We have carried out a thorough study of current and noise
for spin-polarized and spin-entangled electrons in a beamsplitter
geometry, including a local spin-orbit interaction
�Rashba and Dresselhaus� in one of the incoming arms. We
have considered incoming leads with one or two channels, as
well as backscattering effects. The channels can be coupled
via the SO interaction for incoming energies near the band
crossing. We have found that the spin-orbit interaction is a
useful mechanism to coherently rotate spin states. Such rotation
can be used to modulate noise signals, thus providing
unique signatures of spin polarization and spin entanglement.
For spin-polarized electrons, noise measurements can
give a direct measure of the degree of polarization along
different directions. For electron pairs, the coupling between
the channels can play an important role. For pairs with incoming
energies near the band crossing injected into one of
the channels, we find an additional modulation due to the
coherent transfer of electrons between the two channels. In
this case, noise measurement allows us to distinguish all the
different triplets states defined along the y direction, in addition
to the singlet. Furthermore, for equal strengths the combined
effect of the Rashba and Dresselhaus interactions can
partially cancel out. In this case, the spin and orbital degrees
of freedom are separable, the interband coupling essentially
disappears, and the propagation of spin states is robust
against scattering off nonmagnetic impurities.
We have also considered the influence of backscattering
in the beam-splitter with a single channel. The main effect is
an additional contribution related to the partition noise due to
the tunnel barrier describing the backscattering. This reduces
the visibility of the oscillations in the shot noise as the SO
rotation angle SO is varied. It also reduces the maximal
noise value found for perfect antibunching of singlets.
We have generalized earlier results for the shot noise of
entangled electrons by allowing the injection of wave packets,
i.e., coherent superpositions of discrete momentum
eigenstates �plane waves�. We have found a general analytical
formula for the two-particle interference visibility �h�2 in
terms of all three relevant energy scales �, �, and �. Our new
result contains and generalizes both the discrete single-level
case and the continuum case.
Finally, we have developed a simple heuristic picture for
the noise based on number operators in the different leads
and the relevant transmission and reflection probability amplitudes
in the beamsplitter. Within this picture we can more
intuitively rederive some of the formulas for the noise—
previously derived within the rigorous scattering formalism
�Sec. III�—in the presence of spin-orbit interaction and backscattering
in the incoming leads.