From Rabi Oscillations to Einstein B Coefficients: Quantum Light–Matter Interaction

Overview

In this MIT OpenCourseWare lecture, the instructor analyzes how light couples two atomic states via a dipole interaction, tracing the evolution of the atomic wavefunction under optical driving. The discussion starts with perturbation theory for a dipole Hamiltonian and distinguishes between monochromatic and broadband light. For a single mode, the system exhibits Rabi oscillations, while a broadband spectrum requires integrating over a spectrum of frequencies.

Two limiting regimes emerge: at very short times the excitation probability grows as t^2, reflecting coherent evolution, and at longer times the growth is linear in time governed by the spectral density at resonance, in line with Fermi's golden rule. The talk then connects this framework to Einstein's B coefficient, the rate equations, and the appearance of spontaneous emission when the electromagnetic field is quantized. The lecture sets the stage for a microscopic derivation of spontaneous emission using the quantized field and the role of the density of states.

Introduction and Context

The lecture revisits how a two-level atom interacts with an external electromagnetic field through a dipole coupling. Beginning with perturbation theory, the presenter differentiates between monochromatic radiation, where the problem reduces to an exact two-level spin-1/2 in a magnetic field, and broadband radiation, where a spectrum of frequencies interacts with the atom via a time-dependent perturbation.

Broadband Light and Spectrum Integration

For broadband light, the Rabi frequency is replaced by an electric field term, and the Rabi frequency squared becomes an integral over the spectral density. The speaker emphasizes a key assumption: the frequencies are uncorrelated, so the integral neglects inter-frequency interference. The probability to be excited is then the convolution of the coherent Rabi oscillations with the spectral density of the field.

Coherent versus Incoherent Regimes

Two limiting cases are discussed. When the spectral bandwidth is very broad and the observation time is very short, the excitation probability scales as the integral of the spectral density times t^2, effectively behaving as a delta function in time that enforces coherence. When time grows beyond the inverse bandwidth, the integral collapses to a delta-like contribution, yielding linear growth in time with the spectral density at zero detuning. This marks the crossover from coherent evolution to irreversible dynamics described by a rate equation.

Einstein Coefficients and Rate Equations

The discussion then turns to Einstein's A and B coefficients. In a thermal equilibrium with a black body field, the rate of change of the excited-state population involves stimulated absorption and emission, as well as spontaneous emission. Isotropy introduces a factor of 1/3 for the projection of the dipole moment on the light polarization, tying the semi-classical dipole picture to the quantum description. The B coefficient relates the stimulated processes to the spectral density at resonance, while the A coefficient emerges when spontaneous emission is included to maintain thermodynamic consistency with Boltzmann and Planck statistics.

Quantum Field Perspective and Spontaneous Emission

To go beyond the semi-classical picture, the field is quantized. The electromagnetic field is described as a collection of harmonic oscillators, each mode described by creation and annihilation operators. The interaction Hamiltonian now contains the dipole moment coupled to the quantized field, enabling absorption and emission processes that depend on the photon occupation number N. Stimulated emission scales with N, while spontaneous emission corresponds to the extra +1 in the emission amplitude, a result that aligns with Einstein’s treatment but now arises from a microscopic quantum-field calculation.

Density of States and Angular Dependence

Spontaneous emission rates must account for the density of photonic final states. The calculation starts with the differential spontaneous emission per solid angle, including the dipole radiation pattern and the polarization projections. Integration over angles yields the 2/3 factor that arises from averaging over dipole orientations and polarizations. The final rate exhibits a cubic dependence on frequency, reflecting both the single-photon field normalization and the photon density of states in three dimensions.

Outlook and Next Topics

The lecturer outlines what remains to be added to the theory, notably spontaneous emission damping and saturation, and previews the next steps toward a full microscopic treatment of spontaneous emission via field quantization. The course will also explore the role of dimensionality on mode density and spontaneous emission, as well as the historical context of Einstein’s AB coefficients and the emergence of the quantized electromagnetic field.