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Sandbox Physics

Q060 · Release / detect / compare

Bloch Oscillations

Tilt a periodic lattice and scrub through the motion. Compare a moving packet with a breathing single-site state, then add disorder to disturb the revival.

Interactive modelBloch Oscillations
Evolution time · model00
Mean position · 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

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 / WHAT THE ATOM EVOLVES THROUGH

Change the slope.

Vn/E∗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/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

Why a lattice turns motion around

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
Can a constant force produce a return?
One-sentence intuition
The periodic band changes velocity; a single-site input reveals breathing instead of centroid motion.

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.

One-band motion

E(k)=−2Jcos⁡(ka),ℏk˙=FE(k)=-2J\cos(ka),\quad \hbar\dot k=F

In the clean infinite chain, velocity follows the slope of the periodic dispersion. Continuing across the zone changes the sign of velocity.

Bloch period

TB=2πℏ∣F∣aT_B=\frac{2\pi\hbar}{|F|a}

The clean single-band recurrence is an analytic reference. With disorder, zero force or significant edge effects it is not a predicted recurrence of the full model.

Full-state return

F(t)=∣⟨ψ(0)∣ψ(t)⟩∣2\mathcal F(t)=|\langle\psi(0)|\psi(t)\rangle|^2

A model-only fidelity: returning in position alone cannot recover the phases required to measure this quantity. A single-site state breathes even while its centroid stays fixed.

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.