  In the context of the readout scheme for gravitational-wave detectors, the
"double balanced homodyne detection" proposed in [K.~Nakamura and
M.-K.~Fujimoto, arXiv:1709.01697.] is discussed in detail. This double balanced
homodyne detection enables us to measure the expectation values of the photon
creation and annihilation operators. Although it has been said that the
operator $\hat{b}_{\theta}:=\cos\theta\hat{b}_{1}+\sin\theta\hat{b}_{2}$ can be
measured through the homodyne detection in literature, we first show that the
expectation value of the operator $\hat{b}_{\theta}$ cannot be measured as the
linear combination of the upper- and lower-sidebands from the output of the
balanced homodyne detection. Here, the operators $\hat{b}_{1}$ and
$\hat{b}_{2}$ are the amplitude and phase quadrature in the two-photon
formulation, respectively. On the other hand, it is shown that the above double
balanced homodyne detection enables us to measure the expectation value of the
operator $\hat{b}_{\theta}$ if we can appropriately prepare the complex
amplitude of the coherent state from the local oscillator. It is also shown
that the interferometer set up of the eight-port homodyne detection realizes
our idea of the double balanced homodyne detection. We also evaluate the
noise-spectral density of the gravitational-wave detectors when our double
balanced homodyne detection is applied as their readout scheme. Some
requirements for the coherent state from the local oscillator to realize the
double balanced homodyne detection are also discussed.
