Spontaneous Emission in Quantum Optics MIT OCW Vacuum Rabi Oscillations and Cavity QED

Summary

This MIT OpenCourseWare lecture explores spontaneous emission from a two level atom coupled to the quantized electromagnetic field. Starting with a semi classical picture, the instructor shows how coupling to all vacuum modes and summing over directions yields the Einstein E coefficient and the natural linewidth of an excited state. By converting to atomic units, the talk derives order of magnitude estimates for lifetimes and explains the crucial Omega3 dependence that pushes hyperfine transitions to extremely long lifetimes. The lecture then discusses the fully quantized Hamiltonian, the role of degeneracy and matrix elements, and moves toward the Jaynes-Cummings model in a one mode cavity where vacuum Rabi oscillations reveal coherent evolution driven by light matter coupling even without a driving laser.

Overview

The lecture from MIT OpenCourseWare starts with the problem of spontaneous emission and explains how a quantized electromagnetic field interacting with an atom gives rise to the Einstein E coefficient and the natural linewidth. The discussion contrasts a semi classical dipole interaction with a full quantum treatment of the field and emphasizes how summing over all vacuum modes with the appropriate density of states leads to the observable decay rate.

Semi Classical vs Quantized Field

In the semi classical picture a dipole interacting with a classical field can describe basic Rabi flopping, but spontaneous emission requires the quantized nature of the field. The speaker then derives that spontaneous emission is caused by the atom exchanging energy with the quantized vacuum and that the rate is proportional to the cube of the transition frequency, with a small ratio to the energy spacing of the oscillator, illustrating its weak nature in typical atomic systems.

One Dimensional Density of States and Omega Dependence

The lecturer highlights how dimensionality changes the omega dependence. In one dimension, the omega cube from the density of states becomes a linear omega factor, leaving a remaining omega dependence that arises from the atomic matrix element. This discussion reinforces why specific transitions behave differently when the surrounding photonic environment is altered.

Degeneracy and Matrix Elements

Degeneracy factors enter absorption and stimulated emission differently. A key point is that in a generic microscopic derivation, degeneracy can distribute line strengths among several final states, and the proper accounting is essential for consistent line strengths and transition rates across different pathways.

Fully Quantized Hamiltonian and the Jaynes-Cummings Model

The core of the lecture builds the fully quantized Hamiltonian for a two level atom interacting with a single mode of the electromagnetic field inside a cavity. The interaction comprises lowering and raising operators for the atom and the photon field, giving rise to the Jaynes-Cummings model when one mode is resonant. The single photon Rabi frequency appears as a natural coupling strength, connecting the atomic raising and lowering operators with the annihilation and creation operators of the field.

Cavity QED and Vacuum Rabi Oscillations

In a high quality cavity with strong coupling, the rate of emission into the cavity mode can dominate over decay to other modes. This regime yields vacuum Rabi oscillations, a coherent exchange of excitation between the atom and the cavity field in the absence of a driving laser. The speaker explains how detuning splits the interacting states and how the rotating wave approximation simplifies the dynamics by focusing on the resonant co rotating terms, while off shell counter rotating terms contribute to shifts rather than transitions in many situations.

Outlook

The lecture closes with a note on how fully quantized treatments encapsulate radiation reaction, vacuum fluctuations, and Lamb shifts, and sets the stage for further exploration of light forces, cooling, and more complex multi level systems in cavity QED.