In the context of the FLASHLIGHT survey, we obtained deep narrow band images of 15 $zsim2$ quasars with GMOS on Gemini-South in an effort to measure Ly$alpha$ emission from circum- and inter-galactic gas on scales of hundreds of kpc from the central quasar. We do not detect bright giant Ly$alpha$ nebulae (SB~10$^{-17}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ at distances >50 kpc) around any of our sources, although we routinely ($simeq47$%) detect smaller scale <50 kpc Ly$alpha$ emission at this SB level emerging from either the extended narrow emission line regions powered by the quasars or by star-formation in their host galaxies. We stack our 15 deep images to study the average extended Ly$alpha$ surface brightness profile around $zsim2$ quasars, carefully PSF-subtracting the unresolved emission component and paying close attention to sources of systematic error. Our analysis, which achieves an unprecedented depth, reveals a surface brightness of SB$_{rm Lyalpha}sim10^{-19}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ at $sim200$ kpc, with a $2.3sigma$ detection of Ly$alpha$ emission at SB$_{rm Lyalpha}=(5.5pm3.1)times10^{-20}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ within an annulus spanning 50 kpc <R< 500 kpc from the quasars. Assuming this Ly$alpha$ emission is powered by fluorescence from highly ionized gas illuminated by the bright central quasar, we deduce an average volume density of $n_{rm H}=0.6times10^{-2}$ cm$^{-3}$ on these large scales. Our results are in broad agreement with the densities suggested by cosmological hydrodynamical simulations of massive ($Msimeq10^{12.5}M_odot$) quasar hosts, however they indicate that the typical quasars at these redshifts are surrounded by gas that is a factor of ~100 times less dense than the (~1 cm$^{-3}$) gas responsible for the giant bright Ly$alpha$ nebulae around quasars recently discovered by our group.
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