Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit

We study the sensitivity and resolution of phase measurement in a Mach-Zehnder interferometer with two-mode squeezed vacuum ($\overline{n}$ photons on average). We show that superresolution and sub-Heisenberg sensitivity is obtained with parity detection. In particular, in our setup, dependence of the signal on the phase evolves $\overline{n}$ times faster than in traditional schemes, and uncertainty in the phase estimation is better than $1/\overline{n}$, and we saturate the quantum Cramer-Rao bound.

Figure 1

The Mach-Zehnder interferometer used in the calculations. The two-mode squeezed vacuum input state $|{\psi }_{\overline{n}}⟩$ is indicated together with the intermediate state $|\psi ⟩={\widehat{P}}_{\phi }\widehat{U}|{\psi }_{\overline{n}}⟩$. Vertical dash-dotted lines indicate places where two measurements ${\widehat{\mu }}_{AB}$ and ${\widehat{\Pi }}_A$ are to be implemented.

Figure 2

Measured signals at the output of the MZI with coherent light ($\overline{n}=100$) and TMSV ($\overline{n}=10$) inputs against accumulated phase difference. Solid and dashed lines are the outputs of the parity measurement for TMSV and coherent light, respectively. The dotted line, given for comparison, is a scaled down output of intensity difference measurement on the output of the MZI fed with coherent light. TMSV with much smaller photon number performs as well as coherent light.

Figure 3

Sensitivity of phase estimation obtained with the parity measurement at $\phi =0$ (dashed) against average total photon number. Dots are sensitivity estimation based on quantum Fisher information for integer values of $\overline{n}$. Shaded area is between dotted lines $1/\overline{n}$ and $1/\sqrt{\overline{n}}$. The solid line is for the Hofmann limit discussed in the text. Inset: Sensitivity against actual values of accumulated phase difference. Solid lines for TMSV with $\overline{n}=5$ and $\overline{n}=25$; dashed line for coherent light with $\overline{n}=25$.

Figure 4

Phase estimation sensitivity $\Delta \phi$ for the state $\widehat{\rho }(n,\theta )$ with $n=2$ (top four) and $n=5$ (bottom four) against $\theta$. Solid lines represent sensitivity with the parity measurement which saturates the limit set by the quantum Fisher information (dots for a few selected values of $\theta$). Dashed lines represent HL sensitivity estimation based on the averaged total photon number: $1/\overline{n}$. Dot-dashed lines give Hofmann’s estimation of sensitivity discussed in the text.