21. Two-photon Excitation II and Coherence I

The professor begins the lecture by discussing the importance of understanding two-photon processes in atomic physics. He emphasizes that most interactions with atoms involve two-photon events due to the scattering of a photon, which is often mistaken for absorption and re-emission. The lecture aims to clarify this misconception and introduces the perturbation theory approach to two-photon absorption, highlighting the significance of considering both photons from different optical fields (omega 1 and omega 2) and their role in energy state transitions. The professor also discusses the complexity of the interaction without approximations, which can lead to a large number of terms, but focuses on the simplification achieved through the rotating wave approximation and the consideration of resonant terms.

This section delves deeper into the specifics of two-photon absorption, explaining how the process involves both photons contributing to the energy transition of the atom. The professor illustrates how, without approximations, the number of terms to consider can be vast, but the use of the rotating wave approximation and the focus on near-resonant intermediate states can significantly simplify the analysis. The transition probability is discussed using Fermi's Golden Rule, and the concept of a two-photon Rabi frequency is introduced as a product of single-photon Rabi frequencies, divided by the energy detuning. The summary underscores the importance of resonant terms in simplifying the understanding of two-photon processes.

The lecture continues with an exploration of Raman processes, which are significant in experiments and can involve different states such as vibrational or hyperfine states of a molecule or atom. The professor explains the historical distinction between Stokes and anti-Stokes processes and how the use of laser beams in experiments eliminates the need to consider which state is higher in energy. The discussion parallels the treatment of two-photon absorption, emphasizing the analogy and the transition to resonant terms in the context of Raman scattering. The section concludes with a deeper look at the Raman process within the framework of perturbation theory, highlighting the selection of resonant terms and the transition to a stimulated emission process.

The professor introduces the concept of two-photon emission, where one photon is emitted spontaneously and the other in a stimulated manner. The discussion uses the previously established concept of the two-photon Rabi frequency and extends it to include the Einstein A coefficient for spontaneous emission. The summary explains how the rate of two-photon emission can be calculated using the probability of the intermediate state and the spontaneous emission rate from that state. The section also touches on the importance of considering the density of states and the frequency dependence in the calculation of spontaneous emission rates.

This part of the lecture takes a conceptual leap into considering two-photon processes at a higher level, specifically discussing the idea of virtual states created by the first photon and the subsequent absorption of a second photon. The professor uses the analogy of a 'dressed' atom to explain how the atom's state is altered by the interaction with the first photon, creating a stepping stone to the final state. The discussion also covers the concept of spontaneous emission in the context of a two-photon process, emphasizing the beauty and simplicity of the concepts learned and their direct applicability to seemingly complex phenomena.

The focus shifts to the practical applications of Raman processes in precision spectroscopy, particularly in the context of alkali atoms and the challenges associated with broad linewidths of excited states. The professor explains how two-photon processes can provide access to very narrow resonances, which are crucial for precision work in atomic physics. The summary also touches on the historical significance of Raman processes and their importance in the discovery of new photon frequencies, leading to the Nobel Prize-winning work of Raman.

The lecture explores the effects of recoil and Doppler shifts in two-photon processes, discussing how these phenomena can influence the line shape in spectroscopy. The professor explains how the momentum transfer in two-photon processes can be manipulated to minimize Doppler broadening, a technique vital for high-precision spectroscopy. The section delves into the specifics of how counter-propagating laser beams can lead to Doppler-free two-photon absorption, and how this method is used in hydrogen spectroscopy to achieve remarkable precision in measurements.

Building on the discussion of Doppler shifts, the professor introduces the concept of the second-order Doppler effect and its implications for precision measurements in optical clocks. The summary explains how even after eliminating the first-order Doppler broadening, the second-order effect becomes the limiting factor, necessitating the use of cryogenic temperatures to achieve the desired precision. The section also highlights the importance of understanding and accounting for the thermal energy distribution of atoms in high-precision experiments.

The final part of the lecture touches on the challenges and advancements in two-electron calculations, particularly in relation to helium and its potential use in fundamental physics and the determination of constants. The professor discusses the ongoing efforts to push the precision of these calculations and the current status of using hydrogen for testing quantum electrodynamics and fundamental constant measurements. The summary underscores the importance of selecting simple systems for these precise measurements and the ongoing quest for higher precision in atomic calculations.

The lecture concludes with an introduction to the concept of coherence in atomic physics, setting the stage for a deeper exploration in subsequent sessions. The professor outlines the different manifestations of coherence, including coherence within a single atom, between different atoms, and in systems like Bose-Einstein condensates. The summary provides a preliminary definition of coherence as the existence of a well-defined phase between two or more amplitudes and the importance of interference for its observation. The section also emphasizes the relevance of coherence to precise energy level measurements, a cornerstone of atomic spectroscopy.

In this segment, the professor poses questions to the audience about the nature of spontaneous emission and its relationship with coherence. The summary discusses the unitary transformation of the wave function during spontaneous emission and the potential randomness introduced by the measurement process or by averaging over states. The section aims to clarify the fundamental understanding of phase information in spontaneously emitted photons and sets the stage for further exploration of coherence and its limitations in the next lecture.

The lecture segment focuses on the fundamental aspects of spontaneous emission and its impact on phase coherence. The professor explains how an atom, initially excited by a laser pulse, carries the phase of that laser. Upon spontaneous emission, the atom transitions to the ground state, and a photon is emitted. The quantum state of the atom is perfectly transferred to the photon field, maintaining the coherence of the system. The summary highlights the importance of this process in understanding the limitations of phase retrieval in spontaneous emission and the fundamental nature of coherence in quantum systems.