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Coherence in Classical Electromagnetism and Quantum Optics

Hanne-Torill Mevik

Published Jan 1, 2009• 2 citations
OpenAlex

Abstract

This thesis is a study of coherence theory in light in classical\nelectromagnetism and quantum optics. %The coherence is quantified\nSpecifically two quantities are studied: The degree of first-order\ntemporal coherence, which quantifies the field-field coherence, and\nthe degree of second-order coherence, quantifying the intensity-intensity\ncoherence. In the first part of the thesis these concepts are applied to classical\nelectric fields; to both the ideal plane wave and to chaotic light.\nWe then study how they can be measured using two interferometer\ntechnologies from optical astronomy, specifically with the\nMichelson stellar interferometer and the intensity interferometer.\n\nIn the second part we define the\nquantum degrees of first- and second-order coherence. These are calculated\nfor light in a quantum coherent state, in a Fock state and for light\nin a mixed thermal state. The results for the coherent state and\nthe thermal state are found to be analogous to\nthose obtained for the ideal plane wave and chaotic light, respectively,\nfrom the classical coherence theory seen in the first part.\n\nWe proceed to investigate the properties of the three-level laser with\nthe aim of showing that far above threshold it develops similar photon\nstatistics and values for the degrees of first- and second-order coherence,\nto light in a coherent state. The mechanism of phase-drift in the laser\nis also looked into. Subsequently the Mølmer-model is discussed, where it is\ndemonstrated that the coherent state is not a necessary construct, but\nmerely a convenient one, in describing phenomena in quantum optics.

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This thesis is a study of coherence theory in light in classical\nelectromagnetism and quantum optics. %The coherence is quantified\nSpecifically two quantities are studied: The degree of first-order\ntemporal coherence, which quantifies the field-field coherence, and\nthe degree of second-order coherence, quantifying the intensity-intensity\ncoherence. In the first part of the thesis these concepts are applied to classical\nelectric fields; to both the ideal plane wave and to chaotic light.\nWe then study how they can be measured using two interferometer\ntechnologies from optical astronomy, specifically with the\nMichelson stellar interferometer and the intensity interferometer.\n\nIn the second part we define the\nquantum degrees of first- and second-order coherence. These are calculated\nfor light in a quantum coherent state, in a Fock state and for light\nin a mixed thermal state. The results for the coherent state and\nthe thermal state are found to be analogous to\nthose obtained for the ideal plane wave and chaotic light, respectively,\nfrom the classical coherence theory seen in the first part.\n\nWe proceed to investigate the properties of the three-level laser with\nthe aim of showing that far above threshold it develops similar photon\nstatistics and values for the degrees of first- and second-order coherence,\nto light in a coherent state. The mechanism of phase-drift in the laser\nis also looked into. Subsequently the Mølmer-model is discussed, where it is\ndemonstrated that the coherent state is not a necessary construct, but\nmerely a convenient one, in describing phenomena in quantum optics.
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In the history of optical science, we were satisfied with the measurement of mean intensity of light, namely the measurement of first-order coherence of light, for optical studies and observations, until the middle of 20th century when two astrophysicists Hanbury Brown and Twiss (HBT) introduced their “intensity interferometer”. “Correlation” emphasizes its statistical nature, while “coherence” emphasizes its interference nature. The chapter extends the concept of second-order coherence to Nth-order coherence (N > 2) and analyzes a few simple measurements on the third-order temporal and spatial coherence of thermal field. It also calculates the second-order temporal coherence function and the second-order spatial coherence function of thermal field by assuming idealized photodetectors and correlation measurement electronics.
Introduction In this chapter we will put the concept of coherence on a more mathematical basis. This will involve the formal definition of a number of functions that are used to describe the coherence properties of optical fields. These include the analytic signal, various correlation functions, and the degree of coherence for describing both temporal and spatial coherence. We shall see that the degree of temporal coherence is quantitatively related to the lineshape function and that the degree of spatial coherence between two points is determined by the size, intensity distribution, and location of an illuminating source. We will use the wave equation, and special solutions to the wave equation called Green's functions, to show how spatial coherence varies from point to point. The chapter will conclude with a discussion of how the intensity fluctuations of a source depend on its coherence properties, and we will examine a specific scheme, namely the Hanbury Brown–Twiss experiment, by means of which this relationship is studied. This discussion will involve a discussion of “photon statistics,” namely the time variation of the “detection” of photons from a source. In a quantum-mechanical context, square-law detectors respond to these quantized excitations of the optical field, which we call photons. In classical coherence theory it is advantageous to represent the real electromagnetic field by a complex quantity, both for its mathematical simplicity and because it serves to emphasize that coherence theory deals with phenomena that are sensitive to the “envelope” or average intensity of the field.
Interferometers provide a highly sensitive means to investigate and exploit the coherence properties of light in metrology applications. However, interferometers come in various forms and exploit different properties of the optical states within. In this paper, we introduce a classification scheme that characterizes any interferometer based on the number of involved nonlinear elements by studying their influence on single-photon and photon-pair states. Several examples of specific interferometers from these more general classes are discussed, and the theory describing the expected first-order and second-order coherence measurements for single-photon and single-photon-pair input states is summarized and compared. These theoretical predictions are then tested in an innovative experimental setup that is easily able to switch between implementing an interferometer consisting of only one or two nonlinear elements. The resulting singles and coincidence rates are measured in both configurations and the results are seen to fit well with the presented theory. The measured results of coherence are tied back to the presented classification scheme, revealing that our experimental design can be useful in gaining insight into the properties of the various interferometeric setups containing different degrees of nonlinearity.
We explore the concepts of coherence and entanglement as they apply to both the classical and quantum natures of light. In the classical domain, we take inspiration from the tools and concepts developed in foundational quantum mechanics and quantum information science to gain a better understanding of classical coherence theory of light with multiple degrees of freedom (DoFs). First, we use polarization and spatial parity DoFs to demonstrate the notion of classical entanglement, and show that Bell's measure can serve as a useful tool in distinguishing between classical optical coherence theory. Second, we establish a methodical yet versatile approach called 'optical coherency matrix tomography' for reconstructing the coherency matrix of an electromagnetic beam with multiple DoFs. This technique exploits the analogy between this problem in classical optics and that of tomographically reconstructing the density matrix associated with multipartite quantum states in quantum information science. Third, we report the first experimental measurements of the 4 x 4 coherency matrix associated with an electromagnetic beam in which polarization and a spatial DoF are relevant, ranging from the traditional two-point Young's double slit to spatial parity and orbital angular momentum modes. In the quantum domain, we use the modal structure of classical fields to develop qubits and structure Hilbert spaces for use in quantum information processing. Advancing to three-qubit logic gates is an important step towards the success of optical schemes for quantum computing. We experimentally implement a variety of two- and three- qubit, linear and deterministic, single-photon, controlled, quantum logic gates using polarization and spatial parity qubits. Lastly, we demonstrate the implementation of two-qubit single-photon logic using polarization and orbital angular momentum qubits.