---
title: "Molecular Alignment Revivals Interactive Lab | Sandbox Physics"
description: "Why do molecules line up again long after the laser pulse has passed?"
canonical_url: "https://sandboxphysics.com/labs/molecular-alignment-revivals/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/molecular-alignment-revivals/"
last_modified: "2026-10-27"
---

Quantum time: a kicked rotor

# Molecular Alignment Revivals

**Why do molecules line up again long after the laser pulse has passed?** Because the short pulse stamps phases onto the molecule's rotational states, and those phases realign later. The pulse itself does not line the molecules up right away. Every rotational energy is a whole-number multiple of one basic step, so the states come back into phase at revival times long after the pulse, and the molecular axes briefly line up again.

Kick an ideal molecular rotor with a short laser pulse. Turn the three-dimensional distribution of the molecular axis to view it, find when the alignment revives, and slide a second pulse along the timeline. Then collect independent axis readouts and compare their alignment with the model.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The molecule is an isolated, linear rigid rotor that starts in its rotational ground state. The laser kicks are instantaneous, nonresonant and polarized along the same axis. Thermal mixtures, centrifugal distortion, ionization and detector calibration are left out. The surface shows how likely the molecular axis is to point in each direction; it is not a molecule or an orbit.

AXIS PROBABILITY

PULSE 01 DRAG PULSE 02

POLARIZATION AXIS

Radius: square root of axis density. Schematic, not an orbit.

$t/T_{\mathrm{rev}}=0$

Preparing the spectrum…

Pause at one moment and collect data

### Which way did the axes point?

Each ideal readout uses a fresh identical preparation. Collecting pauses time.

Recorded trials $0$

Return probability · model $1$

Alignment · records $\text{—}$

Total probability · model $1$

MODEL / RECORDS / PINNED

Horizontal coordinate $u=\cos\theta$

Teal: model. Gold: recorded frequency density. Dashed: pinned model. Equal-width bins.

ALIGNMENT / AFTER THE PULSES

The dashed baseline is one third for an isotropic ensemble. Parallel and antiparallel axes are equivalent; this is alignment, not orientation.

$0$

Model, units and what the picture means

The molecules start in the isotropic rotational ground state. Two optional collinear kicks add a phase proportional to the square of the axis projection on the polarization direction. Free evolution includes angular momenta from zero to 48. Pulse strength is dimensionless; it is not the laser power.

$$
E_J=BJ(J+1),\quad T_{\mathrm{rev}}=\frac{\pi\hbar}{B}
$$

All times are in units of the revival period. Records are drawn from the displayed 384-bin density, normalized for an ideal detector. Finite resolution, a thermal ensemble and instrument response are not modeled. Display brightness uses a fixed square-root gain.

Original research：[Seideman (1999)](https://doi.org/10.1103/PhysRevLett.83.4971)

Physics tutorial

## Molecular Alignment Revivals

**Background** Kick an ideal molecular rotor with a short laser pulse. Turn the three-dimensional distribution of the molecular axis to view it, find when the alignment revives, and slide a second pulse along the timeline. Then collect independent axis readouts and compare their alignment with the model.

**Why it matters** Watch coherent time evolution unfold, then test it against a finite number of measurements.

Start with the essentials

- **Focus question**: Why do molecules line up again long after the laser pulse has passed?
- **One-sentence intuition**: The short pulse acts on the phase first. The rotational states it creates interfere later, after the light has gone.

### Core mathematical model

#### The energy spectrum

$$
E_J=BJ(J+1),\quad T_{\mathrm{rev}}=\frac{\pi\hbar}{B}
$$

Time is shown as a fraction of this exact revival period. The rotor's rotational constant is given in energy units.

#### The impulsive kick

$$
U_P=\exp(iP\cos^2\theta),\quad P=\frac{\Delta\alpha}{4\hbar}\int\mathcal E^2(t)\,dt
$$

The pulse is nonresonant, and the polarizability anisotropy is taken as positive. The field in the formula is the envelope of the rapidly oscillating real field, and both pulses share one polarization axis.

#### What the measurements estimate

$$
A=\langle\cos^2\theta\rangle,\quad A_{\mathrm{isotropic}}=\frac13
$$

The sample mean of the squared projection onto the axis estimates the alignment. The two ends of the molecule stay equivalent, and the average signed direction stays zero.

### Common difficulties

#### What the picture shows

**Typical misconception** The molecule is an isolated, linear rigid rotor that starts in its rotational ground state. The laser kicks are instantaneous, nonresonant and polarized along the same axis. Thermal mixtures, centrifugal distortion, ionization and detector calibration are left out. The surface shows how likely the molecular axis is to point in each direction; it is not a molecule or an orbit.

**Better mental model** In each direction, the radius is the square root of the angular probability density. The surface is neither the outline of a molecule nor a trajectory. Thermal populations and a real probe would need a different model.

### Run the experiment

1. 01

   #### Predict, then jump

   Compare the instant of the pulse, a short time afterward and one full period.

   **What to observe:** The pulse does not line up the axes at once. Free evolution afterward turns its phase imprint into a visible change in the distribution.

2. 02

   #### Change one thing at a time

   Turn on the second pulse and drag its arrival time. Pin one sequence before you change it.

   **What to observe:** Compare the whole time trace, not just one striking frame.

3. 03

   #### Measure and compare

   Collect 500 independent readouts. Pin the sample, change the time, restore the sample and export the raw CSV.

   **What to observe:** The counts fluctuate. The standard error covers only sampling noise, and the teal reference comes from the model.

Physical conclusion

The molecule is an isolated, linear rigid rotor that starts in its rotational ground state. The laser kicks are instantaneous, nonresonant and polarized along the same axis. Thermal mixtures, centrifugal distortion, ionization and detector calibration are left out. The surface shows how likely the molecular axis is to point in each direction; it is not a molecule or an orbit.

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)