Dissertations / Theses on the topic 'Quantum trajectories'

In quantum mechanics, the process of measurement is intrinsically probabilistic. As a result, continuously monitoring a quantum system will randomly perturb its natural unitary evolution. An accurate measurement record documents this stochastic evolution and can be used to reconstruct the quantum trajectory of the system state in a single experimental iteration. We use weak measurements to track the individual quantum trajectories of a superconducting qubit that evolves under the competing influences of continuous weak measurement and Rabi drive. We analyze large ensembles of such trajectories to examine their characteristics and determine their statistical properties. For example, by considering only the subset of trajectories that evolve between any chosen initial and final states, we can deduce the most probable path through quantum state space. Our investigation reveals the rich interplay between measurement dynamics, typically associated with wavefunction collapse, and unitary evolution. Our results provide insight into the dynamics of open quantum systems and may enable new methods of quantum state tomography, quantum state steering through measurement, and active quantum control.

Quantum trajectories provide a fundamental description of the measurement of individual quantum systems. As such, they have wide impact, and application, in the emerging field of quantum technology. Importantly, they also give a mechanism for developing our understanding of the nature of quantum mechanics. In Part I of this thesis, we develop and apply quantum trajectory methods, with a focus upon experimentally relevant optomechanical systems. By solving the stochastic master equation for sufficiently simple bosonic systems, and subsequently finding the positive operator-valued measure (POVM), we are able to conduct a detailed study of the use of parametric amplification for quantum state tomography of nonclassical optomechanical states of motion. Homodyne tomography is a cornerstone experimental tool, and an analysis of its convergence is carried out in the presence of realistic imperfections. We complete Part I by conducting a detailed preparatory analysis of superfluid optomechanical systems possessing vorticity. Part II of this thesis investigates the correspondence between open classical and open quantum systems. We prove a result shows that open quantum systems are, in general, harder to track than open classical systems. We couch this result in terms of physically realisable ensembles (PREs), which can describe the dynamics of a monitored, $D$-dimensional, quantum system obeying a master equation that has reached equilibrium. Associated with the PRE is a measurement scheme that leads to quantum trajectories in which the system evolution consists purely of jumps between the states that are members of the PRE. The occupation of the $K$, generally non-orthogonal, states in the ensemble can be used to track the system. The number of states in the ensemble, $K$, represents the amount of memory that is required to do so. In comparison, a classical $D$-dimensional system requires occupation of the $D$ states to be tracked. After first developing analysis tools that make feasible the discovery of PREs in $D>2$, we prove our main result that there are quantum systems that have a minimal sized PRE with $K>D$.

Thesis: S.M., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2019
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 85-88).
The transport of atomic ions trapped within a harmonic potential arises necessarily in the course of building a trapped ion quantum computer. We may define this problem in terms of a differential equation and its corresponding boundary conditions to satisfy which are sufficient to guarantee the motional quantum state of the ion is unaltered. However, the solution space to this problem is uncountably large, and the various solutions differ in many qualitative and quantitative aspects. We present an easily-computed functional of transport trajectories with intuitively interpretable terms which may be used to compare solutions to the quantum harmonic transport problem, but does not require an expensive quantum-mechanical simulation of the ion dynamics. Furthermore, we prove the convexity of this cost function under easily satisfied conditions in a Fourier Series parameterization of the problem. We then numerically optimize the cost function to discover optimal trajectories for the quantum harmonic transport problem.
by Samuel Adam Buercklin.
S.M.
S.M. Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science

All fields of physics - be it nuclear, atomic and molecular, solid state, or optical - offer examples of systems which are strongly influenced by the environment of the actual system under investigation. The scope of what is called "the environment" may vary, i.e., how far from the system of interest an interaction between the two does persist. Typically, however, it is much larger than the open system itself. Hence, a fully quantum mechanical treatment of the combined system without approximations and without limitations of the type of system is currently out of reach. With the single assumption of the environment to consist of an internally thermalized set of infinitely many harmonic oscillators, the seminal work of Stockburger and Grabert [Chem. Phys., 268:249-256, 2001] introduced an open system description that captures the environmental influence by means of a stochastic driving of the reduced system. The resulting stochastic Liouville-von Neumann equation describes the full non-Markovian dynamics without explicit memory but instead accounts for it implicitly through the correlations of the complex-valued noise forces. The present thesis provides a first application of the Stockburger-Grabert stochastic Liouville-von Neumann equation to the computation of the dynamics of anharmonic, continuous open systems. In particular, it is demonstrated that trajectory based propagators allow for the construction of a numerically stable propagation scheme. With this approach it becomes possible to achieve the tremendous increase of the noise sample count necessary to stochastically converge the results when investigating such systems with continuous variables. After a test against available analytic results for the dissipative harmonic oscillator, the approach is subsequently applied to the analysis of two different realistic, physical systems. As a first example, the dynamics of a dissipative molecular oscillator is investigated. Long time propagation - until thermalization is reached - is shown to be possible with the presented approach. The properties of the thermalized density are determined and they are ascertained to be independent of the system's initial state. Furthermore, the dependence on the bath's temperature and coupling strength is analyzed and it is demonstrated how a change of the bath parameters can be used to tune the system from the dissociative to the bound regime. A second investigation is conducted for a dissipative tunneling scenario in which a wave packet impinges on a barrier. The dependence of the transmission probability on the initial state's kinetic energy as well as the bath's temperature and coupling strength is computed. For both systems, a comparison with the high-temperature Markovian quantum Brownian limit is performed. The importance of a full non-Markovian treatment is demonstrated as deviations are shown to exist between the two descriptions both in the low temperature cases where they are expected and in some of the high temperature cases where their appearance might not be anticipated as easily.

This thesis is concerned with the application and extension of semiclassical methods, involving correlated trajectories, that were recently developed to explain the observed universal statistics of classically chaotic quantum systems. First we consider systems that depend on an external parameter that does not change the symmetry of the system. 'Ve study correlations between the spectra at different values of the param~ter, a scaled distance x apart, via the parametric spectral form factor K(r, x). Using a semiclassical periodic orbit expansion, we obtain a small r expansion that agrees with random matrix theory for systems with and without time reversal symmetry. Then we consider correlations of the Wigner time delay in open systems. We study a form factor K (r, x, y, M) that depends on the number of scattering channels M, the non-symmetry breaking parameter difference x and also a symmetry breaking parameter y. TheWigner time delay can be expressed semiclassically in terms of the trapped periodic orbits of the system, and using a periodic orbit expansion we obtain several terms in the small r expansion of the form factor that are identical to those calculated from random matrix theory. The Wigner time delay can also be expressed in terms of scattering trajectories that enter and leave the system. Starting from this picture, we derive all terms in the periodic orbit formula and therefore show how the two pictures of the time delay are related on a semiclassical level. A new type of trajectory correlation is derived which recreates the terms from the trapped periodic orbits. This involves two trajectories approaching the same trapped periodic orbit closely - one trajectory approaches the orbit and follows it for several traversals, while its partner approaches in almost the same way but follows the periodic orbit an additional number of times.

Es conocido que a escalas nanométricas se debe tratar con en el problema de muchas partículas a la hora de estudiar dispositivos electrónicos. Es estos escenarios, la ecuación de Schrödinger dependiente del tiempo para muchas partículas solo se puede resolver para unos pocos grados de libertad. En este sentido, diferentes formalismos han sido desarrollados en la literatura (tales como time-dependent Density Functional Theory, Green's functions técnicas o Quantum Monte Carlo técnicas) para tratar sistemas cuánticos de muchos electrones. Estas aproximaciones modelizan de forma razonable el transporte electrónico en sistemas de muchas partículas. Una propuesta alternativa ha sido desarrollada por el Dr. Oriols para descomponer la ecuación de Schrödinger de N-partículas en un sistema de N-ecuaciones de Schrödinger para una sola partícula usando trayectorias (cuánticas) de Bohm. Basado en esta propuesta se presenta un 3D, general, versátil y dependiente del tiempo simulador de transporte de dispositivos electrónicos llamado BITLLES (Bohmian Interacting Transport for non-equiLibrium eLEctronic Structures). Las novedades que aporta el simulador BITLLES se basan en dos puntos. El primero, éste representa un modelo de transporte cuántico de electrones para muchas partículas en el cual se tiene en cuenta de forma explicita las correlaciones de Coulomb y de intercambio entre electrones usando trayectorias de Bohm. En segundo lugar, el simulador proporciona una completa información de los momentos de la corriente (i.e., DC, AC, fluctuaciones o incluso momentos mayores). A continuación resumimos las contribuciones que esta tesis aporta al desarrollo del simulador BITLLES. De esta forma, introducimos de forma explicita la interacción de intercambio entre electrones. En este contexto, mostramos como la interacción de intercambio es la responsable final para determinar la corriente total a través del sistema. Además presentamos una nueva aproximación para estudiar sistemas de muchas partículas donde los espines de los electrones tienen diferente orientación. Hasta donde llega nuestro conocimiento, es la primera vez que la interacción de intercambio es introducida de forma práctica en un simulador de transporte de electrones. Además presentamos la computación de la corriente total dependiente del tiempo en un contexto de alta frecuencia donde se tienen que tener en cuenta las variaciones del campo eléctrico dependientes del tiempo (i.e., la corriente de desplazamiento) para asegurar la conservación de la corriente. También discutimos el cálculo de la corriente total (conducción más desplazamiento) usando los teoremas de Ramo-Shockley-Pellegrini. Diferentes capacidades del simulador BITLLES como AC y fluctuaciones de la corriente se presentan para el diodo túnel resonante. También hemos usado el simulador BITLLES para testear un nuevo tipo de dispositivo nanoeléctronico diseñado para procesar señales dentro del espectro de los THz. Hemos llamado a este dispositivo Driven Tunneling Device. Se trata de un dispositivo de tres terminales donde la conductancia entre el drain y el source se controla por el terminal del gate el cual oscila a frecuencias de THz. También presentamos ejemplos prácticos de la funcionalidad de este dispositivo como un rectificador y un multiplicador de frecuencia. Finalmente, hemos desarrollado una aproximación numérica para resolver la ecuación de Schrödinger usando el modelo de tight-binding con el propósito de mejorar la descripción de la estructura de bandas del simulador BITLLES.
It is known that at nanoscale regime we must deal with the many-particle problem in order to study electronic devices. In this scenario, the time-dependent many-particle Schrödinger equation is only directly solvable for very few degrees of freedom. However, there are many electrons (degrees of freedom) in any electron device. In this sense, many-particle quantum electron formalisms (such as time-dependent Density Functional Theory, Green's functions techniques or Quantum Monte Carlo techniques) have been developed in the literature to provide reasonable approximations to model many-particle electron transport. An alternative proposal has been developed by Dr. Oriols to decompose the N-particle Schrödinger equation into a N-single particle Schrödinger equation using Bohmian trajectories. Based on this proposal a general, versatile and time-dependent 3D electron transport simulator for nanoelectronic devices, named BITLLES (Bohmian Interacting Transport for non-equiLibrium eLEctronic Structures) is presented. The novelty of the BITLLES simulator is based on two points. First, it presents a many-particle quantum electron transport model taking into account explicitly the Coulomb and exchange correlations among electrons using Bohmian trajectories. Second, it provides full information of the all current distribution moments (i.e. DC, AC, fluctuations and even higher moments). We summarize the important contributions of this thesis to the development of BITLLES simulator. Thus, we introduce explicitly the exchange correlations among electrons. In this context, we show how exchange interaction is the final responsible for determining the total current across the system. We also present a new approximation to study many-particle systems with spin of different orientations. Some practical examples are studied taking into account the exchange interaction. To the best of our knowledge, it is the first time that the exchange interaction is introduced explicitly (imposing the exchange symmetry properties directly into the many-particle wavefunction) in practical electron transport simulators. We present the computation of the time-dependent total current in the high-frequency regime where one has to compute time-dependent variations of the electric field (i.e. the displacement current) to assure current conservation. We discuss the computation of the total (conduction plus displacement) current using Bohmian trajectories and the Ramo-Shockley-Pellegrini theorems. Different capabilities of BITLLES simulator such as AC and current fluctuations are presented for Resonant Tunneling Devices. We have used the BITLLES simulator to test a new type of nanoelectronic device designed to process signals at THz regime named Driven Tunneling Device. It is a three terminal device where the drain-source conductance is controlled by a gate terminal that can oscillate at THz frequencies. We also present practical examples on the functionality of this device such as rectifier and frequency multiplier. Finally, we have developed a numerical approximation to solve the Schrödinger equation using tight-binding model to improve the band structure description of the BITLLES simulator.

De nombreux phénomènes de physique quantique ne peuvent être compris que par l'analyse des systèmes ouverts. Un appareil de mesure, par exemple, est un système macroscopique en contact avec un système quantique. Ainsi, tout modèle d'expérience doit prendre en compte les dynamiques propres aux systèmes ouverts. Ces dynamiques peuvent être complexes : l'interaction du système avec son environnement peut modifier ses propriétés, l'interaction peu créer des effets de mémoire dans l'évolution du système, . . . Ces dynamiques sont particulièrement importantes dans l'étude des expériences d'optique quantique. Nous sommes aujourd'hui capables de manipuler individuellement des particules. Pour cela la compréhension et le contrôle de l'influence de l'environnement est crucial. Dans cette thèse nous étudions d'un point de vue théorique quelques procédures communément utilisées en optique quantique. Avant la présentation de nos résultats, nous introduisons et motivons l'utilisation de la description markovienne des systèmes quantiques ouverts. Nous présentons a la fois les équations maîtresses et le calcul stochastique quantique. Nous introduisons ensuite la notion de trajectoire quantique pour la description des mesures indirectes continues. C'est dans ce contexte que l'on présente les résultats obtenus au cours de cette thèse. Dans un premier temps, nous étudions la convergence des mesures non destructives. Nous montrons qu'elles reproduisent la réduction du paquet d'onde du système mesuré. Nous montrons que cette convergence est exponentielle avec un taux fixe. Nous bornons le temps moyen de convergence. Dans ce cadre, en utilisant les techniques de changement de mesure par martingale, nous obtenons la limite continue des trajectoires quantiques discrètes. Dans un second temps, nous étudions l'influence de l'enregistrement des résultats de mesure sur la préparation d'état par ingénierie de réservoir. Nous montrons que l'enregistrement des résultats de mesure n'a pas d'influence sur la convergence proprement dite. Cependant, nous trouvons que l'enregistrement des résultats de mesure modifie le comportement du système avant la convergence. Nous retrouvons une convergence exponentielle avec un taux équivalent au taux sans enregistrement. Mais nous trouvons aussi un nouveau taux de convergence correspondant a une stabilité asymptotique. Ce dernier taux est interprété comme une mesure non destructive ajoutée. Ainsi l'état du système ne converge qu'après un temps aléatoire. A partir de ce temps la convergence peut être bien plus rapide. Nous obtenons aussi une borne sur le temps moyen de convergence
Many quantum physics phenomena can only be understood in the context of open system analysis. For example a measurement apparatus is a macroscopic system in contact with a quantum system. Therefore any experiment model needs to take into account open system behaviors. These behaviors can be complex: the interaction of the system with its environment might modify its properties, the interaction may induce memory effects in the system evolution, ... These dynamics are particularly important when studying quantum optic experiments. We are now able to manipulate individual particles. Understanding and controlling the environment influence is therefore crucial. In this thesis we investigate at a theoretical level some commonly used quantum optic procedures. Before the presentation of our results, we introduce and motivate the Markovian approach to open quantum systems. We present both the usual master equation and quantum stochastic calculus. We then introduce the notion of quantum trajectory for the description of continuous indirect measurements. It is in this context that we present the results obtained during this thesis. First, we study the convergence of non demolition measurements. We show that they reproduce the system wave function collapse. We show that this convergence is exponential with a fixed rate. We bound the mean convergence time. In this context, we obtain the continuous time limit of discrete quantum trajectories using martingale change of measure techniques. Second, we investigate the influence of measurement outcome recording on state preparation using reservoir engineering techniques. We show that measurement outcome recording does not influence the convergence itself. Nevertheless, we find that measurement outcome recording modifies the system behavior before the convergence. We recover an exponential convergence with a rate equivalent to the rate without measurement outcome recording. But we also find a new convergence rate corresponding to an asymptotic stability. This last rate is interpreted as an added non demolition measurement. Hence, the system state converges only after a random time. At this time the convergence can be much faster. We also find a bound on the mean convergence time

Cette thèse décrit une série d’expériences mettant en lumière l’action en retour de la mesure et la décohérence pour un système quantique ouvert élémentaire, le qubit supraconducteur. Ces observations sont rendues possibles grâce au développement récent d’amplificateurs Josephson proches de la limite quantique. L’information extraite du système peut être utilisée dans des boucles de rétroaction quantique. Pour stabiliser un état arbitraire prédéterminé du qubit, une mesure projective est réalisée périodiquement et une boucle de rétroaction permet de corriger les erreurs détectées. En se substituant à l'environnement et en réalisant une mesure hétérodyne continue de la fluorescence du qubit, nous reconstituons des trajectoires quantiques individuelles lors de sa relaxation. En conditionnant cette détection au résultat d'une mesure projective postérieure, nous déterminons les weak values du signal de fluorescence. En formant une boucle de rétroaction continue à partir de ce signal, nous stabilisons également un état arbitraire du qubit. Enfin, nous observons dans une dernière expérience la dynamique quantique Zénon d'un mode micro-onde, induite par son couplage au qubit
This thesis presents a series of experiments highlighting measurement back action and decoherence in a basic open quantum system, the superconducting qubit. These observations are enabled by recent advances in amplification close to the quantum limit using Josephson circuits. The information extracted from the system can then be used as input in quantum feedback. A stroboscopic projective readout is performed and a feedback loop is used to correct for detected errors, thus stabilizing an arbitrary predetermined state of the qubit. When monitoring continuously the environment of the qubit by heterodyne detection of its fluorescence, we reconstruct individual quantum trajectories during relaxation. Conditioning this detection to the outcome of a following projective measurement, we access the weak values of the fluorescence signal. Included in a continuous feedback loop, this detection is also used to stabilize an arbitrary state of the qubit. Finally, a last experiment witnesses quantum Zeno dynamics of a resonant microwave mode, entailed by its coupling to the qubit

Methods for the study of nuclear quantum dynamics can be categorised by the nature of the basis set expansion they employ. The wavefunction can be expanded in a static set of time-independent basis functions, the time evolution being described solely via the expansion coefficients. Alternatively, basis functions can be propagated in time, along with the coefficients, via equations of motion for their parameters. Time-independent basis sets are plagued by exponential scaling, while the equations of motion for time-dependent basis functions are challenging to integrate and, if not derived variationally, can violate energy conservation laws. This work presents a novel basis set sampling method which represents a compromise between these two categories. A set of sampling trajectories, evolving on the potential energy surface of the system, are used to place basis functions in regions of phase space, relevant to wavefunction propagation. These functions then act as a time-independent basis set, the wavefunction being evolved via exact, variational equations of motion for the expansion coefficients. This approach is applied to a challenging quantum dynamics benchmark, namely the relaxation dynamics of photoexcited pyrazine, and yields highly encouraging results. In order to address divergence from exact dynamics at longer timescales, which is attributed to the classical sampling trajectories being a sound approximation to quantum propagation of the wavefunction only in the short-time limit, a modification of this method is proposed. Shorter iterations of trajectory sampling and wavefunction propagation are used, linked by a minimisation algorithm that continuously optimises the basis set, preventing unfavourable scaling. This adaptive sampling approach is again applied to the pyrazine benchmark with a significant increase in performance and accuracy. Highly encouraging results are also obtained for a quantum tunnelling benchmark system, which are improved upon even further, and at little extra cost, by the use of path integral sampling trajectories.

The study of the dynamics of open quantum systems sheds light on dissipative processes in quantum mechanics. Any system under continuous measurement is open and the act of measuring induces abrupt changes of the system’s state (collapses). The evolution conditioned to measurement records generates the so-called quantum trajectories. A continuous (unconditioned) evolution of the system is recovered by averaging over a large number of trajectories. Historically this kind of evolution has been the main focus of theoretical investigations. In this dissertation we consider both conditional and unconditional dynamics of quantum optical systems. Unconditioned dynamics is studied through the collision model paradigm. The formalism is described in detail and used for describing generic systems featuring many quantum emitters coupled to a usually one-dimensional field. The negligible-delay regime is widely explored. Collision models are used to unveil the mechanisms underlying the decoherence-free evolution regime typical of these systems, which has received considerable attention in the last years. Then we investigate conditioned dynamics by broadening the study of statistics of quantum trajectories. Specifically, we exploit the information about the emission’s full-counting statistics from large deviations to define a nonclassicality witness. Finally we come back to collision models in order to extend the theory of biased quantum trajectories from Lindblad-like dynamics to sequences of arbitrary dynamical maps, providing at once a transparent physical interpretation.

An experiment currently underway at BYU is designed to test whether the size of a free electron wave packet affects the character of scattered radiation. Using a semi-classical argument wherein the wave packet is treated as a diffuse charge distribution, one would expect strong suppression of radiation in the direction perpendicular to the propagating field as the wave packet grows in size to be comparable to the wavelength of the driving field. If one disallows the interaction of the wave packet with itself, as is the case when calculating the rate of emission using QED, then regardless of size, the electron wave packet radiates with the strength of a point-like emitter. In support of this experiment, we explore a variety of physical parameters that impact the rate of scattered photons. We employ a classical model to characterize the exposure of electrons to high-intensity laser light in a situation where the electrons are driven by strong ponderomotive gradients. Free electrons are modeled as being donated by low-density helium, which undergoes strong-field ionization early on in the pulse or during a pre-pulse. When exposed to relativistic intensities (i.e. intensities sufficient to cause a Lorentz drift at a significant fraction of c), free electrons experience a Lorentz drift that causes redshifting of the scattered 800 nm laser light. This redshift can be used as a key signature to discern light scattered from the more intense regions of the focus. We characterize the focal volume of initial positions leading to significant redshifting, given a peak intensity of 2 x 10^18 W/cm 2 , which is sufficient to cause a redshift in scattered light of approximately 100 nm. Under this scenario, the beam waist needs to be larger than several wavelengths for a pulse duration of 35 fs in order to ensure free electrons remain in the focus sufficiently long to experience intensities near the peak pulse intensity despite strong ponderomotive gradients. We compute the rate of redshifted scattered photons from an ensemble of electrons distributed throughout the focus and relate the result to the scattered-photon rate of a single electron. We also estimate to what extent the ionization process may produce unwanted light in the redshifted spectral region that may confound the measurement of light scattered from electrons experiencing intensities greater than 1.5 x 10^18 W/cm^2.

Orientador: Antonio Vidiella Barranco
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Fisica "Gleb Wataghin"
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Resumo: A presente tese é dedicada à utilização de conhecidos sistemas quânticos em aplicações de interesse em óptica e informação quântica. Motivados pelos recentes avanços experimentais em sistemas formados por íons aprisionados interagindo com lasers e na eletrodinâmica quântica de cavidades, nós focamos grande parte de nossas propostas nestes sistemas. Mais especificamente, nós estudamos a interação de íons e campos quantizados na chamada eletrodinâmica quântica de cavidades com íons aprisionados. Neste contexto, iniciamos nossos trabalhos com uma proposta de geração de superposições mesoscópicas no movimento do íon. Uma vez que tais superposições são muito sensíveis à decoerência, incluímos perdas na cavidade para tratar uma situação mais realista. Através da observação de quantum jumps, ou fóton-contagens fora da cavidade, mostramos um esquema de geração de estados com características quânticas muito similares aos encontrados no caso da cavidade ideal, sem perdas. Neste aspecto, encontramos um modo de usar a dissipação a nosso favor, fato de grande interesse experimental devido às imperfeições dos espelhos reais. Apresentamos também uma proposta de implementação de uma interação do tipo Kerr em íons como uma alternativa ao uso de cristais não-lineares que apresentam baixíssima eficiência para esse tipo de efeito. Essa proposta abre novas possibilidades para o uso de íons em medidas não demolidoras e computação quântica. Nossos estudos na área de eletrodinâmica quântica com íons aprisionados terminam com a análise dos efeitos do movimento do íon na dinâmica das transições multi-fotônicas. Esse é um estudo mais fundamental e está relacionado com o entendimento da interação da radiação com a matéria. Na última parte desta tese são apresentados resultados sobre o uso de sistemas de muitos corpos para a distribuição de informação quântica. O objetivo de se estudar estes sistemas mais complexos é a busca de implementação de protocolos quânticos em larga escala. Neste sentido, poderíamos pensar numa cadeia de osciladores harmônicos acoplados como ocorre em sistemas típicos da física da matéria condensada. Em particular, nós estudamos como aumentar a eficiência na transmissão de emaranhamento nestas cadeias. Propusemos um esquema que funciona como um tipo de quantum data bus, ou ônibus quântico para transportar e distribuir emaranhamento com alta eficiência
Abstract: This thesis is concerned with the use of firmly established quantum systems for applications in quantum optics and quantum information. Having been driven by recent experimental advances in laser-manipulated trapped ions and cavity quantum electrodynamics, we concentrated more on proposals to be implemented in those systems. Being more specific, we have studied the interaction between trapped ions and quantized fields in the so-called cavity quantum electrodynamics with trapped ions. In this context, we began with a proposal to generate mesoscopic superpositions in the motion of the ion. Since these superpositions are extremely sensitive to decoherence, we have included cavity losses in order to make the situation slightly more realistic. We showed that the observation of quantum jumps, or photon detection outside the cavity, would generate quantum states with properties close to that generated in the ideal lossless case. In spite of the normally destructive effect of dissipation, we found a way to use it in our favor which turns out to be of great experimental importance due to always present mirror imperfections. We also showed how to mimic cross-Kerr nonlinearities in the cavity-ion system as a feasible alternative to the use of nonlinear crystals whose intensity of that non-linearity is too weak. This proposal opens up new possibilities for the use of trapped ions in non-demolition measurements and quantum computing. We finish our work in cavity electrodynamics with trapped ions with the study of the effect of the ionic motion on the dynamics of multiphotonic transitions. This is a more fundamental issue that is related to the understanding of matter-field interaction. In the last part of this thesis, we present results on the use of many-body systems for quantum information distribution. It was our goal to study more complex systems for the implementation of quantum protocols in large scale. In this sense, one could think of a chain of coupled harmonic oscillators as commonly found in condensed matter physics. Particularly, we dealt with the efficiency of entanglement transmission through the chain, trying to improve it. We ended up with a scheme which acts as a quantum data bus able to transport and distribute entanglement around quite efficiently
Doutorado
Física
Doutor em Ciências

The de Broglie-Bohm causal interpretation of quantum mechanics is discussed, and applied to the hydrogen atom in several contexts. Prominent critiques of the causal program are noted and responses are given; it is argued that the de Broglie-Bohm theory is of notable interest to physics. Using the causal theory, electron trajectories are found for the conventional Schrödinger, Pauli and Dirac hydrogen eigenstates. In the Schrödinger case, an additional term is used to account for the spin; this term was not present in the original formulation of the theory but is necessary for the theory to be embedded in a relativistic formulation. In the Schrödinger, Pauli and Dirac cases, the eigenstate trajectories are shown to be circular, with electron motion revolving around the z-axis. Electron trajectories are also found for the 1s-2p0 transition problem under the Schrödinger equation; it is shown that the transition can be characterized by a comparison of the trajectory to the relevant eigenstate trajectories. The structures of the computed trajectories are relevant to the question of the possible evolution of a quantum distribution towards the standard quantum distribution (quantum equilibrium); this process is known as quantum relaxation. The transition problem is generalized to include all possible transitions in hydrogen stimulated by semi-classical radiation, and all of the trajectories found are examined in light of their implications for the evolution of the distribution to the standard distribution. Several promising avenues for future research are discussed.

The emergence of hydrodynamic features in off-equilibrium (1 + 1)-dimensional integrable quantum systems has been the object of increasing attention in recent years. In this Master Thesis, we combine Thermodynamic Bethe Ansatz (TBA) techniques for finite-temperature quantum field theories with the Generalized Hydrodynamics (GHD) picture to provide a theoretical and numerical analysis of Zamolodchikov’s staircase model both at thermal equilibrium and in inhomogeneous generalized Gibbs ensembles. The staircase model is a diagonal (1 + 1)-dimensional integrable scattering theory with the remarkable property of roaming between infinitely many critical points when moving along a renormalization group trajectory. Namely, the finite-temperature dimensionless ground-state energy of the system approaches the central charges of all the minimal unitary conformal field theories (CFTs) M_p as the temperature varies. Within the GHD framework we develop a detailed study of the staircase model’s hydrodynamics and compare its quite surprising features to those displayed by a class of non-diagonal massless models flowing between adjacent points in the M_p series. Finally, employing both TBA and GHD techniques, we generalize to higher-spin local and quasi-local conserved charges the results obtained by B. Doyon and D. Bernard [1] for the steady-state energy current in off-equilibrium conformal field theories.

Cette thèse est consacrée à l’étude des trajectoires quantiques issues de la théorie desmesures continues en mécanique quantique non relativiste. On y présente de nouveaux résultatsthéoriques ainsi que des exemples d’applications. Sur le front théorique, on étudie principalementla limite de mesure «forte» dans laquelle on met en évidence l’émergence de sauts quantiques etd’échardes quantiques, deux phénomènes dont on précise la statistique. Hors de la limite forte, onpropose une méthode d’extraction optimale d’information pour un registre de qubits. Sur le frontdes applications, on introduit une méthode originale de contrôle utilisant l’intensité de la mesurecomme unique variable et on explique la transition balistique-diffusif dans les marches aléatoiresquantiques ouvertes; deux sous produits de l’étude théorique préalable des situations de mesureforte. On s’intéresse aussi au problème de la gravité semi-classique et montre que la théorie desmesures continues peut permettre d’en construire un modèle cohérent à la limite newtonienne. Onsuggère enfin quelques extensions possibles de la théorie à l’estimation a posteriori et d’éventuellesgénéralisations des résultats théoriques à des situations de mesures répétées discrètes. Dans laprésentation des résultats, l’accent est mis davantage sur l’explicitation des liens entre les multiplespoints de vue possibles sur les trajectoires quantiques (parallèles avec la théorie classique du filtrageet les modèles de collapse objectif utilisés dans les fondements) que sur la rigueur mathématique
This thesis is devoted to the study of the quantum trajectories obtained from thetheory of continuous measurement in non relativistic quantum mechanics. New theoretical resultsas well as examples of applications are presented. On the theoretical front, we study mostly thelimit of «strong» measurement where we put forward the emergence of quantum jumps and quantumspikes, two phenomena we characterize in detail. Out of the strong measurement limit, weinvestigate a method to extract information from a register of qubits optimally. On the applicationfront, we introduce an original method to control quantum systems exploiting only the freedomof changing the measurement intensity and we explain the transition between a ballistic and adiffusive behavior in open quantum random walks; two byproduct of the theoretical study of thestrong measurement regime. We further study the problem of semi-classical gravity and show thatcontinuous measurement theory allows to construct a consistent model in the Newtonian regime.We eventually suggest possible extensions of the formalism to a posteriori estimation and hint atgeneralizations of the results for the strong measurement limit in the wider context of discreterepeated measurements. In the course of our presentation, we emphasize the link with other approachesto the theory of continuous measurement (parallels with stochastic filtering and collapsemodels in foundations) rather than aim for mathematical rigor

La reconstruction d'états quantiques, ou tomographie, joue un rôle central dans les technologies quantiques, afin de caractériser les opérations effectuées et d'extraire de l'information sur les états résultats de traitements d'information quantique. Les méthodes répandues de tomographie reposent généralement sur des mesures idéales, effectuées une seule fois sur chaque préparation de l'état d'intérêt. Dans ce travail, nous utilisons une nouvelle méthode, appelée tomographie par trajectoires, qui consiste à enregistrer, pour chaque réalisation de l'état, la trajectoire quantique suivie par le système à l'aide d'une série de mesures successives du système, en présence d'imperfections expérimentales et de décohérence. On extrait alors plus d'information sur l'état à reconstruire et on est capable, à partir d'un ensemble de mesures accessibles données, de créer des mesures plus générales. À l'aide des techniques de l'électrodynamique quantique en cavité, nous avons préparé des états intriqués de photons micro-onde délocalisés sur deux modes distants. Nous avons ensuite reconstruit ces états par tomographie par trajectoires, dans un espace de Hilbert de grande dimension. Nous montrons que cette méthode permet de reconstruire l'état, de développer des stratégies de mesure adaptées pour accélérer l'extraction d'information sur les cohérences quantiques d'intérêt et qu'elle fournit une estimation de l'incertitude sur les coefficients de la matrice densité reconstruite
Quantum state estimation, or tomography, is a key component of quantum technologies, allowing to characterise quantum operations and to extract information on the results of quantum information processes. The usual tomography techniques rely on ideal, single-shot measurements of the unknown state. In this work, we use a new approach, called trajectory quantum tomography, where the quantum trajectory of each realization of the state is recorded through a series of measurements, including experimental imperfections and decoherence. This strategy increases the extracted amount of information and allows to build new measurements for a set of feasible measurements.Using the tools of cavity quantum eletrodynamics, we have prepared entangled states of microwave photons spread on two separated modes. We have then performed a trajectory tomography of these states, in a large Hilbert space. We have proved that this method allows to estimate the state, to develop faster strategies for extracting information on specific coherences of the state and to compute error bars on the components of the estimated density matrix

On s'intéresse à la dynamique électronique de systèmes atomiques soumis à une impulsion laser brève et intense ou à l'impact d'un ion positivement chargé. On procède alors à une comparaison détaillée des descriptions classique et quantique de ces interactions. Sur la base de cette comparaison, on développe une méthode auto-cohérente de trajectoires quantiques, basée sur l'approche hydrodynamique de Bohm. Cette méthode permet d'obtenir des observables très précises tout en conservant le caractère illustratif des méthodes de trajectoires classiques
We are interested in the electronic dynamic of atomic system under influence of a short and intense laser pulse or induced by impact of positively charged ion. We then proceeds in a deeper comparative study of classical and quantal description of these interactions. On the basis of this study, we developped a self-consistent quantum trajectory method, based on the hydrodynamical formulation of Bohm. This method allow to obtain very precise observable while retaining the illustrative character of classical trajectory method

This PhD thesis contains results about two different main topics. The first part deals with the application of continuously monitored quantum systems to high precision quantum metrology. A continuous in time measurement on a quantum system is a kind indirect measurement, which only weakly perturbs the system and leaves room for it to evolve under its dynamics. This time-continuous measurement allows one to collect information about some interesting parameter characterizing the dynamics. In this thesis we show how to apply the theory of quantum parameter estimation to continuously monitored quantum systems. In particular, we study the estimation of a magnetic field applied to an ensemble of two level atoms; we show that by continuously monitoring the system we can obtain a quadratic scaling of the precision with the number of atoms, in two different physical settings (dynamically generated entanglement or initial entanglement). In the second part we study different aspects of nonclassicality of continuous variable quantum systems (bosonic degree of freedoms). They can be described by distributions (in particular, the Wigner function) on a classical phase space, which however can take negative values, the hallmark of nonclassicality. In this context, states with a Gaussian distribution are very useful and very well studied; however, on a fundamental level they must be considered classical. We present several studies connected to the vast topic of non-Gaussian states, starting from an application to parameter estimation, as a link to the first part. We study the relationships between anharmonic Hamiltonians and the nonclassicality of their ground states; we also explore the connection between a quantum effect called `backflow of probability' and the negativity of the Wigner function. We end by showing that quantum operations made out of Gaussian building blocks give rise to a well-defined resource theory of Wigner negativity and non-Gaussianity.

The technique of quantum trajectories (stochastic Schrödinger equations or Monte Carlo wave functions) for open systems is generalized to the non-Markovian regime. I consider a microscopic model of an open system consisting of a boson field coupled linearly (with an excitation preserving coupling) to a localized system. The model allows for a field with an arbitrary dispersion relation and an arbitrary mode-dependent coupling to the system. The trajectories are formulated as continuous measurements of the output field from the system. For a general dispersive field these measurements must be distributed in space for this formulation to be possible. The result of this formulation is a non-Markovian equation for the system conditioned on the measurements. A method of numerically simulating this equation has been determined and implemented in some test cases. Numerical simulation is possible if one can introduce a finite memory time for the evolution of the reduced system. As an illustration, the method is applied to the spectral detection of the emission from a driven two-level atom and also to an atom radiating into an electromagnetic field where the free space modes of the electromagnetic field are altered by the presence of a cavity. In both cases the non-Markovian behaviour arises from the uncertainty in the time of emission of a photon that is later detected (or reabsorbed), although, in the second case, the non-Markovian behaviour is intrinsic to the system environment coupling whereas, in the spectral detection case, it is a consequence of the choice of measurement process. The generalization of the techniques of quantum trajectories to the non-Markovian regime promises to make a range of open system problems where the Born-Markov approximation is invalid tractable to numerical simulation.
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Commonly held opinion is that particle trajectory descriptions are incompatible with quantum mechanics. Louis de Broglie (1926) first proposed a way to include trajectories in quantum mechanics, but the idea was abandoned until David Bohm (1952) re-invented and improved the theory. Bohm interprets the particle trajectories as physically real; for example, an electron actually is a particle moving on a well defined trajectory with a position and momentum at all times. By design, Bohm's trajectories never make predictions that differ from standard quantum mechanics, and their existence cannot be experimentally verified. Three new methods to obtain Bohm's particle trajectories are presented. The methods are non-dynamical, and utilize none of Bohm's equations of motion; in fact, two of the methods have no equations for a particle's trajectory. Instead, all three methods use only the evolving probability density ρ=ψ*ψ to extract the trajectories. The first two methods rest upon probability conservation and density sampling, while the third method employs the informational or geometrical construction of centroidal Voronoi tessellations. In one-dimension all three methods are proved to be equivalent to Bohm's particle trajectories. For higher dimensional configuration spaces, the first two methods can be used in limited situations, but the last method can be applied in all cases. Typically, the resulting higher dimensional non-dynamical trajectories are also identical to Bohm. Together the three methods point to a new interpretation of Bohm's particle trajectories, namely, the Bohm trajectories are simply a kinematic portrayal of the evolution of the probability density. In addition, the new methods can be used to measure Schrödinger's wave function and Planck's constant.
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Quantum trajectories are investigated within the complex quantum Hamilton-Jacobi formalism. A unified description is presented for complex quantum trajectories for one-dimensional time-dependent and time-independent problems. Complex quantum trajectories are examined for the free Gaussian wave packet, the coherent state in the harmonic potential, and the the barrier scattering problems. We analyze the variations of the complex-valued kinetic energy, the classical potential, and the quantum potential along the complex quantum trajectories. For one-dimensional time-independent scattering problems, we demonstrate general properties and similar structures of the complex quantum trajectories and the quantum potentials. In addition, it is shown that a quantum vortex forms around a node in the wave function in complex space, and the quantized circulation integral originates from the discontinuity in the real part of the complex action. Although the quantum momentum field displays hyperbolic flow around a node, the corresponding Polya vector field displays circular flow. Moreover, local topologies of the quantum momentum function and the Polya vector field are thoroughly analyzed near a stagnation point or a pole (including circular, hyperbolic, and attractive or repulsive structures). The local structure of the quantum momentum function and the Polya vector field around a stagnation point are related to the first derivative of the quantum momentum function. However, the magnitude of the asymptotic structures for these two fields near a pole depends only on the order of the node in the wave function. Finally, quantum interference is investigated and it leads to the formation of the topological structure of quantum caves in space-time Argand plots. These caves consist of the vortical and stagnation tubes originating from the isosurfaces of the amplitude of the wave function and its first derivative. Complex quantum trajectories display helical wrapping around the stagnation tubes and hyperbolic deflection near the vortical tubes. Moreover, the wrapping time for a specific trajectory is determined by the divergence and vorticity of the quantum momentum field. The lifetime for interference features is determined by the rotational dynamics of the nodal line in the complex plane. Therefore, these results demonstrate that the complex quantum trajectory method provides a novel perspective for analysis and interpretation of quantum phenomena.
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We have directed much attention towards developing quantum trajectory methods which can accurately predict the transmission probabilities for a variety of quantum mechanical barrier scattering processes. One promising method involves solving the complex quantum Hamilton-Jacobi equation with the Derivative Propagation Method (DPM). We present this method, termed complex valued DPM (CVDPM(n)). CVDPM(n) has been successfully employed in the Lagrangian frame to accurately compute transmission probabilities on 'thick' one dimensional Eckart and Gaussian potential surfaces. CVDPM(n) is able to reproduce accurate results with a much lower order of approximation than is required by real valued quantum trajectory methods, from initial wave packet energies ranging from the tunneling case (E[subscript o]=0) to high energy cases (twice the barrier height). We successfully extended CVDPM(n) to two-dimensional problems (one translational degree of freedom representing an Eckart or Gaussian barrier coupled to a vibrational degree of freedom) in the Lagrangian framework with great success. CVDPM helps to explain why barrier scattering from "thick" barriers is a much more well posed problem than barrier scattering from "thin" barriers. Though results in these two cases are in very good agreement with grid methods, the search for an appropriate set of initial conditions (termed an 'isochrone) from which to launch the trajectories leads to a time-consuming search problem that is reminiscent of the rootsearching problem from semi-classical dynamics. In order to circumvent the isochrone problem, we present CVDPM(n) equations of motion which are derived and implemented in the arbitrary Lagrangian-Eulerian frame for a metastable potential as well as the Eckart and Gaussian surfaces. In this way, the isochrone problem can be circumvented but at the cost of introducing other computational difficulties. In order to understand why CVDPM may give better transmission probabilities than real valued counterparts, much attention we have been studying and applying numerical analytic continuation techniques to visualize complex-extended wave packets as well as the complex-extended quantum potential. Numerical analytic continuation techniques have also been used to analytically continue a discrete real-valued potential into the complex plane for CVDPM with very promising results.

碩士
國立成功大學
航空太空工程學系碩博士班
98
In recent years, analysis and control of quantum chaos is increasingly important, but the lack of the concept of trajectory makes it impossible to analyze quantum chaos by the methods used in classical chaos. The aim of this thesis is to connect the Newton’s world to the quantum world by the complex mechanics so that quantum chaos can be analyzed and controlled by the complex-extended Newtonian mechanics. Through the bridge of complex mechanics, in this thesis we model quantum motions for 2D charged anisotropic harmonic oscillator by complex-valued dynamic equations based on which quantum chaos can be analyzed by using well-known methods used in classical chaos. With the established quantum dynamic model, we then apply the sliding-mode control method to control the chaotic quantum behavior of the considered quantum system. The simulation results show that chaotic motions can be changed into periodic motions by the proposed chaos control and meanwhile, chaos synchronization can be achieved in the presence of variations of initial conditions. Several signatures of chaos are introduced here to justify the chaos to periodicity process under the sliding-mode control law.

All fields of physics - be it nuclear, atomic and molecular, solid state, or optical - offer examples of systems which are strongly influenced by the environment of the actual system under investigation. The scope of what is called "the environment" may vary, i.e., how far from the system of interest an interaction between the two does persist. Typically, however, it is much larger than the open system itself. Hence, a fully quantum mechanical treatment of the combined system without approximations and without limitations of the type of system is currently out of reach. With the single assumption of the environment to consist of an internally thermalized set of infinitely many harmonic oscillators, the seminal work of Stockburger and Grabert [Chem. Phys., 268:249-256, 2001] introduced an open system description that captures the environmental influence by means of a stochastic driving of the reduced system. The resulting stochastic Liouville-von Neumann equation describes the full non-Markovian dynamics without explicit memory but instead accounts for it implicitly through the correlations of the complex-valued noise forces. The present thesis provides a first application of the Stockburger-Grabert stochastic Liouville-von Neumann equation to the computation of the dynamics of anharmonic, continuous open systems. In particular, it is demonstrated that trajectory based propagators allow for the construction of a numerically stable propagation scheme. With this approach it becomes possible to achieve the tremendous increase of the noise sample count necessary to stochastically converge the results when investigating such systems with continuous variables. After a test against available analytic results for the dissipative harmonic oscillator, the approach is subsequently applied to the analysis of two different realistic, physical systems. As a first example, the dynamics of a dissipative molecular oscillator is investigated. Long time propagation - until thermalization is reached - is shown to be possible with the presented approach. The properties of the thermalized density are determined and they are ascertained to be independent of the system's initial state. Furthermore, the dependence on the bath's temperature and coupling strength is analyzed and it is demonstrated how a change of the bath parameters can be used to tune the system from the dissociative to the bound regime. A second investigation is conducted for a dissipative tunneling scenario in which a wave packet impinges on a barrier. The dependence of the transmission probability on the initial state's kinetic energy as well as the bath's temperature and coupling strength is computed. For both systems, a comparison with the high-temperature Markovian quantum Brownian limit is performed. The importance of a full non-Markovian treatment is demonstrated as deviations are shown to exist between the two descriptions both in the low temperature cases where they are expected and in some of the high temperature cases where their appearance might not be anticipated as easily.:1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Theory of Open Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Influence Functional Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Quantum Brownian Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Stochastic Unraveling of the Influence Functional . . . . . . . . . . . . . . . 20 2.4 Improved Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.1 Modified Dynamic Response . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.2 Guide Trajectory Transformation . . . . . . . . . . . . . . . . . . . . 24 2.5 Obtaining Properly Correlated Stochastic Samples from Filtered White Noise 24 3 Unified Stochastic Trajectory Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1 Semiclassical Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.1 Guide Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.1.2 Real Coherent State Center Coordinates . . . . . . . . . . . . . . . . 31 3.1.3 Propagation Scheme Including Stochastic Forces . . . . . . . . . . . 32 3.2 Stochastic Bohmian Mechanics with Complex Action . . . . . . . . . . . . . 33 3.2.1 Hydrodynamic Formulation of Bohmian Mechanics . . . . . . . . . . 33 3.2.2 Bohmian Mechanics with Complex Action . . . . . . . . . . . . . . . 34 3.2.3 Stochastic BOMCA Trajectories . . . . . . . . . . . . . . . . . . . . 38 3.3 Noise Distribution Preserving Removal of Adverse Samples . . . . . . . . . . 39 4 Dissipative Harmonic Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.1 Reservoir Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Harmonic Oscillator Analytic Expectation Values . . . . . . . . . . . . . . . 42 4.2.1 Ohmic Bath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2.2 Drude Regularized Bath . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3 Sampling Strategies and Analytic Comparison . . . . . . . . . . . . . . . . . 44 4.4 Limits of the Markovian Approximation . . . . . . . . . . . . . . . . . . . . 45 5 Dissipative Vibrational Dynamics of Diatomics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.1 Molecular Morse Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.2 Anharmonic Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.3 Transient Non-Markovian Effects . . . . . . . . . . . . . . . . . . . . . . . . 53 5.4 Trapping by Dissipation and Thermalization . . . . . . . . . . . . . . . . . . 53 6 Tunneling with Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.1 Eckart Barrier Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.2 Dissipative Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.3 Investigation of Markovianity . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Appendix A Conventions for Constants, Reservoir Kernels, and Influence Phases 69 Appendix B Stochastic Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 B.1 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 71 B.2 Position Verlet Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 B.3 Runge-Kutta Fourth Order Scheme . . . . . . . . . . . . . . . . . . . . . . . 73 Appendix CMorse Oscillator Expectation Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Appendix DPrerequisites of a Successful Stochastic Propagation . . . . . . . . . . . . . . 79 D.1 Hubbard-Stratonovich Transformation . . . . . . . . . . . . . . . . . . . . . 79 D.2 Kernels of the Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 D.2.1 Quadratic Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 D.2.2 Quartic Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 D.2.3 Strictly Ohmic Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . 89 D.3 Guide Trajectory Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 90 D.3.1 Quadratic Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 D.3.2 Quartic Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 D.3.3 Strictly Ohmic Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . 92 D.4 Computation of Matrix Elements and Expectation Values . . . . . . . . . . 92 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

博士
國立成功大學
航空太空工程學系碩博士班
97
On the basis of quantum Hamilton mechanics, several issues are addressed in this dissertation. First of all, we study the multi-path behavior of quantum systems by virtue of the complex trajectory interpretation of quantum mechanics. It is shown that Feynman’s path-integral trajectories can be represented by the complex trajectories and then parameterized within the framework of quantum Hamilton mechanics. Next, two simplified physical systems, a 1D harmonic oscillator and a 2D charged anisotropic harmonic oscillator in a uniform magnetic field, are demonstrated to exhibit chaos from the viewpoint of particle-like behavior. While conventional quantum mechanics and Bohmian mechanics both predict that 1D harmonic oscillator shows no signature of chaotic behavior, we find that in quantum Hamilton mechanics this system exhibits both regular and chaotic behavior, depending on the composition of wavefunctions and on the particle’s initial position. We continue to investigate chaotic behavior in a 2D charged anisotropic harmonic oscillator. Even the possibility of chaos in eigenstates has been ruled out from Bohm’s trajectory interpretation, we still find obvious chaotic features in eigenstates of this 2D quantum oscillator. The territory of quantum chaos indeed can be enlarged via the complex-extended dynamics. Finally, we point out that the complex chaotic dynamics may be the origin of the probability interpretation of quantum mechanics. In view of the generality of quantum chaos, it is impossible to predict the final states from the initial states for quantum systems. However, the statistical invariability of chaotic behavior offers another route for us to understand quantum systems. In this dissertation, the comparison between the distribution of complex quantum trajectories on the real coordinate space and the theoretic probability density function determined from the wavefunction shows that a chaotic quantum particle which seems to move irregularly is indeed guided by the wavefunction and attempts to appear somewhere with a statistical regularity. We also examine this tendency for Bohmian trajectories, however, no analogue of the complex trajectories can be found.

High-order terahertz sideband generation (HSG), recently discovered experimentally in semiconductors, is an extreme nonlinear optical phenomenon with physics similar to high-order harmonic generation (HHG) but in a much lower frequency regime. A key concept in understanding the HSG and HHG is the quantum trajectories, where the quantum evolution of particles under strong fields can be essentially captured by a small number of quantum trajectories that satisfy the stationary phase condition of the Dirac-Feynmann path integral. However, in contrast to HHG in atoms and molecules, HSG in semiconductors can have interesting effects due to nontrivial “vacuum” states of band materials. A rich structure of the Bloch states in condensed matter systems would lead to a variety of phase effects in extreme nonlinear optics.
In this thesis, we show that in semiconductors with nontrivial gauge structures in the energy bands, the curved quantum trajectory of an electron-hole pair under a strong elliptically polarized terahertz field can accumulate a geometric phase. In particular, the geometric phase becomes the famous gauge invariant Berry phase for a cyclic trajectory. Taking monolayer MoS₂ as a model system, we show that the Berry phase appears as the Faraday rotation angle in the pulse emission from the material under short-pulse excitation. This finding reveals the Berry phase effect in the extreme nonlinear optics regime for the first time.
We further apply the Berry phase dependent quantum trajectory theory to biased bilayer graphene under strong elliptically polarized terahertz fields. The biased bilayer graphene with Bernal stacking has similar Bloch band features and optical properties to the monolayer MoS₂, such as the time-reversal related valleys and valley contrasting optical selection rule. However, the biased bilayer graphene has much larger Berry curvature than that in monolayer MoS₂, which leads to a large Berry phase of the quantum trajectory and in turn a giant Faraday rotation of the optical emission (∼ 1 rad for a THz field with frequency 1 THz and strength 8 kV/cm). This surprisingly big angle shows that the Faraday rotation can be induced more efficiently by the Berry curvature in momentum space than by the magnetic field in real space. It provides opportunities to use bilayer graphene and THz lasers for ultrafast electro-optical devices.
Finally, we study the geometric phase of a quantum wavepacket driven adiabatically along a trajectory in a parameterized state space. Inherent to quantum evolutions, the wavepacket can not only accumulate a quantum phase but may also experience dephasing, or quantum diffusion. We show that the diffusion of quantum trajectories can also be of geometric nature as characterized by the imaginary part of the geometric phase. Such an imaginary geometric phase results from the interference of geometric phase dependent fluctuations around the quantum trajectory. As a specific example, we again study the quantum trajectories of HSG in monolayer MoS₂. We find that while the real part of the geometric phase leads to the Faraday rotation of the linearly polarized light that excites the electron-hole pair, the imaginary part manifests itself as the polarization ellipticity of the terahertz sidebands which can be measured experimentally. The discovery of the geometric quantum diffusion extends the concept of geometric phases.
最近，在實驗上發現了半導體中的一個極端非線性光學現象，即高次太赫茲邊帶產生(HSG)。它是原子与分子系统里的高次谐波产生(HHG)在太赫茲頻域的一個推广。HSG与HHG的關鍵物理過程均可用量子轨道理论解释，其中粒子的路徑積分描述的量子演化由若干滿足穩相近似條件的量子軌道主導。但是HHG与HSG之間存在着本質區別，即半導體的“真空態”可以具備一些非平凡的拓撲結構，從而給極端非線性光學领域帶來許多有趣的物理效應。
在這篇論文中，我們發現在強橢圓偏振太赫茲場作用下的具有非平凡规范結構的半導體中，電子空穴對的量子軌道可以積累一個非零的幾何相。特別地，如果我們考慮週期量子軌道，這個幾何相便成為著名的規範不變的Berry相。我們取單層MoS₂為模型系統，發現在光脉衝激勵下的材料中的光信號經歷一個法拉第旋轉，而且轉角由量子軌道的Berry相給出。這個發現首次揭示了極端非線性光學領域內的Berry相效應。
我們進一步將含Berry相效應的量子軌道理論應用于強橢圓偏振太赫茲場作用下的雙層石墨烯中。Bernal堆疊的雙層石墨烯与單層MoS₂具有某些相似的能帶結構与光學性質，例如兩者都具有兩個時間反演對稱的谷，且兩個谷內具有不同的躍遷選擇定則。但是雙層石墨烯有遠遠大於單層MoS₂的Berry曲率，從而其內的量子軌道也會積累一個遠遠大於單層MoS₂的Berry相。這個Berry相可以導致光信號巨大的法拉第旋轉（在頻率1THz以及場強8kV/cm的太赫茲場下約為1rad）。這個傳統方法下所無法產生的巨大法拉第旋轉說明比起實空間內的磁場，動量空間內的Berry曲率可以更加有效地誘發光信號的法拉第旋轉。我們的結果可以促使雙層石墨烯以及太赫茲激光在超快光電設備中的應用。
最後，我們考慮具有非平凡規範結構的參數空間內的量子波包在絕熱驅動下的量子演化。在演化過程中，這個波包不僅可以獲得一個量子相位，而且會經歷退相干（即量子擴散）。我們發現波包的一部分量子擴散具有幾何性質，而且這部分量子擴散可以表示為一個复幾何相的虛部。這個复幾何相可以通過量子軌道附近的帶有幾何相的量子路徑的相干來解釋。作為例子，我們研究了強橢圓偏振太赫茲場作用下的單層MoS₂中的量子軌道的复幾何相。我們發現此幾何相的實部誘發光的法拉第旋轉，而虛部則表現為邊帶光信號的橢圓偏振度，並且進而可以從實驗上進行測量。我們關於虛幾何相的研究拓展了幾何相這一概念的新領域。
Yang, Fan = 強太赫茲場下半導體中的量子軌道的Berry相 / 楊帆.
Thesis Ph.D. Chinese University of Hong Kong 2014.
Includes bibliographical references (leaves 71-75).
Abstracts also in Chinese.
Title from PDF title page (viewed on 13, September, 2016).
Yang, Fan = Qiang tai he zi chang xia ban dao ti zhong de liang zi gui dao de Berry xiang / Yang Fan.
Detailed summary in vernacular field only.
Detailed summary in vernacular field only.
Detailed summary in vernacular field only.
Detailed summary in vernacular field only.

As technological advances allow us to peer into and beyond microscopic phenomena, new theoretical developments are necessary to facilitate this exploration. In particular, the potential afforded by utilizing quantum resources promises to dramatically affect scientific research, communications, computation, and material science. Control theory is the field dedicated to the manipulation of systems, and quantum control theory pertains to the manoeuvring of quantum systems. Due to the inherent sensitivity of quantum ensembles to their environment, time-optimal solutions should be found in order to minimize exposure to external sources. Furthermore, the complexity of the Schr\"odinger equation in describing the evolution of a unitary operator makes the analytical discovery of time-optimal solutions rare, motivating the development of numerical algorithms. The seminal result of classical control is the Pontryagin Maximum Principle, which implies that a restriction to bounded control amplitudes admits two classifications of trajectories: bang-bang and singular. Extensions of this result to generalized control problems yield the same classification and hence arise in the study of quantum dynamics. While singular trajectories are often avoided in both classical and quantum literature, a full optimal synthesis requires that we analyze the possibility of their existence. With this in mind, this treatise will examine the issue of time-optimal quantum control. In particular, we examine the results of existing numerical algorithms, including Gradient Ascent Pulse Engineering and the Kaya-Huneault method. We elaborate on the ideas of the Kaya-Huneault algorithm and present an alternative algorithm that we title the Real-Embedding algorithm. These methods are then compared when used to simulate unitary evolution. This is followed by a brief examination on the conditions for the existence of singular controls, as well possible ideas and developments in creating geometry based numerical algorithms.

We introduce the de Broglie-Bohm causal interpreation of quantum mechanics and compare it to the standard interpretation of quantum mechanics, the Copenhagen interpretation. We examine the possibility of experimentally distinguishing between the two theories, as well as the potential for the causal interpretation to more easily bridge the gap between the physics of the quantum and classical worlds. We then use the causal interpretation to construct a deterministic model of the helium atom in which the two electrons move along trajectories through space and time about a stationary nucleus. The dynamics are governed by the non-relativistic Schrödinger equation and the spin vectors of both electrons are assumed to be constant along their respective trajectories. We examine the Bohmian trajectories associated with (approximations to) eigenstates of the helium Hamiltonian as well as the trajectories associated with some non-eigenstates. We also compute an approximation to the ground state energy of the helium atom using a representation of the helium wavefunction in terms of hydrogenic eigenfunctions which is motivated by a perturbation approach.

In this thesis we study the Dicke model outside the rotating wave approximation (RWA), by employing phase space techniques and the quantum trajectory theory. We present a review of the basic models of open systems in quantum optics and present an experimental proposition justifying the model to be studied. We use the phase space approach to study, among other subjects, entanglement, squeezing and fluctuations across a quantum phase transition. Three different phase space representations are used and their strengths and weaknesses compared. The quantum trajectory theory is applied to visualise the global quantum fluctuations and to learn how different measurement schemes will affect the creation of entanglement.
The University of Auckland, Department of Physics.

碩士
國立成功大學
航空太空工程學系碩博士班
101
The purpose of this paper is to establish a state-space for quantum systems, according to optimal stochastic control, and to verify that a quantum system is actually an optimized stochastic system. The proposed optimization process comprises two steps: complexification and randomization, and the outcome of the process shows that a quantum motion is actually a complex Brownian motion. After our study, it becomes clear that a quantum path is random and fractal, and is governed by a stochastic differential equation, from which a quantum paths can be solved for electronic motions in hydrogen atom and for orbital angular momentum and quantum dynamics in diatomic molecules.

A novel method for integrating the time-dependent Schrödinger equation is presented. The moving boundary truncation (MBT) method is a time-dependent adaptive method that can significantly reduce the number of grid points needed to perform accurate wave packet propagation while maintaining stability. Hydrodynamic quantum trajectories are used to adaptively define the boundaries and boundary conditions of a fixed grid. The result is a significant reduction in the number of grid points needed to perform accurate calculations. A variety of model potential energy surfaces are used to evaluate the method. Excellent agreement with fixed boundary grids was obtained for each example. By moving only the boundary points, stability was increased to the level of the full fixed grid. Variations of the MBT method are developed which allow it to be applied to any potential energy surface and used with any propagation method. A variation of MBT is applied to the collinear H+H₂ reaction (using a LEPS potential) to demonstrate the stability and accuracy. Reaction probabilities are calculated for the three dimensional non-rotating O(³P)+H₂ and O(³P)+HD reactions to demonstrate that the MBT can be used with a variety of numerical propagation techniques.
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