Journal of Low Temperature Physics

, Volume 175, Issue 5–6, pp 633–654 | Cite as

Recent Progress in Quantum Simulation Using Superconducting Circuits

  • G. S. Paraoanu


Quantum systems are notoriously difficult to simulate with classical means. Recently, the idea of using another quantum system—which is experimentally more controllable—as a simulator for the original problem has gained significant momentum. Amongst the experimental platforms studied as quantum simulators, superconducting qubits are one of the most promising, due to relative straightforward scalability, easy design, and integration with standard electronics. Here I review the recent state-of-the art in the field and the prospects for simulating systems ranging from relativistic quantum fields to quantum many-body systems.


Josephson devices Digital and analog quantum simulation Many-body systems Quantum fields Circuit QED 



Financial support from the Academy of Finland (project 263457, and the Center of Excellence “Low Temperature Quantum Phenomena and Devices” project 250280) and FQXi is gratefully acknowledged.


  1. 1.
    R. Feynman, Simulating physics with computers. Int. J. Theor. Phys. 21, 467488 (1982)MathSciNetGoogle Scholar
  2. 2.
    D. Deutsch, Quantum theory, the Church-Turing principle and the universal quantum computer. Proc. R. Soc. Lond. A 400, 97 (1985)zbMATHMathSciNetADSGoogle Scholar
  3. 3.
    S. Lloyd, Universal quantum simulators. Science 273, 1073 (1996)zbMATHMathSciNetADSGoogle Scholar
  4. 4.
    M.A. Nielsen, I.C. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000)zbMATHGoogle Scholar
  5. 5.
    G.S. Paraoanu, Quantum computing: theoretical possibility versus practical possibility. Phys. Perspect. 13, 359 (2011)MathSciNetADSGoogle Scholar
  6. 6.
    I. Buluta, F. Nori, Quantum simulators. Science 326, 108 (2009)ADSGoogle Scholar
  7. 7.
    I. Buluta, S. Ashhab, F. Nori, Natural and artificial atoms for quantum computation. Rep. Prog. Phys. 74, 104401 (2011)ADSGoogle Scholar
  8. 8.
    J.I. Cirac, P. Zoller, Goals and opportunities in quantum simulation. Nat. Phys. 8, 264 (2012)Google Scholar
  9. 9.
    I.M. Georgescu, S. Ashhab, F. Nori, Quantum simulation. Rev. Mod. Phys. 86, 153 (2014)ADSGoogle Scholar
  10. 10.
    P. Hauke, F.M. Cucchietti, L. Tagliacozzo, I. Deutsch, M. Lewenstein, Can one trust quantum simulators? Rep. Prog. Phys. 75, 082401 (2012)ADSGoogle Scholar
  11. 11.
    M. Lewenstein et al., Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond. Adv. Phys. 56, 243 (2007)ADSGoogle Scholar
  12. 12.
    Y. Makhlin, G. Schön, A. Shnirman, Quantum-state engineering with Josephson-junction devices. Rev. Mod. Phys. 73, 357 (2001)ADSGoogle Scholar
  13. 13.
    M.H. Devoret, A. Wallraff, J.M. Martinis, Superconducting qubits: a short review. arXiv:0411174 Google Scholar
  14. 14.
    J. Clarke, F. Wilhelm, Superconducting quantum bits. Nature 453, 1031 (2008)ADSGoogle Scholar
  15. 15.
    G. Wendin, V.S. Shumeiko, Superconducting quantum circuits, qubits and computing, in Handbook of Theoretical and Computational Nanotechnology, ed. by M. Rieth, W. Schommers (American Scientific Publishers, Los Angeles, 2006), pp. 223–309Google Scholar
  16. 16.
    R.J. Schoelkopf, S.M. Girvin, Wiring up quantum systems. Nature 451, 664 (2008)ADSGoogle Scholar
  17. 17.
    J.Q. You, F. Nori, Atomic physics and quantum optics using superconducting circuits. Nature 474, 589 (2011)ADSGoogle Scholar
  18. 18.
    M.H. Devoret, R.J. Schoelkopf, Superconducting circuits for quantum information: an outlook. Science 339, 1169 (2013)MathSciNetADSGoogle Scholar
  19. 19.
    R.C. Bialczak, M. Ansmann, M. Hofheinz, E. Lucero, M. Neeley, A.D. O’Connell, D. Sank, H. Wang, J. Wenner, M. Steffen, A.N. Cleland, J.M. Martinis, Quantum process tomography of a universal entangling gate implemented with Josephson phase qubits. Nat. Phys. 6, 409 (2010)Google Scholar
  20. 20.
    A. Dewes, F.R. Ong, V. Schmitt, R. Lauro, N. Boulant, P. Bertet, D. Vion, D. Esteve, Characterization of a two-transmon processor with individual single-shot qubit readout. Phys. Rev. Lett. 108, 057002 (2012)ADSGoogle Scholar
  21. 21.
    A.O. Niskanen, K. Harrabi, F. Yoshihara, Y. Nakamura, S. Lloyd, J.S. Tsai, Quantum coherent tunable coupling of superconducting qubits. Science 316, 723 (2007)ADSGoogle Scholar
  22. 22.
    J. Majer, J.M. Chow, J.M. Gambetta, J. Koch, B.R. Johnson, J.A. Schreier, L. Frunzio, D.I. Schuster, A.A. Houck, A. Wallraff, A. Blais, M.H. Devoret, S.M. Girvin, R.J. Schoelkopf, Coupling superconducting qubits via a cavity bus. Nature 449, 443–447 (2007)ADSGoogle Scholar
  23. 23.
    R.C. Bialczak, M. Ansmann, M. Hofheinz, M. Lenander, E. Lucero, M. Neeley, A.D. O’Connell, D. Sank, H. Wang, M. Weides, J. Wenner, T. Yamamoto, A.N. Cleland, J.M. Martinis, Quantum coherent tunable coupling of superconducting qubits. Phys. Rev. Lett. 106, 060501 (2011)ADSGoogle Scholar
  24. 24.
    C. Rigetti, A. Blais, M. Devoret, Protocol for universal gates in optimally biased superconducting qubits. Phys. Rev. Lett. 94, 240502 (2005)ADSGoogle Scholar
  25. 25.
    G.S. Paraoanu, Microwave-induced coupling of superconducting qubits. Phys. Rev. B 74, 140504(R) (2006)ADSGoogle Scholar
  26. 26.
    J. Li, K. Chalapat, G.S. Paraoanu, Entanglement of superconducting qubits via microwave fields: classical and quantum regimes. Phys. Rev. B 78, 064503 (2008)ADSGoogle Scholar
  27. 27.
    C. Rigetti, M. Devoret, Fully microwave-tunable universal gates in superconducting qubits with linear couplings and fixed transition frequencies. Phys. Rev. B 81, 134507 (2010)ADSGoogle Scholar
  28. 28.
    J.M. Chow, J.M. Gambetta, A.W. Cross, S.T. Merkel, C. Rigetti, M. Steffen, Microwave-activated conditional-phase gate for superconducting qubits. New J. Phys. 15, 115012 (2013)ADSGoogle Scholar
  29. 29.
    D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve, M.H. Devoret, Manipulating the quantum state of an electrical circuit. Science 296, 886 (2002)ADSGoogle Scholar
  30. 30.
    I. Chiorescu, Y. Nakamura, C.J.P.M. Harmans, J.E. Mooij, Coherent quantum dynamics of a superconducting flux qubit. Science 299, 1869 (2003)ADSGoogle Scholar
  31. 31.
    J. Claudon, F. Balestro, F.W.J. Hekking, O. Buisson, Coherent oscillations in a superconducting multilevel quantum system. Phys. Rev. Lett. 93, 187003 (2004)ADSGoogle Scholar
  32. 32.
    J.M. Martinis, S. Nam, J. Aumentado, C. Urbina, Rabi oscillations in a large Josephson-junction qubit. Phys. Rev. Lett. 89, 117901 (2002)ADSGoogle Scholar
  33. 33.
    G.S. Paraoanu, Running-phase state in a Josephson washboard potential. Phys. Rev. B 72, 134528 (2005)ADSGoogle Scholar
  34. 34.
    J. Walter, E. Tholén, D.B. Haviland, J. Söstrand, Pulse and hold strategy for switching current measurements. Phys. Rev. B 75, 094515 (2007)ADSGoogle Scholar
  35. 35.
    A. Wallraff, D.I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S.M. Girvin, R.J. Schoelkopf, Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature 431, 162 (2004)ADSGoogle Scholar
  36. 36.
    A. Blais, R.-S. Huang, A. Wallraff, S.M. Girvin, R.J. Schoelkopf, Cavity quantum electrodynamics for superconducting electrical circuits: an architecture for quantum computation. Phys. Rev. A 69, 062320 (2004)ADSGoogle Scholar
  37. 37.
    N. Katz, M. Ansmann, R.C. Bialczak, E. Lucero, R. McDermott, M. Neeley, M. Steffen, E.M. Weig, A.N. Cleland, J.M. Martinis, A.N. Korotkov, Coherent state evolution in a superconducting qubit from partial-collapse measurement. Science 312, 1498 (2006)ADSGoogle Scholar
  38. 38.
    G.S. Paraoanu, Interaction-free measurements with superconducting qubits. Phys. Rev. Lett. 97, 180406 (2006)ADSGoogle Scholar
  39. 39.
    N. Katz, M. Neeley, M. Ansmann, R.C. Bialczak, M. Hofheinz, E. Lucero, A. O’Connell, H. Wang, A.N. Cleland, J.M. Martinis, A.N. Korotkov, Reversal of the weak measurement of a quantum state in a superconducting phase qubit. Phys. Rev. Lett. 101, 200401 (2008)ADSGoogle Scholar
  40. 40.
    A.N. Jordan, A.N. Korotkov, Uncollapsing the wavefunction by undoing quantum measurements. Contemp. Phys. 51, 125 (2010)ADSGoogle Scholar
  41. 41.
    G.S. Paraoanu, Generalized partial measurements. Europhys. Lett. 93, 64002 (2011)ADSGoogle Scholar
  42. 42.
    G.S. Paraoanu, Partial measurements and the realization of quantum-mechanical counterfactuals. Found. Phys. 41, 1214 (2011)zbMATHMathSciNetADSGoogle Scholar
  43. 43.
    M.D. Reed, L. DiCarlo, B.R. Johnson, L. Sun, D.I. Schuster, L. Frunzio, R.J. Schoelkopf, High-fidelity readout in circuit quantum electrodynamics using the Jaynes–Cummings nonlinearity. Phys. Rev. Lett. 105, 173601 (2010)ADSGoogle Scholar
  44. 44.
    F. Mallet, F.R. Ong, A. Palacios-Laloy, F. Nguyen, P. Bertet, D. Vion, D. Esteve, Single-shot qubit readout in circuit quantum electrodynamics. Nat. Phys. 5, 791 (2009)Google Scholar
  45. 45.
    D. Risté, C.C. Bultink, K.W. Lehnert, L. DiCarlo, Feedback control of a solid-state qubit using high-fidelity projective measurement. Phys. Rev. Lett. 109, 240502 (2012)ADSGoogle Scholar
  46. 46.
    R. Vijay, C. Macklin, D.H. Slichter, S.J. Weber, K.W. Murch, R. Naik, A.N. Korotkov, I. Siddiqi, Quantum feedback control of a superconducting qubit: Persistent Rabi oscillations. Nature 490, 77 (2012)ADSGoogle Scholar
  47. 47.
    P. Campagne-Ibarcq, E. Flurin, N. Roch, D. Darson, P. Morfin, M. Mirrahimi, M.H. Devoret, F. Mallet, B. Huard, Persistent control of a superconducting qubit by stroboscopic measurement feedback. Phys. Rev. X 3, 021008 (2013)Google Scholar
  48. 48.
    G. de Lange, D. Risté, M.J. Tiggelman, C. Eichler, L. Tornberg, G. Johansson, A. Wallraff, R.N. Schouten, L. DiCarlo, Reversing quantum trajectories with analog feedback. arXiv:1311.5472
  49. 49.
    A.L. Rakhmanov, A.M. Zagoskin, S. Savel’ev, F. Nori, Quantum metamaterials: electromagnetic waves in a Josephson qubit line. Phys. Rev. B 77, 144507 (2008)ADSGoogle Scholar
  50. 50.
    A.M. Zagoskin, Quantum Engineering: Theory and Design of Quantum Coherent Structures (Cambridge University Press, Cambridge, 2011). Ch. 6.1Google Scholar
  51. 51.
    A. Das, B.K. Chakrabarti, Quantum annealing and analog quantum computation. Rev. Mod. Phys. 80, 1061 (2008)zbMATHMathSciNetADSGoogle Scholar
  52. 52.
    H.F. Trotter, On the product of semi-groups of operators. Proc. Am. Math. Soc. 10, 545 (1959)zbMATHMathSciNetGoogle Scholar
  53. 53.
    M. Suzuki, Decomposition formulas of exponential operators and Lie exponentials with some applications to quantum mechanics and statistical physics. J. Math. Phys. 26, 601 (1985)zbMATHMathSciNetADSGoogle Scholar
  54. 54.
    M. Neeley, R.C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, A.D. O’Connell, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A.N. Cleland, J.M. Martinis, Generation of three-qubit entangled states using superconducting phase qubits. Nature 467, 570 (2010)ADSGoogle Scholar
  55. 55.
    A. Fedorov, L. Steffen, M. Baur, M.P. da Silva, A. Wallraff, Implementation of a Toffoli gate with superconducting circuits. Nature 481, 170 (2012)ADSGoogle Scholar
  56. 56.
    J.M. Chow, J.M. Gambetta, A.D. Córcoles, S.T. Merkel, J.A. Smolin, C. Rigetti, S. Poletto, G.A. Keefe, M.B. Rothwell, J.R. Rozen, M.B. Ketchen, M. Steffen, Universal quantum gate set approaching fault-tolerant thresholds with superconducting qubits. Phys. Rev. Lett. 109, 060501 (2012)ADSGoogle Scholar
  57. 57.
    M.D. Reed, L. DiCarlo, S.E. Nigg, L. Sun, L. Frunzio, S.M. Girvin, R.J. Schoelkopf, Realization of three-qubit quantum error correction with superconducting circuits. Nature 482, 382 (2012)ADSGoogle Scholar
  58. 58.
    U. Las Heras, A. Mezzacapo, L. Lamata, S. Filipp, A. Wallraff, E. Solano, Digital quantum simulation of spin systems in superconducting circuits. arXiv:1311.7626
  59. 59.
    K.R. Brown, R.J. Clark, I.L. Chuang, Limitations of quantum simulation examined by simulating a pairing hamiltonian using nuclear magnetic resonance. Phys. Rev. Lett. 97, 050504 (2006)ADSGoogle Scholar
  60. 60.
    M.A. Sillanpää, J. Li, K. Cicak, F. Altomare, J.I. Park, R.W. Simmonds, G.S. Paraoanu, P.J. Hakonen, Autler-Townes effect in a superconducting three-level system. Phys. Rev. Lett. 103, 193601 (2009)ADSGoogle Scholar
  61. 61.
    M. Baur, S. Filipp, R. Bianchetti, J.M. Fink, M. Göppl, L. Steffen, P.J. Leek, A. Blais, A. Wallraff, Measurement of Autler-Townes and Mollow Transitions in a strongly driven superconducting qubit. Phys. Rev. Lett. 102, 243602 (2009)ADSGoogle Scholar
  62. 62.
    A.A. Abdumalikov Jr, O. Astafiev, A.M. Zagoskin, YuA Pashkin, Y. Nakamura, J.S. Tsai, Electromagnetically induced transparency on a single artificial atom. Phys. Rev. Lett. 104, 193601 (2010)ADSGoogle Scholar
  63. 63.
    W.R. Kelly, Z. Dutton, J. Schlafer, B. Mookerji, T.A. Ohki, J.S. Kline, D.P. Pappas, Direct observation of coherent population trapping in a superconducting artificial atom. Phys. Rev. Lett. 104, 163601 (2010)ADSGoogle Scholar
  64. 64.
    M. Neeley, M. Ansmann, R.C. Bialczak, M. Hofheinz, E. Lucero, A.D. O’Connell, D. Sank, H. Wang, J. Wenner, A.N. Cleland, M.R. Geller, J.M. Martinis, Emulation of a quantum spin with a superconducting phase qudit. Science 325, 722725 (2009)Google Scholar
  65. 65.
    J. Li, G.S. Paraoanu, K. Cicak, F. Altomare, J.I. Park, R.W. Simmonds, M.A. Sillanpää, P.J. Hakonen, Decoherence, Autler-Townes effect, and dark states in two-tone driving of a three-level superconducting system. Phys. Rev. B. 84, 104527 (2011)ADSGoogle Scholar
  66. 66.
    J. Li, G.S. Paraoanu, K. Cicak, F. Altomare, J.I. Park, R.W. Simmonds, M.A. Sillanpää, P.J. Hakonen, Dynamical Autler-Townes control of a phase qubit. Sci. Rep. 2, 645 (2012)ADSGoogle Scholar
  67. 67.
    J. Li, M.P. Silveri, K.S. Kumar, J.M. Pirkkalainen, A. Vepsäläinen, W.C. Chien, J. Tuorila, M.A. Sillanpää, P.J. Hakonen, E.V. Thuneberg, G.S. Paraoanu, Motional averaging in a superconducting qubit. Nat. Commun. 4, 1420 (2013)ADSGoogle Scholar
  68. 68.
    J.S. Pedernales, R. Di Candia, D. Ballester, E. Solano, Quantum simulations of relativistic quantum physics in circuit QED. New J. Phys. 15, 055008 (2013)MathSciNetADSGoogle Scholar
  69. 69.
    B. Doucot, L.B. Ioffe, J. Vidal, Discrete non-Abelian gauge theories in Josephson-junction arrays and quantum computation. Phys. Rev. B 69, 214501 (2004)ADSGoogle Scholar
  70. 70.
    S.P. Jordan, K.S.M. Lee, J. Preskill, Quantum algorithms for quantum field theories. Science 336, 1130 (2013)ADSGoogle Scholar
  71. 71.
    P. Lähteenmäki, G.S. Paraoanu, J. Hassel, P.J. Hakonen, Dynamical Casimir effect in a Josephson metamaterial. Proc. Natl. Acad. Sci. USA 110, 4234–4238 (2013)ADSGoogle Scholar
  72. 72.
    C.M. Wilson, G. Johansson, A. Pourkabirian, M. Simoen, J.R. Johansson, T. Duty, F. Nori, P. Delsing, Observation of the dynamical Casimir effect in a superconducting circuit. Nature 479, 376–379 (2011)ADSGoogle Scholar
  73. 73.
    V.V. Dodonov, V.I. Man’ko, O.V. Man’ko, Correlated states in quantum electronics (resonant circuit). J. Sov. Laser Res. 10, 413 (1989)Google Scholar
  74. 74.
    E. Yablonovitch, J.P. Heritage, D.E. Aspnes, Y. Yafet, Virtual photoconductivity. Phys. Rev. Lett. 63, 976 (1989)ADSGoogle Scholar
  75. 75.
    E. Yablonovitch, Accelerating reference frame for electromagnetic waves in a rapidly growing plasma: Unruh-Davies-Fulling-DeWitt radiation and the nonadiabatic Casimir effect. Phys. Rev. Lett. 62, 1742 (1989)ADSGoogle Scholar
  76. 76.
    J. Li, G.S. Paraoanu, Generation and propagation of entanglement in driven coupled-qubit systems. New J. Phys. 11, 113020 (2009)ADSGoogle Scholar
  77. 77.
    S. Felicetti, M. Sanz, L. Lamata, G. Romero, G. Johansson, P. Delsing, E. Solano, Dynamical Casimir effect entangles artificial atoms. arXiv:1402.4451
  78. 78.
    T. Fujii, S. Matsuo, N. Hatakenaka, S. Kurihara, A. Zeilinger, Quantum circuit analog of the dynamical Casimir effect. Phys. Rev. B 84, 174521 (2011)ADSGoogle Scholar
  79. 79.
    P.D. Nation, M.P. Blencowe, A.J. Rimberg, E. Buks, Analogue Hawking radiation in a dc-SQUID array transmission line. Phys. Rev. Lett. 103, 087004 (2009)ADSGoogle Scholar
  80. 80.
    C. Barceló, S. Liberati, M. Viser, Analogue gravity. Living Rev. Rel. 14, 3 (2011)Google Scholar
  81. 81.
    G.E. Volovik, The Universe in a Helium Droplet (Clarendon Press, Oxford, 2003)zbMATHGoogle Scholar
  82. 82.
    P.D. Nation, J.R. Johansson, M.P. Blencowe, F. Nori, Stimulating uncertainty: amplifying the quantum vacuum with superconducting circuits. Rev. Mod. Phys. 84, 1 (2012)ADSGoogle Scholar
  83. 83.
    J.D. Bekenstein, Black holes and entropy. Phys. Rev. D. 7, 2333 (1973)MathSciNetADSGoogle Scholar
  84. 84.
    S.W. Hawking, Breakdown of predictability in gravitational collapse. Phys. Rev. D 14, 2460 (1976)MathSciNetADSGoogle Scholar
  85. 85.
    A. Almheiri, D. Marolf, J. Polchinski, J. Sully, Black holes: complementarity or firewalls? J. High Energy Phys. 02, 062 (2013)MathSciNetADSGoogle Scholar
  86. 86.
    S.L. Braunstein, Black hole entropy as entropy of entanglement, or it’s curtains for the equivalence principle, [arXiv:0907.1190v1] published as S. L. Braunstein, S. Pirandola and K. Życzkowski, Better Late than Never: Information Retrieval from Black Holes. Phys. Rev. Lett. 110, 101301 (2013)Google Scholar
  87. 87.
    N. Friis, A.R. Lee, K. Truong, C. Sabín, E. Solano, G. Johansson, I. Fuentes, Relativistic quantum teleportation with superconducting circuits. Phys. Rev. Lett. 110, 113602 (2013)ADSGoogle Scholar
  88. 88.
    H.J. Briegel, D.E. Browne, W. Dür, R. Raussendorf, M. Van den Nest, Measurement-based quantum computation. Nat. Phys. 5, 19 (2009)Google Scholar
  89. 89.
    D.E. Bruschi, C. Sabín, P. Kok, G. Johansson, P. Delsing, I. Fuentes, Towards universal quantum computation through relativistic motion. arXiv:1311.5619
  90. 90.
    T. Banks, L. Susskind, J. Kogut, Strong-coupling calculations of lattice gauge theories: (1 + 1)-dimensional exercises. Phys. Rev. D 13, 10431053 (1976)Google Scholar
  91. 91.
    D. Marcos, P. Rabl, E. Rico, P. Zoller, Superconducting circuits for quantum simulation of dynamical gauge fields. Phys. Rev. Lett. 111, 110504 (2013)ADSGoogle Scholar
  92. 92.
    P.W. Anderson, Localized magnetic states in metals. Phys. Rev. 124, 41 (1961)MathSciNetADSGoogle Scholar
  93. 93.
    J.J. García-Ripoll, E. Solano, M.A. Martin-Delgado, Quantum simulation of Anderson and Kondo lattices with superconducting qubits. Phys. Rev. B 77, 024522 (2008)ADSGoogle Scholar
  94. 94.
    J. Ghosh, Simulating Anderson localization via a quantum walk on a one-dimensional lattice of superconducting qubits. Phys. Rev. A 89, 022309 (2014)ADSGoogle Scholar
  95. 95.
    M.J. Hartmann, Strongly interacting polaritons in coupled arrays of cavities. Nat. Phys. 2, 849 (2006)Google Scholar
  96. 96.
    A. Greentree, C. Tahan, J. Cole, L. Hollenberg, Quantum phase transitions of light. Nat. Phys. 2, 856 (2006)Google Scholar
  97. 97.
    M.J. Hartmann, F.G.S.L. Brandaõ, M.B. Plenio, Quantum many-body phenomena in coupled cavity arrays. Laser Photon. Rev. 2, 527 (2008)Google Scholar
  98. 98.
    D.G. Angelakis, M.F. Santos, S. Bose, Photon blockade induced Mott transitions and XY spin models in coupled cavity arrays. Phys. Rev. A 76, 031805 (2007)ADSGoogle Scholar
  99. 99.
    J. Jin, D. Rossini, R. Fazio, M. Leib, M.J. Hartmann, Photon solid phases in driven arrays of nonlinearly coupled cavities. Phys. Rev. Lett. 110, 163605 (2013)ADSGoogle Scholar
  100. 100.
    S. Schmidt, J. Koch, Circuit QED lattices: towards quantum simulation with superconducting circuits. Annalen der Physik 525, 395–412 (2013)ADSGoogle Scholar
  101. 101.
    S. Schmidt, D. Gerace, A.A. Houck, G. Blatter, H.E. Türeci, Nonequilibrium delocalization-localization transition of photons in circuit quantum electrodynamics. Phys. Rev. B 82, 100507 (2010)ADSGoogle Scholar
  102. 102.
    J. Raftery, D. Sadri, S. Schmidt, H.E.Türeci, A. A. Houck, Observation of a dissipation-induced classical to quantum, transition. arXiv:1312.2963
  103. 103.
    Gh-S Paraoanu, S. Kohler, F. Sols, A.J. Leggett, The Josephson plasmon as a Bogoliubov quasiparticle. J. Phys. B 34, 4689 (2001)ADSGoogle Scholar
  104. 104.
    A. Nunnenkamp, J. Koch, S.M. Girvin, Synthetic gauge fields and homodyne transmission in Jaynes–Cummings lattices. New J. Phys. 13, 095008 (2011)ADSGoogle Scholar
  105. 105.
    J. Koch, A.A. Houck, K. Le Hur, S.M. Girvin, Time-reversal symmetry breaking in circuit-QED based photon lattices. Phys. Rev. A 82, 043811 (2010)ADSGoogle Scholar
  106. 106.
    Gh-S Paraoanu, Persistent currents in a circular array of Bose-Einstein condensates. Phys. Rev. A 67, 023607 (2003)ADSGoogle Scholar
  107. 107.
    F. Mei, V.M. Stojanović, I. Siddiqi, L. Tian, Analog superconducting quantum simulator for Holstein polarons. Phys. Rev. B 88, 224502 (2013)ADSGoogle Scholar
  108. 108.
    T. Holstein, Studies of polaron motion: part II. The “small” polaron. Ann. Phys. (NY) 8, 343 (1959)ADSGoogle Scholar
  109. 109.
    A.J. Heeger, S.A. Kivelson, J.R. Schrieffer, W.-P. Su, Solitons in conducting polymers. Rev. Mod. Phys. 60, 781 (1988)ADSGoogle Scholar
  110. 110.
    K. Hannewald, V.M. Stojanović, J.M.T. Schellekens, P.A. Bobbert, G. Kresse, J. Hafner, Theory of polaron bandwidth narrowing in organic molecular crystals. Phys. Rev. B 69, 144302 (2004)Google Scholar
  111. 111.
    V.M. Stojanovic, M. Vanević, Quantum-entanglement aspects of polaron systems. Phys. Rev. B 78, 214301 (2008)ADSGoogle Scholar
  112. 112.
    D.J.J. Marchand, G. De Filippis, V. Cataudella, M. Berciu, N. Nagaosa, N.V. Prokof’ev, A.S. Mishchenko, P.C.E. Stamp, Sharp transition for single polarons in the one-dimensional Su-Schrieffer-Heeger model. Phys. Rev. Lett. 105, 266605 (2010)ADSGoogle Scholar
  113. 113.
    F. Herrera, K.W. Madison, R.V. Krems, M. Berciu, Investigating polaron transitions with polar molecules. Phys. Rev. Lett. 110, 223002 (2013)ADSGoogle Scholar
  114. 114.
    V. M. Stojanović, M. Vanević, E. Demler, L. Tian, Transmon-based simulator of nonlocal electron-phonon coupling: a platform for observing sharp small-polaron transitions. arXiv:1401.4783
  115. 115.
    A.M. Zagoskin, S. Savel’ev, F. Nori, Modeling an adiabatic quantum computer. Phys. Rev. Lett. 98, 120503 (2007)ADSGoogle Scholar
  116. 116.
    M.W. Johnson, M.H.S. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A.J. Berkley, J. Johansson, P. Bunyk, E.M. Chapple, C. Enderud, J.P. Hilton, K. Karimi, E. Ladizinsky, N. Ladizinsky, T. Oh, I. Perminov, C. Rich, M.C. Thom, E. Tolkacheva, C.J.S. Truncik, S. Uchaikin, J. Wang, B. Wilson, G. Rose, Quantum annealing with manufactured spins. Nature 473, 194 (2011)ADSGoogle Scholar
  117. 117.
    A. Perdomo-Ortiz, N. Dickson, M. Drew-Brook, G. Rose, A. Aspuru-Guzik, Finding low-energy conformations of lattice protein models by quantum annealing. Sci. Rep. 2, 571 (2012)ADSGoogle Scholar
  118. 118.
    S. Boixo, T.F. Rønnow, S.V. Isakov, Z. Wang, D. Wecker, D.A. Lidar, J.M. Martinis, M. Troyer, Quantum annealing with more than one hundred qubits. arXiv:1304.4595
  119. 119.
    T.F.Rønnow, Z. Wang, J. Job, S. Boixo, S.V. Isakov, D. Wecker, J.M. Martinis, D.A. Lidar, and M. Troyer, Defining and detecting quantum speedup. arXiv:1401.2910
  120. 120.
    K.L. Pudenz, T. Albash, D.A. Lidar, Error corrected quantum annealing with hundreds of qubits. arXiv:1307.8190
  121. 121.
    M. Tavis, F.W. Cummings, Exact solution for an n-molecule-radiation-field hamiltonian. Phys. Rev. 170, 379 (1968)ADSGoogle Scholar
  122. 122.
    J.M. Fink, Dressed collective qubit states and the Tavis–Cummings model in circuit QED. Phys. Rev. Lett. 103, 083601 (2009)ADSGoogle Scholar
  123. 123.
    G.S. Paraoanu, Realism and single-quanta nonlocality. Found. Phys. 41, 734 (2011)zbMATHMathSciNetADSGoogle Scholar
  124. 124.
    L. Heaney, A. Cabello, M.F. Santos, V. Vedral, Extreme nonlocality with one photon. New J. Phys. 13, 053054 (2011)ADSGoogle Scholar
  125. 125.
    P. Macha, G. Oelsner, J.-M. Reiner, M. Marthaler, S. André, G. Schön, U. Huebner, H.-G. Meyer, E. Il’ichev, A.V. Ustinov, Implementation of a quantum metamaterial. arXiv:1309.5268

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.O. V. Lounasmaa LaboratoryAalto University School of ScienceEspooFinland

Personalised recommendations