QUANTUM MOTION / INTERFERENCE PUTS ON THE BRAKES

Quantum Kicks & Dynamical Localization

Deliver a train of instantaneous cosine kicks. Compare the quantum momentum distribution with a classical ensemble. Tune the interval from suppressed diffusion to resonance, or make every second kick undo the first.

An isolated delta-kicked rotor in one fixed quasimomentum sector. 2048 Fourier modes, 80 kicks; a separate 8192-point classical ensemble. No pulse duration, thermal momentum spread, spontaneous emission or interactions. Finite-time suppression is not a proof of asymptotic localization.

QUANTUMCLASSICAL

KICK0

TUNE INTERVAL

VIEW RANGE

-64

p/(2\hbar k_L)=m+\beta

64

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

Preparing the sequence…

ONE PREPARATION. MANY INDEPENDENT READOUTS.

How much momentum remains?

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

$m+\beta$

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

SPREAD / EVERY STEP

$\langle(m+\beta)^2\rangle\quad\text{vs.}\quad N$

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

A zero integer-momentum initial state with a fixed quasimomentum offset. The cosine kick precedes free propagation. The classical comparator starts with uniform angle and the same momentum. Its histogram uses unit momentum bins; its moment uses unbinned momenta.

q_{n+1}=q_n+\kappa\sin\theta_n,\quad\theta_{n+1}=\theta_n+\tau q_{n+1}\pmod{2\pi}

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：Ullah et al. (2012)

Start with the essentials

Focus question: Keep kicking. The wave can stop spreading.
One-sentence intuition: Coherent kicks can interfere destructively over many steps; at resonance their phases line up and momentum spreads ballistically.

Core mathematical model

One complete step

U=e^{-i\tau(\hat m+\beta)^2/2}e^{-i\kappa\cos\hat\theta}

Kick first, then propagate freely. Integer momentum labels and the fixed offset are dimensionless; the kick interval is four pi times the displayed ratio.

A checkable limit

r=1,\ \beta=0:\quad P_m(N)=J_m^2(N\kappa),\quad\langle m^2\rangle=\frac{N^2\kappa^2}{2}

At primary resonance the free operator is the identity. At half this interval and zero offset, two kicks restore the initial momentum state.

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 isolated delta-kicked rotor in one fixed quasimomentum sector. 2048 Fourier modes, 80 kicks; a separate 8192-point classical ensemble. No pulse duration, thermal momentum spread, spontaneous emission or interactions. Finite-time suppression is not a proof of asymptotic localization.

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
Choose resonance, then nudge the interval or the quasimomentum.
What to observe: The exact two-kick return requires zero quasimomentum and the half-period setting.
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.