---
title: "Reading an electric string from atomic occupations"
description: "Follow an optical-lattice paper from atomic occupations to gauge observables, then calculate the strong-mass configuration-energy crossing."
canonical_url: "https://sandboxphysics.com/stories/lattice-gauge/optical-lattice-mapping/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/lattice-gauge/optical-lattice-mapping/"
last_modified: "2026-10-27"
---

[Lattice gauge fields and string breaking](https://sandboxphysics.com/stories/lattice-gauge/)

L4 · Chapter 7

# Reading an electric string from atomic occupations

Sandbox Physics · 2026-10-27

An atom microscope counts atoms in individual wells. Reading that image as an electric string requires a map from occupations to link fields and from field differences to charge. Open Liu et al., arXiv:2411.15443v1, and follow Figures 1, 2 and 5 in that order.

## Before you begin

You need site-by-site Gauss-law addition and the competition between electric energy and pair mass. This paper uses spin-one-half quantum links, so its parameters need their own definitions.

[Review the preceding chapter](https://sandboxphysics.com/stories/lattice-gauge/physical-subspace/)

- Count atomic wells, gauge links and matter sites separately.
- Read field reversal together with the appearance of charge.
- Estimate why longer strings need a smaller field bias to break.

## Figure 1 uses several different lengths

In Figure 1b, $L$ counts atomic wells. The corresponding gauge description has $L+1$ gauge positions and $L+2$ matter positions, including two fixed end charges. Their separation is $d=L+1$, but only $\mathcal L=L-1$ internal gauge links reverse between the string and broken-string configurations. Those are different counts.

Each quantum link has two states. The paper defines $E_{\ell,\ell+1}=(-1)^\ell S^z_{\ell,\ell+1}$, with spin components plus or minus one half. The integer rotor used earlier has a larger link space. Both descriptions can satisfy Gauss’s law, but they assign different states and energies to a link.

The paper writes $\tilde t$ for the matter–gauge hopping energy scale, $m$ for mass and $h$ for the background bias. Divide both mass and bias by that same hopping scale in the worked example.

$$
S^z=\pm\frac12,\qquad E^2=\frac14
$$

Electric-field squared is constant in this two-state space. The tunable string tension comes from a linear background-field term, with a different role from the rotor’s electric-energy coefficient.

Count positions using Figure 1 before inserting a length into the energy estimate.

| Atomic wells | Gauge positions | Matter positions including ends | Reversed internal links |
| --- | --- | --- | --- |
| $L=6$ | 7 | 8 | 5 |
| $L=8$ | 9 | 10 | 7 |
| $L=10$ | 11 | 12 | 9 |

Reading: Ying Liu et al. · Sandbox Physics FR04 finite Hamiltonian

## Figure 2 compares separately prepared final states

The optical lattice holds rubidium atoms in wells. Appendix A maps neighboring atomic occupations onto two link-spin states; Gauss’s law then determines matter charge. Atomic population, link spin and gauge-matter population therefore need separate labels.

Appendix A restricts this mapping to the near-resonant condition $U\approx\Delta\gg J$. Here $U$ is the interaction energy of double occupation, $\Delta$ is the neighboring-well energy offset and $J$ is atomic hopping. Tuning the offset near the double-occupation cost selects processes such as $|1,1\rangle\leftrightarrow|2,0\rangle$ and suppresses off-resonant hopping. This atomic hopping parameter has a different meaning from the electric-energy coefficient with the same letter in the next paper.

Preparation begins with one atom in each well, $|111\ldots1\rangle$. Figure 2b raises mass from a negative value towards zero over 75 milliseconds, then changes mass and background field together over 150 milliseconds. The microscope reads the final occupations. Each field setting in Figures 2c and 2d uses another preparation. Moving along that horizontal axis does not follow a single released string through time.

For six wells, the illustrated atomic patterns are $|202020\rangle$ for the string and $|120201\rangle$ for the broken string. Both contain six atoms. Creating a pair of encoded matter charges rearranges the same six rubidium atoms.

Reading: Ying Liu et al. · Sandbox Physics FR04 finite Hamiltonian

## Use charge to interpret the field reversal

Read the internal fields in Figure 2c, then the charges at the same mass in Figure 2d. Newly appearing opposite charges connect the field reversal to screening. A local check follows directly from Gauss’s law: the difference between neighboring fields should match the matter charge between them.

Figure 3 averages the $L-1$ internal links and takes an absolute value. That step removes direction information, so return to the signed local fields when identifying the dominant orientation. The experiment also rejects shots that fail total atom-number or Gauss-law tests. This postselection makes the displayed mean conditional on the retained shots.

$$
\bar E=\frac{1}{L-1}\sum_{\ell=1}^{L-1}\langle E_{\ell,\ell+1}\rangle
$$

The brackets average repeated preparations at a fixed setting. The outer sum averages internal links. Opposite field orientations can have the same absolute spatial mean.

Reading: Ying Liu et al. · Sandbox Physics FR04 finite Hamiltonian

## Estimate the slope in Figure 5 from configuration energies

At large positive mass, first neglect hopping-induced mixing. Creating a matter pair costs $2m$. Reversing $\mathcal L$ internal links into the orientation favored by the background releases $h\mathcal L$. Taking the intact string as the energy reference gives the difference below.

Choose $m/\tilde t=4$ for a worked zero-hopping estimate. Five internal links in a six-well chain give $h_c/\tilde t=8/5=1.60$. Seven links give $8/7\approx1.14$, and nine give $8/9\approx0.889$. A longer string can release energy from more links, so a weaker bias can pay for the pair.

Figure 5 instead extracts critical fields from experimental observables. Our values are newly calculated configuration-energy crossings. Finite hopping mixes configurations; system size and preparation speed also affect the observed conversion. Comparing the fitted experimental slopes requires the paper’s specified observable, selection and fitting procedure.

$$
\Delta\mathcal E=2m-h(L-1),\qquad \frac{h_c}{m}\approx\frac{2}{L-1}
$$

A positive difference favors the intact string. A negative difference favors the screened configuration. The approximate crossing describes this strong-mass configuration comparison.

Reading: Ying Liu et al. · Sandbox Physics FR04 finite Hamiltonian

## Carry the mapping to the next paper

Write down what the apparatus measures before writing the model quantity inferred from it. Count fixed endpoints, internal links and the positions entering each average. Record the initial state, parameter ramp and readout time together. These choices tell you whether to compare a time trajectory or a family of approximately ground-state configurations.

A useful next question is how lower mass or a faster ramp broadens the conversion between the two configurations. Answering it starts with one explicit convention for staggered links, spin raising and lowering, and matter charge, followed by a finite matrix that preserves each Gauss generator. The next chapter completes an independent calculation for a four-site Hamiltonian with a directly specified spin representation.

## Try the prediction

### [Infer a screening pair from field differences](https://sandboxphysics.com/labs/gauss-law-builder/)

1. Choose An allowed string and check the endpoint sources and internal flux.
2. Choose A screening pair. Locate the weakened field segment and infer the signs of the matter charges from its ends.
3. Compare that step with Figures 2c and 2d: field changes and charge positions must agree in both. Read the atomic occupations using the paper’s separate two-state link mapping.

## Exercises and explanations

1. Why does the ten-well energy release use $9$ links rather than the source separation $11$?

   Read the explanation

   The boundary links do not reverse with the newly created screening pair. Only nine internal links change orientation and contribute to the energy difference; the source separation also includes two endpoint segments.

2. At $m/\tilde t=3$, where is the eight-well configuration-energy crossing?

   Read the explanation

   Seven internal links reverse. Equating the pair cost of six to the released field energy gives $h_c/\tilde t=6/7\approx0.857$. Extracting an experimental critical field still requires the specified measured observable.

3. Can the mean field of postselected shots establish that the original experiment had no violations?

   Read the explanation

   No. It describes the shots that passed the tests. Assessing device errors also requires the original shot count, pass fraction and violation distribution.

[Previous: Projecting out violations or penalizing them](https://sandboxphysics.com/stories/lattice-gauge/physical-subspace/) [Next: Independently recalculating a four-site pair-production curve](https://sandboxphysics.com/stories/lattice-gauge/reproducible-comparison/)

## Sources and calculation scope

1. [Sandbox Physics FR04 finite Hamiltonian, conventions and calculation scope](https://github.com/yangzichao/sandboxphysics/blob/main/docs/frontier/fr04/model-specification.md)

   Defines the course’s zero-based staggered charge, open boundaries, integer-flux hard cutoff, source geometry and deliberate gauge-breaking error. The accompanying paper-reproduction and numerical-validation notes specify which finite-model calculations were checked independently. This is our model definition and calculation record.

2. [Ying Liu et al., String breaking mechanism in a lattice Schwinger model simulator, arXiv:2411.15443v1 (23 November 2024)](https://arxiv.org/html/2411.15443v1)

   Equation (1), Figures 1–3 and 5, and Appendix A fix the spin-one-half quantum-link Hamiltonian, optical-lattice mapping, static sources and measured observables. The reading guide recalculates the zero-hopping configuration-energy relation behind Figure 5 for atomic chain lengths 6, 8 and 10; it does not reproduce the full quantum-link Hamiltonian. Figure 2 supplies the experimental protocol context. The arXiv page displays a perpetual non-exclusive license. No measured points, source figures, fitted slopes or experimental ramp dynamics are reproduced.