14. Atom-light Interactions III

The script begins with an introduction to the topic of light atom interaction, emphasizing the importance of understanding the dynamical evolution of atomic wave functions under the influence of optical fields. The professor recaps the discussion on the matrix element and the coupling between two states, highlighting the perturbation theory with a dipole Hamiltonian. The lecture also touches upon the distinction between monochromatic and broadband cases, leading to the concept of Rabi oscillations and their perturbative limit. The goal is to explore how the system evolves with the coupling of atomic states using optical fields.

This paragraph delves deeper into the dynamics of Rabi oscillations, particularly in the context of broadband light interactions. The professor explains the mathematical formulation of these oscillations when considering a spectrum of frequencies, and the assumptions made regarding the lack of correlation between different frequencies. The lecture discusses the probability of finding an atom in the excited state and the implications of a flat and broad spectral density. The transition from perturbative Rabi oscillations to a delta function in the spectral density is also explored, leading to two distinct cases for the behavior of the system.

The script examines the time-dependent behavior of light-atom systems, focusing on the evolution from short to long timescales. It discusses the peculiar nature of the Rabi oscillation function and its implications for the excitation probability. The professor illustrates how at very short times, the system behaves as if it were under monochromatic light, with the probability of excitation being proportional to t squared times the total intensity. However, as time progresses, the system transitions to a regime where the spectral density dominates, leading to a different behavior in the excitation probability. This paragraph also addresses questions from the audience regarding the bandwidth and its impact on the system's evolution.

The professor introduces Einstein's a and b coefficients, discussing their significance in atomic physics and their role in understanding the interaction between atoms and light. The lecture provides a detailed explanation of how these coefficients are derived using perturbation theory for broadband light, leading to the formulation of rate equations. The connection between the b coefficient and the spectral density at the resonance frequency is established, and the transition from coherent evolution to rate equations is discussed. The script also touches on the historical context of these coefficients and their derivation before the development of quantum mechanics.

This paragraph summarizes the transition from coherent evolution, characterized by Rabi oscillations, to dissipative processes described by rate equations. The professor contrasts the two regimes: one involving a single final state of the atom and a single mode of the electromagnetic field, and the other involving many final states and a continuum of external field modes. The irreversibility of the time evolution in the latter case is highlighted, as well as the absence of spontaneous emission in the current discussion, which will be addressed in future lectures.

The script shifts focus to spontaneous emission, which is identified as a key missing element in the current understanding of light-atom interactions. The professor outlines the agenda for the next part of the lecture, which includes discussing Einstein's a and b coefficients in the context of spontaneous emission, using modern formalism of quantized electromagnetic fields, and eventually deriving the a coefficient microscopically. The historical significance of Einstein's work is reiterated, and the intention to connect these concepts with the quantization of the electromagnetic field is highlighted.

The professor provides a detailed analysis of Einstein's a and b coefficients, discussing the assumptions and derivations behind these coefficients. The lecture explores the equilibrium between the electronic structure of an atom and the photon field, and how Einstein's treatment of these coefficients predated the development of quantum mechanics. The script also delves into the mechanical effects of light, such as laser cooling, and how these effects are related to the equilibrium between the motional degree of the atom and the radiation field.

This paragraph presents the mathematical formulation behind Einstein's a and b coefficients. The professor explains how these coefficients are derived from the rate equation for the atoms and the spectral density of black-body radiation. The lecture discusses the equilibrium solution of the rate equation and how it leads to the determination of the a and b coefficients. The script also addresses the importance of these coefficients in establishing thermal equilibrium and the role of the Boltzmann factor in this context.

The script explores the connection between Einstein's a and b coefficients and the quantization of the electromagnetic field. The professor discusses how the coefficients can be understood in terms of photon numbers, leading to the realization that stimulated emission is essentially n times the photon number, with an additional factor for spontaneous emission. The lecture also highlights the importance of the plus 1 factor in establishing thermal equilibrium and the role of the Bose-Einstein distribution in this context.

The professor discusses the impact of dimensionality on spontaneous emission, explaining how the density of modes changes in one and two dimensions. The lecture encourages students to consider how the summation of all modes contributes to spontaneous emission and how this understanding can clarify the changes that occur in different dimensions. The script also mentions the agenda for the next chapter, which involves the quantization of the radiation field and the exploration of its implications for the interaction between photons and light.

The script concludes with an overview of the quantization of the electromagnetic field. The professor outlines the process of quantization, from the vector potential to the introduction of creation and annihilation operators. The lecture emphasizes the analogy between the electromagnetic field and harmonic oscillators, and how this leads to the quantum description of the field. The script also touches on the importance of understanding the interaction between light and atoms in the most fundamental and microscopic way.

This paragraph focuses on the quantum description of the electromagnetic field, starting with the assumption of plane waves and deriving the relationship between the electric field and the vector potential. The professor introduces the concepts of q and p, analogous to position and momentum in a harmonic oscillator, and then transitions to the quantization of these quantities. The lecture explains how the energy of the field is expressed in terms of creation and annihilation operators, leading to the identification of the Hamiltonian for the field and the introduction of the photon number operator.

The script describes the process of deriving the Hamiltonian for the interaction between light and atoms, starting with the dipole Hamiltonian and incorporating the quantized electric field. The professor explains how the electric field becomes an operator acting on the quantum state of the electromagnetic field and how the dipole matrix element connects the atomic states. The lecture also discusses the importance of the electric field of a single photon and its role in the interaction Hamiltonian.

This paragraph examines the matrix elements of the interaction Hamiltonian and their implications for the rates of absorption and emission. The professor discusses how the matrix elements are non-vanishing for transitions that differ by plus or minus one photon, corresponding to absorption and emission, respectively. The lecture also explains how the rates are determined by the matrix elements squared and the density of states, leading to the conclusion that the rate of emission is proportional to n plus 1, while the rate of absorption is proportional to n.

The script presents a microscopic derivation of Einstein's a coefficient for spontaneous emission, starting with Fermi's Golden Rule and considering the matrix element for the transition from an excited atomic state to a ground state with one photon emitted. The professor explains how the density of states and the dipole matrix element contribute to the rate of spontaneous emission and how the integration over the solid angle is affected by the dipole pattern. The lecture concludes with the derivation of the a coefficient, including its dependence on the cube of the frequency and the factor of 3 in the denominator due to the average over the dipole pattern.