Fig. 1: The schematic of classical and quantum double lock-in amplifiers.
a The classical double lock-in amplifier. Vs(t) = S(t) + N(t) is the input signal, where \(S(t)=A\sin (\omega t+\beta )\) is the target signal submerged within the noise N(t). Vr1,2(t) are the two orthogonal reference signals. The amplitude A, frequency ω, and phase β can be extracted after mixing with a multiplier and filtering by integration. b The quantum double lock-in amplifier. There are two identical quantum mixers. For each quantum mixer, the coupling between the probe and the signal is described by \({\hat{H}}_{{{{{{{{\rm{int}}}}}}}}}=\frac{\hslash }{2}M(t){\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}\) where M(t) = S(t) + N(t) includes the target signal S(t) and the noise N(t). The mixing modulations \({\hat{H}}_{{{{{{{{\rm{ref1}}}}}}}}}=\frac{\hslash }{2}{\Omega }_{{{{{{{{\rm{PDD}}}}}}}}}(t){\hat{\sigma }}_{{{{{{{{\rm{x}}}}}}}}}\) and \({\hat{H}}_{{{{{{{{\rm{ref2}}}}}}}}}=\frac{\hslash }{2}{\Omega }_{{{{{{{{\rm{CP}}}}}}}}}(t){\hat{\sigma }}_{{{{{{{{\rm{x}}}}}}}}}\) (implemented by the periodic dynamical decoupling (PDD) and Carr-Purcell (CP) sequences respectively), which do not commute with \({\hat{H}}_{{{{{{{{\rm{int}}}}}}}}}\), are analog to the two reference signals Vr1(t) and Vr2(t). Each mixer obeys the Hamiltonian \(\hat{H}={\hat{H}}_{{{{{{{{\rm{int}}}}}}}}}+{\hat{H}}_{{{{{{{{\rm{ref1,2}}}}}}}}}\), which can be regarded as a single quantum lock-in amplifier. The mixing process is achieved by non-commutating operations, and the filtering process is realized by time-evolution. The combination of the two quantum lock-in amplifiers forms a quantum double lock-in amplifier, which can extract the complete characteristics of the target signal \(S(t)=A\sin (\omega t+\beta )\).
