Comparing nonlinear optomechanical coupling in membrane-in-the-middle and single-cavity systems

We begin by revisiting the linear and intrinsic nonlinear optomechanical coupling that occurs in single cavity optomechanical systems. The optical mode couples to an external field with rate κ_ex. The mechanical dissipation rate is Γ_m. Starting from the Hamiltonian of equation (1), moving to a frame rotating at the laser drive frequency ω_L and introducing the laser detuning Δ = ω_L − ω_c, the quantum Heisenberg–Langevin equations of motion can be derived. These govern the dynamics of the operators in the open quantum system [16] and read

$$\begin{equation}\dot {\hat{a}}=-\frac{\kappa }{2}\hat{a}+i\left({\Delta}+{g}_{0}\hat{x}\right)\hat{a}+\sqrt{{\kappa }_{\text{ex}}}{\hat{a}}_{\text{in}}+\sqrt{{\kappa }_{0}}{\hat{f}}_{\text{in}},\end{equation} \tag{ 2a }$$

$$\begin{equation}\dot {\hat{x}}=2{\times}{{\Omega}}_{m}\hat{p},\end{equation} \tag{ 2b }$$

$$\begin{equation}\dot {\hat{p}}=-\frac{{{\Omega}}_{m}}{2}\hat{x}-{{\Gamma}}_{m}\hat{p}+{g}_{0}{\hat{a}}^{{\dagger}}\hat{a}-\frac{{\hat{F}}_{\text{in}}}{m{{\Omega}}_{m}{x}_{\text{zpf}}}+\sqrt{{{\Gamma}}_{m}}{\hat{P}}_{\text{in}},\end{equation} \tag{ 2c }$$

where we have used the unitless mechanical position and momentum operators, $\hat{x}=\left(\hat{b}+{\hat{b}}^{{\dagger}}\right)$ and $\hat{p}=i\left(\hat{b}{}^{{\dagger}}-\hat{b}\right)/2$, respectively. We have introduced ${\hat{a}}_{\text{in}},{\hat{f}}_{\text{in}}$, for the optical input fields through the external channel and quantum fluctuations that drive the system and cause intrinsic decay, with rates κ_ex and κ₀, respectively, fulfiling κ₀ + κ_ex = κ, where κ is the total decay rate. The field ${\hat{P}}_{\text{in}}$ introduces mechanical fluctuations associated with coupling to a thermal bath whereas ${\hat{F}}_{\text{in}}$ accounts for coherent mechanical drive fields (${\hat{H}}_{\mathrm{d}}=-\hat{x}{\hat{F}}_{\text{in}}/\left(m{{\Omega}}_{m}{x}_{\text{zpf}}\right)$). Also, ${x}_{\text{zpf}}\equiv \sqrt{1/\left(2m{{\Omega}}_{m}\right)}$ is the mechanical zero point motion for the mechanical oscillator with effective mass m. In our calculations, we reduce these equations to the semiclassical, nonlinear equations of motion in the mean-field approximation $\left\langle \hat{x}\hat{a}\right\rangle \approx xa$, denoting $\left\langle \hat{a}\right\rangle =a$ and $\left\langle \hat{x}\right\rangle =x$. Assuming no external mechanical forces ($\left\langle {\hat{F}}_{\text{in}}\right\rangle =0$) and incoherent (e.g., thermal) input fluctuations, $\left\langle {P}_{\text{in}}\right\rangle =0,\left\langle {f}_{\text{in}}\right\rangle =0$, we arrive to:

$$\begin{equation}\ddot {x}=-{{\Omega}}_{m}^{2}x-{{\Gamma}}_{m}\dot {x}+{{\Omega}}_{m}{g}_{0}\vert a{\vert }^{2},\end{equation} \tag{ 3a }$$

$$\begin{equation}\dot {a}=i\left(\tilde {{\Delta}}+{g}_{0}x\right)a+\sqrt{{\kappa }_{\text{ex}}}{a}_{\text{in}}.\end{equation} \tag{ 3b }$$

Here we, for convenience, absorbed the optical decay rate as an imaginary part of the complex detuning $\tilde {{\Delta}}$: $\kappa =2\enspace \mathrm{I}\mathrm{m}\left(\tilde {{\Delta}}\right)$.

In taking the mean-field approximation, our treatment does no longer take into account quantum fluctuations in either the optical or mechanical modes, being formally valid within the classical regime. However, some estimations can still be made regarding quantum-limited processes. In particular, when inferring the mechanical position $\hat{x}$ from a measurement of optical output field ${\hat{a}}_{\text{out}}$, shot noise fundamentally limits the available information about $\langle \hat{x}\rangle$. The resulting measurement imprecision on $\hat{x}$, expressed as a displacement noise spectral density ${\bar{S}}_{\text{xx}}^{\text{imp}}\left(\omega \right)$, can be estimated by comparing the shot noise spectral density to the classical signal at the optical output generated by transduction of coherent mechanical motion [39], the latter of which can be calculated from the mean-field equations following our treatment below. In addition, the imprecision-force inequality ${\bar{S}}_{\text{xx}}^{\text{imp}}\left(\omega \right)\cdot {\bar{S}}_{\text{FF}}\left(\omega \right){\leqslant}{\hslash }^{2}/4$ provides a lower bound on the backaction force power spectral density ${\bar{S}}_{\text{FF}}\left(\omega \right)$, given the imprecision noise spectral density ${\bar{S}}_{\text{xx}}^{\text{imp}}\left(\omega \right)$ [40], from which the magnitude of backaction on the mechanical resonator can be estimated. We stress, however, that our equations can not be used to give general predictions for quantum dynamics, as the mean field approximation breaks down for mechanical or optical states that can not be approximated as large, coherent amplitudes and for ${g}_{0}/\kappa { >}1$, the condition hallmarking the SPSC regime.

The steady state solutions follow from setting $\ddot {x},\dot {a}=0$ in equation (3a):

$$\begin{equation}\bar{a}=i\frac{\sqrt{{\kappa }_{\text{ex}}}}{\bar{{\Delta}}}{a}_{\text{in}},\end{equation} \tag{ 4a }$$

$$\begin{equation}\bar{x}=\frac{{g}_{0}}{{{\Omega}}_{m}}\vert \bar{a}{\vert }^{2}.\end{equation} \tag{ 4b }$$

Here, $\bar{{\Delta}}=\tilde {{\Delta}}+{g}_{0}\bar{x}$, which still contains $\bar{x}$. However, we will assume that the optical power is limited such that the static displacement of the resonator is much smaller than the linewidth, ${g}_{0}\bar{x}\ll \kappa$, such that $\bar{{\Delta}}\approx \tilde {{\Delta}}$. This sets an upper limit for a few 100 intracavity photons in photonic crystal cavities [41], while for other systems it is much less restricting.

We will evaluate the optical sidebands created by coherent mechanical motion of a specific amplitude X₀, described by $x=\bar{x}+{X}_{0}\enspace \mathrm{cos}\left({{\Omega}}_{m}t\right)$. This ansatz neglects mechanical damping by assuming Γ_m ≪ Ω_m. For the optical field, we look for a perturbative solution of the form [33]:

$$\begin{equation}a\left(t\right)=\bar{a}+\sum _{\zeta ={\pm}}{A}_{\zeta }^{\left(1\right)}\enspace {\text{e}}^{\text{i}\zeta {{\Omega}}_{m}t}+{A}_{\zeta }^{\left(2\right)}\enspace {\text{e}}^{\text{i}\zeta 2{{\Omega}}_{m}t}.\end{equation} \tag{ 5 }$$

By collecting terms in the mean-field EOM with the same time dependence, we can solve for the first-order coefficients:

$$\begin{equation}{A}_{{\pm}}^{\left(1\right)}=\frac{{g}_{0}\bar{a}}{{\pm}{{\Omega}}_{m}-\bar{{\Delta}}}\frac{{X}_{0}}{2}.\end{equation} \tag{ 6 }$$

And, using this result, we can also retrieve second-order coefficients

$$\begin{equation}{A}_{{\pm}}^{\left(2\right)}=\frac{{g}_{0}{A}_{{\pm}}^{\left(1\right)}}{{\pm}2{{\Omega}}_{m}-\bar{{\Delta}}}\frac{{X}_{0}}{2}=\frac{{g}_{0}^{2}\bar{a}}{\left({\pm}2{{\Omega}}_{m}-\bar{{\Delta}}\right)\left({\pm}{{\Omega}}_{m}-\bar{{\Delta}}\right)}{\left(\frac{{X}_{0}}{2}\right)}^{2}.\end{equation} \tag{ 7 }$$

In the approach we take above, the hierarchy of higher-order sidebands has been truncated assuming the cavity resonance frequency shift resulting from mechanical motion is negligible compared to the optical linewidth, i.e. ${g}_{0}\bar{x}{< }\kappa$, in which case every higher-order sideband can be treated as a perturbation of the previous. Indeed, various applications of quadratic coupling rely on this more practically reached coupling regime [9, 10, 14, 15]. Because current optomechanical devices are not in the SPSC regime, the condition g₀x < κ holds for most devices, although this is not fulfiled by exceptions with large g₀ and mechanical amplitudes [36].

Having applied our approach to single cavities, we now move to the MIM system. Our starting point is the standard Hamiltonian of the MIM system in the rotating frame of an input laser field detuned from two optical modes by $\mathrm{R}\mathrm{e}\enspace {\tilde {{\Delta}}}_{i}={\omega }_{L}-{\omega }_{c,i}$, with loss rates $2\enspace \mathrm{I}\mathrm{m}\enspace {\tilde {{\Delta}}}_{i}={\kappa }_{i}$ that are coupled to a single mechanical membrane, displaced from the equilibrium position by $\hat{x}$. In the basis of the physical cavities with annihilation operators ${\hat{a}}_{i}$ (i = {1, 2}) the system is governed by the Hamiltonian

$$\begin{equation}\hat{H}={{\Omega}}_{m}{\hat{b}}^{{\dagger}}\hat{b}+{\hat{H}}_{\text{OM}}+{\hat{H}}_{J}+\sum _{i}{\hat{H}}_{{\kappa }_{i}},\end{equation} \tag{ 8 }$$

where optomechanical coupling reads

$$\begin{equation}{\hat{H}}_{\text{OM}}=-\left({{\Delta}}_{1}+{g}_{0,1}\hat{x}\right){\hat{a}}_{1}^{{\dagger}}{\hat{a}}_{1}-\left({{\Delta}}_{2}-{g}_{0,2}\hat{x}\right){\hat{a}}_{2}^{{\dagger}}{\hat{a}}_{2},\end{equation} \tag{ 9 }$$

and the optical inter-cavity coupling is characterized by

$$\begin{equation}{\hat{H}}_{J}=-J\left({\hat{a}}_{1}^{{\dagger}}{\hat{a}}_{2}+{\hat{a}}_{2}^{{\dagger}}{\hat{a}}_{1}\right),\end{equation} \tag{ 10 }$$

where J is the rate of inter-cavity coupling. Coupling to input/output channels via Hamiltonians ${\hat{H}}_{{\kappa }_{i}}$ is assummed to occur to separate environments, (e.g., single-mode waveguides) with rates κ_ex,i. Because the optical cavities are coupled, equation (8) can be expressed in terms of the optical supermodes that arise. In conditions of equal cavity frequency Δ₁ = Δ₂ ≡ Δ, these are given by ${\hat{a}}_{e,o}=\left({\hat{a}}_{1}{\pm}{\hat{a}}_{2}\right)/\sqrt{2}$. These supermodes are also depicted in figure 1(a). Assuming equal optomechanical couplings g_0,1 = g_0,2 ≡ g₀, the system Hamiltonian in this basis reads $\hat{H}={{\Omega}}_{m}{\hat{b}}^{{\dagger}}\hat{b}+{\sum }_{s=e,o}{\omega }_{s}{\hat{a}}_{s}^{{\dagger}}{\hat{a}}_{s}+{\hat{H}}_{\text{OM}}$ with ω_e,o = −Δ ∓ J, with an optomechanical interaction:

$$\begin{equation}{\hat{H}}_{\text{OM}}=-{g}_{0}\hat{x}\left({\hat{a}}_{e}^{{\dagger}}{\hat{a}}_{o}+{\hat{a}}_{o}^{{\dagger}}{\hat{a}}_{e}\right).\end{equation} \tag{ 11 }$$

Here, we want to emphasize the fact that optomechanical coupling has now become cross-mode, i.e. the Hamiltonian contains terms $\propto \hat{x}{\hat{a}}_{e}^{{\dagger}}{\hat{a}}_{o}+\mathrm{H}.\mathrm{c}.$, whereas it previously contained self-mode terms, $\propto \hat{x}{\hat{a}}_{1}^{{\dagger}}{\hat{a}}_{1},\hat{x}{\hat{a}}_{2}^{{\dagger}}{\hat{a}}_{2}$.

The frequencies of these optical supermodes can be found by treating this mechanical position as a quasi-static parameter analogous to the Born–Oppenheimer approximation of molecular physics ($\hat{x}{\mapsto}x$). This is only valid for mechanical motion that is slow with respect to the optical coupling rate, or J ≫ Ω_m, which is not true for a number of experimental implementations [7, 19, 42]. Using this adiabatic approximation allows for diagonalization of the system Hamiltonian in equation (8) [17], yielding the x-dependent eigenfrequencies in figure 1(b). Still assuming equal frequency of both optical cavities, this dependence is approximately quadratic and given by ${\omega }_{e,o}^{\text{ad.}}\left(x\right)\approx -{\Delta}\mp \left(J+{g}_{0}^{\left(2\right)}{x}^{2}\right)$, or, equivalently, the effective quadratic coupling Hamiltonian

$$\begin{equation}{\hat{H}}^{\text{ad.}}=-{\Delta}\left({\hat{a}}_{e}^{{\dagger}}{\hat{a}}_{e}+{\hat{a}}_{o}^{{\dagger}}{\hat{a}}_{o}\right)-\left(J+{g}_{0}^{\left(2\right)}{\hat{x}}^{2}\right)\left({\hat{a}}_{e}^{{\dagger}}{\hat{a}}_{e}-{\hat{a}}_{o}^{{\dagger}}{\hat{a}}_{o}\right),\end{equation} \tag{ 12 }$$

with effective quadratic coupling ${g}_{0}^{\left(2\right)}={g}_{0}^{2}/2J$. Through the adiabatic approximation, the threefold interaction that combines optical supermodes and the mechanical operator in equation (11) is thus reduced to a pair of effectively quadratic interactions. Furthermore, by assuming one of the optical supermodes to be off-resonant, the standard approach removes its effect completely, yielding an effective, quadratic Hamiltonian that involves only a certain superposition of the cavity fields. It is this form of the Hamiltonian that drew attention to the MIM system as a platform for strong quadratic optomechanical coupling. This adiabatic limit, however, breaks down as optical Rabi oscillations occur at scales that compare with mechanical oscillations, i.e. where the supermode splitting is resonant with the mechanical mode (2J ≈ Ω_m). In this limit, optical and mechanical degrees of freedom need to be treated on the same footing, via numerical methods or effective Hamiltonians that are perturbative in g₀/κ [43, 44]. Moreover, as described in the introduction, it was quickly recognised that the effective Hamiltonian in equation (12) does not fully describe the system, because the linear cross-mode coupling is no longer included [32, 45].

Our goal now is to provide a description of nonlinear dynamics in the MIM system that does not have the shortcomings of the adiabatic approximation. To do so, we apply the same perturbative approach as with the single cavity to the full model in equation (9). Our mean-field equations of motion are:

$$\begin{equation}\ddot {x}=-{{\Omega}}_{m}^{2}x-{{\Gamma}}_{m}\dot {x}+{{\Omega}}_{m}\left({g}_{0,1}\vert {a}_{1}{\vert }^{2}-{g}_{0,2}\vert {a}_{2}{\vert }^{2}\right)+\frac{{F}_{\text{in}}}{m{x}_{\text{xpf}}},\end{equation} \tag{ 13a }$$

$$\begin{equation}{\dot {a}}_{1}=i\left({\tilde {{\Delta}}}_{1}+{g}_{0,1}x\right){a}_{1}+iJ{a}_{2}+\sqrt{{\kappa }_{\text{ex,}\;1}}{a}_{\text{in},1},\end{equation} \tag{ 13b }$$

$$\begin{equation}{\dot {a}}_{2}=i\left({\tilde {{\Delta}}}_{2}-{g}_{0,2}x\right){a}_{2}+iJ{a}_{1}+\sqrt{{\kappa }_{\text{ex,}\;2}}{a}_{\text{in},2}.\end{equation} \tag{ 13c }$$

Here the optical decay rates ${\kappa }_{i}=2\enspace \mathrm{I}\mathrm{m}\enspace {\tilde {{\Delta}}}_{i}$ are included in the complex detunings ${\tilde {{\Delta}}}_{i}$. We have added the term $\propto {F}_{\text{in}}=\left\langle {\hat{F}}_{\text{in}}\right\rangle$ to represent external classical forces acting on the resonator, which will be of use later on. We first find steady state values for a₁, a₂ and x:

$$\begin{equation}{\bar{a}}_{1,2}=i\frac{\left({\bar{{\Delta}}}_{2,1}{\xi }_{1,2}-J{\xi }_{2,1}\right)}{{\bar{{\Delta}}}_{1}{\bar{{\Delta}}}_{2}-{J}^{2}},\end{equation} \tag{ 14a }$$

$$\begin{equation}\bar{x}=\frac{{g}_{0,1}\vert {a}_{1}{\vert }^{2}-{g}_{0,2}\vert {a}_{2}{\vert }^{2}}{{{\Omega}}_{m}}.\end{equation} \tag{ 14b }$$

Where ${\bar{{\Delta}}}_{i}={\tilde {{\Delta}}}_{i}{\pm}{g}_{0,i}\bar{x}$ is the detuning to the cavity resonance that has been displaced by mean mechanical position $\bar{x}$ and incoming photon population ${\xi }_{i}=\sqrt{{\kappa }_{\text{ex,}\;\text{i}}}{a}_{\text{in},i}$. Similarly to the discussion of the single-cavity intrinsic nonlinearity, we propose an ansatz

$$\begin{equation}{a}_{i}={\bar{a}}_{i}+\sum _{\zeta ={\pm}}{A}_{i,\zeta }^{\left(1\right)}\enspace {\text{e}}^{\text{i}\zeta {{\Omega}}_{m}t}+{A}_{i,\zeta }^{\left(2\right)}\enspace {\text{e}}^{\text{i}\zeta 2{{\Omega}}_{m}t}.\end{equation} \tag{ 15 }$$

We then derive explicit expressions for the first-order coefficients,

$$\begin{equation}{A}_{1,{\pm}}^{\left(1\right)}=\frac{\left({\pm}{{\Omega}}_{m}-{\bar{{\Delta}}}_{2}\right){g}_{0,1}{\bar{a}}_{1}-J{g}_{0,2}{\bar{a}}_{2}}{\left({\pm}{{\Omega}}_{m}-{\bar{{\Delta}}}_{1}\right)\left({\pm}{{\Omega}}_{m}-{\bar{{\Delta}}}_{2}\right)-{J}^{2}}\frac{{X}_{0}}{2},\end{equation} \tag{ 16a }$$

$$\begin{equation}{A}_{2,{\pm}}^{\left(1\right)}=-\frac{\left({\pm}{{\Omega}}_{m}-{\bar{{\Delta}}}_{1}\right){g}_{0,2}{\bar{a}}_{2}-J{g}_{0,1}{\bar{a}}_{1}}{\left({\pm}{{\Omega}}_{m}-{\bar{{\Delta}}}_{1}\right)\left({\pm}{{\Omega}}_{m}-{\bar{{\Delta}}}_{2}\right)-{J}^{2}}\frac{{X}_{0}}{2},\end{equation} \tag{ 16b }$$

as well as the second order coefficients

$$\begin{equation}{A}_{1,+}^{\left(2\right)}=\frac{\left(2{{\Omega}}_{m}-{\bar{{\Delta}}}_{2}\right){g}_{0,1}{A}_{1,+}^{\left(1\right)}-J{g}_{0,2}{A}_{2,+}^{\left(1\right)}}{\left(2{{\Omega}}_{m}-{\bar{{\Delta}}}_{1}\right)\left(2{{\Omega}}_{m}-{\bar{{\Delta}}}_{2}\right)-{J}^{2}}\frac{{X}_{0}}{2},\end{equation} \tag{ 17a }$$

$$\begin{equation}{A}_{1,-}^{\left(2\right)}=\frac{\left(-2{{\Omega}}_{m}-{\bar{{\Delta}}}_{2}\right){g}_{0,1}{A}_{1,+}^{\left(1\right)}-J{g}_{0,2}{A}_{2,+}^{\left(1\right)}}{\left(-2{{\Omega}}_{m}-{\bar{{\Delta}}}_{1}\right)\left(-2{{\Omega}}_{m}-{\bar{{\Delta}}}_{2}\right)-{J}^{2}}\frac{{X}_{0}}{2},\end{equation} \tag{ 17b }$$

$$\begin{equation}{A}_{2,+}^{\left(2\right)}=-\frac{\left(2{{\Omega}}_{m}-{\bar{{\Delta}}}_{1}\right){g}_{0,2}{A}_{2,+}^{\left(1\right)}-J{g}_{0,1}{A}_{1,+}^{\left(1\right)}}{\left(2{{\Omega}}_{m}-{\bar{{\Delta}}}_{1}\right)\left(2{{\Omega}}_{m}-{\bar{{\Delta}}}_{2}\right)-{J}^{2}}\frac{{X}_{0}}{2},\end{equation} \tag{ 17c }$$

$$\begin{equation}{A}_{2,-}^{\left(2\right)}=-\frac{\left(-2{{\Omega}}_{m}-{\bar{{\Delta}}}_{1}\right){g}_{0,2}{A}_{2,+}^{\left(1\right)}-J{g}_{0,1}{A}_{1,+}^{\left(1\right)}}{\left(-2{{\Omega}}_{m}-{\bar{{\Delta}}}_{1}\right)\left(-2{{\Omega}}_{m}-{\bar{{\Delta}}}_{2}\right)-{J}^{2}}\frac{{X}_{0}}{2}.\end{equation} \tag{ 17d }$$

Here, we impose the quasi-static limit (2J ≫ Ω_m) in the general solutions above and assume optical mode splitting to be resolved in frequency (2J ≫ κ_i). Without loss of generality, we drive the input of cavity 1 only, close to the even optical supermode, resulting in ${\bar{a}}_{1}\approx {\bar{a}}_{2}=\bar{a}$ according to equation (14), but such that the 2Ω_m sideband is on resonance, namely $\mathrm{R}\mathrm{e}\left(\bar{{\Delta}}\right)=2{{\Omega}}_{m}-J$ (see equation (17)). We will assume a sideband resolved system with Ω_m > κ, which is the more interesting regime for the MIM system, as we will discuss later in subsection 3.3.

The quasi-static diagonalization shows that photonic eigenmodes acquire a dependence on x. For κ₁ ≠ κ₂, this in addition yields an effective x-dependent supermode decay rate (also known as dissipative coupling [45, 46]), leading to information about $\hat{x}$ leaking from the cavity. In a similar but distinct effect, the two optical supermodes also become coupled through their dissipation into the same optical channel for κ₁ ≠ κ₂ [38, 45]. However, for clarity of our discussion, we will neglect both of these effects by assuming identical optical cavities (g_0,1 = g_0,2 ≡ g₀, Δ₁ = Δ₂ ≡ Δ, and κ₁ = κ₂ ≡ κ).

Under the conditions above, the relevant first-order sideband amplitudes reduce to

$$\begin{equation}{A}_{1,+}^{\left(1\right)}=\frac{{g}_{0}}{{{\Omega}}_{m}+J-\bar{{\Delta}}}\frac{\bar{a}{X}_{0}}{2}=-{A}_{2,+}^{\left(1\right)}.\end{equation} \tag{ 18 }$$

Here we see that this first sideband amplitude has a resonance only at the odd optical mode, or for $\mathrm{R}\mathrm{e}\left(\bar{{\Delta}}\right)={{\Omega}}_{m}+J$. Because this resonance frequency is far from the (even mode) input frequency (see figure 2(a)), first sideband generation is suppressed. This is a signature of the inter-mode optomechanical coupling between supermodes in equation (11): if the even mode is populated, the mechanical mode scatters light from the carrier into the odd mode. In figure 2(a), we illustrate this situation. In our perturbative picture, the second sidebands at ±2Ω_m are seen as being scattered from the first sidebands by the mechanical mode. Because of the cross-mode coupling the second sidebands are again in the even mode. For our choice of detuning, this means the positive frequency second sideband is on resonance with the even mode and has amplitude

$$\begin{equation}{A}_{1,+}^{\left(2\right)}={A}_{2,+}^{\left(2\right)}=\frac{{g}_{0}{A}_{1,+}^{\left(1\right)}}{2{{\Omega}}_{m}-J-\bar{{\Delta}}}\frac{{X}_{0}}{2}\approx \frac{{g}_{0}^{2}}{2J}\frac{a}{-i\kappa /2}{\left(\frac{{X}_{0}}{2}\right)}^{2},\end{equation} \tag{ 19 }$$

which is depicted in figure 2(a). Note that a quadratic optomechanical interaction, which in practice involves the adiabatic elimination of the supermode off-resonant with the input field (${\hat{a}}_{o}$ in this case), yields the same result for the effective quadratic coupling as in the adiabatic diagonalisation (see equation (12)), namely ${g}_{0}^{\left(2\right)}={g}_{0}^{2}/2J$. We conclude that our approach gives the correct quadratic coupling found in the quasi-static approach, but now as a manifestation of the intrinsic optomechanical nonlinearities of cavities 1 and 2, as recognised by [11].

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We now use our model to describe general transduction in the MIM system. In particular, we show how transduction of motion to Ω_m and 2Ω_m optical sidebands changes with tunnelling rate J and input laser detuning Δ. In doing so, we will first assume only one optical supermode is excited by the input field, even when this field is not on resonance with that mode. This assumption makes the following discussion more clear and in fact can also be achieved in experiment by exciting the MIM system through both input ports with a particular relative phase. For example, using a_in,1 = a_in,2 allows excitation of only the even optical mode, regardless of optical detuning.

When discussing the dynamics of the MIM system, two distinct situations can be distinguished, namely, (i) a constant input power (P_in ≡ ∑_iω_c,i|a_in,i|²) or (ii) a constant cavity photon number (${\bar{n}}_{\mathrm{c}}\equiv {\sum }_{i}\vert {\bar{a}}_{i}{\vert }^{2}$). The latter scenario allows isolating optomechanical effects, including the strength of nonlinear transduction, from purely optical cavity input effects, i.e. the enhancement of cavity occupation for a resonant input field. Moreover, cavity occupation is often the limiting factor in the experiment, due to nonlinear effects and heating [47]. However, it could also be advantageous to minimise the input power that is required to achieve a certain cavity photon number in certain scenarios. Thus, we will discuss both situations in the following.

The amplitude of the −Ω_m and +2Ω_m sidebands of the supermodes, ${A}_{o,-}^{\left(1\right)}$ and ${A}_{e,+}^{\left(2\right)}$, for even input (a_in,1 = a_in,2) are shown in figure 3 for constant P_in (panels figures 3(a) and (b)) and constant ${\bar{n}}_{\mathrm{c}}$ (panels figures 3(c) and (d)). These amplitudes are defined as ${A}_{o,-}^{\left(1\right)}=\frac{1}{\sqrt{2}}\left({A}_{1,-}^{\left(1\right)}-{A}_{2,-}^{\left(1\right)}\right)$ and ${A}_{e,+}^{\left(2\right)}=\frac{1}{\sqrt{2}}\left({A}_{1,+}^{\left(2\right)}+{A}_{2,+}^{\left(2\right)}\right)$. The amplitudes are normalised to the optimum first sideband or second sideband amplitude that would be obtained in a single cavity for the same P_in or ${\bar{n}}_{\mathrm{c}}$, which occur at $\mathrm{R}\mathrm{e}\enspace \bar{{\Delta}}={\pm}{{\Omega}}_{m}$. From equations (6) and (7), these read

$$\begin{equation}{A}_{+}^{\left(1\right)}\left({\Delta}={{\Omega}}_{m}\right)\equiv {A}_{\text{ref}}=i\frac{{g}_{0}\bar{a}}{\kappa }{X}_{0},\end{equation} \tag{ 20a }$$

$$\begin{equation}{A}_{+}^{\left(2\right)}\left({\Delta}={{\Omega}}_{m}\right)\equiv {A}_{\text{ref}}^{\left(2\right)}=\frac{i}{\kappa }\frac{{g}_{0}^{2}\bar{a}}{{{\Omega}}_{m}-i\kappa /2}\frac{{X}_{0}^{2}}{2},\end{equation} \tag{ 20b }$$

with $\bar{a}=\sqrt{{\bar{n}}_{\mathrm{c}}}$ or $\bar{a}=\sqrt{{\kappa }_{\text{ex}}}{a}_{\text{in}}/\left(\kappa /2-i{{\Omega}}_{m}\right)$ for constant ${\bar{n}}_{\mathrm{c}}$ or P_in, respectively. We choose to display the −Ω_m first order and +2Ω_m second order sidebands, because these show special double resonance conditions for the even mode illumination condition, as discussed below. Note that, because an annihilation operator is counterrotating in time, the +Ω_m sidebands ${A}_{i,+}^{\left(1\right)}$ are actually rotating at frequency ω_L − Ω_m in the lab frame.

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From figure 3(a), we observe strong first-order sideband generation in the odd mode either when the carrier is on resonance with the even mode ($\mathrm{R}\mathrm{e}\left(\bar{{\Delta}}\right)=-J$), or when the first sideband is on resonance with the odd mode ($\mathrm{R}\mathrm{e}\left(\bar{{\Delta}}\right)=J-{{\Omega}}_{m}$). Where these two resonant conditions are simultaneously met, we see a resulting enhancement of first sideband generation [38] and the sideband amplitude exceeds ${A}_{\text{ref}}^{\left(1\right)}$, the largest amplitude possible in a single cavity. Moving to figure 3(c), we now keep the cavity photon number ${\bar{n}}_{\mathrm{c}}$ constant, instead of the input power. In this case, we see that the resonance of the carrier no longer results in large sideband amplitude. The sideband amplitude no longer exceeds ${A}_{\text{ref}}^{\left(1\right)}$, that of a single cavity, anywhere and we can not recognise an enhancement anymore. We conclude that the enhancement observed in figure 3(a) does not result from enhanced couplings inside the cavity, but from a higher cavity acceptance of input light.

Moving to the second-order sideband amplitude in figure 3(b), we see resonance lines that correspond to either carrier resonance or +Ω_m sideband resonance. Wherever the first positive sideband amplitude is large (not shown), the second sideband amplitude rises accordingly. However, an additional resonance is observed for the second-order sideband in figure 3(b), where the second sideband is on resonance with the even mode ($\mathrm{R}\mathrm{e}\left(\bar{{\Delta}}\right)=-J+2{{\Omega}}_{m}$). Figures 3(b) and (d) show identical dependencies, except for the line followed by the carrier resonance ($\mathrm{R}\mathrm{e}\left(\bar{{\Delta}}\right)=-J$), which is not observed for constant ${\bar{n}}_{\mathrm{c}}$. Of special interest is the crossing of two resonance lines in the plots for quadratic transduction in figures 3(b) and (d), corresponding to the doubly resonant condition $\mathrm{R}\mathrm{e}\left(\bar{{\Delta}}\right)=3{{\Omega}}_{m}/2$ and 2J = Ω_m. For these conditions, both the first and the second sidebands are on resonance with their respective optical mode, as we have sketched in figure 2(b). At these points we find the strongest generation of second-order (nonlinear) sidebands, the maxima for ${A}_{e,+}^{\left(2\right)}$, which are larger than possible in a single cavity, (denoted ${A}_{\text{ref}}^{\left(2\right)}$). Unlike with the enhanced first sideband, this effect does not disappear when considering a fixed ${\bar{n}}_{\mathrm{c}}$.

This resonance effect has been described before by Ludwig et al [1] through a perturbative expansion of the threefold interaction between ${\hat{a}}_{o},{\hat{a}}_{o}$ and $\hat{x}$ in equation (11). This leads to an effective nonlinear interaction Hamiltonian that is enhanced for 2J − Ω_m ≪ κ, namely ${\hat{H}}_{\text{OM}}^{\text{eff}}\sim {g}_{0}^{2}\left(1/\left(2J-{{\Omega}}_{m}\right)+1/\left(2J+{{\Omega}}_{m}\right)\right)\left({\hat{a}}_{e}^{{\dagger}}{\hat{a}}_{e}-{\hat{a}}_{o}^{{\dagger}}{\hat{a}}_{o}\right){\hat{x}}^{2}$. However, the magnitude of this interaction and its dependency on parameters such as κ was not discussed. This and related works [35, 48] have investigated the implications of this enhancement for specific quantum applications at the strong single-photon optomechanical coupling level (g₀ > κ) and weak driving/low cavity occupation regime. In these works, it was demonstrated that the coupled cavity system had a significant advantage over a single cavity system [11], but single-photon strong coupling was still needed to produce the sought-after nonclassical effects.

If we were to excite using odd input light conditions (a_in,1 = −a_in,2), the roles of odd and even modes would be interchanged (not shown) and the same resonance conditions found on the +Ω_m and −2Ω_m sidebands. From a more practical perspective, using only single-port excitation of our MIM system would result in a $\bar{{\Delta}}$-dependence convoluted with the detuning-dependent excitation of the supermodes.

To understand exactly what the MIM system offers over a single cavity system in terms of optomechanical nonlinearity, it is important to calculate how large the enhanced second sideband amplitude is and how this depends on system parameters. To do so, we compare optimum second sideband amplitude from equation (17), i.e. at the double resonance condition described above, to optimum second sideband from a single cavity, as described in equation (20). For this, we introduce a metric that combines both sidebands of the same order, namely

$$\begin{equation}{\mathcal{A}}_{s}^{\left(1\right)}=\vert {A}_{s,-}^{\left(1\right)}\vert +\vert {A}_{s,+}^{\left(1\right)}\vert ,\end{equation} \tag{ 21a }$$

$$\begin{equation}{\mathcal{A}}_{s}^{\left(2\right)}=\vert {A}_{s,-}^{\left(2\right)}\vert +\vert {A}_{s,+}^{\left(2\right)}\vert ,\end{equation} \tag{ 21b }$$

where s = {e, o}. As shown in appendix A, this metric is proportional to the homodyne signal amplitude at Ω_m or 2Ω_m in the optimum optical quadrature. This metric can also be applied to the single-cavity case using equations (6) and (7), to obtain the reference values ${\mathcal{A}}_{\text{ref}}^{\left(1,2\right)}$.

In figure 4, we highlight the differences between mechanical transduction in a single cavity and in an MIM system. In figure 4(a), we see the characteristic MIM supermode frequency dependence on the static mechanical displacement $\bar{x}$. When static displacement is large (figure 4(b)), the two cavities have frequencies that differ by more than 2J and we effectively recover the limit of two uncoupled cavities, whereas a zero static displacement gives the coupled cavity MIM system (figure 4(c)). In figures 4(b) and (c), we look at sidebands generated in a sideband-resolved MIM system corresponding to these crosscuts. For this, we assume a drive of the even mode and plot quantities ${\mathcal{A}}_{o}^{\left(1\right)}$ and ${\mathcal{A}}_{e}^{\left(2\right)}$. The horizontal dashed lines are the single cavity limits ${\mathcal{A}}_{\text{ref}}^{\left(1\right)}\approx {A}_{\text{ref}}^{\left(1\right)}$ and ${\mathcal{A}}_{\text{ref}}^{\left(2\right)}\approx {A}_{\text{ref}}^{\left(2\right)}$ for first and second sidebands as calculated previously. All plotted values are now normalised by ${A}_{\text{ref}}^{\left(1\right)}$, which is done to give an idea of the relative size of first and second sidebands for currently available system parameters. In figure 4(b), we reveal that transduction for the uncoupled cavities adheres to the single cavity limits, as expected. Moving to the coupled cavity system in figure 4(c), we see that the second sideband amplitude now surpasses the single cavity limit.

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In figure 4(d), we plot the ratio of ${\mathrm{max}}_{{\Delta}}\left({\mathcal{A}}_{e}^{\left(2\right)}\left({\Delta}\right)\right)$ and ${\mathrm{max}}_{{\Delta}}\left({\mathcal{A}}_{\text{ref}}^{\left(2\right)}\left({\Delta}\right)\right)$ as a measure of the enhancement of nonlinearity for different values of sideband resolution Ω_m/κ. For sideband-unresolved systems (Ω_m < κ), we see the nonlinearity is, at most, equally strong in the MIM and the single-cavity system. This is in line with previous works that have compared nonlinear measurement of mechanical motion in the two systems in the sideband-unresolved regime and found no enhancement of the MIM system over a linearly coupled, single cavity [12, 49]. However, for sideband-resolved systems (Ω_m > κ), the enhancement increases with sideband resolution. Figure 4(d) demonstrates that the MIM system can only feature larger quadratic transduction than in a single cavity when it is sideband-resolved. The absence of enhancement for a sideband-unresolved system (Ω_m ≪ κ) can be attributed to the fact that, in a single cavity, carrier, first and second sidebands are already resonantly enhanced due to the large spectral overlap. We derive an expression for the enhancement factor in the case of a sideband resolved system. We look at the case of constant ${\bar{n}}_{\mathrm{c}}$, driving of the even mode and finally large sideband resolution Ω_m ≫ κ to simplify the expression. We find

$$\begin{equation}\left\vert \frac{{\mathrm{max}}_{{\Delta}}\left({\mathcal{A}}_{o}^{\left(2\right)}\left({\Delta}\right)\right)}{{\mathrm{max}}_{{\Delta}}\left({\mathcal{A}}_{\text{ref}}^{\left(2\right)}\left({\Delta}\right)\right)}\right\vert \approx 2\frac{{{\Omega}}_{m}}{\kappa }.\end{equation} \tag{ 22 }$$

Equation (22) demonstrates that, for a sideband-resolved system, the MIM system enhancement of nonlinearity is given by the degree of sideband resolution. The result from equation (22) plotted as a red dashed line in figure 4(d). Next, in figure 4(e), we display the enhancement of the MIM system over a single cavity for second sideband amplitude, namely ${\mathcal{A}}^{\left(2\right)}/{A}_{\text{ref}}^{\left(2\right)}$. We see that, at the double resonance condition, the enhancement peaks to the value of 2Ω_m/κ.

We predict that enhanced nonlinear transduction could be experimentally observed in the currently available MIM systems. As we have seen, this requires a sideband-resolved system Ω_m/κ > 1 where the double resonance condition can be fulfiled (2J = Ω_m). In membrane systems, sideband resolution reaches Ω_m/κ ≈ 10 [7, 18, 50]. For these systems, the requirement 2J = Ω_m can be fulfiled by a reduction (increase) of J (Ω_m) of less than a factor 2, both J and Ω_m being in the range of hundreds of kHz. Related coupled microtoroid resonators platforms [42] feature tunable inter-cavity coupling 2J ⩾ Ω_m/2 and Ω_m/κ ≈ 10, satisfying all conditions to observe enhancement of optomechanical nonlinearity. An additional implementation of a coupled-cavity system was proposed for 2D optomechanical crystals [41], of which it was recently shown that individual cavities could reach Ω_m/κ ≈ 28 [47].

For experiments in which readout of the mechanical energy $\sim {\hat{x}}^{2}$ is desired, maximising the ratio of first to second-order sideband amplitude is crucial. This is because first sidebands carry information about $\hat{x}$ and their creation is thus inevitably associated with a linear quantum backaction that changes the mechanical state of the system [32, 45].

As a figure of merit, we calculate the optimal ratio of the different sidebands, $\iota =\vert {\mathcal{A}}_{e}^{\left(2\right)}\vert /\vert {\mathcal{A}}_{o}^{\left(1\right)}\vert$. From the equations (equation (17)), it can be derived this value is highest at the double resonance condition, which we find to be

$$\begin{equation}\iota {\leqslant}\frac{{g}_{0}{X}_{0}}{\kappa }.\end{equation} \tag{ 23 }$$

Using equation (7), we easily see that this is the same limit as can be found in a single cavity. In other words, equation (23) indicates that the MIM system does not allow for more selective generation of the second over first sideband as compared to a single cavity. In figure 4(e), we have plotted this sideband ratio as a function of Δ for 2J = Ω_m and ground-state-level coherent amplitude X₀ = 1. We see it also peaks at the double resonance condition, where it is limited by g₀/κ. Although our calculations are classical, this limit reminds of the results by Miao et al on QND measurements of mechanical energy. As we will briefly discuss at the end of subsection 4.1, this is no coincidence, as the calculation underlying equation (23) is indeed closely related to an analysis of quantum measurement noise limits.

Finally, we want to highlight another feature of the MIM system. Next to the (limited) enhancement of optomechanical nonlinearity, the MIM systems offer a simple method for separation of different sidebands, as they occur in orthogonal modes. Separation can be attained by a beam splitter (cf figure 1(a)), even if the different sidebands are too close in frequency for the use of other filtering techniques. The degree of filtering this offers, though, is reduced when the cavity is not perfectly balanced, e.g., g_0,1 ≠ g_0,2 or κ₁ ≠ κ₂, because the different sidebands are no longer output into orthogonal modes.

Our approach starts again from the semiclassical equations of motion equation (13a) and is similar to that of Jayich et al [17]. A related method is used to determine DBA effects in single cavities [16]. The aim is to find the susceptibility χ(ω) of the mechanical resonator to an external force, given by the real amplitude F_in(t) = F₀  cos(ωt). We solve for a mechanical motion that is strictly real, but can have an arbitrary phase that we account for by letting ${X}_{0}\in \mathbb{C}$, i.e. $x\left(t\right)=\left({X}_{0}\enspace {\text{e}}^{\text{i}\omega t}+{X}_{0}^{{\ast}}\enspace {\text{e}}^{-\text{i}\omega t}\right)/2$. Note that this means information about both mechanical quadratures is now caught in the complex nature of X₀. We thus want to rewrite the mechanical EOM in the form X₀(ω) = χ(ω)F₀.

For mechanical coherent motion given by X₀, we can write down the generated first sidebands using our previous equation (16) and thus expand |a_i|² in equation (13a) in terms of X₀. In the present case we observe that the sidebands ${A}_{i,-}^{\left(1\right)}$ at −Ω_m actually depend on ${X}_{0}^{{\ast}}$ instead of X₀. By collecting all terms with the same time dependence, we can derive:

$$\begin{equation}\chi {\left(\omega \right)}^{-1}={x}_{\text{zpf}}m\left[-{\omega }^{2}+{{\Omega}}_{m}^{2}+i{{\Gamma}}_{m}\omega +{{\Omega}}_{m}\left({g}_{0,1}{\beta }_{1,+},-{g}_{0,2}{\beta }_{2,+}\right)\right],\end{equation} \tag{ 24a }$$

$$\begin{equation}{\beta }_{i,+}={\bar{a}}_{i}{\tilde {A}}_{i,-}^{{\ast}}+{\bar{a}}_{i}^{{\ast}}{\tilde {A}}_{i,+},\end{equation} \tag{ 24b }$$

and where ${\tilde {A}}_{i,-}=2{A}_{i,-}^{\left(1\right)}/{X}_{0}^{{\ast}}$ and ${\tilde {A}}_{i,+}=2{A}_{i,+}^{\left(1\right)}/{X}_{0}$.

One of the striking features of a Hamiltonian with quadratic optomechanical coupling, as in equation (12), is that the optical cavity occupation ${\bar{n}}_{\mathrm{c}}=\left\langle {\hat{a}}_{\mathrm{c}}^{{\dagger}}{\hat{a}}_{\mathrm{c}}\right\rangle$ directly changes the mechanical frequency by acting as an additional potential well for the resonator [20, 37]. This can be seen from the Hamiltonian equation (12) to be:

$$\begin{equation}{{\Omega}}_{\text{eff}}={{\Omega}}_{m}+2{g}_{0}^{\left(2\right)}{\bar{n}}_{\mathrm{c}}.\end{equation} \tag{ 25 }$$

We shall refer to this effect as the static optical spring effect. Here, we show this effect can be described as a consequence of DBA, in which form it is much easier to include other DBA effects that can not be recovered from the quadratic coupling Hamiltonian, but are present in the MIM system.

By inserting Ω_eff = Ω_m + δΩ and Γ_eff = Γ_m + δΓ into the susceptibility for ${\bar{n}}_{\mathrm{c}}=0$, and comparing to equation (24a), we can find expressions for these shifts to be

$$\begin{equation}\delta {\Omega}=\frac{1}{2}\enspace \mathrm{R}\mathrm{e}\left({g}_{0,1}{\beta }_{1,+}-{g}_{0,2}{\beta }_{2,+}\right),\end{equation} \tag{ 26a }$$

$$\begin{equation}\delta {{\Gamma}}_{m}=\mathrm{I}\mathrm{m}\left({g}_{0,1}{\beta }_{1,+}-{g}_{0,2}{\beta }_{2,+}\right).\end{equation} \tag{ 26b }$$

Now, we assume that the drive is close to resonance of the even supermode and, as before, that the two cavities are identical. In the adiabatic limit 2J ≫ Ω_m, ${\tilde {A}}_{i,{\pm}}\approx {g}_{0}{\bar{a}}_{i}/\left(2J\right)$ and β_1,+ ≈ −β_2,+. Combining these findings, we recover the quadratic coupling approximation: $\delta {\Omega}={g}_{0}^{2}{\bar{n}}_{\mathrm{c}}/J$ by identifying ${g}_{0}^{\left(2\right)}={g}_{0}^{2}/2J$.

We see that the static optical spring effect can be regarded as a consequence of DBA, which considers only first sidebands, and thus is not a consequence of nonlinear optomechanical coupling. To be precise, for J ≫ Ω_m, the static optical spring effect is equal in magnitude to the optical spring effect in a single cavity with a laser detuned from optical resonance by J. The only difference is that, due to the multimode nature of MIM system, the carrier can be on resonance with one of the supermodes while the sidebands are far from resonance (i.e. to the other supermode), allowing for an optical spring effect with lower input power, an idea related to that presented by Grudinin et al [42]. The reduction in input power is given by Δ₀/κ, where Δ₀ is the desired detuning from resonance for the particular application. To suppress unwanted DBA heating or cooling, it is generally taken larger than Ω_m [51]. This is one of the applications in which the MIM could outperform a single cavity: optical tuning the mechanical resonance through the optical spring effect using a detuned laser to suppress DBA heating or cooling in a sideband-resolved system.

In order to further discuss optically-induced mechanical frequency and linewidth in the MIM system, we depict the relative modifications δΩ_m/Ω_m and δΓ_m/Γ_m as a function of J and $\bar{{\Delta}}$ for constant input power in figure 5. In figure 5(a) we can see that, in the adiabatic regime 2J > Ω_m, we find the static optical spring effect peaking at each of the supermode resonances in a way that closely resembles results from Lee et al [37]. Approaching the regime where 2J ≈ Ω_m, the size of the optical spring increases since both sideband and carrier can be on resonance with one of the supermodes. A strong transition is found for 2J = Ω_m, where one of the sidebands crosses the resonance and flips the sign of the spring effect. In figure 5(b), we can see the optically-induced change in linewidth. The effect is again most substantial when both the carrier and one of the first sidebands are on resonance. Comparing to the standard optical spring effect, the linewidth change falls of more quickly when sidebands are not on resonance, which is already well known for single cavities [16].

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In previous work [17], Jayich et al extensively studied dynamical backaction as a function of inter-cavity detuning, here given by δ = ω_c,1 − ω_c,2. They noted a lack of backaction for δ = 0 in the adiabatic regime. It is argued that backaction vanishes completely because of the fact that the first derivative of supermode frequency vanishes at δ = 0, suppressing linear coupling (see figure 1(b)). Here, however, we have revealed that the DBA does not vanish completely. For δ = 0 and in the adiabatic regime, the first sideband amplitude is suppressed, as discussed in section 3.1, but is not identically zero. This fact is important because we have shown that second sideband amplitude (and thus nonlinear transduction) is suppressed when the first sideband amplitude is suppressed. Conversely, the generation of nonlinear transduction is associated with the presence of DBA.

This last statement can be interpreted as a classical analogue of previous results concerning the QND measurement of mechanical Fock states using quadratic optomechanical coupling [32, 45]. These authors showed that, as a result of the linear cross-mode coupling of the MIM system, the light field's vacuum fluctuations would destroy a mechanical Fock state before it could be measured through the effective ${\hat{x}}^{2}$-coupling, unless the SPSC condition was fulfiled. An expression for quantum backaction was found by calculating the susceptibility of the optical modes to the input quantum fluctuations, leading to a result similar to ${A}_{i,{\pm}}^{\left(1\right)}$ in equation (16), from which we can extract the susceptibility of the optical modes to mechanically-induced fluctuations. It is therefore also not surprizing that we recover that the ratio of second to the first sideband is limited by the same SPSC condition g₀/κ > 1. Indeed, the ratio of second to first sideband amplitude is closely related to the ratio between the amount of information on ${\hat{x}}^{2}$ leaving the cavity and the quantum backaction, as the latter is directly related to the amount of information on $\hat{x}$ (i.e. the linear transduction) that leaves the cavity [40].

In parametric squeezing, the spring constant of a resonator is modulated at twice the mechanical frequency, which results in a quadrature-dependent amplification or damping of the resonator [52]. Such a scheme has previously been used in electromechanical [53, 54] and linearly coupled optomechanical [51, 55] systems. In a quadratically coupled optomechanical system, it is possible to directly alter the mechanical spring constant using the optical field, which can be exploited to implement this scheme [8]. In fact, we find that the parametric squeezing effect lies at the heart of the two-phonon OMIT-like effect reported for the MIM system [13]. This can be seen from the fact that this OMIT effect works by amplifying thermal fluctuations in only one particular mechanical quadrature, attenuating motion in the opposite quadrature. We now set out to compare the parametric driving effect in the MIM system to a single cavity system.

To include cavity modulation, we start from an even intracavity field given by:

$$\begin{equation}{\bar{a}}_{i}=a\left(1+{\epsilon}\enspace {\mathrm{e}}^{\mathrm{i}2{{\Omega}}_{m}t}\right),\end{equation} \tag{ 27 }$$

where the constant ${\epsilon}\in \mathbb{C}$, assumed to be || ≪ 1, controls the modulation phase and amplitude. Our approach shares ingredients in common with that by Rugar and Grütter [52]. We assume a force with fixed phase F_in(t) = F₀  cos(Ω_mt) and allow $x\left(t\right)=\mathrm{R}\mathrm{e}\left\{{X}_{0}\enspace {\text{e}}^{\text{i}{{\Omega}}_{m}t}\right\}$ as previously. The modulation sideband controlled by gives an additional component to |a_i|²(±Ω_m), that shows up in equation (13a). In figure 6(a), we sketch the sidebands that are created and the associated contribution to the radiation pressure force. After making the dependence on X₀ explicit, the EOM from equation (13a) implies:

$$\begin{equation}{{\Omega}}_{m}\left[i{{\Gamma}}_{m}-\left({g}_{0,1}{\beta }_{1,+}-{g}_{0,2}{\beta }_{2,+}\right)\right]\frac{{X}_{0}}{2}=\left[{{\Omega}}_{m}\left({g}_{0,1}{\beta }_{1,-}-{g}_{0,2}{\beta }_{2,-}\right)\right]\frac{{X}_{0}^{{\ast}}}{2}+\frac{{F}_{0}}{2{x}_{\text{zpf}}m},\end{equation} \tag{ 28 }$$

where now ${\beta }_{i,-}={\bar{a}}_{i}{\epsilon}{E}_{i}^{{\ast}}+{\bar{a}}_{i}^{{\ast}}{\tilde {A}}_{i,-}$, and the amplitudes for the sidebands generated from the modulation tone with amplitude a by mechanical motion read

$$\begin{equation}{E}_{1}=-{\epsilon}\frac{J{g}_{0,2}{\bar{a}}_{2}+\left({\bar{{\Delta}}}_{2}+{{\Omega}}_{m}\right){g}_{0,1}{\bar{a}}_{1}}{\left({\bar{{\Delta}}}_{1}+{{\Omega}}_{m}\right)\left({\bar{{\Delta}}}_{2}+{{\Omega}}_{m}\right)-{J}^{2}},\end{equation} \tag{ 29a }$$

$$\begin{equation}{E}_{2}={\epsilon}\frac{-J{g}_{0,1}{\bar{a}}_{1}+\left({\bar{{\Delta}}}_{1}+{{\Omega}}_{m}\right){g}_{0,2}{\bar{a}}_{2}}{\left({\bar{{\Delta}}}_{1}+{{\Omega}}_{m}\right)\left({\bar{{\Delta}}}_{2}+{{\Omega}}_{m}\right)-{J}^{2}}.\end{equation} \tag{ 29b }$$

Since the modulation tone is displaced by 2Ω_m from the carrier, its sidebands have a different dependence on $\bar{{\Delta}}$ than the amplitudes ${\tilde {A}}_{i,{\pm}}$. X₀ can be retrieved by combining equation (28) with its complex conjugate, to give

$$\begin{equation}{X}_{0}=\frac{{c}^{{\ast}}+d}{\vert c{\vert }^{2}-\vert d{\vert }^{2}}\frac{{F}_{0}}{{x}_{\text{zpf}}m}\end{equation} \tag{ 30 }$$

with

$$\begin{equation}\begin{aligned}\hfill c& =i{{\Gamma}}_{m}{{\Omega}}_{m}-{{\Omega}}_{m}\left({g}_{0,1}{\beta }_{1,+}-{g}_{0,2}{\beta }_{2,+}\right),\hfill \\ \hfill d& ={{\Omega}}_{m}\left({g}_{0,1}{\beta }_{1,-}-{g}_{0,2}{\beta }_{2,-}\right).\hfill \end{aligned}\end{equation} \tag{ 31 }$$

In figure 6(b), we display an example of the mechanical response changing with the phase of . When changing the phase of , d changes with similar phase, altering |X₀|. This is quadrature-dependent amplification of motion: depending on the relative phase of the modulation tone and the force F_in, the mechanical amplitude |X₀| of the system will differ from a system with no optomechanical coupling, which we indicate by $\vert {\bar{X}}_{0}\vert$.

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Now, from equation (30), we can make some observations on parametric driving in the MIM system. The amplitude of the enhanced mechanical quadrature depends on first sideband amplitudes, of which we have determined that these are not enhanced in the MIM system with respect to a single cavity for constant cavity number. In other words, although the MIM promises enhanced nonlinear coupling, the parametric drive per cavity photon is not larger than in a single cavity.

A system with multiple optical modes, such as the MIM, could, however, help to reduce the required input power, as was shown previously in the context of linear position measurement [38] and phonon lasing [42]. Here, we propose a similar use that is particularly useful in optical parametric driving. A schematic example of the idea is shown in figure 6(c). In an optomechanical parametric driving scheme, it is often desirable to have the carrier far detuned from the cavity to suppress DBA heating or cooling of the resonator [51]. This means a considerable input power is needed to reach an appreciable intracavity photon number. In the application we envision, the carrier is on resonance with one of the two supermodes. In that case, the sidebands can be far off-resonant given that 2J ≫ Ω_m, while requiring much less input power.