  We observed B1600+434 and B1608+656 with CXO/ACIS, detecting both quasar
images in B1600+434 and three of four images in B1608+656. We did not detect
significant X-ray emission from nearby galaxy groups or clusters associated
with each lens galaxy. The upper limits on the X-ray luminosity of any cluster
within 4 arcmin of each lens and at each lens redshift are of ~2\times10^{42}
and ~6\times10^{42} erg/s for B1600+434 and B1608+656, respectively. The
radio-loud source quasars have power-law photon indices of \Gamma=1.9\pm0.2 and
\Gamma=1.4\pm0.3 and X-ray luminosities of 1.4^{+0.2}_{-0.1}\times10^{45} and
2.9^{+0.7}_{-0.4}\times10^{44} erg/s for B1600+434 and B1608+656, respectively
before correcting for the magnification. We detected a differential absorption
column density of \Delta N_H ~ 3\times10^{21} cm^{-2} between the two images of
B1600+434, roughly consistent with expectations from differential extinction
estimates of \Delta E(B-V)=0.1 mag and a standard dust-to-gas ratio. The
differential absorption observed in gravitational lenses may serve as an
important probe to study the gas content in high redshift galaxies since it can
separate the absorbing column originating from the lens galaxy and those
intrinsic to quasars. We also detected 157 serendipitous X-ray sources in the
two Chandra fields and identified the brighter optical counterparts using the
SDSS and DPOSS surveys.
