Fig. 4: Extraction of a strong target signal.
a The fast Fourier transform (FFT) spectrum of \({\langle {\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}\rangle }_{{{{{{{{\rm{n}}}}}}}}}^{{{{{{{{\rm{sum}}}}}}}}}\) versus (τm − τ) with even positive integers n up to nm = 400. Given τ = τm, the FFT spectrum just has four peaks and one can use it to determine the lock-in point. b The inverse participation ratio (IPR) versus (τm − τ). The maximum of IPR can be used to determine the lock-in point. c The inset for the local amplification region in (b) and denotes the shift of lock-in point D. d The FFT of \({\langle {\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}\rangle }_{{{{{{{{\rm{n}}}}}}}}}^{{{{{{{{\rm{sum}}}}}}}}}\) at τm = τ. The four peaks locate at \({\omega }_{{{{{{{{\rm{FFT}}}}}}}}}^{{{{{{{{\rm{CP}}}}}}}}}/\omega =\frac{2}{\omega }A| \sin (\beta )| =0.637\), \({\omega }_{{{{{{{{\rm{FFT}}}}}}}}}^{{{{{{{{\rm{PDD}}}}}}}}}/\omega =\frac{2}{\omega }A| \cos (\beta )| =1.103\), \({\overline{\omega }}_{{{{{{{{\rm{FFT}}}}}}}}}^{{{{{{{{\rm{PDD}}}}}}}}}/\omega =(\pi -{\omega }_{{{{{{{{\rm{FFT}}}}}}}}}^{{{{{{{{\rm{PDD}}}}}}}}}/\omega )\) and \({\overline{\omega }}_{{{{{{{{\rm{FFT}}}}}}}}}^{{{{{{{{\rm{CP}}}}}}}}}/\omega =(\pi -{\omega }_{{{{{{{{\rm{FFT}}}}}}}}}^{{{{{{{{\rm{CP}}}}}}}}}/\omega )\) respectively. e The variation of the shift D versus the maximum sensing scanning time nm. Here, we choose A = 2, β = −π/6, and ω = π.
