Q030 · Reading ion motion from sidebands

Trapped-Ion Sideband Spectroscopy

Why does the red sideband disappear when a trapped ion reaches its motional ground state? Driving the red sideband takes one quantum of motion away from the ion, so it fails once the ion is in its motional ground state: there is nothing left to take. The blue sideband adds a quantum and still works. Comparing the two sideband strengths over many repeated measurements tells you how cold the motion is.

Drive a trapped ion on either side of its resonance and compare many repeated readings of its internal state. Then use cooling pulses to remove its 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.

Ion and motion envelope enlarged · glow shows excitation probability

Probe duration

t/t_*=0

Try a comparison

Choose a preparation, then compare red and blue.

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

The model has one effective harmonic mode of motion and one addressed transition. The carrier and the first sidebands are resolved and use exact displacement matrix elements. After each cooling pulse the ion resets ideally, with no recoil. A separate classical RF/DC mode integrates the micromotion and the loss through the aperture, but it does not recalibrate the harmonic spectrum. Heating and the photon statistics of fluorescence are left out. All records are simulated.

02 / Follow the exchange

The red and blue sidebands respond differently.

Excited ion0%

Motional ground state before probe0%

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

Drag to inspect any probe duration. Autoplay repeats the early model response without taking measurements.

Quantitative readout and count inference

Mean motional occupation after preparation · model

\text{—}

Thermal estimate · from paired counts

\text{—}

Prepared oscillator distribution

03 / Oscillator occupation

Which motion levels remain?

\epsilon_{\rm tail}=0

Saved counts and captured settings

04 / Save the evidence

A matched red–blue comparison

Acquire a comparison to fill the table. The table keeps the settings it was taken with, even if you change the controls.

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.

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

The sideband and cavity protocols use an arbitrary reference time unit. The separate RF protocol below uses physical SI units. Geometry is schematic, with enlarged atoms and mode envelopes.

The apparatus dial sets the same model carrier drive as the slider; pulse area also depends on probe duration. Its angle is a user interface, with no manufacturer power calibration. Moving the view does not change trap or spectroscopy calibration.

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

Paul trap motion and stability

05 / Build the trap

Bounded motion can still hit an aperture.

Integrate one classical test ion in a declared linear quadrupole. The stability map tests the ideal equations; the trajectory tests only this launch for this observation time. Neither establishes capture.

Voltages, launch and observation window

RF amplitude / V

20

Endcap DC / V

5

RF frequency / MHz

5

Launch RF phase / deg

0

Initial radial position / µm

15

Other radial position / µm

5

Initial axial position / µm

20

Initial radial velocity / m/s

0

Absorbing radius / µm

50

Observation / RF periods

80

Recorded time cursor

Captured Mathieu parameters

a_r=0,\quad q_r=0

Position · simulated coordinate

r=0\,\mu\mathrm m

Radial monodromy trace

\operatorname{tr}M=2

Within-period radial residual RMS

\text{—}

Radial motion and period means

Blue: integrated radial coordinate. Amber: complete-period means. The residual includes secular change within a period; it is not a unique intrinsic/excess micromotion separation.

Ideal Mathieu stability section

Blue: both radial axes and axial motion bounded. Red: unstable. Gray: boundary or degenerate. The marker uses the captured voltages; the grid is a numerical section, not an electrode acceptance map.

Restore checkpoint

Field, geometry and what the results can show

\Phi=\Phi_{\rm rf}+\Phi_{\rm dc}

\begin{gathered}\Phi_{\rm rf}=\frac{V_{\rm rf}(x^2-y^2)}{2r_0^2}\\{}\cdot\cos(\Omega t+\phi)\end{gathered}

\begin{gathered}\Phi_{\rm dc}=\frac{\kappa U_{\rm dc}}{z_0^2}D\\D=z^2-\frac{x^2+y^2}{2}\end{gathered}

\begin{gathered}a_x=a_y=-\frac{4e\kappa U_{\rm dc}}{m z_0^2\Omega^2}\\a_z=-2a_x\end{gathered}

\begin{gathered}q_x=-q_y=\frac{2eV_{\rm rf}}{m r_0^2\Omega^2}\\q_z=0\end{gathered}

\begin{gathered}\ddot u_i=-\frac{\Omega^2}{4}A_i(t)u_i\\A_i(t)=a_i+2q_i\cos(\Omega t+\phi)\end{gathered}

Positive charge; mass 40 atomic mass units. Radial scale 0.5 mm; axial scale 1 mm; endcap factor 0.2. Coordinates form a right-handed frame: axial points along the rods; the radial pair points upward and across the stage. Geometry is schematic; ion displacement is enlarged tenfold relative to the electrode schematic. The 200 µm axial limit and adjustable radial cylinder are absorbing diagnostic boundaries inside the near-axis field, not electrode surfaces.

Velocity Verlet uses 192 steps per RF period; the Floquet matrix uses independent RK4 integration. Trace within a numerical tolerance of either boundary is withheld as marginal, including free drift. Escape time is bracketed by adjacent integration steps. Survival is finite and depends on initial energy, phase and aperture. No cooling, gas collision, space charge, electrode anharmonicity, stray fields or dissipative loading is modeled.

This classical RF protocol and the harmonic quantum sideband protocol are separate experiments. Changing the RF fields does not recalibrate the sideband frequency and does not produce spectroscopy counts. All coordinates here are simulated, not detected ion positions.

Berkeland et al. · 1998 · equations 1–6 (NIST collection) · Wineland et al. · linear trap and sideband manipulation

Start with the essentials

Focus question: What changes when the oscillator contains no quanta?
One-sentence intuition: A model curve and a finite set of measurements 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

The red sideband removes one phonon and the blue sideband adds one. Detuning is measured from 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 motion starts in a thermal distribution. Cooling updates the probability of every phonon number; the result is never replaced by 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 model keeps the full displacement matrix element for an isolated first sideband. At small displacement it approaches the square-root dependence on phonon number.

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 state reading is a new Bernoulli trial. A red-sideband drive cannot excite an ion in its motional ground state.

Red-to-blue ratio for a thermal state

\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}}

This works only for a thermal single-mode state probed with matched pulses. The point estimate is hidden after cooling, or when the two pointwise Wilson intervals overlap. That rule is not a confidence interval for the occupation.

Common difficulties

The glow is not a path

Typical misconceptionThe glowing object traces the path of a single particle.

Better mental modelThe fixed marker only shows where the particle sits. Brightness shows a probability or a mean occupation. Only the recorded tiles are actual samples.

Limits of the model

Typical misconceptionThis is a calibrated real apparatus.

Better mental modelThe sideband experiment uses a harmonic pseudopotential, resolved transitions and an ideal cooling reset with no recoil. The separate classical RF experiment follows the driven motion and the loss through a finite aperture, but its fields do not recalibrate the sideband model. Neither mode simulates the stream of arriving photons or the actual loading of the trap.

Run the experiment

01
Prepare
Try motion in the ground state, then warm motion.
What to observe: The whole response curve changes between the two cases.
02
Probe
Drag the time cursor and read out 200 fresh preparations.
What to observe: The preview never produces data by itself.
03
Compare
Acquire a matched pair of red and blue sideband readings.
What to observe: You can export the raw CSV and recompute the state frequencies yourself. Saved comparisons keep the settings they were taken with.