Look things up here. The live experiments — the workshop — are in the hub ↗.
A lookup surface, not assigned reading: each card is a link target the hub and the exercises point into at the moment you need it. One thin card per anchor — definition, the load-bearing equations, the one thing it does not imply, and where to read more. Cards assume the course's working level (logs, 1/√N, basic quantum optics).
▶ workshop: the trade-off triangle
A sensor's performance is bounded by three competing figures of merit: sensitivity (the smallest change it can resolve, set by its noise floor), dynamic range (the ratio of the largest to smallest values it reads without saturating), and bandwidth (the band of input frequencies it follows faithfully). These pull against one another, so a real sensor is a chosen operating point on the triangle — not a maximum of all three at once.
More sensitivity is not strictly better — it is bought against dynamic range and bandwidth.
| Sensor | Sensitivity | Dynamic range | Bandwidth |
|---|---|---|---|
| Dark-adapted eye | ≈ single photon | ~10⁶ | ~10–60 Hz |
| MEMS accelerometer | ~10⁻³ m/s² | 0–80 m/s² | < 100 Hz |
| Ion optical clock | ~1 part in 10¹⁸ | one line ± few MHz | ~1/hour |
From the 2025 Sensing I notes. No row wins all three corners.
▶ workshop: the Allan / PSD explorer
Noise is read three complementary ways. The mean and standard error summarise central value and precision; the power spectral density (PSD) shows how variance is spread over frequency; and the Allan deviation measures stability versus averaging time τ, separating white, flicker (pink), and random-walk noise by their characteristic slopes. The Allan curve typically falls, flattens, then rises — a "bathtub" whose minimum marks the best averaging time.
A minimum in the Allan deviation fixes the optimal averaging time; it does not by itself identify the physical noise source.
| Noise | PSD slope (vs f) | Allan slope (vs τ) | Meaning |
|---|---|---|---|
| White | 0 | −½ | averaging always helps |
| Flicker / pink | −1 | 0 | a floor averaging can't beat |
| Random walk / drift | −2 | +½ | averaging eventually hurts |
The bathtub minimum is where the −½ and +½ branches cross.
▶ workshop: the 1/√N wall (counting → the SQL)
The standard quantum limit (SQL), or shot-noise limit, is the precision floor set by counting statistics of independent quanta — photons, atoms, or repeated trials. A phase or field estimated from N independent contributions has an uncertainty that falls only as 1/√N. It is the floor that squeezing (anchor 5) and entanglement (anchor 4) set out to beat.
The 1/√N scaling is not uniquely quantum — it follows from independent classical counting statistics.
See where 1/√N comes from — independent counting — in the anchor-3 demo; and the SQL as the vacuum's uncertainty disk in the squeezing widget (set squeezing to 0).
▶ workshop: the quantum ladder
Quantum metrology is a ladder of resources. A coherent probe gives the SQL, 1/√N; squeezing lowers the prefactor to ξ/√N (ξ<1); multipartite entanglement — GHZ (Greenberger–Horne–Zeilinger) and NOON states — reaches the Heisenberg limit, 1/N; and entanglement protected by error correction aims to keep 1/N in a noisy world. Each rung trades a stronger resource for greater fragility.
A higher rung is not automatically better once decoherence and overhead are counted — the resource–fragility trade.
| Level | Key idea | Platform | Scaling |
|---|---|---|---|
| 1–2 Coherent | Ramsey / spin-echo superposition | ion, NV, atom clock | 1/√N |
| 2S Squeezed | redistribute noise | spin-squeezed clock; LIGO light | ξ/√N, ξ<1 |
| 3 Entangled | GHZ / NOON, Heisenberg | 10-ion GHZ; photonic NOON | 1/N |
| 4 + QEC | protect against noise | NV / ions (emerging) | retains 1/N |
The level table from the 2025 Sensing II notes.
▶ workshop: squeezing & back-action
Back-action is the unavoidable disturbance a measurement imposes on the conjugate observable (Heisenberg). Squeezing reshapes quantum noise: it narrows one quadrature below the vacuum (the SQL) at the cost of fattening the conjugate one — and it can be shaped across frequency. It lowers the noise floor only where the measurement actually reads the squeezed quadrature; a small misalignment lets the huge anti-squeezed variance leak back in.
Squeezing does not beat the Heisenberg uncertainty product; it redistributes variance between conjugate quadratures (and across frequency).
Phase space: the vacuum disk squeezed into an ellipse (area conserved) — see it live in the phase-space widget, and watch the noise pop back above the SQL as you misalign θ.
Transduction is the chain of physical couplings that turns the quantity of interest into a readable signal — typically motion (phonons) ↔ spin ↔ light (photons). Quantum sensors live or die on these chains: every rung of the ladder needs one to prepare and to read out the probe (e.g. a trapped ion's tiny displacement is mapped to its spin, then to fluorescence photons).
A transduction chain is only as good as its noisiest, least efficient link.
| Photons | Phonons (ion motion) | |
|---|---|---|
| Conjugate pair | E- and B-field quadratures | position x, momentum p |
| Cooling to vacuum | T ≈ 300 K is cold enough | active cooling < 1 mK (sideband) |
| Squeezing | OPA / OPO crystal | parametric drive at 2× trap freq. |
| Readout | homo-/heterodyne | motion → spin → fluorescence |
Condensed from the 2025 Sensing II photons-vs-phonons table.
Background, not part of the six core cards above — read these only if you want to see where the anchors live in real hardware. This interlude leans on three platforms. The point of each primer: none is exotic — each is just the building blocks you already know, a spin-½ and a harmonic oscillator, dressed in hardware. The NV centre and the gravitational-wave detector the Quantum Hardware course meets only in passing; trapped atomic ions are the home platform of the group hosting this course — and the one where both abstractions live natively in a single atom, a textbook spin-½ alongside the ion's own quantized motion, with the tools to wire them together.
shows up in: noise & coherence · the ladder · transduction
A point defect in diamond — a nitrogen atom beside a missing carbon (a vacancy) — trapping unpaired electrons. Its ground state is an electronic spin triplet (S = 1) with a 2.87 GHz zero-field splitting; it is optically initialised and read out, and it works at room temperature.
Apply a magnetic field to Zeeman-split the ms = ±1 sublevels, isolate one transition (e.g. {0 ↔ −1}) and drive it with microwaves: you now have a clean two-level system — the same spin-½ behind Ramsey, coherence and the SQL (anchors 2–4). Reading it out by spin-dependent fluorescence (ODMR) is exactly the optical transduction of anchor 6.
It is really spin-1, not spin-½ — the third level and the host nuclear-spin bath are precisely what limit (and what you can exploit for) coherence.
shows up in: the SQL · back-action & squeezing
A kilometre-scale Michelson laser interferometer (LIGO, Virgo, GEO600) — enhanced with Fabry-Pérot arm cavities and recycling — that registers the differential strain of space-time: wave amplitudes near 1 part in 1021, from an amplitude-spectral-density noise floor near 10−23/√Hz, as a passing wave stretches one arm and squeezes the other.
The quantum-limited measurement reads the quadratures of the light field, and a single mode of light is a quantum harmonic oscillator. Shot noise, radiation-pressure back-action, the standard quantum limit, and squeezed-vacuum injection are all the same quadrature / phase-space physics as the squeezing widget (anchors 3 & 5). The suspended mirrors are mechanical oscillators too — so the detector couples two harmonic oscillators (optomechanics).
"One oscillator" is a simplification — the real quantum-noise budget is multimode and frequency-dependent, which is why LIGO needs frequency-dependent squeezing.
shows up in: noise & coherence · the ladder · back-action & squeezing · transduction
One or more atomic ions held in a radio-frequency (Paul) trap, whose oscillating field makes a time-averaged, near-harmonic well (a static electrostatic 3D trap is forbidden by Earnshaw's theorem; a Penning trap instead adds a static magnetic field). The ions are laser-cooled — Doppler first, then on a resolved sideband — to near the motional ground state.
Unlike a qubit wired to a separate cavity, here the oscillator is the ion's own quantized motion — both halves are native and exceptionally clean. Two long-lived internal levels (optical/metastable, or a hyperfine/Zeeman pair) make a textbook spin-½, the same one behind Ramsey, coherence and the ladder. The ions' motion is a set of quantized normal modes — three per ion — each a harmonic oscillator with phonons and quadratures, home to ground-state cooling and motional squeezing.
A laser tuned to a motional sideband couples spin to phonon: the red sideband (Jaynes–Cummings) removes a phonon as it flips the spin; the blue (anti-JC) adds one. That one knob gives sideband cooling, maps spin ↔ motion (transduction), and — using the shared mode as a quantum bus — drives entangling gates (the Mølmer–Sørensen gate, robust because the motion is only virtually excited).
"Two clean levels and one perfect oscillator" is a limit, not the device — the pseudopotential is only near-harmonic, and excess micromotion plus anomalous heating from electrode-surface electric-field noise are what really bound performance.