Q030 · Drive / cool / count

Trapped-Ion Sideband Spectroscopy

Drive a trapped ion on either side of its resonance. Compare repeated state readings, then remove motion one quantum at a time.

Model excited fraction $\text{—}$

Recorded at this time $\text{—}$

01 / PREPARE THE EXPERIMENT

Take one quantum of motion away.

Compare the red and blue sidebands. At zero motion, the red pulse has nothing to remove.

Schematic trap · stationary motion envelope · no trajectory is shown

Probe duration

t/t_*=0

TRY A CONTRAST

Every shot starts fresh. The example includes 200 simulated state readings.

Drive transition

Carrier drive

\Omega_0t_*

1

Initial mean phonon number

\bar n_0

1

Cooling cycles

N_{\rm cool}

0

Detuning, model & readout

Local detuning

\Delta t_*

0

Lamb–Dicke parameter

\eta

0.22

State-readout availability

\epsilon

1

Sampling seed

s

31030

One effective harmonic mode and one addressed transition. Resolved carrier and first sidebands use exact displacement matrix elements. Cooling pulses have an ideal recoil-free reset; radio-frequency micromotion, heating and fluorescence photon statistics are omitted. All records are simulated.

02 / FOLLOW THE EXCHANGE

Two sidebands tell different stories.

Excited ion0%

Motional ground state before probe0%

Red: red sideband. Blue: blue sideband. Gray: selected transition. Dots and 95% Wilson bars: recorded outcomes.

Drag across the plot or use the time slider. Playing changes the preview only.

Prepared mean motion · model

\text{—}

Thermal estimate · from paired counts

\text{—}

03 / THE OSCILLATOR

Which motion levels remain?

\epsilon_{\rm tail}=0

04 / KEEP THE EVIDENCE

A matched red–blue comparison

Acquire a comparison to fill the table. It keeps its captured settings when controls change.

Probe	Read / total	Excited

Model and measurement boundaries

Ideal binary state discrimination when available; unread trials are retained. No fluorescence photon arrival stream is simulated. Every trial uses an independent preparation. Model number distributions are not reconstructed from these binary counts.

Thermal preparation in one effective harmonic mode. First-sideband and carrier dynamics assume resolved transitions. The trap frequency is fixed at twenty reference inverse times. Cooling uses resonant red pulses and a perfect recoil-free reset, so the cooled distribution is generally not thermal.

Time uses an arbitrary reference unit. Geometry is schematic, with enlarged atoms and mode envelopes.

Primary sources: Leibfried et al. · 2003 · Vybornyi et al. · 2023

Start with the essentials

Focus question: What changes when the oscillator contains no quanta?
One-sentence intuition: A model curve and a finite collection of observations are different kinds of evidence.

Core mathematical model

Resolved sidebands

\omega_L=\omega_0+s\nu+\Delta,\quad s\in\{-1,0,1\},\quad\nu t_*=20

Red removes a phonon; blue adds one. Detuning is local to the selected transition. The carrier changes only the internal state.

Thermal preparation

p_n=\frac{\bar n_0^n}{(1+\bar n_0)^{n+1}}

The initial distribution is thermal. Cooling propagates every number probability and does not replace the result with a fitted thermal state.

Sideband coupling

\Omega_{n,n+1}=\Omega_0e^{-\eta^2/2}\frac{\eta}{\sqrt{n+1}}L_n^1(\eta^2)

The full displacement matrix element within an isolated first sideband is retained. At small displacement it approaches the square-root number law.

State reading

P_e(t)=\sum_n p_n\frac{\Omega_{n,n+s}^2}{\Omega_{n,n+s}^2+\Delta^2}\sin^2\!\left(\frac t2\sqrt{\Omega_{n,n+s}^2+\Delta^2}\right)

Each observed state is a new Bernoulli trial. A red drive cannot excite a motionless ground-state ion.

Matched thermal comparison

\frac{P_{\rm red}}{P_{\rm blue}}=\frac{\bar n}{1+\bar n},\quad\widehat{\bar n}=\frac{\hat P_{\rm red}}{\hat P_{\rm blue}-\hat P_{\rm red}}

Requires a thermal single-mode preparation and matched pulses. The point estimate is suppressed after cooling or if the two pointwise Wilson intervals overlap. This is not an occupation confidence interval.

Common difficulties

Readout is not a trajectory

Typical misconceptionThe glowing object shows a single particle path.

Better mental modelThe fixed marker locates the particle; brightness encodes a probability or mean occupation. Only recorded tiles are samples.

Approximation boundary

Typical misconceptionThis is a calibrated apparatus.

Better mental modelThe ion uses a harmonic pseudopotential, resolved transitions and ideal recoil-free cooling reset. No micromotion, motional heating or photon arrival stream is modeled.

Run the experiment

01
Prepare
Try ground motion, then warm motion.
What to observe: The contrast changes the entire response curve.
02
Probe
Drag the time cursor and read 200 fresh preparations.
What to observe: The preview never generates data on its own.
03
Compare
Acquire a matched sideband pair.
What to observe: Export the raw CSV and reproduce the state frequencies. Saved comparisons keep their original settings.