Beat shot noise — for free?

The vacuum's uncertainty disk is the standard quantum limit. Squeeze one quadrature below it — and watch the price you pay in the other.

The instrument — squeeze, then misalign the measurement

Squeezing:

Squeezing

squeezed-quadrature variance, in dB below shot noise

Measurement angle θ

0° = measure the squeezed quadrature · 90° = the anti-squeezed one

① Phase space (X, P). Dashed circle = SQL (vacuum). The squeezed-state ellipse tilts by θ against the fixed horizontal detection axis; the bar is its shadow on that axis (grey drop-lines), not its radius — inside the circle beats SQL, outside is worse.

② Noise vs measurement angle (dB vs the SQL). Below 0 = beats shot noise; above 0 = worse. The more you squeeze, the narrower the winning window.

Try this Squeeze the state, then misalign the measurement

The question: if squeezing pushes the noise below the shot-noise limit, why not just squeeze as hard as possible? Crank the squeezing, then nudge the measurement angle θ and watch plot ②.

Got a feel for it? Step to 2 · Notice →

Notice this What did squeezing actually buy?

…

Explain this

At θ = 0 more squeezing always looks better. Rotate the angle a few degrees with 20 dB on, and the noise shoots above the SQL. Why does harder squeezing make the measurement more fragile, not just better — and what did you never actually remove?

Reveal a one-paragraph answer

Heisenberg fixes the area of the uncertainty patch: ΔX·ΔP = ½, always. Squeezing doesn't remove noise — it moves it, thinning one quadrature by ξ = e^−r and fattening the other by 1/ξ. If your measurement axis sits exactly on the thin direction, you win by ξ. But the fat direction holds enormous anti-squeezed noise (1/ξ² in variance), so a small angle error — a little phase jitter — leaks that fat variance in and you lose. The harder you squeeze, the fatter the other quadrature, the less misalignment you can tolerate. There is an optimal finite squeezing set by how well you control the phase (and, in a real interferometer, by optical loss, which mixes vacuum back in regardless of angle) — which is why LIGO runs ~6 dB, not 20.

Show the math

For a squeezed vacuum with the squeezed quadrature at X, the variance measured along an axis at angle θ is V(θ)/σ₀² = e^−2rcos²θ + e^+2rsin²θ, with the squeeze strength S (dB) giving e^−2r = 10^−S/10. At θ = 0 the variance is 10^−S/10 (i.e. −S dB, a factor 10^S/10 below the SQL); at θ = 90° it is 10^+S/10 (+S dB, a factor 10^S/10 above). The product of the two quadrature widths is e^−r·e^+r = 1 — the area is conserved (Heisenberg). The SQL itself (S = 0) is the vacuum disk, and it is the same 1/√N shot-noise floor you met as the white-noise wall (anchor 2) and the SQL line (anchor 4).

Now connect it

This disk is the wall. The dashed vacuum circle is the standard quantum limit (anchor 3) — the very same 1/√N you met as the Allan white-noise floor (anchor 2) and the SQL line on the ladder (anchor 4). Anchor 3 shows where that 1/√N comes from; this page shows the one way under it.
This is the ladder's “2S” rung. Measuring the squeezed quadrature gives ξ/√N, ξ < 1 — that is exactly how squeezing bends the SQL line down on the quantum ladder (anchor 4).
Back-action (anchor 5). You bought X-precision by inflating P — you can't beat both. Measuring only one quadrature (stroboscopic / back-action evasion) is how you dodge the disturbance into the part you don't read.

What good looks like:

Reports the sign flip, not just "worse": at θ = 10°, 6 dB still beats the SQL while 20 dB has crossed above it (positive dB = worse than shot noise) — harder squeezing lost the race the gentler setting won.
Names the mechanism via area conservation (ΔX·ΔP = ½): more squeezing fattens the anti-squeezed quadrature, so the same small misalignment leaks more variance back in. You moved noise, you never removed it.
Draws the design conclusion: there is an optimal finite squeezing set by how well you control the phase — which is why LIGO runs ~6 dB, not 20.

Logbook (one entry): Tried · Stuck (where/how long) · Hub resource used · Changed my mind because… · Still unclear.

Exercise check. Set 20 dB (greedy) and slide θ to 10°. Read the noise-vs-SQL number, and compare it to 6 dB at the same angle. Say in one line why “squeeze harder” is the wrong instinct.

Step 1 of 4