  A transmission electron microscope that takes advantage of superconducting
quantum circuitry is proposed. The microscope is designed to improve image
contrast of radiation-sensitive weak phase objects, in particular biological
specimens. The objective in this setting is to measure the phase shift of the
probe electron wave to a precision $\Delta\theta$ within the number of
electrons $N$ that does not destroy the specimen. In conventional electron
microscopy $\Delta\theta$ scales as $\sim 1/N^{1/2}$, which falls short of the
Heisenberg limit $\sim 1/N$. To approach the latter by using quantum
entanglement, we propose a design that involves a Cooper pair box placed on the
surface of an electrostatic electron mirror in the microscope. Significant
improvement could be attained if inelastic scattering processes are
sufficiently delocalized.
