Adaptive phase estimation with two-mode squeezed vacuum and parity measurement

A proposed phase-estimation protocol based on measuring the parity of a two-mode squeezed-vacuum state at the output of a Mach-Zehnder interferometer shows that the Cramér-Rao sensitivity is sub-Heisenberg [P. M. Anisimov et al., Phys. Rev. Lett. 104, 103602 (2010)]. However, these measurements are problematic, making it unclear if this sensitivity can be obtained with a finite number of measurements. This sensitivity is only for a phase near zero, and in this region there is a problem with ambiguity because measurements cannot distinguish the sign of the phase. Here, we consider a finite number of parity measurements and show that an adaptive technique gives a highly accurate phase estimate regardless of the phase. We show that the Heisenberg limit is reachable, where the number of trials needed for mean photon number $\overline{n}=1$ is approximately 100. We show that the Cramér-Rao sensitivity can be achieved approximately, and the estimation is unambiguous in the interval ($-\pi /2,\pi /2$).

Figure 1

Two-mode squeezed-vacuum states are generated at the input of the Mach-Zehnder interferometer (MZI) by an optical parametric amplifier (OPA), where $\phi$ is the phase being measured and $\theta$ is a controllable phase. The parity signal at the output of the MZI is then measured with photon-number-resolving detectors. Since TMSV states always have even photon numbers, the parity signal can be detected by performing photon counting at only one output.

Figure 2

Probability distribution for the phase $\phi$ for an example measurement record, generated for an actual phase of $\phi =0.15$ and the control phase $\theta =0$. The record consists of $M=512$ parity detections for $\overline{n}=3$, of which 466 are even.

Figure 3

Probability distribution for the phase $\phi$ for an example measurement record, generated for an actual phase of $\phi =0.15$ and $\theta$ is chosen adaptively. The record consists of $M=512$ parity detections for $\overline{n}=3$. There is no ambiguity in the sign of $\phi$.

Figure 4

Distribution of estimates for $\phi$, where $M=256$ detections were used to obtain each estimate. The plot contains results from $J=20000$ measurement records which were numerically generated for $\phi =0.5$ and $\overline{n}=3$. The distribution has a MSE $\mathrm{\Delta }{\phi }^2=3.81\times {10}^{-4}$.

Figure 5

Ratio of the phase MSE to the Heisenberg limit vs the measurement record length $M$, for TMSV states with a range of mean photon numbers. Error bars are shown only if they are larger than the marker size. The HL is plotted for comparison (it is 1 because all values are shown as a ratio to the HL).

Figure 6

Ratio of the phase MSE to the quantum Cramér-Rao (QCR) bound vs the measurement record length $M$, for TMSV states with a range of mean photon numbers. The QCR bound is shown for comparison. The error bars are shown only if they are larger than the size of the markers.

Figure 7

Ratio of the phase MSE to the Heisenberg limit is plotted against the measurement record length $M$ for $\overline{n}=1$ and a range of levels of loss. The HL for $\eta =1$ is plotted for comparison. The error bars are not shown if they are smaller than the markers.

Figure 8

Same as Fig. 7, but for $\overline{n}=3$.

Figure 9

Ratio of the phase MSE to the Heisenberg limit vs the measurement record length $M$ for $\overline{n}=1$ and a range of dark count rates. The HL for $r=0$ is plotted for comparison. The error bars are not shown if they are smaller than the markers.

Figure 10

Same as Fig. 9, but for $\overline{n}=3$.