Dissertationen zum Thema „Classical oscillator“

The existence of a perfectly isolated quantum system is impossible. In reality, no quantum system is completely isolated from its surroundings, so every quantum system is open to some extent. The dynamics of any open quantum system is described by Lindblad equation [1].

La spectroscopie de dichroïsme circulaire (CD) est une des techniques fondamentales en biologie structurale qui permet la détermination du contenu en structures secondaires d'une protéine. Le rayonnement synchrotron a considérablement augmenté l’utilité de la méthode, car il permet de travailler avec une gamme spectrale étendue et à meilleure intensité. Le développement de modèles permettant d’établir une relation entre la structure d’une protéine et son spectre CD d’une manière efficace n’a pourtant pas suivi l’évolution technique et l’analyse de spectres CD de protéines entières reste un défi sur le plan théorique. Dans ce contexte, nous avons développé un modèle "minimaliste" pour la spectroscopie CD des protéines, où chaque atome C-alpha de la chaîne principale porte un oscillateur de Lorentz classique, i.e. une charge mobile qui est tenue par un potentiel quadratique. Les oscillateurs sont couplés par un potentiel coulombien et leurs déplacements suivent les tangentes locales respectives de la courbe spatiale décrite par les atomes C-alpha. Le système d'oscillateurs est couplé à une onde électromagnétique plane décrivant la source de lumière et le phénomène d'absorption est modélisé par des forces de friction. Nous montrons que le modèle reproduit correctement le phénomène CD d'une chaîne polypeptidique hélicoïdale et en particulier son signe en fonction de l'orientation de la chaîne. Comme première application, nous présentons l'ajustement du modèle au spectre CD d'un polypeptide composé de 15 résidus qui se plie sous forme d'une hélice alpha. La transférabilité de ces paramètres est ensuite évaluée pour la myoglobine, une protéine de 153 résidus contenant 8 hélices alpha
Circular dichroism (CD) spectroscopy is one of the fundamental techniques in structural biology that allows us to investigate the secondary structure of proteins. Synchrotron radiation has considerably increased the usefulness of the method because it allows to work with a wider range of spectrum and much greater signal-to-noise ratios. The development of a theoretical model to establish a relationship between the structure of a protein and its CD spectra in an efficient manner proved to be a complex task. The calculation of the CD spectra of large molecules, such as protein, remains a challenge, due to the size and flexibility of the molecules. In this context, we have developed a “minimal” model to explain the CD spectroscopy of proteins, which associates each C-alpha position on the protein backbone with a classical Lorentz oscillator i.e. a mobile charge attaches to a corresponding atom by a quadratic potential. The coupling between charges is through the Coulomb potential and their displacements follow the direction of the respective local tangents to the Calpha space curve. This system is coupled to a planar electromagnetic wave describing the light source and the absorption phenomenon is modeled by frictional forces. We show that the model correctly reproduces the CD phenomenon of a helical polypeptide chain and in particular its sign depending on the orientation of the chain. At first, we have fitted a model to CD spectra of a polypeptide chain of 15 residues folded into alpha helix. The transferability of these parameters is then evaluated with myoglobin, a protein of 153 residues containing eight alpha helices

Une mémoire quantique réversible permettant de stocker et relire de l'information quantique est une composante majeure dans la mise en œuvre de nombreux protocoles d'information quantique. Comme la lumière est un porteur de l'information quantique fiable sur des longues distances, et comme les atomes offrent la possibilité d'obtenir de longues durées de stockage, le recherche actuelle sur la création d'une mémoire quantique se concentre sur la transfert des fluctuations quantiques de la lumière sur des cohérences atomiques. Le travail réalisé durant cette thèse porte sur le développement d'une mémoire quantique pour la lumière comprimée, utilisant un ensemble d'atomes froids de Césium stock'es dans un piege magnéto-optique. Nos deux principaux objectifs étaient le développement d'une source de lumière non-classique, et le développement d'un milieu atomique pour le stockage de celle-ci. Tout d'abord, nous commençons par présenter la construction d'un oscillateur paramétrique optique qui utilise un cristal non-linéaire de PPKTP. Cet OPO fonctionne comme source d'états de vide comprime résonant avec la raie D2 du Césium. Nous caractérisons ces états grâce à une reconstruction par tomographie quantique, en utilisant une approche de vraisemblance maximale. Ensuite, nous examinons une nouvelle expérience qui nous permet d'utiliser comme milieu de stockage des atomes froids de Césium dans un piège magneto-optique récemment développé. Car cette expérience exige l'utilisation de nouveaux outils et techniques, nous discutons le développement de ceux-ci, et comment ils ont contribue à notre progression vers le stockage des états quantiques dans nos atomes des Césium, et finalement vers l'intrication de deux ensembles atomiques.

Hamilton-Jacobi theory provides a powerful method for extracting the equations of motion out of some given systems in classical mechanics. On occasion it allows some systems to be solved by the method of separation of variables. If a system with n degrees of freedom has 2n - 1 constants of the motion that are polynomial in the momenta, then that system is called superintegrable. Such a system can usually be solved in multiple coordinate systems if the constants of the motion are quadratic in the momenta. All superintegrable two dimensional Hamiltonians of the form H = (p_x)sup2 + (p_y)sup2 + V(x,y), with constants that are quadratic in the momenta were classified by Kalnins et al [5], and the coordinate systems in which they separate were found. We discuss Hamilton-Jacobi theory and its development from a classical viewpoint, as well as superintegrability. We then proceed to use the theory to find equations of motion for some of the superintegrable Hamiltonians from Kalnins et al [5]. We also discuss some of the properties of the Poisson algebra of those systems, and examine the orbits.

Thesis (S.B.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2013.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 37-38).
This thesis explores four nonlinear classical models of neural oscillators, the Hodgkin- Huxley model, the Fitzhugh-Nagumo model, the Morris-Lecar model, and the Hindmarsh-Rose model. Analysis techniques for nonlinear systems were used to develop a set of observers and perform synchronization analysis on the aforementioned neural systems. By using matrix analysis techniques, a study of biological background and motivation, and MATLAB simulation with mathematical computation, it was possible to do a preliminary contraction and nonlinear control systems structural study of these classical neural oscillator models. Neural oscillation and signaling models are based fundamentally on the biological function of the neuron, with behavior mediated through the channeling of ions across a cell membrane. The variable assumed to be measured for this study is the voltage or membrane potential, which could be measured empirically through the use of a neuronal force-clamp system. All other variables were estimated by using the partial state and full state observers developed here. Preliminary observer rate convergence analysis was done for the Fitzhugh-Nagumo system, and preliminary synchronization analysis was done for both the Fitzhugh-Nagumo and the Hodgkin- Huxley systems. It was found that by using a variety of techniques and mathematical matrix analyses methods (e.g. diagonal dominance or other norms), it was possible to develop a case-by-case nonlinear control systems approach to each particular system as a biomathematical entity.
by Ranjeetha Bharath.
S.B.

In the mid-1920s, the great Albert Einstein proposed that at extremely low temperatures, a gas of bosonic particles will enter a new phase where a large fraction of them occupy the same quantum state. This state would bring many of the peculiar features of quantum mechanics, previously reserved for small samples consisting only of a few atoms or molecules, up to a macroscopic scale. This is what we today call a Bose-Einstein condensate. It would take physicists almost 70 years to realize Einstein's idea, but in 1995 this was finally achieved. The research on Bose-Einstein condensates has since taken many directions, one of the most exciting being to study their behavior when they are placed in optical lattices generated by laser beams. This has already produced a number of fascinating results, but it has also proven to be an ideal test-ground for predictions from certain nonlinear lattice models. Because on the other hand, nonlinear science, the study of generic nonlinear phenomena, has in the last half century grown out to a research field in its own right, influencing almost all areas of science and physics. Nonlinear localization is one of these phenomena, where localized structures, such as solitons and discrete breathers, can appear even in translationally invariant systems. Another one is the (in)famous chaos, where deterministic systems can be so sensitive to perturbations that they in practice become completely unpredictable. Related to this is the study of different types of instabilities; what their behavior are and how they arise. In this thesis we compare classical and quantum mechanical nonlinear lattice models which can be applied to BECs in optical lattices, and also examine how classical nonlinear concepts, such as localization, chaos and instabilities, can be transfered to the quantum world.

After giving a brief background on the subject we introduce in section two the Phase-Integral Method of Fröman & Fröman in terms of the platform function of Yngve and Thidé. In section three we derive a different form of the radial Bohr-Sommerfeld condition in terms of the apsidal angle of the corresponding classical motion. Using the derived expression, we then show how easily one can calculate the exact energy eigenvalues of the hydrogen atom and the isotropic three-dimensional harmonic oscillator, we also derive an expression for higher order quantization condition. In section four we derive an expression for the angular frequencies of a restricted (0≤φ≤β) soap bubble and also give a proposition concerning the parameters l and m of the associated Legendre differential equation.
Vi använder Fröman & Frömans Fas-Integral Metod tillsammans med Yngve & Thidés plattformfunktion för att härleda kvantiseringsvilkoret för högre ordningar. I sektion tre skriver vi Bohr-Sommerfelds kvantiseringsvillkor på ett annorlunda sätt med hjälp av den så kallade apsidvinkeln (definierad i samma sektion) för motsvarande klassiska rörelse, vi visar också hur mycket detta underlättar beräkningar av energiegenvärden för väteatomen och den isotropa tredimensionella harmoniska oscillatorn. I sektion fyra tittar vi på en såpbubbla begränsad till området 0≤φ≤β för vilket vi härleder ett uttryck för dess (vinkel)egenfrekvenser. Här ger vi också en proposition angående parametrarna l och m tillhörande den associerade Legendreekvationen.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior
Nesse trabalho apresentamos as soluÃÃes clÃssicas e quÃnticas de duas classes de osciladores harmÃnicos dependentes de tempo, a saber: (a) o oscilador log-periÃdico e (b) o oscilador tipo Caldirola-Kanai. Para a classe (a) estudamos os seguintes osciladores: (I) $m(t)=m_0frac{t}{t_0}$, (II) $m(t)=m_0$ e (III) $m(t)=m_0ajust{frac{t}{t_0}}^2$. Nesses trÃs casos $omega(t)=omega_0frac{t_0}{t}$. Para a classe (b) estudamos o oscilador (IV) de Caldirola-Kanai onde $omega(t)=omega_0$ e $m(t)=m_0 ext{Exp}ajust{gamma t}$ e osciladores com $omega(t)=omega_0$ e $m(t)=m_0ajust{1+frac{t}{t_0}}^alpha$, para (V) $alpha=2$ e (VI) $alpha=4$. Para obter as soluÃÃes clÃssicas de cada oscilador resolvemos suas respectivas equaÃÃes de movimento e analisamos o comportamento de $q(t)$, $p(t)$ assim como do diagrama de fase $q(t)$ vs $p(t)$. Para obter as soluÃÃes quÃnticas usamos uma transformaÃÃo unitÃria e o mÃtodo dos invariantes quÃnticos de Lewis e Riesenfeld. A funÃÃo de onda obtida à escrita em termos de uma funÃÃo $ ho$, que à soluÃÃo da equaÃÃo de Milne-Pinney. Ainda, para cada sistema resolvemos a respectiva equaÃÃo de Milne-Pinney e discutimos como o produto da incerteza evolui no tempo.
In this work we present the classical and quantum solutions of two classes of time-dependent harmonic oscillators, namely: (a) the log-periodic and (b) the Caldirola-Kanai-type oscillators. For class (a) we study the following oscillators: (I) $m(t)=m_0frac{t}{t_0}$, (II) $m(t)=m_0$ and (III) $m(t)=m_0ajust{frac{t}{t_0}}^2$. In all three cases $omega(t)=omega_0frac{t_0}{t}$. For class (b) we study the Caldirola-Kanai oscillator (IV)where $omega(t)=omega_0$ and $m(t)=m_0 ext{exp}ajust{gamma t}$ and the oscillator with $omega(t)=omega_0$ and $m(t)=m_0ajust{1+frac{t}{t_0}}^alpha$, for $alpha=2$ (V) and $alpha=4$ (VI). To obtain the classical solution for each oscillator we solve the respective equation of motion and analyze the behavior of $q(t)$, $p(t)$ as well as the phase diagram $q(t)$ vs $p(t)$. To obtain the quantum solutions we use a unitary transformation and the Lewis and Riesenfeld quantum invariant method. The wave functions obtained are written in terms of a function ($ ho$) which is solution of the Milne-Pinney equation. Futhermore, for each system we solve the respective Milne-Pinney equation and discuss how the uncertainty product evolves with time.

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The simultaneously incidence of two light beams on one or more atoms causes destructive interference of these beams in atomic states causing cancellation of the absorption of one of the incident beams and this phenomenon is called Electromagnetically Induced Transparency (EIT). The main objective of this work is to show that the Electromagnetically Induced Transparency, which is usually studied in the quantum context, can be modeled classically as a function of coupled harmonic oscillators subject to an external force and dissipation. To will establish the classical equivalence, it will be presented the theory of the EIT in diferent quantum systems and also the theory of classic harmonic oscillators. Analogies will be performed comparing the classical and quantum equations of motion obtained for each scheme. For this, we perform the equivalence of EIT in quantum systems as atoms in three levels in Λ configuration, two-level atoms plus a cavity mode and the cavity optomechanics with the classical system of two coupled harmonics oscillator forced and damped. We also analyze the equivalence of two diferent quantum systems: three level atoms plus one cavity mode and four levels atoms in free space with a classical system composed by three coupled harmonic oscilators, forced and damped, in diferent configurations.
A incidência simultânea de dois feixes luminosos em um ou mais átomos provoca a interferência destrutiva desses feixes em um dos estados atômicos causando o cancelamento da absorção de um dos feixes incidentes e esse fenômeno é denominado Transparência Eletromagneticamente Induzida (\Electromagnetically Induced Transparency", EIT). O objetivo principal deste trabalho é mostrar que a Transparência Eletromagneticamente Induzida, que é normalmente estudada no contexto quântico, pode ser modelada classicamente em função de osciladores harmônicos amortecidos forçados e acoplados. Para que a equivalência clássica seja bem fundamentada, será apresentada a teoria da EIT em diversos sistemas quânticos e também a teoria dos osciladores harmônicos clássicos. As equivalências serão realizadas comparando as equações de movimento clássicas e quânticas obtidas para cada regime. Para isso, vamos realizar a equivalência da EIT em sistemas quânticos de átomos de três níveis em configuração Λ, dois níveis atômicos mais um modo da cavidade e a optomecânica de cavidades com o sistema clássico de dois osciladores harmônicos amortecidos forçados e acoplados. Logo após, será analisada a equivalência de dois sistemas quânticos compostos por átomos de três níveis mais um modo da cavidade e átomos de quatro níveis com os sistemas clássicos de três osciladores harmônicos amortecidos forçados e acoplados em diferentes configurações.

In this thesis, I demonstrate the dynamic way in which musical processes can be described as metaphors. Using Steve Larson’s three main metaphors (gravity, inertia, and magnetism) as a starting point, I propose additional metaphors (friction, repulsion, momentum, wave, orbit, and oscillation) to analyze the first movements of Mozart’s Piano Concerto No. 20 in D minor, K 466 and Schumann’s Piano Concerto in A minor, op. 54. These metaphors provide a means to discuss points of convergence and divergence between the Classical style and the early-Romantic style. Additionally, most theorists of the energeticist tradition only discuss motion through prose; I introduce a way to represent these metaphors as musical examples. By focusing on the listener’s experience through musical motion, the model proposed in this thesis is useful, not only for the theorist, but for all who wish to communicate ideas about music in a dynamic way.

by Tsui Wan-lam.
Chinese title in romanization: Zi you dian zi ji guang fang da qi ji zhen dang qi di jing dian li lun.
Thesis (Ph.D.)--Chinese University of Hong Kong, 1987.
Bibliography: leaves 93-95.

Thermostated time evolutions are on a firm ground and widely used in classical molecular dynamics (MD) simulations. Hamilton´s equations of motion are supplemented by time-dependent pseudofriction terms that convert the microcanonical isoenergetic time evolution into a canonical isothermal time evolution, thus permitting the calculation of canonical ensemble averages by time averaging. However, similar methods for quantum MD schemes are still lacking. Given the rich dynamical behavior of ultracold trapped quantum gases depending on the value of the s-wave scattering length, it is timely to investigate how classical thermostating methods can be combined with powerful approximate quantum dynamics schemes to deal with interacting quantum systems at finite temperature. In this work, the popular method of Nose and Hoover to create canonically distributed positions and momenta in classical MD simulations is generalized to a genuine quantum system of infinite dimensionality. We show that for the quantum harmonic oscillator, the equations of motion in terms of coherent states may be modified in a Nose-Hoover manner to mimic the coupling of the system to a thermal bath and create a quantum canonical ensemble. The method is developed initially for a single particle and then generalized to the case of an arbitrary number of identical quantum particles, involving entangled distribution functions. The resulting isothermal equations of motion for bosons and fermions contain additional terms leading to Bose-attraction and Pauli-blocking, respectively. Questions of ergodicity are discussed for different coupling schemes. In the many-particle case, the superiority of the Nose-Hoover technique to a Langevin approach is demonstrated. In addition, the work contains an investigation of the Grilli-Tosatti thermostating method applied to the harmonic oscillator, and calculations for quantum wavefunctions moving with a time-invariant shape in a harmonic potential.