Experimental setup for levitation

Our experimental setup is displayed in Fig. 1. A silica nanoparticle (green) of radius R ≈ 90 nm, mass m = 6.4 × 10⁻¹⁸ kg and refractive index n_r = 1.45 is placed in a cavity (purple) by an optical tweezers trap (yellow) with wavelength $$ \lambda $$_t = 2π/k_t = 1064 nm, power P_t ≃ 150 mW, numerical aperture NA = 0.8, and optical axis (z) perpendicular to the cavity axis (y). The trap is linearly polarised along the axis defined as $$ {ϵ}_{\theta }={ϵ}_x\cos \theta $$ (see inset in Fig. 1).

Fig. 1

Experimental setup.

An optical tweezers trap (yellow) levitates a silica nanoparticle inside a high finesse cavity. The trapping field is locked relatively to the cavity resonance ω_cav using Pound-Drever-Hall locking with a detuning Δ = ω_t − ω_cav. A 3D piezo stage positions the particle precisely inside the cavity at variable y₀. The inset displays the linear trap polarisation axis along $$ {ϵ}_{\theta }={ϵ}_x\cos \theta $$. The rate of coherently scattered photons into the cavity mode (purple) depends on y₀, ϵ_θ and Δ. The transmitted cavity output field is monitored on a photodiode (CO) and the forward scattered trapping light is used to detect the particle motion (see “Experimental setup” section).

The nanoparticle’s eigenfrequencies Ω_x,y,z = 2π × (172 kHz, 197 kHz, 56 kHz) are non-degenerate due to tight focusing. The trap is mounted on a nano-positioning stage allowing for precise 3D placement of the particle inside the low loss, high finesse Fabry-Pérot cavity with a cavity linewidth κ ≈ 2π × 10 kHz, cavity finesse F = 5.4 × 10⁵ and free spectral range Δω_FSR = πc/L_c = 2π × 5.4 GHz. The relative detuning Δ = ω_t − ω_c between the trap and the cavity resonance is tunable. The intracavity photon number n_cav is estimated from the transmitted cavity power P_out (CO in Fig. 1), and the particle position displacement is measured by interfering the scattered light with a co-propagating reference beam³⁹. In CS, scattering events from the detuned trapping field, locked at Δ, populate the cavity. This contrasts the approach of actively driving the cavity^3,7,24. A particle in free space, solely interacting with the trapping light, Raman scatters photons into free space and the energy difference between incident and emitted light equals ±ℏΩ_m with m = x, y, z. In this case, photon up and down conversion are equally probable⁴⁰. The presence of an optical cavity alters the density of states of electromagnetic modes and enhances the CS into the cavity modes through the Purcell effect. If trap photons are red (blue) detuned with respect to the cavity resonance, the cavity enhances photon up (down) conversion and net cooling (heating) takes place.

Coherent scattering theory

In order to estimate the corresponding optomechanical coupling strength in CS, we follow ref. ³⁶. The interaction Hamiltonian for a polarisable particle interacting with an electric field E(R) is given by $$ {\widehat{\mathrm{H}}}_{\mathrm{int}}=-\frac{1}{2}\alpha {\mathbf{E}}^2\left(\mathbf{R}\right)$$ with the particle polarizability $$ \alpha =4\pi {ϵ}_0{R}^3\frac{n_{\mathrm{r}}^2-1}{n_{\mathrm{r}}^2+2}$$ and vacuum permittivity ϵ₀. The total electric field consists of the trap ($$ {\mathbf{E}}_{\mathrm{tr}}\left(\mathbf{R}\right)$$), cavity (E_cav(R)) and free space electromagnetic modes (E_free(R)) yielding the interaction Hamiltonian

In the following, we use the simplified interaction Hamiltonian given by Eq. (2) where we separate the parts contributing to the optomechanical coupling due to CS $$ {\widehat{\mathrm{H}}}_{\mathrm{CS}}$$, active driving $$ {\widehat{\mathrm{H}}}_{\mathrm{DR}}$$, and population of the intracavity field $$ {\widehat{\mathrm{H}}}_{\mathrm{CAV}}$$ (see “Interaction Hamiltonian and power spectral densities” section).

For the measurements presented here, the trap is x-polarised with θ = 0 (see inset Fig. 1). This simplifies $$ {\widehat{\mathrm{H}}}_{\mathrm{CS}}$$ to $$ {\widehat{\mathrm{H}}}_{\mathrm{CS}}=-\hslash \left[g_y\left({\widehat{a}}^{†}+\widehat{a}\right)\left({\widehat{b_y}}^{†}+\widehat{b_y}\right)+g_z\left({\widehat{a}}^{†}-\widehat{a}\right)\left({\widehat{b_z}}^{†}+\widehat{b_z}\right)\right]$$, where $$ \widehat{a}$$ ($$ {\widehat{a}}^{†}$$) is the photon annihilation (creation) operator and $$ \widehat{b}$$ ($$ {\widehat{b}}^{†}$$) is the phonon annihilation (creation) operator. The CS optomechanical coupling strengths g_y,z are

The optical cavity resonance frequency shift caused by a particle located at maximum intensity of the intracavity standing wave is $$ G_{\perp }=\alpha E_0\sqrt{\frac{{\omega }_{\mathrm{c}}}{2\hslash {ϵ}_0{V}_{\mathrm{c}}}}$$ with cavity mode volume $$ V_{\mathrm{c}}=\pi {\mathrm{w}}_{\mathrm{c}}^2{L}_{\mathrm{c}}/4$$, cavity waist w_c, cavity length L_c, and ω_c = 2πc/$$ \lambda $$_c The trap electric field is $$ E_0=\sqrt{\frac{4P_{\mathrm{t}}}{\pi {ϵ}_{0}c{\mathrm{w}}_x{\mathrm{w}}_y}}$$ with trap waists w_x and w_y.

Due to the intracavity standing wave, the optomechanical coupling strength has a sinusoidal dependence on y₀ with opposite phase for g_y and g_z. In contrast, g_x = 0 if θ = 0.

For clarity, we limit the discussion to coupling along the cavity axis (y), such that Ω_m = Ω_y and g = g_y. Similar results can be obtained for the other directions x, z with the same level of control.

The maximum expected coupling strength from CS is $$ g_y^{\max }=G_{\perp }{k}_{\mathrm{c}}{y}_{\mathrm{zpf}}=2\pi \times 31.7\mathrm{kHz}$$ for our parameters. However, we displace the particle by δz ≈ 40 μm from the cavity centre for better experimental stability. Hence, our expected optomechanical coupling strength is reduced by ≈30% down to $$ g_y^{\mathrm{th}}=2\pi \times 22.6\mathrm{kHz}$$, enabling the SCR with g_y > κ. Despite the fact that this value is a factor of ≈3 lower than previously reported⁹, the deep SRC with g > κ remains unaccomplished.

Transition to the SCR

In the weak coupling regime g < κ, the Lorentzian-shaped spectra of our mechanical oscillator displays a single peak at its resonance frequency Ω_m. When g increases, the energy exchange rate between optical and mechanical mode grows until the SCR is reached at g > κ/4 (ref. ³³). In the SCR, the optical and mechanical mode hybridise, which gives rise to two new eigenmodes at shifted eigenfrequencies Ω_± (see Eq. (9)). At this point, the energy exchange in between the optical and mechanical mode is faster than the decoherence rate of each individual mode. The hybridised eigenmode frequencies

As can be seen from Eq. (3), we control g_y through various parameters like the trap power P_t, the particle position y₀ and the polarisation angle θ. The optical coupling rate Γ_opt depends additionally on the trap detuning Δ and is maximised at Δ = − Ω_m to $$ {\mathrm{\Gamma }}_{\mathrm{opt}}=4g_y^2/\kappa $$ (refs. ^7[,12). While P_t and Δ only influence the magnitude of the coupling strength, y₀ and θ change also the nature of the coupling from 1D to potentially 3D³⁶. For simplicity, we focus on varying Δ and y₀ in the following measurements and keep P_t, θ and Γ_m = 2π × 0.8 kHz, corresponding to p = 1.4 mBar, fixed (see “Experimental setup” section). The range of Δ is limited due to instabilities in the experiment.

Observation of strong coupling

Figure 2 left panel displays the experimental position power spectral density (PSD) versus Δ for different y₀. Throughout the remaining part of the manuscript, we fit our PSD to Eq. (9), if not stated differently. From this fit, we can extract the hybridised modes Ω_± that are separated by 2g_y. We cover a total distance of δy₀ ≈ $$ \lambda $$_c/4 and change the optomechanical coupling strength, and therefore also the NMS, from (a) g_y/2π = 22.8 kHz, (b) 15.4 kHz, (c) 4.6 kHz and (d) 0 kHz, exploring the entire range from strong coupling to zero coupling. The right panel shows the fit, which is in good agreement with the data. We observe two eigenmodes Ω_± with an exceptional NMS of 2g_y ≈ 4.6κ at Δ = − Ω_m, corresponding to 20% of the bare mechanical eigenfrequency, once the system enters the SCR at g > κ/4 (ref. ³³). For g_y = 0, we observe only the mechanical mode with slightly increased frequency Ω_m = 2π × 200 kHz due to the additional trapping potential supplied by the cavity field (see Fig. 2d). In Fig. 2a and b, we observe an additional NMS in the y-mode, which stems from a second cross polarised optical mode. Note that, throughout all our measurements (see Figs. 2–4), the second NMS is the largest source for discrepancies between experiment and theory (for more details, see Supplementary Information). We also attribute the NMS of the x-mode at Ω/Ω_m = 0.89 to the second optical mode as observed in Fig. 2d, since the x-mode should be decoupled from the first mode (g_x = 0 if θ = 0).

Fig. 2

Normal mode splitting.

Particleʹs position power spectral density PSD(Ω) versus Δ for different y₀ and therefore various g_y. Experimental data are displayed on the left, and theory on the right. The bare mechanical (optical) modes correspond to horizontal (diagonal) lines. a Maximum normal mode splitting of 2g_y is observed at Δ = − Ω_m yielding a value of g_y = 2π × 22.8 kHz = 2.3κ, where y₀ ≈ λ_c/4 is close to the intensity minimum (see Eq. (3)). b When the particle is moved by δy₀ ≈ 0.12λ_c, the coupling reduces to g_y = 2π × 15.4 kHz = 1.5κ. c Normal mode splitting is still visible at δy₀ ≈ 0.2λ_c, yielding g_y = 2π × 4.6 kHz = 0.46κ. d At the intensity maximum, corresponding to a shift of δy₀ ≈ λ_c/4 and g_y = 0 kHz, the normal mode splitting vanishes and we only see a shift of δΩ_m ≈ 2π × 5 kHz in the mechanical frequency due to the increased intracavity photon number (see Supplementary Fig. 1). In general, we observe a good agreement between experimental data and theory. We attribute discrepancies to a second cross polarised cavity mode inducing a second normal mode splitting (for more details, see Supplementary Information).

Fig. 3

Power spectral density versus cavity detuning Δ.

a–c Experiment (purple) and theory (yellow, dashed) fitted to Eq. (9) at Δ = −2π × 293 kHz ≈ −1.5Ω_m (a), Δ = −2π × 225 kHz ≈ −1.2Ω_m (b) and Δ = −2π × 205 kHz ≈ −Ω_m (c). The optomechanical coupling strength g_y grows with increasing Δ. Optical and mechanical modes start to hybridise clearly at Δ≥ −1.5Ω_m. We attribute the discrepancy between data and theory to the second optical mode (see Supplementary Informtation). d Hybridised eigenmodes Ω_± versus Δ at the intensity minimum (y₀ ≈ λ_c/4). Maximum normal mode splitting of 2g_y with g_y = 2π × 22.8 kHz = 2.3κ occurs at Δ = −Ω_m. The black line fits the data to Eq. (4), while the inner (outer) edges of the grey area correspond to a fit using solely to the upper (lower) branch Ω₋ (Ω₊).

Fig. 4

Normal mode splitting versus particle position y₀.

a Experiment and b theory according to Eq. (9). Particle position power spectral density PSD(Ω) at the optimal Δ = − 2π × 193 kHz ≈ −Ω_m along y₀ is shown. The hybridised modes split by 2g_y. The white dashed line displays Ω_±/Ω_m = 1 ± g_y/Ω_m where g_y follows Eq. (3). The mechanical mode at Ω/Ω_m ≈ 0.89 corresponds to the mechanical x−mode. The data and fit show very good agreement. c ∣g_y∣ at Δ ≈ −Ω_m versus y₀. Maximum and minimum coupling are separated by δy₀ = λ_c/4 as expected by Eq. (3). Black dashed line fits to the absolute value of Eq. (3) with a maximum $$ g_y^{\max }\approx -2.3\kappa $$ and the grey shaded area corresponds to 3σ_std of the fit. The dotted lines indicate the positions used in Fig. 2.

Figure 3a–c displays the particle’s position PSD at different Δ while it is located at the intensity minimum, corresponding to the position of maximum coupling g_y = 2.3κ, displayed in Fig. 2a. Our theory (yellow) captures the data (purple) well. In Fig. 3a, the optical mode and mechancial mode begin to hybridise into new eigenmodes at Δ = −1.5Ω_m which is confirmed by a second peak appearing at Ω ≈ 2π × 300 kHz. The hybridisation becomes stronger as Δ approaches the cavity resonance and the NMS is maximised at Δ ≈ −Ω_m as shown in Fig. 3c. The dependence of the new eigenmodes Ω_± on Δ is shown in Fig. 3d, displaying clearly the expected avoided crossing of 2g_y. The solid line is a fit to Eq. (4). The edges of the shaded area represent the upper and lower limit of the fit, which we obtain by fitting only the upper branch (yellow) or the lower branch (purple), respectively.

As already discussed previously, our experiment allows to change the optomechanical coupling by changing various experimental parameters, which stands in contrast to many other experimental platforms. Figure 4 displays this flexibility to reach the SCR by demonstrating the position dependence of g_y at optimal detuning Δ ≈ −Ω_m extracted from the data Fig. 2a–d. The experimental and theoretical position PSDs versus y₀ are depicted in Fig. 4a and b. The mode at Ω/Ω_m ≈ 0.89 corresponds to the decoupled x-mode. The dashed line highlights the theoretical frequency of the eigenmodes Ω_±/Ω_m following Eq. (4). In both experiment and theory, we observe the expected sinusoidal behaviour predicted by Eq. (3). Figure 4c depicts ∣g_y∣ = (Ω₊ − Ω₋)/2 (circles) extracted from Fig. 4a. The dashed line represents the fit to the absolute value of Eq. (3) yielding $$ g_y^{\exp }=2\pi \times \left(22.8\pm 0.2\right)$$kHz which coincides well with the theoretical value of $$ g_y^{\mathrm{th}}=2\pi \times 22.6$$ kHz. The measured period coincides with the expected period of $$ \lambda $$_c/4. The shaded area corresponds to 3σ_std of the fit.

Experimental setup

The experimental setup is displayed in Fig. 5. A silica nanoparticle is loaded at ambient pressure into a long range single beam trap and transferred to a more stable, short range optical tweezers trap⁴² (with wavelength $$ \lambda $$_t = 1064 nm, power P ≃ 150 mW, focusing lens NA = 0.8) inside a vacuum chamber. Due to the tight focusing, the nanoparticle non-degenerate eigenfrequencies are Ω_x,y,z = 2π × (172 kHz, 197 kHz, 56 kHz), respectively. The optical tweezers are mounted on a 3D nanometre resolution piezo system allowing for precise 3D positioning inside a high finesse Fabry-Perot cavity (with cavity finesse F = 540,000, free spectral range FSR = 2π × 5.4 GHz).

Fig. 5

Extended experimental setup.

A 1064 nm Mephisto laser (yellow) traps a silica particle of d = 177 nm inside a high finesse cavity (purple). The trap light is locked at a variable detuning Δ + FSR from the cavity resonance via the Pound-Drever-Hall technique by detecting the error signal on a photodiode (PDH). The particle motion is detected in backreflection (BR) and balanced forward detection (FS). The intracavity field is estimated from the transmitted power detected on a photodiode (CO).

In order to control the detuning Δ = ω_t − ω_c between the cavity resonance ω_c and the trap field ω_t, we use a weak cavity field for locking the cavity via the Pound-Drever-Hall technique (PDH) on the TEM01 mode minimising additional heating effects through the photon recoil heating of the cavity lock field. The PDH errorsignal acts on the internal laser piezo and an external AOM (not shown). We separate lock and trap light in frequency space by one free spectral range (FSR) such that the total detuning between lock and trap yields ω_t = ω − FSR − Δ. The variable EOM modulation FSR + Δ is provided by a signal generator. The intracavity power can be deduced from the transmitted cooling light observed on a photodiode behind the cavity (CO).

All particle information shown is gained in forward balanced detection interfering the scattered light field and the non-interacting part of the trap beam as shown in Fig. 5. The highly divergent trap light is collected using a lens (NA = 0.8). We use three balanced detectors (FS) to monitor the oscillation of the particle in all three degrees of freedom.

The data time traces are acquired at 1 MHz acquisition rate. Each particle position PSD is obtained by averaging over N  = 25 samples of which each one is calculated from individual 40 ms time traces, corresponding to a total measurement time of t = 1 s.

We keep the pressure stable at p = 1.4 mbar. The thermal bath couples as

In the measurements presented, we cool our particle’s centre of mass motion to T = 235, corresponding to a reduction of the phonon occupation by roughly 20%. The theoretically expected heating rate due to the residual gas accounts fully for the experimentally observed heating rate.

Interaction Hamiltonian and power spectral densities

Following ref. ³⁶, the relevant contributions to the CS interaction Hamiltonian for θ = 0 are given by

The mechanical susceptibility χ is given as³³