VIOLENCIA SEXUAL CONTRA LAS MUJERES EN LOS MONTES DE MARÍA.

|1〉 ≡ |2,+1〉 |0〉 ≡ |1,-1〉

Normalized atom number

MW + RF pulse duration τ [ms] 0 1 2 3 4 5 6 7 8 0.0 0.2 0.4 0.6 0.8 1.0

Figure 4.6: Normalized atom numbers Ni/(N0 +N1) in state |ii obtained

by integrating the absorption images of Fig. 4.5.

the microwave is applied to waveguiding structures on the chip instead of being radiated from outside the vacuum chamber, as explained in chapter5. Below we use Rabi oscillations for internal-state manipulation. With the help of a resonant π-pulse (ΩRτ =π, ∆0 = 0), the atoms can be transferred

from|0i to |1i. The measured transition probability is >95%. With a π/2- pulse (ΩRτ = π/2, ∆0 = 0), an equal superposition √1₂(|0i+eiϕ|1i) can be

prepared, the relative phaseϕdepends on the phase offset of the two-photon drive. Before we investigate trap and coherence lifetimes of such states near the surface, we describe our calibration of the atom-surface distance.

4.6

Calibration of the trap-surface distance

An investigation of surface effects requires an accurate calibration of the distance d between the chip surface and the center of the trap. The trap- wire distancedw can be inferred from a simulation of the trapping potential,

but d = dw −he depends on the thickness he of the epoxy layer above the

wires, see Fig. 4.4(b), which was not well controlled during chip fabrication. We therefore use two different methods to directly measure d.

Distance calibration by imaging with a reflected probe beam

In the first technique [147], the imaging beam is tilted by an angle of 2◦ towards the mirror on the chip surface, so that both a direct and a mirror image of the atoms are simultaneously visible in the absorption image, see Fig. 4.7. The distance between the cloud centers equals 2d. Due to the

4 Coherence near the surface: an atomic clock on a chip

finite resolution of the imaging system, this technique is limited to d > 10µm. We take images such as Fig. 4.7 at several values ofd and determine corresponding values ofdw from a simulation of the trapping potentials. This

yieldshe = 25.2±2.2 µm.

48 µm D 0.2 0.1 0.0

Figure 4.7: Absorption image taken with a tilted imaging beam. Pixel size is 10.6 µm×10.6 µm. A fit to the density distribution with two Gaussians (not shown) gives d= 24±1 µm for this picture.

Distance calibration from trap loss in the Casimir-Polder potential

In the second technique, the surface is located by measuring the loss of atoms in the attractive Casimir-Polder surface potential, compare section1.9.2and [44]. Starting with N atoms (all in state |0i) in a trap atd =d0 sufficiently

far from the surface, we move the trap to a distance d < d0 by increasing

Bb,y. We hold the atoms there for a time th, then move back to the original

location, and determine the remaining number of atoms Nr. Figure 4.8(b)

shows the remaining atom fractionχ=Nr/N as a function of d.

We model the observed loss of atoms in the Casimir-Polder surface poten- tial as a sudden truncation of the tail of the Boltzmann energy distribution due to the finite trap depth Vb, see Fig. 4.8(a). The theoretical value for the

remaining fraction of atoms is χ= 1−exp(−Vb/kBT), where Vb is the trap

depth atd=dw−he. The model containsheas the only free parameter. The

temperature T = 200 nK of the cloud used for this measurement, the atom number N = 1.48×104_{, the trap frequencies (f}

x, fy, fz) = (37,279,332) Hz,

and the scaling dw(Bb,y) are determined independently. The chip surface is

modeled as a perfect conductor, Vb is determined using Eq. (1.45) for the

surface potential.

Because of the Casimir-Polder potential, the trap disappears at finite d. The corresponding shift in d is 1.1 µm if one compares with a model where the Casimir-Polder potential is neglected. If the optical method to determine he described in the previous section was more accurate, we could use this for

4.6 Calibration of the trap-surface distance a) b) 0 5 10 0 200 400 600 z [µm] E / kB [n K ] e − E / kB T V(z) d Vb χ trap-surface distanced [µm] re m a in in g fr a ct io n χ 0 2 4 6 8 0 0.2 0.4 0.6 0.8 1

Figure 4.8: Loss of atoms in the Casimir-Polder surface potential. (a) The surface potential decreases the trap depth, atoms in the tail of the Boltzmann distribution are lost. (b) Remaining fraction of atoms χ as a function of d for a hold time th = 10 ms. Measured data (black dots) in comparison with

a fit to our model (blue line, see text). The red arrow indicates the trap with smallestd in which coherence measurements were made.

turn the situation around and use the Casimir-Polder potential to calibrate the trap-surface distance. From a fit of our model to the data, we determine he = 27.1 µm with a statistical error of ±0.1 µm, consistent with the value

of he obtained from the absorption images. Figure 4.8(b) shows the fit and

the resulting calibration ofd. Including errors in the other model parameters and the trap simulations, we estimate an uncertainty in d of ±0.8 µm. The arrow in Fig 4.8 corresponds to the trap at d = 4.9 µm, the smallest d at which coherence measurements were performed.

A hold time of th = 10 ms is short enough to avoid systematic errors

in the calibration due to other surface-related loss mechanisms, which affect the atoms only on a much longer time scale (see section4.7 below). We also neglect in our model the collisional rethermalization of the atoms during th

and the associated additional loss through evaporation [44]. In our case it is reasonable to include only the sudden loss since th < γel−1 = 40 ms, whereγel

is the elastic collision rate in the trap. Note that for significantly longer th,

we do indeed observe surface evaporation, as expected (see section4.7). We also considered the possibility that the surface potential is dominated by 87_{Rb adsorbates, as discussed in section 1.9.3. More than 10}8 _adatoms

4 Coherence near the surface: an atomic clock on a chip

trapped atomic cloud would be required to shift the measured value of he

by 0.1 µm. This corresponds to 104 _{deposited atom clouds, while we have}

intentionally deposited only 102 clouds in the course of the distance cali- bration. Even more deposited adatoms would be required if we consider a less localized adatom distribution, which is more realistic because of surface diffusion [107]. This leads us to the conclusion that the surface potential is not dominated by adsorbates, although some caution is in order because the number and distribution of atoms deposited unintentionally on the surface is not known.

With the atom-surface distance calibrated as described, we can search for possible effects of the surface on trap and coherence lifetimes.