Q059 · Release, detect and compare

Anderson Localization

Can a wave get stuck without a wall? Yes, by interference alone. In a disordered lattice the wave scatters many times, and the paths leading away from the start interfere destructively. The atom stops spreading, and its position distribution stays localized with no wall anywhere.

Release one atom into a disordered lattice and compare how far it spreads. Change the disorder landscape, and build up the position distribution one detection at a time.

Evolution time · model $0$

Width · model $0$

01 / Release a matter wave

Roughness interrupts the spread.

Compare the clean lattice with a rough one. Then change the disorder sample: does the packet still stay nearby?

Enlarged lattice · blue glow = model probability, amber ring = last detected site

Move through the evolution

tE_*/\hbar=0

Try a comparison

Tunneling

J/E_*

1

Disorder strength

W/E_*

5

Preparation & detection

Disorder smoothing

\ell/a

0

Initial width · zero is one site

\sigma_0/a

0

Initial phase gradient

k_0a/\pi

0

Disorder realization

r

17

Detection probability

\eta

1

Sampling seed

s

2026

Moving time preserves measurements. Changing preparation starts a new sample; save the old one first. Each detection uses a newly prepared atom.

The model is a one-band tight-binding chain of 81 sites with open ends, holding one noninteracting particle. Position records are ideal simulated detections of the site the atom is on. The model has no tunneling between bands and no calibrated optical imaging, and it does not prove localization in an infinite system.

02 / Each click gives one position

Build the distribution, click by click.

Model reference

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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The participation number counts how many sites the model state effectively occupies. It is not a fitted localization length.

04 / The potential the atom moves through

Inspect the roughness.

$V_n/E_*$

Drag upward to increase disorder. Blue sites are lower in energy. Smoothing keeps the ensemble variance fixed; the range of each sample can still change.

Quasimomentum · model only $ka/\pi$

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 / Save a sample for comparison

Ask the same question under 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\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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Start with the essentials

Focus question: What stops the spreading?
One-sentence intuition: One snapshot is not enough: compare whole time records and several independent disorder samples.

Core mathematical model

The finite chain

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, and a positive force lowers the site energies toward the right. Changing the tunneling leaves the reference energy fixed.

Initial state

\psi_n(0)\propto\exp[-(na)^2/(4\sigma_0^2)+ik_0an]

The discrete Gaussian is normalized on the finite chain. Zero width puts the atom on the central site instead. Every shot starts again from the same initial state.

Position statistics

\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 measured frequencies are kept apart. The measured mean uses only detected positions, and a standard error is given once at least two positions are available.

Ballistic spreading in a clean chain

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

A state that starts on the central site of an infinite clean chain spreads ballistically. The finite open chain agrees until its ends start to matter.

Participation number

P_{\rm eff}=(\sum_n p_n^2)^{-1}

It counts how many sites the modeled wave effectively occupies. It is not a fitted exponential length, and it does not classify an infinite system.

Correlated disorder

\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 set how smooth the landscape is. The ensemble variance stays fixed, and no single realization is rescaled. This noise is not speckle.

Common difficulties

A finite run cannot show an infinite-system phase

Typical misconceptionA narrow packet proves the phase of an infinite system.

Better mental modelThe run lasts a finite time on a finite chain, and the result depends on the disorder sample. The Lab infers no localization length, mobility edge or many-body phase.

Only one band is modeled

Typical misconceptionIf a strong force works here, it works in a real optical lattice at any strength.

Better mental modelThis model has no second band. A real instrument may leak atoms into higher bands, and without a specified band gap no numerical force limit is stated.

Run the experiment

01
Predict first
Compare the first two presets before looking at the data.
What to observe: Each preset includes 200 detections, all clearly marked as simulated.
02
Follow the motion in time
Drag time or the probability map, then record 17 times.
What to observe: Playback keeps the data and adds no trials. The scan prepares an independent atom at every time.
03
Test the explanation
Save, change the landscape or initial width, scan again and export the raw records.
What to observe: The saved sample keeps its settings. In the export, a missed detection has no site attached.