Quantum motion / Interference stops the spreading

Quantum Kicks & Dynamical Localization

If you keep kicking a quantum rotor, can its momentum stop spreading? It can. Over many kicks the phases interfere destructively, so the quantum momentum spread stops growing, while a classical ensemble kicked the same way keeps diffusing. At special resonant kick intervals the phases line up instead, and the spread grows in proportion to the number of kicks.

Hit a rotor with a train of instantaneous cosine kicks and compare its quantum momentum distribution with a classical ensemble. Tune the interval between kicks from suppressed diffusion to resonance, or make every second kick undo the one before.

The model is an isolated delta-kicked rotor in one fixed quasimomentum sector, computed with 2048 Fourier modes over 80 kicks. A separate classical ensemble uses 8192 points. Pulse duration, thermal momentum spread, spontaneous emission and interactions are left out. Suppressed spreading over a finite time does not prove localization at long times.

QUANTUMCLASSICAL

KICK0

TUNE INTERVAL

VIEW RANGE

-64

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

64

Probability landscape · height is the square root of probability · both rows share one scale

Preparing the sequence…

Many independent readouts of one preparation

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 after every step

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

The second moment is taken about zero, not about the mean, so it includes drift as well as width. The teal and amber curves are computed independently.

The 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 approximations

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: If you keep kicking a quantum rotor, can its momentum stop spreading?
One-sentence intuition: Over many steps, coherent kicks can cancel one another by interference. At resonance their phases line up, and the momentum spreads ballistically.

Core mathematical model

One complete step

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

Each step is a kick followed by free motion. The integer momentum labels and the fixed offset are dimensionless, and the kick interval equals four pi times the displayed ratio.

A limit you can check exactly

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 the primary resonance, the free-evolution operator is the identity. At half this interval with zero offset, two kicks bring back 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 standard error of the sample. These records cannot reconstruct the quantum phase.

Common difficulties

A probability landscape

Typical misconceptionEach peak is an individual particle, or the trajectory one particle follows.

Better mental modelThe model is an isolated delta-kicked rotor in one fixed quasimomentum sector, computed with 2048 Fourier modes over 80 kicks. A separate classical ensemble uses 8192 points. Pulse duration, thermal momentum spread, spontaneous emission and interactions are left out. Suppressed spreading over a finite time does not prove localization at long times.

Run the experiment

01
Watch quantum and classical part ways
Play the sequence. Compare the teal quantum ridge with the amber classical ridge.
What to observe: Both rows use the same horizontal scale and the same square-root height scale. For a quantitative comparison, use the linear plots below.
02
Change only the phase
Choose resonance, then nudge the interval or the quasimomentum.
What to observe: An exact return after two kicks needs zero quasimomentum and the half-period setting.
03
Record a distribution
Collect 500 outcomes and pin the preparation. Change the experiment, restore the pinned sample and export its raw CSV.
What to observe: The histogram and the second moment come only from recorded outcomes; the classical ensemble is a separate reference.