Dynamical backaction in an ultrahigh-finesse fiber-based microcavity

The use of low-dimensional objects in the field of cavity optomechanics is limited by their low scattering cross section compared to the size of the optical cavity mode. Fiber-based Fabry-P\'{e}rot microcavities can feature tiny mode cross sections and still maintain a high finesse, boosting the light-matter interaction and thus enabling the sensitive detection of the displacement of minute objects. Here we present such an ultrasensitive microcavity setup with the highest finesse reported so far in loaded fiber cavities, $\mathcal{F} = 195\,000$. We are able to position-tune the static optomechanical coupling to a silicon nitride membrane stripe, reaching frequency pull parameters of up to $\mathrm{\lvert G/2\pi\rvert=1}\,\mathrm{GHz\, nm^{-1}}$. We also demonstrate radiation pressure backaction in the regime of an ultrahigh finesse up to $\mathcal{F}=165\,000$.

objects. Reducing the mode volume increases the light-matter coupling [22]. This boosts the frequency pull parameter G = − ∂ωcav ∂z , which translates the displacement of the mechanical resonator to a frequency shift of the cavity. Although the frequency pull parameter is set by the geometry and the optical properties of the system, the cavity finesse F boosts the circulating photon number n circ and therefore the effective coupling g = √ n circ g 0 = √ n circ Gx zpf .
A large finesse also contributes to maintain small cavity linewidths κ = ω fsr F despite the large free spectral range ω fsr of small mode volume cavities. This is necessary to enhance the single photon cooperativity C 0 = 4g 2 0 κΓ m , a key parameter for optomechanical experiments [1]. In addition, the finesse increases the magnitude of phase fluctuations in the output field and, thus, improves the sensitivity of the measurement.
The required cavity specifications can be fulfilled with fiber-based Fabry-Pérot microcavities (FFPCs), which have been pioneered in the cavity quantum electrodynamics community [21,23,24] but have successfully been adapted to optomechanical [25] and ion-trapping systems [26]. Although FFPC-based optomechanical systems are nowadays being studied by several groups [27][28][29], their operation is to date enabled by measures to suppress their extreme sensitivity towards frequency fluctuations. This is conveniently accomplished by constraining the finesse to not exceed values of a few 10 000 and/or by operating at cryogenic temperatures where the Brownian thermal noise of the FFPC mirrors is suppressed.
In this paper, we introduce a platform for cavity optomechanics that is optimized for ultrasensitive optical detection of single-digit nanometer-size mechanical objects. This is achieved with a FFPC that features a small mode cross section while maintaining the highest finesse reported so far in loaded FFPCs, exceeding that of the ground-breaking work in Ref. [25] by almost one order of magnitude. The stable operation of the cavity is enabled by a very rigid cavity gluing scheme combined with a thorough acoustic shielding. This unlocks the previously unaccessible realm of precision sensing at room temperature using ultrahigh-finesse FFPC systems.
In the following, we present a proof-of-principle demonstration of the FFPC platform, demonstrating radiation pressure backaction using a free-standing stoichiometric silicon nitride (Si 3 N 4 ) membrane stripe which is inserted into the cavity. We note, however, that the choice of the mechanical resonator in this resonator-in-the-middle scheme is flexible, and conveniently allows for the investigation of, for example, tethered membranes [20,30], nanowires [29] or low-dimensional materials such as carbon nanotubes (CNTs) [ or two-dimensional crystals [34]. In particular, CNTs are discussed as one possible path towards quantum optomechanics at room temperature [32].

II. METHODS
The measurements presented in this work are performed on a stoichiometric high-stress Si 3 N 4 membrane (Norcada) etched in wide stripes of dimensions 30 µm × 500 µm × 30 nm.
The sample frame is mechanically cleaved to obtain a U shape which allows the membrane stripes to be inserted into the fiber cavity. Figure 1(a) shows a micrograph of the sample. The mechanical mode of interest is the second harmonic flexural eigenmode (n = 2) with a mechanical resonance frequency, linewidth and mechanical quality factor in vacuum (10 −6 mbar) of Ω 0 /2π = 932.58 kHz, Γ 0 /2π = 4.7 Hz, and Q = 197 000, respectively. We calculate the effective mass of the sample m eff = 0.54 ng from its geometry, yielding zero point fluctuations of x zpf = 5.8 fm. The sample is mounted on a five-axis nanopositioner tower (Attocube ANP101 series) placed close to the FFPC inside a vacuum chamber that enables optical access. To eliminate any source of vibrations the vacuum is maintained at 10 −6 mbar with an ion pump and the whole chamber is placed inside an acoustically shielded box. A 780-nm home-build external cavity diode laser [36] stabilized on a rubidium cell is used for interferometric readout of the sample position.

III. FIBER-BASED MICROCAVITY
The microcavity consists of two mirrors concavely shaped on fiber end faces by CO 2 laser ablation, with resulting curvatures of 191 µm and 140 µm for the single-mode (SM) input and multi-mode (MM) output fiber, respectively, which yield a near-planar cavity configuration [37]. The mirror spacing is L = 43.  and the 780-nm reflection is measured at PD3 (see Fig. 1(b)).
We characterize the reflection and transmission response of the cavity by fixing the lock laser frequency and scanning the cavity length across a TEM 00 resonance. Phase-modulated sidebands enable the voltage sent to the cavity piezos to be converted into frequency units.
Careful calibration of the gains and losses in the system allows the normalized scattering parameters |S 21 | 2 = P t /P in and |S 11 | 2 = P r /P in to be extracted from the input power P in and the power in reflection P r and transmission P t in front of the cavity ( Fig. 2(b)). We fit the normalized transmission and reflection to extract the total cavity losses κ and the input couplings κ e,SM and κ e,MM of the two mirrors. Due to the imperfect alignment of the mirror surface with respect to the fiber core, light reflected from the cavity is filtered spatially, resulting in an asymmetrical reflection lineshape. This effect is specific to FFPCs and effectively lowers the off-resonant reflection [24]. We take the asymmetry into account by adding an heuristic phase factor V bg exp (iφ) to S 11 , but a detailed description of this phenomenon can be found in Ref. [24]. The final fit formulas read as follows: (1) Note that we integrate losses due to imperfect mode matching (especially significant for fiber-based cavities with off-centered mirrors) into the input coupling to comply with the established notation for open cavities. We measure a cavity length (mirror spacing) of L = 43.8 µm by sweeping the probe tone across multiple free spectral ranges. From the cavity length we obtain a free spectral range of ω fsr /2π = 3.42 THz. The mode profile can be calculated from the mirror properties. The resulting mode volume is as small as V ≈ 277 λ 3 with a mode waist of 5.2 µm. The finesse of the empty cavity is F = ω fsr κ = 204 000, which is almost an oder of magnitude higher than that of Ref. [25].

IV. OPTOMECHANICAL COUPLING AND DYNAMICAL BACKACTION
To map out the dispersive coupling experimentally, that is, the frequency pull parameter G, we scan both piezos symmetrically to change the cavity length while the membrane is moved along the cavity axis [15] (z direction, cf. are extracted from this measurement. The resulting linear coupling ( Fig. 3(b)) can be as large as |G/2π| = 1 GHz nm −1 , corresponding to a single-photon coupling rate of |g 0 /2π| = |G/2π| x zpf = 5.8 kHz. The quadratic coupling is substantial only when the sample is close to the nodes or antinodes of the intracavity field. This is when the quadratic coupling reaches values of up to |G 2 /2π| = 10 MHz nm −2 (Fig. 3(c)). We can numerically reproduce this modulation via the transfer matrix method [38]. The membrane stripe is simulated as a dielectric slab with a refractive index of n = 1.725 + (3.55 × 10 −5 )i (see Appendix A), in accordance with the values reported in the literature [39,40]. The shape and magnitude of both the linear and quadratic coupling agree well with the numerical simulations.
Any absorption in the Si 3 N 4 mechanical resonator at 1550 nm, intrinsic and/or due to residues from fabrication, leads to a modulation of the cavity losses with respect to the sample position [16,31,41]. However, this modulation is small due the small value of the imaginary part of the refractive index [40,42]. With the sample placed at the node of the cavity, we measure linewidths of 17.5 MHz, corresponding to a loaded finesse of F = 195 000, almost unchanged from the empty cavity. Away from the node, the linewidth stays below position we can extract the dissipative coupling G κ = ∂κ ∂z (see Appendix A) and we obtain a value of the order of |G κ /2π|= 0.1 MHz nm −1 . The dissipative coupling is four orders of magnitude smaller than the dispersive coupling.
We demonstrate dynamical backaction in our system by measuring the optical spring effect [43,44] from the cavity and the second harmonic of the Si 3 N 4 stripe (Ω 0 /2π = 932.5 kHz). For this measurement the sample is placed 20 nm away from a cavity node, corresponding to G = 0.18 GHz nm −1 or g 0 = 1 kHz according to the data in Fig. 3. The cavity length is stabilized on the lock tone and the probe tone is scanned across the cavity resonance. For each probe detuning ∆ p we record the mechanical power spectral density (PSD), yielding the thermally induced vibrations of the membrane stripe. As their Lorentzian shape is obscured by small yet unavoidable frequency fluctuations, a Voigt fit is employed to extract the mechanical resonance frequency Ω m and effective linewidth Γ m (see Appendix B for details).
A detuned drive of the cavity optomechanical system exerts dynamical backaction on the mechanical resonator leading to a modification of the mechanical resonance frequency expressed as Ω m 2 = Ω 2 0 + δ(Ω 2 ) (Ref. [45]). Including a dissipative coupling term, this frequency shift is given by with the effective dispersive coupling g and the effective dissipative coupling g κ = √ n circ g 0,κ , respectively [41].
The mechanical frequency shift depends on the circulating photon number through g = √ n circ g 0 and g κ = √ n circ g 0,κ , the probe detuning ∆ p , and κ. However, the circulating photon number also depends on the probe detuning. In order to be able to fit Equation 2 to the measured mechanical frequency shift, we need to know the exact photon number.
To this end, for every detuning step, we calculate the photon number n circ from the probe transmission and the launched probe power P in = 7 µW (Fig. 4(a), black circles). For all presented measurements, we sweep the detuning from positive to negative values. The circulating photon number response appears bistable. First, it increases slightly and at  The cavity linewidth κ = 62 MHz that we obtain from the fit in Fig. 4(b) is three times larger than the cavity linewidth calculated from the resonant transmission κ = 21 MHz. We assume that this is due to noise in the detuning from cavity length fluctuations. Fluctuations in the cavity length lead to fluctuations of the effective detuning, resulting in a mismatch between the theoretical and experimental values of the mechanical frequency shift. However, if one neglects photothermal effects inside the mirror coatings and radiation pressure effects on the cavity mirrors, these fluctuations are independent of the photon number. In our measurements we effectively average the mechanical response and the mechanical frequency matches on average the theoretical value ( Fig. 4(b)). This occurs because the fluctuations are symmetric with respect to the applied effective detuning.
The optical damping arising from dynamical backaction is given by [41] Γ opt = 2g 2 κ 2 so that the effective mechanical linewidth Γ m results in Γ m = Γ 0 + Γ opt . In the presence of optomechanical backaction, the thermal force driving the mechanical resonator remains constant, but the effective mechanical linewidth is altered. To measure the mode temperature, we extract the effective mechanical damping Γ m from the Voigt fit of the measured mechanical spectra and obtain the effective mode temperature T eff from the phonon bath The obtained linewidths are plotted in Fig. 5(a) together with the theoretical curve (black line) with the parameters from the fit of the optical spring measurement (g 0 = 575 Hz, κ = 62 MHz) and neglecting the dissipative coupling term. Figure 5(b) displays the effective mode temperature calculated from the measured linewidths. The theory predicts a negative effective mechanical damping, that is, heating of the mechanical mode, for the positive detunings where we measure a plateau of the photon number. In the measurements we observe an effective damping close to zero which corresponds to self-sustained oscillations.
This measurement is limited by the bandwidth of the measurement and the quality of the Voigt profile fits that we use to extract the mechanical linewidth. For negative detunings, we observe optomechanical cooling of the mode from room temperature down to T eff = 12 K, which is verified by additional measurements where we sweep the probe power (see Appendix C).

V. CONCLUSION
In conclusion, we present an extremely sensitive FFPC setup with a small mode volume and an ultrahigh finesse exceeding 200 000. We perform a full characterization of the setup and we demonstrate dynamical backaction on the second harmonic flexural mode of a stoichiometric Si 3 N 4 membrane stripe. All measurements are supported by theoretical models, allowing the parameters of the system to be quantitatively extracted. The finesse of the loaded cavity of up to F = 195 000 is exceptionally high for a loaded FFPC system, exceeding that from Ref. [25] by more than one order of magnitude. We report an optomechanical coupling strength of |g 0 /2π| = 1 kHz while values up to |g 0 /2π| = 5.8 kHz are accessible at a different position of the membrane stripe in the cavity mode. Furthermore, we show self-sustained oscillations of the mechanical mode and optomechanical cooling from room temperature down to an effective mode temperature of T eff = 12 K.
Following the proposal in Ref. [31], the use of single-walled CNTs [27] in this FFPC is very promising. A dispersive coupling strength around |g 0 /2π| = 25 kHz is expected, which would place the system deep inside the strong coupling regime.In addition, the outstanding sensitivity of CNTs [46] enables ultrasensitive cavity optomechanics at room temperature. Owing to the versatility of the presented setup other interesting nanoscale mechanical structures such as nanowires, nanorods, or two-dimensional materials such as h-BN can be studied.
By additionally exploiting optical dipole transitions in the said materials, the realization of hybrid optomechanical systems seems to be in reach.
The data supporting the findings of this study are available online [47].   This validates our approximation to extract the cavity linewidth from the normalized onresonant transmission.
The dissipative coupling can be calculated from the linewidths obtained both from the measurements and from the numerical simulations (see Fig. 7(b)). The exact shape of the measured coupling is hard to extract and depends strongly on how smoothly the derivative is calculated. Nevertheless, both the measurement and simulations show a dissipative coupling of the order of 0.1 MHz nm −1 . Thus, for the measurements presented in this letter, the dissipative coupling is four orders of magnitude smaller than dispersive coupling.

Appendix B: Mechanical Response
As soon as we lock the cavity, any cavity length fluctuations will alter the detuning and consequently the optical spring effect, which is translated into mechanical frequency noise. As this noise is of Gaussian origin, the resulting mechanical spectra follows a Voigt distribution -a convolution of a Lorentzian peak with a Gaussian noise distribution -very common in spectroscopy where lines appear Doppler broadened [49]. The Voigt profile V (ω, σ, Γ) is given by V (ω, σ, Γ) = G(ω, σ) L(ω, Γ = 2).
Here G(ω, σ) denotes the Gaussian distribution with standard deviation σ and L(ω, Γ = 2) is the textbook Cauchy-Lorentz distribution with the half-width at half maximum Γ = 2.
Voigt profiles can also be used to fit peaks that are too narrow to be resolved by the finite bandwidth of the measurement devices [50]. In Fig. 8 (b) we lock the cavity with the lock tone blue detuned. As expected, the optical spring effect shifts the mechanical resonance to slightly higher frequencies. The expected narrowing of the effective mechanical linewidth is obscured by broadening of the resonance ( Fig. 8 (b) We perform a power sweep for a fixed probe detuning to confirm our observations of optomechanical cooling. According to Eqs. 2 and 3 we expect both the mechanical resonance frequency and the effective mechanical linewidth to increase linearly with the number of cavity photons. In contrast to the measurements in the main text, we consider the fundamental harmonic flexural mode in the following (n = 1, Ω 0 /2π = 468.2 kHz, Γ 0 /2π = 10.8 Hz, Q = 43 000). As a result of its lower mechanical quality factor this mode is less prone to start self-oscillating. We place the sample at a position of around z = −30 nm (below a node of the cavity field) resulting in a cavity linewidth of κ/2π = 22.3 MHz and stabilize the cavity in a way that the lock tone is blue detuned.
We fix a probe detuning of ∆ p /κ = 0.56 and we scan the probe power from P in = 0.2 µW to P in = 18 µW. That corresponds to a number of circulating photons due to the probe tone of up to n circ = 160 000. The photon number due to the lock tone is constant during the measurement, n circ ∼ 90 000. We select a large lock detuning to minimize backaction from the lock tone. At the same time, a large lock photon number ensures a strong feedback signal that boosts the lock quality.
We record the mechanical spectra of both the 780-nm interferometer and the X quadrature extracted from the heterodyne signal in transmission. The mechanical PSDs are fitted with Voigt profiles to obtain the mechanical resonances and the effective mechanical dampings.  The mode self-oscillates for photon numbers below n circ = 30 000, where the mechanical resonance frequency remains constant and the observed effective damping is close to zero.
The linear regression allows to estimate the optical damping induced by the lock tone to be of the order of Γ opt /2π = −40 Hz.