Following a released electric-flux string in time

Each energy component accumulates its own phase

Diagonalizing the finite Hamiltonian gives energies $\mathcal E_k$ and states $|k\rangle$ . In this new basis, the Hamiltonian multiplies each vector by one energy. Project the initial state onto these vectors, rotate each component by its energy-dependent phase, then add them back.

The overlap $\langle k|\psi(0)\rangle$ is the initial amplitude along energy state $k$ . Conjugate that energy vector’s components, multiply them by the corresponding initial components, and sum the products.

Relative phases change measurement probabilities. A dimming link represents a changing quantum expectation over flux configurations. A single measurement would give one allowed configuration. It need not resemble a smoothly fading classical rope.

|\psi(t)\rangle=\sum_k e^{-i\mathcal E_k t}|k\rangle\langle k|\psi(0)\rangle

With reduced Planck constant one, energy times time is a phase angle. Exact diagonalization solves the retained finite model; numerical rounding and diagonalization residuals still need checking.

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A resonance you can calculate by hand

Four sites with endpoint sources and electric cutoff one have exactly two physical configurations: the initial string and the state that screens its middle link. Subtract the initial string energy. Their diagonal entries are zero and $\Delta=2m-g^2/2$ ; the allowed hop between them is $-w$ .

Choose $m=1/4$ , $g^2=1$ and $w=1$ . The detuning vanishes and pair probability becomes $\sin^2t$ . Transfer is complete at $t=\pi/2$ , and the initial string returns at $t=\pi$ . These full oscillations are exact for this two-state truncated model.

H_{\mathrm{two}}=\begin{pmatrix}0&-w\\-w&\Delta\end{pmatrix},\qquad \Omega=\sqrt{\Delta^2+4w^2}

Detuning is the diagonal configuration-energy difference. Hopping mixes the configurations. The gap between the two eigenenergies sets the oscillation frequency.

P_{\mathrm{pair}}(t)=\frac{4w^2}{\Omega^2}\sin^2\left(\frac{\Omega t}{2}\right)

The state starts as the intact string. Detuning reduces the transfer amplitude. Raising the electric cutoff adds configurations and changes the problem.

How two charges fluctuate together

Two mean charges do not tell us how their fluctuations occur together. Multiply the two charges within each configuration, then average with its probability. Subtract the product of the separate means to obtain the connected correlation. The Lab observes site one and the second-to-last site, which are $1$ and $4$ on a six-site chain.

Use the four-site two-state example to calculate it. Let $p$ be the probability of the pair configuration. Its middle charges are minus one and plus one; the intact string has zero matter charge at both sites. The separate means are $-p$ and $p$ , but the joint mean is $-p$ . Their covariance is $-p(1-p)$ . At resonance and $t=\pi/4$ , both configurations have probability one half, so the covariance is minus one quarter.

The negative sign describes opposite charge fluctuations. After complete transfer both charges are certain, so their connected fluctuation returns to zero. This observable describes charge statistics. A classical mixture can give the same covariance.

C_{ab}(t)=\langle Q_aQ_b\rangle-\langle Q_a\rangle\langle Q_b\rangle,\qquad (a,b)=(1,N-2)

The joint and separate means use the same evolving state. Moving a source leaves the two observation sites fixed. Changing chain length changes the second-to-last index.

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Longer chains have more routes

Six or eight sites allow pairs to appear at different positions and move further. When one link weakens, inspect its neighboring charges. Their changes must obey the local Gauss relation. A spatial field profile often reveals more than a single average.

The closed system conserves energy. A decrease in electric energy can enter the mass term and the expectation of the hopping term. Hopping energy contains coherence between configurations; multiplying the visible pair count by mass does not account for it all.

A return can result from interference among a few energy levels or from propagation to a boundary and back. To identify a finite-volume echo, compare the same local region at different chain lengths and look for a boundary-dependent time shift. A matched local comparison at two chain lengths tests how the boundary contributes to that return.

Define what the pair count is relative to

At half filling and zero total matter charge, occupied even sites and holes on odd sites have equal total counts. The sum of even-site occupations is our bare pair number relative to the staggered product reference. Its expectation can be fractional.

That initial product state is not the interacting vacuum after hopping is turned on. It can develop bare pairs even without external sources. A source-induced increment therefore requires a second, source-free evolution with the same parameter and initial-state convention. Subtracting those curves defines an increment relative to that chosen quench baseline.

N_{\mathrm{pair}}(t)=\sum_{n\ \mathrm{even}}\langle\hat n_n(t)\rangle,\qquad \Delta N_{\mathrm{pair}}=N_{\mathrm{pair}}^{\mathrm{sources}}-N_{\mathrm{pair}}^{\mathrm{no\ sources}}

The first quantity is a bare pair count. The second uses a specified source-free quench as reference. Screening itself also requires the corresponding local field and charge profiles.

Check what must remain unchanged

State norm must remain one, total energy must keep its initial value and every $\langle\hat G_n^2\rangle$ must remain zero. The square matters: positive and negative violations can cancel in a mean residual, but both contribute positively to its square.

The next chapter adds cutoff and size comparisons. A poorly truncated Hamiltonian can conserve its own energy perfectly. Conservation checks the evolution within the chosen model; convergence checks whether omitted degrees of freedom change the answer.

Try the prediction

Start with a solvable oscillation, then lengthen the string

Set four sites, electric cutoff one and fixed endpoint sources, with mass 0.25, electric coupling one and hopping one. Compare the central field and pair count near $t=\pi/2$ and $t=\pi$ .
With the same two-state setup, open Numerical diagnostics and full record. Compare the covariance near $t=\pi/4$ with its value near complete transfer, using the pair probability at the actual sampled time.
Return to the light-matter preset, find a link that dims strongly, and explain its change using neighboring charges.
Compare heavy matter with frozen matter. One detunes transitions and the other removes hopping; explain the different curves.

Exercises and explanations

Why is the connected correlation zero when the pair is certain in the two-state example?
Read the explanation
The two charges are fixed at minus one and plus one. Their joint mean and the product of their means both equal minus one. Subtracting gives zero: the pair exists, but these charges no longer fluctuate.
For $\Delta=2$ and $w=1$ , what is the largest pair probability in the two-state model?
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Here $\Omega^2=4+4=8$ . The probability prefactor is $4/8=1/2$ , so the maximum is one half. Detuning prevents complete transfer.
What does $\langle G_n\rangle=0$ together with $\langle G_n^2\rangle>0$ tell you?
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The state or mixture contains locally violating configurations whose signed residuals cancel on average. A state entirely in the zero-residual physical sector has zero squared residual as well.

Sources and calculation scope

Sandbox Physics FR04 finite Hamiltonian, conventions and calculation scope
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.
E. A. Martinez et al., Real-time dynamics of lattice gauge theories with a few-qubit quantum computer, Nature 534, 516–519 (2016); arXiv:1605.04570v1 (15 May 2016)
Methods equation (3), Figure 2a and the main-text density definition fix the four-site ideal continuous-Hamiltonian calculation: electric energy equal to hopping, mass equal to half the hopping, zero boundary field and the bare spin vacuum. The associated vacuum-persistence observable is defined in the main text and used in Figure 3; no Figure 3 scan is reproduced. DOI: 10.1038/nature18318. The arXiv page displays a perpetual non-exclusive license; no source figures, author code or measured datasets are redistributed.
Sandbox Physics FR04 original numerical references and reproduction scripts
benchmarks-browser.json and benchmarks-independent.json retain the separately calculated raw observables; benchmarks-summary.json records their differences, numerical environment, cutoff and finite-size checks. scripts/frontier/fr04/reproduce-benchmarks.mjs and independent-reference.py regenerate the original deterministic records. No experimental data are included.