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.

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 01DRAG PULSE 02

POLARIZATION AXIS

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)

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 misconceptionThe 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 modelIn 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

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.
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.
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.