Effects of phase fluctuations on phase sensitivity and visibility of path-entangled photon Fock states

We study effects of phase fluctuations on phase sensitivity and visibility of a class of robust path-entangled photon Fock states (known as $m{m}^{\prime }$ states) as compared to the maximally path-entangled NOON states in the presence of realistic phase fluctuations such as turbulence noise. Our results demonstrate that the $m{m}^{\prime }$ states, which are more robust than the NOON state against photon loss, perform equally well when subject to such fluctuations. We derive the quantum Fisher information with the phase-fluctuation noise and show that the phase sensitivity with parity detection for both of the above states saturates the quantum Cramér-Rao bound in the presence of such noise, suggesting that parity detection is an optimal detection strategy.

Figure 1

Schematic diagram of a simplified Mach-Zehnder interferometer with the modes $a$ and $b$ for the $m{m}^{\prime }$ and NOON states as the input. The source and detector in the interferometer are represented by the respective boxes. Effects of the phase fluctuations due to the turbulence noise are represented by $\Delta \varphi$ in the upper path b of the interferometer. The upper beam passes through a phase shifter $\varphi$, and the phase acquired depends on the total number of photons $\Delta m=m-m^{\prime }$ (or $N$) passing throughout the upper path. Transformed parity detection is used as the detection scheme at both of the two modes at stage III inside the interferometer.

Figure 2

Phase sensitivity $\delta \varphi$ of the $m{m}^{\prime }$ state $|5::1\rangle$, or the NOON state $|4::0\rangle$, having the same phase information, as a function of phase shift $\varphi$ from a two-mode interferometer for different values of $\Gamma$: $\Gamma =0.1$ (curved dashed line, blue online), $\Gamma =0.3$ (curved black double-dotted line), and $\Gamma =0.5$ (curved dotted line, purple online). The Heisenberg limit ($1/N$) and the shot noise limit ($1/\sqrt{N}$) of the phase sensitivity for the NOON state are shown by the red (gray) solid line and the black dashed line, respectively, for comparison.

Figure 3

Minimum phase sensitivity $\delta {\varphi }_{\min }$ of the $m{m}^{\prime }$ state $|5::1\rangle$ or the NOON state $|4::0\rangle$ as a function of $\Gamma$. The shot noise limits (SNL) and the Heisenberg limits (HL) of the phase sensitivity for both the states are also shown for comparison.

Figure 4

Visibility $V$ of the $m{m}^{\prime }$ state for different $\Delta m$ (for different $N$ in case of NOON states with the same phase information) as a function of $\Gamma$. The visibility $V$ is plotted for $N$(or $\Delta m)=2$ (solid line, blue online), $N$(or $\Delta m)=4$ (dashed line, red online), $N$(or $\Delta m)=6$ (double dotted black line), and $N$(or $\Delta m)=8$ (dot-dashed line, purple online). We see that the visibility drops faster for larger values of $\Delta m$ (or $N$).

Figure 5

Two fictitious beam splitters are introduced to Fig. 1 to mimic the loss of photons from the system into the environment. After tracing out the environment modes $v_b$ and $v_a$, the system results in a mixed state at stage I.