Insensitivity points and performance of open quantum interferometers under number-conserving & non-conserving Lindblad dynamics

TL;DR

The paper analyzes how environmental noise, modeled by Lindblad dynamics with number-conserving and non-conserving operators, creates insensitivity points in a two-mode atom interferometer and how these points depend on input states and particle number. By solving analytically for N=1,2 and numerically for N>2, it shows that insensitivity-point locations are independent of noise strength, while overall sensitivity degrades with increasing noise; crucially, the Cramér-Rao bound favors particle non-conserving noise across all N. The study compares N0, TF, and NOON inputs, revealing state-dependent density of insensitivity points and different scaling behavior with N. These results guide design and operation of realistic interferometric sensors, highlighting regimes to avoid insensitivity points and the potential advantages of non-conserving noise channels for ultimate precision.

Abstract

We investigate the phase sensitivity of a linear two-mode atom interferometer subject to environmental noise, modeled within the framework of open quantum systems with both number-conserving and non-conserving Lindblad operators. Considering several input states, we first study the cases N=1,2 (N number of particles) and perform numerical simulations for N>2. The sensitivity as a function of the holding time can display divergence points where phase estimation becomes impossible, to which we refer as insensitivity points. We characterize their behavior as the input state, particle number, and noise operator are varied, and we find that their positions are independent of the noise intensity. Moreover, while our fixed measurement scheme may favor number-conserving noise at small N (i.e., having better sensitivity), the Cramér-Rao bound reveals that particle non-conserving noise yields strictly lower achievable sensitivity for all particle numbers.

Figures (9)

• Figure 1: The sensitivity of the interferometer (red lines) for input state $\ket{1,0}$, noise operators $\hat{S}_{z}$, $\hat{S}_{\pm}$ and $N=1$ is plotted as a function of $T_H$ for $\delta=0.5$ and different values of $\gamma$. The blue lines show the noiseless case function $1/T_H$.
• Figure 2: The sensitivity of the interferometer (red lines) for input state $NOON$, noise operators $\hat{S}_{z}$, $\hat{S}_{\pm}$ and $N=1$ is plotted as a function of $T_H$ for $\delta=0.5$ and different values of $\gamma$. The blue lines show the noiseless case function $1/T_H$.
• Figure 3: The sensitivity of the interferometer for input state $\ket{2,0}$ is plotted as a function of the holding time $T_H$ for $\delta=0.5$ and different values of $\gamma$. The yellow lines show the case of noise operator $\hat{S}_z$ while the blue lines the case of noise operator $\hat{S}_{\pm}$. The purple lines show the (noise-free case) function $1/\sqrt{2}T_H$ which represents the shot-noise limit.
• Figure 4: The sensitivity of the interferometer for input states $\ket{1,1}$, $NOON$ with $N=2$ is plotted as a function of $T_H$ for $\delta=0.5$ and different values of $\gamma$. The yellow lines show the case of noise operator $\hat{S}_z$ while the blue lines the case of noise operator $\hat{S}_{\pm}$. The purple lines show the (noise-free case) function $1/2T_H$ which represents the Heisenberg limit.
• Figure 5: The sensitivity $\Delta \delta$ of the atom interferometer for the $N0=\ket{N,0}$ (upper four), $TF=\ket{N/2,N/2}$ (middle four) and $NOON$ (lower four) input states is plotted as function of the holding time $T_H$ for $\delta=0.5$. For each initial state the first plot has fixed $\gamma=0$ and $\Delta \delta$ is plotted at different $N$. The second plot shows $\gamma \neq 0$ plots at different $N$. In the third graph $N=4$ is fixed and different plots differ in their $\gamma \neq 0$ parameter. In the previous three plots we always fixed the Lindblad operator to $\hat{S}_z$. The fourth plot has $N=4$ and $\gamma=0.01$, both fixed, and $\Delta \delta$ is plotted for different Lindblad operators $\hat{S}_-$, $\hat{S}_+$ and $\hat{S}_z$. We see the insensitivity points mostly appear only for the noisy cases (with the exception of the $\ket{N/2,N/2}$ case). Increasing $\gamma$ never results in shifts of the insensitivity points positions in $T_H$. Comparing different Lindblad operators we see that the $\hat{S}_-$ and $\hat{S}_+$ noise operators yield the best sensitivity behaviour for every initial state.