Legislación para Proteger a los Niños y Niñas contra la Violencia

In this section we use a numerical simulation of the Gross-Pitaevskii equation2.26in 1D to study the excitation of a BEC. In a first step, the ground state wave function for the repulsive gas in the potential U(z) = Um +Us is found by imaginary time propagation. Then we perform time integration using a split-step Fourier method to

5.8 Simulation of cloud excitation 119

Figure 5.17.: Time evolution of a BEC with N = 2000 in a coupling trap with ωz/2π = 10 kHz (top) andωz/2π = 5 kHz (bottom) at d= 1.5 µm for th = 3 ms.

The cantilever amplitude is set to a= 50 nm (top) and a= 70 nm (bottom). Axes are in dimensionless units (z0 =

p

~/mωz, the color scale indicates the normalized

probability density |ψ|2_{). Horizontal lines show the position of the barrier (grey),} the matching layer (dark grey), and the cantilever surface (black).

obtain the temporal dynamics of the wave function in the time dependent potential U(z, t). The code is adapted from [82, 270]. To avoid artificial Bragg scattering from the computational grid at high momentum in the attractive potential, it is necessary to introduce absorbing boundary conditions. We use the concept of a perfectly matched layer [271, 272] and implement an imaginary potential rather than an imaginary spatial coordinate. The idea is based on the fact that in a region, where a potential has a sizable imaginary contribution, matter waves are exponentially damped. The imaginary region is added to the potential in the form

U(z) = (Um+UCP+Uad)×exp −iπ 4 1−tanhz−zl dz . (5.21)

This implements a smooth transition from a purely real potential for z zl to an imaginary potential for z zl within a matching layer of thickness dz. For reason- ably largedz (∼50−200 nm), artificial reflections are very small and incident waves are completely absorbed. The position zl is chosen such that the deBroglie wave- length remains much larger than the grid spacing,λdB(zl) =~/

p

2m(µc−U(zl)) (zi+1−zi).

6 8 10 12 14 16 18 20 0 0.5 1 cantilever frequency ωm/2π [kHz] χ 6 8 10 12 14 16 18 10−8 10−6 10−4 10−2 100 ω*/2π[kHz] loss a = 100 nm a = 40 nm

Figure 5.18.: BEC loss as a function of cantilever or trap frequency (trap param- eters as in fig. 5.17). Upper panel: Fraction of remaining atoms χ as a function of ωm for fixed ωz,0 = 2π×10 kHz. Lower panel: Logarithmic plot of the contrast

C = 1−χ. For comparison with experimental data we transform the axis to a frequencyω∗ =ω2_z,0/ωm which is analog to the trap frequency in a scan with fixed

ωm and variedωz.

ωz and the breathing mode at ωm = 2ωz. The dipole oscillation amplitude rises linearly in time until it saturates at an amplitude where the rim of the cloud reaches the trap barrier. Excitation of internal dynamics is visible from the modulation of the peak density. The breathing mode at twice the trap frequency increases also rather linearly in amplitude and saturates at a similar amplitude as the c.o.m. mode. For long times, dephasing leads to interference and finally to a decay of the breathing mode. The wave function temporarily shows only minor dynamics and is transformed into an excited state with∼4 nodes. With adequate timing, this could enable the preparation Fock states.

The atom number remaining after a coupling interval is simply evaluated by in- tegrating over the density distribution. Simulating the coupling loss for different excitation frequencies yields a loss spectrum as shown in Fig. 5.18. For better con- trol of the trap parameters, the cantilever frequency is varied rather than the trap frequency. This avoids e.g. unwanted changes in U0, , or the contribution from

tunneling, and thus permits to compare the strength of the resonances. The trap parameters are ωz,0/2π = 10.0 kHz and d= 1.5 µm. The shifted trap frequency for

small amplitude oscillations is ωz/2π= 9.434 kHz. The c.o.m. and breathing mode resonance frequencies for medium oscillation amplitude are shifted by−450 Hz and

5.8 Simulation of cloud excitation 121 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 cantilever amplitude [nm rms] C

Figure 5.19.: Comparison of measured and simulated contrast as a function of cantilever amplitude for the c.o.m. mode. The parameters for the simulation are identical to those of the experiment without adjustments.

the classical simulation but deviating from experimental findings. The resonances of the 3D collective BEC modes are not reproduced (as expected), because the coupling to the other dimensions is not included. An interesting feature is the additional res- onance in the spectrum for the stronger drive at ωm ≈8 kHz. The cloud dynamics shows that several collective modes are excited at these subharmonic frequencies. This might explain the observed resonance lineshape in experiments for large drive and small atom-surface distance. Such modes do not exist in an harmonic trap and are also absent in the classical simulation for the anharmonic potential.

A central benefit of the simulation is to provide insight about the sensitivity limit of the used readout scheme. The simulation requires only the trap parameters and the cantilever amplitude as input, and has no free parameters to adjust. The only uncertainty comes from the error bars from the measurement of the trap frequency (see chapter4.4.3). In Fig.5.19, the simulated coupling loss as a function of the can- tilever amplitude is compared to the measurement of chapter 5.11. Measurements and simulation agree surprisingly well, and especially the sensitivity predicted by the simulation is very close to our experimental findings. This is surprising, since we model a pure BEC in a 1D trap, whereas the experiment deals with partially condensed clouds with thermal fraction in a 3D trap. Possible effects could be e.g. nonlinear coupling to other trap axes, variation of ωz along the trap axis, depen- dence of the barrier height on the axial distance to the trap center, and damping of condensate motion by the thermal cloud. However, it seems that these effects play a minor role for the loss.