2.7 Approaching the quantum ground state
2.7.3 Assessing measurement backaction
The strong suppression of thermal noise in the measurements presented here allows an interesting analysis of the data from the perspective of quantum measurement. As discussed in section 2.4.1, fundamentally, the achieved to- tal uncertainty in the measurement of the displacement of the mechanical oscillator is limited by two effects: imprecision and backaction. The impreci- sion in our experiment is given by detection shot noise, in the detuned case discussed here it is described by equation (2.181).
On the other hand, the light used to measure the mechanical oscillator exerts backaction on it. Inevitably, intracavity photon number fluctuations give rise to a fluctuating force the spectrum of which can be derived to read [10] ¯ SF Fba,qn(Ω) =!2g02¯a2κ ∆¯ 2 + (κ/2)2 +Ω2 ¯ ∆4 + 2 ¯∆2((κ/2)2−Ω2) + ((κ/2)2 +Ω2)2. (2.183)
These force fluctuations are usually referred to as quantum backaction, sim- plifying to ¯ SF Fba,qn(Ωm) ≈ 2g02Pinηc! ωΩ2 m (2.184) in the resolved-sideband regime (|∆¯| = Ωm . κ). Note that the spectra of
imprecision and backaction noise, as well as their possible correlationSxF(Ω) reflect properties of the measurement device (the cavity pumped by a laser field)—independent of the mechanical oscillator.
Other sources of measurement backaction include excess noise in the intracavity photon number, which may arise from laser frequency fluctua- tions [8, 236, 237]. This effect can be ruled out in this work as the employed Ti:S is known to exhibit only quantum fluctuations at the Fourier frequencies of interest. Another possible source of measurement backaction is heating of the torus due to light absorption. This increases the temperature of the structure, and raises the level of Langevin force fluctuations driving the me- chanical oscillator.
A series of cooling measurements (figure 2.50) indeed reveals a devia- tion from the simple relation (2.147), which can however be reproduced by introducing a heating term
Tm =
Γm(T") Γm(T") +Γdba
100 100 1000 Mechanical linewidth (kHz) 300 30 Mechanical linewidth (kHz) T e mp e ra tu re (K) Ph o n o n o ccu p a to in Ph o n o n o ccu p a to in Ωm/2π=65.1 MHz 1 0.5 100 200 300 100 200 300 2 1 T e mp e ra tu re (K) Ωm/2π=121.7 MHz
Figure 2.50: Resolved-sideband cooling of two samples with frequencies of Ωm/2π =
65.1 MHz (left panel) and Ωm/2π = 121.7 MHz (right panel). The graphs show the
mechanical linewidthΓeff (abscissa) versus the derived mode temperatureTm (ordinate)
during a cooling run. Open points correspond to measurements with the laser tuned close to the optical resonance (no optical cooling), and filled points to measurements with the laser tuned close to the lower sideband. When varying the power of the cooling laser, both linewidth and mode temperature are changed. Deviation from the simple cooling behavior (blue line) is attributed to an increase of the structures temperature (red dashed line), taken into account in a more elaborate model (green dashed line). Scatter is due to varying operation conditions, uncertainty in phonon occupation for each point is<30%. Figure from ref. [10].
with T" = T +∆Tabs, where ∆Tabs is proportional to the power circulating
in the cavity. In addition, we have taken also the temperature dependence of the mechanical damping Γm(T") into account. For the 65 MHz-oscillator
below 2 K, with [11] dΓm/dT ≈2π16 kHz/K, we find a heating of about 5 K
per Watt of circulating power. Similar values were extracted from studies of optical multi-stability at low temperature at a wavelength of 1.5µm, cor- roborating the attribution of the observed backaction effect to laser-induced heating.
A comparison with a second cooling run with a different sample with
Ωm/2π = 121.7 MHz and Qm = 2,200, but a broader WGM resonance with κ/2π = 155 MHz emphasizes the importance of the resolved-sideband regime for the efficiency of cooling in the presence of laser absorption: A significantly more pronounced heating effect prevents reaching occupations below "n# = 100, in spite of the higher mechanical frequency (figure 2.50).
From the data of the "n# ≈ 63-cooling run we can now extract quanti- tative values of the corresponding backaction force fluctuations. As a very conservative upper limit, we may use the total thermal force fluctuations
¯
SF Fthe(Ω) = 2meffΓmkBT", and find a value of
p¯
Sthe
F F(Ωm) = 8 fN/
√
Hz. If we consider only the temperatureraise ∆Tabs caused by the laser absorption as
the backaction of the measurement, a lower value ofpS¯ba
F F(Ωm) = 4 fN/
√
Hz is found. In these assessments, we benefit from the low occupation which al-
lows us to extract the effect of measurement backaction, as it large enough to be observed on top of the background of the thermal noise.
It is interesting to compare these findings with fundamental limits. In- deed, quantum mechanics imposes an inequality on imprecision and back- action noise, which, for the particular case of an optical measurement of a mechanical oscillator’s displacement, can be written as simple as
¯
Sxxim(Ω)·S¯baF F(Ω) ≥
!2
4 . (2.186)
This relation can be considered a manifestation of the Heisenberg uncertainty principle in the context of continuous position measurement [94].
Taking the force noise extracted from our data, and the experimental imprecision ofpS¯im
xx(Ωm)≈1.4 am/
√
Hz achieved in the same measurement, an upper limit from the backaction-imprecision product of
q ¯ Sim xx(Ωm)·S¯F Fthe(Ωm)≈220 ! 2
is found. Considering only the absorption-induced heating as a backaction mechanism, an even lower value of
q ¯ Sim xx(Ωm)·S¯F Fba (Ωm)≈100 ! 2
is found from our experiments. This is an order of magnitude lower than the values achieved with nanomechanical oscillators cooled in dilution refrig- erators: Readout with an atomic point contact [255] achieved a backaction- imprecision product of 1700±400!/2, while measurements using a sSET [246]
have achieved a value15 of ∼800!/2.