QUANTUM CONTRASTS / CURRENT AGAINST MOMENTUM

Quantum Backflow on a Ring

Superpose two strictly positive angular momenta on a ring. Move an observation gate through the interference pattern, find negative probability current and erase it by removing coherence. Record angle and momentum on separate fresh preparations.

An exact two-mode particle on an ideal field-free ring. The raised band shows angular probability density; arrows show current, not particle tracks. The gate is a non-invasive model probe. Angle samples do not directly measure current. Ring currents are not subject to the straight-line backflow bound.

POSITIVENEGATIVE

CURRENT AT GATE

0

Height: density, fivefold gain · arrows: current direction · positive is counterclockwise from above

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ASK THE STATE. KEEP THE ANSWERS.

Two measurements. Fresh preparations.

Collect independent angle and angular-momentum outcomes. The signed current remains a separate model prediction.

Raw records

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Current · at gate

0

Sector occupancy · sample

\text{—}

Mean momentum · model

1

ANGLE DISTRIBUTION / MOVE THE GATE

$\theta\ (\mathrm{rad})$

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Teal: exact density. Outlined dots: measured bin frequency divided by bin width. The gold gate begins a quarter-circle sector.

SIGNED CURRENT / MODEL

\text{—}

Teal and coral: forward and reverse current. Amber: the same momentum populations with no coherence. The occupancy error is one binomial standard error. Position counts alone do not determine these curves.

Model, units and what is actually measured

\psi(\theta,s)=\frac{\cos\eta\,e^{i\theta-is/2}+e^{i\phi}\sin\eta\,e^{im\theta-im^2s/2}}{\sqrt{2\pi}}

s=\frac{\hbar t}{MR^2},\quad j=\operatorname{Im}(\psi^*\partial_\theta\psi),\quad \frac{t}{T_b}=\frac{s(m^2-1)}{4\pi}

The displayed formula is the pure state. Coherence scales only the off-diagonal density-matrix terms. Both angular momenta are strictly positive, one and the selected upper integer in units of the reduced Planck constant. Positive angle is counterclockwise from above. The time slider covers one beat of the density, not necessarily one full revolution. Current is per dimensionless time; density is per radian.

\frac{d}{ds}\int_{\theta_g}^{\theta_g+\pi/2}\rho\,d\theta=j(\theta_g)-j(\theta_g+\pi/2)

The gold gate does not absorb or collapse the wave. A finite sector has two boundaries, so its occupancy change is not the flux through one gate alone. The dephased reference uses the same mode populations. Position and momentum records use different fresh preparations, never simultaneous values for one particle. No physical detector geometry, potential, magnetic flux, interactions or optimized maximum-backflow state.

Goussev · Quantum backflow in a ring, equations 5–14 and 21

Start with the essentials

Focus question: All momenta point forward. Some current flows back.
One-sentence intuition: The current contains cross terms between positive-momentum waves. Those terms can locally outweigh their positive individual currents.

Core mathematical model

Evolution

E_m=\frac{\hbar^2m^2}{2MR^2},\quad m_1=1,\quad m_2\in\{2,\ldots,8\}

Each angular-momentum component is positive. The controllable relative phase changes interference without changing those populations.

Remove coherence

\rho_{12}\longmapsto v\rho_{12},\quad 0\leq v\leq1

This declared noise channel preserves the diagonal populations. It is not a model of energy loss.

What is recorded

P_{[a,b]}=\int_a^b|\psi(\theta)|^2\,d\theta

Angle samples estimate sector occupancy. Angular-momentum samples come from a different ensemble. Neither supplies a particle trajectory.

Common difficulties

A model is not a detector

Typical misconceptionThe moving density peak traces a particle, and a gate measures its velocity.

Better mental modelAn exact two-mode particle on an ideal field-free ring. The raised band shows angular probability density; arrows show current, not particle tracks. The gate is a non-invasive model probe. Angle samples do not directly measure current. Ring currents are not subject to the straight-line backflow bound.

Run the experiment

01
Find reverse flow
Choose Against the flow. Move the gate to a coral arrow or press Find reverse flow.
What to observe: The local current can be negative while the mean angular momentum stays positive.
02
Collect evidence
Collect 500 independent angle outcomes and 500 separate angular-momentum outcomes. Inspect the sector frequency.
What to observe: All angular momenta remain positive. The angle histogram approaches the density, not the current curve.
03
Remove the interference
Choose Erase coherence. Keep the same momentum weights and inspect the current.
What to observe: The dephased mixture has a uniform positive current. Equal coherent weights also remove negative current for this two-mode family.