Apéndices Apéndice A
5. Experiencia significativa que la haya marcado
cal oscillators
In this section the properties of the employed nanomechanical oscillators will be charac- terized in somewhat more detail. For the nanomechanical strings and membranes intrinsic mechanical quality factors ranging from 104 to >105 are found (for frequencies of 5 MHz
to 15 MHz) in vacuum (pressure<10−5mbar) depending on the cleanliness of the samples.
These are similar to the values reported elsewhere for nanostrings [128, 131]. To our knowl- edge, however, similarly small membranes have not been examined by other researchers so far. For much larger membranes (edge lengths ≥500µm) considerably higher mechanical Q has been found [37, 136]. The measured mechanical quality factors are most likely lim- ited by two level systems within the amorphous silicon nitride [143, 144] (see also section 3 for an in-depth analysis of different mechanical damping mechanisms). In the following, we will focus on the mode structure of both nanomechanical strings and membranes.
The eigenmodes of a thin membrane (thickness t, edge lengths ly and lz, mass density
2.3 Transduction and actuation of nanomechanical motion 59
b, a,
Figure 2.17: Modes of a 40×0.7×0.1µm3 nanomechanical string. a, Out-of-plane (tri- angles) and in-plane mechanical modes (circles). The out-of-plane mode family follows a harmonic frequency ladder whereas the in-plane mode family increases faster in frequency owing to the larger in-plane cross-sectional inertia. Displacing the toroid along the string’s axis allows recording the nanomechanical string’s spatial mode patterns, as shown in b: peak squared displacement of the second (upper panel) and fourth (lower panel) out-of- plane mode normalized to the fundamental mode. Good agreement with Eq. (2.20) (dashed lines) is found, when the finite sampling length of the microresonator ¯ly (cf. Eq. 2.14) is
taken into account (full lines).
ρ) dominated by its internal tensile stress S (force per cross-sectional area) are given by
Ω(j,k)m /(2π) = 1 2 s S ρ q j2/l2 y +k2/lz2. (2.36)
Fig. 2.16 shows the measured frequencies of the first 28 eigenmodes of a 50×40×0.03µm3 membrane (up to order (7, 2)) along with the expected frequencies according to Eq. (2.36). For the latter, the ratio pS/ρ is adjusted in a least square fit to the measured val- ues. Excellent agreement is found between theory and experiment with the membrane frequencies fully described by the strain model. From this fit one obtains a value of
S/ρ = (1.0± 1)×105m2/s2. Using the density of ρ = (2.5±0.5) kg/m3 [37] for low-
stress silicon nitride, this yields a tensile stress of (260±70) MPa.
For the strained nanomechanical strings, however, deviations from the purely strain- dominated model are found. Their frequencies for arbitrary intrinsic stress are given by Eq. (2.20). Fig. 2.17a shows the nine lowest order eigenmodes of a 40×0.7× .1µm3 nanomechanical string. The out-of-plane modes follow a linear dependence on mode num- ber, here up to the fifth mode at 25 MHz since their motion is dominated by the internal tensile stress, i.e. S (jπt/L)2Y /12 (cf. Eq. 2.20). The spatial displacement patterns of these modes can be measured by displacing the toroid microresonator with respect to the centre of the nanostring. Fig. 2.17b shows the squared oscillation amplitude of the
second and fourth out-of-plane modes normalized to the fundamental out-of-plane mode. The measured amplitudes are in good agreement with the theoretical expectation taking into account the transverse optical sampling length ¯ly (see Eq. 2.14). In fact, mini-
mizing the signal of even eigenmodes constitutes a simple way to find the centre of the nanostrings and thus optimize the transduction of the fundamental mode. The measured out-of-plane modes allow inferring the ratio S/ρ = (1.6±0.1)105m2/s2. For LPCV de-
posited high-stress silicon-nitride a density of (3±0.5)103kg/m3 is expected, see Ref. [131]
and references therein. Thus, the resulting tensile stress in this particular sample is given by S = (0.5±0.1) GPa. Values up to S = 0.8 GPa were, however, obtained using other samples. The measurement is moreover also sensitive to in-plane modes of the string (a higher order optical mode is used for this measurement). Owing to the high aspect-ratio of the nanomechanical oscillators, these exhibit considerably different frequencies and in particular do not follow linear dispersion, as shown in Fig. 2.17a. This is due to the in- creased value of the cross-sectional inertia (by a factor (w/t)2) for modes oscillating along the longer axis of the string. With this additional mode family, the Young’s modulus of the employed strings can be calculated. Using the above value for density and stress in order to fit expression (2.20) to this mode family (see Fig. 2.17a), one obtains a Young’s modulus of Y = (55±9) GPa which is lower than the value measured in Ref. [131] for higher stress (S≈1.2 GPa) LPCVD silicon nitride (Y ≈200 GPa).
2.3.5
Vacuum optomechanical coupling rate
In the previous section, we followed the convention of normalizing the nanomechanical oscil- lator’s displacement by the overlap of optical and mechanical modes. This is a very intuitive approach and has its origin in Fabry-Perot type cavities [139] following the aforementioned conceptual identity of static and dynamic optomechanical coupling coefficientsgfor all me- chanical modes. Using integrated structures, however, the notion of static coupling rates is far less intuitive. In complex structures, e.g. photonic crystal cavities with very complex three-dimensional displacement patterns, the choice of a single one-dimensional coordinate becomes less obvious and thus the separation of coupling coefficient and displacement be- comes somewhat arbitrary. A quantity that is, however, a-priori well-defined is the cavity frequency noise spectrum Sω induced by the mechanical oscillator’s motion. In order to
characterize the optomechanical coupling without the necessity of introducing a coupling coefficient g and a corresponding effective mass that both depend on the definition of the displacement coordinate, the zero-point coupling rate g0 = g ×xzpf (cf. section 1.1.1) is
ideally suited as it is independent of the actual definition of the measured displacement. It can be expressed solely in terms of the cavity frequency noise spectrum and thus avoids the ambiguity in the definition of coupling coefficient and corresponding displacement. It can be written as [3] g0 = s ~Ωm 2kBT Z ∞ 0 Sss ω[Ω] dΩ 2π , (2.37)
2.3 Transduction and actuation of nanomechanical motion 61
~ g0
Figure 2.18: Experimental determination of the vacuum optomechanical coupling rate. The (single-sided) frequency noise induced by the fundamental mode of a nanomechanical string is fitted with a Lorentzian. The integral of the corresponding trace (shaded area) directly yields the vacuum optomechanical coupling rate g0/2π according to Eq. (2.37),
without having to introduce effective masses or coupling coefficients.
where the superscript inSss
ω stresses the fact that here, the experimentally acquired single-
sided spectrum is used9. Provided that the mechanical mode thermalizes to its surrounding bath (which can be easily verified e.g. by measuring the frequency spectrumSω at varying
input power levels), this quantity is thus most easily experimentally accessible. Besides the cavity frequency noise spectrum Sω[Ω] only the mechanical oscillator’s eigenfrequency Ωm
is required which can be simply extracted from the measured spectrum. Fig. 2.18 shows a calibrated frequency noise spectrum Sω[Ω] (see appendix B.3.2 for details on the calibra-
tion), reflecting the Brownian motion of a nanomechanical string (30×0.7×0.1µm3). Fit- ting the measured trace and directly integrating the Lorentzian frequency noise spectrum (without the constant measurement background) yields a value of g0/2π = (520±50) Hz.
The error bar reflects both the uncertainty in the frequency noise calibration and in the Lorentzian fit. Using the conventions introduced in the previous chapters, a zero-point motion of xzpf ∼ 20 fm and correspondingly a coupling coefficient of g/2π ∼ 30 MHz/nm
is obtained. In contrast to these, the evaluated zero-point coupling g0 is independent of
the particular definition of the oscillator mass and thus yields an unambiguous measure for the optomechanical interaction strength. Here, g0 is larger than the typical mechanical
dissipation rates Γm/2π ∼ 100 Hz. However, it is considerably smaller than the typical
optical dissipation rates κ/2π ∼ 5 MHz. Compared to other cavity optomechanical sys- tems one finds a huge variation of attainable coupling rates, ranging from g0/2π ∼1 Hz in
moveable mirror setups (e.g. Ref. [39]) up to g0/2π ∼100 kHz in optomechanical crystals
[53]. All systems, however, share the common feature that the optomechanical coupling rate is orders of magnitude smaller than the optical cavity decay rateκ. Thus, in order to reach the regime of strong optomechanical coupling, which leads to normal mode splitting [36, 145], a strong intracavity field ¯a, enhancing the vacuum optomechanical coupling rate to an effective coupling rate ¯a×g0, is required. Strong coupling was first experimentally