Sub-SQL imprecision

Having characterized the different background noise contributions, now the actual measure- ments of nanomechanical motion will be presented. It is interesting to note that according to Eq. (2.38), all sources of both classical and quantum-mechanical noise are uniformly suppressed by increasing the coupling coefficientg. This is depicted in Fig. 2.26, where the measurement imprecision for a nanomechanical string (30×0.7×0.1µm3_{, Ω}

m = 8.3 MHz,

Qm = 300000, meff = 3·10−15kg) is shown as a function of optomechanical coupling for

two different input power levels. For large enough coupling coefficients, the imprecision is lowered below the SQL.

As a second variable of the measurements, the input power can be used to vary the shot noise contribution S_ωshot, which as opposed to the other contributions to Sx depends

on the optical input power Sshot

ω ∝ 1/P (cf. Eq. 1.39). Correspondingly, it is limiting for

low input power levels. For large enough power levels, it can however be reduced below the classical noise sources, as already shown in Fig. 2.25.

The first measurements with an imprecision below the SQL for a nanomechanical oscilla- tor were obtained using a 1550 nm Erbium-doped fibre laser combined with Pound-Drever- Hall detection, as described in section 2.5.1.2. Employing a κ/2π = 50 MHz resonance, a coupling coefficient of g/2π = 3.8 MHz/nm and 65µW input power, an imprecision of

Sx = (0.5−+0.50.25)SxSQLis achieved for a 4.9·10

−15_{kg nanomechanical string (Ω}

m/2π= 8 MHz,

Qm= 400000) [4]. This measurement represents the first time that a nanomechanical oscil-

lator is measured with an imprecision below the SQL. In absolute terms, the imprecision corresponds to a displacement noise floor of √Sx = 570·10−18m/

√

Hz. The error bar in this measurement however amounts to 3 dB. The exceptional dynamic range of>60 dB in addition to the nanomechanical oscillator’s narrow linewidth leads to a washing out of the peak of the spectrum in the presence of small drifts of the mechanical resonance frequency

Figure 2.26: Measurement imprecision for a nanomechanical string as a function of coupling coefficient. The measurement imprecision reduces with increased coupling g, irrespective of its origin. It can be lowered below the SQL for both P = 1µW (red) and P = 8µW (blue), using homodyne detection. The black lines are guides to the eye. Inset: FEM simulation of the fundamental mode of a nanomechanical string.

on the Hz-level. Thus, the mechanical quality factor (and hence the level of the SQL for the respective measurement) can only approximately be determined from the data which leads to a 3 dB error bar.

As mentioned earlier, in order to increase the optomechanical coupling the experiment was transferred to shorter wavelength and smaller cavities. This allowed an increase of the optomechanical coupling by more than one order of magnitude. Consequently the power needed to reach the SQL (cf. Eq. 1.51) could be reduced by two orders of magnitude. These measurements were performed with the titanium-sapphire laser system presented in section 2.5.1.1. Fig. 2.27a shows a spectrum acquired with a coupling coefficient of

g/2π = 40 MHz/nm and a cavity resonance with a critically coupled linewidth of κ/2π = 20 MHz (Q0 = 7.2·107). Already 1µW of input power is sufficient to obtain an imprecision

below the SQL, Sx = (0.47±0.2)SxSQL 12. By increasing the input power, the shot-noise

limited imprecision can be lowered to values > 10 dB below the SQL, using only 8µW, as depicted in Fig. 2.27b. Due to the thermal instability of toroid microresonators in connection with residual vibrations in the optomechanical system, however, the coupling coefficient has to be slightly reduced for this measurement (to g/2π = 15 MHz/nm) in order to allow a stable laser lock. Consequently, the higher power measurement is partly limited by thermorefractive frequency noise (Sthr

ω ∼(2π14)2Hz at Ω∼8 MHz).

12 _{The mechanical quality factor is evaluated using small coupling. Data whose fitted quality}

factor deviates by more than 15% from the low coupling value are discarded. This leads to the reduced error bar compared to the previous measurement.

2.5 Quantum measurements 77

>10 dB a,

b,

Figure 2.27: Measurement imprecision below the SQL for a nanomechanical string (30×0.7×0.1µm3_{, Q}

m = 300000, meff = 3·10−15kg). Spectra (blue, single-sided) and

corresponding fits (red) of the nanomechanical oscillator’s fundamental mode at 8.35 MHz. a, The measurement imprecision (grey, acquired by removing the nanomechanical oscillator from the microresonator, i.e. at g = 0) lies 3 dB below the SQL (dashed line). 8.4 MHz: in-plane mode of the nanomechanical oscillator, 8 MHz: calibration marker, 9 MHz: me- chanical modes of the microtoroid. b, Using higher input power (P = 8µW), the shot-noise level (grey, measured by detuning the laser from cavity resonance) is lowered more than 10 dB below the SQL (dashed line).

Figure 2.28: Measurement sensitivity (blue line) for an ideal measurement as a function of normalized coupling power. Both shot-noise and quantum backaction (grey) whose values are also given for an impedance matched measurement (dashed grey lines) contribute. The full squares correspond to the measured shot-noise of the traces in Fig. 2.27 whereas the stars mark the corresponding measured total imprecision. The measured shot-noise is fitted by the black line. The empty squares denote the calculated quantum backaction, corresponding to approximately nqba = 10 and nqba = 20 noise quanta, respectively.

The measurements shown here are so far the only measurements of nanomechanical motion enabling an imprecision at and below the SQL at room temperature (cf. Fig. 2.2). With the coupling coefficients g/2π > 200 MHz/nm demonstrated in section 2.2.1 even an imprecision deeply below the SQL would be possible. As mentioned above, the thermal bistability of the toroid microresonator as well as residual vibrations prevent from obtaining a sufficiently stable laser lock in that parameter range for the continuous measurements shown here. By increasing the stability of the employed experimental apparatus (e.g. by exchanging the employed turbomolecular with a vibration free ion-getter pump) and thus the accessible optomechanical coupling, considerable further improvements should be possible.