QUANTUM MOTION / CHOOSE BOTH WAYS

Quantum Walk & Phase Traps

Turn the quantum coin, shift the initial phase and place a phase defect at the origin. Compare coherent spreading with a initially dephased, measured-at-every-step reference, then build the final distribution from independent position records.

An ideal discrete coined walk on a line, up to 64 steps. No hardware loss or atom imaging model. The classical reference measures coin and position initially and after every step; only the balanced coin gives the ordinary unbiased random walk.

QUANTUMCLASSICAL

STEP0

TURN PHASE

VIEW RANGE

-64

x/a

64

Probability sculpture · height uses square root · both rows share one scale

Preparing the sequence…

ONE PREPARATION. MANY INDEPENDENT READOUTS.

Where does it land?

Collecting pauses the sequence. Each outcome comes from a fresh identical preparation.

Recorded outcomes

0

Second moment · quantum

0

Second moment · classical

0

Second moment · records

\text{—}

PROBABILITY / LINEAR SCALE

$x/a$

Teal: quantum model. Amber: classical reference. White-centered dots: recorded frequency. Purple dashes: pinned quantum model.

SPREAD / EVERY STEP

$\langle(x/a)^2\rangle\quad\text{vs.}\quad N$

Second moment is measured about zero, not about the mean. It includes drift as well as width. Teal and amber are independently computed.

A WHOLE SEQUENCE AT A GLANCE

Horizontal: position or momentum. Down: steps. Click to revisit a step. This is probability, not a trajectory.

Model, units and approximation boundaries

One ideal walker begins at the origin in an equal-weight coin superposition. The two internal states shift left and right after a real coin rotation. A phase defect acts only at the origin. The classical comparator dephases position and coin initially and after every step; it retains coin memory when unbalanced.

|\psi_0\rangle=|0\rangle\otimes\frac{|L\rangle+e^{i\phi}|R\rangle}{\sqrt2}

The 3D view groups adjacent bins when needed. Both models share the same square-root height gain and a fixed horizontal range per preparation. The distribution below shows probabilities per native bin. Measurement outcomes sample the quantum model only; no phase reconstruction is claimed.

Primary research：Karski et al. (2009)

Start with the essentials

Focus question: One walker. Two wavefronts. A phase that changes the route.
One-sentence intuition: Add amplitudes before squaring. Paths that rejoin can reinforce or cancel; measuring every step removes those interferences.

Core mathematical model

One complete step

U=D_\chi S(C_\alpha\otimes I),\quad C_\alpha=\begin{pmatrix}\cos\alpha&\sin\alpha\\\sin\alpha&-\cos\alpha\end{pmatrix}

Coin zero shifts one site left; coin one shifts right. The local phase acts on both coin components at the origin after the shift.

A checkable limit

\alpha=\frac\pi4,\ \phi=\frac\pi2,\ \chi=0:\quad\langle x^2\rangle\sim\left(1-\frac1{\sqrt2}\right)N^2

The balanced symmetric quantum walk spreads ballistically; the completely measured walk has second moment equal to the step count. The displayed finite sequence need not equal the asymptotic coefficient exactly.

What is sampled

\widehat{M_2}=\frac1n\sum_{j=1}^{n}q_j^2

Each final readout comes from a fresh preparation. The reported uncertainty is one sample standard error. These records do not reconstruct quantum phase.

Common difficulties

A probability landscape

Typical misconceptionThe peaks are not individual particles or trajectories.

Better mental modelAn ideal discrete coined walk on a line, up to 64 steps. No hardware loss or atom imaging model. The classical reference measures coin and position initially and after every step; only the balanced coin gives the ordinary unbiased random walk.

Run the experiment

01
Watch the separation
Play the sequence. Compare the teal quantum ridge with the amber classical ridge.
What to observe: The two rows share one horizontal scale and one square-root height scale. The linear plots below carry the quantitative comparison.
02
Change only the phase
Keep the coin balanced. Turn the initial phase, then add a phase at the origin.
What to observe: The measured reference is insensitive to either phase; the coherent walk is not.
03
Record a distribution
Collect 500 outcomes, pin the preparation, change the experiment and restore the pinned sample. Export its raw CSV.
What to observe: The histogram and second moment come only from recorded outcomes; the classical ensemble is a separate reference.