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(Ch.5), we also monitored the probability of finding a hole at the second next neigh- bors (light blue regions and data points in Fig.6.4). As both yielded the same hole probability of 6(2)%, we attribute all holes to thermal excitations and conclude that the probability of addressing a neighboring atom is indiscernibly small.

We fitted the hole probability p0(δx) of the addressed site with a flat-top model function, given by p0(δx) = A 2 erf δx+σa/2 σs +erf −δx−σa/2 σs +B. (6.3)

Here, erf(x) = 2/√πR₀xe−τ2dτis the error function,σadenotes the full-width at half-

maximum of the flat-top profile andσs the edge sharpness. The offset B is kept fixed

at the thermal contribution of 6%. We chose this model function since our HS1-pulses produce a flat-top population transfer profile, the edges of which are dominated by randomly fluctuating quantities (beam pointing and magnetic fields) following Gaus- sian statistics. The addressing fidelity is defined asF= A/(1−B)taking into account that the maximum hole probability pmax₀ = A+B also includes holes from thermal defects. These yield a hole with probabilityBat unsuccessfully addressed sites which occur with probability 1−F, such thatpmax₀ =F+ (1−F)B. From the fit, we derived a spin-flip fidelity of 95(2)%, a full-width at half-maximum of σa = 330(10)nm and

an edge sharpness of σs = 50(10)nm [Fig.6.4(c)]. These values correspond to 60%

and 10% of the addressing beam diameter, demonstrating that our method reaches sub-diffraction-limited resolution, well below the lattice spacing.

The observed maximum spin-flip fidelity is currently limited by the population transfer efficiency of our microwave sweep. The edge sharpness σs originates from

the beam pointing error of . 0.1alat and from variations in the magnetic bias field. The latter causes frequency fluctuations of∼5 kHz, which translate into an effective pointing error of 0.05alat at the maximum slope of the addressing beam profile. The resolutionσacould in principle be further reduced by a narrower microwave sweep,

at the cost of a larger sensitivity to the magnetic field fluctuations. A larger addressing beam power would reduce this sensitivity, but we observed that this deformed the lattice potential, due to the imperfectσ−-polarization, allowing neighboring atoms to tunnel to the addressed sites.

6.5 Positioning of the addressing beam

Calibration of the addressing beam position

To move the addressing laser beam in the object plane, we changed the angle of the beam entering from the reverse direction into the microscope objective using a two- axis piezo mirror (S-334 Physik Instrumente GmbH & Co. KG, Karlsruhe). The device has an angular resolution of 5µrad, corresponding to a theoretical resolution in the

a

c

Pointing offset

δ

0.2 0.4 0 -0.2 -0.4 0.8 0.6 0.4 0.2 0 1 FWHM σa = 330(10) nm Hole probability p 0 Fidelity F = 0.95(2) y x

δ

b

Figure 6.4: Addressing fidelity. The spin-flip probability was measured by sequentially ad- dressing a series of 16 neighboring sites along the ylattice axis [red circles in (a)] in a Mott insulator with unity filling. The red data points in (c) show the resulting hole probability

p0(δx) as a function of the pointing offset δx, as defined in (b). Each point was obtained

by averaging over 4-7 pictures (total 50-100 addressed lattice sites), taking only those sites into account which lie well within a Mott shell with unity filling. The displayed error bars show the 1σstatistical uncertainty, given by the Clopper-Pearson confidence limits. The data

was fitted by a flat-top model function [Eq. (6.3)] and yields a full-width at half-maximum

σa = 330(10)nm, an edge sharpness of σs = 50(10)nm, and a peak fidelity of 95(2)%. The

offset was fixed at the 6(2)% probability of thermally activated holes as deduced from the next neighboring and second next neighboring sites [blue shaded regions in (a),(b) and blue points in (c)].

6.5 Positioning of the addressing beam

object plane of 10 nm' 0.02alat. In order to position the addressing laser beam onto the atoms with high precision, we measured calibration functions that translate the two control voltages of the piezo mirror into image coordinates. This calibration was performed by replacing the far detuned addressing laser beam by a near resonant molasses beam that follows the identical beam path. Using in addition the x and

y molasses beams, we took a fluorescence image of a large thermal atom cloud in the vertical lattice alone and observed a strongly enhanced signal at the position of the focused beam [see Fig.6.5(c)]. We determined the position of this fluorescence maximum with an uncertainty of 0.2 pixels in our images, corresponding to 0.05alat = 25 nm in the object plane [see Fig.6.5(d)]. The long term drifts of the addressing beam position are on the order of 0.1alat per hour, which we took into account by regular recalibration of the beam position.

Lattice phase feedback

In order to compensate slow phase drifts of the optical lattice, we applied a feedback on the position of the addressing beam. We determined the two lattice phases (offsets of the position of the lattice sites) along the x and y direction after each realization of the experiment by fitting the position of isolated atoms (see Fig.6.5(a), the isolated atom is marked by a circle). Averaging over the positions of typically 1-5 isolated atoms per image allowed us to determine the lattice phase to better than 0.05alat. For the determination of the phase, we used the lattice constant alat and the lattice angles determined from a fluorescence image with many isolated atoms (Sec.4.4). We find that the phases slowly drift over many lattice sites [see Fig.6.5(b)], showing an oscillation with an amplitude of 3alat and a period of about 80 min. The same period is found in the drift of the temperature above the experimental table. The amplitude of 3◦C should make a difference in wavelength of ∆λlat/λlat = 3·10−6 due to the change in the refractive index of the air. The beam path in air from the vacuum chamber to the retro-reflecting mirror is about L =2·100 mm =2·105λlat, so the change in length is ∆L = (∆λlat/λlat)· L = 3·10−6·2·105λlat = 0.6λlat = 1.2alat. The fact that the phases in x and y direction drift along further supports this interpretation and a better temperature stabilization is certainly necessary. Since our phase drifts were slower than 0.04alat between two successive realizations of the experiment ('25 s cycle time), we used the lattice phase from the previous image to correct the addressing beam position. The feedback was done by adding appropriate offsets to the piezo control voltages.

Table6.1 summarizes the positioning errors. We estimate that we can position the addressing beam with respect to the lattice with a total error of ∼ 0.1alat = 50 nm, consistent with the edge sharpnessσs =50(10)nm in our fidelity curve (see Fig.6.4).

0

Time (min)

100 200 300 400

Lattice phase (lattice sites)

-8 -6 -4 -2 0 2 xlattice y lattice 0 20 40 60 −0.2 0 0.2 0.4 0.6 Time (min)

Addressing beam position (pixel)

x lattice y lattice b d a c 2 µm 5 µm x y

Figure 6.5: Positioning of the addressing beam. (a) The lattice phase is determined from the position of isolated atoms, which are placed intentionally. The phase is evaluated online and fed forward to the addressing beam. (b) The lattice phases drift over several sites over the day. The maximal drift between two successive sequences is however only 0.4alat. (c) The

position of the addressing beam is determined from the increased fluorescence at its position, when it is replaced by an additional molasses beam. The increase in signal is so large that the fluorescence from the rest of the cloud is not visible on this scale. The feature to the left is stray light from this beam. (d) The drifts of the addressing beam position are smaller than 0.5a_lat.