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Deist E, Lu YH, Ho J, Pasha MK, Zeiher J, Yan Z, Stamper-Kurn DM. Mid-Circuit Cavity Measurement in a Neutral Atom Array. PHYSICAL REVIEW LETTERS 2022; 129:203602. [PMID: 36462020 DOI: 10.1103/physrevlett.129.203602] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/31/2022] [Revised: 08/19/2022] [Accepted: 10/05/2022] [Indexed: 06/17/2023]
Abstract
Subsystem readout during a quantum process, or mid-circuit measurement, is crucial for error correction in quantum computation, simulation, and metrology. Ideal mid-circuit measurement should be faster than the decoherence of the system, high-fidelity, and nondestructive to the unmeasured qubits. Here, we use a strongly coupled optical cavity to read out the state of a single tweezer-trapped ^{87}Rb atom within a small tweezer array. Measuring either atomic fluorescence or the transmission of light through the cavity, we detect both the presence and the state of an atom in the tweezer, within only tens of microseconds, with state preparation and measurement infidelities of roughly 0.5% and atom loss probabilities of around 1%. Using a two-tweezer system, we find measurement on one atom within the cavity causes no observable hyperfine-state decoherence on a second atom located tens of microns from the cavity volume. This high-fidelity mid-circuit readout method is a substantial step toward quantum error correction in neutral atom arrays.
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Affiliation(s)
- Emma Deist
- Department of Physics, University of California, Berkeley, California 94720, USA
- Challenge Institute for Quantum Computation, University of California, Berkeley, California 94720, USA
| | - Yue-Hui Lu
- Department of Physics, University of California, Berkeley, California 94720, USA
- Challenge Institute for Quantum Computation, University of California, Berkeley, California 94720, USA
| | - Jacquelyn Ho
- Department of Physics, University of California, Berkeley, California 94720, USA
- Challenge Institute for Quantum Computation, University of California, Berkeley, California 94720, USA
| | - Mary Kate Pasha
- Department of Physics, University of California, Berkeley, California 94720, USA
- Challenge Institute for Quantum Computation, University of California, Berkeley, California 94720, USA
| | - Johannes Zeiher
- Department of Physics, University of California, Berkeley, California 94720, USA
- Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany
- Munich Center for Quantum Science and Technology (MCQST), 80799 Munich, Germany
| | - Zhenjie Yan
- Department of Physics, University of California, Berkeley, California 94720, USA
- Challenge Institute for Quantum Computation, University of California, Berkeley, California 94720, USA
| | - Dan M Stamper-Kurn
- Department of Physics, University of California, Berkeley, California 94720, USA
- Challenge Institute for Quantum Computation, University of California, Berkeley, California 94720, USA
- Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
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2
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Pérez-González B, Gómez-León Á, Platero G. Topology detection in cavity QED. Phys Chem Chem Phys 2022; 24:15860-15870. [PMID: 35758058 DOI: 10.1039/d2cp01806c] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
We explore the physics of topological lattice models immersed in c-QED architectures for arbitrary coupling strength with the photon field. We propose the use of the cavity transmission as a topological marker and study its behaviour. For this, we develop an approach combining the input-output formalism with a Mean-Field plus fluctuations description of the setup. We illustrate our results with the specific case of a fermionic Su-Schrieffer-Heeger (SSH) chain coupled to a single-mode cavity. Our findings confirm that the cavity can indeed act as a quantum sensor for topological phases, where the initial state preparation plays a crucial role. Additionally, we discuss the persistence of topological features when the coupling strength increases, in terms of an effective Hamiltonian, and calculate the entanglement entropy. Our approach can be applied to other fermionic systems, opening a route to the characterization of their topological properties in terms of experimental observables.
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Affiliation(s)
- Beatriz Pérez-González
- Instituto de Ciencia de Materiales de Madrid, ICMM-CSIC, Calle Sor Juana Inés de la Cruz, n°3, 28049 Madrid, Spain.
| | - Álvaro Gómez-León
- Instituto de Física Fundamental, IFF-CSIC, Calle Serrano 113b, 28006 Madrid, Spain
| | - Gloria Platero
- Instituto de Ciencia de Materiales de Madrid, ICMM-CSIC, Calle Sor Juana Inés de la Cruz, n°3, 28049 Madrid, Spain.
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3
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Du FF, Shi ZR. Robust hybrid hyper-controlled-not gates assisted by an input-output process of low-Q cavities. OPTICS EXPRESS 2019; 27:17493-17506. [PMID: 31252708 DOI: 10.1364/oe.27.017493] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/09/2019] [Accepted: 05/14/2019] [Indexed: 06/09/2023]
Abstract
The two or more degrees of freedoms (DOFs) of photon systems are very useful in hyperparallel photonic quantum computing to accomplish more quantum logic gate operations with less resource, and depress photonic dissipation noise in quantum information processing. We present some flexible and adjustable schemes for hybrid hyper-controlled-not (hyper-CNOT) gates assisted by low-Q cavities, on the two-photon systems in both the spatial-mode and the polarization DOFs. These hybrid spatial-polarization hyper-CNOT gates consume less quantum resource and are more robust against photonic dissipation noise, compared with the integration of two cascaded CNOT gates in one DOF. Besides, simultaneous counter-propagation of two photons economize extremely the operation time in the whole process of our schemes. Moreover, these quantum logic gates are more feasible for fast quantum operations in the weak-coupling region of the low-Q cavities with current experimental technology, which are much different from strong-coupling cases of the high-Q ones.
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4
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Bao DQ, Zhu CJ, Yang YP, Agarwal GS. Sensing single atoms in a cavity using a broadband squeezed light. OPTICS EXPRESS 2019; 27:15540-15547. [PMID: 31163749 DOI: 10.1364/oe.27.015540] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/26/2019] [Accepted: 05/06/2019] [Indexed: 06/09/2023]
Abstract
We investigate a single atom cavity-QED system directly driven by a broadband squeezed light. We demonstrate how the squeezed radiation can be used to sense the presence of a single atom in a cavity. This happens by transferring one of the photons from the field in a state with an even number of photons to the atom and thereby populating an odd number of Fock states. Specifically, the presence of the atom is sensed by remarkable changing in the presence of one photon and the loss of squeezing of the cavity field. A complete study of quantum fluctuations and the excitation of multiphoton transitions is given.
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Delić U, Reisenbauer M, Grass D, Kiesel N, Vuletić V, Aspelmeyer M. Cavity Cooling of a Levitated Nanosphere by Coherent Scattering. PHYSICAL REVIEW LETTERS 2019; 122:123602. [PMID: 30978033 DOI: 10.1103/physrevlett.122.123602] [Citation(s) in RCA: 16] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/21/2018] [Indexed: 06/09/2023]
Abstract
We report three-dimensional (3D) cooling of a levitated nanoparticle inside an optical cavity. The cooling mechanism is provided by cavity-enhanced coherent scattering off an optical tweezer. The observed 3D dynamics and cooling rates are as theoretically expected from the presence of both linear and quadratic terms in the interaction between the particle motion and the cavity field. By achieving nanometer-level control over the particle location we optimize the position-dependent coupling and demonstrate axial cooling by two orders of magnitude at background pressures of 6×10^{-2} mbar. We also estimate a significant (>40 dB) suppression of laser phase noise heating, which is a specific feature of the coherent scattering scheme. The observed performance implies that quantum ground state cavity cooling of levitated nanoparticles can be achieved for background pressures below 1×10^{-7} mbar.
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Affiliation(s)
- Uroš Delić
- Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
- Institute for Quantum Optics and Quantum Information (IQOQI), Boltzmanngasse 3, A-1090 Vienna, Austria
| | - Manuel Reisenbauer
- Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
| | - David Grass
- Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
| | - Nikolai Kiesel
- Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
| | - Vladan Vuletić
- Department of Physics and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Markus Aspelmeyer
- Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
- Institute for Quantum Optics and Quantum Information (IQOQI), Boltzmanngasse 3, A-1090 Vienna, Austria
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6
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Dinardo BA, Anderson DZ. A technique for individual atom delivery into a crossed vortex bottle beam trap using a dynamic 1D optical lattice. THE REVIEW OF SCIENTIFIC INSTRUMENTS 2016; 87:123108. [PMID: 28040917 DOI: 10.1063/1.4972250] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We describe a system for loading a single atom from a reservoir into a blue-detuned crossed vortex bottle beam trap using a dynamic 1D optical lattice. The lattice beams are frequency chirped using acousto-optic modulators, which causes the lattice to move along its axial direction and behave like an optical conveyor belt. A stationary lattice is initially loaded with approximately 6000 atoms from a reservoir, and the conveyor belt transports them 1.1 mm from the reservoir to a bottle beam trap, where a single atom is loaded via light-assisted collisions. Photon counting data confirm that an atom can be delivered and loaded into the bottle beam trap 13.1% of the time.
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Affiliation(s)
- Brad A Dinardo
- JILA, University of Colorado, and National Institute of Standards and Technology, Boulder, Colorado 80309-0440, USA
| | - Dana Z Anderson
- JILA, University of Colorado, and National Institute of Standards and Technology, Boulder, Colorado 80309-0440, USA
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7
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He S, Su SL, Wang DY, Sun WM, Bai CH, Zhu AD, Wang HF, Zhang S. Efficient shortcuts to adiabatic passage for three-dimensional entanglement generation via transitionless quantum driving. Sci Rep 2016; 6:30929. [PMID: 27499169 PMCID: PMC4976372 DOI: 10.1038/srep30929] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/26/2016] [Accepted: 07/11/2016] [Indexed: 11/09/2022] Open
Abstract
We propose an effective scheme of shortcuts to adiabaticity for generating a three-dimensional entanglement of two atoms trapped in a cavity using the transitionless quantum driving (TQD) approach. The key point of this approach is to construct an effective Hamiltonian that drives the dynamics of a system along instantaneous eigenstates of a reference Hamiltonian to reproduce the same final state as that of an adiabatic process within a much shorter time. In this paper, the shortcuts to adiabatic passage are constructed by introducing two auxiliary excited levels in each atom and applying extra cavity modes and classical fields to drive the relevant transitions. Thereby, the three-dimensional entanglement is obtained with a faster rate than that in the adiabatic passage. Moreover, the influences of atomic spontaneous emission and photon loss on the fidelity are discussed by numerical simulation. The results show that the speed of entanglement implementation is greatly improved by the use of adiabatic shortcuts and that this entanglement implementation is robust against decoherence. This will be beneficial to the preparation of high-dimensional entanglement in experiment and provides the necessary conditions for the application of high-dimensional entangled states in quantum information processing.
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Affiliation(s)
- Shuang He
- Department of Physics, College of Science, Yanbian University, Yanji, Jilin 133002, China
| | - Shi-Lei Su
- Department of Physics, College of Science, Yanbian University, Yanji, Jilin 133002, China.,School of Physical Science &Engineering and Key Laboratory of Materials Physics of Ministry of Education of China, Zhengzhou University, Zhengzhou 450052 China
| | - Dong-Yang Wang
- Department of Physics, College of Science, Yanbian University, Yanji, Jilin 133002, China
| | - Wen-Mei Sun
- Department of Physics, College of Science, Yanbian University, Yanji, Jilin 133002, China
| | - Cheng-Hua Bai
- Department of Physics, College of Science, Yanbian University, Yanji, Jilin 133002, China
| | - Ai-Dong Zhu
- Department of Physics, College of Science, Yanbian University, Yanji, Jilin 133002, China
| | - Hong-Fu Wang
- Department of Physics, College of Science, Yanbian University, Yanji, Jilin 133002, China
| | - Shou Zhang
- Department of Physics, College of Science, Yanbian University, Yanji, Jilin 133002, China
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8
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Universal quantum gates for photon-atom hybrid systems assisted by bad cavities. Sci Rep 2016; 6:24183. [PMID: 27067992 PMCID: PMC4828709 DOI: 10.1038/srep24183] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/26/2015] [Accepted: 03/22/2016] [Indexed: 12/01/2022] Open
Abstract
We present two deterministic schemes for constructing a CNOT gate and a Toffoli gate on photon-atom and photon-atom-atom hybrid quantum systems assisted by bad cavities, respectively. They are achieved by cavity-assisted photon scattering and work in the intermediate coupling region with bad cavities, which relaxes the difficulty of their implementation in experiment. Also, bad cavities are feasible for fast quantum operations and reading out information. Compared with previous works, our schemes do not need any auxiliary qubits and measurements. Moreover, the schematic setups for these gates are simple, especially that for our Toffoli gate as only a quarter wave packet is used to interact the photon with each of the atoms every time. These atom-cavity systems can be used as the quantum nodes in long-distance quantum communication as their relatively long coherence time is suitable for multi-time operations between the photon and the system. Our calculations show that the average fidelities and efficiencies of our two universal hybrid quantum gates are high with current experimental technology.
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9
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10
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Cao C, Wang C, He LY, Zhang R. Atomic entanglement purification and concentration using coherent state input-output process in low-Q cavity QED regime. OPTICS EXPRESS 2013; 21:4093-4105. [PMID: 23481943 DOI: 10.1364/oe.21.004093] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
We investigate an atomic entanglement purification protocol based on the coherent state input-output process by working in low-Q cavity in the atom-cavity intermediate coupling region. The information of entangled states are encoded in three-level configured single atoms confined in separated one-side optical micro-cavities. Using the coherent state input-output process, we design a two-qubit parity check module (PCM), which allows the quantum nondemolition measurement for the atomic qubits, and show its use for remote parities to distill a high-fidelity atomic entangled ensemble from an initial mixed state ensemble nonlocally. The proposed scheme can further be used for unknown atomic states entanglement concentration. Also by exploiting the PCM, we describe a modified scheme for atomic entanglement concentration by introducing ancillary single atoms. As the coherent state input-output process is robust and scalable in realistic applications, and the detection in the PCM is based on the intensity of outgoing coherent state, the present protocols may be widely used in large-scaled and solid-based quantum repeater and quantum information processing.
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Affiliation(s)
- Cong Cao
- State Key Laboratory of Information Photonics and Optical Communications , Beijing University of Posts and Telecommunications, Beijing 100876, China
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11
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Zhang H, McConnell R, Cuk S, Lin Q, Schleier-Smith MH, Leroux ID, Vuletić V. Collective state measurement of mesoscopic ensembles with single-atom resolution. PHYSICAL REVIEW LETTERS 2012; 109:133603. [PMID: 23030090 DOI: 10.1103/physrevlett.109.133603] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/06/2012] [Indexed: 06/01/2023]
Abstract
We demonstrate single-atom resolution, as well as detection sensitivity more than 20 dB below the quantum projection noise limit, for hyperfine-state-selective measurements on mesoscopic ensembles containing 100 or more atoms. The measurement detects the atom-induced shift of the resonance frequency of an optical cavity containing the ensemble. While spatially varying coupling of atoms to the cavity prevents the direct observation of a quantized signal, the demonstrated measurement resolution provides the readout capability necessary for atomic interferometry substantially below the standard quantum limit and down to the Heisenberg limit.
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Affiliation(s)
- Hao Zhang
- Department of Physics, MIT-Harvard Center for Ultracold Atoms, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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12
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Bochmann J, Mücke M, Guhl C, Ritter S, Rempe G, Moehring DL. Lossless state detection of single neutral atoms. PHYSICAL REVIEW LETTERS 2010; 104:203601. [PMID: 20867026 DOI: 10.1103/physrevlett.104.203601] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/15/2010] [Indexed: 05/29/2023]
Abstract
We introduce lossless state detection of trapped neutral atoms based on cavity-enhanced fluorescence. In an experiment with a single 87Rb atom, a hyperfine-state-detection fidelity of 99.4% is achieved in 85 μs. The quantum bit is interrogated many hundreds of times without loss of the atom while a result is obtained in every readout attempt. The fidelity proves robust against atomic frequency shifts induced by the trapping potential. Our scheme does not require strong coupling between the atom and cavity and can be generalized to other systems with an optically accessible quantum bit.
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Affiliation(s)
- J Bochmann
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany
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13
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Kang S, Choi Y, Lim S, Kim W, Kim JR, Lee JH, An K. Continuous control of the coupling constant in an atom-cavity system by using elliptic polarization and magnetic sublevels. OPTICS EXPRESS 2010; 18:9286-9302. [PMID: 20588776 DOI: 10.1364/oe.18.009286] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/29/2023]
Abstract
Atom-cavity coupling constant is a key parameter in cavity quantum electrodynamics for describing the interaction between an atom and a quantized electromagnetic field in a cavity. This paper reports a novel way to tune the coupling constant continuously by inducing an averaging of the atomic dipole moment over degenerate magnetic sublevels with elliptic polarization of the cavity field. We present an analytic solution of the stationary-state density matrix for this system with consideration of F -> F +1 hyperfine transition under a weak excitation condition. We rigorously show that the stationary-state emission spectra of this system can be approximated by that of a non-degenerate two-level atom with an effective coupling constant as a function of the elliptic angle of the cavity field only. A precise condition for this approximation is derived and its physical meaning is interpreted in terms of a population-averaged transition strength and its variance. Our results can be used to control the coupling constant in cavity quantum electrodynamics experiments with a degenerate two-level atom with magnetic sublevels. Possible applications of our results are discussed.
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Affiliation(s)
- Sungsam Kang
- Department of Physics and Astronomy, Seoul National University, Seoul, 151-742, Korea
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14
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Leibrandt DR, Labaziewicz J, Vuletić V, Chuang IL. Cavity sideband cooling of a single trapped ion. PHYSICAL REVIEW LETTERS 2009; 103:103001. [PMID: 19792300 DOI: 10.1103/physrevlett.103.103001] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/01/2009] [Indexed: 05/28/2023]
Abstract
We report a demonstration and quantitative characterization of one-dimensional cavity cooling of a single trapped (88)Sr(+) ion in the resolved-sideband regime. We measure the spectrum of cavity transitions, the rates of cavity heating and cooling, and the steady-state cooling limit. The cavity cooling dynamics and cooling limit of 22.5(3) motional quanta, limited by the moderate coupling between the ion and the cavity, are consistent with a simple model [Phys. Rev. A 64, 033405 (2001)] without any free parameters, validating the rate equation model for cavity cooling.
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Affiliation(s)
- David R Leibrandt
- Department of Physics & Center for Ultracold Atoms Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139, USA.
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15
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Weber B, Specht HP, Müller T, Bochmann J, Mücke M, Moehring DL, Rempe G. Photon-photon entanglement with a single trapped atom. PHYSICAL REVIEW LETTERS 2009; 102:030501. [PMID: 19257335 DOI: 10.1103/physrevlett.102.030501] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/11/2008] [Indexed: 05/27/2023]
Abstract
An experiment is performed where a single rubidium atom trapped within a high-finesse optical cavity emits two independently triggered entangled photons. The entanglement is mediated by the atom and is characterized both by a Bell inequality violation of S=2.5, as well as full quantum-state tomography, resulting in a fidelity exceeding F=90%. The combination of cavity-QED and trapped atom techniques makes our protocol inherently deterministic--an essential step for the generation of scalable entanglement between the nodes of a distributed quantum network.
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Affiliation(s)
- B Weber
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany
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16
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Abstract
Quantum networks provide opportunities and challenges across a range of intellectual and technical frontiers, including quantum computation, communication and metrology. The realization of quantum networks composed of many nodes and channels requires new scientific capabilities for generating and characterizing quantum coherence and entanglement. Fundamental to this endeavour are quantum interconnects, which convert quantum states from one physical system to those of another in a reversible manner. Such quantum connectivity in networks can be achieved by the optical interactions of single photons and atoms, allowing the distribution of entanglement across the network and the teleportation of quantum states between nodes.
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17
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Ye J, Kimble HJ, Katori H. Quantum State Engineering and Precision Metrology Using State-Insensitive Light Traps. Science 2008; 320:1734-8. [DOI: 10.1126/science.1148259] [Citation(s) in RCA: 293] [Impact Index Per Article: 18.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/02/2022]
Affiliation(s)
- Jun Ye
- JILA, National Institute of Standards and Technology (NIST) and University of Colorado, Boulder, CO 80309–0440, USA
- Norman Bridge Laboratory of Physics 12-33, California Institute of Technology, Pasadena, CA 91125, USA
- Department of Applied Physics, School of Engineering, University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan
| | - H. J. Kimble
- JILA, National Institute of Standards and Technology (NIST) and University of Colorado, Boulder, CO 80309–0440, USA
- Norman Bridge Laboratory of Physics 12-33, California Institute of Technology, Pasadena, CA 91125, USA
- Department of Applied Physics, School of Engineering, University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan
| | - Hidetoshi Katori
- JILA, National Institute of Standards and Technology (NIST) and University of Colorado, Boulder, CO 80309–0440, USA
- Norman Bridge Laboratory of Physics 12-33, California Institute of Technology, Pasadena, CA 91125, USA
- Department of Applied Physics, School of Engineering, University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan
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