Abstract

We introduce a passively-aligned, flexure-tuned cavity optomechanical system in which a membrane is positioned microns from one end mirror of a Fabry-Perot optical cavity. By displacing the membrane through gentle flexure of its silicon supporting frame (i.e., to ∼80 m radius of curvature (ROC)), we gain access to the full range of available optomechanical couplings, finding also that the optical spectrum exhibits none of the abrupt discontinuities normally found in “membrane-in-the-middle” (MIM) systems. More aggressive flexure (3 m ROC) enables >15 μm membrane travel, milliradian tilt tuning, and a wavelength-scale (1.64 ± 0.78 μm) membrane-mirror separation. We also provide a complete set of analytical expressions for this system’s leading-order dispersive and dissipative optomechanical couplings. Notably, this system can potentially generate orders of magnitude larger linear dissipative or quadratic dispersive strong coupling parameters than is possible with a MIM system. Additionally, it can generate the same purely quadratic dispersive coupling as a MIM system, but with significantly suppressed linear dissipative back-action (and force noise).

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

In the field of optomechanics, radiation forces have provided previously inaccessible control over mechanical objects of all sizes, with systems increasingly often exhibiting quantum properties of motion and light [1]. Within this context, the so-called “membrane-in-the-middle” (MIM) geometries [2, 3] have enjoyed great success, owing in large part to (i) separating the tasks of fabricating high quality mirrors and low-noise mechanical elements, and (ii) the exquisite properties of Si3N4, which is capable of achieving extremely low force noise [4–6], high Q × f products [4, 7], and optical losses compatible with ultrahigh cavity finesse ∼ 106 [5, 8, 9]. To date, MIM systems exhibiting linear dispersive coupling (wherein the membrane’s displacement modulates the cavity’s resonant frequency linearly) have been used to laser-cool millimeter-scale membranes to the quantum realm [10, 11] and the back-action limit [12], observe quantum radiation pressure noise [13], squeeze light [14,15], approach the standard quantum limit [16], and transfer information between microwave and optical carriers [17,18].

Since the membrane inherently interacts with two cavity modes, it is also possible to generate purely quadratic dispersive coupling [2, 9], providing access to stable optical springs [19] applicable to enhancing mechanical Q-factors [20–22] or tuning the shape of a mechanical mode [23], and unique squeezing opportunities [24]. Additionally, the existence of purely quadratic dispersive coupling suggests the possibility of quantum nondemolition (QND) readout of the membrane’s phonon number states [2,25]. However, even with a lossless single-port cavity, the linear dissipative coupling (wherein the membrane displacement modulates the cavity’s decay rate linearly) imposes the yet-illusive requirement of single-photon strong coupling to resolve an energy eigenstate before it is demolished [26,27]. On the other hand, such dissipative coupling could enable ground-state cooling in the “fast-cavity” limit [28], provide more squeezing opportunities [29], and generate stable optical springs (studied thus far in the Michelson-Sagnac geometry [30,31]).

Finally, fiber-coupled micro-mirrors have enabled MIM systems to achieve orders of magnitude stronger optomechanical coupling by reducing the cavity length from centimeters to ∼ 80 μm [32]; one outstanding goal is to create a stable, wavelength-scale MIM cavity, thereby achieving a per-photon force comparable to that normally associated with nanoscale photonic/phononic crystals – which themselves have achieved impressive results (e.g., [1, 33–37]) – though deep within the fast-cavity limit. Wavelength-scale membrane-membrane separations (currently limited to ∼ 8 μm [38]) are also desirable for nested cavity systems, wherein the optomechanical coupling can be enhanced [39], potentially even to the single-photon strong coupling limit [40].

Here, we present a simple geometry in which a membrane is positioned very close to one end mirror of a Fabry-Perot cavity. In Sec. 2, we derive analytical expressions for the leading-order optomechanical couplings, notably finding that this “membrane-at-the-edge” (MATE) geometry could be used to alleviate the requirement of single-photon strong coupling for QND readout, or to generate larger optomechanical couplings. We then demonstrate and characterize a simple, passively aligned, flexure-tuned MATE system. As discussed in Sec. 3, gently flexing the membrane’s silicon supporting frame (to 80 m radius of curvature (ROC)) allows us to smoothly tune the form of the optomechanical coupling through its full range. Advantageously, we observe a smooth variation in the cavity response over wavelength-scale membrane displacements, with none of the discontinuities normally found in MIM systems (consistent with our model). In Sec. 4, we show how aggressive flexure (3 m ROC) enables large (> 15 μm) membrane travel and a mirror-membrane separation comparable to the laser wavelength. By applying pressure at a single point, we also tune the membrane’s tilt (relative to the mirror surface) by 0.7 mrad over the full travel range while protecting the membrane from collision with the mirror. Importantly, this monolithic geometry poses significantly fewer technical challenges than alignment with multi-axis stages, and, for small ROC, reduces susceptibility to dust at small separations, as in convex lens induced confinement (CLIC) systems [41]. In addition to the aforementioned advantages, this work paves the way toward flexure-tuned fiber-coupled Fabry-Perot MIM systems and membrane-membrane cavities at the wavelength scale.

2. Optomechanics with a membrane at the edge

A MATE system exhibits both quantitative and qualitative differences from a MIM system, the most essential of which are captured by a 1D scattering model drawn in Fig. 1(a). Extending the formalism of [25] (Appendix A), we consider a cavity comprising two fixed end mirrors (gray) separated by a length L, with reflection (transmission) coefficients −rj (itj). A membrane (blue) having reflection (transmission) coefficients rm (tm) is positioned in between. For high-finesse cavities (rj → 1), the right-moving (Aj) and left-moving (Bj) fields of each sub-cavity differ only by a phase, and, for long cavities Lλ (where λ = 2π/k is the light’s wavelength and k is its wavenumber), the cavity’s resonant frequencies are

ωMIMckMIM=NωFSR+ωFSRπ(arccos[(1)N+1|rm|cos(2kNΔx)]ϕr)
for a MIM geometry (small displacements Δx from the cavity center), and
ωMATEckMATE=NωFSR+ωFSRπarctan[cos(ϕr)+|rm|cos(2kNΔx)sin(ϕr)|rm|sin(2kNΔx)],
for a MATE geometry (small displacements Δx from either end mirror), where kMIM and kMATE are the resonant wavenumbers, ωFSR = πc/L is the empty cavity’s free spectral range, ϕr is the phase change for light reflected from the membrane, and kN = πN/L is the Nth empty-cavity resonance, with N ≫ 1. These frequencies are plotted in Fig. 1(b) for the MIM system and Fig. 1(c) for the MATE system. In both cases, the membrane divides the cavity into left and right sub-cavity modes (each having resonant frequencies indicated by black dashed lines) and hybridizes them via its transmission. This produces the shown networks of avoided crossings that define the dispersive optomechanical coupling.

 

Fig. 1 Optical resonances for a membrane in a cavity. (a) 1D model, comprising two fixed end mirrors (gray), with reflection (transmission) coefficients −rj (itj), and a thin, flexible dielectric membrane (blue) having reflection (transmission) coefficients rm (tm). Aj (Bj) indicate the right-moving (left-moving) field amplitudes just outside the membrane. (b)–(c) Dependence of the cavity resonance’s detuning (normalized by the free spectral range (FSR)) on membrane displacement Δx from (b) the cavity center (x=L/2) and (c) the first mirror (x=0), with rm values (from light to dark) −0.4, −0.6, −0.8, −0.931, and −0.9977 (rm’s phase chosen to highlight avoided crossings as in [2]). Horizontal dashed lines represent empty cavity resonant frequencies (rm=0), while other dashed lines represent the left (negatively sloped) and right (positively sloped) sub-cavity resonances when tm=0. Insets qualitatively show the field distribution of these modes. (d)–(e) Dependence of cavity’s energy decay rate κ on membrane displacement Δx from (d) the cavity center and (e) the first mirror, normalized by the empty cavity value for a single-port cavity (t2=0, r2=−1).

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At the same time, the x-dependence of the power landing on each end mirror (see inset sketches in Fig. 1(b)–(c)) leads to an x-dependent cavity decay rate

κ=(1|rm|2)c|t1|2+(1+2|rm|cos(2kx+ϕr)+|rm|2)c|t2|22x(1|rm|2)+2(Lx)(1+2|rm|cos(2kx+ϕr)+|rm|2).
for any location x in the cavity, where k (here) is the resonant wavenumber. This expression is plotted for the “single-port” (|r1| < 1, r2 → −1) case in Fig. 1(d) with k = kMIM, and in Fig. 1(e) with k = kMATE and the membrane positioned a distance ΔxL from the first “input” mirror. This highlights the aforementioned linear dissipative coupling (non-zero xκ) occurring at positions where the dispersive coupling is purely quadratic. As discussed in Appendix A, however, xκ at these locations can be suppressed by a factor 2Δx/L ≪ 1 simply by positioning the membrane a distance Δx from the second “backstop” mirror. This similarly suppresses the added radiation force noise, thereby relaxing the requirement of single-photon strong coupling for QND phonon number readout (the subject of a forthcoming quantum treatment); this suppression can also benefit the pursuit of low-noise optically trapped (levitated) systems exploiting quadratic dispersive coupling. On the other hand, the linear dissipative “strong coupling” parameter (xκ)xzpf/κ (a unitless figure of merit that, when greater than one, indicates the strong coupling regime, where xzpf is the membrane’s zero-point motion) is twice that of a MIM system when the membrane is instead positioned near the first “input” mirror.

The large asymmetry of the sub-cavities enables two additional enhancements. First, while the value x2ω at the extrema of ωMATE is identical to that of the MIM system, moving the membrane away from the extrema (toward the steep side of the sawtooth in Fig. 1(c)) provides an increase in x2ω by a factor as large as ∼ 1/|tm|3 in the low-|tm| limit. This comes at the expense of increased κ, but nonetheless corresponds to an increase in the corresponding strong coupling parameter (x2ω)xzpf2/κ by ∼ 1/|tm|. Second, again by moving the membrane away from the extrema, the dissipative strong coupling parameter (xκ)xzpf/κ can be similarly increased by a factor ∼ 1/|tm|. Note these enhancements are most relevant for cavities incorporating a high-reflectivity mechanical element, such as a structured membrane [42,43] (unpatterned SiN structures can only produce few percent enhancement, while higher-index materials (e.g., Si) can achieve ∼50%), but the same formalism is relevant for coupled transverse modes [9] having very different linear couplings and nearfield systems on chip [44].

Finally, as a practical concern, we emphasize that the x-dependence of ωMATE is approximately identical for adjacent optical modes, meaning the full spectrum of cavity modes should exhibit no abrupt avoided crossings with higher-order transverse modes. This will yield consistent cavity performance over a large range of Δx, easing the tasks of translating the membrane while locked (e.g., to tune the optomechanical coupling) and / or tracking large-amplitude vibrations.

3. Optical resonances of flexural MATE system

The flexural MATE system under study is drawn in Fig. 2(a). The left sub-cavity comprises a flat input mirror (M1) with four 0.2-mm-diameter, 21-μm-thick photoresist (AZ 40XT photoresist, spin coated at 3000 rpm, baked on a hot plate at 120°C for 240 s, exposed at 450 mJ/cm2 UV, baked on a hot plate at 120°C for 120 s, and developed in AZ 300 MIF for 180 s) spacers positioned a distance 11 mm from the mirror center. A 24 mm × 24 mm, 0.65-mm-thick silicon chip supporting a 1 mm × 1 mm Si3N4 membrane of thickness 88±3 nm is pressed against the spacers by three piezo-driven ∼mm-diameter hemispherical “pushers” positioned a radius 3 mm from the mirror center. We choose these “thick” photoresist spacers to later test the durability of the chip under extreme flexure (Sec. 4), and to reduce sensitivity to dust particles that might preclude compression to small membrane-mirror separations. The right sub-cavity boundary is defined by a high-reflectivity backstop mirror (M2) a distance L = 10 cm from M1, that can be swept with a second set of piezo actuators.

 

Fig. 2 Optical response of flexural MATE system. (a) Diagram of measurement system and MATE cavity (not to scale), comprising an input mirror M1, 21 μm-thick spacers, a membrane, piezo-actuated pushers, and a “backstop” mirror M2. 10 mW of laser light (λ = 1550 nm) from a fiber collimator C passes through a 50:50 splitter, is reflected from M1, and collected with a photodiode PD. The length L = 10 cm between M1 and M2 is tuned with piezo actuators on M2; the inset shows the fractional reflected power (normalized by the off-resonance value 2.5 mW) as a function of the laser’s detuning from an L-swept cavity resonance. The violet curve is a “typical” Lorentzian fit used to extract the cavity’s energy decay rate κ. (b) Same measurement for a range of L at varied membrane displacements Δx. The white dashed line shows part of a simultaneous fit to three modes’ resonant frequencies (Eq. 2) used to eliminate piezo nonlinearities, and the arrow shows the location and direction of the sweep in (a). Higher-order transverse modes appear as faint resonances. (c) Spectrum predicted by a 1D transfer matrix model, with parameters estimated as described in the main text. (d) Dependence of cavity decay rate κ on membrane position for the central TEM00 (brightest) resonance in (b). The solid line is κ(x) obtained by numerically solving the 1D model, and the black part highlights the region used for our main fits (including the piezo corrections). The dashed line is the fit empty cavity decay rate 2π/(|t1|2 + |t2|2 + S1) (agrees with our measured value prior to incorporating the membrane). (e) Dependence of resonant reflection on membrane position for the middle TEM00 resonance in (b). The solid line shows the fit result as in (d).

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We characterize the optical modes of the cavity system by reflecting 5 mW of laser light (λ = 1550 nm) from M1 while sweeping M2. On resonance, the cavity’s internal losses (including M2’s transmission) reduce the reflected power, as shown in Fig. 2(a). For convenience, we have mapped the backstop mirror displacement onto laser detuning and normalized the reflected power by its off-resonance value. The present cavity is overcoupled (|t1|2 dominates the cavity loss), verified with heterodyne analysis, and (more simply) by noting the initial reduction in resonantly reflected power when moving an absorber into the cavity mode. Figure 2(b) shows the same measurement (now swept along the vertical axis) over a wide range of backstop and membrane positions. Following the method of [5], we simultaneously fit the resonant detunings of 3 modes to the expression for ωMATE in Eq. 2, including a fourth-order polynomial stretching function along both axes to compensate for piezo nonidealities (part of this fit is shown as a white dashed curve). Assuming the membrane’s refractive index nSiN = 2.0, this provides an additional estimate of the membrane’s thickness d = 81.347 ± 0.008 nm. Despite the qualitative differences between the MATE and MIM systems’ optical mode spectra, this approach still systematically underestimates d, as noted in MIM systems [5,9].

The locations, depths, and widths of the observed resonances can be quantitatively captured with our 1D model, the results of which are shown in Fig. 2(c). We model M1’s internal loss (absorption and scattering) with a single-pass power attenuation factor eS1 (where S1 is a real-valued constant), at the mirror-air interface inside the cavity [42], while M2’s internal loss is combined with its transmission into a larger effective |t2| (for reflection measurements, it is not useful to distinguish the two losses). We also include a “mode-matching” parameter defined as the fractional input power that actually couples to the TEM00 cavity modes [45]; the other fraction (1 − ) simply reflects from the cavity while resonant with the TEM00 modes, but couples to higher-order transverse modes at other detunings (additional faint resonances in Fig. 2(b)). We assume the membrane has negligible optical losses [5,8,9], a refractive index nSiN = 2.0, and thickness d = 88 nm as measured with a white-light interferometer. Finally, by simultaneously fitting the observed dependence of linewidth (Fig. 2(d)) and resonant reflection (Fig. 2(e)) on membrane position, we estimate = 0.75 ± 0.05, |t1|2 = 7.5 ± 0.3 × 10−3, S1 = 8.0 ± 0.8 × 10−4, and M2 total loss |t2|2 = 6 ± 1 × 10−4. Our estimated value of |t1|2 is consistent with M1’s specified power reflectivity (> 0.99), and we infer that this simple photolithography protocol is compatible with cavity finesse of at least 2π/S1 ∼ 8000. For higher-finesse systems, we recommend harder spacers fabricated with a less invasive method, such as deposition through a shadow mask.

In addition to validating the model, Fig. 2 illustrates that it is straightforward to tune the dissipative and dispersive optomechanical coupling over the full range by flexure, which is perhaps not surprising: in the chosen geometry, the required membrane displacement of 775 nm can be achieved with 80 m ROC, which is comparable to the natural curvature (∼100 m) induced by coating one side of the Si wafer with 100 nm of stoichiometric Si3N4. We also emphasize that the cavity mode frequency, decay rate, and reflected power are all observed to be smooth functions of membrane position; this is a significant departure from MIM systems, wherein avoided crossings with higher-order transverse modes tend to cause a multitude of sudden changes or discontinuities in these quantities as the membrane is swept over a similar range, even in well-aligned systems [5,46,47]. Furthermore, our simple 1D model, which uses a single value of to describe the entire data set, quantitatively captures the observed (smooth) oscillations in Figs. 2(d)–(e), highlighting that MATE resonances are well-approximated by a single transverse mode.

4. Large flexure

The chip can also be flexed far enough to contact M1, and, by engaging a single pusher, make modest adjustments to the tilt while protecting the membrane from collision. Figure 3 summarizes our observations over the full range of displacements. To measure the membrane-mirror separation and tilt, we move the photodiode inside the cavity to measure the transmission through the stack. As discussed in Appendix B, the transmitted power Pt depends on the membrane-mirror distance x0 (at center of the incident beam) and tilt θ as

Pt|t1|2|tm|21+|r1|2|rm|22|r1||rm|cos(2kx0+ϕ){1k2θ2σ2|r1||rm|(1+|r1|2|rm|2)cos(2kx0+ϕ)+|r1||rm|(cos(4kx0+2ϕ)3)[1+|r1|2|rm|22|r1||rm|cos(2kx0+ϕ)]2},
where k = 2π/λ is the laser wavenumber, σ = 100 μm is the measured beam radius, and ϕ = ϕ1 + ϕm is the sum of reflection coefficient phases for M1 (ϕ1) and the membrane (ϕm). For fixed λ = 1550 nm and swept x0, we observe transmission peaks every λ/2, allowing simple measurement of relative displacements, but the longitudinal index l0 of the first observed peak is not known initially. Centering x0 at one such peak, we can roughly estimate x0 and θ by sweeping λ over a range that is small compared to M1’s bandwidth, producing spectra like those inset to Fig. 3; qualitatively, reducing x0 widens the peaks while increasing θ reduces the height, as per Eq. 4. To more accurately estimate l0, we simultaneously fit all transmission spectra as follows. We fix the mirror M1’s phase to ϕ1 = π (varying this within reason does not significantly affect our estimate), the membrane thickness d = 88 nm, and Si3N4 index nSiN = 2. We assume θθ0AΔx0 for a geometrical constant A and initial tilt θ0 (at mode l0), and treat the global constants |r1|2, θ0, and A, along with each spectrum’s peak wavelength as free parameters. We then select the (integer) value of l0 that minimizes χ2. Doing so yields l0 = 24 ± 1, |r1|2 = 0.9935 ± 0.0001, θ0 = 0.18 ± 0.08 mrad, and A = 0.040 ± 0.006 mrad/μm, corresponding to the solid curves in Fig. 3. Uncertainties are dominated by a conservative assumed range of possible membrane thickness d = 88 ± 3 (combined range from our white light interferometer and ellipsometer), which defines the shaded region in Fig. 3(top). For tilt, the dominant uncertainties arise from both the the membrane thickness and the exact details of M1’s coating (the dark shaded region includes the same fit with leading wavelength-dependence of r1). The ∼10% uncertainty in the beam spot-size corresponds to a (smaller) uncertainty as is shown by the light shaded area. As a consistency check, the orange points show the results of individual transmission spectra fits assuming |r1|2 = 0.9935 with θ and l0 as free parameters; this also gives a sense of each spectrum’s ability to resolve these quantities.

 

Fig. 3 Large flexure. Length x (top) and tilt θ (bottom) as a function of the mode index l. The starting index l0 is estimated from a fit to the transmission spectra (upper inset) described in the main text. The uncertainty on l0 is ±1, as represented by the shaded area. Below l=8, the transmission data is too broad to constrain x, and “×” symbols represent the passing of a bright fringe for further flexure. The lower inset shows the interference pattern (1310 nm light) of the mirror-membrane system while maximally flexed; dotted square represents the approximate location and size of the membrane. The dark shaded area corresponds to the uncertainty in membrane thickness and M1’s coating (included via transfer matrix calculation). The tilt was calculated assuming the measured beam radius σ = 100 μm, and the light shading corresponds to a conservative upper bound (σ = 110 μm) on the actual value at the membrane.

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Below l = 8, the spectra are too broad to reliably estimate the cavity length, but we count 5 more fringes (“×” symbols) before the chip contacts M1. At our furthest flexure (l = 3), we estimate x0 = 1.64 ± 0.78 μm. The tilt is observed to vary linearly by 0.5 milliradians over 12.4 μm (16 modes) displacement, as expected and consistent with the 3 mm lateral offset between the membrane and engaged pusher. Extrapolating to l = 3 implies a total added tilt of ∼0.7 milliradians, and the maximally displaced geometry corresponds to radius of curvature Rframe = 3 m.

These estimates are corroborated by shining a widely-collimated, low-coherence λ = 1310 nm laser on the input mirror, and collecting the reflected interference image with a CCD, an image of which is shown in Fig. 3(inset); this image is taken with the chip contacting M1, and the approximate size and location of the membrane is indicated by a dashed box. The membrane is ∼3 fringes from the center, suggesting a mirror-membrane separation of ∼2 μm, a few-mm offset from the point of minimal separation, and a tilt ∼ 1 milliradians, consistent with the above estimates.

5. Conclusions

We have demonstrated a passively aligned, flexure-tuned membrane-at-the-edge optomechanical system. Gentle flexure (80 m ROC) of the silicon chip accesses the full range of available optomechanical couplings, and extreme flexure (3 m ROC) accesses a wavelength-scale separation between a membrane and flat mirror. This monolithic, flexural approach defines a rigid mirror-membrane gap while still allowing for in situ adjustment. By applying pressure asymmetrically, we protect the membrane from contacting the mirror while keeping the tilt to within a milliradian (even over the large membrane travel), which is sufficient to avoid significantly reducing the optomechanical coupling [9]. On the other hand, for the chosen 3 mm lateral offset between the membrane and point of closest approach, we can use flexure to tune the tilt by as much as 0.7 mrad. If applied to a MIM system, this enables one to make fine in situ adjustments to the membrane-mediated coupling between transverse cavity modes.

The relationship between the membrane’s mechanical quality factor and the frame’s ROC (in a vacuum system) remains to be seen. However, we expect flexure will not play an important role for gentle distortions of Fig. 2, and, for high-Q trampolines [5,6], we expect the dominant effect of flexure to be increased tension in the tethers, since chip distortions will essentially serve to separate the point-like clamps rather than redistributing stress along the edge of a membrane. Note the strain associated with our most aggressive flexure is only 10−4, which would increase the Si3N4 stress by tens of MPa, thereby changing the resonant frequency at the percent scale.

Within the context of MATE systems, these results are of interest for generating low-noise, purely quadratic dispersive coupling, and stronger forms of quadratic dispersive or linear dissipative couplings, while (at the same time) suppressing abrupt changes in the cavity mode as the membrane moves. Equally interestingly, it should be straightforward to realize a rigid, tunable, MIM system at wavelength scale by positioning a small back mirror (e.g., a fiber mirror [32, 48]) within the silicon etch pit. Finally, we suggest that, by instead depositing spacers on the supporting frame of a second membrane, it should be straightforward to create wavelength-scale two-membrane cavities useful for (among other things) increasing the coupling in nested cavity geometries.

Appendices

A. Membrane in cavity: scattering model in 1D

In this section, we present a simple 1D scattering model describing the membrane-cavity system shown in Fig. 1(a). Section A.1 reviews the high-finesse (closed-cavity) approximation, Sec. A.2 applies this to relate cavity length, membrane position, and the wavenumber (frequency) of resonant light, providing analytical expressions in the MIM and MATE limits. Incorporating a small end mirror transmission, we then derive expressions for the cavity’s dissipation in Sec. A.3. Finally, Secs. A.4 and A.5 derive expressions for the dissipative and dispersive optomechanical couplings, discussing the advantages of the MATE system. Of note, the linear dissipative coupling can be suppressed by many orders of magnitude at locations of purely quadratic dispersive coupling, providing a potentially simpler pathway toward QND readout of mechanical energy [2,25] in a single-port cavity, but without the requirement [27] of strong single-photon optomechanical coupling. Additionally, for low-transmission (tm) membranes (e.g., Refs. [6,42,43]), the single-photon strong quadratic dispersive and linear dissipative coupling parameters can both be enhanced by a factor ∼ 1/|tm|.

A.1. Setup: high-finesse cavity-membrane system

We consider the Fabry-Perot geometry drawn in Fig. 1(a), comprising two end mirrors (gray) separated by a distance L, and a partially reflective membrane (blue) positioned a distance x from the first mirror. The right (left) moving free-space field amplitudes Aj (Bj) at the membrane surfaces are related by the membrane’s coefficients of transmission (tm) and reflection (rm) as

A2=tmA1+rmB2
B1=tmB2+rmA1.
For the case of a high-finesse cavity, the end mirror reflectivities r ≈ −1, meaning the fields are also related by the round trips to the end mirrors and back as
A1=B1e2ikx
B2=A2e2ik(Lx),
where k = 2π/λ is the light’s wavenumber, and λ is its wavelength. This closed-cavity (high-finesse) approximation enables us to derive the cavity’s resonant frequency, dissipation, and corresponding optomechanical couplings.

A.2. Resonant conditions

Eliminating the fields Aj and Bj from the above four equations yields a transcendental equation constraining the cavity length, membrane position, and wavelength:

(tm2rm2)eikLeikL=2rmcos(2kxkL).
This is “readily” solved numerically for a membrane of arbitrary (complex-valued) reflection and transmission coefficients. However, if we assume a lossless membrane (|tm|2 + |rm|2 = 1, which is a good approximation for Si3N4 at infrared wavelengths [8, 9]), unitarity then imposes that |A1|2 + |B2|2 = |A2|2 + |B1|2. Expressing tm = |tm|et and rm = |rm|er (for real-valued phases ϕt and ϕr), this means e2i(ϕtϕr) = −1, and the above transcendental simplifies to
cos(kL+ϕr)=|rm|cos(2kxkL).
Expanding the cosines enables one to solve for the resonant length
L=1karctan[cos(ϕr)+|rm|cos(2kx)sin(ϕr)|rm|sin(2kx)].

To derive analytical expressions for the resonant frequency (i.e., the convenient quantity for calculating dispersive coupling) requires further approximation. We now consider the “long cavity” limit Lλ/2, such that the membrane perturbs the resonant wavenumber k from that of the Nth empty cavity resonance kN = 2π/λN = /L (where N ≫ 1, and λN is the Nth empty cavity resonance wavelength) by a comparatively small amount ΔkkN as

k=kN+Δk.
In this limit, it is possible to derive analytical expressions for the resonant k (and frequency ω = ck) of MIM and MATE systems.

MIM:

Summarizing the method of [25], we approximate Eq. 10 for small displacements ΔxL from the cavity center L/2. To leading order,

kΔxkNΔx+ΔkΔxkNΔx,
such that Eq. 12 in Eq. 10 yields a resonant wavenumber
kMIMNπL+1L(arccos[(1)N+1|rm|cos(2kNΔx)]ϕr).
and frequency
ωMIMNωFSR+ωFSRπ(arccos[(1)N+1|rm|cos(2kNΔx)]ϕr),
where ωFSR = πc/L is the empty cavity’s free spectral range (FSR). Figure 1(b) shows the x-dependence of ωMIM for several values of N and rm. Note for these plots we have followed the convention of [2], choosing rm = −|rm| and tm = i|tm|, to facilitate the visualization of avoided crossings. The choice of phase ϕr and ϕt shifts all modes by at most a free spectral range, and does not affect our conclusions. The convention of [25], e.g., is to assume an infinitesimally thin membrane of varied index. This represents a closer approximation to the perturbation from a thin dielectric slab [9,49], but for other structures, such as photonic crystal reflectors [42,43], the phases ϕr and ϕt can vary significantly from either approximation.

MATE:

If the membrane is instead positioned a distance ΔxL from the left (input) mirror,

kΔx=kNΔx+ΔkΔxkNΔx
such that Eq. 12 in Eq. 11 yields the wavenumber
kMATENπL+1Larctan[cos(ϕr)+|rm|cos(2kNΔx)sin(ϕr)|rm|sin(2kNΔx)],
and resonant frequency
ωMATENωFSR+ωFSRπarctan[cos(ϕr)+|rm|cos(2kNΔx)sin(ϕr)|rm|sin(2kNΔx)].
This is plotted for several values of rm and N in Fig. 1(c).

Comparison:

The spectra of MIM and MATE resonances comprise a network of transmission-mediated avoided crossings between the left sub-cavity mode frequencies (having negative slope) and right sub-cavity mode frequencies (having positive slope). For the MATE case, the left sub-cavity is comparatively infinitesimal, leading to large dispersive optomechanical coupling, especially in the low-tm limit. As discussed in Sec. A.5, the coupling strength is practically limited by tm, which prevents light from remaining in the left sub-cavity indefinitely.

In order to derive expressions for the strong dispersive optomechanical coupling parameters, we must first calculate the dissipation of these cavity modes.

A.3. Cavity dissipation

The quantities |Aj|2 and |Bj|2 calculated from Eqs. 58 are proportional to the power landing on the end mirrors, which, combined with the left (right) end mirror transmissions |t1|2 (|t2|2), enables an estimate of the cavity’s energy decay rate κ as follows. Power circulating in the left (right) sub-cavity P1 (P2) corresponds to stored energy

E=2xcP1+2(Lx)cP2,
which will leak out of the end mirrors at a rate
tE=P1|t1|2P2|t2|2.
We can eliminate P2 from these equations by plugging Eqs. 78 into Eqs. 56 and taking the ratio
|A2B1|2=P2P1=1+|rm|2+2|rm|cos(2kx+ϕr)1|rm|2.
Substituting this into the energy (Eq. 19) and rearranging,
P1=cE2(x+(Lx)1+|rm|2+2|rm|cos(2kx+ϕr)1|rm|2)1,
which yields a leak rate (Eq. 20)
tE=κE,
with
κ=(1|rm|2)c|t1|2+(1+2|rm|cos(2kx+ϕr)+|rm|2)c|t2|22x(1|rm|2)+2(Lx)(1+2|rm|cos(2kx+ϕr)+|rm|2).
Note this expression is valid for any position x, and can be used to normalize the dissipative (Sec. A.4) and dispersive (Sec. A.5) couplings for either geometry.

A.4. Dissipative optomechanical coupling

Here we use Eq. 24 to extract analytical expressions for the linear dissipative strong coupling parameter [28,50,51]

B˜1κdκdxxzpf,
where xzpf=/(2mΩ) is the membrane’s zero-point motion, m is its mass and Ω is its resonant frequency. Reminding ourselves of the resonant wavenumber’s position dependence k(x), the general analytical expressions for are cumbersome. However, for a single-port cavity (|t1|2 ≈ 0 or |t2|2 ≈ 0, required, e.g., for dissipative ground state cooling [28]) and a high-reflectivity membrane (|tm| ≪ 1), the expressions simplify dramatically.

MIM:

If t2 = 0, the dissipative coupling (Eqs. 25 and 24) becomes

B˜MIM=2|rm|L|rm|+cos(2kMIMx+ϕr)+(kMIM+xdkMIMdx)Lsin(2kMIMx+ϕr)1+|rm|cos(2kMIMx+ϕr)xzpf.
In the limit |tm| ≪ 1 with kMIM from Eq. 14, this takes on maximal values
B˜MIM,max|tm|1332kNxzpf1|tm|,
occurring at positions
ΔxMIM,max(B˜)|tm|112kN(jπ+(1)j+N+1|tm|3)
that are sightly shifted slightly away from Δx = /4 to favor the right-cavity (positive xω) mode, where κ is somewhat lower but xκ has not yet decreased by much.

MATE:

If the membrane is instead positioned near the input mirror (keeping t2 = 0), the approximation of Eq. 16 reduces the dissipative coupling (Eq. 25) to

B˜input2|rm|L|rm|+cos(2kNΔx+ϕr)+2kNLsin(2kNΔx+ϕr)1+|rm|2+2|rm|cos(2kNΔx+ϕr)xzpf.
For a high-reflectivity membrane, this takes on a maximal value
B˜MATE,max|tm|14kNxzpf1|tm|2
at locations
ΔxMATE,max(B˜)|tm|114kN(2π(2j+1)2ϕr±|tm|2).

Comparison:

Importantly, the MATE geometry enables a dissipative coupling enhancement as high as

B˜MATE,maxB˜MIM,max|tm|18331|tm|.
We emphasize that this gain is for the strong dissipative optomechanical coupling parameter, and note that these membrane positions also have non-zero linear dispersive coupling xω in general. Positions at which xω vanishes are discussed at the end of the next section.

A.5. Dispersive optomechanical coupling

Here we present expressions for the linear and quadratic dispersive couplings, along with their associated strong coupling parameters.

MIM:

Equation 15 leads to a linear (GMIM(1)) and quadratic (GMIM(2)) dispersive couplings

GMIM(1)xωMIM=(1)N+12ωFSRkNπ|rm|sin(2kNΔx)1|rm|2cos2(2kNΔx)
GMIM(2)x2ωMIM=(1)N+14ωFSRkN2π|rm|(1|rm|2)cos(2kNΔx)(1|rm|2cos2(2kNΔx))3/2,
with extremal values
GMIM,max(1)=±2ckNL|rm|
GMIM,max(2)=±4ckN2L|rm|1|rm|2|tm|1±4ckN2L1|tm|
at locations
ΔxMIM,max(1)=(2j+1)λ8
ΔxMIM,max(2)=jλ4,
respectively (j ∈ ℤ). For this geometry, the maximal values for quadratic coupling occur when the membrane is at a node or antinode of the empty cavity field, and the maximal linear coupling occurs at the midpoints in between. Also, the maximal values for each occur when the coupling of the other is zero, as can be seen in Fig. 1(b).

MATE:

Equation 18 leads to linear (GMATE(1)) and quadratic (GMATE(2)) dispersive couplings

GMATE(1)xωMATE=2kNπωFSR|rm|(|rm|+cos(2kNΔx+ϕr))|rm|2+2|rm|cos(2kNΔx+ϕr)+1
GMATE(2)x2ωMATE=4kN2πωFSR|rm|(1|rm|2)sin(2kNΔx+ϕr)(|rm|2+2|rm|cos(2kNΔx+ϕr)+1)2,
with extremal values
GMATE,max(1)=2ckNL|rm|1|rm||tm|14ckNL1|tm|2
GMATE,max(2)=GMATE(2)(ΔxMATE,max(2))|tm|1±183ckN2L1|tm|4
at locations
ΔxMATE,max(1)=(2j+1)πϕr2kN
ΔxMATE,max(2)=12kN(2πjϕr±2arctan{6|rm|+|rm|4+34|rm|2+1(1|rm|)2})
respectively (j ∈ ℤ). Note the maximal quadratic coupling does not occur at the extrema in ωMATE (where GMATE(1)=0). At those “purely” quadratic locations
ΔxMATE,pure(2)=±arccos{|rm|}ϕr+2πj2kN,
the coupling is identical to that of the MIM system (Eq. 36).

Comparison:

As noted elsewhere [52], the maximal linear coupling is larger than that of the MIM system by a factor

GMATE,max(1)GMIM,max(1)=11|rm||tm|12|tm|2,
but it comes at the expense of larger κ, due to the associated cavity mode’s higher intensity in the (shorter) cavity (see inset of Fig. 1(c)). As such, there is no difference in the maximum possible strong coupling parameter ā(1) = −G(1)xzpf/κ [52], and
A˜max,MIM(1)=A˜max,MATE(1)|tm|18kNxzpf|t1|2|tm|2.
On the other hand, the maximal quadratic coupling can be enhanced by a larger factor
GMATE,max(2)GMIM,max(2)|tm|19231|tm|3.
This improvement also comes with increased κ, but the strong coupling parameter (A˜(2)=G(2)xzpf2/κ) can still be improved, as
A˜max,MATE(2)A˜max,MIM(2)|tm|14331|tm|.
We note the caveat that these MATE expressions assume the limit 4Δx/L ≪ |tm|2, such that the membrane’s transmission limits the coupling rates (i.e., not the mirror-membrane separation).

Purely quadratic dispersive coupling:

When the membrane resides at locations having purely quadratic dispersive coupling (to lowest order), the remaining position dependence of the cavity mode still represents a linear back-action that can preclude quantum nondemolition (QND) readout of mechanical energy eigenstates [26], even in an ideal one-port cavity [27]. For a MIM system with t2 = 0 (no restrictions on membrane reflectivity), Eqs. 38 and 26 yield a dissipative coupling

B˜MIM,pure=±2kNxzpf|rm||tm|
at these “purely quadratic” points. For the MATE system with the membrane near the input mirror, the dissipative coupling at these points is
B˜MATE,pure=±4kNxzpf|rm||tm|
which is twice as large as for the MIM system. More compellingly, if the membrane is positioned near the back mirror, where LxL, the dissipative coupling at these points
B˜MATE,pure=±4kNxzpfΔxMATE,pureL|rm||tm|,
a reduction by the factor 2ΔxMATE,pure/L relative to the MIM case discussed in [26,27], thereby suppressing the unwanted linear back-action by many orders of magnitude.

B. Fabry-Perot cavity with tilted end mirror

Here we derive an expression for the power transmitted through a wedged Fabry-Perot cavity comprising mirror M1 and the membrane in Fig. 1(a). Section B.1 reviews a 1D empty cavity (parallel mirrors, plane waves), and Section B.2 treats a wedged cavity as a continuum of such 1D cavities, all having different lengths.

B.1. Empty cavity in 1D

For the mirror-membrane cavity in Fig. 1(a), we can relate the transmitted field A2 to the input field Ein from the left by summing all possible paths the light can take to exit through the membrane. Specifically, every path is transmitted through M1, travels the length x of the cavity, and can make any number j of round trips (reflecting off both mirrors each time) before transmitting through the membrane. Summing all such paths yields

A2=Ein(t1tmeikx)j=0[r1rme2ikx]j
=Eint1tmeikx1r1rme2ikx,
where k is the incident light’s wavenumber, and we have yet made no assumption about r1, t1, rm and tm.

B.2. Empty cavity in 2D with tilt

In this section, we derive the transmission through a mirror-membrane stack when the two surfaces are not parallel, such that the cavity length varies with the transverse position y as x = x0 + θy, where x0 is the nominal separation and θ is the slope. For convenience, consider an incident Gaussian beam having power density 2πσ2e2y2/σ2 (i.e., normalized to total power 1), with beam radius σ. At a given transverse location y, Eq. 54 then yields a fractional power transmission

pt|A2/Ein|2=|t1|2|tm|21+|r1|2|rm|22|r1||rm|cos(2kx0+2kθy+ϕ)2πσ2e2y2σ2,
where we now employ the convention r1 = |r1|e1 (rm = |rm|em), with ϕ1 (ϕm) being the real-valued phase of M1 (the membrane), and ϕϕ1 + ϕm.

To estimate the total fractional transmission Pt, we can now integrate Eq. 55 over y. To simplify, we assume the tilt is sufficiently small that 2kθy ≪ 1, and expand the previous expression over the whole beam, yielding

pt|t1|2|tm|21+|r1|2|rm|22|r1||rm|cos(2kx0+ϕ)2πσ2e2y2σ2{14k2θ2y2|r1||rm|(1+|r1|2|rm|2)cos(2kx0+ϕ)+|r1||rm|(cos(4kx0+2ϕ)3)[1+|r1|2|rm|22|r1||rm|cos(2kx0+ϕ)]2},
where we have left out the term ∝ y for brevity since it will vanish by symmetry. Integrating over y yields
Pt|t1|2|tm|21+|r1|2|rm|22|r1||rm|cos(2kx0+ϕ){1k2θ2σ2|r1||rm|(1+|r1|2|rm|2)cos(2kx0+ϕ)+|r1||rm|(cos(4kx0+2ϕ)3)[1+|r1|2|rm|22|r1||rm|cos(2kx0+ϕ)]2}.
Most relevantly, this expression describes how tilt reduces and broadens the resonant transmission peaks (occurring at positions satisfying 2kx0 + ϕ ≈ 2πl for integer l), as shown in Fig. 3(inset).

Funding

NSERC (RGPIN 418459-2012, 2018-05635); FRQNT (NC-172619); CFI (228130); CRC (235060); INTRIQ; Centre for the Physics of Materials at McGill.

Acknowledgments

We thank Raphael Saint-Gelais, Yannik Fontana, Erika Janitz, Sasa Ristic and John Li for useful discussions regarding fabrication and optical characterization. VD acknowledges support from the NSERC CGS-M and FRQNT-B1 scholarships. VD and SB acknowledge support from FRQNT-B2 scholarships. SB and CR acknowledge support from Schulich fellowships.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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References

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  1. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
    [Crossref]
  2. J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
    [Crossref] [PubMed]
  3. M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77, 033819 (2008).
    [Crossref]
  4. A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
    [Crossref] [PubMed]
  5. C. Reinhardt, T. Müller, A. Bourassa, and J. Sankey, “Ultralow-noise SiN trampoline resonators for sensing and optomechanics,” Phys. Rev. X 6, 021001 (2016).
  6. R. A. Norte, J. P. Moura, and S. Gröblacher, “Mechanical resonators for quantum optomechanics experiments at room temperature,” Phys. Rev. Lett. 116, 147202 (2016).
    [Crossref] [PubMed]
  7. Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nano. 12, 776–783 (2017).
    [Crossref]
  8. D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble, “Cavity optomechanics with stoichiometric SiN films,” Phys. Rev. Lett. 103, 207204 (2009).
    [Crossref]
  9. J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
    [Crossref]
  10. T. P. Purdy, P.-L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, “Optomechanical Raman-ratio thermometry,” Phys. Rev. A 92, 031802 (2015).
    [Crossref]
  11. M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Børkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
    [Crossref]
  12. R. W. Peterson, T. P. Purdy, N. S. Kampel, R. W. Andrews, P. L. Yu, K. W. Lehnert, and C. A. Regal, “Laser cooling of a micromechanical membrane to the quantum backaction limit,” Phys. Rev. Lett. 116, 063601 (2016).
    [Crossref] [PubMed]
  13. T. P. Purdy, R. W. Peterson, and C. A. Regal, “Observation of radiation pressure shot noise on a macroscopic object,” Science 339, 801–804 (2013).
    [Crossref] [PubMed]
  14. W. H. P. Nielsen, Y. Tsaturyan, C. B. Møller, E. S. Polzik, and A. Schliesser, “Multimode optomechanical system in the quantum regime,” Proc. Natl. Acad. Sci. 114, 62–66 (2017).
    [Crossref]
  15. T. P. Purdy, P.-L. L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X 3, 031012 (2013).
  16. M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, “Measurement-based quantum control of mechanical motion,” Nature 563, 53–58 (2018).
    [Crossref] [PubMed]
  17. R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10, 321–326 (2014).
    [Crossref]
  18. A. P. Higginbotham, P. S. Burns, M. D. Urmey, R. W. Peterson, N. S. Kampel, B. M. Brubaker, G. Smith, K. W. Lehnert, and C. A. Regal, “Harnessing electro-optic correlations in an efficient mechanical converter,” Nat. Phys. 14, 1038–1042 (2018).
    [Crossref]
  19. D. Lee, M. Underwood, D. Mason, A. B. Shkarin, S. W. Hoch, and J. G. E. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Comm. 6, 6232 (2014).
    [Crossref]
  20. K. K. Ni, R. Norte, D. J. Wilson, J. D. Hood, D. E. Chang, O. Painter, and H. J. Kimble, “Enhancement of mechanical Q factors by optical trapping,” Phys. Rev. Lett. 108, 214302 (2012).
    [Crossref] [PubMed]
  21. D. E. Chang, K.-K. Ni, O. Painter, and H. J. Kimble, “Ultrahigh-Q mechanical oscillators through optical trapping,” New J. Phys. 14, 45002 (2012).
    [Crossref]
  22. T. Müller, C. Reinhardt, and J. Sankey, “Enhanced optomechanical levitation of minimally supported dielectrics,” Phys. Rev. A 91, 053849 (2015).
    [Crossref]
  23. A. Barasheed, T. Müller, and J. Sankey, “Optically defined mechanical geometry,” Phys. Rev. A 93, 053811 (2016).
    [Crossref]
  24. A. Nunnenkamp, K. Børkje, J. G. E. Harris, and S. M. Girvin, “Cooling and squeezing via quadratic optomechanical coupling,” Phys. Rev. A 82, 021806 (2010).
    [Crossref]
  25. A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
    [Crossref]
  26. H. Miao, S. Danilishin, T. Corbitt, and Y. Chen, “Standard quantum limit for probing mechanical energy quantization,” Phys. Rev. Lett. 103, 100402 (2009).
    [Crossref] [PubMed]
  27. Y. Yanay, J. Sankey, and A. Clerk, “Quantum backaction and noise interference in asymmetric two-cavity optomechanical systems,” Phys. Rev. A 93, 063809 (2016).
    [Crossref]
  28. F. Elste, S. M. Girvin, and A. A. Clerk, “Quantum noise interference and backaction cooling in cavity nanomechanics,” Phys. Rev. Lett. 102, 207209 (2009).
    [Crossref] [PubMed]
  29. D. Kilda and A. Nunnenkamp, “Squeezed light and correlated photons from dissipatively coupled optomechanical systems,” J. Opt. 18, 014007 (2016).
    [Crossref]
  30. S. P. Tarabrin, H. Kaufer, F. Y. Khalili, R. Schnabel, and K. Hammerer, “Anomalous dynamic backaction in interferometers,” Phys. Rev. A 88, 023809 (2013).
    [Crossref]
  31. A. Sawadsky, H. Kaufer, R. M. Nia, S. P. Tarabrin, F. Y. Khalili, K. Hammerer, and R. Schnabel, “Observation of generalized optomechanical coupling and cooling on cavity resonance,” Phys. Rev. Lett. 114, 043601 (2015).
    [Crossref] [PubMed]
  32. N. Flowers-Jacobs, S. Hoch, J. Sankey, A. Kashkanova, A. Jayich, C. Deutsch, J. Reichel, and J. Harris, “Fiber-cavity-based optomechanical device,” Appl. Phys. Lett. 101, 221109 (2012).
    [Crossref]
  33. P. Arrangoiz-Arriola, E. A. Wollack, Z. Wang, M. Pechal, W. Jiang, T. P. McKenna, J. D. Witmer, and A. H. Safavi-Naeini, “Resolving the energy levels of a nanomechanical oscillator,” arXiv:1902.04681 (2019).
  34. S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).
  35. I. Marinković, A. Wallucks, R. Riedinger, S. Hong, M. Aspelmeyer, and S. Gröblacher, “Optomechanical Bell test,” Phys. Rev. Lett. 121, 220404 (2018).
    [Crossref]
  36. R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473–477 (2018).
    [Crossref] [PubMed]
  37. S. Hong, R. Riedinger, I. Marinković, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, “Hanbury Brown and Twiss interferometry of single phonons from an optomechanical resonator,” Science 358, 203–206 (2017).
    [Crossref] [PubMed]
  38. B. Nair, A. Naesby, and A. Dantan, “Optomechanical characterization of silicon nitride membrane arrays,” Opt. Lett. 42, 1341 (2017).
    [Crossref] [PubMed]
  39. P. Piergentili, L. Catalini, M. Bawaj, S. Zippilli, N. Malossi, R. Natali, D. Vitali, and G. D. Giuseppe, “Two-membrane cavity optomechanics,” New J. Phys. 20, 083024 (2018).
    [Crossref]
  40. J. Li, A. Xuereb, N. Malossi, and D. Vitali, “Cavity mode frequencies and strong optomechanical coupling in two-membrane cavity optomechanics,” J. Opt. 18, 084001 (2016).
    [Crossref]
  41. S. R. Leslie, A. P. Fields, and A. E. Cohen, “Convex lens-induced confinement for imaging single molecules,” Anal. Chem. 82, 6224–6229 (2010).
    [Crossref] [PubMed]
  42. C. Stambaugh, H. Xu, U. Kemiktarak, J. Taylor, and J. Lawall, “From membrane-in-the-middle to mirror-in-the-middle with a high-reflectivity sub-wavelength grating,” Annalen der Physik 527, 81–88 (2015).
    [Crossref]
  43. X. Chen, C. Chardin, K. Makles, C. Caër, S. Chua, R. Braive, I. Robert-Philip, T. Briant, P.-F. Cohadon, A. Heidmann, T. Jacqmin, and S. Deléglise, “High-finesse Fabry-Perot cavities with bidimensional Si3N4 photonic-crystal slabs,” Light. Sci. & Appl. 6, e16190 (2017).
    [Crossref]
  44. T. K. Paraiso, M. Kalaee, L. Zang, H. Pfeifer, F. Marquardt, and O. Painter, “Position-squared coupling in a tunable photonic crystal optomechanical cavity,” Phys. Rev. X 5, 041024 (2015).
  45. C. Hood, H. Kimble, and J. Ye, “Characterization of high-finesse mirrors: Loss, phase shifts, and mode structure in an optical cavity,” Phys. Rev. A 64, 033804 (2001).
    [Crossref]
  46. J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Nonlinear optomechanical couplings: Tools for dealing with solid mechanical objects in the quantum regime,” in Optics InfoBase Conference Papers (2010).
  47. M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” J. Opt. 15, 025704 (2011).
    [Crossref]
  48. D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hänsch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12, 065038 (2010).
    [Crossref]
  49. J. C. Sankey, A. M. Jayich, B. M. Zwickl, C. Yang, and J. G. E. Harris, “Improved “position-squared” readout using degenerate cavity modes,” in ICAP Proceedings (2008), pp. 131–149.
  50. T. Weiss, C. Bruder, and A. Nunnenkamp, “Strong-coupling effects in dissipatively coupled optomechanical systems,” New J. Phys. 15, 045017 (2013).
    [Crossref]
  51. T. Weiss and A. Nunnenkamp, “Quantum limit of laser cooling in dispersively and dissipatively coupled optomechanical systems,” Phys. Rev. A 88, 023850 (2013).
    [Crossref]
  52. T. P. Purdy, R. W. Peterson, P. L. Yu, and C. A. Regal, “Cavity optomechanics with Si3N4 membranes at cryogenic temperatures,” New J. Phys. 14, 115021 (2012).
    [Crossref]

2018 (6)

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref] [PubMed]

A. P. Higginbotham, P. S. Burns, M. D. Urmey, R. W. Peterson, N. S. Kampel, B. M. Brubaker, G. Smith, K. W. Lehnert, and C. A. Regal, “Harnessing electro-optic correlations in an efficient mechanical converter,” Nat. Phys. 14, 1038–1042 (2018).
[Crossref]

M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, “Measurement-based quantum control of mechanical motion,” Nature 563, 53–58 (2018).
[Crossref] [PubMed]

I. Marinković, A. Wallucks, R. Riedinger, S. Hong, M. Aspelmeyer, and S. Gröblacher, “Optomechanical Bell test,” Phys. Rev. Lett. 121, 220404 (2018).
[Crossref]

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473–477 (2018).
[Crossref] [PubMed]

P. Piergentili, L. Catalini, M. Bawaj, S. Zippilli, N. Malossi, R. Natali, D. Vitali, and G. D. Giuseppe, “Two-membrane cavity optomechanics,” New J. Phys. 20, 083024 (2018).
[Crossref]

2017 (5)

X. Chen, C. Chardin, K. Makles, C. Caër, S. Chua, R. Braive, I. Robert-Philip, T. Briant, P.-F. Cohadon, A. Heidmann, T. Jacqmin, and S. Deléglise, “High-finesse Fabry-Perot cavities with bidimensional Si3N4 photonic-crystal slabs,” Light. Sci. & Appl. 6, e16190 (2017).
[Crossref]

S. Hong, R. Riedinger, I. Marinković, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, “Hanbury Brown and Twiss interferometry of single phonons from an optomechanical resonator,” Science 358, 203–206 (2017).
[Crossref] [PubMed]

B. Nair, A. Naesby, and A. Dantan, “Optomechanical characterization of silicon nitride membrane arrays,” Opt. Lett. 42, 1341 (2017).
[Crossref] [PubMed]

W. H. P. Nielsen, Y. Tsaturyan, C. B. Møller, E. S. Polzik, and A. Schliesser, “Multimode optomechanical system in the quantum regime,” Proc. Natl. Acad. Sci. 114, 62–66 (2017).
[Crossref]

Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nano. 12, 776–783 (2017).
[Crossref]

2016 (7)

C. Reinhardt, T. Müller, A. Bourassa, and J. Sankey, “Ultralow-noise SiN trampoline resonators for sensing and optomechanics,” Phys. Rev. X 6, 021001 (2016).

R. A. Norte, J. P. Moura, and S. Gröblacher, “Mechanical resonators for quantum optomechanics experiments at room temperature,” Phys. Rev. Lett. 116, 147202 (2016).
[Crossref] [PubMed]

R. W. Peterson, T. P. Purdy, N. S. Kampel, R. W. Andrews, P. L. Yu, K. W. Lehnert, and C. A. Regal, “Laser cooling of a micromechanical membrane to the quantum backaction limit,” Phys. Rev. Lett. 116, 063601 (2016).
[Crossref] [PubMed]

D. Kilda and A. Nunnenkamp, “Squeezed light and correlated photons from dissipatively coupled optomechanical systems,” J. Opt. 18, 014007 (2016).
[Crossref]

A. Barasheed, T. Müller, and J. Sankey, “Optically defined mechanical geometry,” Phys. Rev. A 93, 053811 (2016).
[Crossref]

Y. Yanay, J. Sankey, and A. Clerk, “Quantum backaction and noise interference in asymmetric two-cavity optomechanical systems,” Phys. Rev. A 93, 063809 (2016).
[Crossref]

J. Li, A. Xuereb, N. Malossi, and D. Vitali, “Cavity mode frequencies and strong optomechanical coupling in two-membrane cavity optomechanics,” J. Opt. 18, 084001 (2016).
[Crossref]

2015 (7)

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

T. K. Paraiso, M. Kalaee, L. Zang, H. Pfeifer, F. Marquardt, and O. Painter, “Position-squared coupling in a tunable photonic crystal optomechanical cavity,” Phys. Rev. X 5, 041024 (2015).

C. Stambaugh, H. Xu, U. Kemiktarak, J. Taylor, and J. Lawall, “From membrane-in-the-middle to mirror-in-the-middle with a high-reflectivity sub-wavelength grating,” Annalen der Physik 527, 81–88 (2015).
[Crossref]

T. Müller, C. Reinhardt, and J. Sankey, “Enhanced optomechanical levitation of minimally supported dielectrics,” Phys. Rev. A 91, 053849 (2015).
[Crossref]

A. Sawadsky, H. Kaufer, R. M. Nia, S. P. Tarabrin, F. Y. Khalili, K. Hammerer, and R. Schnabel, “Observation of generalized optomechanical coupling and cooling on cavity resonance,” Phys. Rev. Lett. 114, 043601 (2015).
[Crossref] [PubMed]

T. P. Purdy, P.-L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, “Optomechanical Raman-ratio thermometry,” Phys. Rev. A 92, 031802 (2015).
[Crossref]

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Børkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
[Crossref]

2014 (3)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

D. Lee, M. Underwood, D. Mason, A. B. Shkarin, S. W. Hoch, and J. G. E. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Comm. 6, 6232 (2014).
[Crossref]

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10, 321–326 (2014).
[Crossref]

2013 (5)

S. P. Tarabrin, H. Kaufer, F. Y. Khalili, R. Schnabel, and K. Hammerer, “Anomalous dynamic backaction in interferometers,” Phys. Rev. A 88, 023809 (2013).
[Crossref]

T. P. Purdy, R. W. Peterson, and C. A. Regal, “Observation of radiation pressure shot noise on a macroscopic object,” Science 339, 801–804 (2013).
[Crossref] [PubMed]

T. P. Purdy, P.-L. L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X 3, 031012 (2013).

T. Weiss, C. Bruder, and A. Nunnenkamp, “Strong-coupling effects in dissipatively coupled optomechanical systems,” New J. Phys. 15, 045017 (2013).
[Crossref]

T. Weiss and A. Nunnenkamp, “Quantum limit of laser cooling in dispersively and dissipatively coupled optomechanical systems,” Phys. Rev. A 88, 023850 (2013).
[Crossref]

2012 (4)

T. P. Purdy, R. W. Peterson, P. L. Yu, and C. A. Regal, “Cavity optomechanics with Si3N4 membranes at cryogenic temperatures,” New J. Phys. 14, 115021 (2012).
[Crossref]

K. K. Ni, R. Norte, D. J. Wilson, J. D. Hood, D. E. Chang, O. Painter, and H. J. Kimble, “Enhancement of mechanical Q factors by optical trapping,” Phys. Rev. Lett. 108, 214302 (2012).
[Crossref] [PubMed]

D. E. Chang, K.-K. Ni, O. Painter, and H. J. Kimble, “Ultrahigh-Q mechanical oscillators through optical trapping,” New J. Phys. 14, 45002 (2012).
[Crossref]

N. Flowers-Jacobs, S. Hoch, J. Sankey, A. Kashkanova, A. Jayich, C. Deutsch, J. Reichel, and J. Harris, “Fiber-cavity-based optomechanical device,” Appl. Phys. Lett. 101, 221109 (2012).
[Crossref]

2011 (1)

M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” J. Opt. 15, 025704 (2011).
[Crossref]

2010 (4)

D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hänsch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12, 065038 (2010).
[Crossref]

S. R. Leslie, A. P. Fields, and A. E. Cohen, “Convex lens-induced confinement for imaging single molecules,” Anal. Chem. 82, 6224–6229 (2010).
[Crossref] [PubMed]

J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
[Crossref]

A. Nunnenkamp, K. Børkje, J. G. E. Harris, and S. M. Girvin, “Cooling and squeezing via quadratic optomechanical coupling,” Phys. Rev. A 82, 021806 (2010).
[Crossref]

2009 (3)

F. Elste, S. M. Girvin, and A. A. Clerk, “Quantum noise interference and backaction cooling in cavity nanomechanics,” Phys. Rev. Lett. 102, 207209 (2009).
[Crossref] [PubMed]

D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble, “Cavity optomechanics with stoichiometric SiN films,” Phys. Rev. Lett. 103, 207204 (2009).
[Crossref]

H. Miao, S. Danilishin, T. Corbitt, and Y. Chen, “Standard quantum limit for probing mechanical energy quantization,” Phys. Rev. Lett. 103, 100402 (2009).
[Crossref] [PubMed]

2008 (3)

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77, 033819 (2008).
[Crossref]

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
[Crossref]

2001 (1)

C. Hood, H. Kimble, and J. Ye, “Characterization of high-finesse mirrors: Loss, phase shifts, and mode structure in an optical cavity,” Phys. Rev. A 64, 033804 (2001).
[Crossref]

Andrews, R. W.

R. W. Peterson, T. P. Purdy, N. S. Kampel, R. W. Andrews, P. L. Yu, K. W. Lehnert, and C. A. Regal, “Laser cooling of a micromechanical membrane to the quantum backaction limit,” Phys. Rev. Lett. 116, 063601 (2016).
[Crossref] [PubMed]

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10, 321–326 (2014).
[Crossref]

Arrangoiz-Arriola, P.

P. Arrangoiz-Arriola, E. A. Wollack, Z. Wang, M. Pechal, W. Jiang, T. P. McKenna, J. D. Witmer, and A. H. Safavi-Naeini, “Resolving the energy levels of a nanomechanical oscillator,” arXiv:1902.04681 (2019).

Aspelmeyer, M.

I. Marinković, A. Wallucks, R. Riedinger, S. Hong, M. Aspelmeyer, and S. Gröblacher, “Optomechanical Bell test,” Phys. Rev. Lett. 121, 220404 (2018).
[Crossref]

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473–477 (2018).
[Crossref] [PubMed]

S. Hong, R. Riedinger, I. Marinković, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, “Hanbury Brown and Twiss interferometry of single phonons from an optomechanical resonator,” Science 358, 203–206 (2017).
[Crossref] [PubMed]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

Barasheed, A.

A. Barasheed, T. Müller, and J. Sankey, “Optically defined mechanical geometry,” Phys. Rev. A 93, 053811 (2016).
[Crossref]

Barg, A.

Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nano. 12, 776–783 (2017).
[Crossref]

Bawaj, M.

P. Piergentili, L. Catalini, M. Bawaj, S. Zippilli, N. Malossi, R. Natali, D. Vitali, and G. D. Giuseppe, “Two-membrane cavity optomechanics,” New J. Phys. 20, 083024 (2018).
[Crossref]

Bereyhi, M. J.

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref] [PubMed]

Bhattacharya, M.

M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77, 033819 (2008).
[Crossref]

Biancofiore, C.

M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” J. Opt. 15, 025704 (2011).
[Crossref]

Børkje, K.

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Børkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
[Crossref]

A. Nunnenkamp, K. Børkje, J. G. E. Harris, and S. M. Girvin, “Cooling and squeezing via quadratic optomechanical coupling,” Phys. Rev. A 82, 021806 (2010).
[Crossref]

Bourassa, A.

C. Reinhardt, T. Müller, A. Bourassa, and J. Sankey, “Ultralow-noise SiN trampoline resonators for sensing and optomechanics,” Phys. Rev. X 6, 021001 (2016).

Braive, R.

X. Chen, C. Chardin, K. Makles, C. Caër, S. Chua, R. Braive, I. Robert-Philip, T. Briant, P.-F. Cohadon, A. Heidmann, T. Jacqmin, and S. Deléglise, “High-finesse Fabry-Perot cavities with bidimensional Si3N4 photonic-crystal slabs,” Light. Sci. & Appl. 6, e16190 (2017).
[Crossref]

Briant, T.

X. Chen, C. Chardin, K. Makles, C. Caër, S. Chua, R. Braive, I. Robert-Philip, T. Briant, P.-F. Cohadon, A. Heidmann, T. Jacqmin, and S. Deléglise, “High-finesse Fabry-Perot cavities with bidimensional Si3N4 photonic-crystal slabs,” Light. Sci. & Appl. 6, e16190 (2017).
[Crossref]

Brubaker, B. M.

A. P. Higginbotham, P. S. Burns, M. D. Urmey, R. W. Peterson, N. S. Kampel, B. M. Brubaker, G. Smith, K. W. Lehnert, and C. A. Regal, “Harnessing electro-optic correlations in an efficient mechanical converter,” Nat. Phys. 14, 1038–1042 (2018).
[Crossref]

Bruder, C.

T. Weiss, C. Bruder, and A. Nunnenkamp, “Strong-coupling effects in dissipatively coupled optomechanical systems,” New J. Phys. 15, 045017 (2013).
[Crossref]

Burns, P. S.

A. P. Higginbotham, P. S. Burns, M. D. Urmey, R. W. Peterson, N. S. Kampel, B. M. Brubaker, G. Smith, K. W. Lehnert, and C. A. Regal, “Harnessing electro-optic correlations in an efficient mechanical converter,” Nat. Phys. 14, 1038–1042 (2018).
[Crossref]

Caër, C.

X. Chen, C. Chardin, K. Makles, C. Caër, S. Chua, R. Braive, I. Robert-Philip, T. Briant, P.-F. Cohadon, A. Heidmann, T. Jacqmin, and S. Deléglise, “High-finesse Fabry-Perot cavities with bidimensional Si3N4 photonic-crystal slabs,” Light. Sci. & Appl. 6, e16190 (2017).
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F. Elste, S. M. Girvin, and A. A. Clerk, “Quantum noise interference and backaction cooling in cavity nanomechanics,” Phys. Rev. Lett. 102, 207209 (2009).
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X. Chen, C. Chardin, K. Makles, C. Caër, S. Chua, R. Braive, I. Robert-Philip, T. Briant, P.-F. Cohadon, A. Heidmann, T. Jacqmin, and S. Deléglise, “High-finesse Fabry-Perot cavities with bidimensional Si3N4 photonic-crystal slabs,” Light. Sci. & Appl. 6, e16190 (2017).
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X. Chen, C. Chardin, K. Makles, C. Caër, S. Chua, R. Braive, I. Robert-Philip, T. Briant, P.-F. Cohadon, A. Heidmann, T. Jacqmin, and S. Deléglise, “High-finesse Fabry-Perot cavities with bidimensional Si3N4 photonic-crystal slabs,” Light. Sci. & Appl. 6, e16190 (2017).
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N. Flowers-Jacobs, S. Hoch, J. Sankey, A. Kashkanova, A. Jayich, C. Deutsch, J. Reichel, and J. Harris, “Fiber-cavity-based optomechanical device,” Appl. Phys. Lett. 101, 221109 (2012).
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D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hänsch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12, 065038 (2010).
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M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” J. Opt. 15, 025704 (2011).
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A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
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A. Nunnenkamp, K. Børkje, J. G. E. Harris, and S. M. Girvin, “Cooling and squeezing via quadratic optomechanical coupling,” Phys. Rev. A 82, 021806 (2010).
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F. Elste, S. M. Girvin, and A. A. Clerk, “Quantum noise interference and backaction cooling in cavity nanomechanics,” Phys. Rev. Lett. 102, 207209 (2009).
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A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
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J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
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Giuseppe, G. D.

P. Piergentili, L. Catalini, M. Bawaj, S. Zippilli, N. Malossi, R. Natali, D. Vitali, and G. D. Giuseppe, “Two-membrane cavity optomechanics,” New J. Phys. 20, 083024 (2018).
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R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473–477 (2018).
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I. Marinković, A. Wallucks, R. Riedinger, S. Hong, M. Aspelmeyer, and S. Gröblacher, “Optomechanical Bell test,” Phys. Rev. Lett. 121, 220404 (2018).
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S. Hong, R. Riedinger, I. Marinković, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, “Hanbury Brown and Twiss interferometry of single phonons from an optomechanical resonator,” Science 358, 203–206 (2017).
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A. Sawadsky, H. Kaufer, R. M. Nia, S. P. Tarabrin, F. Y. Khalili, K. Hammerer, and R. Schnabel, “Observation of generalized optomechanical coupling and cooling on cavity resonance,” Phys. Rev. Lett. 114, 043601 (2015).
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S. P. Tarabrin, H. Kaufer, F. Y. Khalili, R. Schnabel, and K. Hammerer, “Anomalous dynamic backaction in interferometers,” Phys. Rev. A 88, 023809 (2013).
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Hänsch, T. W.

D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hänsch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12, 065038 (2010).
[Crossref]

Harris, J.

N. Flowers-Jacobs, S. Hoch, J. Sankey, A. Kashkanova, A. Jayich, C. Deutsch, J. Reichel, and J. Harris, “Fiber-cavity-based optomechanical device,” Appl. Phys. Lett. 101, 221109 (2012).
[Crossref]

J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
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J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Nonlinear optomechanical couplings: Tools for dealing with solid mechanical objects in the quantum regime,” in Optics InfoBase Conference Papers (2010).

Harris, J. G. E.

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Børkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
[Crossref]

D. Lee, M. Underwood, D. Mason, A. B. Shkarin, S. W. Hoch, and J. G. E. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Comm. 6, 6232 (2014).
[Crossref]

A. Nunnenkamp, K. Børkje, J. G. E. Harris, and S. M. Girvin, “Cooling and squeezing via quadratic optomechanical coupling,” Phys. Rev. A 82, 021806 (2010).
[Crossref]

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
[Crossref]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

J. C. Sankey, A. M. Jayich, B. M. Zwickl, C. Yang, and J. G. E. Harris, “Improved “position-squared” readout using degenerate cavity modes,” in ICAP Proceedings (2008), pp. 131–149.

Heidmann, A.

X. Chen, C. Chardin, K. Makles, C. Caër, S. Chua, R. Braive, I. Robert-Philip, T. Briant, P.-F. Cohadon, A. Heidmann, T. Jacqmin, and S. Deléglise, “High-finesse Fabry-Perot cavities with bidimensional Si3N4 photonic-crystal slabs,” Light. Sci. & Appl. 6, e16190 (2017).
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Higginbotham, A. P.

A. P. Higginbotham, P. S. Burns, M. D. Urmey, R. W. Peterson, N. S. Kampel, B. M. Brubaker, G. Smith, K. W. Lehnert, and C. A. Regal, “Harnessing electro-optic correlations in an efficient mechanical converter,” Nat. Phys. 14, 1038–1042 (2018).
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Hoch, S.

N. Flowers-Jacobs, S. Hoch, J. Sankey, A. Kashkanova, A. Jayich, C. Deutsch, J. Reichel, and J. Harris, “Fiber-cavity-based optomechanical device,” Appl. Phys. Lett. 101, 221109 (2012).
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Hoch, S. W.

D. Lee, M. Underwood, D. Mason, A. B. Shkarin, S. W. Hoch, and J. G. E. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Comm. 6, 6232 (2014).
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Hofer, S. G.

S. Hong, R. Riedinger, I. Marinković, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, “Hanbury Brown and Twiss interferometry of single phonons from an optomechanical resonator,” Science 358, 203–206 (2017).
[Crossref] [PubMed]

Hong, S.

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473–477 (2018).
[Crossref] [PubMed]

I. Marinković, A. Wallucks, R. Riedinger, S. Hong, M. Aspelmeyer, and S. Gröblacher, “Optomechanical Bell test,” Phys. Rev. Lett. 121, 220404 (2018).
[Crossref]

S. Hong, R. Riedinger, I. Marinković, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, “Hanbury Brown and Twiss interferometry of single phonons from an optomechanical resonator,” Science 358, 203–206 (2017).
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Hood, C.

C. Hood, H. Kimble, and J. Ye, “Characterization of high-finesse mirrors: Loss, phase shifts, and mode structure in an optical cavity,” Phys. Rev. A 64, 033804 (2001).
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Hood, J. D.

K. K. Ni, R. Norte, D. J. Wilson, J. D. Hood, D. E. Chang, O. Painter, and H. J. Kimble, “Enhancement of mechanical Q factors by optical trapping,” Phys. Rev. Lett. 108, 214302 (2012).
[Crossref] [PubMed]

Hunger, D.

D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hänsch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12, 065038 (2010).
[Crossref]

Jacqmin, T.

X. Chen, C. Chardin, K. Makles, C. Caër, S. Chua, R. Braive, I. Robert-Philip, T. Briant, P.-F. Cohadon, A. Heidmann, T. Jacqmin, and S. Deléglise, “High-finesse Fabry-Perot cavities with bidimensional Si3N4 photonic-crystal slabs,” Light. Sci. & Appl. 6, e16190 (2017).
[Crossref]

Jayich, A.

N. Flowers-Jacobs, S. Hoch, J. Sankey, A. Kashkanova, A. Jayich, C. Deutsch, J. Reichel, and J. Harris, “Fiber-cavity-based optomechanical device,” Appl. Phys. Lett. 101, 221109 (2012).
[Crossref]

J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
[Crossref]

J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Nonlinear optomechanical couplings: Tools for dealing with solid mechanical objects in the quantum regime,” in Optics InfoBase Conference Papers (2010).

Jayich, A. M.

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
[Crossref]

J. C. Sankey, A. M. Jayich, B. M. Zwickl, C. Yang, and J. G. E. Harris, “Improved “position-squared” readout using degenerate cavity modes,” in ICAP Proceedings (2008), pp. 131–149.

Jiang, L.

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Børkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
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Jiang, W.

P. Arrangoiz-Arriola, E. A. Wollack, Z. Wang, M. Pechal, W. Jiang, T. P. McKenna, J. D. Witmer, and A. H. Safavi-Naeini, “Resolving the energy levels of a nanomechanical oscillator,” arXiv:1902.04681 (2019).

Kalaee, M.

T. K. Paraiso, M. Kalaee, L. Zang, H. Pfeifer, F. Marquardt, and O. Painter, “Position-squared coupling in a tunable photonic crystal optomechanical cavity,” Phys. Rev. X 5, 041024 (2015).

Kampel, N. S.

A. P. Higginbotham, P. S. Burns, M. D. Urmey, R. W. Peterson, N. S. Kampel, B. M. Brubaker, G. Smith, K. W. Lehnert, and C. A. Regal, “Harnessing electro-optic correlations in an efficient mechanical converter,” Nat. Phys. 14, 1038–1042 (2018).
[Crossref]

R. W. Peterson, T. P. Purdy, N. S. Kampel, R. W. Andrews, P. L. Yu, K. W. Lehnert, and C. A. Regal, “Laser cooling of a micromechanical membrane to the quantum backaction limit,” Phys. Rev. Lett. 116, 063601 (2016).
[Crossref] [PubMed]

T. P. Purdy, P.-L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, “Optomechanical Raman-ratio thermometry,” Phys. Rev. A 92, 031802 (2015).
[Crossref]

T. P. Purdy, P.-L. L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X 3, 031012 (2013).

Karuza, M.

M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” J. Opt. 15, 025704 (2011).
[Crossref]

Kashkanova, A.

N. Flowers-Jacobs, S. Hoch, J. Sankey, A. Kashkanova, A. Jayich, C. Deutsch, J. Reichel, and J. Harris, “Fiber-cavity-based optomechanical device,” Appl. Phys. Lett. 101, 221109 (2012).
[Crossref]

Kaufer, H.

A. Sawadsky, H. Kaufer, R. M. Nia, S. P. Tarabrin, F. Y. Khalili, K. Hammerer, and R. Schnabel, “Observation of generalized optomechanical coupling and cooling on cavity resonance,” Phys. Rev. Lett. 114, 043601 (2015).
[Crossref] [PubMed]

S. P. Tarabrin, H. Kaufer, F. Y. Khalili, R. Schnabel, and K. Hammerer, “Anomalous dynamic backaction in interferometers,” Phys. Rev. A 88, 023809 (2013).
[Crossref]

Kemiktarak, U.

C. Stambaugh, H. Xu, U. Kemiktarak, J. Taylor, and J. Lawall, “From membrane-in-the-middle to mirror-in-the-middle with a high-reflectivity sub-wavelength grating,” Annalen der Physik 527, 81–88 (2015).
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Khalili, F. Y.

A. Sawadsky, H. Kaufer, R. M. Nia, S. P. Tarabrin, F. Y. Khalili, K. Hammerer, and R. Schnabel, “Observation of generalized optomechanical coupling and cooling on cavity resonance,” Phys. Rev. Lett. 114, 043601 (2015).
[Crossref] [PubMed]

S. P. Tarabrin, H. Kaufer, F. Y. Khalili, R. Schnabel, and K. Hammerer, “Anomalous dynamic backaction in interferometers,” Phys. Rev. A 88, 023809 (2013).
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Kilda, D.

D. Kilda and A. Nunnenkamp, “Squeezed light and correlated photons from dissipatively coupled optomechanical systems,” J. Opt. 18, 014007 (2016).
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Kimble, H.

C. Hood, H. Kimble, and J. Ye, “Characterization of high-finesse mirrors: Loss, phase shifts, and mode structure in an optical cavity,” Phys. Rev. A 64, 033804 (2001).
[Crossref]

Kimble, H. J.

K. K. Ni, R. Norte, D. J. Wilson, J. D. Hood, D. E. Chang, O. Painter, and H. J. Kimble, “Enhancement of mechanical Q factors by optical trapping,” Phys. Rev. Lett. 108, 214302 (2012).
[Crossref] [PubMed]

D. E. Chang, K.-K. Ni, O. Painter, and H. J. Kimble, “Ultrahigh-Q mechanical oscillators through optical trapping,” New J. Phys. 14, 45002 (2012).
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D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble, “Cavity optomechanics with stoichiometric SiN films,” Phys. Rev. Lett. 103, 207204 (2009).
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Kippenberg, T. J.

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
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M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
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Lawall, J.

C. Stambaugh, H. Xu, U. Kemiktarak, J. Taylor, and J. Lawall, “From membrane-in-the-middle to mirror-in-the-middle with a high-reflectivity sub-wavelength grating,” Annalen der Physik 527, 81–88 (2015).
[Crossref]

Lee, D.

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Børkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
[Crossref]

D. Lee, M. Underwood, D. Mason, A. B. Shkarin, S. W. Hoch, and J. G. E. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Comm. 6, 6232 (2014).
[Crossref]

Lehnert, K. W.

A. P. Higginbotham, P. S. Burns, M. D. Urmey, R. W. Peterson, N. S. Kampel, B. M. Brubaker, G. Smith, K. W. Lehnert, and C. A. Regal, “Harnessing electro-optic correlations in an efficient mechanical converter,” Nat. Phys. 14, 1038–1042 (2018).
[Crossref]

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I. Marinković, A. Wallucks, R. Riedinger, S. Hong, M. Aspelmeyer, and S. Gröblacher, “Optomechanical Bell test,” Phys. Rev. Lett. 121, 220404 (2018).
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M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, “Measurement-based quantum control of mechanical motion,” Nature 563, 53–58 (2018).
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[Crossref]

Smith, G.

A. P. Higginbotham, P. S. Burns, M. D. Urmey, R. W. Peterson, N. S. Kampel, B. M. Brubaker, G. Smith, K. W. Lehnert, and C. A. Regal, “Harnessing electro-optic correlations in an efficient mechanical converter,” Nat. Phys. 14, 1038–1042 (2018).
[Crossref]

Stambaugh, C.

C. Stambaugh, H. Xu, U. Kemiktarak, J. Taylor, and J. Lawall, “From membrane-in-the-middle to mirror-in-the-middle with a high-reflectivity sub-wavelength grating,” Annalen der Physik 527, 81–88 (2015).
[Crossref]

Steinmetz, T.

D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hänsch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12, 065038 (2010).
[Crossref]

Tarabrin, S. P.

A. Sawadsky, H. Kaufer, R. M. Nia, S. P. Tarabrin, F. Y. Khalili, K. Hammerer, and R. Schnabel, “Observation of generalized optomechanical coupling and cooling on cavity resonance,” Phys. Rev. Lett. 114, 043601 (2015).
[Crossref] [PubMed]

S. P. Tarabrin, H. Kaufer, F. Y. Khalili, R. Schnabel, and K. Hammerer, “Anomalous dynamic backaction in interferometers,” Phys. Rev. A 88, 023809 (2013).
[Crossref]

Taylor, J.

C. Stambaugh, H. Xu, U. Kemiktarak, J. Taylor, and J. Lawall, “From membrane-in-the-middle to mirror-in-the-middle with a high-reflectivity sub-wavelength grating,” Annalen der Physik 527, 81–88 (2015).
[Crossref]

Thompson, J. D.

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
[Crossref]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

Tombesi, P.

M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” J. Opt. 15, 025704 (2011).
[Crossref]

Tsaturyan, Y.

M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, “Measurement-based quantum control of mechanical motion,” Nature 563, 53–58 (2018).
[Crossref] [PubMed]

Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nano. 12, 776–783 (2017).
[Crossref]

W. H. P. Nielsen, Y. Tsaturyan, C. B. Møller, E. S. Polzik, and A. Schliesser, “Multimode optomechanical system in the quantum regime,” Proc. Natl. Acad. Sci. 114, 62–66 (2017).
[Crossref]

Underwood, M.

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Børkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
[Crossref]

D. Lee, M. Underwood, D. Mason, A. B. Shkarin, S. W. Hoch, and J. G. E. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Comm. 6, 6232 (2014).
[Crossref]

Urmey, M. D.

A. P. Higginbotham, P. S. Burns, M. D. Urmey, R. W. Peterson, N. S. Kampel, B. M. Brubaker, G. Smith, K. W. Lehnert, and C. A. Regal, “Harnessing electro-optic correlations in an efficient mechanical converter,” Nat. Phys. 14, 1038–1042 (2018).
[Crossref]

Uys, H.

M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77, 033819 (2008).
[Crossref]

Vitali, D.

P. Piergentili, L. Catalini, M. Bawaj, S. Zippilli, N. Malossi, R. Natali, D. Vitali, and G. D. Giuseppe, “Two-membrane cavity optomechanics,” New J. Phys. 20, 083024 (2018).
[Crossref]

J. Li, A. Xuereb, N. Malossi, and D. Vitali, “Cavity mode frequencies and strong optomechanical coupling in two-membrane cavity optomechanics,” J. Opt. 18, 084001 (2016).
[Crossref]

M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” J. Opt. 15, 025704 (2011).
[Crossref]

Wallucks, A.

I. Marinković, A. Wallucks, R. Riedinger, S. Hong, M. Aspelmeyer, and S. Gröblacher, “Optomechanical Bell test,” Phys. Rev. Lett. 121, 220404 (2018).
[Crossref]

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473–477 (2018).
[Crossref] [PubMed]

S. Hong, R. Riedinger, I. Marinković, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, “Hanbury Brown and Twiss interferometry of single phonons from an optomechanical resonator,” Science 358, 203–206 (2017).
[Crossref] [PubMed]

Wang, Z.

P. Arrangoiz-Arriola, E. A. Wollack, Z. Wang, M. Pechal, W. Jiang, T. P. McKenna, J. D. Witmer, and A. H. Safavi-Naeini, “Resolving the energy levels of a nanomechanical oscillator,” arXiv:1902.04681 (2019).

Weiss, T.

T. Weiss and A. Nunnenkamp, “Quantum limit of laser cooling in dispersively and dissipatively coupled optomechanical systems,” Phys. Rev. A 88, 023850 (2013).
[Crossref]

T. Weiss, C. Bruder, and A. Nunnenkamp, “Strong-coupling effects in dissipatively coupled optomechanical systems,” New J. Phys. 15, 045017 (2013).
[Crossref]

Wilson, D. J.

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref] [PubMed]

K. K. Ni, R. Norte, D. J. Wilson, J. D. Hood, D. E. Chang, O. Painter, and H. J. Kimble, “Enhancement of mechanical Q factors by optical trapping,” Phys. Rev. Lett. 108, 214302 (2012).
[Crossref] [PubMed]

D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble, “Cavity optomechanics with stoichiometric SiN films,” Phys. Rev. Lett. 103, 207204 (2009).
[Crossref]

Witmer, J. D.

P. Arrangoiz-Arriola, E. A. Wollack, Z. Wang, M. Pechal, W. Jiang, T. P. McKenna, J. D. Witmer, and A. H. Safavi-Naeini, “Resolving the energy levels of a nanomechanical oscillator,” arXiv:1902.04681 (2019).

Wollack, E. A.

P. Arrangoiz-Arriola, E. A. Wollack, Z. Wang, M. Pechal, W. Jiang, T. P. McKenna, J. D. Witmer, and A. H. Safavi-Naeini, “Resolving the energy levels of a nanomechanical oscillator,” arXiv:1902.04681 (2019).

Xu, H.

C. Stambaugh, H. Xu, U. Kemiktarak, J. Taylor, and J. Lawall, “From membrane-in-the-middle to mirror-in-the-middle with a high-reflectivity sub-wavelength grating,” Annalen der Physik 527, 81–88 (2015).
[Crossref]

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Børkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
[Crossref]

Xuereb, A.

J. Li, A. Xuereb, N. Malossi, and D. Vitali, “Cavity mode frequencies and strong optomechanical coupling in two-membrane cavity optomechanics,” J. Opt. 18, 084001 (2016).
[Crossref]

Yanay, Y.

Y. Yanay, J. Sankey, and A. Clerk, “Quantum backaction and noise interference in asymmetric two-cavity optomechanical systems,” Phys. Rev. A 93, 063809 (2016).
[Crossref]

Yang, C.

J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
[Crossref]

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
[Crossref]

J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Nonlinear optomechanical couplings: Tools for dealing with solid mechanical objects in the quantum regime,” in Optics InfoBase Conference Papers (2010).

J. C. Sankey, A. M. Jayich, B. M. Zwickl, C. Yang, and J. G. E. Harris, “Improved “position-squared” readout using degenerate cavity modes,” in ICAP Proceedings (2008), pp. 131–149.

Ye, J.

C. Hood, H. Kimble, and J. Ye, “Characterization of high-finesse mirrors: Loss, phase shifts, and mode structure in an optical cavity,” Phys. Rev. A 64, 033804 (2001).
[Crossref]

Yu, P. L.

R. W. Peterson, T. P. Purdy, N. S. Kampel, R. W. Andrews, P. L. Yu, K. W. Lehnert, and C. A. Regal, “Laser cooling of a micromechanical membrane to the quantum backaction limit,” Phys. Rev. Lett. 116, 063601 (2016).
[Crossref] [PubMed]

T. P. Purdy, R. W. Peterson, P. L. Yu, and C. A. Regal, “Cavity optomechanics with Si3N4 membranes at cryogenic temperatures,” New J. Phys. 14, 115021 (2012).
[Crossref]

Yu, P.-L.

T. P. Purdy, P.-L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, “Optomechanical Raman-ratio thermometry,” Phys. Rev. A 92, 031802 (2015).
[Crossref]

Yu, P.-L. L.

T. P. Purdy, P.-L. L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X 3, 031012 (2013).

Zang, L.

T. K. Paraiso, M. Kalaee, L. Zang, H. Pfeifer, F. Marquardt, and O. Painter, “Position-squared coupling in a tunable photonic crystal optomechanical cavity,” Phys. Rev. X 5, 041024 (2015).

Zippilli, S.

P. Piergentili, L. Catalini, M. Bawaj, S. Zippilli, N. Malossi, R. Natali, D. Vitali, and G. D. Giuseppe, “Two-membrane cavity optomechanics,” New J. Phys. 20, 083024 (2018).
[Crossref]

Zwickl, B.

J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
[Crossref]

J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Nonlinear optomechanical couplings: Tools for dealing with solid mechanical objects in the quantum regime,” in Optics InfoBase Conference Papers (2010).

Zwickl, B. M.

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
[Crossref]

J. C. Sankey, A. M. Jayich, B. M. Zwickl, C. Yang, and J. G. E. Harris, “Improved “position-squared” readout using degenerate cavity modes,” in ICAP Proceedings (2008), pp. 131–149.

Anal. Chem. (1)

S. R. Leslie, A. P. Fields, and A. E. Cohen, “Convex lens-induced confinement for imaging single molecules,” Anal. Chem. 82, 6224–6229 (2010).
[Crossref] [PubMed]

Annalen der Physik (1)

C. Stambaugh, H. Xu, U. Kemiktarak, J. Taylor, and J. Lawall, “From membrane-in-the-middle to mirror-in-the-middle with a high-reflectivity sub-wavelength grating,” Annalen der Physik 527, 81–88 (2015).
[Crossref]

Appl. Phys. Lett. (1)

N. Flowers-Jacobs, S. Hoch, J. Sankey, A. Kashkanova, A. Jayich, C. Deutsch, J. Reichel, and J. Harris, “Fiber-cavity-based optomechanical device,” Appl. Phys. Lett. 101, 221109 (2012).
[Crossref]

J. Opt. (3)

D. Kilda and A. Nunnenkamp, “Squeezed light and correlated photons from dissipatively coupled optomechanical systems,” J. Opt. 18, 014007 (2016).
[Crossref]

M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” J. Opt. 15, 025704 (2011).
[Crossref]

J. Li, A. Xuereb, N. Malossi, and D. Vitali, “Cavity mode frequencies and strong optomechanical coupling in two-membrane cavity optomechanics,” J. Opt. 18, 084001 (2016).
[Crossref]

Light. Sci. & Appl. (1)

X. Chen, C. Chardin, K. Makles, C. Caër, S. Chua, R. Braive, I. Robert-Philip, T. Briant, P.-F. Cohadon, A. Heidmann, T. Jacqmin, and S. Deléglise, “High-finesse Fabry-Perot cavities with bidimensional Si3N4 photonic-crystal slabs,” Light. Sci. & Appl. 6, e16190 (2017).
[Crossref]

Nat. Comm. (1)

D. Lee, M. Underwood, D. Mason, A. B. Shkarin, S. W. Hoch, and J. G. E. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Comm. 6, 6232 (2014).
[Crossref]

Nat. Nano. (1)

Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nano. 12, 776–783 (2017).
[Crossref]

Nat. Phys. (3)

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10, 321–326 (2014).
[Crossref]

A. P. Higginbotham, P. S. Burns, M. D. Urmey, R. W. Peterson, N. S. Kampel, B. M. Brubaker, G. Smith, K. W. Lehnert, and C. A. Regal, “Harnessing electro-optic correlations in an efficient mechanical converter,” Nat. Phys. 14, 1038–1042 (2018).
[Crossref]

J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
[Crossref]

Nature (3)

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, “Measurement-based quantum control of mechanical motion,” Nature 563, 53–58 (2018).
[Crossref] [PubMed]

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473–477 (2018).
[Crossref] [PubMed]

New J. Phys. (6)

P. Piergentili, L. Catalini, M. Bawaj, S. Zippilli, N. Malossi, R. Natali, D. Vitali, and G. D. Giuseppe, “Two-membrane cavity optomechanics,” New J. Phys. 20, 083024 (2018).
[Crossref]

T. P. Purdy, R. W. Peterson, P. L. Yu, and C. A. Regal, “Cavity optomechanics with Si3N4 membranes at cryogenic temperatures,” New J. Phys. 14, 115021 (2012).
[Crossref]

T. Weiss, C. Bruder, and A. Nunnenkamp, “Strong-coupling effects in dissipatively coupled optomechanical systems,” New J. Phys. 15, 045017 (2013).
[Crossref]

D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hänsch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12, 065038 (2010).
[Crossref]

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
[Crossref]

D. E. Chang, K.-K. Ni, O. Painter, and H. J. Kimble, “Ultrahigh-Q mechanical oscillators through optical trapping,” New J. Phys. 14, 45002 (2012).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (10)

C. Hood, H. Kimble, and J. Ye, “Characterization of high-finesse mirrors: Loss, phase shifts, and mode structure in an optical cavity,” Phys. Rev. A 64, 033804 (2001).
[Crossref]

T. Weiss and A. Nunnenkamp, “Quantum limit of laser cooling in dispersively and dissipatively coupled optomechanical systems,” Phys. Rev. A 88, 023850 (2013).
[Crossref]

T. Müller, C. Reinhardt, and J. Sankey, “Enhanced optomechanical levitation of minimally supported dielectrics,” Phys. Rev. A 91, 053849 (2015).
[Crossref]

A. Barasheed, T. Müller, and J. Sankey, “Optically defined mechanical geometry,” Phys. Rev. A 93, 053811 (2016).
[Crossref]

A. Nunnenkamp, K. Børkje, J. G. E. Harris, and S. M. Girvin, “Cooling and squeezing via quadratic optomechanical coupling,” Phys. Rev. A 82, 021806 (2010).
[Crossref]

S. P. Tarabrin, H. Kaufer, F. Y. Khalili, R. Schnabel, and K. Hammerer, “Anomalous dynamic backaction in interferometers,” Phys. Rev. A 88, 023809 (2013).
[Crossref]

Y. Yanay, J. Sankey, and A. Clerk, “Quantum backaction and noise interference in asymmetric two-cavity optomechanical systems,” Phys. Rev. A 93, 063809 (2016).
[Crossref]

M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77, 033819 (2008).
[Crossref]

T. P. Purdy, P.-L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, “Optomechanical Raman-ratio thermometry,” Phys. Rev. A 92, 031802 (2015).
[Crossref]

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Børkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
[Crossref]

Phys. Rev. Lett. (8)

R. W. Peterson, T. P. Purdy, N. S. Kampel, R. W. Andrews, P. L. Yu, K. W. Lehnert, and C. A. Regal, “Laser cooling of a micromechanical membrane to the quantum backaction limit,” Phys. Rev. Lett. 116, 063601 (2016).
[Crossref] [PubMed]

D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble, “Cavity optomechanics with stoichiometric SiN films,” Phys. Rev. Lett. 103, 207204 (2009).
[Crossref]

R. A. Norte, J. P. Moura, and S. Gröblacher, “Mechanical resonators for quantum optomechanics experiments at room temperature,” Phys. Rev. Lett. 116, 147202 (2016).
[Crossref] [PubMed]

F. Elste, S. M. Girvin, and A. A. Clerk, “Quantum noise interference and backaction cooling in cavity nanomechanics,” Phys. Rev. Lett. 102, 207209 (2009).
[Crossref] [PubMed]

A. Sawadsky, H. Kaufer, R. M. Nia, S. P. Tarabrin, F. Y. Khalili, K. Hammerer, and R. Schnabel, “Observation of generalized optomechanical coupling and cooling on cavity resonance,” Phys. Rev. Lett. 114, 043601 (2015).
[Crossref] [PubMed]

I. Marinković, A. Wallucks, R. Riedinger, S. Hong, M. Aspelmeyer, and S. Gröblacher, “Optomechanical Bell test,” Phys. Rev. Lett. 121, 220404 (2018).
[Crossref]

H. Miao, S. Danilishin, T. Corbitt, and Y. Chen, “Standard quantum limit for probing mechanical energy quantization,” Phys. Rev. Lett. 103, 100402 (2009).
[Crossref] [PubMed]

K. K. Ni, R. Norte, D. J. Wilson, J. D. Hood, D. E. Chang, O. Painter, and H. J. Kimble, “Enhancement of mechanical Q factors by optical trapping,” Phys. Rev. Lett. 108, 214302 (2012).
[Crossref] [PubMed]

Phys. Rev. X (4)

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

T. P. Purdy, P.-L. L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X 3, 031012 (2013).

C. Reinhardt, T. Müller, A. Bourassa, and J. Sankey, “Ultralow-noise SiN trampoline resonators for sensing and optomechanics,” Phys. Rev. X 6, 021001 (2016).

T. K. Paraiso, M. Kalaee, L. Zang, H. Pfeifer, F. Marquardt, and O. Painter, “Position-squared coupling in a tunable photonic crystal optomechanical cavity,” Phys. Rev. X 5, 041024 (2015).

Proc. Natl. Acad. Sci. (1)

W. H. P. Nielsen, Y. Tsaturyan, C. B. Møller, E. S. Polzik, and A. Schliesser, “Multimode optomechanical system in the quantum regime,” Proc. Natl. Acad. Sci. 114, 62–66 (2017).
[Crossref]

Rev. Mod. Phys. (1)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

Science (3)

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref] [PubMed]

T. P. Purdy, R. W. Peterson, and C. A. Regal, “Observation of radiation pressure shot noise on a macroscopic object,” Science 339, 801–804 (2013).
[Crossref] [PubMed]

S. Hong, R. Riedinger, I. Marinković, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, “Hanbury Brown and Twiss interferometry of single phonons from an optomechanical resonator,” Science 358, 203–206 (2017).
[Crossref] [PubMed]

Other (3)

J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Nonlinear optomechanical couplings: Tools for dealing with solid mechanical objects in the quantum regime,” in Optics InfoBase Conference Papers (2010).

J. C. Sankey, A. M. Jayich, B. M. Zwickl, C. Yang, and J. G. E. Harris, “Improved “position-squared” readout using degenerate cavity modes,” in ICAP Proceedings (2008), pp. 131–149.

P. Arrangoiz-Arriola, E. A. Wollack, Z. Wang, M. Pechal, W. Jiang, T. P. McKenna, J. D. Witmer, and A. H. Safavi-Naeini, “Resolving the energy levels of a nanomechanical oscillator,” arXiv:1902.04681 (2019).

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Figures (3)

Fig. 1
Fig. 1 Optical resonances for a membrane in a cavity. (a) 1D model, comprising two fixed end mirrors (gray), with reflection (transmission) coefficients −rj (itj), and a thin, flexible dielectric membrane (blue) having reflection (transmission) coefficients rm (tm). Aj (Bj) indicate the right-moving (left-moving) field amplitudes just outside the membrane. (b)–(c) Dependence of the cavity resonance’s detuning (normalized by the free spectral range (FSR)) on membrane displacement Δx from (b) the cavity center (x=L/2) and (c) the first mirror (x=0), with rm values (from light to dark) −0.4, −0.6, −0.8, −0.931, and −0.9977 (rm’s phase chosen to highlight avoided crossings as in [2]). Horizontal dashed lines represent empty cavity resonant frequencies (rm=0), while other dashed lines represent the left (negatively sloped) and right (positively sloped) sub-cavity resonances when tm=0. Insets qualitatively show the field distribution of these modes. (d)–(e) Dependence of cavity’s energy decay rate κ on membrane displacement Δx from (d) the cavity center and (e) the first mirror, normalized by the empty cavity value for a single-port cavity (t2=0, r2=−1).
Fig. 2
Fig. 2 Optical response of flexural MATE system. (a) Diagram of measurement system and MATE cavity (not to scale), comprising an input mirror M1, 21 μm-thick spacers, a membrane, piezo-actuated pushers, and a “backstop” mirror M2. 10 mW of laser light (λ = 1550 nm) from a fiber collimator C passes through a 50:50 splitter, is reflected from M1, and collected with a photodiode PD. The length L = 10 cm between M1 and M2 is tuned with piezo actuators on M2; the inset shows the fractional reflected power (normalized by the off-resonance value 2.5 mW) as a function of the laser’s detuning from an L-swept cavity resonance. The violet curve is a “typical” Lorentzian fit used to extract the cavity’s energy decay rate κ. (b) Same measurement for a range of L at varied membrane displacements Δx. The white dashed line shows part of a simultaneous fit to three modes’ resonant frequencies (Eq. 2) used to eliminate piezo nonlinearities, and the arrow shows the location and direction of the sweep in (a). Higher-order transverse modes appear as faint resonances. (c) Spectrum predicted by a 1D transfer matrix model, with parameters estimated as described in the main text. (d) Dependence of cavity decay rate κ on membrane position for the central TEM00 (brightest) resonance in (b). The solid line is κ(x) obtained by numerically solving the 1D model, and the black part highlights the region used for our main fits (including the piezo corrections). The dashed line is the fit empty cavity decay rate 2π/(|t1|2 + |t2|2 + S1) (agrees with our measured value prior to incorporating the membrane). (e) Dependence of resonant reflection on membrane position for the middle TEM00 resonance in (b). The solid line shows the fit result as in (d).
Fig. 3
Fig. 3 Large flexure. Length x (top) and tilt θ (bottom) as a function of the mode index l. The starting index l0 is estimated from a fit to the transmission spectra (upper inset) described in the main text. The uncertainty on l0 is ±1, as represented by the shaded area. Below l=8, the transmission data is too broad to constrain x, and “×” symbols represent the passing of a bright fringe for further flexure. The lower inset shows the interference pattern (1310 nm light) of the mirror-membrane system while maximally flexed; dotted square represents the approximate location and size of the membrane. The dark shaded area corresponds to the uncertainty in membrane thickness and M1’s coating (included via transfer matrix calculation). The tilt was calculated assuming the measured beam radius σ = 100 μm, and the light shading corresponds to a conservative upper bound (σ = 110 μm) on the actual value at the membrane.

Equations (57)

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ω MIM c k MIM = N ω FSR + ω FSR π ( arccos [ ( 1 ) N + 1 | r m | cos ( 2 k N Δ x ) ] ϕ r )
ω MATE c k MATE = N ω FSR + ω FSR π arctan [ cos ( ϕ r ) + | r m | cos ( 2 k N Δ x ) sin ( ϕ r ) | r m | sin ( 2 k N Δ x ) ] ,
κ = ( 1 | r m | 2 ) c | t 1 | 2 + ( 1 + 2 | r m | cos ( 2 k x + ϕ r ) + | r m | 2 ) c | t 2 | 2 2 x ( 1 | r m | 2 ) + 2 ( L x ) ( 1 + 2 | r m | cos ( 2 k x + ϕ r ) + | r m | 2 ) .
P t | t 1 | 2 | t m | 2 1 + | r 1 | 2 | r m | 2 2 | r 1 | | r m | cos ( 2 k x 0 + ϕ ) { 1 k 2 θ 2 σ 2 | r 1 | | r m | ( 1 + | r 1 | 2 | r m | 2 ) cos ( 2 k x 0 + ϕ ) + | r 1 | | r m | ( cos ( 4 k x 0 + 2 ϕ ) 3 ) [ 1 + | r 1 | 2 | r m | 2 2 | r 1 | | r m | cos ( 2 k x 0 + ϕ ) ] 2 } ,
A 2 = t m A 1 + r m B 2
B 1 = t m B 2 + r m A 1 .
A 1 = B 1 e 2 i k x
B 2 = A 2 e 2 i k ( L x ) ,
( t m 2 r m 2 ) e i k L e i k L = 2 r m cos ( 2 k x k L ) .
cos ( k L + ϕ r ) = | r m | cos ( 2 k x k L ) .
L = 1 k arctan [ cos ( ϕ r ) + | r m | cos ( 2 k x ) sin ( ϕ r ) | r m | sin ( 2 k x ) ] .
k = k N + Δ k .
k Δ x k N Δ x + Δ k Δ x k N Δ x ,
k MIM N π L + 1 L ( arccos [ ( 1 ) N + 1 | r m | cos ( 2 k N Δ x ) ] ϕ r ) .
ω MIM N ω FSR + ω FSR π ( arccos [ ( 1 ) N + 1 | r m | cos ( 2 k N Δ x ) ] ϕ r ) ,
k Δ x = k N Δ x + Δ k Δ x k N Δ x
k MATE N π L + 1 L arctan [ cos ( ϕ r ) + | r m | cos ( 2 k N Δ x ) sin ( ϕ r ) | r m | sin ( 2 k N Δ x ) ] ,
ω MATE N ω FSR + ω FSR π arctan [ cos ( ϕ r ) + | r m | cos ( 2 k N Δ x ) sin ( ϕ r ) | r m | sin ( 2 k N Δ x ) ] .
E = 2 x c P 1 + 2 ( L x ) c P 2 ,
t E = P 1 | t 1 | 2 P 2 | t 2 | 2 .
| A 2 B 1 | 2 = P 2 P 1 = 1 + | r m | 2 + 2 | r m | cos ( 2 k x + ϕ r ) 1 | r m | 2 .
P 1 = c E 2 ( x + ( L x ) 1 + | r m | 2 + 2 | r m | cos ( 2 k x + ϕ r ) 1 | r m | 2 ) 1 ,
t E = κ E ,
κ = ( 1 | r m | 2 ) c | t 1 | 2 + ( 1 + 2 | r m | cos ( 2 k x + ϕ r ) + | r m | 2 ) c | t 2 | 2 2 x ( 1 | r m | 2 ) + 2 ( L x ) ( 1 + 2 | r m | cos ( 2 k x + ϕ r ) + | r m | 2 ) .
B ˜ 1 κ d κ d x x zpf ,
B ˜ MIM = 2 | r m | L | r m | + cos ( 2 k MIM x + ϕ r ) + ( k MIM + x d k MIM d x ) L sin ( 2 k MIM x + ϕ r ) 1 + | r m | cos ( 2 k MIM x + ϕ r ) x zpf .
B ˜ MIM , max | t m | 1 3 3 2 k N x zpf 1 | t m | ,
Δ x MIM , max ( B ˜ ) | t m | 1 1 2 k N ( j π + ( 1 ) j + N + 1 | t m | 3 )
B ˜ input 2 | r m | L | r m | + cos ( 2 k N Δ x + ϕ r ) + 2 k N L sin ( 2 k N Δ x + ϕ r ) 1 + | r m | 2 + 2 | r m | cos ( 2 k N Δ x + ϕ r ) x zpf .
B ˜ MATE , max | t m | 1 4 k N x zpf 1 | t m | 2
Δ x MATE , max ( B ˜ ) | t m | 1 1 4 k N ( 2 π ( 2 j + 1 ) 2 ϕ r ± | t m | 2 ) .
B ˜ MATE , max B ˜ MIM , max | t m | 1 8 3 3 1 | t m | .
G MIM ( 1 ) x ω MIM = ( 1 ) N + 1 2 ω FSR k N π | r m | sin ( 2 k N Δ x ) 1 | r m | 2 cos 2 ( 2 k N Δ x )
G MIM ( 2 ) x 2 ω MIM = ( 1 ) N + 1 4 ω FSR k N 2 π | r m | ( 1 | r m | 2 ) cos ( 2 k N Δ x ) ( 1 | r m | 2 cos 2 ( 2 k N Δ x ) ) 3 / 2 ,
G MIM , max ( 1 ) = ± 2 c k N L | r m |
G MIM , max ( 2 ) = ± 4 c k N 2 L | r m | 1 | r m | 2 | t m | 1 ± 4 c k N 2 L 1 | t m |
Δ x MIM , max ( 1 ) = ( 2 j + 1 ) λ 8
Δ x MIM , max ( 2 ) = j λ 4 ,
G MATE ( 1 ) x ω MATE = 2 k N π ω FSR | r m | ( | r m | + cos ( 2 k N Δ x + ϕ r ) ) | r m | 2 + 2 | r m | cos ( 2 k N Δ x + ϕ r ) + 1
G MATE ( 2 ) x 2 ω MATE = 4 k N 2 π ω FSR | r m | ( 1 | r m | 2 ) sin ( 2 k N Δ x + ϕ r ) ( | r m | 2 + 2 | r m | cos ( 2 k N Δ x + ϕ r ) + 1 ) 2 ,
G MATE , max ( 1 ) = 2 c k N L | r m | 1 | r m | | t m | 1 4 c k N L 1 | t m | 2
G MATE , max ( 2 ) = G MATE ( 2 ) ( Δ x MATE , max ( 2 ) ) | t m | 1 ± 18 3 c k N 2 L 1 | t m | 4
Δ x MATE , max ( 1 ) = ( 2 j + 1 ) π ϕ r 2 k N
Δ x MATE , max ( 2 ) = 1 2 k N ( 2 π j ϕ r ± 2 arctan { 6 | r m | + | r m | 4 + 34 | r m | 2 + 1 ( 1 | r m | ) 2 } )
Δ x MATE , pure ( 2 ) = ± arccos { | r m | } ϕ r + 2 π j 2 k N ,
G MATE , max ( 1 ) G MIM , max ( 1 ) = 1 1 | r m | | t m | 1 2 | t m | 2 ,
A ˜ max , MIM ( 1 ) = A ˜ max , MATE ( 1 ) | t m | 1 8 k N x zpf | t 1 | 2 | t m | 2 .
G MATE , max ( 2 ) G MIM , max ( 2 ) | t m | 1 9 2 3 1 | t m | 3 .
A ˜ max , MATE ( 2 ) A ˜ max , MIM ( 2 ) | t m | 1 4 3 3 1 | t m | .
B ˜ MIM , pure = ± 2 k N x zpf | r m | | t m |
B ˜ MATE , pure = ± 4 k N x zpf | r m | | t m |
B ˜ MATE , pure = ± 4 k N x zpf Δ x MATE , pure L | r m | | t m | ,
A 2 = E in ( t 1 t m e i k x ) j = 0 [ r 1 r m e 2 i k x ] j
= E in t 1 t m e i k x 1 r 1 r m e 2 i k x ,
p t | A 2 / E in | 2 = | t 1 | 2 | t m | 2 1 + | r 1 | 2 | r m | 2 2 | r 1 | | r m | cos ( 2 k x 0 + 2 k θ y + ϕ ) 2 π σ 2 e 2 y 2 σ 2 ,
p t | t 1 | 2 | t m | 2 1 + | r 1 | 2 | r m | 2 2 | r 1 | | r m | cos ( 2 k x 0 + ϕ ) 2 π σ 2 e 2 y 2 σ 2 { 1 4 k 2 θ 2 y 2 | r 1 | | r m | ( 1 + | r 1 | 2 | r m | 2 ) cos ( 2 k x 0 + ϕ ) + | r 1 | | r m | ( cos ( 4 k x 0 + 2 ϕ ) 3 ) [ 1 + | r 1 | 2 | r m | 2 2 | r 1 | | r m | cos ( 2 k x 0 + ϕ ) ] 2 } ,
P t | t 1 | 2 | t m | 2 1 + | r 1 | 2 | r m | 2 2 | r 1 | | r m | cos ( 2 k x 0 + ϕ ) { 1 k 2 θ 2 σ 2 | r 1 | | r m | ( 1 + | r 1 | 2 | r m | 2 ) cos ( 2 k x 0 + ϕ ) + | r 1 | | r m | ( cos ( 4 k x 0 + 2 ϕ ) 3 ) [ 1 + | r 1 | 2 | r m | 2 2 | r 1 | | r m | cos ( 2 k x 0 + ϕ ) ] 2 } .

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