The following work is motivated by the conceptual problems associated with the wave-particle duality and the notion of the photon. Two simple classical models for radiation from individual emitters are compared, one based on sines with random phasejumps, another based on pulse trains. The sum signal is calculated for a varying number of emitters. The focus lies on the final signal's statistical features quantified by means of the temporal coherence function and the temporal coherence length. We show how these features might be used to experimentally differentiate between the models. We also point to ambiguities in the definition of the temporal coherence length.
The following work is motivated by the conceptual problems associated with the wave-particle duality and the notion of the photon. Two simple classical models for radiation from individual emitters are compared, one based on sines with random phasejumps, another based on pulse trains. The sum signal is calculated for a varying number of emitters. The focus lies on the final signal's statistical features quantified by means of the temporal coherence function and the temporal coherence length. We show how these features might be used to experimentally differentiate between the models. We also point to ambiguities in the definition of the temporal coherence length.
The following work is motivated by the conceptual problems associated with the wave‐particle duality and the notion of the photon. Two simple classical models for radiation from individual emitters are compared, one based on sines with random phasejumps, another based on pulse trains. The sum signal is calculated for a varying number of emitters. The focus lies on the final signal’s statistical features quantified by means of the temporal coherence function and the temporal coherence length. We show how these features might be used to experimentally differentiate between the models. We also point to ambiguities in the definition of the temporal coherence length.
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.
The following Master’s thesis is concerned with several aspects of the wave-particle duality of light. It is loosely divided in three parts. In the first part we consider historical, theoretical and experimental aspects of the duality problem. We explain how the notion of duality has developed through the last 400 years. We discuss theoretical underpinnings of the duality emodied by Maxwell’s electromagnetic theory, quantization of electromagnetic modes, Fock’s states and coherent states. We critically review several experiments which serve to demonstrate the corpuscular or undulatory behaviour of light and matter; in particular we present how the photoelectric effect and the Compton effect can be explained using the undulatory model, and we critically review Grangier, Roger and Aspect correlation experiment.\n\nIn the second part we describe two illustrative experiments on the duality of light conducted at Quantum Optics Laboratory at University of Oslo. The results of the experiment allow us to discuss how coincidence measurements can be used to exhibit the corpuscular behaviour of light, and how Mach-Zender interferometry performed at very low intensity can be used to exhibit the undulatory behaviour at\nthe (assumed) single-photon level. In addition, in the second part we review elements of theories closely associated with the experiment and the experimental setup: optical coherence, photocount and photon\nstatistics, beam splitter models and Gaussian beams. A proposition for extending the semiclassical model is given, and shortcomings of the present beam splitter models are discussed.\n\nIn the third part of the thesis we consider first Afshar’s experiment and some of the critical response that it has been met with. Then we discuss how the wave-particle duality is to be understood in the\nstandard interpretation of quantum mechanics, and how it could possibly be explained using either an alternative model for light or an alternative interpretation of quantum mechanics, and what difficulties such explanations present.
We examine the problem of determining the phase difference between two optical fields, first for classical and later for quantum fields, by reference to two simple measurement schemes that yield the sine and/or cosine of the phase difference between classical fields. We show that certain difficulties exist even within the framework of semiclassical radiation theory when the field is very weak, and particularly when amplitude and phase fluctuations are correlated. We find that a clear distinction has to be made between the measured values of the sine or cosine and the values that can be inferred from a series of repeated measurements. A corresponding distinction can be made also for a quantum field, although the interpretation is not the same. The dynamical variables chosen to represent the cosine and sine that emerge from the discussion of the measurement schemes commute when the sine and cosine are obtained together, but not when the measurement yields one or the other. These sine and cosine operators have well-defined values only when there is a large dispersion of the photon number. We arrive at expressions for the moments of the measured and of the inferred sines and cosines that differ from most previous treatments. The expressions are applied to optical fields in several different quantum states. Only for the Fock state and for the so-called phase state, which was treated recently at some length by Pegg and Barnett [Phys. Rev. A 39, 1665 (1989)], do the measured and the inferred moments coincide. Our analysis of the problem of phase measurement leads to the conclusion that the appropriate dynamical variables for the measured sine and cosine depend on the measurement scheme, and that different schemes correspond to different operators.
The spectral and temporal response of a set of N quantum emitters embedded in a photonic cavity is studied. Quantum mechanically, such systems can be described by the Tavis-Cummings (TC) model of N two-level systems coupled to a single light mode. Here we compare the full quantum solution of the TC model for different numbers of quantum emitters with its semiclassical limit after a pulsed excitation of the cavity mode. Considering different pulse amplitudes, we find that the spectra obtained from the TC model approach the semiclassical one for an increasing number of emitters N. Furthermore, they match very well for small pulse amplitudes. While we observe a very good agreement in the temporal dynamics for photon numbers much smaller than N, considerable deviations occur in the regime of photon numbers similar to or larger than N, which are linked to collapse and revival phenomena. Wigner functions of the light mode are calculated for different scenarios to analyze the quantum state of the light field. We find strong deviations from a coherent state even if the dynamics of the expectation values are still well described by the semiclassical limit. For higher pulse amplitudes, Wigner functions similar to those of Schr\"odinger cat states between two or more quasicoherent contributions build up.
We study angular correlations of the radiation emitted by spatially extended sets of atoms. Such correlations result from the fact that the photons are identical quantum particles. We show that the symmetry (or the antisymmetry) alone of the wave function for identical, indistinguishable particles always leads to significant angular correlations and that in full agreement with our intuition, bosons tend to travel in the same direction while fermions tend to travel in opposite directions. In order to quantify these intuitive notions, we introduce---as a measure of angular correlations---the average value of the cosine of the angle between the directions of the momenta of two particles. The angular correlation coefficient is calculated for simple, model wave functions and the results are compared with those obtained from an exactly soluble model describing radiating harmonic oscillators. We find that even though the interaction modifies the exact values of the angular correlation coefficient, the gross features can be obtained from quantum statistics, i.e., from the symmetry of the many-photon wave functions. Moreover, the angular correlations as measured by the average cosine exhibit an almost universal character, at least for a small number of emitting atoms. Finally, we compare angular correlations obtained in quantum electrodynamics with the correlations resulting from the phenomenologically introduced randomness of the phases in the classical description of the electromagnetic field.