Asymmetry-Based Quantum Backaction Suppression in Quadratic Optomechanics

As the field of optomechanics advances, quadratic dispersive coupling (QDC) promise an increasingly feasible class of qualitatively new functionality. However, the leading QDC geometries also generate linear dissipative coupling, and an associated quantum radiation force noise that is detrimental to QDC applications. Here, we propose a simple modification that dramatically reduces this noise without altering the QDC strength. We identify optimal regimes of operation, and discuss advantages within the examples of optical levitation and nondestructive phonon measurement.

As the field of optomechanics advances, quadratic dispersive coupling (QDC) promise an increasingly feasible class of qualitatively new functionality. However, the leading QDC geometries also generate linear dissipative coupling, and an associated quantum radiation force noise that is detrimental to QDC applications. Here, we propose a simple modification that dramatically reduces this noise without altering the QDC strength. We identify optimal regimes of operation, and discuss advantages within the examples of optical levitation and nondestructive phonon measurement.
Introduction.-The field of optomechanics [1] explores the forces exerted by light, and increasingly accesses the quantum regime with an eye toward sensing, information, and fundamental tests of macroscopic quantum motion. To date, the vast majority of experimental breakthroughs -room-temperature generation of broadband squeezed light [2], measurement near the Heisenberg limit or below the standard quantum limit [3,4], quantum information transduction [5], and creation of exotic quantum motion [6], to list just a few -have been achieved in systems having so-called linear dispersive coupling (LDC), wherein an optical resonance frequency depends linearly on the displacement of a mechanical element. As the field advances, systems exhibiting purely quadratic dispersive coupling (QDC), wherein the optical frequency depends on the square of mechanical displacement, promise a wide and increasingly feasible range of applications that are qualitatively different, such as quantum nondemolition (QND) readout of phonon number [7][8][9] or shot noise [10], generation of exotic quantum states [11][12][13] or entanglement [14], non-reciprocal photon control [15], photon [16] or phonon [17] blockade, and stable center-ofmass [18,19] or torsional [20] optical traps for geometry tuned [21] and ultrahigh-Q mechanical systems.
Here we propose an optomechanical geometry that can * vincent.dumont@mail.mcgill.ca dramatically reduce QRFN without compromising the strength of QDC. Specifically, our system exploits two non-identical sub-cavities, a situation that can be realized, e.g., by simply displacing the membrane in conventional MIM setups. A previous classical wave analysis showed that this configuration can exhibit reduced linear dissipative coupling [38], suggesting the possibility of suppressing QRFN in the quantum regime. Here we present a full quantum analysis to verify this conjecture and quantify its limits. We first derive the equations of motion, revealing the quantum mechanical origin of QDC, then quantify the QFRN with full consideration of fundamental sources of quantum noise. In the ideal regime of negligible internal loss and a single-port cavity (i.e., where one mirror is perfectly reflective), we show that QFRN can be suppressed by many orders of magnitude by reducing the length of one sub-cavity. When internal loss is non-negligible, our analysis identifies the optimal configuration that minimizes QFRN, achieving more than two orders of magnitude suppression in a realistic system. We then discuss how this can be applied to cavity-assisted optical levitation, achieving noise below that of free-space traps (and MIM systems), and to improving the resolvability of QND phonon number readout.
Quantum model.-For illustrative purposes, we focus on the membrane-cavity geometry shown in Fig. 1(a); a detailed derivation including similar expressions for general systems can be found in the supplementary materials [39]. Our setup consists of a cavity of length L partitioned into two sub-cavities by a membrane with (field) transmission |t m | 1. Including the quantum mechanical displacement of the membranex, the length of the sub-cavities 1 and 2 are respectively L 1 +x and L 2 −x, where L 1 and L 2 ≡ L − L 1 are chosen such that the sub-cavity frequencies are degenerate at frequency ω 0 = N 1 πc/L 1 = N 2 πc/L 2 for some integers N 1 and N 2 . The Hamiltonian of the sub-cavity photonic fields is given byĤ opt =Ĥ 1 +Ĥ 2 +Ĥ c , where, at the leading order ofx, the photonic energies arê H j = ω 0 +x(−1) j ω 0 /L j â † jâ j for j = 1, 2, withâ j being the photon annihilation operator. The transmissive membrane couples the two sub-cavities viaĤ c = The sub-cavities are coupled to each other through the membrane at rate J. Drive and detection are conducted through the external coupling at rates κ ext j , while photons are lost at rates κ int j . (b) Cavity transmission for varied membrane position ∆x and drive detuning ∆ ≡ ωin − ω0. In this example, |tm| 2 = 7 × 10 4 ppm, mirror 1 transmission |t1| 2 = 6 × 10 4 ppm, back mirror transmission |t2| 2 = 4 × 10 4 ppm, and asymmetry L1/L2 = 100. Grey dashed lines show the uncoupled (J = 0) sub-cavity frequencies ωin = ωj ≡ ω0 1 + (−1) j ∆x/Lj ; orange dashed lines show eigenfrequencies ω±.
. In contrast to a MIM setup, the LDC of photonic eigenmodes is non-vanishing when L 1 = L 2 . To obtain purely QDC, the mean position of the membrane needs to be slightly displaced to a quadratic point ( Fig. 1(b)). To illustrate the idea, we first define the membrane's quantum motionẑ around the classical displacement ∆x, i.e.
x ≡ ∆x +ẑ where ∆x L j . The photonic Hamiltonian can be generally re-written aŝ whereĤ p involves only photonic fields, and is the radiation force [40]. At any ∆x, the eigenmodes ofĤ p can be expressed succinctly asâ ± (∆x) = cos(θ ± )â 1 + sin(θ ± )â 2 , where the amplitudes satisfy cot(2θ ± ) = ±Lω 0 ∆x/ √ L 1 L 2 c|t m |. In terms of the eigenmodes,Ĥ p = ω +â † The radiation force can also be expressed in terms of the eigenmodes asF opt = h +â † on the composition of eigenmodes, and thus ∆x. The dispersive optomechanical coupling in Eq. (1) is generally linear (i.e., the adiabatic eigenfrequencies mostly depend linearly on position to leading order), with the exception of the "quadratic points" where the frequency of one eigenmode exhibits QDC to leading order inẑ (i.e., h ± = 0 at ∆x ± ). For simplicity, we focus on ∆x + hereafter, since the physics of interest is identical for ∆x − . The leading coherent optomechanical effect onâ + will be a tunnelling withâ − , but, due to the gap of eigenfrequencies in Eq. (3),â − can be adiabatically eliminated, and the dynamics ofâ + is governed by a pure QDC:Ĥ Importantly, our derivation shows that the overall QDC strength is determined by the cavity length L but not individual sub-cavity lengths L 1 and L 2 , in agreement with classical calculations. Force noise from dissipative coupling.-While the effective adiabatic Hamiltonian in Eq. (5) is purely quadratic inẑ, the full dynamics give rise to linear dissipative backaction. In particular, the cavity is coupled to an external environment for control and readout, and is also subjected to internal loss. In the membrane-cavity setup, the external coupling and internal loss rates are, respectively, where t j is the end-mirror field transmission, and T j is the round-trip internal photon loss fraction for each subcavity. These environmental couplings create fluctuation in the sub-cavity fields that generates QRFN, with power spectral density S F F (ω) = e iωt δF (t)δF (0) dt (where δF ≡F opt − F opt is the optical force fluctuation [41]) we will now quantify.
Ideal single-port cavity.-To illustrate the potential for improvement, we first consider a single-port cavity (κ int 2 = κ ext 2 = 0). In the presence of a drive, S F F (ω) can be calculated using the linearized Heisenberg-Langevin equations of motion [41]. Ifâ + mode is driven through mirror 1 at frequency ω in , the QRFN power spectral density becomes where |ā + | 2 is the mean photon number in theâ + mode, is the total loss rate of sub-cavity j, is the susceptibility, andχ lm are matrix elements of the eigenmode susceptibilitỹ In the 'large-gap' limit 2c|t m |/L κ 1 , κ 2 , |ω|, and when the drive is resonant withâ + (ω in = ω + ), Eq. (7) reduces to where theâ + decay rate κ + (∆x + ) = κ 1 L 1 /L. The advantage of our setup is now clear: when the membrane is positioned near the back mirror, (L 2 L 1 ≈ L), QRFN is suppressed by a factor (2L 2 /L) 2 . In a centimeter-scale cavity with wavelength-scale membrane-mirror separation [38], this suppression factor can reach (2L 2 /L) 2 ∼ 10 −8 when compared to the MIM geometry. We stress that the quadratic coupling in Eq. (5) remains unchanged in this limit.
Lossy cavity.-Practical photonic resonators suffer from internal losses that play an important role. While loss in sub-cavity 1 simply rescales κ 1 in Eq. (9), the loss in sub-cavity 2 (i.e. κ 2 = 0) leads to QRFN spectral density where the dissipation rate ofâ + andâ − modes are, respectively, κ + = κ 1 L 1 /L + κ 2 L 2 /L and κ − = κ 1 L 2 /L + κ 2 L 1 /L. In a modified form of the resolved-sideband regime, wherein the mechanical frequency Ω m κ + Lκ 2 /4L 1 κ − [39], the force noise in Eq. (10) then simplifies to Because κ 2 scales inversely with L 2 (due to the changing round-trip time) in Eq. (6), force noise cannot be suppressed indefinitely by shrinking the second sub-cavity. Instead, S F F (Ω M ) reaches a minimum at an optimal first sub-cavity length The markers indicates the optimal membrane position derived in Eq. (12). Inset: optimal reduction of force noise relative to the MIM setup. The orange shaded area indicates the unresolved sideband regime, and the dotted line corresponds to the resolved-sideband approximation in Eq. (14).
Notably, this is also where the sub-cavities have equal dissipation (κ 1 = κ 2 ). At quadratic points near this optimal position, When comparing with the force noise of the MIM setup (L 1 = L/2), the noise is then suppressed by a factor S min where the last step is in the typical near-single-port limit . Eq. (14) is our main result: even with internal loss, QRFN can be significantly suppressed by simply placing the mechanical mirror away from the mid-point, without reducing the QDC strength (c.f. Eq. (5)).
For completeness, we note that the force noise in Eq. (10) can also be minimized outside the resolved sideband regime, yielding where To get a sense of the practical performance of our scheme in realistic laser cavities with dielectric mirrors, Fig. 2 shows these results for a variety of parameters. Importantly, this setup can suppress QRFN by more than two orders of magnitude relative to the conventional MIM configuration. Our scheme is most advantageous in the single-port limit, with diminishing returns as the system enters the deeply unresolved sideband regime.
This main result directly benefits all QDC applications requiring low force noise; in the remaining text, we discuss this within the illustrative examples of optical levitation and QND phonon measurement.
Optical levitation.-By placing a minimally supported reflector inside a cavity at a quadratic point, the reflector's motion collinear with the cavity axis can be optically trapped. For a resolved-sideband cavity driven at frequency ω in = ω + ≈ ω 0 , the dispersive optical spring constant at ∆x + is where ω /2 = 2ω 2 0 /c|t m |L is the QDC strength in Eq. (5). This spring constant is identical to that of a standing wave in free space for the same circulating powerP circ . However, a free space trap's QRFN S FS F F ≈ 8 ω inPin /c 2 , meaning our optimal membrane-cavity system has a relative force noise (from Eq. 13) where the last expression is in the nearly-single-port limit. This means that, as long as most of the backcavity light leaves through the membrane (i.e. |t m | 2 |t 2 | 2 , T 2 ), QRFN can be significantly suppressed relative to free space (and MIM, as per Fig. 2 and Eq. (14)). Furthermore,P circ in a cavity system is achieved with much less input power, making it far easier to realize a quantum-limited light source that actually reaches this limit. QND phonon measurement.-In the resolvedsideband regime, QDC naturally measures the timeaveraged mechanical energy, enabling quantum nondemolition (QND) readout of phonon number [7,9,39], which is usually proposed assuming a near-single-port cavity. We quantify the quantum-limited performance of such measurements with the ratio of measurement rate Γ meas to backaction rate Γ ba,n [35], a figure of merit that exceeds one when it is possible to resolve phonon number state n before QRFN causes a jump. For our setup, when a + mode is driven on resonance, this ratio becomes (again assuming the large-gap limit, and a modified resolvedsideband limit Ω m /κ Γ meas Γ ba,n = 64 2n + 1 Resolvable fluctuation x 2 res for QND phonon measurement of the ground state (n = 0). Our scheme exhibits clear advantage for a wide range of front mirror transmission |t1| 2 (black and purple), and provides additional improvement when optimizing |t1| 2 opt (red). Further reducing |t1| 2 provides no benefit due to reduced collection efficiency. In all cases, the system parameters are T1 = T2 = 1 ppm, |t2| 2 = 0 ppm (a loss-limited Bragg stack), |tm| 2 = 10 4 ppm, and L = 10 cm, ωin = ω+ in the resolved-sideband limit. The dotted vertical line indicates L1 = L/2.
where g j = ω 0 x zpf /L j is the single-photon optomechanical coupling rate for sub-cavity j, and x zpf = /2mΩ m is the zero-point fluctuation of the membrane (mass m). We also define a "resolvable variance" x 2 res , which is the value of x 2 zpf the membrane must have in order to achieve unity Γ meas /Γ ba,n .
The parenthetical factor κ ext 1 L 1 /κ + L captures the cavity mode's input coupling efficiency, while the bracketed factor g 1 g 2 /κ − κ + characterizes the single-photon strong coupling in our asymmetric system.
For single-port approaches, Eq. (19) yields dramatic improvements over the MIM approach, as shown in Fig. 3. In fact, this improvement is again the same large ratio that controls the force noise suppression in Eq. (14); this is not surprising, since Γ ba,n ∝ S F F , and the quadratic coupling strength is independent of position. We emphasize the practical benefits of performing these measurements with larger |t 1 | 2 , notably boosting the collected signal above detector noise and easing the laser-lock process. In systems operating far from the single-port regime, our analysis also identifies an optimal input mirror transmission [39] that is neither single-port nor balanced. This further improves x 2 res as shown by the red curve in Fig. 3. Summary.-We propose an optomechanical setup that dramatically reduces quantum radiation force noise without affecting the quadratic dispersive coupling strength. For the illustrative membrane-cavity geometry, this is im-plemented by simply relocating the membrane toward the mirror with higher reflectivity. Our full quantum analysis identifies optimal configurations, and we demonstrate its advantage in optical levitation and nondemolition phonon measurement. Owing to its ease of implementation and our universal desire to control noise, we expect this proposal will immediately impact all optomechanical experiments aiming to exploit quadratic dispersive coupling, moving forward. We −∞ e +iωt X † (t)dt; thus, X † (ω) refers to the Fourier transform of the time domain variable X † (t). Note also that X † (ω) = [X(−ω)] † in this convention.

Supplementary Information Appendix A: Backaction Model
Here we derive the expressions for the quantum radiation pressure force noise (QRFN) in an asymmetric cavity optomechanical system. We begin with the general optical equations of motion, eigenmodes, and quadratic coupling in Sec. A 1, then focus on geometries having purely quadratic dispersive coupling in Sec. A 2, showing that asymmetry can lead to a large reduction of force noise in Sec. A 3 at the optimal membrane position. Finally, we discuss quadratic optical trapping (Sec. A 4), and quantum non-demolition (QND) mechanical energy measurement Sec. A 5, both of which benefit from reduced force noise.

Equations of Motion, Eigenmodes, Dissipation and Optomechanical Coupling
General Equations of Motion: To model the optical dynamics, we first write down the input-output equations of motion [41] for sub-cavity operatorsâ 1 andâ 2 in the frame rotating at an external drive frequency ω in . These are whereâ in j is the external drive applied to sub-cavity j, κ ext j is the associated (power) coupling rate, κ int j is the subcavity's internal loss rate, κ j = κ ext j + κ int j is the total sub-cavity decay rate, G j is the sub-cavity's optomechanical coupling, and ∆ = ω in − ω 0 is the drive's detuning relative to the frequency ω 0 at which modesâ 1 andâ 2 are degenerate, which we define to occur at mechanical displacement ∆x ≡ 0.
Optical Eigenmodes: The system's eigenfrequencies are obtained by diagonalizing the Heisenberg equations of motion (ȧ j = − i [â,Ĥ opt ]), and have associated eigenmodes that can be written succinctly asâ where the amplitudes satisfy Dispersive Couplings: When G 1 = G 2 , the eigenfrequencies of Eq. A5 exhibit an avoided crossing structure, with linear dispersive coupling (LDC) which, if G 1 < 0 and G 2 > 0, becomes zero at "quadratic points" 8 The frequencies at these extrema are corresponding to an avoided gap 4J √ −G 1 G 2 /(G 2 − G 1 ). The quadratic dispersive coupling is generally which, at ∆x ± , simplifies to The expressions (as with others involving G j and J in this document) still represent a general system with two optical modes linearly coupled to one mechanical mode. For a membrane in a cavity, where G j = (−1) j ω 0 /L j and J = c|t m |/2 √ L 1 L 2 (see Appendix B), with sub-cavity length L j and membrane (amplitude) transmission t m , this becomes where L = L 1 + L 2 is the total cavity length. Eigenmode Decay Rates: Combined with the sub-cavity (power) losses κ j = κ int j + κ ext j (note these exclude J), the amplitudes attached toâ 1 andâ 2 in Eq. A6 permit the calculation of power decay rates for eigenmodesâ ± , whereκ ≡ (κ 1 + κ 2 )/2 and ∆κ ≡ (κ 1 − κ 2 )/2. At the quadratic points, For the case of a membrane-cavity system, we can write these (choosing ∆x + ) in terms of the end mirror power transmissions |t j | 2 and sub-cavity round-trip power losses T j : where we have used κ int j = cT j /2L j and κ ext j = c|t j | 2 /2L j . At this location, κ + is identical to the empty cavity decay rate κ 0 due to the fact that sub-cavity circulating powers are identical. This is true for any quadratic point we choose.
Dissipative Coupling: From Eq. A14, the linear dissipative coupling is which, at the quadratic point, becomes with corresponding single-photon strong coupling parameter in the single-port limit (κ 2 → 0). We can now see that this coupling can be greatly reduced in the limit |G 2 | |G 1 |. For a single-port membrane-cavity, e.g.,B which diminishes as L 2 /L, suggesting a commensurate reduction of QRFN achieved by simply moving the membrane toward the back mirror.
Equations of Motion at Quadratic Point: Finally, we write down the equations of motion for the eigenmodeŝ a ± themselves at a quadratic point ∆x = ∆x + . First, defining for convenience, the equations of motion in the eigenmode basis (Eqs. A6) becomė where δ + ≡ ω in − ω + = ∆ + 2Jαβ is the detuning from the "+" resonance, and ∆κ ≡ (κ 1 − κ 2 )/2 (as above). These reduce to those of Ref.

Measurement Backaction with Quadratic Dispersive Coupling
Optical Susceptibilities: In the frequency domain, the equations of motion in the original sub-cavity basisâ j (Eq. A1) can be written where I is the identity matrix, andâ in j comprises any drive and / or fluctuations (including thermal). We can define a susceptibility matrixχ is the optical susceptibility of the uncoupled (J = 0) sub-cavity j, andχ ij is shorthand notation for the matrix elements. At the quadratic point ∆x = ∆x + , the sub-cavity susceptibilities become Radiation Force Noise: At the quadratic point ∆x + , the optical force operatorF opt = −dĤ opt /dx reduces tô The corresponding power spectral density can be written as [41][46] where δF opt (ω) is the Fourier transform of the fluctuations δF opt (t) ≡F opt (t) − F opt about the mean (i.e. timeaveraged) F opt . Similarly, expressingâ j (t) =ā j + δâ j (t) in terms of fluctuations δâ j (t) about the meanā j , we can linearize the optical force operator for small δâ j , yielding such that (similar to Ref. [36]) its Fourier transform becomes with coefficients whereχ nm are the matrix elements of the optical susceptibilityχ (Eq. A29). Assuming the usual input noise correlators [41] wheren in j is the mean thermal occupation of the input port bath; the same relations hold for the noise operators of the loss channels by changing subscript "in" → "int" throughout (though these ports are considered purely thermal). If the baths are in their ground state (n in j =n int j ≈ 0), the force noise simplifies to In this work, we are particularly interested in the force noise when one eigenmode, having purely quadratic dispersive coupling, is driven by a single port. For example, if ∆x = ∆x + and the system is driven byā in 1 (withā in 2 = 0), the steady-state amplitudes simplify toā Together with Eq. A6, the ratioā 1 /ā 2 now allows us to expressā j in terms ofā + as which, finally, allows us to write the force noise (Eq. A42) in terms of |ā + | 2 : In the large-gap limit, where 4Jαβ κ 1 , κ 2 , |ω|, |δ + |, such that only the "+" mode is involved, at zero detuning this simplifies to If we instead consider a single-port cavity (κ 2 = 0 and κ 1 = κ ext 1 ), Eq. A47 becomes which reduces to in the large-gap limit. This expression agrees with the force noise calculated from the simple assumption of dissipative coupling [27] whenB is given by Eq. A22.

Force Noise Reduction
To see how asymmetric sub-cavity LDC can reduce quantum radiation force noise (QRFN), we now consider a membrane-cavity style system. In the large-gap limit, the QRFN (Eq. A49) is with decay rates at the quadratic point ∆x + . At the mechanical frequency Ω m , this be rewritten This expression can be minimized with respect to L 1 for fixed cavity length, by substituting the expressions for the decay rates (Eqs. A18 and A19), yielding where we defined noting B ∈ [0, 1] with B = 1 corresponding to the "single-port" limit, where |t 1 | 2 + T 1 |t 2 | 2 + T 2 . This minimum is found a distance from the input mirror. "Modified" resolved-sideband limit: In the limit Ω m /κ + Lκ 2 /4L 1 κ − and Ω m /κ + 1/2 -both of which are succinctly captured by √ BΩ m κ + at L 1,min -the minimal QRFN reduces to where the second line is evaluated for small but finite |t 2 | 2 + T 2 |t 1 | 2 + T 1 (B ≈ 1). In the latter case, the QRFN is limited simply by the distance between the lossless back mirror and its closest quadratic point (wavelength scale).

Application: Optical Trapping
The optical spring generated at a quadratic point has the same strength as a free-space trap [47] per watt incident on the membrane, though with significantly less input power due to the cavity enhancement. As discussed below, our scheme realizes this with lower QRFN than can be achieved in free space (or with a MIM system).
If we write the optical Hamiltonian in the "+" mode basis, and expand its frequency ω + to second order in mechanical displacementx, we find which allows us to directly identify the spring constant where we have substituted in the quadratic dispersive coupling ∂ 2 x ω + from Eq. A13 and considered (for the time being) only the non-dynamical part of the spring (i.e.,â + ≈ā + ). The optical spring can then be expressed as a function of the (mean) circulating powerP using ω in ≈ ω 0 , which is the same expression as for a free-space trap with a retro-reflected beam [47]. When approaching the single-port limit (|t 2 | 2 + T 2 |t 1 | 2 + T 1 ), the minimal force noise at the optimal membrane position is regardless of whether the system is sideband-resolved, where is the force noise associated with a free-space trap in the limit |t m | 2 1. S FS F F can be obtained noting that, for a highly reflective membrane, the force applied to each side is F ≈ 2P circ /c, yielding a force noise of S F F = 4S PcircPcirc /c 2 . For shot noise, the power spectral density is S PcircPcirc = ω inPcirc and since shot noise is uncorrelated at each side of the membrane, it adds in quadrature, yielding the above equation. Importantly, Eq. A66 shows that the force noise can be improved by a factor 2|t m | 2 /(|t 2 | 2 + T 2 ); together with the comparative ease of realizing shot-noise-limited input light at lower powers, our approach presents a significant advantage over free-space traps.

Application: QND Measurement
Here we present the potential advantages our technique provides within the context of quantum nondemolition (QND) phonon number measurements.

a. Backaction Rate
The rate at which S F F adds or removes a phonon from a mechanical oscillator containing n phonons is [10,41] With a resonant drive (δ + = 0) and in the resolved sideband regime (Ω m κ + ), this is where we assume ( In a quadratically-coupled optomechanical system, phonons each produce a shift in the resonant frequency of the optical mode [7][8][9]. In a resonantly driven cavity, this will produce a shift in the phase of the reflected light. Here we derive the phonon measurement rate for an asymmetric optomechanical system with quadratic dispersive coupling using homodyne detection.

Input-Output Relations
To quantify how the phase of the reflected light depends on phonon number, we first relate the mean reflected output fieldā out 1 to the mean input fieldā in 1 (assuming we only address sub-cavity 1) using the standard input-output relation [41]ā out 1 whereā 1 is the mean field in sub-cavity 1; the second line is obtained from Eq. A43. In the large gap limit, where 4Jαβ κ 1 , κ 2 , |ω|, |δ + |, which simplifies the valueχ 11 given by Eq. A29, this becomes where, in the second line, we defined the (real) amplitude reflection coefficient and the (real) reflected phase φ ≡ arctan δ + β 2 κ ext 1 δ 2 For a resonant drive frequency ω in (detuning δ + = 0), this phase changes as We can also relate the mean field in the "+" mode to the input field using Eqs. A6, A43 and A44: so that the output field (Eq. A70) can be written in terms of "+" mode as In the large-gap limit, this reduces toā (A77)

Measurement Rate with Homodyne Detection
To derive the measurement rate, we consider an ideal homodyne measurement (all the photons are collected, the laser is shot-noise limited, and the detection scheme has identical arms) where a local-oscillator ("LO"; mean optical powerP LO ) is combined at a beamsplitter with a measurement beam leaving the cavity (mean powerP out 1 ) with the aim of resolving its fluctuating phase δφ(t). Specifically, suppose one beamsplitter input has classical LO field E LO = √ 2P LO cos(ω in t + ∆φ) with some fixed phase ∆φ, and the other has the signal field E out 1 = 2P out 1 cos(ω in t + δφ(t)), both oscillating at frequency ω in . The homodyne signal, obtained by subtracting the photocurrents measured at each of the beamsplitter outputs, is then where A is a gain relating optical power to photocurrent. Tuning the LO phase to maximize the sensitivity to phase (e.g. ∆φ = 0) and Taylor expanding sin yields At the same time, the combined shot noise power spectral density S sn II from the two (subtracted) photocurrents provides a noise floor and, for large LO (P LO P out 1 ), Furthermore, since a frequency fluctuation δω + of the cavity produces phase δφ ≈ (dφ/dω + )δω + (dφ/dω + derived as above), the frequency noise floor where, in the last step, we substituted in Eqs. A80-A81 and expressed the mean signal powerP out 1 = ω in |ā out 1 | 2 in term of the reflected photon rate |ā out 1 | 2 , in accordance with Ref.
[41]. To convert this result to the measurement time t meas required to resolve a frequency shift ∆ω + , we compare the noise floor to the frequency shift, requiring in order to resolve the frequency shift with unity signal-to-noise ratio. For a single phonon jump, the induced frequency shift is yielding a measurement rate (A85) In the large gap limit, using Eq. A12 for d 2 ω + /dx 2 , and Eqs. A74 and A77 derived in Sec. A 5 b to substitute dφ/dω + andā out,1 respectively, this measurement rate simplifies to The correction term β 2 κ in 1 /κ + accounts for the proportion of photons in the "+" mode leaving through the input port.
For the case of a truly single-port cavity (κ 2 = 0 and κ 1 = κ ext 1 ), the measurement rate is Note for a membrane-cavity system, κ + , ∂ 2 x ω + , and thus Γ meas , do not depend upon at which quadratic point the membrane is positioned, allowing one to tune the ratio G 1 /G 2 without affecting the measurement rate.

c. Backaction-Limited Number State Resolution
The measurement rate, which is derived in Sec. A 5 b below, is yielding a figure of merit Γ meas Γ ba,n = 64 2n + 1 describing how well a phonon number state n can be measured before backaction causes a jump, where we define the usual optomechanical coupling rate g i ≡ x zpf |G i |. In the single-port cavity limit (κ 2 = 0), this ratio becomes Γ meas Γ ba,n = 64 2n + 1 (A91) The advantage of our approach becomes clearer if we consider a membrane-cavity system in which the membrane is a distance L 1 from the first mirror and L 2 from the second (total length L = L 1 + L 2 ). If the sub-cavity crossing frequency ω 0 = N 1 πc/L 1 = N 2 πc/L 2 where N i = 2L i /λ is the (integer) mode index of each sub-cavity at wavelength λ, and the coupling rates G 1 = −ω 0 /L 1 and G 2 = +ω 0 /L 2 , the ratio becomes Γ meas Γ ba,n = 64 2n + 1 with κ 0 being the empty cavity power decay rate (Eq. A18). The second line is expressed in terms of the half-cavity single-photon optomechanical coupling rate for the MIM system g MIM ≡ 2x zpf ω 0 /L to facilitate a direct comparison: by reducing L 2 (moving the membrane toward the back mirror), the usual strong coupling requirement g MIM /κ 0 > 1 for measuring a state before QRFN destroys it [36] is relaxed by a factor L/2L 2 , which can be as large as ∼ L/λ at the quadratic point nearest the back mirror. For fixed mirror losses (round trip power loss T j = T , say) we can improve the fidelity by a less dramatic factor, and there is some advantage to be gained by bringing |t 1 | 2 closer to T . To calculate the optimal L 1,min and t 1,opt , we first substitute the expression for the decay rates (Eqs. A18 and A19), and optomechanical coupling g i = x zpf ω 0 /L j in the ratio Eq. A90; the optimal membrane position is then where the force noise is minimal and the ratio Γ meas Γ ba,n L1,min = 256 2n + 1 is maximal; L 1,min is the same as before (Eq. A57) since the measurement rate is independent of the membrane position within the cavity. We can also compare with the same ratio for the membrane-in-the-middle, Γ meas Γ ba,n L/2 = 1024 2n + 1 finding our approach yields an improvement Γ meas Γ ba,n L1,min Γ meas Γ ba,n L/2 = 1 4 As expected, this is the same improvement as we found for the force noise.
With the membrane at the optimal L 1,min , we can also calculate the the optimal input mirror transmission that maximizes the ratio, yielding Γ meas Γ ba,n L1,min,|t1| 2 opt = 256 2n + 1 This optimized QND "fidelity" is a factor Γ meas Γ ba,n L1,min,|t1| 2 opt Γ meas Γ ba,n L1,min,matched larger than that of a traditional matched cavity (|t 1 | set equal to |t 2 |). Equation A99 can be viewed as a prefactor that generalizes the existing "standard quantum limit" [35] to the case of asymmetric systems. The largest gains occur in systems maximing the back-mirror reflectivity, which is just a statement that one needs to open the input mirror enough to get a reasonable fraction of the cavity light out, and up to 3 dB improvement can also be achieved with transmission-dominated mirrors (T |t j | 2 ), though this assumes we ignore the information in the light leaving the back mirror.

Appendix B: Hopping Rate for a Membrane-in-Cavity
In this section, we derive the classical equations of motion for the electric fields for a cavity with a membrane inside it, extending the formalism of Refs. [49][50][51]. This allows for a simple derivation of EOMs and direct access to the hopping rate J between two sub-cavities separated by a partial reflector.
Consider the Fabry-Perot cavity in Fig. 4, comprising two end mirrors of field reflection (transmission) coefficients r j (t j ) partitioned by a third mirror (membrane) having field reflection (transmission) coefficient r m (t m ), such that the sub-cavity lengths L j sum to total length L = L 1 + L 2 . The right-moving field amplitude A 2 just to the right of the membrane at a time t is related to the left and right incoming amplitudes A 1 and B 2 as Following this wave a round-trip time τ 2 = 2L 2 /c later in sub-cavity 2 (c is the speed of light), we find a returning field in a frame rotating at the laser frequency ω in . Substituting Eq. B1 in Eq. B2, we obtain B 2 (t + τ 2 ) = r m r 2 e iωinτ2 B 2 (t) + t m r 2 e iωinτ2 A 1 (t) (B3) For small τ 2 [52], we can approximate the time derivative of B 2 aṡ Following the same method for the left sub-cavity field, we finḋ As a matter of convention, we then rescale the fields as such that |α j | 2 represents the number of photons in sub-cavity j. In a matrix form, these coupled equations then become in agreement with the more careful derivation of Ref. [53].
To recover the equations obtained from input-output theory, we make additional approximations. First, for simplicity, we now make a common unitarity preserving choice for the phase on all the coefficients, such that r m = −|r m | and t m = i|t m |. We furthermore assume that the end mirrors have high reflectivity such that r j ≈ −1 + |t j | 2 /2, and write the laser frequency as where ∆ ≡ ω in − ω 0 is the detuning from the crossing frequency (defining ∆x ≡ 0), ω j (∆x) is the resonant frequency of sub-cavity j, and G j = ∂ x ω j is the usual linear dispersive coupling. Note that, by definition, the sub cavities have an integer N j = 2L/λ j half wavelengths, such that ω j (∆x)τ j = 2πN j . Again that the detuning is much smaller than the round-trip rate of each sub-cavity (|∆| ∼ κ j 1/τ j ), that the membrane is displaced by much less than a half wavelength (∆x λ/2, or, equivalently, |G j ∆x| 1/τ j ), and keeping only the terms to leading order in |t m |, |t 1 | 2 , |t 2 | 2 , and ∆, with κ j = c|t j | 2 /(2L j ). Most importantly, we can immediately identify the off-diagonal elements as the hopping rate