This chapter discusses light–matter interactions from a semiclassical point of view. By expanding the electromagnetic field into a Taylor series we derive the multipolar interaction potential and particle-field Hamiltonian. Then, using the Green function formalism, we calculate the fields of an oscillating dipole and, based on Poynting’s theorem, derive a general expression for the rate of energy dissipation in an arbitrary environment. This expression leads to the concept of local density of states (LDOS) and provides a direct link to spontaneous emission and atomic decay rates. The rate of energy dissipation of an oscillating dipole is also used to derive the absorption cross-section in terms of the polarizability. By accounting for radiation reaction and scattering losses, we obtain a compact expression for the dynamic polarizability. Dipole radiation and atomic decay rates can be enhanced via LDOS engineering, and the enhancement factor is referred to as the Purcell factor. We show that if the LDOS gets enhanced in a certain frequency range, it must be reduced in other frequency ranges, a feature described by the LDOS sum rule. After discussing the properties of a single dipole, we continue with analyzing the interaction between multiple dipoles. We derive the interaction potential and calculate the energy transfer rate between dipoles. For short distances we recover the famous Förster energy transfer formula. If the interaction energy becomes sufficiently large, we enter the regime of strong coupling, which gives rise to hybridized and delocalized modes, level splittings, and entanglement.
This chapter discusses light–matter interactions from a semiclassical point of view. By expanding the electromagnetic field into a Taylor series we derive the multipolar interaction potential and particle-field Hamiltonian. Then, using the Green function formalism, we calculate the fields of an oscillating dipole and, based on Poynting’s theorem, derive a general expression for the rate of energy dissipation in an arbitrary environment. This expression leads to the concept of local density of states (LDOS) and provides a direct link to spontaneous emission and atomic decay rates. The rate of energy dissipation of an oscillating dipole is also used to derive the absorption cross-section in terms of the polarizability. By accounting for radiation reaction and scattering losses, we obtain a compact expression for the dynamic polarizability. Dipole radiation and atomic decay rates can be enhanced via LDOS engineering, and the enhancement factor is referred to as the Purcell factor. We show that if the LDOS gets enhanced in a certain frequency range, it must be reduced in other frequency ranges, a feature described by the LDOS sum rule. After discussing the properties of a single dipole, we continue with analyzing the interaction between multiple dipoles. We derive the interaction potential and calculate the energy transfer rate between dipoles. For short distances we recover the famous Förster energy transfer formula. If the interaction energy becomes sufficiently large, we enter the regime of strong coupling, which gives rise to hybridized and delocalized modes, level splittings, and entanglement.
This chapter presents the electron oscillator model of light-matter interaction. It models atomic response to the fields in an electromagnetic wave after a classical driven charged (dipole) oscillator. It is used to calculate atomic polarization and dielectric susceptibility and derive an equation of motion for atomic polarization for a collection of atoms. By replacing the oscillator number density in this equation with the level population density difference and interpreting oscillator frequency with transition frequency and decay rate with spontaneous emission rate a quantum mechanically correct equation for atomic polarization is obtained. Poynting theorem on energy exchange between light and atoms in an EM wave then yields quantum mechanically correct rate equations for population densities. These equations together with the equation for atomic polarization provide a complete description of atomic response to light. Modifications to the atomic response due to finite atomic lifetimes, collisions, atomic motion, local variation of resonance frequency and other factors are also discussed in terms of lineshape function and cross-section.
Using the electromagnetic response function of an electric dipole located within a dielectric geometry, we derive the mathematical equivalence between the classical response and quantum mechanical resonant dipole–dipole interaction between two quantum objects (atoms, quantum dots, etc). Cooperative spontaneous emission likewise emerges from this equivalence. We introduce a practical numerical technique using finite difference time domain for calculating both dipole–dipole interaction and collective spontaneous emission in confined dielectric structures, where strong light–matter coupling might arise. This method is capable of obtaining resonant dipole–dipole interaction over a wide range of frequencies in a single run. Our method recaptures the results of quantum mechanical second order perturbation theory for weak light–matter coupling. In strong coupling situations such as near a photonic band edge, second order Rayleigh–Schrödinger perturbation theory leads to divergences, and instead Brillouin–Wigner perturbation theory is required. This is equivalent to the use of a variational wavefunction to describe the exciton transfer between initial and final states. We introduce a system of coupled classical oscillators, that describes resonant dipole–dipole interaction and vacuum Rabi splitting in the strong-coupling regime, and that provides an effective numerical scheme based on the finite difference time domain method. This includes the effects of quantum entanglement and the correlation of quantum fluctuations. We discuss the crossover to Forster energy transfer when quantum correlations between the dipoles are damped by strong environmental interactions.
Radiation can change the energy state of matter via three optical processes: spontaneous emission, absorption and stimulated emission. In spontaneous emission, an excited atomic or molecular state decays to a lower energy state with the emission of a photon of energy equal to the difference in the quantum state energies. Photon absorption is the reverse process, with a photon causing excitation from a lower to an excited quantum state. Stimulated emission was first proposed by Einstein in a paper published in 1917. He attempted to balance spontaneous emission and photon absorption, but could not get the correct balance without the postulated process of stimulated emission (see Section 4.2). With stimulated emission, an excited quantum state is stimulated to decay to a lower quantum state by an impinging photon. The existing photon survives with another identical photon being created (so that energy is conserved). The rates at which radiative transitions occur can be calculated using quantum mechanics. The time-dependent Schrodinger equation is used with the system wavefunction comprising time-varying linear combinations of the two wavefunctions for the lower and upper quantum states. In semi-classical treatments, an atom is treated quantum mechanically with the radiation field treated classically. There is a perturbation in energy associated with, say, the electric field of the radiation interacting with the electric dipole of the atom, though other interactions can be significant if the electric dipole interaction energy is small. In absorption, for example, a differential equation representing the change in ‘weighting’ of the wavefunction from the lower to upper state is obtained. A ‘time constant’ for the speed of change represents the rate of the radiative process. We examine the quantum mechanical nature of radiation absorption in Section 10.1 before discussing (Section 10.3) another important parameter arising with radiative transitions between discrete bound quantum states: the lineshape function. Quantum Theory of the Atom–Radiation Interaction An expression for the Einstein B-coefficient for photo-absorption is derived using quantum mechanics in this section. Einstein determined the detailed balance relationship between the rates of spontaneous emission, photo-absorption and stimulated emission when light interacts with an atom by considering an atomic system of two energy levels which are in equilibrium with a black-body radiation field. We use a semi-classical treatment where the atom is treated quantum mechanically and the radiation field is treated as a classical oscillating electric field.
Abstract The atom-field interaction is treated in semiclassical radiation theory, starting from the transformation from the minimal coupling Hamiltonian to the electric dipole form used extensively in quantum optics. The Heisenberg and density matrix approaches are developed and applied to two-state atoms, Bloch equations, Rabi oscillations, Maxwell-Bloch equations, and transition rates for absorption and emission. Einstein’s theory of blackbody radiation based on momentum fluctuations and dissipation is reviewed. The Einstein fluctuation formula is derived and used to introduce wave-particle duality, Hanbury Brown-Twiss correlations, and photon bunching.
In this chapter a quantum mechanical treatment of atomic response to the fields of an electromagnetic wave is presented. Absorption and emission of energy by atoms involves transition between their quantized energy levels driven by an interaction Hamiltonian. Schrodinger equation is used to derive the equations of motion for the density matrix of the atom, which is better suited to include the effects of mechanisms other than the interaction Hamiltonian – collisions, level decays, level shifts, etc. that also affect the state of the atom. Using the density matrix equation of motion and focusing on two atomic levels, induced dipole moment for a single two-level atom and polarization and susceptibility for a collection of two-level atoms, including the effects of level decays, and collisions, are calculated. Equations of motion for level populations are also derived. These expressions naturally predict stimulated emission as well as absorption. Expressed in terms of spontaneous emission rate they confirm the validity of equations derived in Chapter 5 using the dipole oscillator model of atomic response. The limit in which quantum mechanical equations for atomic response reduce to the classical dipole oscillator equations is also discussed.
Devoting some thought to the interactions between light and matter quickly conjure a myriad of different possibilities; different models for light and matter, different possible interaction Hamiltonians, different simplifying approximations and different initial conditions. This of course means that light-matter interactions has just as many uses and surprises yet to be explored. In this thesis we will approach light-matter interactions from three different perspectives: 1) How can light-matter interactions be used to manipulate quantum fields into exotic energy distributions, 2) How well does a classical light-matter approximation carry through to quantum light-matter interactions and 3) In a relativistic theory, what are the causal consequences of approximations commonly used to reduce the analytical and numerical complexity of quantum fields. \n \nMy first aim in this thesis is to develop an operationally feasible procedure for generating exotic energy distributions. Current proposals for generating negative energy densities include using the dynamical Casimir effects or by squeezed vacua, which require relativistically accelerating mirrors or non-linear crystals. Instead we seek an optimal protocol that does not require relativistically accelerating elements and exploits the coherent control of a detector. The quantum energy teleportation (QET) protocol is ideal for this task, and we present a QET protocol acting on a relativistic scalar field optimised for the generation of negative energy densities. We show that QET can be used to generate local negative energy densities using stationary qubits. In addition we discuss the consequences of detector smearings on the (QET generated) energy distribution, providing simple and intuitive guidelines for sculpting negative energy distributions. We also show that this protocol is capable of generating regions with an arbitrary amount of negative energy, with the total amount of negative energy ∆E increasing as the negative energy well’s width ∆r is decreased by ∆E ∼ ∆r⁻³ . However, this is accompanied by increasingly large (∼ ∆r⁻³ ) positive energy peaks on either side of the well. We further find this energy-distance scaling saturates the quantum interest conjecture, suggesting the near optimality of QET for generating negative energy densities. \n \nMy second aim is to determine the validity of the dipole model of light-matter interaction with a quantum EM field. This model, classically derived from the more fundamental minimal model, is dependent on the existence of a dominant mode, whose wavelength is required to be much larger than the size of the atom it is interacting with (dipole approximation criterion). Quantisation of the EM field result in vacuum fluctuations without a dominant wavelength. Past works have attempted to overcome this by using point-like atoms, however this introduces UV divergences in the response of the atoms and is inadequate for describing light-matter interactions. The full influence of gauge considerations and quantum behaviour in light matter interaction is quite involved and complex, particularly when the atomic nucleus’ mass is considered finite and the centre of mass degrees of freedom become relevant. We consider effective models (infinite nucleus mass), which neglect the centre of mass degrees of freedom and their additional complexity, whilst allowing accurate investigations of the electronic orbital behaviour. Here we will attempt to determine under what circumstances the dipole and minimal models agree; and hope to clarify and extend the dipole approximation validity criteria for general use in quantum fields. \n \nUsing the 'dressed state' formalism to remove gauge issues, we compare the transition probabilities of Hydrogen-like atoms under both models to determine the validity of the dipole model. We show that for atomic transitions with an initial EM vacuum state, both models noticeably disagree for short interaction times (i.e. shorter than the light-crossing time of the atom). We find the transition rates predicted by both models begin to converge for longer interaction times, particularly when considering vacuum excitations. Vacuum emissions have the additional requirement that the atomic energy gap must satisfy the dipole approximation criterion due to single mode approximation effects. In addition, we find that shrinking the atom (by increasing the atom’s proton number) does not improve the accuracy of the dipole model, a result of the infinite number of UV modes in the vacuum fluctuations. We determine that the dipole model can be used with a quantum EM field, provided any intrinsically dominant modes, e.g. atomic energy gap (for vacuum emissions) or excited EM modes, satisfy the dipole approximation criterion and the interaction time is longer than the light crossing time of the atom. \n \nOur final aim is to determine the causal consequences of the commonly used rotating wave approximation (RWA) and determine in what regimes it may be accurately used. The RWA removes terms from the (detector-field) interaction Hamiltonian that do not conserve excitation number, e.g. field creation operators tensored with SU(2) raising operators. This is justified by noting that these terms oscillate quickly in time and therefore, for long interaction times (T), will integrate to zero, where long interaction time is defined by the RWA criterion ΩT≫1, where Ω is the detector’s energy gap. We wish to determine the extent of the non-locality and causality violation introduced into this originally local and causal theory; as well as determine in what regime the RWA model can be used as a faithful substitute for the unapproximated UDW model. We quantify these violations by studying the non-locality of the RWA interaction Hamiltonian, inspecting the acausal expectation values of field amplitude squared (ϕ²) and the local energy density (T₀₀); and quantifying the superluminal communication in a 2 qubit communication protocol. \n \nWe verify and extend the previous results by showing the RWA interaction Hamiltonian has a 1/r² polynomial non-locality, proportional to the vacuum Wightman function of the field. We also show that this non-locality translates into causality violations of 1/d⁴ and 1/d⁶ for the expectation values ϕ² and T₀₀ respectively; where d is measured from the surface of the detector’s lightcone and this polynomial causality violation is independent of time. We also find similar 1/d² non-localities in the channel capacity of the 2 qubit communication protocol, with non-local influence in the space-like and time-like regions of the first detector’s interaction. We show that in setups that are not causally sensitive, e.g. measuring field observables within the bulk of a source detector’s light-cone, then the RWA criterion ΩT≫1 is sufficient for the RWA model with interaction strength λ to converge to the unapproximated UDW model with interaction strength λ/2. We establish that this factor of 1/2 comes from a mathematically unsound commutation of limits and integrals. \n \nIn the case of cavity fields, we introduced a numerical trick to make infinite mode sums computationally possible without requiring single or few mode approximations. Using this trick we found the RWA non-local behaviour to be generally similar to the free space case. We find the RWA model introduces some severe polynomial non-localities and causality violations, but if used to model light-matter interactions well within the bulk of the detector’s light-cone and ΩT≫1 then it is a good approximation for the half strength UDW model.