---
title: "Bose–Hubbard · Count the Fluctuations Interactive Lab | Sandbox Physics"
description: "What happens to atoms in a lattice as their repulsion grows?"
canonical_url: "https://sandboxphysics.com/labs/bose-hubbard-lattice/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/bose-hubbard-lattice/"
last_modified: "2026-10-27"
---

Quantum matter / Bosons

# Bose–Hubbard · Count the Fluctuations

**What happens to atoms in a lattice as their repulsion grows?** The number of atoms on each site fluctuates less, because a second boson on the same site now costs more energy. The atoms get pinned to their sites and lose phase coherence between them, and the sharp interference peak in their momentum distribution fades.

Prepare atoms in a small lattice and count the atoms on every site, then release independent copies of the same state to read out their momentum. As the repulsion grows, the atom number on each site fluctuates less and the interference peak fades.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The lattice has two to four sites. The calculation uses the complete single-band basis at fixed total atom number and prepares either the ground state or a canonical thermal state. Atom counting and single-atom momentum measurements are ideal, with no imaging loss. What you see is a gradual change in a finite system, not a thermodynamic phase boundary.

Site-by-site view of the lattice

$L=4$

One atom Empty site Schematic · one orbital per site

Measure

Each shot starts with a freshly prepared state.

Sites with two or more atoms $0\%$

Sites with exactly one atom $0\%$

Model probabilities · averaged over every site or bond.

From prediction to recorded outcomes

## Which patterns keep appearing?

Select a row to inspect it above. Gold is the model probability; teal is its frequency in your snapshots.

Each snapshot comes from a fresh preparation

## The same state gives different outcomes each time.

Recent snapshots Select to replay

Read across a snapshot from left to right. A gap is a real empty site; two dots share an orbital. These are independent outcomes, not frames of atoms moving.

Measured site by site

## Count first. Average second.

Teal bars: recorded mean. Whiskers: one standard error. Gold marks: exact model expectation.

$M=0$ $\text{—}$ $\text{—}$

Pair count is averaged per site; the neighbor readout is the raw spin product, separate from connected covariance below. Estimates use raw rows only. Error bars describe independent preparations, not independent sites. A single preparation has no estimated standard error.

Per-site numerical values

| Site | Measured mean | Sample variance | Model mean |
| --- | --- | --- | --- |

Measured momentum

## Does the central peak survive?

$ka/\pi$

Teal: frequencies in 24 equal bins. Gold: bin-integrated model probabilities, not fitted to records. The flat Wannier envelope and one-zone readout are idealizations; one atom is recorded per independent preparation.

## Keep a comparison

| Trial | Readout | Result |
| --- | --- | --- |

Latest twelve records shown. CSV retains every outcome and setting; fermion occupations list all up sites, then all down sites. Save up to six samples and select one to restore it.

Exact small system · explicit limits

$D=0$

All number configurations in the selected sector are included. There is no per-site occupation truncation beyond the fixed total boson number. The residual checks numerical diagonalization, not the accuracy of the single-band physical model.

$\begin{gathered}x_i=i-(L-1)/2\\V_i=Vx_i^2+Fa,x_i\end{gathered}$

Sites run left to right in spacing units. The middle-bond multiplier changes the central bond of four sites, or the first bond of a shorter chain. All energies use a fixed reference energy. Open boundaries; no loading ramp, higher bands, imaging point-spread function, reservoir or two-dimensional bulk limit. At zero temperature, degenerate ground states are equally mixed.

[Greiner et al. · 2002](https://doi.org/10.1038/415039a) [Mazurenko et al. · 2017](https://arxiv.org/html/1612.08436v1) [Companion experiment](https://sandboxphysics.com/labs/fermi-hubbard-spin-correlations/)

Physics tutorial

## Bose–Hubbard · Count the Fluctuations

**Background** Prepare atoms in a small lattice and count the atoms on every site, then release independent copies of the same state to read out their momentum. As the repulsion grows, the atom number on each site fluctuates less and the interference peak fades.

**Why it matters** Prepare a few sites again and again, count the atoms each time, and the counts reveal many-body physics.

Start with the essentials

- **Focus question**: What happens to atoms in a lattice as their repulsion grows?
- **One-sentence intuition**: Even with the total atom number fixed, the number on each site can fluctuate. Strong repulsion makes atoms avoid sharing a site and weakens single-particle coherence.

### Core mathematical model

#### Open-chain Hamiltonian

$$
\begin{gathered}H=-J\sum_i(b_i^\dagger b_{i+1}+\mathrm{h.c.})\\+\frac U2\sum_i n_i(n_i-1)+\sum_i V_i n_i\end{gathered}
$$

Each site has a single orbital. Energies are given in a fixed reference unit, so changing the hopping does not change the unit.

#### Momentum probability density

$$
p(q)=\frac{1}{2\pi N}\sum_{ij}\langle b_i^\dagger b_j\rangle e^{iq(i-j)}
$$

Each independent preparation contributes one randomly chosen atom, read out within one Brillouin zone. The Wannier envelope is assumed flat.

#### Local number fluctuation

$$
\operatorname{Var}(n_i)=\langle n_i^2\rangle-\langle n_i\rangle^2
$$

The table uses the unbiased sample variance. A single snapshot cannot estimate a fluctuation.

### Common difficulties

#### A few sites are not an infinite system

**Typical misconception** A few sites are enough to locate a macroscopic phase transition.

**Better mental model** These exact calculations on small systems show gradual crossovers and correlations, not thermodynamic critical points.

#### Calculated curves and counted atoms

**Typical misconception** The calculated curve is reconstructed from the counted atoms.

**Better mental model** Reference curves come straight from the density matrix. The bars, means and covariances use only the raw samples.

### Run the experiment

1. 01

   #### Start with a control

   Acquire the free case and save it. Increase repulsion without changing the filling.

   **What to observe:** Compare the variance on each site, then acquire an independent momentum sample.

2. 02

   #### Break the pattern

   Remove one atom, then raise the central confinement.

   **What to observe:** The same repulsion need not produce the same correlations at another filling or trap.

3. 03

   #### Test a limit

   Set tunneling to zero. Compare two and four sites and export complete records.

   **What to observe:** The degenerate ground states are mixed with equal weights, so no single ordered snapshot is picked.

Physical conclusion

The lattice has two to four sites. The calculation uses the complete single-band basis at fixed total atom number and prepares either the ground state or a canonical thermal state. Atom counting and single-atom momentum measurements are ideal, with no imaging loss. What you see is a gradual change in a finite system, not a thermodynamic phase boundary.

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)