temporal variation of astrophysical observable quantities over a few decades is becoming more and more realistic. This kind of real-time observations can be called the “[*real-time cosmology*]{}”. The most typical real-time observable is the [*redshift drift*]{}, which can give a direct measurement for the expansion rate (namely, the Hubble parameter) of the universe in a specific range of redshift.

The approach of measuring the redshift drift was first proposed by Sandage, who suggested a direct measurement of the redshift variation for the extra–galactic sources [@sandage]. At that time, obviously, such a measurement was out of reach with the technological limitation of the day. Then, the method was further improved by Loeb, who suggested a more realistic way of measuring the redshift drift using Lyman-$\alpha$ absorption lines of the distant quasars (QSOs) to detect the redshift variation [@Loeb:1998bu]. Loeb concluded that the signal would be detectable when 100 quasars can be observed over 10 years with a 10-meter class telescope. Thus, the method of redshift drift measurement is also referred to as the “Sandage-Loeb" (SL) test.

Based on the SL test, the scheduled European Extremely Large Telescope (E-ELT), a giant 40-meter class optical telescope, is equiped with a high-resolution spectrograph to 