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Sandbox Physics

Q014 · Prepare / couple / read out

Coherent Double Well Transfer

Raise the central barrier or tilt the wells. Watch probability move between two sides; test a closing pulse, reconstruct relative coherence and expose the limits of a two-state model.

Interactive modelCoherent Double Well Transfer
Model initial energy gap—\text{—}
Model right-side probability—\text{—}
Recorded evidence & model checks

SIMULATED INDEPENDENT PREPARATIONS

Does the measured transfer follow two states?

Blue points are measured right-side fractions with 95% Wilson intervals. Gray is the spatial model; dashed orange is the unnormalized two-state projection. The lower bars count both phase analyzers, including the subspace complement.

Measured coherence and phase

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Model norm / two-state leakage

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Initial spatial gap / projected gap

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Basis and boundary checks

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The outer walls at positions minus three and plus three are physical. The reference localized states come from the unbiased lowest doublet and retain small opposite-side tails. Their relative phase is a basis coherence, not the phase at a point. A sudden closing pulse can populate higher modes; the two-state projection discards that amplitude without renormalizing it.

Sources: MIT 8.04 · 11 · Feynman III · 8

Recorded groups
Latest 12 events; CSV includes all events and settings.

Physics tutorial

Transfer is a phase-sensitive beat

BackgroundConfinement changes the allowed spatial states.

Why it mattersConnect a potential you can change to records you can actually estimate from.

Start with the essentials

Focus question
When does the two-state picture fail?
One-sentence intuition
Numerical states, model references and measured outcomes have different roles.

Core mathematical model

Units

X=x/ℓ,μ=m/m∗,E=Em∗ℓ2/ℏ2,τ=ℏt/(m∗ℓ2)X=x/\ell,\quad \mu=m/m_*,\quad \mathcal E=Em_*\ell^2/\hbar^2,\quad \tau=\hbar t/(m_*\ell^2)

Choose a reference mass and length. These units apply to controls, axes and CSV. The double-well mass ratio is fixed at one.

Spatial approximation

uj(X)=2/Lsin⁡ ⁣[jπ(X+L/2)L],Hjk=j2π22μL2δjk+∫ujVuk dXu_j(X)=\sqrt{2/L}\sin\!\left[\frac{j\pi(X+L/2)}{L}\right],\quad H_{jk}=\frac{j^2\pi^2}{2\mu L^2}\delta_{jk}+\int u_jVu_k\,dX

All retained sine modes evolve. Potential integrals are analytic; the displayed matrix residual measures the eigensolver, not basis or boundary error.

Double-well potential

V(X,τ)=B(τ)1∣X∣<b/2+ϵ2sgn⁡X,−3<X<3V(X,\tau)=B(\tau)\mathbf1_{|X|<b/2}+\frac{\epsilon}{2}\operatorname{sgn}X,\quad -3<X<3

The outer walls are infinite. Positive bias raises the right side. A closing protocol abruptly changes the barrier to twelve at the selected time; changing controls starts a new preparation.

Localized reference pair

∣L⟩=∣0⟩+∣1⟩2,∣R⟩=∣0⟩−∣1⟩2,∣ψ(0)⟩=1−r∣L⟩+eiϕr∣R⟩|L\rangle=\frac{|0\rangle+|1\rangle}{\sqrt2},\quad |R\rangle=\frac{|0\rangle-|1\rangle}{\sqrt2},\quad |\psi(0)\rangle=\sqrt{1-r}|L\rangle+e^{i\phi}\sqrt r|R\rangle

Use the lowest two unbiased spatial eigenstates, choosing signs to put the left state mostly on the left. Small opposite-side tails remain. The narrow-packet challenge instead projects a Gaussian centered at minus 1.7 with position standard deviation 0.28 into the sine basis and normalizes that preparation.

Coherent time evolution

∣ψ(τ)⟩=e−iHτ∣ψ(0)⟩,∣ψ(τ>τc)⟩=e−iHc(τ−τc)e−iHτc∣ψ(0)⟩|\psi(\tau)\rangle=e^{-iH\tau}|\psi(0)\rangle,\qquad |\psi(\tau>\tau_c)\rangle=e^{-iH_c(\tau-\tau_c)}e^{-iH\tau_c}|\psi(0)\rangle

Eigenphases evolve exactly within the retained spatial basis, without time stepping. The state is continuous at a quench; energy can change by the work done on the barrier. There is no environment or continuous measurement.

Two-state comparison

H2=PHP,P=∣L⟩⟨L∣+∣R⟩⟨R∣,PR(τ)=12+[PR(0)−12]cos⁡[(E1−E0)τ]H_2=P H P,\quad P=|L\rangle\langle L|+|R\rangle\langle R|,\qquad P_R(\tau)=\tfrac12+[P_R(0)-\tfrac12]\cos[(\mathcal E_1-\mathcal E_0)\tau]

The cosine relation is exact for the symmetric, static, left-localized reference preparation. The projected model uses the same physical side observable and does not renormalize discarded amplitude. Bias, narrow states and sudden barrier changes can expose its limitations.

Phase-sensitive measurements

∣±θ⟩=∣L⟩±eiθ∣R⟩2,p+−p−=2Re⁡(e−iθaL∗aR)|\pm_\theta\rangle=\frac{|L\rangle\pm e^{i\theta}|R\rangle}{\sqrt2},\quad p_+-p_-=2\operatorname{Re}(e^{-i\theta}a_L^*a_R)

Real and imaginary analyzers use angles zero and ninety degrees on separate fresh preparations. The complement of this reference pair is recorded as a third outcome. Left/right counts alone cannot reconstruct coherence.

Common difficulties

A cloud is not a path

Typical misconceptionAn animated dot crosses the wall like a classical particle.

Better mental modelThe animation shows coherent probability density. Each recorded event uses a freshly prepared state.

More events do not fix the model

Typical misconceptionSampling noise is the only source of error.

Better mental modelFinite basis, artificial boundaries and omitted detector physics need separate checks.

Run the experiment

  1. 01

    Transfer

    Play the default state and sample nine times.

    What to observe: The gray reference and observed fractions can be compared.
  2. 02

    Hold

    Try a tall, wide barrier; then tune the closing time.

    What to observe: A closed barrier need not freeze a quenched state perfectly.
  3. 03

    Read phase

    Sample both coherence analyzers at the same time.

    What to observe: A phase estimate requires measurable coherence.
  4. 04

    Break the approximation

    Choose the narrow packet and compare the two model curves.

    What to observe: Higher modes can carry substantial probability.