2 MARCO TEÓRICO
2.4 LAS TECNOLOGÍAS DE LA INFORMACIÓN Y
Let us first discuss the classical sources of frequency noise leading to measurement impre- cision, given by thermorefractive noise Sωthr and Brownian motion of the microresonator
ω2/R2Sµ
x. A broadband measurement illustrating these is shown in Fig. 2.25.
2.5.2.1 Thermorefractive noise
The thermodynamic temperature fluctuations within a given volume element scale inversely proportional to its size [150]. By means of a temperature dependent refractive index, these temperature fluctuations lead to a fluctuating phase of electromagnetic waves when propagating through the volume element. In an electromagnetic resonator this effect thus causes fluctuations of its resonance frequency: thermorefractive frequency noise, which particularly plays a role for small mode-volume resonators. The fundamental temperature fluctuationsST[Ω] within the cavity mode volume lead to thermorefractive frequency noise
given by
Sωthr[Ω] = (ω/n·dn/dT)2×ST[Ω]. (2.39)
For spherical optical WGM microresonators an analytic model of the temperature fluc- tuation spectrum ST[Ω] has been developed [150]. This model is adapted to the toroid
microresonators employed in this work [2]. To this end, the optical mode profile is ap- proximated by a Gaussian ellipse with semi-axes rz given by Eq. (1.73) and rx given by
b = 0.77R/`2/3 11. From Eq. (1.75) (see also Ref. [151]), the angular mode number ` of
the fundamental optical mode can be deduced:
`+ 1.8558`1/3+1 2 r R r − P n √ n2−1 =nk0R . (2.40)
11 This corresponds to the equivalent mode radius when the Bessel function in Eq. (1.72)
2.5 Quantum measurements 73 100 K 30 K 3 K 300 K b, a,
Figure 2.25: Broadband frequency noise (single-sided) of a toroid microresonator. a, The predominant noise source is thermorefractive noise. A fit according to Eq. (2.41) yields good agreement (red line). The equivalent frequency noise caused by shot-noise (grey) is much lower due to sufficiently high input power (Pin ∼ 10µW). The mechanical modes
of the microresonator (pink dashed line) have negligible off-resonant contribution (8 MHz: calibration marker), here. b, Thermorefractive cavity frequency noise is expected to con- tinuously reduce for lower temperatures, particularly taking the temperature dependence of silica’s material parameters into account. For 30 K, a suppression of 25 dB compared to room temperature is expected. Also the toroid mechanical background noise reduces with temperature despite the concomitant decrease of mechanical quality factors (see sec- tion 3.5.2 for details) which is explicitly taken into account, here. The right axis shows the corresponding noise floor in displacement units, assuming a coupling coefficient of
g/2π= 50 MHz/nm.
The optical mode volume is given by V = 2π2rxrzR. Defining a cut-off time τ via τ−1 =
(4/π)1/3D(1/r2
x + 1/r2z), where D denotes the thermal diffusivity of silica one arrives at
the following approximation for the (single-sided) thermorefractive frequency noise for a fundamental mode: Sωthr[Ω] = ω n dn dT 2 ×(16π) 1/3k BT2τ V ρC√Ωτ 1 (1 + (Ωτ)3/4)2 , (2.41)
where C and ρ are the heat capacity (per unit mass) and density of silica. The approxi- mation is valid for Fourier frequencies Ω/2π D/(2πr2)∼40 kHz.
The resonator used for obtaining the data shown in Fig. 2.25 has a major radius of
R = 18.4µm and a minor radius of r = 1.8µm. For a temperature of T = 300 K, the probe wavelength of λ = 853 nm and the refractive index of silica n = 1.45, expression (2.41) and the constant shot-noise background of Sshot
ω [Ω] = (2π6
√
Hz)2 are fitted to the
measured data by keeping all parameters fixed and using only the mode axes rx and rz
as fit parameters. Increasing rz by×1.42 and decreasingrx by×0.75 (thus increasing the
T dn/dT (K−1) C (J/kgK) k (J/msK)
300 K 8.7·10−6 730 1.38 100 K 3.5·10−6 260 0.64
30 K 1.2·10−6 60 0.26
Table 2.1: Material parameters of silica from Refs. [153] and [154].
Fig. 2.25. The parameter adjustments are attributed to the deviation of the actual mode profile from its Gaussian approximation and the simplified boundary conditions assumed in the analytic approximation.
In total, to a level of∼10% good agreement between measurement and model is found. Thus, thermorefractive noise which may be of relevance also for other nano-optomechanical transducers [49, 152] is well understood in our system. This allows an extrapolation of its behaviour to lower temperatures. In addition to its direct temperature dependence, expression (2.41) also indirectly depends on temperature via the temperature dependent material parameters dTdn,C and D=k/(ρC), wherek is the thermal conductivity of silica. In order to extrapolate the temperature dependence we fit the tabulated values of these parameters [153, 154] via polynomials. Table 2.1 shows values of dTdn, C and k for a few representative temperatures. The overall temperature dependence of Eq. (2.41) predicts a steady reduction of thermorefractive noise for lower temperatures. In Fig. 2.25b, the thermorefractive noise contributions expected for 100 K and 30 K are compared to the room temperature data. At 30 K the level of thermorefractive noise is reduced by approximately 25 dB compared to the room temperature value. Thus, already at 30 K its contribution to the measurement imprecision would be negligible. Since the temperature dependence of the refractive index of silica is known only above 30 K we cannot estimate its exact quantitative behavior for lower temperatures. Most likely it will, however, continue to decrease further for lower temperatures and in similarly sensitive measurements [8] no evidence for thermorefractive frequency noise in toroid microresonators was found.
2.5.2.2 Toroid mechanical modes
As can be seen in Fig. 2.25, also mechanical modes of the toroid contribute to the measure- ment background. Other than thermorefractive noise, these are, however, peaked around their respective resonance frequencies. By appropriate choice of the cavity geometry, no mechanical mode will be present below 12 MHz, i.e. the frequency range of interest, here. Thus, it is essentially the low frequency tail of all mechanical modes present in the toroid that contributes to the background noise in the frequency band of interest. Since the low frequency tail of a mechanical mode scales as Sx
Ω→0
∝ Γm/Ω4m it is desirable to maximize
both the quality factors and the resonance frequencies of the toroid modes. The latter can be simply accomplished by minimizing the size of the cavity and the undercut of the silica disk. This requirement thus nicely coincides with the properties needed for large optome- chanical coupling. In order to obtain high mechanical quality factors of the mechanical modes, the rotational symmetry of the toroid as well as the supporting silicon pillar are cru-
2.5 Quantum measurements 75
cial since asymmetries always give rise to low quality factor mechanical modes. Satisfying symmetry can be achieved although the resonators are fabricated at the edge of their chip support by careful microfabrication (see appendix B.1.1 for details) and thus the current measurements are not limited by mechanical noise of the microresonator. In chapter 3, the mechanical modes of the toroid microresonator will be analyzed more extensively. As will also be shown there, the mechanical quality factor of the intrinsic silica microcavity modes deteriorates at lower temperatures (see section 3.5.2), as opposed to the mechanical quality factor of silicon nitride nanomechanical oscillators [143]. The micromechanical background noise expected for different cryogenic temperatures taking into account the known temper- ature dependence of the mechanical quality factors of toroid microresonators (see section 3 for details) is depicted in Fig. 2.25b. Despite the reduced quality factors, this noise source also reduces sufficiently fast with temperature and should not limit the sensitivity to val- ues above the shot-noise level. Consequently, at low temperatures shot-noise should be the only relevant contribution to the measurement imprecision and an imprecision at the level of 10−17m/√Hz should be feasible (assuming a coupling coefficient g/2π = 50 MHz/nm),
as shown in Fig. 2.25b.