Many-Body Quantum Lock-In Amplifier

Achieving high-precision detection of time-dependent signals within noise is a ubiquitous issue in physics and a critical task in metrology. Lock-in amplifiers are detectors that can extract alternating signals within extreme noise via a known carrier frequency. Here we present a protocol for achieving an entanglement-enhanced lock-in amplifier via use of many-body multipulse quantum interferometry. The many-body quantum lock-in amplifier is implemented by application of a periodic multi-$\pi$-pulse sequence during the interrogation. Our analytic results show that, by our choosing suitable input states and readout operations, the frequency and amplitude of an unknown alternating field can be simultaneously extracted via population measurements. The lock-in point can be determined via the symmetry of the signal during a single interrogation time or the time-averaged signals for multiple interrogation times. We find that the measurement signal at the lock-in point is independent of the interrogation time. In particular, if we input spin cat states and apply interaction-based readout operations, the measurement precisions for frequency and amplitude can both approach the Heisenberg limit. Moreover, our many-body quantum amplifier is also robust with regard to extreme stochastic noise. Our study paves a new way for measuring time-dependent signals with many-body quantum systems, and provides a feasible method for achieving Heisenberg-limited detection of alternating signals.

Popular Summary

The high-precision detection of time-dependent signals is a ubiquitous issue in physics and metrology. Lock-in amplifiers are detectors that can extract alternating signals with a known carrier frequency from noise. The quantum analog of the classical lock-in amplifier with a single particle has been demonstrated and used for frequency measurement, magnetic field sensing, and so on. However, most existing studies of quantum lock-in amplifiers concentrate on single-particle systems. It is well known that quantum entanglement can be exploited to increase measurement precision. How to achieve an entanglement-enhanced quantum lock-in amplifier is still an open question. Here we present a protocol for achieving an entanglement-enhanced quantum lock-in amplifier by use of many-body quantum interferometry.

Our many-body quantum lock-in amplifier is realized by our applying a periodic multipulse sequence during the signal accumulation. By our choosing suitable input states and readout operations, the frequency and amplitude of an unknown alternating signal can be simultaneously extracted via population measurements. In particular, if we input spin cat states and apply interaction-based readout operations, the measurement precisions for frequency and amplitude can both approach the Heisenberg limit. Moreover, our many-body quantum amplifier is robust with regard to extreme stochastic noise.

Our study provides a new way for achieving Heisenberg-limited detection of time-dependent signals in many-body quantum systems. Based on state-of-the-art techniques, it would be beneficial for the development of practical entanglement-enhanced quantum technologies including atomic clocks, magnetometers, and weak-force detectors.

Figure 1

A classical lock-in amplifier and a many-body quantum lock-in amplifier. (a) The classical lock-in amplifier. $V_s(t)$ is the target signal submerged in extreme noise. $V_r(t)$ is a known reference signal. Inputting the two signals through the classical lock-in amplifier, one can extract the signal $V_s(t)$. (b) The many-body quantum lock-in amplifier. The coupling between the probe and the signal is described by ${\hat{H}}_{\mathrm{int}}=M(t){\hat{J}}_z$, with $M(t)=S(t)+N_o(t)$, where $S(t)$ is the target signal and $N_o(t)$ is the stochastic noise. The oscillating modulation term ${\hat{H}}_{\mathrm{mix}}$, which is analogous to the reference signal $V_r(t)$, does not commute with ${\hat{H}}_{\mathrm{int}}$. Thus, the quantum probe, obeying the Hamiltonian $\hat{H}={\hat{H}}_{\mathrm{int}}+{\hat{H}}_{\mathrm{mix}}$, can be used to realize a quantum lock-in amplifier for extraction of the signal $S(t)$.

Figure 2

The general protocol for a many-body quantum lock-in amplifier for detecting an ac magnetic field. (a) The many-body quantum interferometry consists of three stages: (i) preparation, (ii) signal interrogation, and (iii) readout. In the initialization stage, an input state $|\mathrm{\Psi }{⟩}_{\mathrm{in}}$ is prepared. Then the input state undergoes an interrogation stage for signal accumulation. At this stage, the system state interacts with the magnetic field, and a train of $\pi$ pulses with specific equidistant spacing ${\tau }_i$ is applied. In the readout stage, a certain unitary operation $\hat{U}$ is applied for recombination, and the half-population difference is measured. Here the pulse spacing ${\tau }_i$ ($i=1,2,\dots ,M$) is adjusted to find the frequency locking point $\omega ={\omega }_e$. (b) The half-population difference versus the detuning $\omega -{\omega }_e$ can give the frequency locking point $\omega ={\omega }_e$. Here, for illustration, we choose a GHZ state with $N=2$, $T=4\pi$, $\omega =20\pi$, and $\gamma B=2\pi$.

Figure 3

The lock-in signal of the many-body quantum lock-in amplifier. The measurement signal $J_z$ versus the detuning $\omega -{\omega }_e$ for (a),(d) the SCS, (b),(e) the spin cat state ${|\mathrm{\Psi }(\theta =\pi /8)⟩}_{\mathrm{cat}}$, and (c),(f) the GHZ state ${|\mathrm{\Psi }(\theta =0)⟩}_{\mathrm{cat}}$ for different evolution times $T$. The measurement signals $J_z$ are symmetric (or antisymmetric) with respect to the lock-in point $\omega -{\omega }_e=0$ and they are both consistent with the analytic expressions [Eqs. (28), (38), and (41)]. The time-averaged signal $\widetilde{J_z}=\frac{1}{T_2-T_1}{\int }_{T_1}^{T_2}{J}_{z}dt$ versus the detuning $\omega -{\omega }_e$ for (g) the SCS, (h) the spin cat state ${|\mathrm{\Psi }(\theta =\pi /8)⟩}_{\mathrm{cat}}$, and (i) the GHZ state ${|\mathrm{\Psi }(\theta =0)⟩}_{\mathrm{cat}}$, with $T_1=\pi$ and $T_2=3\pi$. The time-averaged signal $\widetilde{J_z}$ is symmetric (or antisymmetric) with respect to the lock-in point $\omega -{\omega }_e=0$. The analytic results (dashed red line) fit well with the numerical results (solid blue line) in a wide range around the lock-in point. Here $\omega =20\pi$, $\gamma B=2\pi$, and $N=6$.

Figure 4

Log-log scaling of the optimal measurement precisions (a) $\mathrm{\Delta }\omega$ and (b) $\mathrm{\Delta }B$ versus total particle number. The circles are the results for a SCS with ${\hat{U}}_I=e^{-i\frac{\pi }{2}{\hat{J}}_y}$. The squares and triangles denote the results for the spin cat state ${|\mathrm{\Psi }(\theta =\pi /8)⟩}_{\mathrm{cat}}$ and the GHZ state ${|\mathrm{\Psi }(\theta =0)⟩}_{\mathrm{cat}}$, respectively, with ${\hat{U}}_I=e^{i\frac{\pi }{2}{\hat{J}}_x}{e}^{i\frac{\pi }{2}{\hat{J}}_z^2}{e}^{i\frac{\pi }{2}{\hat{J}}_x}$. The lines are the corresponding fitting curves. Here $T=2\pi$, $\omega =20\pi$, $|(\omega -{\omega }_e)/\omega |\in [0,0.008]$, and $\gamma B\in [0,2\pi ]$.

Figure 5

The robustness of the many-body quantum amplifier against stochastic noise. The measurement precision $\mathrm{\Delta }\omega$ versus versus the detuning $\omega -{\omega }_e$ for (a) the SCS, (b) the spin cat state ${|\mathrm{\Psi }(\theta =\pi /8)⟩}_{\mathrm{cat}}$, and (c) the GHZ state ${|\mathrm{\Psi }(\theta =0)⟩}_{\mathrm{cat}}$. The solid blue lines are the results without stochastic noise ($\eta =0$). The dashed red lines and the dotted green lines are the results under stochastic noise with $\eta =5\gamma B$ and $\eta =25\gamma B$, respectively. Here $T=2\pi$, $\omega =20\pi$, $\gamma B=2\pi$, $N=6$, and $N_o(t)\in [-\eta ,\eta ]$. These results are averaged 20 times.

Figure 6

Variations of the two phases ${\mathrm{\Phi }}_1$ and ${\mathrm{\Phi }}_2$ versus $\omega -{\omega }_e$ for (a) $\omega =10\pi$, (b) $\omega =15\pi$, and (c) $\omega =20\pi$. Here $T=2\pi$ and $\gamma B=1$.

Figure 7

Variation of half-population difference measurement versus time in the Schrödinger picture,in the interaction picture, and according to the analytic results. (a) Individual particles in the SCS, (b) the spin cat state ${|\mathrm{\Psi }(\theta =\pi /8)⟩}_{\mathrm{cat}}$, and (c) the GHZ state ${|\mathrm{\Psi }(\theta =0)⟩}_{\mathrm{cat}}$. Here $\omega =20\pi$, $\gamma B=2\pi$, $N=2$, and ${\tau }_1=\pi /(\omega +0.001\omega )$.

Figure 8

Variation of half-population difference measurement versus time in the Schrödinger picture, in the interaction picture, and according to the analytic results. (a) Individual particles in the SCS, (b) the spin cat state ${|\mathrm{\Psi }(\theta =\pi /8)⟩}_{\mathrm{cat}}$, and (c) the GHZ state ${|\mathrm{\Psi }(\theta =0)⟩}_{\mathrm{cat}}$. Here $\omega =20\pi$, $\gamma B=2\pi$, $N=2$, and ${\tau }_1=\pi /(\omega -0.005\omega )$.

Figure 9

Variation of the total signal versus time. (a) Total signal $S(t)+N_o(t)$ with (a) $\eta =0$, (b) $\eta =5\gamma B$, and (c) $25\gamma B$. Here $\omega =20\pi$ and $N_o(t)\in [-\eta ,\eta ]$.

Figure 10

Husimi distributions for the spin cat state with different $\theta$. (a) Husimi distributions for (a) the GHZ state ${|\mathrm{\Psi }(\theta =0)⟩}_{\mathrm{cat}}$, (b) the spin cat state ${|\mathrm{\Psi }(\theta =\pi /8)⟩}_{\mathrm{cat}}$, and (c) the spin cat state ${|\mathrm{\Psi }(\theta =\pi /4)⟩}_{\mathrm{cat}}$. Here $N=20$.

Figure 11

Variation of measurement precisions for individual particles and entangled particles for fixed $N$. (a) $\mathrm{\Delta }\omega /\omega$ and (d) $\gamma \mathrm{\Delta }B/\omega$ for a SCS with ${\hat{U}}_I=e^{-i\frac{\pi }{2}{\hat{J}}_y}$. (b) $\mathrm{\Delta }\omega /\omega$ and (e) $\gamma \mathrm{\Delta }B/\omega$ for a spin cat state with ${\hat{U}}_I=e^{i\frac{\pi }{2}{\hat{J}}_x}{e}^{i\frac{\pi }{2}{\hat{J}}_z^2}{e}^{i\frac{\pi }{2}{\hat{J}}_x}$ and $\theta =\pi /8$. (c) $\mathrm{\Delta }\omega /\omega$ and (f) $\gamma \mathrm{\Delta }B/\omega$ for a GHZ state with ${\hat{U}}_I=e^{i\frac{\pi }{2}{\hat{J}}_x}{e}^{i\frac{\pi }{2}{\hat{J}}_z^2}{e}^{i\frac{\pi }{2}{\hat{J}}_x}$. Here $T=2\pi$, $\omega =20\pi$, and $N=20$.

Figure 12

Variation of the measurement signal $J_z$ versus detuning $\omega -{\omega }_e$ for individual particles and entangled particles for fixed $B$. The dotted green line show the results for a SCS with ${\hat{U}}_I=e^{-i\frac{\pi }{2}{\hat{J}}_y}$. The dashed orange line shows the results for a spin cat state with ${\hat{U}}_I=e^{i\frac{\pi }{2}{\hat{J}}_x}{e}^{i\frac{\pi }{2}{\hat{J}}_z^2}{e}^{i\frac{\pi }{2}{\hat{J}}_x}$ and $\theta =\pi /8$. The solid blue line show the results for a GHZ state with ${\hat{U}}_I=e^{i\frac{\pi }{2}{\hat{J}}_x}{e}^{i\frac{\pi }{2}{\hat{J}}_z^2}{e}^{i\frac{\pi }{2}{\hat{J}}_x}$. These results indicate that the expectation of the half-population difference is symmetric for detuning $\omega -{\omega }_e$ and the lock-in point located at $\omega -{\omega }_e=0$, i.e., $\omega ={\omega }_e$. Here $T=2\pi$, $\omega =20\pi$, $\gamma B=2\pi$, $N=6$, and $k=0$.

Figure 13

Variation of the measurement precisions for individual particles and entangled particles for fixed $N$. (a) $\mathrm{\Delta }\omega /\omega$ and(d) $\gamma \mathrm{\Delta }B/\omega$ for a SCS with ${\hat{U}}_I=e^{-i\frac{\pi }{2}{\hat{J}}_y}$. (b) $\mathrm{\Delta }\omega /\omega$ and (e) $\gamma \mathrm{\Delta }B/\omega$ for a spin cat state with ${\hat{U}}_I=e^{i\frac{\pi }{2}{\hat{J}}_x}{e}^{i\frac{\pi }{2}{\hat{J}}_z^2}{e}^{i\frac{\pi }{2}{\hat{J}}_x}$ and $\theta =\pi /8$. (c) $\mathrm{\Delta }\omega /\omega$ and (f) $\gamma \mathrm{\Delta }B/\omega$ for a GHZ state with ${\hat{U}}_I=e^{i\frac{\pi }{2}{\hat{J}}_x}{e}^{i\frac{\pi }{2}{\hat{J}}_z^2}{e}^{i\frac{\pi }{2}{\hat{J}}_x}$. Here $T=2\pi$, $\omega =20\pi$, $N=20$, and $k=0$.

Figure 14

Log-log plots of the scaling of the optimal measurement precision versus total particle number for (a) frequency $\omega$ and (b) amplitude $B$. The circles show the results for a SCS with ${\hat{U}}_I=e^{-i\frac{\pi }{2}{\hat{J}}_y}$ and the lines are the corresponding fitting curves. The squares show the results for a spin cat state with ${\hat{U}}_I=e^{i\frac{\pi }{2}{\hat{J}}_x}{e}^{i\frac{\pi }{2}{\hat{J}}_z^2}{e}^{i\frac{\pi }{2}{\hat{J}}_x}$ and $\theta =\pi /8$ and the lines are the corresponding fitting curves. The triangles show the results for a GHZ state with ${\hat{U}}_I=e^{i\frac{\pi }{2}{\hat{J}}_x}{e}^{i\frac{\pi }{2}{\hat{J}}_z^2}{e}^{i\frac{\pi }{2}{\hat{J}}_x}$ and the lines are the corresponding fitting curves. Here $T=2\pi$, $\omega =20\pi$, $|(\omega -{\omega }_e)/\omega |\in [0,0.008]$, $\gamma B\in [0,2\pi ]$, and $k=-1$.