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Sandbox Physics

Q056 · Prepare / image / compare

Bose–Hubbard Lattice

Turn up repulsion, take occupation snapshots, and watch number fluctuations shrink. Four sites make the many-body state small enough to inspect.

Interactive modelBose–Hubbard Lattice
Accepted snapshots—\text{—}
Mean number variance · model—\text{—}

02 / TAKE A PICTURE

One preparation. Four sites.

Numbers are ideal atom counts. The tiles are decoded readings, not fluorescence brightness.

Mean number variance · recorded

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On-site atom pairs · recorded

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03 / REPEAT TO SEE THE PATTERN

How much does each site fluctuate?

Select a site to inspect its count distribution.

Blue bars: recorded frequencies. Amber ticks: model probabilities. Thin bars: pointwise 95% Wilson intervals.

Connected number covariance · from the same snapshots

Blue: negative. Amber: positive. Zero: white. Select a cell for its recorded and model values.

MODEL DIAGNOSTIC · NOT RECONSTRUCTED FROM THE IMAGES

Normalized coherence across the middle link. Number-only snapshots cannot determine this off-diagonal quantity.

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04 / KEEP A FAIR COMPARISON

Save. Change. Measure again.

Saving freezes this sample; the next capture starts a new one. Up to eight comparisons.

Scientific model & measurement boundaries

Four sites with open ends. Only nearest neighbors tunnel; the middle link can be weakened. The energy rises from left to right for positive tilt. One arbitrary reference energy sets every control. A complete canonical density matrix is used, including all eigenstates. At zero temperature, degenerate ground states are equally mixed.

Four bosons; all 35 occupation states are retained, including four atoms on one site. Reduced fluctuations indicate a finite-system Mott-like crossover, not a thermodynamic phase boundary.

Each trial starts from a new equilibrium preparation. Availability discards an entire image independently of its state; unread trials remain in the CSV. Ideal occupation readout differs from parity-only fluorescence microscopy. No optical point-spread function, light-assisted collisions, preparation dynamics or detector error is modeled.

The apparatus shows an enlarged four-site sample, collection objective and camera along a shared vertical axis. It illustrates readout geometry, not a calibrated optical design.

Zhang & Dong · 2011 · Cheuk et al. · 2016 · Bakr et al. · 2009

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Physics tutorial

Count fluctuations in a tiny lattice

BackgroundFour sites expose many-body statistics without a thermodynamic-limit claim.

Why it mattersA convincing picture is not yet a correlation measurement.

Start with the essentials

Focus question
Can repulsion lock the occupation?
One-sentence intuition
Every histogram and recorded correlation uses the same joint snapshots.

Core mathematical model

Bose–Hubbard Hamiltonian

H=−∑i=13Ji(bi†bi+1+h.c.)+U2∑ini(ni−1)+∑iViniH=-\sum_{i=1}^3J_i(b_i^\dagger b_{i+1}+\mathrm{h.c.})+\frac U2\sum_i n_i(n_i-1)+\sum_i V_i n_i

All four-boson configurations are retained. Site labels increase from left to right.

Number covariance

Cij=⟨ninj⟩−⟨ni⟩⟨nj⟩C_{ij}=\langle n_i n_j\rangle-\langle n_i\rangle\langle n_j\rangle

The diagonal is the number variance. Fixed total number requires every row to sum to zero. The empirical estimator uses the sample mean and divides by the number of readable shots.

Coherence needs a different measurement

g23(1)=⟨b2†b3⟩⟨n2⟩⟨n3⟩g_{23}^{(1)}=\frac{\langle b_2^\dagger b_3\rangle}{\sqrt{\langle n_2\rangle\langle n_3\rangle}}

The model computes this off-diagonal observable; site-count images alone cannot reconstruct it.

Prepare equilibrium

ρ=Z−1e−H/(kBT),Vi=F(i−5/2),J1=J3=J, J2=rJ\rho=Z^{-1}e^{-H/(k_BT)},\quad V_i=F(i-5/2),\quad J_1=J_3=J,\ J_2=rJ

Canonical ensemble in the fixed number sector. Zero temperature equally mixes degenerate ground states; changing controls prepares a new state, not a dynamical ramp.

A whole-image probability

P(n)=⟨n∣ρ∣n⟩P(\mathbf n)=\langle\mathbf n|\rho|\mathbf n\rangle

Sample a joint Fock configuration, never independent site marginals. The full image either reads successfully or is marked unread.

Common difficulties

Finite size

Typical misconceptionA four-site result is a bulk phase diagram.

Better mental modelIt is a finite-system crossover. No thermodynamic critical point, long-range order or superconductivity is inferred.

Readout contract

Typical misconceptionThe colored tiles reproduce a calibrated microscope image.

Better mental modelThey encode ideal number-resolved readings, with spin resolution for fermions. Real parity imaging, optical resolution and light-assisted loss require additional models.

Run the experiment

  1. 01

    Prepare

    Start with the first preset, then choose its contrasting partner.

    What to observe: Each preset includes 200 labeled simulated shots.
  2. 02

    Inspect

    Select a snapshot, a site and a correlation cell.

    What to observe: Single-shot configurations differ even at identical settings.
  3. 03

    Compare

    Save, adjust temperature or interaction, collect again and export.

    What to observe: The CSV preserves unread trials and captured settings.