In the the double-slit experiment, nonclassical paths are Feynman paths that go through both slits. Prior work with atom cavities as which-way detectors in the double-slit experiment has shown these paths to be experimentally inaccessible. In this paper, we show how such a setup can indeed detect nonclassical paths with 1% probability if one considers a different type of nonclassical path than previously investigated. We also show how this setup can be used to erase and restore the coherence of the nonclassical paths. Finally, we also show how atom cavities may be used to implement a exact measure of Born-rule violation [Quach, Which-way double-slit experiments and Born-rule violation, Phys. Rev. A 95, 042129 (2017)], which up until now has only been a formal construct.
In the the double-slit experiment, nonclassical paths are Feynman paths that go through both slits. Prior work with atom cavities as which-way detectors in the double-slit experiment has shown these paths to be experimentally inaccessible. In this paper, we show how such a setup can indeed detect nonclassical paths with 1% probability if one considers a different type of nonclassical path than previously investigated. We also show how this setup can be used to erase and restore the coherence of the nonclassical paths. Finally, we also show how atom cavities may be used to implement a exact measure of Born-rule violation [Quach, Which-way double-slit experiments and Born-rule violation, Phys. Rev. A 95, 042129 (2017)], which up until now has only been a formal construct.
In which-way double-slit experiments with perfect detectors, it is assumed that having a second detector at the slits is redundant because it will not change the interference pattern. We, however, show that if higher-order or nonclassical paths are accounted for, the presence of the second detector will have an effect on the interference pattern. Accounting for these nonclassical paths also means that the Sorkin parameter in triple-slit experiments is only an approximate measure of Born-rule violation. Using the difference between single and double which-way detectors, we give an alternative parameter which is an exact measure of Born-rule violation.
In a double slit interference experiment, the wave function at the screen with both slits open is not exactly equal to the sum of the wave functions with the slits individually open one at a time. The three scenarios represent three different boundary conditions and as such, the superposition principle should not be applicable. However, most well-known text books in quantum mechanics implicitly and/or explicitly use this assumption that is only approximately true. In our present study, we have used the Feynman path integral formalism to quantify contributions from nonclassical paths in quantum interference experiments that provide a measurable deviation from a naive application of the superposition principle. A direct experimental demonstration for the existence of these nonclassical paths is difficult to present. We find that contributions from such paths can be significant and we propose simple three-slit interference experiments to directly confirm their existence.
Quantum optics experiments are currently the most advanced techniques to under- stand, verify and simulate quantum phenomena. However, to access all the perfor- mance available in quantum states of light, one needs to address fundamental opera- tional limits. In quantum mechanics, the measurement strategy affects the quantum state; therefore, to access all the degrees of freedom available in the quantum states, one must implement the optimal feasible measurement. In this thesis, I investigate how to perform more precise measurements in optics, namely slit-interference and image resolution, by exploiting the quantum mechanical nature of light. A complete description of multi-slit interference must include nonclassical paths, Feyn- man paths that goes through two or more slits. Prior work with atomic interference in the double-slit experiment with cavities as which-way detectors, has shown these paths to be experimentally inaccessible. In this thesis I show how such a setup can detect nonclassical paths with 1% probability, if different nonclassical paths are in- cluded. I also show how this setup can be used to erase and restore the coherence of the nonclassical paths. In the same chapter I demonstrate how the same setup could implement an exact measure of Born-rule violation. And in the last part I debate about the figures of merit in the literature to test the Born-rule. During more than one century, there was a fundamental limit on image resolution; due to diffraction effects in finite detectors apertures, one cannot resolve two incoher- ent sources very close to each other, e.g. stars. In the last decade, the formalism of quantum information allowed new proposals for sub-diffraction limited resolution or super-resolving measurements. Nevertheless, these measurements are susceptible to misalignment. In this thesis, I suggest alternative measurement strategies to incorpo- rate misalignment in super-resolution imaging, showing that sub-diffraction limited resolution is still possible. The proposed measurements can be implemented using linear optical transformations and offer an advantage in the case of estimation and discrimination of two incoherent point sources allowing one to quantify the mitigat- ing effects of misalignment. Moreover, I propose a collective measurement strategy, on two or more photons, that estimates the separation between two incoherent point sources and is oblivious to misalignment. In an optics experiment, the quantum state verification relies on tomography measure- ments on copies of the prepared state. The error in tomography experiments is called confidence region, and it defines the region in which the quantum state is found with the desired probability. There are different methods to compute confidence regions; in this thesis, I analyze the capability of the known methods by resolving two nearby quantum states using a finite amount of measurement data. Actualmente, los experimentos en óptica cuántica son las técnicas más avanzadas para entender, verificar y simular los fenómenos cuánticos. Sin embargo, para obtener el máximo desempeño de los estados cuánticos, es necesario, primeramente, resolver algunas limitaciones teóricas fundamentales. En mecánica cuántica, el procedimiento de medición afecta el estado cuántico; por tanto, es necesario elaborar una estrategia de medición con posibilidad de ejecución práctica que permita acceder a todos los grados de libertad disponibles de los estados cuánticos. En esta tesis, yo uso la naturaleza cuántica de la luz para investigar, de forma teórica, cómo realizar mediciones ópticas más precisas, por ejemplo: medir con mejor precision los patrones de interferencia producidos por rendijas múltiples y obtener imágenes con mejor resolución. Una descripción detallada de la interferencia de múltiples rendijas debería incluir caminos no-clásicos, como los caminos de Feynman que pasan por dos o más rendijas. En trabajos anteriores, se demostró que la interferencia de átomos en la doble-rendija con cavidades ópticas, no es posible detectar los caminos no-clásicos. En esta tesis muestro cómo seria viable detectar con la misma configuración diferentes caminos no-clásicos con un 1% de probabilidad. También muestro, cómo se puede usar esta configuración para detectar o borrar la coherencia de los caminos no-clásicos. En el mismo capítulo, demuestro cómo esta configuración podría ser implementada para obtener una medida exacta de violación de la regla de Born. Finalmente, en la última parte discuto sobre las figuras de mérito en la literatura, para contrastar la regla de Born. Durante más de un siglo, fue asumida la existencia de un límite fundamental en la resolución de imágenes; dicho límite fue asociado a las aberturas finitas de los detectores que producen efectos de difracción e impiden distinguir dos fuentes de luz incoherentes muy cercanas entre sí, como por ejemplo, dos estrellas. En la última década, con el desarrollo de la información cuántica surgieron nuevas propuestas para medir fuentes de luz con una precisión mejorada. Sin embargo, estas medidas son sensibles a desalineamiento del instrumento de medida. En esta tesis, propongo nuevas estrategias de medición para incorporar el propio desalineamiento en la súperresolución de imágenes, y cuantifico los diferentes limites en resolución en presencia de tal desalineamiento. Las medidas que propongo pueden ser implementadas utilizando transformaciones ópticas lineales y ofrecen una ventaja en el caso de estimación y discriminación de dos fuentes de luz incoherentes. Además, propongo otra estrategia de medición colectiva, actuando en dos o más fotones, mediante la cual es posible estimar la separación entre dos fuentes de luz incoherentes, sin que la medida sea afectada por el desalineamiento. En un experimento de óptica, la verificación del estado cuántico se basa en la estadística de la tomografía del estado cuántico preparado. La incertidumbre en los experimentos de tomografía se denomina la región de confianza y define una región en que se encuentra el estado cuántico con la probabilidad deseada. Existen diferentes métodos para calcular las regiones de confianza; En esta tesis, analizo la capacidad de los métodos conocidos para resolver dos estados cuánticos cercanos utilizando una cantidad finita de datos de medición.
I argue that the marquis characteristics of the quantum-mechanical double-slit experiment (point detection, random distribution, Born rule) can be explained using Schroedinger's equation alone, if one takes into account that, for any atom in a detector, there is a small but nonzero gap between its excitation energy and the excitation energies of all other relevant atoms in the detector (isolated-levels assumption). To illustrate the point I introduce a toy model of a detector. The form of the model follows common practice in quantum optics and cavity QED. Each detector atom can be resonantly excited by the incoming particle, and then emit a detection signature (e.g., bright flash of light) or dissipate its energy thermally. Different atoms have slightly different resonant energies per the isolated-levels assumption, and the projectile preferentially excites the atom with the closest energy match. The toy model permits one easily to estimate the probability that any atom is resonantly excited, and also that a detection signature is produced before being overtaken by thermal dissipation. The end-to-end detection probability is the product of these two probabilities, and is proportional to the absolute-square of the incoming wavefunction at the atom in question, i.e. the Born rule. I consider how closely a published neutron interference experiment conforms to the picture developed here; I show how this paper's analysis steers clear of creating a scenario with local hidden variables; I show how the analysis steers clear of the irreversibility implicit in the projection postulate; and I discuss possible experimental tests of this paper's ideas. Hopefully, this is a significant step toward realizing the program of solving the measurement problem within unitary quantum mechanics envisioned by Landsman, among others.
I argue that the marquis characteristics of the quantum-mechanical double-slit experiment (point detection, random distribution, Born rule) can be explained using Schroedinger's equation alone, if one takes into account that, for any atom in a detector, there is a small but nonzero gap between its excitation energy and the excitation energies of all other relevant atoms in the detector (isolated-levels assumption). To illustrate the point I introduce a toy model of a detector. The form of the model follows common practice in quantum optics and cavity QED. Each detector atom can be resonantly excited by the incoming particle, and then emit a detection signature (e.g. bright flash of light) or dissipate its energy thermally. Different atoms have slightly different resonant energies per the isolated-levels assumption, and the projectile preferentially excites the atom with the closest energy match. The toy model permits one easily to estimate the probability that any atom is resonantly excited, and also that a detection signature is produced before being overtaken by thermal dissipation. The end-to-end detection probability is the product of these two probabilities, and is proportional to the absolute square of the incoming wavefunction at the atom in question, i.e. the Born rule. I consider how closely a published neutron-interference experiment conforms to the picture developed here; I show how this paper's analysis steers clear of creating a scenario with local hidden variables; I show how the analysis steers clear of the irreversibility implicit in the projection postulate; and I discuss possible experimental tests of this paper's ideas. Hopefully, this is a significant step toward realizing the program of solving the measurement problem within unitary quantum mechanics envisioned by Landsman, among others.
We argue that the double-slit experiment can be understood much better by considering it as an experiment whereby
one uses electrons to study the set-up rather than an experiment whereby we use a set-up to study the behaviour of electrons.
We also show that the concept of undecidability (like e.g. occurs in G\odel's theorem) can be used in an intuitive way to make sense of the double-slit experiment
and the quantum rules for calculating coherent and incoherent probabilities. We meet here a situation where the electrons always behave in a fully
deterministic way, while the detailed design of the set-up may render the question about the way they move through the set-up experimentally undecidable.
It is very important to make a distinction in quantum mechanics
between the determinism of nature (Einstein)
and the decidability of a question within an experimental set-up (Bohr).
The former is about the absolute truth of an answer to a yes or no question, and follows binary logic
(true or false),
the latter about what an experimental set-up
can decide and tell about the truth of that answer, and follows ternary logic (true, false or undecidable).
Binary and ternary logic are incompatible.
The viewpoints of Bohr and Einstein are thus operating on different levels and it is only by confusing these two levels
that these two viewpoints seem to be irreconcilable.
We show that the expression ψ1+ψ2 for the wave function of the double-slit experiment is numerically correct, but logically flawed. It has to be replaced in the interference region
by the logically correct expression ψ1′+ψ2′, which has the same numerical value as
ψ1+ψ2, such that ψ1′+ψ2′=ψ1+ψ2,
but with ψ1′=2ψ1+ψ2e4π=ψ1 and
ψ2′=2ψ1+ψ2e−4π=ψ2.
Here ψ1′ and ψ2′ are the correct
contributions from the slits to the total wave function ψ1′+ψ2′.
We have then $p = |\psi'_{1} + \psi'_{2}|^{2} = |\psi'_{1}|^{2} + |\psi'_{2}|^{2} = p_{1}+p_{2} $ such that the paradox
that quantum mechanics (QM) would not follow the traditional rules of probability calculus disappears.
The paradox is rooted in the wrong intuition that ψ1 and ψ2 would be the true physical contributions to
ψ1′+ψ2′=ψ1+ψ2 like in the case of waves in a water tank.
The solution proposed here is not {\em ad hoc} but based on an extensive analysis of the geometrical meaning of spinors within group representation theory
and its application to QM. Working further on the argument one can even show that an interference pattern is the only way to satisfy simultaneously
two conditions: The condition obeying binary logic (in the spirit of Einstein) that the electron has only two mutually exclusive options to get to the detector (viz. going through slit S1 or going through slit S2) and the condition obeying ternary logic (in the spirit of Bohr)
that the question which one of these two options the electron has taken is experimentally undecidable.
A very important element in the analysis is the problem of the non-existence of {\em common} probability distributions for different experimental set-ups.
And this is a recurrent theme in other situations that are felt as paradoxical.
The CHSH inequality used in the experiments of Aspect {\em et al.}
clashes with quantum mechanics because it
is based on the assumption that there would exist a {\em common} probability distribution for the
hidden variables in the different experiments that have to be made to determine the various quantities that intervene in the inequality.
That common probability distribution for measured quantities does not exist is well known from quantum mechanics itself (because the operators that come into play do not commute and therefore do not have common eigenstates, i.e.
common probability amplitudes). But this non-existence of common probability distributions applies also to certain variables in the set-ups of the different experiments and the latter fact
can also be explained by purely classical reasoning.