Figure 1

Schematic diagram of the experimental setup. A He:Ne laser is divided into two beams, one acts as the local oscillator and the other represents the signal beam. The signal is modulated at 4 MHz with a defined modulation depth to control the average number of photons in the generated state. One of the mirrors in the signal path is piezo mounted to obtain a variable phase difference between the two beams. The data are recorded by a homodyne detector whose difference photocurrent is demodulated and then acquired by a computer after a low pass filter. We also show the typical homodyne samples obtained for coherent signals of different amplitudes by varying the phase of the local oscillator (these are used to check the calibration of the piezo, which is performed using signals with a larger number of photons).Reuse & Permissions

Figure 2

Typical examples of homodyne samples measured at fixed optimal $\theta$, together with the corresponding Bayesian a posteriori distribution for the phase shift. The phase diffusion is $\Delta =\pi /6$ rad, and the yellow areas denote the portions of the data used to infer the phase shifts.Reuse & Permissions

Figure 3

The noise ratio $K_M=\left(\mathrm{Var}\left[{\phi }_{\mathrm{B}}\right]M{H}_{\alpha }\right)$ as a function of the number of data $M$ and for different values of the number of photons $N$ and the noise parameter $\Delta$. Blue circles: $N=0.90$, $\Delta =\pi /18$ rad; red squares: $N=0.90$, $\Delta =\pi /9$ rad; yellow diamonds: $N=4.12$, $\Delta =\pi /18$ rad; green triangles: $N=4.12$, $\Delta =\pi /9$ rad.Reuse & Permissions

Figure 4

Variance $V_M=M\mathrm{Var}\left[{\phi }_{\mathrm{B}}\right]$ of the Bayesian estimator from homodyne data after $M=100$ measurements (points), together with the CR bound for coherent states (solid lines) and for optimized Gaussian states (dashed lines). The top panel shows the behavior of $V_{100}$ as a function of $\Delta$ for different values of the number of photons (top red lines/squares: $N=0.90$; bottom blue lines/circles: $N=14.11$). The bottom panel shows $V_{100}$ as a function of the number of photons $N$ and for different values of the noise (top blue lines/squares: $\Delta =\pi /9$ rad; bottom green lines/circles: $\Delta =\pi /18$ rad).Reuse & Permissions