The noise pushed below the floor

Point a perfectly stabilised laser at a photodiode and the current still fluctuates. The fluctuation is not a fault of the laser or the detector. It is what light made of photons does when those photons arrive independently: their number in any interval scatters by its own square root, and the resulting shot noise sets how finely the brightness, or the phase, of a steady beam can be read. A measurement made with a million photons is limited to a part in a thousand, with a billion to a part in thirty thousand, and the only way to improve it was to use more light.

That floor has a precise origin and it is not the one usually given. A steady laser beam is in a coherent state, and a coherent state’s noise is a disc of fixed size in the plane of the field’s two quadratures — the same size as the noise of the vacuum, carried along with the field’s mean value. Shot noise is the vacuum’s noise, seen against a bright beam. The uncertainty principle fixes the area of that disc and says nothing about its shape.

Light whose noise disc has been squeezed into an ellipse of the same area is quieter than the vacuum in one quadrature and louder in the other. It was first made in 1985. It is now injected into the gravitational-wave detectors, and everything about why that works, and why it is so difficult, can be read off the shape of the noise.

The noise of light as a shape #

A single mode of light at one frequency is described by two numbers: the part of its field that is in step with a reference wave, and the part a quarter-cycle out of step. Those are its two quadratures. Plotted as a point, the distance from the centre is the amplitude and the angle is the phase, and a measurement of the field returns a point that scatters around the mean.

The vacuum is a round cloud at the origin: no mean field, and the same noise in every direction. Steady laser light is the same round cloud moved away from the origin — a mean field with the vacuum’s noise attached. Squeezed light is an ellipse. The one drawn is squeezed by 4 decibels, which makes it narrower than the vacuum by a factor of 1.58 in one direction and wider by the same factor in the other, and the samples were checked against those widths.

The direction of the squeeze decides what is quiet. Pointed along the line from the origin, the ellipse is narrow in amplitude: the light’s brightness fluctuates less than any steady laser’s, and its phase fluctuates more. Pointed across that line, it is narrow in phase. The product of the two widths is the vacuum’s, which is the price the uncertainty principle sets and which squeezing does not evade — it spends it.

Where the vacuum gets in #

The claim that shot noise is the vacuum’s noise has a concrete meaning inside an interferometer, and it is the observation that made squeezing useful. An interferometer splits a laser beam, sends the halves along two arms and recombines them, and it is operated so that almost no light leaves by one of its output ports — the dark port. A beam splitter has two inputs as well as two outputs, and the input matching the dark port is open to the room. Through it enters nothing but the vacuum.

In 1981 Carlton Caves showed that the photon-counting noise of an interferometer and the radiation-pressure noise on its mirrors both come from the vacuum fluctuations entering by that open port, beating against the bright laser light — and that replacing the vacuum there with squeezed vacuum would reshape both. Nothing about the laser needs to change. Squeezing an interferometer means squeezing the darkness that leaks into it.

The squeeze has to be looked at from the right direction #

A measurement of one quadrature is made by mixing the light with a strong reference beam and choosing the reference’s phase, which sets which quadrature is read. The quiet quadrature is quiet only along its own direction, and the noise the measurement sees depends on the angle between the two.

Pointed exactly along the quiet quadrature, 12 decibels of squeezing shows as noise 12 decibels below a steady laser’s — a factor of sixteen in noise power. Turned by 14.1°, the loud quadrature has leaked in enough to bring the noise back up to the laser’s level, and beyond that the measurement is noisier than it would have been with no squeezing at all. The more strongly the light is squeezed, the louder its loud quadrature, and the narrower the angle within which the squeeze can be seen.

That turns the stability of the reference into a limit. A reference phase that wanders at random by one degree costs a third of a decibel; by three degrees, it turns twelve decibels into 9.7; by six, into 6.3. The best squeezing ever measured, fifteen decibels in 2016, needed the measured angle held to a small fraction of a degree, and every other source of phase noise in the apparatus had to be controlled to match.

Every lost photon lets the vacuum back in #

The second enemy is loss, and it is less obvious. A beam splitter that discards a fraction of the light does not simply discard a fraction of its noise. It has a second, unused input port, and through that port enters the vacuum — with its ordinary, unsqueezed noise — which is mixed into the output in exactly the proportion the light was lost.

So the measured noise is the surviving fraction of the squeezed noise plus the lost fraction of ordinary noise, and heavily squeezed light approaches a floor fixed entirely by the loss. With 95 per cent of the light reaching the detector, no amount of squeezing can show more than 13.0 decibels. With 80 per cent, 7.0. With 60 per cent, 4.0.

That is why squeezing is an engineering discipline of optical loss as much as of nonlinear optics. Every mirror, every lens surface, every photodiode’s quantum efficiency and every imperfect match between the shape of the squeezed beam and the shape of the reference counts against it. It is also why squeezing is such a sensitive test of an apparatus: a squeezing level is a direct measurement of the total loss between the source and the detector, including losses nobody had catalogued.

Photons that come in pairs #

The noise picture has a counterpart in photon numbers, and it explains where squeezing comes from. Squeezed light is made by a process that takes photons from a pump beam at twice the optical frequency and splits each into two at the optical frequency — the optical version of a swing pumped at twice its natural rate, which amplifies motion in one phase and suppresses it in the other.

Squeezed vacuum — squeezed light with no mean field, which is what is injected into a detector — contains photons only in even numbers. At 10 decibels it holds none 57.5 per cent of the time, two 19.2 per cent, four 9.7 per cent, and never one or three. The distribution was checked to sum to one with the right average. Thermal light with the same average falls off steadily from none; a laser’s numbers scatter around their mean.

The pairs are the squeezing. Two photons created together have fluctuations that are correlated, and in one quadrature those correlations cancel the vacuum’s noise while in the other they reinforce it. The same pairs, fed into a network of beam splitters, are what the largest multi-photon interference experiments use as their source, and the same correlations are why those experiments are as hard to simulate as they are.

Quiet brightness, and photons that arrive in order #

Amplitude-squeezed light has a property in photon counting that connects it to the other kind of non-classical light. Its brightness fluctuates less than a steady laser’s, so the number of photons counted in a fixed interval scatters by less than its own square root — sub-Poissonian light, in which the photons arrive more evenly spaced than independent arrivals would allow. That is a statistical cousin of the refusal of a single photon to trigger two detectors, and it has the same classical bound: no classical wave, however it fluctuates, can produce counts less noisy than Poisson statistics.

The most direct way to make amplitude-squeezed light does not use pairs at all. A semiconductor laser driven by a current that is itself extraordinarily quiet — held steady by a large series resistance, whose own noise at room temperature is a tiny fraction of the shot noise of the current it carries — emits photons almost as regularly as the electrons arrive, because each injected electron produces one photon with high probability. The regularity of the pump is copied into the light, and the light’s intensity noise falls below the shot-noise level of a laser of the same power.

Quieter than the vacuum, in other hands #

The same trade is made wherever a measurement is limited by the noise of a quantum state rather than of an apparatus, and it has spread well beyond optics.

Atomic clocks read the phase of a collection of atoms, and a collection of independent atoms has projection noise — the atomic counterpart of shot noise — which the best clocks are now limited by. Entangling the atoms so that their collective spin is squeezed, quiet in the direction the clock reads and loud in the other, has been used to run optical clocks below that projection noise limit.

In the microwave range, the amplifiers used to read superconducting circuits are themselves squeezing devices: a circuit built around a Josephson junction, pumped at twice its signal frequency, amplifies one quadrature and suppresses the other, exactly as the optical pair source does. A search for the hypothesised dark-matter particle called the axion used squeezed microwave vacuum in this way to scan its frequency range about twice as fast as the same apparatus could without it, which is squeezing turned from a demonstration into a saving of years.

Squeezing a gravitational-wave detector #

A gravitational-wave detector reads the tiny change in the length of its arms as a change in the phase of the light that has travelled them. Its sensitivity at high frequency is limited by the shot noise of that light — by the photon-counting noise in the phase — and the obvious fix, more laser power, runs into a second quantum noise: the photons’ momentum kicks on the mirrors, the pressure light exerts, which fluctuates with the photon number and shakes the mirrors at low frequency.

The two noises trade against each other, and the point where they are equal defines the standard quantum limit: the least noise an interferometer with uncorrelated light can reach at a given frequency, however its power is chosen.

Squeezing the injected light in phase lowers the counting noise by the squeezing factor. It also, unavoidably, makes the amplitude louder by the same factor, and amplitude noise is what drives radiation pressure. In the model drawn, 6 decibels of phase squeezing lowers the noise above the crossover by four times in power and raises it below by four times, so that below 99 hertz the squeezed detector is worse than the unsqueezed one. The standard quantum limit is not beaten; the noise curve has only been slid along it.

The way past is to make the squeeze rotate with frequency: quiet in phase at high frequencies, where counting noise dominates, and quiet in amplitude at low frequencies, where radiation pressure does. The third curve is that arrangement, and it is lowered by the full six decibels everywhere, dipping below the standard quantum limit on both sides of the crossover. The rotation is produced by reflecting the squeezed light off a long optical cavity before it enters the detector, which delays and turns each frequency component by a different amount.

The history follows the curves. The GEO600 detector began injecting squeezed light in 2010. During their third observing run, from 2019, both LIGO detectors injected phase-squeezed light and gained about three decibels at high frequency, and found the radiation-pressure penalty at low frequency where the model says it should be. In 2020 the LIGO team reported measuring quantum noise below the standard quantum limit over a band of frequencies, by exploiting the correlations squeezing creates between the light and forty-kilogram mirrors. From 2023 both detectors used filter cavities three hundred metres long to rotate the squeeze with frequency, and gained sensitivity across their band instead of trading one end of it against the other.

What the model and the figures assume #

The squeezed states are ideal. Every ellipse has exactly the vacuum’s area. Real squeezed light is always somewhat larger than that minimum — a mixed state, because the same losses that limit the measured squeezing also degrade the state as it is made — and the loud quadrature is louder than the quiet one is quiet.

The detector model has two noises and nothing else. A real interferometer’s quantum noise depends on its arm cavities, its signal-recycling configuration and the frequency response of every optic, and its total noise is dominated at low frequency by seismic and thermal noise that squeezing does not touch. The model reproduces the trade that squeezing makes, not the shape of any real detector’s sensitivity.

Loss is a single number. In practice loss is distributed along the path, frequency-dependent inside cavities, and joined by phase noise, by backscattered light and by the mismatch between the squeezed beam and the interferometer’s own mode. Each degrades squeezing in its own way, and the measured improvement in a detector is the result of all of them together.

One mode drawn, a band of modes squeezed #

The noise clouds are drawn for one mode of light at one frequency. Real squeezed light is squeezed over a band of frequencies, each sideband pair its own mode, and the squeezing at audio frequencies that a gravitational-wave detector needs is a very different technical problem from squeezing at megahertz, because at low frequencies every classical noise in the laser and the apparatus is larger and has to be removed before the quantum noise is even visible.

Nor do the pictures show time. A squeezed state measured once gives one point in the cloud; the ellipse only appears after many measurements. Whether a single measurement “was” squeezed is not a question the state answers, in exactly the way a single arrival in one arrival at a time is not an interference pattern.

Still open: how far below the limit a detector can go #

Frequency-dependent squeezing has taken the detectors below the standard quantum limit over part of their band. How much further is limited now by optical loss — the floor of the third figure — and by phase noise in the filter cavities, and the next generation of proposed detectors plans for around ten decibels of detected squeezing, which requires losses in the injection path of a few per cent at most.

Beyond squeezing there are schemes that read the light differently rather than preparing it differently: measuring a combination of quadratures that varies with frequency at the output, or building the interferometer so that the radiation-pressure noise cancels out of its signal by design. Each evades the same limit by a different route, and none has yet been shown to work at full scale in a detector whose other noises are low enough for the quantum noise to be the limit. Which of them will set the sensitivity of detectors in the 2030s is a question about engineering loss as much as about quantum mechanics.

The habit worth carrying away is to ask of any noise floor what shape it assumes. The uncertainty principle bounds an area, and a limit derived from it is a limit only for the shape of noise that was assumed. Shot noise is the vacuum’s noise drawn as a circle, and a measurement that needs only one quadrature is free to push the circle into an ellipse — at a cost paid in the other quadrature, in fragility against every lost photon, and in the precision with which the measurement must point at the quiet direction.