems, and we introduce the "quantum non-expander" problem, proving that it is QMA-complete. A quantum expander is a type of rapidly-mixing quantum channel; we show that estimating its mixing time is a co-QMA-complete problem.
by Adam Darryl Bookatz.
Ph. D.
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Ziraldo, Simone. "Thermalization and relaxation after a quantum quench in disordered Hamiltonians". Doctoral thesis, SISSA, 2013. http://hdl.handle.net/20.500.11767/4817.
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In the present thesis we study the unitary dynamics and the thermalization properties of free-fermion-like Hamiltonians after a sudden quantum quench in presence of disorder. With analytical and numerical arguments, we show that the existence of a stationary state and its description with a generalized Gibbs ensemble (GGE) depend crucially on the observable considered (local versus extensive, one-body versus many-body) and on the localization properties of the final Hamiltonian. We then show an extension of the Wang-Landau algorithm which allows the computation of weighted distributions associated to quantum quenches, like the diagonal and the GGE ensemble expectation-value distributions. We present results on three one-dimensional models, the Anderson model, a
disordered one-dimensional fermionic chain with long-range hopping, and the disordered Ising/XY spin chain.
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Freund, Silvia [Verfasser] y Stefan [Akademischer Betreuer] Teufel. "Effective Hamiltonians for magnetic Bloch bands / Silvia Freund ; Betreuer: Stefan Teufel". Tübingen : Universitätsbibliothek Tübingen, 2013. http://d-nb.info/1162844701/34.
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Richter, Steffen, Tom Michalsky, Lennart Fricke, Chris Sturm, Helena Franke, Marius Grundmann y Rüdiger Schmidt-Grund. "Maxwell consideration of polaritonic quasi-particle Hamiltonians in multi-level systems". American Institute of Physics, 2015. https://ul.qucosa.de/id/qucosa%3A31195.
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We address the problem of the correct description of light-matter coupling for excitons and cavity
photons in the case of systems with multiple photon modes or excitons, respectively. In the literature,
two different approaches for the phenomenological coupling Hamiltonian can be found:
Either one single Hamiltonian with a basis whose dimension equals the sum of photonic modes and
excitonic resonances is used. Or a set of independent Hamiltonians, one for each photon mode, is
chosen. Both are usually used equivalently for the same kind of multi-photonic systems which cannot
be correct. However, identifying the suitable Hamiltonian is difficult when modeling experimental
data. By means of numerical transfer matrix calculations, we demonstrate the scope of
application of each approach: The first one holds only for the coupling of a single photon state to
several excitons, while in the case of multiple photon modes, separate Hamiltonians must be used
for each photon mode.
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Fagerlund, Alexander y Chistopher Ekman. "Finite-dimensional PT-symmetric Hamiltonians with an application to neutrino oscillations". Thesis, KTH, Skolan för teknikvetenskap (SCI), 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-275690.
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In this report, we first briefly summarize Hermitian quantum mechanics before moving on to the non-Hermitian case. We then review PT-symmetric quantum mechanics with a focus on finite-dimensional systems, and include a novel generalization of a perturbative calculation of the C-operator. After briefly covering the basics of neutrino oscillations, we perturbatively examine a PT-symmetric addition to the neutrino oscillation Hamiltonian. We examine the effects of the addition with two different definitions of transition probabilities. However, probability is not conserved to first order with either definition. Further, we note that the effect of the chosen perturbation is to shift the transition probabilities by some phase, and to change the amplitudes of the transition probabilities.
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Kirtschig, Frank. "Topological k.p Hamiltonians and their applications to uniaxially strained Mercury telluride". Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2017. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-226489.
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Topological insulators (TIs) are a new state of quantum matter that has fundamentally challenged our knowledge of insulator and metals. They are insulators in the bulk, but metallic on the edge. A TI is characterized by a so-called topological invariant. This characteristic integer number is associated to every mapping between two topological spaces and can be defined for an electronic system on the lattice. Due to the bulk-edge correspondence a non-trivial value leads to topologically protected edge states. To get insight into the electronic characteristics of these edge/surface states, however, an effective continuum theory is needed. Continuum models are analytical and are also able to model transport. In this thesis we will address the suitability of continuum low-energy theories to describe the topological characteristics of TIs. The models which are topologically well-defined are called topological k.p Hamiltonians. After introducing a necessary background in chapter 1 and 2, we will discuss in the methodological chapter 3 the strategies that have to be taken into account to allow for studying topological surface states. In chapter 4 we will study two different model classes associated to a spherical basis manifold. Both have an integer topological invariant, but one shows a marginal bulk-edge correspondence. In chapter 5 we will study a different continuum theory where the basis manifold corresponds to a hemisphere. We then apply all these ideas to a time-reversal invariant TI -- uniaxially strained Mercury Telluride (HgTe). We determine the spin textures of the topological surface states of strained HgTe using their close relations with the mirror Chern numbers of the system and the orbital composition of the surface states. We show that at the side surfaces with $C_{2v}$ point group symmetry an increase in the strain magnitude triggers a topological phase transition where the winding number of the surface state spin texture is flipped while the four topological invariants characterizing the bulk band structure are unchanged. In the last chapter we will give a summary.
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Miller, Steven Michael. "Scattering and Van der Waals dynamics of Hâ†2-OH(#CHI#'2#PI#)". Thesis, University of Cambridge, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.309145.
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Wijewardena, Udagamge. "Iterative method of solving schrodinger equation for non-Hermitian, pt-symmetric Hamiltonians". DigitalCommons@Robert W. Woodruff Library, Atlanta University Center, 2016. http://digitalcommons.auctr.edu/dissertations/3194.
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PT-symmetric Hamiltonians proposed by Bender and Boettcher can have real energy spectra. As an extension of the Hermitian Hamiltonian, PT-symmetric systems have attracted a great interest in recent years. Understanding the underlying mathematical structure of these theories sheds insight on outstanding problems of physics. These problems include the nature of Higgs particles, the properties of dark matter, the matter-antimatter asymmetry in the universe, and neutrino oscillations. Furthermore, PT-phase transition has been observed in lasers, optical waveguides, microwave cavities, superconducting wires and circuits. The objective of this thesis is to extend the iterative method of solving Schrodinger equation used for an harmonic oscillator systems to Hamiltonians with PT-symmetric potentials. An important aspect of this approach is the high accuracy of eigenvalues and the fast convergence. Our method is a combination of Hill determinant method [8] and the power series expansion. eigenvalues and the fast convergence. One can transform the Schrodinger equation into a secular equation by using a trial wave function. A recursion structure can be obtained using the secular equation, which leads to accurate eigenvalues. Energy values approach to exact ones when the number of iterations is increased. We obtained eigenvalues for a set of PT-symmetric Hamiltonians.
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Gaim, Wolfgang [Verfasser]. "Higher Order Semiclassical Approximations for Hamiltonians with Operator-Valued Symbols / Wolfgang Gaim". Tübingen : Universitätsbibliothek Tübingen, 2021. http://d-nb.info/1235399141/34.
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Stagno, Gabriele Vittorio. "Quantum cosmology in loop quantum gravity : 2-vertex spinfoam amplitudes and effective hamiltonians". Thesis, Aix-Marseille, 2018. http://www.theses.fr/2018AIXM0190.
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Nous avons d'abord étudié les amplitudes de transition entre deux réseaux de spin limite avec diagramme dipolaire, fournis par l'EPRL spinfoam Lorentz modèle avec 2 sommets non-simpliciaux. Une évaluation systématique de ces amplitudes de transition ont été abordées, en identifiant lesquels étaient pertinents pour processus physiques. Grande échelle de spin le comportement et les corrélations entre la limite initiale et finale ont été exquis analytiquement pour un modèle simplifié et numériquement pour le modèle complet, en trouvant que les contributions de différents graphiques peuvent être organisées selon à leur comportement de mise à l'échelle dans une hiérarchie qui est également conservée à de petites pirouettes et bien capturée déjà par un modèle simplifié introduit. Ensuite, la dynamique cosmologique efficace dans la gravité de la boucle réduite quantique a été abordée au moyen d'une nouvelle schéma de régularisation basé sur des états préparés dans une superposition de graphes. De nouveaux hamiltoniens efficaces ont été calculés en montrant d'abord que les schémas de régularisation précédents introduits en boucle de cosmologie quantique (LQC) peuvent être obtenus dans ce nouveau modèle puis étendre le domaine de validité de notre schéma au cas non isotropique (Bianchi I). Pour le cas isotrope, le nouvel hamiltonien effectif génère une dynamique différente de celui fourni par LQC: le scénario de rebond symétrique est remplacé par un évolution qui est quasi stationnaire dans la phase de pré-rebond, puis d'accord avec le LQC un, soutenant la conjecture pour l'émergence de l'univers rebondissant à être un caractéristique générale du secteur isotrope de QRLGWe first studied the transition amplitudes in loop quantum gravity (LQG) between two dipole spin-networks, as provided by the EPRL spinfoam model with 2 non-simplicial vertices. A systematic evaluation of these transition amplitudes has been discussed, identifying which ones were relevant for physical processes. Large scale spin behavior and correlations between the initial and final states has been evaluated, analytically for a simplified model and numerically for the full one, finding that the contributions of different graphs can be organized according to their behavior of setting to the scale in a hierarchy that is also preserved at small pirouettes and well captured already by a simplified model introduced.Beside, the effective quantum cosmological dynamics has been addressed for both isotropic and non isotropic models within the framework of Quantum reduced loop gravity (QRLG), a gauge fixed version of LQG. Dynamics has been addressed by means of a new regularization scheme based on states prepared in a superposition of graphs. New Hamiltonians have been computed, showing the usual regularization schemes introduced in loop quantum cosmology (LQC) naturally fit in this new scheme. Then we extended the domain of validity of our model to the non-isotropic case (Bianchi I spacetime). Both for isotropic and non isotropic cases, the new Hamiltonians generate a dynamics which is different from the one provided by LQC: in the isotropic case, the symmetric big bounce scenario is replaced by an evolution which is quasi stationary in the pre bounce phase and then follows the usual expansion. For Bianchi I an intriguing accelerated phase replaces the stationary one
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Moriya, Paulo Hisao. "Manipulação do pulso superradiante via interações atômicas". Universidade de São Paulo, 2012. http://www.teses.usp.br/teses/disponiveis/76/76131/tde-07052012-135135/.
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O fenômeno da superradiância é caracterizado por um processo de ordenamento das transições dos dipolos atômicos em amostras excitadas, moderadamente densas, decorrente das correlações induzidas entre os átomos desenvolvidas pela radiação coerente emitida pelos próprios átomos. O processo superradiante que é iniciado a partir de uma total desordem em t = 0 atinge um ordenamento máximo em um tempo τ α N-1, gerando um pulso de radiação de intensidade seguindo a lei do sech2 e com pico proporcional à N2, e em seguida os dipolos relaxam para um equilíbrio desordenado. Neste trabalho, tratamos a interação de dois modos de uma cavidade, ωa e ωb, e uma amplificação, com um sistema de N átomos de dois níveis, com frequência de transição atômica ω0 de forma que interaja ressonantemente com ωa e dispersivamente com ωb, responsável pelo acoplamento entre os átomos. Para enterdemos como a lei do sech2 será afetada pela interação direta entre os átomos, utilizamos o método das perturbações via de pequenas rotações não-lineares para obtermos o hamiltoniano efetivo do sistema com uma forma mais explícita da interação dipolar entre os átomos. Por fim, após escrevermos a equação mestra do sistema, utilizamos a aproximação de campo médio e o método dos invariantes de Lewis-Riesenfeld para chegar aos principais aspectos deste fenômeno no sistema.The superradiant phenomena is characterized by atomic dipoles ordering process in excited samples moderately denses, that occours due to the atomic induced correlations developed not directly but by the coherent radiation emitted by atoms themselves. The superradiant process evolves from a total disorder at t = 0, attain a maximum order in a time τ α N-1 creating a radiation pulse whose intensity follows the sech2 law and its peak is proportional to N2, thereafter the dipoles relax to a disordered equilibrium state. In this essay, we deal with the interaction between two cavity modes ωa and ωb and a classical pump with a system of N two-level atoms, whose atomic transition frequencies ω0. We consider a resonant interaction between atoms and mode ωa and a dispersive coupling of atoms with mode ωb, which couple the atomic sample, and the classical pump. In order to obtain how sech2 law changes, we use the method of nonlinear small rotations to obtain effective Hamiltonian, expliciting dipolar interaction between atoms. Finally, after write the effective master equation, we use the mean-field approximation and Lewis and Riesenfeld method to obtain the mean features of this phenomena to our system.
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Abreu, Jean Faber Ferreira de. "Quantum games from biophysical Hamiltonians and a sub-neuronal optimization criterion of the information". Laboratório Nacional de Computação Científica, 2006. http://www.lncc.br/tdmc/tde_busca/arquivo.php?codArquivo=108.
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The Theory of Games is a mathematical formalism used to analyze conflicts between two or more parts. In those conflicts, each part has a group of actions (strategies) that aids them in the optimization of their objectives. The objectives of the players are the rewards (payoffs) given according to their chosen strategy. By quantizing a game, advantages in operational efficiency and in the stability of the game solutions are demonstrated. In a quantum game, the strategies are operators that act on an isolated system. A natural issue is to consider a game in an open system. In this case the strategies are changed by Kraus operators which represent a natural measurement of the environment. We want to find the necessary physical conditions to model a quantum open system as a game. To analyze this issue we applied the formalism of Quantum Operations on the Fröhlich system and we described it as a model of Quantum Game. The interpretation is a conflict among different configurations of the environment which, by inserting noise in the main system exhibits regimes of minimum loss of information. On the other hand, the model of Fröhlich has been used to describe the biophysical dynamics of the neuronal microtubules. By describing the model of Fröhlich in the Quantum Game formalism, we have shown that regimes of stability may exist even under physiological conditions. From the evolutionary point of view, the Theory of Games can be the key to describe the natural optimization at sub-neuronal levels.A Teoria de Jogos (TJs) é um formalismo matemático usado para analisar situações de conflitos entre duas ou mais partes. Nesses conflitos, cada parte possui um conjunto de ações (estratégias) que auxilia na otimização de seus objetivos. Os objetivos dos jogadres são as recompensas (payoffs) que cada um recebe de acordo com a estratégia adotada. Ao se quantizar um jogo, mostra-se ganhos em eficiência operacional e ganhos na estabilidade das soluções. Em um jogo quântico (JQ), as estratégias são operadores que atuam num sistema isolado. Uma questão natural é considerar um jogo num sistema aberto. Nesta situação as estratégias são trocadas por operadores de Kraus que representam uma medida natural do ambiente. Nosso interesse é encontrar as condições físicas necessáriaas para modelarmos um sistema quântico aberto como um jogo. Para analisar essa questão aplicamos o formalismo de Operações Quânticas (OQs) sobre o sistema de Fröhlich e o apresentamos como um modelo de JQ. A interpretação é um conflito entre diferentes configurações do ambiente que, ao inserirem ruído no sistema principal, exibem regiões de mínima perda de informação. O modelo de Fröhlich vem sendo usado para descrever a dinâmica biofísica dos microtúbulos neuronais. Ao estruturamos o modelo de Fröhlich nos JQs, mostramos que as regiões de estabilidade podem existir sob condições fisiológicas. Usando o aspecto evolucionista, a TJs pode ser a chave para a descrição de processos de otimização da informação em nível sub-neuronal.
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Okan, Osman Burak. "Merging quadratic programming with kernel smoothing for automated cluster expansions of complex lattice Hamiltonians". Thesis, Massachusetts Institute of Technology, 2008. http://hdl.handle.net/1721.1/44383.
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Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Materials Science and Engineering, 2008.Includes bibliographical references (p. 46-48).
We present a general outline for automating cluster expansions of configurational energetics in systems with crystallographic order and well defined space group symmetry. The method presented herein combines constrained optimization techniques of positive-definitive quadratic forms with the mathematical tool of Tikhonov regularization (kernel smoothing) for automated expansions of an arbitrary general physical property without compromising the underlying physics. Throughout the thesis we treat formation energy as the fundamental physical observable to expand on since the predominant application of cluster expansions is the extraction of robust approximations for configurational energetics in alloys and oxides. We therefore present the implementational aspects of the novel algorithmic route on a challenging material system NaxCoO2 and reconstruct the corresponding GGA ground state line with arbitrary precision in the formation energy-configuration space. The mathematical arguments and proofs, although discussed for cases with arbitrary spin assignments and multiple candidate species for single site occupancy, are eventually formulated and illustrated for binary systems. Various numerical challanges and the way they are resolved in the framework of kernel smoothing are addressed in detail as well. However, the applicability of the procedure described herein is more universal and can be tailored to probe different observables without resorting to modifications in the algorithmic implementation or the fundemantal mathematical construction. The effectiveness in recovering correct physics shall than be solely tied to the presence of superposable nature (of the physical property of interest) of local atomic configurations or lackthereof.
by Osman Burak Okan.
S.M.
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Fassari, Silvestro. "Spectral properties of relativistic and non-relativistic Krönig- Penney Hamiltonians with short-range impurities". Diss., Virginia Polytechnic Institute and State University, 1989. http://hdl.handle.net/10919/54524.
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In this work, we investigate the spectrum of the non-relativistic Krönig-Penney Hamiltonian Hα= -d²/dx² +αΣm∈Zδ(-(2m+1)π) perturbed by a short-range potential λW and the spectrum of its relativistic counterpart obtained by replacing the Schrödinger Hamiltonian Hα with its relativistic analogue H̅α.
The interesting feature of both spectra is that they have gaps and that bound states may occur in such gaps as a consequence of the presence of the short-range potential representing the impurity. Such bound states, often called "impurity states" in the solid state physics literature. are important with regard to the conductivity properties of solids
We show the existence of such bound states of Hα + λW in each sufficiently remote gap of its essential spectrum if the integral of W is different from zero and the 1 + 𝛅-moment of W is finite for some 𝛅 > 0. Furthermore, if the potential has a constant sign we prove that there is only one bound state in each sufficiently remote gap.
We shall see that in the relativistic case one may have more than one bound state in each remote gap under the same assumptions on W. Nevertheless, we shall see that such additional bound states cannot appear in the range of energies of solid state physics.Ph. D.
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Liu, Yimin. "Theory of vibronic coupling in impurity systems". Thesis, University of Nottingham, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.282583.
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Jorba-Cuscó, Marc. "Periodic time dependent Hamiltonian systems and applications". Doctoral thesis, Universitat de Barcelona, 2019. http://hdl.handle.net/10803/666655.
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A dynamical system is one that evolves with time. This definition is so diffuse that seems to be completely useless, however, gives a good insight of the vast range of applicability of this field of Mathematics has. It is hard to track back in the history of science to find the origins of this discipline. The works by Fibonacci, in the twelfth century, concerning the population growth rate of rabbits can be already considered to belong to the above mentioned field. Newton's legacy changed the prism through the humankind watched the universe and established the starting shot of several areas of knowledge including the study of difierential equations. Newton's second law relates the acceleration, the second derivative of the position of a body with the net force acting upon it. The formulation of the law of universal gravitation settled the many body problem, the fundamental question around the field of celestial mechanics has grown. Newton itself solved the two body problem, providing an analytical proof of Kepler's laws. In the subsequent years a number of authors, among of them Euler and Lagrange, exhausted Newton's powerful ideas but none of them was able to find a closed solution of the many body problem. By the end of the nineteenth century, Poincaré changed again the point of view: The French mathematician realized that the many body problem could not be solved in the sense his predecessors expected, however, many other fundamental questions could be addressed by studying the solutions of not quantitatively but by means of their geometrical and topological properties. The ideas that bloomed in Poincaré's mind are nowadays a source of inspiration for modern scientist facing problems located along all the spectrum of human knowledge.
Poincaré understood that invariant structures organize the long term behaviour of the solutions of the system. Invariant objects are, therefore, the skeleton of the dynamics. These
invariant structures and their linear normal behaviour are to be analyzed carefully and this shall lead to a good insight on global aspects of the phase space.
For nonintegrable systems the task of studying invariant objects and their stability is, in general, a problem which is hard to be handled rigorously. Usually, the hypotheses needed to prove specific statements on the solutions of the systems reduce the applicability of the results. This is especially relevant in physical problems: Indeed, we cannot, for instance, choose the mass of Sun to be suficiently small.
The advent of the computers changed the way to undertake studies of dynamical systems. The task of writing programs for solving, numerically, problems related to specific examples is, at the present time, as important as theoretical studies. This has two main consequences: On the first hand, more involved models can be chosen to study real problems and this allow us to understand better the relation between abstract concepts and physical phenomena. Secondly, even when facing fundamental questions on dynamics, the numerical studies give us data from which build our theoretical developments. Nowadays, a large number of commercial (or public) software packages helps scientist to study simple problems avoiding the tedious work to master numerical algorithms and programming languages. These programs are coded to work in the largest possible number of different situations, therefore, they do not have the eficiency that programs written specifically for a certain purpose have. Some of the computations presented in this dissertation cannot be performed by using commercial software or, at least, not in a reasonable amount of time. For this reason, a large part of the work presented here has to do with coding and debugging programs to perform numerical computations. These programs are written to be highly eficient and adapted to each problem. At the same time, the design is done so that specific blocks of the code can be used for other computations, that is, there exist a commitment between eficiency and reusability which is hard to achieve without having full control on the code.
Under these guiding principles we undertake the study of applied dynamical systems according to the following stages: From a particular problem we get a simple model, then perform a number of numerical experiments that permits us to understand the invariant objects of the system, with that information, we can isolate the relevant phenomena and identify the key elements playing a role on it. Next, we try to find an even simpler model in which we can develop theoretical arguments and produce theorems that, with more effort, can be generalized or related to other problems which, in principle, seem to be difierent to the original one. Paraphrasing Carles Simó, from a physical problem we can take the lift to the abstract world, use theoretical arguments, come out with conclusions and, finally, lift down to the real world and apply these conclusions to specific problems (maybe not only the original one).
This methodology has been developed in the last decades over the world when it turned out to outstand among the most powerful approaches to cope with problems in applied mathematics. The group of Dynamical Systems from Barcelona has been one of the bulwarks of this development from the late seventies to the present days.
Following the guidelines presented in the previous section, we concern with several problems, mostly from the field of celestial mechanics but we also deal with a phenomenon coming from high energy physics. All these situations can be modeled by means of periodically time dependent Hamiltonian systems. To cope with those investigations, we develop software which can be used to perform computations in any periodically perturbed Hamiltonian system. We split the contents of this dissertation in two parts. The first one is devoted to general tolos to handle periodically time dependent Hamiltonians, even though we fill this first part with a number of illustrating examples, the goal is to keep the exposition in the abstract setting.
Most of the contents of Part I deal with the development of software used to be applied in the second part. Some of the software has not been applied to the specific contents of Part
II, this is left for future work. We also devote a whole chapter to some theoretical issues that, while are motivated by physical problems, they fall out of the category of periodic time dependent Hamiltonians. This splitting of contents has the intention of reecting, somehow, the basic methodological principles presented in the previous paragraph, keeping separated the abstract and the physical world but keeping in mind the lift.
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Maurice, Rémi. "Zero-field anisotropic spin hamiltonians in first-Row transition metal complexes : theory, models and applications". Doctoral thesis, Universitat Rovira i Virgili, 2011. http://hdl.handle.net/10803/37363.
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Aquest treball presenta l’estudi teòric de l’anisotropia magnètica en complexos de metalls de transició, combinant esquemes de càlcul multiconfiguracionals relativistes amb derivacions analítiques basades en la teoria del camp del lligand, el que permet racionalitzar a través de conceptes senzills els resultats quantitatius obtinguts i interpretar les propietats estudiades.
Es desenvolupa primer una metodologia per extreure els paràmetres d’anisotropia en complexos mononuclears de metalls de transició. El mètode es basa en assignar els resultats d’un càlcul ab initio d’alt nivell a un Hamiltonià model mitjançant la teoria d’Hamiltonians efectius. Aquesta metodologia s’aplica a complexos de Ni(II), Co(II) i Mn(III) i es comprova que és aplicable de forma general a complexos mononuclears.
S’estén després la metodologia a complexos binuclears, pels quals l’Hamiltonià model usualment utilitzat té una base menys rigorosa. L’Hamiltonià efectiu obtingut per un complex binuclear de Ni(II) introdueix una nova parametrització amb termes addicionals de les interaccions anisotròpiques en sistemes polinuclears. Es tracta d’un procediment universal que proporciona valors precisos i a més és capaç de contrastar la consistència interna dels Hamiltonians models existents.
Per racionalitzar les correlacions magnetoestructurals dels paràmetres d’anisotropia en complexos de Ni(II) i Mn(III), es descriuen els mecanismes electrònics bàsics en base a consideracions de la teoria del camp del lligand. Aquest procediment proporciona regles senzilles per augmentar l’anisotropia, que poden ser aplicades en el disseny de nous materials.
Finalment, s’estudien les interaccions anisotròpiques simètriques i antisimètriques en compostos binuclears de Cu(II), interaccions de gran importància per explicar les propietats d’alguns materials d’interès tecnològic. Les interaccions antisimètriques s’extreuen a partir de càlculs ab initio d’estructura electrònica per primer cop en aquest treball. Es concentra l’atenció d’aquesta part en dos sistemes: el conegut complex binuclear de Cu(II) amb quatre ponts acetat, i l’òxid de coure en el que recentment s’ha evidenciat una fase ferroelèctrica.
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Schnells, Vera [Verfasser], Martin [Gutachter] Greiter, Haye [Gutachter] Hinrichsen y Friedrich [Gutachter] Reinert. "Fractional Insulators and their Parent Hamiltonians / Vera Schnells ; Gutachter: Martin Greiter, Haye Hinrichsen, Friedrich Reinert". Würzburg : Universität Würzburg, 2019. http://d-nb.info/1193423813/34.
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Farid, Amro M. (Amro Mohsen) 1978. "Compensation of incoherent errors in the precise implementation of effective Hamiltonians for quantum information processing". Thesis, Massachusetts Institute of Technology, 2002. http://hdl.handle.net/1721.1/89348.
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Schmidt, Julian [Verfasser] y Stefan [Akademischer Betreuer] Teufel. "Interior-Boundary Conditions as a Direct Description of QFT Hamiltonians / Julian Schmidt ; Betreuer: Stefan Teufel". Tübingen : Universitätsbibliothek Tübingen, 2019. http://d-nb.info/1205002065/34.
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Mehringer, Josef [Verfasser] y Edgardo [Akademischer Betreuer] Stockmeyer. "Spectral and dynamical properties of certain quantum hamiltonians in dimension two / Josef Mehringer. Betreuer: Edgardo Stockmeyer". München : Universitätsbibliothek der Ludwig-Maximilians-Universität, 2015. http://d-nb.info/1080122257/34.
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Ezzahra, Lembarki Fatima. "Periodic orbits of differential systems via the averaging theory with special emphasis on Hamiltonian systems". Doctoral thesis, Universitat Autònoma de Barcelona, 2016. http://hdl.handle.net/10803/400392.
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Calci, Angelo Verfasser], Robert [Akademischer Betreuer] Roth y Jochen [Akademischer Betreuer] [Wambach. "Evolved Chiral Hamiltonians at the Three-Body Level and Beyond / Angelo Calci. Betreuer: Robert Roth ; Jochen Wambach". Darmstadt : Universitäts- und Landesbibliothek Darmstadt, 2014. http://d-nb.info/1110901984/34.
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Calci, Angelo [Verfasser], Robert Akademischer Betreuer] Roth y Jochen [Akademischer Betreuer] [Wambach. "Evolved Chiral Hamiltonians at the Three-Body Level and Beyond / Angelo Calci. Betreuer: Robert Roth ; Jochen Wambach". Darmstadt : Universitäts- und Landesbibliothek Darmstadt, 2014. http://nbn-resolving.de/urn:nbn:de:tuda-tuprints-40698.
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Neto, Flávio de Oliveira. "Hamiltoniano Intensity Dependent na teoria do laser". Universidade de São Paulo, 2016. http://www.teses.usp.br/teses/disponiveis/76/76131/tde-04052016-155606/.
Texto completoResumen
Tem-se como intuito desse projeto a construção e o desenvolvimento de um Hamiltoniano intensity dependent, cuja interação entre radiação-matéria dependa do número de fótons que residem dentro da cavidade do laser. O Hamiltoniano de Jaynes Cummings é tradicionalmente conhecido por descrever a interação radiação-matéria, e através de uma modificação efetuada no mesmo, criando um Hamiltoniano não-linear em termos dos operadores de criação e aniquilação, pretende-se obter uma nova distribuição do número de fótons dentro de tal cavidade, bem como uma nova estatística em relação ao modelo usual de laser. Para tal,faz-se uso de um modelo de átomo de dois níveis para a descrição da matéria dentro da cavidade, bem como conhecimentos de informação e óptica quântica para o desenvolvimento e análise dos resultados obtidos, como o fator Q de Mandel, juntamente com aplicações de Hamiltonianos não lineares, necessários para o entendimento do projeto. Finalmente, discute-se as consequências da nova distribuição; suas semelhanças e suas diferenças em relação à tradicional, focando nos papéis dos parâmetros do laser.The purpose of this work is the construction and development of an intensity dependent Hamiltonian, whose interaction between radiation and matter depends on the photon number inside the laser cavity. The Jaynes Cummings\'s Hamiltonian is traditionally known because of its description of the radiation-matter interaction, and through a modification on this hamiltonian, building a non-linear hamiltonian in terms of the creation and anihilation operators, we intend to obtain a new distribuition of the photon number inside the laser cavity, as well as a new statistics regarding the usual laser model. In order to do it, we use two levels atom model to describe the matter inside the cavity, as well as knowledge of quantum optical and quantum information to develop and analyze the obtained results, like the Mandel Q parameter, along with the non-linear hamiltonian applications, necessary to understand this project. Finally, we discuss the consequences of its new distribution, its similarities and diferences about the traditionals, focusing on the roles of the laser parameters.
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Pacha, Andújar Juan Ramón. "On the quasiperiodic hamiltonian andronov-hopf bifurcation". Doctoral thesis, Universitat Politècnica de Catalunya, 2002. http://hdl.handle.net/10803/5830.
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Aquest treball es situa dintre del marc dels sistemes dinàmics hamiltonians de tres graus de llibertat. Allà considerem famílies d'òrbites periòdiques amb una transició estable-complex inestable: sigui L el paràmetre que descriu la família i suposarem que per a valors del paràmetre més petits que un cert valor crític, L', els multiplicadors característics de les òrbites periòdiques corresponents hi són sobre el cercle unitat, quan L=L' aquests col·lisionen per parelles conjugades (òrbita ressonant o crítica) i per L > L', abandonen el cercle unitat cap al pla complex (col·lisió de Krein amb signatura oposada). El canvi d'estabilitat subseqüent es descriu a la literatura com "transició estable a complex inestable". Tanmateix, a partir d'estudis numèrics sobre certes aplicacions simplèctiques (n'esmentarem D. Pfenniger, Astron. Astrophys. 150, 97-111, 1985), és coneguda l'aparició (sota condicions d'incommensurabilitat) de fenòmens de bifurcació quasi-periòdica, en particular, el desplegament de famílies de tors 2-dimensionals. A més aquesta bifurcació s'assembla a la (clàssica) bifurcació d'Andronov-Hopf, en el sentit de què hi sorgeixen objectes linealment estables (tors-2D el·líptics) "al voltant" d'objectes inestables de dimensionalitat més baixa (òrbites periòdiques), i recíprocament, n'apareixen tors inestables (hiperbòlics) "al voltant" d'òrbites periòdiques linealment estables.Nostre objectiu és entendre la dinàmica local en un entorn de l'òrbita periòdica ressonant per tal de provar, analíticament, l'existència dels tors invariants bifurcats segons l'esquema descrit dalt. Això el portem a terme mitjançant l'anàlisi següent:
(i) Primer de tot obtenim d'una manera constructiva (això és, donant algorismes) una forma normal ressonant en un entorn de l'òrbita periòdica crítica. Aquesta forma normal la portem fins a qualsevol ordre arbitrari r. Així doncs, mostrem que el hamiltonià inicial es pot posar com la suma de la forma normal (integrable) més una resta no integrable. A partir d'aquí, podem estudiar la dinàmica de la forma normal, prescindint dels altres termes i, amb aquest tractament (formal) del problema, som capaços d'identificar els paràmetres que governen tant l'existència de la bifurcació com la seva tipologia (directa, inversa). Cal, remarcar que el que es fa fins aquí, no és només un procés qualitatiu, ja que a més ens permet derivar parametritzacions molt acurades dels tors no pertorbats.
(ii) A continuació, calculem acotacions "òptimes" per a la resta. D'aquesta manera, esperem provar que un bon nombre de tors (en sentit de la mesura) es preserven quan s'afegeix la pertorbació.
(iii) Finalment, apliquem mètodes KAM per establir que la majoria (veure comentari dalt) dels tors bifurcats sobreviuen. Aquests mètodes es basen en la construcció d'un esquema de convergència quadràtica capaç de contrarestar l'efecte dels petits divisors que apareixen quan s'aplica teoria de pertorbacions per trobar solucions quasi-periòdiques. En el nostre cas, a més, resulta que alguna de les condicions "típiques" que s'imposen sobre les freqüències (intrínseques i normals) dels tors no pertorbats, no estan ben definides per als tors bifurcats, de manera que ens ha calgut desenvolupar un tractament més específic.
keywords: Bifurcation problems, perturbations, normal forms, small divisors, KAM theory.
Classificació AMS: 37J20, 37J25, 37J40
This work is placed into the context of the three-degree of freedom Hamiltonian systems, where we consider families of periodic orbits undergoing transitions stable-complex unstable. More precisely: Let L be the parameter of the family and assuming that, for values of L smaller than some critical value say, L', the characteristic multipliers of the periodic orbits lie on the unit circle, when L=L' they colllide pairwise (critical or resonant periodic orbit) and, for L > L' leave the unit circle towards the complex plane (Krein collision with opposite signature).
From numerical studies on some concrete symplectic maps (for instance, D. Pfennniger, Astron. Astrophys. 150, 97-111, 1985) it is known the rising (under certain irrationality conditions), of quasi-periodic bifurcation phenomena, in particular, the appearance of unfolded 2D invariant tori families. Moreover, the bifurcation takes place in a way that resembles the classical Andronov-Hopf one, in the sense that either stable invariant objects (elliptic tori) unfold "around" linear unstable periodic orbits, or conversely, unstable invariant structures (hyperbolic tori) appear "surrounding" stable periodic orbits.
Our objective is, thus, to understand the (local) dynamics in a neighbourhood of the critical periodic orbit well enough to prove analytically, the existence of such quasi-periodic solutions together with the bifurcation pattern described above. This is carried out through three steps:
(i) First, we derive, in a constructive way (i. e., giving algorithms), a resonant normal form around the critical periodic orbit up to any arbitrary order r. Whence, we show that the initial raw Hamiltonian can be casted --through a symplectic change--, into an integrable part, the normal form itself, plus a (non-integrable) remainder. From here, one can study the dynamics of the normal form, skipping the remainder off. As a result of this (formal) approach, we are able to indentify the parameters governing both, the presence of the bifurcation and its type (direct, inverse). We remark that this is not a merely qualitative process for, in addition, accurate parametrizations of the bifurcated families of invariant tori are derived in this way.
(ii) Beyond the formal approach, we compute "optimal" bounds for the remainder of the normal form, so one expects to prove the preservation of a higher (in the measure sense) number of invariant tori --than, indeed, with a less sharp estimates--.
(iii) Finally, we apply KAM methods to establish the persistence of (most, in the measure sense) of the bifurcated invariant tori. These methods involve the design of a suitable quadratic convergent scheme, able to overcome the effect of the small divisors appearing in perturbation techniques when one looks for quasi-periodic solutions. In this case though, some of the "typical" conditions that one imposes on the frequencies (intrinsic and normal) of the unperturbed invariant tori do not work, due to the proximity to parabolic tori, so one is bound to sketch specific tricks.
keywords: Bifurcation problems, perturbations, normal forms, small divisors, KAM theory
AMS classification: 37J20, 37J25, 37J40
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O'Hara, Michael James. "Adiabatic quantum computation noise in the adiabatic theorem and using the Jordan-Wigner transform to find effective Hamiltonians /". College Park, Md.: University of Maryland, 2008. http://hdl.handle.net/1903/8113.
Texto completoResumen
Thesis (Ph. D.) -- University of Maryland, College Park, 2008.Thesis research directed by: Applied Mathematics and Scientific Computation Program. Title from t.p. of PDF. Includes bibliographical references. Published by UMI Dissertation Services, Ann Arbor, Mich. Also available in paper.
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Hogan-O'Neill, Jason. "Interface effects in superconductors : self-consistent solution of the Bogoliubov-de Gennes equations via the recursion method". Thesis, University of Bristol, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.302164.
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Morgan, Samuel Alexander. "A gapless theory of Bose-Einstein condensation in dilute gases at finite temperature". Thesis, University of Oxford, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.302178.
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Stosiek, Matthias [Verfasser], Ferdinand [Akademischer Betreuer] Evers y Jaroslav [Akademischer Betreuer] Fabian. "Self-consistent-field ensembles of disordered Hamiltonians: Efficient solver and application to superconducting films / Matthias Stosiek ; Ferdinand Evers, Jaroslav Fabian". Regensburg : Universitätsbibliothek Regensburg, 2020. http://d-nb.info/1216703604/34.
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Agostinelli, Chiara. "Edge states and zero modes in quadratic fermionic models on a 1-D lattice". Master's thesis, Alma Mater Studiorum - Università di Bologna, 2015. http://amslaurea.unibo.it/9344/.
Texto completoResumen
In una formulazione rigorosa della teoria quantistica, la definizione della
varietà Riemanniana spaziale su cui il sistema è vincolato gioca un ruolo fondamentale.
La presenza di un bordo sottolinea l'aspetto quantistico del sistema: l'imposizione
di condizioni al contorno determina la discretizzazione degli autovalori del Laplaciano,
come accade con condizioni note quali quelle periodiche, di Neumann o
di Dirichlet. Tuttavia, non sono le uniche possibili. Qualsiasi condizione al
bordo che garantisca l'autoaggiunzione dell' operatore Hamiltoniano è ammissibile.
Tutte le possibili boundary conditions possono essere catalogate a partire
dalla richiesta di conservazione del flusso al bordo della varietà.
Alcune possibili condizioni al contorno, permettono l'esistenza di stati legati
al bordo, cioè autostati dell' Hamiltoniana con autovalori negativi, detti edge
states.
Lo scopo di questa tesi è quello di investigare gli effetti di bordo in sistemi unidimensionali
implementati su un reticolo discreto, nella prospettiva di capire
come simulare proprietà di edge in un reticolo ottico.
Il primo caso considerato è un sistema di elettroni liberi. La presenza di edge
states è completamente determinata dai parametri di bordo del Laplaciano discreto.
Al massimo due edge states emergono, e possono essere legati all' estremità destra o sinistra della catena a seconda delle condizioni al contorno.
Anche il modo in cui decadono dal bordo al bulk e completamente determinato
dalla scelta delle condizioni.
Ammettendo un' interazione quadratica tra siti primi vicini, un secondo tipo
di stati emerge in relazione sia alle condizioni al contorno che ai parametri del
bulk. Questi stati sono chiamati zero modes, in quanto esiste la possibilità che
siano degeneri con lo stato fondamentale.
Per implementare le più generali condizioni al contorno, specialmente nel caso
interagente, è necessario utilizzare un metodo generale per la diagonalizzazione,
che estende la tecnica di Lieb-Shultz-Mattis per Hamiltoniane quadratiche a
matrici complesse.
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Bieske, Evan John y n/a. "The Electronic Spectroscopy of Neutral and Ionic Clusters". Griffith University. School of Science, 1989. http://www4.gu.edu.au:8080/adt-root/public/adt-QGU20051109.112502.
Texto completoResumen
This thesis is concerned with weakly bound neutral and ionic clusters. Spectra of the region near the S1fS0 electronic origin of four neutral van der Waals molecules - aniline-argon, phenol-argon, chlorobenzene-argon and fluorobenzene-argon - were obtained using resonance enhanced multiphoton ionization (REMPI). These spectra indicate that Fermi resonances between van der Waals stretching and bending motions are important in these molecules. Effective Hamiltonians are constructed that describe well the low frequency vibrations. In order to better discuss the low frequency van der Waals motions of aromatics bound to one and two rare gas atoms a simple model for the vibrations is developed. The model enables expression of van der Waals frequencies in terms of fundamental molecular properties and enables facile comparison of effective force constants in a variety of van der Waals molecules. The model is successfully employed to explain van der Waals vibrational structure associated with the origin region of aniline-(argon)2 using van der Waals potential parameters derived from the aniline-(argon)1 spectrum. REMPI and emission spectra of larger clusters of aniline and argon are also reported and discussed. Using atom-atom potentials, equilibrium structures for aniline-(argon)n (n=l, 2, 3) are calculated. The calculations prove useful in the analysis of the spectra.The BfX transitions of the cation complexes fluorobenzene+-argon and chlorobenzene+-argon have been investigated. The cations were prepared by resonance enhanced multiphoton ionization of the neutral van der Waals molecules. A time delayed tunable dye laser was then used to dissociate the cations, loss of an argon atom being the dominant process. When the second laser was tuned to a cation resonance the dissociation cross section increased markedly, allowing characterization of BfX transition. The resulting spectra are presented and discussed.
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Andre, Daniel Batista. "Weyl expansion for multicomponent wave equations". Thesis, University of Bristol, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.310887.
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Bieske, Evan John. "The Electronic Spectroscopy of Neutral and Ionic Clusters". Thesis, Griffith University, 1989. http://hdl.handle.net/10072/367202.
Texto completoResumen
This thesis is concerned with weakly bound neutral and ionic clusters. Spectra of the region near the S1fS0 electronic origin of four neutral van der Waals molecules - aniline-argon, phenol-argon, chlorobenzene-argon and fluorobenzene-argon - were obtained using resonance enhanced multiphoton ionization (REMPI). These spectra indicate that Fermi resonances between van der Waals stretching and bending motions are important in these molecules. Effective Hamiltonians are constructed that describe well the low frequency vibrations. In order to better discuss the low frequency van der Waals motions of aromatics bound to one and two rare gas atoms a simple model for the vibrations is developed. The model enables expression of van der Waals frequencies in terms of fundamental molecular properties and enables facile comparison of effective force constants in a variety of van der Waals molecules. The model is successfully employed to explain van der Waals vibrational structure associated with the origin region of aniline-(argon)2 using van der Waals potential parameters derived from the aniline-(argon)1 spectrum. REMPI and emission spectra of larger clusters of aniline and argon are also reported and discussed. Using atom-atom potentials, equilibrium structures for aniline-(argon)n (n=l, 2, 3) are calculated. The calculations prove useful in the analysis of the spectra.The BfX transitions of the cation complexes fluorobenzene+-argon and chlorobenzene+-argon have been investigated. The cations were prepared by resonance enhanced multiphoton ionization of the neutral van der Waals molecules. A time delayed tunable dye laser was then used to dissociate the cations, loss of an argon atom being the dominant process. When the second laser was tuned to a cation resonance the dissociation cross section increased markedly, allowing characterization of BfX transition. The resulting spectra are presented and discussed.Thesis (PhD Doctorate)
Doctor of Philosophy (PhD)
School of Science
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