1
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Lewis-Swan RJ, Castro JCZ, Barberena D, Rey AM. Exploiting Nonclassical Motion of a Trapped Ion Crystal for Quantum-Enhanced Metrology of Global and Differential Spin Rotations. PHYSICAL REVIEW LETTERS 2024; 132:163601. [PMID: 38701452 DOI: 10.1103/physrevlett.132.163601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/04/2023] [Revised: 02/23/2024] [Accepted: 03/18/2024] [Indexed: 05/05/2024]
Abstract
We theoretically investigate prospects for the creation of nonclassical spin states in trapped ion arrays by coupling to a squeezed state of the collective motion of the ions. The correlations of the generated spin states can be tailored for quantum-enhanced sensing of global or differential rotations of subensembles of the spins by working with specific vibrational modes of the ion array. We propose a pair of protocols to utilize the generated states and demonstrate their viability even for small systems, while assessing limitations imposed by spin-motion entanglement and technical noise. Our work suggests new opportunities for the preparation of many-body states with tailored correlations for quantum-enhanced metrology in spin-boson systems.
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Affiliation(s)
- R J Lewis-Swan
- Homer L. Dodge Department of Physics and Astronomy, The University of Oklahoma, Norman, Oklahoma 73019, USA
- Center for Quantum Research and Technology, The University of Oklahoma, Norman, Oklahoma 73019, USA
| | - J C Zuñiga Castro
- Homer L. Dodge Department of Physics and Astronomy, The University of Oklahoma, Norman, Oklahoma 73019, USA
- Center for Quantum Research and Technology, The University of Oklahoma, Norman, Oklahoma 73019, USA
| | - D Barberena
- JILA, NIST, and Department of Physics, University of Colorado, Boulder, Colorado, USA
- Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado, USA
| | - A M Rey
- JILA, NIST, and Department of Physics, University of Colorado, Boulder, Colorado, USA
- Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado, USA
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2
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Rico A, Huber F. Entanglement Detection with Trace Polynomials. PHYSICAL REVIEW LETTERS 2024; 132:070202. [PMID: 38427872 DOI: 10.1103/physrevlett.132.070202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/21/2023] [Accepted: 12/18/2023] [Indexed: 03/03/2024]
Abstract
We provide a systematic method for nonlinear entanglement detection based on trace polynomial inequalities. In particular, this allows us to employ multipartite witnesses for the detection of bipartite states, and vice versa. We identify pairs of entangled states and witnesses for which linear detection fails, but for which nonlinear detection succeeds. With the trace polynomial formulation a great variety of witnesses arise from immanant inequalities, which can be implemented in the laboratory through the randomized measurements toolbox.
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Affiliation(s)
- Albert Rico
- Faculty of Physics, Astronomy and Applied Computer Science, Institute of Theoretical Physics, Jagiellonian University, 30-348 Kraków, Poland
| | - Felix Huber
- Faculty of Physics, Astronomy and Applied Computer Science, Institute of Theoretical Physics, Jagiellonian University, 30-348 Kraków, Poland
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3
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Prasannan N, Sperling J, Brecht B, Silberhorn C. Direct Measurement of Higher-Order Nonlinear Polarization Squeezing. PHYSICAL REVIEW LETTERS 2022; 129:263601. [PMID: 36608182 DOI: 10.1103/physrevlett.129.263601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/19/2022] [Revised: 09/20/2022] [Accepted: 12/05/2022] [Indexed: 06/17/2023]
Abstract
We report on nonlinear squeezing effects of polarization states of light by harnessing the intrinsic correlations from a polarization-entangled light source and click-counting measurements. Nonlinear Stokes operators are obtained from harnessing the click-counting theory in combination with angular-momentum-type algebras. To quantify quantum effects, theoretical bounds are derived for second- and higher-order moments of nonlinear Stokes operators. The experimental validation of our concept is rendered possible by developing an efficient source, using a spectrally decorrelated type-II phase-matched waveguide inside a Sagnac interferometer. Correlated click statistics and moments are directly obtained from an eight-time-bin quasi-photon-number-resolving detection system. Macroscopic Bell states that are readily available with our source show the distinct nature of nonlinear polarization squeezing in up to eighth-order correlations, matching our theoretical predictions. Furthermore, our data certify nonclassical correlations with high statistical significance, without the need to correct for experimental imperfections and limitations. Also, our nonlinear squeezing can identify nonclassicality of noisy quantum states which is undetectable with the known linear polarization-squeezing criterion.
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Affiliation(s)
- Nidhin Prasannan
- Integrated Quantum Optics Group, Institute for Photonic Quantum Systems (PhoQS), Paderborn University, Warburger Straße 100, 33098 Paderborn, Germany
| | - Jan Sperling
- Theoretical Quantum Science, Institute for Photonic Quantum Systems (PhoQS), Paderborn University, Warburger Straße 100, 33098 Paderborn, Germany
| | - Benjamin Brecht
- Integrated Quantum Optics Group, Institute for Photonic Quantum Systems (PhoQS), Paderborn University, Warburger Straße 100, 33098 Paderborn, Germany
| | - Christine Silberhorn
- Integrated Quantum Optics Group, Institute for Photonic Quantum Systems (PhoQS), Paderborn University, Warburger Straße 100, 33098 Paderborn, Germany
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4
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Kala V, Filip R, Marek P. Cubic nonlinear squeezing and its decoherence. OPTICS EXPRESS 2022; 30:31456-31471. [PMID: 36242226 DOI: 10.1364/oe.464759] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/24/2022] [Accepted: 07/17/2022] [Indexed: 06/16/2023]
Abstract
Squeezed states of the harmonic oscillator are a common resource in applications of quantum technology. If the noise is suppressed in a nonlinear combination of quadrature operators below threshold for all possible up-to-quadratic Hamiltonians, the quantum states are non-Gaussian and we refer to the noise reduction as nonlinear squeezing. Non-Gaussian aspects of quantum states are often more vulnerable to decoherence due to imperfections appearing in realistic experimental implementations. Therefore, a stability of nonlinear squeezing is essential. We analyze the behavior of quantum states with cubic nonlinear squeezing under loss and dephasing. The properties of nonlinear squeezed states depend on their initial parameters which can be optimized and adjusted to achieve the maximal robustness for the potential applications.
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5
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Podhora L, Lachman L, Pham T, Lešundák A, Číp O, Slodička L, Filip R. Quantum Non-Gaussianity of Multiphonon States of a Single Atom. PHYSICAL REVIEW LETTERS 2022; 129:013602. [PMID: 35841581 DOI: 10.1103/physrevlett.129.013602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/18/2021] [Revised: 03/22/2022] [Accepted: 05/25/2022] [Indexed: 06/15/2023]
Abstract
Quantum non-Gaussian mechanical states are already required in a range of applications. The discrete building blocks of such states are the energy eigenstates-Fock states. Despite progress in their preparation, the remaining imperfections can still invisibly cause loss of the aspects critical for their applications. We derive and apply the most challenging hierarchy of quantum non-Gaussian criteria on the characterization of single trapped-ion oscillator mechanical Fock states with up to 10 phonons. We analyze the depth of these quantum non-Gaussian features under intrinsic mechanical heating and predict their requirement for reaching quantum advantage in the sensing of a mechanical force.
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Affiliation(s)
- L Podhora
- Department of Optics, Palacký University, 17. listopadu 12, 77146 Olomouc, Czech Republic
| | - L Lachman
- Department of Optics, Palacký University, 17. listopadu 12, 77146 Olomouc, Czech Republic
| | - T Pham
- Institute of Scientific Instruments of the Czech Academy of Sciences, Královopolská 147, 612 64 Brno, Czech Republic
| | - A Lešundák
- Institute of Scientific Instruments of the Czech Academy of Sciences, Královopolská 147, 612 64 Brno, Czech Republic
| | - O Číp
- Institute of Scientific Instruments of the Czech Academy of Sciences, Královopolská 147, 612 64 Brno, Czech Republic
| | - L Slodička
- Department of Optics, Palacký University, 17. listopadu 12, 77146 Olomouc, Czech Republic
| | - R Filip
- Department of Optics, Palacký University, 17. listopadu 12, 77146 Olomouc, Czech Republic
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6
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Xu K, Zhang YR, Sun ZH, Li H, Song P, Xiang Z, Huang K, Li H, Shi YH, Chen CT, Song X, Zheng D, Nori F, Wang H, Fan H. Metrological Characterization of Non-Gaussian Entangled States of Superconducting Qubits. PHYSICAL REVIEW LETTERS 2022; 128:150501. [PMID: 35499907 DOI: 10.1103/physrevlett.128.150501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/29/2021] [Accepted: 03/15/2022] [Indexed: 06/14/2023]
Abstract
Multipartite entangled states are significant resources for both quantum information processing and quantum metrology. In particular, non-Gaussian entangled states are predicted to achieve a higher sensitivity of precision measurements than Gaussian states. On the basis of metrological sensitivity, the conventional linear Ramsey squeezing parameter (RSP) efficiently characterizes the Gaussian entangled atomic states but fails for much wider classes of highly sensitive non-Gaussian states. These complex non-Gaussian entangled states can be classified by the nonlinear squeezing parameter (NLSP), as a generalization of the RSP with respect to nonlinear observables and identified via the Fisher information. However, the NLSP has never been measured experimentally. Using a 19-qubit programmable superconducting processor, we report the characterization of multiparticle entangled states generated during its nonlinear dynamics. First, selecting ten qubits, we measure the RSP and the NLSP by single-shot readouts of collective spin operators in several different directions. Then, by extracting the Fisher information of the time-evolved state of all 19 qubits, we observe a large metrological gain of 9.89_{-0.29}^{+0.28} dB over the standard quantum limit, indicating a high level of multiparticle entanglement for quantum-enhanced phase sensitivity. Benefiting from high-fidelity full controls and addressable single-shot readouts, the superconducting processor with interconnected qubits provides an ideal platform for engineering and benchmarking non-Gaussian entangled states that are useful for quantum-enhanced metrology.
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Affiliation(s)
- Kai Xu
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Yu-Ran Zhang
- Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
- RIKEN Center for Quantum Computing (RQC), Wako-shi, Saitama 351-0198, Japan
| | - Zheng-Hang Sun
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Hekang Li
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Pengtao Song
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Zhongcheng Xiang
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Kaixuan Huang
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Hao Li
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Yun-Hao Shi
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Chi-Tong Chen
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Xiaohui Song
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Dongning Zheng
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Franco Nori
- Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
- RIKEN Center for Quantum Computing (RQC), Wako-shi, Saitama 351-0198, Japan
- Physics Department, University of Michigan, Ann Arbor, Michigan 48109-1040, USA
| | - H Wang
- Interdisciplinary Center for Quantum Information, State Key Laboratory of Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Department of Physics, Zhejiang University, Hangzhou 310027, China
| | - Heng Fan
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- Beijing Academy of Quantum Information Sciences and CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China
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7
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Baamara Y, Sinatra A, Gessner M. Scaling Laws for the Sensitivity Enhancement of Non-Gaussian Spin States. PHYSICAL REVIEW LETTERS 2021; 127:160501. [PMID: 34723607 DOI: 10.1103/physrevlett.127.160501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/24/2021] [Accepted: 09/14/2021] [Indexed: 06/13/2023]
Abstract
We identify the large-N scaling of the metrological quantum gain offered by over-squeezed spin states that are accessible by one-axis twisting, as a function of the preparation time. We further determine how the scaling is modified by relevant decoherence processes and predict a discontinuous change of the quantum gain at a critical preparation time that depends on the noise. Our analytical results provide recipes for optimal and feasible implementations of quantum enhancements with non-Gaussian spin states in existing experiments, well beyond the reach of spin squeezing.
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Affiliation(s)
- Youcef Baamara
- Laboratoire Kastler Brossel, ENS-Université PSL, CNRS, Sorbonne Université, Collège de France, 24 Rue Lhomond, 75005 Paris, France
| | - Alice Sinatra
- Laboratoire Kastler Brossel, ENS-Université PSL, CNRS, Sorbonne Université, Collège de France, 24 Rue Lhomond, 75005 Paris, France
| | - Manuel Gessner
- Laboratoire Kastler Brossel, ENS-Université PSL, CNRS, Sorbonne Université, Collège de France, 24 Rue Lhomond, 75005 Paris, France
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8
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Sorelli G, Gessner M, Walschaers M, Treps N. Optimal Observables and Estimators for Practical Superresolution Imaging. PHYSICAL REVIEW LETTERS 2021; 127:123604. [PMID: 34597062 DOI: 10.1103/physrevlett.127.123604] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/26/2021] [Accepted: 08/06/2021] [Indexed: 06/13/2023]
Abstract
Recent works identified resolution limits for the distance between incoherent point sources. However, it remains unclear how to choose suitable observables and estimators to reach these limits in practical situations. Here, we show how estimators saturating the Cramér-Rao bound for the distance between two thermal point sources can be constructed using an optimally designed observable in the presence of practical imperfections, such as misalignment, cross talk, and detector noise.
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Affiliation(s)
- Giacomo Sorelli
- Laboratoire Kastler Brossel, Sorbonne Université, ENS-Université PSL, CNRS, Collége de France, 4 Place Jussieu, F-75252 Paris, France
| | - Manuel Gessner
- Laboratoire Kastler Brossel, Sorbonne Université, ENS-Université PSL, CNRS, Collége de France, 4 Place Jussieu, F-75252 Paris, France
| | - Mattia Walschaers
- Laboratoire Kastler Brossel, Sorbonne Université, ENS-Université PSL, CNRS, Collége de France, 4 Place Jussieu, F-75252 Paris, France
| | - Nicolas Treps
- Laboratoire Kastler Brossel, Sorbonne Université, ENS-Université PSL, CNRS, Collége de France, 4 Place Jussieu, F-75252 Paris, France
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9
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Yadin B, Fadel M, Gessner M. Metrological complementarity reveals the Einstein-Podolsky-Rosen paradox. Nat Commun 2021; 12:2410. [PMID: 33893281 PMCID: PMC8065158 DOI: 10.1038/s41467-021-22353-3] [Citation(s) in RCA: 20] [Impact Index Per Article: 6.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/12/2020] [Accepted: 03/11/2021] [Indexed: 02/02/2023] Open
Abstract
The Einstein-Podolsky-Rosen (EPR) paradox plays a fundamental role in our understanding of quantum mechanics, and is associated with the possibility of predicting the results of non-commuting measurements with a precision that seems to violate the uncertainty principle. This apparent contradiction to complementarity is made possible by nonclassical correlations stronger than entanglement, called steering. Quantum information recognises steering as an essential resource for a number of tasks but, contrary to entanglement, its role for metrology has so far remained unclear. Here, we formulate the EPR paradox in the framework of quantum metrology, showing that it enables the precise estimation of a local phase shift and of its generating observable. Employing a stricter formulation of quantum complementarity, we derive a criterion based on the quantum Fisher information that detects steering in a larger class of states than well-known uncertainty-based criteria. Our result identifies useful steering for quantum-enhanced precision measurements and allows one to uncover steering of non-Gaussian states in state-of-the-art experiments.
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Affiliation(s)
- Benjamin Yadin
- grid.4563.40000 0004 1936 8868School of Mathematical Sciences and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, UK ,grid.4991.50000 0004 1936 8948Wolfson College, University of Oxford, Oxford, UK
| | - Matteo Fadel
- grid.6612.30000 0004 1937 0642Department of Physics, University of Basel, Basel, Switzerland
| | - Manuel Gessner
- grid.462844.80000 0001 2308 1657Laboratoire Kastler Brossel, ENS-Université PSL, CNRS, Sorbonne Université, Collège de France, Paris, France
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10
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Ren Z, Li W, Smerzi A, Gessner M. Metrological Detection of Multipartite Entanglement from Young Diagrams. PHYSICAL REVIEW LETTERS 2021; 126:080502. [PMID: 33709723 DOI: 10.1103/physrevlett.126.080502] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/07/2020] [Accepted: 01/13/2021] [Indexed: 06/12/2023]
Abstract
We characterize metrologically useful multipartite entanglement by representing partitions with Young diagrams. We derive entanglement witnesses that are sensitive to the shape of Young diagrams and show that Dyson's rank acts as a resource for quantum metrology. Common quantifiers, such as the entanglement depth and k-separability are contained in this approach as the diagram's width and height. Our methods are experimentally accessible in a wide range of atomic systems, as we illustrate by analyzing published data on the quantum Fisher information and spin-squeezing coefficients.
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Affiliation(s)
- Zhihong Ren
- Institute of Theoretical Physics and Department of Physics, State Key Laboratory of Quantum Optics and Quantum Optics Devices, Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
- Laboratoire Kastler Brossel, ENS-Université PSL, CNRS, Sorbonne Université, Collège de France, 24 Rue Lhomond, 75005 Paris, France
| | - Weidong Li
- Institute of Theoretical Physics and Department of Physics, State Key Laboratory of Quantum Optics and Quantum Optics Devices, Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
| | - Augusto Smerzi
- Institute of Theoretical Physics and Department of Physics, State Key Laboratory of Quantum Optics and Quantum Optics Devices, Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
- QSTAR, INO-CNR, and LENS, Largo Enrico Fermi 2, 50125 Firenze, Italy
| | - Manuel Gessner
- Laboratoire Kastler Brossel, ENS-Université PSL, CNRS, Sorbonne Université, Collège de France, 24 Rue Lhomond, 75005 Paris, France
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11
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Gessner M, Smerzi A, Pezzè L. Multiparameter squeezing for optimal quantum enhancements in sensor networks. Nat Commun 2020; 11:3817. [PMID: 32733031 PMCID: PMC7393128 DOI: 10.1038/s41467-020-17471-3] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/28/2020] [Accepted: 06/12/2020] [Indexed: 11/09/2022] Open
Abstract
Squeezing currently represents the leading strategy for quantum enhanced precision measurements of a single parameter in a variety of continuous- and discrete-variable settings and technological applications. However, many important physical problems including imaging and field sensing require the simultaneous measurement of multiple unknown parameters. The development of multiparameter quantum metrology is yet hindered by the intrinsic difficulty in finding saturable sensitivity bounds and feasible estimation strategies. Here, we derive the general operational concept of multiparameter squeezing, identifying metrologically useful states and optimal estimation strategies. When applied to spin- or continuous-variable systems, our results generalize widely-used spin- or quadrature-squeezing parameters. Multiparameter squeezing provides a practical and versatile concept that paves the way to the development of quantum-enhanced estimation of multiple phases, gradients, and fields, and for the efficient characterization of multimode quantum states in atomic and optical sensor networks.
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Affiliation(s)
- Manuel Gessner
- Laboratoire Kastler Brossel, ENS-PSL Université, CNRS, Sorbonne Université, Collège de France, 24 Rue Lhomond, 75005, Paris, France.
| | - Augusto Smerzi
- QSTAR, CNR-INO and LENS, Largo Enrico Fermi 2, 50125, Firenze, Italy
| | - Luca Pezzè
- QSTAR, CNR-INO and LENS, Largo Enrico Fermi 2, 50125, Firenze, Italy
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13
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Evrard A, Makhalov V, Chalopin T, Sidorenkov LA, Dalibard J, Lopes R, Nascimbene S. Enhanced Magnetic Sensitivity with Non-Gaussian Quantum Fluctuations. PHYSICAL REVIEW LETTERS 2019; 122:173601. [PMID: 31107084 DOI: 10.1103/physrevlett.122.173601] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/18/2019] [Indexed: 06/09/2023]
Abstract
The precision of a quantum sensor can overcome its classical counterpart when its constituents are entangled. In Gaussian squeezed states, quantum correlations lead to a reduction of the quantum projection noise below the shot noise limit. However, the most sensitive states involve complex non-Gaussian quantum fluctuations, making the required measurement protocol challenging. Here we measure the sensitivity of nonclassical states of the electronic spin J=8 of dysprosium atoms, created using light-induced nonlinear spin coupling. Magnetic sublevel resolution enables us to reach the optimal sensitivity of non-Gaussian (oversqueezed) states, well above the capability of squeezed states and about half the Heisenberg limit.
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Affiliation(s)
- Alexandre Evrard
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France
| | - Vasiliy Makhalov
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France
| | - Thomas Chalopin
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France
| | - Leonid A Sidorenkov
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France
| | - Jean Dalibard
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France
| | - Raphael Lopes
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France
| | - Sylvain Nascimbene
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France
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