Squeezed light ──
pushing AM noise into FM $\Delta X_1\cdot\Delta X_2\ge1/4$ holds because ── the dynamics preserves phase-space area (numerically $\det M=1.0000000000$).
★★ So shrink one quadrature to $e^{-r}$ and the other must stretch to $e^{+r}$. 6 dB of squeezing multiplies the observable volume by 8.
★ And 5 % loss caps you at 13 dB ── attenuation mixes in vacuum fluctuations, and there is no getting them out.

Prerequisites: Episodes 9, 22, 23, 32, 34, 35Computation: kenshou/asshuku.js

Episode 35 looked at the freedoms of phase (angle, path). This episode is about the noise of phase.
Split a monochromatic beam into two quadratures ── $E(t)=X_1\cos\omega_0t+X_2\sin\omega_0t$, with $X_1$ the amplitude side (AM) and $X_2$ the phase side (FM) (the same split as Episode 32).

The floor quantum mechanics imposes $$\Delta X_1\cdot\Delta X_2\ge\frac14$$

The same shape as Episode 9's $\Delta t\cdot\Delta\omega\ge1/2$ ── because they are Fourier conjugates

★ Which invites the obvious thought ── if the product is fixed, can one be reduced by pushing it into the other? That is squeezed light.

01Counting the squeezing

One real number $r$ describes it $$\Delta X_1=\tfrac12e^{-r},\qquad \Delta X_2=\tfrac12e^{+r}\qquad\Longrightarrow\qquad \text{the product is always }\tfrac14$$

$r$	$\Delta X_1$	$\Delta X_2$	Product	Squeezing [dB]
0	0.500000	0.500000	0.25000000	0.00
0.5	0.303265	0.824361	0.25000000	−4.34
1.151	0.158160	1.580676	0.25000000	−10.00
1.727	0.088909	2.811879	0.25000000	−15.00

The heart of §01

The product is 0.25 everywhere ── uncertainty is not violated. Only the allocation changed.
★ For scale: LIGO (since 2019) about 3 dB, recently about 6 dB; the laboratory record about 15 dB.
In audio terms ── moving noise with an equalizer. The total cannot change, but which quadrature carries it can.

02★★ AM and FM sidebands ── continuing Episode 32

Put a pair of sidebands $\Omega$ above and below the carrier $\omega_0$. Written complex relative to the carrier, the perturbation collapses to one expression.

Two sidebands, combined $$P(t)=2a\,e^{i\theta}\cos(\Omega t+\delta),\qquad \theta=\frac{\theta_++\theta_-}{2},\quad \delta=\frac{\theta_+-\theta_-}{2}$$

★ Only the mean $\theta$ decides AM versus FM; the difference $\delta$ only sets the timing

$\theta_+$	$\theta_-$	$\theta$	Amplitude swing	Phase swing	What it is
0	0	0	0.100000	0.000000	★ pure AM
$\pi/2$	$-\pi/2$	0	0.100000	0.000000	★ pure AM (only $\delta$ differs)
$\pi/2$	$\pi/2$	$\pi/2$	0.002494	0.099669	★ pure FM
$\pi$	0	$\pi/2$	0.002494	0.099669	★ pure FM (only $\delta$ differs)
$\pi/4$	$\pi/4$	$\pi/4$	0.070534	0.070945	a 45° mixture

The heart of §02

★★★ Squeezed light ＝ correlating the sideband pair so that AM-side noise moves to the FM side.

Episode 32 read the Higgs as AM and the Goldstone as FM. The same split is here a design choice: which quadrature carries the noise.

03★ What does the squeezing ── phase-sensitive amplification

Squeezing is made by nonlinearity (Episode 23): shake the spring constant at twice the frequency, $\ddot x+\omega_0^2(1+h\cos2\omega_0t)x=0$.
To separate growth from decay cleanly we look at the eigenvalues of the one-period map (the Floquet matrix $M$).

$h$	$\det M$	$\|\mu_+\|$	$\|\mu_-\|$	Product	Growth rate	Predicted $h\omega_0/4$
0	1.0000000000	1.00000000	1.00000000	1.000000	0.000000	0.000000
0.05	1.0000000000	1.04004817	0.96149392	1.000000	0.012499	0.012500
0.10	1.0000000000	1.08168155	0.92448651	1.000000	0.024993	0.025000
0.20	1.0000000000	1.16987402	0.85479289	1.000000	0.049942	0.050000

The heart of §03 ── ★★★ the core of this episode

$\det M=1$ ＝ phase-space area is conserved ＝ $\Delta X_1\cdot\Delta X_2$ is conserved.

★ Shrink one quadrature to $e^{-r}$ and the other must stretch to $e^{+r}$. Uncertainty is unbreakable because the dynamics preserves area.
The growth rate matches $h\omega_0/4$ to four digits.

★ In Episode 34 nothing grew because the shaking frequency was not right. Parametric amplification needs exactly twice ── the same reason a child on a swing crouches twice per swing.

04How much it buys for gravitational waves

Squeezing [dB]	Noise amplitude	Range	★ Observable volume
0	1.00000	1.00×	1.00×
3	0.70795	1.41×	2.82×
6	0.50119	2.00×	★ 7.94×
10	0.31623	3.16×	31.6×

The heart of §04

★★ 6 dB of squeezing doubles the range and multiplies the observable volume by 8. "Eight times as much universe" ── without changing a mirror or a laser.

★ But not for free (Episode 22's standard quantum limit) ── reduce the phase noise and the amplitude noise grows, and that amplitude noise kicks the mirrors (radiation-pressure noise). Radiation pressure dominates at low frequency and shot noise at high, so the squeezing angle has to be rotated with frequency (frequency-dependent squeezing).

05★ The limit is loss ── attenuation mixes in the vacuum

Passing through an element of transmission $\eta$ $$S=\eta\,e^{-2r}+(1-\eta)$$

The second term is vacuum fluctuation mixed in ── it enters in place of the light that was lost

Transmission $\eta$	Loss	At $r=1.5$ [dB]	★ Limit as $r\to\infty$ [dB]
1.00	0 %	−13.03	(no limit)
0.99	1 %	−12.27	−20.00
0.95	5 %	−10.12	★ −13.01
0.90	10 %	−8.39	−10.00
0.50	50 %	−2.80	−3.01

The heart of §05

★★ With 5 % loss, 13 dB is the ceiling however hard you try. With 10 %, 10 dB ── this is the experimental wall.
★ In wave language ── attenuation is an operation that mixes in vacuum fluctuations. As with Episode 22's monotonically decaying decoherence, once diluted it cannot be undone (a phenomenon of the imaginary axis).
★ So squeezing experiments are a fight against loss ── "amplify the signal instead" does not help, because amplification adds noise too.

Exercises

"Reducing noise without violating uncertainty" sounds contradictory. Where is the consistency?
Show the answer
Only the noise of the quadrature you measure is reduced. As §03 showed, $\det M=1$, so the product of the two quadrature noises is fixed. Reduce one and the other grows.
★ An interferometer measures only one quadrature ── so the excess can be dumped into the one it never looks at. Try to measure everything at once and the gain disappears.
★★ This is exactly Episode 22's "measurement is a choice of basis". Only the allocation is yours to choose ── the same conclusion again.
§03 showed $\det M=1$ is the essential point. What happens in a system with friction?
Show the answer
$\det M<1$ and the area shrinks. Classically that is phase-space contraction ── dissipation ── which sounds attractive.
★ Quantum mechanically it does not work ── as §05 showed, dissipation always brings vacuum fluctuation in with it (the fluctuation–dissipation theorem). Whatever is removed is replaced by noise.
★★ The area cannot be shrunk; only reshaped. Episode 12's "dissipation lives in $\sin(\alpha\pi/2)$" reappears here as the ceiling on squeezing.
If 6 dB gives 8× the volume, why not aim for 20 dB?
Show the answer
Loss will not allow it. By §05, 20 dB requires under 1 % loss ── across every optic, a kilometre-scale round trip, and the detector.
★ And there is the radiation-pressure trade-off (Episode 22) ── squeeze the phase side too hard and the amplitude side kicks the mirrors, ruining low-frequency sensitivity.
★★ Which is why real detectors use frequency-dependent squeezing ── rotating the squeezing angle with frequency (amplitude side at low frequency, phase side at high), achieved by adding one filter cavity.

Summary　Uncertainty was about the total, not the allocation

$\Delta X_1\cdot\Delta X_2\ge1/4$ constrains a product. So one can be reduced by pushing it into the other.

★ The sideband language makes it clear ── the perturbation collapses to $P(t)=2ae^{i\theta}\cos(\Omega t+\delta)$, and only the mean phase $\theta$ decides AM versus FM (the difference $\delta$ is just timing). Squeezed light means correlating the sideband pair so AM-side noise moves to the FM side ── Episode 32's AM/FM becomes a design principle.

★★★ And the core was $\det M=1$.
The Floquet eigenvalues satisfy $|\mu_+||\mu_-|=1.000000$, with the growth rate matching $h\omega_0/4$ to four digits.
Because phase-space area is conserved, shrinking one quadrature must stretch the other.

★ Uncertainty is unbreakable because the dynamics preserves area ── and Episode 34's failed parametric amplification is explained: the shaking has to be at exactly twice the frequency.

The payoff is large ── 6 dB doubles the range and multiplies the observable volume by 8, without changing a mirror or a laser. Though the standard-quantum-limit trade-off means the squeezing angle must rotate with frequency.

★ And the limit is loss ── 5 % caps you at 13 dB, 10 % at 10 dB. Attenuation mixes in vacuum fluctuations, and once diluted there is no undoing it, nor any recovering it by amplification (which adds noise of its own).

Next episode So far the medium has been neat and uniform. ★ Next time we break that.
★ Send a wave into a disordered medium and, past a threshold, it stops propagating (Anderson localization). It halts with no absorption at all.
★★ And it depends on dimension ── in 1D and 2D it always halts; only 3D has a mobility edge. Episode 30's "dimension and order" comes back once more.

This document is Episode 36 of the "Thinking in Waves" series. The computations are in kenshou/asshuku.js. On the numbers here: the squeezing table, the sideband analysis, the Floquet eigenvalues, the sensitivity and volume, and the effect of loss were all computed for this article. Squeezed states and $\Delta X_1\Delta X_2=1/4$, generating squeezing by parametric amplification, the loss formula $S=\eta e^{-2r}+(1-\eta)$, and the use of squeezing (and frequency-dependent squeezing) in LIGO are all well-known results and not discoveries of this article ── what this article did was translate them into "how AM and FM noise is allocated" and show numerically that $\det M=1$ is exactly the conservation of the uncertainty product. §03 is a classical Mathieu-equation analogy; it does not solve the quantum generation of squeezing (degenerate parametric down-conversion) itself, though the structure — two quadratures scaled by $e^{\pm r}$ with area preserved — is the same. The reported squeezing levels (LIGO 3–6 dB, laboratory ~15 dB) are literature values, not verified here. The "8× volume" in §04 assumes sensitivity scales inversely with distance and that sources are uniformly distributed. ── To print, use your browser's Print → Save as PDF.