1
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van de Stolpe GL, Kwiatkowski DP, Bradley CE, Randall J, Abobeih MH, Breitweiser SA, Bassett LC, Markham M, Twitchen DJ, Taminiau TH. Mapping a 50-spin-qubit network through correlated sensing. Nat Commun 2024; 15:2006. [PMID: 38443361 PMCID: PMC10914733 DOI: 10.1038/s41467-024-46075-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/09/2023] [Accepted: 02/09/2024] [Indexed: 03/07/2024] Open
Abstract
Spins associated to optically accessible solid-state defects have emerged as a versatile platform for exploring quantum simulation, quantum sensing and quantum communication. Pioneering experiments have shown the sensing, imaging, and control of multiple nuclear spins surrounding a single electron spin defect. However, the accessible size of these spin networks has been constrained by the spectral resolution of current methods. Here, we map a network of 50 coupled spins through high-resolution correlated sensing schemes, using a single nitrogen-vacancy center in diamond. We develop concatenated double-resonance sequences that identify spin-chains through the network. These chains reveal the characteristic spin frequencies and their interconnections with high spectral resolution, and can be fused together to map out the network. Our results provide new opportunities for quantum simulations by increasing the number of available spin qubits. Additionally, our methods might find applications in nano-scale imaging of complex spin systems external to the host crystal.
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Affiliation(s)
- G L van de Stolpe
- QuTech, Delft University of Technology, PO Box 5046, 2600, GA Delft, The Netherlands
- Kavli Institute of Nanoscience Delft, Delft University of Technology, PO Box 5046, 2600, GA Delft, The Netherlands
| | - D P Kwiatkowski
- QuTech, Delft University of Technology, PO Box 5046, 2600, GA Delft, The Netherlands
- Kavli Institute of Nanoscience Delft, Delft University of Technology, PO Box 5046, 2600, GA Delft, The Netherlands
| | - C E Bradley
- QuTech, Delft University of Technology, PO Box 5046, 2600, GA Delft, The Netherlands
- Kavli Institute of Nanoscience Delft, Delft University of Technology, PO Box 5046, 2600, GA Delft, The Netherlands
| | - J Randall
- QuTech, Delft University of Technology, PO Box 5046, 2600, GA Delft, The Netherlands
- Kavli Institute of Nanoscience Delft, Delft University of Technology, PO Box 5046, 2600, GA Delft, The Netherlands
| | - M H Abobeih
- QuTech, Delft University of Technology, PO Box 5046, 2600, GA Delft, The Netherlands
- Kavli Institute of Nanoscience Delft, Delft University of Technology, PO Box 5046, 2600, GA Delft, The Netherlands
| | - S A Breitweiser
- Quantum Engineering Laboratory, Department of Electrical and Systems Engineering, University of Pennsylvania, 200 South 33rd Street, Philadelphia, PA, 19104, USA
| | - L C Bassett
- Quantum Engineering Laboratory, Department of Electrical and Systems Engineering, University of Pennsylvania, 200 South 33rd Street, Philadelphia, PA, 19104, USA
| | - M Markham
- Element Six Innovation, Fermi Avenue, Harwell Oxford, Didcot, Oxfordshire, OX11 0QR, UK
| | - D J Twitchen
- Element Six Innovation, Fermi Avenue, Harwell Oxford, Didcot, Oxfordshire, OX11 0QR, UK
| | - T H Taminiau
- QuTech, Delft University of Technology, PO Box 5046, 2600, GA Delft, The Netherlands.
- Kavli Institute of Nanoscience Delft, Delft University of Technology, PO Box 5046, 2600, GA Delft, The Netherlands.
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2
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Mayländer M, Thielert P, Quintes T, Vargas Jentzsch A, Richert S. Room Temperature Electron Spin Coherence in Photogenerated Molecular Spin Qubit Candidates. J Am Chem Soc 2023. [PMID: 37337625 DOI: 10.1021/jacs.3c04021] [Citation(s) in RCA: 5] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/21/2023]
Abstract
One of the main challenges in the emerging field of molecular spintronics is the identification of new spin qubit materials for quantum information applications. In this regard, recent work has shown that photoexcited chromophore-radical systems are promising candidates to expand our repertoire of suitable candidate molecules. Here, we investigate a series of three chromophore-radical compounds composed of a perylene diimide (PDI) chromophore and a stable 2,2,6,6-tetramethylpiperidin-1-oxyl (TEMPO) radical by transient electron paramagnetic resonance (EPR) techniques. We explore the influence of isotope labeling of the TEMPO moiety on the EPR spectra and electron spin coherence times of the molecular quartet states generated upon photoexcitation and illustrate that (i) a coherent manipulation of the spin state is possible in these systems even at room temperature and that (ii) a spin coherence time of 0.7 μs can be achieved under these conditions. This demonstration of electron spin coherence at ambient temperatures paves the way for practical applications of such systems in functional molecular devices.
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Affiliation(s)
- Maximilian Mayländer
- Institute of Physical Chemistry, University of Freiburg, Albertstraße 21, 79104 Freiburg, Germany
| | - Philipp Thielert
- Institute of Physical Chemistry, University of Freiburg, Albertstraße 21, 79104 Freiburg, Germany
| | - Theresia Quintes
- Institute of Physical Chemistry, University of Freiburg, Albertstraße 21, 79104 Freiburg, Germany
| | - Andreas Vargas Jentzsch
- SAMS Research Group, Université de Strasbourg, CNRS, Institut Charles Sadron UPR 22, 67000 Strasbourg, France
| | - Sabine Richert
- Institute of Physical Chemistry, University of Freiburg, Albertstraße 21, 79104 Freiburg, Germany
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3
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Taran G, Bonet E, Moreno-Pineda E, Ruben M, Wernsdorfer W. Thermalization of Nuclear Spins in Lanthanide Molecular Magnets. Inorg Chem 2023. [PMID: 37220076 DOI: 10.1021/acs.inorgchem.3c00530] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 05/25/2023]
Abstract
Single-molecule magnets (SMMs) distinguish themselves in the field of quantum magnetism through the ability to combine fundamental research with promising applications. The evolution of quantum spintronics in the last decade exemplifies the potential held by molecular-based quantum devices. Notably, the readout and manipulation of the nuclear spin states embedded in a lanthanide-based SMM hybrid device were employed in proof of principle studies of quantum computation at the single-molecule level. In the quest for further understanding of the relaxation behavior in SMMs for their integration in novel applications, herein, we study the relaxation dynamics of the 159Tb nuclear spins in a diluted molecular crystal employing the recently acquired understanding of the nonadiabatic dynamics of TbPc2 molecules. Through numerical simulation, we find that phonon-modulated hyperfine interaction opens a direct relaxation channel between the nuclear spins and the phonon bath. The mechanism is of potential importance for the theory of spin bath and the relaxation dynamics of the molecular spins.
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Affiliation(s)
- Gheorghe Taran
- Physikalisches Institut, Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany
| | - Edgar Bonet
- Néel Institute, CNRS, 25 rue des Martyrs, Grenoble 38042, France
| | - Eufemio Moreno-Pineda
- Depto. de Química-Física, Facultad de Ciencias Naturales, Exactas y Tecnología, Universidad de Panamá, Panamá 0824, Panamá
- Grupo de Investigación de Materiales, Facultad de Ciencias Naturales, Exactas y Tecnología, Universidad de Panamá, Panamá 0824, Panamá
| | - Mario Ruben
- Centre Européen de Sciences Quantiques (CESQ) within the Institut de Science et d'Ingénierie Supramoléculaires (ISIS), 8 allée Gaspard Monge, BP 70028, 67083 Strasbourg Cedex, France
- Institute of Nanotechnology (INT), Karlsruhe Institute of Technology (KIT), Hermann-von-Helmholtz-Plats 1, D-76344 Eggenstein-Leopoldshafen, Germany
- Institute for Quantum Materials and Technology (IQMT), Karlsruhe Institute of Technology (KIT), Hermann-von-Helmholtz-Platz 1, D-76344 Eggenstein-Leopoldshafen, Germany
| | - Wolfgang Wernsdorfer
- Physikalisches Institut, Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany
- Institute of Nanotechnology (INT), Karlsruhe Institute of Technology (KIT), Hermann-von-Helmholtz-Plats 1, D-76344 Eggenstein-Leopoldshafen, Germany
- Institute for Quantum Materials and Technology (IQMT), Karlsruhe Institute of Technology (KIT), Hermann-von-Helmholtz-Platz 1, D-76344 Eggenstein-Leopoldshafen, Germany
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4
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Davis EJ, Ye B, Machado F, Meynell SA, Wu W, Mittiga T, Schenken W, Joos M, Kobrin B, Lyu Y, Wang Z, Bluvstein D, Choi S, Zu C, Jayich ACB, Yao NY. Probing many-body dynamics in a two-dimensional dipolar spin ensemble. NATURE PHYSICS 2023; 19:836-844. [PMID: 37323805 PMCID: PMC10264245 DOI: 10.1038/s41567-023-01944-5] [Citation(s) in RCA: 5] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 03/20/2021] [Accepted: 01/05/2023] [Indexed: 06/17/2023]
Abstract
The most direct approach for characterizing the quantum dynamics of a strongly interacting system is to measure the time evolution of its full many-body state. Despite the conceptual simplicity of this approach, it quickly becomes intractable as the system size grows. An alternate approach is to think of the many-body dynamics as generating noise, which can be measured by the decoherence of a probe qubit. Here we investigate what the decoherence dynamics of such a probe tells us about the many-body system. In particular, we utilize optically addressable probe spins to experimentally characterize both static and dynamical properties of strongly interacting magnetic dipoles. Our experimental platform consists of two types of spin defects in nitrogen delta-doped diamond: nitrogen-vacancy colour centres, which we use as probe spins, and a many-body ensemble of substitutional nitrogen impurities. We demonstrate that the many-body system's dimensionality, dynamics and disorder are naturally encoded in the probe spins' decoherence profile. Furthermore, we obtain direct control over the spectral properties of the many-body system, with potential applications in quantum sensing and simulation.
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Affiliation(s)
- E. J. Davis
- Department of Physics, University of California, Berkeley, CA USA
| | - B. Ye
- Department of Physics, University of California, Berkeley, CA USA
| | - F. Machado
- Department of Physics, University of California, Berkeley, CA USA
- Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA USA
| | - S. A. Meynell
- Department of Physics, University of California, Santa Barbara, CA USA
| | - W. Wu
- Department of Physics, University of California, Berkeley, CA USA
- Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA USA
| | - T. Mittiga
- Department of Physics, University of California, Berkeley, CA USA
- Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA USA
| | - W. Schenken
- Department of Physics, University of California, Santa Barbara, CA USA
| | - M. Joos
- Department of Physics, University of California, Santa Barbara, CA USA
| | - B. Kobrin
- Department of Physics, University of California, Berkeley, CA USA
- Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA USA
| | - Y. Lyu
- Department of Physics, University of California, Berkeley, CA USA
| | - Z. Wang
- Department of Physics, University of California, Berkeley, CA USA
- Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA USA
| | - D. Bluvstein
- Department of Physics, Harvard University, Cambridge, MA USA
| | - S. Choi
- Department of Physics, University of California, Berkeley, CA USA
| | - C. Zu
- Department of Physics, University of California, Berkeley, CA USA
- Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA USA
- Department of Physics, Washington University, St. Louis, MO USA
| | | | - N. Y. Yao
- Department of Physics, University of California, Berkeley, CA USA
- Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA USA
- Department of Physics, Harvard University, Cambridge, MA USA
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5
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Shen Y, Wang P, Cheung CT, Wrachtrup J, Liu RB, Yang S. Detection of Quantum Signals Free of Classical Noise via Quantum Correlation. PHYSICAL REVIEW LETTERS 2023; 130:070802. [PMID: 36867814 DOI: 10.1103/physrevlett.130.070802] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2022] [Revised: 12/08/2022] [Accepted: 01/09/2023] [Indexed: 06/18/2023]
Abstract
Extracting useful signals is key to both classical and quantum technologies. Conventional noise filtering methods rely on different patterns of signal and noise in frequency or time domains, thus limiting their scope of application, especially in quantum sensing. Here, we propose a signal-nature-based (not signal-pattern-based) approach which singles out a quantum signal from its classical noise background by employing the intrinsic quantum nature of the system. We design a novel protocol to extract the quantum correlation signal and use it to single out the signal of a remote nuclear spin from its overwhelming classical noise backgrounds, which is impossible to be accomplished by conventional filter methods. Our Letter demonstrates the quantum or classical nature as a new degree of freedom in quantum sensing. The further generalization of this quantum nature-based method opens a new direction in quantum research.
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Affiliation(s)
- Yang Shen
- Department of Physics and the IAS Center for Quantum Technologies, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
| | - Ping Wang
- College of Education for the future, Beijing Normal University, Zhuhai 519087, China
- Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China
- Centre for Quantum Coherence and The Hong Kong Institute of Quantum Information Science and Technology, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China
| | - Chun Tung Cheung
- Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China
| | - Jörg Wrachtrup
- 3. Physikalisches Institut, Integrated Quantum Science and Technology (IQST), University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
- Max Planck Institute for Solid State Research, Stuttgart, Germany
| | - Ren-Bao Liu
- Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China
- Centre for Quantum Coherence and The Hong Kong Institute of Quantum Information Science and Technology, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China
| | - Sen Yang
- Department of Physics and the IAS Center for Quantum Technologies, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
- Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China
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6
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Savytskyy R, Botzem T, Fernandez de Fuentes I, Joecker B, Pla JJ, Hudson FE, Itoh KM, Jakob AM, Johnson BC, Jamieson DN, Dzurak AS, Morello A. An electrically driven single-atom "flip-flop" qubit. SCIENCE ADVANCES 2023; 9:eadd9408. [PMID: 36763660 PMCID: PMC9916988 DOI: 10.1126/sciadv.add9408] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 07/13/2022] [Accepted: 01/09/2023] [Indexed: 06/18/2023]
Abstract
The spins of atoms and atom-like systems are among the most coherent objects in which to store quantum information. However, the need to address them using oscillating magnetic fields hinders their integration with quantum electronic devices. Here, we circumvent this hurdle by operating a single-atom "flip-flop" qubit in silicon, where quantum information is encoded in the electron-nuclear states of a phosphorus donor. The qubit is controlled using local electric fields at microwave frequencies, produced within a metal-oxide-semiconductor device. The electrical drive is mediated by the modulation of the electron-nuclear hyperfine coupling, a method that can be extended to many other atomic and molecular systems and to the hyperpolarization of nuclear spin ensembles. These results pave the way to the construction of solid-state quantum processors where dense arrays of atoms can be controlled using only local electric fields.
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Affiliation(s)
- Rostyslav Savytskyy
- School of Electrical Engineering and Telecommunications, UNSW Sydney, Sydney, NSW 2052, Australia
| | - Tim Botzem
- School of Electrical Engineering and Telecommunications, UNSW Sydney, Sydney, NSW 2052, Australia
| | | | - Benjamin Joecker
- School of Electrical Engineering and Telecommunications, UNSW Sydney, Sydney, NSW 2052, Australia
| | - Jarryd J. Pla
- School of Electrical Engineering and Telecommunications, UNSW Sydney, Sydney, NSW 2052, Australia
| | - Fay E. Hudson
- School of Electrical Engineering and Telecommunications, UNSW Sydney, Sydney, NSW 2052, Australia
| | - Kohei M. Itoh
- School of Fundamental Science and Technology, Keio University, Kohoku-ku, Yokohama, Japan
| | - Alexander M. Jakob
- School of Physics, University of Melbourne, Melbourne, VIC 3010, Australia
| | - Brett C. Johnson
- School of Physics, University of Melbourne, Melbourne, VIC 3010, Australia
| | - David N. Jamieson
- School of Physics, University of Melbourne, Melbourne, VIC 3010, Australia
| | - Andrew S. Dzurak
- School of Electrical Engineering and Telecommunications, UNSW Sydney, Sydney, NSW 2052, Australia
| | - Andrea Morello
- School of Electrical Engineering and Telecommunications, UNSW Sydney, Sydney, NSW 2052, Australia
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7
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Adam D, Bouton Q, Nettersheim J, Burgardt S, Widera A. Coherent and Dephasing Spectroscopy for Single-Impurity Probing of an Ultracold Bath. PHYSICAL REVIEW LETTERS 2022; 129:120404. [PMID: 36179201 DOI: 10.1103/physrevlett.129.120404] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/10/2021] [Accepted: 08/24/2022] [Indexed: 06/16/2023]
Abstract
We report Ramsey spectroscopy on the clock states of individual Cs impurities immersed in an ultracold Rb bath. We record both the interaction-driven phase evolution and the decay of fringe contrast of the Ramsey interference signal to obtain information about bath density or temperature nondestructively. The Ramsey fringe is modified by a differential shift of the collisional energy when the two Cs states superposed interact with the Rb bath. This differential shift is directly affected by the mean gas density and the details of the Rb-Cs interspecies scattering length, affecting the phase evolution and the contrast of the Ramsey signal. Additionally, we enhance the temperature dependence of the phase shift preparing the system close to a low-magnetic-field Feshbach resonance where the s-wave scattering length is significantly affected by the collisional (kinetic) energy. Analyzing coherent phase evolution and decay of the Ramsey fringe contrast, we probe the Rb cloud's density and temperature. Our results point at using individual impurity atoms as nondestructive quantum probes in complex quantum systems.
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Affiliation(s)
- Daniel Adam
- Department of Physics and Research Center OPTIMAS, Technische Universität Kaiserslautern, Kaiserslautern 67663, Germany
| | - Quentin Bouton
- Department of Physics and Research Center OPTIMAS, Technische Universität Kaiserslautern, Kaiserslautern 67663, Germany
- Laboratoire de Physique des Lasers, CNRS, UMR 7538, Université Sorbonne Paris Nord, F-93430 Villetaneuse, France
| | - Jens Nettersheim
- Department of Physics and Research Center OPTIMAS, Technische Universität Kaiserslautern, Kaiserslautern 67663, Germany
| | - Sabrina Burgardt
- Department of Physics and Research Center OPTIMAS, Technische Universität Kaiserslautern, Kaiserslautern 67663, Germany
| | - Artur Widera
- Department of Physics and Research Center OPTIMAS, Technische Universität Kaiserslautern, Kaiserslautern 67663, Germany
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8
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Quantum nonlinear spectroscopy of single nuclear spins. Nat Commun 2022; 13:5318. [PMID: 36085280 PMCID: PMC9463177 DOI: 10.1038/s41467-022-32610-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/18/2021] [Accepted: 08/04/2022] [Indexed: 11/30/2022] Open
Abstract
Conventional nonlinear spectroscopy, which use classical probes, can only access a limited set of correlations in a quantum system. Here we demonstrate that quantum nonlinear spectroscopy, in which a quantum sensor and a quantum object are first entangled and the sensor is measured along a chosen basis, can extract arbitrary types and orders of correlations in a quantum system. We measured fourth-order correlations of single nuclear spins that cannot be measured in conventional nonlinear spectroscopy, using sequential weak measurement via a nitrogen-vacancy center in diamond. The quantum nonlinear spectroscopy provides fingerprint features to identify different types of objects, such as Gaussian noises, random-phased AC fields, and quantum spins, which would be indistinguishable in second-order correlations. This work constitutes an initial step toward the application of higher-order correlations to quantum sensing, to examining the quantum foundation (by, e.g., higher-order Leggett-Garg inequality), and to studying quantum many-body physics. Signals that look the same from their low-order correlations can often be distinguished by looking at higher-order ones. Here, the authors exploit the sensitivity of quantum nonlinear spectroscopy to fourth-order correlations to identify Gaussian noises, random-phased AC fields, and quantum spins.
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9
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Libersky M, McKenzie RD, Silevitch DM, Stamp PCE, Rosenbaum TF. Direct Observation of Collective Electronuclear Modes about a Quantum Critical Point. PHYSICAL REVIEW LETTERS 2021; 127:207202. [PMID: 34860035 DOI: 10.1103/physrevlett.127.207202] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/12/2021] [Revised: 07/12/2021] [Accepted: 10/13/2021] [Indexed: 06/13/2023]
Abstract
We directly measure the low energy excitation modes of the quantum Ising magnet LiHoF_{4} using microwave spectroscopy. Instead of a single electronic mode, we find a set of collective electronuclear modes, in which the spin-1/2 Ising electronic spins hybridize with the bath of spin-7/2 Ho nuclear spins. The lowest-lying electronuclear mode softens at the approach to the quantum critical point, even in the presence of disorder. This softening is rapidly quenched by a longitudinal magnetic field. Similar electronuclear structures should exist in other spin-based quantum Ising systems.
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Affiliation(s)
- M Libersky
- Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, California 91125, USA
| | - R D McKenzie
- Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada
| | - D M Silevitch
- Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, California 91125, USA
| | - P C E Stamp
- Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada
- Pacific Institute of Theoretical Physics, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada
| | - T F Rosenbaum
- Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, California 91125, USA
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10
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Intrinsic and induced quantum quenches for enhancing qubit-based quantum noise spectroscopy. Nat Commun 2021; 12:6528. [PMID: 34764276 PMCID: PMC8586144 DOI: 10.1038/s41467-021-26868-7] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/13/2021] [Accepted: 10/26/2021] [Indexed: 11/18/2022] Open
Abstract
Quantum sensing protocols that exploit the dephasing of a probe qubit are powerful and ubiquitous methods for interrogating an unknown environment. They have a variety of applications, ranging from noise mitigation in quantum processors, to the study of correlated electron states. Here, we discuss a simple strategy for enhancing these methods, based on the fact that they often give rise to an inadvertent quench of the probed system: there is an effective sudden change in the environmental Hamiltonian at the start of the sensing protocol. These quenches are extremely sensitive to the initial environmental state, and lead to observable changes in the sensor qubit evolution. We show how these new features give access to environmental response properties. This enables methods for direct measurement of bath temperature, and for detecting non-thermal equilibrium states. We also discuss how to deliberately control and modulate this quench physics, which enables reconstruction of the bath spectral function. Extensions to non-Gaussian quantum baths are also discussed, as is the application of our ideas to a range of sensing platforms (e.g., nitrogen-vacancy (NV) centers in diamond, semiconductor quantum dots, and superconducting circuits).
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11
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Onizhuk M, Galli G. PyCCE: A Python Package for Cluster Correlation Expansion Simulations of Spin Qubit Dynamics. ADVANCED THEORY AND SIMULATIONS 2021. [DOI: 10.1002/adts.202100254] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Affiliation(s)
- Mykyta Onizhuk
- Department of Chemistry University of Chicago Chicago IL 60637 USA
| | - Giulia Galli
- Department of Chemistry University of Chicago Chicago IL 60637 USA
- Pritzker School of Molecular Engineering University of Chicago Chicago IL 60637 USA
- Materials Science Division and Center for Molecular Engineering Argonne National Laboratory Lemont IL 60439 USA
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12
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Huangfu Y, Jing J. High-capacity and high-power collective charging with spin chargers. Phys Rev E 2021; 104:024129. [PMID: 34525586 DOI: 10.1103/physreve.104.024129] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/04/2021] [Accepted: 08/06/2021] [Indexed: 11/07/2022]
Abstract
Quantum battery works as a micro- or nanodevice to store and redistribute energy at the quantum level. Here we propose a spin-charger protocol, in which the battery cells are charged by a finite number of spins through a general Heisenberg XY interaction. Under the isotropic interaction, the spin-charger protocol is endowed with a higher capacity in terms of the maximum stored energy than the conventional protocols, where the battery is charged by a continuous-variable system, e.g., a cavity mode. By tuning the charger size, a tradeoff between the maximum stored energy and the average charging power is found in comparison to the cavity-charger protocol in the Tavis-Cummings model. Quantum advantage of our protocol is manifested by the scaling behavior of the optimal average power with respect to the battery size, in comparing the collective charging scheme to its parallel counterpart. We also discuss the detrimental effect on the charging performance from the anisotropic interaction between the battery and the charger, the nonideal initial states for both of them, and the crosstalk among the charger spins. A strong charger-charger interaction can be used to decouple the battery and the charger. Our findings about the advantages of the spin-charger protocol over the conventional cavity-charger protocols, including the high capacity of energy storage and the superior power law in the collective charging, provide an insight to exploit an efficient quantum battery based on the spin-spin-environment model.
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Affiliation(s)
- Yong Huangfu
- Department of Physics, Zhejiang University, Hangzhou 310027 Zhejiang, China
| | - Jun Jing
- Department of Physics, Zhejiang University, Hangzhou 310027 Zhejiang, China
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13
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Coherent control of a donor-molecule electron spin qubit in silicon. Nat Commun 2021; 12:3323. [PMID: 34083543 PMCID: PMC8175606 DOI: 10.1038/s41467-021-23662-3] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/08/2019] [Accepted: 05/07/2021] [Indexed: 11/11/2022] Open
Abstract
Donor spins in silicon provide a promising material platform for large scale quantum computing. Excellent electron spin coherence times of \documentclass[12pt]{minimal}
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\begin{document}$${T}_{2}^{* }=268$$\end{document}T2*=268 μs with fidelities of 99.9% have been demonstrated for isolated phosphorus donors in isotopically pure 28Si, where donors are local-area-implanted in a nanoscale MOS device. Despite robust single qubit gates, realising two-qubit exchange gates using this technique is challenging due to the statistical nature of the dopant implant and placement process. In parallel a precision scanning probe lithography route has been developed to place single donors and donor molecules on one atomic plane of silicon with high accuracy aligned to heavily phosphorus doped silicon in-plane gates. Recent results using this technique have demonstrated a fast (0.8 ns) two-qubit gate with two P donor molecules placed 13 nm apart in natSi. In this paper we demonstrate a single qubit gate with coherent oscillations of the electron spin on a P donor molecule in natSi patterned by scanning tunneling microscope (STM) lithography. The electron spin exhibits excellent coherence properties, with a \documentclass[12pt]{minimal}
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\begin{document}$${T}_{2}$$\end{document}T2 decoherence time of 298 ± 30 μs, and \documentclass[12pt]{minimal}
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\begin{document}$${T}_{2}^{* }$$\end{document}T2* dephasing time of 295 ± 23 ns. Multi-donor molecules in Si provide a promising qubit platform, offering advantages over single donor qubits in terms of performance and fabrication. Here, the authors report a single qubit gate and long coherence times in a P donor molecule qubit in natural Si patterned by scanning tunneling microscope hydrogen lithography.
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14
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Shumilin AV, Smirnov DS. Nuclear Spin Dynamics, Noise, Squeezing, and Entanglement in Box Model. PHYSICAL REVIEW LETTERS 2021; 126:216804. [PMID: 34114866 DOI: 10.1103/physrevlett.126.216804] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/09/2020] [Revised: 03/24/2021] [Accepted: 04/27/2021] [Indexed: 06/12/2023]
Abstract
We obtain a compact analytical solution for the nonlinear equation for the nuclear spin dynamics in the central spin box model in the limit of many nuclear spins. The total nuclear spin component along the external magnetic field is conserved and the two perpendicular components precess or oscillate depending on the electron spin polarization, with the frequency, determined by the nuclear spin polarization. As applications of our solution, we calculate the nuclear spin noise spectrum and describe the effects of nuclear spin squeezing and many body entanglement in the absence of a system excitation.
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Affiliation(s)
| | - D S Smirnov
- Ioffe Institute, 194021 St. Petersburg, Russia
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15
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Manshouri H, Hoseinpour A, Zarei M. Quantum Boltzmann equation for fermions: An attempt to calculate the NMR relaxation and decoherence times using quantum field theory techniques. Int J Clin Exp Med 2021. [DOI: 10.1103/physrevd.103.096020] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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16
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Wu W, Zhang ZZ. Controllable dynamics of a dissipative two-level system. Sci Rep 2021; 11:7188. [PMID: 33785788 PMCID: PMC8010078 DOI: 10.1038/s41598-021-86553-z] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/18/2021] [Accepted: 03/12/2021] [Indexed: 11/09/2022] Open
Abstract
We propose a strategy to modulate the decoherence dynamics of a two-level system, which interacts with a dissipative bosonic environment, by introducing an ancillary degree of freedom. It is revealed that the decay rate of the two-level system can be significantly suppressed under suitable steers of the assisted degree of freedom. Our result provides an alternative way to fight against decoherence and realize a controllable quantum dissipative dynamics.
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Affiliation(s)
- Wei Wu
- Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou, 730000, Gansu, China.
| | - Ze-Zhou Zhang
- Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou, 730000, Gansu, China
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17
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Soetbeer J, Millen M, Zouboulis K, Hülsmann M, Godt A, Polyhach Y, Jeschke G. Dynamical decoupling in water-glycerol glasses: a comparison of nitroxides, trityl radicals and gadolinium complexes. Phys Chem Chem Phys 2021; 23:5352-5369. [PMID: 33635938 DOI: 10.1039/d1cp00055a] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Our previous study on nitroxides in o-terphenyl (OTP) revealed two separable decoherence processes at low temperatures, best captured by the sum of two stretched exponentials (SSE) model. Dynamical decoupling (DD) extends both associated dephasing times linearly for 1 to 5 refocusing pulses [Soetbeer et al., Phys. Chem. Chem. Phys., 2018, 20, 1615]. Here we demonstrate an analogous DD behavior of water-soluble nitroxides in water-glycerol glass by using nitroxide and/or solvent deuteration for component assignment. Compared to the conventional Hahn experiment, we show that Carr-Purcell and Uhrig DD schemes are superior in resolving and identifying active dephasing mechanisms. Thereby, we observe a partial coherence loss to intramolecular nitroxide and trityl nuclei that can be alleviated, while the zero field splitting-induced losses for gadolinium labels cannot be refocused and contribute even at the central transition of this spin-7/2 system. Independent of the studied spin system, Uhrig DD leads to a characteristic convex dephasing envelope in both protonated water-glycerol and OTP glass, thus outperforming the Carr-Purcell scheme.
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Affiliation(s)
- Janne Soetbeer
- Laboratory of Physical Chemistry, ETH Zürich, Vladimir-Prelog-Weg 2, CH-8093 Zürich, Switzerland.
| | - Marthe Millen
- Laboratory of Physical Chemistry, ETH Zürich, Vladimir-Prelog-Weg 2, CH-8093 Zürich, Switzerland.
| | - Konstantin Zouboulis
- Laboratory of Physical Chemistry, ETH Zürich, Vladimir-Prelog-Weg 2, CH-8093 Zürich, Switzerland.
| | - Miriam Hülsmann
- Bielefeld University, Department of Chemistry, Universitätsstr. 25, D-33615 Bielefeld, Germany
| | - Adelheid Godt
- Bielefeld University, Department of Chemistry, Universitätsstr. 25, D-33615 Bielefeld, Germany
| | - Yevhen Polyhach
- Laboratory of Physical Chemistry, ETH Zürich, Vladimir-Prelog-Weg 2, CH-8093 Zürich, Switzerland.
| | - Gunnar Jeschke
- Laboratory of Physical Chemistry, ETH Zürich, Vladimir-Prelog-Weg 2, CH-8093 Zürich, Switzerland.
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18
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Crosse JA. Local-Field Corrections as a Regularization Method for the Spin-Boson Model. Sci Rep 2019; 9:5216. [PMID: 30914667 PMCID: PMC6435645 DOI: 10.1038/s41598-019-41303-0] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/14/2018] [Accepted: 03/01/2019] [Indexed: 11/09/2022] Open
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19
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Kveder M, Rakvin B, You J. A quantum many body model for the embedded electron spin decoherence in organic solids. J Chem Phys 2019; 151:164124. [DOI: 10.1063/1.5124561] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Affiliation(s)
- Marina Kveder
- Division of Physical Chemistry, Ruder Bošković Institute, Bijenička 54, 10000 Zagreb, Croatia
| | - Boris Rakvin
- Division of Physical Chemistry, Ruder Bošković Institute, Bijenička 54, 10000 Zagreb, Croatia
| | - Jiangyang You
- Division of Physical Chemistry, Ruder Bošković Institute, Bijenička 54, 10000 Zagreb, Croatia
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20
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Jeschke G. Quo vadis EPR? JOURNAL OF MAGNETIC RESONANCE (SAN DIEGO, CALIF. : 1997) 2019; 306:36-41. [PMID: 31345773 DOI: 10.1016/j.jmr.2019.07.008] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/18/2019] [Revised: 03/21/2019] [Accepted: 07/08/2019] [Indexed: 06/10/2023]
Abstract
Complexity of paramagnetic catalysts and materials increases, and the same applies to systems targeted by integrative structural biology. Hence, EPR spectroscopists must find ways to enhance information content of their data. I argue that a third major wave of method development in EPR spectroscopy, which is triggered by recent advances in digital electronics and computing, can achieve this. Transfer of NMR methods to EPR will go on, but part of the new EPR methodology will depend on completely new concepts.
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Affiliation(s)
- Gunnar Jeschke
- ETH Zurich, Lab. Phys. Chem., Vladimir-Prelog-Weg 2, 8093 Zurich, Switzerland.
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21
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Wang P, Chen C, Peng X, Wrachtrup J, Liu RB. Characterization of Arbitrary-Order Correlations in Quantum Baths by Weak Measurement. PHYSICAL REVIEW LETTERS 2019; 123:050603. [PMID: 31491311 DOI: 10.1103/physrevlett.123.050603] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/16/2019] [Indexed: 06/10/2023]
Abstract
Correlations of fluctuations are the driving forces behind the dynamics and thermodynamics in quantum many-body systems. For qubits embedded in a quantum bath, the correlations in the bath are key to understanding and combating decoherence-a critical issue in quantum information technology. However, there is no systematic method for characterizing the many-body correlations in quantum baths beyond the second order or the Gaussian approximation. Here we present a scheme to characterize the correlations in a quantum bath to arbitrary order. The scheme employs a weak measurement of the bath via the projective measurement of a central system. The bath correlations, including both the "classical" and the "quantum" parts, can be reconstructed from the correlations of the measurement outputs. The possibility of full characterization of many-body correlations in a quantum bath forms the basis for optimizing quantum control against decoherence in realistic environments, for studying the quantum characteristics of baths, and for the quantum sensing of correlated clusters in quantum baths.
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Affiliation(s)
- Ping Wang
- Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China
| | - Chong Chen
- Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China
| | - Xinhua Peng
- Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
- CAS Key Laboratory of Microscale Magnetic Resonance and Synergetic Innovation Center of Quantum Information & Quantum Physics, University of Science and Technology of China, Hefei 230026, China
| | - Jörg Wrachtrup
- 3rd Institute of Physics, Research Center SCoPE and IQST, University of Stuttgart, 70569 Stuttgart, Germany
- Max Planck Institute for Solid State Research, 70569 Stuttgart, Germany
| | - Ren-Bao Liu
- Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China
- The Hong Kong Institute of Quantum Information Science and Technology, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China
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22
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Gu B, Franco I. When can quantum decoherence be mimicked by classical noise? J Chem Phys 2019; 151:014109. [DOI: 10.1063/1.5099499] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Bing Gu
- Department of Chemistry, University of Rochester, Rochester, New York 14627, USA
| | - Ignacio Franco
- Department of Chemistry, University of Rochester, Rochester, New York 14627, USA
- Department of Physics, University of Rochester, Rochester, New York 14627, USA
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23
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You J, Carić D, Rakvin B, Štefanić Z, Užarević K, Kveder M. Matrix material structure dependence of the embedded electron spin decoherence. J Chem Phys 2019; 150:164124. [DOI: 10.1063/1.5090215] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Jiangyang You
- Division of Physical Chemistry, Ruđer Bošković Institute, Bijenička 54, 10000 Zagreb, Croatia
| | - Dejana Carić
- Division of Physical Chemistry, Ruđer Bošković Institute, Bijenička 54, 10000 Zagreb, Croatia
| | - Boris Rakvin
- Division of Physical Chemistry, Ruđer Bošković Institute, Bijenička 54, 10000 Zagreb, Croatia
| | - Zoran Štefanić
- Division of Physical Chemistry, Ruđer Bošković Institute, Bijenička 54, 10000 Zagreb, Croatia
| | - Krunoslav Užarević
- Division of Physical Chemistry, Ruđer Bošković Institute, Bijenička 54, 10000 Zagreb, Croatia
| | - Marina Kveder
- Division of Physical Chemistry, Ruđer Bošković Institute, Bijenička 54, 10000 Zagreb, Croatia
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24
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Abstract
In this work, we establish an exact relation which connects the heat exchange between two systems initialized in their thermodynamic equilibrium states at different temperatures and the Rényi divergences between the initial thermodynamic equilibrium state and the final nonequilibrium state of the total system. The relation tells us that the various moments of the heat statistics are determined by the Renyi divergences between the initial equilibrium state and the final nonequilibrium state of the global system. In particular the average heat exchange is quantified by the relative entropy between the initial equilibrium state and the final nonequilibrium state of the global system. The relation is applicable to both finite classical systems and finite quantum systems.
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Affiliation(s)
- Bo-Bo Wei
- School of Physics and Energy, Shenzhen University, Shenzhen 518060, China
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25
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Jing J, Wu LA. Decoherence and control of a qubit in spin baths: an exact master equation study. Sci Rep 2018; 8:1471. [PMID: 29367640 PMCID: PMC5784175 DOI: 10.1038/s41598-018-19977-9] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/08/2017] [Accepted: 01/08/2018] [Indexed: 11/21/2022] Open
Abstract
In spin-based nanosystems for quantum information processing, electron spin qubits are subject to decoherence due to their interactions with nuclear spin environments. In this paper, we present an exact master equation for a central spin-1/2 system in time-dependent external fields and coupled to a spin-half bath in terms of hyperfine interaction. The master equation provides a unified description for free and controlled dynamics of the central spin and is formally independent of the details and size of spin environments. Different from the previous approaches, the master equation remains exact even in the presence of external control fields. Using the parameters for realistic nanosystems with nonzero nuclear spins, such as GaAs, we investigate the Overhauser's effect on the decoherence dynamics of the central spin under different distributions of bath-spin frequencies and system-bath coupling strengths. Furthermore, we apply the leakage elimination operator, in a nonperturbative manner, to this system to suppress the decoherence induced by hyperfine interaction.
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Affiliation(s)
- Jun Jing
- Department of Physics, Zhejiang University, Hangzhou, 310027, Zhejiang, China
- Department of Theoretical Physics and History of Science, The Basque Country University (EHU/UPV), PO Box 644, 48080, Bilbao, Spain
| | - Lian-Ao Wu
- Department of Theoretical Physics and History of Science, The Basque Country University (EHU/UPV), PO Box 644, 48080, Bilbao, Spain.
- Ikerbasque, Basque Foundation for Science, 48011, Bilbao, Spain.
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26
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Gong B, Tu T, Zhou ZQ, Zhu XY, Li CF, Guo GC. Environment spectrum and coherence behaviours in a rare-earth doped crystal for quantum memory. Sci Rep 2017; 7:18030. [PMID: 29269863 PMCID: PMC5740147 DOI: 10.1038/s41598-017-18229-6] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/30/2017] [Accepted: 12/07/2017] [Indexed: 11/09/2022] Open
Abstract
We theoretically investigate the dynamics of environment and coherence behaviours of the central ion in a quantum memory based on a rare-earth doped crystal. The interactions between the central ion and the bath spins suppress the flip-flop rate of the neighbour bath spins and yield a specific environment spectral density S(ω). Under dynamical decoupling pulses, this spectrum provides a general scaling for the coherence envelope and coherence time, which significantly extend over a range on an hour-long time scale. The characterized environment spectrum with ultra-long coherence time can be used to implement various quantum communication and information processing protocols.
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Affiliation(s)
- Bo Gong
- Key Lab of Quantum Information, Chinese Academy of Sciences, University of Science and Technology of China, Hefei, 230026, China
| | - Tao Tu
- Key Lab of Quantum Information, Chinese Academy of Sciences, University of Science and Technology of China, Hefei, 230026, China.
| | - Zhong-Quan Zhou
- Key Lab of Quantum Information, Chinese Academy of Sciences, University of Science and Technology of China, Hefei, 230026, China.,Department of Physics and Astronomy, University of California at Los Angeles, California, 90095, USA
| | - Xing-Yu Zhu
- Key Lab of Quantum Information, Chinese Academy of Sciences, University of Science and Technology of China, Hefei, 230026, China.,Department of Physics and Astronomy, University of California at Los Angeles, California, 90095, USA
| | - Chuan-Feng Li
- Key Lab of Quantum Information, Chinese Academy of Sciences, University of Science and Technology of China, Hefei, 230026, China.
| | - Guang-Can Guo
- Key Lab of Quantum Information, Chinese Academy of Sciences, University of Science and Technology of China, Hefei, 230026, China
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Szańkowski P, Ramon G, Krzywda J, Kwiatkowski D, Cywiński Ł. Environmental noise spectroscopy with qubits subjected to dynamical decoupling. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2017; 29:333001. [PMID: 28569239 DOI: 10.1088/1361-648x/aa7648] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
A qubit subjected to pure dephasing due to classical Gaussian noise can be turned into a spectrometer of this noise by utilizing its readout under properly chosen dynamical decoupling (DD) sequences to reconstruct the power spectral density of the noise. We review the theory behind this DD-based noise spectroscopy technique, paying special attention to issues that arise when the environmental noise is non-Gaussian and/or it has truly quantum properties. While we focus on the theoretical basis of the method, we connect the discussed concepts with specific experiments, and provide an overview of environmental noise models relevant for solid-state based qubits, including quantum-dot based spin qubits, superconducting qubits, and NV centers in diamond.
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Affiliation(s)
- P Szańkowski
- Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland
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