  To quantify the effect of decoherence in quantum measurements, it is
desirable to measure not merely the square modulus of the spatial wavefunction,
but the entire density matrix, whose phases carry information about momentum
and how pure the state is. An experimental setup is presented which can measure
the density matrix (or equivalently, the Wigner function) of a beam of
identically prepared charged particles to an arbitrary accuracy, limited only
by count statistics and detector resolution. The particles enter into an
electric field causing simple harmonic oscillation in the transverse direction.
This corresponds to rotating the Wigner function in phase space. With a
slidable detector, the marginal distribution of the Wigner function can be
measured from all angles. Thus the phase-space tomography formalism can be used
to recover the Wigner function by the standard inversion of the Radon
transform. By applying this technique to for instance double-slit experiments
with various degrees of environment-induced decoherence, it should be possible
to make our understanding of decoherence and apparent wave-function collapse
less qualitative and more quantitative.
