INTERPRETACIÓN DE LOS DATOS

and heating rates in the trap tuned to this mode frequency, the maximal SNR is comparable to that of the c.o.m. mode, but for longer hold time th = 20 ms. The comparable value of the optimum coupling signal for the two studied modes is rather a coincidence. It is a consequence of the trap frequency scaling of the parasitic loss, and thus depends on the trap geometry, the atom number Nr, the technical current noise level, and also on the value of the cantilever eigenfrequency.

5.6. Readout sensitivity

So far we have shown that resonant excitation of atomic motion via surface forces can lead to significant trap loss after a few ms of interaction time. This demonstrates that the atoms can be used for readout of cantilever motion. To determine the achievable sensitivity of the readout method we use parameters that maximize the SNR and measure the signal as a function of the cantilever amplitude.

Figure 5.11 shows measurements of the contrast as a function of the cantilever amplitude for the c.o.m. mode. For comparison, the signal for an off resonant trap with ωz = 4 kHz is shown. The measurements yield a minimum resolvable r.m.s. cantilever amplitude of arms = 13±4 nm for SNR=1 without averaging, where the

error is dominated by the uncertainty of the cantilever amplitude (see chapter4.1.2). As introduced already in the previous chapter, we identify two origins for the observed sensitivity limit. First, the lifetime of the atoms in the trap set by para- sitic loss limits the coupling duration and thus the achievable excitation for small amplitudes. This is indicated by the measurements shown in Fig. 5.10. Second, trap anharmonicity leads to dephasing of the cloud oscillations and thereby to a maximum cloud amplitude for a given cantilever amplitude. A detailed picture of the dynamics of the cloud and the influence of the trap anharmonicity is found by numerical simulations which we discuss in chapter 5.8.

We also perform coupling measurements on the dielectric back side of the can- tilever in a trap with comparable trap frequency and depth U0. We observe an

approximately linear dependence C ∝ a on both sides as long as the contrast does not saturate, i.e. for C <1. The coupling signal can thus be quantified by the value C/a, and we find it to be a factor β = 3.2±0.6 smaller on the back side than on the metallized side. Since the origin of the excitation is the modulation of δzt, one can conclude that C ∝ δzt. With Eq. 5.1, the contrast is thus determined by the coupling strength parameter, C/a ∝ = (mω2

z)

−1_∂2_U

s/∂z2, and thereby related to the curvature of the surface potential. In chapter 5.4 we use this result together with static measurements to quantitatively infer the absolute strength of Uson both sides of the cantilever.

0 50 100 150 0 0.2 0.4 0.6 0.8 1 cantilever amplitude a [nm rms] C

Figure 5.11.: Contrast C for the c.o.m. mode as a function of the cantilever amplitude a. The smallest detectable cantilever amplitude is arms = 13±4 nm

for SNR=1 without averaging. For measurements on the metallized side (dark

blue) we find C/a = 1.1×10−2nm−1 while on the dielectric backside (light blue), C/a= 3.5×10−3nm−1, a factorβ= 3.2±0.6 smaller. For comparison, the contrast for an off-resonant trap withωz/2π= 4 kHz is shown (red). The dotted line indicates

the rms noise of the measurement.

Improvement of the sensitivity

We have used trap loss as the simplest way to detect BEC dynamics induced by the coupling. For trap loss to occur, the cantilever has to drive the BEC to large amplitude oscillations with ∼ 103 phonons. Achieving such cloud amplitudes with small cantilever amplitudes is hindered by the strong trap anharmonicity close to the barrier, and by the finite trap lifetime. By contrast, BEC amplitudes down to the single phonon level could be observed by direct imaging of the motion. A coherent state|αi of the c.o.m. mode of N = 100 atoms with α= 1 released from a relaxed detection trap withωz = 2π×100 Hz has an amplitude of

p

2_~ωz/mN αt= 400 nm after t = 4 ms time-of-flight. This is about 10% of the BEC radius and could be resolved by absorption imaging with improved spatial resolution. From a simulation of the cloud excitation (see chapter5.8) we estimate that arms= 0.2 nm

would excite the BEC to α = 1 within th = 20 ms and could thus be detected. This would be sufficient to resolve the thermal motion of our cantilever, which has a relatively large effective mass Meff = 5 × 10−12 kg and correspondingly small

r.m.s. thermal amplitude ath = p

kBT /Meffω2m = 0.4 nm, where T = 300 K is the cantilever temperature. Using similar cantilevers with comparable ωm but smaller Meff [194, 52], the thermal motion would be detectable already with the presently

used technique.

Furthermore, one can harness the result that a stronger surface potential leads to a stronger coupling. Stronger potentials could be generated e.g. by electrostatic