Q023 · Closed path / reference arm

Berry Phase Interferometer

How far does an interference fringe shift when a spin is carried around a loop? By a geometric phase equal to half the solid angle of the loop. A spin that slowly follows a field around a cone picks up this phase no matter how long the trip takes. Compared with a reference arm that shares the ordinary dynamical phase, it shows up as a fringe shift. Run the loop backwards and the shift reverses.

Turn an effective field once around a cone while a spin follows it. Recombine that spin with a stationary reference arm, scan the analyzer and read the phase from the counts at the output ports. Then reverse the loop or run it faster.

Model interferometric phase $\text{—}$

Model spin leakage $\text{—}$

Make a prediction

After a closed loop, is the interference exactly where it started?

The orange arrow is the field; blue is the spin. Close the loop, combine with the reference arm, then scan the analyzer to read a phase.

Predict first, then compare and collect evidence.

Preview time (each measurement runs the full protocol)

Reverse winding directionEqual-energy stationary referenceCone opening angle

\theta

60^{\circ}

Time for one circuit

T

60

Field strength / energy gap

B

2

Drag the cone-angle handle or use its slider. Analyzer phase preserves the scan; changing the loop or reference starts a new record.

Model details and sampling

Analyzer phase

\alpha

0^{\circ}

Trials at each setting

N

400

Reproducible sampling seed

s

1984

Energy uses a chosen scale; time uses the corresponding inverse angular frequency. Values are dimensionless. Sampling is reproducible and capped at 30,000 trials.

The two-level evolution in the rotating field is solved exactly, and the paths are split and recombined ideally. A stationary reference with the same energy cancels the dynamical phase that both arms share in the adiabatic limit. At finite speed the measured shift also contains nonadiabatic corrections, so it is not automatically the Berry phase. The sphere shows parameter space and state space, not the orbit of a particle.

Recorded evidence and model limits

Simulated independent trials

Recover a phase from port counts

Blue dots and their pointwise 95% Wilson intervals come from the recorded trials. The gray curve is the finite-time model. The intervals are not simultaneous bands and carry no guarantee under optional stopping. Preview and replay never add trials.

Current setting: fraction / 95% range

\text{—}

Phase from four measured quadratures

\text{—}

Adiabatic geometric reference

\text{—}

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.

Start with the essentials

Focus question: When the loop runs slower and in the opposite direction, which part of the fringe shift remains?
One-sentence intuition: When the spin follows the field slowly in its lower state, the geometric phase is half the signed solid angle of the loop. A reference arm with the same energy cancels the shared dynamical phase, but corrections from finite speed still show up in the measurement.

Core mathematical model

A field turning around a cone

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)

The field makes one full turn in one period. Positive winding increases the azimuthal angle, and the field strength sets the energy gap. The spin starts pointing opposite the field. Energy and time use the same dimensionless units as the Landau–Zener passage Lab.

Exact evolution used in this Lab

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 holds at any loop speed. The geometric-phase formula is only a reference for slow loops; the simulated signal does not use it.

Geometric phase of the lower state

\gamma_-=\tfrac12\Omega=s\pi(1-\cos\theta)\pmod{2\pi},\qquad s\in\{-1,1\}

The field path encloses a cap around the north pole, counted with a sign. Reversing the winding flips that sign. Every displayed phase is given as its principal value between minus pi and plus pi. At the poles the geometric phase is zero, up to whole turns.

Overlap with the reference arm

C=\langle\psi_0|U(T)|\psi_0\rangle e^{-iBT/2}

The exponential factor belongs to the stationary reference with equal energy: in the adiabatic limit its lower state picks up the same positive dynamical phase. A frozen reference has no such factor. Removing this factor does not cancel every dynamical effect at finite speed.

The fringe at the output

P_+(\alpha)=\tfrac12\left[1+\operatorname{Re}(Ce^{-i\alpha})\right],\qquad V=|C|

The paths are split evenly and recombined ideally. The output measures the path, not the spin. If the two arms end with spins pointing in different directions, the fringe contrast drops; the final spin leakage is shown separately.

Infer phase from four count fractions

\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, and all four settings must be sampled independently. A conservative rule of thumb hides the phase as unresolved when the contrast is below three standard errors. The Lab gives no calibrated phase interval, does not fit the model phase and does not force the contrast estimate into the physical range.

Common difficulties

The arrow hides the phase

Typical misconceptionIf the Bloch arrow comes back to where it started, the phase must be zero.

Better mental modelThe arrow cannot show the overall phase of the spinor. Interfering the state with a reference arm reveals it.

Loop speed matters

Typical misconceptionThe measured phase is the same however fast the loop runs.

Better mental modelThe Berry phase assumes the spin follows the field adiabatically. At finite speed the exact evolution can leak out of the lower state and pick up other phase contributions.

Phase conventions do not move the fringe

Typical misconceptionChoosing a different phase convention for the eigenvectors shifts the count fringe.

Better mental modelIf the preparation and the evolution are transformed consistently, the overlap of the two arms does not depend on the gauge. The analyzer phase is different: it is a real relative phase that you set on the instrument.

Run the experiment

01
Follow the field around
Replay the slow loop. Compare the orange field with the blue spin.
What to observe: The spin stays almost exactly opposite the field all the way around, yet its relative phase can still change.
02
Read the phase
Keep the equal-energy reference and acquire a full fringe scan.
What to observe: The reported phase comes from the raw port counts at four analyzer phases.
03
Reverse the loop and change the cone
Reverse winding, then change the cone angle and repeat the scan.
What to observe: For a slow loop, the phase follows the signed solid angle, up to whole turns.
04
Go too fast
Rush the loop, or freeze the reference. Compare the leakage with the measured phase.
What to observe: Closing the path of the Hamiltonian is not enough on its own to make the measured phase purely geometric.