  Atomic clocks can measure the gravitational redshift predicted by general
relativity with great accuracy and for height differences as little as 1 cm.
All existing experiments, however, involve the comparison of two independent
clocks at different locations rather than a single clock in a delocalized
quantum superposition. Here we present an interferometry scheme employing
group-II-type atoms, such as Sr or Yb, capable of measuring the gravitational
time dilation in a coherent superposition of atomic wave packets at two
different heights. In contrast to other recent proposals, there is no need for
pulses that can efficiently diffract both internal states. Instead, the scheme
relies on very simple atom optics for which high-diffraction efficiencies can
be achieved with rather mild requirements on laser power. Furthermore, the
effects of vibration noise are subtracted by employing a simultaneous Rb
interferometer that acts as an inertial reference. Remarkably, the recently
commissioned VLBAI facility in Hannover, a 10-meter atomic fountain that can
simultaneously operate Yb and Rb atoms and enables up to 2.8 s of free
evolution time, meets all the requirements for a successful experimental
implementation.
