LIGO's quantum response to squeezed states

LIGO’s quantum response to squeezed states

L. McCuller et al.

Phys. Rev. D 104, 062006 – Published 13 September 2021

Abstract

Gravitational wave interferometers achieve their profound sensitivity by combining a Michelson interferometer with optical cavities, suspended masses, and now, squeezed quantum states of light. These states modify the measurement process of the LIGO, VIRGO and GEO600 interferometers to reduce the quantum noise that masks astrophysical signals; thus, improvements to squeezing are essential to further expand our gravitational view of the Universe. Further reducing quantum noise will require both lowering decoherence from losses as well more sophisticated manipulations to counter the quantum back-action from radiation pressure. Both tasks require fully understanding the physical interactions between squeezed light and the many components of km-scale interferometers. To this end, data from both LIGO observatories in observing run three are expressed using frequency-dependent metrics to analyze each detector’s quantum response to squeezed states. The response metrics are derived and used to concisely describe physical mechanisms behind squeezing’s simultaneous interaction with transverse-mode selective optical cavities and the quantum radiation pressure noise of suspended mirrors. These metrics and related analysis are broadly applicable for cavity-enhanced optomechanics experiments that incorporate external squeezing, and—for the first time—give physical descriptions of every feature so far observed in the quantum noise of the LIGO detectors.

Received 26 May 2021
Accepted 19 July 2021

DOI:https://doi.org/10.1103/PhysRevD.104.062006

Physics Subject Headings (PhySH)

Optical interferometry Optomechanics Quantum measurements Quantum optics Squeezing of quantum noise

Gravitational wave detectors

Gravitation, Cosmology & AstrophysicsAtomic, Molecular & Optical

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Vol. 104, Iss. 6 — 15 September 2021

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Figure 1
This simplified diagram of the interferometer layout shows the propagation of the source laser (solid red) and squeezed beam (dashed burgundy). At (a), the squeezed beam is sourced from a parametric amplifier cavity and circulated to the interferometer with a Faraday isolator. At (b), the squeezing field reflects from the interferometer. Depending on the frequency and transverse beam profile, the states partially transit the interferometer cavities, but also partially reflect promptly. The squeezing that enters the interferometer symmetrically is beam split inside the signal recycling cavity, coherently resonates in both arms, and recombines again at the beam splitter, effectively experiencing the two branches as a single linear coupled cavity. Injected at a different port, the red laser field carries substantial laser power and is symmetrically split to pump the arm cavities. Differential length signals are sourced by modulating the circulating pump field, creating a phase-quadrature field that resonates in the same effective linear cavity as the squeezing. The signal is emitted at (b), stacking with the squeezing that reflects at (b). The transverse beam profile (mode) of the signal and squeezing is then selected using the output mode cleaning cavity at (c). Ultimately, the signal and noise are read as time series in photodetectors at (d). This effect of coherent interference between prompt and cavity-circulated squeezing from this sequence is formulated, measured, and analyzed in the following sections.
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Figure 2
This figure plots the total quantum and classical noise measured in the LIGO detectors in displacement amplitude spectral density units. The black trace plots a reference measurement of the total noise without injected squeezing at 0.25 Hz resolution over 1.5 hr integration for LLO and 1.1 hr for LHO. The orange shot-noise measurement shows the displacement calibration, $\sqrt{G (Ω)}$ , in amplitude density units. Subtracting the shot-noise level from the reference yields the gray data points, which have been rebinned using a median statistic applied after the subtraction and with a logarithmic bin spacing. The subtraction primarily shows the classical noise but also contains QRPN. Multiple measurements are taken at varied squeezing angles, with 5 of 12 plotted for Livingston (LLO) and 5 of 34 plotted for Hanford (LHO), using the same median rebinning method as the gray subtraction. The variation in the data error bars results from the binning span of each data point, $Δ F$ , and the measurement integration time, $Δ T$ . The error of the measured spectra is relative to the total noise and proportional to $1 / \sqrt{Δ T Δ F}$ . The squeezing angle of $- 3.9 °$ and 1.4° data sets at LLO used $\sim 1 Hr$ integration, and the remainder used 15 min each. The squeezing angle 4.5° data set at LHO used $\sim 1 Hr$ integration, while all others use 2 minutes each. The squeezing level $e^{\pm 2 r}$ is constant over all angles, but different between the two sites. This accounts for the difference in the yellow, $\sim 30 °$ , data set at each site.
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Figure 3
This figure shows the data of Fig. 2 processed as per Sec. 3 for each LIGO site, with 9 of 12 shown for LLO and 9 of 34 for LHO. The processing subtracts away the classical noise determined from the unsqueezed reference data set. The top panels show the relative noise change $Q_{k} (Ω_{i})$ of Eq. (22) computed using $Γ_{k} (Ω_{i})$ from the exact interferometer model of Appendix pp5 using the parameters of Table 1. The top panel includes dots with error bars for the processed data and lines for the best-fit $Q_{k} (Ω_{i})$ . The middle panel shows the best-fit frequency-dependent loss as data points, with error bars propagated through the fit. For LLO, two sets of loss data points are shown, corresponding to interferometer models with different readout angles $ζ$ . The loss plots also show $1 - η (Ω)$ as computed from the exact matrix model, along with a phenomenological fit against the model of Eq. (85) of Sec. 6. The phenomenological fit assumes frequency-independent losses from the input and output squeezing path with a frequency-dependent addition attributed to transverse mismatch. The bottom panels show the frequency-dependent fit to the observed squeezing angle $θ_{k} (Ω_{i})$ , using the convention of $θ (2 π \cdot 3 kHz) = 0$ . It also plots $θ (Ω)$ as computed using the exact matrix model. For the LLO data, the $ζ = 0 °$ model is typically assumed for Michelson-like interferometers such as LIGO; however, the model at that readout angle implies losses at low frequencies that are not favored by the $η (Ω)$ models explored in this paper. Alternatively, the $ζ \approx - 13 °$ model is consistent with both the fitted losses and the fitted squeezing angles.
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Figure 4
The two-photon transformation matrices experienced by squeezing through the sequence of Fig. 1. The effective linear coupled cavity, including the optomechanical effect of radiation pressure, is collected and computed into the transformation $H_{R}$ . The middle cavity is the signal recycling cavity and the rightmost cavity represents the coherent combination of both arms. Each cavity adds losses from each mirror. For simplicity, these are collected into round-trip cavity loss contributions, $Λ_{R, s}$ , and $Λ_{R, s}$ that inject standard optical vacuum into the cavities, circulating and transforming into the loss terms $T_{R, μ}$ while lowering the efficiency $η_{R}$ . Transformations of the squeezing at the input and output are included with the terms, $H_{I}$ , $H_{O}$ and any additive vacuum contributions, $T_{I, μ}$ , $T_{O, μ}$ .
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Figure 5
Propagation of the squeezed beam and unsqueezed higher-order transverse beam modes from source to readout. The stages (a)–(d) correspond to the components in Fig. 1, depicting the matrix math of Eqs. (73)–(76). $ϒ_{I}$ represents the transverse mismatch loss of the squeezing to interferometer, and $ϒ_{O}$ is the mismatch of the squeezing to readout via the output mode cleaner (OMC). These mismatches cause beam-splitter-like mixing between the LG0 and LG1+ modes through Eq. (72). $ψ_{I}$ , $ψ_{O}$ , $ψ_{G}$ are unmeasured phasing terms of the interferometer and output mismatch and of the Gouy-phase advance from the beam propagating to the output mode cleaner.
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