Fig. 6: Numerical results of the quantum double lock-in amplifier via five-level double-Λ coherent population trapping (CPT) in ⁸⁷Rb.

a Locking of a weak target signal via the normalized common excited-state population ρ₅₅ versus (τ_m − τ). The numerical results of ρ₅₅/a (dashed red line) fit well with the sum of excited-state populations given by two independent Λ configurations (solid blue line), and they are both consistent with the analytical approximation (dotted green line). Here, B₀ = 1 nT, n = 200, ω = 2π × 50 kHz (τ = 10 μs), β = − π/6, and T_Ω = 2 μs. b Locking of a strong target signal via the IPR versus (τ_m − τ). The inverse participation ratio (IPR) approach its maximum at the lock-in point τ_m = τ. c The fast Fourier transform (FFT) spectrum of \({\tilde{\rho }}_{{{{{{{{\rm{55,n}}}}}}}}}\) for τ_m = τ. The first two peaks are 0.587 and 0.968 which are very close to the theoretical ones \({\omega }_{{{{{{{{\rm{FFT}}}}}}}}}^{{{{{{{{\rm{CP}}}}}}}}}/\omega =2A| \sin (\beta )| /\omega =0.561\) and \({\omega }_{{{{{{{{\rm{FFT}}}}}}}}}^{{{{{{{{\rm{PDD}}}}}}}}}/\omega =2A| \cos (\beta )| /\omega =0.971\). Here, B₀ = 2 μT, n = 2, 4,⋯, 400, ω = 2π × 50 kHz(τ = 10 μs), β = − π/6, and T_Ω = 2 μs.