Quantum optics offers a huge variety of exciting phenomena. Many of them are still in their infancy and especially when it comes to implementing devices using these effects for more than a proof of principle demonstration still many things have to be investigated and understood. In this thesis I discuss the role of correlations in some areas of quantum optics and in some cases compare it to classical optics. Four papers form the core of the thesis. In the first paper, I propose a new measure for entanglement. This measure is based on correlations between two states. I show, how this measure relates to another measure, the concurrence. It turns out that the measure is a bijective map of the concurrence for a pure state of two qubits. I motivate why the new measure is useful if one wants to implement it experimentally. I discuss its behaviour for the case of two qubits and show its properties when dealing with pure and with mixed states. The second paper extends the result of the first one to the case where one has higher-dimensional states than qubits. In the third paper I look at phase super-resolution. I show that it can be interpreted as a purely classical effect and I analyse what is needed and what is not needed to achieve it. Specifically, I show that quantum correlations in terms of entanglement is not needed to demonstrate phase super-resolution. By doing so I propose how one could achieve arbitrarily high phase super-resolution. Finally, the last paper looks at the efficiency of quantum lithography and quantum imaging. It shows, that some basic assumptions in the original proposals of quantum lithography seems unfounded and that, as a consequence, the efficiency is poor. I give formulæ for the explicit scaling behaviour when changing the number of photons in a mode or when changing the number of pixels. The effect of the results on the future of quantum lithography is discussed as well.
Quantum optics offers a huge variety of exciting phenomena. Many of them are still in their infancy and especially when it comes to implementing devices using these effects for more than a proof of principle demonstration still many things have to be investigated and understood. In this thesis I discuss the role of correlations in some areas of quantum optics and in some cases compare it to classical optics. Four papers form the core of the thesis. In the first paper, I propose a new measure for entanglement. This measure is based on correlations between two states. I show, how this measure relates to another measure, the concurrence. It turns out that the measure is a bijective map of the concurrence for a pure state of two qubits. I motivate why the new measure is useful if one wants to implement it experimentally. I discuss its behaviour for the case of two qubits and show its properties when dealing with pure and with mixed states. The second paper extends the result of the first one to the case where one has higher-dimensional states than qubits. In the third paper I look at phase super-resolution. I show that it can be interpreted as a purely classical effect and I analyse what is needed and what is not needed to achieve it. Specifically, I show that quantum correlations in terms of entanglement is not needed to demonstrate phase super-resolution. By doing so I propose how one could achieve arbitrarily high phase super-resolution. Finally, the last paper looks at the efficiency of quantum lithography and quantum imaging. It shows, that some basic assumptions in the original proposals of quantum lithography seems unfounded and that, as a consequence, the efficiency is poor. I give formulæ for the explicit scaling behaviour when changing the number of photons in a mode or when changing the number of pixels. The effect of the results on the future of quantum lithography is discussed as well.
In this dissertation we begin with a brief introduction to quantum optics concentrating on the topics of the noise of quantum optical states, quantum estimation theory, quantum interferometry and the atom-field interaction. This background is necessary for understanding the discussions in later chapters. In particular, quantum interferometry, which is optical interferometry when the light source is a quantum mechanical state, plays a central role in this dissertation. In Chapter 2 we discuss the phase estimation sensitivity of quantum metrology when photon loss is present. In Chapter 3 we extend the discussion to include the phase fluctuation of the system caused by the environment. We model our metrological system with the Mach-Zehnder interferometer (MZI) and use a light field in the symmetric number-path entangled state as the source. In both chapters we use the parity operator as the detection scheme and show that it is optimal under pure dephasing. In Chapter 4 we discuss the application of quantum optical states in remote sensing and propose a new scheme for a quantum radar. Again, our scheme consists of a MZI and a coherent light source. It is shown that using only coherent states of light and quantum homodyne detection, super-resolving ranging and angle determination are achievable. Chapter 5 is devoted to the generation of a super-resolving single-photon number-path entangled state which may serve as a proof-of-principle prototype for quantum lithography. The repeated implementation of MZIs is shown to be able to remove photons coherently from both modes of a symmetric number-path entangled state with arbitrarily high photon number. Lastly, in Chapter 6 we introduce the phenomenon known as polarization self-rotation and discuss its potential in generating a squeezed vacuum state, which has a huge impact in quantum interferometry.
This thesis fathoms the possibilities and limitations of measurements in quantum optics. After a short introduction to the general aspects of measurements in quantum mechanics, it presents three examples of a measurement process in concrete applications to quantum optics. For one, while operators describing quantities of interest might have a simple theoretical form, it is often hard to actually implement a measurement of their expectation values experimentally. This situation is encountered in the first part which is devoted to the measurement of quantum statistical properties of a light beam. In contrast to classical optics, these statistics are not fully determined by first- and second-order correlations. Instead, higher-order correlations, usually described by non-Hermitian operators, are necessary to characterize the quantum state completely. A particular example is the N-photon Fock state, for which only a measurement of the Nth-order correlations allows for a full characterization. Here, we present a new method how to measure an important class of Nth-order correlations for arbitrary N in a single spatial mode with two polarizations with limited experimental resources. The second part is again centered around the realization of specific measurement operators for photon correlation measurements. However, in contrast to the previous chapter, these measurements are not an end in their own, but are designed to create specific quantum states as a consequence of the measurement process: A suitable measurement of photons emitted by a number of atoms allows to transfer the atoms into long-lived entangled states. We find that with this method it is possible to entangle two remote atomic qubits to an arbitrary and well-defined degree. Furthermore, we show how to generate two families of quantum states in an arbitrary number of remote atomic qubits: one family consists of all -- symmetric and non-symmetric -- total angular momentum eigenstates in N remote qubits, the other is a family of cluster states. It is also systematically investigated whether the technique of quantum state engineering by projective measurements allows for the creation of any arbitrary state. We find explicit solutions for the case of two and three qubits, while it is shown that no solution exist for N > 6. We conclude the section by proving that with some changes in the setup, this measurement-based quantum state engineering technique allows to create the same states among the scattered photons themselves rather than among the emitting atoms. The last part is devoted to the amount of information one is able to gain about a given system when performing a measurement. The corresponding limits of information are investigated at the example of the duality that arises in a two-way interferometer between the visibility of the interference pattern and the knowledge about the path of the interfering object. We show that there exist correlations between the phase difference the interfering object acquires on its way through the interferometer and the amount of which-way information retrievable from certain observables of the which-way detector. In this way, the which-way information becomes a phase-dependent quantity. In particular, we find that for certain values of the phase shift, the amount of extractable which-way information can be larger than allowed by phase-independent measurement. This property is put to use to find the optimal observable for the which-way detector readout in dependence on the phase of the interfering object. Following this strategy, we are able to gain more which-way information than previously thought possible.
This thesis summarizes investigations on quantum correlations in the fluorescence light of single trapped atoms. The results found have applications in the broad field of quantum information science: it is shown how to overcome classical imaging boundaries using a new developed scheme of quantum imaging, how to project and engineer correlated quantum states in the ground states of multi-level atoms by detecting their fluorescence light and how to prove the quantum nature of these correlations using fundamental tests of quantum information science. Without the need of further introduction to these topics it is interesting to note at this early point that all correlations observed are indeed of quantum nature and show in particular entanglement features, while initially the system under consideration is entirely uncorrelated and the only intervention comes along with the measurement process itself. Therefore, this thesis is embedded into a current field of research activities in the area of quantum information science which is commonly referred to as measurement-induced entanglement. In the first part of this thesis, the correlated photon signal as measured in the fluorescence light of single-photon emitters is at the focus of investigations. It is shown that this correlated signal can be used to overcome classical signals in terms of resolution when applied to image processing. In the second part, the fluorescence signal of a system of multi-level atoms is used to project correlated quantum states into the long-lived ground levels of the atoms. Hereby, a broad variety of entangled quantum states can be generated, ranging from symmetric states like the W-states and GHZ-states to arbitrary total angular momentum eigenstates. Finally, the last part of this thesis takes a closer look at the fundamental physical processes inherent in all schemes and applications investigated throughout the thesis. There, amongst others we investigate Bell-type inequalities that are able to prove the quantum nature of the correlations.
The insights offered by quantum mechanics to the field of optical metrology are many-fold, with non-classical states of light themselves used to make sensors that surpass the sensitivity of sensors using classical states of light. Unfortunately, this advantage, referred to often as "super-sensitivity" is notoriously fragile, and even the slightest experimental imperfections may greatly reduce the efficacy of the non-classical sensors, sometimes completely obviating their advantage. In my thesis I have shown that the performance of an otherwise ideal two-photon interferometer, which exploits entanglement between photons to make super-sensitive measurements of phase, is crippled by the slightest introduction of decoherence between modes of the interferometer. I have shown further that such drastic reduction in sensitivity can also appear in classical measurement problems, specifically that the recently developed methods of estimating the separation between a pair of point sources are rendered less effective when the ideal assumption of complete spatial incoherence is relaxed. Towards overcoming these and other issues, I have designed new configurations that use ancillary optical degrees of freedom, a tool-set that has recently garnered much interest in the field of quantum optics. In the context of two-photon interferometry, I have shown that by coupling polarization to the spatial-structure of the two photon state used to probe phase it is possible to obviate the need for a reference phase, even in the context of decoherence and imperfections in the interferometer. In the context of two-point resolution, I have developed an anisotropic imaging system that performs the function of an image-inversion interferometer and is inherently stable, offering an attractive implementation of recently developed methods of sub-Rayleigh imaging. I have further shown both theoretically and experimentally that the same anisotropic image-inversion interferometer is useful in measuring spatially encoded phases, both in the context of classical illumination as well as quantum-aided two-photon super-sensing. In both cases, the ability to perform interferometric measurements of the spatial structure of an electric field without splitting beam paths forms a bridge between conception and implementation of precision-sensing measurement strategies. Finally, I have shown that binary interferometric method based on the common-path anisotropic imaging system that I introduced, are able to measure both phase gradients and transverse beam tilts with a sensitivity beating conventional systems that are used both commercially and in research laboratories.
In this thesis, I present experimental and theoretical work which assesses and develops a range of tools that are required for performing quantum information processing, particularly in photonic systems. I investigate the three degrees of freedom of a single photon—its polarisation, spatialmomentum and time-frequency distributions. For polarisation, I show that using wave plates to implement arbitrary, single-qubit rotations is more complicated than is commonly appreciated. In the spatial-momentum and time-frequency domains, I develop ways to perform tomographic analysis of quantum states, and I report the first demonstrations of these techniques. In the time-frequency domain, the tomography technique utilises entanglement in the photon polarisation as a resource to store and provide access to the time-frequency information. In my first two experiments, I use spontaneous parametric down-conversion to produce entanglement between pairs of single photons in all three degrees of freedom. I demonstrate the first characterisation of entanglement in spatial modes and the time-frequency domain, the first quantitative measurement of entangled qutrit states, and the highest quality entangled states yet measured in both polarisation and spatial modes. I also report the first realisation of complete hyperentanglement, and a full, black-box tomography of a 36-dimensional two-photon state—the largest system to be characterised in this way to date. In my final experiment, I model and implement a new architecture for a controlled-Z gate which is much simpler to align than previous implementations. This gate requires only one non-classical interference condition, the visibility of which is the main limitation to its performance. I show that the gate operates effectively as a means of both creating entanglement and discriminating between the four elements in a basis of maximally entangled, Bell-type states. Indeed, its observed performance as a Bell analyser would be sufficient to build a quantum state teleporter which would guarantee that the recipient would be left with a better copy of an unknown input state than any eavesdropper. In a series of numerical simulations, I investigate some of the practicalities that arise when using tomographic reconstruction techniques, including how to estimate errors, which measurements to make and what is actually the optimal reconstruction. In particular, I show that tomographies perform better when based on the results from over-complete sets of measurements. Finally, I discuss the important issue of how to compare two processes, particularly when trying to assess the quality of a measured process by comparing it to some expected ideal. Judging possible candidate measures against a set of experimentally and theoretically motivated
This thesis divides very naturally into three chapters, reflecting the three separate areas I have worked in throughout my PhD studies. During my PhD, I published two papers, one relating to the work in the first chapter, and one to that of the second. At the time of submission, the project which the third chapter relates to was still ongoing, with plans for a future publication. The first chapter discusses work done on the real space renormalisation group (RG) for Ising and Potts models in 2 dimensions. Taking inspiration from Hasenbusch's work, a new framework for carrying out the RG is developed, its computational implementation discussed in some detail, as well as the results for a variety of systems and the implications of these results. The numerical scaling of the procedure and the consequences of this for future work are also covered. The results documented in chapter two are purely theoretical and presented in closed form. The research conducted is to do with properties of free interfaces. The second chapter is by and large critical of previous assumptions which have been made about the so-called wave vector dependent, in order to attempt to experimentally measure it and use these measurements to make further predictions. Using some simple toy models, many of these assumptions are shown to be false. In a sense, the goal of the research presented in the chapter two, is not to motivate further research, but to dissuade research in a direction we consider to be misguided, due to the faulty assumptions it is based on. The third chapter covers a small subset of a project concerned with studying correlations within the so-called Abelian Manna Model. The majority of the project involves (computational) Monte Carlo simulations of the dynamics of such systems, but as these results are not ready to present at the time of writing, the chapter is mainly concerned with some analytical results which were derived in order to validate our models for small systems, explain certain quirky phenomena arising from our simulations, and help quantify errors. Finally, there is an appendix which expands upon various topics from the first two chapters.