Spectrally Multimode Squeezing Detection

• Spectrally multimode squeezing detection is defined as measuring squeezing distributed across numerous independent frequency modes produced via nonlinear processes like SPDC, FWM, and OPO.
• Detection methods include broadband correlation functions, photon-number-resolving techniques, pulse-shaped homodyne, and camera-enabled schemes that provide parallel, mode-resolved readout.
• Experimental results demonstrate mode counts from a few tens up to hundreds, emphasizing the need for precise mode matching and basis selection to fully reveal the underlying quantum correlations.

Spectrally multimode squeezing detection is the measurement and reconstruction of squeezing that is distributed over many orthogonal frequency modes within a single optical field rather than confined to one narrowband mode. In practice, the relevant states arise from broadband spontaneous parametric down-conversion, four-wave mixing, optical parametric oscillators, micro-resonators, and related nonlinear platforms, and are naturally described in Schmidt or supermode bases rather than by a single quadrature variance. The subject has evolved from loss-insensitive broadband correlation-function measurements and direct photon-number-resolving detection of ultrabroadband squeezed vacuum to pulse-shaped homodyne readout, camera-enabled massively parallel homodyne, optical-parametric-amplification-based direct detection, and broadband up-conversion followed by visible-light camera spectrometry [(Christ et al., 2010); (Wakui et al., 2013); (Yoon et al., 9 Jun 2026); (Kalash et al., 6 Aug 2025); (Presutti et al., 2024)].

1. Modal structure of spectrally multimode squeezed light

The central structural fact is that broadband squeezing is typically multimode even when it occupies a single spatial beam. For collinear type-0 SPDC with a symmetric joint-spectral distribution, the squeezing operator can be decomposed into independent broadband spectral modes,

$$\hat{\bm S}=\exp\!\left[\sum_k r_k(\hat A_k^{\dagger 2}-\hat A_k^2)\right],\qquad |\mathbf r\rangle=\hat{\bm S}|0\rangle,$$

with one squeezing parameter $r_k$ per spectral mode. In the underlying joint-spectral representation,

$$\frac{i}{\hbar}A^* f^*(\omega_s,\omega_i)=\sum_k r_k\,\phi_k(\omega_s)\phi_k(\omega_i),$$

so the physical content of the multimode state is encoded in the mode functions $\phi_k(\omega)$ and the modal weights $r_k$ rather than in a single optical frequency channel (Wakui et al., 2013).

For twin-beam parametric processes, the same structure appears as a Schmidt-mode factorization,

$$\hat{U}_{OPA}=\exp\!\left[\sum_k r_k \hat{A}_k^\dagger \hat{B}_k^\dagger - h.c.\right] =\bigotimes_k \hat{S}^{ab}_k(-r_k), \qquad r_k=B\lambda_k,\ \sum_k \lambda_k^2=1.$$

This form makes explicit that broadband PDC or FWM generates a tensor product of independent squeezers or two-mode squeezers indexed by $k$, with an overall gain $B$ and a normalized modal distribution $\lambda_k$ (Christ et al., 2010).

In more general Gaussian frequency-domain treatments, the full transformation is written as

$$\hat a_\text{out} = C \hat a_\text{in} + S \hat a_\text{in}^\dagger,$$

and Bloch–Messiah decomposition produces orthogonal supermodes with independent squeezing parameters $r_k$0. In a broadband visible-light demonstration based on a degenerate optical parametric amplifier followed by adiabatic frequency conversion, this decomposition was used to predict about 600 supermodes with substantial occupation over more than 50 THz (Presutti et al., 2024).

This modal viewpoint is operationally decisive. Spectrally multimode squeezing detection is therefore not only a noise-measurement problem but also a basis-selection problem: a detector must either project onto the correct supermodes, infer them from correlation data, or map their quadrature fluctuations onto directly measurable intensities.

2. Broadband observables and loss-robust inference

The earliest general strategy was to avoid full multimode homodyne tomography and instead exploit broadband normalized intensity correlations. With detector gates much longer than the pulse duration, the measured observable is the total photon number summed over all occupied spectral modes,

$r_k$1

In the low-gain regime for a multimode twin-beam state,

$r_k$2

so the second-order broadband autocorrelation directly yields the effective Schmidt number $r_k$3. The broadband cross-correlation $r_k$4 then yields the overall gain $r_k$5, with $r_k$6 in the low-gain limit. Because these are normalized correlation functions, the method is described as loss insensitive and experimentally compatible with standard avalanche photodiodes and large detection windows (Christ et al., 2010).

Direct photon-number-resolving detection subsequently provided a more explicit route to the same multimode structure. In an ultrabroadband telecom experiment, a collinear type-0 SPDC source in PPKTP was pumped by a 767.5 nm field generated from a 1535 nm pulsed laser, producing a squeezed vacuum with about 150 nm FWHM around 1.5 $r_k$7m and covering the S, C, and L bands. A superconducting transition-edge sensor with about $r_k$8 efficiency at 1535 nm and near-flat response over much of the telecom band measured the photon-number distribution of the entire ultrabroadband pulse (Wakui et al., 2013).

The reconstructed photon-number distributions showed strong even-odd oscillations after loss correction, which is the standard squeezed-vacuum hallmark. The same dataset supported Klyshko nonclassicality violations for even $r_k$9, notably $$\frac{i}{\hbar}A^* f^*(\omega_s,\omega_i)=\sum_k r_k\,\phi_k(\omega_s)\phi_k(\omega_i),$$0, and yielded factorial-moment correlation functions

$$\frac{i}{\hbar}A^* f^*(\omega_s,\omega_i)=\sum_k r_k\,\phi_k(\omega_s)\phi_k(\omega_i),$$1

The measured $$\frac{i}{\hbar}A^* f^*(\omega_s,\omega_i)=\sum_k r_k\,\phi_k(\omega_s)\phi_k(\omega_i),$$2 and $$\frac{i}{\hbar}A^* f^*(\omega_s,\omega_i)=\sum_k r_k\,\phi_k(\omega_s)\phi_k(\omega_i),$$3 deviated from the single-mode squeezed-vacuum relations and, under the Christ-et-al. analysis with thermal mode weights $$\frac{i}{\hbar}A^* f^*(\omega_s,\omega_i)=\sum_k r_k\,\phi_k(\omega_s)\phi_k(\omega_i),$$4, gave

$$\frac{i}{\hbar}A^* f^*(\omega_s,\omega_i)=\sum_k r_k\,\phi_k(\omega_s)\phi_k(\omega_i),$$5

The corresponding effective mode number was

$$\frac{i}{\hbar}A^* f^*(\omega_s,\omega_i)=\sum_k r_k\,\phi_k(\omega_s)\phi_k(\omega_i),$$6

with the uncertainty in $$\frac{i}{\hbar}A^* f^*(\omega_s,\omega_i)=\sum_k r_k\,\phi_k(\omega_s)\phi_k(\omega_i),$$7 implying a range of roughly $$\frac{i}{\hbar}A^* f^*(\omega_s,\omega_i)=\sum_k r_k\,\phi_k(\omega_s)\phi_k(\omega_i),$$8. The experimentally reported “several tens” of squeezers therefore meant on the order of two dozen independent broadband squeezing modes, with uncertainty extending from the mid-teens to nearly a hundred (Wakui et al., 2013).

These two strands—broadband normalized correlations and direct photon-number statistics—established the basic fact that multimode squeezing can be quantified without scanning a local oscillator through every spectral component. They also introduced a durable methodological theme: the full spectral Hilbert space can be accessed indirectly through integrated observables, provided the modal model linking those observables to the underlying supermodes is explicit.

3. Homodyne readout: from mode-selective projection to simultaneous many-mode measurement

Balanced homodyne detection remains the canonical quadrature measurement. In the strong-local-oscillator limit,

$$\frac{i}{\hbar}A^* f^*(\omega_s,\omega_i)=\sum_k r_k\,\phi_k(\omega_s)\phi_k(\omega_i),$$9

with

$\phi_k(\omega)$0

For spectrally multimode states, the critical additional requirement is that the local oscillator select the desired spectral supermode. In practice, this is achieved by spectral shaping of the LO pulse (Kouadou et al., 6 Nov 2025).

A direct realization of this idea was reported for a single-pass type-0 PPKTP waveguide pumped by 397.5 nm pulses derived from a 22-fs, 795 nm Ti:sapphire laser running at 156 MHz. By shaping the LO into Hermite–Gauss spectral supermodes, the experiment individually measured 21 squeezed spectral modes up to HG$\phi_k(\omega)$1. The first mode had a width of about 18 nm, the LO shaping bandwidth was about 40 nm FWHM, and the measured squeezing at a 50 MHz sideband frequency decreased with mode order, from HG$\phi_k(\omega)$2 at $\phi_k(\omega)$3 dB squeezing and $\phi_k(\omega)$4 dB antisqueezing to HG$\phi_k(\omega)$5 at $\phi_k(\omega)$6 dB and $\phi_k(\omega)$7 dB. The same source was also measured pulse by pulse with a wideband detector and fast oscilloscope, reproducing the multimode structure up to HG$\phi_k(\omega)$8. In an 8-band “frexel” basis of 5 nm per band, the reconstructed covariance matrix revealed strong anti-diagonal correlations, and 114 of the 127 possible bipartitions were found entangled by the PPT criterion (Kouadou et al., 2022).

Detector bandwidth then became a separate bottleneck. Wideband room-temperature homodyne detectors based on the OPA856 transimpedance architecture were engineered to preserve the spectral multimode properties of pulsed squeezed light at repetition rates up to 150 MHz. In the telecom implementation, the source was centered at 1560 nm with 64 fs pulses at 100 MHz; the detector achieved 15 dB shot-noise clearance over an 80 MHz range at $\phi_k(\omega)$9 mW, a common-mode rejection ratio of 61 dB, and an effective electronic quantum efficiency of $r_k$0. Using 10 Gsamples/s oscilloscope acquisition over 1 ms, the experiment extracted pulse-resolved quadratures for five spectral modes from HG$r_k$1 to HG$r_k$2, showing that homodyne detection can be simultaneously pulse-resolved and mode-resolved in the telecom regime (Kouadou et al., 6 Nov 2025).

The most explicit answer to the homodyne scalability problem was camera-enabled homodyne detection. In that architecture, a broadband LO and the multimode quantum field interfere on a 50:50 beam splitter, and the two outputs are spectrally dispersed onto a 1D CCD array so that each frequency-bin mode is mapped to a pixel pair. The quadrature of each mode is extracted from the gain-balanced, weighted, normalized difference of the corresponding pixel counts. Using a Teledyne BLAZE 400HRX CCD camera with 95% quantum efficiency, the experiment measured 60 optical frequency-bin modes simultaneously with less than 2 nW LO power per mode, a six-orders-of-magnitude reduction relative to conventional homodyne implementations. The detector was shot-noise limited, with an example linearity of $r_k$3 and $r_k$4, clearance between 24 dB and 28 dB across all 60 modes, and negligible crosstalk as evidenced by an almost identity vacuum covariance matrix. Applied to a synchronously pumped optical parametric oscillator, it directly observed squeezing and entanglement in 60 optical modes, with all 30 symmetric/antisymmetric mode pairs satisfying the Duan inseparability criterion; it also verified GHZ- and cluster-type states using nullifier variances and van Loock–Furusawa tests (Yoon et al., 9 Jun 2026).

The main homodyne-based architectures can be summarized as follows.

Detection paradigm	Primary readout	Representative demonstrated scale
Broadband correlation / photon counting	$r_k$5, $r_k$6, Klyshko tests	Effective $r_k$7, roughly $r_k$8 modes
Pulse-shaped balanced homodyne	Supermode-projected quadrature variance	21 squeezed spectral modes; pulse-resolved up to HG$r_k$9
Wideband pulse-by-pulse homodyne	Per-pulse quadratures	Repetition rates up to 150 MHz; five telecom modes HG$$\hat{U}_{OPA}=\exp\!\left[\sum_k r_k \hat{A}_k^\dagger \hat{B}_k^\dagger - h.c.\right] =\bigotimes_k \hat{S}^{ab}_k(-r_k), \qquad r_k=B\lambda_k,\ \sum_k \lambda_k^2=1.$$0–HG$$\hat{U}_{OPA}=\exp\!\left[\sum_k r_k \hat{A}_k^\dagger \hat{B}_k^\dagger - h.c.\right] =\bigotimes_k \hat{S}^{ab}_k(-r_k), \qquad r_k=B\lambda_k,\ \sum_k \lambda_k^2=1.$$1
Camera-enabled homodyne	Pixel-pair quadratures and covariance matrices	60 optical modes simultaneously; 24–28 dB clearance

Taken together, these experiments show a transition from sequential, mode-matched homodyne detection toward parallel spectral readout in which the detector itself supplies the multiplexing.

4. All-optical amplification and camera-based broadband direct detection

A distinct detection philosophy is to convert quadrature fluctuations into intensities before any electronic readout. In phase-sensitive optical parametric amplification, the output mean photon number depends on the input quadrature variance as

$$\hat{U}_{OPA}=\exp\!\left[\sum_k r_k \hat{A}_k^\dagger \hat{B}_k^\dagger - h.c.\right] =\bigotimes_k \hat{S}^{ab}_k(-r_k), \qquad r_k=B\lambda_k,\ \sum_k \lambda_k^2=1.$$2

and in the high-gain regime this reduces to a direct mapping of quadrature variance to output intensity. A spatial multimode proof-of-principle implemented this idea using high-gain PDC followed by a second parametric amplification stage and camera-based covariance decomposition, obtaining the eight strongest spatial modes with highest squeezing and anti-squeezing values of $$\hat{U}_{OPA}=\exp\!\left[\sum_k r_k \hat{A}_k^\dagger \hat{B}_k^\dagger - h.c.\right] =\bigotimes_k \hat{S}^{ab}_k(-r_k), \qquad r_k=B\lambda_k,\ \sum_k \lambda_k^2=1.$$3 dB and $$\hat{U}_{OPA}=\exp\!\left[\sum_k r_k \hat{A}_k^\dagger \hat{B}_k^\dagger - h.c.\right] =\bigotimes_k \hat{S}^{ab}_k(-r_k), \qquad r_k=B\lambda_k,\ \sum_k \lambda_k^2=1.$$4 dB, respectively (Barakat et al., 2024).

The spectral-domain extension was then reported as the first experimental demonstration of spectrally multimode squeezing detection using OPA. Broadband squeezed vacuum was generated by collinear degenerate type-I PDC in a 3 mm BiBO crystal pumped by an 18 ps, 1 kHz laser at 354.67 nm with pulse energy about 70 $$\hat{U}_{OPA}=\exp\!\left[\sum_k r_k \hat{A}_k^\dagger \hat{B}_k^\dagger - h.c.\right] =\bigotimes_k \hat{S}^{ab}_k(-r_k), \qquad r_k=B\lambda_k,\ \sum_k \lambda_k^2=1.$$5J. At the lower squeezer gain, the experiment reported $$\hat{U}_{OPA}=\exp\!\left[\sum_k r_k \hat{A}_k^\dagger \hat{B}_k^\dagger - h.c.\right] =\bigotimes_k \hat{S}^{ab}_k(-r_k), \qquad r_k=B\lambda_k,\ \sum_k \lambda_k^2=1.$$6, more than 500 squeezed spectral modes over 37 nm, and an expected fundamental-mode squeezing of about $$\hat{U}_{OPA}=\exp\!\left[\sum_k r_k \hat{A}_k^\dagger \hat{B}_k^\dagger - h.c.\right] =\bigotimes_k \hat{S}^{ab}_k(-r_k), \qquad r_k=B\lambda_k,\ \sum_k \lambda_k^2=1.$$7 dB. The same crystal, pumped more strongly, acted as the multimode OPA detector with gain $$\hat{U}_{OPA}=\exp\!\left[\sum_k r_k \hat{A}_k^\dagger \hat{B}_k^\dagger - h.c.\right] =\bigotimes_k \hat{S}^{ab}_k(-r_k), \qquad r_k=B\lambda_k,\ \sum_k \lambda_k^2=1.$$8; to maintain phase stability and suppress dispersion, the analysis was restricted to an 8 nm window around degeneracy, which still contained more than 60 spectral modes. After amplification, the light was coupled into a single-mode fiber, spectrally resolved by a diffraction grating, and imaged onto an sCMOS camera with 13 pm resolution (Kalash et al., 6 Aug 2025).

In that experiment, the dark-fringe spectrum remained below the amplified-vacuum spectrum across the entire measured window, and the normalized integrated intensity varied from about $$\hat{U}_{OPA}=\exp\!\left[\sum_k r_k \hat{A}_k^\dagger \hat{B}_k^\dagger - h.c.\right] =\bigotimes_k \hat{S}^{ab}_k(-r_k), \qquad r_k=B\lambda_k,\ \sum_k \lambda_k^2=1.$$9 dB to $k$0 dB with pump phase. Because direct mode sorting was unavailable, the modal decomposition was reconstructed from the spectral intensity covariance,

$k$1

followed by coherent-mode decomposition and an overlap calculation between squeezer and amplifier modes. The final reconstructed squeezing stayed nearly constant from about $k$2 dB to $k$3 dB across modes up to roughly the 60th mode, establishing genuinely simultaneous spectral multimode readout (Kalash et al., 6 Aug 2025).

A different route to broadband direct spectral detection uses frequency conversion rather than parametric preamplification. In a visible-light system based on a 4 cm MgO-doped lithium niobate waveguide degenerate optical parametric amplifier followed by a 3 cm poled KTP adiabatic frequency-conversion stage, infrared squeezing near 1550 nm was upconverted to around 620 nm and measured with an EMCCD camera-based spectrometer. The conversion bandwidth exceeded 45 THz, the estimated conversion efficiency was 92.5%, the EMCCD had about 95% quantum efficiency at 620 nm, and the spectrometer achieved about 0.12 nm per pixel over a 60 nm window. The experiment simultaneously squeezed and detected more than 400 modes, reported a lower bound $k$4 in the infrared and about 433 modes in the visible, and observed approximately 700 visible photons per shot. By shaping the conversion pump spectrum $k$5, it also demonstrated programmable spectral correlations, albeit with partial rather than full programmability because the available pump bandwidth was less than 5 THz (Presutti et al., 2024).

The conceptual basis of these direct-detection schemes is consistent with multimode phase-sensitive amplification as a loss-tolerant premeasurement stage. In the multimode PSA analysis of seeded second-order amplification, phase-sensitive amplification of squeezed states was shown to help overcome detection loss and noise, and the authors explicitly drew a time/frequency-domain analogy to the near-field strategy that optimizes overlap with coupled signal-idler eigenmodes (Frascella et al., 2021).

5. Hidden squeezing, complex covariance, and basis mismatch

A persistent difficulty in spectrally multimode squeezing detection is that not all squeezing is visible to standard homodyne readout. In continuous fields with frequency-dependent correlations, homodyne detection measures

$k$6

so it only accesses the real part of the AM–PM covariance. When the off-diagonal covariance element is complex, the imaginary part encodes unequal-time or temporally phased correlations that ordinary homodyne is blind to. In cavity optomechanics, this “complex squeezing” appears naturally in ponderomotive squeezing because the mechanical susceptibility is generally complex; near mechanical resonance, the AM–PM correlation can become purely imaginary, meaning that squeezing exists but is hidden from standard homodyne. Synodyne detection addresses this by using a two-tone local oscillator,

$k$7

which accesses the full complex covariance and, in the force-sensing problem studied there, allows the back-action contribution to be cancelled so that sensitivity is limited only by the mechanical oscillator’s thermal occupation (Buchmann et al., 2016).

The same hidden-versus-detectable distinction reappears in integrated nonlinear photonics. For synchronously pumped Si and SiN micro-resonators, the multimode Gaussian output is diagonalized by frequency-dependent “morphing supermodes” defined through the analytical Bloch–Messiah decomposition

$k$8

The optimal squeezing at analysis frequency $k$9 is given by the singular values in $B$0, but standard homodyne only measures

$B$1

with a real LO profile $B$2. Because the supermodes in $B$3 are generally complex and frequency dependent, part of the optimal squeezing is hidden: it is present in the state but not fully accessible to a real, frequency-independent LO. Pump-spectrum engineering, for example $B$4, and dispersion engineering toward $B$5 were proposed as ways to reduce the complexity of the supermodes and make more of the squeezing detectable with standard homodyne (Gouzien et al., 2022).

Basis mismatch also limits more conventional broadband PDC detection. In a type-II PDC supermode analysis, ideal homodyne with an LO shaped to the signal and idler supermodes retrieves the full multimode squeezing, while an unshaped LO based on the pump spectrum measures only one effective mode,

$B$6

For one representative case with $B$7 and $B$8, the unshaped detection captured only about $B$9. This establishes a recurrent point of confusion in the literature: weakly observed squeezing need not imply weak generation; it can reflect incomplete projection onto the true supermode basis (Hosseinidehaj et al., 2017).

The notion of “hidden squeezing” is therefore not a semantic refinement but a measurement-theoretic distinction. Spectrally structured quantum states may contain stronger squeezing than standard homodyne, a fixed LO, or an inadequately engineered measurement basis can reveal.

6. Networked operation, emerging sources, and entanglement-level interpretation

Spectrally multimode squeezing detection has moved beyond laboratory-local characterization into distributed and networked settings. In a deployed-fiber demonstration, a PPLN ridge waveguide pumped by a 771 nm CW field generated broadband SPDC from which two telecom modes, C43 at 1542.94 nm and C45 at 1541.35 nm, were selected. Each squeezed mode was frequency multiplexed with a local oscillator at C44, a 1310 nm RF-over-fiber reference, and conventional telecom traffic. Joint homodyne detection after separate transmission produced $\lambda_k$0 dB squeezing and $\lambda_k$1 dB anti-squeezing after two 5-km spools with coexistence, $\lambda_k$2 dB with RFoF-synchronized post-processing, and $\lambda_k$3 dB and $\lambda_k$4 dB over deployed campus fibers of about 250 m and 1.2 km. The reported conclusion was that the conventional traffic caused no statistically significant degradation, beyond multiplexing loss, and that synchronized post-processed homodyne can recover joint squeezing at separate locations (Chapman et al., 2023).

New source classes further broaden the detection problem. In a second-order multimode cavity with cascaded three-wave mixing and $\lambda_k$5-factor engineering, bright amplitude squeezing exceeding 10 dB below shot noise was predicted and analyzed for discrete frequency modes above threshold, together with multimode squeezing in several selected modes and long-range amplitude-noise correlations. For some mode pairs, the normalized sum or difference noise lay more than 20 dB below the uncorrelated twin-beam level. Because the relevant observables are output amplitude-noise spectra and mode-resolved intensity correlations in a synthetic frequency dimension, the work suggests that future spectrally multimode detection will increasingly have to address bright, discrete, and strongly coupled frequency-bin structures rather than only vacuum supermodes (Pontula et al., 2024).

Pure-Kerr parametrically driven cavity solitons provide an even more strongly frequency-dependent case. There the output transfer matrix is decomposed by analytical Bloch–Messiah methods into supermodes whose identity changes with the analysis frequency $\lambda_k$6. Below threshold, the system exhibits single-mode squeezing at the central mode and two-mode squeezing in spectral pairs; above threshold, it supports “quantum dispersive waves,” interpreted as the quantum analog of soliton Cherenkov radiation, with squeezing localized near zero-dispersion crossings. For experimentally routine parameters and 99% overcoupling, the reported squeezing reached up to 20 dB and was limited mainly by intrinsic loss and coupling losses. This suggests that future detectors will need to track not only spectral mode index but also the analysis-frequency dependence of the optimal measurement basis (Mendez et al., 5 May 2026).

At the covariance-matrix level, multimode squeezing detection also functions as entanglement detection. For an $\lambda_k$7-mode continuous-variable state with covariance matrix $\lambda_k$8, the bosonic multi-mode squeezing coefficient

$\lambda_k$9

satisfies $$\hat a_\text{out} = C \hat a_\text{in} + S \hat a_\text{in}^\dagger,$$0 for separable states, so $$\hat a_\text{out} = C \hat a_\text{in} + S \hat a_\text{in}^\dagger,$$1 certifies inseparability. The criterion is mode agnostic: the modes may be spectral, spatial, temporal, or otherwise labeled, provided the covariance matrix is experimentally accessible (Gessner et al., 2017).

The resulting landscape is methodologically plural rather than convergent toward a single detector design. Broadband correlation functions remain attractive because they are loss insensitive and cheap; direct photon-number-resolving detectors reveal multimode structure through nonclassical statistics; shaped and camera-enabled homodyne provide basis-selective or massively parallel quadrature access; OPA-based schemes map quadratures into intensities and relax mode-matching constraints; up-conversion plus visible-light cameras opens very large spectral Hilbert spaces to room-temperature imaging hardware. The common technical problem across these approaches is the same: spectrally multimode squeezing is present in a structured covariance or modal decomposition, and successful detection requires a measurement basis that is broad enough, efficient enough, and scalable enough to expose that structure rather than average it away.