In-Depth Notes on Light-Matter Interaction

Focus on photons and their interaction with materials.
Presentation based on a pre-print from "Optical Nanospectroscopy" by C. Höppener et al.

Max Planck (1858-1947) proposed a model to explain the spectrum of thermal radiation.
Key observations:
- A material emits a broad spectrum of electromagnetic radiation at temperature T.
- Intensity increases with temperature, shifting maximum spectral intensity towards shorter wavelengths (visible light).
Planck regarded energy exchanges as discrete (photons) with energy ΔEi = ħωi, leading to the concept of Planck's constant, h ≈ 6.626 × 10^-34 J·s.

Consider a one-dimensional cavity with two mirrors separated by distance L:
- Standing waves are formed, characterized by intensity maxima in the cavity.
- The resonance condition states that wavelengths fit such that L = n(λ/2) where n is a positive integer.
For three-dimensional cavities:
- Define wave numbers kn, leading to equations for three dimensions:
- k1 = (π/L)·n1
- k2 = (π/L)·n2
- k3 = (π/L)·n3
- Resulting wavelengths relationship: λn = 2L/n.

Spectral mode density n(ω) determines the number of modes at a frequency ω:
- In a cubic resonator, N(ω < ωm) is computed based on k-space geometry.
- Use the equation:
- n(ω < ωm) = (1/3)(ωm^3) / (π^2c^3).
Average energy per mode at thermal equilibrium:
- Following Planck’s law leading to the black-body radiation law, given by: ρ(ω, T) = n(ω)·W(T, ω).

Transition types:
- Absorption: Photon excites electron, transferring energy matching the resonance condition (E2 - E1 = ħω).
- Stimulated Emission: Excited electron returns to lower energy state, emitting a photon.
- Spontaneous Emission: Photon emission without external trigger.
Coefficients:
- Transition probabilities => P12, P21, spontaneous decay A21.
- Relate Einstein coefficients B12, B21 through these probabilities.

For a two-level system interacting with a light field:
- Change in populations of excited/ground states modeled using N1 and N2.
- Deduces the population ratio: N1/N2 ties back to Boltzmann distribution.
- Steady-state leads to a definitive equation that equates probabilities, revealing dependence on photon density.

Combines classical electrodynamics and quantum mechanics for a two-level atom excited by light.
Governs behavior when light interacts classically with atomic wave functions under specific conditions:
- Time evolution is dictated by Schrödinger’s equation modified for the interaction with a light wave.
Introduction of Rabi oscillations reflects state transitions depending on interaction strength and frequency.

In solids, interactions between valence and conduction bands:
- Absorption from lower states populates higher states, generating electron-hole pairs (related to the band gap energy).
- Reverse processes lead to emission (luminescence).

Excitons: Bound states of electrons and holes (Frenkel excitons and Wannier-Mott excitons).
- Charge transfer excitons result from intermolecular interactions.
Phonons: Lattice vibrations; energy carriers in solids and interact with electons.

Describe localized surface plasmon resonances (LSPRs) in nanoparticles:
- Relate to free carriers oscillating under electromagnetic fields, with properties influenced by particle size and surrounding medium.
- Applications extend to sensing and photothermal therapy.

The principles outlined are foundational for understanding nanotechnology and quantum optics.
Theoretical developments and experimental validations of these models continue to shape technological advances in science and engineering.