A logic ion reports the state of a spectroscopy ion

Two-Ion Quantum Logic Spectroscopy

How can you take the spectrum of one ion by watching the light from another ion beside it? Through the motion the two ions share in one trap. After the probe laser tries to excite the first ion, one pulse turns any excitation into a quantum of that shared vibration, and a second pulse hands it to the other ion, whose fluorescence shows whether it arrived. Repeating this at each probe frequency traces out the first ion's line. If the vibration is lost between pulses, the line fades.

Excite the spectroscopy ion, pass its excitation through one shared quantum of motion to the logic ion, and count the logic ion's fluorescence. Calibrate the bright and dark counts first, then find the center of the line.

Prepare a state and count the outcomes

How can you take the spectrum of one ion by watching the light from another ion beside it?

Source line shift · excluded from fit

\delta_0/\mathrm{kHz}

4

Calibrated rectangular probe duration

T/\mathrm{\mu s}

20

Both sideband pulse areas

\theta/\pi

1

Delay between mappings · illustrative lifetime one millisecond

t_d/\mathrm{ms}

0

Source bright mean · calibrated from counts

\bar n_B

18

Trials per probe frequency

N

100

Inspect source sequence stage · keeps records

j

4

Inspect retained scan trial · keeps records

i

0

Inspection keeps every record. Acquisition changes clear current data; a saved comparison remains in memory.

Both ions are synthetic two-level systems that share one motional mode, which starts cold. The red-sideband pulses are coherent, phonon loss uses an illustrative channel, and trapping and preparation are ideal. The Lab does not solve for the RF electrode field, leaves out thermal modes, spontaneous decay and clock systematics, and makes no precision claim. The fluorescence spectrum shows how much population was transferred, not coherence or transfer fidelity. The enlarged apparatus and the 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.

Reproducible synthetic observations

A spectrum read through the auxiliary ion.

Ready to collect simulated data.

A new run starts with 64 preparations known to be bright and 64 known to be dark, then an interleaved scan over 41 frequencies. Each trial records the detected photon count, not a hidden label for the ion’s state. The spectral estimator does not use the source line shift, sideband pulse areas, delay or source photon means. The acquisition is sped up.

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 the records

Claim only what the measurement supports

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 if the acquisition is incomplete, the references overlap, the transferred response is weak or the fit lands at the edge of the search range. 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.

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

At each candidate center the transfer amplitude is optimized again. The plot caps the relative deviance at forty.

Last eight raw records

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

Index	Preparation	Detuning	Photons

Model and sources

Excitation by a rectangular pulse

\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}

In the formula, detunings are frequencies in hertz; the scan axis shows kilohertz. The probe duration is calibrated and supplied as a known input.

Passing the excitation to the logic ion

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

Two resonant red-sideband pulses pass the excitation through the shared motional mode. If the spectroscopy ion was not excited, nothing changes. The source evolution shown on screen is computed with a density matrix.

How much population is transferred

\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}

Between the pulses, an illustrative zero-temperature amplitude-damping channel acts on the motional mode. Its lifetime is a chosen parameter, not a value measured on an apparatus.

Probability of a dark reading

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

Counts from known bright and dark references set the threshold and the two classification rates. A binomial likelihood then fits the line center and the transfer amplitude to the scan counts. The source shift and pulse areas are not given to the fit.

Schmidt et al. · Spectroscopy Using Quantum Logic (2005)

Three-Photon GHZ Correlation Bench

Start with the essentials

Focus question: How can you take the spectrum of one ion by watching the light from another ion beside it?
One-sentence intuition: Keep what is assumed about the source apart from what is estimated from the recorded data.

Core mathematical model