  Although time is one of the fundamental notions in physics, it does not have
a unique description. In quantum theory time is a parameter ordering the
succession of the probability amplitudes of a quantum system, while according
to relativity theory each system experiences in general a different proper
time, depending on the system's world line, due to time to time dilation. It is
therefore of fundamental interest to test the notion of time in the regime
where both quantum and relativistic effects play a role, for example, when
different amplitudes of a single quantum clock experience different magnitudes
of time dilation. Here we propose a realization of such an experiment with a
single electron in a Penning trap. The clock can be implemented in the
electronic spin precession and its time dilation then depends on the radial
(cyclotron) state of the electron. We show that coherent manipulation and
detection of the electron can be achieved already with present day technology.
A single electron in a Penning trap is a technologically ready platform where
the notion of time can be probed in a hitherto untested regime, where it
requires a relativistic as well as quantum description.

General Relativity and Quantum Cosmology;Quantum Physics