22. Coherence II

The script begins with an introduction to the concept of coherence in physics, specifically within the context of MIT OpenCourseWare's educational resources. The professor emphasizes the importance of the final chapter on coherence, promising exciting topics ahead. The discussion is set to explore coherence in single atoms and then between atoms, starting with spontaneous emission and its common misconceptions. The emphasis is on the unitary time evolution of the system, highlighting that spontaneous emission is not as random as often believed due to the presence of a specific Hamiltonian operator.

This paragraph delves into the fundamental aspects of spontaneous emission, challenging the assumption of randomness in the process. The professor explains that spontaneous emission is a unitary process without random variables, influenced by the Hamiltonian operator. The concept of vacuum Rabi oscillation is introduced as a simple system to understand spontaneous emission, where an atom in the ground state and a cavity in the vacuum state can lead to a coherent superposition created by a laser pulse. The lecture aims to clarify the misconceptions and explore the phase of the spontaneously emitted photon, emphasizing the role of the laser's well-defined phase in the process.

The script discusses phase space plots for the photon field, explaining how these plots can be used to understand the evolution of a harmonic oscillator and the states of photons. It covers how the phase of a coherent state can be determined and the implications for the phase measurement of photons. The professor explains that the phase is best defined when there is an equal superposition of states, and how the Heisenberg uncertainty relation applies to phase measurement. The summary also touches on the limitations of phase measurement accuracy when dealing with single photons versus a large number of photons.

This section describes a homodyne experiment to measure the phase of the photon field, which involves interfering the emitted photon with a local oscillator, typically the laser used to excite the atom. The script clarifies that fluctuations in the measured phase are not due to Hamiltonian fluctuations but arise from the quantum nature of the states involved. The professor introduces the concept of a Mach-Zehnder interferometer to illustrate how the homodyne measurement is performed, emphasizing that while a sharp phase value cannot be obtained, the fluctuations are intrinsic to the quantum state.

The script explores the relationship between coherence and phase information in atomic systems, particularly after a π pulse excitation. It discusses how at t=0, there is no coherence or phase information, but as time progresses and the atom decays to a 50% excited state, a phase appears, although it is random. The ensemble of atoms represents a collection of wave functions with random phase phi, leading to no coherence in the statistical operator. The dipole moment's ensemble average is zero, but the d squared value remains, indicating the presence of a photon. The script also touches on the concept of vacuum fluctuations and their potential role in phase uncertainty.

This paragraph introduces the concept of superradiance, where two atoms excited to an excited state can emit photons with correlated phases due to the same random vacuum fluctuations triggering the spontaneous emission. The script discusses how the absolute phase is random, but the relative phase between two atoms shows a correlation. It also addresses the question of whether the two atoms need to be within an optical wavelength of each other for this effect, explaining that while it is a simplified example, superradiance can also occur in extended samples with coherence in a smaller solid angle.

The script discusses coherence in two-level systems, using the example of spin precession in a magnetic field to illustrate the concept. It explains how the precession of spin in the xy plane is a manifestation of coherence within an atom and how the phase of the superposition state determines the spin's pointing direction. The summary also covers how the expectation value for the x spin changes over time due to the coherent time evolution of the amplitudes, contrasting this with the case of no coherence where the expectation value for the spin components vanishes.

This section introduces quantum beat spectroscopy as a method to measure narrow level spacings without narrow band excitation. The script explains how a broadband source can be used to excite a coherent superposition of energy levels, leading to quantum beats that can be observed as oscillations in the emission spectrum over time. The summary highlights how this technique allows for the measurement of level spacings that are much narrower than the Doppler width, providing a sub-Doppler technique to exploit coherence and obtain detailed atomic structure.

The script explores the concept of delayed detection in spectroscopy, questioning whether spectral resolution narrower than natural line widths can be achieved by measuring only the long-lived atoms. It explains the mathematical concept behind Fourier transform and how it applies to the situation of quantum beats with exponential decay. The summary clarifies that while delayed detection can theoretically provide higher resolution, it comes at the cost of an exponentially smaller signal, and the actual resolution is dependent on the knowledge or reproducibility of the phase at the start of the measurement.

This paragraph discusses the introduction of coherence in three-level systems, particularly focusing on the lambda type system where two ground states are connected through an excited state. The script explains how such systems can exhibit fundamentally new effects, such as lasing without population inversion, electromagnetically induced transparency, slowing light, and quantum mechanical memories for quantum computation. The summary also touches on the practical applications of these systems and the importance of understanding the differences between various three-level systems for their implementation.

The script delves into the concept of optical pumping and its relation to coherence in three-level systems. It explains how with two laser fields, it is possible to pump all atoms into a state that does not scatter light or react with the light, a phenomenon known as electromagnetically induced transparency. The summary describes the Hamiltonian for the system and the conditions under which destructive interference can occur, leading to a dark state that is a coherent superposition of two ground states. The script sets the stage for further exploration of this phenomenon in the subsequent class.