20. Line Broadening IV and Two-photon Excitation I

The script begins with an introduction to MIT OpenCourseWare, a platform offering free educational resources under a Creative Commons license. The professor encourages support to maintain this service, directing interested parties to ocw.mit.edu for donations and additional materials. The course's agenda includes discussions on line shifts and broadening, with a focus on collisional narrowing (Dicke narrowing) and a brief overview of the spectrum of emitted light by an atom and collisional broadening. The professor emphasizes the importance of understanding these topics, even though they are not covered in-depth, as they provide foundational knowledge in atomic physics and are relevant to modern physics concepts.

This paragraph delves into the concept of line broadening in atomic spectra, specifically discussing collisional broadening and its relation to coherence time. The professor uses the analogy of an atom interacting with a laser beam and the impact of collisions with buffer gas on coherence. The coherence time is identified as the time during which the atom maintains phase-coherent interaction with the laser beam. The mean free pass is highlighted as a significant parameter, with the coherence time estimated by the time it takes for an atom to diffuse more than a wavelength, leading to a loss of phase coherence. The paragraph concludes with a quantitative estimate of the line width based on the diffusion constant and the wave vector of light.

The professor introduces a theoretical framework to understand Dicke narrowing, a phenomenon where the spectral line width of atoms in a gas can be significantly reduced under certain conditions. The discussion revolves around the impact of buffer gas on the diffusion of atoms and how this relates to the narrowing of spectral lines. The mean free path's role in determining line width is explored, with the conclusion that if the mean free path is much smaller than the wavelength, Dicke narrowing occurs. The lecture also addresses the limitations of the model and when it applies, leading to a deeper understanding of the conditions necessary for observing Dicke narrowing.

This section discusses the use of the correlation function to analyze the line shape of spectral lines in the context of Dicke narrowing. The professor explains how the line width is related to the Fourier transform of the correlation function, which describes the phase of the drive field experienced by atoms at different times. The diffusion of particles is characterized by a random walk, and the correlation function is calculated by convoluting the probability distribution of this random walk. The result is an exponentially decaying function, indicative of a Lorentzian line shape when analyzed through Fourier transformation. The width of this Lorentzian is derived, providing a clear understanding of how the line shape is influenced by the diffusion constant and the wave vector of light.

The professor poses a question regarding the spectrum of diffusive motion and challenges the students to reconcile the theoretical prediction of a Lorentzian line shape with an intuitive model suggesting a bimodal distribution. The discussion highlights the importance of considering both the diffusive and ballistic motion of atoms and how these motions contribute to the observed line shape. The professor clarifies that the diffusive propagator is only valid after the first collision, and before that, the motion is ballistic, leading to Doppler broadening. The lecture emphasizes the importance of understanding the limitations of models and the relevance of experimental conditions in interpreting spectral data.

This paragraph explores the concept of the fluorescent spectrum of an atom when excited by light. The professor discusses the different aspects of spectroscopy, including the case of a motionless atom where the expected line shape would be a Lorentzian or power-broadened Lorentzian. The lecture also introduces the idea of fixing the detuning and analyzing the emitted light's frequency, leading to different possible line shapes. The discussion is framed within the context of perturbative excitation, where the laser power is assumed to be very low, and the focus is on the intrinsic line widths of the electronic transition.

The lecture continues by examining the impact of laser power on the spectral analysis of an atom. The professor discusses the scenario where the laser is on resonance but has a low power, leading to a delta function at the laser frequency due to energy conservation. The discussion then shifts to the case of higher laser power, where the Rabi frequency becomes significant, and the atom can undergo Rabi oscillations before being damped. This section introduces the concept of sidebands in the spectrum, which are indicative of the internal dynamics of the atom at the Rabi frequency, and the importance of understanding the difference between transient and continuous wave (CW) responses in spectroscopy.

This paragraph delves into the topic of line broadening in the context of two-photon processes. The professor presents different scenarios for the line shapes observed when an atom is subjected to higher laser powers, leading to the appearance of sidebands split by the Rabi frequency. The discussion explores whether these sidebands are sharp delta functions or broadened by natural line widths, and the conditions under which different broadening effects occur. The lecture concludes with the understanding that at very low powers, the central peak is sharp with small broadened sidebands, while at very high powers, all peaks are broadened, and the central elastic component disappears.

The professor introduces pressure broadening, a phenomenon where the spectral line width of an atom increases with the pressure of the surrounding medium. Two models are discussed: one where collisions lead to a quenching of the excited state, and another where collisions cause phase jumps in the atomic oscillator. The lecture explores the microscopic mechanisms behind these effects, including the interaction potential between atoms and the impact of this potential on the phase and frequency of the atomic oscillator. The discussion concludes with an understanding of how the line shape can provide information about both the interruption of coherence and the molecular potential in the wings of the spectral line.

The lecture concludes with a transition to the topic of two-photon excitations, motivated by the natural occurrence of such processes at higher laser powers and the practical applications of exciting atoms to high-lying levels with lower frequency lasers. The professor discusses the need for an intermediate state in two-photon processes within the dipole approximation and introduces the concept of a virtual state. The lecture also touches on the equivalence of different descriptions of light scattering and sets the stage for a deeper exploration of two-photon processes in the subsequent classes.

In the final paragraph, the professor outlines the calculation of two-photon excitation using second-order perturbation theory. The process involves an intermediate state, referred to as a virtual state, and the calculation is presented in a step-by-step manner. The lecture highlights the importance of considering both the forward and reverse order of photon absorption in the calculation. The summary of the calculation process is left incomplete due to time constraints, but it sets the foundation for understanding the amplitude in the final state resulting from two-photon absorption.