We present three-dimensional particle-in-cell (3D-PIC) simulations of laser wake field acceleration (LWFA) of electrons in the bubble regime, using the code ILLUMINATION}. The study is motivated by the new Light Wave Synthesizer (LWS), currently under construction at Max-Planck-Institut für Quantenoptik (MPQ), which is expected to produce short (sub-10 fs), high power (multi-terawatt) laser pulses. They will be short enough for optimal drive of bubbles in dense gas jet targets at laser energies as low as 20 mJ and to generate efficiently very bright, quasi-monoenergetic, low-emittance bunches of 10–100 MeV electrons. It is shown how the results change when plasma density, laser amplitude and focusing are varied. The density and the spectral evolution of bubble evolution are presented for a reference case in much detail in terms of a movie. In particular, the formation of peaked spectra with an energy spread in the range of 10% are discussed qualitatively. Also plasma dynamics at the laser front and the rear vertex of the bubble as well as the periodic structures in the stem of accelerated electrons are addressed.