Quantum-Enhanced Multi-Parameter Sensing in a Single Mode

TL;DR

The paper tackles the problem of estimating two incompatible observables with precision beyond the standard quantum limit by using modular-variable sensing in a single bosonic mode. It introduces and experimentally implements grid states to measure small displacements in position and momentum, and demonstrates a metrological gain up to 5.1 dB over the simultaneous SQL, with adaptive Bayesian quantum phase estimation achieving 2.6 dB below SQL. In addition, it realizes number–phase NP states as a new quantum sensing resource and reports a metrological gain of 3.1 dB over their SQL, highlighting a versatile approach to multi-parameter sensing in a single mode. The results suggest significant potential for phase-insensitive force sensing, error-corrected metrology, and scalable quantum sensor networks, marking a step toward unprecedented precision in fundamental physics and quantum technologies.

Abstract

Precision metrology underpins scientific and technological advancements. Quantum metrology offers a pathway to surpass classical sensing limits by leveraging quantum states and measurement strategies. However, measuring multiple incompatible observables suffers from quantum backaction, where measurement of one observable pollutes a subsequent measurement of the other. This is a manifestation of Heisenberg's uncertainty principle for two non-commuting observables, such as position and momentum. Here, we demonstrate measurements of small changes in position and momentum where the uncertainties are simultaneously reduced below the standard quantum limit (SQL). We measure $\textit{modular observables}$ using tailored, highly non-classical states that ideally evade measurement backactions. The states are deterministically prepared in the single mode of the mechanical motion of a trapped ion using an optimal quantum control protocol. Our experiment uses grid states to measure small changes in position and momentum and shows a metrological gain of up to 5.1(5)~dB over the simultaneous SQL. Using an adaptive-phase estimation algorithm with Bayesian inference, we estimate these displacements with a combined variance of 2.6(1.1)~dB below the SQL. Furthermore, we examine simultaneously estimating $\textit{number}$ and $\textit{phase}$, which are the polar counterparts of position and momentum. This is performed by preparing a novel quantum resource -- number-phase states -- and we demonstrate a metrological gain over their SQL. The combination of quantum control and multi-parameter quantum metrology marks a significant step towards unprecedented precision with applications ranging from fundamental physics to advanced quantum technologies.

Figures (4)

• Figure 1: Multi-parameter quantum enhanced sensing.(a) The uncertainty in simultaneous position--momentum measurements is bounded by the canonical commutation relation. (b) Their modular counterparts can instead be made to commute, allowing for estimation of $\hat{x}_{[2\pi/l_x]}$ and $\hat{p}_{[2\pi/l_p]}$ with uncertainties simultaneously below the SQL, where $l_s = l_x = l_p = \sqrt{2\pi}$ are the modulus lengths. (c) The eigenstates of modular position--momentum are grid states (shown is the Wigner function), which can sense displacements in position by $\epsilon_x$ and displacements in momentum by $\epsilon_p$ by measuring commuting operators $\hat{S}_x$ and $\hat{S}_p$. (d) Physical grid states are prepared in the bosonic mode of a trapped ion. An ancilla qubit encoded in the electronic ground state of the ion couples to the bosonic mode via a laser interaction (pink beam), allowing for measurement of position and momentum observables. (e) A Quantum Phase Estimation (QPE) circuit is used for multi-parameter estimation of $\epsilon_x$ and $\epsilon_p$. After preparing the sensing state and undergoing an unknown displacement, multiple rounds of a QPE sub-routine (blue box) are applied, where each round applies conditional $\hat{S}_x$ or $\hat{S}_p$, which gives one bit of phase estimation, $\mathrm{m}_x$ or $\mathrm{m}_p$. Since $\hat{S}_x$ and $\hat{S}_p$ commute, they can be applied alternately $N_s$ times, ideally performing many rounds of backaction-evading measurements.
• Figure 2: Metrological gain of grid states for multi-parameter displacement sensing.(a) Experimentally reconstructed characteristic function of prepared grid states with target parameters $\Delta$ of (i) 0.61, (ii) 0.37 and (iii) 0.30. Increasing energy results in increased squeezing along position and momentum. Crosses in (iii) show points of the characteristic function that correspond to values of the visibility parameters $\eta_x = \mathrm{Re}[\hbox{\Large$\chi$}(i \sqrt{\pi})]$ and $\eta_p = \mathrm{Re}[\hbox{\Large$\chi$}(- \sqrt{\pi})]$. (b) Multi-parameter variance, $\mathrm{V}(\epsilon_x) + \mathrm{V}(\epsilon_p)$, of grid states with increasing average phonon number, $\langle \hat{n} \rangle$. Variances (red circles) are calculated from the classical Fisher information of the experimentally measured probability distributions $\mathrm{P}_x$ and $\mathrm{P}_p$. The sensing signal is varied in the range $\{\epsilon_{x}, \epsilon_{p}\} \in [0, 1.4]$. Error bars correspond to one standard deviation calculated from quantum projection noise. Dashed red line is the expected multi-parameter variance of the target grid states. The simultaneous standard quantum limit (SQL*) corresponds to the multi-parameter variance of a coherent state from heterodyne measurement (see SM). The simultaneous lower bound (LB*) plots the minimum uncertainty from the quantum Fisher information, $1/(2\langle \hat{n} \rangle + 1)$Genoni2013optimaljointDuivenvoorden2017. Dashed purple line corresponds to the multi-parameter variance from heterodyne detection, with a single-mode squeezed state squeezed in one quadrature and anti-squeezed in the other. Heterodyne detection is equivalent to double homodyne detection: the state is split by a 50:50 beam splitter, with position measured on one output and momentum on the other.
• Figure 3: Quantum Phase Estimation (QPE) with grid states.(a) The multi-parameter variance of a grid state with $\Delta = 0.41$ is measured for increasing measurement repetitions $M \in [8, 128]$ with $N_S=1$ QPE sub-routine repetitions. Estimates ($\tilde{\epsilon}_x, \tilde{\epsilon}_p)$ of an unknown displacement $\hat{D}((\epsilon_x + i \epsilon_p)/\sqrt{2})$ are obtained from a QPE algorithm using a non-adaptive routine (red circles), where the control phases ($\theta_{x,p}$) are pre-determined prior to the experiment. Inset plots the individual variances, and shares the same x- and y-axes as the main plot. Dashed line plots SQL*, which is equal to $2/M$. The non-adaptive variance is below SQL* at $M=128$ measurements. The variance is further reduced by using an adaptive protocol (black star), where the control phases are optimised in real-time. (b) Zoom-in examination at $M=128$, where adaptive QPE outperforms both SQL* and the non-adaptive measurement. Error bars correspond to one standard deviation calculated from quantum projection noise.
• Figure 4: Metrological gain of number--phase states for multi-parameter number and phase sensing.(a, b) Theoretical Fock distribution and characteristic function of a number--phase (NP) state with spacing $N=4$, Fock cutoff $F=18$ and offset of 2 (lowest occupied Fock state is $\ket{n = 2}$). The energy of this state is constrained by damping the Fock coefficients with a sine envelope (dotted grey line), giving $\langle \hat{n} \rangle = 10$. The NP state has exact rotational phase-space symmetry of $l_n = 2\pi/4$ and approximate translational Fock symmetry of $l_\phi=4$, making it an exact eigenstate of $\hat{S}_n$ and an approximate eigenstate of $\hat{S}_\phi$. This spacing in phase and number can be used to simultaneously sense small unknown rotations by $\epsilon_\phi$ in phase space and shifts by $\epsilon_n$ in Fock space. (c) Experimentally reconstructed characteristic function of the above NP state. (d) The variances of number and phase are obtained from the classical Fisher information extracted from reconstructed probability distributions. Dashed red line is the expected multi-parameter variance of the target NP state. Insets (i., ii.) plot the individual variances for number and phase, with $\mathrm{SQL}_n = 2 \langle \hat{n}\rangle + 1$ and $\mathrm{SQL}_\phi = (2 \langle \hat{n}\rangle)^{-1} + 3 (8 \langle \hat{n}\rangle^2)^{-1}$, which correspond to a coherent state subjected to heterodyne measurements (see SM). SQL* is the sum of SQL for number and phase after rescaling by $(2\langle \hat{n} \rangle)^{-1}$ and $2 \langle \hat{n} \rangle$, respectively, such that both variances are equal to 1 and $\text{SQL*}=2$ at large $\langle n \rangle$. The combined variances (red squares) are the total of $\mathrm{V}(\epsilon_n)$ and $\mathrm{V}(\epsilon_\phi)$ after appropriately rescaling the covariance matrix. Error bars correspond to one standard deviation calculated from quantum projection noise.