HOW SCIENCE WORKS · CONDENSED MATTER PHYSICS
Find your route through Anderson localisation
Anderson localisation occurs when a coherent wave is repeatedly scattered by static disorder and the many paths interfere. Return paths can reinforce while outward propagation is suppressed. The wave remains spatially localised even without a conventional confining wall.
Chapters 1–3.2. Work through the model
Chapters 4–5.3. Evaluate the evidence
Chapters 6–8.
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A mechanism-focused enrichment guide for curious students, parents and teachers. Worked numbers are explicitly hypothetical; research links identify supporting scientific studies. This is not an examination-syllabus claim or a laboratory operating procedure.
Chapter index
Mechanism · Chapters 1–3
Model · Chapters 4–5
Evidence and limits · Chapters 6–8
In a periodic structure, wave states can extend across the system. Random variations in site energy, spacing or refractive index scatter the wave and create a large family of paths between two points.
Because waves carry phase, intensities cannot simply be added path by path. Amplitudes interfere, and time-reversed loops have closely related phases in a symmetric system.
Weak disorder can reduce transport while still allowing diffusive spreading. Stronger coherent multiple scattering can make return amplitudes reinforce so effectively that transmission falls exponentially with distance.
This is not a classical particle merely bouncing more often. Localisation is a wave-interference effect and can weaken when phase coherence is lost.
A localised state has a characteristic localisation length ξ. Far from its centre, amplitude typically decays approximately exponentially; intensity falls with twice the exponent.
The value of ξ depends on disorder, energy or frequency, dimensionality and symmetry. In three dimensions a mobility edge can separate localised from extended states.
Suppose intensity follows I(x) = I₀ exp(−2x/ξ). At x = 2ξ, I/I₀ = exp(−4), about 0.018. Only about 1.8% of the central intensity remains in this idealised tail.
This demonstrates an exponential profile. It does not prove Anderson localisation because absorption and leakage can also reduce transmitted intensity.
| Observation | Localisation interpretation | Alternative |
|---|---|---|
| Exponential spatial tail | Finite localisation length | Absorption or leakage |
| Suppressed spreading | Interference limits transport | Finite observation time |
| Strong return probability | Constructive return paths | Classical trapping |
| Size-dependent transmission | Scaling towards localisation | Band gap or contact loss |
Absorption removes wave energy and usually converts it to other forms. Anderson localisation can suppress spatial transport while energy remains in the wave field until separate leakage or dissipation occurs.
Time-resolved measurements help separate them. Absorption changes survival, while localisation changes spatial spreading and transmission statistics.
Researchers vary disorder, size, dimension and coherence, then measure spatial profiles, transmission distributions, diffusion, return probability or wave-packet width. Scaling with size is especially informative.
A narrow output spot can mislead. Loss, finite apertures, nonlinear self-trapping and band gaps are alternatives that require controls.
A Physical Review Letters experiment with an atomic kicked rotor reported evidence of two-dimensional Anderson localisation through localised matter-wave dynamics.
Related behaviour has been studied with electrons, light, microwaves, sound and ultracold atoms. These platforms test the same interference principle under different controls.
Interactions, nonlinearity, moving disorder and environmental decoherence can change or destroy the ideal noninteracting picture. Finite experiments may also mimic a stopped packet before longer-time spreading appears.
The lesson is counter-intuitive: randomness does not always enhance spreading. When repeated scattering preserves phase, disorder can organise interference into confinement.
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