![ Expected spectra calculated from $S_k(t)$ in the case of $J T_2 = 22$. (a) $\theta =  0^\circ$, (b) $\theta = 50^\circ$ for $S_{\bigcirc}(t)$, (c) $\theta = 50^\circ$ for $S_{\circ}(t)$. The frequency differences of these peaks are $J$. []{data-label="fig_fid_th"}](Th_spectra){width="8.6cm"}

Engineered Noisy Environment {#TH_E_E}
----------------------------

Our idea to engineer the environment can be seen from Fig. \[fig:model\]. If System I directly interacts with Markovian Environment, it decays exponentially. If it interacts with Markovian Environment through System II, it shows various decay behaviors since System II acts as a memory of an engineered environment formed by System II and the Markovian Environment [@kondo2016; @Iwakura2017; @Kondo2018; @kondo2007; @PhysRevA.55.2290; @PhysRevLett.120.030402].

The spins chain ($\circ-\bigcirc-\circ$ in Fig. \[fig:q\_circuit\](c)) is regarded as the sensor where the side spins (two $\circ$’s) accumulate the phase under the external field and this phase is measured via the center spin ($\bigcirc$), as we discussed in Sect. \[TH\_M\_S\]. The side spins ($\circ$’s) are surrounded by two sets of three spins (three $\bullet$’s) that we call environment spins. We regard the side spins (two $\circ$’s) as two independent System I’s, while we do the two sets of $\bullet$’s as two System II’s. These environment spins with Markovian Environment outside them act as 