Estado de la cuestión
4.3 Análisis de datos
The preceding measurements were performed using λ= 935 nm radiation from a low power diode laser. After propagating through the optical beam path (Section 5.3.3), we were restricted in the power available from this laser to moderate input powers of several tens of microwatts. We also observed that cooling to an occupation number of ¯n66<104 with this laser resulted in a distorted
thermal noise peak, as evidenced in the blue peak in figure 9.2. As discussed above, we suspect that this behavior is due to the relatively high frequency noise spectral density of the diode laser at 5 MHz (∼10−16 m/√Hz when expressed as apparent cavity length noise; Section 8.8.1.1).
As soon as the preliminary results in [14] were finalized, we switched to a high-power, low- noise titanium-sapphire laser (ti-sapph). The measured frequency noise on this laser at 5 MHz is
Figure 9.3: Demonstration of cavity cooling [14] of the (6,6) (red circles) and (1,1) (grey squares) modes with frequencies {Ω11/2π,Ω66/2π} = {0.80 MHz,4.83 MHz} and initial quality factors
{Q11, Q66} = {5.8×104,1.5×106}, where Qij is measured via ringdown. For these data the cavity input power is fixed so that hPout(∆ = 0)i ≈ 10 µW. The representative error bars show the uncertainty due to our knowledge of{gm, ηij, mij} (Section 8.7.4). Panel (a): measured mode temperature (points) and theoretically expected cooling (lines) as a function of detuning ∆ from the cavity resonance. Panel (b): Corresponding measured effective mechanical linewidth and theoretical expectation. In both models, we assumeκ/2κ2= 1 when computing the intracavity photon number
(Eq. 9.19b).
∼10−19m/√Hz (expressed as an apparent cavity length noise) (Section 8.8.1.2). We maneuvered
the wavelength of this laser to the opposite side of the mirror coating curve (Figure 6.3), λ≈ 810 nm, to maintain a moderate finesse of F ≈ 104 at which the cavity length could be stabilized.
For the following measurements, the cavity linewidth was measured to beκ≈8.5 MHz (HWHM) when the membrane was positioned to maximizegm. The absolute value ofgmalso changes at this wavelength for two reasons: because of the tighter intracavity standing wave spacing (g0 ∝λ−1)
and because of the increased reflectivity of the film. (According to Eq. 3.3 the reflectivity of a 50 nm film changes from 0.423 to 0.465 for an index of 2.0.) The net increase in the optomechanical coupling over the λ = 935 nm field is (935/810)×(0.465/0.423) = 1.27. The intracavity photon number for a given input power also decreases by a factor of λ(Eq. 9.19b). The optical damping rate for a given input power (Eq. 9.24) then changes fractionally by a factor ∼ λ−1g2
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measurement model:Figure 9.4: Stronger damping of the (6,6) mode with a low noise Ti-Sapph laser operating at 810 nm. Here the detuning has been fixed at ∆/(2π) =−κ/(2π) =−8.5 MHz. The effective damping rate is obtained from a Lorentzian fit to the apparent cavity length noise spectrumSL (Eq. 9.29). The points at higher power have been obtained from a fit to multiple Lorentzian, including the off-resonant displacement of nearby vibrational modes, as shown in Figure 9.6. Note that the model assumesκ/(2κ2) = 1.
(935/810)×(0.465/0.423)2×(13.75/8.5)≈2.2.
In Figure 9.4, we fix the input field detuning at ∆ = −κand change the optical power from hPout(∆ =−κ)i= 0.1µW to 100µW, beyond which the cavity length servo is unstable (a problem that we have yet to solve). We then compare the measured damping rate of the (6,6) mode to the weak cooling model (Eq. 9.4), using κ/2κ2 = 1. We observe a systematic discrepancy of
≈ 50% consistent with our uncertainties in{gm, η66, m6} and the predicted value of κ/2κ2 = 1.3
(Eq. 9.20). At large optical powers the off-resonant displacement noise of the (6,6) mode begins to overlap significantly with neighboring vibrational modes. We then had to fit to multiple Lorentzians to estimate the linewidth, an example of which is shown in Figure 9.6. At the highest power, the observed optical damping rate is Γ66
opt≈10 kHz, which suggests a final occupation number of ¯
n66≈1.3×106∗3.2/104≈410 phonons.
In Figure 9.5, we vary both the the detuning and power in order to minimize and maximize damping of the (6,6) mode. As a consistency check on the calibration ofSb66b66(Ω), we plot both the damping rate inferred from the width of a Lorentzian fit against the effective temperature inferred the area beneath the Lorentzian fit. We also compare these measurements to the weak cooling
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0.1 W measurement 100 W model for:Figure 9.5: Demonstration of optomechanical cooling of the (6,6) vibrational membrane mode (Ωm/2π = 4.83 MHz) from room temperature to approximately 100 mK, corresponding to a re- duction in occupation number from 1.3×106 to ≈500. Input power has been varied so that the
transmitted power on resonance, hPout(∆ = 0)i, ranges from 0.1 µW to 100µW. Detuning is also varied from ∆> 100 MHz to ∆≈Ω66 in order to minimize and maximize the damping rate. We
plot the measure damping rate vs. the measured effective temperature (green points) and compare to the model (red curve) Tef f66 = 298 K ×Γ66/(Γ66+ Γ66opt), using Γ66/2π= 3.2 Hz. Data points
at low temperature were fit to multiple Lorentzians to include off-resonant displacement of nearby vibrational modes, as shown in Figure 9.6.
model (red line) for whichT66
ef f = 298K×Γ66/(Γ66+ Γ66opt), with Γ66 = 3.2 Hz. High-temperature
measurements are omitted because of the difficulty in obtaining the narrow natural linewidth of the oscillator — this appears to be due to thermal drift of the peak frequency. For temperatures
T66
ef f .100 K, the discrepancy between the linewidth- and area-inferred temperature is<50%, with the lowest measured temperatures coinciding with ¯n66 ≈400− −500. The growing discrepancy at
occupation numbers ¯n66<1000 is believed to be due to the encroachment of off-resonant vibration
from neighboring thermal noise peaks, as discussed in the next paragraph.
When optically damping the (6,6) mode to Γ66opt ∼ 10 kHz, we begin to observe an elevated displacement noise background due to the off-resonant vibration of neighboring modes. For the low temperature point in Figure 9.5, the optical damping rate and effective temperature were obtained from a multiple-Lorentzian fit including vibrational modes (6,6), (3,8):(8:3), and (5,7):(7,5). An example is shown in Figure 9.6. Here we plot the multimode spectrum in terms the apparent
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model for = 430
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(6,6) (3,8):(8,3) (5,7):(7,5)Figure 9.6: Multimode cooling: neighboring thermal noise peaks begin to overlap when Γopt is on the order of the mode frequency splitting. Here we show the thermal noise in the vicinity of the (6,6) mode when an optical damping rate of Γ66
opt≈10 kHz has been applied. We express the noise in terms of apparent cavity length displacement,SL(Ω) (Eq. 9.29). The dotted line is a multiple-Lorentzian fit to modes{(6,6),(3,8),(8,3),(5,7),(7,5)}which assumes that their noise adds incoherently. This fit is used to obtain the the value of Γ66optandTef f66 for one of the bottom-right points in Figure 9.4. As a consistency check, we use the weak cooling model (9.17) with input power as a free parameter to generate the solid black curve. The model is constrained to give an occupation number of of ¯n66=
430 for the (6,6) mode, corresponding to the temperature inferred from the measured values of Γ66
opt and Γ66. Quality factors and overlap factors for each mode have been independently determined for
the model.
cavity length displacementSL(Ω) =Pijη
2
ij(gm2/g02)Sbij(Ω) (see Eq. 7.28). In the figure, the dotted line corresponds to a five-Lorentzian fit with the linewidth and area left as free variables for each peak. This line is used to estimate the linewidth and temperature of a point in Figures 9.4 and 9.5. Also shown is a model for multimode cooling based on an incoherent sum of five noise peaks, all damped according to Eq. 9.17. To generate this model, we separately measure the intrinsic mechanical quality factor of each mode using a HeNe and a piezo shaker (Section 5.2.3.2); this gives {Q66, Q38, Q83, Q57, Q75} = {1.5,1.5,1.4,0.84,0.71} ×106. From the transverse position of
the optical beam, which was remeasured at a position (x0, y0) = (40.5µm,120µm) we determine
the different overlap factors for each mode as {η66, η38, η83, η57, η75} ={0.63,0.11,0.36,0.10,0.11}.
In the model we also assume that each mode is decoupled, so the displacement noise peaks add incoherently. The model is constrained so that ¯n66= 430, corresponding to the upper bound for the
temperature estimate obtained from the linewidth measurement in Figure 9.6.
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Figure 9.7: Measurement (green) and model (red) of multimode cooling of the membrane compared to measurement (black) and model (gray) of end-mirror substrate noise. Noise is expressed as apparent cavity length displacement, SL(Ω) (Eq. 9.29). Measured cooling data corresponds to the data in Figure 9.6. Modeled cooling data represents an extension of the model in Figure 9.6, which is bootstrapped to ¯n66 ≈430 based on the linewidth inferred from a experimental fit to the
measurement. Substrate noise curves correspond to measurement and model with the membrane removed, as discussed in Section 8.8.2. Note that substrate noise and off-resonant vibration of the membrane both provide a broadband displacement noise background of∼10−17− −10−18 m/√Hz.
This is significantly higher than the off-resonant background of just the (6,6) mode, shown in dashed red. The broadband noise background is expected to match the thermal noise of (6,6) mode if cooled to ¯n66<100 using current parameters.
displacement noise (green) against the same multimode model (red) extending to a span from 100 kHz to 5 MHz. For the extended model we assume that Qij = 1.5×106 for all modes other than those characterized specifically, {(6,6),(3,8),(8,3),(5,7),(7,5)}. Natural frequencies are extrapo- lated from Ω66/2π= 4.83 MHz and overlap factors are inferred from (x0, y0) = (40.5µm,120µm).
The dashed red line shows the contribution from only the (6,6) mode to highlight the fact that the background is dominated by the off-resonant contribution from other membrane modes. In the next chapter, we argue that placing the optical mode near the center of the membrane can significantly reduce this “off-resonant” background because in this caseηij vanishes for all even-ordered modes (e.g., (2,2) and (2,3)). The remaining displacement noise background is anticipated to be the ther- mal noise of the end-mirror substrates, as discussed in Section 8.8.2. To illustrate this point, in Figure 9.7 we have added the measurement (black) and model (gray) of the end-mirror substrate noise for the bare cavity, as discussed in Section 8.8.2. We note that this background is significant
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Figure 9.8: (a) Calculated achievable phonon number as a function of coupling strength geff for
parameters described in the text. The solid line is for a cavity withκ= 2π×5 MHz and the dashed line for κ = 2π×1 MHz. The green line would result for κ = 2π×5 MHz upon increasing Qm to 4×107, which may be possible for a thinner SiN film. (b) Emergence of normal-mode splitting
as the cavity linewidth is varied for fixed mechanical frequency ωm = 2π×5 MHz and coupling
geff = 2π×1 MHz. The exact spectrum Sb(ω) is displayed in this density plot where zero weight
corresponds to the blue shading and white to maximum mechanical response. The lines mark the position of the parameters used in (a).
for occupation numbers of < 100 for the (6,6) mode. Ways around this problem include using a smaller or higher mechanical quality membrane, cryogenic pre-cooling of the apparatus, or active feedback. We address this subject in detail in the next chapter.