Determination of the vacuum optomechanical coupling rate using frequency noise calibration

The strength of optomechanical interactions in a cavity optomechanical system can be quantified by a vacuum coupling rate $\vcr$ analogous to cavity quantum electrodynamics. This single figure of merit removes the ambiguity in the frequently quoted coupling parameter defining the frequency shift for a given mechanical displacement, and the effective mass of the mechanical mode. Here we demonstrate and verify a straightforward experimental technique to derive the vacuum optomechanical coupling rate. It only requires applying a known frequency modulation of the employed electromagnetic probe field and knowledge of the mechanical oscillator's occupation. The method is experimentally verified for a micromechanical mode in a toroidal whispering-gallery-resonator and a nanomechanical oscillator coupled to a toroidal cavity via its near field.


Introduction
Optomechanical systems are being explored for a variety of studies, ranging from exploring quantum limits of displacement measurements [1,2], ground-state cooling of mechanical oscillators [3,4] to low-noise radiation-pressure driven oscillators [5]. In this new research field of cavity optomechanics [6,7], rapid progress towards these goals has recently been enabled by the conception and realization of a large variety of nano-and micro-optomechanical systems, which implement parametric coupling of a mechanical oscillator to an optical or microwave resonator (for an overview of recently studied systems, see e. g. reference [8]). Thereby, the resonance frequency ω c of the electromagnetic cavity is changed upon a mechanical displacement x according to ω c (x) = ω c (0) + Gx, with the coupling parameter The resulting optomechanical interaction can be described by the interaction Hamiltonian [9],Ĥ wherex = x zpf (â † m +â m ) is the mechanical displacement operator,n o =â † oâo the photon occupation of the optical cavity, andâ o ,â † o ,â m andâ † m are the usual ladder operators of the optical (o) and mechanical (m) degrees of freedom. The root-mean-square amplitude of the mechanical oscillator's zero-point fluctuations is determined by its resonance frequency Ω m and its effective mass m eff as discussed in more detail below. The canonical example of such a system would be a Fabry-Pérot resonator, one mirror of which is mounted as a mass on a spring moving as a whole along the symmetry axis of the cavity. In this case, the coupling parameter is given by G = −ω c /L, where L is the cavity length, and the oscillator's effective mass simply is the moving mirror's mass. In most real optomechanical systems, however, internal mechanical modes constitute the mechanical degree of freedom of interest (for simplicity, our discussion will be restricted to a single mechanical mode only). In general, the displacement of such a mode has to be described by a three-dimensional vector field u( r). For every volume element located at r, it yields the displacement u( r) from its equilibrium position. In order to retain the simple one-dimensional description of equation (2), it is then necessary to map the vector field u( r) to a scalar displacement x.
A frequently adapted procedure to address this mapping is to deliberately choose a coupling parameter G adequate to the overall geometry of the system, such as G = −ω c /L for a Fabry-Pérot-type cavity of length L [10][11][12], or G = −ω c /R for a whispering-gallery-mode (WGM) resonator of major radius R [8,13]. The equivalent displacement x is then defined by the measured resonance frequency shift ∆ω c induced by the displacement field u( r), i. e. x ≡ ∆ω c /G. Following a different approach, the equivalent displacement x is defined, for example, as the displacement of the structure at a given position, x ≡ | u( r 0 )| or the maximum displacement of a mode [14][15][16][17]. Then, the coupling parameter G is calculated via the relation x = ∆ω c /G. In some cases [14,17,18], the coupling parameter is expressed as the effective length L −1 eff = 1 ω c dω c dx of a resonator for which |G| = ω c /L eff . In this approach, the coupling parameter may be defined individually for each mechanical mode. In all cases, the only physically significant relation is the induced resonance frequency shift ∆ω c upon a displacement u( r), which in a perturbation theory treatment, can be calculated This expression is uniquely determined by the distribution of the dielectric constant ε and the (unperturbed) electric field E and can, in most cases, be linearized in the small displacements u and reduced to a surface integral over the relevant resonator boundaries [10,20]. In contrast to the unambiguous relation (4), the equivalent displacement x which is a-priori not a well-defined quantity, may be defined in an arbitrary way as discussed above. An important consequence of this ambiguity in the choice of x is the necessity to introduce an effective mass m eff of the mechanical mode [10,21] which retains the correct relation for the energy U stored in the mechanical mode given a displacement x, Evidently, rescaling x → αx implies G → G/α and m eff → m eff /α 2 , so that the frequently quoted figures of merit G and m eff are individually only of very limited use. It is possible to eliminate this ambiguity by referring to the vacuum optomechanical coupling rate using terminology analogous to cavity quantum electrodynamics [22]. As expected from the Hamiltonian description (2), all optomechanical dynamics can be fully described in terms of this parameter. As a consequence, parameters such as the cooling rate or optical spring effect can be derived upon knowledge of g 0 . For instance, the cooling rate in the resolved sideband regime is given by Γ = 4 n o g 2 0 /κ, while the power to reach the SQL scales with κ 2 /g 2 0 with the cavity loss rate κ. Furthermore, the regime where g 0 n o κ, Γ m (Γ m is the mechanical damping rate) denotes the strong optomechanical coupling regime, in which parametric normal mode splitting occurs [3,23,24].

Determination of the vacuum optomechanical coupling rate
Our intention here is to detail and verify an experimental technique to determine g 0 in a simple and straightforward manner for an arbitrary optomechanical system. The technique we present relies on the fluctuation-dissipation theorem, from which the double-sided (symmetrized) spectral density of displacement fluctuations of a mechanical oscillator in contact with a thermal bath at temperature T can be derived [10,25,26], where m eff is the effective mass corresponding to the choice of the normalization of x, as derived from (5), and the approximation is valid for large mechanical occupation numbers n m ≈ k B T /hΩ m 1. The displacement fluctuations are not directly experimentally accessible, but cause fluctuations of the cavity resonance frequency ω c according to If such a spectrum can be measured, it is straightforward to integrate it over Fourier frequencies resulting in the interesting relation which implies that the knowledge of the mechanical occupation number and the cavity resonance frequency fluctuation spectrum are sufficient to directly derive g 0 . The former is usually a straightforward task (provided dynamical backaction [27,28] is ruled out, for example, by low probing powers), whereas the latter has to be implemented using a phase-sensitive measurement scheme as shown in figure 1, since the required changes of the optical resonance frequency change the phase of the internal and hence output field. Fig. 1. Generic scheme for the determination of the vacuum optomechanical coupling rate. The cavity optomechanical system is probed with a monochromatic source of electromagnetic waves which is frequency-modulated by a known amount at a Fourier frequency close to the mechanical resonance frequency. A phase-sensitive detector generates a signal I, which is analyzed with a spectrum analyzer. As two possible examples, a homodyne detector measures the phase difference between the arms in an interferometer, one of which accommodates the optomechanical system (shutter S open). If the shutter S is closed, the transmission signal I of an optomechanical system is amplitude-dependent when the driving wave is detuned with respect to the optical resonance.
Various methods for such measurements are available, such as direct detection when the probing laser is detuned, frequency modulation [29], polarization [30], or homodyne spectroscopy [31]. In general, cavity resonance frequency fluctuations are transduced into the signal in a frequency-dependent manner, such that the spectrum of the measured signal (e.g. a photocurrent) I is given by where ψ is the phase of the intracavity field, which gets modulated upon a change of the resonance frequency. Importantly, to exploit the relation (9), it is necessary to convert the spectrum S II (Ω) back into a frequency noise spectrum, so the function K(Ω) ≡ K(Ω, ∆, η c , κ, P) (where ∆ = ω l − ω c is the detuning from resonance, and η c = κ ex /κ is the coupling parameter quantifying the ratio of the cavity loss rate due to input coupling κ ex compared to the total loss rate κ) has to be calibrated in absolute terms. Experimentally, this can be accomplished by frequency-modulating the source that is used for measuring S II [31][32][33]. Under correct experimental conditions (cf. appendix) the phase modulation of the input field undergoes the exactly same transduction, so that S II (Ω) = K(Ω)S φ φ (Ω), where S φ φ (Ω) is the spectrum of input field phase modulation. For the typical case of monochromatic input field phase modulation (E l ∝ e −iφ 0 cos Ω mod t ), the equivalent spectral density is given by S φ φ (Ω) = 2π 1 2 (δ (Ω − Ω mod ) + δ (Ω + Ω mod )) φ 2 0 /2. A laboratory spectrum analyzer will, of course, not resolve this spectrum but convolute it with its bandwidth filter lineshape F(Ω), for example a Gaussian centered at Ω = 0. If a timedomain trace is analyzed using a discrete Fourier transform, the filter function would be given by the squared modulus of the Fourier transform of the time-domain windowing function. We then have S meas where a factor of 2 was introduced to account for the fact that most analyzers conventionally display single-sided spectra (this factor is irrelevant for our final result, however). The filter function is usually normalized and can be characterized by the effective noise bandwidth (ENBW), for which essentially absorbing differences in hardware or software implementations of spectral analysis [34]. Note, however, that this description is still idealized. In reality, correction factors may apply depending on details of the signal processing (such as averaging of logarithmic noise power readings) in the respective analyzer used, see e.g. [35]. If the effective noise bandwidth is chosen much narrower than all spectral features in S ψψ (Ω), the measured spectrum is approximately Assuming that the signal at Ω mod is dominated by laser modulation, the transduction function K can then be calibrated in absolute terms at this frequency, as in the relation all parameters except K(Ω mod ) are known. The spectral shape of K is then needed to derive the spectrum S ψψ of interest. This task can be accomplished either by repeating the measurement for all modulation frequencies of interest, or by theoretical analysis, relating it to the conditions under which the measurements are taken. As the simplest example, we may consider the case of direct detection, where the signal is the power P "transmitted" through the optomechanical system. For a given input power P in , we find (see appendix) the transduction function: We assumed in this expression sufficiently low pump powers such that effects related to optomechanically induced tranparency (OMIT) [36] are negligible. For the more advanced homodyne spectroscopy we calculate (see appendix), Advantageously, this method can also be applied with the laser tuned to resonance (∆ = 0), ruling out effects of dynamical backaction [27,28] even for larger laser powers. In this special case Another interesting case is ∆ = ±Ω m in the resolved sideband regime [33] with Ω m κ, where We would finally like to emphasize that in many experimental situations, the transduction function K(Ω) is sufficiently constant over the range of frequencies, in which the mechanical response in (9) is significant (several mechanical linewidths around Ω m ). In this case, it can be simply approximated to be flat, K(Ω) ≈ K(Ω m ) ≈ K(Ω mod ), and we get from equations (9), (10), (14) and (15) The last fraction in expression (20) can be interpreted as the ratio of the integrals of the displacement thermal noise and the calibration peak as broadened by the filter function. Note also that according to its definition the ENBW is given in direct frequency and not angular frequency as is Γ m .

Application of the method
As two examples, we have applied the discussed method to two different optomechanical systems shown in figure 2. We first consider the case of a doubly-clamped strained silicon nitride beam (30 × 0.7 × 0.1 µm 3 ) placed in the near-field of a silica toroidal resonator [37]. This scheme allows sensitive transduction of the nanomechanical motion with an imprecision below the level of the standard quantum limit [38]. Note that the coupling rate in such a configuration depends on the strength of the evanescent field at the location of the nanomechanical oscillator, which increases exponentially if the oscillator is approached to the toroid. The data in figure 2 show the single-sided cavity frequency noise spectral density S meas νν (Ω) ≡ S meas II (Ω) · (Ω/2π) 2 /K(Ω) recorded using the homodyne technique. The spectrum was calibrated by laser frequency modulation at the Fourier frequency of 8 MHz. The spectrum needs not be processed further, since the transduction function for homodyne detection with ∆ = 0 is suffciently flat around the mechanical peak, Γ m κ. However, the cavity frequency noise spectrum as measured here contains also contributions from other noise mechanisms present in the silica toroidal oscillator, in particular thermorefractive noise [32] known to dominate frequency noise at lower ( 10 MHz) Fourier frequencies. The integration of the spectrum is therefore exclusively performed on the Lorentzian corresponding to the mechanical mode (gray area), which can be easily isolated by fitting the noise spectrum with a model accommodating all known noise mechanisms including thermorefractive noise [38,39]. The result of this evaluation is δ ω 2 c = (2π · 530 kHz) 2 , and the corresponding vacuum coupling rate is given by g 0 = 2π · 420 Hz.
Silica WGM microresonators such as spheres or toroids actually support structure-inherent micromechanical modes themselves [39,40]. Figure 2 also shows an equivalent cavity frequency noise spectrum measured on such a cavity using direct detection. Again we neglect the frequency-dependence of K due to the very narrow mechanical peak (Γ m κ, Ω m ). For the Fig. 2. Measured (single-sided) frequency noise spectra S meas νν (Ω) (blue lines, ν ≡ ω c /2π) induced by a nanomechanical beam oscillator (8.3 MHz) coupled to a silica microtoroid [38] (left) and a radial breathing mode of a silica microtoroid (right), calibrated using laser frequency modulation at frequencies of 8 and 71.38 MHz, respectively. Orange lines are Lorentzian fits, the gray area underneath them yield the optomechanical coupling rates multiplied with the occupation number of the oscillator. evaluation of g 0 , we remove backgrounds due to thermal noise from other modes, thermorefractive fluctuations or shot noise, by fitting a Lorentzian model with a flat background term. Only the peak amplitude of the Lorentzian is then used with the relation (20). A vacuum coupling rate of g 0 = 2π · 570 Hz is obtained, which is interestingly very similar to the vacuum coupling rate of the nanomechanical beam.

Conclusion
In summary we have presented a simple method which allows direct measurements of the vacuum coupling rate g 0 in optomechanical systems using frequency-modulation. We have applied the method to two different devices, namely doubly-clamped strained silicon nitride beam (g 0 /2π ≈ 10 3 Hz) and the radial breathing mode in a silica toroidal resonator (g 0 /2π ≈ 10 3 Hz). It is interesting to note how widely this coupling rate can vary in different optomechanical systems, for example, we estimate g 0 /2π ∼ 10 −2 Hz for (present) microwave systems [15], g 0 /2π ∼ 10 0 Hz in lever-based optical systems [12,24], and up to g 0 /2π ∼ 10 5 Hz in integrated optomechanical crystals [17]. Together with the optical and mechanical dissipation rates and resonance frequencies (which do also vary widely), the coupling rate provides a convenient way to assess the potential of optomechanical platforms for the different phenomena and experiments within the setting of cavity optomechanics.