The pointer basis and the feedback stabilization of quantum systems

The dynamics for an open quantum system can be `unravelled' in infinitely many ways, depending on how the environment is monitored, yielding different sorts of conditioned states, evolving stochastically. In the case of ideal monitoring these states are pure, and the set of states for a given monitoring forms a basis (which is overcomplete in general) for the system. It has been argued elsewhere [D. Atkins et al., Europhys. Lett. 69, 163 (2005)] that the `pointer basis' as introduced by Zurek and Paz [Phys. Rev. Lett 70, 1187(1993)], should be identified with the unravelling-induced basis which decoheres most slowly. Here we show the applicability of this concept of pointer basis to the problem of state stabilization for quantum systems. In particular we prove that for linear Gaussian quantum systems, if the feedback control is assumed to be strong compared to the decoherence of the pointer basis, then the system can be stabilized in one of the pointer basis states with a fidelity close to one (the infidelity varies inversely with the control strength). Moreover, if the aim of the feedback is to maximize the fidelity of the unconditioned system state with a pure state that is one of its conditioned states, then the optimal unravelling for stabilizing the system in this way is that which induces the pointer basis for the conditioned states. We illustrate these results with a model system: quantum Brownian motion. We show that even if the feedback control strength is comparable to the decoherence, the optimal unravelling still induces a basis very close to the pointer basis. However if the feedback control is weak compared to the decoherence, this is not the case

Some contributions to filtering theory with applications in financial modelling

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Two main groups of filtering algorithms are characterised and developed. Their applicability is demonstrated using actuarial and financial time series data. The first group of algorithms involved hidden Markov models (HMM), where the parameters of an asset price model switch between regimes in accordance with the dynamics of a Markov chain. We start with the known HMM filtering set-up and extend the framework to the case where the drift and volatility have independent probabilistic behaviour. In addition, a non-normal noise term is considered and recursive formulae in the online re-estimation of model parameters are derived for the case of students’ t-distributed noise. Change of reference probability is employed in the construction of the filters. Both extensions are then tested on financial and actuarial data. The second group of filtering algorithms deals with sigma point filtering techniques. We propose a method to generate sigma points from symmetric multivariate distributions. The algorithm matches the first three moments exactly and the fourth moment approximately; this minimises the worst case mismatch using a semidefinite programming approach. The sigma point generation procedure is in turn applied to construct algorithms in the latent state estimation of nonlinear time series models; a numerical demonstration of the procedure’s effectiveness is given. Finally, we propose a partially linearised sigma point filter, which is an alternative technique for the optimal state estimation of a wide class of nonlinear time series models. In particular, sigma points are employed for generating samples of possible state values and then a linear programming-based procedure is utilised in the update step of the state simulation. The performance of the filtering technique is then assessed on simulated, highly non-linear multivariate interest rate process and is shown to perform significantly better than the extended Kalman filter in terms of computational time

Levitation and control of particles with internal degrees of freedom

Levitodynamics is a fast growing field that studies the levitation and manipulation of micro- and nanoobjects, fuelled by both fundamental physics questions and technological applications. Due to the isolated nature of trapped particles, levitated systems are highly decoupled from the environment, and offer experimental possibilities that are absent in clamped nanomechanical oscillators. In particular, a central question in quantum physics is how the transition between the classical and quantum world materializes, and levitated objects represent a promising avenue to study this intermediate regime. In the last years, most levitation experiments have been restricted to optically trapped silica nanoparticles in vacuum, controlling the particle's position with intensity modulated laser beams. However, the use of optical traps severely constrains the experiments that can be performed, because few particle materials can withstand the optical absorption and resulting heating in vacuum. This completely prevents the use of objects with internal degrees of freedom, which---coupled to mechanical variables---offer a clear path towards the study of quantum phenomena at the macroscale. In this thesis, we address these issues by considering other types of trap and feedback schemes, achieving excellent control on the dynamics of optically active nanoparticles. With stochastic calculus, simulations and experiments, we study the dynamics of trapped particles in different regimes, considering also a hybrid quadrupole-optical trapping scheme. Then, using a Paul trap of our own design, we demonstrate the trapping, interrogation and feedback cooling of a nanodiamond hosting a single NV center in vacuum, a clear candidate to perform quantum physics experiments at the single spin level. Finally, we discuss and implement an optimal controller to cool the center of mass motion of an optically levitated nanoparticle. The feedback is realized by exerting a Coulomb force on a charged particle with a pair of electrodes, and thus requires no optics.La levitodinàmica és un camp de la física en ràpida expansió que estudia la levitació i manipulació de micro- i nano-objectes, empesa per la possibilitat de solucionar trencaclosques de física fonamental i de desenvolupar noves aplicacions tecnològiques. Gràcies al gran aïllament de les partícules en levitació, l’evolució dels sistemes levitodinàmics està molt desacoplada del seu entorn. Per consegüent, permeten fer experiments que no serien possibles en nanooscil·ladors mecànics sobre substrat. En particular, una qüestió central en física consisteix en entendre com es produeix la transició entre els mons clàssic i quàntic; els objectes en levitació permeten estudiar aquest règim intermedi de manera innovadora. En els últims anys, la majoria d’experiments de levitodinàmica s’han limitat a atrapar òpticament partícules de sílice en el buit, tot controlant la posició de la partícula amb feixos làser modulats. Tot i així, l’ús de trampes òptiques suposa un obstacle a l’hora d’exportar aquests experiments a règims més diversos perquè, a baixes pressions, pocs materials són capaços de suportar les altes temperatures resultants de l’absorció de llum làser. Això impedeix l’ús d’objectes amb graus de llibertat interns, que –acoplats a variables mecàniques– suposen un full de ruta clar per estudiar fenòmens quàntics a escala macroscòpica En aquesta tesi, adrecem aquestes qüestions tot considerant altres tipus de trampa i tècniques de feedback, i assolim un control excel·lent de la dinàmica de nanopartícules òpticament actives en levitació. Mitjançant càlcul estocàstic, simulacions i experiments, estudiem la dinàmica de les partícules en règims diversos, àdhuc considerant un esquema híbrid de trampa de Paul-òptica. A continuació, utilitzant una trampa de Paul, demostrem experimentalment l’atrapament, interrogació i feedback-cooling en el buit d’un nanodiamant que conté un únic NV− center, un clar candidat per a la realització d’experiments de física quàntica amb un únic spin. Finalment, estudiem i implementem un controlador òptim per a refredar el centre de massa d’una partícula òpticament levitada. El feedback es realitza exercint una força de Coulomb sobre una partícula carregada positivament mitjançant un parell d’elèctrodes, i per tant no requereix elements òptic

Unraveling the Thousand Word Picture: An Introduction to Super-Resolution Data Analysis

Super-resolution microscopy provides direct insight into fundamental biological processes occurring at length scales smaller than light’s diffraction limit. The analysis of data at such scales has brought statistical and machine learning methods into the mainstream. Here we provide a survey of data analysis methods starting from an overview of basic statistical techniques underlying the analysis of super-resolution and, more broadly, imaging data. We subsequently break down the analysis of super-resolution data into four problems: the localization problem, the counting problem, the linking problem, and what we’ve termed the interpretation problem