## Introduction

The dawn of gravitational-wave (GW) astronomy began with the historic detection of binary black hole coalescence in 20151 and several more detection events that followed in subsequent years2. The latest observation of gravitational waves was from a binary neutron star inspiral. It was succeeded by observations of a broad spectrum of electromagnetic counterparts3,4, demonstrating that gravitational-wave astronomy is invaluable for understanding the Universe5,6. Further increasing the sensitivity of gravitational-wave observatories (GWOs) is of utmost importance to maximize the scientific output of combined multi-messenger astronomical observations.

Gravitational-wave observatories, such as Advanced LIGO7, Advanced Virgo8, GEO6009, and KAGRA10, are based on the Michelson interferometer topology (see Fig. 1), where the incoming gravitational wave changes the relative optical path length of two interferometer arms. The ability of the observatories to measure gravitational waves is limited by various disturbances that also change the differential path length or manifest themselves as such. The main noise source at signal frequencies above ~50 Hz in the current generation of GWOs is the quantum uncertainty of the light field, which results in shot noise (photon counting noise)7,11. Noise at lower frequencies has contributions of several origins, such as Brownian motion of the mirror surfaces and suspensions or quantum radiation pressure noise, which comes from mirrors’ random motion due to quantum fluctuations of light power12,13. All these noise sources contribute to the photocurrent of the photodiode placed on the signal port of the detector. The observatory’s sensitivity to the GW signal, i.e., its ability to discriminate between the GW signal and noise is given by the observatory’s signal-to-noise ratio (SNR). For a given set of parameters of the detector, e.g., the interferometer’s arm length or laser power, its sensitivity is ultimately limited by its “quantum Cramer-Rao bound” (QCRB)14. For continuous signals, the QCRB at every frequency is determined by the radiation pressure force exerted on the test mass by quantum fluctuations of the light field in a pure state15. The higher is this quantum radiation-pressure force on the mirror, the higher is the uncertainty in its momentum, and thus the better can the position be resolved, according to the Heisenberg uncertainty relation16. The QCRB represents an ultimate limit on the sensitivity of the detector, which can only be saturated in the absence of decoherence and under the optimal measurement protocol. For example, reaching this limit at low frequencies in GWDs requires implementing the back-action evasion measurement scheme, e.g., a variational readout17, such that the sensitivity is fully limited by shot noise.

Generally, the QCRB is decreased when the squeezed states of light are added to the interferometer configuration (such that the quantum uncertainty in the amplitude of the light field is increased)12,18,19. The injection of squeezed states has become a well-established technique for GWOs20,21,22.

A classical way to lower the QCRB is to increase the signal response of the detector by optical resonators, as implemented already in the first generation of GWOs23. The resonance buildup of optical energy in resonators increases the radiation pressure force16 and hence lowers the QCRB. The current (second) generation design includes Fabry-Perot cavities in the arms and a signal-extraction (SE) cavity on the dark port of the detector24. Resonators, however, significantly lower the QCRB only at frequencies below the resonator’s linewidth, i.e., they reduce the observatory’s detection bandwidth25. The injection of squeezed states, mentioned above, is not able to counteract the loss of bandwidth due to resonators.

The issue of detector bandwidth has become crucial in the era of multi-messenger astronomy3. Information about the physics of extremal nuclear matter is hidden in the waveforms of gravitational waves radiated from the post-merger remnants of binary neutron star systems26. Obtaining this information is important for unraveling the physics of compact astrophysical objects—the engines that drive gamma-ray bursts, the origin of heavy elements and possible modifications to general relativity27,28. These waveforms have typical frequencies above 1 kHz, where the sensitivity of current observatories degrades due to limited bandwidth.

Over the past 20 years, the challenge of increasing the bandwidth without changing the peak sensitivity at low frequencies has become one of the cornerstones for the design of future GWOs29,30. Previous concepts involved unstable optomechanical or atomic systems operating in the so-called “negative dispersion” regime31,32,33,34,35.

In this work, we propose a new and all-optical concept without instabilities that targets the same goal, i.e., arbitrary expansion of the detection bandwidth, given a sufficiently low quantum decoherence. This quantum-expanded signal extraction concept is based on the optical parametric amplification process inside the interferometer, which allows an increase in the quantum fluctuations in the amplitude of the light by introducing quantum correlations, thereby providing a new knob to turn for reducing the QCRB. Due to the optical coupling between the cavities, the quantum uncertainty at high frequencies gets squeezed such that it compensates for the reduction in signal enhancement due to the cavity linewidth. At low frequencies, neither the signal nor the quantum noise changes, which maintains the existing sensitivity in an optimized state for observing the pre-merger stages of binary coalescence. Our approach is fully compatible with other enhancements to the detector design, such as injection of frequency-dependent squeezed light and variational readout17,36,37,38.

Placing an optical parametric amplifier inside the detector has been considered for other purposes before, i.e., for increasing the low-frequency39 or mid-frequency40,41,42 sensitivity, yet all-optical quantum expansion of bandwidth has never been proposed so far.

## Results

### Hamiltonian of the quantum expander

In the future, GW interferometers will operate with the signal port at the dark fringe. In this operating condition, all of the light power pumped into the interferometer is reflected towards the laser source. The only light that leaves the interferometer through the dark signal port corresponds to the signal caused by the dynamic change in the differential arm length, e.g., due to a GW. The zero-point fluctuation that enters the dark signal port defines the shot noise of the interferometer.

With respect to the quantum noise and the signal, the interferometer topology can be conceptually represented by a simpler system of two coupled cavities:43 the arm cavity with optical mode $$\widehat a$$, and the signal-extraction cavity, which is formed by the front mirror of the arm cavity and the signal-extraction mirror, with optical mode $$\widehat a_q$$; see Fig. 2a. The two modes are coupled through the partially reflective front mirror of the arm cavity, with a coupling frequency ωs, which depends on the front mirror’s reflectivity. For illustrative purposes, we limit the discussion to the interaction of these two modes, while the complete description should include the modes of the next longitudinal resonances of the arm cavity, separated by one free spectral range. In this approximation, the system can be described by the standard Hamiltonian for coupled harmonic oscillators: $$\widehat H/\hbar = \omega _0\widehat a^\dagger \widehat a + \omega _0\widehat a_q^\dagger \widehat a_q + \omega _s(\widehat a_q^\dagger \widehat a + \widehat a^\dagger \widehat a_q)$$. If the system is continuously excited at one of the normal frequencies, ω0 ± ωs, the excitation energy is equally split between the two modes $$\widehat a$$ and $$\widehat a_q$$. However, when one of the modes (e.g., $$\widehat a_q$$) is continuously excited at ω0, the complete energy is redistributed into the other mode (e.g., $$\widehat a$$). In this way, when mode $$\widehat a_q$$ is open to the environment and driven by the incoming zero-point fluctuation, its noise components are strongly suppressed at sideband frequencies ω0 ± Ω, Ω ωs, and all the energy at these frequencies goes into the arm cavity mode $$\widehat a$$. For large values of Ω, the noise becomes resonant inside the SE cavity as well, reaching its resonance maximum at ωs, as shown in Fig. 2b. It is this particular resonant structure of the coupled system that we take advantage of to enhance the sensitivity of the detector at high frequencies as follows.

We propose to place an optical parametric amplifier, e.g., a χ(2) nonlinear crystal, inside the SE cavity. The parametric process amplifies the fluctuations in one quadrature of the mode $$\widehat a_q$$ and suppress the fluctuations in its conjugate counterpart. Depending on the sideband frequency Ω, the amplification strength varies due to the presence of the coupled cavity structure. At frequencies around ω0, the excitation of mode $$\widehat a_q$$ is suppressed, so the parametric process is inefficient, and almost no squeezing is produced. At the same time, the SE cavity is resonant for higher frequencies Ω ~ ωs, so the crystal produces a high squeeze factor. The suppression of shot noise at frequencies 0  Ω ωs happens exactly at the same rate as the reduction in the signal amplification due to the detector bandwidth (see Fig. 2c). The two processes compensate each other, and the SNR remains constant; thus, the bandwidth is expanded (see Fig. 2d).

The quantum expansion effect can be demonstrated in more detail by formulating a complete Hamiltonian of the model two-mode system (for a general analysis of the system, see the Supplementary Material):

$$\widehat H = \widehat H_0 + \widehat H_{{\mathrm{int}}} + \widehat H_\gamma + \widehat H_x - F_{{\mathrm{GW}}}x$$
(1)
$$\widehat H_0 = \hbar \omega _0\widehat a^\dagger \widehat a + \hbar \omega _0\widehat a_q^\dagger \widehat a_q$$
(2)
$$\widehat H_{{\mathrm{int}}} = \hbar \omega _s\widehat a_q^\dagger \widehat a + \frac{1}{2}\hbar \kappa \beta e^{-2i\omega _0t}\widehat a_q^\dagger \widehat a_q^\dagger e^{i\phi } + h.c.$$
(3)
$$\widehat H_\gamma\,=\,i\hbar \sqrt {2\gamma } \mathop {\int }\limits_{ - \infty }^\infty \left( {\widehat a_q^\dagger (\omega )\widehat a_{{\mathrm{in}}}(\omega ) - \widehat a_{{\mathrm{in}}}^\dagger (\omega )\widehat a_q(\omega )} \right)d\omega$$
(4)
$$\widehat H_x\,=\,- \widehat F_{rp}\widehat x\,=\,- \hbar G_0\widehat a^\dagger \widehat a\widehat x$$
(5)

where $$\widehat a,\widehat a_q$$ are the arm cavity and SE cavity modes, respectively, and ω0 is their natural resonance frequency; $$\omega _s\,=\,c\sqrt {T_{{\mathrm{ITM}}}/(4L_{{\mathrm{SE}}}L_{{\mathrm{arm}}})}$$ is the coupling rate between two cavities, TITM is the transmission of the front mirror of the arm cavity; LSE, Larm are the lengths of the SE and arm cavity, respectively; γ = cTSE/(4LSE) is the coupling rate of the SE mode to the continuum of input modes $$\widehat a_{{\mathrm{in}}}$$; x is the displacement of the test mass partially in reaction to the GW tidal force FGW; the mirror motion x is coupled via the radiation-pressure force $$\widehat F_{{\mathrm{rp}}}$$ to the cavity mode with strength G0 = ω0/Larm; and κ is the coupling strength due to 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. Quantum expansion affects only the high-frequency sensitivity, which is dominated by shot noise. This justifies us to ignore in this simple model the effects of the quantum radiation pressure on the dynamics of the test mass, effectively assuming an infinite mass of the mirrors. The displacement of the mirrors x in this approximation is caused only by the GW strain h0 = x/Larm. In addition, note that the expression for the coupling frequency ωs only applies when ωsωFSR ≡ c/(2Larm) and has to be modified when the neighboring longitudinal resonances of the arm cavity are taken into account.

### Input-output relation

The light field in the coupled system can be expressed in terms of the input fields by solving the Hamiltonian above. We write the input-output relations for the amplitude and phase quadrature of the light (denoted by the upper indices (1, 2) correspondingly), representing the field leaving the detector $$\widehat a_{{\mathrm{out}}}^{(1)}$$, the field inside the SE cavity $$\widehat a_q^{(2)}$$, and the field inside the arm cavity $$\widehat a^{(1)}$$ in terms of the input noise fields $$\widehat a_{{\mathrm{in}}}^{(1,2)}$$:

$$\begin{array}{lcc}\widehat a_{{\mathrm{out}}}^{\left( 1 \right)}\left( {\mathrm{\Omega }} \right) = \widehat a_{{\mathrm{in}}}^{\left( 1 \right)}\left( {\mathrm{\Omega }} \right)\frac{{\left( {\gamma - \chi } \right){\mathrm{\Omega }} + i\left( {{\mathrm{\Omega }}^2 - \omega _s^2} \right)}}{{\left( {\gamma + \chi } \right){\mathrm{\Omega }} - i\left( {{\mathrm{\Omega }}^2 - \omega _s^2} \right)}} \\ + \, h_0\left( {\mathrm{\Omega }} \right)\frac{{2iG\sqrt \gamma \omega _s}}{{\left( {\gamma + \chi } \right){\mathrm{\Omega }} - i\left( {{\mathrm{\Omega }}^2 - \omega _s^2} \right)}}\end{array}$$
(6)
$$\begin{array}{lcc}\widehat a_q^{\left( 2 \right)}\left( {\mathrm{\Omega }} \right) = \widehat a_{{\mathrm{in}}}^{\left( 2 \right)}\left( {\mathrm{\Omega }} \right)\frac{{\sqrt {2\gamma } {\mathrm{\Omega }}}}{{\left( {\gamma + \chi } \right){\mathrm{\Omega }} - i\left( {{\mathrm{\Omega }}^2 - \omega _s^2} \right)}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \, h_0\left( {\mathrm{\Omega }} \right)\frac{{iG\omega _s}}{{\left( {\gamma + \chi } \right){\mathrm{\Omega }} - i\left( {{\mathrm{\Omega }}^2 - \omega _s^2} \right)}}\end{array}$$
(7)
$$\widehat a^{\left( 1 \right)}\left( {\mathrm{\Omega }} \right)\,=\,\widehat a_{{\mathrm{in}}}^{\left( 2 \right)}\left( {\mathrm{\Omega }} \right)\frac{{i\sqrt {2\gamma } \omega _s}}{{\left( {\gamma - \chi } \right){\mathrm{\Omega }}-i\left( {{\mathrm{\Omega }}^2 - \omega _s^2} \right)}}$$
(8)

where we linearized the system dynamics and introduced an effective parametric gain χ = κβ , effective signal coupling strength $$G\,=\,\sqrt {2P_cL_{{\mathrm{arm}}}\omega _0/(\hbar c)}$$ and optical power inside the arm cavity $$P_c = \hbar \omega _0\bar a^2$$, with $$\overline a$$ being an average amplitude of the mode $$\widehat a$$. The phase of the pump field was chosen such that the parametric process squeezes the signal quadrature inside the SE cavity, i.e., ϕ = π/2. Several features are provided by these equations. First, when we remove the crystal (i.e., χ = 0) in the typical operational range of GWOs Ω ωs, the input-output relation Eq. (6) reduces to the standard relation for a baseline GWO43,44, with the detection bandwidth given by $$\gamma _{{\mathrm{baseline}}} = \omega _s^2/\gamma = cT_{{\mathrm{ITM}}}/(T_{{\mathrm{SE}}}L_{{\mathrm{arm}}})$$. Second, the noise term in Eq. (7) is strongly suppressed at zero sideband frequency, as we described above in the example with two coupled modes: $$\widehat a_q^{(2)}(0) = h(0)G/\omega _s$$; therefore, virtually no squeezing is produced at low frequencies. The noise in the output in Eq. (6) at low frequencies is defined by the vacuum field reflected directly off the SE mirror. Third, when the sideband frequency matches the normal mode frequency, Ω = ωs, the signal mode takes the form $$\widehat a_{{\mathrm{out}}}^{(1)}(\omega _s) = \widehat a_{{\mathrm{in}}}^{(1)}(\gamma - \chi )/(\gamma + \chi ) + 2ih_0(\omega _s)G\sqrt \gamma /(\gamma + \chi )$$. This equation shows that for a parametric gain close to the threshold (χ → γ), the noise term becomes almost infinitely squeezed19, but the signal becomes deamplified by at most a factor of 2. Despite the signal deamplification, ideally, the SNR in this case can become infinite, as we show below by computing the sensitivity of the quantum-expanded observatory.

### Sensitivity spectrum

The noise spectral density of the GWO with a quantum expander, normalized to a GW strain amplitude h of unity, can be obtained from Eqs. (68) and further approximated in the typical regime where GWOs operate, γωs Ω, as follows:

$$\begin{array}{lcc}S_h\left( {\mathrm{\Omega }} \right) = \frac{{\hbar c}}{{8\omega _0L_{{\mathrm{arm}}}P_c}}\frac{{\left( {{\mathrm{\Omega }}^2 - \omega _s^2} \right)^2 + \left( {\gamma - \chi } \right)^2{\mathrm{\Omega }}^2}}{{\gamma \omega _s^2}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \approx \frac{{\hbar c}}{{8\omega _0L_{{\mathrm{arm}}}P_c}}\frac{{\gamma _q^2 + {\mathrm{\Omega }}^2}}{{\gamma \omega _s^2}}\left( {\gamma - \chi } \right)^2\end{array}$$
(9)

with the new detection bandwidth defined as $$\gamma _q = \omega _s^2/(\gamma - \chi )$$. Without the quantum expansion, χ = 0, the baseline sensitivity decreases with increasing frequency, limited by the detector’s bandwidth $$\gamma _{{\mathrm{baseline}}} = \omega _s^2/\gamma$$:

$$\begin{array}{lcc}S_h^{{\mathrm{baseline}}}\left( {\mathrm{\Omega }} \right) = \frac{{\hbar c}}{{8\omega _0L_{{\mathrm{arm}}}P_c}}\frac{{\left( {{\mathrm{\Omega }}^2 - \omega _s^2} \right)^2 + \gamma ^2{\mathrm{\Omega }}^2}}{{\gamma \omega _s^2}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \approx \frac{{\hbar c}}{{8\omega _0L_{{\mathrm{arm}}}P_c}}\frac{{\gamma _{{\mathrm{baseline}}}^2 + {\mathrm{\Omega }}^2}}{{\gamma \omega _s^2}}\gamma ^2\end{array}$$
(10)

The detection bandwidth γq can ideally be expanded infinitely (in the two-mode approximation) by a factor of γ/(γ − χ) → ∞ when the squeezing approaches the threshold point χ = γ. At this point, the sensitivity is given by

$$S_h\left( {\mathrm{\Omega }} \right)\,=\,\frac{{\hbar c}}{{8\omega _0L_{{\mathrm{arm}}}P_c}}\frac{{\omega _s^2}}{\gamma }$$
(11)

which is approximately frequency independent under Ω ωs as a result of the expanded bandwidth γq. In reality, even in the lossless case, the bandwidth is still limited by the next longitudinal resonance of the arm cavity and the detector’s reduced response when the detector’s arm length is comparable to the gravitational wavelength.

The effect of the quantum expander on a baseline GWO is shown in Fig. 3. To produce this figure, we compute the sensitivity based on the transfer matrix approach (as presented in the Supplementary Material), which better describes the high-frequency behavior in the longer detectors, i.e., when ωs ~ ωFSR. It also takes into account the effects of quantum radiation pressure noise, quantum decoherence (see the Discussion section for more details), the next free spectral ranges of the cavities and the response function of the detector to gravitational waves.

### Quantum Cramer-Rao bound

The sensitivity of any GWO is ultimately limited by its QCRB $$S_{h}^{\mathrm{QCRB}}(\Omega)$$15. The conditions for reaching this bound are that (i) the quantum radiation pressure noise is evaded, and (ii) the upper and lower optical sidebands generated by the GW are equal in amplitude15. Naturally, there is also a typical requirement of absence of optical decoherence and technical noises. The quantum expander configuration does not affect the QRPN and allows satisfying condition (i) at low frequencies by well-known back-action evading techniques (e.g., variational readout, see the Discussion below). We prove that condition (ii) is satisfied by directly computing the QCRB in the case of GW detectors, defined as follows:15

$$S_h^{\rm{QCRB}}(\Omega ) = \frac{{{\hbar ^2}}}{{2L_{\rm{arm}}^2{S_{\rm{FF}}}(\Omega )}} = \frac{{\hbar c}}{{4{\omega _0}{L_{\rm{arm}}}{P_c}}}\frac{1}{{{S_{\rm{aa}}}(\Omega )}}$$
(12)

where SFF(Ω) is the single-sided spectrum of the radiation-pressure force $$\widehat F_{{\mathrm{rp}}}$$ and Saa(Ω) is the noise spectrum of the arm cavity field, which one can compute from Eq. (9):

$$S_{aa}\left( {\mathrm{\Omega }} \right) = \frac{{2\gamma \omega _s^2}}{{\left( {\gamma - \chi } \right)^2{\mathrm{\Omega }}^2 + \left( {{\mathrm{\Omega }}^2 - \omega _s^2} \right)^2}}$$
(13)

Therefore, the limit on the sensitivity is given by the QCRB in the following form:

$$S_x^{{\mathrm{QCRB}}}\left( {\mathrm{\Omega }} \right) = \frac{{\hbar c}}{{4\omega _0L_{{\mathrm{arm}}}P_c}}\frac{{\left( {{\mathrm{\Omega }}^2 - \omega _s^2} \right)^2 + \left( {\gamma - \chi } \right)^2{\mathrm{\Omega }}^2}}{{2\gamma \omega _s^2}}$$
(14)

which is identical to Eq. (9). The sensitivity becomes unbounded (QCRB goes to zero) at the parametric threshold χ = γ at frequency Ω = ωs.

This calculation demonstrates that the quantum expander strongly reduces the QCRB at high frequencies compared to the baseline GWO and that the expanded detector does reach its lowered QCRB (in the case of the typical assumption of zero photon loss).

## Discussion

### Quantum decoherence

Non-classical light is sensitive to decoherence, i.e., to optical loss, which destroys the inherent quantum correlations45. Losses occur inside the detector, as well as on the readout and have multiple contributions. Any squeezed light application and QRPN suppression technique is limited by optical loss, and the proposed scheme is not an exception. The quantum expander relies on squeezing operation inside the interferometer to compensate for the decrease in the signal amplification due to the finite cavity linewidth. The higher the squeeze factor is, the more it is susceptible to optical loss. The effect of different readout losses is shown in Fig. 3. In the current generation of GWOs, the optical readout loss is on the order of 10%46, and in next observatory generation, 3–5% might be achievable47. With advanced techniques, which have been proposed48,49 but have yet to be explored experimentally, the readout loss could conceivably be reduced to as small as 0.5%. Introducing a nonlinear crystal inside the detector will increase the internal loss; however, there is no fundamental reason why such a crystal should have a bulk or surface absorption higher than that of the interferometer central beam splitter. Fortunately, the crystal needs to be placed at a position where the light power is comparatively low. There are other technical issues associated with placing a crystal inside the detector, such as the need for intense second-harmonic pump light, the requirements on the crystal size being larger than the beam diameter and the potential influence of the parametric amplification process on the interferometer’s control and stabilization systems. These issues are crucial for the practical detector design, yet rather technical in nature, so their detailed discussion can be found in the Supplementary Material. We believe that the added loss due to a crystal can be relatively small (see, e.g., the discussion in ref. 39) and the additional technical challenges can be overcome, and we consider our work to be a strong motivation for detailed experimental research and development.

### Combination with external-squeezing injection

All observatories of the current generation are already operating with external-squeezing injection. When the quantum expander is combined with external-squeezing injection, the overall squeeze factor at high frequencies increases further. This makes the requirements for low optical loss more stringent. The benefit from quantum expansion in combination with external squeezing depends not only on the amount of loss but also on the places where it occurs. There exists an optimal parametric gain in the quantum expander that maximizes the sensitivity by balancing the signal deamplification in the parametric process and squeeze factor39. Ultimately, every specific design of the detector has to be optimized with respect to optical parameters to be able to maximally benefit from quantum expansion. An in-depth discussion of the physical and technical aspects of combining the quantum expander with external-squeezing injection can be found in the Supplementary Material.

We envision the quantum expander to benefit future generations of GWOs when technological progress allows lowering of optical losses and detectors become longer and overall more sensitive, e.g., in the extensions of the third generation of observatories (Einstein Telescope and Cosmic Explorer) and beyond.

### Combination with other QND techniques

Quantum-expanded signal extraction will further reduce the shot noise at high frequencies without affecting the established improvement factor from external squeezing. The quantum noise at low frequencies (QRPN) will remain unchanged. This distinguishes our approach from other designs targeting the high-frequency sensitivity44,50,51. The QRPN can be suppressed independently using already developed approaches using frequency-dependent squeezing, variational readout or quantum non-demolition measurements17,36,37,38. In Fig. 3, we illustrate this point by combining the quantum expander with the variational readout technique, which requires an additional filter cavity (assumed lossless here) on the output of the GWO.

### Potential astrophysical applications

Currently, all GWOs maximize the SNR at frequencies around 100 Hz, where signals of compact binary inspirals can be observed. Merger and post-merger signals from binary neutron stars are expected to be at frequencies of ~1–3 kHz, where the sensitivity of the detectors significantly deteriorates due to the detection bandwidth. Quantum expansion of the detection bandwidth allows an increase in the sensitivity exactly at these frequencies, as shown in Fig. 3. To illustrate the potential of the quantum expander, we compute the SNR of a particular model of the post-merger signal52,53 (see the details in the Supplementary Materials) and demonstrate the improvement in detection rates on the histogram in Fig. 4: from a 9% chance of a single loud event surpassing the detection threshold after a full year of data acquisition in a baseline GWO, to ~76% and almost 100% for quantum-expanded detectors with 3 and 0.5% optical loss, respectively.

We envision the quantum expander as a potential tool allowing GW observatories to contribute to a better understanding of the physics of ultra-dense quantum matter in neutron stars and astrophysics of compact objects in general. We anticipate that other metrological54,55, as well as optomechanical56,57 experiments will benefit from our approach of using a coupled-cavity system with an internal parametric amplifier for bandwidth expansion.

## Materials and methods

The Hamiltonian of the quantum expander, presented in the Results section, is constructed based on the input-output relations for the noise and the signal58,59. For this derivation, we assume the effects of radiation-pressure noise and optical losses to be negligible, although the full model used to plot Fig. 3 includes both of these effects and is presented in the Supplementary Materials. The results of the derivation of the input-output relations, which we present in the full extent in the Supplementary Materials, allow us to obtain the shot-noise limited output field containing the noise an(Ω) and signal terms Xout(Ω):

$$a_{{\mathrm{out}}}^{(c)}\left({\mathrm{\Omega }}\right) = a_n\left( {\mathrm{\Omega }} \right) + X_{{\mathrm{out}}}\left( {\mathrm{\Omega }} \right)$$
(15)
$$\begin{array}{lcc}a_n\left({\mathrm{\Omega}}\right) = - \frac{{e^{2i\varphi }e^{2i\Omega \tau _{{\mathrm{SE}}}}(e^{2i\Omega \tau _{{\mathrm{arm}}}} - R_i) + e^{2q}(e^{2i\Omega \tau _{{\mathrm{arm}}}}R_i - 1)}}{{e^{2q}(e^{2i\Omega \tau _{{\mathrm{arm}}}}R_i - 1) + e^{2i\varphi }e^{2i\Omega \tau _{{\mathrm{SE}}}}(e^{2i\Omega \tau _{{\mathrm{arm}}}} - R_i)R_s}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,a_{{\mathrm{in}}}^{(c)}\left( {\mathrm{\Omega}} \right)\end{array}$$
(16)
$$\begin{array}{lcc}X_{{\mathrm{out}}}\left( {\mathrm{\Omega }} \right)\,=\,\frac{{2ik_p\bar ae^{i\varphi }e^{i\Omega \tau _{SE}}e^{i\Omega \tau _{{\mathrm{arm}}}}e^qT_iT_s}}{{e^{2q}(e^{2i\Omega \tau _{{\mathrm{arm}}}}R_i\,-\,1)\,+\,e^{2i\varphi }e^{2i\Omega \tau _{{\mathrm{SE}}}}(e^{2i\Omega \tau _{{\mathrm{arm}}}} - R_i)R_s}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\left( {\mathrm{\Omega}} \right)\end{array}$$
(17)

where $$R_{i,s}\,=\,\sqrt {R_{{\mathrm{ITM,SE}}}} ,T_{i,s} = \sqrt {T_{{\mathrm{ITM,SE}}}}$$ are, respectively, the amplitude reflectivity and transmissivity of the input test mirror and signal-extraction mirror; q is an amplification factor on the single pass through the crystal; τarm,SE = Larm,SE/c is the single trip time in arm cavity of length Larm and SE cavity of length LSE, with c being the speed of light; φ = π/2 is the tuning of the SE cavity with respect to the arm cavity; x = h0Larm is a small displacement of the end mirror due to the GW strain h0; $$\overline a$$ is the average amplitude of field inside the arm cavity and kp is the wave vector of the carrier light field.

These expressions can be simplified by making several approximations that are realistic for GWOs. We assume that Ωτarm 1 and ΩτSE 1, so $$e^{i\Omega \tau _{{\mathrm{arm,SE}}}}\,\approx\,1\,+\,i\Omega \tau _{{\mathrm{arm}},{\mathrm{SE}}}$$; Ti,s 1, so $$R_i \approx 1 - T_i^2/2\,=\,1\,-\,2\gamma _{{\mathrm{arm}}}\tau _{{\mathrm{arm}}}$$ and $$R_s\,\approx\,1\,-\,T_s^2/2\,=\,1\,-\,2\gamma \tau _{{\mathrm{arm}}}$$, where γarm, γ are the arm cavity and the signal-extraction cavity linewidth, respectively; and the single-pass optical gain is small: q 1, so eq ≈ 1 + q = 1 + χτSE, where χ is an effective parametric gain. In these assumptions, we obtain Eq. (6) for the output noise and signal. This allows us to construct the Hamiltonian presented in Eqs. (15).

We would like to point out the limits of this approximation: it is valid only until coupling and signal frequencies are much smaller than the free spectral range of the arm cavity: Ω, ωsωFSR ≡ c/2Larm. This condition sets a limit on the transmissivity of the ITM: $$T_i^2 \ll L_{{\mathrm{SE}}}/L_{{\mathrm{arm}}}$$, and restricts the applicability of the derived simplified equations to a detector with a relatively short arm length (e.g., Advanced LIGO). A longer detector (such as the baseline GWO chosen as a reference in Figs. 24) would require a more sophisticated expression with the higher longitudinal resonances of the arm cavity taken into account. The assumption of a small transmission of the SE mirror is often not valid in real designs, leading to additional contributions in the noise spectrum. We perform the full analysis, which avoids these limitations, in Section 5 of the Supplementary Materials.