Optical homodyne detection in view of the joint probability distribution

Optical homodyne detection is examined in view of the joint probability distribution. The usual view is that the relative phase between independent laser fields is localized by photon-number measurements in interference experiments such as homodyne detection. This is why operationally coherent states for laser fields are used in the description of homodyne detection and optical quantum-state tomography. Here, we elucidate these situations by considering the joint probability distribution and the invariance of homodyne detection under the phase transformation of optical fields.

Figure 1

Schematic diagram of homodyne detection. PD represents a photodetector, BS a beam splitter, and PS a phase shifter.

Figure 2

Typical results of the empirical measure ${\Lambda }_{{\mathrm{x̃}}_M}(x)$ (solid lines) for the squeezed state signal $\mathrm{\rho ̂}=|r,\beta \rangle \langle r,\beta |$, where $M=10\hspace{0.5em}000$ outcomes are generated for each operation of repeated homodyne detections. The parameters for the LO and signal are taken as $\alpha =\sqrt{15}$, $\beta e^{-r}=\sqrt{3}$, and $r=-1$. The resolution of the quadrature is given by $\Delta x=1/(\sqrt{2}\alpha )=1/\sqrt{30}$. The resultant random phases ${\phi }_0$ are estimated from the average ${\mathrm{x̄}}_M$ of the outcomes as ${\phi }_0=3.05$, $1.94$, and $0.43\hspace{0.5em}\mathrm{rad}$ from left to right. The quadrature distributions $q_{{\phi }_0}(x)$ (dashed lines) are also plotted for these values of ${\phi }_0$, showing good agreement with ${\Lambda }_{{\mathrm{x̃}}_M}(x)$.