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Adhikari S, Smit R, Orrit M. Future Paths in Cryogenic Single-Molecule Fluorescence Spectroscopy. THE JOURNAL OF PHYSICAL CHEMISTRY. C, NANOMATERIALS AND INTERFACES 2024; 128:3-18. [PMID: 38229590 PMCID: PMC10788914 DOI: 10.1021/acs.jpcc.3c06564] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 10/02/2023] [Revised: 11/23/2023] [Accepted: 11/28/2023] [Indexed: 01/18/2024]
Abstract
In the last three decades, cryogenic single-molecule fluorescence spectroscopy has provided average-free understanding of the photophysics and of fundamental interactions at molecular scales. Furthermore, they propose original pathways and applications in the treatment and storage of quantum information. The ultranarrow lifetime-limited zero-phonon line acts as an excellent sensor to local perturbations caused either by intrinsic dynamical degrees of freedom, or by external perturbations, such as those caused by electric fields, elastic and acoustic deformations, or light-induced dynamics. Single aromatic hydrocarbon molecules, being sensitive to nanoscale probing at nanometer scales, are potential miniaturized platforms for integrated quantum photonics. In this Perspective, we look back at some of the past advances in cryogenic optical microscopy and propose some perspectives for future development.
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Affiliation(s)
| | - Robert Smit
- Huygens−Kamerlingh
Onnes Laboratory, Leiden University, 2300 RA Leiden, The Netherlands
| | - Michel Orrit
- Huygens−Kamerlingh
Onnes Laboratory, Leiden University, 2300 RA Leiden, The Netherlands
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2
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Sharifi Z, Dobinson M, Hajisalem G, Shariatdoust MS, Frencken AL, van Veggel FCJM, Gordon R. Isolating and enhancing single-photon emitters for 1550 nm quantum light sources using double nanohole optical tweezers. J Chem Phys 2021; 154:184204. [PMID: 34241038 DOI: 10.1063/5.0048728] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Single-photon sources are required for quantum technologies and can be created from individual atoms and atom-like defects. Erbium ions produce single photons at low-loss fiber optic wavelengths, but they have low emission rates, making them challenging to isolate reliably. Here, we tune the size of gold double nanoholes (DNHs) to enhance the emission of single erbium emitters, achieving 50× enhancement over rectangular apertures previously demonstrated. This produces enough enhancement to show emission from single nanocrystals at wavelengths not seen in our previous work, i.e., 400 and 1550 nm. We observe discrete levels of emission for nanocrystals with low numbers of emitters and demonstrate isolating single emitters. We describe how the trapping time is proportional to the enhancement factor for a given DNH structure, giving us an independent way to measure the enhancement. This shows a promising path to achieving single emitter sources at 1550 nm.
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Affiliation(s)
- Zohreh Sharifi
- Department of Electrical and Computer Engineering, University of Victoria, Victoria, British Columbia V8P 5C2, Canada
| | - Michael Dobinson
- Department of Electrical and Computer Engineering, University of Victoria, Victoria, British Columbia V8P 5C2, Canada
| | - Ghazal Hajisalem
- Department of Electrical and Computer Engineering, University of Victoria, Victoria, British Columbia V8P 5C2, Canada
| | - Mirali Seyed Shariatdoust
- Department of Electrical and Computer Engineering, University of Victoria, Victoria, British Columbia V8P 5C2, Canada
| | - Adriaan L Frencken
- Centre for Advanced Materials & Related Technologies (CAMTEC), University of Victoria, Victoria, British Columbia V8W 2Y2, Canada
| | - Frank C J M van Veggel
- Centre for Advanced Materials & Related Technologies (CAMTEC), University of Victoria, Victoria, British Columbia V8W 2Y2, Canada
| | - Reuven Gordon
- Department of Electrical and Computer Engineering, University of Victoria, Victoria, British Columbia V8P 5C2, Canada
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3
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Kagan CR, Bassett LC, Murray CB, Thompson SM. Colloidal Quantum Dots as Platforms for Quantum Information Science. Chem Rev 2020; 121:3186-3233. [DOI: 10.1021/acs.chemrev.0c00831] [Citation(s) in RCA: 49] [Impact Index Per Article: 9.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/21/2023]
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Alizadehkhaledi A, Frencken AL, van Veggel FCJM, Gordon R. Isolating Nanocrystals with an Individual Erbium Emitter: A Route to a Stable Single-Photon Source at 1550 nm Wavelength. NANO LETTERS 2020; 20:1018-1022. [PMID: 31891509 DOI: 10.1021/acs.nanolett.9b04165] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
Single-photon emitters based on individual atoms or individual atomic-like defects are highly sought-after components for future quantum technologies. A key challenge in this field is how to isolate just one such emitter; the best approaches still have an active emitter yield of only 50% so that deterministic integration of single active emitters is not yet possible. Here, we demonstrate the ability to isolate individual erbium emitters embedded in 20 nm nanocrystals of NaYF4 using plasmonic aperture optical tweezers. The optical tweezers capture the nanocrystal, whereas the plasmonic aperture enhances the emission of the Er and allows the measurement of discrete emission rate values corresponding to different numbers of erbium ions. Three separate synthesis runs show near-Poissonian distribution in the discrete levels of emission yield that correspond to the expected ion concentrations, indicating that the yield of active emitters is approximately 80%. Fortunately, the trap allows for selecting the nanocrystals with only a single emitter, and so this gives a route to isolating and integrating single emitters in a deterministic way. This demonstration is a promising step toward single-photon quantum information technologies that utilize single ions in a solid-state medium, particularly because Er emits in the low-loss fiber-optic 1550 nm telecom band.
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Affiliation(s)
- Amirhossein Alizadehkhaledi
- Department Electrical and Computer Engineering , University of Victoria , Victoria , British Columbia V8W 2Y2 , Canada
- Centre for Advanced Materials & Related Technologies (CAMTEC) , University of Victoria , Victoria , British Columbia V8W 2Y2 , Canada
| | - Adriaan L Frencken
- Department of Chemistry , University of Victoria , Victoria , British Colombia V8W 2Y2 , Canada
- Centre for Advanced Materials & Related Technologies (CAMTEC) , University of Victoria , Victoria , British Columbia V8W 2Y2 , Canada
| | - Frank C J M van Veggel
- Department of Chemistry , University of Victoria , Victoria , British Colombia V8W 2Y2 , Canada
- Centre for Advanced Materials & Related Technologies (CAMTEC) , University of Victoria , Victoria , British Columbia V8W 2Y2 , Canada
| | - Reuven Gordon
- Department Electrical and Computer Engineering , University of Victoria , Victoria , British Columbia V8W 2Y2 , Canada
- Centre for Advanced Materials & Related Technologies (CAMTEC) , University of Victoria , Victoria , British Columbia V8W 2Y2 , Canada
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5
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Phenicie CM, Stevenson P, Welinski S, Rose BC, Asfaw AT, Cava RJ, Lyon SA, de Leon NP, Thompson JD. Narrow Optical Line Widths in Erbium Implanted in TiO 2. NANO LETTERS 2019; 19:8928-8933. [PMID: 31765161 DOI: 10.1021/acs.nanolett.9b03831] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
Atomic and atomlike defects in the solid state are widely explored for quantum computers, networks, and sensors. Rare earth ions are an attractive class of atomic defects that feature narrow spin and optical transitions that are isolated from the host crystal, allowing incorporation into a wide range of materials. However, the realization of long electronic spin coherence times is hampered by magnetic noise from abundant nuclear spins in the most widely studied host crystals. Here, we demonstrate that Er3+ ions can be introduced via ion implantation into TiO2, a host crystal that has not been studied extensively for rare earth ions and has a low natural abundance of nuclear spins. We observe efficient incorporation of the implanted Er3+ into the Ti4+ site (>50% yield) and measure narrow inhomogeneous spin and optical line widths (20 and 460 MHz, respectively) that are comparable to bulk-doped crystalline hosts for Er3+. This work demonstrates that ion implantation is a viable path to studying rare earth ions in new hosts and is a significant step toward realizing individually addressed rare earth ions with long spin coherence times for quantum technologies.
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Huang TY, Grote RR, Mann SA, Hopper DA, Exarhos AL, Lopez GG, Kaighn GR, Garnett EC, Bassett LC. A monolithic immersion metalens for imaging solid-state quantum emitters. Nat Commun 2019; 10:2392. [PMID: 31160564 PMCID: PMC6546684 DOI: 10.1038/s41467-019-10238-5] [Citation(s) in RCA: 33] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/16/2018] [Accepted: 04/15/2019] [Indexed: 11/09/2022] Open
Abstract
Quantum emitters such as the diamond nitrogen-vacancy (NV) center are the basis for a wide range of quantum technologies. However, refraction and reflections at material interfaces impede photon collection, and the emitters’ atomic scale necessitates the use of free space optical measurement setups that prevent packaging of quantum devices. To overcome these limitations, we design and fabricate a metasurface composed of nanoscale diamond pillars that acts as an immersion lens to collect and collimate the emission of an individual NV center. The metalens exhibits a numerical aperture greater than 1.0, enabling efficient fiber-coupling of quantum emitters. This flexible design will lead to the miniaturization of quantum devices in a wide range of host materials and the development of metasurfaces that shape single-photon emission for coupling to optical cavities or route photons based on their quantum state. Photon collection from quantum emitters is difficult, and their scale requires the use of free-space optical measurement setups which prevent packaging of quantum devices. Here, the authors design and fabricate a metasurface that acts as an immersion lens to collect and collimate the emission of an individual emitter.
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Affiliation(s)
- Tzu-Yung Huang
- Quantum Engineering Laboratory, Department of Electrical and Systems Engineering, University of Pennsylvania, 200 S. 33rd Street, Philadelphia, PA, 19104, USA
| | - Richard R Grote
- Quantum Engineering Laboratory, Department of Electrical and Systems Engineering, University of Pennsylvania, 200 S. 33rd Street, Philadelphia, PA, 19104, USA.,Rockley Photonics Inc., 234 E. Colorado Blvd, Suite 600, Pasadena, CA, 91101, USA
| | - Sander A Mann
- Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG, Amsterdam, The Netherlands.,Photonics Initiative, Advanced Science Research Center, City University of New York, New York, NY, 10031, USA
| | - David A Hopper
- Quantum Engineering Laboratory, Department of Electrical and Systems Engineering, University of Pennsylvania, 200 S. 33rd Street, Philadelphia, PA, 19104, USA.,Department of Physics and Astronomy, University of Pennsylvania, 209 S. 33rd Street, Philadelphia, PA, 19104, USA
| | - Annemarie L Exarhos
- Quantum Engineering Laboratory, Department of Electrical and Systems Engineering, University of Pennsylvania, 200 S. 33rd Street, Philadelphia, PA, 19104, USA.,Department of Physics, Lafayette College, Easton, PA, 18042, USA
| | - Gerald G Lopez
- Singh Center for Nanotechnology, University of Pennsylvania, 3205 Walnut St., Philadelphia, PA, 19104, USA
| | - Garrett R Kaighn
- Quantum Engineering Laboratory, Department of Electrical and Systems Engineering, University of Pennsylvania, 200 S. 33rd Street, Philadelphia, PA, 19104, USA
| | - Erik C Garnett
- Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG, Amsterdam, The Netherlands
| | - Lee C Bassett
- Quantum Engineering Laboratory, Department of Electrical and Systems Engineering, University of Pennsylvania, 200 S. 33rd Street, Philadelphia, PA, 19104, USA.
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7
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Zhong T, Kindem JM, Bartholomew JG, Rochman J, Craiciu I, Verma V, Nam SW, Marsili F, Shaw MD, Beyer AD, Faraon A. Optically Addressing Single Rare-Earth Ions in a Nanophotonic Cavity. PHYSICAL REVIEW LETTERS 2018; 121:183603. [PMID: 30444379 DOI: 10.1103/physrevlett.121.183603] [Citation(s) in RCA: 39] [Impact Index Per Article: 5.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/20/2018] [Indexed: 05/26/2023]
Abstract
We demonstrate optical probing of spectrally resolved single Nd^{3+} rare-earth ions in yttrium orthovanadate. The ions are coupled to a photonic crystal resonator and show strong enhancement of the optical emission rate via the Purcell effect, resulting in near radiatively limited single photon emission. The measured high coupling cooperativity between a single photon and the ion allows for the observation of coherent optical Rabi oscillations. This could enable optically controlled spin qubits, quantum logic gates, and spin-photon interfaces for future quantum networks.
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Affiliation(s)
- Tian Zhong
- Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
- Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
- Institute for Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA
| | - Jonathan M Kindem
- Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
- Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
| | - John G Bartholomew
- Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
- Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
| | - Jake Rochman
- Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
- Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
| | - Ioana Craiciu
- Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
- Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
| | - Varun Verma
- National Institute of Standards and Technology, 325 Broadway, MC 815.04, Boulder, Colorado 80305, USA
| | - Sae Woo Nam
- National Institute of Standards and Technology, 325 Broadway, MC 815.04, Boulder, Colorado 80305, USA
| | - Francesco Marsili
- Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California 91109, USA
| | - Matthew D Shaw
- Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California 91109, USA
| | - Andrew D Beyer
- Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California 91109, USA
| | - Andrei Faraon
- Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
- Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
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8
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Kovalyuk V, Ferrari S, Kahl O, Semenov A, Shcherbatenko M, Lobanov Y, Ozhegov R, Korneev A, Kaurova N, Voronov B, Pernice W, Gol'tsman G. On-chip coherent detection with quantum limited sensitivity. Sci Rep 2017; 7:4812. [PMID: 28684752 PMCID: PMC5500578 DOI: 10.1038/s41598-017-05142-1] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/20/2016] [Accepted: 05/23/2017] [Indexed: 12/02/2022] Open
Abstract
While single photon detectors provide superior intensity sensitivity, spectral resolution is usually lost after the detection event. Yet for applications in low signal infrared spectroscopy recovering information about the photon's frequency contributions is essential. Here we use highly efficient waveguide integrated superconducting single-photon detectors for on-chip coherent detection. In a single nanophotonic device, we demonstrate both single-photon counting with up to 86% on-chip detection efficiency, as well as heterodyne coherent detection with spectral resolution f/∆f exceeding 1011. By mixing a local oscillator with the single photon signal field, we observe frequency modulation at the intermediate frequency with ultra-low local oscillator power in the femto-Watt range. By optimizing the nanowire geometry and the working parameters of the detection scheme, we reach quantum-limited sensitivity. Our approach enables to realize matrix integrated heterodyne nanophotonic devices in the C-band wavelength range, for classical and quantum optics applications where single-photon counting as well as high spectral resolution are required simultaneously.
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Affiliation(s)
- Vadim Kovalyuk
- Department of Physics, Moscow State Pedagogical University, Moscow, 119992, Russia
- Institute of Nanotechnology, Karlsruhe Institute of Technology, Karlsruhe, 76132, Germany
| | - Simone Ferrari
- Institute of Nanotechnology, Karlsruhe Institute of Technology, Karlsruhe, 76132, Germany
- Institute of Physics, University of Münster, Münster, 48149, Germany
| | - Oliver Kahl
- Institute of Nanotechnology, Karlsruhe Institute of Technology, Karlsruhe, 76132, Germany
- Institute of Physics, University of Münster, Münster, 48149, Germany
| | - Alexander Semenov
- Department of Physics, Moscow State Pedagogical University, Moscow, 119992, Russia
- Moscow Institute of Physics and Technology, Moscow, 141700, Russia
| | - Michael Shcherbatenko
- Department of Physics, Moscow State Pedagogical University, Moscow, 119992, Russia
- Moscow Institute of Physics and Technology, Moscow, 141700, Russia
| | - Yury Lobanov
- Department of Physics, Moscow State Pedagogical University, Moscow, 119992, Russia
- Moscow Institute of Physics and Technology, Moscow, 141700, Russia
| | - Roman Ozhegov
- Department of Physics, Moscow State Pedagogical University, Moscow, 119992, Russia
| | - Alexander Korneev
- Department of Physics, Moscow State Pedagogical University, Moscow, 119992, Russia
- Moscow Institute of Physics and Technology, Moscow, 141700, Russia
| | - Nataliya Kaurova
- Department of Physics, Moscow State Pedagogical University, Moscow, 119992, Russia
| | - Boris Voronov
- Department of Physics, Moscow State Pedagogical University, Moscow, 119992, Russia
| | - Wolfram Pernice
- Institute of Physics, University of Münster, Münster, 48149, Germany
| | - Gregory Gol'tsman
- Department of Physics, Moscow State Pedagogical University, Moscow, 119992, Russia.
- National Research University Higher School of Economics, Moscow, 101000, Russia.
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9
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Quantum decoherence dynamics of divacancy spins in silicon carbide. Nat Commun 2016; 7:12935. [PMID: 27679936 PMCID: PMC5056425 DOI: 10.1038/ncomms12935] [Citation(s) in RCA: 40] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/27/2016] [Accepted: 08/17/2016] [Indexed: 11/08/2022] Open
Abstract
Long coherence times are key to the performance of quantum bits (qubits). Here, we experimentally and theoretically show that the Hahn-echo coherence time of electron spins associated with divacancy defects in 4H-SiC reaches 1.3 ms, one of the longest Hahn-echo coherence times of an electron spin in a naturally isotopic crystal. Using a first-principles microscopic quantum-bath model, we find that two factors determine the unusually robust coherence. First, in the presence of moderate magnetic fields (30 mT and above), the 29Si and 13C paramagnetic nuclear spin baths are decoupled. In addition, because SiC is a binary crystal, homo-nuclear spin pairs are both diluted and forbidden from forming strongly coupled, nearest-neighbour spin pairs. Longer neighbour distances result in fewer nuclear spin flip-flops, a less fluctuating intra-crystalline magnetic environment, and thus a longer coherence time. Our results point to polyatomic crystals as promising hosts for coherent qubits in the solid state.
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Laplane C, Zambrini Cruzeiro E, Fröwis F, Goldner P, Afzelius M. High-Precision Measurement of the Dzyaloshinsky-Moriya Interaction between Two Rare-Earth Ions in a Solid. PHYSICAL REVIEW LETTERS 2016; 117:037203. [PMID: 27472133 DOI: 10.1103/physrevlett.117.037203] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/01/2016] [Indexed: 06/06/2023]
Abstract
We report on a direct measurement of the pairwise antisymmetric exchange interaction, known as the Dzyaloshinsky-Moriya interaction (DMI), in a Nd^{3+}-doped YVO_{4} crystal. To this end, we introduce a broadband electron spin resonance technique coupled with an optical detection scheme which selectively detects only one Nd^{3+}-Nd^{3+} pair. Using this technique we can fully measure the spin-spin coupling tensor, allowing us to experimentally determine both the strength and direction of the DMI vector. We believe that this ability to fully determine the interaction Hamiltonian is of interest for studying the numerous magnetic phenomena where the DMI interaction is of fundamental importance, including multiferroics. We also detect a singlet-triplet transition within the pair, with a highly suppressed magnetic-field dependence, which suggests that such systems could form singlet-triplet qubits with long coherence times for quantum information applications.
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Affiliation(s)
- Cyril Laplane
- Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland
| | | | - Florian Fröwis
- Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland
| | - Philippe Goldner
- PSL Research University, Chimie ParisTech, CNRS, Institut de Recherche de Chimie Paris, 75005 Paris, France
| | - Mikael Afzelius
- Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland
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Jamonneau P, Hétet G, Dréau A, Roch JF, Jacques V. Coherent Population Trapping of a Single Nuclear Spin Under Ambient Conditions. PHYSICAL REVIEW LETTERS 2016; 116:043603. [PMID: 26871331 DOI: 10.1103/physrevlett.116.043603] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/28/2015] [Indexed: 06/05/2023]
Abstract
We demonstrate coherent population trapping of a single nuclear spin in a room-temperature solid. To this end, we exploit a three-level system with a Λ configuration in the microwave domain, which consists of nuclear spin states addressed through their hyperfine coupling to the electron spin of a single nitrogen-vacancy defect in diamond. Moreover, the Λ-scheme relaxation is externally controlled through incoherent optical pumping and separated in time from consecutive coherent microwave excitations. Such a scheme allows us (i) to monitor the sequential accumulation of population into the dark state and (ii) to reach a novel regime of coherent population trapping dynamics for which periodic arrays of dark resonances can be observed, owing to multiple constructive interferences. This Letter offers new prospects for quantum state preparation, information storage in hybrid quantum systems, and metrology.
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Affiliation(s)
- P Jamonneau
- Laboratoire Aimé Cotton, CNRS, Université Paris-Sud, ENS Cachan, Université Paris-Saclay91405 Orsay Cedex, France
| | - G Hétet
- Laboratoire Aimé Cotton, CNRS, Université Paris-Sud, ENS Cachan, Université Paris-Saclay91405 Orsay Cedex, France
- Laboratoire Pierre Aigrain, CNRS, Université Pierre et Marie Curie, Université Paris Diderot and Ecole Normale Supérieure, 75005 Paris, France
| | - A Dréau
- Laboratoire Aimé Cotton, CNRS, Université Paris-Sud, ENS Cachan, Université Paris-Saclay91405 Orsay Cedex, France
| | - J-F Roch
- Laboratoire Aimé Cotton, CNRS, Université Paris-Sud, ENS Cachan, Université Paris-Saclay91405 Orsay Cedex, France
| | - V Jacques
- Laboratoire Aimé Cotton, CNRS, Université Paris-Sud, ENS Cachan, Université Paris-Saclay91405 Orsay Cedex, France
- Laboratoire Charles Coulomb, Université de Montpellier and CNRS, 34095 Montpellier, France
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