1
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Li Y, Wang C, Tang Y, Liu YC. Time Crystal in a Single-Mode Nonlinear Cavity. PHYSICAL REVIEW LETTERS 2024; 132:183803. [PMID: 38759188 DOI: 10.1103/physrevlett.132.183803] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/10/2023] [Revised: 01/22/2024] [Accepted: 04/08/2024] [Indexed: 05/19/2024]
Abstract
Time crystal is a class of nonequilibrium phases with broken time-translational symmetry. Here, we demonstrate the time crystal in a single-mode nonlinear cavity. The time crystal originates from the self-oscillation induced by a linear gain and is stabilized by a nonlinear damping. We show in the time crystal phase there are sharp dissipative gap closing and pure imaginary eigenvalues of the Liouvillian spectrum in the thermodynamic limit. Dynamically, we observe a metastable regime with the emergence of quantum oscillation, followed by a dissipative evolution with a timescale much longer than the oscillating period. Moreover, we show there is a dissipative phase transition at the Hopf bifurcation, which can be characterized by the photon number fluctuation in the steady state. These results pave a new promising way for further experiments and deepen our understanding of time crystals.
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Affiliation(s)
- Yaohua Li
- State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China
| | - Chenyang Wang
- State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China
| | - Yuanjiang Tang
- State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China
| | - Yong-Chun Liu
- State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China
- Frontier Science Center for Quantum Information, Beijing 100084, China
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2
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Sasaki K, Abe E. Suppression of Pulsed Dynamic Nuclear Polarization by Many-Body Spin Dynamics. PHYSICAL REVIEW LETTERS 2024; 132:106904. [PMID: 38518331 DOI: 10.1103/physrevlett.132.106904] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/17/2023] [Accepted: 02/06/2024] [Indexed: 03/24/2024]
Abstract
We study a mechanism by which nuclear hyperpolarization due to the polarization transfer from a microwave-pulse-controlled electron spin is suppressed. From analytical and numerical calculations of the unitary dynamics of multiple nuclear spins, we uncover that, combined with the formation of the dark state within a cluster of nuclei, coherent higher-order nuclear spin dynamics impose limits on the efficiency of the polarization transfer even in the absence of mundane depolarization processes such as nuclear spin diffusion and relaxation. Furthermore, we show that the influence of the dark state can be partly mitigated by introducing a disentangling operation. Our analysis is applied to the nuclear polarizations observed in ^{13}C nuclei coupled with a single nitrogen-vacancy center in diamond [Randall et al., Science 374, 1474 (2021)SCIEAS0036-807510.1126/science.abk0603]. Our Letter sheds light on collective engineering of nuclear spins as well as future designs of pulsed dynamic nuclear polarization protocols.
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Affiliation(s)
- Kento Sasaki
- Department of Physics, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
| | - Eisuke Abe
- RIKEN Center for Quantum Computing, Wako, Saitama 351-0198, Japan
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3
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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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4
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Hesselmeier E, Kuna P, Takács I, Ivády V, Knolle W, Son NT, Ghezellou M, Ul-Hassan J, Dasari D, Kaiser F, Vorobyov V, Wrachtrup J. Qudit-Based Spectroscopy for Measurement and Control of Nuclear-Spin Qubits in Silicon Carbide. PHYSICAL REVIEW LETTERS 2024; 132:090601. [PMID: 38489642 DOI: 10.1103/physrevlett.132.090601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/23/2023] [Accepted: 01/17/2024] [Indexed: 03/17/2024]
Abstract
Nuclear spins with hyperfine coupling to single electron spins are highly valuable quantum bits. Here we probe and characterize the particularly rich nuclear-spin environment around single silicon vacancy color centers (V2) in 4H-SiC. By using the electron spin-3/2 qudit as a four level sensor, we identify several sets of ^{29}Si and ^{13}C nuclear spins through their hyperfine interaction. We extract the major components of their hyperfine coupling via optical detected nuclear magnetic resonance, and assign them to shells in the crystal via the density function theory simulations. We utilize the ground-state level anticrossing of the electron spin for dynamic nuclear polarization and achieve a nuclear-spin polarization of up to 98±6%. We show that this scheme can be used to detect the nuclear magnetic resonance signal of individual spins and demonstrate their coherent control. Our work provides a detailed set of parameters and first steps for future use of SiC as a multiqubit memory and quantum computing platform.
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Affiliation(s)
- Erik Hesselmeier
- 3rd Institute of Physics, IQST, and Research Centre SCoPE, University of Stuttgart, Stuttgart, Germany
| | - Pierre Kuna
- 3rd Institute of Physics, IQST, and Research Centre SCoPE, University of Stuttgart, Stuttgart, Germany
| | - István Takács
- Eötvös Loránd University, Egyetem tér 1-3, H-1053 Budapest, Hungary
- MTA-ELTE Lendület "Momentum" NewQubit Research Group, Pázmány Péter, Sétány 1/A, 1117 Budapest, Hungary
| | - Viktor Ivády
- Eötvös Loránd University, Egyetem tér 1-3, H-1053 Budapest, Hungary
- MTA-ELTE Lendület "Momentum" NewQubit Research Group, Pázmány Péter, Sétány 1/A, 1117 Budapest, Hungary
- Department of Physics, Chemistry and Biology, Linköping University, Olaus Magnus väg, 583 30 Linköping, Sweden
| | - Wolfgang Knolle
- Department of Sensoric Surfaces and Functional Interfaces, Leibniz-Institute of Surface Engineering (IOM), Permoserstraße 15, 04318 Leipzig, Germany
| | - Nguyen Tien Son
- Department of Physics, Chemistry and Biology, Linköping University, Olaus Magnus väg, 583 30 Linköping, Sweden
| | - Misagh Ghezellou
- Department of Physics, Chemistry and Biology, Linköping University, Olaus Magnus väg, 583 30 Linköping, Sweden
| | - Jawad Ul-Hassan
- Department of Physics, Chemistry and Biology, Linköping University, Olaus Magnus väg, 583 30 Linköping, Sweden
| | - Durga Dasari
- 3rd Institute of Physics, IQST, and Research Centre SCoPE, University of Stuttgart, Stuttgart, Germany
| | - Florian Kaiser
- Materials Research and Technology (MRT) Department, Luxembourg Institute of Science and Technology (LIST), 4422 Belvaux, Luxembourg
- University of Luxembourg, 41 rue du Brill, L-4422 Belvaux, Luxembourg
| | - Vadim Vorobyov
- 3rd Institute of Physics, IQST, and Research Centre SCoPE, University of Stuttgart, Stuttgart, Germany
| | - Jörg Wrachtrup
- 3rd Institute of Physics, IQST, and Research Centre SCoPE, University of Stuttgart, Stuttgart, Germany
- Max Planck Institute for solid state physics, Heisenbergstraße 1, 70569 Stuttgart, Germany
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5
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Gong R, Du X, Janzen E, Liu V, Liu Z, He G, Ye B, Li T, Yao NY, Edgar JH, Henriksen EA, Zu C. Isotope engineering for spin defects in van der Waals materials. Nat Commun 2024; 15:104. [PMID: 38168074 PMCID: PMC10761865 DOI: 10.1038/s41467-023-44494-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/22/2023] [Accepted: 12/14/2023] [Indexed: 01/05/2024] Open
Abstract
Spin defects in van der Waals materials offer a promising platform for advancing quantum technologies. Here, we propose and demonstrate a powerful technique based on isotope engineering of host materials to significantly enhance the coherence properties of embedded spin defects. Focusing on the recently-discovered negatively charged boron vacancy center ([Formula: see text]) in hexagonal boron nitride (hBN), we grow isotopically purified h10B15N crystals. Compared to [Formula: see text] in hBN with the natural distribution of isotopes, we observe substantially narrower and less crowded [Formula: see text] spin transitions as well as extended coherence time T2 and relaxation time T1. For quantum sensing, [Formula: see text] centers in our h10B15N samples exhibit a factor of 4 (2) enhancement in DC (AC) magnetic field sensitivity. For additional quantum resources, the individual addressability of the [Formula: see text] hyperfine levels enables the dynamical polarization and coherent control of the three nearest-neighbor 15N nuclear spins. Our results demonstrate the power of isotope engineering for enhancing the properties of quantum spin defects in hBN, and can be readily extended to improving spin qubits in a broad family of van der Waals materials.
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Affiliation(s)
- Ruotian Gong
- Department of Physics, Washington University, St. Louis, MO, 63130, USA
| | - Xinyi Du
- Department of Physics, Washington University, St. Louis, MO, 63130, USA
| | - Eli Janzen
- Tim Taylor Department of Chemical Engineering, Kansas State University, Manhattan, KS, 66506, USA
| | - Vincent Liu
- Department of Physics, Harvard University, Cambridge, MA, 02138, USA
| | - Zhongyuan Liu
- Department of Physics, Washington University, St. Louis, MO, 63130, USA
| | - Guanghui He
- Department of Physics, Washington University, St. Louis, MO, 63130, USA
| | - Bingtian Ye
- Department of Physics, Harvard University, Cambridge, MA, 02138, USA
| | - Tongcang Li
- Department of Physics and Astronomy, Purdue University, West Lafayette, IN, 47907, USA
- Elmore Family School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, 47907, USA
| | - Norman Y Yao
- Department of Physics, Harvard University, Cambridge, MA, 02138, USA
| | - James H Edgar
- Tim Taylor Department of Chemical Engineering, Kansas State University, Manhattan, KS, 66506, USA
| | - Erik A Henriksen
- Department of Physics, Washington University, St. Louis, MO, 63130, USA
- Institute of Materials Science and Engineering, Washington University, St. Louis, MO, 63130, USA
| | - Chong Zu
- Department of Physics, Washington University, St. Louis, MO, 63130, USA.
- Institute of Materials Science and Engineering, Washington University, St. Louis, MO, 63130, USA.
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6
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Chen YH, Zhang X. Realization of an inherent time crystal in a dissipative many-body system. Nat Commun 2023; 14:6161. [PMID: 37789006 PMCID: PMC10547780 DOI: 10.1038/s41467-023-41905-3] [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: 10/30/2022] [Accepted: 09/21/2023] [Indexed: 10/05/2023] Open
Abstract
Time crystals are many-body states that spontaneously break translation symmetry in time the way that ordinary crystals do in space. While experimental observations have confirmed the existence of discrete or continuous time crystals, these realizations have relied on the utilization of periodic forces or effective modulation through cavity feedback. The original proposal for time crystals is that they would represent self-sustained motions without any external periodicity, but realizing such purely self-generated behavior has not yet been achieved. Here, we provide theoretical and experimental evidence that many-body interactions can give rise to an inherent time crystalline phase. Following a calculation that shows an ensemble of pumped four-level atoms can spontaneously break continuous time translation symmetry, we observe periodic motions in an erbium-doped solid. The inherent time crystal produced by our experiment is self-protected by many-body interactions and has a measured coherence time beyond that of individual erbium ions.
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Affiliation(s)
- Yu-Hui Chen
- Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurements of Ministry of Education, School of Physics, Beijing Institute of Technology, 100081, Beijing, China
- Beijing Key Laboratory of Nanophotonics & Ultrafine Optoelectronic Systems, School of Physics, Beijing Institute of Technology, 100081, Beijing, China
| | - Xiangdong Zhang
- Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurements of Ministry of Education, School of Physics, Beijing Institute of Technology, 100081, Beijing, China.
- Beijing Key Laboratory of Nanophotonics & Ultrafine Optoelectronic Systems, School of Physics, Beijing Institute of Technology, 100081, Beijing, China.
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7
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He G, Ye B, Gong R, Liu Z, Murch KW, Yao NY, Zu C. Quasi-Floquet Prethermalization in a Disordered Dipolar Spin Ensemble in Diamond. PHYSICAL REVIEW LETTERS 2023; 131:130401. [PMID: 37832016 DOI: 10.1103/physrevlett.131.130401] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/15/2023] [Accepted: 08/18/2023] [Indexed: 10/15/2023]
Abstract
Floquet (periodic) driving has recently emerged as a powerful technique for engineering quantum systems and realizing nonequilibrium phases of matter. A central challenge to stabilizing quantum phenomena in such systems is the need to prevent energy absorption from the driving field. Fortunately, when the frequency of the drive is significantly larger than the local energy scales of the many-body system, energy absorption is suppressed. The existence of this so-called prethermal regime depends sensitively on the range of interactions and the presence of multiple driving frequencies. Here, we report the observation of Floquet prethermalization in a strongly interacting dipolar spin ensemble in diamond, where the angular dependence of the dipolar coupling helps to mitigate the long-ranged nature of the interaction. Moreover, we extend our experimental observation to quasi-Floquet drives with multiple incommensurate frequencies. In contrast to a single-frequency drive, we find that the existence of prethermalization is extremely sensitive to the smoothness of the applied field. Our results open the door to stabilizing and characterizing nonequilibrium phenomena in quasiperiodically driven systems.
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Affiliation(s)
- Guanghui He
- Department of Physics, Washington University, St. Louis, Missouri 63130, USA
| | - Bingtian Ye
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
- Department of Physics, University of California, Berkeley, California 94720, USA
| | - Ruotian Gong
- Department of Physics, Washington University, St. Louis, Missouri 63130, USA
| | - Zhongyuan Liu
- Department of Physics, Washington University, St. Louis, Missouri 63130, USA
| | - Kater W Murch
- Department of Physics, Washington University, St. Louis, Missouri 63130, USA
- Institute of Materials Science and Engineering, Washington University, St. Louis, Missouri 63130, USA
| | - Norman Y Yao
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
- Department of Physics, University of California, Berkeley, California 94720, USA
- Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
| | - Chong Zu
- Department of Physics, Washington University, St. Louis, Missouri 63130, USA
- Institute of Materials Science and Engineering, Washington University, St. Louis, Missouri 63130, USA
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8
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Gu Y, Zhuang WF, Chai X, Liu DE. Benchmarking universal quantum gates via channel spectrum. Nat Commun 2023; 14:5880. [PMID: 37735170 PMCID: PMC10514318 DOI: 10.1038/s41467-023-41598-8] [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: 02/11/2023] [Accepted: 09/12/2023] [Indexed: 09/23/2023] Open
Abstract
Noise remains the major obstacle to scalable quantum computation. Quantum benchmarking provides key information on noise properties and is an important step for developing more advanced quantum processors. However, current benchmarking methods are either limited to a specific subset of quantum gates or cannot directly describe the performance of the individual target gate. To overcome these limitations, we propose channel spectrum benchmarking (CSB), a method to infer the noise properties of the target gate, including process fidelity, stochastic fidelity, and some unitary parameters, from the eigenvalues of its noisy channel. Our CSB method is insensitive to state-preparation and measurement errors, and importantly, can benchmark universal gates and is scalable to many-qubit systems. Unlike standard randomized schemes, CSB can provide direct noise information for both target native gates and circuit fragments, allowing benchmarking and calibration of global entangling gates and frequently used modules in quantum algorithms like Trotterized Hamiltonian evolution operator in quantum simulation.
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Affiliation(s)
- Yanwu Gu
- Beijing Academy of Quantum Information Sciences, Beijing, 100193, China.
- State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing, 100084, China.
| | - Wei-Feng Zhuang
- Beijing Academy of Quantum Information Sciences, Beijing, 100193, China
| | - Xudan Chai
- Beijing Academy of Quantum Information Sciences, Beijing, 100193, China
- State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing, 100084, China
| | - Dong E Liu
- Beijing Academy of Quantum Information Sciences, Beijing, 100193, China.
- State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing, 100084, China.
- Frontier Science Center for Quantum Information, Beijing, 100184, China.
- Hefei National Laboratory, Hefei, 230088, China.
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9
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Ravindranath V, Chen X. Robust Oscillations and Edge Modes in Nonunitary Floquet Systems. PHYSICAL REVIEW LETTERS 2023; 130:230402. [PMID: 37354400 DOI: 10.1103/physrevlett.130.230402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/09/2022] [Accepted: 05/11/2023] [Indexed: 06/26/2023]
Abstract
We explore oscillatory behavior in a family of periodically driven spin chains which are subject to a weak measurement followed by postselection. We discover a transition to an oscillatory phase as the strength of the measurement is increased. By mapping these spin chains to free fermion models, we find that this transition is reflected in the opening of a gap in the imaginary direction. Interestingly, we find a robust, purely real, edge π mode in the oscillatory phase. We establish a correspondence between the complex bulk spectrum and these edge modes. These oscillations are numerically found to be stable against interactions and disorder.
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Affiliation(s)
- Vikram Ravindranath
- Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA
| | - Xiao Chen
- Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA
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10
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Gong R, He G, Gao X, Ju P, Liu Z, Ye B, Henriksen EA, Li T, Zu C. Coherent dynamics of strongly interacting electronic spin defects in hexagonal boron nitride. Nat Commun 2023; 14:3299. [PMID: 37280252 DOI: 10.1038/s41467-023-39115-y] [Citation(s) in RCA: 5] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Grants] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/07/2022] [Accepted: 05/26/2023] [Indexed: 06/08/2023] Open
Abstract
Optically active spin defects in van der Waals materials are promising platforms for modern quantum technologies. Here we investigate the coherent dynamics of strongly interacting ensembles of negatively charged boron-vacancy ([Formula: see text]) centers in hexagonal boron nitride (hBN) with varying defect density. By employing advanced dynamical decoupling sequences to selectively isolate different dephasing sources, we observe more than 5-fold improvement in the measured coherence times across all hBN samples. Crucially, we identify that the many-body interaction within the [Formula: see text] ensemble plays a substantial role in the coherent dynamics, which is then used to directly estimate the concentration of [Formula: see text]. We find that at high ion implantation dosage, only a small portion of the created boron vacancy defects are in the desired negatively charged state. Finally, we investigate the spin response of [Formula: see text] to the local charged defects induced electric field signals, and estimate its ground state transverse electric field susceptibility. Our results provide new insights on the spin and charge properties of [Formula: see text], which are important for future use of defects in hBN as quantum sensors and simulators.
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Affiliation(s)
- Ruotian Gong
- Department of Physics, Washington University, St. Louis, MO, 63130, USA
| | - Guanghui He
- Department of Physics, Washington University, St. Louis, MO, 63130, USA
| | - Xingyu Gao
- Department of Physics and Astronomy, Purdue University, West Lafayette, IN, 47907, USA
| | - Peng Ju
- Department of Physics and Astronomy, Purdue University, West Lafayette, IN, 47907, USA
| | - Zhongyuan Liu
- Department of Physics, Washington University, St. Louis, MO, 63130, USA
| | - Bingtian Ye
- Department of Physics, Harvard University, Cambridge, MA, 02138, USA
- Department of Physics, University of California, Berkeley, CA, 94720, USA
| | - Erik A Henriksen
- Department of Physics, Washington University, St. Louis, MO, 63130, USA
- Institute of Materials Science and Engineering, Washington University, St. Louis, MO, 63130, USA
| | - Tongcang Li
- Department of Physics and Astronomy, Purdue University, West Lafayette, IN, 47907, USA
- Elmore Family School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, 47907, USA
| | - Chong Zu
- Department of Physics, Washington University, St. Louis, MO, 63130, USA.
- Institute of Materials Science and Engineering, Washington University, St. Louis, MO, 63130, USA.
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11
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Šafránek D, Rosa D, Binder FC. Work Extraction from Unknown Quantum Sources. PHYSICAL REVIEW LETTERS 2023; 130:210401. [PMID: 37295083 DOI: 10.1103/physrevlett.130.210401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/07/2022] [Accepted: 04/07/2023] [Indexed: 06/12/2023]
Abstract
Energy extraction is a central task in thermodynamics. In quantum physics, ergotropy measures the amount of work extractable under cyclic Hamiltonian control. As its full extraction requires perfect knowledge of the initial state, however, it does not characterize the work value of unknown or untrusted quantum sources. Fully characterizing such sources would require quantum tomography, which is prohibitively costly in experiments due to the exponential growth of required measurements and operational limitations. Here, we therefore derive a new notion of ergotropy applicable when nothing is known about the quantum states produced by the source, apart from what can be learned by performing only a single type of coarse-grained measurement. We find that in this case the extracted work is defined by the Boltzmann and observational entropy in cases where the measurement outcomes are, or are not, used in the work extraction, respectively. This notion of ergotropy represents a realistic measure of extractable work, which can be used as the relevant figure of merit to characterize a quantum battery.
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Affiliation(s)
- Dominik Šafránek
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon - 34126, Korea
| | - Dario Rosa
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon - 34126, Korea
- Basic Science Program, Korea University of Science and Technology (UST), Daejeon-34113, Korea
| | - Felix C Binder
- School of Physics, Trinity College Dublin, Dublin 2, Ireland
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12
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Krishna M, Solanki P, Hajdušek M, Vinjanampathy S. Measurement-Induced Continuous Time Crystals. PHYSICAL REVIEW LETTERS 2023; 130:150401. [PMID: 37115890 DOI: 10.1103/physrevlett.130.150401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/06/2022] [Accepted: 03/13/2023] [Indexed: 06/19/2023]
Abstract
Strong measurements usually restrict the dynamics of measured finite dimensional systems to the Zeno subspace, where subsequent evolution is unitary due to the suppression of dissipative terms. Here, we show qualitatively different behavior induced by the competition between strong measurements and the thermodynamic limit, inducing a time-translation symmetry breaking phase transition resulting in a continuous time crystal. We consider an undriven spin star model, where the central spin is subject to a strong continuous measurement, and qualify the dynamic behavior of the system in various parameter regimes. We show that above a critical value of measurement strength, the magnetization of the thermodynamically large ancilla spins, along with the central spin, develops limit-cycle oscillations.
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Affiliation(s)
- Midhun Krishna
- Department of Physics, Indian Institute of Technology-Bombay, Powai, Mumbai 400076, India
| | - Parvinder Solanki
- Department of Physics, Indian Institute of Technology-Bombay, Powai, Mumbai 400076, India
| | - Michal Hajdušek
- Keio University Shonan Fujisawa Campus, 5322 Endo, Fujisawa, Kanagawa 252-0882, Japan
- Keio University Quantum Computing Center, 3-14-1 Hiyoshi, Kohoku, Yokohama, Kanagawa 223-8522, Japan
| | - Sai Vinjanampathy
- Department of Physics, Indian Institute of Technology-Bombay, Powai, Mumbai 400076, India
- Centre of Excellence in Quantum Information, Computation, Science and Technology, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
- Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore, Singapore
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13
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Liu S, Zhang SX, Hsieh CY, Zhang S, Yao H. Discrete Time Crystal Enabled by Stark Many-Body Localization. PHYSICAL REVIEW LETTERS 2023; 130:120403. [PMID: 37027857 DOI: 10.1103/physrevlett.130.120403] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/15/2022] [Revised: 11/11/2022] [Accepted: 02/15/2023] [Indexed: 06/19/2023]
Abstract
Discrete time crystals (DTCs) have recently attracted increasing attention, but most DTC models and their properties are only revealed after disorder average. In this Letter, we propose a simple disorder-free periodically driven model that exhibits nontrivial DTC order stabilized by Stark many-body localization (MBL). We demonstrate the existence of the DTC phase by analytical analysis from perturbation theory and convincing numerical evidence from observable dynamics. The new DTC model paves a new promising way for further experiments and deepens our understanding of DTCs. Since the DTC order does not require special quantum state preparation and the strong disorder average, it can be naturally realized on the noisy intermediate-scale quantum hardware with much fewer resources and repetitions. Moreover, in addition to the robust subharmonic response, there are other novel robust beating oscillations in the Stark-MBL DTC phase that are absent in random or quasiperiodic MBL DTCs.
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Affiliation(s)
- Shuo Liu
- Institute for Advanced Study, Tsinghua University, Beijing 100084, China
- Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong 518057, China
| | - Shi-Xin Zhang
- Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong 518057, China
| | - Chang-Yu Hsieh
- Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong 518057, China
| | - Shengyu Zhang
- Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong 518057, China
| | - Hong Yao
- Institute for Advanced Study, Tsinghua University, Beijing 100084, China
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14
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Klein J, Song Z, Pingault B, Dirnberger F, Chi H, Curtis JB, Dana R, Bushati R, Quan J, Dekanovsky L, Sofer Z, Alù A, Menon VM, Moodera JS, Lončar M, Narang P, Ross FM. Sensing the Local Magnetic Environment through Optically Active Defects in a Layered Magnetic Semiconductor. ACS NANO 2023; 17:288-299. [PMID: 36537371 DOI: 10.1021/acsnano.2c07655] [Citation(s) in RCA: 10] [Impact Index Per Article: 10.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/17/2023]
Abstract
Atomic-level defects in van der Waals (vdW) materials are essential building blocks for quantum technologies and quantum sensing applications. The layered magnetic semiconductor CrSBr is an outstanding candidate for exploring optically active defects because of a direct gap, in addition to a rich magnetic phase diagram, including a recently hypothesized defect-induced magnetic order at low temperature. Here, we show optically active defects in CrSBr that are probes of the local magnetic environment. We observe a spectrally narrow (1 meV) defect emission in CrSBr that is correlated with both the bulk magnetic order and an additional low-temperature, defect-induced magnetic order. We elucidate the origin of this magnetic order in the context of local and nonlocal exchange coupling effects. Our work establishes vdW magnets like CrSBr as an exceptional platform to optically study defects that are correlated with the magnetic lattice. We anticipate that controlled defect creation allows for tailor-made complex magnetic textures and phases with direct optical access.
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Affiliation(s)
- Julian Klein
- Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts02139, United States
| | - Zhigang Song
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts02138, United States
- College of Letters and Science, UCLA, Los Angeles, California90095, United States
| | - Benjamin Pingault
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts02138, United States
- QuTech, Delft University of Technology, 2600 GADelft, The Netherlands
| | - Florian Dirnberger
- Department of Physics, City College of New York, New York, New York10031, United States
| | - Hang Chi
- Francis Bitter Magnet Laboratory, Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, Massachusetts02139, United States
- U.S. Army CCDC Army Research Laboratory, Adelphi, Maryland20783, United States
| | - Jonathan B Curtis
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts02138, United States
- College of Letters and Science, UCLA, Los Angeles, California90095, United States
| | - Rami Dana
- Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts02139, United States
| | - Rezlind Bushati
- Department of Physics, City College of New York, New York, New York10031, United States
- Department of Physics, The Graduate Center, City University of New York, New York, New York10016, United States
| | - Jiamin Quan
- Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, Texas78712, United States
- Photonics Initiative, CUNY Advanced Science Research Center, New York, New York10031, United States
- Department of Electrical Engineering, City College of the City University of New York, New York, New York10031, United States
- Physics Program, Graduate Center, City University of New York, New York, New York10026, United States
| | - Lukas Dekanovsky
- Department of Inorganic Chemistry, University of Chemistry and Technology Prague, Technická 5, 166 28Prague 6, Czech Republic
| | - Zdenek Sofer
- Department of Inorganic Chemistry, University of Chemistry and Technology Prague, Technická 5, 166 28Prague 6, Czech Republic
| | - Andrea Alù
- Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, Texas78712, United States
- Photonics Initiative, CUNY Advanced Science Research Center, New York, New York10031, United States
- Department of Electrical Engineering, City College of the City University of New York, New York, New York10031, United States
- Physics Program, Graduate Center, City University of New York, New York, New York10026, United States
| | - Vinod M Menon
- Department of Physics, City College of New York, New York, New York10031, United States
- Department of Physics, The Graduate Center, City University of New York, New York, New York10016, United States
| | - Jagadeesh S Moodera
- Francis Bitter Magnet Laboratory, Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, Massachusetts02139, United States
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts02139, United States
| | - Marko Lončar
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts02138, United States
| | - Prineha Narang
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts02138, United States
- College of Letters and Science, UCLA, Los Angeles, California90095, United States
| | - Frances M Ross
- Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts02139, United States
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15
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Miessen A, Ollitrault PJ, Tacchino F, Tavernelli I. Quantum algorithms for quantum dynamics. NATURE COMPUTATIONAL SCIENCE 2023; 3:25-37. [PMID: 38177956 DOI: 10.1038/s43588-022-00374-2] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/13/2021] [Accepted: 11/12/2022] [Indexed: 01/06/2024]
Abstract
Among the many computational challenges faced across different disciplines, quantum-mechanical systems pose some of the hardest ones and offer a natural playground for the growing field of quantum technologies. In this Perspective, we discuss quantum algorithmic solutions for quantum dynamics, reporting on the latest developments and offering a viewpoint on their potential and current limitations. We present some of the most promising areas of application and identify possible research directions for the coming years.
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Affiliation(s)
| | - Pauline J Ollitrault
- IBM Quantum, IBM Research - Zurich, Rüschlikon, Switzerland
- QC Ware, Palo Alto, CA, USA
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16
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Bull K, Hallam A, Papić Z, Martin I. Tuning between Continuous Time Crystals and Many-Body Scars in Long-Range XYZ Spin Chains. PHYSICAL REVIEW LETTERS 2022; 129:140602. [PMID: 36240421 DOI: 10.1103/physrevlett.129.140602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/10/2022] [Accepted: 08/18/2022] [Indexed: 06/16/2023]
Abstract
Persistent oscillatory dynamics in nonequilibrium many-body systems is a tantalizing manifestation of ergodicity breakdown that continues to attract much attention. Recent works have focused on two classes of such systems: discrete time crystals and quantum many-body scars (QMBS). While both systems host oscillatory dynamics, its origin is expected to be fundamentally different: discrete time crystal is a phase of matter which spontaneously breaks the Z_{2} symmetry of the external periodic drive, while QMBS span a subspace of nonthermalizing eigenstates forming an su(2) algebra representation. Here, we ask a basic question: is there a physical system that allows us to tune between these two dynamical phenomena? In contrast to much previous work, we investigate the possibility of a continuous time crystal (CTC) in undriven, energy-conserving systems exhibiting prethermalization. We introduce a long-range XYZ spin model and show that it encompasses both a CTC phase as well as QMBS. We map out the dynamical phase diagram using numerical simulations based on exact diagonalization and time-dependent variational principle in the thermodynamic limit. We identify a regime where QMBS and CTC order coexist, and we discuss experimental protocols that reveal their similarities as well as key differences.
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Affiliation(s)
- Kieran Bull
- School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom
| | - Andrew Hallam
- School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom
| | - Zlatko Papić
- School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom
| | - Ivar Martin
- Material Science Division, Argonne National Laboratory, Argonne, Illinois 08540, USA
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17
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Huang B, Leung TH, Stamper-Kurn DM, Liu WV. Discrete Time Crystals Enforced by Floquet-Bloch Scars. PHYSICAL REVIEW LETTERS 2022; 129:133001. [PMID: 36206415 DOI: 10.1103/physrevlett.129.133001] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/20/2022] [Accepted: 08/29/2022] [Indexed: 06/16/2023]
Abstract
We analytically identify a new class of quantum scars protected by spatiotemporal translation symmetries, dubbed Floquet-Bloch scars. They are distinguished from previous (quasi-)static scars by a rigid spectral pairing only possible in Floquet systems, where strong interaction and drivings equalize the quasienergy corrections to all scars and maintain their spectral spacings against generic bilinear perturbations. Scars then enforce the spatial localization and rigid discrete time crystal (DTC) oscillations as verified numerically in a trimerized kagome lattice model relevant to recent cold atom experiments. Our analytical solutions offer a potential scheme to understand the mechanisms for more generic translation-invariant DTCs.
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Affiliation(s)
- Biao Huang
- Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China
| | - Tsz-Him Leung
- Department of Physics, 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
| | - W Vincent Liu
- Department of Physics and Astronomy and IQ Initiative, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
- Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China
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18
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Zhang X, Jiang W, Deng J, Wang K, Chen J, Zhang P, Ren W, Dong H, Xu S, Gao Y, Jin F, Zhu X, Guo Q, Li H, Song C, Gorshkov AV, Iadecola T, Liu F, Gong ZX, Wang Z, Deng DL, Wang H. Digital quantum simulation of Floquet symmetry-protected topological phases. Nature 2022; 607:468-473. [PMID: 35859194 PMCID: PMC9300455 DOI: 10.1038/s41586-022-04854-3] [Citation(s) in RCA: 10] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2021] [Accepted: 05/11/2022] [Indexed: 11/09/2022]
Abstract
Quantum many-body systems away from equilibrium host a rich variety of exotic phenomena that are forbidden by equilibrium thermodynamics. A prominent example is that of discrete time crystals1–8, in which time-translational symmetry is spontaneously broken in periodically driven systems. Pioneering experiments have observed signatures of time crystalline phases with trapped ions9,10, solid-state spin systems11–15, ultracold atoms16,17 and superconducting qubits18–20. Here we report the observation of a distinct type of non-equilibrium state of matter, Floquet symmetry-protected topological phases, which are implemented through digital quantum simulation with an array of programmable superconducting qubits. We observe robust long-lived temporal correlations and subharmonic temporal response for the edge spins over up to 40 driving cycles using a circuit of depth exceeding 240 and acting on 26 qubits. We demonstrate that the subharmonic response is independent of the initial state, and experimentally map out a phase boundary between the Floquet symmetry-protected topological and thermal phases. Our results establish a versatile digital simulation approach to exploring exotic non-equilibrium phases of matter with current noisy intermediate-scale quantum processors21. Signatures of non-equilibrium Floquet SPT phases with a programmable superconducting quantum processor are observed in which the discrete time translational symmetry only breaks at the boundaries and not in the bulk.
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Affiliation(s)
- Xu Zhang
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
| | - Wenjie Jiang
- Center for Quantum Information, IIIS, Tsinghua University, Beijing, China
| | - Jinfeng Deng
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
| | - Ke Wang
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
| | - Jiachen Chen
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
| | - Pengfei Zhang
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
| | - Wenhui Ren
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
| | - Hang Dong
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
| | - Shibo Xu
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
| | - Yu Gao
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
| | - Feitong Jin
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
| | - Xuhao Zhu
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
| | - Qiujiang Guo
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China.,Alibaba-Zhejiang University Joint Research Institute of Frontier Technologies, Hangzhou, China
| | - Hekang Li
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China.,Alibaba-Zhejiang University Joint Research Institute of Frontier Technologies, Hangzhou, China
| | - Chao Song
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China.,Alibaba-Zhejiang University Joint Research Institute of Frontier Technologies, Hangzhou, China
| | - Alexey V Gorshkov
- Joint Quantum Institute and Joint Center for Quantum Information and Computer Science, University of Maryland and NIST, College Park, MD, USA
| | - Thomas Iadecola
- Department of Physics and Astronomy, Iowa State University, Ames, IA, USA.,Ames Laboratory, Ames, IA, USA
| | - Fangli Liu
- Joint Quantum Institute and Joint Center for Quantum Information and Computer Science, University of Maryland and NIST, College Park, MD, USA.,QuEra Computing Inc., Boston, MA, USA
| | - Zhe-Xuan Gong
- Department of Physics, Colorado School of Mines, Golden, CO, USA.,National Institute of Standards and Technology, Boulder, CO, USA
| | - Zhen Wang
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China. .,Alibaba-Zhejiang University Joint Research Institute of Frontier Technologies, Hangzhou, China.
| | - Dong-Ling Deng
- Center for Quantum Information, IIIS, Tsinghua University, Beijing, China. .,Shanghai Qi Zhi Institute, Shanghai, China.
| | - H Wang
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China.,Alibaba-Zhejiang University Joint Research Institute of Frontier Technologies, Hangzhou, China
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19
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Dumitrescu PT, Bohnet JG, Gaebler JP, Hankin A, Hayes D, Kumar A, Neyenhuis B, Vasseur R, Potter AC. Dynamical topological phase realized in a trapped-ion quantum simulator. Nature 2022; 607:463-467. [PMID: 35859195 DOI: 10.1038/s41586-022-04853-4] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/15/2021] [Accepted: 05/11/2022] [Indexed: 11/09/2022]
Abstract
Nascent platforms for programmable quantum simulation offer unprecedented access to new regimes of far-from-equilibrium quantum many-body dynamics in almost isolated systems. Here achieving precise control over quantum many-body entanglement is an essential task for quantum sensing and computation. Extensive theoretical work indicates that these capabilities can enable dynamical phases and critical phenomena that show topologically robust methods to create, protect and manipulate quantum entanglement that self-correct against large classes of errors. However, so far, experimental realizations have been confined to classical (non-entangled) symmetry-breaking orders1-5. In this work, we demonstrate an emergent dynamical symmetry-protected topological phase6, in a quasiperiodically driven array of ten 171Yb+ hyperfine qubits in Quantinuum's System Model H1 trapped-ion quantum processor7. This phase shows edge qubits that are dynamically protected from control errors, cross-talk and stray fields. Crucially, this edge protection relies purely on emergent dynamical symmetries that are absolutely stable to generic coherent perturbations. This property is special to quasiperiodically driven systems: as we demonstrate, the analogous edge states of a periodically driven qubit array are vulnerable to symmetry-breaking errors and quickly decohere. Our work paves the way for implementation of more complex dynamical topological orders8,9 that would enable error-resilient manipulation of quantum information.
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Affiliation(s)
- Philipp T Dumitrescu
- Center for Computational Quantum Physics, Flatiron Institute, New York, NY, USA.
| | | | | | | | | | - Ajesh Kumar
- Department of Physics, University of Texas at Austin, Austin, TX, USA
| | | | - Romain Vasseur
- Department of Physics, University of Massachusetts, Amherst, MA, USA
| | - Andrew C Potter
- Department of Physics, University of Texas at Austin, Austin, TX, USA. .,Department of Physics and Astronomy, and Quantum Matter Institute, University of British Columbia, Vancouver, British Columbia, Canada.
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20
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Kongkhambut P, Skulte J, Mathey L, Cosme JG, Hemmerich A, Keßler H. Observation of a continuous time crystal. Science 2022; 377:670-673. [PMID: 35679353 DOI: 10.1126/science.abo3382] [Citation(s) in RCA: 11] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/02/2022]
Abstract
Time crystals are classified as discrete or continuous depending on whether they spontaneously break discrete or continuous time translation symmetry. While discrete time crystals have been extensively studied in periodically driven systems, the experimental realization of a continuous time crystal is still pending. We report the observation of a limit cycle phase in a continuously pumped dissipative atom-cavity system, that is characterized by emergent oscillations in the intracavity photon number. The phase of the oscillation found to be random for different realizations, and hence this dynamical many-body state breaks continuous time translation symmetry spontaneously. Furthermore, the observed limit cycles are robust against temporal perturbations and therefore demonstrate the realization of a continuous time crystal.
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Affiliation(s)
- Phatthamon Kongkhambut
- Zentrum für Optische Quantentechnologien and Institut für Laser-Physik, Universität Hamburg, 22761 Hamburg, Germany
| | - Jim Skulte
- Zentrum für Optische Quantentechnologien and Institut für Laser-Physik, Universität Hamburg, 22761 Hamburg, Germany.,The Hamburg Center for Ultrafast Imaging, 22761 Hamburg, Germany
| | - Ludwig Mathey
- Zentrum für Optische Quantentechnologien and Institut für Laser-Physik, Universität Hamburg, 22761 Hamburg, Germany.,The Hamburg Center for Ultrafast Imaging, 22761 Hamburg, Germany
| | - Jayson G Cosme
- National Institute of Physics, University of the Philippines, Diliman, Quezon City 1101, Philippines
| | - Andreas Hemmerich
- Zentrum für Optische Quantentechnologien and Institut für Laser-Physik, Universität Hamburg, 22761 Hamburg, Germany.,The Hamburg Center for Ultrafast Imaging, 22761 Hamburg, Germany
| | - Hans Keßler
- Zentrum für Optische Quantentechnologien and Institut für Laser-Physik, Universität Hamburg, 22761 Hamburg, Germany
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21
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Sarkar S, Dubi Y. Emergence and Dynamical Stability of a Charge Time-Crystal in a Current-Carrying Quantum Dot Simulator. NANO LETTERS 2022; 22:4445-4451. [PMID: 35580301 DOI: 10.1021/acs.nanolett.2c00976] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/15/2023]
Abstract
Periodically driven open quantum systems that never thermalize exhibit a discrete time-crystal behavior, a nonequilibrium quantum phenomenon that has shown promise in quantum information processing applications. Measurements of time-crystallinity are currently limited to (magneto-) optical experiments in atom-cavity systems and spin-systems making it an indirect measurement. We theoretically show that time-crystallinity can be measured directly in the charge-current from a spin-less Hubbard ladder, which can be simulated on a quantum-dot array. We demonstrate that one can dynamically tune the system out and then back on a time-crystal phase, proving its robustness against external forcings. These findings motivate further theoretical and experimental efforts to simulate the time-crystal phenomena in current-carrying nanoscale systems.
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Affiliation(s)
- Subhajit Sarkar
- Department of Chemistry, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel
- School of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel
| | - Yonatan Dubi
- Department of Chemistry, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel
- Ilse Katz Center for Nanoscale Science and Technology, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel
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22
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Abobeih MH, Wang Y, Randall J, Loenen SJH, Bradley CE, Markham M, Twitchen DJ, Terhal BM, Taminiau TH. Fault-tolerant operation of a logical qubit in a diamond quantum processor. Nature 2022; 606:884-889. [PMID: 35512730 PMCID: PMC9242857 DOI: 10.1038/s41586-022-04819-6] [Citation(s) in RCA: 27] [Impact Index Per Article: 13.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/10/2021] [Accepted: 04/28/2022] [Indexed: 11/24/2022]
Abstract
Solid-state spin qubits is a promising platform for quantum computation and quantum networks1,2. Recent experiments have demonstrated high-quality control over multi-qubit systems3–8, elementary quantum algorithms8–11 and non-fault-tolerant error correction12–14. Large-scale systems will require using error-corrected logical qubits that are operated fault tolerantly, so that reliable computation becomes possible despite noisy operations15–18. Overcoming imperfections in this way remains an important outstanding challenge for quantum science15,19–27. Here, we demonstrate fault-tolerant operations on a logical qubit using spin qubits in diamond. Our approach is based on the five-qubit code with a recently discovered flag protocol that enables fault tolerance using a total of seven qubits28–30. We encode the logical qubit using a new protocol based on repeated multi-qubit measurements and show that it outperforms non-fault-tolerant encoding schemes. We then fault-tolerantly manipulate the logical qubit through a complete set of single-qubit Clifford gates. Finally, we demonstrate flagged stabilizer measurements with real-time processing of the outcomes. Such measurements are a primitive for fault-tolerant quantum error correction. Although future improvements in fidelity and the number of qubits will be required to suppress logical error rates below the physical error rates, our realization of fault-tolerant protocols on the logical-qubit level is a key step towards quantum information processing based on solid-state spins. By using a five-qubit error-correcting code with a recently discovered flag protocol, a logical qubit that is operated fault-tolerantly is realized based on solid-state spin qubits in diamond.
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Affiliation(s)
- M H Abobeih
- QuTech, Delft University of Technology, Delft, The Netherlands.,Kavli Institute of Nanoscience Delft, Delft University of Technology, Delft, The Netherlands
| | - Y Wang
- QuTech, Delft University of Technology, Delft, The Netherlands
| | - J Randall
- QuTech, Delft University of Technology, Delft, The Netherlands.,Kavli Institute of Nanoscience Delft, Delft University of Technology, Delft, The Netherlands
| | - S J H Loenen
- QuTech, Delft University of Technology, Delft, The Netherlands.,Kavli Institute of Nanoscience Delft, Delft University of Technology, Delft, The Netherlands
| | - C E Bradley
- QuTech, Delft University of Technology, Delft, The Netherlands.,Kavli Institute of Nanoscience Delft, Delft University of Technology, Delft, The Netherlands
| | | | | | - B M Terhal
- QuTech, Delft University of Technology, Delft, The Netherlands.,JARA Institute for Quantum Information, Forschungszentrum Juelich, Juelich, Germany
| | - T H Taminiau
- QuTech, Delft University of Technology, Delft, The Netherlands. .,Kavli Institute of Nanoscience Delft, Delft University of Technology, Delft, The Netherlands.
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23
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Frey P, Rachel S. Realization of a discrete time crystal on 57 qubits of a quantum computer. SCIENCE ADVANCES 2022; 8:eabm7652. [PMID: 35235347 PMCID: PMC8890700 DOI: 10.1126/sciadv.abm7652] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 10/10/2021] [Accepted: 01/07/2022] [Indexed: 06/14/2023]
Abstract
Unconventional dynamical phases that violate ergodicity have been a subject of extensive research in recent years. A periodically driven system is naively expected to lose all memory of its initial state due to thermalization, yet this can be avoided in the presence of many-body localization. A discrete time crystal represents a driven system whose local observables spontaneously break time translation symmetry and retain memory of the initial state indefinitely. Here, we report the observation of a discrete time crystal on a chain consisting of 57 superconducting qubits on a state-of-the-art quantum computer. We probe random initial states and compare the cases of vanishing and finite disorder to distinguish many-body localization from prethermal dynamics. We further report results on the dynamical phase transition between the discrete time crystal and a thermal regime, which is observed via critical fluctuations in the system's subharmonic frequency response and a substantial speedup of spin depolarization.
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24
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Ruskuc A, Wu CJ, Rochman J, Choi J, Faraon A. Nuclear spin-wave quantum register for a solid-state qubit. Nature 2022; 602:408-413. [PMID: 35173343 DOI: 10.1038/s41586-021-04293-6] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/29/2021] [Accepted: 11/29/2021] [Indexed: 11/09/2022]
Abstract
Solid-state nuclear spins surrounding individual, optically addressable qubits1,2 are a crucial resource for quantum networks3-6, computation7-11 and simulation12. Although hosts with sparse nuclear spin baths are typically chosen to mitigate qubit decoherence13, developing coherent quantum systems in nuclear-spin-rich hosts enables exploration of a much broader range of materials for quantum information applications. The collective modes of these dense nuclear spin ensembles provide a natural basis for quantum storage14; however, using them as a resource for single-spin qubits has thus far remained elusive. Here, by using a highly coherent, optically addressed 171Yb3+ qubit doped into a nuclear-spin-rich yttrium orthovanadate crystal15, we develop a robust quantum control protocol to manipulate the multi-level nuclear spin states of neighbouring 51V5+ lattice ions. Via a dynamically engineered spin-exchange interaction, we polarize this nuclear spin ensemble, generate collective spin excitations, and subsequently use them to implement a quantum memory. We additionally demonstrate preparation and measurement of maximally entangled 171Yb-51V Bell states. Unlike conventional, disordered nuclear-spin-based quantum memories16-24, our platform is deterministic and reproducible, ensuring identical quantum registers for all 171Yb3+ qubits. Our approach provides a framework for utilizing the complex structure of dense nuclear spin baths, paving the way towards building large-scale quantum networks using single rare-earth ion qubits15,25-28.
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Affiliation(s)
- Andrei Ruskuc
- Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA, USA.,Kavli Nanoscience Institute, California Institute of Technology, Pasadena, CA, USA.,Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA
| | - Chun-Ju Wu
- Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA, USA.,Kavli Nanoscience Institute, California Institute of Technology, Pasadena, CA, USA.,Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA.,Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA, USA
| | - Jake Rochman
- Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA, USA.,Kavli Nanoscience Institute, California Institute of Technology, Pasadena, CA, USA.,Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA
| | - Joonhee Choi
- Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA. .,Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA, USA.
| | - Andrei Faraon
- Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA, USA. .,Kavli Nanoscience Institute, California Institute of Technology, Pasadena, CA, USA. .,Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA.
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25
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Stability of the Discrete Time-Crystalline Order in Spin-Optomechanical and Open Cavity QED Systems. PHOTONICS 2022. [DOI: 10.3390/photonics9020061] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
Discrete time crystals (DTC) have been demonstrated experimentally in several different quantum systems in the past few years. Spin couplings and cavity losses have been shown to play crucial roles for realizing DTC order in open many-body systems out of equilibrium. Recently, it has been proposed that eternal and transient DTC can be present with an open Floquet setup in the thermodynamic limit and in the deep quantum regime with few qubits, respectively. In this work, we consider the effects of spin damping and spin dephasing on the DTC order in spin-optomechanical and open cavity systems in which the spins can be all-to-all coupled. In the thermodynamic limit, it is shown that the existence of dephasing can destroy the coherence of the system and finally lead the system to its trivial steady state. Without dephasing, eternal DTC is displayed in the weak damping regime, which may be destroyed by increasing the all-to-all spin coupling or the spin damping. By contrast, the all-to-all coupling is constructive to the DTC in the moderate damping regime. We also focus on a model which can be experimentally realized by a suspended hexagonal boron nitride (hBN) membrane with a few spin color centers under microwave drive and Floquet magnetic field. Signatures of transient DTC behavior are demonstrated in both weak and moderate dissipation regimes without spin dephasing. Relevant experimental parameters are also discussed for realizing transient DTC order in such an hBN optomechanical system.
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26
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Randall J, Bradley CE, van der Gronden FV, Galicia A, Abobeih MH, Markham M, Twitchen DJ, Machado F, Yao NY, Taminiau TH. Many-body-localized discrete time crystal with a programmable spin-based quantum simulator. Science 2021; 374:1474-1478. [PMID: 34735218 DOI: 10.1126/science.abk0603] [Citation(s) in RCA: 23] [Impact Index Per Article: 7.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/02/2022]
Abstract
[Figure: see text].
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Affiliation(s)
- J Randall
- QuTech, Delft University of Technology, PO Box 5046, 2600 GA Delft, Netherlands.,Kavli Institute of Nanoscience Delft, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
| | - C E Bradley
- QuTech, Delft University of Technology, PO Box 5046, 2600 GA Delft, Netherlands.,Kavli Institute of Nanoscience Delft, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
| | - F V van der Gronden
- QuTech, Delft University of Technology, PO Box 5046, 2600 GA Delft, Netherlands.,Kavli Institute of Nanoscience Delft, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
| | - A Galicia
- QuTech, Delft University of Technology, PO Box 5046, 2600 GA Delft, Netherlands.,Kavli Institute of Nanoscience Delft, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
| | - M H Abobeih
- QuTech, Delft University of Technology, PO Box 5046, 2600 GA Delft, Netherlands.,Kavli Institute of Nanoscience Delft, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
| | - 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
| | - F Machado
- Department of Physics, University of California, Berkeley, CA 94720, USA.,Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
| | - N Y Yao
- Department of Physics, University of California, Berkeley, CA 94720, USA.,Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
| | - T H Taminiau
- QuTech, Delft University of Technology, PO Box 5046, 2600 GA Delft, Netherlands.,Kavli Institute of Nanoscience Delft, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
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27
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Mi X, Ippoliti M, Quintana C, Greene A, Chen Z, Gross J, Arute F, Arya K, Atalaya J, Babbush R, Bardin JC, Basso J, Bengtsson A, Bilmes A, Bourassa A, Brill L, Broughton M, Buckley BB, Buell DA, Burkett B, Bushnell N, Chiaro B, Collins R, Courtney W, Debroy D, Demura S, Derk AR, Dunsworth A, Eppens D, Erickson C, Farhi E, Fowler AG, Foxen B, Gidney C, Giustina M, Harrigan MP, Harrington SD, Hilton J, Ho A, Hong S, Huang T, Huff A, Huggins WJ, Ioffe LB, Isakov SV, Iveland J, Jeffrey E, Jiang Z, Jones C, Kafri D, Khattar T, Kim S, Kitaev A, Klimov PV, Korotkov AN, Kostritsa F, Landhuis D, Laptev P, Lee J, Lee K, Locharla A, Lucero E, Martin O, McClean JR, McCourt T, McEwen M, Miao KC, Mohseni M, Montazeri S, Mruczkiewicz W, Naaman O, Neeley M, Neill C, Newman M, Niu MY, O'Brien TE, Opremcak A, Ostby E, Pato B, Petukhov A, Rubin NC, Sank D, Satzinger KJ, Shvarts V, Su Y, Strain D, Szalay M, Trevithick MD, Villalonga B, White T, Yao ZJ, Yeh P, Yoo J, Zalcman A, Neven H, Boixo S, Smelyanskiy V, Megrant A, Kelly J, Chen Y, Sondhi SL, Moessner R, Kechedzhi K, Khemani V, Roushan P. Time-Crystalline Eigenstate Order on a Quantum Processor. Nature 2021; 601:531-536. [PMID: 34847568 PMCID: PMC8791837 DOI: 10.1038/s41586-021-04257-w] [Citation(s) in RCA: 33] [Impact Index Per Article: 11.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/30/2021] [Accepted: 11/17/2021] [Indexed: 11/10/2022]
Abstract
Quantum many-body systems display rich phase structure in their low-temperature equilibrium states1. However, much of nature is not in thermal equilibrium. Remarkably, it was recently predicted that out-of-equilibrium systems can exhibit novel dynamical phases2–8 that may otherwise be forbidden by equilibrium thermodynamics, a paradigmatic example being the discrete time crystal (DTC)7,9–15. Concretely, dynamical phases can be defined in periodically driven many-body-localized (MBL) systems via the concept of eigenstate order7,16,17. In eigenstate-ordered MBL phases, the entire many-body spectrum exhibits quantum correlations and long-range order, with characteristic signatures in late-time dynamics from all initial states. It is, however, challenging to experimentally distinguish such stable phases from transient phenomena, or from regimes in which the dynamics of a few select states can mask typical behaviour. Here we implement tunable controlled-phase (CPHASE) gates on an array of superconducting qubits to experimentally observe an MBL-DTC and demonstrate its characteristic spatiotemporal response for generic initial states7,9,10. Our work employs a time-reversal protocol to quantify the impact of external decoherence, and leverages quantum typicality to circumvent the exponential cost of densely sampling the eigenspectrum. Furthermore, we locate the phase transition out of the DTC with an experimental finite-size analysis. These results establish a scalable approach to studying non-equilibrium phases of matter on quantum processors. A study establishes a scalable approach to engineer and characterize a many-body-localized discrete time crystal phase on a superconducting quantum processor.
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Affiliation(s)
- Xiao Mi
- Google Research, Mountain View, CA, USA
| | - Matteo Ippoliti
- Department of Physics, Stanford University, Stanford, CA, USA
| | | | | | | | | | | | | | | | | | - Joseph C Bardin
- Google Research, Mountain View, CA, USA.,Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA, USA
| | | | | | | | - Alexandre Bourassa
- Google Research, Mountain View, CA, USA.,Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL, USA
| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | - Alan Ho
- Google Research, Mountain View, CA, USA
| | | | | | | | | | - L B Ioffe
- Google Research, Mountain View, CA, USA
| | | | | | | | | | | | | | | | - Seon Kim
- Google Research, Mountain View, CA, USA
| | | | | | - Alexander N Korotkov
- Google Research, Mountain View, CA, USA.,Department of Electrical and Computer Engineering, University of California, Riverside, CA, USA
| | | | | | | | - Joonho Lee
- Google Research, Mountain View, CA, USA.,Department of Chemistry, Columbia University, New York, NY, 10027, USA
| | - Kenny Lee
- Google Research, Mountain View, CA, USA
| | | | | | | | | | | | - Matt McEwen
- Google Research, Mountain View, CA, USA.,Department of Physics, University of California, Santa Barbara, CA, USA
| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | - Yuan Su
- Google Research, Mountain View, CA, USA
| | | | | | | | | | | | | | - Ping Yeh
- Google Research, Mountain View, CA, USA
| | | | | | | | | | | | | | | | - Yu Chen
- Google Research, Mountain View, CA, USA
| | - S L Sondhi
- Department of Physics, Princeton University, Princeton, NJ, USA.,Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford, OX1 3PU, United Kingdom
| | - Roderich Moessner
- Max-Planck-Institut für Physik komplexer Systeme, 01187, Dresden, Germany
| | | | - Vedika Khemani
- Department of Physics, Stanford University, Stanford, CA, USA.
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