Optomechanical tension and crumpling of resonant membranes

TL;DR

This paper predicts that resonant illumination can modify the mechanical tension of membranes like graphene, potentially causing crumpling or instability due to optomechanical effects, with implications for controlling membrane states.

Contribution

It introduces the concept of optomechanical tension in resonant membranes and analyzes how optical fields can induce crumpling and instability.

Findings

Optical illumination can induce anisotropic tension in membranes.

Spectral detuning determines whether tension is positive or negative.

Negative tension can lead to membrane crumpling.

Abstract

We predict that illumination by a plane electromagnetic wave of optically resonant membranes, such as graphene or monolayers of transition metal dichalcogenides, directly affects their mechanical tension. The induced optomechanical tension is anisotropic and, depending on the spectral detuning from the resonance, can be both positive and negative. In the latter case, it can overcome the bending rigidity of the membrane leading to transition to the crumpled phase. The instability caused by optomechanical heating of flexural vibrations is also considered.

Equations

$$\displaystyle\mathcal{L}_{\text{int}}=\int{\rm d}\bm{\rho}\,\hat{R}\widetilde{\bm{P}}(\bm{\rho},t)\cdot\big{\{}\bm{E}_{\perp}[\bm{\rho},h(\bm{\rho},t),t]$$

$$\displaystyle+(1-\dot{h}^{2})^{1/2}\bm{e}_{z}E_{z}[\bm{\rho},h(\bm{\rho},t),t]+\dot{h}\,\bm{e}_{z}\times\bm{B}[\bm{\rho},h(\bm{\rho},t),t]\big{\}}\,,$$

$${\tilde{\Omega }}_{\mathbf{q}}^2={\Omega }_{\mathbf{q}}^2+\Sigma (\mathbf{q},\Omega )/\rho ,$$

$$\displaystyle\Sigma_{\text{aS}}(\bm{q},\Omega)=\frac{|E_{0}|^{2}}{2\pi c}\Bigl{\{}\omega\,{\rm Im\,}r(\omega)\left[1+(cq/\omega)^{2}\cos^{2}\varphi\right]$$

$$\displaystyle+{\rm i}\omega_{\text{aS}}\Big{[}\frac{r_{p}(\omega_{\text{aS}},\theta_{\text{aS}})}{\cos\theta_{\text{aS}}}\cos^{2}\varphi+r_{s}(\omega_{\text{aS}},\theta_{\text{aS}})\cos\theta_{\text{aS}}\sin^{2}\varphi$$

$$\displaystyle\hskip 42.67912pt-|r(\omega)|^{2}(1-\sin^{2}\varphi\sin^{2}\theta_{\text{aS}})/\cos\theta_{\text{aS}}\Big{]}\Bigr{\}},$$

$$\sigma_{\text{om}}(\varphi)=\frac{c|E_{0}|^{2}}{\pi\omega}\mathop{\mathrm{missing}}{Im}\nolimits r(\omega)\left[\cos^{2}\varphi+\cos 2\varphi\mathop{\mathrm{missing}}{Re}\nolimits r(\omega)\right]$$

$$\displaystyle\gamma_{\text{om}}=\frac{|E_{0}|^{2}}{2\pi\rho c}\left\{|r(\omega)|^{2}-\mathop{\mathrm{missing}}{Re}\nolimits\frac{{\rm d}}{{\rm d}\omega}\left[\omega r(\omega)\right]\right\}$$

$$\displaystyle\sigma_{\text{om}}^{\parallel}=\frac{c|E_{0}|^{2}}{\pi\omega}\mathop{\mathrm{missing}}{Im}\nolimits r\mathop{\mathrm{missing}}{Re}\nolimits t,\quad\sigma_{\text{om}}^{\perp}=-\frac{c|E_{0}|^{2}}{\pi\omega}\mathop{\mathrm{missing}}{Im}\nolimits r\mathop{\mathrm{missing}}{Re}\nolimits r\,,$$

$$r(\omega )=\frac{\mathrm{i}{\Gamma }_0}{{\omega }_x-\omega -\mathrm{i}({\Gamma }_0+\Gamma )},$$

$$\mathcal{F}=({\sigma }_{\text{om}}{q}^2{h}^2+\kappa q^4{h}^2+2p|h|)L^2$$

$${\mathcal{F}}_{\text{min}}^{\ast }=-\frac{8{\pi }^2{p}^2{\kappa }^2}{{\sigma }_{\text{om}}^3}=-\frac{64{\pi }^3{\omega }^3{\kappa }^2}{c^3|E_0{|}^2}\frac{(\text{Re}r{)}^2}{(\text{Im}r{)}^3}$$

$${\mathcal{L}}_{\text{int}}^{(1)}=\int \mathbf{P}\cdot \left[h\frac{\partial \mathbf{E}}{\partial z}+\dot{h}({\mathbf{e}}_z\times \mathbf{B})-\mathbf{s}\times \mathbf{B}\right]d\mathbf{\rho },$$

$${\mathcal{L}}_{\text{int}}^{(2)}=\int \mathbf{P}\cdot \left[\frac{h^2}{2}\frac{{\partial }^2\mathbf{E}}{\partial z^2}+h\dot{h}{\mathbf{e}}_z\times \frac{\partial \mathbf{B}}{\partial z}-h\mathbf{s}\times \frac{\partial \mathbf{E}}{\partial z}-\dot{h}(\mathbf{s}\cdot \mathbf{B}){\mathbf{e}}_z+\mathbf{s}\times (\mathbf{s}\times \mathbf{E})-\frac{{\dot{h}}^2}{2}{E}_z{\mathbf{e}}_z\right]d\mathbf{\rho },$$

$${\mathbf{\Lambda }}^{(1)}({\omega }^{\prime },{\Omega }_1,\omega )=\mathrm{i}{h}_1\left[k_z-\frac{{\Omega }_1}{{\omega }^{\prime }}\mathbf{k}\otimes {\mathbf{e}}_z-\frac{\omega }{{\omega }^{\prime }}({\mathbf{e}}_z\otimes \mathbf{q}-\mathbf{q}\otimes {\mathbf{e}}_z)\right],$$

$${\mathbf{\Lambda }}^{(2)}({\omega }^{\prime },{\Omega }_1,{\Omega }_2,\omega )=h_1{h}_2\left(-k_z^2+\frac{{\Omega }_1+{\Omega }_2}{{\omega }^{\prime }}{k}_z\mathbf{k}\otimes {\mathbf{e}}_z+\frac{\omega {\Omega }_1{\Omega }_2}{{\omega }^{\prime }}{\mathbf{e}}_z\otimes {\mathbf{e}}_z\right)+\frac{\omega }{{\omega }^{\prime }}\left(\frac{{\mathbf{s}}_1\otimes {\mathbf{s}}_2+{\mathbf{s}}_2\otimes {\mathbf{s}}_1}{2}-{\mathbf{s}}_1\cdot {\mathbf{s}}_2\right),$$

$$-\frac{\mathrm{i}{h}_1{\mathbf{s}}_2}{{\omega }^{\prime }}\times \left[(\omega +{\Omega }_1)k_z-{\Omega }_1\mathbf{k}\otimes {\mathbf{e}}_z\right]-\frac{\mathrm{i}{h}_2{\mathbf{s}}_1}{{\omega }^{\prime }}\times \left[(\omega +{\Omega }_2)k_z-{\Omega }_2\mathbf{k}\otimes {\mathbf{e}}_z\right],$$

$${\Sigma }_{\text{aS,out}}=-[{\mathbf{\Lambda }}^{(1)}({\omega }_{\mathrm{aS}},\Omega ,\omega )\mathbf{A}{]}^{\ast }\cdot \mathcal{P}({\omega }_{\mathrm{aS}}){\mathbf{\Lambda }}^{(1)}({\omega }_{\mathrm{aS}},\Omega ,\omega )\mathbf{A},$$

$${\Sigma }_{\text{aS,in}}=-[{{\mathbf{\Lambda }}^{(1)}}^T(-\omega ,\Omega ,-{\omega }_{\mathrm{aS}})\mathcal{P}(\omega )\mathbf{A}{]}^{\ast }\cdot \mathbf{D}({\omega }_{\mathrm{aS}}){{\mathbf{\Lambda }}^{(1)}}^T(-\omega ,\Omega ,-{\omega }_{\mathrm{aS}})\mathcal{P}(\omega )\mathbf{A},$$

$${\Sigma }_{\text{aS,2}}=-[\mathcal{P}(\omega )\mathbf{A}{]}^{\ast }\cdot {\mathbf{\Lambda }}^{(2)}(\omega ,-\Omega ,\Omega ,\omega )\mathcal{A},$$

$${\Sigma }_{\text{S,out}}=-[{\mathbf{\Lambda }}^{(1)}({\omega }_{\mathrm{S}},-\Omega ,\omega )\mathbf{A}{]}^{\ast }\cdot {\mathcal{P}}^{\ast }({\omega }_{\mathrm{S}}){\mathbf{\Lambda }}^{(1)}({\omega }_{\mathrm{S}},-\Omega ,\omega )\mathbf{A},$$

$${\Sigma }_{\text{S,in}}=-[{{\mathbf{\Lambda }}^{(1)}}^T(-\omega ,-\Omega ,-{\omega }_{\mathrm{S}})\mathcal{P}(\omega )\mathbf{A}{]}^{\ast }\cdot {\mathbf{D}}^{\ast }({\omega }_{\mathrm{S}}){{\mathbf{\Lambda }}^{(1)}}^T(-\omega ,-\Omega ,-{\omega }_{\mathrm{S}})\mathcal{P}(\omega )\mathbf{A},$$

$${\Sigma }_{\text{S,2}}=-{\mathcal{A}}^{\ast }\cdot {{\mathbf{\Lambda }}^{(2)}}^T(\omega ,-\Omega ,\Omega ,\omega )\mathcal{P}(\omega )\mathbf{A},$$

$${\mathcal{P}}_{\mathbf{k}}(\omega )={\mathcal{P}}_t\left(1-\frac{{\mathbf{k}}_{\perp }\otimes {\mathbf{k}}_{\perp }}{k_{\perp }^2}\right)+{\mathcal{P}}_l\frac{{\mathbf{k}}_{\perp }\otimes {\mathbf{k}}_{\perp }}{k_{\perp }^2},{\mathcal{P}}_l=\frac{\omega }{2\pi \mathrm{i}}\frac{r_p(\omega ,\theta )}{\cos \theta },{\mathcal{P}}_t=\frac{\omega }{2\pi \mathrm{i}}{r}_s(\omega ,\theta )\cos \theta$$

$${\mathbf{D}}_{\mathbf{k}}(\omega )=-\frac{4\pi }{(\omega +\mathrm{i}0{)}^2-{\mathbf{k}}^2}\left(1-\frac{\mathbf{k}\otimes \mathbf{k}}{{\omega }^2}\right),$$

$${\mathbf{\Lambda }}^{(1)}\mathbf{A}=\mathrm{i}\left[k_z\mathbf{A}+\left(\frac{\omega }{{\omega }^{\prime }}{\mathbf{k}}_{\text{aS}\perp }-{\mathbf{k}}_{\perp }\right)A_z\right]+(\dots ){\mathbf{e}}_z$$

$$({\mathbf{\Lambda }}^{(1)}\mathbf{A}{)}_{l^{\prime }}$$

$$({\mathbf{\Lambda }}^{(1)}\mathbf{A}{)}_{t^{\prime }}$$

$$\displaystyle\Sigma_{\text{aS,out}}=\frac{{\rm i}\omega^{2}\omega_{\rm aS}}{2\pi}\Big{[}r_{s}(\omega_{\rm aS},\theta_{\rm aS})\cos\theta_{\rm aS}\Big{|}A_{s}\cos\theta\cos\phi_{\rm aS}-A_{p}\sin\phi_{\rm aS}\Big{|}^{2}$$

$$\displaystyle+\frac{r_{p}(\omega_{\rm aS},\theta_{\rm aS})}{\cos\theta_{\rm aS}}\Big{|}A_{s}\cos\theta\sin\phi_{\rm aS}+A_{p}(\cos\phi_{\rm aS}-\sin\theta\sin\theta_{\rm aS})\Big{|}^{2}\Big{]}\,.$$

$$\mathbf{j}=\mathcal{P}\mathbf{A}=\frac{\omega }{2\pi \mathrm{i}}(r_s\cos \theta A_s{\mathbf{e}}_t-r_p{A}_p{\mathbf{e}}_l).$$

$${{\mathbf{\Lambda }}^{(1)}}^T\mathbf{j}=k_z^{\prime }\mathbf{j}-\left[\left({\mathbf{k}}^{\prime }-\frac{{\omega }_{\text{aS}}}{\omega }\mathbf{k}\right)\cdot \mathbf{j}\right]{\mathbf{e}}_z.$$

$$({\mathbf{\Lambda }}^T\mathbf{j}{)}^{\ast }\cdot {\mathbf{D}}^R({\mathbf{\Lambda }}^T\mathbf{j})=-\frac{4\pi }{(\omega +\mathrm{i}0{)}^2-{\mathbf{k}}^2}\left\{{\left|\left({\mathbf{k}}^{\prime }-\frac{{\omega }_{\text{aS}}}{\omega }\mathbf{k}\right)\cdot \mathbf{j}\right|}^2+k_z^{\prime 2}\left(|\mathbf{j}{|}^2-\frac{|\mathbf{k}\cdot \mathbf{j}{|}^2}{{\omega }^2}\right)\right\},$$

$$2K\left(|\mathbf{j}{|}^2-\frac{|\mathbf{k}\cdot \mathbf{j}{|}^2}{{\omega }^2}\right)+\frac{2\pi \mathrm{i}}{k_z^{\prime }}\left[k_z^{\prime 2}\left(|\mathbf{j}{|}^2-\frac{|\mathbf{k}\cdot \mathbf{j}{|}^2}{{\omega }^2}\right)+{\left|\left({\mathbf{k}}^{\prime }-\frac{{\omega }_{\text{aS}}}{\omega }\mathbf{k}\right)\cdot \mathbf{j}\right|}^2\right],$$

$${\Sigma }_{\text{aS,in}}=-2K{\left(\frac{\omega }{2\pi }\right)}^2{\cos }^2\theta \left(|r_s(\omega ,\theta )A_s{|}^2+|r_p(\omega ,\theta )A_p{|}^2\right)-\mathrm{i}\frac{{\omega }^2{\omega }_{\mathrm{aS}}}{2\pi }\cos {\theta }_{\mathrm{aS}}$$

$$\times \left[{\cos }^2\theta \left(|r_s(\omega ,\theta )A_s{|}^2+|r_p(\omega ,\theta )A_p{|}^2\right)+\frac{{\left|\cos \theta \sin {\theta }_{\mathrm{aS}}\sin {\varphi }_{\mathrm{aS}}{r}_s(\omega ,\theta )A_s-(\sin \theta -\sin {\theta }_{\mathrm{aS}}\cos {\varphi }_{\mathrm{aS}})r_p(\omega ,\theta )A_p\right|}^2}{{\cos }^2{\theta }_{\mathrm{aS}}}\right],$$

Full text

Introduction.

Membranes demonstrate rich mechanical phenomena, including non-Hookean elasticity, crumpling phase transitions and nonlinear dynamical structures Nelson et al. (2004). While these effects have been initially discussed for biological membranes, man-made atom-thin membranes from graphene and other novel two-dimensional crystals have recently attracted a lot of attention Nicholl et al. (2015); Liu and Wu (2016); Khestanova et al. (2016); Doussal and Radzihovsky (2018). Understanding their mechanical behavior is crucial for charge and thermal Balandin (2011); Cepellotti and Marzari (2016); Song et al. (2018); Kang et al. (2018) transport applications. However, despite the ongoing experimental Seol et al. (2010); Nicholl et al. (2015); Ackerman et al. (2016) and theoretical Gornyi et al. (2015); Lee et al. (2015); Cepellotti and Marzari (2016) efforts, the mechanics of the monoatomic membranes remains intriguing. For instance, even the sign of the Poisson ratio for graphene is not completely clear Burmistrov et al. (2018a, b). Another family of two-dimensional materials, the transition metal dichalcogenides (TMDCs), have strong optical resonances due to the huge exciton binding energy Wang et al. (2018) and are, therefore, promising for resonant optomechanics Bertolazzi et al. (2011); Castellanos-Gomez et al. (2013); Liu and Wu (2016).

Here, we predict that mechanical properties of membranes can be strongly affected by light. We show that the optical illumination by a plane wave leads not only to the well-known pressure of light, but also directly controls the membrane tension. In a thin membrane, electric field induces polarization that is parallel to the membrane surface. When the membrane shape is modulated, the polarization $\bm{P}$, induced by the normally incident light, is not aligned with the electric field $\bm{E}$. This leads to appearance of the torque $\bm{K}=\bm{P}\times\bm{E}$. If the membrane polarizability is positive, the torque tends to decrease the amplitude of the modulation, see Fig. 1(a). This is equivalent to the tension that flattens the membrane. The proposed optomechanical tension is strongly anisotropic and depends on the light polarization, in stark contrast to the well-known radiation pressure and thermally-induced forces Inoue et al. (2017). In the opposite case, when the membrane polarizability is negative, the torque tends to increase the amplitude of the modulation, see Fig. 1(b). We demonstrate that such negative optomechanical tension can overcome the intrinsic bending rigidity of the membrane and leads to its transition to the crumpled phase. Contrary to the pearling instability induced by the localized optical tweezers Bar-Ziv and Moses (1994); Bar-Ziv et al. (1998) and thermal surface structures Sipe et al. (1983); Tsibidis et al. (2012); Paradisanos et al. (2014), the proposed effect takes place under normal illumination by a homogeneous plane wave and is due to the coherent light-membrane interaction.

Theory of light-membrane optomechanical interaction.

While the origin of the optomechanical tension is well captured by the naïve approach illustrated in Fig. 1, the rigorous calculation is highly non-trivial due to the modification of the electromagnetic field by the membrane itself, i.e., the difference between $\bm{E}$ and $\bm{E}_{0}$ in Fig. 1. Such radiative corrections lead to the renormalization of the light–membrane interaction, and become highly important in the most interesting case of strong resonant polarizability, realized, e.g., in the TMDC membranes. Our aim is to put forward a rigorous theory of the interaction between the electromagnetic field and the flexural vibrations of the optically-resonant membrane and derive the universal expression for the optomechanical tension via the light reflection coefficient from the membrane, directly applicable to experiments.

The straightforward way is to calculate the electromagnetic field by solving the Maxwell equations and accounting for the membrane flection $h$ as a perturbation Poshakinskiy and Poddubny (2019) and then use the Maxwell stress tensor to determine the linear-in-$h$ force acting on the membrane. However, such approach turns out to be quite intricate due to the multiple discontinuities and singularities of the electromagnetic field components at the membrane surface that need to be carefully handled in the stress tensor, see Sec. S3 of the Supplementary Materials. Instead, we use the equivalent Lagrangian formalism, that yields the same results in a significantly more compact fashion and easily handles all the radiative corrections. To construct the Lagrangian that describes the interaction between the polarization of the deformed membrane and the electromagnetic field, we start from the action $S_{\text{int}}=\int\widetilde{\bm{E}}\cdot\widetilde{\bm{P}}\,{\rm d}\widetilde{\bm{r}}\,{\rm d}\widetilde{t}$, where $\widetilde{\bm{E}}$, $\widetilde{\bm{P}}$, $\widetilde{\bm{r}}$ and $\widetilde{t}$ are the electric field, polarization, coordinate and time, respectively, in the reference frame moving and rotating with the membrane. Then, we express the field $\widetilde{\bm{E}}$ through the fields ${\bm{E}}$, $\bm{B}$ in the frame at rest by the subsequent rotation, the spatial shift, and the Lorentz boost and obtain $S_{\text{int}}=\int\mathcal{L}_{\text{int}}{\rm d}t$, where ${\rm d}t=(1-\dot{h}^{2})^{-1/2}{\rm d}\widetilde{t}$, the interaction Lagrangian reads

[TABLE]

$\bm{e}_{z}$ is a unitary vector along the undeformed membrane normal $z$, $\bm{\rho}=(x,y)$ are the in-plane coordinates, $E_{z}$ and $\bm{E}_{\perp}$ are the electric field components along $z$-axis and perpendicular to it, $\dot{h}=\partial h/c\,\partial t$, and $\hat{R}$ is the matrix of rotation by the angle $\arctan|\bm{\nabla}h|$ around the axis $(\partial h/\partial y,-\partial h/\partial x,0)$. The Lagrangian Eq. (1) can be decomposed into the Taylor series over the membrane displacement, $\mathcal{L}_{\text{int}}=\sum_{m=0}^{\infty}\mathcal{L}_{\text{int}}^{(m)}$ with $\mathcal{L}_{\text{int}}^{(m)}\propto h^{m}$, see the Supplementary Materials for the explicit expressions. The $h$-independent term $\mathcal{L}_{\text{int}}^{(0)}=\int d\bm{\rho}\,\widetilde{\bm{P}}(\bm{\rho},t)\cdot\bm{E}(\bm{\rho},t)$ stands for the interaction of light with the undeformed membrane, that we fully take into account. The remaining terms describe the coupling between vibrations, polarization and light, and are treated as a perturbation.

Optomechanical correction to the dispersion of flexural vibrations.

The optical excitation changes the mechanical properties of the membrane, which is revealed in the modified flexural phonon frequency $\tilde{\Omega}_{\bm{q}}$,

[TABLE]

where $\Omega_{\bm{q}}=\sqrt{(\sigma q^{2}+\kappa q^{4})/\rho}$ is the dispersion of flexural phonons in the absence of optical pump, $\sigma$ is the membrane tension, $\kappa$ is the effective bending rigidity Aronovitz and Lubensky (1988); Le Doussal and Radzihovsky (1992); Gornyi et al. (2015), $\rho$ is the membrane mass per unit area, and $\Sigma(\bm{q},\Omega)$ is the phonon self-energy correction due to the optomechanical interaction. To calculate the latter in the non-equillibrium conditions of optical excitation, we exploit the Keldysh Green’s function technique Ivanov and Keldysh (1982); Poshakinskiy et al. (2016); Poshakinskiy and Poddubny (2017).

In the second order in the optomechanical interaction, the relevant self-energy diagrams are of the three types shown in Fig. 2. They include either one vertex corresponding to the second-order interaction $\mathcal{L}_{\text{int}}^{(2)}$, panel (a), or two vertices corresponding to the first-order interaction $\mathcal{L}_{\text{int}}^{(1)}$, panels (b) and (c). The explicit form for $\mathcal{L}_{\text{int}}^{(1,2)}$ and for the diagrams of Fig. 2 is given in the Supplementary Sec. S1. Note that we take the light-matter interaction $\mathcal{L}_{\text{int}}^{(0)}$ in all orders by using the dressed polarization operators (filled bubbles) and the dressed light Green’s function (thick wavy line) in contribution (a). In contributions (b) and (c), the bare light Green’s function is used, because the vertex $\mathcal{L}_{\text{int}}^{(1)}$ is odd under $z$-direction inversion, while the light Green’s function is even, and the dressing vanishes Poshakinskiy and Poddubny (2019). For the normally incident monochromatic wave with the electric field amplitude $\bm{E}_{0}$ and frequency $\omega$, evaluation of the three diagrams in Fig. 2 and three more diagrams with the inverted phonon lines yields $\Sigma_{\text{om}}=\Sigma_{\text{aS}}(\bm{q},\Omega)+\Sigma_{\text{S}}(\bm{q},\Omega)$, where

[TABLE]

$\Sigma_{\text{S}}(\bm{q},\Omega)=\Sigma_{\text{aS}}^{*}(-\bm{q},-\Omega)$, $\omega_{\text{aS}}=\omega+\Omega$ is the frequency of the wave anti-Stokes-scattered by the phonon under consideration, $\sin\theta_{\text{aS}}=q/(\omega+\Omega)$ is the corresponding scattering angle, $r_{s,p}(\omega,\theta)=2\pi{\rm i}\omega\chi(\omega)/[c\cos^{\pm 1}\theta-2\pi{\rm i}\omega\chi(\omega)]$ and $r(\omega)=r_{s,p}(\omega,0)$ are the coefficients of light reflection from the undeformed membrane at the oblique and normal incidence for two polarizations, and $\varphi$ is the angle between $\bm{q}$ and $\bm{E}_{0}$.

At small $\bm{q}$ and $\Omega$, the optomechanical self-energy has the form $\Sigma_{\text{om}}(\bm{q},\Omega)=\sigma_{\text{om}}(\varphi)\,q^{2}-2{\rm i}\rho\Omega\gamma_{\text{om}}$. According to Eq. (2),

[TABLE]

describes the optomechanical correction to the membrane tension and

[TABLE]

is the correction to the flexural phonon damping 111A similar expression for the optomechanical damping rate correction in the system of coupled waveguide and vibrating resonant particle was derived in Poshakinskiy and Poddubny (2019).

The optomechanical tension, Eq. (4), contains two terms of different physical origin; neither can be derived from the conventional radiation pressure force. The first term is due to the torque of the light-induced dipole polarization discussed previously. It is linear in the reflection coefficient; therefore, it dominates when the membrane polarizability is weak and $r\ll 1$. The second term in Eq. (4) originates from the Ampère force acting upon the displacement current in the deformed membrane, see Sec. S2 of the Supplementary Materials for details. Being quadratic in the reflection coefficient, this term becomes significant in case of strong reflection, $r\sim 1$, which can be realized at resonances.

The obtained optomechanical tension Eq. (4), induced by the linearly polarized light, is strongly anisotropic. The tension along ($\varphi=0$) and perpendicular ($\varphi=\pi/2$) to the electric field of the incident light is given by

[TABLE]

where $t=1+r$ is the transmission coefficient. We note that $0\leq\text{Re\,}t\leq 1$ and $-1\leq\text{Re\,}r\leq 0$ in the absence of gain. Therefore, the sign of both tension components is determined by the sign of \mathop{\mathrm{missing}}{Im}\nolimits r. In case of unpolarized or circularly polarized excitation, the optomechanical tension is isotropic and given by $\langle\sigma_{\text{om}}\rangle=(\sigma_{\text{om}}^{\parallel}+\sigma_{\text{om}}^{\perp})/2$.

Frequency and polarization dependence.

To be specific, we consider the membrane with the reflection coefficient of the form

[TABLE]

which is realized, e.g., for the TDMC membranes in the vicinity of the exciton resonance frequency $\omega_{x}$. Here, $\Gamma_{0}$ and $\Gamma$ are the radiative and nonradiative broadening of the resonance, respectively. The latter is governed by the quality of the structure Robert et al. (2016). Figure 3(a) shows the frequency dependence of the optomechanical tension components for the resonant membrane with small losses, $\Gamma<\Gamma_{0}$. The tension is positive below the resonance and negative above it. Interestingly, in the vicinity of the resonance, $|\omega-\omega_{x}|<\Delta^{*}=\sqrt{\Gamma_{0}^{2}-\Gamma^{2}}$, the perpendicular tension is larger than the parallel one, for $|\omega-\omega_{x}|=\Delta^{*}$ the tension is isotropic, while for $|\omega-\omega_{x}|>\Delta^{*}$ the parallel tension dominates. The angular dependence of the tension for these three cases is shown in Fig. 3(b). The case of resonance with significant losses, $\Gamma>\Gamma_{0}$ is illustrated in Figs. 3(c),(d). For all detunings, the optomechanical tension is predominantly along the electric field. Indeed, in such case the reflection coefficient is small and the optomechanical tension is determined by the first term of Eq. (4).

For the light with intensity of 1 W/$\mu$m2, we estimate that the optomechanical tension can reach $0.2$ mN/m. Such a correction to the membrane tension shall lead to a measurable change of the vibration eigenfrequencies for a typical suspended membrane with mechanical quality factor $Q\gtrsim 10^{3}$ pretensioned to $\sigma\sim 10\,\rm mN/m$ Bertolazzi et al. (2011); Liu and Wu (2016); Castellanos-Gomez et al. (2013). The characteristic polarization and frequency dependence of the optomechanical tension can be used to distinguish it from the radiation pressure and heating due to the light absorption.

Finally, we analyze the optomechanical correction to the decay rate of flexural vibrations in resonant membranes. In the vicinity of the resonance, the reflection coefficient $r$ changes rapidly and Eq. (5) simplifies to \gamma_{\text{om}}=-\omega|E_{0}|^{2}\mathop{\mathrm{missing}}{Re}\nolimits({\rm d}r/{\rm d}\omega)/(2\pi\rho c). For the resonant reflection coefficient of the form of Eq. (7), the correction to the decay rate is shown in Fig. 4(a). Under the excitation with frequency below the resonance, the damping of the membrane flexural vibrations is increased, while it is suppressed when the frequency is above the resonance. This agrees with the concept of optomechanical cooling and heating effects Aspelmeyer et al. (2014); Poshakinskiy and Poddubny (2019).

Optomechanical instabilities.

When the optomechanical tension or the optomechanical damping are negative, the flat membrane can become unstable. First, we discuss the instability caused by the optomechanical heating, $\gamma_{\text{om}}<0$. The intrinsic phonon decay rate decreases for small phonon frequencies $\Omega$, while $\gamma_{\text{om}}$ remains constant. Hence, the total damping of low-frequency vibrations is negative. The fluctuations of the membrane shape with the corresponding wave vectors are amplified, promoting a self-oscillatory or chaotic behavior Aspelmeyer et al. (2014). Assuming that the low-frequency vibrations in the free membrane have the intrinsic quality factor of the order of unity due to the strong anharmonicity Lee et al. (2015); Burmistrov and Kachorovskii , we estimate that, under 1 W/$\mu$m2 excitation, such an instability is realized for the phonons with wave vectors $q\lesssim 0.2\,\mu$m*-1*, which requires the membrane with the lateral dimensions $L\gtrsim 20\,\mu$m.

Now we discuss the possibility of the crumpling instability caused by $\sigma_{\text{om}}<0$. To crumple the membrane in a typical suspended mechanical resonator, one should overcome the pretension $\sigma\sim 10~{}\rm mN/m$ by the negative optomechanical tension. This requires the light intensity $\sim 30\,\rm W/\mu m^{2}$ that seems to be reasonably below the optical damage threshold in the regime of pulsed excitation Roberts et al. (2011); Paradisanos et al. (2014). However, this consideration ignores the effect of the pressure of light. In the limit of small deformations, the deflection induced by the pressure of light can be estimated as $h_{0}\sim|E|^{2}L^{2}/\sigma$. For the 30-$\rm W/\mu m^{2}$ excitation and the lateral size $L\sim 1~{}\rm\mu m$, we find $h_{0}\sim 10~{}\mu{\rm m}$, which is clearly beyond the small flexure limit. This means that the membrane shape is controlled by the pressure of light directly, rather than by the proposed optomechanical tension. The undesired light pressure can be suppressed by placing the membrane in an antinode of a standing electromagnetic wave in a cavity or just by considering smaller flakes.

An alternative to the suspended pretensioned membranes is provided by the membranes freely lying on a substrate. In this case the membrane shape is determined by a competition of the optomechanical crumpling, the radiation pressure, and the membrane rigidity. The potential energy change due to the modulation of the membrane shape with the amplitude $h$ and the wave vector $q$ can be estimated as

[TABLE]

where $p=-\text{Re\,}r\,|E_{0}|^{2}/\pi$ is the radiation pressure, $\kappa\sim 1\,$eV is the membrane bending rigidity, and we assume $\sigma_{\text{om}}<0$. For the wave vectors $q<\sqrt{-\sigma_{\text{om}}/\kappa}$ the dependence $\mathcal{F}(h)$ is sketched in Fig. 4(b). While at large $h$ the optomechanical crumpling dominates, there is a light-pressure barrier of the height $\mathcal{F}^{*}=-L^{2}p^{2}/(\sigma_{\text{om}}q^{2}+\kappa q^{4})$ that separates the flat ($h=0$) and crumpled ($h\to\infty$) phases. The minimal barrier $4Sp^{2}\kappa/\sigma_{\text{om}}^{2}$ is realized for the wave vector $q^{*}=\sqrt{-\sigma_{\text{om}}/2\kappa}$. Considering the smallest membrane that supports such a modulation, $L=\pi/q^{*}$, we finally obtain

[TABLE]

for unpolarized light. The barrier height is inversely proportional to the light intensity. When $\mathcal{F}^{*}_{\text{min}}$ becomes comparable with the temperature, the thermal fluctuations push the membrane over the barrier and the transition to the crumpled phase occurs. In the resonant conditions, we estimate the intensity required for the crumpling at room temperature as 0.5 W/$\mu$m2 for $\hbar\omega=0.1\,$eV. The moderate detuning from the resonance, which leads to a decrease of $(\text{Re}\,r)^{2}/(\text{Im}\,r)^{3}\sim 1/(\omega-\omega_{x})$, can be used to further facilitate the crumpling. For the polarized excitation, the barrier height depends on the wave vector direction. In the vicinity of the resonance (away from the resonance), the crumpling starts in the direction perpendicular (parallel) to the electric field, see Fig. 3.

Our theoretical findings indicate the huge potential of membranes for the fusion of nonlinear mechanics with the resonant nano-optics. While the optomechanics of non-resonant membranes is already a mature field with recent successful demonstration of the spontaneous symmetry-breaking buckling of a membrane as a whole Buchmann et al. (2012); Xu et al. (2017), proposed optomechanical crumpling harnesses the non-homogeneous membrane deformations. Even though our consideration is oversimplified by ignoring the strongly non-Hookean elasticity of the membranes Gornyi et al. (2015) as well as the effect of adhesion Khestanova et al. (2016), in our opinion it clearly indicates that the optomechanical tension and crumpling is potentially important for realistic membranes. The effect could be also enhanced when the wave vector of the incident or scattered light is in resonance with the surface waves, supported by the membrane.

Moreover, our results are not limited to graphene and TMDC membranes in the optical frequency range. Since the optomechanical tension Eq. (4) is inversely proportional to the light wavelength, one could expect interesting physics at the lower frequencies, in the THz or even radio frequency spectral range. Optomechanical tension can be used for visualization of fundamental physics of phase transitions and effective mechanical field theories, tailoring the heat and charge transport as well as modification of the light-matter coupling at the nanoscale Camerer et al. (2011); Vochezer et al. (2018). Other potential platforms for the proposed effect range from the biological membranes, where the tensions of $10^{-5}\rm N/m$ are routinely studied by optical tweezers  Hochmuth et al. (1996); Pontes et al. (2017) to the structured solar sails, now attracting a lot of attention Atwater et al. (2018); Ilic and Atwater (2019).

S1 S1. Optomechanical self-energy for flexural vibrations

S1 First- and second-order optomechanical interaction vertices

Decomposing the Lagrangian (1) into the Taylor series over the membrane displacement $h(\bm{\rho})$, we obtain the linear-in-$h$ optomechanical interaction

[TABLE]

and the $h$-quadratic interaction

[TABLE]

where $\bm{s}=(\partial h/\partial y,-\partial h/\partial x,0)$. Next, we express the optomechanical action $S_{\text{om}}^{(1,2)}=\int\mathcal{L}_{\text{int}}^{(1,2)}$ in terms of the vector potential and the current determined by $\bm{E}=-(\partial\bm{A}/\partial\bm{t})$ and $\bm{j}=\partial\bm{P}/\partial\bm{t}$, and switch to the Fourier space. We obtain then $S_{\text{om}}^{(1,2)}=-\bm{P}(\omega^{\prime})\cdot\bm{\Lambda}^{(1,2)}(\omega^{\prime},\omega)\bm{A}(\omega)$, where the first-order Poshakinskiy and Poddubny (2019) and the second-order optomechanical vertices read

[TABLE]

where $h_{1,2}$, $\bm{q}_{1,2}$, and $\Omega_{1,2}$ are the amplitudes, wave vectors, and frequencies of the involved vibrations, $\bm{s}_{1,2}={\rm i}h_{1,2}\bm{q}_{1,2}\times\bm{e}_{z}$, $\bm{k}$ is the wave vector corresponding to $\bm{A}$, and $\omega^{\prime}=\omega+\Omega_{1}(+\Omega_{2})$ is assumed for the first(second)-order vertex.

S2 Evaluation of the optomechanical self-energy correction

We consider here the general case when the pump wave at frequency $\omega$ is incident at the angle $\theta$ with respect to the membrane normal. For clarity, we set $\hbar=c=1$.

The self-energy correction to the flexural phonon dispersion is given by the three diagrams depicted in Fig. 2,

[TABLE]

and three more diagrams with the inverted direction of pump laser lines, which correspond to Stokes scattering and yield

[TABLE]

where $\bm{A}=A_{s}\bm{e}_{s}+A_{p}\bm{e}_{p}$ is the vector potential of the incident light, $\bm{e}_{s}$ and $\bm{e}_{p}$ are the unitary vectors representing $s$- and $p$-polarization, $\bm{\mathcal{A}}=(1+\bm{D\mathcal{P}})\bm{A}$ is the vector potential dressed by the interaction with layer polarization, $\bm{\mathcal{P}}$ is the dressed polarization operator of the layer,

[TABLE]

with $\bm{k}_{\perp}=(k_{x},k_{y})$ being the in-plane wave vector and $\cos\theta=\sqrt{\omega^{2}-k^{2}_{\perp}}/k_{\perp}$, $\bm{D}$ is retarded Green’s function of the bare photon

[TABLE]

$\omega_{\rm aS(S)}=\omega\pm\Omega$ is the frequency of anti-Stokes(Stokes)-scattered wave.

S2.1 Calculation of $\Sigma_{\mathrm{aS,out}}$, diagram Fig. 2(b)

First, we calculate the action of the first-order optomechanical interaction operator on the incident field,

[TABLE]

where $\bm{k}_{\text{aS}\perp}={\bm{k}}_{\perp}+\bm{q}$ and the ellipsis stands for the out-of-plane contribution that does not enter the subsequent calculations. Then, we decompose $\bm{\Lambda}^{(1)}\bm{A}$ into the components parallel and perpendicular to $\bm{k}_{\text{aS}\perp}$, $\bm{\Lambda}^{(1)}\bm{A}=(\bm{\Lambda}^{(1)}\bm{A})_{l^{\prime}}\bm{e}_{l^{\prime}}+(\bm{\Lambda}^{(1)}\bm{A})_{t^{\prime}}\bm{e}_{t^{\prime}}$. The components read

[TABLE]

where $\theta_{\text{aS}}=\arcsin k_{\text{aS}\perp}/\omega_{\text{aS}}$ and $\phi_{\text{aS}}$ is the angle between $\bm{k}_{\perp}$ and $\bm{k}_{\text{aS}\perp}$, see Fig. S1, and the basis of $\bm{e}_{l^{\prime}}$ and $\bm{e}_{t^{\prime}}$ is determined in such way that $\bm{e}_{t}=\bm{e}_{s,\text{aS}}$ and $(\bm{e}_{l}\times\bm{e}_{t})\cdot\bm{e}_{z}>0$. Then, we apply the $\bm{\mathcal{P}}_{\bm{k}_{\text{aS}}}(\omega_{\text{aS}})$ polarization operator, Eq. (S11), and find

[TABLE]

S2.2 Calculation of $\Sigma_{\mathrm{aS,in}}$, diagram Fig. 2(c)

First, we calculate the polarization current induced by the incident wave,

[TABLE]

Application of the first-order optomechanical vertex yields

[TABLE]

Then, using the Green function $\bm{D}_{\bm{k}^{\prime}}(\omega_{\text{aS}})$, Eq. (S12), we obtain

[TABLE]

which has to be summed over $k_{z}^{\prime}$ yielding

[TABLE]

where now $k_{z}^{\prime}=\sqrt{\omega_{\text{aS}}^{2}-\bm{k}_{\text{aS}\perp}^{2}}$ and $K=\int dk_{z}$ is a diverging constant. Finally, we get the self energy correction

[TABLE]

which can be alternatively rewritten as

[TABLE]

S2.3 Calculation of $\Sigma_{\mathrm{aS,2}}$, diagram Fig. 2(a)

For $\bm{h}_{1}=\bm{h}_{2}^{*}$ and $\Omega_{1}=-\Omega_{2}=\Omega$, we get $\bm{s}_{1}=\bm{s}_{2}^{*}={\rm i}h_{1}[\bm{q}\times\bm{e}_{z}]$ and the second-order optomechanical interaction vertex Eq. (S4) reduces to

[TABLE]

where the ellipsis replaces the out-of plane components that do not enter the following calculations. Then, we obtain

[TABLE]

where we introduced $\bm{q}\cdot\bm{e}_{l}=q\cos\alpha$ and $\bm{q}\cdot\bm{e}_{t}=q\sin\alpha$. The imaginary part of $\Sigma_{\text{aS,2}}$ is irrelevant, since it is cancelled by $\text{Im\,}\Sigma_{\text{S,2}}=-\text{Im\,}\Sigma_{\text{aS,2}}$. Note that the singular term $\sim K$ cancels out when the contributions $\Sigma_{\text{in,aS}}$ and $\Sigma_{\text{aS,2}}$ are summed.

When the normal incidence is considered, $\theta=0$, the summation of the three contributions, $\Sigma_{\text{aS,out}}$, $\Sigma_{\text{aS,2}}$ and $\Sigma_{\text{aS,2}}$, Eqs. (S2.1), (S21), and (S2.3), yields Eq. (3) of the main text.

S2 S2. The physical origins of the optomechanical tension

In this section we present an alternative derivation of the optomechanical tension, Eq. (4). The goal is to clarify the microscopic origin of the two contributions. To this end, we consider a static modulation of the membrane shape, $\Omega=0$, and calculate the optical forces that appear. We stress that the optomechanical tension force cannot be derived from the conventional radiation pressure force. Indeed, the latter is a function of the local membrane tilt angle $\bm{\nabla}h$, while the former should be proportional to $\nabla^{2}h$.

The first term in Eq. (4) is due to the torque acting upon the membrane polarization, as illustrated in Fig. 1. The dipole polarization induced by normally incident light in the flat membrane reads $\bm{P}=\chi\bm{E}$, where $\bm{E}$ is electric field in the membrane. Membrane flexion leads to an appearance of the $z$ component of the polarization, proportional to the tilt,

[TABLE]

The torque acting on the polarization, $\bm{K}=2\,\text{Re}(\bm{P}\times\bm{E}^{*})$, is spatially inhomogeneous, and leads to the force

[TABLE]

acting per unit area of the membrane. This corresponds to the action of the effective tension $\sigma_{\text{om},\alpha\beta}={\,\text{Re\,}}\chi\,(E^{*}_{\alpha}E_{\beta}+E_{\alpha}E^{*}_{\beta})$. Finally, we express the polarizability via the reflection coefficient $r$ as $\chi=rc/[2\pi{\rm i}\omega(1+r)]$, link the field $\bm{E}$ to the field of the incident wave $\bm{E}_{0}$ by $\bm{E}=(1+r)\bm{E}_{0}$ and obtain

[TABLE]

reproducing the first term of Eq. (4).

To reproduce the other contribution to the optomechanical tension, we evaluate the Ampère force acting on the membrane. Due to the membrane flexion, the incident electric field, acting on the membrane, changes as

[TABLE]

Suppose now the membrane flexion has the form of harmonic modulation $h(\bm{\rho})=h_{0}{\rm e}^{{\rm i}\bm{q}\cdot\bm{\rho}}+\text{c.c.}$. The induced current then reads, linearly in $\chi$,

[TABLE]

To account for radiative corrections, the bare polarizability $\chi$ has to be replaced by the dressed one, which is easily expressed via the light reflection coefficients. Since the current has an in-plane wave vector, the reflection coefficient for oblique incidence should be used,

[TABLE]

where $\bm{E}_{0,\parallel}$ and $\bm{E}_{0,\perp}$ are the components of $\bm{E}_{0}$ parallel and perpendicular to $\bm{q}$. The Ampère force acting on the current is given by $f_{z}=(2/c)\text{Re\,}[\bm{j}\times\bm{B}(h)^{*}]_{z}$, where

[TABLE]

is the magnetic field at the layer. Then, up to the terms linear in $h$, we get

[TABLE]

While the first term describes the homogeneous light pressure, the second, being proportional to $h$, is a contribution to the optomechanical back-action force. Next, we suppose that $\theta\ll 1$, use the expansion

[TABLE]

and obtain the quadratic in $q$ contribution

[TABLE]

which reproduces the second term of Eq. (4).

S3 S3. Calculation of the optomechanical tension from the Maxwell stress tensor

Yet another approach to calculate the optomechanical tension is provided by the momentum conservation law in classical electrodynamics. Namely, we solve the Maxwell equations directly accounting the membrane flection $h$ as a perturbation Poshakinskiy and Poddubny (2019). Next, we use the Maxwell stress tensor to determine the linear-in-$h$ force acting on the membrane. The whole procedure is rather straightforward but requires accurate handling of the multiple discontinuities and singularities of the electromagnetic field components at the membrane surface. The details are given below.

S1 Light scattering on the deformed membrane

Our aim is to solve the wave equation for the electric field

[TABLE]

The general geometry of the problem is shown in Fig. S1. The vibration of the membrane surface

[TABLE]

with the frequency $\Omega$ modifies the spatial distribution of the permittivity tensor $\varepsilon(\bm{r})$, and thus affects the displacement $\bm{D}=\varepsilon\bm{E}$, leading to the appearance of Stokes and anti-Stokes scattered waves, oscillating at the frequencies $\omega_{\text{S}}=\omega-\Omega$, $\omega_{\text{aS}}=\omega+\Omega$.

We choose the $x$ axis along the phonon wave vector $\bm{q}$. Since the calculation of the optical force is rather lengthy, below we reproduce only the result for the normal incidence ($\theta=0$).

In order to derive the explicit expression for the permittivity tensor of the deformed membrane, we introduce the the normals $\bm{n}(x,y)$ to the membrane surface

[TABLE]

The permittivity tensor is then compactly expressed as

[TABLE]

where $\chi$ is the in-plane susceptibility of the membrane while the out-of-plane susceptibility is supposed zero. The projection operator $\delta_{ij}-n_{i}n_{j}$ takes into account that the considered thin membrane has the electric dipole polarization only along its curved surface. Using the explicit form of $h=h_{0}{\rm e}^{{\rm i}qx-{\rm i}\Omega t}+\rm c.c.$ we expand this operator up to the linear-in-$h$ terms,

[TABLE]

Finally, expanding $\delta(z-h)=\delta^{\prime}(z)-h\delta(z)$, we derive the permittivity tensor,

[TABLE]

The linear-in-$h$ terms in the displacement vector $\bm{D}_{\text{aS(S)}}$ have the form $\bm{D}^{(1)}_{\text{aS(S)}}=\varepsilon^{(0)}_{\omega\pm\Omega}\bm{E}^{(1)}_{\text{aS(S)}}+\varepsilon^{(1)}_{\omega\pm\Omega}\bm{E}^{(0)}$. Next, we substitute the displacement vector $\bm{D}=\bm{D}^{(0)}{\rm e}^{-{\rm i}\omega t}+\bm{D}^{(1)}_{\rm S}{\rm e}^{-{\rm i}\omega_{\rm S}t}+\bm{D}^{(1)}_{\rm aS}{\rm e}^{-{\rm i}\omega_{\rm aS}t}$ into the wave equation Eq. (S35) and solve for the electric field.

S2 Explicit results for the scattered electric fields

In this section we present the explicit form of the electric field, scattered on the deformed membrane, in the zeroth and first order in the membrane deformation $h$. The field can be obtained from the solution of the Maxwell equations in the previous section. That yields the same results as in Poshakinskiy and Poddubny (2019) but neglects the possible difference between $\chi(\omega)$ and $\chi(\omega)_{\text{aS(S)}}$. The latter however is not important for the optomechanical tension force that is defined for $\Omega=0$.

We introduce the basis vectors for electric field of $s$- and $p$-polarized waves

[TABLE]

for the incident wave and

[TABLE]

for the anti-Stokes (aS) and Stokes (S) scattered waves. The incident wave is a superposition of two light polarizations

[TABLE]

The electric field, describing light scattering from the flat undeformed membrane is

[TABLE]

Here, we have introduced an auxiliary operator $\hat{Z}\equiv{\rm diag}\>(1,1,\mathop{\mathrm{sign}}\nolimits z)$ accounting for the altering sign along $z$ and set $c=1$ for simplicity. The linear-in-$h$ anti-Stokes (Stokes) scattering can be described by the Jones matrix Poshakinskiy and Poddubny (2019)

[TABLE]

where $\theta_{\text{aS(S)}}$ is the angle between $\bm{k}_{\text{aS(S)}}$ and $\bm{z}$, $\phi_{\text{aS(S)}}$ is the angle between $\bm{k}_{\text{aS(S)}\perp}$ and $\bm{q}$, see Fig. S1, and

[TABLE]

are the reflection coefficients for the two polarizations. In case of normal incidence $r_{p}=r_{s}=r$ and $\phi_{\text{aS(S)}}=0$.

Then, the Jones matrix is diagonal and the electric field of the anti-Stokes (Stokes) scattered light assumes the form

[TABLE]

where we introduced the operator

[TABLE]

to account for the alternating sign of the $z$ component of $\bm{e}_{p,\text{aS(S)}\hookleftarrow}$ and $\bm{e}_{p,\text{aS(S)}\rightarrow}$, see Fig. S1.

The explicit form of the magnetic field is readily evaluated from Eqs. (S43) and (S45).

S3 Derivation of the optical force

Once the electromagnetic field around the trembling membrane is found, we proceed to the calculation of the optical forces. We start with evaluation of the momentum density of the electromagnetic field

[TABLE]

and the Maxwell stress tensor of the form Pfeifer et al. (2007)

[TABLE]

that determines the flux of the field momentum. The force acting on the matter in a small volume $dV$ is expressed as

[TABLE]

Then, the $z$-component of the surface force density $f_{z}$, which describes the back-action for flexural vibrations, reads

[TABLE]

The electromagnetic fields $\bm{E},\bm{D},\bm{H}$, the vibration amplitude $h$ and the components of the stress-energy tensor are real. Thus, we need to calculate only, e.g., the anti-Stokes component of the optical force $f_{\rm aS}=f_{\Omega}\propto\exp(-{\rm i}\Omega t)$ to obtain the linear-in-$h$ contribution to the optical force $f^{(1)}=f_{\Omega}+f_{-\Omega}^{*}$ acting on the membrane. As the force $f$ is the sum of the products $\sum_{ij}\int{\rm d}zF_{i}F_{j}$, where $\bm{F}=\bm{F}^{(0)}+\bm{F}^{(1)}_{\rm aS}+\bm{F}^{(1)}_{\rm S}+\rm c.c.$, its anti-Stokes component assumes the general form

[TABLE]

where $F_{i}$ stands for the Cartesian component of any of the fields $\bm{E},\bm{D}$ or $\bm{H}$.

For normal light incidence and phonon propagating along $x$, the second term in Eq. (S48) vanishes due to the uniformity along $y$ ($\partial_{y}\to 0$). There exist 10 products to be evaluated in total. Nonzero contributions to the optical force $f_{z}$ come only from singular terms $\propto\delta(z)$ in $\sigma_{xz}$ or $S_{z}$: $\{E_{x}D_{z},H_{x}H_{z},\quad E_{x}H_{y},E_{y}H_{x}\},$ and discontinuous terms $\propto\mathop{\mathrm{sign}}\nolimits(z)$ in $\sigma_{zz}$: $\{E_{z}D_{z},E_{x}D_{x},E_{y}D_{y},\quad H_{z}H_{z},H_{x}H_{x},H_{y}H_{y}\}$, that we present below.

Singular terms in $E_{x}D_{z}$ and $H_{x}H_{z}$ (components of $\sigma_{xz}$):

[TABLE]

Singular terms in $E_{x}H_{y}$ and $E_{y}H_{x}$ (components of $S_{z}$):

[TABLE]

Discontinuous terms in $E_{i}D_{i}$ and $H_{i}H_{i}$ (components of $\sigma_{zz}$):

[TABLE]

After the integration over ${\rm d}z$ we finally derive the optical force

[TABLE]

Substituting

[TABLE]

we obtain

[TABLE]

where $\Sigma_{\text{om}}$ matches the one determined by Eq. (3) of the main text that was calculated using the diagrammatic approach of Section S1. Specifically, in the static limit of vanishing phonon frequency, $\Omega\to 0$, when $\omega_{\rm aS}=\omega_{\rm S}\equiv\omega,\theta_{\rm aS}=\theta_{\rm S}$, we recover Eq. (4) from the main text.