Q060 · Release, detect and compare

Bloch Oscillations

Why does a steady push make a particle in a lattice swing back and forth instead of speeding up? Because in a periodic lattice, a steady push raises the quasimomentum until it wraps around the band, and there the velocity reverses. A wave packet keeps swinging and never speeds away. A state that starts on a single site breathes in and out instead, and disorder spoils the return.

Tilt a periodic lattice and drag through time to follow the motion. Compare a moving wave packet with a state that starts on one site and breathes in place, then add disorder to disturb the revival.

Evolution time · model $0$

Mean position · model $0$

01 / Release a matter wave

The force pushes one way, yet the packet keeps coming back.

Move through time. The packet turns around even though the force keeps pointing the same way.

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

Force to the right

Fa/E_*

0.8

Preparation & detection

Disorder strength

W/E_*

0

Disorder smoothing

\ell/a

0

Initial width · zero is one site

\sigma_0/a

3

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

Returning is different from standing still.

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.

Mean position over time · model line and recorded dots

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Revival fidelity compares the full state with the initial state. Position images alone cannot reconstruct it.

04 / The potential the atom moves through

Change the slope.

$V_n/E_*$

Drag upward for a force to the right: the site energy falls to the right. A slope does not change the physical positions of the sites.

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: Can a constant force make a particle come back?
One-sentence intuition: The periodic band makes the velocity change as the force drives the momentum; a state started on a single site breathes instead of moving its center.

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.

Motion in one band

E(k)=-2J\cos(ka),\quad \hbar\dot k=F

In a clean infinite chain, the velocity follows the slope of the periodic band. As the momentum continues across the zone, the velocity changes sign.

Bloch period

T_B=\frac{2\pi\hbar}{|F|a}

This is the analytic return time for a clean single band. With disorder, zero force or strong edge effects, it no longer predicts when the full model returns.

Return of the full state

\mathcal F(t)=|\langle\psi(0)|\psi(t)\rangle|^2

This fidelity comes only from the model, because position detections cannot recover the phases needed to measure it. A single-site state breathes even while its center stays put.

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 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.