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Sandbox Physics

Q023 · Closed path / reference arm

Berry Phase Interferometer

Turn an effective field around a cone. Recombine the evolving spin with a stationary reference arm, scan the analyzer and recover the phase from recorded port counts. Reverse the loop or speed it up.

Interactive modelBerry Phase Interferometer
Model interferometric phase—\text{—}
Model spin leakage—\text{—}
Recorded evidence and model limits

SIMULATED INDEPENDENT TRIALS

Recover a phase from port counts

Blue dots and pointwise 95% Wilson intervals come from the recorded trials. Gray is the finite-time model. These are not simultaneous bands or an optional-stopping guarantee. Preview and replay never collect trials.

Current setting: fraction / 95% range

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Phase from four measured quadratures

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Adiabatic geometric reference

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The phase estimate uses four analyzer settings. A conservative contrast threshold suppresses unresolved phases; it is not a phase confidence interval. Equal-energy reference cancellation is exact only for the common adiabatic dynamical term. Fast loops can leak into the other spin state and shift the fringe away from the geometric reference.

All measured settings
Latest 12 outcomes; CSV retains every trial and the preparation settings.

Physics tutorial

Measure a loop through interference

BackgroundA spin begins in the lower eigenstate of an effective field. One interferometer arm turns the field around a cone; the reference arm either keeps the starting field or freezes its spin.

Why it mattersA state-space arrow cannot show an overall phase. A reference and a recombination measurement can.

Start with the essentials

Focus question
Which part of the fringe shift survives when the loop is made slower and its direction is reversed?
One-sentence intuition
For a slowly followed lower spin state the geometric phase is half the signed solid angle. Equal-energy reference evolution removes the common dynamical phase, while finite-speed corrections remain measurable.

Core mathematical model

Closed cone

H/E0=B2n⋅σ,n=(sin⁡θcos⁡ωτ,sin⁡θsin⁡ωτ,cos⁡θ)H/E_0=\tfrac B2\boldsymbol n\cdot\boldsymbol\sigma,\quad\boldsymbol n=(\sin\theta\cos\omega\tau,\sin\theta\sin\omega\tau,\cos\theta)

One full turn has duration set by the period control. Positive winding advances azimuth; field strength sets the gap. Initial spin points opposite the field. Energy and time use the same dimensionless convention as the passage Lab.

Exact evolution used in this Lab

U(τ)=e−iωτσz/2exp⁡ ⁣[−iτ2(Bsin⁡θ σx+(Bcos⁡θ−ω)σz)]U(\tau)=e^{-i\omega\tau\sigma_z/2}\exp\!\left[-\frac{i\tau}{2}\left(B\sin\theta\,\sigma_x+(B\cos\theta-\omega)\sigma_z\right)\right]

This rotating-frame solution applies at any configured speed. The geometric formula is a separate slow-loop reference, not substituted into the signal.

Lower-state geometric reference

γ−=12Ω=sπ(1−cos⁡θ)(mod2π),s∈{−1,1}\gamma_-=\tfrac12\Omega=s\pi(1-\cos\theta)\pmod{2\pi},\qquad s\in\{-1,1\}

The field path encloses the signed north-pole cap. Reversing winding changes its sign. All displayed phases use the principal value from negative to positive pi. At the poles the geometric phase is zero modulo a full turn.

Reference-arm overlap

C=⟨ψ0∣U(T)∣ψ0⟩e−iBT/2C=\langle\psi_0|U(T)|\psi_0\rangle e^{-iBT/2}

The factor shown is present for the equal-energy stationary reference. Its lower state accumulates the same positive dynamical phase in the adiabatic limit. A frozen reference omits this factor. No cancellation of all finite-speed dynamical effects is claimed.

A measured fringe

P+(α)=12[1+Re⁡(Ce−iα)],V=∣C∣P_+(\alpha)=\tfrac12\left[1+\operatorname{Re}(Ce^{-i\alpha})\right],\qquad V=|C|

Ideal balanced path splitting and recombination are assumed. The spin is not measured at the output. Nonparallel final spin states reduce path interference contrast; final spin leakage is shown separately.

Infer phase from four count fractions

ϕ^=atan2⁡ ⁣(P^90−P^270,P^0−P^180)\widehat\phi=\operatorname{atan2}\!\left(\widehat P_{90}-\widehat P_{270},\widehat P_0-\widehat P_{180}\right)

The subscripts are analyzer angles in degrees. All four independently sampled settings are required. A conservative three-standard-error contrast heuristic hides unresolved phases. There is no calibrated phase interval, fit to the model phase or forced physical projection of the contrast estimate.

Common difficulties

Arrow versus phase

Typical misconceptionA Bloch arrow returning proves zero phase.

Better mental modelAn overall spinor phase is invisible in the arrow. Interfering with a reference reveals it.

Geometry versus speed

Typical misconceptionAny loop speed gives the same measured phase.

Better mental modelThe Berry reference assumes adiabatic following. The exact finite-speed signal can leak and acquire other phase contributions.

Gauge versus instrument

Typical misconceptionChanging an eigenvector phase convention changes the count fringe.

Better mental modelThe physical overlap of both arms is gauge invariant when preparation and evolution are transformed consistently. The analyzer phase is an actual relative phase control.

Run the experiment

  1. 01

    Close the field path

    Replay the slow loop. Compare the orange field with the blue spin.

    What to observe: The lower spin nearly follows opposite the field, yet its relative phase can change.
  2. 02

    Read the phase

    Keep the equal-energy reference and acquire a full fringe scan.

    What to observe: Raw port counts at four analyzer phases determine the reported phase.
  3. 03

    Reverse and reshape

    Reverse winding, then change the cone angle and repeat the scan.

    What to observe: Slow-loop phase tracks the signed solid angle modulo a full turn.
  4. 04

    Break the approximation

    Rush the loop, or freeze the reference. Compare leakage and measured phase.

    What to observe: A closed Hamiltonian path alone does not make the observed phase purely geometric.