Quantum double lock-in amplifier

Precisely measuring weak alternating signals amidst strong noise is crucial across scientific research and applications. Lock-in amplifiers excel at extracting such signals, typically using a reference signal to isolate the target frequency. However, when the signal's initial phase is unknown, a single reference signal is insufficient to extract complete signal characteristics (amplitude, frequency, and phase). Double lock-in amplifiers address this by employing two orthogonal reference signals. Quantum lock-in amplifiers, utilizing quantum control, have shown promise, but existing methods often assume a known initial phase. This research proposes a general protocol for a quantum double lock-in amplifier to address the unknown initial phase challenge, and demonstrates its experimental feasibility using a five-level system to enhance efficiency and robustness.

Classical lock-in amplifiers use a mixing process involving multiplication of the input signal with a reference signal followed by low-pass filtering to extract the target signal. For signals with unknown initial phases, double lock-in amplifiers use two orthogonal reference signals. Quantum lock-in techniques have been previously demonstrated for various applications like frequency measurement, magnetic field sensing, and weak-force detection. These often leverage dynamical decoupling sequences (like Carr-Purcell and periodic dynamical decoupling) as reference signals, but existing approaches fall short when the initial phase is unknown. This study aims to address this limitation by developing a quantum double lock-in amplifier.

The proposed quantum double lock-in amplifier uses double quantum interferometry with two orthogonal periodic multi-pulse sequences (e.g., PDD and CP sequences) acting as orthogonal reference signals. The protocol involves two quantum mixers, where the interaction between the probe and the signal is described by a Hamiltonian. Dynamical decoupling sequences are employed as reference signals, making them analogous to the classical orthogonal reference signals (sin(ωt) and cos(ωt)). The mixing is achieved via non-commuting operations, and the filtering is accomplished through time-evolution. The paper details a practical implementation using a five-level double-Λ coherent population trapping (CPT) system in ⁸⁷Rb atoms. Each Λ system acts as a quantum mixer, and PDD and CP sequences serve as orthogonal reference signals. This system offers advantages, including reduced measurement time and mitigation of time-dependent systematic errors compared to two-level systems. The impact of finite pulse length and stochastic noise is analyzed to ensure experimental feasibility. Analytical and numerical methods are utilized to assess the performance for both weak and strong signals, analyzing the symmetry of the combined measurement signal for weak signals and Fast Fourier Transform (FFT) analysis for strong signals. A five-level double-Λ CPT system in ⁸⁷Rb atoms is used to demonstrate the experimental implementation, leveraging the system's ability to efficiently generate orthogonal pulse sequences and detect signals via fluorescence or transmission spectroscopy. The Lindblad master equation is used to model the system's dynamics, considering factors like finite pulse length and stochastic noise.

The paper successfully demonstrates a quantum double lock-in amplifier protocol. For weak signals (A<1), the sum of the measurement signals from the PDD and CP sequences exhibits symmetry at the lock-in point, allowing for the extraction of the target signal's parameters. For strong signals (A≥1), the FFT of the combined measurement signal shows a bisinusoidal oscillation at the lock-in point, allowing for amplitude and phase extraction. The five-level double-Λ CPT system in ⁸⁷Rb provides a practical implementation. The system significantly reduces measurement time compared to two-level systems, and mitigates time-dependent systematic errors. The robustness of the system to experimental imperfections, specifically finite pulse length and stochastic noise, is demonstrated. Numerical simulations show that the quantum double lock-in amplifier maintains performance even with finite pulse lengths (T₀ ≤ 0.4τ) and in the presence of significant noise (SNR ≥ -20dB for weak signals and SNR > -10dB for strong signals). The analytical approximations presented accurately predict the system's behavior.

The proposed quantum double lock-in amplifier successfully addresses the challenge of measuring alternating signals with unknown initial phases. The use of two orthogonal dynamical decoupling sequences enables the extraction of complete signal characteristics. The experimental implementation using a five-level system offers practical advantages, including reduced measurement time and error mitigation. The robustness demonstrated against experimental imperfections highlights the practical potential of this approach for quantum sensing applications. The results open up possibilities for improved sensitivity and accuracy in various quantum sensing technologies.

This research provides a comprehensive framework for building a quantum double lock-in amplifier, offering a quantum analog to its classical counterpart. The protocol's effectiveness is demonstrated analytically and numerically, highlighting its robustness. The use of a five-level system provides a practical and efficient implementation that reduces measurement time and avoids additional errors. Future work could explore the application of this method to different quantum systems and the optimization of pulse sequences for specific noise environments.

The study primarily focuses on white noise. While robustness to this common noise source is demonstrated, other noise sources may influence the system's performance, warranting further investigation. The analytical calculations use approximations (like Dyson expansion), limiting their strict accuracy for some parameter regimes. The specific five-level system used for demonstration is one example, and the adaptability to other physical platforms requires further study. Finally, detailed error analysis and comparison with state-of-the-art classical methods would further strengthen the results.