DISPERSIVE PROBES / CAUSAL CONTROL

Cavity Photon Counting & Feedback

Send successive Ramsey probe atoms through one cavity. Retain their binary state records, infer a photon-number distribution, then use delayed feedback and a fresh verification sequence to test its stability.

COUNT / INFER / REPAIR

Let the atoms carry the evidence.

Prepared number · excluded from inference

n_{\rm source}

4

Feedback target

n_\star

4

Phase shift per photon

\Phi/\mathrm{rad}

0.67

Calibrated Ramsey contrast

C

0.92

Calibrated readout efficiency

\eta

0.8

Zero-temperature energy decay rate

\kappa/\mathrm{s}^{-1}

0

Probe separation

\Delta t/\mathrm{ms}

5

Actuation latency in probe intervals

L

4

Calibrated jump-actuator success

q

0.85

Feedback enabled

f_{\rm on}

0

Probe atoms per acquisition

N_{\rm probe}

160

Changing any setting starts a fresh cavity and clears current records. A saved run remains available. Probe phases cycle through four quadratures.

Synthetic number-diagonal model on zero through nine photons. Calibrated Ramsey contrast and missed detections; exact zero-temperature photon loss between probes. The declared stochastic single-photon actuator is an educational replacement for coherent displacement, with clamping at both number boundaries. No phase coherence, thermal excitation, multi-atom packets, atomic transit dynamics or historical fidelity is simulated. Non-demolition refers to the probe, not cavity loss or feedback.

Atoms travel left to right through both Ramsey zones and the gap between the cavity mirrors, then enter the state detector. The stored microwave axis is perpendicular to the atom beam. Motion, dimensions and magnification are schematic; no displayed particle is a recorded event.

SAME CAVITY / SUCCESSIVE ATOMS

A binary record. A number distribution.

Prepare atomFirst Ramsey pulseDispersive cavitySecond Ramsey pulseState readout

Continuing control after verification prepares a fresh cavity. The number filter begins with equal weights from zero to nine. It receives observed atom states, phase settings and calibrated transitions. Prepared number and hidden actuator success never enter inference.

Ready. Predict first, then acquire.

Space: acquire or stop. Right arrow: one atom. Escape: stop. Shortcuts apply only when the workbench has focus; ordinary controls retain their native keys.

RAW ATOM STREAM

A missed readout remains a record.

$k,\quad j_k\in\{0,1,\varnothing\}$

Teal: control atoms. Violet: fresh verification atoms. Gray: missed detection. Phase is retained in every raw row; the plot shows the most recent three hundred probes.

NUMBER KNOWLEDGE / NO HIDDEN TRUTH

Keep every plausible number.

$n,\quad p(n\mid\text{retained atom records})$

Control posteriorFresh verificationSaved control posterior

CAUSAL CONTROL / FINITE DELAY

Each pulse follows earlier evidence.

$k,\quad p_k(n_\star),\quad d_k\in\{-1,0,+1\}$

Green pulses add one photon probabilistically; gold pulses remove one. The controller propagates the calibrated probability, never the hidden success. A single queued command waits for its full delay. Decisions require eight probes and a posterior peak above fifty-five percent.

RECORD-ONLY METRICS

Retained probes

\text{—}

Control posterior mean

\text{—}

Control target probability

\text{—}

Control uncertainty / bits

\text{—}

Fresh verifier target probability

\text{—}

Saved target probability

\text{—}

Posterior target probability is conditional on this calibrated number model. It is not observed preparation fidelity or a phase-coherence measurement. Verification begins from a fresh flat prior, cancels all queued pulses and freezes the control posterior. Cavity loss continues during verification.

LATEST RAW RECORDS / ALL ROWS IN CSV

Probe	Purpose	Phase / rad	State	Pulse	Issued after

SAVE / REPLAY / EVIDENCE

No saved run. Saving preserves the exact random state, record streams, posteriors and queued pulses in memory.

Zero temperature is assumed. Number is bounded from zero through nine; attempted boundary jumps stay at the boundary. Feedback is a stochastic number-jump teaching model, rather than the coherent microwave displacement of the historical apparatus.

Guerlin et al. · QND counting (2007)Sayrin et al. · real-time feedback (2011)Ramsey frequency measurement Compare fluorescence state readout

Start with the essentials

Focus question: Can a stream of atoms count and stabilize light while the photons remain inside?
One-sentence intuition: Keep four things separate: stored photons, detected atoms, inferred number probabilities and commanded energy changes. More certain photon number does not mean preserved optical phase.

Core mathematical model

One atom, partial information

P(j\mid n,\varphi)=\frac{1+(-1)^jC\cos(n\Phi-\varphi)}{2}

The two observed energy states have complementary Ramsey probabilities. Alternating the analysis phase supplies both quadratures, but cannot undo exact number-period aliases.

Only records update knowledge

p_k(n)=\frac{P(j_k\mid n,\varphi_k)\,\widetilde p_k(n)}{\sum_mP(j_k\mid m,\varphi_k)\,\widetilde p_k(m)}

The predicted distribution includes known loss and commanded stochastic actuation. The filter never receives the prepared or simulated number. A missed atom leaves the predicted distribution unchanged.

Independent photon survival

\begin{gathered}P(m\mid n,\Delta t)\\=\binom{n}{m}s^m(1-s)^{n-m},\\s=e^{-\kappa\Delta t}\end{gathered}

The exact zero-temperature loss map allows several photons to disappear between probes. Its mean decays exponentially. This Lab omits thermal upward jumps.

Declared educational actuator

\mathcal A_{d}(p)_m=\sum_np_n\left[(1-q)\delta_{m,n}+q\delta_{m,\operatorname{clip}(n+d,0,9)}\right]

A commanded addition or removal succeeds with known probability. The controller propagates this entire map, never the hidden success. Commands take effect only after the chosen delay. This is not the coherent displacement used by Sayrin and colleagues.

Common difficulties

Non-demolition has a scope

Typical misconceptionA QND cavity cannot lose photons.

Better mental modelThe diagonal probe preserves photon number. Cavity damping and the feedback actuator exchange energy. Their transitions are part of both the simulator and the filter.

Knowledge is not coherence

Typical misconceptionA sharp number posterior proves that the full field state has been preserved.

Better mental modelThe displayed state contains number populations only. It cannot represent phase coherence or cat-state interference. Number measurements generally disturb phase; no phase-preservation claim is made.

Fresh verification has a clock

Typical misconceptionThe verifier measures a perfectly frozen final photon number.

Better mental modelVerification uses fresh outcomes with a fresh prior, but shares the same evolving cavity. Loss continues, all commanded pulses are canceled, and the controller posterior remains frozen for comparison.

Run the experiment

01
Ask the atoms
Choose Count without loss and acquire a sequence. Compare the binary record, posterior and missing readouts.
What to observe: A detected atom changes knowledge; the dispersive probe itself does not change the simulated photon number.
02
Keep an ambiguity
Select Parity alias, acquire, and inspect all surviving posterior peaks. Then try Weak / missed probes.
What to observe: Identical likelihoods retain multiple number candidates. A high posterior mean is not a precise number measurement.
03
Close the causal loop
Select Repair photon loss. Observe commands being queued after a record and applied at later probe indices. Save a run, then compare with Late controller.
What to observe: Inference can lag a loss. A slow actuator may respond to an obsolete estimate; target probability is conditional knowledge, not observed fidelity.
04
Verify with new atoms
Stop, then acquire fresh verification. Queued commands are canceled, feedback is frozen and a separate flat-prior filter processes only the new atoms.
What to observe: The verification posterior is independent of control records conditional on the physical field and calibrated model. The cavity still loses photons during verification, so this does not measure a frozen final state.
05
Keep a reproducible record
Save, change settings, restore and export. Stop midway to retain a partial acquisition; use one-atom steps to inspect the sequence.
What to observe: Restoration retains parameters, random generator, both posteriors, every raw outcome and the pending command queue. CSV exports observed records and commands, never hidden photon truth.