Q027 · Reading an atom from its photon clicks

Fluorescence & Quantum Jumps

When the detector goes quiet, has the atom really gone dark? Probably, but not certainly. The longer the silence lasts, the more likely the atom has jumped into a dark shelving state, a level that does not scatter the laser light. The detector misses some photons, though, so part of any silence is just missed light. All you can give is a probability, worked out from the pattern of clicks.

Record the fluorescence clicks from one atom and work out, from the clicks alone, how likely it is to be shelved. Then reveal the simulated trajectory and try a repump that fires after a delay.

Recorded clicks $N=0$

Inferred dark probability $\text{—}$

01 / The experimental setup

One atom produces a stream of clicks.

Excite the atom; collect fluorescence at right angles.

Drag to rotate · scroll to zoomAtom enlarged · glow replays detected clicks

Recorded time

Choose an experiment

Incoherent pump rate

r

2

Detection efficiency

\eta

0.65

Detector & model settings

Background rate per detector

b

0.005

Fluorescence decay rate

\gamma

1

Shelving rate

s

0.04

Dark-state recovery rate

k

0.025

Record duration

T

400

Silence trigger

t_{\rm trigger}

15

Feedback latency

t_{\rm delay}

5

Reproducible seed

q

2735

Repump after recorded silence

Changing a setting loads a new simulated record. Replay preserves every timestamp. Time and playback speed use illustrative reference units.

Record management

Before you measure

Does a silent counter prove the atom is dark?

The atom is a three-level Lindblad model with incoherent pumping, and its state stays diagonal. The model includes emission, shelving, missed photons and independent background clicks. It leaves out coherent Rabi dynamics, atomic motion and the details of any particular ion species. The conditional probabilities hold for this monitoring model.

02 / Follow the evidence

When the clicks stop, how does the inference change?

Read a quiet interval

—

Reveal the simulated state at this time

Click or drag on the record to inspect a moment.

Top: recorded clicks. Below: inferred dark probability. Gray dashed: an ensemble reference without feedback. Orange: known repump pulses, or the hidden trajectory when Reveal is enabled.

Numerical results & model checks

Conditional dark probability at cursor

\text{—}

Longest recorded silence / clicks

\text{—}

Open-loop steady populations

\text{—}

Ensemble comparison

\text{—}

Raw records & scientific boundary

Each run starts in the ground state. The filter never sees the hidden transitions or which photons were missed. A click can come from background, and a silence can come from loss. Feedback responds to the recorded silence: it schedules one ideal pulse, then waits for a later click before it can fire again. The pulse can arrive after the atom has already recovered. This does not imply that quantum trajectories have a unique interpretation.

The detector dial and efficiency slider set the same model probability for registering an emitted photon. Turning it loads a fresh simulated record. Its angle is a user interface without manufacturer calibration; replay brightness still comes only from detected clicks.

Primary references: Wineland · Nobel lecture · Kimble et al. · 1977 · quED · HBT

Start with the essentials

Focus question: When the detector goes quiet, has the atom gone dark?
One-sentence intuition: Draw conclusions from the recorded clicks first, and only then look at the simulated state.

Core mathematical model

Units of time and rate

\tau=t/t_*,\qquad r,\gamma,s,k,b,\Phi\ \text{in}\ t_*^{-1}

The reference time is arbitrary and is not tied to any atom or real lifetime. All times, rates and CSV values are given in this unit.

Diagonal Lindblad dynamics

\dot{\rho}=\sum_{a}\mathcal D[L_a]\rho,\quad\mathcal D[L]\rho=L\rho L^\dagger-\tfrac12\{L^\dagger L,\rho\}

The jump operators below keep a diagonal state diagonal. The simulation samples exact exponential waiting times between jumps, and the model has no coherent Hamiltonian drive.

Four physical transitions

L_r=\sqrt r\,|e\rangle\langle g|,\quad L_\gamma=\sqrt\gamma\,|g\rangle\langle e|,\quad L_s=\sqrt s\,|d\rangle\langle e|,\quad L_k=\sqrt k\,|g\rangle\langle d|

Only the fluorescence is observed, and the detector misses some of it. A jump to the dark state interrupts the bright cycle of excitation and emission.

Population master equation

\dot{\mathbf p}=Q\mathbf p,\quad Q=\begin{pmatrix}-r&\gamma&k\\r&-\gamma-s&0\\0&s&-k\end{pmatrix}

Each column sums to zero. The ensemble curve starts in the ground state and applies only when feedback is off. Below the record you can check it against an average of 200 trajectories.

Inference from silence and clicks

\dot{\widetilde{\mathbf p}}=(Q-\eta\gamma|g\rangle\langle e|-bI)\widetilde{\mathbf p},\quad\mathbf p_{+}=\frac{\eta\gamma p_e|g\rangle+b\mathbf p}{\eta\gamma p_e+b}

Normalizing the no-click likelihood gives the state probabilities, given that the detector stayed silent. A click might be background, so it does not always put the inferred atom back in the ground state. The filter never sees the hidden trajectory.

A delayed repump

\mathbf p\longmapsto(p_g+p_d,\ p_e,\ 0)^{\mathsf T}

When a recorded silence lasts longer than the trigger time, one ideal repump is scheduled after the chosen latency. A click in the meantime does not cancel it. The pulse moves the dark population to the ground state and leaves the excited state alone. The controller rearms after the next click.

Common difficulties

A gap in the clicks does not prove shelving

Typical misconceptionA gap in the clicks proves the atom is shelved.

Better mental modelThe conditional likelihood has to account for missed photons and background clicks.

Run the experiment

01
Predict
Does a silence prove the atom is shelved? Predict, then try the missed-photon case.
What to observe: A quiet detector does not have to mean a dark atom.
02
Record
Collect a record, then drag the time cursor and watch the inferred probability.
What to observe: Make your own inference first, and only then reveal the simulated state.
03
Add feedback
Turn on feedback and vary its latency. Then turn it off to run the ensemble check.
What to observe: The controller responds only to recorded clicks; it cannot see the hidden jumps.

Fluorescence & Quantum Jumps

When the clicks stop, how does the inference change?

Read a quantum jump record

Core mathematical model

Units of time and rate

Diagonal Lindblad dynamics

Four physical transitions

Population master equation

Inference from silence and clicks

A delayed repump

Common difficulties

A gap in the clicks does not prove shelving

Run the experiment

Predict

Record

Add feedback

Related experiments and articles

Clocks, traps and single-quantum control