Quantum backaction evading measurements of a silicon nitride membrane resonator

In the basic optomechanical measurement, a single coherent pump tone is applied to the cavity at the frequency ω_d, inducing a pump photon number n_c. The pump detuning is defined as Δ = ω_d − ω_c. The pumping leads to an enhanced (effective) optomechanical coupling

$$\begin{equation}G={g}_{0}\sqrt{{n}_{\text{c}}}.\end{equation} \tag{ 5 }$$

The strength of interaction is also conveniently expressed as the cooperativity parameter, for which we use two definitions that differ by the interpretation of the mechanical oscillator's damping rate. The intrinsic cooperativity

$$\begin{equation}{C}_{0}=\frac{4{G}^{2}}{\kappa \gamma }\end{equation} \tag{ 6 }$$

describes the situation where the oscillator exhibits its intrinsic damping rate γ. In preparation for discussing the experiments, we also define a generic cooperativity C, where the oscillator can be further damped up to the total damping γ_eff via sideband cooling (see equation (11a) below):

$$\begin{equation}C=\frac{4{G}^{2}}{\kappa {\gamma }_{\mathrm{e}\mathrm{ff}}}.\end{equation} \tag{ 7 }$$

The further damping can occur via additional cooling tone, which is thus another process on top of the main measurement process considered in the following. Without the cooling tone, C = C₀.

2.1.1. Unresolved sidebands

The unresolved sidebands situation κ ≫ ω_m is a basic model for a position measurement. A fast cavity reacts to the oscillator without a time delay, and the position x is measured by the cavity P_c quadrature. The oscillator momentum suffers a backaction from X_c, which further leaks into x in the dynamics. As a consequence, the backaction affects position and momentum in the same manner, given by the variances in units of quanta:

$$\begin{equation}\langle {x}^{2}\rangle =\langle {p}^{2}\rangle =\frac{1}{2}+{n}_{\text{m}}^{\mathit{\text{T}}}+C(1+2{n}_{\text{c}}^{\mathit{\text{T}}}).\end{equation} \tag{ 8 }$$

The terms on the rhs correspond, respectively, to vacuum, thermal, and backaction noise, with the quantum contribution to the backaction given by

$$\begin{equation}\langle {x}^{2}{\rangle }_{\mathit{qba}}=\langle {p}^{2}{\rangle }_{\mathit{qba}}={n}_{\mathit{qba}}=C.\end{equation} \tag{ 9 }$$

On top of the quantum backaction, the term $2C{n}_{\text{c}}^{\mathit{\text{T}}}$ in equation (8) is an additional backaction due to classical cavity noise.

2.1.2. Resolved sidebands

We now treat the opposite case, κ ≪ ω_m. Here, the cavity reacts only slowly to the mechanics, and the dominant consequence becomes the dynamical backaction on the mechanical oscillator. In particular, this implies the possibility to carry out sideband cooling of the mechanics down to the ground state when the pump detuning is set at the red sideband, viz, Δ = −ω_m. We obtain

$$\begin{equation}\langle {x}^{2}\rangle =\langle {p}^{2}\rangle =\frac{1}{1+C}\left[\frac{1}{2}+{n}_{\text{m}}^{\mathit{\text{T}}}+C\left(\frac{1}{2}+{n}_{\text{c}}^{\mathit{\text{T}}}\right)\right].\end{equation} \tag{ 10 }$$

The term $\left(\frac{1}{2}+{n}_{\text{m}}^{\mathit{\text{T}}}\right){(1+C)}^{-1}$ is the result of the sideband cooling process, where the effective mechanical damping is the sum of the intrinsic damping, and of the 'optical' damping γ_opt that is linear in pumping power:

$$\begin{equation}{\gamma }_{\mathrm{e}\mathrm{ff}}=\gamma +{\gamma }_{\mathrm{o}\mathrm{p}\mathrm{t}},\end{equation} \tag{ 11a }$$

$$\begin{equation}{\gamma }_{\mathrm{o}\mathrm{p}\mathrm{t}}=\frac{4{G}^{2}}{\kappa }.\end{equation} \tag{ 11b }$$

We emphasize that in equation (10) and elsewhere in this section, the cooperativity should typically be interpreted as C = C₀. Although sideband cooling produces enhanced damping, a possible additional damping is assumed to be due to another process. When the sideband cooling is started from a high temperature, the final occupation number of the mechanical oscillator becomes

$$\begin{equation}{n}_{\text{m}}\approx \frac{\gamma }{{\gamma }_{\mathrm{e}\mathrm{ff}}}{n}_{\text{m}}^{\mathit{\text{T}}}+{n}_{\text{c}}^{\mathit{\text{T}}}.\end{equation} \tag{ 12 }$$

Backaction is generally associated with heating, however, a quantum backaction is also associated with the measurement under sideband cooling conditions, although an overall effect is a strong cooling. One can consider the measurement as a measurement of an effective number of phonons ${n}_{\text{eff}}=\frac{1}{C+1}\left(\frac{1}{2}+{n}_{\text{m}}^{\mathit{\text{T}}}\right)$. Cavity quantum noise generates an increased effective number of phonons:

$$\begin{equation}{\langle {x}^{2}\rangle }_{qba,\mathrm{e}\mathrm{ff}}=\frac{C}{2(1+C)},\end{equation} \tag{ 13 }$$

and the effect of the quantum backaction on the measured energy of the initial bath (independently from dynamical backaction) is:

$$\begin{equation}{\langle {x}^{2}\rangle }_{\mathit{qba}}={n}_{\mathit{qba}}=\frac{C}{2}.\end{equation} \tag{ 14 }$$

The reason for only a half the backaction as compared to equation (9) is that in equation (9), the backaction comes from each of the two frequency bands of the cavity field at which the spectrum is projected.

Position cannot be measured in a continuous and QND manner, but the quadrature amplitudes, as defined in equation (4), can be. The idea is to modulate the interaction between the oscillator and the detector (cavity in this case) in synchrony with the motion [20, 42, 43]. In optomechanics this is accomplished as depicted in figure 1(b). Two equal-power coherent tones are applied exactly at the mechanical sidebands, at the frequencies ω_± = ω_c ± ω_m. The complex amplitudes of the fields at these frequencies are denoted by |α|exp(iθ_±). We move the cavity operators to a frame rotating at the mean frequency of the two pumps (that is, the cavity frequency), and the mechanics to a frame rotating at ω_m. In this frame, we define the phase difference of the tones as θ = (θ₊ − θ₋)/2. After linearization, and supposing a good-cavity situation κ ≪ ω_m, which allows us to neglect terms oscillating at ±2ω_m, equation (2) becomes

$$\begin{equation}H=\vert G\vert \left[\mathrm{exp}(i\theta ){a}^{{\dagger}}+\mathrm{exp}(-i\theta )a\right]\left({b}^{{\dagger}}+b\right)=2G{X}_{\text{c}}^{\theta }X,\end{equation} \tag{ 15 }$$

where the rotated cavity quadrature is ${X}_{\text{c}}^{\theta }={X}_{\text{c}}\,\mathrm{cos}\,\theta +{Y}_{\text{c}}\,\mathrm{sin}\,\theta$. Notably, all the dynamical variables are now quadrature amplitudes. The effective coupling is G = g₀|α|. The mechanical quadrature X is now a constant of motion since [X, H] = 0, while the unmeasured P quadrature absorbs all the backaction.

The total number of quanta in the mechanical quadratures becomes

$$\begin{equation}\begin{aligned}\hfill & \langle {X}^{2}\rangle =\frac{1}{2}+{n}_{\text{m}}^{\mathit{\text{T}}},\hfill \\ \hfill & \langle {P}^{2}\rangle =\frac{1}{2}+{n}_{\text{m}}^{\mathit{\text{T}}}+2C+4C{n}_{\text{c}}^{\mathit{\text{T}}}.\hfill \end{aligned}\end{equation} \tag{ 16 }$$

The backaction is hence entirely projected onto the P quadrature:

$$\begin{equation}\begin{aligned}\hfill {\langle {P}^{2}\rangle }_{\mathit{qba}}& =2C,\hfill \\ \hfill {n}_{\mathit{qba}}& =C.\hfill \end{aligned}\end{equation} \tag{ 17 }$$

The quantum backaction on the quadratures at a given cooperativity is thus different in all the three example models, equations (9), (14) and (17). The total backaction on the oscillator is the same in equations (9) and (17), although this can be seen as incidental. The backaction in the two-tone BAE model is most closely related to the resolved-sideband case discussed in appendix 2.1.2. In the BAE case, at a given cooperativity, there is double the power, and thus a double total backaction.

In experiment, the signal is in the incoherently emitted field from the cavity. In microwave measurements, one measures the spectrum of the voltage fluctuations, reproducing the mechanical X spectrum via cavity transduction, see equation (A21). The actual measured spectra contains also a white noise contribution by the noise added by the amplifier. The detected mechanical spectrum can be understood as an effective mechanical spectrum S_X,eff[ω], knowledge of which is degraded by the noise sources dominated by the amplifier noise (equations (A22) and (A23)).

We also note variants of the single-quadrature detection idea, where collective motion of two oscillators can be measured in BAE fashion [4, 24, 27, 44, 45].

The oscillating membrane is a standard 0.5 mm square window from Norcada, made of 50 nm thick SiN with a high pre-stress of 900 MPa. It is metallized in the center by 20 nm Al in the form of a 200 μm square.

For coupling to microwave mode, we use flip-chip technique similar to [28, 29, 46]. Antenna pattern, with 120 nm of Al metallization, is realized on a separate high-purity Si chip. The membrane chip is glued on the antenna chip from one corner using a drop of fast epoxy (see figure 2(a)). Without any support posts between the chips, or removal of Si from the antenna chip outside the metallization, the natural gap between the chips becomes on the order 500 nm. The gap is determined by irregularities, or possibly bending of the membrane chip resulting from the fact that the stressed SiN is removed from the backside. The dimensions yield an estimated g₀/2π ≃ 10 Hz.

Zoom In Zoom Out Reset image size

As depicted in figure 2(b), the chip assembly is embedded in a 3D microwave cavity [47] made of oxygen-free copper that is annealed at 850 °C for 8 h, with a small air flow at 2 × 10⁻³ mBar. We found that (non-superconducting) Cu cavities provided better thermalization of the mechanics in comparison to superconducting Al cavities, where the mode usually does not thermalize below $\sim 100$ mK. The reason is likely bad thermal conductivity of the superconductor.

Because the membranes are highly sensitive to vibration noise of the pulse tube in contemporary cryogen-free dilution refrigerators, the measurements are conducted in a wet liquid-He based dilution refrigerator, which has a base temperature of about 25 mK.

The mechanical mode (the lowest flexural mode) has the frequency ω_m/2π ≃ 707.4 kHz. Its damping rate measured with ringdown is γ/2π ≃ 8.8 mHz, corresponding to quality factor Q_m ≃ 80 millions. The cavity resonance is found at ω_c/2π ≃ 4.517 GHz. At temperatures below around 300 mK, the cavity has the internal and external loss rates κ_i/2π ≃ 156 kHz, and κ_e/2π ≃ 145 kHz, which sum up to κ/2π ≃ 301 kHz.

At room temperature, the Cu cavities were found to have a Q of 2000–4000. We expect that well-annealed Cu exhibits a bulk conductivity enhancement of 2–3 orders of magnitude when cooled down below 4 K. The observed increase of Q is much less than anticipated from the enhanced conductivity. We therefore believe that conductive surface oxides limit the Q. Nonetheless, even though the cavity is not superconducting, we can approach the good-cavity situation ω_m ≫ κ.

Under red-sideband measurement, the mechanical mode temperature is proportional to the area of the recorded mechanical spectral peak. In cryogenic optomechanics experiments, the mechanical mode may be excited to energies higher than expected based on the refrigerator temperature. This is associated to low thermal conductivity, (technical) heating by the pump irradiation, or external vibration noise. In the present case, the mechanical mode is found to thermalize rather well, down to 30...40 mK when the cryostat temperature is varied, as seen in figure 3 (squares). An important calibration point in the ensuing analysis is given by the equilibrium mode occupation ${n}_{\text{m}}^{\mathit{\text{T}},0}$ at a given refrigerator temperature. In the forthcoming experiments, we operate at the base temperature of 25 mK. We determine ${n}_{\text{m}}^{T,0}$ as the average of the two lowest-temperature data points of the single-tone measurement in figure 3, obtaining ${n}_{\text{m}}^{T,0}\simeq 1100$. The rest of the calibration procedure is detailed in appendix B.

Zoom In Zoom Out Reset image size

Standard sideband cooling is performed by irradiating the cavity with a strong red-sideband tone. This measurement is used to benchmark the performance of the device. It also allows to calibrate mechanical parameters such as the effective mechanical damping rate.

As depicted in equation (12), sideband cooling allows in the limit of large effective damping γ_eff, cooling the mechanical mode close to the ground state where n_m = 0. This has been demonstrated in many different types of mechanical oscillators, including SiN membranes [30, 31].

We show sideband cooling close to the ground-state in figure 4, reaching n_m ≃ 0.3 at C ≃ 10⁴ where γ_opt/2π ≃ 90 Hz, with an outlier data point probably due to a building vibration. At higher powers, we observe significant technical heating of the mechanical bath, displayed also in figure 4. This heating is seen in many microwave optomechanical experiments [2, 4, 25, 39, 48]. The exact mechanism is unknown. Technical heating of the cavity mode, however, is nearly absent as seen in figure 4. This is in stark contrast to results obtained with samples we measured in superconducting Al cavities, where the cavity mode can heat up to tens of quanta. This hints that the poor thermal conductivity of superconductors has a role in cavity heating.

Zoom In Zoom Out Reset image size

To properly reach quantum backaction evasion using the scheme in figure 1(b), the frequency separation of the pump tones need to equal twice the mechanical frequency within at least a few tens of percent of the mechanical linewidth. The mHz intrinsic mechanical linewidth of the membranes makes this nearly unreachable, especially as the mechanical frequency was observed to jump intermittently at the scale of γ. Therefore, we use an additional cooling tone. The cooling tone cooperativity was chosen such that we obtain γ_eff/2π ≃ 3.1 Hz ≫ γ, which relieves the above problem. The frequency of the cooling tone was incommensurately spaced from the BAE tones (detuned from the red sideband by δ = 400 Hz in the experiment), and thus the cooling process can be treated independently of the BAE process. In terms of the mechanical spectrum, equation (A18), the bath temperature becomes replaced by the mode temperature, viz ${n}_{\text{m}}^{T}\to {n}_{\text{m}}$ (equation (12)).

Same as with sideband cooling, we need a calibration point for ${n}_{\text{m}}^{T}$ (or n_m). Although under purely red-detuned tone irradiation, thermalization and technical heating were very favorable, this may not hold true under more complicated driving. The calibration is obtained by performing a temperature sweep as shown in figure 3 (circles). The integrated mechanical peak, the flux (equation (A26)), in the output spectrum gives the energy of the mechanical mode. There are non-characterized amounts of both gain and attenuation in the output cabling, and thus we cannot obtain absolute calibration of the mode temperature using equation (A26). We assume a good thermalization at several 100 mK temperatures, and use a linear fit of the peak area in this region as the basis of the calibrating ${n}_{\text{m}}^{T}$ (and n_m) at the base temperature where the final measurements are performed. The bath temperature is finally inferred from equation (B8).

In the measurement, we compensated for the temperature dependence of the mechanical frequency by correcting the pump tone frequencies based on a calibration of ω_m at each temperature. As seen in figure 3, the thermalization in the BAE scheme is comparable to that with a single-tone pumping, only with a modest thermal decoupling close to the base temperature.

We now describe the BAE measurement at the base temperature, aiming at showing evasion of QBA that would appear as an increased occupation of the mechanics. Like in the temperature sweep, we used a strong cooling tone broadening the linewidth, here up to γ_eff/2π ≃ 2.7 Hz. The relevant cooperativity is then that including the additional damping, equation (7). In figure 5, we first display in panel (a) the detected output spectra converted into effective mechanical spectra. An important benchmarking of BAE is that it does not influence the spectrum, including linewidth, of the measured X quadrature. This is verified in figure 5(b), where the spectral broadening resulting from the additional sideband cooling is also shown as a reference. In spite of some scatter at low powers, the linewidth is essentially unaffected by a strong BAE measurement. We then proceed to presenting the output flux in figure 5(c) (see equations (A26) and (B10)). The flux first grows linearly with power, but then overshoots the linear trend, which is associated to the technical heating of the mechanics.

Zoom In Zoom Out Reset image size

Finally, in figure 5(d) we display the main result. Using the calibration procedure summarized in equations (B10) and (B11), we obtain the fluctuations of the measured X quadrature. The overshoot seen above in panel (c) now appears as heating of X at cooperativities C ≳ 10.

As discussed above, there is no clear model-independent way to define what an expected QBA in an optomechanical measurement should be. When treating the two-tone BAE measurement, the difficulty is that the cooperativity is not in direct correspondence to the cooperativity in single-tone measurements. A possible comparison point comes from the QBA on the P quadrature, equaling 2C, in the BAE measurement. If this would be equally distributed between the two quadratures, each would receive a QBA equal to C. This is also the same QBA as in the bad-cavity system, equation (9). We make the worst-case comparison, and benchmark against all the discussed models. As seen in the figure, the X fluctuations remain well below the expected QBA contributions from all the models. It is hence justified to claim that our measurement truly evades the quantum backaction. Against the most reasonable comparison, equation (9), the backaction is evaded by 9 ± 2 quanta at cooperativity C ≃ 10. This implies the oscillator fluctuations amount to only ≃60% of that with QBA contribution included.

The best measured effective mechanical variance becomes $\langle {X}_{\mathrm{X},\mathrm{e}\mathrm{ff}}^{2}\rangle \simeq 8.4\pm 2.5$ quanta, also at C ≃ 10. This is mostly limited by the mechanical noise, and to a lesser extent contributed by the amplifier noise.

We start from the equations of motion, equation (A2), derived from the Hamiltonian in equation (A1). As a reminder, the cavity mode is transferred to the frame rotating with the pump tone.

We also define the susceptibilities of the mechanics and of the cavity as ${\chi }_{\text{m}}^{-1}=\frac{\gamma }{2}-i\omega$ and ${\chi }_{\text{c}}^{-1}=\frac{\kappa }{2}-i\omega$.

We transfer the cavity operators to the rotating frame at the frequency ω_d of the pump (e.g., x_c → X_c, while the mechanics stays in the lab frame). Under a strong pumping of, say the X_c quadrature of the cavity (making a choice of phase), the pumping induces a cavity field ${X}_{\text{c}}\to \bar{{X}_{\text{c}}}+{X}_{\text{c}}$, where $\bar{{X}_{\text{c}}}$ is the large coherent field and X_c describes small fluctuations. With this change to equation (2), the system is linearized as

$$\begin{equation}H=-\frac{{\Delta}}{2}\left({X}_{\text{c}}^{2}+{P}_{\text{c}}^{2}\right)+\frac{{\omega }_{\text{m}}}{2}\left({x}^{2}+{p}^{2}\right)+2G{X}_{\text{c}}x.\end{equation} \tag{ A1 }$$

The input–output theory equations of motion in the frequency domain become

$$\begin{equation}{\chi }_{\text{c}}^{-1}{X}_{\text{c}}=-{\Delta}{P}_{\text{c}}+\sqrt{{\kappa }_{\text{e}}}{X}_{\text{c,in}}+\sqrt{{\kappa }_{\text{i}}}{X}_{\text{I,in}},\end{equation} \tag{ A2a }$$

$$\begin{equation}{\chi }_{\text{c}}^{-1}{P}_{\text{c}}={\Delta}{X}_{\text{c}}-2Gx+\sqrt{{\kappa }_{\text{e}}}{P}_{\text{c,in}}+\sqrt{{\kappa }_{\text{i}}}{P}_{\text{I,in}},\end{equation} \tag{ A2b }$$

$$\begin{equation}{\chi }_{\text{m}}^{-1}x={\omega }_{\text{m}}p+\sqrt{\gamma }{x}_{\text{in}},\end{equation} \tag{ A2c }$$

$$\begin{equation}{\chi }_{\text{m}}^{-1}p=-{\omega }_{\text{m}}x-2G{X}_{\text{c}}+\sqrt{\gamma }{p}_{\text{in}}.\end{equation} \tag{ A2d }$$

The input noise operators associated to the losses have the white spectral densities $\langle z(\omega )z(-\omega )\rangle =\frac{1}{2}+{n}_{z}^{T}$ for all z = X_{c, in}, P_{c, in}, X_I,in, P_I,in, x_in, p_in. ${n}_{z}^{T}$ is the thermal occupation of a given bath, in particular ${n}_{\text{m}}^{T}={\left[\mathrm{exp}(\hslash {\omega }_{\text{m}}/{k}_{\text{B}}T)-1\right]}^{-1}$ is the occupation of the mechanical oscillator at the temperature T. The occupation number relates to the variances of the dimensionless position and momentum as ${n}_{\text{m}}^{T}+\frac{1}{2}=\langle {b}^{{\dagger}}b\rangle +\frac{1}{2}=\frac{1}{2}\langle {x}^{2}\rangle +\frac{1}{2}\langle {p}^{2}\rangle$, and similarly for the quadrature amplitudes. We assume the cavity external bath is at zero temperature, while there can be internal heating of the cavity described by ${n}_{\text{I}}^{T}$. The cavity heating is most conveniently given by the cavity thermal occupation as ${n}_{\text{c}}^{T}={n}_{\text{I}}^{T}\frac{{\kappa }_{\text{i}}}{\kappa }$.

A.1.1. Bad-cavity, ω_m ≪ κ

In the equations of motion, equation (A2), we take Δ = 0, and approximate χ_c = 2/κ. The solution for the mechanics can be written in the form

$$\begin{equation}x={\tilde{X}}_{x}{x}_{\text{in}}+{\tilde{X}}_{\text{p}}{p}_{\text{in}}+{\tilde{X}}_{ba}{X}_{\text{c,in}}+{\tilde{X}}_{\mathit{baI}}{X}_{\text{c,inI}},\end{equation} \tag{ A3a }$$

$$\begin{equation}p={\tilde{P}}_{\text{p}}{p}_{\text{in}}+{\tilde{P}}_{x}{x}_{\text{in}}+{\tilde{P}}_{ba}{X}_{\text{in}}+{\tilde{P}}_{\mathit{baI}}{X}_{\text{c},\text{inI}}.\end{equation} \tag{ A3b }$$

The transduction coefficients that are not explicitly written down are equal to zero. The result is

$$\begin{equation}{\tilde{X}}_{x}={\tilde{\mathit{\text{P}}}}_{\text{p}}=i\sqrt{\gamma }\omega {\chi }_{\text{m}}^{\prime },\end{equation} \tag{ A4a }$$

$$\begin{equation}{\tilde{X}}_{\text{p}}=-{\tilde{P}}_{x}=\sqrt{\gamma }{\omega }_{\text{m}}{\chi }_{\text{m}}^{\prime },\end{equation} \tag{ A4b }$$

$$\begin{equation}{\tilde{X}}_{ba}=-\sqrt{4C\gamma }\beta {\omega }_{\text{m}}{\chi }_{\text{m}}^{\prime },\end{equation} \tag{ A4c }$$

$$\begin{equation}{\tilde{X}}_{\mathit{baI}}=-\sqrt{4C\gamma }\alpha {\omega }_{\text{m}}{\chi }_{\text{m}}^{\prime },\end{equation} \tag{ A4d }$$

$$\begin{equation}{\tilde{P}}_{ba}=i\sqrt{4C\gamma }\beta \omega {\chi }_{\text{m}}^{\prime },\end{equation} \tag{ A4e }$$

$$\begin{equation}{\tilde{P}}_{\mathit{baI}}=i\sqrt{4C\gamma }\alpha \omega {\chi }_{\text{m}}^{\prime }.\end{equation} \tag{ A4f }$$

Out of these, all other than X_ba(I), P_ba(I) are those of non-interacting oscillators. Here, we defined

$$\begin{equation}{\chi }_{\text{m}}^{\prime }=\frac{1}{{\left(\frac{\gamma }{2}-i\omega \right)}^{2}+{\omega }_{\text{m}}^{2}},\end{equation} \tag{ A5 }$$

and

$$\begin{equation}\begin{aligned}\hfill \alpha & =\sqrt{\frac{{\kappa }_{\text{i}}}{\kappa }},\hfill \\ \hfill \beta & =\sqrt{\frac{{\kappa }_{\text{e}}}{\kappa }}.\hfill \end{aligned}\end{equation} \tag{ A6 }$$

For evaluating the spectra, we define the following quantities:

$$\begin{equation}\begin{aligned}\hfill & {\mathcal{L}}_{\pm }=\frac{1}{{(\omega \mp {\omega }_{\text{m}})}^{2}+{\left(\frac{\gamma }{2}\right)}^{2}},\hfill \\ \hfill & {S}_{0}=\frac{\gamma }{2}\left({\mathcal{L}}_{+}+{\mathcal{L}}_{-}\right).\hfill \end{aligned}\end{equation} \tag{ A7 }$$

S₀ is a double Lorentzian, peaking at the positive and negative mechanical frequencies.

The following holds for the absolute value of the quantity in equation (A5):

$$\begin{equation}\vert {\chi }_{\text{m}}^{\prime }{\vert }^{2}=\frac{{\mathcal{L}}_{+}+{\mathcal{L}}_{-}}{4{\omega }_{\text{m}}^{2}}=\frac{1}{2{\omega }_{\text{m}}^{2}\gamma }{S}_{0}.\end{equation} \tag{ A8 }$$

The spectral density of x can be written as

$$\begin{equation}{S}_{x}=\langle x(\omega )x(-\omega )\rangle ={S}_{x}^{0}+{S}_{x}^{T}+{S}_{x}^{ba}.\end{equation} \tag{ A9 }$$

Here, the terms of the rhs are the spectra divided into contributions by the vacuum noise, oscillator thermal occupation, cavity external vacuum noise, and oscillator internal thermal occupation. These are explicitly given by

$$\begin{equation}{S}_{x}^{0}=\frac{\gamma }{2}{\mathcal{L}}_{+},\end{equation} \tag{ A10a }$$

$$\begin{equation}{S}_{x}^{T}={S}_{0}{n}_{\;\text{m}\;}^{T},\end{equation} \tag{ A10b }$$

$$\begin{equation}{S}_{x}^{ba}=C\left(1+2{n}_{\text{c}}^{T}\right){S}_{0}.\end{equation} \tag{ A10c }$$

The vacuum noise spectrum exhibits a peak only at positive frequencies.

The fluctuation energy in x becomes

$$\begin{equation}\langle {x}^{2}\rangle =\frac{1}{2\pi }\int \mathrm{d}\omega \,{S}_{x}=\frac{1}{2}+{n}_{\text{m}}^{T}+C(1+2{n}_{\text{c}}^{T}),\end{equation} \tag{ A11 }$$

which is the result stated in equation (8). The spectral density and energy in p are the same as those of x.

A.1.2. Good-cavity, ω_m ≫ κ

We assume the sideband cooling conditions: Δ = −ω_m. Consequently, in contrast to equation (A3), the dynamical quantities depend on the inputs to all others.

For the mechanics, we write

$$\begin{equation}\begin{aligned}\hfill x& ={\tilde{X}}_{x}{x}_{\text{in}}+{\tilde{X}}_{\text{p}}{p}_{\text{in}}+{\tilde{X}}_{ba}{X}_{\text{c},\text{in}}\hfill \\ \hfill & \quad +{\tilde{X}}_{\mathit{baI}}{X}_{\text{c},\text{inI}}+{\tilde{X}}_{\mathit{baP}}{P}_{\text{c,}\text{in}}+{\tilde{X}}_{\mathit{baPI}}{P}_{\text{c},\text{inI}},\hfill \\ \hfill p& ={\tilde{P}}_{x}{x}_{\text{in}}+{\tilde{P}}_{\text{p}}{p}_{\text{in}}+{\tilde{P}}_{ba}{X}_{\text{c},\text{in}}\hfill \\ \hfill & \quad +{\tilde{P}}_{\mathit{baI}}{X}_{\text{c},\text{inI}}+{\tilde{P}}_{\mathit{baP}}{P}_{\text{c,}\text{in}}+{\tilde{P}}_{\mathit{baPI}}{P}_{\text{c},\text{inI}}.\hfill \end{aligned}\end{equation} \tag{ A12 }$$

The backaction terms to the position, as an example, become

$$\begin{equation}\begin{aligned}\hfill {\tilde{X}}_{ba}& =-\frac{2iG\beta {\omega }_{\text{m}}}{\sqrt{\kappa }}{\chi }_{\text{e}}^{\prime },\hfill \\ \hfill {\tilde{X}}_{\mathit{baI}}& =-\frac{2iG\alpha {\omega }_{\text{m}}}{\sqrt{\kappa }}{\chi }_{\text{e}}^{\prime },\hfill \\ \hfill {\tilde{X}}_{\mathit{baP}}& =\frac{2G\beta \omega }{\sqrt{\kappa }}{\chi }_{\text{e}}^{\prime },\hfill \\ \hfill {\tilde{X}}_{\mathit{baPI}}& =\frac{2G\alpha \omega }{\sqrt{\kappa }}{\chi }_{\text{e}}^{\prime },\hfill \end{aligned}\end{equation} \tag{ A13 }$$

Here, in contrast to equation (A5), the susceptibilities exhibit the effective mechanical linewidth, equation (11a):

$$\begin{equation}{\chi }_{\text{e}}^{\prime }=\frac{1}{{\left(\frac{{\gamma }_{\mathrm{e}\mathrm{f}\mathrm{f}}}{2}-i\omega \right)}^{2}+{\omega }_{\text{m}}^{2}}.\end{equation} \tag{ A14 }$$

The energy in the position and momentum is evaluated in the same manner as in the previous section, obtaining equation (10).

The equations of motion for the quadratures, based on equation (15), are written as

$$\begin{equation}\begin{aligned}\hfill & \dot{{X}_{\text{c}}}=-\frac{\kappa }{2}X+\sqrt{{\kappa }_{\text{e}}}{X}_{\text{c,}\text{in}}+\sqrt{{\kappa }_{\text{I}}}{X}_{\text{c},\text{inI}},\hfill \\ \hfill & \dot{{P}_{\text{c}}}=2GX-\frac{\kappa }{2}{P}_{\text{c}}+\sqrt{{\kappa }_{\text{e}}}{P}_{\text{c,}\text{in}}+\sqrt{{\kappa }_{\text{i}}}{P}_{\text{c},\text{inI}},\hfill \\ \hfill & \dot{X}=-\frac{\gamma }{2}X+\sqrt{\gamma }{X}_{\text{in}},\hfill \\ \hfill & \dot{P}=-2G{X}_{\text{c}}-\frac{\gamma }{2}P+\sqrt{\gamma }{P}_{\text{in}}.\hfill \end{aligned}\end{equation} \tag{ A15 }$$

Since the system is defined in the frame rotating with the mechanics, the mechanical spectrum now consists of a single Lorentzian at zero frequency. Physically, the two pumps drop sidebands toward the cavity center frequency, where the two sidebands overlap and interfere.

The system dynamics is written as a function of the non-zero transduction coefficients from the inputs, or between the dynamical variables (the term containing ${\tilde{P}}_{\text{c},x}$):

$$\begin{equation}\begin{aligned}\hfill & {X}_{\text{c}}={\tilde{X}}_{\text{c}}{X}_{\text{c,}\text{in}}+{\tilde{X}}_{\text{I}}{X}_{\text{c,}\text{inI}},\hfill \\ \hfill & {P}_{\text{c}}={\tilde{P}}_{\text{c},x}X+{\tilde{P}}_{\text{c}}{P}_{\text{c,}\text{in}}+{\tilde{P}}_{\text{cI}}{P}_{\text{c,}\text{inI}},\hfill \\ \hfill & X={\tilde{X}}_{x}{X}_{\text{in}}\hfill \\ \hfill & P={\tilde{P}}_{\text{p}}{P}_{\text{in}}+{\tilde{P}}_{ba}{X}_{\text{c,}\text{in}}+{\tilde{P}}_{\mathit{baI}}{X}_{\text{c,}\text{inI}},\hfill \end{aligned}\end{equation} \tag{ A16 }$$

where

$$\begin{equation}\begin{aligned}\hfill {\tilde{X}}_{x}& ={\tilde{P}}_{\text{p}}=\sqrt{\gamma }{\chi }_{\text{m}},\hfill \\ \hfill {\tilde{P}}_{ba}& =-2G\sqrt{{\kappa }_{\text{e}}}{\chi }_{\text{m}}{\chi }_{\text{c}},\hfill \\ \hfill {\tilde{P}}_{\mathit{baI}}& =-2G\sqrt{{\kappa }_{\text{i}}}{\chi }_{\text{m}}{\chi }_{\text{c}},\hfill \\ \hfill {\tilde{P}}_{\text{c},x}& =2G{\chi }_{\text{c}}.\hfill \end{aligned}\end{equation} \tag{ A17 }$$

As in the other cases, we suppose χ_c = 2/κ. One now easily arrives at the mechanical quadrature spectra:

$$\begin{equation}\begin{aligned}\hfill {S}_{X}[\omega ]& =\frac{\gamma }{{\omega }^{2}+{(\gamma /2)}^{2}}\left(\frac{1}{2}+{n}_{\text{m}}^{T}\right),\hfill \\ \hfill {S}_{\mathit{\text{P}}}[\omega ]& ={S}_{X}[\omega ]+\frac{16{G}^{2}}{\left[{\omega }^{2}+{(\gamma /2)}^{2}\right]\kappa }\left(\frac{1}{2}+{n}_{\text{c}}^{T}\right),\hfill \end{aligned}\end{equation} \tag{ A18 }$$

and the energy stated in equation (16).

We now proceed with the output field from the cavity, with only the P_c quadrature containing any information on the mechanics. The field emitted by the sample is

$$\begin{equation}\begin{aligned}\hfill {P}_{\text{c,}\text{out}}& =\sqrt{{\kappa }_{\text{e}}}{P}_{\text{c}}-{P}_{\text{c,}\text{in}}\hfill \\ \hfill & =\sqrt{{\kappa }_{\text{e}}}{\chi }_{\text{c}}[2GX+\sqrt{{\kappa }_{\text{e}}}{P}_{\text{c,}\text{in}}+\sqrt{{\kappa }_{\text{i}}}{P}_{\text{c,}\text{inI}}]-{P}_{\text{c,}\text{in}}.\hfill \end{aligned}\end{equation} \tag{ A19 }$$

Before detection at room temperature, the signal is amplified by the total power gain A ≫ 1. In addition, one has to also consider the noise added by the microwave amplifier, which typically is the dominant noise source. It is characterized by the noisy field N(t) and the corresponding white spectral density N_amp. At the detection (quantities marked with d superscripts) at room temperature, the signal is

$$\begin{equation}{P}_{\text{c,}\text{out}}^{d}=A\left({P}_{\text{c,}\text{out}}+N\right).\end{equation} \tag{ A20 }$$

The main studied quantity, the recorded output spectrum, becomes

$$\begin{equation}\begin{aligned}\hfill \langle {P}_{\text{c,}\text{out}}^{d}[\omega ]{P}_{\text{c,}\text{out}}^{d}[-\omega ]\rangle & \equiv {S}_{\mathrm{o}\mathrm{u}\mathrm{t}}^{d}[\omega ]=A{S}_{\mathrm{o}\mathrm{u}\mathrm{t}}[\omega ]\hfill \\ \hfill & =A\left[16{G}^{2}\frac{{\kappa }_{\text{e}}}{{\kappa }^{2}}{S}_{X}[\omega ]+\frac{4{\kappa }_{\text{e}}}{\kappa }{n}_{\text{c}}^{T}+{S}_{\text{in}}+{N}_{\text{amp}}\right],\hfill \end{aligned}\end{equation} \tag{ A21 }$$

with ${S}_{\text{in}}=\frac{1}{2}$. Equation (A21) can be re-organized to define the effective recorded mechanical spectrum:

$$\begin{equation}{S}_{\mathrm{X},\mathrm{e}\mathrm{ff}}[\omega ]\equiv \frac{{\kappa }^{2}}{{\kappa }_{\text{e}}}\frac{{S}_{\mathrm{o}\mathrm{u}\mathrm{t}}^{d}[\omega ]}{16{G}^{2}A}={S}_{X}[\omega ]+{S}_{\mathrm{i}\mathrm{m}\mathrm{p}}.\end{equation} \tag{ A22 }$$

Here, the imprecision set by the noise sources is

$$\begin{equation}{S}_{\mathrm{i}\mathrm{m}\mathrm{p}}=\frac{\kappa }{4{G}^{2}}{n}_{\text{c}}^{T}+\frac{{\kappa }^{2}}{{\kappa }_{\text{e}}}\frac{1}{32{G}^{2}}+\frac{{\kappa }^{2}}{{\kappa }_{\text{e}}}\frac{{N}_{\text{amp}}}{16{G}^{2}}.\end{equation} \tag{ A23 }$$

It is convenient to express the effective mechanical spectrum and imprecision in units quanta, which offers clear understanding of if one is close to the quantum level of detection. Therefore, we scale the effective spectrum by γ/4:

$$\begin{equation}{S}_{\mathrm{X},\mathrm{e}\mathrm{ff}}^{n}[\omega ]\equiv \frac{\gamma }{4}{S}_{\mathrm{X},\mathrm{e}\mathrm{ff}}[\omega ],\end{equation} \tag{ A24 }$$

and similarly for the imprecision noise spectrum, to obtain the quantity n_imp in units of quanta.

The resonant value

$$\begin{equation}\langle {X}_{\mathrm{X},\mathrm{e}\mathrm{ff}}^{2}\rangle \equiv {S}_{\mathrm{X},\mathrm{e}\mathrm{ff}}^{n}[{\omega }_{\text{m}}]=\frac{1}{2}+{n}_{\text{m}}^{T}+{n}_{\mathrm{i}\mathrm{m}\mathrm{p}},\end{equation} \tag{ A25 }$$

thus equals the total mechanical variance plus the imprecision noise.

The above discussion of the output field holds for single-quadrature detection. In the experiment, instead, the total spectrum is recorded. As a result, the imprecision becomes twice that quoted in equation (A23).

The measured flux is the integrated mechanical peak in equation (A21):

$$\begin{equation}{n}_{\mathrm{o}\mathrm{u}\mathrm{t}}^{d}=A\frac{4C\gamma {\kappa }_{\text{e}}}{\kappa }\langle {X}^{2}\rangle .\end{equation} \tag{ A26 }$$

Following the temperature sweep, we calibrate the effective interaction strength G as a function of the generator power setting P. The cavity pump photon number n_c is proportional to the power, which relates to G according to equation (5). However, a large fraction of the microwave power is lost on its way to the sample. The power loss is described by the calibration constant $\mathcal{J}$, such that the gain defined in equation (5) is written as

$$\begin{equation}{G}^{2}=\mathcal{J}P.\end{equation} \tag{ B1 }$$

The quantity that is the most straightforward to measure is the optical damping rate, equation (11b), which together with equation (B1) becomes

$$\begin{equation}{\gamma }_{\mathrm{o}\mathrm{p}\mathrm{t}}=\frac{4\mathcal{J}P}{\kappa }.\end{equation} \tag{ B2 }$$

We perform a linear fit to the experimentally determined γ_opt as a function of the generator power, obtaining

$$\begin{equation}{\gamma }_{\mathrm{o}\mathrm{p}\mathrm{t}}=\mathcal{L}P,\end{equation} \tag{ B3 }$$

and comparison with equation (B2) yields the calibration coefficient:

$$\begin{equation}\mathcal{J}=\frac{\mathcal{L}\kappa }{4}.\end{equation} \tag{ B4 }$$

The cooperativity is obtained similarly:

$$\begin{equation}C=\frac{{\gamma }_{\mathrm{o}\mathrm{p}\mathrm{t}}}{\gamma }=\frac{\mathcal{L}}{\gamma }P.\end{equation} \tag{ B5 }$$

The measured flux can be written as

$$\begin{equation}{n}_{\mathrm{o}\mathrm{u}\mathrm{t}}^{d}=A\frac{4C\gamma {\kappa }_{\text{e}}}{\kappa }\frac{\gamma {k}_{\text{B}}}{{\gamma }_{\mathrm{e}\mathrm{ff}}\hslash {\omega }_{\text{m}}}T.\end{equation} \tag{ B6 }$$

The flux is fitted linearly to the high-temperature data points where a good thermalization is obtained:

$$\begin{equation}{n}_{\mathrm{o}\mathrm{u}\mathrm{t}}^{d}=\mathcal{H}T=\mathcal{H}\frac{{n}_{\text{m}}^{T}\hslash {\omega }_{\text{m}}}{{k}_{\text{B}}},\end{equation} \tag{ B7 }$$

where $\mathcal{H}$ is the calibration coefficient to be obtained from the fit. This is inverted to yield the bath temperature:

$$\begin{equation}{n}_{\text{m}}^{T}=\frac{{n}_{\mathrm{o}\mathrm{u}\mathrm{t}}^{d}}{\mathcal{H}}\frac{{k}_{\text{B}}}{\hslash {\omega }_{\text{m}}}.\end{equation} \tag{ B8 }$$

To infer the mode temperature during the BAE power sweep, we proceed as follows. We use the first form of equation (B6), and fit the measured flux linearly with the cooperativity:

$$\begin{equation}{n}_{\mathrm{o}\mathrm{u}\mathrm{t}}^{d}=\mathcal{N}C,\end{equation} \tag{ B9 }$$

where $\mathcal{N}$ is the calibration coefficient. In the ideal BAE case, a linear relation holds between the flux and cooperativity because the measured quadrature is not affected. In the practical case, the linear dependence may well be violated because of technical heating or other issues. If, and when, we find a linear regime at least at the lowest powers where the signal is observable, we know that the mode temperature stays constant in that power regime. The energy of the measured quadrature in this regime is denoted by ${\langle {X}^{2}\rangle }_{0}$, and this value is obtained from the sideband cooling (figure 4) at a given cooling cooperativity. Combining equations (B6) and (B9) gives

$$\begin{equation}\begin{aligned}\hfill \mathcal{N}& =A\frac{4\gamma {\kappa }_{\text{e}}}{\kappa }{\langle {X}^{2}\rangle }_{0},\hfill \\ \hfill {n}_{\mathrm{o}\mathrm{u}\mathrm{t}}^{d}& =\mathcal{N}\frac{\langle {X}^{2}\rangle }{{\langle {X}^{2}\rangle }_{0}}C,\hfill \\ \hfill \langle {X}^{2}\rangle & ={n}_{\mathrm{o}\mathrm{u}\mathrm{t}}^{d}{\langle {X}^{2}\rangle }_{0}\frac{1}{\mathcal{N}C}.\hfill \end{aligned}\end{equation} \tag{ B10 }$$

The effective mechanical spectra and imprecision are obtained similarly based on equation (A22):

$$\begin{equation}{S}_{\mathrm{X},\mathrm{e}\mathrm{ff}}[\omega ]={S}_{\mathrm{o}\mathrm{u}\mathrm{t}}^{d}{\langle {X}^{2}\rangle }_{0}\frac{1}{\mathcal{N}C}.\end{equation} \tag{ B11 }$$

All the error bars represent 2σ confidence limits. The uncertainty of the given numerical values typically is influenced by the statistical uncertainty from fitting a given peak area, and by error propagation in the chain of calibrations. In the thermal sweeps (figure 3), which are the starting point of the calibrations, only the uncertainties of the peak areas are used. For the performance of the BAE, uncertainties of the final values are evaluated by error propagation in the pertaining equations (B10) and (B11), using peak area errors for ${n}_{\mathrm{o}\mathrm{u}\mathrm{t}}^{d}$, and curve fitting error for $\mathcal{N}$. As discussed in the main text, a worst-case estimate is made for the uncertainty of ${\langle {X}^{2}\rangle }_{0}$.