Abstract
Quantum lock-in amplifiers have been proposed to extract an alternating signal from a strong noise background. However, due to the typical target signal has unknown initial phase, it is challenging to extract complete information about the signal’s amplitude, frequency, and initial phase. Here, we present a general protocol for achieving a quantum double lock-in amplifier by employing two quantum mixers operating under orthogonal pulse sequences. To demonstrate the practical implementation, we discuss the experimental feasibility using a five-level double-Λ coherent population trapping system with Rb atoms. Here, each Λ structure acts as a quantum mixer, and two applied dynamical decoupling sequences serve as orthogonal reference signals. Notably, the system significantly reduces the total measurement time by nearly half and mitigates time-dependent systematic errors compared to conventional two-level systems. Furthermore, our quantum double lock-in amplifier is robust against experimental imperfections. This study establishes a pathway to alternating signal measurement, thereby facilitating the development of practical quantum sensing technologies.
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Introduction
The precise measurement of weak alternating signals in the presence of a noisy background is of great significance in both fundamental scientific research and practical applications. Typically, the desired signal is obscured by the noise, making detection challenging. To achieve a high signal-to-noise ratio, it is necessary to minimize the influence of noise while amplifying the response to the target signal. Lock-in amplifiers have been widely used in various fields1,2,3,4,5,6 due to their ability to extract time-dependent alternating signals from highly noisy backgrounds. In essence, a lock-in amplifier employs a mixing process by multiplying the input signal with a reference signal, followed by detection through an adjustable low-pass filter. The filter effectively eliminates contributions from signals that do not share the same frequency as the reference signal, thereby selectively rejecting all other frequency components.
Typically, when dealing with time-dependent alternating signals of known initial phase, conventional lock-in techniques using a single reference signal can effectively extract the characteristics of the target signal. However, when the initial phase is unknown7,8, extracting the complete characteristics, including amplitude, frequency, and initial phase, becomes challenging with a single reference signal alone. To address this issue, a double lock-in amplifier, also known as phase-sensitive detector9, has been proposed and widely used for signal measurement. In a double lock-in amplifier, the target signal is mixed with two orthogonal reference signals. The resulting signals from the two mixers pass through separate low-pass filters, generating two output signals that enable the extraction of the complete characteristics of the target signal. Quantum lock-in measurements, leveraging advancements in quantum control techniques, have been successfully demonstrated and widely employed for frequency measurement10,11, magnetic field sensing10, vector light shift detection12, and weak-force detection13. The key to realizing a quantum lock-in amplifier lies in identifying quantum analogs of mixing and filtering operations. By utilizing quantum probes, these processes can be achieved through non-commutative operations and time-evolution, respectively. Similar to the implementation of a quantum mixer7,10,11,12, dynamical decoupling sequences can be employed as reference signals to realize a quantum lock-in amplifier. Notably, Carr-Purcell (CP) and periodic dynamical decoupling (PDD) sequences have been widely utilized in various quantum lock-in amplifiers, ranging from single-particle systems10 to many-body systems11. However, these existing schemes only cater to target signals with known initial phases. Extracting the complete characteristics of a target signal using a single PDD or CP sequence becomes challenging when the initial phase is unknown. Similar to the classical double lock-in amplifier, the question arises as to whether a quantum counterpart can be developed to extract the complete characteristics of a target signal. Furthermore, how can a quantum double lock-in amplifier be realized using currently available experimental techniques?
In this article, we propose a general protocol for implementing a quantum double lock-in amplifier by combining double quantum interferometry with two orthogonal periodic multi-pulse sequences, characterized by orthogonal filter functions. In our protocol, each quantum interferometry using a specific periodic multi-pulse sequence serves as a single quantum lock-in amplifier. Specifically, we select the PDD and CP sequences as the two orthogonal periodic multi-pulse sequences. Additionally, the XY4-N sequences14,15,16,17,18, with appropriate delays, can also be employed to realize the quantum double lock-in amplifier. Further, to demonstrate the feasibility of our protocol, we illustrate its implementation using a five-level double-Λ coherent population trapping (CPT) system consisting of 87Rb atoms with PDD and CP pulse sequences. By appropriately adjusting the detuning, the five-level double-Λ system can be split into two Λ systems19,20,21, serving as the quantum mixers for the quantum double lock-in amplifier. The utilization of the five-level system offers significant advantages, including a reduction of nearly half the total measurement time and the avoidance of additional time-dependent systematic errors compared to conventional two-level systems. Furthermore, we analyze the impact of finite pulse length and stochastic noise to demonstrate the experimental feasibility of our approach. Numerical results indicate that the quantum double lock-in amplifier exhibits robustness against these imperfections. Our scheme provides a practical pathway for accurately measuring the complete characteristics of an alternating signal within a strong noise background.
Results
General protocol
In this section, we introduce the general protocol of a quantum double lock-in amplifier, which aims to extract the complete characteristics of a target signal within strong noise background. In general, a conventional classical lock-in amplifier cannot effectively extract the phase information of the target signal. However, a classical double lock-in amplifier can solve this problem. By mixing the input signal VS(t) = S(t) + N(t) with two orthogonal reference signals \({V}_{{{{{{{{\rm{r1}}}}}}}}}(t)=\sin ({\omega }_{m}t)\) and \({V}_{{{{{{{{\rm{r2}}}}}}}}}(t)=\cos ({\omega }_{m}t)\) respectively and integrating the two mixed signals over a certain time, the target signal can be extracted, see Fig. 1a. Here, \(S(t)=A\sin (\omega t+\beta )\) is the target signal submerged in the noise N(t) and all three parameters (A, ω, β) are unknown to be measured and unchanged in measurements. The two multipliers, which are described by \({{{{{{{{\rm{V}}}}}}}}}_{{{{{{{{\rm{r1}}}}}}}}}^{{{{{{{{\rm{mix}}}}}}}}}(t)={V}_{{{{{{{{\rm{S}}}}}}}}}(t)\times {V}_{{{{{{{{\rm{r1}}}}}}}}}(t)\) and \({{{{{{{{\rm{V}}}}}}}}}_{{{{{{{{\rm{r2}}}}}}}}}^{{{{{{{{\rm{mix}}}}}}}}}(t)={V}_{{{{{{{{\rm{S}}}}}}}}}(t)\times {V}_{{{{{{{{\rm{r2}}}}}}}}}(t)\), are used for mixing input and reference signals. The integrator is used to filter out the components whose frequencies are different from the reference frequency ωm. One can find that, at the lock-in point ωm = ω, the two output signals are given as \(I=\frac{AT}{2}\cos (\beta )\) and \(Q=\frac{AT}{2}\sin (\beta )\). Therefore, at the lock-in point, one can obtain \(A=2\sqrt{{I}^{2}+{Q}^{2}}/T\) and \(\beta =\arctan (Q/I)\) for the target signal22,23. In particular, if the noise spectral components are far from the reference frequency ωm, the noise effects will be averaged out through the integration (see Supplementary Note 1 for more details).
In analogy to a classical double lock-in amplifier, a quantum double lock-in amplifier can be realized by using two orthogonal multi-pulse sequences, which act the role of two orthogonal reference signals. Generally, to realize the quantum double lock-in amplifier, one can choose any two multi-pulse sequences whose filtering functions are orthogonal. The spacing of the adjacent π pulses is τm and one can define the carrier frequency ωm = π/τm. As the applied multi-pulse sequences (acting as the reference signal) are non-commutating with the target signal7,10,11, one can mix the input signal and the reference signal. And then the following time-evolution filters out the noise spectral components different from the reference frequency ωm.
To illustrate our protocol, we consider two individual two-level systems such as NV center7,8,14 and single trapped ion10 whose energy levels are labeled by \(\left\vert \uparrow \right\rangle\) and \(\left\vert \downarrow \right\rangle\). For each two-level system, the coupling between the probe and the external signal is described by the Hamiltonian \({\hat{H}}_{{{{{{{{\rm{int}}}}}}}}}=\frac{\hslash }{2}M(t){\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}\) with the Pauli operators \({\hat{\sigma }}_{{{{{{{{\rm{x,y,z}}}}}}}}}\). The external signal M(t) = S(t) + N(t) consists of the target signal \(S(t)=A\sin (\omega t+\beta )\) and the stochastic noise N(t). Here, we consider the mixing term \({\hat{H}}_{{{{{{{{\rm{ref}}}}}}}}}=\frac{\hslash }{2}\Omega (t){\hat{\sigma }}_{{{{{{{{\rm{x}}}}}}}}}\), which does not commute with \({\hat{H}}_{{{{{{{{\rm{int}}}}}}}}}\). Thus, the whole Hamiltonian reads
The time-evolution obeys the Schrödinger equation,
where \({\left\vert \Psi (t)\right\rangle }_{{{{{{{{\rm{S}}}}}}}}}\) denotes the system state. In the interaction picture with respecting to \({\hat{H}}_{{{{{{{{\rm{ref}}}}}}}}}\) (See Supplementary Note 2 for more details), the time-evolution obeys
with \(\alpha (t)=\int\nolimits_{0}^{t}\Omega ({t}^{{\prime} })d{t}^{{\prime} }\). The instantaneous state at time t reads
where \(\hat{{{{{{{{\mathcal{T}}}}}}}}}\) denotes the time-ordering operator, and \({\phi }_{{{{{{{{\rm{z}}}}}}}}}(t)=\int\nolimits_{0}^{t}{\omega }_{{{{{{{{\rm{z}}}}}}}}}({t}^{{\prime} })d{t}^{{\prime} }\) and \({\phi }_{{{{{{{{\rm{y}}}}}}}}}(t)=\int\nolimits_{0}^{t}{\omega }_{{{{{{{{\rm{y}}}}}}}}}({t}^{{\prime} })d{t}^{{\prime} }\) with angular frequencies \({\omega }_{{{{{{{{\rm{z}}}}}}}}}(t)=M(t)\cos (\alpha (t))\) and \({\omega }_{{{{{{{{\rm{y}}}}}}}}}(t)=M(t)\sin (\alpha (t))\). In our scheme, the time-dependent modulation Ω(t) is designed as a sequence of π pulses with equidistant spacings. This technique is mature and has been widely used in quantum sensing for measuring an oscillating signal in the presence of noises10,11,13,14,24,25,26. In experiments, the π pulse sequences can usually be approximated as square waves,
where TΩ denotes the pulse length. In the limit of TΩ → 0, the π-pulse sequences can be described by hard pulses
with δ(t) being the Dirac δ function, Np denoting the pulse number, τm describing the spacing of the adjacent π pulses, and the parameter λ determining the relative phase with respect to the target signal. Here, as an example for realizing our quantum double lock-in amplifier, we choose two orthogonal sequences: a PDD sequence (λ = 0) and a CP sequence (λ = 1/2)15,27, which can be easily realized with current experimental techniques10,11,15 (see Supplementary Note 2 for more details). In addition to the above sequences, the XY4-N sequences with suitable delay can also be applied to realize the quantum double lock-in amplifier. For hard π pulses, one can easily find \(\alpha (t)=\int\nolimits_{{{{{{{{\rm{0}}}}}}}}}^{t}\Omega ({t}^{{\prime} })d{t}^{{\prime} }={N}_{p}\pi\), therefore we have \(\sin [\alpha (t)]=0\). Initializing each quantum system into the state \({\left\vert \Psi \right\rangle }_{{{{{{{{\rm{in}}}}}}}}}=(\left\vert \uparrow \right\rangle +\left\vert \downarrow \right\rangle )/\sqrt{2}\), in the interaction picture, the instantaneous state at time tn = nτm becomes
with n chosen as a even number to suppress DC noise15. For the results determined by the density matrix \(\rho =\left\vert \Psi \right\rangle \left\langle \Psi \right\vert\), the global phase \({e}^{-i{\phi }_{n}/2}\) can be ignored. Back to the Schrödinger picture10,11, the output state is
with the pulse number Np. For PDD and CP sequences, the total accumulation phases are
and
respectively (see Supplementary Note 2 for more details). Here τ = π/ω is the half period of the target signal. Obviously, given β = 0 (β = π/2), \({\phi }_{n}^{{{{{{{{\rm{PDD}}}}}}}}}\) is symmetric (anti-symmetric) and \({\phi }_{n}^{{{{{{{{\rm{CP}}}}}}}}}\) is anti-symmetric (symmetric) with respect to the lock-in point τm = τ, see Fig. 2. However, the symmetry of \({\phi }_{n}^{{{{{{{{\rm{PDD(CP)}}}}}}}}}\) will be destroyed when β ≠ 0 (or β ≠ π/2), see Fig. 2. Different from the classical lock-in amplifier, which defines the lock-in point based on the reference frequency ωm, the lock-in point of our scheme is defined according to the half period of target signal τ10.
In the stage of signal extraction, an unitary operation \(U={e}^{-i\frac{\pi }{4}{\hat{\sigma }}_{{{{{{{{\rm{y}}}}}}}}}}\) is applied for recombination and the readout state becomes \({\left\vert \Psi \right\rangle }_{{{{{{{{\rm{re}}}}}}}}}=\left[-i\sin (\frac{{\phi }_{n}^{{\prime} }}{2})\left\vert \uparrow \right\rangle +\cos (\frac{{\phi }_{n}^{{\prime} }}{2})\left\vert \downarrow \right\rangle \right]\) with \({\phi }_{n}^{{\prime} }={(-1)}^{{N}_{p}}{\phi }_{n}\). Hence the final probability of the probe in the state \(\left\vert \uparrow \right\rangle\) are \({P}_{\uparrow ,n}^{{{{{{{{\rm{PDD}}}}}}}}}=[1-\cos ({\phi }_{n}^{{{{{{{{\rm{PDD}}}}}}}}})]/2\) and \({P}_{\uparrow ,n}^{{{{{{{{\rm{CP}}}}}}}}}=[1-\cos ({\phi }_{n}^{{{{{{{{\rm{CP}}}}}}}}})]/2\) respectively. The corresponding expectations of z-component Pauli operator are \({\langle {\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}\rangle }_{{{{{{{{\rm{n}}}}}}}}}^{{{{{{{{\rm{PDD}}}}}}}}}=-\cos ({\phi }_{n}^{{{{{{{{\rm{PDD}}}}}}}}})\) and \({\langle {\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}\rangle }_{{{{{{{{\rm{n}}}}}}}}}^{{{{{{{{\rm{CP}}}}}}}}}=-\cos ({\phi }_{n}^{{{{{{{{\rm{CP}}}}}}}}})\) respectively. For β = 0 (or β = π/2), \({P}_{\uparrow ,n}^{{{{{{{{\rm{PDD(CP)}}}}}}}}}\) and \({\langle {\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}\rangle }_{{{{{{{{\rm{n}}}}}}}}}^{{{{{{{{\rm{PDD(CP)}}}}}}}}}\) are both symmetric with respect to the lock-in point τ = τm. Thus through modulating the pulse repetition period τm, one can determine the lock-in point from the pattern symmetry, and the amplitude can be extracted from Eq. (9) or Eq. (10) via a fitting procedure7,10,11,15,28. However, when β ≠ 0 (or β ≠ π/2), the phase \({\phi }_{n}^{{{{{{{{\rm{PDD(CP)}}}}}}}}}\) is not symmetric with respect to the lock-in point τ = τm. Thus the symmetry of \({P}_{\uparrow ,n}^{{{{{{{{\rm{PDD(CP)}}}}}}}}}\) and \({\langle {\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}\rangle }_{{{{{{{{\rm{n}}}}}}}}}^{{{{{{{{\rm{PDD(CP)}}}}}}}}}\) are destroyed, see Fig. 3. This means that one cannot determine the lock-in point from the spectrum and extract the target signal S(t) only by means of a single PDD or CP sequence. Below, we introduce how to solve this issue via the quantum double lock-in amplifier. To analyze our protocl analytically, we divide the target signals into two types: (i) the weak signals of \(\frac{A}{\omega }\le \frac{1}{2n}\) and (ii) the strong signals of \(\frac{A}{\omega } > \frac{1}{2n}\).
For weak signals, i.e., \(\frac{A}{\omega }\le \frac{1}{2n}\), we choose the sum of \({P}_{\uparrow ,n}^{{{{{{{{\rm{PDD}}}}}}}}}\) and \({P}_{\uparrow ,n}^{{{{{{{{\rm{CP}}}}}}}}}\) as a measurement signal to recover the spectrum symmetry, that is,
Obviously, \({P}_{\uparrow ,n}^{{{{{{{{\rm{sum}}}}}}}}}\) is symmetric with respect to the lock-in point τm = τ again, see Fig. 3e, f. Thus, one can determine the lock-in point from the symmetric pattern of \({P}_{\uparrow ,n}^{{{{{{{{\rm{sum}}}}}}}}}\) versus (τm − τ), which can be obtained by adjusting the spacing τm of adjacent π pulses. Our analytical results are well consistent with the numerical ones, even for the case of \(\frac{A}{\omega }=\frac{1}{2n}\). Once the value of ω is determined via the the lock-in point τm = τ, one can extract A from Eq. (11) via a fitting procedure26,29. Moreover, due to the value of ω and A are both extracted, one can determine β from \({P}_{\uparrow ,n}^{{{{{{{{\rm{PDD}}}}}}}}}\) or \({P}_{\uparrow ,n}^{{{{{{{{\rm{CP}}}}}}}}}\). Thus the complete characteristics of the target signal can be obtained within our scheme.
For strong signals, i.e., \(\frac{A}{\omega } > \frac{1}{2n}\), the Taylor expansion is no longer valid and one cannot obtain the target signal via the measurement signal \({P}_{\uparrow ,n}^{{{{{{{{\rm{sum}}}}}}}}}\). To resolve this problem, we choose the sum of \({\langle {\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}\rangle }_{{{{{{{{\rm{n}}}}}}}}}^{{{{{{{{\rm{PDD}}}}}}}}}\) and \({\langle {\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}\rangle }_{{{{{{{{\rm{n}}}}}}}}}^{{{{{{{{\rm{CP}}}}}}}}}\) as a new measurement signal, that is,
At the lock-in point τm = τ, it reads
which is an exactly bisinusoidal oscillation. By using a fast Fourier transform (FFT), one can determine the lock-in point by judging whether \({\langle {\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}\rangle }_{{{{{{{{\rm{n}}}}}}}}}^{{{{{{{{\rm{sum}}}}}}}}}\) has bisinusoidal oscillatory pattern. In comparison to \({P}_{\uparrow ,n}^{{{{{{{{\rm{sum}}}}}}}}}\), the FFT of \({\langle {\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}\rangle }_{{{{{{{{\rm{n}}}}}}}}}^{{{{{{{{\rm{sum}}}}}}}}}\) does not contain the zero-frequency component and it is easy to determine the lock-in point from the FFT. In our analysis, we consider a series of measurement results \({\langle {\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}\rangle }_{{{{{{{{\rm{n}}}}}}}}}^{{{{{{{{\rm{sum}}}}}}}}}\) for different evolution times tn = nτm. As shown in Fig. 4a, we give the FFT spectrum of \({\langle {\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}\rangle }_{{{{{{{{\rm{n}}}}}}}}}^{{{{{{{{\rm{sum}}}}}}}}}\) versus τm. When τm = τ, the FFT spectrum has only four peaks and therefore we can determine the lock-in point via its inverse participation ratio (IPR)30,31,32,33,34, which is defined as
with \({F}_{k}={\sum}_{{{{{{{{\rm{n}}}}}}}} = {2},{{{{{{{{\rm{even}}}}}}}}}}^{{n}_{m}}{\langle {\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}\rangle }_{{{{{{{{\rm{n}}}}}}}}}^{{{{{{{{\rm{sum}}}}}}}}}{e}^{-i\left(\frac{2\pi }{{n}_{m}}nk\right)}\), ∣Fk∣ is the FFT amplitude corresponding to FFT angular frequency \({\omega }_{k}=k\frac{2\pi \omega }{{n}_{m}}\,\)(k = 1, 2,⋯, nm/2) and \({t}_{{n}_{m}}={n}_{m}{\tau }_{m}\) is the maximum sensing time. After some algebra, when ω(τm − τ) ≪ 1 and nm → ∞, we have IPR = 1/4 for τm = τ and IPR = 0 for τm ≠ τ (see Supplementary Note 3 for more details). Given τm = τ, the four peaks appear at \({\omega }_{{{{{{{{\rm{FFT}}}}}}}}}^{{{{{{{{\rm{CP}}}}}}}}}=2A| \sin (\beta )|\), \({\omega }_{{{{{{{{\rm{FFT}}}}}}}}}^{{{{{{{{\rm{PDD}}}}}}}}}=2A| \cos (\beta )|\), \({\overline{\omega }}_{{{{{{{{\rm{FFT}}}}}}}}}^{{{{{{{{\rm{PDD}}}}}}}}}=(\pi \omega -{\omega }_{{{{{{{{\rm{FFT}}}}}}}}}^{{{{{{{{\rm{PDD}}}}}}}}})\) and \({\overline{\omega }}_{{{{{{{{\rm{FFT}}}}}}}}}^{{{{{{{{\rm{CP}}}}}}}}}=(\pi \omega -{\omega }_{{{{{{{{\rm{FFT}}}}}}}}}^{{{{{{{{\rm{CP}}}}}}}}})\) respectively, see Fig. 4c. In which, the two peaks at \({\omega }_{{{{{{{{\rm{FFT}}}}}}}}}^{{{{{{{{\rm{CP}}}}}}}}}\) and \({\omega }_{{{{{{{{\rm{FFT}}}}}}}}}^{{{{{{{{\rm{PDD}}}}}}}}}\) correspond to the two oscillation frequencies of the measurement signal \({\langle {\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}\rangle }_{n}^{{{{{{{{\rm{sum}}}}}}}}}{| }_{{\tau }_{m} = \tau }\). Thus, the values of A and β are given as
and
which is similar to a classical double lock-in amplifier. Moreover, one can determine the sign of parameters β from \({\langle {\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}\rangle }_{{{{{{{{\rm{n}}}}}}}}}^{{{{{{{{\rm{PDD}}}}}}}}}\) or \({\langle {\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}\rangle }_{{{{{{{{\rm{n}}}}}}}}}^{{{{{{{{\rm{CP}}}}}}}}}\). The sign of β can be uniquely determined by the derivation of \({\langle {\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}\rangle }_{n}^{{{{{{{{\rm{PDD}}}}}}}}}\) or \({\langle {\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}\rangle }_{n}^{{{{{{{{\rm{CP}}}}}}}}}\) with respect to the pulse repetition period τm for three different modulation periods τm = (τ − dτ), τm = τ and τm = (τ + dτ), in which dτ is the spacing of two adjacent pulse repetition periods (see Supplementary Note 2 for more details). Considering the coherence time of a quantum system is limited in experiments, we numerically calculate the systems under moderate nm satisfying \({n}_{m}\le \frac{{T}_{2}}{\tau }\) with the coherence time T27,10,19. It is shown that IPR ≈ 1/4 for τm = τ and IPR ≈ 0 for τm ≠ τ, thus we can still extract the frequency ω by finding the maximum of IPR, see Fig. 4b. Further, given the measurement signal \({\langle {\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}\rangle }_{{{{{{{{\rm{n}}}}}}}}}^{{{{{{{{\rm{sum}}}}}}}}}{| }_{{\tau }_{m} = \tau }\), we can extract A and β via Eqs. (15) and (16), see Fig. 4d.
In experiments, the coherence time T2 is finite and thus the value of nm should satisfy T2 > nmτ. Given the value of nm, the frequency ωFFT can only take discrete values \({\omega }_{k}=k\frac{2\pi \omega }{{n}_{m}}\)(k = 1, 2,⋯, nm/2), and thus the lock-in point may be shifted. In Fig. 4c, we show the frequency shift D of the lock-in point. We numerically calculate how the frequency shift D varies with nm and find that the envelop of D decreases with nm due to the resolution ratio of FFT frequency ωFFT is \(({\omega }_{{{{{{{{\rm{k+1}}}}}}}}}-{\omega }_{k})=\frac{2\pi \omega }{{n}_{m}}\propto 1/{n}_{m}\), see Fig. 4e.
Experimental feasibility
Our quantum double lock-in amplifier can be realized via a double-Λ CPT system, in which each Λ system is employed as a quantum mixer. As shown in Fig. 5a, the double-Λ CPT system can be realized by simultaneously coupling two independent two-level systems (\(\{\left\vert 1\right\rangle ,\left\vert 2\right\rangle \}\) and \(\{\left\vert 3\right\rangle ,\left\vert 4\right\rangle \}\)) with a common excited state (\(\left\vert 5\right\rangle\)), which has been extensively observed in alkali atoms such as Cs3,35,36 and Rb19,37,38. Moreover, such a system can be divided into two single-Λ systems by splitting the Zeeman sublevels of their ground states19,39, which provide the two independent physical channels. For every single-Λ structure, the population of the excited state reflects the coherence information of two ground states19,20,21. Therefore, for this double-Λ CPT system, the population of the common excited state includes two groups of ground states coherence information, and it may act the role of the sum of two-level systems with PDD and CP sequence modulation19. The simultaneous coupling of these two physical channels can be achieved by lin∣∣lin CPT scheme, in which two CPT light fields are linearly polarized to the same direction and orthogonal to the applied magnetic field19,20,21.
In general, for a single-Λ system, the CPT lasers will pump the atoms into a dark state which is a coherent superposition of two ground states20,36,40. To achieve our quantum double lock-in amplifier, we can simultaneously prepare two dark states as the initial states of two physical channels by a CPT pulse with just one laser and a fiber-coupled electro-optic modulator (EOM) for different tones and such a CPT system has been realized in several laboratories41. After the initialization via CPT pulse, the prepared two dark states can independently accumulate phases under the control of PDD and CP sequences, respectively. During the signal interrogation process, the applied PDD and CP sequences only couple the ground states in the same quantum mixer, see Fig. 5a. Due to the action times of PDD and CP sequence pulses are different, one can utilize just one microwave synthesizer or just a laser beam to realize π pulses via the microwave or Raman lasers, which are both very mature with current experimental technology42,43,44. In comparison to the protocol via separately applying different sequences on the same two-level system7,10,11,14,26, in our protocol with a five-level system, the two orthogonal sequences are applied simultaneously and thus one can save near a half of the total measurement times. Meanwhile, the relative phase of the two orthogonal sequences in five-level system can be tuned easily. Then through applying the CPT pulse, the coherent property of ground states becomes proportional to the population of the common excited state, which can be experimentally detected by fluorescence or transmission spectrum19,20,21,45,46. In particular, the summed signal in our quantum double lock-in amplifier can be directly obtained by only one single measurement on the common excited-state population. In the following, we show an example of quantum double lock-in amplifier via a five-level double-Λ system constructed by the D1 line of 87Rb atoms. The involved energy levels include two groups of magneto-sensitive states \(\{\left\vert 1\right\rangle =\left\vert F=1,{m}_{F}=-1\right\rangle ,\left\vert 2\right\rangle =\left\vert F=2,{m}_{F}=-1\right\rangle \}\) and \(\{\left\vert 3\right\rangle =\left\vert F=1,{m}_{F}= 1\right\rangle ,\left\vert 4\right\rangle = \left\vert F= 2,{m}_{F}= 1\right\rangle \}\) coupling with a common excited state \(\left\vert 5\right\rangle =\left\vert {F}^{{\prime} }=1,{m}_{F}=0\right\rangle\), as shown in Fig. 5a. Their eigen frequencies are respectively labeled as ωj(j = 1, 2, 3, 4, 5). The total decay rate from the excited state (\(\left\vert 5\right\rangle\)) to the four ground states is Γ = 2π × 5.746 MHz. To perform the simultaneous coupling, the CPT pulse should contain four frequency components (ωL1, ωL2, ωL3, ωL4) with Rabi frequencies (Ω1, Ω2, Ω3, Ω4), which can be generated by modulating a single laser with a fiber-coupled EOM. For convenience, one can set the four Rabi frequencies Ωj = Ω are real and the four decay rates γj = Γ/4 (j = 1, 2, 3, 4)19,20. The time-evolution obeys the Lindblad master equation20,
where \({\hat{L}}_{j}=\left\vert j\right\rangle \left\langle 5\right\vert\) is the Lindblad operator, and \(\hat{\rho }\) is the density matrix. In the rotating frame, the Hamiltonian matrix is given as
where Δ1 = (ω2 + ωL2) − (ω1 + ωL1) and Δ2 = (ω4 + ωL4) − (ω3 + ωL3) are the two-photon detunings, and δ1 = [ω5 − (ω2 + ωL2 + ω1 + ωL1)/2] and δ2 = [(ω4 + ωL4 + ω3 + ωL3)/2 − ω5] are the average detunings.
In order to realize the quantum double lock-in amplifier, we input two dark states \(\left\vert {D}_{{{{{{{{\rm{12}}}}}}}}}\right\rangle =(\left\vert 1\right\rangle -\left\vert 2\right\rangle )/\sqrt{2}\) and \(\left\vert {D}_{{{{{{{{\rm{34}}}}}}}}}\right\rangle =(\left\vert 3\right\rangle -\left\vert 4\right\rangle )/\sqrt{2}\) as two probe states which can be prepared via CPT procedure. In experiments, one can prepare the two dark states via setting a big gap between the average detunings δ1 and δ2 which satisfy the far detuning condition: δ2 = δ1 ≫ {Ω, Δ1,2}, see Fig. 5b. Thus the corresponding density matrix reads \({\hat{\rho }}_{D}=\frac{1}{2}\left(\left\vert {D}_{{{{{{{{\rm{12}}}}}}}}}\right\rangle \left\langle {D}_{{{{{{{{\rm{12}}}}}}}}}\right\vert +\left\vert {D}_{{{{{{{{\rm{34}}}}}}}}}\right\rangle \left\langle {D}_{{{{{{{{\rm{34}}}}}}}}}\right\vert \right)\). During the signal interrogation process, due to the two physical channels have different resonance frequencies and there is a strong bias field magnetic field Bbias, the two dark states \(\left\vert {D}_{{{{{{{{\rm{12}}}}}}}}}\right\rangle\) and \(\left\vert {D}_{{{{{{{{\rm{34}}}}}}}}}\right\rangle\) will independently accumulate phases \({\phi }_{n}^{{{{{{{{\rm{PDD}}}}}}}}}\) and \({\phi }_{n}^{{{{{{{{\rm{CP}}}}}}}}}\) under the corresponding PDD and CP sequences (see Supplementary Note 4 for more details).
In our calculations, the two physical channels \(\{\left\vert 1\right\rangle ,\left\vert 2\right\rangle \}\) and \(\{\left\vert 3\right\rangle ,\left\vert 4\right\rangle \}\) are decoupled in the whole process due to the far detuning condition, and the detunings are set as δ1 = δ2 = 2π × 1 MHz. At time tn = nτm before detection, the density matrix is \({\hat{\rho }}_{f}=\left(\left\vert {\Psi }_{{{{{{{{\rm{12}}}}}}}}}({t}_{n})\right\rangle \left\langle {\Psi }_{{{{{{{{\rm{12}}}}}}}}}({t}_{n})\right\vert + \left\vert {\Psi }_{{{{{{{{\rm{34}}}}}}}}}({t}_{n})\right\rangle \left\langle {\Psi }_{{{{{{{{\rm{34}}}}}}}}}({t}_{n})\right\vert \right)/2\) with \(\left\vert {\Psi }_{{{{{{{{\rm{12}}}}}}}}}({t}_{n})\right\rangle =(\left\vert 1\right\rangle -{e}^{i{\phi }_{n}^{{{{{{{{\rm{PDD}}}}}}}}}}\left\vert 2\right\rangle )/\sqrt{2}\) and \(\left\vert {\Psi }_{{{{{{{{\rm{34}}}}}}}}}({t}_{n})\right\rangle =(\left\vert 3\right\rangle -{e}^{i{\phi }_{n}^{{{{{{{{\rm{CP}}}}}}}}}}\left\vert 4\right\rangle )/\sqrt{2}\). At last, a CPT query pulse of 2 μs is imposed to obtain the population of the common excited state. Here, the transmission of CPT light can be converted as the transmission signal (TS) via a photodetector19,20, which is proportional to (1 − ρ55), i.e., the absorption is proportional to the excited-state population ρ55. Moreover, the common excited-state population can be expressed as
where ℜ(x) denotes the real part of x and ρij,n is the density matrix element at time tn reflecting the coherence of ρ12,n and ρ34,n (see Supplementary Note 4 for more details). Further, according to Eq. (19), for a weak signal, we have the common excited-state population
which is proportional to the measurement signal \({P}_{\uparrow ,n}^{{{{{{{{\rm{sum}}}}}}}}}\) given in Eq. (11). According to Eq. (20), one can extract the weak signal from the common excited-state population ρ55,n, which is proportional to the CPT light absorption and can be detected via a photodetector19,47. For a strong signal, we define
which is proportional to \({\langle {\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}\rangle }_{n}^{{{{{{{{\rm{sum}}}}}}}}}\) given in Eq. (12). At the point of τm = τ, we have \({\tilde{\rho }}_{{{{{{{{\rm{55,n}}}}}}}}}\approx -\frac{{\Omega }^{2}}{2{\Gamma }^{2}}\left[\cos \left(\frac{2A}{\omega }\cos (\beta )\cdot n\right)+\cos \left(\frac{2A}{\omega }\sin (\beta )\cdot n\right)\right]\), which can be used to extract the strong signal (see Supplementary Note 4 for more details).
Therefore, according to Eqs. (19) and (20), one can successfully extract the target signal from the population of the common excited-state in a five-level double-Λ CPT system. In experiment, one can obtain the common excited-state population ρ55,n and further obtain \({\tilde{\rho }}_{{{{{{{{\rm{55,n}}}}}}}}}\) by adjusting the spacing τm of adjacent π pulses. In our consideration, we chose \(\{\left\vert 1\right\rangle ,\left\vert 2\right\rangle \}\) and \(\{\left\vert 3\right\rangle ,\left\vert 4\right\rangle \}\) to form two groups of magneto-sensitive transitions with gyromagnetic ratios γ12 = −γ34 = γg = −(g2 − g1)μB/ℏ = −1.0014 × 2π × 1.4 MHz/G, where μB is the Bohr magneton, {g1, g2} are the Landég factors for ground states F = {1, 2}, and μB/ℏ = 2π × 1.4 MHz/G and (g2 − g1) = 1.0014. Below, for a target AC magnetic field in form of \({B}_{0}\sin (\omega t+\beta )\), we use A = γgB0 to denote its amplitude. Based upon currently available experimental techniques, we set the frequency ω = 2π × 50 kHz, the initial phase β = −π/6, the π pulse length TΩ = 2 μs, and the Rabi frequencies \(\Omega =0.035\times \frac{\Gamma }{4}\) for simulation18,19,42,48,49. To split the Zeeman sublevels, a bias magnetic field Bbias = 0.143 mT is applied while the two-photon detunings are set as Δ1 = Δ2 ≈ 0, leading to the average detunings δ1 = δ2 ≈ 2π × 1 MHz.
Our numerical results from the Lindblad master equation (17) are well consistent with the analytical ones given by Eqs. (19) and (21), see Fig. 6. For a weak target signal, such as a magnetic field of B0 = 2 × 10−9 T satisfying \(\frac{A}{\omega }\le \frac{1}{2n}\), one can obtain the information of the target signal by measuring ρ55,n, see Fig. 6a. Based upon the numerical simulation of Lindblad master equations, the normalized common excited-state population (ρ55/a with the normalization coefficient \(a \, \approx \, \frac{{\Omega }^{2}}{{\Gamma }^{2}}\)) given by a five-level double-Λ system (red dashed line) fit well with the sum of excited-state populations given by two independent Λ systems (blue line). Moreover, these numerical results are both well consistent with the analytical ones of Eq.(19) (green dotted line). Thus one can determine the lock-in point to extract the frequency ω from the common excited-state population. In addition, one can determine the initial phase via only performing extra independent CP or PDD measurements for three specific modulation periods τm = (τ − dτ), τm = τ and τm = (τ + dτ) (see Supplementary Note 2 for more details). For a strong target signal, such as a magnetic field B0 = 2 × 10−6 T satisfying \(\frac{A}{\omega } > \frac{1}{2n}\), one can determine the value of ω via the IPR given by the FFT spectrum of the measurement signal \({\tilde{\rho }}_{{{{{{{{\rm{55,n}}}}}}}}}\), see in Fig. 6b. Therefore one can extract A and β from the FFT spectrum of \({\tilde{\rho }}_{{{{{{{{\rm{55,n}}}}}}}}}\) at the lock-in point τm = τ, see Fig. 6c. Similarly, we find the numerical results are well consistent with the analytical ones, see Fig. 6b, c. The little deviation of numerical between analytical results is because that the pulse is not a ideal Dirac δ function but a square pulse of a finite length TΩ = 2 μs. Overall, comparing with the protocol with two-level systems, our protocol with five-level double-Λ system can dramatically reduce the total measurement times and avoid additional time-dependent systematic errors (see Supplementary Note 4 for more details). In addition, a single NV center in diamond is a system with electron spin (S = 1) and nuclear spin (I = 1), thus it may be able to realize the double-Λ five-level via delicate design50,51.
Robustness
In experiments, there are many imperfections that may influence the lock-in signal. Below we discuss two key imperfections: the finite pulse length TΩ and the stochastic noises18,48,49. Firstly, we consider square pules and analyze the influence of their pulse length TΩ on our scheme. According to Eq. (4), for a weak target signal, we can ignore the time-ordering operator \(\hat{{{{{{{{\mathcal{T}}}}}}}}}\), that is, \({\left\vert \Psi (t)\right\rangle }_{{{{{{{{\rm{I}}}}}}}}}={e}^{-i\frac{1}{2}\left[{\phi }_{{{{{{{{\rm{z}}}}}}}}}(t){\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}+{\phi }_{{{{{{{{\rm{y}}}}}}}}}(t){\hat{\sigma }}_{{{{{{{{\rm{y}}}}}}}}}\right]}{\left\vert \Psi (0)\right\rangle }_{{{{{{{{\rm{I}}}}}}}}}\) with \({\left\vert \Psi (0)\right\rangle }_{{{{{{{{\rm{I}}}}}}}}}=\frac{\left\vert \uparrow \right\rangle +\left\vert \downarrow \right\rangle }{\sqrt{2}}\). Hence, after an unitary operation \(U={e}^{-i\frac{\pi }{4}{\hat{\sigma }}_{{{{{{{{\rm{y}}}}}}}}}}\) for readout, the population in the state \(\left\vert \uparrow \right\rangle\) reads
The analytical results of Eq. (22) are well consistent with the corresponding numerical ones (see Supplementary Note 5 for more details). To show the influences of the pulse length TΩ, we numerically calculate the common exited-state population ρ55,n in double-Λ system with TΩ = {0, 0.2 τ, 0.4τ}. It indicates that the finite pulse length TΩ almost has no effects on the lock-in point, see Fig. 7a. For a strong target signal, one can still extract the frequency via the periodicity of the measurement signal \({\langle {\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}\rangle }_{n}^{{{{{{{{\rm{sum}}}}}}}}}\). When τm = τ, we have \(\hat{H}(t+2{\tau }_{m})=\hat{H}(t)\) and the IPR reaches its maximum. We also numerically verify that the pulse length TΩ indeed does not affect the lock-in point, as shown in Fig. 7b. Moreover, we numerically calculate the FFT spectrum for different TΩ at the lock-in point τm = τ, see Fig. 7c. In addition, in order to estimate the influence of the pulse length TΩ, we numerically calculate the relative deviation of the amplitude B0 and the phase β for different TΩ. They are \(\delta {B}_{0}^{{{{{{{{\rm{est}}}}}}}}}=\left({B}_{0}^{{{{{{{{\rm{est}}}}}}}}}-{B}_{0}\right)\) and \(\delta | \beta {| }^{{{{{{{{\rm{est}}}}}}}}}=\left(| \beta {| }^{{{{{{{{\rm{est}}}}}}}}}-| \beta | \right)\). In general, the pulse length need satisfy 0 < TΩ < τ to avoid the mixing term become a continuous driving. Here, B0 and ∣β∣ denote the exact values, and \({B}_{0}^{{{{{{{{\rm{est}}}}}}}}}\) and ∣β∣est correspond to the estimated values according to Eqs. (15) and Eq. (16), respectively. Our numerical results show that the effect of pulse length TΩ is small enough to be ignored when TΩ ≤ 0.4 τ, as shown in Fig. 7d.
Below we will illustrate the robustness of our quantum double lock-in amplifier against stochastic noise. In experiments, one of the most common and most dominant types of noise is white noise. The white noise is a random signal equably distributed in the whole frequency domain and it is also called f0 noise for its constant power spectral density. Due to the PDD and CP sequences can enhance the part with frequency \({f}_{k}=\frac{(2k+1)}{2{\tau }_{m}}(k=0,1,2,\cdots )\)15, the signal-to-noise ratio (SNR) can be effectively improved by these two sequences (see more details in Supplementary Note 8). Here, we consider the signal and the noise couple to the probe through the same channel, that is, the system obeys the Hamiltonian \(\hat{H}(t,{T}_{\Omega },\sigma )=\frac{\hslash }{2}A\left[\sin (\omega t+\beta )+{N}_{(0,\sigma )}(t)\right]{\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}+\frac{\hslash }{2}\Omega (t){\hat{\sigma }}_{{{{{{{{\rm{x}}}}}}}}}\). Here, N(0, σ)(t) = σN(0,1)(t) denotes the Gaussian random noise with the time-averaged value \(\overline{{N}_{(0,\sigma )}(t)}=0\), and σ = N/S is the standard deviation. Thus, we have the corresponding SNR = − 10lg(N/S) dB52,53. To illustrate the robustness of our scheme, we numerically calculate the two measurement signals versus the modulation period τm under different noise strengths. For a weak target signal, we calculate the population ρ55,n versus the difference (τm − τ) for different noise strengths σ = 0, σ = 100, and σ = 300, as shown in Fig. 8a. The stochastic noise almost do not affect the lock-in point even under a large noise strength σ = 100. Our protocol is still hold for SNR ≥ −20 dB after averaging over 20 times. For a strong target signal, we calculate the IPR versus (τm − τ) and the FFT spectra of \({\tilde{\rho }}_{{{{{{{{\rm{55,n}}}}}}}}}\) for different noise strengths σ = 0, σ = 5, and σ = 10, see Fig. 8b. Our results show that the maximum IPR and the FFT amplitude decrease with the noise strength. When SNR≥ −10 dB, one can still determine the lock-in point via the IPR and extract the target signal via performing FFT on \({\tilde{\rho }}_{{{{{{{{\rm{55,n}}}}}}}}}\) (averaging over 20 times), see Fig. 8c. In addition, our scheme is also robust against decoherence phenomenon causing by the population decay, the depolarization noise arising from the optical pumping effect (see Supplementary Note 6 for more details) and 50 Hz noise which originates from the electrical power for a lab (see Supplementary Note 7 for more details).
In general, different quantum systems have different environments and therefore different noise sources. For trapped-ion system, except for the white noise and 50 Hz noise, the 150 Hz noise is also one of the most common noise and can limit the ultimate measurement precision54,55. For the NV centers, the photon shot noise always exists and can affect the the measurement8. Moreover, since the NV centers near the surface, the interaction with external materials and spins also can affect the ultimate measurement precision56,57. Their influences within our scheme are worthy studied in future.
Discussion
In conclusion, we propose a comprehensive framework for implementing a quantum double lock-in amplifier utilizing two orthogonal dynamical decoupling sequences: PDD and CP sequences. This scheme establishes a quantum counterpart to the classical double lock-in amplifier. We mathematically derive a general formula for measuring an AC signal in the presence of strong noise using our quantum double lock-in amplifier, which relies on two quantum mixers subjected to orthogonal modulations. Our protocol enables the extraction of the complete characteristics, including frequency, amplitude, and initial phase, of the target AC signal. In the case of a weak target signal, the lock-in point can be determined by exploiting the symmetry of the combined measurement signal, allowing for subsequent extraction of the amplitude and, the phase through a fitting procedure. For a strong target signal, the target signal can be extracted from the FFT spectrum of the combined measurement signal. We also compare the measurement sensitivity of the amplitude A achieved by two methods for weak and strong signals at the lock-in point (see supplementary Note 9). In the previous scheme8, the frequency is measured via sweeping MW strength Ω firstly, and then measure the amplitude and the phase via Rabi oscillation. Differently, our scheme measures the frequency via modulating the pulse repetition period τm and can extract complete information about the signal’s amplitude, frequency, and initial phase simultaneously.
Furthermore, we illustrate the realization of a quantum double lock-in amplifier using a five-level double-Λ CPT system of 87Rb as an example. In this five-level double-Λ CPT system, the common excited state population can be measured to obtain two measurement signals: ρ55,n and \({\tilde{\rho }}_{{{{{{{{\rm{55,n}}}}}}}}}\). Compared to implementations using two-level systems, our experimental proposal based on a five-level system can significantly reduce the total measurement time and avoid additional time-dependent systematic errors. Moreover, the system control complexity does not increase appreciably and all necessary techniques are compatible with current experimental capabilities. Our scheme also demonstrates strong robustness against finite pulse length and stochastic noise. Owing to the highly developed quantum control methods, various physical systems are well-suited for realizing quantum double lock-in amplifiers. These include Bose condensed atoms58,59,60,61, trapped ions62,63,64,65,66, nitrogen-vacancy centers in diamond56,67,68,69,70, doped spins in semiconductors71,72, and even “artificial atoms” like superconducting qubits27,73,74 and so on. Meanwhile, how to realize the five-level structure in other quantum systems for achieving our scheme still have open questions, and should be carefully analyzed and studied in future. Furthermore, our protocol could enable the development of practical quantum sensors, such as magnetometers11,75,76,77,78, atomic clocks79,80,81, weak-force detectors13, and noise spectroscopy detector27,81,82.
Methods
Calculation of the common excited-state population
According to Eqs. (17) and (18), we can obtain the time-evolution of the density matrix ρ (see Supplementary Note 4 for more details). After the adiabatic elimination (\(t \, \gg \, \frac{1}{\Gamma }\) and \(\frac{\partial }{\partial t}{\rho }_{{{{{{{{\rm{55}}}}}}}}}=0\)) and assuming the four Rabi frequencies satisfy Ωj = Ω and {Ω, δ} ≪ Γ, we can derive the common excited-state population
In our consideration, ρ13, ρ14, ρ23, and ρ24 are almost 0 before detection, hence the population ρ55 can be approximately given as Eq. (19). Due to the two Λ configurations \(\left\{\left\vert 1\right\rangle ,\left\vert 2\right\rangle ,\left\vert 5\right\rangle \right\}\) and \(\left\{\left\vert 3\right\rangle ,\left\vert 4\right\rangle ,\left\vert 5\right\rangle \right\}\) have their density matrix elements satisfying \({\rho }_{{{{{{{{\rm{21}}}}}}}}}^{{\prime} }=2{\rho }_{{{{{{{{\rm{21}}}}}}}}}\) and \({\rho }_{{{{{{{{\rm{34}}}}}}}}}^{{\prime} }=2{\rho }_{{{{{{{{\rm{34}}}}}}}}}\), we have
According to the above equation, we find that the five-level double Λ configuration can be divided to such two Λ configurations.
Dyson expansion of quantum evolution
In order to simplify the calculation of the time-ordering operator \(\hat{{{{{{{{\mathcal{T}}}}}}}}}\) in Eq. (4), we consider the case of \(At\le \frac{\pi }{2}\) and utilize the Dyson expansion
with \(\hat{H}({t}^{{\prime} })=\frac{\hslash }{2}\left[{\omega }_{{{{{{{{\rm{z}}}}}}}}}({t}^{{\prime} }){\hat{\sigma }}_{{{{{{{{\rm{z}}}}}}}}}+{\omega }_{{{{{{{{\rm{y}}}}}}}}}({t}^{{\prime} }){\hat{\sigma }}_{{{{{{{{\rm{y}}}}}}}}}\right]\). In units of ℏ = 1, one can obtain
and
Thus, through using \(\{{\hat{\sigma }}_{{{{{{{{\rm{I}}}}}}}}},{\hat{\sigma }}_{j}\}={\hat{\sigma }}_{{{{{{{{\rm{I}}}}}}}}}{\hat{\sigma }}_{j}+{\hat{\sigma }}_{j}{\hat{\sigma }}_{{{{{{{{\rm{I}}}}}}}}}=2{\delta }_{{{{{{{{\rm{i,j}}}}}}}}}(i,j=x,y,z)\), the time-ordering operator \(\hat{{{{{{{{\mathcal{T}}}}}}}}}\) in Eq. (26) can be removed when \(\phi \le 2n\frac{A}{\omega }\le 1\) corresponding to \(At\le \frac{\pi }{2}\).
Data availability
Data that support the figures within this paper and other findings of this study are available from the corresponding authors upon reasonable request.
Code availability
The code is available from the corresponding authors upon reasonable request.
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Acknowledgements
This work is supported by the National Key Research and Development Program of China (Grant No. 2022YFA1404104), the National Natural Science Foundation of China (Grant No. 12025509, Grant No. 12305022), and the Key-Area Research and Development Program of Guangdong Province (Grant No. 2019B030330001).
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C.L. and J.H. conceived the project. S.C. and M.Z. contributed equally to this work and designed the protocol. All authors discussed the results and contributed to compose and revise the manuscript. C.L. supervised the project.
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Communications Physics thanks Guoqing Wang, Martina Esposito, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.
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Chen, S., Zhuang, M., Fang, R. et al. Quantum double lock-in amplifier. Commun Phys 7, 189 (2024). https://doi.org/10.1038/s42005-024-01687-6
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DOI: https://doi.org/10.1038/s42005-024-01687-6
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