  In the context of the FLASHLIGHT survey, we obtained deep narrow band images
of 15 $z\sim2$ 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 $z\sim2$ 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$_{\rm
Ly\alpha}\sim10^{-19}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ at $\sim200$ kpc,
with a $2.3\sigma$ detection of Ly$\alpha$ emission at SB$_{\rm
Ly\alpha}=(5.5\pm3.1)\times10^{-20}$ 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 $n_{\rm H}=0.6\times10^{-2}$ cm$^{-3}$ on these large scales. Our results
are in broad agreement with the densities suggested by cosmological
hydrodynamical simulations of massive ($M\simeq10^{12.5}M_\odot$) 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.
