Ultrahigh-Q mechanical oscillators through optical trapping

The coupling of a high-Q mode of a micro-mechanical oscillator to an optical cavity has emerged as a promising route toward observing quantum behavior at macroscopic scales [1]. This opto-mechanical interaction is being used, for example, to optically cool a mechanical mode toward its quantum ground state [1–3]. Ground-state cooling requires that the product of the quality factor Q_m and the frequency f_m of the mechanical mode exceed k_BT_bath/h, where h is Planck's constant. For a room-temperature bath, this condition is marginally satisfied only in a small number of current experiments [4]. The ratio hQ_mf_m/k_BT_bath also determines the quantum coherence time of the system relative to the mechanical period. Significant increases in this ratio beyond that required for ground state cooling are thus critical to prepare and detect non-classical states of motion in most optomechanical schemes (e.g., as in [5, 6]) or to store quantum information for long periods of time [7]. Mechanical systems exhibiting extremely high quality factors also offer novel opportunities for precision measurement and force detection [8, 9]. The difficulty of improving Q–frequency products, however, is highlighted by the fact that a number of systems [10, 11] already exhibit quality factors that are approaching fundamental material limits [12, 13].

In this paper, we propose a class of systems that should enable one to obtain unprecedented Q–frequency products. The approach is based on optically 'trapping' a tethered membrane with low natural mechanical frequency in the anti-node of a strong optical standing wave (see figure 1(a)) [14]. While there are many possible realizations, here we focus on a pendulum geometry where a relatively large disc is supported by a single thin tether. The dielectric disc is attracted to the anti-node of the field, leading to an optical stiffening of its flexural modes. Under realistic conditions, the re-normalized mode frequencies can be significantly enhanced over the values expected under material stresses alone. Of particular interest is the 'center-of-mass' (CM) mode, where the disc oscillates in the optical potential with negligible flexural motion. We show that this motion exhibits an extremely large ratio of potential energy stored in the optical field to strain energy, U_opt/U_mech. This is important because the optical potential is 'lossless', and the result is a correspondingly large increase in the Q–frequency product over a conventional mechanical system due to the dilution of internal friction.

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Our approach to achieving long coherence times builds on previous proposals, which suggested that the highly isolated CM mode of an optically levitated nanosphere [15, 16] can enable quantum opto-mechanics in room-temperature environments [17, 18]. Compared to the nanosphere, our approach has two significant advantages. Firstly, the nanosphere scatters light omni-directionally, leading to motional heating via photon recoil. Suppression of recoil heating to reach the quantum regime requires spheres with sub-wavelength volumes, V/λ³ ≪ 1 [17]. In contrast, the planar membrane primarily couples the counter-propagating components of the trapping beam, strongly reducing recoil heating even for large systems. Secondly, these membranes can be fabricated using well-established techniques that have already yielded excellent mechanical and optical properties in a number of experiments [4, 19]. Our proposal thus shows how the ideas of optical levitation can be brought to bear upon 'conventional' and practically deployable mechanical systems to yield remarkable coherence times.

This paper is organized as follows. In section 1, we develop a model for the optical forces acting on thin membranes trapped in free space, and identify the ratio of optical energy to strain energy (U_opt/U_mech) as a relevant figure of merit for enhancing Q–frequency products. We then apply these results to one particular form of dissipation, thermoelastic damping. Under realistic conditions, we show that Q–frequency product enhancements of three orders of magnitude are possible, limited by practical dimensions of the system and trapping beam. In section 2, we modify our analysis to account for trapping in a Fabry–Perot cavity. In particular, the diffraction of the optical field around the membrane causes a distortion of the cavity mode, and the optical forces must be calculated self-consistently. Scattering of the field also gives rise to photon recoil heating, an additional source of decoherence of the mechanical motion. We find that tapering or apodizing the edges of the membrane can significantly reduce photon scattering and is crucial for achieving long quantum coherence times. In section 3, the advantages of our scheme over other optical spring or trapping proposals are discussed. We conclude the paper in section 4.

We begin by considering the mechanical modes of a free, thin circular membrane with thickness d and radius a in the absence of any tethers. In equilibrium, the membrane is situated at z = 0, in the anti-node of an optical standing wave E(z)∝cos kz (see figure 1(a)). The optical field polarizes the dielectric disc, yielding a gradient force trap around z = 0. In the absence of any internal forces, the optical field traps a thin membrane (d ≪ λ) with a restoring frequency given by $\omega _{\mathrm {opt}}(r)= (\frac {2k^{2}I(r)(\epsilon -1)}{{\rho }c})^{1/2}$, where r is the radial coordinate. Here I(r) is the beam intensity profile (assumed to be rotationally symmetric) in the direction transverse to z, k = 2π/λ is the optical wavevector, ρ is the mass density and is the dielectric constant. Now including the internal stresses, the mechanical displacement field from equilibrium, ζ(x,y), obeys

$$\begin{equation} \frac{\partial^2\zeta}{{\partial}t^2}= -\omega_{\mathrm{opt}}^{2}(r)\zeta-\frac{Ed^2}{12\rho(1-\sigma^{2})}\nabla^{4}\zeta, \end{equation} \tag{ 1 }$$

subject to free boundary conditions [20] (also see appendix A). E,σ denote Young's modulus and Poisson's ratio, respectively. Here and in the following, ∇² is understood to be the Laplacian in the transverse plane. Due to the rotational symmetry, we seek solutions of the form ζ(x,y) = f(r) cos mθ e^−iω_mt. The spatial modes are indexed by the number of nodal diameters and circles, (m,n) (see figure 1(b)). For our numerical results, we take material parameters E = 270 GPa, σ = 0.25, ρ = 2.7 g cm⁻³,  = 4 corresponding to stoichiometric silicon nitride [4], and an operating wavelength of λ = 1 μm. For a free disc without optical forces, the fundamental (2,0) flexural mode has natural frequency $\omega _{\mathrm {m},\mathrm {nat}}^{(2,0)}/2\pi \approx 0.25\frac {d}{a^2}\sqrt {\frac {E}{\rho (1-\sigma ^2)}}$, or ω^(2,0)_m,nat/2π ≈ 1.3 MHz for a disc of dimensions a = 10 μm, d = 50 nm. There is also a trivial solution corresponding to the CM or (0,0) mode with constant f(r) and zero frequency.

For a uniform intensity, I(r) = I₀, the natural radial functions f^(m,n)_nat(r) remain eigenmodes of equation (1), but with re-normalized mechanical frequencies given by $\omega _{\mathrm {m}}^{(m,n)}(I_0)=\sqrt {\omega _{\mathrm {opt}}^2(I_0)+(\omega _{\mathrm {m},\mathrm {nat}}^{(m,n)}){}^2}$. Thus, the CM mode is now a nontrivial solution with frequency ω^CM_m = ω_opt, but still retains a uniform spatial profile. The frequencies of all the flexural modes increase as well, with the CM mode remaining the lowest in frequency (see figure 2(a)).

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The absence of energy stored in internal strains for the CM motion has the important implication of eliminating dissipation due to internal friction. Instead, the energy is stored in an optical potential that contributes no losses (but can contribute a recoil heating force, as described later). So far, we have neglected the tether and have assumed a spatially uniform trapping intensity (i.e. infinite beam waist), whereas taking these factors into account prevents complete suppression of internal strain. To quantify how finite strain energy influences the mechanical quality factor, we consider the effect of thermoelastic damping on our system. We focus on thermoelastic damping, because (i) it can be analytically modeled [12, 20], (ii) it is a fundamental limit even for perfectly fabricated devices [12] and (iii) a number of micro-mechanical systems are approaching this limit [10, 11].

Thermoelastic damping arises because realistic materials have a nonzero coefficient of thermal expansion. The flexural motion creates local volume changes that then lead to temperature gradients and heat flow. Mechanical energy must be expended to drive this heat flow, leading to a finite Q_m. We make two simplifying assumptions to the general thermoelastic equations [12, 20], which are well justified in our system. Firstly, we assume that the thermoelastic coupling is weak, so that the spatial modes determined by equation (1) are not altered to lowest order. Secondly, the strains vary most rapidly along the thin direction of the disc, and thus we ignore the relatively small transverse temperature gradients. The temperature field is given by T(x,y,z,t) = T_bath + ΔT(x,y,z,t), where ΔT satisfies the driven heat equation

$$\begin{equation} \left(c_{V}\frac{\partial}{{\partial}t}-\kappa_{\mathrm{th}}\frac{\partial^2}{{\partial}z^2}\right)\Delta{T}=\frac{E{\alpha}T_{\mathrm{bath}}z}{3(1-2\sigma)}\frac{\partial}{\partial{t}}\nabla^{2}\zeta \end{equation} \tag{ 2 }$$

with boundary conditions ∂ΔT/∂z = 0 at z = ± d/2. Here c_V is the heat capacity per unit volume, κ_th is the thermal conductivity and α is the volumetric thermal expansion coefficient (we take c_V = 2 J cm⁻³ K,κ_th = 20 W m K and α = 4.8 × 10⁻⁶ K⁻¹ for SiN). The work done in driving the heat flow over one cycle is

$$\begin{equation} \Delta{W}\approx -\frac{\kappa_{\mathrm{th}}}{T_{\mathrm{bath}}}\int_{0}^{2\pi/\omega_{\mathrm{m}}}\mathrm{d}t\int\,\mathrm{d}^{3}\textbf{r} \Delta{T}(\textbf{r})(\partial^2\Delta{T}/{\partial}z^2), \end{equation} \tag{ 3 }$$

and the thermoelastically limited quality factor is Q_m,th = 2π(U_opt + U_mech)/ΔW, where U_opt and U_mech are the energies stored in the optical field and strains, respectively. To good approximation, one finds that the thermoelastically limited Q–frequency product is given by $Q_{\mathrm {m},\mathrm {th}}f_{\mathrm {m}}=\frac {45\kappa _{\mathrm {th}}}{{\pi }Ed^{2}T_{\mathrm {bath}}\alpha ^{2}}\frac {1-\sigma }{1+\sigma }(1+U_{\mathrm {opt}}/U_{\mathrm {mech}})$ (see appendix B). Thus, the storage of energy in the optical field leads directly to an enhancement of the Q–frequency product. As a useful comparison, in the absence of optical trapping, an unstressed stoichiometric SiN film of thickness d = 50 nm would have a Q–frequency product limited to Q_m,thf_m ≈ 4 × 10¹³ Hz at room temperature, which only marginally exceeds the fundamental limit k_BT_bath/h ≈ 6 × 10¹² Hz needed for ground-state cooling. The Q–frequency product is also proportional to the number of coherent oscillations that the system can undergo before a single phonon is exchanged with the thermal bath, N^(osc)_th = Q_m,thf_mh/(2πk_BT_bath) (N^(osc)_th ∼ 1 for the conventional membrane described above). A large value is critical for preparing and observing quantum superposition or entangled states [5, 6].

We now examine thermoelastic damping of our free disc. Clearly, if ω_opt(r) is spatially uniform, the CM mode has no internal strains and experiences zero thermoelastic dissipation. A finite beam waist creates inhomogeneous optical forces that internal stresses must compensate for, which mixes CM and internal motion together (we still refer to this mixed mode as the 'CM'). The mixing becomes significant when the variation in ω_opt(r) across the disc overtakes the natural rigidity of the system (as characterized by the natural fundamental frequency), and can be avoided by using sufficiently large beam waists or low intensities. In this regime, significant enhancements to the Q–frequency product should result.

These effects are illustrated in figure 3(a). For concreteness, we assume that the trapping beam has a Gaussian profile with waist w, I(r) = I₀ e^−2r2/w2  (for now, we ignore possible corrections due to distortion as the beam diffracts around the disc). In figure 3(a), we plot the ratio U_opt/U_mech for the CM mode as a function of its frequency, which is varied through the peak intensity I₀. We use the same disc dimensions as before and a waist of w = 35 μm. The energy ratio monotonically decreases, reflecting the increased inhomogeneity in the optical potential. In the inset, a complementary process is illustrated, where the beam waist w is varied, while the peak intensity I₀ is fixed such that ω_opt(I₀)/2π = 1 MHz. The energy ratio increases indefinitely with w/a and approaches infinity in the plane-wave limit.

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We now consider the realistic pendulum geometry shown in figure 1(a), where the tether provides an extremely weak restoring force for the 'CM' motion of the disc. We first present a simplified analysis that isolates the role of the tether in the mode spectrum and Q–frequency product. Specifically, we treat the membrane as a perfectly rigid point particle of mass M, which experiences an optical restoring force with frequency ω_opt, while internal stresses alone act on the tether. Then, for a tether of length L whose long axis is situated along x, the displacement field ϕ(x,t) (where 0 ⩽ x ⩽ L) satisfies the beam equation [20]

$$\begin{equation} \frac{\partial^{2}\phi}{\partial t^2}=-\frac{Eb^2}{12\rho}\frac{\partial^{4}\phi}{\partial x^4}. \end{equation} \tag{ 4 }$$

Here b denotes the width of the tether (assumed to be square in cross-section). The beam is clamped at x = 0, ϕ(0,t) = ∂_xϕ(0,t) = 0, while at x = L the boundary conditions are given by ∂²_xϕ(L,t) = 0 and M∂²_tϕ(L,t) = −Mω²_optϕ(L,t) + Eb⁴∂³_xϕ(L,t)/12. The last equation describes the acceleration of the membrane due to optical restoring forces and the shear force imparted by the tether.

It is straightforward to solve for the system eigenmodes and the results are summarized here. For large mass ratios between the membrane and tether, M/m_t → ∞, the modes usually consist of a CM mode for the membrane with frequency $\omega _{\mathrm {m}}^{\mathrm {CM}}\approx \sqrt {\omega _{\mathrm {p}}^2+\omega _{\mathrm {opt}}^2}$ and a set of discrete tether modes with frequencies $\omega _n\approx (\frac {(n+1/4)\pi }{{\beta }L})^2$, where β = (12ρ/Eb²)^1/4. The CM mode spectrum is understood as a low-frequency 'pendulum' mode (with natural frequency $\omega _{\mathrm {p}}\approx \sqrt {Eb^{4}/4ML^3}$, where ω_p ≪ ω_opt,ω_n for our systems of interest) whose frequency can be strongly re-normalized by the optical force, while the tether mode spectrum results from the heavy membrane essentially acting as a second clamp. This description holds except near degeneracies ω_opt ≈ ω_n, where coupling between the tether and membrane motions yields an avoided crossing whose width decreases with increasing mass ratio M/m_t. This result is illustrated in figure 2(b), for a mass ratio of M/m_t = 125 (corresponding to the disc size considered earlier attached to a tether of length L = 50 μm and width b = 50 nm). In figure 3(b), we plot the energy ratios U_opt/U_mech for the CM motion as a function of ω_opt. Here, the strain energy is completely attributable to the tether, as we take the membrane to be a rigid object. The energy ratio is dramatically reduced near the avoided crossings, while away from these crossings, the energy ratio plateaus to a value near U_opt/U_mech ∼ 8M/m_t.

For a realistic tethered system (as in figure 1(a)) where the membrane is not perfectly rigid, mode mixing between the tether and membrane and mixing between the CM and internal membrane motion will occur simultaneously. We have numerically solved the full stress–strain equations for such a system using COMSOL, a commercial finite-element simulation package. A characteristic mode spectrum is plotted in figure 2(c) as a function of the peak trapping intensity I₀, for parameters a = 10 μm, L = 50 μm, d = b = 50 nm and w = 35 μm. Away from avoided crossings, the modes can clearly be characterized as tether modes (gray points) or membrane modes (color). For our particular choice of beam waist size, the tethers themselves experience optical restoring forces, leading to a slight optical stiffening of tether modes that have displacements along the optical propagation axis. Comparing the membrane modes, the CM mode lies lower in frequency than the flexural modes, as in the case of a free disc. A nearly degenerate torsional (1,0) mode also exists, which in principle should have no opto-mechanical coupling to the cavity field and can be ignored.

In figure 3(c), we plot the energy ratio U_opt/U_mech for the CM mode of our tethered structure as a function of its frequency. The features displayed here are clearly a combination of those appearing in the limiting cases of a free disc and a rigid disc attached to a tether. In particular, large plateaus in U_opt/U_mech appear away from avoided crossings with tether modes. The plateau heights can be higher than the limit U_opt/U_mech ∼ 8M/m_t predicted by the simple model, since the tether experiences an optical spring force as well. The decrease in the plateau heights with increasing CM frequency is associated with increased mixing between pure CM and internal membrane motion. For the realistic geometry considered here, an enhancement in the Q–frequency product of the order of ∼10³ compared to a conventional system can be realized at a frequency of ω_m/2π ∼ 1 MHz. For a thermoelastically limited system, this corresponds to a coherence time of N^(osc)_th ∼ 10³.

Although we have focused on thermoelastic processes, which can be exactly modeled, we emphasize that our conclusions are qualitatively correct for any internal dissipative process. For example, a number of other mechanical systems phenomenologically suffer from frequency-independent dissipation, which may be due to effects such as tunneling in amorphous solids [13] or surface mechanisms [21]. Such systems can be characterized by a complex, frequency-independent Young's modulus E = E_r + iE_i, where the imaginary component accounts for dissipation of strain energy. In this case, the quality factor in the presence of optical trapping behaves like Q_m,E = (E_r/E_i)(1 + U_opt/U_mech), again indicating the importance of storing energy in the lossless optical potential.

So far, we have assumed that the trapping beam has a Gaussian profile. For cavity opto-mechanics [1], it will be necessary to trap the membrane within a Fabry–Perot cavity, as illustrated in figure 1(a). For example, here, a relatively strong beam could be used for trapping, while a second, weaker beam with a nonzero intensity gradient at the trap position would facilitate cooling of the CM motion or quantum state transfer processes [17, 18]. The membrane scatters and diffracts the cavity light, which introduces two important effects. Firstly, the mode will no longer be Gaussian, and the new optical mode accommodated by the cavity mirrors and the corresponding optical forces must be determined. Secondly, photon scattering out of the cavity reduces cavity finesse, and the associated random momentum kicks ('photon recoil') imparted on the membrane lead to additional decoherence.

To quantify these effects, we begin by calculating the cavity modes in the presence of the membrane using a modified Fox–Li propagation technique [22]. Here, the electric field is treated within the scalar paraxial approximation, and thus it is completely described by its transverse profile E(x,y). This approximation is justified by noting that the disc should primarily diffract light at small angles $\theta \lesssim (ka)^{-1}$ around the z-axis, where k = 2π/λ is the optical wavevector. Within this approximation, free propagation over a distance z is accounted for by a phase shift in the Fourier transform of the field profile, $\tilde {E}(k_x,k_y){\rightarrow }\mathrm {e}^{\mathrm {i}kz-\mathrm {i}(k_x^2+k_y^2)z/(2k)}\tilde {E}(k_x,k_y)$. In our case, we are interested in systems with rotational symmetry, and thus the transforms are implemented through the quasi-discrete Hankel transform described in [23]. Reflection off a circular mirror with radius of curvature R_c and reflectance R_m is characterized by the real-space transformation $E(x,y){\rightarrow }\sqrt {R_{\mathrm {m}}}E(x,y)\exp\kern-2pt \left (2\mathrm {i}\,k(R_{\mathrm {c}}-\sqrt {R_{\mathrm {c}}^2-(x^2+y^2)})\right )$. Similarly, at the membrane location, the wave front can undergo reflection and transmission. In the case of an infinite dielectric membrane of uniform thickness d, the Fresnel equations yielding the thickness-dependent reflection and transmission coefficients r(d),t(d) can be exactly solved. Following the Fox–Li technique, for a finite-size membrane or a membrane of non-uniform thickness, we apply these expressions to approximate the scattering amplitude and phase shift locally. For example, a field E(x,y) incident on the membrane undergoes the transformation E(x,y) → t(d(x,y))E(x,y) upon transmission. Note that an initial wave front incident on the membrane thus splits into two wave fronts (a reflected and a transmitted field), and we keep track of the multiple scattered fields to all orders to calculate the field buildup or cavity eigenmodes. In contrast, the original technique of [22] only accounts for transmission. Thus, our approach properly captures the effects of the reflected amplitude and back-scattered angle. Furthermore, our modified technique reveals specifically at what frequencies resonances should occur.

We now discuss the effect of the membrane on the cavity finesse. To speed up calculations (taking advantage of the rotational symmetry) and given the relatively small size of the tether, only the central disc is treated here. As realistic parameters, we consider a membrane placed symmetrically in the center of an optical cavity of length L = 1.99 cm with spherical mirrors having radii of curvature R_c = 1 cm and perfect reflectivity (such that we can identify the contribution κ_mem to the cavity linewidth due to scattering from the membrane). The transverse extent of the spherical mirror surfaces is r_m = 0.95 mm, i.e. all portions of the beam front with x² + y² > r²_m are scattered out and set to zero upon reflection at the mirror. Accounting for a realistic mirror extent is important in determining the scattering rate, as a fully enclosing mirror would be a closed system and would exhibit no losses. An empty cavity in this configuration yields a Gaussian mode of waist w₀ ≈ 15 μm in the center. In figure 4(a), we plot the membrane-limited cavity finesse $\mathcal {F}_{\mathrm {mem}}\equiv {\pi }c/{\kappa _{\mathrm {mem}}}L$ for a membrane of uniform thickness d = 30 nm and varying radius a (black circles). Clearly, cavity losses are negligible when the nominal waist is small compared to the disc radius, $w_0/a\lesssim 1$. In the regime w₀/a ≳ 1, however, the finesse rapidly drops, which is attributable to scattering by the hard edges of the disc. This effect is strongly reduced by 'softening' or apodizing the disc edge [24]. In figure 4(a) (red circles), we also plot the finesse for a membrane whose thickness d(r) = d₀(1 − (r/a)²)² tapers down to zero at the edge, where d₀ = 30 nm is the maximum thickness. Remarkably, the apodization can improve the cavity finesse by several orders of magnitude. The modification of the cavity modes by the membrane is illustrated in figure 4(b), where we plot the transverse profile at the membrane position for some representative apodized disc sizes.

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We find the CM eigenmodes of the apodized disc using equation (A.3), which is a generalization of equation (1) to a disc of non-uniform thickness. The optical potential ω_opt(r) is now evaluated using the modified cavity mode profiles. The thermoelastic limit is subsequently calculated using equations (2) and (3). In figure 4(c), the number of oscillations N^(osc)_th due to thermoelastic damping is plotted (in black). Here the circulating intra-cavity power is chosen such that the CM frequency is fixed at ω_m/2π = 0.5 MHz.

We next consider the effect of photon recoil heating. We assume that each scattered photon contributes the maximum possible momentum kick of ℏk along the z-axis, giving rise to a momentum diffusion process d〈p²_z〉/dt = (ℏk)²R_sc [17], where R_sc is the photon scattering rate. Converting this expression into a jump rate, it can be shown that the number of coherent oscillations before a jump in the phonon number can be written as

$$\begin{equation} N^{(\mathrm{osc})}_{\mathrm{sc}}=\frac{1}{2\pi}\frac{V}{V_{\mathrm{c}}}\frac{\omega_0}{\kappa}\frac{\omega_{\mathrm{m}}^2}{k^{2}I_{\mathrm{max}}/{\rho}c}. \end{equation} \tag{ 5 }$$

Here I_max is the maximum cavity intensity (which in general does not need to be at the center of the membrane due to mode distortion), ω₀ = ck, V is the volume of the disc and V_c is the cavity mode volume. Assuming that the cavity mode is not significantly distorted and that the beam waist w₀ ≳ a such that the entire membrane experiences the optical force, one can approximate $\frac {\omega _{\mathrm {m}}^2}{k^{2}I_{\mathrm {max}}/{\rho }c}\sim 1$ and V_c ∼ πw²₀L/4. In this case, the number of coherent oscillations scales roughly as $N^{(\mathrm {osc})}_{\mathrm {sc}}\sim \frac {kV}{w_0^2}\mathcal {F}_{\mathrm {mem}}$. Note that this result is purely geometric in nature and also scales directly with the cavity finesse (which itself depends on V ). In figure 4(c), we plot N^(osc)_sc for the apodized disc (in red). Combining the effects of thermoelastic damping and recoil heating, the total number of coherent oscillations is given by N^(osc)_tot = (N^(osc)−1_sc + N^(osc)−1_th)⁻¹ (blue curve). It can be seen that an apodized disc of radius r ∼ 9 μm can support a coherence time of N^(osc)_tot ∼ 2000 in a room-temperature environment. Remarkably, the coherence time of this system is comparable to that of a much smaller levitated nanosphere of radius r ∼ 25 nm [17], or a conventional 1 MHz oscillator with Q_m exceeding 10⁸ at a bath temperature of 1 K. In appendix D, we provide an example set of parameters for this system, demonstrating that it can be cooled to the quantum ground state.

Our analysis so far presents the limitations imposed on the system due to the trapping mode. If a separate mode at a nearby frequency is used for optomechanical coupling and cooling, one should expect similar effects (e.g. finesse limitation and mode distortion) for this secondary mode as well. We note, however, that it should also be possible to separately optimize the parameters such as finesse, beam waist and effective cavity length for the two modes by using significantly different wavelengths or compound cavity geometries.

We emphasize that our approach to reach the quantum regime, which relies upon achieving ultrahigh-Q factors, is fundamentally different from other 'optical spring' proposals based on optical backaction forces [25, 26]. In the latter case, the linear coupling of the mechanical displacement to the intra-cavity intensity can yield a dynamic optical spring effect. This effect, however, is accompanied by significant Raman scattering of the optical pump field, which causes phonons to be rapidly removed and added to the system. While this does not preclude ground state cooling [26], our analysis below demonstrates that it imposes severe limitations on the quantum coherence time and makes it difficult to prepare, e.g., quantum superposition states. In contrast, in our scheme, the phonons are truly long-lived excitations.

Specifically, we consider the dynamic optical spring resulting from a mechanical degree of freedom whose displacement is linearly coupled to the optical cavity frequency. The corresponding Hamiltonian for such a system in a rotating frame is [27]

$$\begin{equation} H_{\mathrm{int}}=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega_{\mathrm{m}}^{2}\hat{z}^2-\hbar\delta\hat{a}^{\dagger}\hat{a}-{\hbar}\omega'\hat{z}\hat{a}^{\dagger}\hat{a}-\hbar\Omega_L(\hat{a}+\hat{a}^{\dagger}). \end{equation} \noindent \tag{ 6 }$$

Here $\hat {a}$ is the annihilation operator for the optical mode, $\hat {z},\hat {p}$ are the position and momentum operators corresponding to the mechanical resonator, ω_m is the natural mechanical frequency, δ = ω_L − ω₀ is the frequency detuning between an external pump field driving the optical cavity and the optical resonance frequency ω₀ (when the mechanical resonator is in equilibrium), Ω_L is the driving amplitude and ω' is the optical cavity frequency shift per unit mechanical displacement. In addition to the Hamiltonian terms, the optical cavity is assumed to have losses characterized by a linewidth κ.

For weak opto-mechanical coupling, it is customary to linearize the optical cavity dynamics around the classical steady-state value $\langle \hat{a}\rangle =\alpha ={\rm i}\,\Omega _{L}/(\kappa /2-{\rm i}\,\delta )$ (here we have incorporated a steady-state shift of the optical resonance frequency into our definition of the detuning δ), and eliminate the cavity to yield an effective susceptibility χ(ω) of the mechanical displacement in response to an external force f(ω) [27]. Specifically, one finds that

$$\begin{equation} \chi(\omega)^{-1}=\omega_{\mathrm{m}}^2-\omega^2+\frac{16\omega_{\mathrm{m}}\delta\Omega_{\mathrm{m}}^2}{4\delta^2+(\kappa-2\mathrm{i}\omega)^2}. \end{equation} \noindent \tag{ 7 }$$

Here we have defined an effective opto-mechanical driving amplitude Ω_m = gα, and $g=\omega 'z_{\mathrm {zp}}=\omega '\sqrt {\frac {\hbar }{2m\omega _{\mathrm {m}}}}$ is the optical cavity frequency shift per unit mechanical zero-point uncertainty. In the perturbative limit, this expression can be written in terms of the susceptibility of a simple oscillator with an effective linewidth and frequency that is modified due to opto-mechanical interactions, χ(ω)⁻¹ ≈ ω²_m,eff − ω² − i ωΓ_eff. The effective linewidth and mechanical frequency shift can be interpreted as resulting from optically induced cooling (or heating) and a dynamic optical spring constant, respectively. In the relevant regime of large detuning δ ≫ ω,κ, and when the optical spring is dominant compared to the natural mechanical frequency, the effective mechanical frequency is given by [27, 28]

$$\begin{equation} \omega_{\mathrm{m},\mathrm{eff}}\approx 2\Omega_{\mathrm{m}}\sqrt{\frac{\omega_{\mathrm{m}}}{\delta}}. \end{equation} \tag{ 8 }$$

The damping rate is given by

$$\begin{equation} \Gamma_{\mathrm{eff}}\approx\Omega_{\mathrm{m}}^2\kappa\frac{\omega_{\mathrm{m}}}{\omega_{\mathrm{m},\mathrm{eff}}}\left[\frac{1}{(\kappa/2)^2+(\delta+\omega_{\mathrm{m},\mathrm{eff}})^2}-\frac{1}{(\kappa/2)^2+(\delta-\omega_{\mathrm{m},\mathrm{eff}})^2}\right], \end{equation} \tag{ 9 }$$

which is interpreted as the difference between anti-Stokes and Stokes scattering rates. Note that for positive detuning δ > 0, the opto-mechanical interaction yields an increase in the mechanical frequency but an anti-damping force (Γ_eff < 0). One can achieve simultaneous stiffening and cooling by employing multiple beams with different amplitudes and detunings [26], but for our purposes it is sufficient to consider only the beam that leads to stiffening.

We wish to consider the ratio of the effective mechanical frequency to the rate of decoherence Γ_d induced by optical Raman scattering, which is given by the sum (and not difference) of the anti-Stokes and Stokes scattering rates. This is an important point—although the removal and addition of a phonon by two Raman scattering events has no net effect in terms of energy, it does destroy quantum coherence. In the relevant limit of large detuning and dominant optical spring effect, one finds that

$$\begin{equation} \frac{\omega_{\mathrm{m},\mathrm{eff}}}{\Gamma_{\mathrm{d}}}\approx \frac{2\delta}{\kappa}. \end{equation} \tag{ 10 }$$

This result states that the cavity must be driven very far off resonance in order to yield a frequency shift that is much larger than the decoherence rate. Operating at large detuning in turn requires extremely large cavity input powers to get an appreciable optical spring effect. As an example, we consider the dynamic spring constant for a realistic geometry, such as a Fabry–Perot cavity of length L = 1 cm and cavity finesse $\mathcal {F}=10^5$ (with the cavity linewidth given by $\kappa ={\pi }c/(\mathcal {F}L)=2\pi {\times }150\,\mathrm {kHz}$ for our specific parameters). The optical driving amplitude is related to the input power P_i through $\Omega _L=\sqrt {{\kappa }P_{\mathrm {i}}/(2\hbar \omega _L)}$ for perfect in-coupling efficiency, while the opto-mechanical interaction strength is of order ω' ∼ ω₀/L. The operating wavelength is taken to be λ = 1 μm. We also assume that the SiN membrane has a radius a = 10 μm and that it undergoes pure CM motion (such that its effective motional mass is the same as the physical mass). Then, an input power of P_i ∼ 2 kW is required if one wants to achieve the number of coherent oscillations of $N^{(\mathrm {osc})}=\frac {\omega _{\mathrm {m},\mathrm {eff}}}{2\pi \Gamma _{\mathrm {d}}}{\sim }10^3$ and an effective mechanical frequency of ω_m,eff ∼ 2π × 1 MHz. This corresponds to an input intensity of ∼10 W μm⁻² for a beam focused to a size comparable to the membrane radius.

In contrast, in our static trapping scheme, a comparable mechanical frequency and coherence time can be achieved for an intra-cavity intensity of ∼0.1 W μm⁻², and the cavity can be driven resonantly to facilitate the intra-cavity field buildup. Because the static trap results in the membrane being trapped at an anti-node, there is no linear opto-mechanical coupling for the trapping field and the lowest-order opto-mechanical coupling is quadratic in nature. The anti-Stokes and Stokes scattering rates in this case (at frequencies ω_L ± 2ω_m) have been calculated in [29] and are extremely rare for our realistic systems (occurring at a sub-Hz level), leading to negligible decoherence.

Furthermore, regardless of the trapping scheme used, our analysis properly captures the role that strong, spatially non-uniform optical forces have in mixing internal motion, which is neglected in lowest-order opto-mechanical models but is relevant to most flexural systems. In the case of the dynamic optical spring, for example, our model shows that the opto-mechanical coupling strength g decreases significantly at the large intensities needed to observe strong optical spring effects (see appendix C).

We also point out the proposal in [30] that is qualitatively similar to ours, which involves the optical levitation of a macroscopic mirror using dipole forces. We believe that their conclusion of the feasibility of levitating such a macroscopic system is based on a number of erroneous assumptions. Specifically, their expression for the trapping force is derived from the optical polarizability of a sub-wavelength particle. In this regime, both the trapping force and mass increase linearly with the volume of the particle, and thus the CM oscillation frequency remains size independent for a fixed intensity. However, for objects greater than a wavelength, the ratio of trapping force to mass decreases (inversely with thickness, for the case of a planar structure [31]), and thus the authors of [30] greatly underestimate the power requirements for trapping a macroscopic mirror. In contrast, our membranes remain thin along the trapping axis, which allows for sub-wavelength formulae to hold. Furthermore, their calculation of the recoil heating rate only accounts for scattering from thermal density fluctuations (this mechanism is important for guided modes in a low-loss fiber) [32]. However, this ignores the large scattering cross-section of the externally illuminated dielectric object. Indeed, it has been shown previously that wavelength-scale particles already suffer severe decoherence times (N^(osc)_sc ∼ 1) due to this effect [17]. This mechanism is suppressed in our scheme because of our planar geometry and the use of a cavity mode to trap, which enables the cavity mirrors to re-capture much of the scattered light.

We have described a technique that allows the ideas of optical levitation to be applied to conventional, scalable mechanical systems, yielding Q–frequency products significantly higher than what material properties would nominally dictate. Although we have focused on thermoelastic losses in the above calculations, we expect similar improvements for any other internal damping mechanism. The key idea is that it is possible to circumvent natural material limits of damping by storing energy in a lossless optical field rather than the internal strain. By making the ratio of these energies large, U_opt/U_mech ≫ 1, any internal losses can be suppressed by a corresponding degree. This fundamental observation allows one to design a novel class of mechanical systems that can be fabricated and deployed using conventional techniques, yet yield Q–frequency products that are several orders of magnitude higher than previous systems. We believe that this work will stimulate further investigation into the relationship between optical forces and material dissipation in a number of systems where the mechanical motion can be strongly renormalized by light [26, 33]. Furthermore, we anticipate that such studies will open up interesting possibilities for quantum manipulation of mechanical systems in room-temperature environments.

The authors thank Dal Wilson and Richard Norte for many helpful discussions. DEC acknowledges support from the NSF (grant no. PHY-0803371), the Gordon and Betty Moore Foundation through Caltech's Center for the Physics of Information (CPI), and Fundació Privada Cellex Barcelona. KN acknowledges support from the CPI. HJK and OJP acknowledge support from the DARPA ORCHID program. HJK also acknowledges support from the NSF and DoD National Security Science and Engineering Faculty Fellowships (NSSEFF) program.

Here we derive the equation of motion for a thin non-uniform disc of thickness d(r), which has a reflection symmetry around z = 0 (such that the surface of the plate is located at z = ± d(r)/2). We are interested in the situation where the thickness is much less than the characteristic transverse size (e.g. the radius a of a circular disc), such that its degree of freedom along the thin direction can be effectively eliminated and the flexural motion can be described by a two-dimensional displacement field ζ(x,y). The equation of motion for ζ(x,y) can be obtained by a generalization of the derivation for a uniform disc given in [20]. Specifically, the energy associated with the displacement field ζ(x,y) is given by

$$\begin{eqnarray} &&\fl U_{\mathrm{mech}} = \frac{E}{24(1-\sigma^2)}\int\,\mathrm{d}x\,\mathrm{d}y\, \mathrm{d}(x,y)^3\left[\left(\zeta_{xx}+\zeta_{yy}\right)^2+2(1-\sigma)\left((\zeta_{xy})^2-\zeta_{xx}\zeta_{yy}\right)\right]. \end{eqnarray} \tag{ A.1 }$$

Here E,σ are the Young's modulus and Poisson's ratio, respectively, while $\zeta _{xx}=\frac {\partial ^{2}\zeta }{{\partial }x^2}$, etc. To derive the equilibrium field ζ(x,y) under some external normal pressure P(x,y), we employ the variational principle to minimize the system energy. Under small variations δζ and following some algebra, the variation in U_mech can be written as the sum of an integral over the transverse area of the disc and two integrals over the circumference or edge of the disc,

$$\begin{equation} {\delta}U_{\mathrm{mech}}=\int\,\mathrm{d}x\,\mathrm{d}y\,Z_{1}\delta\zeta+\oint\,\mathrm{d}l\,Z_{2}\frac{\partial\delta\zeta}{{\partial}n}+\oint\,\mathrm{d}l\,\,Z_{3}\delta\zeta. \end{equation} \tag{ A.2 }$$

Here Z_i are complicated expressions involving ζ and d(r) whose forms are given below, while n denotes the normal to the edge of the disc. The integral over the disc area yields the equilibrium equation of the disc, Z₁ = P(x,y), or the dynamical equation can be obtained by replacing $P(x,y){\rightarrow }-{\rho }d(r)\frac {\partial ^2\zeta }{{\partial }t^2}$. Doing so, and including the effect of external optical forces, one finds that

$$\begin{eqnarray} &&\fl \frac{\partial^2\zeta}{{\partial}t^2} = -\omega_{\mathrm{opt}}^{2}(r)\zeta-\frac{E}{12\rho(1-\sigma^{2})d(r)}\left[\nabla^{2}\left(g(r)\nabla^{2}\zeta\right)-(1-\sigma)\left(\zeta_{yy}g_{xx}+\zeta_{xx}g_{yy}-2\zeta_{xy}g_{xy}\right)\right].\nonumber\\ \end{eqnarray} \noindent \tag{ A.3 }$$

Here we have defined g(r) = d(r)³, and ∇² is understood to be the Laplacian in the transverse plane. As described in the main text, $\omega _{\mathrm {opt}}(r)=(\frac {2k^{2}I(r)(\epsilon -1)}{{\rho }c})^{1/2}$. For a disc with free boundary conditions at the edge, the quantities δζ and ∂δζ/∂n are arbitrary on the boundary, so the coefficients Z_2,3 should vanish, yielding the two boundary conditions. Defining n and l to be the normal and tangential directions to the edge of the disc, and θ as the local angle between x and n, these boundary conditions become

$$\begin{eqnarray} &&\fl 0 = \frac{\partial}{{\partial}n}\left(g\nabla^{2}\zeta\right)+(1-\sigma)\left[\frac{\partial}{{\partial}l}\left(-g\zeta_{xy}\cos 2\theta+(g/2)\sin 2\theta(\zeta_{xx}-\zeta_{yy})\right)\right. \nonumber \\ &&+\left.\cos\theta(\zeta_{yy}g_{x}-\zeta_{xy}g_{y})+\sin\theta(\zeta_{xx}g_{y}-\zeta_{xy}g_{x})\right],\\ &&\fl 0 = \nabla^{2}\zeta +(1-\sigma) \left(2\zeta_{xy}\sin\theta\cos\theta-\zeta_{yy}\cos^{2}\theta-\zeta_{xx}\sin^{2}\theta\right).\nonumber \end{eqnarray} \tag{ A.4 }$$

Solving equation (2) and substituting into equation (3), one finds that

$$\begin{equation} \Delta{W}\approx\frac{\pi\omega_{\mathrm{m}}\alpha^2 E^2 d^5 T_{\mathrm{bath}}}{1080\kappa_{\mathrm{th}}(1-\sigma)^2}\int\,\mathrm{d}x\,\mathrm{d}y\,(\nabla^{2}\zeta)^2. \end{equation} \tag{ B.1 }$$

Let us now compare this quantity with the total strain energy U_mech given in equation (A.1). For simplicity, here we specialize to the case where the disc has a uniform thickness d, such that

$$\begin{equation} U_{\mathrm{mech}}=\frac{Ed^3}{24(1-\sigma^2)}\int\,\mathrm{d}x\,\mathrm{d}y\,\left[(\nabla^{2}\zeta)^2+2(1-\sigma)\left((\zeta_{xy})^2-\zeta_{xx}\zeta_{yy}\right)\right]. \end{equation} \tag{ B.2 }$$

Note that ΔW and U_mech have similar forms, as both involve an integral over the membrane area of the quantity (∇²ζ)². The strain energy contains a second term, however, whose relative importance we characterize now. The second term can in fact be re-written as a line integral around the circumference of the membrane,

$$\begin{equation} \int\,\mathrm{d}x\,\mathrm{d}y\,(\zeta_{xy})^2-\zeta_{xx}\zeta_{yy}=\oint\,\mathrm{d}x\,\zeta_{xy}\zeta_{x}-\oint\,\mathrm{d}y\,\zeta_{yy}\zeta_{x}. \end{equation} \tag{ B.3 }$$

This boundary integral vanishes identically for certain types of shapes or boundary conditions, such as a clamped membrane. For our free disc, this boundary term does not identically vanish, but numerically we can confirm that the boundary contribution is small relative to the total strain energy. To a good approximation then we can write $U_{\mathrm {mech}}\approx \frac {Ed^3}{24(1-\sigma ^2)}\int \,\mathrm {d}x\,\mathrm {d}y\,(\nabla ^{2}\zeta )^2$. This leads to the expression for the thermoelastically limited Q–frequency product given in the main text,

$$\begin{equation} Q_{\mathrm{m},\mathrm{th}}f_{\mathrm{m}}=\frac{45\kappa_{\mathrm{th}}}{{\pi}Ed^{2}T_{\mathrm{bath}}\alpha^{2}}\frac{1-\sigma}{1+\sigma}\left(1+\frac{U_{\mathrm{opt}}}{U_{\mathrm{mech}}}\right). \end{equation} \tag{ B.4 }$$

Our theory of optical trapping of membranes predicts dramatic corrections to the simple model of opto-mechanical interactions given by equation (6), when the optical restoring forces (either static or dynamic) become large compared to the natural ridigity of the membrane. In this scenario of strong optical forces, the mechanical mode shape and thus the opto-mechanical coupling strength g become functions of intensity as well, with g generally decreasing with larger intensity. The origin of this effect is intuitively seen by considering a membrane that interacts with a Gaussian cavity mode whose beam waist w is smaller than the membrane radius a. Then, if the optical restoring forces are large compared to the membrane stiffness, the optical beam in fact resembles a new boundary condition that 'pins' the region $r\lesssim w$ of the membrane into place. This reduces the overlap between the mechanical displacement field and the optical beam, and thus g.

This effect is illustrated in figure C.1 for a free SiN membrane of thickness d = 30 nm and a = 25 μm, interacting with a beam of waist w = 15 μm. In this calculation the membrane is statically trapped, although a similar effect would occur for sufficiently large dynamical backaction forces as well. In figure C.1(a), we calculate the opto-mechanical coupling strength g (to another cavity mode that has the same beam waist but exhibits an intensity gradient at the membrane position) as a function of the CM frequency, which is varied through the intensity of the trapping beam. The value of g is calculated using the expression [4]

$$\begin{equation} g\propto\,z_{\mathrm{zp}}\int\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{e}^{-2(x^2+y^2)/w^2}\zeta(x,y)/\textrm{max}|\zeta|. \end{equation} \tag{ C.1 }$$

In figure C.1(a), we have normalized the obtained value of g with the value g₀ if the motion were purely CM, where the displacement field ζ is constant. At larger frequencies, g dramatically decreases, reflecting the 'pinning' effect that the optical force has on the center of the membrane. This is also directly seen in figure C.1(b), where we plot the displacement field ζ(x,y) for a CM frequency ω_m/2π = 300 kHz.

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In this section, we show that the membrane trapped inside a Fabry–Perot cavity analyzed in section 2 can be cooled to the ground state starting from room temperature under realistic conditions.

To recall, the apodized membrane with radius r = 9 μm is positioned inside a cavity with nominal beam waist w₀ ≈ 15 μm, and we assume that the CM motion is trapped with a frequency of ω_m/(2π) = 0.5 MHz (see table D.1 for a summary of parameters). The corresponding peak circulating intensity of the trapping field is I_max ≈ 0.04 W μm⁻². Such a configuration yields a scattering-limited cavity finesse of $\mathcal {F}_{\mathrm {mem}}\sim 3 \times 10^5$. For our following analysis, we take a more conservative value of $\mathcal {F}=10^5$ for the overall cavity finesse (e.g. the finesse is limited by the cavity mirror transmission). The corresponding cavity linewidth κ/(2π) = 75 kHz is much smaller than ω_m, ensuring that the 'sideband-resolved regime' is reached and efficient optical cooling can take place [27].

The membrane is trapped at an anti-node of the 'primary' mode used for trapping; however, as in [17, 18], a secondary optical cavity mode with a maximum intensity gradient at the membrane position can be used to facilitate linear opto-mechanical coupling and cooling. The theory of cooling in this configuration has been thoroughly analyzed in previous work [17, 18, 34], and the results are briefly summarized here. We assume that the circulating intensity in the secondary mode is a small fraction ξ ≪ 1 compared to that in the primary mode, so that the trapping (cooling) effect of the secondary (primary) mode is negligible (we take ξ = 5 × 10⁻⁵ for our analysis). The optical cooling rate effected by the second mode is obtained from equation (9), with a calculated opto-mechanical coupling strength of g/(2π) = 40 Hz for our particular geometry. We choose a detuning of δ ≈ −ω_m for our secondary or cooling beam, which maximizes the cooling efficiency in the sideband-resolved regime.

Using these parameters, we find that optical cooling can yield a steady-state phonon occupation number and temperature of 〈n_f〉 = 2.5 × 10⁻³ and T_f = 4 μK, respectively, starting from room temperature. Thus, the ground state of the CM motion can be prepared with extremely high fidelity. So far in our analysis, we have assumed that re-heating of the CM motion occurs due to thermoelastic processes and photon recoil. However, it is known that damping and re-thermalization due to collisions with background gas molecules can be a major decoherence effect for levitated systems [17, 18]. In the low-pressure regime, where gas molecules independently collide with the membrane, the energy damping rate for our system can be derived using the techniques of [35] and is given by $\gamma _{\mathrm {g}}=96P_{\mathrm {g}}/(\pi \bar {v}\rho d_0)$, where P_g is the gas pressure, $\bar {v}$ is the mean molecular speed, ρ is the mass density of the membrane and d₀ is its maximum thickness. For our system, the effect of the background gas is negligible compared to other heating sources at pressure levels of P_g ∼ 10⁻¹⁰ torr.