We consider a fully quantized model of spontaneous emission, scattering and absorption, and study propagation of a single photon from an emitting atom to a detector atom both with and without an intervening scatterer. We find an exact quantum analogue to the classical complex analytic signal describing an electromagnetic wave scattered by a medium of charged oscillators. This quantum signal exhibits classical phase delays. We define a time of detection which, in the appropriate limits, exactly matches the predictions of a classically defined delay for light propagating through a medium of charged oscillators. The fully quantized model provides a simple, unambiguous and causal interpretation of delays that seemingly imply speeds greater than c in the region of anomalous dispersion.
We consider a fully quantized model of spontaneous emission, scattering and absorption, and study propagation of a single photon from an emitting atom to a detector atom both with and without an intervening scatterer. We find an exact quantum analogue to the classical complex analytic signal describing an electromagnetic wave scattered by a medium of charged oscillators. This quantum signal exhibits classical phase delays. We define a time of detection which, in the appropriate limits, exactly matches the predictions of a classically defined delay for light propagating through a medium of charged oscillators. The fully quantized model provides a simple, unambiguous and causal interpretation of delays that seemingly imply speeds greater than c in the region of anomalous dispersion.
We introduce a simple model for electromagnetically induced transparency in which all fields are treated quantum mechanically. We study a system of three separated atoms at fixed positions in a one-dimensional multimode optical cavity. The first atom serves as the source for a single spontaneously emitted photon; the photon scatters from a three-level Lambda-configuration atom which interacts with a single-mode field coupling two of the atomic levels; the third atom serves as a detector of the total transmitted field. We find an analytical solution for the quantum dynamics. From the quantum amplitude describing the excitation of the detector atom we extract information that provides exact single-photon analogs to wave delays predicted by semi-classical theories. We also find complementary information in the expectation value of the electric field intensity operator.
Dielectric materials in which both the relative permittivity, ε, and the relative permeability, μ, are negative exhibit striking optical properties, which are a consequence of the fact that the effective index of refraction in such materials is negative. Most studies of the propagation of electromagnetic fields in these media have concentrated on classical fields with well-defined phases. In contrast, I use simple fully-quantized models to illustrate how some of the striking properties of negative-index media are manifested in the propagation of single, spontaneously emitted, photons. Single-photon quantum fields do not have an absolute phase, but the examples in this paper demonstrate a quantum analog of the negative phase accumulated by classical fields as photons propagate through negative-index media. The single photons also exhibit classical reflection and transmission coefficients, as well as the expected group velocity.
The puzzles of quantum mechanics are: 1. what is the medium for the photon and for the electron wave? i.e. what is waving? 2. particle interference pattern 3. observer effect. 4. entanglement. In Quantum Erasure and Double-Slit experiments, how does the emitter know to direct the photon two both slits or only to one slit? And how does the detector know where to detect the photons to form an interference pattern and not a bell-shaped (Gaussian) pattern or vice versa? This paper proposes a few, closely related, simple and subtle laws of nature governing quantum phenomena. Some of the findings in this paper are: 1. A photon is emitted in two steps: the creation of photon path and the emission of the photon 2. Although (almost) all of the energy of the photon is localized in space, the photon extends to infinity. An extremely small energy of the photon, which we call the 'precursor wave' in this paper, extends to infinity. The instant the photon is emitted, its precursor wave creates a path to infinity; it appears instantaneously in all its path to infinity, with infinitely small energy density. The precursor wave serves as the 'highway' for the propagation of the main energy packet. However, to conserve the principle of causality, we assume that the tip of the photon propagates at a speed billions times the speed of light. Thus, there are two speeds associated with the photon: the speed we know as c and the speed at which the photon length increases. 2. A photon will be emitted only towards an absorber. A photon is emitted by the interaction of the emitter and a detector. It is obvious that a photon will not be emitted if there is no emitter. Nor will a photon be emitted if there is no detector . Once a photon is emitted, however, it exists objectively, autonomously. The photon is emitted only after its path is determined by the interaction of the emitter and the absorber, at the instant of emission. 3. The photon energy will be concentrated where there is higher wave amplitude, i.e higher intensity of oscillation of electric and magnetic fields. This results in 'dragging' of the electromagnetic energy by the wave, hence eliminating the need for any medium. It is also proposed that single photon interference is a quantum phenomenon whereas two photon interference is a classical phenomenon. Similar laws hold for the electron. The 'medium' for the electron wave is the electron (the electron mass density field) itself. Where there is higher intensity of electron wave/oscillation, there will be higher electron mass density. This means that the electron wave 'drags' the electron with itself. Therefore, no exotic 'medium' is required for the photon and for the electron wave.
Kinetic equations have played a significant role in optics since the introduction of the radiative transport equation (RTE) by Schuster in the early 20th century. Now they are applied in many scientific fields such as atmospheric physics and biomedical imaging to understand how light interacts with a complex medium. In this thesis we investigate how these equations arise in: 1. models of classical optics in an inhomogeneous disordered scattering medium 2. in fully quantum real space models with a random distribution of atoms In certain spatial and temporal regimes it has been empirically observed that light interacting with a highly disordered scattering medium may be described by a radiative transport equation with a constant scattering coefficient. One way to derive such equations from microscopic principles is to model the medium as a random field which is statistically homogeneous. We show that in a quasi-homogeneous media, where statistical correlations vary rapidly on small length scales and slowly on large length scales, light obeys a radiative transport equation with variable scattering coefficient. In the second part of this thesis we introduce a new model in quantum optics which allows us to treat a quantized electromagnetic field coupled to a medium of two level atoms, both in real space. The rest of the chapter focuses on the interaction of atoms with a field consisting of at most one photon. First we recover a classical result due to Wigner and Weisskopf on the rate of single atom spontaneous emission. Next we show that in a statistically homogeneous distribution of two level atoms the field and atomic amplitude satisfy linear kinetic equations. Finally we show that at sufficiently long times and large distances these amplitudes are asymptotically governed by diffusion equations which may be solved analytically. In the final chapter we use the previous model to analyze a system in which there can be at most two photons. This allows us to study the time evolution of states that initially contain two entangled photons, a phenomenon which has no classical analogue. We show that the various probability amplitudes satisfy linear two particle kinetic equations which again may be asymptotically evaluated at large distances and long times.
When a single photon propagates through a cloud of two-level atoms, part of its energy is temporarily stored as a collective atomic excitation. This raises a fundamental question: is it possible to meaningfully quantify how long the photon spends in the form of an atomic excitation? If so, does this duration depend on whether the photon is ultimately transmitted through the cloud or scattered by the atoms? These questions motivated recent experimental work by some of us [Sinclair et al., PRX Quantum 3, 010314 (2022)], but until now there has been no theoretical prediction for these conditional durations. We answer these questions by extending the concept of a dwell time from quantum tunneling to light–matter interactions, introducing an experimentally measurable “atomic excitation time.” Using the weak-value formalism combined with quantum trajectory theory, we derive analytical expressions for this time conditioned on the photon being transmitted through the cloud or scattered into a side mode. We show that, for transmitted photons, this time is equal to the (spectrally averaged) group delay, even in regimes where the group delay is negative. For scattered photons, the time is given by the sum of an ensemble-averaged group delay and an always-positive Wigner time delay, well known in elastic scattering theory. Finally, we present a toy model that explains how such negative dwell times can arise as a result of quantum interference. This work provides new insight into the complex and surprising histories of photons traveling through absorptive media.
In this work, we present a quantum description of electromagnetic waves propagating through time-dependent homogeneous nondispersive linear media without sources. By using the Coulomb gauge, we show that this description can be perfomed in terms of a time-dependent quantum harmonic oscillator. In addition, we construct conherent states for the quantized electromagnetic waves and evaluate the quantum fluctuations in coordinate and momentum as well as the uncertainty product for each mode of the electromagnetic field.