with the fact that the signal decays quickly, as shown in Fig. \[fig:sen\_rel\](b). The relaxation constant is about 30 ms. The spectra in Fig. \[fig:sen\_rel\](c) were measured when HSs were not decoupled (the same as (b)) but the XY-8 sequence was applied on CC and CSs simultaneously. The signals decay much more slowly than those in (b) which indicates that dynamical decoupling was effective. When a dynamical decoupling technique is applied on a sensor, it cannot detect the DC component but can measure the AC one whose frequency is determined by the decoupling technique [@RevModPhys.89.035002]. Therefore, our results in the Fig. \[fig:sen\_rel\](c) show the potential to realize the entanglement enhanced AC magnetic field sensor under time inhomogeneous noise, as theoretically predicted in [@PhysRevA.84.012103; @chin2012quantum; @tanaka2015proposed; @RevModPhys.88.021002].

![ Measurements under various environments as a function of measurement time $\tau(=3.44~{\rm ms} \times n)$. (a) “without noise” by decoupling all H spins (full-decoupling case), (b) “under noise” without decoupling HSs (selective-decoupling), and (c) “under noise” but the noise is suppressed by a dynamical decoupling technique (XY-8) during measurements. []{data-label="fig:sen_rel"}](noise_dep){width="8.6cm"}

Summary
=======

We successfully modeled an entangled magnetic field sensor under various noisy environments with NMR techniques. In our model, a sensor is a 