Balancing efficiencies by squeezing in realistic eight-port homodyne detection

We address measurements of covariant phase observables (CPOs) by means of realistic eight-port homodyne detectors. We do not assume equal quantum efficiencies for the four photodetectors and investigate the conditions under which the measurement of a CPO may be achieved. We show that balancing the efficiencies using an additional beam splitter allows us to achieve a CPO at the price of reducing the overall effective efficiency, and prove that it is never a smearing of the ideal CPO achievable with unit quantum efficiency. An alternative strategy based on employing a squeezed vacuum as a parameter field is also suggested, which allows one to increase the overall efficiency in comparison to the passive case using only a moderate amount of squeezing. Both methods are suitable for implementation with current technology.

Figure 1

Schematic diagram of the eight-port homodyne detection scheme. The scheme consists of four input modes, four balanced $50$:$50$ beam splitters, a phase shifter which provides a phase-shift of $\frac{\pi }{2}$ on one of the modes and four photodetectors with quantum efficiencies ${\epsilon }_j$, $j=1,2,3,4$, which are not assumed to be equal. The measured quantities are the photon number differences between modes 1 and 3 and between modes 2 and 4, rescaled by the amplitude of the local oscillator, i.e., a strong coherent state $|\sqrt{2}z\rangle$ impinged into mode 4. The signal field in mode 1 is the field under investigation, while the parameter field in mode 2 determines the measured observable. The input mode 3 is left empty so it corresponds to a vacuum field.

Figure 2

Contour lines for the overall efficiency ${\epsilon }_{\mathit{ij}}$ as a function of ${\epsilon }_i$ and ${\epsilon }_j$. The addition of a beam splitter with transparency ${\epsilon }_{\mathrm{bs}}$ can be used to balance the setup and obtain ${\epsilon }_{13}^{\text{'}}={\epsilon }_{24}$.

Figure 3

The ratio $\gamma$ between the effective efficiency achievable by squeezing the parameter field and the corresponding quantity obtained by the insertion of a beam splitter as a function of ${\epsilon }_{13}$ and ${\epsilon }_{24}$.

Figure 4

Parametric plot of the ratio $\gamma$ between the effective efficiency achievable by squeezing the parameter field and the corresponding quantity obtained by the insertion of a beam splitter as a function of the squeezing $a$ needed to achieve the compensation.

Figure 5

The phase distributions of a coherent state $|z\rangle$ with $z=1$ in the case of ideal detectors (higher dashed line), balancing by squeezing (solid line), and balancing by an additional beam splitter (lower dashed line).