From Maxwell to Quantum: Deriving the Atom-Photon Interaction Hamiltonian and the Dipole Approximation

Short Summary

MIT OpenCourseWare presents a rigorous derivation of the Hamiltonian that describes how atoms interact with light. The lecture begins by separating the electromagnetic field into local (Coulomb) and radiation components, then uses a Fourier-space formulation of Maxwell's equations to distinguish longitudinal from transverse fields. In the Coulomb gauge, the transverse field is expressed in terms of normal modes, which are shown to be independent harmonic oscillators that become quantum operators upon quantization. The discussion culminates in the full light-matter Hamiltonian, comprising atomic kinetic energy, Coulomb interactions, and the coupling to the radiation field. A key focus is the Electric Dipole Approximation as the practical workhorse in atomic physics, followed by a more rigorous treatment that includes the quadratic term. The lecture concludes with references for deeper reading.

Introduction

The lecture aims to derive from first principles the Hamiltonian that governs the interaction between atoms and light. It emphasizes that atomic physics centers on how building blocks of matter couple to electromagnetic fields and how this leads to practical tools like quantum gates and laser cooling.

Local vs Radiation Fields

The instructor starts with Maxwell's equations and shows how to rigorously separate the field into local (associated with charges in atoms) and radiation components. Through a Fourier transform, the fields are decomposed into longitudinal and transverse parts, with the longitudinal (Coulomb) field tied to charge densities and the transverse field representing propagating radiation. This separation is crucial for identifying independent degrees of freedom in the quantized theory.

Gauge Choice and Potentials

Using the Coulomb gauge, the divergence of the vector potential is set to zero, which eliminates the longitudinal component of the vector potential. This reduces the problem to the transverse vector potential and its conjugate momentum, simplifying subsequent quantization and making the physics transparent in the radiation sector.

Normal Modes and Quantization

The transverse field is recast in terms of normal modes, which, in classical language, transform a coupled system into a set of decoupled harmonic oscillators. In the quantum treatment, these normal modes are promoted to operators, giving rise to creation and annihilation operators. The discussion highlights that this path—starting from classical normal modes and moving to quantum operators—is a standard route to quantum electrodynamics.

The Electromagnetic Field Hamiltonian

With the field described in terms of its normal modes, the energy of the radiation field is written as a sum over modes, each contributing a quadratic term in the normal-mode amplitude. This mirrors the familiar harmonic oscillator energy structure, reinforcing the connection between field theory and quantum mechanics.

The Atom-Photon Interaction Hamiltonian

The total Hamiltonian is decomposed into three parts: atoms, the radiation field, and the interaction. The interaction term contains three pieces: the cross term between particle momentum and the vector potential, the A-squared term from the square of the vector potential, and the spin coupling to the magnetic field. The speaker emphasizes that these terms are sufficient to understand atomic physics, with the A-squared term sometimes debated in common textbooks. A unitary transformation introduces the dipole form D·E, providing a rigorous route to the Electric Dipole Approximation and clarifying when the A-squared term can be neglected.

Dipole Approximation

The dipole approximation rests on three main ideas: (1) long-wavelength fields compared to atomic size, (2) near-resonant interactions with atomic transitions, and (3) neglecting the A-squared quadratic term in the lowest-order description. The speaker derives how the momentum matrix element between atomic states can be expressed in terms of the position matrix element times the transition frequency, leading to the dipole interaction term -D·E. A careful discussion shows that, under usual atomic-physics conditions, this approximation is robust and that a more rigorous treatment still aligns with the standard dipole Hamiltonian, once the proper field definitions and unitary transformations are accounted for.

Reading and Further Reading

To deepen understanding, the instructor points to Claude Contanucci and collaborators’ books, including the broader treatment of photons and atoms and discussions of various quantization schemes. The goal is to provide both a practical toolkit for atomic physics and a rigorous foundation for the underlying quantum electrodynamics.

Closing

The session ends with a reminder of the next steps, including assignments and the availability of teaching assistants for help. It reinforces the idea that the dipole interaction captures the essential physics of light-matter coupling in most atomic-physics contexts, while a rigorous treatment provides a deeper mathematical grounding.