Squeezing a quantum state along a specific direction has long been recognized as a crucial technique for enhancing the precision of quantum metrology by reducing parameter uncertainty. However, practical quantum metrology often involves the simultaneous estimation of multiple parameters, necessitating the use of high-quality squeezed states along multiple orthogonal axes to surpass the standard quantum limit for all relevant parameters. In addition, a temporally stabilized squeezed state can provide an event-ready probe for parameters, regardless of the initial state, and robust to the timing of the state preparation process once stabilized. In this work, we generate and stabilize a two-mode squeezed state along two secular motional modes in a vibrating trapped ion with reservoir engineering, despite starting from a thermal state of the motion. Leveraging this resource, we demonstrate an estimation of two simultaneous collective displacements along the squeezed axes, achieving improvements surpassing the classical limit by up to 6.9(3) and 7.0(3) decibels (dB), respectively. Our demonstration can be readily scaled to squeezed states with even more modes. The practical implications of our findings span a wide range of applications, including quantum sensing, quantum imaging, and various fields that demand precise measurements of multiple parameters.
Squeezing a quantum state along a specific direction has long been recognized as a crucial technique for enhancing the precision of quantum metrology by reducing parameter uncertainty. However, practical quantum metrology often involves the simultaneous estimation of multiple parameters, necessitating the use of high-quality squeezed states along multiple orthogonal axes to surpass the standard quantum limit for all relevant parameters. In addition, a temporally stabilized squeezed state can provide an event-ready probe for parameters, regardless of the initial state, and robust to the timing of the state preparation process once stabilized. In this work, we generate and stabilize a two-mode squeezed state along two secular motional modes in a vibrating trapped ion with reservoir engineering, despite starting from a thermal state of the motion. Leveraging this resource, we demonstrate an estimation of two simultaneous collective displacements along the squeezed axes, achieving improvements surpassing the classical limit by up to 6.9(3) and 7.0(3) decibels (dB), respectively. Our demonstration can be readily scaled to squeezed states with even more modes. The practical implications of our findings span a wide range of applications, including quantum sensing, quantum imaging, and various fields that demand precise measurements of multiple parameters.
Quantum metrology employs entanglement to enhance measurement precision [V. Giovannetti et al., Quantum-enhanced measurements: Beating the standard quantum limit, Science 306, 1330 (2004)SCIEAS0036-807510.1126/science.1104149, V. Giovannetti et al., Advances in quantum metrology, Nat. Photonics 5, 222 (2011)NPAHBY1749-488510.1038/nphoton.2011.35, and L. Pezzè et al., Quantum metrology with nonclassical states of atomic ensembles, Rev. Mod. Phys. 90, 035005 (2018)RMPHAT0034-686110.1103/RevModPhys.90.035005]. The focus and progress so far have primarily centered on estimating a single parameter. In diverse application scenarios, estimation of more than one single parameter is often required. Joint estimation of multiple parameters can benefit from additional advantages for further enhanced precision. Here we report quantum-enhanced estimation of simultaneous spin rotations around two orthogonal axes, making use of spin-nematic squeezing in an atomic Bose-Einstein condensate. Aided by the F=2 atomic ground hyperfine manifold coupled to the nematic-squeezed F=1 states as an auxiliary field through a sequence of microwave (MW) pulses, multiple spin-1 observables are simultaneously measured, reaching an enhancement factor 3.3 to 6.3 decibels (dB) beyond the classical limit over a wide range of rotation angles. Our work realizes the first quantum enhanced multiparameter estimation using entangled massive particles. The techniques developed and the protocols implemented also highlight the application of two-mode squeezed vacuum states in quantum-enhanced sensing of noncommuting spin rotations simultaneously.
Bosonic systems, such as photons and ultracold atoms, have played a central role in demonstrating quantum-enhanced sensing. Quantum entanglement, through squeezed and GHZ states, enables sensing beyond classical limits. However, such a quantum advantage has so far been confined to two-mode bosonic systems, as analogous multi-mode squeezed states are non-trivial to prepare. Here, we develop a Monte-Carlo based optimization technique which can be used to efficiently engineer a Hamiltonian control-sequence for multi-mode bosonic systems to prepare multi-mode squeezed states. Specifically, we consider a Bose-Einstein condensate in an optical lattice, relevant for applications in gravimetry and gradiometry, and demonstrate that metrologically useful squeezed states can be generated using the Bose-Hubbard Hamiltonian which includes on-site atomic interactions, tunable via Feshbach resonances. By analyzing the distribution (density) of the quantum Fisher information (QFI) over the Hilbert space, we identify a characteristic \textit{intermediate scaling} of the QFI: O(N2L+L2N), which lies between the standard quantum limit (SQL) and the Heisenberg limit (HL) for N atoms in L modes. We show that in general, within the Hilbert space there is a finite, O(1) measure subset of Hilbert space with an intermediate QFI scaling. Therefore, one can find a Hamiltonian control sequence using a Monte Carlo optimization over random control sequences, that produces a state with intermediate scaling of the QFI. We assume an experimentally accessible range of the control parameters in the Hamiltonian resources and show that the intermediate scaling can be readily achieved. Our results indicate that the HL can be approached in quantum gravimetry using realistic experimental parameters for systems with L=O(1) and N≫L.
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 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 number and 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.
Quantum metrology is often constrained by measurement backaction when jointly estimating two non-commuting observables—a canonical example is position and momentum. Here, we experimentally demonstrate a single-mode displacement sensor that estimates small shifts in both position and momentum with precision below the standard quantum limit (SQL). Our approach measures commuting modular position and modular momentum observables, enabled by highly nonclassical grid states (Gottesman–Kitaev–Preskill–type) prepared in the mechanical vibrational mode of a single trapped ion. Using a phase-estimation–based measurement, we extract both parameters in one experiment while evading the backaction that would ordinarily couple the estimates. We further extend the protocol to another naturally non-commuting pair—number and phase—and again observe sub-SQL performance.
High-precision quantum parameter estimation is fundamental to the advancement of quantum metrology. Although reservoir engineering provides a powerful approach to improve estimation by tailoring system-environment interactions, the role of the squeezing phase and correlations arising from the sequential utilization of the same squeezed reservoir remains inadequately explored. In this work, we employ a correlated squeezed-thermal reservoir to enhance the precision of estimating the phase parameter φ and the correlation factor μ, both individually and simultaneously. We show that the squeezing phase Φ is crucial for achieving quantum-enhanced precision, with optimal phase-matching conditions that depend strongly on μ. Specifically, we derive the near-optimal phase-matching relations aimed at maximizing the quantum Fisher information (QFI) for both φ and μ, as well as minimizing the total variance Δsim in joint estimation. Furthermore, we show that the joint estimation variance is dominated by Fφ, which motivates our search for the phase-matching conditions that minimize Δsim. Through the ratio R of variances, we demonstrate that joint estimation conserves quantum resources and maintains high precision when the squeezing phase is optimized for Fφ, despite the inherent incompatibility of the parameters. These findings provide practical insights into reservoir engineering strategies for high-precision quantum sensing and information processing.
The major problem of multiparameter quantum estimation theory is to find an ultimate measurement scheme to go beyond the standard quantum limits that each quasi-classical estimation measurement is limited by. Although, in some specifics quantum protocols without environmental noise, the ultimate sensitivity of a multiparameter quantum estimation can beat the standard quantum limit. However, the presence of noise imposes limitations on the enhancement of precision due to the inevitable existence of environmental fluctuations. Here, we address the motivation behind the usage of Gaussian quantum resources and their advantages in reaching the standard quantum limits under realistic noise. In this context, our work aims to explore the ultimate limits of precision for the simultaneous estimation of a pair of parameters that characterize the displacement channel acting on Gaussian probes and subjected to open dynamics. More precisely, we focus on a general two-mode mixed squeezed displaced thermal state, after reducing it to various Gaussian probes states, like; a two-mode pure squeezed vacuum, two-mode pure displaced vacuum, two-mode mixed displaced thermal, two-mode mixed squeezed thermal. To study the ultimate estimation precision, we evaluate the upper and bottom bound of HCRB in various cases. We find that when the entangled states, two-mode pure squeezed vacuum and two-mode mixed squeezed thermal, are employed as probes states, the upper and bottom bound of HCRB beats the standard quantum limit in the presence of a noisy environment.