We used a BEC of 87Rb with about 45 × 103 atoms in a single-beam dipole trap realized by a CO2 laser (maximum power, 10 W; beam waist, 30 μm). The condensate is cigar-shaped and has dimensions of 80 μm × 6 μm × 6 μm. We then loaded the BEC into a one-dimensional optical lattice created by two blue detuned laser beams (λ = 774 nm; beam waist, 500 μm) crossed at an angle of 90°. The linear polarization of both laser beams was along the same direction, such that the interference pattern was maximally modulated. The resulting lattice has a period of d = 547 nm. The trap frequencies in a lattice site are νr = 165 Hz (transverse direction) and νz = 12 kHz (lattice direction). Each site contains a small, pancake-shaped BEC with about 700 atoms (value in the center of the trap). The total number of lattice sites is about 200. The lattice depth V0 in units of the recoil energy Er = π22/(2md2) (m is the mass of the atom) is given by V0 = 10Er. An electron column, which is implemented in our experimental chamber, provides a focused electron beam, which is used to introduce a well-defined local particle loss as a dissipative process in one site of the lattice. To ensure a homogeneous loss process over the whole extension of the lattice site, we rapidly scanned the electron beam in the transverse direction (3-kHz scan frequency) with a sawtooth pattern. To adjust the dissipation strength γ, we varied the amplitude of the scan pattern. An image of the experimental chamber together with a sketch of the optical trapping configuration is provided in fig. S1.

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