{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T20:37:35Z","timestamp":1788295055801,"version":"build-2803163510"},"reference-count":61,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2016,2,11]],"date-time":"2016-02-11T00:00:00Z","timestamp":1455148800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2016,2,11]],"date-time":"2016-02-11T00:00:00Z","timestamp":1455148800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Sci Rep"],"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Phase transitions have been a research focus in many-body physics over past decades. Cold ions, under strong Coulomb repulsion, provide a repealing paradigm of exploring phase transitions in stable confinement by electromagnetic field. We demonstrate various conformations of up to sixteen laser-cooled <jats:sup>40<\/jats:sup>Ca<jats:sup>+<\/jats:sup> ion crystals in a home-built surface-electrode trap, where besides the usually mentioned structural phase transition from the linear to the zigzag, two additional phase transitions to more complicated two-dimensional configurations are identified. The experimental observation agrees well with the numerical simulation. Heating due to micromotion of the ions is analysed by comparison of the numerical simulation with the experimental observation. Our investigation implies very rich and complicated many-body behaviour in the trapped-ion systems and provides effective mechanism for further exploring quantum phase transitions and quantum information processing with ultracold trapped ions.<\/jats:p>","DOI":"10.1038\/srep21547","type":"journal-article","created":{"date-parts":[[2016,2,11]],"date-time":"2016-02-11T10:28:08Z","timestamp":1455186488000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":40,"title":["Exploring structural phase transitions of ion crystals"],"prefix":"10.1038","volume":"6","author":[{"given":"L. L.","family":"Yan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"W.","family":"Wan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"L.","family":"Chen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"F.","family":"Zhou","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"S. J.","family":"Gong","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"X.","family":"Tong","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"M.","family":"Feng","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2016,2,11]]},"reference":[{"key":"BFsrep21547_CR1","doi-asserted-by":"publisher","first-page":"2931","DOI":"10.1103\/PhysRevLett.59.2931","volume":"59","author":"F Diedrich","year":"1987","unstructured":"Diedrich, F., Peik, E., Chen, J. M., Quint, W. & Walther, H. Observation of a phase transition of stored laser-cooled ions. Phys. Rev. Lett. 59, 2931\u20132934 (1987).","journal-title":"Phys. Rev. Lett."},{"key":"BFsrep21547_CR2","doi-asserted-by":"publisher","first-page":"2935","DOI":"10.1103\/PhysRevLett.59.2935","volume":"59","author":"DJ Wineland","year":"1987","unstructured":"Wineland, D. J., Bergquist, J. C., Itano, W. M., Bollinger, J. J. & Manney, C. H. Atomic-ion Coulomb clusters in an ion trap. Phys. Rev. Lett. 59, 2935\u20132938 (1987).","journal-title":"Phys. Rev. Lett."},{"key":"BFsrep21547_CR3","doi-asserted-by":"publisher","first-page":"686","DOI":"10.1126\/science.279.5351.686","volume":"279","author":"WM Itano","year":"1998","unstructured":"Itano, W. M. et al. Bragg diffraction from crystallized ion plasmas. Science 279, 686\u2013689 (1998).","journal-title":"Science"},{"key":"BFsrep21547_CR4","doi-asserted-by":"publisher","first-page":"2878","DOI":"10.1103\/PhysRevLett.81.2878","volume":"81","author":"M Drewsen","year":"1998","unstructured":"Drewsen, M., Brodersen, C., Hornekaer, L., Hangst, J. S. & Schiffer, J. P. Large ion crystals in a linear Paul trap. Phys. Rev. Lett. 81, 2878\u20132881 (1998).","journal-title":"Phys. Rev. Lett."},{"key":"BFsrep21547_CR5","doi-asserted-by":"publisher","first-page":"603","DOI":"10.1007\/s003400050443","volume":"66","author":"HC N\u00e4gerl","year":"1998","unstructured":"N\u00e4gerl, H. C., Bechter, W., Eschner, J., Schmidt-Kaler, F. & Blatt, R. Ion strings for quantum gates. Appl. Phys. B 66, 603\u2013608 (1998).","journal-title":"Appl. Phys. B"},{"key":"BFsrep21547_CR6","doi-asserted-by":"publisher","first-page":"1751","DOI":"10.1063\/1.873433","volume":"6","author":"TB Mitchell","year":"1999","unstructured":"Mitchell, T. B., Bollinger, J. J., Huang, X.-P., Itano, W. M. & Dubin, D. H. E. Direct observations of the structural phases of crystallized ion plasmas. Phys. Plasmas 6, 1751 (1999).","journal-title":"Phys. Plasmas"},{"key":"BFsrep21547_CR7","doi-asserted-by":"publisher","first-page":"L375","DOI":"10.1088\/0953-4075\/33\/11\/101","volume":"33","author":"M Block","year":"2000","unstructured":"Block, M., Drakoudis, A., Leuthner, H., Seibert, P. & Werth, G. Crystalline ion structures in a Paul trap. J. Phys. B 33, L375\u2013L382 (2000).","journal-title":"J. Phys. B"},{"key":"BFsrep21547_CR8","doi-asserted-by":"publisher","first-page":"033409","DOI":"10.1103\/PhysRevA.75.033409","volume":"75","author":"K Okada","year":"2007","unstructured":"Okada, K. et al. Crystallization of Ca+ ions in a linear rf octupole ion trap. Phys. Rev. A 75, 033409 (2007).","journal-title":"Phys. Rev. A"},{"key":"BFsrep21547_CR9","doi-asserted-by":"publisher","first-page":"043435","DOI":"10.1103\/PhysRevA.86.043435","volume":"86","author":"P Horak","year":"2012","unstructured":"Horak, P., Dantan, A. & Drewsen, M. Optically induced structural phase transitions in ion Coulomb crystals. Phys. Rev. A 86, 043435 (2012).","journal-title":"Phys. Rev. A"},{"key":"BFsrep21547_CR10","doi-asserted-by":"publisher","first-page":"489","DOI":"10.1038\/nature10981","volume":"484","author":"JW Britton","year":"2012","unstructured":"Britton, J. W. et al. Engineered two-dimensional Ising interactions in a trapped-ion quantum simulator with hundreds of spins. Nature (London) 484, 489\u2013492 (2012).","journal-title":"Nature (London)"},{"key":"BFsrep21547_CR11","doi-asserted-by":"publisher","first-page":"213003","DOI":"10.1103\/PhysRevLett.108.213003","volume":"108","author":"BC Sawyer","year":"2012","unstructured":"Sawyer, B. C. et al. Spectroscopy and thermometry of drumhead modes in a mesoscopic trapped-ion crystal using entanglement. Phys. Rev. Lett. 108, 213003 (2012).","journal-title":"Phys. Rev. Lett."},{"key":"BFsrep21547_CR12","doi-asserted-by":"publisher","first-page":"2571","DOI":"10.1038\/ncomms3571","volume":"4","author":"S Mavadia","year":"2013","unstructured":"Mavadia, S. et al. Control of the conformations of ion Coulomb crystals in a Penning trap. Nat. Commun. 4, 2571 (2013).","journal-title":"Nat. Commun."},{"key":"BFsrep21547_CR13","doi-asserted-by":"publisher","first-page":"818","DOI":"10.1103\/PhysRevLett.70.818","volume":"70","author":"JP Schiffer","year":"1993","unstructured":"Schiffer, J. P. Phase transitions in anisotropically confined ionic crystals. Phys. Rev. Lett. 70, 818\u2013821 (1993).","journal-title":"Phys. Rev. Lett."},{"key":"BFsrep21547_CR14","doi-asserted-by":"publisher","first-page":"2753","DOI":"10.1103\/PhysRevLett.71.2753","volume":"71","author":"DHE Dubin","year":"1993","unstructured":"Dubin, D. H. E. Theory of structural phase transitions in a trapped Coulomb crystal. Phys. Rev. Lett. 71, 2753\u20132756 (1993).","journal-title":"Phys. Rev. Lett."},{"key":"BFsrep21547_CR15","doi-asserted-by":"publisher","first-page":"2466","DOI":"10.1103\/PhysRevLett.85.2466","volume":"85","author":"DG Enzer","year":"2000","unstructured":"Enzer, D. G. et al. Observation of power-law scaling for phase transitions in linear trapped ion crystals. Phys. Rev. Lett. 85, 2466\u20132469 (2000).","journal-title":"Phys. Rev. Lett."},{"key":"BFsrep21547_CR16","doi-asserted-by":"publisher","first-page":"095002","DOI":"10.1103\/PhysRevLett.91.095002","volume":"91","author":"N Kjaergaard","year":"2003","unstructured":"Kjaergaard, N. & Drewsen, M. Observation of a structural transition for Coulomb crystals in a linear Paul trap. Phys. Rev. Lett. 91, 095002 (2003).","journal-title":"Phys. Rev. Lett."},{"key":"BFsrep21547_CR17","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1103\/RevModPhys.71.87","volume":"71","author":"DHE Dubin","year":"1999","unstructured":"Dubin, D. H. E. & Neil, T. M. O. Trapped nonneutral plasmas, liquids and crystals (the thermal equilibrium states). Rev. Mod. Phys. 71, 87\u2013172 (1999).","journal-title":"Rev. Mod. Phys."},{"key":"BFsrep21547_CR18","doi-asserted-by":"publisher","first-page":"2290","DOI":"10.1038\/ncomms3290","volume":"4","author":"S Ulm","year":"2013","unstructured":"Ulm, S. et al. Observation of the Kibble-Zurek scaling law for defect formation in ion crystals. Nat. Commun. 4, 2290 (2013).","journal-title":"Nat. Commun."},{"key":"BFsrep21547_CR19","doi-asserted-by":"publisher","first-page":"2291","DOI":"10.1038\/ncomms3291","volume":"4","author":"K Pyka","year":"2013","unstructured":"Pyka, K. et al. Topological defect formation and spontaneous symmetry breaking in ion Coulomb crystals. Nat. Commun. 4, 2291 (2013).","journal-title":"Nat. Commun."},{"key":"BFsrep21547_CR20","doi-asserted-by":"publisher","first-page":"133004","DOI":"10.1103\/PhysRevLett.110.133004","volume":"110","author":"M Mielenz","year":"2013","unstructured":"Mielenz, M. et al. Trapping of topological-structural defects in Coulomb crystals. Phys. Rev. Lett. 110, 133004 (2013).","journal-title":"Phys. Rev. Lett."},{"key":"BFsrep21547_CR21","doi-asserted-by":"publisher","first-page":"093003","DOI":"10.1088\/1367-2630\/15\/9\/093003","volume":"15","author":"H Landa","year":"2013","unstructured":"Landa, H., Reznik, B., Brox, J., Mielenz, M. & Schaetz, T. Structure dynamics and bifurcations of discrete solitons in trapped ion crystals. New J. Phys. 15, 093003 (2013).","journal-title":"New J. Phys."},{"key":"BFsrep21547_CR22","doi-asserted-by":"publisher","first-page":"1387","DOI":"10.1088\/0305-4470\/9\/8\/029","volume":"9","author":"TWB Kibble","year":"1976","unstructured":"Kibble, T. W. B. Topology of cosmic domains and strings. J. Phys. A 9, 1387 (1976).","journal-title":"J. Phys. A"},{"key":"BFsrep21547_CR23","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1016\/0370-1573(80)90091-5","volume":"67","author":"TWB Kibble","year":"1980","unstructured":"Kibble, T. W. B. Some implications of a cosmological phase transition. Phys. Rep. 67, 183\u2013199 (1980).","journal-title":"Phys. Rep."},{"key":"BFsrep21547_CR24","doi-asserted-by":"publisher","first-page":"505","DOI":"10.1038\/317505a0","volume":"317","author":"WH Zurek","year":"1985","unstructured":"Zurek, W. H. Cosmological experiments in superfluid helium. Nature (London) 317, 505\u2013508 (1985).","journal-title":"Nature (London)"},{"key":"BFsrep21547_CR25","doi-asserted-by":"publisher","first-page":"010401","DOI":"10.1103\/PhysRevLett.106.010401","volume":"106","author":"E Shimshoni","year":"2011","unstructured":"Shimshoni, E., Morigi, G. & Fishman, S. Quantum zigzag transition in ion chains. Phys. Rev. Lett. 106, 010401 (2011).","journal-title":"Phys. Rev. Lett."},{"key":"BFsrep21547_CR26","doi-asserted-by":"publisher","first-page":"045017","DOI":"10.1088\/1367-2630\/10\/4\/045017","volume":"10","author":"R Schmied","year":"2008","unstructured":"Schmied, R., Roscilde, T., Murg, V., Porras, D. & Cirac, J. I. Quantum phases of trapped ions in an optical lattice. New J. Phys. 10 045017 (2008).","journal-title":"New J. Phys."},{"key":"BFsrep21547_CR27","doi-asserted-by":"publisher","first-page":"207209","DOI":"10.1103\/PhysRevLett.107.207209","volume":"107","author":"A Bermudez","year":"2011","unstructured":"Bermudez, A., Almeida, J., Schmidt-Kaler, F., Retzker, A. & Plenio, M. B. Frustrated quantum spin models with cold Coulomb crystals. Phys. Rev. Lett. 107, 207209 (2011).","journal-title":"Phys. Rev. Lett."},{"key":"BFsrep21547_CR28","doi-asserted-by":"publisher","first-page":"265703","DOI":"10.1103\/PhysRevLett.105.265703","volume":"105","author":"ZX Gong","year":"2010","unstructured":"Gong, Z. X., Lin, G. D. & Duan, L. M. Temperature-driven structural phase transition for trapped ions and a proposal for its experimental detection. Phys. Rev. Lett. 105, 265703 (2010).","journal-title":"Phys. Rev. Lett."},{"key":"BFsrep21547_CR29","doi-asserted-by":"publisher","first-page":"214408","DOI":"10.1103\/PhysRevB.89.214408","volume":"89","author":"D Podolsky","year":"2014","unstructured":"Podolsky, D. et al. From classical to quantum criticality. Phys. Rev. B 89, 214408 (2014).","journal-title":"Phys. Rev. B"},{"key":"BFsrep21547_CR30","doi-asserted-by":"publisher","first-page":"259","DOI":"10.6028\/jres.103.019","volume":"103","author":"DJ Wineland","year":"1998","unstructured":"Wineland, D. J. et al. Experimental issues in coherent quantum-state manipulation of trapped atomic ions. J. Res. NIST 103, 259 (1998).","journal-title":"J. Res. NIST"},{"key":"BFsrep21547_CR31","doi-asserted-by":"publisher","first-page":"709","DOI":"10.1038\/nature00784","volume":"417","author":"D Kielpinski","year":"2002","unstructured":"Kielpinski, D., Monroe, C. & Wineland, D. J. Architecture for a large-scale ion-trap quantum computer. Nature (London) 417, 709\u2013711 (2002).","journal-title":"Nature (London)"},{"key":"BFsrep21547_CR32","first-page":"419","volume":"5","author":"J Chiaverini","year":"2005","unstructured":"Chiaverini, J. et al. Surface-electrode architecture for ion-trap quantum information processing. Quant. Inf. Comput. 5, 419 (2005).","journal-title":"Inf. Comput."},{"key":"BFsrep21547_CR33","doi-asserted-by":"publisher","first-page":"1227","DOI":"10.1126\/science.1177077","volume":"325","author":"JP Home","year":"2009","unstructured":"Home, J. P. et al. Complete methods set for scalable ion trap quantum information processing. Science 325, 1227\u20131230 (2009).","journal-title":"Science"},{"key":"BFsrep21547_CR34","doi-asserted-by":"publisher","first-page":"1164","DOI":"10.1126\/science.1231298","volume":"339","author":"C Monroe","year":"2013","unstructured":"Monroe, C. & Kim, J. Scaling the ion trap quantum processor. Science 339, 1164\u20131169 (2013).","journal-title":"Science"},{"key":"BFsrep21547_CR35","doi-asserted-by":"publisher","first-page":"3637","DOI":"10.1038\/ncomms4637","volume":"5","author":"RC Sterling","year":"2013","unstructured":"Sterling, R. C. et al. Fabrication and operation of a two-dimensional ion-trap lattice on a high-voltage microchip. Nat. Commun. 5, 3637 (2013).","journal-title":"Nat. Commun."},{"key":"BFsrep21547_CR36","doi-asserted-by":"publisher","first-page":"013114","DOI":"10.1063\/1.3056227","volume":"105","author":"RJ Clark","year":"2009","unstructured":"Clark, R. J., Lin, T., Brown, K. R. & Chuang, I. L. A two-dimensional lattice ion trap for quantum simulation. J. Appl. Phys. 105, 013114 (2009).","journal-title":"J. Appl. Phys."},{"key":"BFsrep21547_CR37","doi-asserted-by":"publisher","first-page":"171110","DOI":"10.1063\/1.4705153","volume":"100","author":"B Szymanski","year":"2012","unstructured":"Szymanski, B. et al. Large two dimensional Coulomb crystals in a radio frequency surface ion trap. Appl. Phys. Lett. 100, 171110 (2012).","journal-title":"Appl. Phys. Lett."},{"key":"BFsrep21547_CR38","doi-asserted-by":"publisher","first-page":"063062","DOI":"10.1088\/1367-2630\/16\/6\/063062","volume":"16","author":"M Ramm","year":"2014","unstructured":"Ramm, M., Pruttivarasin, T. & H\u00e4ffner, H. Energy transport in trapped ion chains. New J. Phys. 16, 063062 (2014).","journal-title":"New J. Phys."},{"key":"BFsrep21547_CR39","doi-asserted-by":"publisher","first-page":"030602","DOI":"10.1103\/PhysRevLett.112.030602","volume":"112","author":"A Ruiz","year":"2014","unstructured":"Ruiz, A., Alonso, D., Plenio, M. B. & del Campo, A. Tuning heat transport in trapped-ion chains across a structural phase transition. Phys. Rev. Lett. 112, 030602 (2014).","journal-title":"Phys. Rev. Lett."},{"key":"BFsrep21547_CR40","doi-asserted-by":"publisher","first-page":"170501","DOI":"10.1103\/PhysRevLett.102.170501","volume":"102","author":"R Stock","year":"2009","unstructured":"Stock, R. & James, D. F. V. Scalable, high-speed measurement-based quantum computer using trapped ions. Phys. Rev. Lett. 102, 170501 (2009).","journal-title":"Phys. Rev. Lett."},{"key":"BFsrep21547_CR41","doi-asserted-by":"publisher","first-page":"210501","DOI":"10.1103\/PhysRevLett.111.210501","volume":"111","author":"BP Lanyon","year":"2013","unstructured":"Lanyon, B. P. et al. Measurement-based quantum computation with trapped ions. Phys. Rev. Lett. 111, 210501 (2013).","journal-title":"Phys. Rev. Lett."},{"key":"BFsrep21547_CR42","doi-asserted-by":"publisher","first-page":"302","DOI":"10.1126\/science.1253742","volume":"18","author":"D Nigg","year":"2014","unstructured":"Nigg, D. et al. Quantum computations on a topologically encoded qubit. Science 18, 302\u2013305 (2014).","journal-title":"Science"},{"key":"BFsrep21547_CR43","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1103\/RevModPhys.87.307","volume":"87","author":"BM Terhal","year":"2015","unstructured":"Terhal, B. M. Quantum error correction for quantum memories. Rev. Mod. Phys. 87, 307 (2015).","journal-title":"Rev. Mod. Phys."},{"key":"BFsrep21547_CR44","first-page":"021041","volume":"4","author":"T Lanting","year":"2014","unstructured":"Lanting, T. et al. Entanglement in a quantum annealing processor. Phys. Rev. X 4, 021041 (2014).","journal-title":"Phys. Rev. X"},{"key":"BFsrep21547_CR45","doi-asserted-by":"publisher","first-page":"053304","DOI":"10.1103\/PhysRevE.92.053304","volume":"92","author":"EM Inack","year":"2015","unstructured":"Inack, E. M. & Pilati, S. Simulated quantum annealing of double-well and multiwell potentials. Phys. Rev. E 92, 053304 (2015).","journal-title":"Phys. Rev. E"},{"key":"BFsrep21547_CR46","doi-asserted-by":"publisher","first-page":"015401","DOI":"10.1103\/PhysRevA.75.015401","volume":"75","author":"KR Brown","year":"2007","unstructured":"Brown, K. R. et al. Loading and characterization of a printed-circuit-board atomic ion trap. Phys. Rev. A 75, 015401 (2007).","journal-title":"Phys. Rev. A"},{"key":"BFsrep21547_CR47","doi-asserted-by":"publisher","first-page":"154006","DOI":"10.1088\/0953-4075\/42\/15\/154006","volume":"42","author":"U Tanaka","year":"2009","unstructured":"Tanaka, U. et al. Design and characterization of a planar trap. J. Phys. B 42, 154006 (2009).","journal-title":"J. Phys. B"},{"key":"BFsrep21547_CR48","doi-asserted-by":"publisher","first-page":"103008","DOI":"10.1088\/1367-2630\/11\/10\/103008","volume":"11","author":"F Splatt","year":"2009","unstructured":"Splatt, F. et al. Deterministic reordering of 40Ca+ ions in a linear segmented Paul trap. New J. Phys. 11, 103008 (2009).","journal-title":"New J. Phys."},{"key":"BFsrep21547_CR49","doi-asserted-by":"publisher","first-page":"013702","DOI":"10.1088\/0256-307X\/30\/1\/013702","volume":"30","author":"L Chen","year":"2013","unstructured":"Chen, L. et al. Demonstration of cold 40Ga+ ions confined in a microscopic surface-electrode ion trap. Chin. Phys. Lett. 30, 013702 (2013).","journal-title":"Chin. Phys. Lett."},{"key":"BFsrep21547_CR50","doi-asserted-by":"publisher","first-page":"073701","DOI":"10.1088\/0256-307X\/30\/7\/073701","volume":"30","author":"W Wan","year":"2013","unstructured":"Wan, W. et al. Manipulation of ions in microscopic surface-electrode ion traps. Chin. Phys. Lett. 30, 073701 (2013).","journal-title":"Chin. Phys. Lett."},{"key":"BFsrep21547_CR51","doi-asserted-by":"publisher","first-page":"033402","DOI":"10.1103\/PhysRevA.78.033402","volume":"78","author":"MG House","year":"2008","unstructured":"House, M. G. Analytic model for electrostatic fields in surface-electrode ion traps. Phys. Rev A. 78, 033402 (2008).","journal-title":"Phys. Rev A."},{"key":"BFsrep21547_CR52","doi-asserted-by":"publisher","first-page":"765","DOI":"10.1007\/s00340-010-4086-7","volume":"100","author":"C Schuck","year":"2010","unstructured":"Schuck, C., Almendros, M., Rohde, F., Hennrich, M. & Eschner, J. Two-color photoionization of calcium using SHG and LED light. J. Appl. Phys. B 100, 765\u2013771 (2010).","journal-title":"J. Appl. Phys. B"},{"key":"BFsrep21547_CR53","doi-asserted-by":"publisher","first-page":"R061402","DOI":"10.1103\/PhysRevA.82.061402","volume":"82","author":"N Akerman","year":"2010","unstructured":"Akerman, N. et al. Single-ion nonlinear mechanical oscillator. Phys. Rev. A 82, R061402 (2010).","journal-title":"Phys. Rev. A"},{"key":"BFsrep21547_CR54","doi-asserted-by":"publisher","first-page":"063401","DOI":"10.1103\/PhysRevA.89.063401","volume":"89","author":"W Wan","year":"2014","unstructured":"Wan, W. et al. Demonstration of motion transduction in a single-ion nonlinear mechanical oscillator, Phys. Rev. A 89, 063401 (2014).","journal-title":"Phys. Rev. A"},{"key":"BFsrep21547_CR55","doi-asserted-by":"publisher","first-page":"263003","DOI":"10.1103\/PhysRevLett.109.263003","volume":"109","author":"H Kaufmann","year":"2012","unstructured":"Kaufmann, H. et al. Precise experimental investigation of eigenmodes in a planar ion crystal. Phys. Rev. Lett. 109, 263003 (2012).","journal-title":"Phys. Rev. Lett."},{"key":"BFsrep21547_CR56","doi-asserted-by":"publisher","first-page":"073001","DOI":"10.1103\/PhysRevLett.114.073001","volume":"114","author":"A Lemmer","year":"2015","unstructured":"Lemmer, A., Cormick, C., Schmiegelow, C. T., Schmidt-Kaler, F. & Plenio, M. B. Two-dimensional spectroscopy for the study of ion Coulomb crystals. Phys. Rev. Lett. 114, 073001 (2015).","journal-title":"Phys. Rev. Lett."},{"key":"BFsrep21547_CR57","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1103\/RevModPhys.75.281","volume":"75","author":"D Leibfried","year":"2003","unstructured":"Leibfried, D., Blatt, R., Monroe, C. & Wineland, D. J. Quantum dynamics of single trapped ions. Rev. Mod. Phys. 75, 281 (2003).","journal-title":"Rev. Mod. Phys."},{"key":"BFsrep21547_CR58","doi-asserted-by":"publisher","first-page":"036401","DOI":"10.1088\/0034-4885\/73\/3\/036401","volume":"73","author":"K-A Brickman Soderberg","year":"2010","unstructured":"Brickman Soderberg, K.-A. & Monroe, C. Phonon-mediated entanglement for trapped ion quantum computing. Rep. Prog. Phys. 73, 036401 (2010).","journal-title":"Rep. Prog. Phys."},{"key":"BFsrep21547_CR59","doi-asserted-by":"publisher","first-page":"8555","DOI":"10.1038\/srep08555","volume":"5","author":"S-T Wang","year":"2014","unstructured":"Wang, S.-T., Shen, C. & Duan, L.-M. Quantum computation under micromotion in a planar ion crystal. Sci. Rep. 5, 8555 (2014).","journal-title":"Sci. Rep."},{"key":"BFsrep21547_CR60","unstructured":"Snyman, J. A. Practical mathematical optimization: An introduction to basic optimization theory and classical and new gradient-based algorithms Springer Publishing, NY (2005)."},{"key":"BFsrep21547_CR61","doi-asserted-by":"publisher","first-page":"1927","DOI":"10.1080\/0026897031000108113","volume":"101","author":"A Ricci","year":"2003","unstructured":"Ricci, A. & Ciccotti, G. Algorithms for Brownian dynamics. Mol. Phys. 101, 1927\u20131931 (2003).","journal-title":"Mol. Phys."}],"container-title":["Scientific Reports"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.nature.com\/articles\/srep21547","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/srep21547.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/srep21547.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,4]],"date-time":"2023-01-04T14:00:20Z","timestamp":1672840820000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.nature.com\/articles\/srep21547"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,2,11]]},"references-count":61,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2016,8,26]]}},"alternative-id":["BFsrep21547"],"URL":"https:\/\/doi.org\/10.1038\/srep21547","relation":{},"ISSN":["2045-2322"],"issn-type":[{"value":"2045-2322","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,2,11]]},"assertion":[{"value":"10 November 2015","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 January 2016","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 February 2016","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing financial interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"21547"}}