  In this paper we address the problem of optimizing an unbalanced Mach-Zehnder
interferometer, for a given pure input state and considering a specific
detection scheme. While the optimum transmission coefficient of the first beam
splitter can be uniquely determined via the quantum Fisher information only
[Phys. Rev. A 105, 012604 (2022)], the second beam splitter transmission
coefficient is detection-scheme dependent, too. We systematically give analytic
solutions for the optimum transmission coefficient of the second beam splitter
for three types of widely used detection schemes. We provide detailed examples
including both Gaussian and non-Gaussian input states, showing when an
unbalanced Mach-Zehnder interferometer can outperform its balanced counterpart
in terms of phase sensitivity.
