Quantum Trajectories

It is shown how Adler's trace dynamics can be applied to Nelon's stochastic mechanics and also to complex classical dynamical systems. Emergent noncommutivity due to the fractal nature of trajectories in stochastic mechanics is closely related to the fact that the forward and backward time derivatives are different. Consequently, non-commuting matrices representing coordinates and momenta can be defined and then trace dynamics becomes applicable. A new variational approach to stochastic mechanics based on trace dynamics is introduced using this approach. First it is shown that the standard variational methods of stochastic mechanics can be generalized to allow for any diffusion constant in a stochastic model of Schrödinger's equation, and that they can also describe dissipative diffusion. Then it is shown that the trace dynamical theory seems to only describe dissipative diffusion unless an extra quantum mechanical potential term is added to the Hamiltonian. This adds to the debate about what action principle should be applied to stochastic mechanics. It also opens a path for applying trace dynamical method to complex systems which have limiting behavior approximated by Brownian motion. The differential space theory of Wiener and Siegel is reconsidered as a useful tool in this framework, and is generalized to stochastic processes as well as deterministic ones for the hidden trajectories of observables. It also is generalized to a larger class of deterministic models then was previously contemplated. It is proposed that the natural measure space for Wiener-Siegel theory is Haar measure for random unitary matrices. A new physical interpretation of the polychotomic algorithm is given.

The "main road" open by de Broglie's and Schrödinger's discovery of matter waves and of their eigenfunctions branched off, as is well known, into different "sub-routes". The most widely accepted one is Standard Quantum Mechanics (SQM), interpreting the time-dependent Schrödinger equation as the basic evolution law of a wave-packet which represents the simultaneous probabilistic permanence of a particle in its full set of eigenstates. Another "sub-route" is offered by Bohm's Mechanics, able to reproduce the same results of SQM, while interpreting the streamlines of the probability current density as the "quantum trajectories" of the moving particles. Reminding that the so-called quasi-optical approximation represents a standard mathematical technique allowing a ray-based treatment of any kind of wave-like features, we present here an exact wavemechanical "sub-route", based on the observation that the time-independent Schrödinger equation (as well as any other Helmholtz-like equation) may be treated, bypassing the quasi-optical approximation, in terms of a Hamiltonian set of rays mutually coupled by an energy-dependent function (which we call "Wave Potential") encoded in the very structure of any Helmholtz-like equation. These rays, reducing to the classical pointparticle trajectories when the Wave Potential is neglected, lend themselves to be interpreted as the exact wavedynamical trajectories and motion laws of classical-looking point-particles associated with the de Broglie-Schrödinger matter waves. The role of the Wave Potential, acting perpendicularly to the momentum of the moving particles, is to "pilot" them without any energy exchange: a property which isn't shared by the wellknown "Quantum Potential" of the Bohmian theory, involving the entire spectrum of possible eigen-energies of a wave-packet. This property turns out to allow the numerical computation of the particle trajectories, which we perform and discuss here for particles moving (under the guiding rule of the Wave Potential) in many different force-fields, such as a constant external field and the fields due to a potential barrier, a potential step and a lenslike potential, respectively.

Contrary to a widespread commonplace, an exact, ray-based treatment holding for any kind of monochromatic wave-like features (such as diffraction and interference) is provided by the structure itself of the Helmholtz equation. This observation allows to dispel-in apparent violation of the Uncertainty Principle-another commonplace, forbidding an exact, trajectory-based approach to Wave Mechanics.

On montre que le comportement des ondes monochromatiques classiques dans les milieux stationnaires est gouverné par une fonction dispersive, qu'on appelle ici "Potentiel d'Onde", codée dans la structure même de l'équation d' Helmholtz, et on présente une description exacte, en termes de rayons hamiltoniens (se réduisant à l'approximation de l'optique géométrique lorsque cette fonction est négligée), qui révèle une dépendence mutuelle entre les rayons et permet aussi le traitement de phénomènes typiquement ondulatoires tels que la diffraction et l'interférence, sous l'action de "pilotage" du Potentiel d'Onde. Puisque l'équation de Schrödinger indépendante du temps (associant le mouvement des particules mono-énergétiques avec des ondes materielles stationnaires monochromatiques) est elle-même une équation d'Helmholtz, le traitement mathématique valable dans le cas classique est étendu, sans recourir à des concepts probabilistes, à la dynamique hamiltonienne quantique de particules ponctiformes mono-énergétiques. Les trajectoires dynamiques exactes des particules sont couplées et pilotées, dans ce cas, par un Potentiel d'Onde exact et dispersif similaire dans sa forme au "Potentiel Quantique" bohmien, dont il ne partage pas, cependant, la nature probabiliste et l'indépendance de l'énergie. Ce résultat, associé au lien entre le Potentiel d'Onde et le Principe d'Incertitude, suggère une nouvelle façon de comprendre la Mécanique Ondulatoire, dans l'esprit originaire de Schrödinger et de Broglie.

Classical and wave-mechanical treatments of undulatory features may be faced in terms of exact Hamiltonian particle trajectories, encoded in the very structure of any Helmholtz-like equation and coupled by an intrinsic "Wave Potential" function determined by the launching conditions. These trajectories are shown to provide the bridge between particles and waves, supplying an insight into the dual nature of the physical world.

Standard Quantum Mechanics (SQM) is the most widely accepted "route" starting from the common ground provided by de Broglie's and Schroedinger's discovery of matter waves and of their eigen-functions and eigen-values. Another "route" is offered by Bohm's Mechanics, able to reproduce the same results of SQM, while interpreting the streamlines of probability current density as a set of (variously interpreted) "quantum trajectories". We exploit here a somewhat different "route", based on the fact that the stationary Schroedinger equation (as well as any other Helmholtz-like equation) may be treated in terms of exact point-particle trajectories, coupled and guided by a stationary, energy dependent function, which we call "Wave Potential", acting normally to the relevant particle momentum and preserving, therefore, the particle energy: a property which cannot be shared by the time-dependent "Quantum Potential" of the Bohmian theory, involving the entire spectrum of eigen-energies. This basic property turns out, indeed, to allow the numerical computation of the particle trajectories in stationary external force fields, and to show that all diffractive processes consist of energy-preserving exchanges, mediated by the Wave Potential, between the longitudinal and transverse momentum components of each single particle. We compute and discuss the trajectories of particles moving, under the guiding action of the Wave Potential, in a number of different external stationary force-fields, such as a constant field and the ones due to a potential barrier, a potential step and a focalizing potential, respectively. The very existence of these exact solutions appears to circumvent the general character of the Uncertainty Principle.

The "main road" open by de Broglie's and Schrödinger's discovery of matter waves and of their eigenfunctions branched off, as is well known, into different "sub-routes". The most widely accepted one is Standard Quantum Mechanics (SQM), interpreting the time-dependent Schrödinger equation as the basic evolution law of a wave-packet which represents the simultaneous probabilistic permanence of a particle in its full set of eigenstates. Another "sub-route" is offered by Bohm's Mechanics, able to reproduce the same results of SQM, while interpreting the streamlines of the probability current density as the "quantum trajectories" of the moving particles. Reminding that the so-called quasi-optical approximation represents a standard mathematical technique allowing a ray-based treatment of any kind of wave-like features, we present here an exact wavemechanical "sub-route", based on the observation that the time-independent Schrödinger equation (as well as any other Helmholtz-like equation) may be treated, bypassing the quasi-optical approximation, in terms of a Hamiltonian set of rays mutually coupled by an energy-dependent function (which we call "Wave Potential") encoded in the very structure of any Helmholtz-like equation. These rays, reducing to the classical pointparticle trajectories when the Wave Potential is neglected, lend themselves to be interpreted as the exact wavedynamical trajectories and motion laws of classical-looking point-particles associated with the de Broglie-Schrödinger matter waves. The role of the Wave Potential, acting perpendicularly to the momentum of the moving particles, is to "pilot" them without any energy exchange: a property which isn't shared by the wellknown "Quantum Potential" of the Bohmian theory, involving the entire spectrum of possible eigen-energies of a wave-packet. This property turns out to allow the numerical computation of the particle trajectories, which we perform and discuss here for particles moving (under the guiding rule of the Wave Potential) in many different force-fields, such as a constant external field and the fields due to a potential barrier, a potential step and a lenslike potential, respectively.

The experimental results of Kocsis et al., Mahler et al. and the proposed experiments of Morley et al. show that it is possible to construct "trajectories" in interference regions in a two-slit interferometer. These results call for a theoretical re-appraisal of the notion of a "quantum trajectory" first introduced by Dirac and in the present paper we reexamine this notion from the Bohm perspective based on Hamiltonian flows. In particular, we examine the short-time propagator and the role that the quantum potential plays in determining the form of these trajectories. These trajectories differ from those produced in a typical particle tracker and the key to this difference lies in the active suppression of the quantum potential necessary to produce Mott-type trajectories. We show, using a rigorous mathematical argument, how the active suppression of this potential arises. Finally we discuss in detail how this suppression also accounts for the quantum Zeno effect.

Recent developments in quantum technology mean that is it now possible to manipulate systems and measure fermion fields (e.g. reservoirs of electrons) at the quantum level. This progress has motivated some recent work on filtering theory for quantum systems driven by fermion fields by Korotkov, Milburn and others. The purpose of this paper is to develop fermion filtering theory using the fermion quantum stochastic calculus. We explain that this approach has close connections to the classical filtering theory that is a fundamental part of the systems and control theory that has developed over the past 50 years.

The experimental results of Kocsis et al., Mahler et al. and the proposed experiments of Morley et al. show that it is possible to construct "trajectories" in interference regions in a two-slit interferometer. These results call for a theoretical re-appraisal of the notion of a "quantum trajectory" first introduced by Dirac and in the present paper we reexamine this notion from the Bohm perspective based on Hamiltonian flows. In particular, we examine the short-time propagator and the role that the quantum potential plays in determining the form of these trajectories. These trajectories differ from those produced in a typical particle tracker and the key to this difference lies in the active suppression of the quantum potential necessary to produce Mott-type trajectories. We show, using a rigorous mathematical argument, how the active suppression of this potential arises. Finally we discuss in detail how this suppression also accounts for the quantum Zeno effect.

The method of quantum trajectories proposed by de Broglie and Bohm is applied to the study of atom diffraction by surfaces. As an example, a realistic model for the scattering of He off corrugated Cu is considered. In this way, the final angular distribution of trajectories is obtained by box counting, which is in excellent agreement with the results calculated by standard S matrix methods of scattering theory. More interestingly, the accumulation of quantum trajectories at the different diffraction peaks is explained in terms of the corresponding quantum potential. This nonlocal potential ''guides'' the trajectories causing a transition from a distribution near the surface, which reproduces its shape, to the final diffraction pattern observed in the asymptotic region, far from the diffracting object. These two regimes are homologous to the Fresnel and Fraunhofer regions described in undulatory optics. Finally, the turning points of the quantum trajectories provide a better description of the surface electronic density than the corresponding classical ones, usually employed for this task.

Contrary to a widespread commonplace, an exact, ray-based treatment holding for any kind of monochromatic wave-like features (such as diffraction and interference) is provided by the structure itself of the Helmholtz equation. This observation allows to dispel-in apparent violation of the Uncertainty Principle-another commonplace, forbidding an exact, trajectory-based approach to Wave Mechanics.

The time-independent Schroedinger and Klein-Gordon equations-as well as any other Helmholtz-like equation-were recently shown to be associated with exact sets of ray-trajectories (coupled by a "Wave Potential" function encoded in their structure itself) describing any kind of wave-like features, such as diffraction and interference. This property suggests to view Wave Mechanics as a direct, causal and realistic, extension of Classical Mechanics, based on exact trajectories and motion laws of point-like particles "piloted" by de Broglie's matter waves and avoiding the probabilistic content and the wave-packets both of the standard Copenhagen interpretation and of Bohm's theory. RÉSUMÉ-On a démontré récemment que les équations indépendantes du temps de Schroedinger et de Klein-Gordon, ainsi que toutes les autres équations d'Helmholtz, sont associées à des systèmes de trajectoires hamiltoniennes couplées par une function (le "potentiel d'onde") codée dans leur structure même, qui permettent de décrire tous le phénomènes ondulatoires, comme le diffraction et l'interférence. Cette propriété suggère d'envisager la Mécanique Ondulatoire comme une extension directe, causale et réaliste de la Mécanique Classique (basée sur les trajectoires exactes de particules ponctiformes pilotées par des ondes materielles monochromatiques) évitant à la fois le probabilisme et les paquets d'ondes de l'interpretation de Copenhagen et de la théorie de Bohm.

Se simuló el lanzamiento bidimensional de una partícula de masa m en un entorno circular cerrado, con cuatro dispersores iguales situados en el centro, utilizando el lenguaje computacional de phyton y el runge kutta de orden 4, de manera que se describiera la trayectoria de la partícula a partir de las condiciones iniciales (r o , v o). Luego mediante los coeficientes de Lyapunov se comprobó que el sistema presenta caos, y se observo la región caótica para un set de parámetros específicos, también se demostró que el grado de dispersión de las partículas (ángulo de salida), depende del parámetro de impacto y de la energía inicial de lanzamiento, y de manera similar se determino la sección eficaz, a partir de la distribución de impacto sobre el contorno de la región circular.

Single-spin measurement is an extremely important challenge, and necessary for the future successful development of several recent spin-based proposals for quantum information processing. Magnetic resonance force microscopy (MRFM) has been suggested as a promising technique for single-spin detection. We discuss how to read out the quantum state of a single spin using the MRFM technique based on cyclic adiabatic inversion (CAI). We include, in our analysis, a measurement device (an optical interferometer) to monitor the position of the cantilever, which then provides us with information of the spin state. We consider various relevant sources of noise and taken into account the effect of spin relaxation on the single-spin detection scheme. We also present a realistic continuous measurement model, and discuss the approximations and conditions to achieve a quantum nondemolition measurement of a single spin by MRFM. Finally we will present some simulation results for the single-spin measurement process.

We present and study a two-particle quantum walk on the line in which the two particles interact via a long-range Coulombian-like interaction. We obtain the spectrum of the system as well as study the type of molecules that form, attending to the bosonic or fermionic nature of the walkers. The usual loss of distinction between attractive and repulsive forces does not entirely apply in our model because of the long-range of the interaction.

En este trabajo resumimos una estructura explícita de cálculo numérico para resolver el formalismo de David Bohm de la Mecánica Cuántica. El programa fue desarrollado en Fortran 77 para correr en equipos PC de capacidad estándar. Mostramos numéricamente la existencia de la inestabilidad cuántica en el potencial de David. Bohm cuando los estados estacionarios son ligeramente perturbados. Para los estados perturbados el espacio de las fases asociado a la partícula virtual presenta aspecto caótico.

Parameter inference of dynamical systems is a challenging task faced by many researchers and practitioners across various fields. In many applications, it is common that only limited variables are observable. In this paper, we propose a method for parameter inference of a system of nonlinear coupled ODEs with partial observations. Our method combines fast Gaussian process based gradient matching (FGPGM) and deterministic optimization algorithms. By using initial values obtained by Bayesian steps with low sampling numbers, our deterministic optimization algorithm is both accurate and efficient.

The approximations of classical mechanics resulting from quantum mechanics are richer than a correspondence of classical dynamical variables with self-adjoint Hilbert space operators. Assertion that classical dynamic variables correspond to self-adjoint Hilbert space operators is disputable and sets unnatural limits on quantum mechanics. Well known examples of classical dynamical variables not associated with self-adjoint Hilbert space operators are discussed as a motivation for the realizations of quantum field theory that lack Hermitian field operators but exhibit interaction.

It is shown how Adler's trace dynamics can be applied to stochastic mechanics and other complex classical dynamical systems. Emergent non-commutivity due to the fractal nature of sample trajectories is closely related to the fact that the forward and backward time derivatives are different for these diffusions. A new variational approach to stochastic mechanics based on trace dynamics is introduced. It is shown that Yasue's method and Guerra and Morato's method can both be generalized to allow for any diffusion constant in a stochastic model of Schrodinger's equation, and that they can all also describe dissipative diffusion. Then it is shown that the trace dynamical theory seems to only describe dissipative diffusion unless an extra quantum mechanical potential term is added to the Hamiltonian. The differential space theory of Wiener and Siegel is reconsidered as a useful tool in this framework, and is generalized to stochastic processes instead of deterministic one...

The conceptual understanding of Quantum Mechanics has been of great interest in physics education in recent years; many investigations have been conducted in order to identify, document and design learning strategies for the assimilation of quantum concepts. In Honduras, an investigation was carried out on the conceptual understanding of non-relativistic quantum mechanics in one spatial dimension that have undergraduate students of the UNAH in a physics course, a specialized physics test was applied to 15 students who had completed the first Quantum Mechanics course, this test was designed by university professors of the United States (USA). In this article we compare the results obtained by UNAH students and students from American Universities. In the USA, the items that relate the infinite square well were chosen with the concepts of: superposition principle, stationary and non-stationary states, measurements and collapse of the wave function, continuous measurements of physical observables, temporal dependence, discrimination of possible functions of wave of a system.

We consider a central fermion strongly interacting with a surrounding mesoscopic bath of fermions which is weakly coupled to a Markovian bath of fermions. The master equation of the system consisting of the central fermion and the mesoscopic bath is derived, and based on this master equation, the reduced dynamics and thermalization of the central fermion are investigated. As probes for studying the dynamics and thermalization, the mean number of the central fermion and the current density are calculated. The observed dynamics reveal that strong coupling strength induces non-Markovian behavior and that in the long time limit thermalization is attained leaving the central fermion in a thermal state that is described by the corresponding Gibbs distribution. Upon attaining the thermal state, the central fermion is found to be entangled with the collective mode of the surrounding mesoscopic bath in the high-temperature regime, whereas in the low-temperature regime, it is found to undergo sudden death of entanglement.

Among the formulations of the theory of quantum measurements in continuous time, quantum trajectory theory is very suitable for the introduction of measurement based feedback and closed loop control of quantum systems. In this paper we present such a construction in the concrete case of a two-level atom stimulated by a coherent, monochromatic laser. In particular, we show how fast feedbackà la Wiseman and Milburn can be introduced in the formulation of the theory. Then, the spectrum of the free fluorescence light is studied and typical quantum phenomena, squeezing and sub-natural line-narrowing, are presented.

Abstract: It is shown how Adler's trace dynamics can be applied to stochastic mechanics and other complex classical dynamical systems. Emergent non-commutivity due to the fractal nature of sample trajectories is closely related to the fact that the forward and backward time derivatives are different for these diffusions. A new variational approach to stochastic mechanics based on trace dynamics is introduced. It is shown that Yasue's method and Guerra and Morato's method can both be generalized to allow for any diffusion constant in ...

After providing a general formulation of Fermion flows within the context of Hudson-Parthasarathy quantum stochastic calculus, we consider the problem of determining the noise coefficients of the Hamiltonian associated with a Fermion flow so as to minimize a naturally associated quadratic performance functional. This extends to Fermion flows results of the authors previously obtained for Boson flows .

Multiplicative white-noise stochastic processes continuously attract the attention of a wide area of scientific research. The variety of prescriptions available to define it difficults the development of general tools for its characterization. In this work, we study equilibrium properties of Markovian multiplicative white-noise processes. For this, we define the time reversal transformation for this kind of processes, taking into account that the asymptotic stationary probability distribution depends on the prescription. Representing the stochastic process in a functional Grassman formalism, we avoid the necessity of fixing a particular prescription. In this framework, we analyze equilibrium properties and study hidden symmetries of the process. We show that, using a careful definition of equilibrium distribution and taken into account the appropriate time reversal transformation, usual equilibrium properties are satisfied for any prescription. Finally, we present a detailed deduction of a covariant supersymmetric formulation of a multiplicative Markovian white-noise process and study some of the constraints it imposes on correlation functions using Ward-Takahashi identities.

La comprensión conceptual de la Mecánica Cuántica ha sido de mucho interés en los últimos años en la educación de la fı́sica, se han realizado numerosas investigaciones con el fin de identificar, documentar y diseñar estrategias de aprendizaje para la asimilación de los conceptos cuánticos. En Honduras se realizó una investigación sobre la comprensión conceptual de la Mecánica Cuántica no relativista en una dimensión espacial que tienen los estudiantes de pregrado de la Carrera de Fı́sica de la UNAH, se aplicó una prueba especializada de fı́sica a 15 estudiantes que habı́an finalizado el curso Mecánica Cuántica I, esta prueba fue diseñada y publicada por profesores de Universidades de Estados Unidos de América. En el presente artı́culo se comparan los resultados obtenidos por los estudiantes de la UNAH y estudiantes de Universidades de EE. UU, se eligieron los ı́tems que relacionan el pozo cuadrado infinito con los conceptos de: principio de superposición, estados estacionarios y no...

In the restricted problem of three bodies when the primaries are triaxial rigid bodies, the necessary and sufficient conditions to find the locations of the three libration collinear points are stated. In addition, the Linear stability of these points is studied for the case of the Euler angles of rotational motion being θ i = 0, ψ i + φ i = π/2, i = 1, 2 accordingly. We underline that the model studied in this paper has special importance in space dynamics when the third body moves in gravitational fields of planetary systems and particularly in a Jupiter model or a problem including an irregular asteroid.

Recent developments in quantum technology mean that is it now possible to manipulate systems and measure fermion fields (e.g. reservoirs of electrons) at the quantum level. This progress has motivated some recent work on filtering theory for quantum systems driven by fermion fields by Korotkov, Milburn and others. The purpose of this paper is to develop fermion filtering theory using the fermion quantum stochastic calculus. We explain that this approach has close connections to the classical filtering theory that is a fundamental part of the systems and control theory that has developed over the past 50 years.

UDC 517.9 The backgrounds of quantum mathematics, a new discipline in mathematical physics, are discussed and analyzed from both historical and analytical points of view. The magic properties of the second quantization method, invented by Fock in 1934, are demonstrated, and an impressive application to the theory of nonlinear dynamical systems is considered. Von Neumann first applied the spectral theory of self-adjoint operators on Hilbert spaces to explain the radiation spectra of atoms and the related matter stability [2] (1926); Fock was the first to introduce the notion of many-particle Hilbert space, named a Fock space, and introduced related creation and annihilation operators acting on it [3] (1932); Weyl understood the fundamental role of the notion of symmetry in physics and developed a physics-oriented group theory; moreover, he showed the importance of different representations of classical matrix groups for physics and studied unitary representations of the Heisenberg-Weyl group related to creation and annihilation operators on a Fock space [4] (1931). At the end of the 20th century, new developments were due to Faddeev with co-workers (quantum inverse spectral theory transform [5], 1978); Drinfeld, Donaldson, and Witten (quantum groups and algebras, quantum topology, and quantum superanalysis [6-8], 1982-1994);

Departing from a two-dimensional hyperbolic system that describes the interaction between some activator and inhibitor substances in chemical reactions, we investigate a general form of that model using a finite-difference approach. The model under investigation is a nonlinear system consisting of two coupled partial differential equations with generalized reaction terms. The presence of two-dimensional diffusive terms consisting of fractional operators of the Riesz type is considered here, using spatial differentiation orders in the set (1, 2]. We impose initial conditions on a closed and bounded rectangle, and a finite-difference methodology based on the use of weighted-shifted Grünwald differences is proposed. Among the most important results of this work, we establish analytically the second-order consistency of our scheme. Moreover, the discrete energy method is employed to prove the stability and the quadratic convergence of the technique. Some numerical simulations obtained through our method show the appearance of Turing patterns and wave instabilities, in agreement with some reports found in the literature on superdiffusive hyperbolic activator-inhibitor systems.

This brief article gives an overview of quantum mechanics as a quantum probability theory. It begins with a review of the basic operator-algebraic elements that connect probability theory with quantum probability theory. Then quantum stochastic processes is formulated as a generalization of stochastic processes within the framework of quantum probability theory. Quantum Markov models from quantum optics are used to explicitly illustrate the underlying abstract concepts and their connections to the quantum regression theorem from quantum optics.

In this paper, our prime objective is to connect the curvature of our observable De Sitter Universe with the spectroscopic study of entanglement of two atoms in an open quantum system (OQS). The OQS considered in our work is made up of two atoms which are represented by Pauli spin tensor operators projected along any arbitrary direction. They mimic the role of a pair of freely falling Unruh De-Witt detectors, which are allowed to non-adiabatically interact with a conformally coupled massless probe scalar field in the De Sitter background. The effective dynamics of the atomic detectors are actually an outcome of their non-adiabatic interaction, which is commonly known as the Resonant Casimir Polder Interaction (RCPI) with the thermal bath. We find from our analysis that the RCPI of two stable entangled atoms in the quantum vacuum states in OQS depends on the De Sitter space-time curvature relevant to the temperature of the thermal bath felt by the static observer. We also find that, in OQS, RCPI produces a new significant contribution appearing in the effective Hamiltonian of the total system and thermal bath under consideration. This will finally give rise to Lamb Spectroscopic Shift, as appearing in the context of atomic and molecular physics. This analysis actually plays a pivotal role to make the bridge between the geometry of our observed Universe to the entanglement in OQS through Lamb Shift atomic spectroscopy. In two atomic OQS, Lamb Shift spectra is characterised by a L −2 decreasing inverse square power law behaviour when inter atomic Euclidean distance (L) is much larger than a characteristic length scale (k) associated with the system, which quantifies the breakdown of a local inertial description within OQS. On the other hand, the RCPI of this two atomic OQS immersed in a thermal bath in the background of Minkowski flat Universe is completely characterised by a temperature independent L −1 decreasing inverse power law. This mimics exactly the same situation where the characteristic length scale k is sufficiently large compared to the interatomic Euclidean distance between the two atoms. Thus, we are strongly aiming to connect the curvature of the background space-time of our Universe to open quantum Lamb Shift spectroscopy by measuring the quantum properties of a two entangled OQS in the atomic experiment.

It has been usually assumed that under very general and common conditions the outcome of a collision experiment does not depend on the properties of the projectiles' beam [1]. However, recent evidence in ionization experiments [2] points to a breakdown of these conditions and a dependence of the collision outcome on the incident beam's coherence properties. These facts open the question of how is the result of a collision affected by the preparation of the projectiles' beam.
This thesis presents a study of this problem analyzing inconsistencies of the standard stationary formulation of the scattering theory [1], and how these can affect the interpretation of projectile's coherence effects in ion impact collision experiments. To get this done, the framework of the De Broglie-Bohm formulation [3, 4] was used, which has recently recovered its lost mommentum due to its ability to deal with novel weak measurement results [5, 6]. In addition, this formulation is an advantageous option to describe a number of physical problems of current interest, such as the method of quantum trajectories developed by Robert E. Wyatt [7], or the study of effects such as the appearance of vortices in multichannel processes [8].

Among the formulations of the theory of quantum measurements in continuous time, quantum trajectory theory is very suitable for the introduction of measurement based feedback and closed loop control of quantum systems. In this paper we present such a construction in the concrete case of a two-level atom stimulated by a coherent, monochromatic laser. In particular, we show how fast feedback \`a la Wiseman and Milburn can be introduced in the formulation of the theory. Then, the spectrum of the free fluorescence light is studied and typical quantum phenomena, squeezing and sub-natural line-narrowing, are presented.

In this article we reconsider a version of quantum trajectory theory based on the stochastic Schrödinger equation with stochastic coefficients, which was mathematically introduced in the '90s, and we develop it in order to describe the non Markovian evolution of a quantum system continuously measured and controlled thanks to a measurement based feedback. Indeed, realistic descriptions of a feedback loop have to include delay and thus need a non Markovian theory. The theory allows to put together non Markovian evolutions and measurements in continuous time in agreement with the modern axiomatic formulation of quantum mechanics. To illustrate the possibilities of such a theory, we apply it to a two-level atom stimulated by a laser. We introduce closed loop control too, via the stimulating laser, with the aim to enhance the "squeezing" of the emitted light, or other typical quantum properties. Note that here we change the point of view with respect to the usual applications of control theory. In our model the "system" is the two-level atom, but we do not want to control its state, to bring the atom to a final target state. Our aim is to control the "Mandel Q-parameter" and the spectrum of the emitted light; in particular the spectrum is not a property at a single time, but involves a long interval of times (a Fourier transform of the autocorrelation function of the observed output is needed).

Dynamics of quantum systems which are perturbed by linear coupling to the reservoir stochastically can be studied in terms of quantum stochastic differential equations (for example, quantum stochastic Liouville equation and quantum Langevin equation). To work it out one needs definition of quantum Brownian motion. Since till very recent times only its boson version has been known, in present paper we demonstrate definition which makes possible consideration also for fermion Brownian motion.

A unified canonical operator formalism for quantum stochastic differential equations, including the quantum stochastic Liouville equation and the quantum Langevin equation both of the Itô and the Stratonovich types, is presented within the framework of Non-Equilibrium Thermo Field Dynamics (NETFD). It is performed by introducing an appropriate martingale operator in the Schrödinger and the Heisenberg representations with fermionic and bosonic Brownian motions. In order to decide the double tilde conjugation rule and the thermal state conditions for fermions, a generalization of the system consisting of a vector field and Faddeev-Popov ghosts to dissipative open situations is carried out within NETFD.

We examine some aspects of the continuous photodetection model for photocounting processes in cavities. First, we work out a microscopic model that describes the field-detector interaction and deduce a general expression for the Quantum Jump Superoperator (QJS), that shapes the detector's post-action on the field upon a detection. We show that in particular cases our model recovers the QJSs previously proposed ad hoc in the literature and point out that by adjusting the detector parameters one can engineer QJSs. Then we set up schemes for experimental verification of the model. By taking into account the ubiquitous non-idealities, we show that by measuring the lower photocounts moments and the mean waiting time one can check which QJS better describes the photocounting phenomenon.

A derivation of Belavkin's stochastic Schrödinger equations is given using quantum filtering theory. We study an open system in contact with its environment, the electromagnetic field. Continuous observation of the field yields information on the system: it is possible to keep track in real time of the best estimate of the system's quantum state given the observations made. This estimate satisfies a stochastic Schrödinger equation, which can be derived from the quantum stochastic differential equation for the interaction picture evolution of system and field together. Throughout the paper we focus on the basic example of resonance fluorescence.

Recent developments in quantum technology mean that is it now possible to manipulate systems and measure fermion fields (e.g. reservoirs of electrons) at the quantum level. This progress has motivated some recent work on filtering theory for quantum systems driven by fermion fields by Korotkov, Milburn and others. The purpose of this paper is to develop fermion filtering theory using the fermion quantum stochastic calculus. We explain that this approach has close connections to the classical filtering theory that is a fundamental part of the systems and control theory that has developed over the past 50 years.

Los indicadores clásicos de caos no son directamente aplicables en la mecánica cuántica ya que el concepto de trayectoria no existe en esta teoría. Por otro lado, a la luz del principio de correspondencia, la mecánica clásica es un caso límite de la mecánica cuántica y es de esperarse que en las propiedades cuánticas del sistema se manifiesten huellas del comportamiento caótico del correspondiente sistema clásico. Se describe el proceso de cuantización para sistemas integrables y no integrables, al igual que las funciones de Wigner y de Husimi, definidas en el espacio de fase clásico. Finalmente, se presenta un experimento reciente en física atómica que hace uso de átomos en estados Rydberg, los cuales son objetos que están en la frontera entre los dominios clásico y cuántico.

Recent developments in quantum technology mean that is it now possible to manipulate systems and measure fermion fields (e.g. reservoirs of electrons) at the quantum level. This progress has motivated some recent work on filtering theory for quantum systems driven by fermion fields by Korotkov, Milburn and others. The purpose of this paper is to develop fermion filtering theory using the fermion quantum stochastic calculus. We explain that this approach has close connections to the classical filtering theory that is a fundamental part of the systems and control theory that has developed over the past 50 years.

The momentum and position observables in an n-mode boson Fock space Γ(C n )
have the whole real line R as their spectrum. But the total number operator N has a discrete
spectrum Z + = {0, 1, 2, · · · }. An n-mode Gaussian state in Γ(C n ) is completely determined
by the mean values of momentum and position observables and their covariance matrix which
together constitute a family of n(2n + 3) real parameters. Starting with N and its unitary
conjugates by the Weyl displacement operators and operators from a representation of the
symplectic group Sp(2n) in Γ(C n ) we construct n(2n + 3) observables with spectrum Z + but
whose expectation values in a Gaussian state determine all its mean and covariance parameters.
Thus measurements of discrete-valued observables enable the tomography of the underlying
Gaussian state and it can be done by using 5 one mode and 4 two mode Gaussian symplectic
gates in single and pair mode wires of Γ(C n ) = Γ(C) ⊗n . Thus the tomography protocol admits
a simple description in a language similar to circuits in quantum computation theory. Such
a Gaussian tomography applied to outputs of a Gaussian channel with coherent input states
permit a tomography of the channel parameters. However, in our procedure the number of
counting measurements exceeds the number of channel parameters slightly. Presently, it is not
clear whether a more efficient method exists for reducing this tomographic complexity.
As a byproduct of our approach an elementary derivation of the probability generating
function of N in a Gaussian state is given. In many cases the distribution turns out to be
infinitely divisible and its underlying L ́evy measure can be obtained. However, we are unable
to derive the exact distribution in all cases. Whether this property of infinite divisibility holds
in general is left as an open problem.

A, classification theory of quantum stationary processes similar to the corresponding theory for classical stationary processes is presented, Our main result is the classification of those pairs of classical stationary processes that admit a joint boson Fock canonical representation.

We extend the Gaussian scale mixture model of dependent subspace source densities to include non-radially symmetric densities using Generalized Gaussian random variables linked by a common variance. We also introduce the modeling of skew in source densities and subspaces using a generalization of the Normal Variance-Mean mixture model. We give closed form expressions for subspace likelihoods and parameter updates in the EM algorithm.

We extend the Gaussian scale mixture model of dependent subspace source densities to include non-radially symmetric densities using Generalized Gaussian random variables linked by a common variance. We also introduce the modeling of skew in source densities and subspaces using a generalization of the Normal Variance-Mean mixture model. We give closed form expressions for subspace likelihoods and parameter updates in the EM algorithm.