Quantum expander for gravitational-wave observatories

The quantum uncertainty of laser light limits the sensitivity of gravitational-wave observatories. Over the past 30 years, techniques for squeezing the quantum uncertainty, as well as for enhancing gravitational-wave signals with optical resonators have been invented. Resonators, however, have finite linewidths, and the high signal frequencies that are produced during the highly scientifically interesting ring-down of astrophysical compact-binary mergers still cannot be resolved. Here, we propose a purely optical approach for expanding the detection bandwidth. It uses quantum uncertainty squeezing inside one of the optical resonators, compensating for the finite resonators’ linewidths while keeping the low-frequency sensitivity unchanged. This quantum expander is intended to enhance the sensitivity of future gravitational-wave detectors, and we suggest the use of this new tool in other cavity-enhanced metrological experiments.


Supplementary Text
In the supplementary text we discuss the experimental feasibility of the quantum expander, with emphasis on the optical loss sources and the beneficial combination of the quantum expander with external squeezing of the light's quantum noise; derive the equations in the main text based on the simplified input-output relations and on the Hamiltonian description of the system; give details on the astrophysical implications of use of quantum expander; provide details of derivation of the full spectral density of the detector with various imperfections based on the optical transfer matrix approach.

S1 Experimental feasibility
In this section we discuss some of the issues of the experimental feasibility. We indicate the main sources of loss and their contribution into the resulting sensitivity, and analyze the achievable benefit from quantum expansion when combined with external squeezed-light injection.

S1.1 Optical loss
As we discuss in the main text, our quantum expander creates squeezing at high frequencies to counteract the effect of the detector's bandwidth. When combined with external squeezing, the  Fig. S1: Relative contribution of different vacuum modes (optical loss) to the overall sensitivity of the detector versus frequency. Input vacuum mode (solid red) defines the main sensitivity level. The rest arises from various sources of loss: loss inside the SE cavity (dashed magenta), detection loss (dot-dashed blue) and arm cavity loss (dotted green). The parameters are taken according to Table S1: internal loss is 1500 ppm single-trip, detection loss is 1%, transmission of the end mirror is 100 ppm (increased relative to Table S1 to emphasize the smallness of its influence on the sensitivity) quantum expander produces a high amount of squeezing at high frequencies, which imposes strict demands on reducing the optical losses. The losses occur inside the detector: inside the arm cavity, and inside the SE cavity; as well as on the readout train: from the SE mirror to the detector. The external squeezing additionally suffers from injection loss. On Fig. S1 we show the contribution of different sources of loss as a function of frequency. We note that the detection loss and loss inside the SE cavity are the most important contributions. The detection loss currently is rather high, but the way to mitigate this loss by parametric amplification was proposed by Caves (1) and recently re-investigated experimentally (2). The idea of this approach is to amplify both the signal and the noise by the same amount before it experiences loss, such that the resulting noise is much above vacuum uncertainty, and the loss does not affect it significantly. The simplest example of it is detecting some signal G embedded in squeezed vacuum, with detection efficiency η: where r is the squeeze factor and q is the Caves' amplification factor. The signal-to-noise ratio is given by SNR = ηe 2q (1 − η(1 − e −2r e 2q )) −1 . Without amplification, q = 0, in the limit of large squeezing e −2r ≈ 0 the SNR is limited to SNR q=0 ≤ η(1 − η) −1 . When the amplification is large, q → ∞, the SNR becomes independent on the loss: SNR q→∞ = e 2r , and only benefits from initial squeezing. The only source of detection loss that cannot be mitigated by Caves' amplification is the loss in the Faraday isolator used for injecting external squeezing.
We assume this to be a limitation in the detection loss, which corresponds to the 0.5% (3) mentioned in the main text.
Internal loss will be increased due to the additional optical surfaces of the nonlinear crystal and the absorption of the crystal. While the actual contribution to the loss from such a crystal requires a separate investigation, we give an estimate based on the squeezing cavity design for the table-top experiments. If the PPKTP crystal will be used, it's absorption is ∼ 100 ppm per cm depending on wavelength (4); the surfaces of the crystal will have to be coated with anti-reflecting coating to minimize the scattering loss. We estimate that the current standard technology can bring this added loss on the level of 200-500 ppm in single-pass.
We would like to emphasize, that not every configuration of the GWO will be able to get a significant benefit from quantum expansion when the external squeezing is in use. Depending on the amount of loss, and amount of external squeezing injected, the benefit will vary. The : An improvement in the sensitivity of the detector by quantum expander, relative to the detector with external squeezing injection, depending on the amount of total loss (internal and readout). The higher is the external squeezing, the more stringent is the loss requirement for being able to benefit from using the quantum expander. The sensitivity depends in a non-trivial way on the losses, which is reflected in the benefit from QE shown on the figure.
reason is an additional de-amplification of the signal in the quantum expander. When the loss is high, the squeezing of the noise by quantum expander in addition to external squeezing might be not significant. However, the parametric process inside the detector reduces the signal, hence the signal-to-noise ratio might even become reduced compared to the detector without quantum expander, if the sub-optimal parametric gain is chosen. There always exists an optimal gain, for which the benefit is maximal. If the loss is high, it might be optimal to amplify the signal (and anti-squeeze the noise), similar to the Caves' amplification discussed above. We demonstrate possible improvements to the sensitivity in Fig. S2. We note, that this specific design is based on  Table S1. the benchmark parameters adopted by the LIGO-Virgo Collaboration, as presented in Table 1, and corresponds to the sensitivity as given in Fig. S3. In reality, the benefit from quantum expansion can be increased by optimizing the optical design (e.g. SE cavity length and mirrors' reflectivities). The optimized sensitivity given by the quantum expander is a topic of future studies.

S1.2 Crystal inside the detector
There are several issues to be taken into account with placing the crystal inside the SE cavity.
First, the aperture of the crystal has to be large to avoid beam clipping at the edges of the crystal. Currently the diameter of the beam inside the SE cavity is ∼ 2 cm (6) In order to plot the spectral densities in the paper we use the following set of parameters of some baseline GW observatory, without choosing a specific design from many possibilities of a 3-G topologies. We note that our double-cavity model uses effective parameters. In order to use this model for the Michelson topology, an effective light power inside the arm cavity has to be used: P c = 2P arm , where P arm is the power inside the arms of the Michelson topology (5) point outside the SE cavity. For comparison, the aperture of typical PPKTP crystals used in the squeezed-light generation is 1×2 mm (4). Either large enough crystals need to be realized or the beam need to be focused inside the SE crystal, which is in principle possible without increasing the number of optics inside the SE cavity.
Second, while absorption and scattering are generally important issues to avoid possible heating and parasitic interferences (7), the crystal will not be illuminated by bright light, because the detector is operated with a destructive interference of carrier light in the SE cavity (dark port condition).
Third, the crystal has to be pumped with frequency doubled light of a sufficient intensity, which would require an additional cavity for the pump light, as it is realized in modern table-top squeezing experiments (8).
It would also be necessary to deliver the pump to the crystal inside the detector and ensure good wavefront matching of the pump and the main beam. Both aspects can be achieved in several ways. As the wavelength of the pump is so different from the fundamental wavelength, it is possible to realize optical coatings, such that an additional cavity is formed by the SEM and ITM for the pump (9). Alternatively, the pump can be brought in by replacing the existing steering mirrors in the SE cavity with dichroic steering mirrors, being transmissive for the frequency doubled pump. In any case, no additional optics inside the main interferometer would be required.
Fourth, there is a need for additional control of the pump amplitude and phase to ensure the optimal squeeze factor and squeeze angle. Finally, the non-linear crystal might affect the control of the main interferometer, as some of the auxiliary beams could be affected by the parametric process, which has to be taken into account when designing the detector's control scheme.
In conclusion, while a non-linear crystal inside the interferometer is technologically challenging, we do not foresee fundamental problems, and expect our proposal for quantum expansion to motivate the future research and development work in this direction.

S2 Astrophysical analysis
In this section, we give an illustrative example to estimate the capability of using the quantum expanders to detect the gravitational waves radiated by neutron star poster-merger remnants.
The method we used here follows the estimation procedure as described in (10,11). We perform a Monte Carlo simulation based on the following assumptions: first, the mass of each individual neutron star in a binary system follows an independent Gaussian distribution centered at 1.33M with variance 0.09M . The distributions of angular sky position, inclination and polarisation angles, and the initial phase of the source are assumed to be flat. The searching range is assumed to be 1 Gpc and the event rate is taken to be ≈ 1 Mpc −3 Myr −1 . Second, the post-merger waveform is assumed to be a parametrized damped oscillation, which depends on the equation of state of a neutron star, and in frequency domain it is given by the equation: where d is the source distance, h p is the peak value of the wave amplitude, Q is the quality factor of the post-merger oscillation,φ 0 , f p are the initial phase and the peak frequency of the waveform, respectively. Among them, h p , Q, f p are parametrized by fitting with the results generated by numerical simulation (12) and they depend on the choice of equation of states.
In the illustrative examples here, we make use of a relatively stiffer equation of state proposed in (13), where Q = 23.3, h p ≈ 5 × 10 −22 , and the peak frequency is given by: where R = 14.42 km is the radius of each neutron star, and m 1,2 are their masses. The parameters a 2 , a 1 , a 0 take the value of 5.503, −0.5495, 0.0157, respectively (13). We define the signal to noise ratio as: where we take the integration range to be f min = 1000 Hz, f max = 4000 Hz. We run 100 Monte-Carlo realizations each with 1000 samples, corresponds to one-year observation. We exclude the binaries with total mass larger than 3.45M since they will collapse into a black hole in a very short period of time, less than one period of post-merger oscillation. For each different interferometer parameter set, we selected out the loudest event in each Monte-Carlo realization, set SNR = 5 as a threshold signal-to-noise ratio and produce the Fig. 4 in the main text.

S3 Input-output relations
In this section we derive the sensitivity based on the input-output formalism. For simplicity in this section we ignore the effects of quantum radiation pressure noise and optical losses.
These will be included in the full transfer matrix description in Sec. S5. Based on the obtained equations we give motivation for writing the Hamiltonian of the system in Sec. S4.
Using the perturbation theory, we decompose the light field into a steady-state amplitude with amplitude A 0 and laser carrier frequency ω 0 and a slowly varying noise amplitude a(t) (see details in (14)): where A is the laser beam cross-section area,h is the reduced Plank constant. It is helpful to consider the input-output relations of our system in the 'two-photon formalism' (15,16), where the amplitude and phase quadrature amplitudesâ (c) andâ (s) of the modulation field at frequency Ω are linked to the optical fieldsâ(ω ± Ω) viâ These operators obey the commutation relation [a x (Ω), a y (Ω )] = [a x (Ω), a y (Ω )] = 2πiδ(Ω + Ω ) .
We make several simplifications to the notation: as we are primarily interested in the phase quadrature, we will omit index (s) in equations below; we also omit the hats on the operators for brevity, although all the fields are quantised; we consider only the noise fields in the frequency domain, so we don't write that in the equations explicitly: e.g.â (s) (Ω) → a. We model the parametric amplification process as a simple linear amplification of amplitude quadrature of the light by some factor e q , without considering the effects of the parametric pump and the finite size of a crystal. In the full model in Sec. 5 we also will introduce the possibility to tune the amplification quadrature. With this in mind we start with writing down the steadystate input-output relations (15,16) for the quantum fluctuations of the phase quadrature of the light field, for the cavity cavity model depicted of Fig. S4. For the detailed explanation of the approach we refer the reader to the review by Danilishin and Khalili (14). We choose the arm cavity to be tuned on resonance, so that for Ω = 0 it has the maximal light power inside.
where R i,s = R ITM,SE , T i,s = T ITM,SE are the amplitude reflectivity and transmissivity of input test mirror and signal-extraction mirror; q is an amplification factor on the single pass through the crystal; τ arm,SE = L arm,SE /c is the single trip time in arm cavity of length L arm and signal extraction cavity of length L SE , with c being the speed of light; ϕ = π/2 is the tuning of the SE cavity with respect to the arm cavity; x is a small displacement of the end mirror due to the GW signal, E is the large classical amplitude of field inside the arm cavity and k p is the wave vector of the carrier light field.
We find a solution to these equation, splitting the output b into the noise part b n and signal where R a (Ω), T (Ω) are the noise and signal optical transfer functions correspondingly.
We can obtain an intuitive expression for these functions by doing several approximations.
We assume Ωτ arm 1, Ωτ SE 1, so e iΩτ arm,SE ≈ 1 + iΩτ arm,SE ; and T i,s 1, so R i ≈ where γ arm , γ are the arm cavity and the signal-extraction cavity linewidth, respectively; a single-pass optical gain is small: q 1, so e q ≈ 1 + q = 1 + χτ SE , where χ is an effective parametric gain.
With these approximations equations (18)(19) can be simplified to where we defined a sloshing frequency ω s = c T 2 i /(4L SE L arm ). Notice that these equations correspond to Eq. (6) in the main text. This helps us to construct a Hamiltonian in the next section, which would correspond to this model system.

S4 Hamiltonian approach
In this section we derive the sensitivity of the detector (Eq.
whereâ,â q are the arm cavity and SE cavity modes, and ω 0 is their natural resonance frequency; is the coupling rate between two cavities, T ITM is the transmission of the front mirror of the arm cavity, L SE , L arm are the lengths of the signal extraction and arm cavity, respectively; γ = cT SE /(4L SE ) is the coupling rate of the SE mode to the continuum of input modesâ in ; x is the displacement of the test mass partially in reaction to the gravitationalwave tidal force F GW ; the mirror motion x is coupled via the radiation-pressure forceF rp to the cavity mode with strength G 0 = ω 0 /L arm ; κ is the coupling strength due to a crystal nonlinearity under a second harmonic pump field with amplitude β and phase φ. The pump field is assumed to be classical and its depletion is neglected. The effect of the back-action noise can be neglected, so displacement of the mirror is coupled only to a GW strain: We obtain the Langevin equations of motion for the cavity modes in the frame rotating at ω 0 and expand the quantum amplitudes into a sum of large classical amplitude and small quantum fluctuation,â → A +â:ȧ where we defined an effective coupling strength of GW signal strain G = 2P c L arm ω 0 /(hc) and optical power inside the arm cavity P c =hω 0ā , withā being an average amplitude of the modeâ; and the effective parametric gain χ = κβ, As we are interested in the spectral properties of the system, we transform into a Fourier domain:ȧ(t) → −iΩâ(Ω). The outgoing light is measured by a homodyne detector, which measures the quadratures of the light, that are defined as: We obtain the input-output relations for the two quadratures by solving Eqs.(27-29): out (Ω) =â (2) in (Ω) From these input-output relations we can obtain the sensitivity, by computing the spectral densities. We define the spectral density of the fieldâ(Ω) as: Then the spectral density the output noiseâ (c) out (Ω) is: where S in (Ω) is the spectral density of incoming light field, which we assume here to be vacuum: S in (Ω) = 1. Assuming that we squeeze the signal quadrature of the light: φ = −π/2, we obtain the following noise spectral density and signal transfer function: The total strain sensitivity is given by the noise normalised to the signal transfer function (subsituting the expression for the effective coupling strength G): which is the equation (9) in the Main text.

S5 Transfer matrix approach to full description
In this section we use the transfer matrix approach (14) to compute the sensitivity of the detector taking into account the radiation pressure noise and optical losses. We start from the same point as in the section S3, but write down the input-output relations as propagation of the field amplitudes in terms of transfer matrices for each optical element. The description is broader than strictly needed to compute the spectral density in the main text (e.g. it includes the effects of dynamical back action), but we find it helpful to use a general approach. The source code that implements these equations can be found in (17).

S5.1 Input-output relations
We describe a two-cavity system, as shown on Fig. S4, in terms of input and output quantum fields. Based on two-photon quadrature amplitudes we define the vectorâ( The signal extraction cavity can rotate the quadratures due to its detuning from resonance. The optical parametric amplification process also squeezes and rotates the quadratures. The effect of the signal recycling cavity can be described as a set of rotations and squeezing operations: where we denote the amplitude reflectivity and transmissivity of the signal recycling and input mirrors by R s,i , T s,i , the power loss inside the cavity (before the crystal) is λ s ; signal recycling cavity global delay τ SE = L SE /c and the phase delay due to the cavity detuning before and after the crystal by φ, ϕ. We now introduce the squeeze angle θ and the rotation matrix and squeezing matrix with q being the single-pass squeeze factor.
For the arm cavity the corresponding set of equations readŝ where k = ω/c is the wave vector of the main field, δ arm is the arm cavity detuning and τ arm = L arm /c is the propagation time with L arm being the length of the arm cavity, and c the speed of light. The field E corresponds to the classical amplitude of the field impinging on the end mirror.
We find the solution to these equations, first for the complex transmissivity and reflectivity of the signal recycling cavitŷ where we defined That provides the input-output relations for the signal extraction cavitŷ where we introduced the transfer matrices for the fields Now we can derive the fields for the arm cavity yieldinĝ where Finally, we find the outgoing field to bê where we defined the transfer matrices: (81)

S5.2 Radiation pressure
The radiation pressure force acting on the mirrors has three contributions. First, there is a constant force due to the classical high-power optical field. It induces a constant shift of the mirror, which can be compensated with classical feedback. Second, there is a dynamical classical part, which is amplified by opto-mechanical parametric amplification and which belongs to the optical spring, and third a fluctuating force due to the uncertainty in the amplitude quadrature of the light. The latter corresponds to the quantum back-action force of the carrier light. Following (5), we assume the input test mass to be fixed, and twice the back action imposed on the back mirror instead (which leads to introduction of effective light power). Such approximation is valid when the transmission of front mirror is small, such that the amplitudes of the fields acting on the front and back mirrors are almost equal (which is the case in our consideration).
where we split the back-action into the noise part F f l (Ω) and position-dependent optical spring force with spring constant K(Ω). Taking into account that F = R e E, we find the equations for these contributions: Without loss of generality we choose the phase of the classical amplitude such that: where the amplitude E is connected to the power in the cavity as P c = 2P arm =hω p |E| 2 , where P arm is a power in the corresponding Michelson interferometer (5).
The equation of motion for the test mass taking into account the radiation pressure force: where χ(Ω) = (−mΩ 2 ) −1 is the mechanical suspeptibility for the free mass m. This allows us to introduce an effective susceptibility: such that x − (Ω) = χ eff (Ω)F f l (Ω).

S5.3 Detection
The presence of optical loss in the readout path, including the detection loss, leads to a loss of quantum correlations due to mixing with vacuum. We model this loss with a beam splitter of power transmissivity η = 1 − λ r and reflectivity (loss) 1 − η = λ r which mixes in vacuum n: The balanced homodyne detection on the outputb at homodyne angle ζ provides the values y(Ω) = {cos ζ, sin ζ} Tb (Ω) = H Tb (Ω) = √ ηH T (−Râ + Tv + L b1n1 + L b2n2 ) + √ ηH T Zx − (Ω) + 1 − ηH T n(Ω), (90) which we renormalize to the differential mirror displacement y = H T (−Râ + Tv + L b1n1 + L b2n2 ) We implement the injection of the squeezing from the outside, by defining an action of the squeezing operation on the input fieldâ as: whereâ vac is the vacuum field before squeezing, and the squeezing matrix with squeeze factor q ext and squeeze angle φ ext is defined as All other fieldsv,n,n 1 ,n 1 are in the vacuum state.
From this we get the spectral density for this output where Finally we normalize the spectral density to the gravitational-wave strain yielding (taking into account the effects of high-frequency corrections (18)

S5.4 Filter cavities
Filter cavities on the can be used to create a necessary frequency dependence of quantum correlations, such that the QRPN is suppressed or evaded completely. There are two scenarios, input filter cavity, where the injected squeezing becomes frequency dependent, and output filter cavity, where the homodyne detection becomes frequency dependent. We follow (14) where γ f is the filter cavity bandwidth, and δ f is it's detuning from resonance. To obtain the spectral corresponding spectral densities it's sufficient to modify the squeeze angle φ ext → φ ext + θ f (Ω) or homodyne angle ζ → ζ − θ f (Ω) in the equations for the spectral density Eq. 94.
The optimal detuning is on the slope of the cavity resonance δ f = γ f , and the exact choice of cavity linewidth depends on the parameters of the detector, including the internal squeezing strength and readout loss.