  Laser plasma acceleration at kilohertz repetition rate has recently been
shown to work in two different regimes, with pulse lengths of either 30 fs or
3.5 fs. We now report on a systematic study in which a large range of pulse
durations and plasma densities were investigated through continuous tuning of
the laser spectral bandwidth. Indeed, two LPA processes can be distinguished,
where beams of the highest quality, with 5.4 pC charge and a spectrum peaked at
2-2.5 MeV are obtained with short pulses propagating in moderate plasma
densities. Through Particle-in-Cell simulations the two different acceleration
processes are thoroughly explained. Finally, we proceed to show the results of
a 5-hour continuous and stable run of our LPA accelerator accumulating more
than $\mathrm{18\times10^6}$ consecutive shots, with 2.6 pC charge and peaked
2.5 MeV spectrum. A parametric study of the influence of the laser driver
energy through PIC simulations underlines that this unprecedented stability was
obtained thanks to micro-scale density gradient injection. Together, these
results represent an important step towards stable laser-plasma accelerated
electron beams at kilohertz repetition rate.
