19. Line Broadening III

The professor begins the lecture by discussing the importance of understanding spectral broadening in the context of quantum systems. They introduce perturbation theory as a framework for analyzing how small disturbances can affect the spectral properties of a system. The lecture focuses on the derivation of spectral broadening using time-dependent perturbations and the significance of the correlation function G in describing these effects. The professor also touches on the quadratic behavior of amplitude in quantum systems and the transition to Fermi's golden rule for the probability of an excited state, indicating a shift from coherent to incoherent processes.

This paragraph delves deeper into the time evolution of quantum systems, emphasizing the transition from quadratic probability amplitudes to linear probabilities as the system moves from coherent to incoherent states. The professor explains the concept of Rabi oscillations and how they relate to the initial conditions of quantum systems. The discussion then shifts to the role of decoherence and the introduction of rate equations, which are derived from the time-dependent perturbation theory. The lecture also covers the connection between the differential equations for amplitude addition and the matrix element, highlighting the transition from constructive interference to incoherent addition as time approaches the coherence time.

The professor explores the connection between spectral broadening and the correlation function of the perturbation. They explain how the correlation function G can be used to describe the excitation rate of a system and how it is related to the spectral properties of the drive field. The paragraph also discusses the time-integrated correlation function and its relation to Fermi's golden rule, emphasizing the importance of the perturbation operator and the correlation time in determining spectral widths. The professor illustrates this with an example of a Fourier transform, showing how it can reveal the resonant components of the drive field and its power spectrum.

In this paragraph, the professor discusses the concept of natural line widths and how they arise from the interplay between coherent drive and spontaneous decay in atoms. They introduce the Optical Bloch equations as a tool for understanding this phenomenon and provide a simple example using a decay rate to model spontaneous decay. The lecture then explains how the correlation function of the matrix element, which includes an exponentially decaying factor, leads to a Lorentzian line shape in the Fourier transform, indicative of natural line widths.

The professor introduces Doppler broadening as a result of the motion of atoms in a field. They explain how the spatial dependence of the matrix element and the need to average over the velocity distribution of the atoms lead to Doppler broadening. The lecture covers the calculation of the correlation function for moving atoms and how it results in a Gaussian profile for the broadened light. The professor also offers a new perspective on Doppler broadening by relating it to the coherence time of the atomic ensemble and the spatial spread of atoms due to their motion.

This paragraph examines the concept of coherence time and its relation to the spread of atomic velocities. The professor explains how the coherence time is the duration over which the atomic ensemble remains coherently driven. They discuss the impact of thermal velocity spread on coherence and how it leads to a loss of constructive interference in amplitude addition. The lecture also explores the idea of atoms as wave packets and how the loss of overlap between the ground and excited state parts of the wave packet relates to the coherence time.

The professor discusses the Lamb-Dicke limit, which occurs when atoms are confined to less than an optical wavelength, effectively eliminating Doppler broadening. They explain how the confinement leads to a very sharp spectral line and compare this to the Mossbauer effect. The lecture explores the line shape of confined particles, particularly in a harmonic oscillator, and how this can lead to a delta function spectral feature in the case of tight confinement. The professor also touches on the importance of this regime for precision spectroscopy.

This paragraph delves into the spectrum of an atom in a harmonic trapping potential, considering both internal and external degrees of freedom. The professor explains how the total energy of the system, including the kinetic energy from the motion of the atom, contributes to the observed spectrum. They introduce the concept of sidebands and how they are related to the harmonic oscillator frequency, leading to a discrete spectrum without Doppler broadening. The lecture also discusses the conditions under which the Doppler effect can be observed and how it relates to the intensity of the spectral peaks.

The professor discusses the conditions for resolving sidebands in the spectrum of a confined atom and the impact of natural line widths on this process. They explain how the number of sidebands and the resolution of these sidebands depend on two parameters: the Lamb-Dicke parameter and the ratio of natural line width to sideband spacing. The lecture also explores the different regimes that can be observed depending on these parameters and their implications for spectroscopy.

In this paragraph, the professor draws an analogy between the Mossbauer effect and the recoil-less emission and absorption of photons in tightly confined particles. They explain how the momentum recoil from photon absorption is transferred to the trap or the entire apparatus, effectively endowing the particle with infinite mass for the purpose of momentum exchange. The lecture explores how this situation relates to the concept of the Mossbauer effect and its implications for the observed spectral lines.

The professor introduces the concept of Dicke narrowing, which describes a situation where an atom in a buffer gas can experience a narrowing of spectral lines due to the specific properties of the buffer gas. They explain how collisions in the buffer gas can prevent the atoms from acquiring random phases with respect to the drive field, leading to a sharper spectral line than would be expected from the natural Doppler width. The lecture outlines the conditions under which Dicke narrowing occurs and its potential applications in high precision spectroscopy.