High-sensitivity monitoring of micromechanical vibration using optical whispering gallery mode resonators

The inherent coupling of optical and mechanical modes in high finesse optical microresonators provide a natural, highly sensitive transduction mechanism for micromechanical vibrations. Using homodyne and polarization spectroscopy techniques, we achieve shot-noise limited displacement sensitivities of 10^(-19) m Hz^(-1/2). In an unprecedented manner, this enables the detection and study of a variety of mechanical modes, which are identified as radial breathing, flexural and torsional modes using 3-dimensional finite element modelling. Furthermore, a broadband equivalent displacement noise is measured and found to agree well with models for thermorefractive noise in silica dielectric cavities. Implications for ground-state cooling, displacement sensing and Kerr squeezing are discussed.


I. INTRODUCTION
The transduction and measurement of small displacements of mechanical oscillators is important in a variety of studies ranging from macroscale gravitational wave detection [1] to micron-scale cantilever-based force measurements [2]. One embodiment that is particularly amenable to measurement of small displacements is the parametric coupling of a high-Q electrical or optical resonance to a mechanical oscillator [1]. In the case of a Fabry-Perot interferometer with a harmonically oscillating end-mirror, this parametric coupling manifests itself as a position-dependent shift of the resonance frequency, thereby allowing transducing the mechanical motion into a change of the phase of a resonant field probing the Fabry-Perot cavity.
However, even for a perfect measurement apparatus, the sensitivity is limited by the laws of quantum mechanics. Fundamentally, any linear measurement process entails that it must also exert a backaction onto the mechanical oscillator, as first discussed by Braginsky [3,4].
Braginksy identified two types of backaction: quantum backaction and dynamic backaction.
Dynamic backaction occurs when the optical interferometer (or cavity) is excited in a detuned manner. In this case the radiation pressure force exerted by photons in the cavity can become viscous and lead to either amplification or cooling of the mechanical motion. This effect is entirely classical and has been first observed in 2005 in the case of radiation-pressure amplification and oscillation [5,6] and in 2006 also for radiation-pressure cooling [7,8,9].
For a recent review see Ref. [10]. For the case of resonant excitation of the cavity or circuit, this effect can however in principle be entirely suppressed. In contrast, quantum backaction cannot be suppressed and arises from the discrete nature of the photons (or electrons) involved in the measurement process. The quantum fluctuations of the intracavity field cause a random force that drives the mechanical oscillator and thereby leads to perturbation of its position. This quantum backaction provides a limit to continuous position measurements and leads to the so called standard quantum limit [3,11]. At the standard quantum limit, the measurement imprecision is equal to the zero point motion of the mechanical oscillator, to which both shot noise and quantum backaction contribute in equal amounts. Over the past decades, significant progress has been made in approaching this quantum limit of motion measurement in the context of both electromechanical and optomechanical experiments.
Important to the detection process is that the optical transducer is typically not only sensitive to a single mechanical mode of interest, but sensitive to any differential change in the cavity's optical path length. Therefore, a variety of mechanical modes of the cavity boundary can contribute and the signal from the mode of interest may be spectroscopically extracted based on its Fourier frequency. The thermal excitation of other modes, and other effective cavity length fluctuations may constitute a measurement background in excess of the quantum shot of the lightused to monitor the mode of interest. Assessing this background therefore proves particularly important if one mode is laser-cooled below the bath temperature as recently achieved [7,9,21,23,24,25,26,27,28,29].
In this letter, we provide a broadband analysis of the noises in the optomechanical transduction of radial displacements in toroidal silica microcavities, revealing in detail the influence of all other mechanical modes. Toroidal microcavities [30] are optical resonators that host both high-quality optical and mechanical modes in one and the same device, and have been used to demonstrate radiation pressure dynamic backaction for the first time [5,6].
Efficient cooling by dynamical backaction [31] has been demonstrated both in the "Doppler" [9] and the resolved-sideband regime [29,32,33], rendering them particularly interesting for the goals of the emerging discipline of cavity quantum optomechanics [10], which pertains to studying quantum phenomena of mesoscopic mechanical oscillators. We report broadband interferometric measurement of their radio-frequency mechanical modes based on parametric coupling to the optical whispering-gallery modes (WGMs). Using adaptations of both the Hänsch-Coulliaud polarization spectroscopy [34] and optical homodyne measurement, displacement sensitivities at the level of 10 −19 m/ √ Hz are achieved over a measurement bandwidth of up to 20 MHz.
We find sparse spectra of mechanical modes which allow obtaining a detailed understanding of the modes using 3-dimensional finite-element simulations. More than 20 mechanical modes are observed between DC−100 MHz comprising radial breathing, flexural and torsional modes. We furthermore identify a broadband noise background which is attributed to thermorefractive noise, as previously observed in silica microspheres [35]. The detailed understanding of these noise processes (both due to mechanical modes and thermorefractive noise) is particularly important for studies such as pondermotive squeezing [36], ground state cooling [32,33], as well as squeezing using the third order Kerr nonlinearity of glass [37].

II. WHISPERING GALLERY MODES FOR OPTICAL MOTION TRANSDUC-TION
Decades of research in the field of gravitational wave astronomy have brought major fundamental [3,11] and technological advances in interferometric transduction of mechanical displacements. More recent efforts [8,20,38] have shown that these techniques are well suited for application to much lighter oscillators at the microscale. Such oscillators are expected to display quantum effects at significantly higher temperatures. In the following, we will briefly review the limits for the interferometric detection of micromechanical oscillations using highquality WGMs [39]. The employed silica toroidal resonators (figure 1) possess ultra-high-Q optical modes which are confined by total internal reflection [30]. In addition, microcavities also exhibit structural resonances giving rise to high frequency vibrational modes [5,6,40].
Mechanical modes which affect the circumference of the cavity shift the optical resonance frequency, and thereby couple to the optical degree of freedom. This was recognized in early experiments that demonstrated the parametric oscillation instability [5,6] and provides, as shown here, a natural way for highly sensitive motion transduction.
A change of the major radius R by a small displacement x induces a shift of the resonance frequency of the whispering gallery modes located in the rim of the toroid by an amount ∆ω 0 /ω 0 = x/R. This induces a change in the properties of a field launched into this mode.
In the usual coupling geometry using a tapered fibre, the field transmitted through the tapered fibre reads [41] where τ ex and τ 0 are the inverse cavity decay rates due to output coupling to the taper and due to other losses [62]. The condition with τ ex = τ 0 is usually referred to as critically coupled or impedance matched. To first order, the transmitted amplitude is not affected by small mechanically-induced resonance frequency shifts ∆ω 0 for a resonant laser ω = ω 0 , the phase of the field, however, is. By comparison with a phase reference in an interferometric measurement, the displacement x can be thereby be detected. For example, the output field may be brought to interference with a strong field E lo at the same frequency ω 0 using a half-transmissive beam splitter. Choosing the appropriate phase of the reference field E lo , the photon fluxes detected at the two output ports of the splitter are determined by Subtraction of these two simultaneously measured fluxes yields a differential signal to first order in x, where κ = τ −1 0 + τ −1 ex is the cavity's total decay rate (i.e. linewidth) and P in / ω and P lo / ω are the photon fluxes corresponding to the fields E in and E lo . The coupling efficiency can take values 0 . . . 1, approaching 0 for undercoupling (τ ex → ∞) and 1 in the case of overcoupling (τ ex → 0), while at critical coupling it is equal to 1/2. Physically this quantity thus describes the probability of an intracavity photon coupling to the output fibre. The fundamental noise source in this detection scheme arises from the quantum phase noise of the light being detected. This leads to a spectral density of flux (and thus signal) fluctuations of P lo / ω in the experimentally desirable limit P lo ≫ P out . The resulting minimum detectable displacement δx min is where the finesse F = c/nRκ of the WGM was introduced, λ = 2πc/nω 0 is the optical wavelength in glass and ∆f the measurement bandwidth. Note that the finesse F is also affected by the coupling η c via F = F 0 (1 − η c ), where F 0 is the "intrinsic" finesse in the undercoupled limit τ ex → ∞, so that the optimum sensitivity is achieved at critical coupling We note that the same result is obtained following a more formal approach, using the linearized quantum Langevin equations (QLE) of the coupled optomechanical system around a stable working point [36,42,43]. This allows in particular calculating the fluctuations in any arbitrary quadrature of the output field. Comparison of the resulting quantum fluctuations and mechanically induced fluctuations in the detected output phase quadrature yields a minimum displacement of Compared to (4), this calculation adds only a correction due to the finite response time of the cavity for Fourier frequencies Ω exceeding the cavity cutoff κ/2. We note that the full expression (5) can also be derived from a classical calculation of the signal, considering the amplitude of the motional sidebands of the field coupling out of the cavity and comparing it with detection shot noise [29]. The detection limit (5) corresponds to a spectral density of measurement imprecision As first pointed out by Braginsky [3], this measurement inevitably exerts backaction on the mechanical device. In the case of an optical transducer, quantum backaction is enforced by the fluctuations of radiation pressure due to a fluctuating intracavity photon flux. From the linearized Langevin equations, the intracavity force fluctuation spectrum can be calculated to have the form It is noted that S x and S F fulfill the expected uncertainty relation where is the susceptibility of the mechanical oscillator and Γ m is its mechanical damping rate. It is important to note that this susceptibility is modified when the optical resonance is excited in a detuned manner [44]. However, for resonant probing as considered here, it is not modified.
The total measurement uncertainty is minimized for an input flux of called the standard quantum limit [3,11] in the case η c = 1. Its peak value at Ω m is In this calculation we have explicitly considered the effect of the coupling conditions to the cavity, which can-as a unique feature-be varied continuously in the experiment by adjusting the gap between the coupling waveguide and the WGM resonator. The SQL is approached most closely in the overcoupled limit τ ex ≪ τ 0 . It is noteworthy that the fibretaper coupling technique to microtoroids can deeply enter this regime, and 100 · τ ex < τ 0 (η c = 99%) has been demonstrated [45]. On the other hand, such a strong coupling reduces the cavity finesse and thus comes at the expense of a higher optimum power P in opt . Working with weaker coupling, such as critical coupling as typically pursued in this work, brings only a moderate penalty as (12) shows, for example, a factor of √ 2 for η c = 1/2.
In our experiment performed at room temperature, the noise induced by quantum backaction is masked by thermal noise due to a fluctuating Langevin force with S th adding a third term to the total displacement noise [36,42], This expression constitutes a description of the spectrum that is measured by analyzing the phase quadrature of the transmitted light past the microresonator [42].

III. EXPERIMENTAL IMPLEMENTATION
Detection of the phase of a light field with quantum-limited sensitivity is a standard task in quantum optics, and several techniques have been developed to achieve this. In the following section, two techniques which were successfully implemented for motion transduction in whispering gallery mode microresonators are described.

A. Homodyne spectroscopy
The most common method for quantum-limited phase measurement is a balanced homodyne receiver [46] as employed in previous optomechanical experiments [20,22,47]. We briefly discuss the experimental protocol used for homodyne spectroscopy. This method is An advantageous feature of the homodyne signal is that its d.c.-component directly pro-vides a dispersive error signal h(∆ω) = 2η c κ ∆ω ∆ω 2 + (κ/2) 2 P cav P LO (16) that can be used for locking the laser to the center of the optical microcavity resonance. At the low signal powers used here, locking is very stable. An example of an experimentally obtained error signal is shown in figure 2(d). Simultaneously, the fluctuations in the differential photocurrent induced by both optical shot noise in the signal and the thermal noises in the cavity displacement can be frequency-analyzed using a high-performance electronic spectrum analyzer. For calibration purposes, we frequency-modulate the laser using a LiNbO 3 phase modulator external to the laser. The frequency modulation is given by δω = βΩ mod for known modulation depth β and frequency Ω mod , and generates the same signal as would be induced by a radius modulation of δx = Rδω/ω [20,29,35,38], independent of cavity linewidth and coupling conditions. If the cavity linewidth is known in addition, the spectra can be absolutely calibrated at all Fourier frequencies, taking into account the reduced sensitivity beyond the cavity cutoff at κ/2.

B. Polarization spectroscopy
A simplified setup may be obtained by co-propagating the local oscillator in the same spatial, but orthogonal polarization mode as compared to the signal beam [29]. Since the WGM modes have predominantly TE or TM character and are not degenerate, this guarantees that the local oscillator is not affected by the cavity. Due to common-mode rejection of many sources of noise in the relative phase between signal and LO (for example, frequency noise in the optical fibre), the passive stability is sufficiently enhanced to enable operation without active stabilization (Figure 3).
Enforcing interference between local oscillator and signal beams then corresponds to polarization analysis of the light (comprising both signal and LO) emerging from the cavity.
While novel in the present context of a tapered fibre coupled microcavity, this is a well established technique to derive a dispersive error signal from the light reflected from a Fabry-Perot type reference cavity, named after their inventors Hänsch and Couillaud [34].
If fibre birefringence is adequately compensated, the error signal reads identical to (16), and a typical trace is shown in figure 3(c). This is used to lock the laser at resonance ∆ω ≡ 0 with a bandwidth of about 10 kHz. Calibration of the spectra may be performed as described in the previous section.
While this approach obviously allows reducing the complexity of the experiment, this arrangement proved more sensitive to slow temperature drifts in the polarization mode dispersion of the fibres employed, due to the large ratio of signal and LO powers, the mag-  [29]. The intrinsic polarization selectivity of WGM renders the introduction of an additional polarizer, mandatory in the original implementation [34], obsolete.
In an earlier experiment with a Fabry-Perot cavity [48], the losses associated with an intracavity polarization element limited the finesse, and therefore the attained sensitivity to

IV. OBSERVATION AND ANALYSIS OF QUANTUM AND THERMAL NOISES
In this section, we present characteristic results obtained with silica microtoroids of typical major radii between 25 µm and 50 µm. Figure 4 shows an example of a broadband measurement using homodyne detection. If the taper is retracted from the proximity of the cavity, quantum shot noise exceeds the electronic noise in the receiver. It was verified that the photocurrent noise √ S I scales with the square root of the total power as expected for shot noise.
When the laser is coupled and locked to a WGM resonance, a substantially different spectrum is observed (Figure 4). Its equivalent displacement noise is calibrated in absolute terms using an a priori known phase modulation at 36 MHz, and taking the cavity cutoff into account. The equivalent displacement noise of the cavity exceeds the shot noise at all frequencies for a high enough power in the signal beam, leading to a background level equivalent to a displacement noise of √ S x ∼ 10 −18 m/ √ Hz. The superimposed sparse spectrum of peaks fits the sum of several Lorentzians which arise from the thermal noise of several mechanical modes, i |χ i (Ω)| 2 S th F,i (Ω). In the following, we discuss the features of the spectrum in more detail.

A. Thermorefractive noise
The broadband, low-frequency background noise is attributed to thermorefractive noise, the fluctuations in refractive index induced by the fluctuations ∆T 2 = k B T 2 /ρc p V of temperature on a microscopic volume V [49]. Here k B denotes Boltzmann's constant, T temperature, ρ density and c p specific heat capacity. This leads to fluctuations of the resonance frequency via both the dependence of the refractive index n on temperature, and the thermal expansion of the material. At room temperature however, the coefficient of thermal expansion α is more than twenty times smaller than dn/dT , so the analysis can be restricted to the resonance frequency fluctuations induced by thermorefractive fluctuations.
Introducing a fluctuating thermal source field in the heat diffusion equation similar to a Langevin approach [50] it is possible to derive the spectrum of refractive index fluctuations sampled by a WGM in a silica microsphere [35]. For high frequencies, an approximate ana- lytic expression, neglecting also the boundary conditions for thermal waves, can be obtained.

The result
was found in good agreement with the experimental data obtained on a silica microspheres [35] between 100 Hz and 100 kHz. Here R is the cavity radius, d and b are transverse mode dimensions and D is the thermal diffusivity of silica. For comparison with the toroid measurement calibrated as effective radial displacement, R S δn/n (Ω) has to be evaluated.
Inserting the material parameters of fused silica and the radius of the employed toroid into the model (18), the data between 100 kHz and 20 MHz can be quantitatively reproduced if no parameters except b and the absolute magnitude are adjusted by factors of order 2 ( Figure 4).
These corrections are justified considering the approximations made in the derivation, and potential differences in surface effects in spheres and toroids. It is interesting to calculate the resulting experimental root-mean-square fluctuations of the cavity's refractive index +∞ −∞ S δn/n (Ω)dΩ to be of order 10 −10 , as it constitutes a detection limit for resonance frequency shifts induced by molecules in the evanescent field [51].
For the purposes of cavity quantum optomechanics, thermorefractive fluctuations constitute a background noise, at room temperature rolling off to a level of ∼ 10 −19 m/ √ Hz at Ω/2π > 50 MHz, where the high-quality radial-breathing modes typically reside. Practically, such experiments are going to be performed in a cryogenic environment, leading to significant changes in the material properties. A level of R S δn/n (Ω) ∼ 10 −20 m/ √ Hz at T ∼ 1 K may be estimated, assuming a reduction of dn/dT to 8 · 10 −6 K −1 as indicated by recent measurements [52]. While other sources of noise, such as thermoelastic and photothermal noises [53,54] are not expected to exceed this value, thorough experimental characterization at cryogenic temperatures is necessary.
Such a study is also an important pre-study for experiments aiming at the demonstration of generating broadband intracavity Kerr squeezing. While above-threshold parametric oscillations have been observed in these cavities [55,56], the room temperature experiments reported here indicate that the thermorefractive noise exceeds the quantum noise of the light in the cavity, as evidenced by the homodyne measurement of the cavity output. It therefore has to be suppressed to achieve squeezing of the cavity field.

B. Mechanical modes
The most prominent features in the noise spectrum are the mechanical modes of the microtoroids. To date, the variety of mechanical modes in silica microcavities (toroids and spheres) has been investigated only to a limited extent [10,29], in most cases by driving modes with particularly strong optomechanical coupling into the parametric oscillatory instability [5,6,40,57,58]. Using a Fabry-Perot cavity, however, it was shown that the Brownian motion of a wealth of intrinsic mechanical modes of a cylindrical mirror can be studied [47]. The high sensitivity methods presented in the previous section enable monitoring the Brownian motion of around twenty different mechanical modes in silica microtoroids over a frequency range spanning from below 1 MHz to above 100 MHz. Figure 5 shows a noise spectrum obtained by Hänsch-Couillaud spectroscopy revealing a total of 16 mechanical modes. In order to indentify the observed peaks with the appropriate mechanical mode patterns, a 3D finite element model (FEM) of the microtoroid is employed and implemented [63]. Extracting the geometry parameters using an optical microscope (accuracy ±5%) and matching observed and simulated mechanical frequencies, the FEM simulation allows identifying all observed peaks in the spectrum. Thus all 16 observed modes were assigned to the corresponding simulated mode patterns as depicted in figure 5. Figure 6 gives an overview of simulated frequencies and the frequencies deduced from the spectrum shown in figure 5 revealing excellent agreement. The relative frequency deviation between measurement and simulation is on average less than 2%. Moreover, almost all simulated modes are observed experimentally.
Only three out of 19 modes (number 6, 13, 18) cannot be observed which may be due to low mechanical Q factors (< 10).
Due to its composite structure comprising several geometric objects microtoroids exhibit a diverse set of eigenmodes. In order to characterize the noise spectra, understanding the complex mode structure of microtoroids is in particular important as all modes contribute to a background noise floor. Indeed, various mode-families can be identified in which the motion of the silica disk, the silica torus, and the silicon pillar partially decouples. The mode showing the strongest optomechanical coupling is the radial breathing mode (mode 14 in figure 5), and most previous work has focused on this mode [5,9,29]. An equivalent mode has been studied in a microdisk structure [59] where it was termed radial contour mode. In contrast, the optomechanical coupling of the torsional mode (mode number 4) where the silica disk shows an in-plane rotation vanishes to first order. Interestingly, this torsional mode can nevertheless be observed experimentally.
One particular mode family that can be identified are the radially symmetric flexural modes (modes 2, 8) in which the motion of the free standing part of the silica disk resembles the modes of a cantilever. The fundamental frequencies of a cantilever of length L can in general be expressed as Ω i /2π = C · √ k i /2π, where C is a material constant and the k i are given by the solutions of [60] cos(k i · L) · cosh(k i · L) + 1 = 0. cantilever length L (13.2 µm and 39.6 µm respectively). Both sets of data allow an accurate single quadratic fit of the fundamental radially symmetric modes. Thus, the latter can indeed be regarded as cantilever modes following a uniform quadratic dispersion even for microtoroids of different sizes. In particular, the quadratic dispersion rules out the presence of radial tensile stress within the silica disk as this would imply a linear dispersion relation.
Another obvious mode family which can be distinguished is characterized by sinusoidal oscillations of the torus itself (modes 1, 3,5,7,11,16). The dispersion diagram of these modes, which we refer to as crown-modes, is depicted in figure 7 for two different samples.
The respective wavelength λ is given by twice the distance between two adjacent nodes of each mode. The frequencies Ω/2π of the crown modes observed in microtoroids of different circumference and torus diameter (2π · 23 µm/5.3 µm and 2π · 45 µm/5.7 µm) allow a simultaneous quadratic fit with the frequencies Ω i /2π following a quadratic dependence on the wave vector k i = 2π/λ i . This uniform dependence shows that in this mode family the silica torus, despite its attachment to the silica disk, behaves effectively like an independent element. The observed quadratic dispersion relation rules out the presence of tensile stress within the torus in radial direction as this would lead to linear dispersion characteristic of a vibrating string. Since the microtoroids undergo a reflow process [30] indeed all potentially present stresses should get relaxed during this fabrication step. As such, the observed quadratic dispersion characteristic for a rigid ring structure confirms this expectation. The study of clamping losses in the radial breathing mode and the optimization of its mechanical Q factors [61] also lead to the observation of very high Q crown modes. For these measurements, the cavities are operated in low-pressure environment (p < 1 mbar) in order to reduce viscous damping, which limited the Q's of the previously discussed modes to values 3000. For such a measurement, figure 8 shows the second and third order crown modes with Q factors exceeding 50'000 at frequencies below 10 MHz. These high Q factors are attributed to low clamping losses which are studied in detail in [61].

V. CONCLUSIONS
In conclusion, we have shown that high-finesse whispering gallery modes are extraordinarily well suited as transducers for micromechanical motion. Sensitivities on the order of 10 −19 m/ √ Hz are achieved, on par with the best reported values [8,22]. The small dimensions of the WGM resonators allow in addition extending the measurement bandwidth by more than an order of magnitude compared to earlier work [8,22]. This enlarged range conicides with the resonance frequencies of the mechanical modes of interest present in the device. For example, the radial breathing mode (RBM), particularly amenable to optomechanical effects due to its small effective mass (m eff ∼ 10 ng), typically exhibits a resonance frequency around 50 MHz. At this frequency, average thermal phonon occupancies below unity are achieved at temperatures ∼ 2 mK. Approaching such temperatures appears feasible using a combination of conventional cryogenics and resolved-sideband cooling [29]. It is interesting that the expected spectrum of even the zero-point fluctuations of this mode peaks at a value of S ZPF x (Ω m ) = Q/m eff Ω 2 m ∼ 10 −19 m/ √ Hz, assuming an effective Q = 100 after cryogenic and laser cooling. Such signal levels may be detected on top of the background of the thermal noises studied here, providing a route towards experimental tests of the theory of quantum measurements on mesoscopic objects. Finally, we note that the advantageous properties of WGM resonators may also be exploited for motion transduction of a mechanical oscillator external to the cavity, for example, by bringing it into the near field of the whispering gallery mode.