Kapitza-Dirac Scattering

Created in July 22, 2025

2025   ·   quantum-mechanics   atom-optics   laser-cooling   diffraction   matter-waves

Definition

Kapitza-Dirac scattering refers to the diffraction of matter waves (electrons, atoms, or molecules) by a standing wave of light, analogous to X-ray diffraction in crystals. Predicted by Pyotr Kapitza and Paul Dirac in 1933, it experimentally confirms the wave nature of particles.

Key Mechanism

  1. Light-Matter Interaction:
    A standing laser wave creates a periodic potential (optical lattice) via the AC Stark shift. The potential \(V(x)\) for a two-level atom is:
    \(V(x) = \frac{\hbar \Omega^2}{4 \Delta} \cos^2(kx)\)
    where \(\Omega\) is the Rabi frequency, \(\Delta\) the detuning, and \(k\) the wavevector.

  2. Diffraction Condition:
    Particles with de Broglie wavelength \(\lambda_{dB}\) satisfying \(2d\sinθ = n\lambda_{dB}\) (Bragg condition, \(d = \lambda_L/2\)) undergo constructive interference, splitting into discrete momentum states \(\pm n\hbar k_L\).

Experimental Setup

  1. Requirements:
    • Ultracold Atoms/Electrons: Typically Bose-Einstein condensates (BECs) or laser-cooled atoms (µK temperatures).
    • Standing Wave: Counter-propagating laser beams forming a 1D optical lattice (wavelength \(\lambda_L\)).
    • Pulse Duration: Short interaction time (µs to ms) to avoid spontaneous emission.
  2. Steps:
    • Cooling & Trapping: Prepare atoms in a magneto-optical trap (MOT) and evaporatively cool to BEC.
    • Pulse Application: Shine the standing wave for a time \(t\) (adjusted to reach \(n\)th-order diffraction).
    • Detection: Time-of-flight imaging reveals diffracted peaks separated by \(2\hbar k_L\).

Diffraction Regimes

  • Raman-Nath (Weak Pulses): Multiple diffraction orders (thin grating approximation).
  • Bragg (Strong Pulses): Selective \(n\)-th order transitions (thick grating).

Applications

  • Atom Interferometry: Precision measurements of \(g\), \(\hbar/m\).
  • Quantum Simulation: Emulating solid-state physics in optical lattices.
  • Matter-Wave Optics: Building atomic beam splitters for quantum computing.

Example Experiment

A \(^{87}Rb\) BEC exposed to a 780 nm standing wave (\(I \sim 10^9\, \text{W/m}^2\), \(\Delta \sim 1\, \text{GHz}\)) for \(t = 10\, \mu\text{s}\) produces \(\pm 2\hbar k_L\) momentum states with 30% efficiency.

Math Insight

The diffraction probability \(P_n\) for \(n\)-th order follows Bessel functions (Raman-Nath) or sinusoidal dependence (Bragg):
\(P_n = J_n^2(\Omega t/2) \quad \text{(Raman-Nath)}\)
\(P_{\pm1} = \sin^2(\Omega t/2) \quad \text{(Bragg)}\)

Note: Modern experiments use optical tweezers or delta-kick cooling to enhance resolution.