Introduction

Semiconductor nanowires are key elements in not only nanophotonics1 and nanoelectronics2 but also in nanomechanics owing to their rich mechanical properties (e.g. small-effective mass and high aspect ratio)3. In particular, a III–V compound semiconductor nanowire is a strong candidate for a hybrid quantum interface among photons, excitons, and phonons because of its highly adjustable inner-layered heterostructure and strain-mediated optomechanical coupling. For example, epitaxial engineering allows us to embed semiconductor low-dimensional structures, such as quantum wells and quantum dots, into a III–V semiconductor nanowire, whose resonant energy can be widely tuned from visible to telecom wavelengths via the bottom-up growth process4,5. The deformation potential and piezoelectric effect in the III–V compound semiconductor enable us to couple these quantum structures to the nanowire’s mechanical motion, namely phonons, via mechanically induced dynamic strain6,7,8. If the mechanical motion of such a quantum-structure-embedded nanowire can be resolved at the level of a few-phonon (or a single phonon) regime, its functionality would be dramatically improved in hybrid quantum interfaces9,10, However, detecting and controlling such a tiny mechanical motion of a nanowire by electrical or optical means are technically difficult because of the nanowire’s small electrostatic capacity, low optical absorbance, and reflectivity.

One way to probe and control a few phonons in a semiconductor nanowire is to introduce a framework called "cavity optomechanics”11 by coupling a high-Q optical (photon) cavity to a nanowire mechanical (phonon) resonator to induce photon–phonon interactions. However, such cavity coupling is quite challenging because of the typical size difference between the optical cavity and nanowire. Therefore, very few demonstrations of cavity optomechanics in a nanowire system have been reported. The previous demonstrations were carried out for Si-based nanowires inserted into Fabry–Pérot cavities12,13. This scheme requires that a nanowire is removed from a substrate and manipulated. Therefore, it is difficult to apply this scheme to an as-grown high-quality nanowire and construct a cavity-optomechanical nanowire system on a chip. An alternative scheme that allows us to apply cavity optomechanics in an as-grown high-quality III–V compound semiconductor nanowire would lead to rich optomechanical operation in the wide range of optical wavelengths as well as to hybrid quantum optoelectromechanics through wavelength-tunable quantum structures.

Here, we demonstrate near-field optomechanical coupling of a movable silica whispering-gallery mode (WGM) microsphere cavity (Q = 1.8 × 105) and an as-grown InP/InAs free-standing nanowire through an evanescent gradient force field (Fig. 1a). This near-field approach allows us to achieve a free access with an optical cavity to a subwavelength-scale mechanical element based on the use of telecom fiber optics14,15,16,17,18,19. It enables us to finely probe the thermal motion of a single nanowire with sub-picometer sensitivity by balanced homodyne interferometry (Fig. 1b) under the unresolved sideband regime, where the optical linewidth (κ) is larger than the mechanical mode frequency (Ω)11. Moreover, the optical force in the evanescent field allows us to tune the frequency, linewidth, Duffing nonlinearity, and vibration axis of the two orthogonalized mechanical modes (Fig. 1c). Our results show that, by further optimizing the fiber-cavity positioning apparatus, quantum-limited displacement detection is available in this nanowire system. Although this nanowire includes multiple quantum wells whose resonant energy is outside the telecom wavelength, resonant quantum structures with telecom wavelength can be prepared quite straightforwardly in the InP/InAs nanowire system by tuning the inner layer thickness via epitaxial engineering. Such a quantum-structure-embedded nanowire and its cryogenic measurement would allow us to resonantly couple the cavity photons with excitonic degrees of freedom, which will open the way to the development of novel hybrid quantum nanomechanical systems. In the following, we demonstrate high-sensitive displacement detection and optical control of mechanical motion in a single nanowire using this near-field cavity optomechanical coupling.

Fig. 1: Conceptual illustration of near-field cavity optomechanics with a free-standing nanowire and a whispering-gallery mode microsphere.
figure 1

a Near-field cavity optomechanical coupling between the nanowire and microsphere via an optical evanescent field. Ω and κ denote the mechanical frequency and optical linewidth, respectively. b Optical probe of mechanical displacement via cavity phase modulation with balanced homodyne interferometry. c Optical control of mechanical response with frequency shifts, linewidth broadening, and eigen-mode rotation via static optical force.

Results

High-sensitivity displacement measurement

The near-field cavity optomechanical system was formed with an epitaxially grown free-standing nanowire and a movable WGM microsphere. Nanowires containing InP/InAs heterostructures in the vertical (growth) direction were epitaxially grown on an InP substrate by metal organic vapor phase epitaxy (see “Methods”). The length and diameter of the nanowires are about 14 μm and 500 nm, respectively, which was determined from a scanning electron micrograph (Fig. 2a). The heterostructure was designed to form a very straight nanowire with the band-gap wavelength remaining below the laser wavelength to avoid direct optical absorption from the 1550-nm probe light. A WGM microsphere with a diameter of 40 μm was fabricated from a silica optical fiber by the standard discharge technique and contacted to a tapered optical fiber, which enables fiber access to the WGM using a telecom-wavelength light. The fiber-coupled microsphere was attached to a plate with glue and placed in a vacuum chamber (~10−5 Pa) together with the nanowire on the InP substrate. The linewidth of the WGM is κ/2π = 1.1 GHz, which corresponds to the quality factor Q = 1.8 × 105 (Fig. 2b). The nanowire-containing InP substrate was put on a three-axis piezo-positioner, which enables an adjustment of the relative gap between the nanowires and microsphere along the x-axis, where we define the x-, y-, and z-axis such that the x-axis corresponds to the radial direction of the WGM as shown in Fig. 2c. Using the piezo-positioner, the microsphere was evanescently coupled to a single nanowire, which was located at the edge of the cleaved surface with the normal axis corresponding to [110].

Fig. 2: Device and setup for near-field cavity optomechanics.
figure 2

a Scanning microscope image of as-grown InP/InAs heterostructured nanowires and a schematic of the internal structure of a nanowire. The crystal orientation for the growth direction [111] and direction of cleaved surface [110] are represented by yellow vectors. The inset shows a cross-sectional image of a nanowire. b Transmission spectrum of the optical cavity with a microscope image of it (inset). c Illustration of the configuration of the microsphere and nanowire with coordinates defined such that the radial direction of the whispering-gallery mode corresponds to the x-axis. d Schematic of optical measurement setup with a fiber-based balanced homodyne interferometer. ECDL external cavity diode laser, EOM electro-optic modulator, BS beam splitter, FL fiber loop.

A balanced homodyne interferometer was constructed to directly readout cavity phase modulation for high-sensitivity displacement measurement (see Fig. 2d). Light from an external diode laser was divided into a probe and a local oscillator. The probe light was coupled to the microsphere via an optical tapered fiber inside the vacuum chamber. After it had passed through the optical cavity, the probe interfered with the local oscillator with a fixed power of 3 mW. The optomechanical signals were detected by a balanced photodetector. The radiofrequency part of the signal, i.e. signal in mechanical motion, was monitored with a spectrum analyzer, and the dc part of the signal was sent to a servo-controller to stabilize the optical phase of the local oscillator via a piezo-loaded mirror. The optical phase of the local oscillator was set on in-phase, i.e., only dispersive optomechanical readout was available. To achieve maximum efficiency, the on-resonance probe light was used in the following experiment.

Figure 3a shows the displacement noise power spectrum (at xy ≈ 500 nm, 0 nm; the definition of the position is given later) measured with an optical probe power of 3 μW. This spectrum was obtained by calibrating the optical phase noise power spectrum with the application of an optical calibration tone from an electro-optic modulator20 and by taking into account the amount of the thermal motion and the nanowire’s effective mass of 2.0 fg. It displays the thermal motion of the two orthogonalized mechanical modes of the nanowire at 1.278 and 1.284 MHz (hereafter referred to as mode 1 and mode 2, respectively). The linewidths of the two modes are 283 and 222 Hz, respectively, which leads to the corresponding mechanical quality factors of 4.5 × 103 and 5.8 × 103. These two mode frequencies show good agreement with those of fundamental flexile modes numerically calculated by the finite element method. This reveals that the two spectra correspond to the nearly degenerate orthogonalized modes of the standing nanowire, where the degeneracy was lifted due to the slight geometric asymmetry. Note that the displacement of the nanowire was measured via linear optomechanical coupling, where the output optical phase quadrature is proportional to the mechanical displacement in the bad cavity regime. The resultant linear optomechanical coupling constants, \({g}_{\omega }^{(1)}\equiv \frac{\partial {\omega }_{{\rm{cav}}}}{\partial x}{x}_{{\rm{zpf}}}\), are 2π × 28.5 and 23.9 Hz for mode 1 and mode 2, respectively, where ωcav is the optical resonance frequency, xzpf is zero-point fluctuation, and x is mechanical displacement. The probe power of 3 μW results in a noise floor level (i.e. displacement sensitivity) of 2.4 × 10−12\({\rm{m}}/\sqrt{{\rm{Hz}}}\). The noise floor level was further reduced as the pump power was increased because the net linear optomechanical coupling is enhanced in proportion to the square of the probe power (see Fig. 3b). The noise floor level of 8.2 × 10−13 \({\rm{m}}/\sqrt{{\rm{Hz}}}\) was achieved with the optical power of 15 μW. This noise floor level is equivalent to the level of thermal motion at 2.8 K. This displacement sensitivity implies that thermal motion can be detected at the liquid helium temperature, where rich quantum optical properties appear in heterostructured nanowires. The displacement sensitivity could be further improved to the standard quantum limit by optimizing the taper-sphere coupling (see “Discussion”).

Fig. 3: Characterization of two-mode mechanical spectra and their sensitivity in thermal motion by balanced homodyne interferometry.
figure 3

a Two-mode power spectral density of thermal motion observed by balanced homodyne interferometry with an optical probe power of 3 μW. The insets show the fundamental flexural mode calculated by the finite element method (the color code corresponds to the amount of displacement, and the black vectors depict the vibration directions). b Two-mode power spectral density of thermal motion with different optical probe powers. The lowest noise floor level, —\({S}_{{\rm{floor}}}=8.2\times 1{0}^{-13}\,{\rm{m}}/\sqrt{{\rm{Hz}}}\)—, was achieved with the optical probe power of 15 μW. c Two-mode power spectral density of thermal motion with respect to y coordinate. d Normalized difference between integral of power spectral density for modes η(y) with respect to y coordinate. The black dots show the experimental data and the red solid curve shows the fitted curve with θ0 = 44.9° and an tilted sweep angle ϕ0 = 1.2°. η has completely different profiles with respect to the initial angle θ0 between the x-axis and the vibrating direction of a mechanical mode in the nanowire (curves with θ0 = 0°, 30°, 60°, and 90° are shown with ϕ0 = 1.2°).

The strength of optomechanical coupling depends on the vibration axis, and is thus, in general, different between the two mechanical modes. In particular, the WGM optical field has a strongly anisotropic optical gradient where the radial direction of the microsphere (x direction defined in our experimental coordinates) has a stronger gradient than the other directions (polar and azimuthal directions). Thus, the vibrating mode along the x direction is better resolved than the orthogonal modes. On the other hand, the two-mode mechanical spectra in our experiment were almost equivalently observed via the displacement measurement as shown in Fig. 3a. This indicates that the initial configuration between the microsphere and nanowire has an almost  ±45° angle between the radial direction of the WGM and the two orthogonalized vibration directions. To quantitatively evaluate this initial angle θ0, we measured two-mode thermomechanical spectra by sweeping the y position (see Fig. 3c). Here, we define y =  0, where the difference among the areas of power spectral density becomes zero. Being symmetric with respect to y = 0, the mechanical spectrum of mode 1 (2) dominantly appeared at the positive (negative) y positions. The initial angle θ0 can be estimated by fitting the difference between the areas of power spectral density η(y) ≡ P1(y) − P2(y), where Pi(y) is the integrated power in the power spectral density in mode i. By fitting the experimental values with the theoretical ones, θ0 is estimated to be 44.9° (see Fig. 3d), where θ0 is extracted with the additional fitting parameter, ϕ0, which corresponds to the angle between y-axis and practical sweep direction. Note that such an initial configuration with θ ≈ 45° originates in the fact that the optical microsphere was accessed at the edge of the cleaved surface with the normal axis corresponding to [110].

Control of mechanical responses via static optical forces

Since both orthogonalized mechanical modes are coupled to the optical cavity mode, the optical force influences their frequency, linewidth, and the vibration axis. This can be understood from the following equations of motion:

$${\ddot{X}}_{1}+{\Gamma }_{1}{\dot{X}}_{1}+{\Omega }_{1}^{2}{X}_{1}={F}_{1}^{{\rm{th}}}+{F}_{1}^{{\rm{opt}}}({X}_{1},{X}_{2}),$$
(1)
$${\ddot{X}}_{2}+{\Gamma }_{2}{\dot{X}}_{2}+{\Omega }_{2}^{2}{X}_{2}={F}_{2}^{{\rm{th}}}+{F}_{2}^{{\rm{opt}}}({X}_{1},{X}_{2}),$$
(2)

where Xi, Γi, and Ωi are the displacement, damping factor, and angular frequency of ith modes (we define Ω2 ≥ Ω1). The two modes are driven by the thermal Langevin force \({F}_{i}^{{\rm{th}}}\) under the optical force field \({F}_{i}^{{\rm{opt}}}({X}_{1},{X}_{2})\) for the vibration direction of mode i. In the case of thermal motion with small mechanical displacements, we can approximate the optical force field as

$${F}_{i}^{{\rm{opt}}}({X}_{1},{X}_{2})\approx {F}_{i}^{0}+{g}_{i1}(x){X}_{1}+{g}_{i2}(x){X}_{2},$$
(3)

where \({F}_{i}^{0}\) is a constant force to change the equilibrium points of vibration (and thus ignored in the following discussion), and \({g}_{ij}(x)\equiv {\partial }_{{X}_{j}}{F}_{i}^{{\rm{opt}}}\) is the force gradient with a derivative along the Xj direction. In this model, the optical force is assumed to be conservative. Note that the optical spring effect, which originates from the linear optomechanical coupling and often appears in a cavity optomechanical system by detuning the cavity11, does not appear under the current homodyne measurement with an on-resonance probe. Thus, the quadratic optomechanical coupling is the dominant source of this optical force21. The strength of this quadratic (dispersive) optomechanical coupling exponentially changes with the gap, i.e., gij(x) is an exponent function for x. Therefore, this leads to an exponential frequency shift of the mechanical modes with the gap. Here, note that gij also leads to the rotation of the vibration axis of the two mechanical modes when the off-diagonal component is a non-zero value (see “Methods”)22. When we include this rotation effect, we can theoretically estimate the x dependence of the two-mode spectra in the frequency domain (Fig. 4a). The theoretical results show an exponential shift of the two mode frequencies for x, which is in good agreement with the experimentally measured shift (Fig. 4b). The only difference between the experiments and theory appears in the linewidth. Experimentally, the linewidth broadens as the gap is reduced. This linewidth broadening is caused by the dissipative optomechanical coupling23, which is not included in our theoretical model [Eq. (3)].

Fig. 4: Two-mode mechanical responses with static optical forces.
figure 4

a Theoretically predicted two-mode power spectral density (PSD) with optical force field in terms of gap x. The black dashed lines show the two-mode frequencies without rotation of eigen-modes. b Experimentally observed two-mode power spectral density with respect to x coordinate. c, d Mechanical frequencies and linewidths with respect to x coordinate. The red and blue circles correspond to them in mode 1 (lower-frequency mode) and mode 2 (higher-frequency mode), respectively.

Now, one can find that the amount of frequency shift and the linewidth are not equivalent between the two mechanical modes (Fig. 4c, d). This difference is due to the rotation of the vibration axis by the optical force. This can be seen in the theoretical results by comparing the case including the rotation effect (color plots in Fig. 4a) and the case excluding it (solid curve in Fig. 4a). The optically induced rotation of the vibration axis pushes the frequency shift in mode 1, whereas it suppresses the frequency shift in mode 2. The angle between the vibration axis and the x-axis, θ, can be quantitatively evaluated by taking into account the ratio of the frequency and the area of the displacement noise power spectrum between the two modes24:

$$\theta \equiv {\tan }^{-1}\left(\frac{\Delta {\Omega }_{1}}{\Delta {\Omega }_{2}}\sqrt{\frac{{P}_{1}}{{P}_{2}}}\right),$$
(4)

where ΔΩi is the frequency shift in mode i. Figure 5a shows the gap (x) dependence of the angle θ at y = 0, which was extracted from Eq. (4). The experimental values (open circles) show good agreement with the theoretical values (solid curve, see Eq. (9)), indicating rotation to θ = 18.9° at x = 0.

Fig. 5: Eigen-mode rotation angle and optomechanical coupling constants.
figure 5

a Angle between x-axis and a vibrating mode in the nanowire. The purple circles are the experimentally observed rotation angles, and the black solid line shows data fitted with a theoretical model. b Linear dispersive optomechanical coupling constants, \({g}_{\omega }^{(1)}\), estimated from the net phase modulation. c Quadratic dispersive optomechanical coupling constants, \({g}_{\omega }^{(2)}\), estimated from the frequency shifts. d Quadratic dissipative optomechanical coupling constants, \({g}_{\kappa }^{(2)}\), estimated from the linewidth broadening. The red (blue) circles corresponds to that of mode 1 (mode 2). The insets are conceptual illustrations of modulation in mechanical responses via each optomechanical coupling (corresponding to the illustration in Fig. 1b, c).

This rotation leads to the difference in the optomechanical coupling coefficients between the two modes, in which the coefficient of mode 1 is increased at the price of mode 2, especially when the gap is small. The linear optomechanical coupling constant \({g}_{\omega }^{(1)}\), which contributes to the displacement sensitivity, was estimated by taking into account the linewidth broadening (see “Methods”). Apparently, \({g}_{\omega }^{(1)}\) in mode 1 gained several hundred hertz as the gap decreased due to the rotation, whereas that in mode 2 converged (see Fig. 5b). Because this rotation aligns the vibration axis in mode 1 so that it becomes parallel to the x-axis, we can achieve optimized displacement sensitivity in mode 1 regardless of the initial angle. Such mode-selective coupling due to the rotation also appears in the quadratic optomechanical coupling coefficients. The quadratic optomechanical coupling coefficients are classified into a dispersive part, \({g}_{\omega }^{(2)}\equiv \frac{1}{2}\frac{{\partial }^{2}{\omega }_{{\rm{cav}}}}{\partial {x}^{2}}{x}_{{\rm{zpf}}}^{2}\), and a dissipative part, \({g}_{\kappa }^{(2)}\equiv \frac{1}{2}\frac{{\partial }^{2}\kappa }{\partial {x}^{2}}{x}_{{\rm{zpf}}}^{2}\). The dispersive (dissipative) part is estimated from the frequency shift (linewidth broadening) of the two modes (see “Methods”). The value of coefficients in mode 1 in the small gap region becomes about ten times larger than that in mode 2. Such mode-selective control of both the mechanical frequency and linewidth could be extended to non-reciprocal operation25 and quantum nondemolition measurement14,26,27 via the quadratic optomechanical coupling.

In addition to the linear mechanical responses (i.e. frequency, linewidth, and vibration axes), the Duffing mechanical nonlinearity can be controlled by using the cavity-induced optical force. The Duffing nonlinearity often appears in strongly driven mechanical resonators as a result of the hardening or softening effect28. Such nonlinearity is externally uncontrollable because it is generally caused by the structural or material properties of resonators. The fourth-order optomechanical coupling, whose coefficient is given by \({g}_{\omega }^{(4)}\equiv \frac{1}{4!}\frac{{\partial }^{4}{\omega }_{{\rm{cav}}}}{\partial {x}^{4}}{x}_{{\rm{zpf}}}^{4}\), with the on-resonance probe allows us to induce and control such Duffing nonlinearity in a free-standing structure like the present nanowire (see Fig. 6a). We performed the driven measurement with a lock-in detector under a strong excitation of the mechanical motion of the nanowire achieved by setting a piezoelectric sheet below the InP substrate (see “Methods”). Figure 6b, c shows the mechanical response measured at two different gaps, x ≈ 100 and 500 nm, respectively. The symmetric mechanical spectra in the large gap x ≈ 500 nm reflect that the free-standing structure of nanowire does not show the Duffing nonlinearity (see Fig. 6b). On the other hand, the asymmetric mechanical spectra in the small gap x ≈ 100 nm with the same driving force indicate that the near-field optical force induces the Duffing nonlinearity via the fourth-order optomechanical coupling (see Fig. 6c). Note that the fourth-order optomechanical coupling coefficient under the on-resonance probe can give rise to only the softening effect (\({g}_{\omega }^{(4)}\,> \, 0\)). This positive sign is reasonable because \({g}_{\omega }^{(2)}\,> \, 0\) as the same even-order coefficient due to the exponential profile of the optical fields. Introducing an additional control light and setting its detuning to a finite value might make it possible to tunably adjust the softening and hardening effects like the optical spring effects can be11. Such an optically tunable Duffing nonlinearity could be used for sensing applications of nanomechanical resonators29,30.

Fig. 6: Optically induced Duffing nonlinearity.
figure 6

a Conceptual illustration of optically induced Duffing nonlinearity via the static fourth-order optomechanical coupling. The purple spectral shape shows the typical Duffing response in contrast to the green Lorentz response. Two-mode mechanical amplitude achieved in the driven lock-in measurement at x ≈ 100 nm (blue) (b) and x ≈ 500 nm (red) (c).

Besides the Duffing nonlinearity, the near-field optomechanical coupling might induce richer mechanical nonlinearities in terms of the even-order mechanical nonlinearity and nonlinear mechanical damping31,32. The former can modulate the equilibrium point of mechanical motion in higher-odd-order optomechanical coupling (i.e. coefficient of \({g}_{\omega }^{(2n-1)}\) with n ≥ 2), and the latter can enrich the dynamics of energy relaxation in higher-order dissipative optomechanical coupling (i.e. coefficient of \({g}_{\kappa }^{(n)}\) with n ≥ 3). Note that introducing a control laser with finite detuning might make it possible to tune the sign of these nonlinear effects. Investigating the nonlinear mechanical properties in near-field cavity optomechanics might open the way to further functionalize the manipulation of nanowire mechanical systems.

Discussion

The optomechanical performance achieved in our experiment could be further improved by optimizing the optical taper-cavity coupling with additional positioners. This optimization would increase the loaded optical Q factor, which is typically larger than 107 in a silica microsphere of the current size. We estimate that improving the optical Q factor to 1.0 × 107 (i.e. κ/2π = 20 MHz) could lead to displacement sensitivity of 2.9 ×  10−15 \({\rm{m}}/\sqrt{{\rm{Hz}}}\), which corresponds to the standard quantum limit in this nanowire mechanical resonator, if a similar level of optomechanical coupling, g0/2π = 100 Hz, is obtained with the probe power of 16 nW. This implies that the Heisenberg-limit displacement measurement in the back-action-dominant regime would be available in this nanowire at room temperature17. Moreover, introducing the ultrahigh-Q microsphere (Q > 108)33 would allow us to bring the system into the resolved-sideband regime (Ω > κ), which would lead to rich optomechanics (e.g. sideband cooling and heating) in semiconductor nanowires.

In principle, this near-field optomechanical approach does not limit the nanowire structure. Therefore, it can be extended to various types of nanowires with different sizes and inner structures, including arrayed nanowires34,35. The extension to epitaxially engineered nanowires with an optical or electronic functionality would also be available. This would thus allow us to construct novel quantum hybrid cavity optomechanical systems based on, for example, nanowire field-effect transistors and quantum dots36,37,38,39, which could lead to highly efficient electro-optic transduction and highly sensitive charge detection through optomechanical coupling40. The near-field cavity optomechanical coupling could also be demonstrated at liquid helium temperatures by carefully designing the cryo-measurement setup. Such a cryogenic operation would open the way to a novel hybrid quantum optoelectromechanical system, in which mechanical displacement (i.e. phonons) is resolved at a level close to the zero-point motion and a small number of electrons confined in the nanowire couple to a small number of photons via the cavity-induced optomechanical coupling. Moreover, coupling a cavity to a quantum-dot-embedded free-standing nanowire would lead to novel nanomechancial systems that show rich physics related to exciton–polariton dynamics41,42.

Methods

Fabrication of InP/InAs heterostructured nanowire

We prepared InP/InAs nanowires in a metal organic vapor phase epitaxy system5. We used bottom-up vapor–liquid–solid (VLS) growth for nanowire synthesis. Indium particles were used to catalyze nanowire growth in the self-catalyzed VLS mode. They were formed on a heated InP (111)B substrate (360 C) by introducing trimethylindium (TMIn) source material for 5 min at a flow rate of 3.03 μmol/min43. We then reduced the temperature to 330 C and started nanowire growth by introducing TMIn and tertbutylarsenic (TBA) or tertbutylphosphine (TBP) simultaneously. The nanowire growth initiated from indium particles distributed on the InP substrate. Owing to the epitaxial relationship between the nanowires and substrates, the 〈111〉-oriented nanowires are vertically aligned on it (Fig. 2a). By alternating source As and P materials, we added multiple InAs layers to InP nanowires. Each nanowire shown in Fig. 2a contains 300 InP/InAs units. For the InP segment growth, the flow rates of TMIn and TBP were 3.03 and 803.6 μmol/min, respectively. For the InAs segment growth, the flow rates of TMIn and TBA were 3.03 and, 26.8 μmol/min, respectively. The growth times for InP and InAs segments in each InP/InAs unit were 20 and 2 s, respectively. The thicknesses of the segments were 35 ± 3 and 7.5 ± 2 nm, respectively.

Estimation of initial angle θ 0

To quantitatively estimate the initial angle θ0 between the vibration direction and x direction (the radial direction of the WGM), we measured the two-mode mechanical spectra by sweeping the y position and calculated the difference in the integral of the power spectral density between the two modes for each y, η(y). Note that the x coordinate, which is defined so that x = 0 on the sphere surface, was far enough away (x ≈ 500 nm) to avoid angle rotation due to the static optical force. By approximating that the two mechanical modes have the same effective masses and the same thermal occupation, the integrated power spectral density of each mode Pi only reflects the linear optomechanical transduction for each y. Since the linear optomechanical transduction is proportional to the derivative of the optical near-field potential along the vibration direction, η(y) is modeled by

$$\eta (y)\equiv \,\,{P}_{1}(y)-{P}_{2}(y)\\ = \, {\int}_{{\rm{mode1}}}{\rm{d}}\omega {S}_{VV}(\omega ,y)-{\int}_{{\rm{mode2}}}\,{\rm{d}}\omega {S}_{VV}(\omega ,y)\\ \approx \, \,\,B({G}_{1}(y)-{G}_{2}(y)),$$
(5)

where Gi(y) ≡ ∂Uopt(xy)/∂Xi with Xi the vibration coordinate in mode i, and B an arbitrary constant. The vibration coordinate can be transformed to \({X}_{1}=x\cos {\theta }_{0}+y\sin {\theta }_{0}\) and \({X}_{2}=-x\sin {\theta }_{0}+y\cos {\theta }_{0}\) with the laboratory coordinate (xy) with the initial angle θ0. The optical near-field potential in the WGM of microsphere is given by

$${U}_{{\rm{opt}}} = \,A| {\psi }_{{\rm{WGM}}}^{2}(x,y){| }^{2}\\ = \,{\left|{j}_{l}(k\sqrt{{(x+R)}^{2}+{y}^{2}}){P}_{m}^{l}\left(\frac{y}{\sqrt{{(x+R)}^{2}+{y}^{2}}}\right)\right|}^{2},$$
(6)

where ψWGM(xy) is the optical near-field of WGMs given by the product of a spherical Bessel function jl and associated Legendre function \({P}_{m}^{l}\) with the integers of l and m, R is the microsphere radius, and k = 2π/λ is the wavenumber of light with the resonance wavelength λ. Here, we fixed l = m = 220 and λ = 1567 nm from the boundary condition of the WGM cavity44. By calculating the derivative of Eq. (6) along xi and substituting it into Eq. (5), we achieved the theoretical model with free parameters θ0, x, and y. To fit this model into the experimental results, we assumed that \(x\to {x}_{0}+\tan {\phi }_{0}y\) with x0 fixed at 500 nm and that ϕ0 is a tilted sweep angle to take into account the imperfect configuration. As a result, we achieved θ0 = 44.9° and 1.2° from the experimental data by fitting with the theoretical model.

Two-mode mechanical spectra with static optical forces

From Eqs. (1) and (2), we analytically derive the two-mode mechanical spectra with Fourier transformation:

$$\left(\begin{array}{ll}{\chi }_{1}(\omega )-{g}_{11}(x)&-{g}_{12}(x)\\ -{g}_{21}(x)&{\chi }_{2}(\omega )-{g}_{22}(x)\end{array}\right)\left(\begin{array}{l}{X}_{1}\\ {X}_{2}\end{array}\right)=\left(\begin{array}{l}{F}_{1}^{{\rm{th}}}\\ {F}_{2}^{{\rm{th}}}\end{array}\right),$$
(7)

where \({\chi }_{i}(\omega )={\Omega }_{i}^{2}-{\omega }^{2}+i\omega {\Gamma }_{i}\). Here, we assume that the static optical force makes an angle of  ±45° in the vibration directions. This assumption of the initial angle reduces to gij(x) = fx(x)/2, where fx(x) is the force gradient along the x direction in the laboratory frame (shown in Fig. 2c). Note that we approximate that fxxFxyFx, xFy, yFy  ≈ 0 by taking into account the WGMs. The matrix is diagonalized as diag[χ+(ω), χ(ω)] with

$${\chi }_{\pm }(\omega )= \, \frac{1}{2}\left[{\chi }_{1}(\omega )+{\chi }_{2}(\omega )-{f}_{x}(x)\right.\\ \,\pm \left.\sqrt{{\left({\chi }_{1}(\omega )-{\chi }_{2}(\omega )\right)}^{2}+{f}_{x}{(x)}^{2}}\right].$$
(8)

Here, we assume that the initial angle between the vibration direction and force direction is 45° and regard the optical force gradient as an exponential function \(A\exp \left(-x/{x}_{{\rm{ev}}}\right)\), \({x}_{{\rm{ev}}}\approx \lambda /(4\pi {\sqrt{{n}^{2}_{{{\rm{SiO}}}_{2}}}}-1)\) with the optical wavelength λ and refractive index of optical cavity \({n}_{{{\rm{SiO}}}_{2}}=1.44\). The constant A = 7 × 10−11 was chosen so that the amount of frequency shift almost corresponded to the experimental data.

The two-mode mechanical spectra were experimentally obtained via the linear optomechanical coupling, whose strength also depends on the vibration direction. The linear optomechanical coupling constants, which are proportional to the optical force, are given by \({F}_{x}\cos \theta\) and \({F}_{x}\sin \theta\) where \({F}_{x}\approx \int {\rm{d}}x{f}_{x}=A{x}_{{\rm{ev}}}\exp (-x/{x}_{{\rm{ev}}})\) and θ is the angle between the vibration direction and the dominant optical force in the radial direction of the WGM cavity expressed by

$$\theta (x)=\frac{\pi }{4}-\frac{{\Omega }_{2}^{2}-{\Omega }_{1}^{2}+\sqrt{{f}_{x}{(x)}^{2}+{({\Omega }_{2}^{2}-{\Omega }_{1}^{2})}^{2}}}{{f}_{x}(x)}.$$
(9)

Note that the contribution of mechanical dissipation is ignored because the Q of the mechanical resonance is high enough with \({\Omega }_{i}{\Gamma }_{i}\ll {\Omega }_{i}^{2}\). The sum of the squares of amplitude \(| {\chi }_{+}^{-1}(\omega ){F}_{x}\cos \theta | +| {\chi }_{-}^{-1}(\omega ){F}_{x}\sin \theta |\) are shown in Fig. 4a with an appropriate noise floor with the signal-to-noise ratio of 102, which almost corresponds to the experimental one. Note that the non-rotating case described by the black dotted lines in Fig. 4a corresponds to the case of g12 = g21 = 0.

Estimation of linear optomechanical coupling constant

The linear optomechanical coupling constant was evaluated by using a calibration method. The dispersive optomechanical coupling constant is given by

$${g}_{\omega }^{(1)}=\sqrt{\frac{\int {\rm{d}}\omega {S}_{VV}^{{\rm{mech}}}(\omega )}{\int {\rm{d}}\omega {S}_{VV}^{{\rm{ref}}}(\omega )}}\frac{{\Omega }_{{\rm{ref}}}\beta }{\int {\rm{d}}\omega {S}_{{{xx}}}(\omega )},$$
(10)

where \({S}_{VV}^{{\rm{mech}}}(\omega )\) and \({S}_{VV}^{{\rm{ref}}}(\omega )\) are experimentally measured power spectral densities with the unit of V2/Hz, and Ωref and β are the reference frequency and phase modulation rate, respectively20. In the case of no damping in a mechanical resonator (pure thermal motion), ∫dωSxx(ω) = 4nM with the phonon number nM. However, if there is an additional damping, ∫dωSxx(ω) should be corrected as 4nMΓ/Γ0, where Γ is measured linewidth due to the additional damping from the quadratic optomechanical coupling and Γ0 is an intrinsic linewidth of mechanical resonator. As a result, we estimated the linear optomechanical coupling constant by

$${g}_{\omega }^{(1)}=\sqrt{\frac{\int {\rm{d}}\omega {S}_{VV}^{{\rm{mech}}}(\omega )}{\int {\rm{d}}\omega {S}_{VV}^{{\rm{ref}}}(\omega )}}\frac{{\Omega }_{{\rm{ref}}}\beta }{4{n}_{M}}\frac{{\Gamma }_{0}}{\Gamma }.$$
(11)

Importantly, the corrected optomechanical coupling with Eq. (11) shows an exponential dependence on the gap distance, which is predicted in theory, whereas the noncorrected coupling does not.

Estimation of static quadratic optomechanical coupling constant

The dispersive and dissipative quadratic optomechanical coupling constants, denoted by \({g}_{\omega }^{(2),i}\) and \({g}_{\kappa }^{(2),i}\), respectively, for the orthogonalized mechanical mode i are defined by

$${g}_{\omega }^{(2),i}=\frac{1}{2}\frac{{\partial }^{2}{\omega }_{{\rm{cav}}}}{\partial {X}_{i}^{2}}{x}_{{\rm{zpf}},i}^{2}=\frac{1}{{n}_{{\rm{cav}}}}{\rm{Re}}\left[\frac{\partial {F}_{{X}_{i}}}{\partial {X}_{i}}\right],$$
(12)
$${g}_{\kappa }^{(2),i}=\frac{1}{2}\frac{{\partial }^{2}\kappa }{\partial {X}_{i}^{2}}{x}_{{\rm{zpf}},i}^{2}=\frac{1}{{n}_{{\rm{cav}}}}{\rm{Im}}\left[\frac{\partial {F}_{{X}_{i}}}{\partial {X}_{i}}\right],$$
(13)

where ω and κ are the cavity frequency and linewidth modified by the mechanical displacement Xi of mode i, and xzpf,i are zero-point fluctuations of mode i. The cavity photon number ncav with zero-detuned probe is given by

$${n}_{{\rm{cav}}}=\frac{4\eta }{\kappa }\frac{{P}_{{\rm{in}}}}{\hslash {\omega }_{0}},$$
(14)

where Pin is the input probe power at the tapered fiber, is Planck’s constant over 2π, ω0 is the cavity resonance frequency, and η ≡ κin/κ with taper-cavity coupling constant κin determines the coupling ratio at which the critical coupling condition corresponds to 1/2. In our experiment, ncav was about 4.0 × 103 with the input optical power of 3.0 μW. Although these expressions are equivalent to the ones in previous work21, we modified them to estimate the coupling constants with the experimental observables by taking into account the intermodal coupling.

Here, we regard the static optical force as a complex force where the real part shows a conservative portion inducing a frequency shift, and its imaginary part shows a non-conservative one inducing linewidth broadening. To estimate the static quadratic optomechanical coupling constant from the frequency shift and linewidth broadening, Eq. (8) is recalculated for an arbitrary rotated angle θ (i.e. \({g}_{11}(x)={f}_{x}(x){\cos }^{2}\theta ,{g}_{12}(x)={g}_{21}(x)={f}_{x}(x)\cos \theta \sin \theta ,{g}_{22}(x)={f}_{x}(x){\sin }^{2}\theta\)) as follows:

$${\chi }_{\pm }(\omega )= \frac{1}{2}\left[{\chi }_{1}(\omega )+{\chi }_{2}(\omega )-{f}_{x}(x)\right.\\ \,\pm \left.\sqrt{{\left({\chi }_{1}(\omega )-{\chi }_{2}(\omega )\right)}^{2}+{f}_{x}{(x)}^{2}-2\left({\chi }_{1}(\omega )-{\chi }_{2}(\omega )\right)\cos 2\theta }\right].$$
(15)

The force gradient is decomposed to real and imaginary parts as fx(x) = fR(x) + ifI(x), where each part can be calculated from Eq. (15) as

$${f}_{R}(x)=-(\Delta {\Omega }_{1}+\Delta {\Omega }_{2}),\,{f}_{I}(x)=-(\Delta {\Gamma }_{1}+\Delta {\Gamma }_{2}),$$
(16)

where

$$\Delta {\Omega }_{1,2}={\rm{Re}}\left[{\chi }_{+,-}(\omega )-{\left.{\chi }_{+,-}(\omega )\right|}_{{f}_{x}\to 0}\right],$$
(17)
$$\Delta {\Gamma }_{1,2}={\rm{Im}}\left[{\chi }_{+,-}(\omega )-{\left.{\chi }_{+,-}(\omega )\right|}_{{f}_{x}\to 0}\right].$$
(18)

By transforming the coordinate to (X1X2) (i.e. \({f}_{{X}_{1}}e\equiv {\partial }_{{X}_{1}}{F}_{{X}_{1}}\approx {f}_{X}{\cos }^{2}\theta ,{f}_{{X}_{2}}\equiv {\partial }_{{X}_{2}}{F}_{{X}_{2}}={f}_{X}{\sin }^{2}\theta\)), we can represent the quadratic optomechanical coupling constant with experimentally measured parameters ΔΩi, ΔΓi, and θ as follows:

$${g}_{\omega }^{(2),1}=-\frac{\Delta {\Omega }_{1}+\Delta {\Omega }_{2}}{{n}_{{\rm{cav}}}}{\cos }^{2}\theta ,\,{g}_{\omega }^{(2),2}=-\frac{\Delta {\Omega }_{1}+\Delta {\Omega }_{2}}{{n}_{{\rm{cav}}}}{\sin }^{2}\theta ,$$
(19)
$${g}_{\kappa }^{(2),1}=-\frac{\Delta {\Gamma }_{1}+\Delta {\Gamma }_{2}}{{n}_{{\rm{cav}}}}{\cos }^{2}\theta ,\,{g}_{\omega }^{(2),2}=-\frac{\Delta {\Gamma }_{1}+\Delta {\Gamma }_{2}}{{n}_{{\rm{cav}}}}{\sin }^{2}\theta ,$$
(20)

which were estimated with the experimentally observed ΔΩi, ΔΓi, and θ for each x in Fig 5c, d.

Driven lock-in measurement for observing Duffing nonlinearity

To confirm that the Duffing nonlinearity is induced by the near-field optical gradient, we performed the driven lock-in measurement as follows. The nanowire mechanical amplitude was electrically driven with a radiofrequency drive signal via a piezoelectric sheet attached to the substrate. A lock-in amplifier was used to generate the radiofrequency drive signal and measure the mechanical amplitude from the output of the balanced photoreceiver (shown in Fig. 2d). The two-mode mechanical spectra (shown in Fig. 6b, c) were acquired by sweeping the frequency of the drive signal from 1.28 to 1.29 MHz.