Quantum contrasts / Current against momentum

Quantum Backflow on a Ring

Can the probability current on a ring flow backward when every momentum points forward? Yes, in places. Superpose two states on a ring that both circulate forward, and the probability current picks up cross terms. In some places those terms outweigh the forward currents of the two states, so the current there runs backward. Remove the coherence and the backflow disappears.

Put a particle on a ring into a superposition of two angular momenta, both strictly positive. Slide an angular window through the interference pattern to find where the probability current runs backwards, then remove the coherence and watch the backflow vanish. Angle and angular momentum are measured on separate, freshly prepared particles.

The model is exact for a particle in two angular-momentum states on an ideal ring with no external field. The height of the band shows the probability density around the ring, and the arrows show probability current, not particle paths. The window is an idealized probe that does not disturb the state. Sampling the angle does not measure the current directly. The backflow bound derived for motion on a straight line does not apply to a ring.

POSITIVENEGATIVE

CURRENT AT GATE

0

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

0

Measure the state, keep every outcome

Angle and angular momentum are measured on separate fresh preparations.

Collect angle and angular-momentum outcomes from separate preparations. The signed current comes from the model, not from these outcomes.

Raw records

0

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 from the 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 only marks a position; it does not absorb the wave or collapse it. A finite sector has two boundaries, so the change in its occupancy is not the flux through one gate alone. The dephased reference has the same mode populations. Position and momentum records come from separate fresh preparations; they are never simultaneous values for one particle. The model has no physical detector geometry, potential, magnetic flux or interactions, and the state is not optimized for maximum backflow.

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

Physics tutorial

Quantum Backflow on a Ring

BackgroundPut a particle on a ring into a superposition of two angular momenta, both strictly positive. Slide an angular window through the interference pattern to find where the probability current runs backwards, then remove the coherence and watch the backflow vanish. Angle and angular momentum are measured on separate, freshly prepared particles.

Why it mattersMomentum, probability density and probability current are different quantities, and each one answers a different question about the particle.

Start with the essentials

Focus question: Can the probability current on a ring flow backward when every momentum points forward?
One-sentence intuition: The current includes cross terms between the two positive-momentum waves. In some places those terms outweigh the forward currents the waves carry on their own.

Core mathematical model

Evolution

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

Both angular-momentum components are positive. Changing their relative phase changes the interference but leaves the population of each state unchanged.

Remove coherence

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

This noise channel keeps the diagonal populations unchanged. It does not model energy loss.

What is recorded

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

Angle samples estimate the probability of finding the particle in a sector. Angular-momentum samples come from a separate set of preparations. Neither kind of sample traces a particle's path.

Common difficulties

The display is a model, not a measurement

Typical misconceptionThe moving density peak is the particle itself, and the gate measures its velocity.

Better mental modelThe model is exact for a particle in two angular-momentum states on an ideal ring with no external field. The height of the band shows the probability density around the ring, and the arrows show probability current, not particle paths. The window is an idealized probe that does not disturb the state. Sampling the angle does not measure the current directly. The backflow bound derived for motion on a straight line does not apply to a ring.

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 current at one spot can be negative even though the mean angular momentum stays positive.
02
Collect measurement data
Collect 500 angle outcomes and another 500 angular-momentum outcomes, each from a fresh preparation. Look at how often the particle lands in the sector.
What to observe: Every measured angular momentum is positive. The angle histogram approaches the probability density, not the current curve.
03
Remove the interference
Choose Erase coherence. Keep the same momentum weights and inspect the current.
What to observe: Without coherence, the mixture carries the same positive current everywhere on the ring. For this family of two-mode states, a coherent superposition with equal weights has no negative current either.