Figure 3

The total variance for multiphase magnetic field estimation. (a) The total variance \(\Delta ^{2}\boldsymbol{\phi} = \operatorname{tr}[Q^{-1}]\) as a function of the interaction time t for different choices of φ as shown in the figure. Here, we use \(\phi _{x} = \phi _{y} = \phi _{z} = \varphi \) for illustration (although in general, these values may differ). The solid curves are exact results, which are given by theoretical analysis, and the dotted curves are obtained from the Stoc.PSR. It can be observed that \(\operatorname{tr}[Q^{-1}]\) varies with t and reaches its minimum at a certain time. More importantly, the results show a good agreement between the Stoc.PSR and the exact theoretical analysis. (b) Plot of the total variance versus t under the time-dependent dephasing noise for various decay rates γ. Here, we fixed \(\varphi = \pi /10\)