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 $\text{—}$

Hold an ion, then let it escape

Initial RF amplitude

V_{\mathrm{start}}/\mathrm V

5

Final RF amplitude

V_{\mathrm{end}}/\mathrm V

80

Endcap voltage

U/\mathrm V

0.1

Auxiliary amplitude

V_{\mathrm{AC}}/\mathrm V

0

Auxiliary frequency

f_{\mathrm{AC}}/\mathrm{kHz}

1000

Source offset

y_0/\mathrm{\mu m}

0

Drag the gold RF feed knob. The live reference ion responds while you hold it. Arrow keys also adjust the initial voltage.

Opening a saved finite-flight capture.

Preparation, scale and model

The live storage preview uses a known reference mass of $40\,\mathrm{u}$ with one positive elementary charge.

RF amplitude is the zero-to-peak difference between rod pairs; auxiliary amplitude is the difference between the slotted rods. Endcap DC is relative to ground.

Rod colours indicate RF polarity. The four finite hyperbolic rods have openings through the two x rods. Annular endcaps confine the axial direction. Ions are created at the declared in-situ positions; markers and the apparatus are enlarged. No capture efficiency, cooling, gas collisions or Coulomb interaction is modeled.

The calibrated RF scan uses precisely controlled axial-plane initial states, two separate mirror attempts and independent known ions. Both attempts run in sequence. The grounded negative-side dump is a loss surface. An ideal zero-dead-time counter registers only the positive collector front.

A 97-point octant grid reconstructs the finite three-dimensional electrode potentials. The central coefficients are exact Float32; outer coefficients use checked Float24 compression. The display replays saved RF-resolved trajectories. New acquisitions integrate the same fields and may take several seconds.

The default mass window applies only to the checked controlled preparation and the measured 32–48 calibration domain. It is a conditional interval, with no point mass, species count or charge-state assignment. Broader sources and altered settings can retain hits while mass conversion stops.

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.

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

Which motion levels remain?

\epsilon_{\rm tail}=0

Saved counts and captured settings

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

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

From escape to a collector pulse

Saved acquisition

Collector hits

1

Mass window

36\text{–}40\,\mathrm{u}/e

Acquisition gate

140\,\mathrm{\mu s}

Independent checks

4/4

Inside the checked 32–48 domain, this observation supports a 36–40 window. Overlapping windows do not determine the number of species.

Simulated collector pulses and known-ion windows

Conditional calibrated intervals

m/Q

Import JSON or CSV

Raw observations and calibration

Capture	Absolute tick	Elapsed ticks	ADC code

Exports include every reference, applied preparation command, voltage readback, birth and pulse timestamp, field version, lineage and retained trajectory. Reset changes the apparatus settings and keeps older captures.

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.

Read the instrument principle, sources and limits

Focus question: What connects motion inside the trap to a calibrated mass window?
One-sentence intuition: The field and mass-to-charge ratio determine motion. Finite openings and flight determine collection, and known ions turn those pulses into checked mass intervals.

Core mathematical model

One potential determines force and work

m\ddot{\mathbf r}=-Q\nabla\Phi,\qquad \frac{d}{dt}(K+Q\Phi)=Q\frac{\partial\Phi}{\partial t}

Positions are in metres, time in seconds and charge in coulombs. Time-dependent electrodes exchange energy with the ion. Actual material surfaces terminate a colliding trajectory.

A constant-amplitude ideal-quadrupole reference

\tau=\frac{\Omega t}{2},\qquad q_x=-\frac{2QV_{\mathrm{RF}}}{mr_0^2\Omega^2}

RF voltage is the zero-to-peak difference between opposite rod pairs. The ideal near-axis potential uses that voltage times the transverse squared-coordinate difference divided by twice the field-radius squared. The other transverse Mathieu parameter has the opposite sign. The finite field also includes slots, endcaps and housing.

Averaged motion at small parameters

\omega_{\mathrm{sec}}\simeq\frac{\Omega}{2}\sqrt{a_x+\frac{q_x^2}{2}}

This small-parameter approximation includes the constant field's local curvature in this expression. Fast micromotion rides on the slow oscillation. A voltage scan approaching instability requires the full time-dependent field and an actual collector trajectory to establish the detection time.

A pulse first supplies an arrival interval

t_{\mathrm{arr}}=(n_{\mathrm{pulse}}-n_{\mathrm{birth}})s_{\mathrm{clk}}

Recorded counters and an independent clock standard supply elapsed time. Actual known-ion windows calibrate mass intervals, and different withheld ions test them. Each pulse shares the same reference error, which remains at the mass-window endpoints.

Common difficulties

Escape and detection

Typical misconceptionCrossing the opening counts an ion.

Better mental modelThe ion can return to the trap or strike the negative dump. Only a terminal event at the positive collector front enters the counter.

What intervals identify

Typical misconceptionTwo coloured mass windows identify two molecular species.

Better mental modelTouching or overlapping windows do not determine species count. Different masses and charge states can share a mass-to-charge ratio. This model reports finite conditional intervals and the counts of actual collector hits.

Run the experiment

01
Find the stored ion
Compare the whole trap with the ion closeup, then pause and change initial RF amplitude.
What to observe: Fast RF motion rides on a slower oscillation. Holding the gold feed knob changes the real reference field; the data plots keep the last acquisition.
02
Separate two ejection methods
Compare resonant excitation, AC off and the default RF mass scan.
What to observe: A suitable auxiliary drive grows the oscillation. The default measurement increases RF amplitude until the actual trajectory leaves the trap. Both must continue to the collector to create a count.
03
Follow the opening to the front face
Use the rod-opening and collector views.
What to observe: An opening permits further flight. The positive collector front registers a pulse; the grounded negative dump registers a loss. The default mirror attempts start 200 microseconds apart, after the complete previous gate.
04
Calibrate a finite mass window
Compare the default window, controlled mixed exposures and the broad mixed source.
What to observe: Each collected event supplies a compatible interval. Windows widen near a reference anchor. A broad source changes the ejection-time distribution and can retain pulses while mass interpretation stops.
05
Keep a reproducible capture
Export JSON or CSV, change settings, reset and import the earlier archive.
What to observe: Readbacks, birth times, scans, pulse timestamps and references remain attached to each acquisition. Reset changes the apparatus settings; a new acquisition integrates fresh finite trajectories.

The red and blue sidebands respond differently.

Which motion levels remain?

A matched red–blue comparison

Bounded motion can still hit an aperture.

Radial motion and period means

Ideal Mathieu stability section

From escape to a collector pulse

Core mathematical model

Resolved sidebands

Thermal preparation

Sideband coupling

State reading

Red-to-blue ratio for a thermal state

Common difficulties

The glow is not a path

Limits of the model

Run the experiment

Prepare

Probe

Compare

Core mathematical model

One potential determines force and work

A constant-amplitude ideal-quadrupole reference

Averaged motion at small parameters

A pulse first supplies an arrival interval

Common difficulties

Escape and detection

What intervals identify

Run the experiment

Find the stored ion

Separate two ejection methods

Follow the opening to the front face

Calibrate a finite mass window

Keep a reproducible capture

Related experiments and articles

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Clocks, traps and single-quantum control

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