---
title: "Two-Ion Quantum Logic Spectroscopy Interactive Lab | Sandbox Physics"
description: "Read one ion through another. Probe a spectroscopy ion, map its excitation through a shared motional quantum and count fluorescence from the auxiliary ion. Calibrate bright and dark records before inferring the spectral center."
canonical_url: "https://sandboxphysics.com/labs/two-ion-quantum-logic-spectroscopy/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/two-ion-quantum-logic-spectroscopy/"
last_modified: "2026-10-19"
---

SPECTROSCOPY ION / SHARED MOTION / LOGIC ION

# Two-Ion Quantum Logic Spectroscopy

Probe a spectroscopy ion, map its excitation through a shared motional quantum and count fluorescence from the auxiliary ion. Calibrate bright and dark records before inferring the spectral center.

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

Synthetic two-level ions and one initially cold transfer mode. Coherent red-sideband pulses with an illustrative phonon-loss channel; ideal confinement and preparation. No RF electrode solution, thermal modes, spontaneous decay, clock systematics or precision claim. The fluorescence spectrum determines transferred population, not coherence or transfer fidelity. Enlarged apparatus and pulse animation are schematic.

The trap axis joins the two ions. The two ions and their separation are enlarged. Sideband beams address the ions in sequence; shared motion is not a photon traveling between them. The objective collects only auxiliary-ion fluorescence toward the counter. Sequence colors show source population expectations, not a measured trajectory.

SYNTHETIC OBSERVATIONS / REPRODUCIBLE RECORDS

### A spectrum read through the auxiliary ion.

Ready to acquire synthetic records.

A new run retains 64 known bright and 64 known dark preparations, followed by an interleaved 41-frequency scan. Each trial records detected photon count, not a hidden ion-state label. Source line shift, sideband pulse areas, delay and source photon means are excluded from the spectral estimator. Acquisition is accelerated.

RETAINED DATA

### Individual fluorescence counts.

$i,\quad n_{\rm det}$

First 240 scan trials, excluding calibration. Green: current detected counts. Gray: exact saved counts. The inspection control selects one of these trials; the full run remains in CSV.

ESTIMATES FROM RECORDS

### Keep the claim within the measurement.

Fitted spectral center

$\text{—}$

Transferred population amplitude

$\text{—}$

Measured bright / dark mean photons

$\text{—}$

Calibrated false / true dark fractions

$\text{—}$

Inspected photon count

$\text{—}$

On-resonance source populations · spectroscopy / motion / logic

$\text{—}$

No line center is reported for incomplete acquisition, overlapping references, a weak transferred response or a fit at the search edge. The profile is a relative likelihood diagnostic, not a calibrated confidence interval.

DATA REDUCTION

### Calibrated dark fractions and fitted line.

$\delta/\mathrm{kHz},\quad\widehat q_D$

Green points: classified dark fraction; brown line: response fitted from these counts with calibrated classification rates. No source center is drawn as a measurement.

### Center likelihood profile.

$\delta_0/\mathrm{kHz},\quad 2(\ell_{\max}-\ell)$

Transfer amplitude is profiled at each candidate center. Relative deviance is capped at forty.

Last eight raw records

Columns: index, preparation, probe detuning in kilohertz, detected photons.

| Index | Preparation | Detuning | Photons |
| --- | --- | --- | --- |

### Model and sources

Synthetic two-level ions and one initially cold transfer mode. Coherent red-sideband pulses with an illustrative phonon-loss channel; ideal confinement and preparation. No RF electrode solution, thermal modes, spontaneous decay, clock systematics or precision claim. The fluorescence spectrum determines transferred population, not coherence or transfer fidelity. Enlarged apparatus and pulse animation are schematic.

Rectangular-pulse excitation

$$
\begin{aligned}P_e(\delta)&=\frac{\Omega^2}{\Omega^2+\Delta^2}\\&\quad\cdot\sin^2\!\left(\frac{T\sqrt{\Omega^2+\Delta^2}}2\right)\\\Delta&=2\pi(\delta-\delta_0),\quad\Omega T=\pi\end{aligned}
$$

Detunings in the formula are frequencies in hertz; the scan displays kilohertz. Probe duration is a calibrated input.

State mapping

$$
\begin{aligned}|e_Sg_L0\rangle&\longrightarrow|g_Sg_L1\rangle\\&\longrightarrow|g_Se_L0\rangle\end{aligned}
$$

Two resonant red-sideband pulses transfer excitation through the shared mode. The unexcited ground branch stays unchanged. The displayed source evolution uses a density matrix.

Population transfer

$$
\begin{aligned}A&=\sin^4(\theta/2)\,e^{-t_d/\tau_m}\\P_{L,e}&=A\,P_e,\quad\tau_m=1\,\mathrm{ms}\end{aligned}
$$

An illustrative zero-temperature amplitude-damping channel acts on the mode between pulses. This lifetime is a chosen parameter, not a measured apparatus value.

Calibrated observation model

$$
\begin{aligned}q_D(\delta)&=f_B+(f_D-f_B)\,\widehat A\,P_e(\delta)\end{aligned}
$$

Bright/dark reference counts determine the threshold and two classification rates. A binomial likelihood fits center and transfer amplitude from scan counts; source shift and pulse areas are excluded.

[Schmidt et al. · Spectroscopy Using Quantum Logic (2005)](https://tf.nist.gov/general/pdf/2053.pdf)

[Three-Photon GHZ Correlation Bench](https://sandboxphysics.com/labs/three-photon-ghz-correlations/)

Physics tutorial

## Two-Ion Quantum Logic Spectroscopy

**Background** Probe a spectroscopy ion, map its excitation through a shared motional quantum and count fluorescence from the auxiliary ion. Calibrate bright and dark records before inferring the spectral center.

**Why it matters** A protocol connects a prepared quantum system to retained detector observations.

Start with the essentials

- **Focus question**: Read one ion through another.
- **One-sentence intuition**: Keep source assumptions separate from estimates made from records.

### Core mathematical model

#### Rectangular-pulse excitation

$$
\begin{aligned}P_e(\delta)&=\frac{\Omega^2}{\Omega^2+\Delta^2}\\&\quad\cdot\sin^2\!\left(\frac{T\sqrt{\Omega^2+\Delta^2}}2\right)\\\Delta&=2\pi(\delta-\delta_0),\quad\Omega T=\pi\end{aligned}
$$

Detunings in the formula are frequencies in hertz; the scan displays kilohertz. Probe duration is a calibrated input.

#### State mapping

$$
\begin{aligned}|e_Sg_L0\rangle&\longrightarrow|g_Sg_L1\rangle\\&\longrightarrow|g_Se_L0\rangle\end{aligned}
$$

Two resonant red-sideband pulses transfer excitation through the shared mode. The unexcited ground branch stays unchanged. The displayed source evolution uses a density matrix.

#### Population transfer

$$
\begin{aligned}A&=\sin^4(\theta/2)\,e^{-t_d/\tau_m}\\P_{L,e}&=A\,P_e,\quad\tau_m=1\,\mathrm{ms}\end{aligned}
$$

An illustrative zero-temperature amplitude-damping channel acts on the mode between pulses. This lifetime is a chosen parameter, not a measured apparatus value.

#### Calibrated observation model

$$
\begin{aligned}q_D(\delta)&=f_B+(f_D-f_B)\,\widehat A\,P_e(\delta)\end{aligned}
$$

Bright/dark reference counts determine the threshold and two classification rates. A binomial likelihood fits center and transfer amplitude from scan counts; source shift and pulse areas are excluded.

### Common difficulties

#### A simulation supplies its law

**Typical misconception** Recovering a source parameter independently confirms the law.

**Better mental model** These synthetic records test a measurement and inference workflow. They are not historical observations, a parameter confidence interval, or external scientific certification.

### Run the experiment

1. 01

   #### Calibrate and scan

   Acquire a complete run; export the raw photon counts.

   **What to observe:** The first 128 records alternate known bright and dark preparations; 41 probe frequencies follow.

2. 02

   #### Break the transfer

   Save a comparison, then turn mapping off or add delay.

   **What to observe:** The auxiliary fluorescence loses its spectral contrast even though the source probe excitation remains.

3. 03

   #### Separate population and coherence

   Inspect the sequence stages and the fluorescence calibration.

   **What to observe:** Stage populations are source predictions; the retained photon counts support population inference only. A Ramsey protocol would be needed to test transferred coherence.

Physical conclusion

Synthetic two-level ions and one initially cold transfer mode. Coherent red-sideband pulses with an illustrative phonon-loss channel; ideal confinement and preparation. No RF electrode solution, thermal modes, spontaneous decay, clock systematics or precision claim. The fluorescence spectrum determines transferred population, not coherence or transfer fidelity. Enlarged apparatus and pulse animation are schematic.

Quantum & fields · Starter path

## Start from a single measurement

Predict outcomes and collect counts, building from one qubit to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare it with the frequency of acquired 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)