Measuring the distances to quasars at high redshifts with strong lensing
  Strongly lensed quasars with time-delay measurements are well known to
provide the "time-delay distances" $D_{\Delta t}=(1+z_L)D_LD_S/D_{LS}$ and the
angular diameter distances to lens galaxies $D_L$. These two kinds of distances
give stringent constraints on cosmological parameters. In this work, we explore
a different use of time-delay observables: Under the assumption of a flat
Universe, strong lensing observations can accurately measure the angular
diameter distances to sources $D_S$. The corresponding redshifts of quasars may
be up to $z_S\sim4$ according to the forecast. The high-redshift distances
would sample the Hubble diagram between SNe Ia and CMB,
cosmological-model-independently providing direct information on the evolution
of the nature of our Universe, for example, the dark energy Equation-of-State
parameter $w(z)$. We apply our method to the existing lensing system SDSS
1206+4332 and get $D_S=2388_{-978}^{+2632}Mpc$ at $z_S=1.789$. We also make a
forecast for the era of LSST. The uncertainty of $D_S$ depends on the redshifts
of lens and source, the uncertainties of $D_{\Delta t}$ and $D_L$, and the
correlation between $D_{\Delta t}$ and $D_L$ as well. Larger correlation would
result in tighter $D_S$ determination.
