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Sandbox Physics

Q059 · Release / detect / compare

Anderson Localization

Release one atom into a rough lattice. Compare spreading, change the landscape, and build the position distribution one detection at a time.

Interactive modelAnderson Localization
Evolution time · model00
Width · model00

02 / ONE CLICK IS ONE POSITION

Build the distribution, click by click.

Mean position · recorded

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Width · recorded

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Read / attempted · this time

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Blue bars: recorded frequencies. Amber: model probability. Thin lines: pointwise 95% Wilson intervals. Position is in lattice spacings; mean error is one empirical standard error. No data at this time means no estimate.

03 / FOLLOW THE WHOLE EVOLUTION

How far does the wave spread?

Model probability · click or drag to choose a time

Time runs downward. Color uses a fixed logarithmic probability scale from one hundred-thousandth to one; dark cells are not proof of zero probability.

Width over time · model line and recorded dots

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Participation counts effectively occupied sites in the model. It is not a fitted localization length.

04 / WHAT THE ATOM EVOLVES THROUGH

Inspect the roughness.

Vn/E∗V_n/E_*

Drag upward to increase disorder. Blue sites are lower in energy. Smoothing holds ensemble variance fixed, not the range of each sample.

Quasimomentum · model only ka/πka/\pi

An 81-point Fourier diagnostic over the first Brillouin zone, separate from the acquired site records. It is not a simulated time-of-flight image.

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05 / KEEP A COMPARISON

Same question. Different conditions.

Up to eight frozen samples. A new acquisition after saving starts another sample.

Scientific model & limits

One particle, 81 sites from −40 to 40, open ends and nearest-neighbor tunneling. All energies use a fixed reference energy; time uses the corresponding reduced-Planck time. The finite Hamiltonian is diagonalized completely and evolved unitarily. No state renormalization hides numerical error.

H=−J∑n(∣n⟩⟨n+1∣+h.c.)+∑n(ϵn−Fan)∣n⟩⟨n∣H=-J\sum_n(|n\rangle\langle n+1|+\mathrm{h.c.})+\sum_n(\epsilon_n-Fan)|n\rangle\langle n|

The initial state is a normalized discrete Gaussian with a phase gradient, or one occupied site at zero width. Disorder starts as independent uniform noise, optionally convolved with a Gaussian kernel normalized by its squared weights. It is not optical speckle or a quasiperiodic potential.

Single band throughout: no band excitation or Landau–Zener leakage is calculated. The Bloch period and clean-chain reference apply without disorder and before boundary effects. A finite-time plateau cannot establish infinite-system Anderson localization or many-body localization.

Every trial freshly prepares the state, evolves to the chosen time, and measures position once. State-independent missed detections remain in the CSV with no hidden site. The microscope is a geometric schematic with enlarged spacing; no fluorescence, resolution or optical calibration is simulated.

Billy et al. · 2008 · Korsch & Mossmann · 2003 · Ben Dahan et al. · 1996

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Physics tutorial

Disorder versus spreading

BackgroundA finite coherent chain connects an explicit Hamiltonian to individual site measurements.

Why it mattersA still picture cannot distinguish confinement, a recurrence, and a boundary reflection.

Start with the essentials

Focus question
What stops the spreading?
One-sentence intuition
Compare entire time records and independent disorder samples.

Core mathematical model

One finite chain

H=−J∑n(∣n⟩⟨n+1∣+h.c.)+∑n(ϵn−Fan)∣n⟩⟨n∣H=-J\sum_n(|n\rangle\langle n+1|+\mathrm{h.c.})+\sum_n(\epsilon_n-Fan)|n\rangle\langle n|

Sites run left to right; a positive force lowers the site energy to the right. The reference energy stays fixed when tunneling changes.

Initial preparation

ψn(0)∝exp⁡[−(na)2/(4σ02)+ik0an]\psi_n(0)\propto\exp[-(na)^2/(4\sigma_0^2)+ik_0an]

Normalize the discrete Gaussian on the finite chain. Zero width selects the central site instead. Each shot prepares the same initial state afresh.

Position statistics

xˉ=a∑nnpn,σx2=a2∑n(n−xˉ/a)2pn\bar x=a\sum_n n p_n,\quad \sigma_x^2=a^2\sum_n(n-\bar x/a)^2p_n

Model probabilities and empirical frequencies remain separate. A measured mean uses accepted positions, with a sample standard error when at least two positions are available.

A spreading benchmark

σx2(t)=2a2(Jt/ℏ)2\sigma_x^2(t)=2a^2(Jt/\hbar)^2

A central-site initial state in an infinite clean chain spreads ballistically. The finite open chain agrees before its ends matter.

Participation

Peff=(∑npn2)−1P_{\rm eff}=(\sum_n p_n^2)^{-1}

Counts effectively occupied sites in the modeled wave. It is neither a fitted exponential length nor an infinite-system classification.

Correlated disorder

ϵn=W∑jwjun+j∑jwj2,uj∼U[−1/2,1/2]\epsilon_n=W\frac{\sum_j w_j u_{n+j}}{\sqrt{\sum_j w_j^2}},\quad u_j\sim\mathcal U[-1/2,1/2]

Gaussian weights give the smoothing control. The ensemble variance stays fixed; a single realization is never rescaled. This noise is not speckle.

Common difficulties

Finite observation

Typical misconceptionA narrow packet proves an infinite-system phase.

Better mental modelFinite time, finite length and sample dependence remain. No localization length, mobility edge or many-body phase is inferred.

One band

Typical misconceptionA high force here validates a real optical lattice at any strength.

Better mental modelThere is no second band in this model. A real instrument may leak to higher bands; no quantitative force limit is asserted without a specified band gap.

Run the experiment

  1. 01

    Predict

    Compare the first two presets before looking at the data.

    What to observe: Each preset includes 200 explicitly simulated detections.
  2. 02

    Follow time

    Drag time or the probability map; record 17 times.

    What to observe: Playback preserves data and adds no trials; the scan prepares independent atoms at every time.
  3. 03

    Test the explanation

    Save, change the landscape or initial width, scan again and export the raw records.

    What to observe: The saved sample retains its settings; missed detections have no hidden site in the export.