  The simultaneous two-parameter estimation problem in single squeezed-light
Mach-Zehnder interferometer with double-port homodyne detection is investigated
in this work. The analytical form of the two-parameter quantum Cramer-Bao bound
defined by the quantum Fisher information matrix is presented, which shows the
ultimate limit of the phase sensitivity will be further approved by the
squeezed vacuum state. It can not only surpass the shot-noise limit, but also
can even surpass the Heisenberg limit when half of the input intensity of the
interferometer is provided by the coherent state and half by the squeezed
light. For the double-port homodyne detection, the classical Fisher information
matrix is also obtained. Our results show that although the classical
Cramer-Rao bound does not saturate the quantum one, it can still asymptotically
approach the quantum Cramer -Bao bound when the intensity of the coherent state
is large enough. Our results also indicate that the squeezed vacuum state
indeed can further improve the phase sensitivity. In addition, when half of the
input intensity of the interferometer is provided by the coherent state and
half by the squeezed light, the phase sensitivity obtained by the double-port
homodyne detection can surpass the Heisenberg limit for a small range of the
estimated phase.
