Quantum double lock-in amplifier

The paper addresses the challenge of measuring weak alternating (AC) signals buried in strong noise when the signal’s initial phase is unknown. Classical single lock-in detection is effective when the phase is known, but cannot recover full signal characteristics (amplitude, frequency, phase) with unknown phase. Classical double lock-in amplifiers overcome this by mixing with two orthogonal references and low-pass filtering to obtain I and Q. In the quantum domain, lock-in approaches using dynamical decoupling sequences have been demonstrated for specific tasks, yet prior schemes typically assume known phase. The research question is whether a quantum analogue of the classical double lock-in can be realized to extract complete signal information with unknown phase using experimentally accessible techniques. The authors propose a quantum double lock-in amplifier realized via two orthogonal dynamical decoupling sequences acting as quantum mixers, enabling full reconstruction of an AC signal in noise.

Lock-in amplification is a standard technique for extracting time-dependent signals from noise via mixing with a reference and filtering. Classical double lock-in (phase-sensitive detection) uses two orthogonal references to retrieve full signal parameters. Quantum lock-in methods leveraging non-commuting control and time evolution have been demonstrated in single and many-body systems for magnetometry, frequency measurements, light shift detection, and force sensing. Dynamical decoupling sequences such as Carr-Purcell (CP) and periodic dynamical decoupling (PDD) serve as quantum references. However, prior quantum implementations generally target signals with known phase and single reference sequences, limiting full parameter extraction when the phase is unknown. The paper builds on these advances to propose a quantum counterpart of double lock-in using orthogonal sequences (e.g., PDD and CP, or XY4-N with appropriate delays).

General protocol: Two quantum mixers (two-level systems or effective qubits) are driven by orthogonal periodic multi-pulse sequences that do not commute with the signal coupling, emulating orthogonal reference channels. For each two-level probe {|↑⟩,|↓⟩}, the signal-plus-noise couples via H_int = M(t)σ_x with M(t) = S(t)+N(t), S(t)=A sin(ωt+β). The mixing modulation H_ref = σ_φ φ(t) does not commute with H_int. The total Hamiltonian is H = H_int + (1/2)[M(t)σ_x, σ_φ φ(t)] and evolution obeys the Schrödinger equation. In the interaction picture with respect to H_ref, the modulation enters via a phase α(t) = ∫ λ(t′)dt′, yielding effective instantaneous frequencies ω_1(t) = M(t)cos α(t) and ω_2(t) = M(t)sin α(t). Pulse sequences: The modulation φ(t) is implemented as an n-pulse sequence with equal spacing T_m=τ_m (π pulses), defining a carrier ω_0=π/τ. Two orthogonal sequences are chosen: PDD (λ=0) and CP (λ=1/2). In the hard-pulse limit (pulse length T_0→0), Ω(t) ≈ Σ_j δ(t−jT_m). The lock-in point is defined by τ_m matching the half-period of the target signal, τ=π/ω. Phase accumulation: Initializing each qubit to (|↑⟩+|↓⟩)/√2, after n pulses the accumulated phases under PDD and CP are derived (see Supplementary Note 2). The total accumulated phases are φ_PDD and φ_CP, with closed-form expressions depending on A, ω, β, and detuning from lock-in through (τ_m−τ). Symmetry properties around τ_m=τ differ for PDD and CP and depend on β; for β=0 (β=π/2) PDD is symmetric (anti-symmetric) and CP is anti-symmetric (symmetric). For generic β the individual spectra lose symmetry, preventing direct lock-in identification using a single sequence. Signal extraction logic:

• Weak signals (2nA/ω ≪ 1 up to near unity): Use measurement probabilities P_PDD=[1−cos(φ_PDD)]/2 and P_CP=[1−cos(φ_CP)]/2. Construct P_sum = P_PDD + P_CP. Analytically, P_sum is symmetric about τ_m=τ, restoring a clear lock-in point; fit to extract ω from τ, then A from the analytic form, and finally β from either P_PDD or P_CP once ω and A are known.
• Strong signals (A ≥ 1): Use z-expectations ŏ^DD=cos φ_PDD and ŏ^CP=cos φ_CP and construct ŏ_sum=ŏ^DD+ŏ^CP. At τ_m=τ, ŏ_sum = −cos[2A cosβ/n] − cos[2A sinβ/n], a bisinusoidal oscillation in n. Determine τ (hence ω) by scanning τ_m and computing the FFT of ŏ_sum across n and identifying the unique four-peak bisinusoidal pattern; quantify with the inverse participation ratio (IPR), which peaks at τ_m=τ (IPR→1/4 as n_m→∞ and →0 off-lock). Then extract A and β via peak frequencies ω_P^DD=2A|sinβ| and ω_P^CP=2A|cosβ| using A=(1/2)√[(ω_P^DD)^2+(ω_P^CP)^2] and |β|=arctan(ω_P^DD/ω_P^CP). The sign of β is determined from derivatives of ŏ^DD or ŏ^CP with respect to T_m evaluated at three nearby τ_m values. Experimental realization with five-level double-Λ CPT in 87Rb: The protocol is implemented using a 5-level scheme with two Λ systems {|1⟩,|2⟩} and {|3⟩,|4⟩} sharing excited state |5⟩ on the D1 line. Two dark states |D_12⟩=(|1⟩−|2⟩)/√2 and |D_34⟩=(|3⟩−|4⟩)/√2 are prepared via CPT with far-detuned average detunings δ_1≈δ_2≫(Ω,Δ_1,Δ_2), ensuring the two channels are decoupled. During interrogation, PDD (on {|1⟩,|2⟩}) and CP (on {|3⟩,|4⟩}) are applied simultaneously; a strong bias magnetic field splits Zeeman sublevels. After evolution t_n=nT_m, a short CPT query pulse maps accumulated coherence to the common excited-state population p_55, measured via fluorescence/transmission. Dynamics are modeled by a Lindblad master equation with Hamiltonian including detunings Δ_1, Δ_2 and couplings Ω_j (set equal for convenience), and Lindblad operators L_j=|j⟩⟨5|. The measured p_55 contains contributions proportional to Re(ρ_12^n) and Re(ρ_34^n), enabling direct access to the summed signal. For weak signals, p_55 ∝ P_sum; for strong signals, a normalized sequence p_55,n yields the bisinusoidal form enabling FFT/IPR analysis. Parameter choices for simulations: Typical values include ω=2π×50 kHz (τ=10 μs), pulse length T_0=2 μs, bias B_bias=0.143 mT, δ_1=δ_2=2π×1 MHz, Δ_1=Δ_2=0, and Rabi frequencies Ω_j=Ω. Weak-signal example B_1≈1 nT with n up to 200; strong-signal example B_0≈2 μT with n up to 400–1400 depending on analysis. Robustness analyses: Finite pulse length T_0 (e.g., 0, 0.2τ, 0.4τ) shows negligible effect on locating the lock-in point and small estimation errors when T_0≤0.4τ. Stochastic Gaussian white noise added to both σ_z and σ_x channels reduces FFT amplitudes and IPR but preserves lock-in identification and parameter extraction down to SNR ≈ −10 dB for strong and ≈ −20 dB for weak signals (with averaging). The method is also analyzed against decoherence and lab 50 Hz noise in supplements.

• A quantum double lock-in amplifier is proposed and analyzed, using two quantum mixers driven by orthogonal dynamical decoupling sequences (PDD and CP) to extract full AC signal parameters (A, ω, β) even with unknown initial phase.
• Lock-in identification: For weak signals, summing probabilities P_sum = P_PDD + P_CP restores symmetry around τ_m=τ, enabling determination of ω from the lock-in point. For strong signals, the summed expectation ŏ_sum exhibits a bisinusoidal pattern at lock-in whose FFT has four characteristic peaks; the inverse participation ratio (IPR) peaks at τ_m=τ (IPR→1/4 as n_m→∞, 0 off-lock), providing a robust lock-in criterion.
• Parameter extraction: At lock-in, the two FFT peak frequencies satisfy ω_P^DD/ω = 2A|sinβ| and ω_P^CP/ω = 2A|cosβ|, yielding A=(1/2)√[(ω_P^DD)^2+(ω_P^CP)^2] and |β|=arctan(ω_P^DD/ω_P^CP). The sign of β is obtained from derivatives of single-channel signals around τ_m.
• Five-level double-Λ CPT implementation in 87Rb enables simultaneous application of the two orthogonal sequences in two Λ channels, reducing total measurement time nearly by half and mitigating time-dependent systematic errors compared with sequential two-level approaches. The common excited-state population p_55 directly provides the summed signal in a single measurement.
• Numerical simulations using a Lindblad master equation show excellent agreement with analytical predictions for both weak and strong signal regimes. Example FFT peaks at τ_m=τ appear at normalized frequencies close to theoretical values (e.g., 0.587 and 0.968 vs. predicted 0.561 and 0.971 for β=−π/6), validating extraction formulas.
• Robustness: Finite pulse length T_0 up to ≈0.4τ does not shift the lock-in point; relative estimation errors δB_est/B_0 and δβ_est/β remain small for T_0≤0.4τ. Under Gaussian white noise, lock-in identification via IPR and parameter extraction via FFT remain feasible for SNR > −10 dB (strong signals) and SNR ≥ −20 dB (weak signals) with averaging.
• Practical considerations: Finite coherence time T_2 limits the maximum evolution n_mT and leads to a discrete frequency grid with a small lock-in frequency shift Δ that decreases with larger n_m; this is quantified and shown to be manageable.

The proposed quantum double lock-in amplifier addresses the core challenge of extracting complete AC signal information when the signal phase is unknown, by emulating classical I/Q detection with two orthogonal quantum reference channels. Non-commuting control (mixing) and time evolution (filtering) implement the quantum analogue of mixing and low-pass filtering. The method provides explicit procedures for both weak and strong signals: symmetry-based lock-in identification and fitting for weak signals, and FFT plus IPR diagnostics for strong signals. Implementing the two orthogonal sequences simultaneously in a five-level double-Λ CPT system yields significant experimental advantages, including reduced acquisition time and minimized time-dependent systematics, while keeping control complexity compatible with present technology. The confirmed robustness to finite pulse width, stochastic noise, and decoherence, together with quantitative criteria (e.g., IPR peak at lock-in, SNR thresholds), demonstrates that full recovery of A, ω, and β is practical. Compared to prior quantum lock-in approaches that rely on a single sequence and typically assume a known phase, this work enables simultaneous extraction of all parameters without sequential phase scans, advancing the capabilities of quantum sensing.

The study establishes a quantum counterpart to the classical double lock-in amplifier using two orthogonal dynamical decoupling sequences (PDD and CP). It provides a general protocol and analytical framework to extract frequency, amplitude, and phase of an AC signal immersed in noise. A practical implementation is detailed using a five-level double-Λ CPT system in 87Rb, where simultaneous operation of the two quantum mixers enables near halving of measurement time and reduces time-dependent systematic errors. Numerical simulations corroborate analytical predictions and confirm robustness to finite pulse lengths and realistic noise. Future research could include experimental realization in atomic CPT platforms and solid-state systems (e.g., NV centers, trapped ions), optimization with alternative orthogonal sequences (e.g., XY4-N), comprehensive sensitivity benchmarking, and detailed studies of colored noise and platform-specific noise sources to further improve precision and robustness.

• Finite coherence time T_2 constrains the maximum interrogation time (n_mT ≤ T_2), leading to a discrete frequency grid and a small shift of the lock-in point; larger n_m mitigates the shift at the cost of coherence.
• The analysis assumes far-detuning conditions to decouple the two Λ channels in the five-level CPT implementation; deviations from this regime can introduce cross-talk and systematic errors.
• Finite pulse length T_0 introduces small deviations from the ideal hard-pulse model; while effects are negligible for T_0 ≤ 0.4τ, larger T_0 degrades accuracy and may shift spectral features.
• Robustness to noise has been demonstrated for Gaussian white noise within certain SNR thresholds (e.g., > −10 dB for strong signals with averaging); performance may degrade under harsher noise or different spectra (e.g., technical 50/150 Hz, photon shot noise) and requires platform-specific mitigation.
• Results are supported by analytical derivations and numerical simulations; full experimental validation and calibration in specific platforms remain to be completed.