### Fabrication of density-modulated membranes

We use the comfortable clamping^{6,49,50,51,52,53} approach to appreciate ultrahigh mechanical high quality elements. Our membrane design is impressed by these pioneered in ref. ^{7}, however we use a special materials for the nanopillars and a special fabrication course of (see Supplementary Info for extra particulars). We fabricated density-modulated PNC membranes by patterning amorphous silicon (aSi) nanopillars on a excessive facet ratio Si_{3}N_{4} membrane. In our PNC membranes, we fabricated pillars with diameters *d*_{pil} = 300–800 nm, thickness of about 600 nm and nearest-neighbour distances *a*_{pil} = 1.0–2.0 μm. Amorphous silicon is grown with plasma-enhanced chemical vapour deposition (PECVD) at a temperature of 300 °C. Electron-beam lithography (FOx16 electron-beam resist) and dry etching (utilizing a plasma of SF_{6} and C_{4}F_{8}) are used to sample pillar arrays in aSi. Dry etching is stopped on a 6-nm layer of HfO_{2} (hafnium oxide) grown with atomic layer deposition (ALD) immediately on prime of Si_{3}N_{4}. HfO_{2} is used as an etch-stop layer as a result of it’s fairly proof against hydrofluoric acid (HF) etching, and the undercut created on the pillar base within the following course of steps is restricted. Undercut minimization is essential to regulate the added dissipation induced by pillar movement (Supplementary Info). We take away the FOx masks and the residual etch-stop layer by dipping the wafer in HF 1% for about 3.5 min.

After patterning the pillars, we encapsulate them in a PECVD Si_{x}N_{y} layer to guard them through the silicon deep etching step. We first develop a skinny (about 20 nm), protecting layer of Al_{2}O_{3} with ALD, to protect the membrane layer from plasma bombardment throughout PECVD. Then, roughly 125 nm of Si_{x}N_{y} is grown at 300 °C, with 40 W of radio-frequency energy thrilling the plasma throughout deposition. This Si_{x}N_{y} layer has been characterised to have a tensile stress of round +300 MPa at room temperature. The layer completely seals the nanopillars throughout immersion in scorching KOH, with out important consumption.

After patterning the pillars on the wafer frontside movie, a thick (about 3 μm) layer of constructive tone photoresist is spun on prime for cover through the bottom lithography course of, which we carry out with an MLA150 laser author (Heidelberg Devices). Optical lithography is adopted by Si_{3}N_{4} dry etching with a plasma of CHF_{3} and SF_{6}. After the resist masks and safety layer removing with *N*-methyl-2-pyrrolidone (NMP) and O_{2} plasma, we deep-etch with KOH from the membrane home windows whereas holding the frontside protected, by putting in the wafer in a watertight PEEK holder during which solely the bottom is uncovered^{6}. KOH 40% at 70 °C is used, and the etch is interrupted when about 30–40 μm of silicon stays. The wafer is then rinsed and cleaned with scorching HCl of the residues shaped throughout KOH etching. Then, the wafer is separated into particular person dies with a dicing noticed, and the method continues chipwise. Chips are once more cleaned with NMP and O_{2} plasma, and the deep-etch is concluded with a second immersion in KOH 40% at a decrease temperature of 55 °C, adopted by cleansing in HCl. From the top of the KOH etching step, the composite membranes are suspended, and nice care have to be taken whereas displacing and immersing the samples in liquid. We dry the samples by transferring them to an ultrapure isopropyl alcohol tub after water rinsing. Isopropyl alcohol has a excessive vapour stress, and rapidly evaporates from the chip interfaces, with few residues left behind.

Lastly, the PECVD nitride and Al_{2}O_{3} layers might be eliminated selectively with moist etching in buffered HF. The chips are loaded in a Teflon service during which they’re vertically mounted and immersed for about 3 min 20 s in BHF 7:1. It’s essential to not etch greater than needed to completely take away the encapsulation movies: membranes turn into extraordinarily fragile and the survival yield drops sharply when their thickness is decreased under round 15 nm. The membranes are then rigorously rinsed, transferred in an ethanol tub and dried in a important level dryer, during which the liquids might be evacuated gently and with little contamination.

### Fabrication and simulation of phononic-crystal-patterned mirrors

The highest and backside mirror substrates are, respectively, fused silica and borosilicate glass, with a high-reflection coating sputtered on one face and an anti-reflection layer coating the opposite face. No layer for the safety of the optical coating is utilized earlier than machining. We use a dicing noticed for glass machining to sample an everyday array of strains into the mirror substrates. The blade is repeatedly cooled by a pressurized water jet through the patterning course of. The utmost minimize depth allowed for our blade is 2.5 mm, and we constrain the designed PNC accordingly. We minimize the flat backside mirror from just one aspect (its thickness is just one mm), and the highest mirror is patterned symmetrically with parallel cuts from either side, as it’s 4 mm thick. The comparatively deep cuts within the prime mirror have to be patterned over a number of passes, with steadily growing depths. After patterning one mirror aspect, the piece is flipped and the opposite aspect is patterned after aligning to the primary cuts, seen by the glass substrate. The strains are organized in a sq. lattice for simplicity, though extra advanced patterns might be machined with the dicing noticed. After the dicing course of, the mirrors are topic to ultrasonic cleansing, whereas immersing first in acetone after which in isopropanol.

We simulate the band diagrams of the unit cells of each the highest and the underside mirrors in COMSOL Multiphysics with the Structural Mechanics module. We optimized the lattice fixed and minimize depths to maximise the bandgap width, whereas centring the bandgap round 1 MHz and ensuring that the remaining glass thickness is ample to take care of an affordable stage of structural stiffness. Particulars of the PNC dimensions are proven within the Supplementary Info. Owing to the finite measurement of the mirrors, we anticipate to watch edge modes throughout the mechanical bandgap frequency vary. The thermal vibrations of those modes penetrate into the PNC construction with exponentially decaying amplitudes. To account for his or her noise contributions, we simulated the frequency noise spectrum of the MIM meeting (particulars proven within the Supplementary Info). The eigenfrequency answer confirmed the existence of edge modes with frequencies throughout the mechanical bandgap, however didn’t predict any important contribution to the cavity frequency noise: the PNC is sufficiently massive to scale back their amplitude on the cavity mode place.

After patterning the PNC constructions on the mirrors, we assembled a cavity with a spacer chip rather than a membrane and noticed that the TE_{00} linewidth with the diced mirrors is similar to that of the unique cavity. This means that our fabrication course of doesn’t trigger measurable extra roughness or harm to the mirror surfaces. Against this, when the meeting was clamped too tightly, extra cavity loss occurred due to important deformation of the PNC mirrors, with a decreased stiffness. We mitigate this detrimental impact within the experiment by gently clamping the MIM cavity, with a spring compression ample to ensure the structural stability of the meeting. We additionally make sure that the cavity mode is well-centred on the underside mirror, to scale back the thermal noise contribution of the higher band-edge modes. For the MIM experiment mentioned in the principle textual content, we didn’t observe any mirror modes throughout the mechanical bandgap of the membrane chip. We are able to distinguish membrane modes from mirror modes by exploiting the truth that the coupling charges of membrane modes differ between completely different cavity resonances, whereas this isn’t the case for mirror modes.

### Nonlinear noise cancellation scheme

At room temperature, the massive thermal noise of the cavity, mixed with the nonlinear cavity transduction response, ends in a nonlinear mixing noise (TIN). This noise might result in extra intracavity photon fluctuations and in addition to extra noise in optical detection. Within the following, we talk about the technique to cancel these results within the fast-cavity restrict (*ω* ≪ *κ*). Theoretical derivations and a dialogue of the impact of a finite *ω*/*κ* ratio might be discovered within the Supplementary Info.

Within the experiment, we pump the cavity on the magic detuning, (2overline{varDelta }/kappa =-1/sqrt{3}), during which the nonlinear photon quantity noise is cancelled, to forestall extra oscillator heating as a consequence of nonlinear classical radiation stress noise. To indicate the quantum correlations resulting in optomechanical squeezing and conduct measurement-based state preparation, we have to carry out measurements at arbitrary optical quadrature angles. Balanced homodyne detection offers the potential of tuning the optical quadrature, but it surely doesn’t provide sufficient levels of freedom to cancel the nonlinear noise in detection. Nevertheless, if the native oscillator is injected from a extremely uneven beam splitter with a really small reflectivity (*r* ≪ 1) and the mixed area is detected on a single photodiode, the photodetection nonlinearity is maintained and presents sufficient levels of freedom to cancel the nonlinear noise in detection^{4} (for a derivation, see Supplementary Info). Particularly, simultaneous tuning of native oscillator amplitude and section allows nonlinear mixing noise cancellation at arbitrary quadrature angles. Within the fast-cavity restrict, the cancellation situation is

$$start{array}{l}left|frac{{overline{a}}_{{rm{sig}}}}{{overline{a}}_{hom }}proper|=2{rm{Re}},left(frac{{{rm{e}}}^{-{rm{i}}theta }}{{(-{rm{i}}overline{varDelta }+kappa /2)}^{2}}proper),left({overline{varDelta }}^{2}+{left(frac{kappa }{2}proper)}^{2}proper) ,=2cos (theta -2arg ({chi }_{{rm{cav}}}(0))),finish{array}$$

the place ({overline{a}}_{hom }approx {overline{a}}_{{rm{sig}}}+r{overline{a}}_{{rm{LO}}}) is the coherent mixture of the sign area ({overline{a}}_{{rm{sig}}}) and the native oscillator area ({overline{a}}_{{rm{LO}}}) (outlined as the sphere earlier than the beam splitter), *θ* = *θ*_{hom} − *θ*_{sig} is the quadrature rotation angle and ({chi }_{{rm{cav}}}(0)={left(kappa /2-ioverline{varDelta }proper)}^{-1}) is the cavity d.c. optical susceptibility.

Within the experiment, to detect a sure quadrature angle whereas cancelling nonlinear noise, we lock the homodyne energy on the corresponding mixed area stage ({I}_{hom }=| {overline{a}}_{hom } ^{2}). We then repeatedly differ the native oscillator energy utilizing a tunable impartial density filter till the noise within the mechanical bandgap is completely cancelled. The extent of blending noise could be very delicate to the native oscillator energy, and subsequently the cancellation level can function a very good indicator of the measured quadrature angle *θ*. Understanding the sphere amplitudes (| {overline{a}}_{hom }| ,| {overline{a}}_{{rm{sig}}}| ) and that (overline{varDelta }=-kappa /(2sqrt{3})), we will reconstruct the measured quadrature angle because the one satisfying the cancellation situation.

An in depth characterization of the nonlinear mixing noise and an evaluation of single-detector homodyne effectivity might be discovered within the Supplementary Info.

### Multimode Kalman filter

The continual place measurement of an oscillator at frequency *Ω*_{m} might be seen as a type of heterodyne measurement of two orthogonal mechanical quadratures of movement (widehat{X}) and (widehat{Y}) that rotate with frequency *Ω*_{m}. IQ demodulation can then be carried out on the mechanical frequency *Ω*_{m}. This ends in two impartial measurement channels of two orthogonal mechanical quadratures with impartial measurement noise.

We work in a parameter regime during which the measurement price is considerably smaller than the frequency of the mechanical mode, such that we will carry out IQ demodulation of the mechanical movement at *Ω*_{m} to acquire the slowly various (widehat{X},widehat{Y}) quadratures. Their evolution is described by decoupled quantum grasp equations^{33}. On this parameter regime, solely thermal coherent states are ready by the measurement course of. These states are basically thermal states displaced from the origin of the section area and belong to the bigger group of Gaussian states.

We function within the fast-cavity restrict *Ω*_{m} ≪ *κ*, so the cavity dynamics are simplified in our modelling. After IQ demodulation, the normalized photocurrent sign is described by

$${bf{i}}

(1)

the place the subscript *i* denotes completely different mechanical modes, ({bf{i}}=left(start{array}{c}{i}_{X} {i}_{Y}finish{array}proper)), ({widehat{{bf{r}}}}_{i}=left(start{array}{c}{widehat{X}}_{i} {widehat{Y}}_{i}finish{array}proper)) and ({rm{d}}{bf{W}}=left(start{array}{c}{rm{d}}{W}_{X} {rm{d}}{W}_{Y}finish{array}proper)). The Wiener increment d*W*_{X,Y}(*t*) = *ξ*(*t*)d*t* is outlined when it comes to a super unit Gaussian white noise course of (langle xi

(2)

the place

$${A}_{i}=left(start{array}{cc}-{varGamma }_{{rm{m}}}^{i},/2 & {varOmega }_{i}-{varOmega }_{{rm{m}}} {varOmega }_{{rm{m}}}-{varOmega }_{i} & -{varGamma }_{{rm{m}}}^{i},/2end{array}proper)$$

and

$${B}_{i}=left(start{array}{cc}{sum }_{j}sqrt{{varGamma }_{{rm{meas}}}^{j}}{C}_{{widehat{X}}_{i}{widehat{X}}_{j}} & {sum }_{j}sqrt{{varGamma }_{{rm{meas}}}^{j}}{C}_{{widehat{X}}_{i}{widehat{Y}}_{j}} {sum }_{j}sqrt{{varGamma }_{{rm{meas}}}^{j}}{C}_{{widehat{Y}}_{i}{widehat{X}}_{j}} & {sum }_{j}sqrt{{varGamma }_{{rm{meas}}}^{j}}{C}_{{widehat{Y}}_{i}{widehat{Y}}_{j}}finish{array}proper).$$

The covariance matrix parts ({C}_{widehat{M}widehat{N}}=langle widehat{M}widehat{N}+widehat{N}widehat{M}rangle /2-langle widehat{M}rangle langle widehat{N}rangle ) evolve as

$$start{array}{l}{dot{C}}_{{widehat{M}}_{i}{widehat{N}}_{j}}=-frac{{varGamma }_{{rm{m}}}^{i}+{varGamma }_{{rm{m}}}^{j}}{2}{dot{C}}_{{widehat{M}}_{i}{widehat{N}}_{j}}+{delta }_{{widehat{M}}_{i},{widehat{N}}_{j}}{varGamma }_{{rm{th}}}^{i}+{delta }_{M,N}sqrt{{varGamma }_{{rm{qba}}}^{i}{varGamma }_{{rm{qba}}}^{j}} ,,+{(-1)}^{{delta }_{M,Y}}({varOmega }_{i}-{varOmega }_{{rm{m}}}){C}_{{widehat{{mathcal{M}}}}_{i}{widehat{N}}_{j}}+{(-1)}^{{delta }_{N,Y}}({varOmega }_{j}-{varOmega }_{{rm{m}}}){C}_{{widehat{M}}_{i}{widehat{{mathcal{N}}}}_{j}} ,-4left(sum _{ok}sqrt{{varGamma }_{{rm{meas}}}^{ok}}{C}_{{widehat{M}}_{i}{widehat{X}}_{ok}}proper)left(sum _{l}sqrt{{varGamma }_{{rm{meas}}}^{l}}{C}_{{widehat{N}}_{j}{widehat{X}}_{l}}proper) ,-4left(sum _{ok}sqrt{{varGamma }_{{rm{meas}}}^{ok}}{C}_{{widehat{M}}_{i}{widehat{Y}}_{ok}}proper)left(sum _{l}sqrt{{varGamma }_{{rm{meas}}}^{l}}{C}_{{widehat{N}}_{j}{widehat{Y}}_{l}}proper),finish{array}$$

(3)

the place (widehat{{mathcal{M}}}) and (widehat{{mathcal{N}}}) are the canonical conjugate observables of (widehat{M}) and (widehat{N}).

Equations (1)–(3) type a closed set of replace equations given the measurement document *i*(*t*), and allow quadrature estimations of an arbitrary variety of modes and their correlations. The thermal occupancy ({bar{n}}_{{rm{c}}{rm{o}}{rm{n}}{rm{d}},i}) of a particular mechanical mode is decided by the quadrature phase-space variances ({V}_{{widehat{X}}_{i}}={C}_{{widehat{X}}_{i}{widehat{X}}_{i}}) and ({V}_{{widehat{Y}}_{i}}={C}_{{widehat{Y}}_{i}{widehat{Y}}_{i}}), that are each equal to ({bar{n}}_{{rm{c}}{rm{o}}{rm{n}}{rm{d}},i}+1/2).

We document the voltage output from the photodetector utilizing an UHFLI lock-in amplifier (Zurich Devices), digitizing the sign at a 14-MHz sampling price for a complete length of two s, and we retailer the info digitally for post-processing. The noise energy spectrum density of the digitized sign is in contrast with that concurrently measured on a real-time spectrum analyser, to rule out signal-to-noise ratio degradation from the digitization noise. Particulars of an extra filtering step are mentioned within the Supplementary Info. After filtering, solely the ten mechanical modes across the defect mode frequency *Ω*_{m} are saved for the multimode state estimation examine.

To carry out the multimode state estimation, we extract the required system parameters of the closest 10 mechanical modes round *Ω*_{m} by becoming the measured spectral noise density. We demodulate the sign at *Ω*_{m} and feed the time-series sign **i**(*t*) to the discretized model of the replace equation (2),

$$Delta langle {widehat{{bf{r}}}}_{i}rangle ={A}_{i}^{{prime} }langle {widehat{{bf{r}}}}_{i}rangle Delta t+2{B}_{i}Delta {bf{W}}

(4)

to trace all of the 20 quadrature expectations at completely different occasions. Right here, ({A}_{i}^{{prime} }=left(start{array}{cc}-{varGamma }_{{rm{m}}}^{{prime} i},/2 & {varOmega }_{i}^{{prime} }-{varOmega }_{{rm{m}}} {varOmega }_{{rm{m}}}-{varOmega }_{i}^{{prime} } & -{varGamma }_{{rm{m}}}^{{prime} i},/2end{array}proper)) incorporates modified mechanical parameters:

$$start{array}{l}{varGamma }_{{rm{m}}}^{{prime} i}={varGamma }_{{rm{m}}}^{i}+2{rm{Re}},left(-frac{1-cos (({varOmega }_{i}-{varOmega }_{{rm{m}}})Delta t)}{Delta t}proper) {varOmega }_{i}^{{prime} }={varOmega }_{i}-{rm{Im}},left(i({varOmega }_{i}-{varOmega }_{{rm{m}}})-frac{{{rm{e}}}^{{rm{i}}({varOmega }_{i}-{varOmega }_{{rm{m}}})Delta t}-1}{Delta t}proper)finish{array}$$

to compensate for the affect of discretization on the state estimation efficiency in contrast with a super steady one.

The evolution of the matrix *B*_{i}, involving 210 impartial covariance matrix parts, might be computed independently from the sampled time-domain information. Due to this fact, we calculate it following equation (3), with an replace price of 140 MHz to mitigate the discretization impact, which is then used for the replace equation (4) on the sampling price of 14 MHz. The verification of the right implementation of the multimode Kalman filter is proven within the Supplementary Info.

To experimentally reconstruct the covariance matrix from the estimated quadrature information, we use the retrodiction technique. The retrodiction technique makes use of the measurement document sooner or later as a separate state estimation outcome. We derived the retrodiction replace equations^{39} and located that they’re similar to the prediction replace equations, besides with destructive mechanical frequencies. Because of this, now we have the next relations between covariance matrix parts estimated by prediction and retrodiction (respectively recognized by the superscripts p and r):

$$start{array}{l}{C}_{{widehat{X}}_{i}{widehat{X}}_{j}}^{{rm{p}}}={C}_{{widehat{X}}_{i}{widehat{X}}_{j}}^{{rm{r}}} ,{C}_{{widehat{Y}}_{i}{widehat{Y}}_{j}}^{{rm{p}}}={C}_{{widehat{Y}}_{i}{widehat{Y}}_{j}}^{{rm{r}}} {C}_{{widehat{X}}_{i}{widehat{Y}}_{j}}^{{rm{p}}}=-{C}_{{widehat{X}}_{i}{widehat{Y}}_{j}}^{{rm{r}}}.finish{array}$$

For every time hint slice (1 ms), we calculate the distinction between the prediction and retrodiction outcomes ({langle widehat{{bf{r}}}rangle }_{{rm{r}}}-{langle widehat{{bf{r}}}rangle }_{{rm{p}}}), and calculate the covariance matrix as

$$C=frac{1}{2}langle langle left({langle widehat{{bf{r}}}rangle }_{{rm{r}}}-{langle widehat{{bf{r}}}rangle }_{{rm{p}}}proper)cdot {left({langle widehat{{bf{r}}}rangle }_{{rm{r}}}-{langle widehat{{bf{r}}}rangle }_{{rm{p}}}proper)}^{prime }rangle rangle $$

the place ⟨⟨⋯⟩⟩ is the statistical common over on a regular basis hint slices, and (widehat{{bf{r}}}=left(cdots ,{widehat{X}}_{i},{widehat{Y}}_{i},cdots proper)). The image^{T} signifies the transposed vector.

For a system consisting of a number of mechanical modes that aren’t sufficiently separated in frequency (∣*Ω*_{i} − *Ω*_{j}∣ not considerably quicker than another charges within the system), cross-correlations between completely different mechanical modes emerge due to frequent measurement imprecision noise and customary quantum backaction pressure. This usually results in greater quadrature variance due to the successfully decreased measurement effectivity of particular person modes. To decouple the mechanical oscillators which can be interacting due to the spectral overlap and the measurement course of, we outline a brand new set of collective motional modes by a symplectic (canonical) transformation of quadrature foundation *U* that diagonalizes the covariance matrix *U*^{†}*CU* = *V* (ref. ^{55}). Because the covariance matrix is actual and symmetric, the weather of *U* are at all times actual, which is required for actual observables. The transformation might be understood as a standard mode decomposition of the collective Gaussian state that preserves the commutation relations, versus typical diagonalization utilizing unitary matrices. That is represented by the requirement of the symplectic transformation *UΩU*^{†} = *Ω*, the place (varOmega =left(start{array}{cc}0 & {I}_{N} -{I}_{N} & 0end{array}proper)) is the N-mode symplectic type and *I*_{N} is the *N* × *N* identification matrix. We discover that within the new quadrature foundation based mostly on the diagonalized covariance matrix, the defect mode is just weakly modified. The transformation coefficients for the defect mode are proven within the Supplementary Info.