What are the frequency, shape, kinematics, and luminosity of Ly alpha envelopes surrounding radio-quiet quasars at high redshift, and is the luminosity of these envelopes related to that of the quasar or not? As a first step towards answering these questions, we have searched for Ly alpha envelopes around six radio-quiet quasars at z similar to 4.5, using deep spectra taken with the FORS2 spectrograph attached to the UT1 of the Very Large Telescope (VLT). Using the multi-slit mode allows us to observe several point spread function stars simultaneously with the quasar, and to remove the point-like emission from the quasar, unveiling the faint underlying Ly alpha envelope with unprecedented depth. An envelope is detected around four of the six quasars, which suggests that these envelopes are very frequent. Their diameter varies in the range 26 less than or similar to d less than or similar to 64 kpc, their surface brightness in the range 3 x 10(-19) less than or similar to mu less than or similar to 2 x 10(-17) erg s(-1) cm(-2) arcsec(-2), and their luminosity in the range 10(42) less than or similar to L(Ly alpha) less than or similar to 1044 erg s(-1). Their shape may be strongly asymmetric. The Ly alpha emission line full width at half maximum (FWHM) is 900 < FWHM < 2200 km s-1 and its luminosity correlates with that of the broad line region (BLR) of the quasar, with the notable exception of BR2237-0607, the brightest object in our sample. The same holds for the relation between the envelope Ly alpha luminosity and the ionizing luminosity of the quasar. While the deep slit spectroscopy presented in this paper is very efficient at detecting very faint Ly alpha envelopes, narrow-band imaging is now needed to measure accurately their spatial extent, radial luminosity profile, and total luminosity. These observables are crucial to help us discriminate between the three possible radiation processes responsible for the envelope emission: (i) cold accretion, (ii) fluorescence induced by the quasar, and (iii) scattering of the BLR photons by cool gas.