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Light of one colour is a wave with a cosine part and a sine part. This page draws those
two parts as a point in a plane, the quadrature plane (it is the phase-space
demo 's position and momentum, scaled). Quantum mechanics will not let that point be sharp: the state of the
light is a cloud, and even the vacuum, with no light at all, is a cloud of radius ½ at the
centre. The cloud turns once per period of the light, and the field you would measure, on
the right, is the cloud's shadow on the horizontal axis: a sine wave with noise on it.
Drag the dot in the plane to set the laser's field and watch the wave grow while
its noise band stays the same width: that is the shot noise of light, and it is the same
for a laser as for the dark. Look for the cloud's angle as seen from the centre: that angle
is how well the wave's phase is known, and it shrinks as the field grows.
The views across the top are the four steps of the story. 2 · Squeezing runs the
light through a crystal that makes the cloud an ellipse, thin one way and fat the other.
3 · Homodyne is how one direction of the cloud is measured, with a second laser whose
phase you turn. 4 · LIGO is the payoff: why an ellipse at the right port lets a
gravitational-wave detector hear further, as a number and as a chirp pulled out of the noise. Each view has experiments below;
the first of each is its opening state.
1. The vacuum's noise is real: the shot noise of light
No light at all. The mean field is zero, and the state is a round cloud of radius
$\Delta X_1=\Delta X_2=\tfrac12$ at the origin: the colours are its Wigner function $W$,
drawn as in the phase-space demo , full red at its largest value, $2/\pi$. On the right is
what a detector would see. Each period a point of the cloud is drawn (the ring on the stick)
and its shadow on the $X_1$ axis is traced as the cloud turns: a sine wave of random size and
phase, inside the grey band $\pm\tfrac12$. That band is the vacuum's noise, the zero-point
motion of the field's oscillator, and when light falls on a photodiode this is what it
records as shot noise. The uncertainty product $\Delta X_1\Delta X_2=0.250$ is the least the
commutator $[X_1,X_2]=i/2$ allows: the vacuum is a minimum-uncertainty state. Its photon
number is exactly 0 (the bar chart): noise, with no photons in it.
2. A laser's light: the classical wave plus the vacuum's noise
$|\alpha|=3$ at phase 0: the same cloud, displaced to the point $\alpha=3$ on the $X_1$
axis. The field is a clean cosine of amplitude 3 with the same $\pm\tfrac12$ of noise on
it. A coherent state is the classical field plus the vacuum's noise, which is why a laser
beam is the most classical light there is and still has shot noise. Its photon number is not
fixed: the bar chart is a Poisson distribution with mean $|\alpha|^2=9.00$ and spread
$\sqrt{9}=3.00$ (the readout gives $\langle n\rangle$ and $\Delta n$). Drag the dot round the
origin: the phase of $\alpha$ is where the cosine starts. Drag it outwards: more photons,
the same noise.
3. More photons, the same noise, a sharper phase: $\Delta\varphi=1/(2|\alpha|)$
The cloud's radius $\tfrac12$ never changes, so the angle it fills as seen from the
origin, which is the uncertainty of the field's phase, shrinks as the mean field grows. The
two dashed lines from the origin touch the cloud: $\pm\Delta\varphi$ with
$\Delta\varphi\approx\Delta X/|\alpha|=1/(2|\alpha|)$. At $|\alpha|=1$ (one photon on average)
$\Delta\varphi=0.50$ rad $=28.6^\circ$; at $|\alpha|=2$ (4 photons) 0.25 rad; at $|\alpha|=4$
(16 photons) 0.125 rad. Four times the photons, half the phase noise:
$\Delta\varphi=1/(2\sqrt N)$. This is the standard quantum limit of a phase measured with
ordinary light, and in view 4 it is what limits an interferometer.
4. Squeeze the vacuum: an ellipse with no field and photons in pairs
The crystal above is pumped at twice the light's frequency. It amplifies one quadrature
of whatever comes in and de-amplifies the other, and here nothing comes in but the vacuum.
Out comes the squeezed vacuum : the circle has become an ellipse,
$\Delta X_1=e^{-r}/2=0.250$ along the squeezed direction and $\Delta X_2=e^{r}/2=1.000$
across it, with $r=\ln2=0.693$, and the product still $\tfrac14$: squeezing moves the uncertainty,
it does not remove it. In decibels, $10\log_{10}e^{2r}=6.0$ dB: the squeezed quadrature's
noise power is a quarter of the vacuum's. The field on the right still has zero mean, but
its noise band breathes twice per period, narrow when the thin side of the ellipse faces the
$X_1$ axis and wide a quarter period later. And the state is not empty:
$\langle n\rangle=\sinh^2r=0.563$ photons on average, in pairs only, because the pump's
photon of energy $2\hbar\omega$ splits into two of $\hbar\omega$ (parametric
down-conversion): the bar chart has no odd numbers.
5. A phase-squeezed laser field
The ellipse at $\alpha=3$, squeezed across the field (angle 90°): narrow in the phase
direction, $\Delta X_2=0.250$, and wide along it, $\Delta X_1=1.000$. On the right the
wave's zero crossings are sharper than a coherent state's and its crests noisier: the band is
narrow where the wave crosses zero and wide at its peaks. The phase uncertainty is
$\Delta\varphi=e^{-r}/(2|\alpha|)=0.083$ rad instead of 0.167: the same light, a twice sharper
phase. The price is amplitude noise, and photon-number noise: $\Delta n=6.1$ against the
Poisson spread of 3.1 (super-Poissonian). Set the angle to 0°, squeezing along the field:
amplitude-squeezed light, $\Delta n=2.0$, fewer than Poisson, sub-Poissonian, which no
classical wave can be.
6. Ten decibels, and what loss does to them
10 dB is $r=1.151$: the squeezed quadrature's noise power is a tenth of the vacuum's,
$\Delta X=0.158$. Now the grey box: loss. Half the light escapes ($\eta=0.5$), and wherever
light can leave, the vacuum can come in: the beam splitter that stands for the loss mixes
the vacuum's round cloud into the ellipse, which rounds out. In units of the vacuum's
variance the squeezed quadrature now reads
$$V=\eta\,e^{-2r}+1-\eta=0.55,$$
only 2.6 dB of squeezing left. The rule: the squeezing you measure can never beat the loss,
$-10\log_{10}(1-\eta)=3.0$ dB for $\eta=0.5$, however hard the crystal is pumped (drag $r$ to
1.5: still only 2.8 dB). That is why squeezers, their photodiodes and LIGO's optics are built
for every last per cent of efficiency. Loss also breaks the pairs: odd photon numbers appear
in the bar chart.
7. The local oscillator's phase chooses what is read
The signal (red: a coherent state with $\alpha=2$) meets a strong local oscillator (the
LO, yellow) on a 50:50 beam splitter. Two photodiodes catch the outputs and the difference
of their currents is the photocurrent on the strip chart: $i_-\propto|\alpha_\mathrm{LO}|X_\theta$
(How it works), the signal's quadrature along the LO's phase $\theta$, the yellow axis in
the plane (drawn in the LO's frame, which turns with the light, so the cloud stands still: the
NMR demo 's rotating frame, for light). At $\theta=0$ it reads $X_1$: samples around the mean 2.00 with noise
$\Delta X=0.500$, the histogram along the axis. Drag the axis to 90°: now it reads $X_2$,
mean 0, the same noise. The LO is the big classical wave that lets a tiny field be measured,
and its phase picks the direction in the plane. The noise against $\theta$ (bottom right) is
flat at the vacuum's level, 0 dB: a coherent state is as noisy in every direction.
8. Scan the LO and find the squeezing
The signal is the squeezed vacuum of experiment 4 ($r=0.693$, 6 dB, thin along $X_1$).
The motor turns the LO's phase steadily, and the photocurrent's noise shrinks and grows twice
per turn: the histogram is thin at $\theta=0^\circ$ and $180^\circ$, fat at $90^\circ$ and
$270^\circ$. The bottom-right plot is the measured noise power against $\theta$ (dots, from
the samples) over the theory curve
$$\Delta X_\theta^2=\tfrac14\left(e^{-2r}\cos^2\theta+e^{2r}\sin^2\theta\right),$$
which dips below the vacuum's 0 dB to $-6.0$ dB and rises to $+6.0$ dB. This is the trace
by which squeezed light was first recognised (Slusher et al. 1985; Wu, Kimble, Hall and Wu
1986): noise below the shot-noise level, in one direction only. Add loss ($\eta=0.5$ in the
signal panel): the dips fill in to $-2.0$ dB.
9. A small phase shift, measured at the standard quantum limit and below it
A phase shift $\varphi$ turns the signal's cloud about the origin, so it moves sideways by
$|\alpha|\sin\varphi\approx|\alpha|\varphi$; the dashed circle is where it was. Reading the
phase quadrature ($\theta=90^\circ$), the photocurrent's mean moves from 0 to 0.300 for
$\varphi=0.1$ rad on $|\alpha|=3$, while its noise is $\Delta X_2=0.500$: the shift is seen
with a signal-to-noise ratio of 0.60 per sample, and the smallest shift seen at ratio 1 is
$$\Delta\varphi_\mathrm{min}=\frac{\Delta X_2}{|\alpha|}=\frac1{2|\alpha|}=0.167\ \mathrm{rad},$$
the standard quantum limit for this much light (averaging $M$ samples, or using $M$ times
the photons, improves it as $1/\sqrt M$). Now press Phase-squeezed in the signal
panel: with 6 dB the noise is 0.268 (the turned ellipse shows a sliver of its wide side; it
would be 0.250 for a tiny shift), the histogram thins, and the same shift is seen at a ratio
of 1.12: $\Delta\varphi_\mathrm{min}=0.083$ rad, twice as good with the same light, or as
good with a quarter of it. Turn the LO to $\theta=0$ to see the price: the amplitude
quadrature is now four times noisier in power.
10. The dark port's vacuum sets the shot noise
Above: LIGO as a Michelson whose arms are 4 km Fabry–Pérot cavities holding 400 kW of
light each (the power-recycling mirror and the arm cavities build that up from a laser of
a few hundred watts). Its output port is kept dark, the two arms' light interfering
destructively there, so that a gravitational wave shows up as the difference of the arm
lengths, read as a phase: a strain $h=10^{-21}$ changes the difference by
$hL=4\times10^{-18}$ m and the dark port's phase by $4\pi hL/\lambda=4.7\times10^{-11}$ rad.
What limits reading so small a phase? Not the laser's own noise, which the beam splitter
sends back out of the bright port. The dark port is dark, which means the vacuum enters it
from outside (the dashed arrow and the round cloud), and that vacuum, beating against the
arms' light at the photodiode, is the shot noise of the readout: the blue curve, flat up to
the arm cavities' bandwidth and rising above it, $2.7\times10^{-24}/\sqrt{\mathrm{Hz}}$ at
100 Hz. Drag the arm power up (under More ): shot noise falls as $1/\sqrt P$, but the
purple curve, radiation pressure, the same vacuum's amplitude quadrature pushing the 40 kg
mirrors about, rises as $\sqrt P$. The two cross on the dotted line, the standard quantum
limit, at 48 Hz for 400 kW: no choice of power gets below it.
11. Inject a squeezed vacuum (2019)
Since the dark port's vacuum is the noise, replace it. A squeezer (an OPO, the crystal of
view 2) sends its squeezed vacuum in through the dark port, phase-squeezed, so that the
quadrature the readout beats against is the quiet one; a Faraday isolator lets it in while
the signal comes out. The blue curve drops at high frequency: with 6 dB from the squeezer
and 35% of the light lost on the way (the isolator, mode matching, the photodiodes), the shot
noise is 2.9 dB lower, about what GEO600 first ran with (2010) and what LIGO and Virgo
used through their third observing run (2019), about 3 dB. The total (thick) follows where
shot noise dominates, above about 100 Hz. Set the loss to 0 and the squeezing to 10 dB to
see what a perfect squeezer would do...
12. Radiation pressure takes it back at low frequency
...and look below 50 Hz: the purple curve has gone up by the same 10 dB. The squeezed
vacuum's phase quadrature is quiet, so its amplitude quadrature is ten times louder, and the
amplitude quadrature is what pushes the mirrors: radiation-pressure noise grows by the factor
the shot noise shrank, and below the frequency where the two cross (48 Hz) the detector is
worse than with no squeezing at all. Drag the angle from 90° to 0°: amplitude squeezing,
which quietens the mirrors and spoils the readout, the opposite trade. No single angle helps
everywhere, because at low frequency the readout sees the mirrors' motion, driven by the
amplitude quadrature, and at high frequency it sees the phase quadrature directly.
13. Frequency-dependent squeezing (2023)
The cure is an angle that changes with frequency: phase-squeezed above about 50 Hz,
amplitude-squeezed below, turning through the crossing. Reflecting the squeezed vacuum off
a detuned filter cavity does exactly that (LIGO's are 300 m long, with a bandwidth of a few
tens of hertz), because the cavity turns the two sidebands of the squeezed field by
different amounts. With the angle following the readout's own combination at every
frequency, the whole quantum-noise curve drops by the full factor: 10 dB everywhere,
radiation pressure included. LIGO and Virgo installed filter cavities for their fourth
observing run (2023). Turn the loss up to a realistic 20%: the gain is 5.5 dB at high
frequency, and still a gain at low.
14. Detect a chirp: the payoff as a number
The plot under the noise curve is $h(t)$ from two neutron stars of 1.4 solar masses spiralling together
40 Mpc away (GW170817's distance), seen face on and overhead: the Newtonian chirp, from
70 Hz up to the innermost stable orbit at 1570 Hz in 5.6 s, louder and faster as it goes.
Its spectrum is the green band on the sensitivity plot, falling as $f^{-7/6}$; the
signal-to-noise ratio a matched filter achieves is the area between it and the noise,
$\rho^2=4\int|\tilde h(f)|^2/S_n(f)\,df$. Without squeezing $\rho=72$; with 6 dB of
frequency-dependent squeezing through 20% loss, $\rho=104$: the same event 1.44 times as
loud, or heard 1.44 times as far, which is a volume of space 3.0 times as big, that many
times more events a year. Drag the distance out until $\rho$ falls to 8, the usual
threshold: 360 Mpc without squeezing, 520 Mpc with. Try the two black holes: heavier, so
louder, and over in 6 s from 20 Hz.
15. The chirp buried in the noise, and the filter that finds it
Under the chirp: eight seconds of what the detector would record, simulated. The noise is
Gaussian, drawn from the total noise curve above (a seeded white noise, each frequency scaled
by $\sqrt{S_h(f)}$: How it works), and the chirp of experiment 14 is added to it, merging at
7 s. Press Raw for the strain itself: the noise's spread is $2.8\times10^{-22}$,
from every frequency at once (the rumble below 20 Hz, the shot noise rising above 1 kHz), and
the chirp, at most $5.0\times10^{-22}$, is nowhere to be seen. Whitened divides each frequency by the
noise there, so the noise comes out flat with spread $\sigma=1$ and the chirp shows as the
filter sees it: at most 2.48 σ high, smaller than the noise everywhere (the green band is the
template lying on it; turn Template off and the chirp is only a slight thickening
of the noise in the last second before 7 s). The matched filter (bottom) slides the template along the data and
adds up how well they agree, each frequency weighted by $1/S_h$: its output $z$ has noise of
spread 1, and where the template lines up with the chirp it leaps to $z=104.3$, against the
expected $\rho=104.2$ of the read-out. It adds up 622 cycles over 5.6 s in step, so a signal
smaller than the noise at every instant stands a hundred times above it. Slide the
template to watch it work. Now turn Squeezed light off: the whitened noise is
the same realisation, sample for sample (only the spectrum that shapes it has changed), the
chirp in it shrinks to 1.59 σ, and the peak falls to 72.3 ($\rho=72.3$): experiment 14's
72 → 104, seen. (With squeezing on, the grey dot marks the peak without it.) New
noise draws another realisation: the peak moves by about ±1.
16. At the reach: a peak of 8, just above the noise
The same two neutron stars at 361 Mpc, the reach without squeezing, and squeezing off:
$\rho=8.0$. The chirp in the whitened data is at most 0.18 σ high, nothing an eye could
find, yet the filter's output stands out where the template lines up: $z=8.1$, while
everywhere else in the 8 s the noise reaches at most 4.0. Why 8: the output's noise has
spread 1 and the 8 s hold a few thousand independent tries, so the noise alone reaches about
4; reaching 8 by chance has a probability of $6\times10^{-16}$ a try, which keeps false
alarms away over a year of data and many templates (real detectors also have glitches that are
not Gaussian, which is why the threshold is not lower). Drag the distance out to 600 Mpc:
$\rho=4.8$, the peak 4.9 against the noise's 4.0, barely; at 1000 Mpc ($\rho=2.9$) the
tallest peak is the noise's, at 3.8 s, and the chirp is lost. Back at 361 Mpc, turn
Squeezed light on: $\rho=11.5$, the peak clears the threshold, and it sinks back
to 8 only at 521 Mpc, the reach with squeezing. Press New noise a few times: the
peak at $\rho=8$ moves by about ±1 (7.7 to 8.9 for the next six realisations), the noise's
largest between 3.9 and 4.8.
How it works
One mode is an oscillator. A single mode of the field, with ladder operator $a$,
has the quadratures
$$X_1=\frac{a+a^\dagger}2,\qquad X_2=\frac{a-a^\dagger}{2i},\qquad [X_1,X_2]=\frac i2,\qquad \Delta X_1\Delta X_2\ge\frac14,$$
the cosine and sine parts of the field, $E(t)\propto X_1\cos\omega t+X_2\sin\omega t$. In the
Heisenberg picture $a(t)=a\,e^{-i\omega t}$, so $X_1(t)=X_1\cos\omega t+X_2\sin\omega t=X_{\omega t}$:
the field at time $t$ is the quadrature at angle $\omega t$, and the cloud turns clockwise.
The phase-space demo 's $x,p$ with $\hbar=m=\omega=1$ are $x=\sqrt2X_1$, $p=\sqrt2X_2$, and its
Wigner function is the same picture, $W(x,p)=W(X_1,X_2)/2$: the vacuum is
$W=(2/\pi)\,e^{-2(X_1^2+X_2^2)}$, a Gaussian of width $\tfrac12$, and every state here is a
Gaussian, so $W$ is never negative.
Coherent and squeezed states. $|\alpha\rangle=D(\alpha)|0\rangle$ with
$D(\alpha)=e^{\alpha a^\dagger-\alpha^*a}$: the vacuum displaced to $\langle X_1\rangle+i\langle X_2\rangle=\alpha$,
photon number Poisson with mean $|\alpha|^2$, phase uncertainty $\Delta\varphi\approx\Delta X/|\alpha|=1/(2|\alpha|)$.
The squeeze operator $S(\xi)=\exp\!\left[\tfrac12(\xi^*a^2-\xi a^{\dagger2})\right]$ with
$\xi=re^{2i\psi}$ makes $\Delta X_\psi=e^{-r}/2$ and $\Delta X_{\psi+90^\circ}=e^{r}/2$, the
product kept; it is what a crystal with a $\chi^{(2)}$ nonlinearity, pumped at $2\omega$, does
to the mode (the parametric interaction $i\hbar\kappa(a^2e^{2i\omega t}-\mathrm{h.c.})$,
$r=\kappa t$). The states here are $D(\alpha)S(\xi)|0\rangle$; their photon numbers come from
the three-term recurrence that $(a-\alpha)\cosh r+(a^\dagger-\alpha^*)e^{2i\psi}\sinh r$
annihilates them, and for $\alpha=0$ only even $n$ survive, with
$P(2m)=\frac{(2m)!}{4^m(m!)^2}\frac{\tanh^{2m}r}{\cosh r}$ and $\langle n\rangle=\sinh^2r$.
Loss $1-\eta$ is a beam splitter with the vacuum at its other port: the covariance becomes
$\eta\Sigma+\tfrac14(1-\eta)I$, so the squeezed quadrature reads $\eta e^{-2r}+1-\eta$ in units
of the vacuum's variance, never below $1-\eta$.
Homodyne detection. With the signal $a$ and the local oscillator in the coherent
state $\beta=|\beta|e^{i\theta}$, a 50:50 beam splitter gives $c=(a+\beta)/\sqrt2$ and
$d=(a-\beta)/\sqrt2$, and the difference of the photocurrents is
$$c^\dagger c-d^\dagger d=\beta^*a+\beta a^\dagger=2|\beta|\,X_\theta,\qquad X_\theta=X_1\cos\theta+X_2\sin\theta .$$
The LO's phase chooses the quadrature; its strength only scales the current, and the LO's own
noise cancels in the difference. For a Gaussian state the samples are Gaussian with mean
$\langle X_\theta\rangle$ and variance $\Delta X_\theta^2=\tfrac14(e^{-2r}\cos^2(\theta-\psi)+e^{2r}\sin^2(\theta-\psi))$,
flat $\tfrac14$ for coherent states. The page draws its samples from that Gaussian (a seeded
generator; the record bins them by $\theta$ and shows each bin's sample variance).
Measuring a phase. A shift $\varphi$ sends $\alpha\to\alpha e^{i\varphi}$, moving
the cloud by $|\alpha|\sin\varphi$ across the field. Read in the phase quadrature against
noise $\Delta X_2$, it is resolved when $|\alpha|\varphi>\Delta X_2$:
$\Delta\varphi_\mathrm{min}=1/(2|\alpha|)=1/(2\sqrt N)$, the standard quantum limit, with the
phase-squeezed state's $e^{-r}/(2|\alpha|)$ below it; squeezing is bounded by the photons it
costs ($\sinh^2r$ of them), and the best any state can do is the Heisenberg limit
$\Delta\varphi\sim1/N$.
LIGO's noise budget. A Michelson with Fabry–Pérot arms of length $L$ and
half-bandwidth $\gamma=2\pi f_c$, mirrors of mass $M$ and circulating power $P$ in each arm,
reading its phase quadrature, has the quantum noise (Kimble, Levin, Matsko, Thorne and
Vyatchanin 2001)
$$S_h=\frac{h_\mathrm{SQL}^2}2\left[\frac{V_2}{\mathcal K}-2C_{12}+\mathcal K\,V_1\right],\qquad
h_\mathrm{SQL}^2=\frac{8\hbar}{M\Omega^2L^2},\qquad \mathcal K=\frac{16\,\omega_0\gamma P}{MLc\,\Omega^2(\gamma^2+\Omega^2)},$$
where $V_1$, $V_2$ and $C_{12}$ are the variances of the amplitude and phase quadratures of
the field entering the dark port, and their covariance, in units of the vacuum's. For the
vacuum ($V_1=V_2=1$, $C_{12}=0$) the first term is shot noise, $\propto1/\sqrt P$, flat below
$f_c$ and rising as $f$ above it; the last is radiation pressure, $\propto\sqrt P$, falling as
$1/f^2$; their product is $h_\mathrm{SQL}^4/4$ whatever $P$, so $h_\mathrm{SQL}$ is their
envelope, touched where $\mathcal K=1$. A squeezed vacuum at angle $\lambda$ from the
amplitude quadrature has $V_1=e^{-2r}\cos^2\lambda+e^{2r}\sin^2\lambda$, $V_2$ the same with
sine and cosine swapped, $C_{12}=(e^{-2r}-e^{2r})\sin\lambda\cos\lambda$, each mixed with the
vacuum by the loss; phase squeezing ($\lambda=90^\circ$) divides the shot noise by $e^{2r}$ and
multiplies radiation pressure by it. The readout's combination $a_2-\mathcal Ka_1$ is the
input quadrature at angle $90^\circ+\arctan\mathcal K$, so a squeezer that keeps $\lambda$ there
at every frequency (a filter cavity) lowers the whole curve by $e^{-2r}$. The classical
curves are fixed models: a seismic wall $\propto f^{-10}$ below 10 Hz, and thermal noise from
the suspensions ($\propto f^{-5/2}$) and the mirror coatings ($\propto f^{-1/2}$). Numbers
here are Advanced-LIGO-like: $L=4$ km, $M=40$ kg, $\lambda=1064$ nm, 400 kW in each arm, and
$f_c=400$ Hz for the bandwidth the signal-recycling mirror gives the arms; the simplified
model has no signal recycling of its own.
The chirp. Two masses in a circular orbit radiate at twice the orbital frequency,
which runs away as the orbit shrinks: for a chirp mass $\mathcal M=(m_1m_2)^{3/5}/(m_1+m_2)^{1/5}$,
$$f(\tau)=\frac1\pi\left(\frac5{256\,\tau}\right)^{3/8}\left(\frac{G\mathcal M}{c^3}\right)^{-5/8},\qquad
h(t)=\frac{4}{d}\left(\frac{G\mathcal M}{c^2}\right)^{5/3}\left(\frac{\pi f}{c}\right)^{2/3}\cos\Phi(t),$$
$\tau$ the time to merger and $d$ the distance, stopped at the innermost stable orbit,
$f_\mathrm{ISCO}=c^3/(6^{3/2}\pi GM)$. Its spectrum $|\tilde h(f)|\propto f^{-7/6}$ is taken by
an FFT of the sampled $h(t)$ (the one FFT on this page), and the optimal signal-to-noise ratio
is $\rho^2=4\int|\tilde h|^2/S_n\,df$; the distance at which $\rho=8$ is the horizon for a
face-on binary, and the volume inside it grows as its cube.
The detector's output and the matched filter. A stretch of $T=8$ s at 4096 samples
a second, with frequency bins $f_k=k\,\Delta f$, $\Delta f=1/T$, and
$\tilde x_k=\Delta t\sum_jx_je^{-2\pi ijk/N}$. The noise is made in the frequency domain: a
seeded white spectrum $w_k$ (complex Gaussian, $\langle|w_k|^2\rangle=1$) shaped by the
noise curve,
$$\tilde n_k=w_k\sqrt{S_h(f_k)/2\Delta f},$$
and transformed back: Gaussian noise whose one-sided spectrum is $S_h$ (the raw data
high-passed at 10 Hz). Changing the squeezing changes $S_h$ and keeps $w_k$: the same
realisation, rescaled. Whitening divides each bin of the chirp's band by
$\sqrt{S_h/2\Delta f}$, so the whitened noise is $w$ itself, of unit variance, and the chirp
$\tilde h/\sqrt{S_h}$. The matched filter with the template $T$ (the chirp at 1 Mpc, merging
at $t=0$) put at every merger time $t_0$:
$$z(t_0)=\frac{4\Delta f}{\sigma}\,\mathrm{Re}\sum_k\frac{\tilde d_k\tilde T_k^*}{S_h(f_k)}\,e^{2\pi if_kt_0},\qquad
\sigma^2=4\Delta f\sum_k\frac{|\tilde T_k|^2}{S_h(f_k)},$$
the sums over the chirp's band. For noise alone $z$ is Gaussian with mean 0 and variance
exactly 1 (each bin's noise power $S_h/2\Delta f$ cancels one $S_h$ of the weight); for the
chirp at distance $d$ merging at $t_0$, $z=\sigma/d$, which is the read-out's
$\rho=\sqrt{4\int|\tilde h|^2/S_h\,df}$ (they agree to 0.01 here). So the output is
$\rho$ plus unit noise when the template lines up, and unit noise elsewhere: the
Cauchy–Schwarz inequality says no other linear filter does better. One inverse FFT gives
$z$ at every $t_0$ at once; the page makes the series once per realisation and spectrum, and
the distance only rescales the chirp's part. The stretch is a loop to the FFT, so the template
slid off the start comes back at the end (the drawing leaves that part out). LIGO's own
filter also maximises over the chirp's phase, taking the modulus of the complex sum; this one
knows the phase and keeps the real part.
New in 2026 for BrainFlux. Source:
github.com/gallicchio/brainflux , MIT licence.