We present results of a Gemini adaptive optics (AO) imaging program to investigate the host galaxies of typical QSOs at z~2. Our aim is to study the host galaxies of typical, L*_qso QSOs at the epoch of peak QSO and star formation activity. The large database of faint QSOs provided by the 2dF QSO Redshift Survey allows us to select a sample of QSOs at z=1.75-2.5 which have nearby (<12 arcsecond separation) bright stars suitable for use as AO guide stars. We have observed a sample of 9 QSOs. The images of these sources have AO corrected full-width at half-maximum of between 0.11 and 0.25 arcseconds. We use multiple observations of point spread function (PSF) calibration star pairs in order to quantify any uncertainty in the PSF. We then factored these uncertainties into our modelling of the QSO plus host galaxy. In only one case did we convincingly detect a host (2QZ J133311.4+001949, at z=1.93). This host galaxy has K=18.5+-0.2 mag with a half-light radius, r_e=0.55+-0.1, equivalent to ~3L*_gal assuming a simple passively evolving model. From detailed simulations of our host galaxy modelling process, we find that for four of our targets we should be sensitive to host galaxies that are equivalent to ~2L*_gal (passively evolved). Our non-detections therefore place tight constraints on the properties of L*_qso QSO host galaxies, which can be no brighter (after allowing for passive evolution) than the host galaxies of L*_qso AGN at low redshift, although the QSOs themselves are a factor of ~50 brighter. This implies that either the fueling efficiency is much greater at high redshift, or that more massive black holes are active at high redshift.
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