## Abstract

Interferometric gravitational wave detectors (such as advanced LIGO) employ high-power solid-state lasers to maximize their detection sensitivity and hence their reach into the universe. These sophisticated light sources are ultra-stabilized with regard to output power, emission frequency and beam geometry; this is crucial to obtain low detector noise. However, even when all laser noise is reduced as far as technically possible, unavoidable *quantum noise* of the laser still remains. This is a consequence of the Heisenberg Uncertainty Principle, the basis of quantum mechanics: in this case, it is fundamentally impossible to simultaneously reduce both the phase noise and the amplitude noise of a laser to arbitrarily low levels. This fact manifests in the detector noise budget as two distinct noise sources—photon shot noise and quantum radiation pressure noise—which together form a lower boundary for current-day gravitational wave detector sensitivities, the standard quantum limit of interferometry. To overcome this limit, various techniques are being proposed, among them different uses of non-classical light and alternative interferometer topologies. This article explains how quantum noise enters and manifests in an interferometric gravitational wave detector, and gives an overview of some of the schemes proposed to overcome this seemingly fundamental limitation, all aimed at the goal of higher gravitational wave event detection rates.

This article is part of a discussion meeting issue ‘The promises of gravitational-wave astronomy’.

## 1. Introduction

At the time of writing, the LIGO Scientific Collaboration has detected three gravitational wave events, all of them mergers of binary black hole systems [1–3]. This feat was accomplished using highly sophisticated versions of the humble Michelson interferometer, a simplified version of which is shown in figure 1.

The highly coherent light emitted by a high-power laser diode pumped solid-state laser is split into two beams on a beamsplitter, a mirror that transmits half of the light power and reflects the other half. The two partial beams hit the respective high-reflectance end mirrors (the test masses of the gravitational wave detector) and return to the beamsplitter, where they interfere. The resulting interference pattern is then detected on a photodetector. Depending on the differential arm-length difference of the interferometer, the interference is either constructive or destructive, or anything in between. In all interferometric gravitational wave detectors the initial arm-length is set such that a condition very close to destructive interference is achieved; this is called ‘operation near the dark fringe’—this operating condition has many advantages (such as lower power on the photodetector, a higher dynamic range and the option of power recycling). In this state, if a gravitational wave passes through the Michelson interferometer, stretching and compressing space–time in the arms differentially, the interference signal on the photodetector will change perceivably. However, for the tiny effect of a gravitational wave to be discernible from all other noise sources the interferometer is subject to, all occurring noise sources have to be drastically reduced. One major contributing factor in terms of noise is the laser light source itself. By means of various different stabilization schemes the technical noise of the employed lasers with regard to power, frequency and beam geometry (mode shape and beam pointing) is routinely reduced as far as technically possible. However, on a quantum level we face a new and fundamental challenge: *quantum noise* of the laser light.

One could argue that quantum noise must be an inordinately small effect. However, the fact today is that the advanced gravitational wave detectors (such as aLIGO, adVIRGO and GEO-HF) currently taking data are already quantum noise-limited for measurement frequencies above a few hundred hertz. And once aLIGO reaches design sensitivity (see figure 2 for the aLIGO noise budget [5]), it will be quantum noise limited for all frequencies above 12 Hz, i.e. virtually over the entire measurement band of all ground-based gravitational wave detectors.

Hence the pertinent question is: What can be done to reduce quantum noise in an interferometric gravitational wave detector ? To answer this question, we need to first understand the origin of quantum noise in laser interferometry, and to then look at how quantum noise manifests in a laser interferometer.

## 2. Quantum noise of light

What is quantum noise with regard to light in an interferometer? Keeping in mind the wave–particle dualism, but choosing for now to describe light as an electromagnetic wave and looking at the electric field, the output of a laser can be described by a sinusoid, given by the red-dashed curve on figure 3*a* (the time series representation of the electromagnetic wave) [6]. Its amplitude squared is proportional to the laser intensity, and its wavelength is inversely proportional to the laser frequency and thereby related to the colour of the emitted laser light. Ideally, the waveform would be a pure sinusoid, but due to quantum mechanics unavoidable quantum noise still exists in the form of amplitude and phase noise on the light; this is shown in figure 3, grossly exaggerated, by the blue graph overlaid on the red sinusoid. Amplitude noise corresponds to power fluctuations, whereas phase noise (instantaneous frequency jitter) manifests in minuscule (and not noticeable by the human eye) variations of the colour of the light. An equivalent way of displaying this is as an arrow (so-called *phasor*) in the phase space picture on figure 3*b*. The uncertainty in both amplitude and phase of the light is clearly visible here; in the fact, the tip of the red arrow can be anywhere within the ball made up by the blue dots.

Now let us examine these uncertainties in more detail: in the phase space picture of a *coherent state*—the light emitted by a well-stabilized laser—the coherent amplitude (related to the laser power) is represented by the arrow, the uncertainty of the state is shown as a shaded ‘ball’. The position probability of the tip of the arrow follows a radially symmetric Gaussian distribution. To quantify the uncertainties in amplitude and phase, we shall define quadrature operators and , which are related to the creation () and annihilation () operator of the quantum mechanical harmonic oscillator according to (e.g. [7])^{1}
2.1and
2.2where is the *amplitude-quadrature operator* and is the *phase-quadrature operator*. While annihilation and creation operators correspond to non-observable quantities, the quadrature operators correspond to *observables*, i.e. their respective values can be determined by a suitable measurement.

It is a fundamental principle in quantum mechanics that the product of the standard deviations of two non-commuting quadrature operators can never be smaller than 1—this is stated by the *Heisenberg Uncertainty Principle*, the foundation of quantum mechanics:
2.3

Coherent states are so-called *minimum uncertainty states*. This means that the product of the standard deviation of the amplitude quadrature and the standard deviation of the phase quadrature is minimized, i.e. (here) equal to 1. It is actually more common to quantify the uncertainty of a variable *X* in terms of its variance *Δ*^{2}*X*, which is the square of the standard deviation *ΔX*. For a coherent state, the size of the variance of the amplitude and the phase quadrature is identical, shown by the fact that the uncertainty distribution is spherically symmetric. This is somewhat of a special case.

The product of the variances of two non-commuting variables can be larger than or equal to 1, but never smaller. The Heisenberg Uncertainty Principle, however, does not state that the variances of the uncertainties of the two non-commuting variables have to be *identical*. The Heisenberg Uncertainty Principle can obviously also be fulfilled if the uncertainties are not equal (e.g. for , or for , or even for ).

So-called *squeezed states* fulfil the Heisenberg Uncertainty Principle in this way. The process of squeezing deforms the round ‘noise ball’ (as shown on figure 3*b*) into a noise ellipse, hence the term squeezed states. This does not imply that the laser mode now has a different shape—it means that the uncertainties in the two quadratures have been redistributed, leading to a change of the photon statistics. For instance, a phase-quadrature-squeezed state (shown on figure 4*b*) has reduced noise (i.e. a smaller standard deviation ) in the phase quadrature at the cost of increased noise (i.e. a larger standard deviation ) in the amplitude quadrature as opposed to the coherent state shown on figure 4*a*. This squeezed state still fulfils the Heisenberg Uncertainty Principle as the area of the distribution in phase space remains larger than (or equal to) 1.

But how can this help the field of gravitational wave detection? For this we have to take a closer look at how quantum noise manifests in a laser interferometer.

## 3. Quantum noise in a laser interferometer

To examine the effect of quantum noise in a laser interferometer (as described in detail in [8]), we will now switch to the photon picture of light, as we are dealing with the effects of the light’s photon statistics. Light can be thought to be made up of a stream of quantized energy packets, so-called photons. The photons in a laser beam are not evenly distributed, but instead their arrival time at a certain position follows the Poisson statistic^{2} (which is the same counting statistic that also governs radioactive decay). It implies that the photons do not arrive in a regular stream, but instead with an irregular arrival pattern, ‘sticking together’ in groups with ‘gaps’ in between the groups. This effect is called photon bunching.

The Poissonian photon statistic causes the quantum noise in an interferometric gravitational wave detector and manifests there in two different and distinct ways, namely as (relative) *shot noise* (readout noise of the interferometer) at high detection frequencies and as *quantum radiation pressure noise* (back-action noise) at low detection frequencies. This is expressed in equations (3.1) and visualized in figure 5.
3.1

The equations (3.1) are given in the units of amplitude spectral density of the effective gravitational wave strain noise which the respective quantum noise component adds to the gravitational wave signal in an interferometer. The strain *h* is defined (as shown in equation (3.2)) as the differential arm-length change Δ*L* divided by the baseline arm-length *L*.
3.2

On the one hand, the photons arriving at a photodetector cause shot noise, which is *detection noise*. In equation (3.1) it is given as relative shot noise (i.e. shot noise divided by the DC light power *P* impinging on the photodetector) and therefore scales inversely with the arm-length *L* of the interferometer and inversely with the square root of the laser power *P*. On the other hand, the same stream of photons impinging on a suspended mirror (such as the test masses of a gravitational wave detector) and thereby transferring momentum causes the mirror to move due to quantum radiation pressure noise. This movement is not static (i.e. the mirror is not just pushed away by a constant distance), but the radiation pressure noise causes the mirror position to fluctuate: sometimes the mirror is pushed away from its position of equilibrium more, sometimes less, depending on the number of photons hitting it per time interval. Now, as the mirror is suspended, the restoring force (here, gravity) causes the mirror to fall back into its position of equilibrium. This movement—the mirror pushing back on the light field—causes phase fluctuations on the light field. Radiation pressure noise is therefore *back-action noise*. In strain sensitivity it again scales inversely with interferometer arm-length *L*, but this time proportionally to the square root of the light power *P*. It is also frequency-dependent (1/*f*^{2}) and scales inversely with the mirror mass *m*. The latter at least is intuitive: a more massive mirror is pushed away less by a fixed light power.

By itself, shot noise and radiation pressure noise are uncorrelated. In the sensitivity of the gravitational wave detector they therefore add as the square root of the sum of their squares, giving rise to the so-called *standard quantum limit* of interferometry. The standard quantum limit is the minimal sum of relative shot noise and radiation pressure noise. For each chosen (fixed) power there is an optimal frequency at which the geometric sum of the two quantum noise terms is minimal. The envelope of these minima as a function of power is a graph that is proportional to 1/*f*—the standard quantum limit of interferometry.

Quantum noise in an interferometer is like two flip sides of one coin: the same cause (the photon statistics) manifests simultaneously in two distinct ways (as detection noise and as back-action noise), which scale conversely with laser power. Hence, there is no simple ‘fix-all’ for this problem—merely turning the laser power up or down will not improve the noise behaviour universally, i.e. result in less noise in the strain sensitivity of the interferometer! While it is possible to lower one or the other quantum noise contribution (as we will see in the next section), it is far from trivial (and classical) to surpass the standard quantum limit.

Current day gravitational wave detectors are already limited by quantum noise in the high-frequency range, where shot noise is the dominant (quantum) noise source, as can be seen on the example of aLIGO in figure 6. The red graph shows the sensitivity of aLIGO during O1 (the first observation run from July 2009 until October 2010). At high frequencies (above a few hundred Hertz) the sensitivity curve rises with frequency—this is caused by shot noise.^{3} Below this frequency range other technical noise sources still dominate, but above a few 100 Hz the gravitational wave detector is shot noise limited—a limitation caused by quantum noise! What (if anything) can be done about this sensitivity limitation?

## 4. Fixed-quadrature squeezing

Injecting fixed-quadrature-squeezed light into the output port of the interferometer (colloquially called ‘plugging the open port’, where vacuum noise would otherwise couple in) is a possibility to selectively reduce quantum noise in the interferometer. Depending on the quadrature squeezed, quantum noise can be reduced either at high frequencies (where shot noise is limiting) or at low frequencies (where quantum radiation pressure noise would dominate, if not for currently other, larger technical noise sources). To reduce shot noise, phase-quadrature-squeezed light needs to be injected; to reduce quantum radiation pressure noise, amplitude-quadrature-squeezed light would need to be injected. As shown in §2, it is not possible to reduce the uncertainties in both quadratures simultaneously due to the Heisenberg Uncertainty Principle.

As the current gravitational wave detectors are quantum noise limited only at high frequencies (where shot noise dominates) it is useful to inject phase-squeezed light to reduce the detector noise floor below the shot noise limit. This is routinely done in the gravitational wave detector GEO 600 (a British–German collaboration), which has been continuously operated with a squeezed light source since 2011 [9,10].

Figure 7 shows the sensitivity development of GEO 600 from 2011 (*a*) to 2016 (*b*). Figure 7*a* shows GEO’s sensitivity without (black) and with (red) 10 dB of injected phase-quadrature-squeezed light, resulting (on average) in a 2 dB (≈1.6-fold) sensitivity increase in 2011. Figure 7*b* shows GEO’s sensitivity without (red) and with (blue) 10 dB of phase-quadrature-squeezed light injected, resulting in a 4.3 dB (≈2.7-fold) sensitivity increase for frequencies above approximately 1 kHz in 2016. Comparing figure 7*a*,*b* it is visible that over time the overall sensitivity of GEO600 was continuously improved, and that additionally the degree of squeezing within the interferometer was increased, even though the degree of injected squeezing remained the same. This was achieved by painstakingly reducing optical losses in the interferometer (the effect of losses on non-classical states of light is explained in §5). While a factor of less than 3 in sensitivity does not seem like a large improvement at first glance, it is worthwhile noting that the observable volume of the universe scales with the cube of the sensitivity increase. Thus, even small sensitivity improvements are pursued tenaciously.

It is obvious from the graphs in figure 7 that phase-quadrature squeezing increases the high-frequency sensitivity of the gravitational wave detector. Maybe unexpectedly (after the treatment in §2), it is also evident that the low-frequency sensitivity is *not decreased*—this is due to the fact that technical noise still dominates the detector’s sensitivity at low frequencies, masking the effect of quantum radiation pressure noise (which is thus not visible). It is therefore beneficial for the overall gravitational wave detector sensitivity to inject phase-quadrature-squeezed light, as it increases the high-frequency sensitivity, where the interferometer actually is quantum noise-limited. This will change once the advanced gravitational wave detectors become quantum radiation pressure noise noise-limited at low frequencies, as their design sensitivity (figure 2) predicts.

Keep in mind, though, that by employing this technique the standard quantum limit is not surpassed: the effect of phase-quadrature squeezing is merely as if one turned up the laser power (but without the drawbacks associated with a power increase, e.g. larger thermal noise due to more deposited thermal energy in coatings and mirror substrates, and an increase of scattered light leading to larger phase noise). Fixed-quadrature squeezing is a way of selectively increasing quantum noise-limited detector sensitivity (at either high frequencies or at low frequencies, depending on the squeezing quadrature) by ‘shifting around’ quantum noise advantageously. In the above case, if GEO 600 were radiation pressure noise-limited, phase-quadrature squeezing would lead to an increase in low-frequency (quantum radiation pressure) noise—we would ‘lose on the swings what we gained on the roundabouts’.

The standard quantum limit cannot be beaten by pure amplitude- or phase-quadrature squeezing. There are, however, other techniques that can.

## 5. Going below the standard quantum limit of interferometry

Reaching sensitivities below the standard quantum limit generally^{4} calls for *quantum non-demolition techniques* [12]. Various different techniques have been proposed in the literature (e.g. in [13]) and a smaller number have been demonstrated (in laboratory-scale, so-called ‘table-top’ experiments). However, at the time of writing no interferometric gravitational wave detector is using sub-SQL techniques, as these are both costly and technically challenging to implement. The most mature concept to date is that of *frequency-dependent squeezing* [8,14]; it has been thoroughly investigated and demonstrated [15,16], though not yet in a large-scale gravitational wave detector.

Intrinsically, amplitude-quadrature fluctuations and phase-quadrature fluctuations of a coherent state are uncorrelated. However, when the light carrying these fluctuations interacts with the interferometer (consisting of suspended mirrors subject to quantum radiation pressure noise, and electrical photodetectors which detect quantum shot noise), this gives rise to a coupling of the two formerly uncorrelated quadratures, and this very interaction can be exploited to surpass the standard quantum limit. In the following, some techniques to reach sensitivities below the standard quantum limit are described.

### (a) Frequency-dependent squeezing

We have discussed that for a reduction of shot noise (already limiting current day gravitational wave detectors at high frequencies) phase-quadrature-squeezed light can be injected, whereas for the reduction of quantum radiation pressure noise (limiting gravitational wave detectors at low frequencies in the near future) amplitude-quadrature-squeezed light would be required. What if it were somehow possible to generate squeezed light that is appropriately squeezed at different frequencies? The result is termed frequency-dependent-squeezing [8,13,14], and it is generated by reflecting fixed-quadrature-squeezed light off a detuned optical resonator: a lossless Fabry–Pérot resonator is used as a frequency-selective filter for the fixed-quadrature-squeezed light (hence called a ‘filter cavity’)^{5} . Filter cavities introduce an appropriate phase shift to the fixed-quadrature-squeezed light. The phase-shifted squeezed light is then coupled into the output of the gravitational wave detector. The (simulated) result is shown in figure 8 [17].

Frequency-dependent squeezing has been experimentally demonstrated at radio frequencies (MHz) in a table-top experiment [15] and recently also at audio frequencies (kHz) more relevant to gravitational wave detection [16].

A major problem with *all* experiments employing non-classical states of light is that the light field has to be (optically) manipulated before it is finally coupled into the interferometer; and even in the interferometer the light passes through many optical elements. Non-classical states of light (such as squeezed states) are however highly susceptible to loss—the degree of non-classicality is rapidly degraded when the light field is subject to optical losses. Unfortunately, losses always occur when light propagates through optical components (e.g. when light is reflected off necessarily imperfect mirrors or passes through resonators or apertures). This can be understood by realizing that any loss channel is a way for vacuum noise to be superimposed on the non-classical state, effectively ‘watering down’ the effect of squeezing and resulting in a less-squeezed state. Therefore, losses have to be avoided and/or minimized in the experiment at all cost. Loss is particularly detrimental in optical resonators (such as the filter cavities), as the light is reflected many times in them, for each pass experiencing loss through absorption and scattering. This is the reason why the prospective filter cavities for future gravitational wave detectors are designed to have a large round-trip length (thereby effectively minimizing the number of reflections needed for a sufficient phase shift), even though it is experimentally very challenging and costly to employ long optical resonators.

That being said, after more than 30 years of experimental work, squeezing is now a very mature technique, and frequency-dependent squeezing is by far the most advanced technique to attempt to surpass the standard quantum limit. The effect of frequency-dependent squeezing can be further enhanced by employing further modifications of the interferometer output, as briefly discussed in the next subsection.

### (b) Modifications of the output port

The output port of the interferometer is where the output light field (exiting the gravitational wave detector) is detected by a photodetector. Currently, the necessary photodetection is accomplished by direct detection with a single photodetector. This so-called DC readout only allows measurement of the intensity of the light field, which is proportional to the square of the amplitude of the light field. However, using *homodyne detection* (see e.g. [18,19]), the output light field can be read out phase-sensitively.

As stated above in this section, there exist no inherent correlations between the amplitude-quadrature fluctuations and the phase-quadrature fluctuations of a coherent state. Interaction of the light with the suspended mirrors of the interferometer can, however, introduce correlations between the quadratures. The coherent state with its (minimal) amplitude-quadrature fluctuations impinges on the suspended mirrors, causing a position change due to quantum radiation pressure noise. Owing to the restoring force (gravity) the mirror falls back into its position of equilibrium, thereby pushing back on the light field and causing back-action noise on the phase quadrature of the light exiting the interferometer. The amplitude-quadrature fluctuations of the input field cause phase-quadrature fluctuations on the output field! The output phase quadrature however also carries the information about the gravitational wave signal. Additional back-action noise on this quadrature therefore degrades the overall signal-to-noise ratio, leading to the decreased low-frequency sensitivity visible in the aLIGO design sensitivity curve shown in figure 2. It is important to note that the cause for the added phase-quadrature fluctuations on the output are the initial amplitude-quadrature fluctuations on the input—there now exists a correlation between the two by virtue of the interaction with the suspended interferometer mirrors. This gives us the means of cancelling this effect again: homodyne detection makes it possible to read out the interferometer output under a specific, advantageous detection angle where the back-action noise destructively interferes with the amplitude-quadrature fluctuations and both are thus exactly cancelled. Furthermore, this detection angle can in principle be chosen frequency-dependently by reflecting either the signal light or the local oscillator light off a filter cavity, allowing for a broadband cancellation of quantum radiation pressure noise. This technique is termed *variational readout* [13]. It has not yet been shown experimentally for various reasons: firstly, current gravitational wave detectors are not yet radiation pressure noise limited; secondly, the technique is much more susceptible to optical loss than modifications of the input field (additionally, input field modifications can also easily be blocked, which output modifications cannot); and lastly, variational readout is incompatible with current readout schemes [20].

### (c) Combinations of (quantum) techniques

In principle, many proposed techniques to enhance the sensitivity of an interferometric gravitational wave detector are compatible and hence combinable. As an example, the technique of ‘squeezed variational’ [13] is a combination of frequency-independent (i.e. fixed-quadrature) squeezing and variational output (homodyne detection with frequency-dependent readout angle): frequency-independent squeezing is injected into the dark port of the interferometer (under a fixed frequency-independent optimal angle of *π*/2); the interferometer output is read out variationally with a homodyne detector using filter cavities to introduce the necessary frequency dependence of the signal.

Variational readout can also be combined with frequency-dependent squeezed input light [21]; in the cited case, a signal recycled interferometer topology is additionally assumed, adding another advanced technique to the mix which further increases the sensitivity. Obviously, the combination of techniques increases the complexity of the optical system significantly, and none of them have been shown experimentally yet. However, in the quest for ever higher detector sensitivity it is crucial to pursue these combined approaches, both theoretically [22] and eventually experimentally.

## 6. Alternative topologies

All techniques described in the above sections are based on either cleverly shifting uncertainties from one quadrature to another and thereby reducing the quantum noise advantageously in the corresponding relevant frequency ranges, or on making the readout scheme insensitive to the back-action effect of quantum noise. An entirely different approach is to build a detector that is inherently less susceptible to quantum noise in the first place. One of these ideas is elucidated here: the *speed meter* [23].

In a Michelson interferometer the standard quantum limit stems from the fact that measurements of the position x for different times do not commute: [*x*(*t*),*x*(*t*′)]≠0. However, in systems with momentum conservation one can, in principle, measure the momentum *p* of an ensemble of mirrors continuously with arbitrary precision because the momenta commute for different times: [*p*(*t*),*p*(*t*′)]=0 [24]—for a free particle the velocity is a quantum non-demolition (QND) variable! In a speed meter interferometer the velocity of the test mass is measured instead of its position, hence a speed meter is a QND interferometer [23].

Figure 9 shows intuitively how a speed meter works: the impinging laser light transfers a momentum ‘kick’ to the test mass, travels further along the path and then delivers an ‘anti-kick’ (with opposite sign), effectively reducing the back-action created by quantum radiation pressure noise! The total acquired phase *ϕ* depends on the position difference for the two measurements. After both position measurements the net momentum transfer is zero as the phase change is linear in and does not depend on the position. This holds for delays much smaller than the gravitational-wave period τ≪T_{GW} (e.g. a gravitational wave at 100 Hz has a wavelength of *λ*_{GW}=3×10^{6} m). If this condition is fulfilled, radiation pressure noise is strongly suppressed—however, shot noise is retained.

Speed meters can be realized in different topologies [25–27], but the most intuitive and thoroughly explored [28] is that of the Sagnac interferometer. An experimental demonstration is still outstanding, but well underway. A good overview of the topic is given in [29,30].

There are numerous other approaches to combat quantum noise in future gravitational wave detectors, such as using ponderomotive squeezing [31], cancelling the quantum radiation pressure noise by destructive interference with an ‘anti-noise process’ [32], or injecting even more exotic combinations of (non-classical) light fields into the interferometer [33,34], but this exceeds the scope of this paper.

## 7. Conclusion

This article presents how current and future interferometric gravitational wave detectors are limited by quantum noise, caused by the photon statistics of the laser light. It describes how quantum noise manifests in the interferometer as photon shot noise and quantum radiation pressure noise, which together form the standard quantum limit of interferometry. Some of the main schemes under investigation to overcome the quantum limit are described: a technique to redistribute the quantum noise advantageously is presented, as are advanced schemes to surpass the standard quantum limit by modifications of the input and/or output port, as well as an interferometer topology less susceptible to quantum radiation pressure noise. The way forward to combat quantum noise in order to reach ever-higher detector sensitivity is not yet carved in stone, but the experimental options are diverse.

## Data accessibility

This article has no additional data.

## Competing interests

I declare I have no competing interests.

## Funding

No funding has been received for this article.

## Acknowledgements

The author thanks S. Hild and S. Danilishin for the use of images, and S. Danilishin for many fruitful discussions.

## Footnotes

One contribution of 11 to a discussion meeting issue ‘The promises of gravitational-wave astronomy’.

↵1 The normalization factors in the definition of the quadratures were chosen for maximal simplicity of the presentation.

↵2 The Poission statistic states that the probability of measuring n photons in a coherent state with mean photon number is .

↵3 Shot noise is per se a frequency-independent (i.e. white) noise term, but in a gravitational wave detector it is weighted by the optical transfer function of the recycled interferometer and hence rises with frequency.

↵4 Though it should be noted here that

*signal recycling*—a classical (i.e. not quantum) technique employed in all advanced detectors—can reach sensitivities below the standard quantum limit, by virtue of the*optical spring*effect [11], the optomechanical coupling between the laser light field and the kilogramme-scale interferometer mirrors.↵5 The detuning of the filter cavity relative to the impinging field has to be small and its linewidth needs to be narrow (on the order of the linewidth of the signal recycling cavity)—the finesse and the length of the filter cavity should be on the order of the Michelson interferometer to realize a QND interferometer with broadband noise reduction. To minimize optical losses filter cavities ought to be long so as to require a minimum number of reflections to achieve the necessary phase shift.

- Accepted December 12, 2017.

- © 2018 The Author(s)

Published by the Royal Society. All rights reserved.