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  • Review Article
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Ultrastrong coupling between light and matter

Nature Reviews Physics volume 1, pages 19–40 (2019)Cite this article

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A Publisher Correction to this article was published on 26 February 2019

This article has been updated

Abstract

Ultrastrong coupling between light and matter has, in the past decade, transitioned from a theoretical idea to an experimental reality. It is a new regime of quantum light–matter interaction, which goes beyond weak and strong coupling to make the coupling strength comparable to the transition frequencies in the system. The achievement of weak and strong coupling has led to increased control of quantum systems and to applications such as lasers, quantum sensing, and quantum information processing. Here we review the theory of quantum systems with ultrastrong coupling, discussing entangled ground states with virtual excitations, new avenues for nonlinear optics, and connections to several important physical models. We also overview the multitude of experimental setups, including superconducting circuits, organic molecules, semiconductor polaritons, and optomechanical systems, that have now achieved ultrastrong coupling. We conclude by discussing the many potential applications that these achievements enable in physics and chemistry.

Key points

  • Ultrastrong coupling (USC) can be achieved by coupling many dipoles to light, or by using degrees of freedom whose coupling is not bounded by the smallness of the fine-structure constant.

  • The highest light–matter coupling strengths have been measured in experiments with Landau polaritons in semiconductor systems and in setups with superconducting quantum circuits.

  • With USC, standard approximations break down, allowing processes that do not conserve the number of excitations in the system, leading to a ground state that contains virtual excitations.

  • Potential applications of USC include fast and protected quantum information processing, nonlinear optics, modified chemical reactions and the enhancement of various quantum phenomena.

  • Now that USC has been reached in several systems, it is time to experimentally explore the new phenomena predicted for this regime and to find their useful applications.

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Fig. 1: Regimes of light–matter interaction.
Fig. 2: Spectrum and ground-state properties of ultrastrongly coupled light–matter systems.
Fig. 3: Experimental systems with ultrastrong light–matter coupling.
Fig. 4: Proposed methods for probing and extracting virtual photons dressing the states of an ultrastrongly coupled system.
Fig. 5: Simulations of ultrastrong coupling.
Fig. 6: Experiments and theory for ultrastrong coupling of an atom to an open waveguide.
Fig. 7: Some potential applications of ultrastrong coupling.

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Change history

  • 26 February 2019

    The following changes have been made to the original article: in the lower-right panel of Fig. 1, opoelectrics has been corrected to optoelectronics; in the Box 1 footnote, rotating wave-approximation has been corrected to rotating-wave approximation; in equation B1.3, an operator symbol has been added to the last term; and in the third paragraph of Box 2, |j→|n⟩ was changed to |j⟩→|n⟩. This has been corrected in the HTML and PDF versions of the article.

References

  1. Purcell, E. M. Spontaneous emission probabilities at radio frequencies. Phys. Rev. 69, 681 (1946).

    Google Scholar 

  2. Kaluzny, Y., Goy, P., Gross, M., Raimond, J. M. & Haroche, S. Observation of self-induced Rabi oscillations in two-level atoms excited inside a resonant cavity: the ringing regime of superradiance. Phys. Rev. Lett. 51, 1175 (1983).

    ADS  Google Scholar 

  3. Meschede, D., Walther, H. & Müller, G. One-atom maser. Phys. Rev. Lett. 54, 551 (1985).

    ADS  Google Scholar 

  4. Thompson, R. J., Rempe, G. & Kimble, H. J. Observation of normal-mode splitting for an atom in an optical cavity. Phys. Rev. Lett. 68, 1132 (1992).

    ADS  Google Scholar 

  5. Weisbuch, C., Nishioka, M., Ishikawa, A. & Arakawa, Y. Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity. Phys. Rev. Lett. 69, 3314 (1992).

    ADS  Google Scholar 

  6. Lodahl, P., Mahmoodian, S. & Stobbe, S. Interfacing single photons and single quantum dots with photonic nanostructures. Rev. Mod. Phys. 87, 347 (2015).

    ADS  MathSciNet  Google Scholar 

  7. Gu, X., Kockum, A. F., Miranowicz, A., Liu, Y.-X. & Nori, F. Microwave photonics with superconducting quantum circuits. Phys. Rep. 718-719, 1–102 (2017).

    ADS  MathSciNet  MATH  Google Scholar 

  8. Dicke, R. H. Coherence in spontaneous radiation processes. Phys. Rev. 93, 99 (1954).

    ADS  MATH  Google Scholar 

  9. Devoret, M. H., Girvin, S. & Schoelkopf, R. Circuit-QED: how strong can the coupling between a Josephson junction atom and a transmission line resonator be? Ann. Phys. 16, 767 (2007).

    MATH  Google Scholar 

  10. Ciuti, C., Bastard, G. & Carusotto, I. Quantum vacuum properties of the intersubband cavity polariton field. Phys. Rev. B 72, 115303 (2005). Early prediction of how ultrastrong light-matter coupling can be achieved experimentally.

    ADS  Google Scholar 

  11. Anappara, A. A. et al. Signatures of the ultrastrong light-matter coupling regime. Phys. Rev. B 79, 201303 (2009). The first experiment to demonstrate ultrastrong light-matter coupling.

    ADS  Google Scholar 

  12. Niemczyk, T. et al. Circuit quantum electrodynamics in the ultrastrong-coupling regime. Nat. Phys. 6, 772 (2010). The first experiment to demonstrate breaking of excitation-number conservation due to counter-rotating terms (and also the first demonstration of ultrastrong coupling in superconducting circuits).

    Google Scholar 

  13. Forn-Díaz, P. et al. Observation of the Bloch-Siegert shift in a qubit-oscillator system in the ultrastrong coupling regime. Phys. Rev. Lett. 105, 237001 (2010).

    ADS  Google Scholar 

  14. Gambino, S. et al. Exploring light–matter interaction phenomena under ultrastrong coupling regime. ACS Photonics 1, 1042 (2014).

    Google Scholar 

  15. Genco, A. et al. Bright polariton coumarin-based OLEDs operating in the ultrastrong coupling regime. Adv. Opt. Mater. 6, 1800364 (2018).

    Google Scholar 

  16. Ashhab, S. & Nori, F. Qubit-oscillator systems in the ultrastrong-coupling regime and their potential for preparing nonclassical states. Phys. Rev. A 81, 042311 (2010). Early analysis of the ground state in the quantum Rabi model, showing that it consists of photonic Schrödinger’s cat states entangled with the atom in the DSC regime and that it exhibits squeezing.

    ADS  Google Scholar 

  17. Galego, J., Garcia-Vidal, F. J. & Feist, J. Cavity-induced modifications of molecular structure in the strong-coupling regime. Phys. Rev. X 5, 041022 (2015).

    Google Scholar 

  18. Herrera, F. & Spano, F. C. Cavity-controlled chemistry in molecular ensembles. Phys. Rev. Lett. 116, 238301 (2016).

    ADS  Google Scholar 

  19. Cirio, M., De Liberato, S., Lambert, N. & Nori, F. Ground state electroluminescence. Phys. Rev. Lett. 116, 113601 (2016).

    ADS  Google Scholar 

  20. Kockum, A. F., Miranowicz, A., Macrì, V., Savasta, S. & Nori, F. Deterministic quantum nonlinear optics with single atoms and virtual photons. Phys. Rev. A 95, 063849 (2017). A systematic development of nonlinear optics in the USC regime.

    ADS  Google Scholar 

  21. Bayer, A. et al. Terahertz light-matter interaction beyond unity coupling strength. Nano. Lett. 17, 6340 (2017). The current record holder for light-matter coupling strength with η = 1.43 and the first experiment to demonstrate effective light-matter decoupling due to the high coupling strength.

    ADS  Google Scholar 

  22. Vahala, K. J. Optical microcavities. Nature 424, 839 (2003).

    ADS  Google Scholar 

  23. Shields, A. J. Semiconductor quantum light sources. Nat. Photonics 1, 215 (2007).

    ADS  Google Scholar 

  24. Salter, C. L. et al. An entangled-light-emitting diode. Nature 465, 594 (2010).

    ADS  Google Scholar 

  25. Haroche, S. Nobel lecture: controlling photons in a box and exploring the quantum to classical boundary. Rev. Mod. Phys. 85, 1083 (2013).

    ADS  Google Scholar 

  26. Georgescu, I. & Nori, F. Quantum technologies: an old new story. Phys. World 25, 16 (2012).

    ADS  Google Scholar 

  27. Degen, C. L., Reinhard, F. & Cappellaro, P. Quantum sensing. Rev. Mod. Phys. 89, 035002 (2017).

    ADS  MathSciNet  Google Scholar 

  28. Wendin, G. Quantum information processing with superconducting circuits: a review. Rep. Prog. Phys. 80, 106001 (2017).

    ADS  MathSciNet  Google Scholar 

  29. De Liberato, S. Virtual photons in the ground state of a dissipative system. Nat. Commun. 8, 1465 (2017).

    ADS  Google Scholar 

  30. De Liberato, S. Light-matter decoupling in the deep strong coupling regime: the breakdown of the Purcell effect. Phys. Rev. Lett. 112, 016401 (2014).

    ADS  Google Scholar 

  31. Sundaresan, N. M. et al. Beyond strong coupling in a multimode cavity. Phys. Rev. X 5, 021035 (2015).

    Google Scholar 

  32. George, J. et al. Multiple Rabi splittings under ultrastrong vibrational coupling. Phys. Rev. Lett. 117, 153601 (2016).

    ADS  Google Scholar 

  33. Bosman, S. J. et al. Multi-mode ultra-strong coupling in circuit quantum electrodynamics. npj Quantum Inf. 3, 46 (2017).

    ADS  Google Scholar 

  34. Gely, M. F. et al. Convergence of the multimode quantum Rabi model of circuit quantum electrodynamics. Phys. Rev. B 95, 245115 (2017).

    ADS  Google Scholar 

  35. Sánchez Muñoz, C., Nori, F. & De Liberato, S. Resolution of superluminal signalling in non-perturbative cavity quantum electrodynamics. Nat. Commun. 9, 1924 (2018).

    ADS  Google Scholar 

  36. De Bernardis, D., Jaako, T. & Rabl, P. Cavity quantum electrodynamics in the nonperturbative regime. Phys. Rev. A 97, 043820 (2018).

    ADS  MathSciNet  Google Scholar 

  37. De Bernardis, D., Pilar, P., Jaako, T., De Liberato, S. & Rabl, P. Breakdown of gauge invariance in ultrastrong-coupling cavity QED. Phys. Rev. A 98, 053819 (2018).

  38. Di Stefano, O. et al. Resolution of gauge ambiguities in ultrastrong-coupling cavity QED. Preprint at http://arxiv.org/abs/1809.08749 (2018).

  39. Jaynes, E. T. & Cummings, F. W. Comparison of quantum and semiclassical radiation theories with application to the beam maser. Proc. IEEE 51, 89 (1963).

    Google Scholar 

  40. Shore, B. W. & Knight, P. L. The Jaynes-Cummings model. J. Mod. Opt. 40, 1195 (1993).

    ADS  MathSciNet  MATH  Google Scholar 

  41. Yoshihara, F. et al. Superconducting qubit-oscillator circuit beyond the ultrastrong-coupling regime. Nat. Phys. 13, 44 (2017). The first experiment to demonstrate deep-strong light-matter coupling.

    Google Scholar 

  42. Braak, D. Integrability of the Rabi model. Phys. Rev. Lett. 107, 100401 (2011). Analytical solution of the quantum Rabi model.

    ADS  Google Scholar 

  43. Braak, D. Solution of the Dicke model for N = 3. J. Phys. B. At. Mol. Opt. Phys. 46, 224007 (2013).

    ADS  MathSciNet  Google Scholar 

  44. Peng, J., Ren, Z., Guo, G., Ju, G. & Guo, X. Exact solutions of the generalized two-photon and two-qubit Rabi models. Eur. Phys. J. D 67, 162 (2013).

    ADS  Google Scholar 

  45. Chilingaryan, S. A. & Rodríguez-Lara, B. M. Exceptional solutions in two-mode quantum Rabi models. J. Phys. B. At. Mol. Opt. Phys. 48, 245501 (2015).

    Google Scholar 

  46. Qin, W. et al. Exponentially enhanced light-matter interaction, cooperativities, and steady-state entanglement using parametric amplification. Phys. Rev. Lett. 120, 093601 (2018).

    ADS  Google Scholar 

  47. Leroux, C., Govia, L. C. G. & Clerk, A. A. Enhancing cavity quantum electrodynamics via antisqueezing: synthetic ultrastrong coupling. Phys. Rev. Lett. 120, 093602 (2018).

    ADS  Google Scholar 

  48. Tavis, M. & Cummings, F. W. Exact solution for an N-molecule-radiation-field Hamiltonian. Phys. Rev. 170, 379 (1968).

    ADS  Google Scholar 

  49. Bloch, F. & Siegert, A. Magnetic resonance for non-rotating fields. Phys. Rev. 57, 522 (1940).

    ADS  Google Scholar 

  50. Tomka, M., Pletyukhov, M. & Gritsev, V. Supersymmetry in quantum optics and in spin-orbit coupled systems. Sci. Rep. 5, 13097 (2015).

    ADS  Google Scholar 

  51. Hagenmüller, D., De Liberato, S. & Ciuti, C. Ultrastrong coupling between a cavity resonator and the cyclotron transition of a two-dimensional electron gas in the case of an integer filling factor. Phys. Rev. B 81, 235303 (2010).

    ADS  Google Scholar 

  52. Zhang, Q. et al. Collective non-perturbative coupling of 2D electrons with high-quality-factor terahertz cavity photons. Nat. Phys. 12, 1005 (2016).

    Google Scholar 

  53. Li, X. et al. Vacuum Bloch-Siegert shift in Landau polaritons with ultra-high cooperativity. Nat. Photonics 12, 324 (2018).

    ADS  Google Scholar 

  54. Casanova, J., Romero, G., Lizuain, I., García-Ripoll, J. J. & Solano, E. Deep strong coupling regime of the Jaynes-Cummings model. Phys. Rev. Lett. 105, 263603 (2010).

    ADS  Google Scholar 

  55. Khurgin, J. B. Excitonic radius in the cavity polariton in the regime of very strong coupling. Solid State Commun. 117, 307 (2001).

    ADS  Google Scholar 

  56. Brodbeck, S. et al. Experimental verification of the very strong coupling regime in a GaAs quantum well microcavity. Phys. Rev. Lett. 119, 027401 (2017).

    ADS  Google Scholar 

  57. Moores, B. A., Sletten, L. R., Viennot, J. J. & Lehnert, K. W. Cavity quantum acoustic device in the multimode strong coupling regime. Phys. Rev. Lett. 120, 227701 (2018).

    ADS  Google Scholar 

  58. Hines, A. P., Dawson, C. M., McKenzie, R. H. & Milburn, G. J. Entanglement and bifurcations in Jahn-Teller models. Phys. Rev. A 70, 022303 (2004).

    ADS  MathSciNet  MATH  Google Scholar 

  59. Hepp, K. & Lieb, E. H. On the superradiant phase transition for molecules in a quantized radiation field: the Dicke maser model. Ann. Phys. (N. Y.) 76, 360 (1973).

    ADS  MathSciNet  Google Scholar 

  60. Wang, Y. K. & Hioe, F. T. Phase transition in the Dicke model of superradiance. Phys. Rev. A 7, 831 (1973).

    ADS  Google Scholar 

  61. Emary, C. & Brandes, T. Chaos and the quantum phase transition in the Dicke model. Phys. Rev. E 67, 066203 (2003).

    ADS  MathSciNet  Google Scholar 

  62. Ashhab, S. & Semba, K. Superradiance phase transition in the presence of parameter fluctuations. Phys. Rev. A 95, 053833 (2017).

    ADS  Google Scholar 

  63. Ashhab, S. Superradiance transition in a system with a single qubit and a single oscillator. Phys. Rev. A 87, 013826 (2013).

    ADS  Google Scholar 

  64. Quattropani, A., Andreani, L. C. & Bassani, F. Quantum theory of polaritons with spatial dispersion: exact solutions. Nuovo Cim. D 7, 55 (1986).

    ADS  Google Scholar 

  65. Jaako, T., Xiang, Z.-L., Garcia-Ripoll, J. J. & Rabl, P. Ultrastrong-coupling phenomena beyond the Dicke model. Phys. Rev. A 94, 033850 (2016).

    ADS  Google Scholar 

  66. Le Boité, A., Hwang, M.-J., Nha, H. & Plenio, M. B. Fate of photon blockade in the deep strong-coupling regime. Phys. Rev. A 94, 033827 (2016).

    ADS  Google Scholar 

  67. Hagenmüller, D. & Ciuti, C. Cavity QED of the graphene cyclotron transition. Phys. Rev. Lett. 109, 267403 (2012).

    ADS  Google Scholar 

  68. Nataf, P. & Ciuti, C. No-go theorem for superradiant quantum phase transitions in cavity QED and counter-example in circuit QED. Nat. Commun. 1, 72 (2010).

    ADS  Google Scholar 

  69. Viehmann, O., von Delft, J. & Marquardt, F. Superradiant phase transitions and the standard description of circuit QED. Phys. Rev. Lett. 107, 113602 (2011).

    ADS  Google Scholar 

  70. Chirolli, L., Polini, M., Giovannetti, V. & MacDonald, A. H. Drude weight, cyclotron resonance, and the Dicke model of graphene cavity QED. Phys. Rev. Lett. 109, 267404 (2012).

    ADS  Google Scholar 

  71. Tufarelli, T., McEnery, K. R., Maier, S. A. & Kim, M. S. Signatures of the A2 term in ultrastrongly coupled oscillators. Phys. Rev. A 91, 063840 (2015).

    ADS  Google Scholar 

  72. García-Ripoll, J. J., Peropadre, B. & De Liberato, S. Light-matter decoupling and A2 term detection in superconducting circuits. Sci. Rep. 5, 16055 (2015).

    ADS  Google Scholar 

  73. Rossi, M. A. C. et al. Probing the diamagnetic term in light-matter interaction. Quantum Sci. Technol. 2, 01LT01 (2017).

    Google Scholar 

  74. RzaŻewski, K., Wódkiewicz, K. & Żakowicz, W. Phase transitions, two-level atoms, and the A2 term. Phys. Rev. Lett. 35, 432 (1975).

    ADS  Google Scholar 

  75. Slyusarev, V. A. & Yankelevich, R. P. On the impossibility of a phase transition to the superradiant state in a thermodynamically equilibrium gauge-invariant system. Theor. Math. Phys. 40, 641 (1979).

    Google Scholar 

  76. Keeling, J. Coulomb interactions, gauge invariance, and phase transitions of the Dicke model. J. Phys. Condens. Matter 19, 295213 (2007).

    Google Scholar 

  77. Vukics, A. & Domokos, P. Adequacy of the Dicke model in cavity QED: a counter-no-go statement. Phys. Rev. A 86, 053807 (2012).

    ADS  Google Scholar 

  78. Baksic, A., Nataf, P. & Ciuti, C. Superradiant phase transitions with three-level systems. Phys. Rev. A 87, 023813 (2013).

    ADS  Google Scholar 

  79. Vukics, A., Grießer, T. & Domokos, P. Elimination of the A-square problem from cavity QED. Phys. Rev. Lett. 112, 073601 (2014).

    ADS  Google Scholar 

  80. Bamba, M. & Ogawa, T. Stability of polarizable materials against superradiant phase transition. Phys. Rev. A 90, 063825 (2014).

    ADS  Google Scholar 

  81. Bamba, M. & Imoto, N. Circuit configurations which may or may not show superradiant phase transitions. Phys. Rev. A 96, 053857 (2017).

    ADS  Google Scholar 

  82. Todorov, Y. & Sirtori, C. Intersubband polaritons in the electrical dipole gauge. Phys. Rev. B 85, 045304 (2012).

    ADS  Google Scholar 

  83. De Liberato, S. & Ciuti, C. Quantum theory of intersubband polarons. Phys. Rev. B 85, 125302 (2012).

    ADS  Google Scholar 

  84. De Liberato, S. & Ciuti, C. Quantum phases of a multimode bosonic field coupled to flat electronic bands. Phys. Rev. Lett. 110, 133603 (2013).

    ADS  Google Scholar 

  85. Askenazi, B. et al. Ultra-strong light-matter coupling for designer Reststrahlen band. New J. Phys. 16, 043029 (2014).

    ADS  Google Scholar 

  86. Askenazi, B. et al. Midinfrared ultrastrong light-matter coupling for THz thermal emission. ACS Photonics 4, 2550 (2017).

    Google Scholar 

  87. Günter, G. et al. Sub-cycle switch-on of ultrastrong light–matter interaction. Nature 458, 178 (2009).

    ADS  Google Scholar 

  88. De Liberato, S., Ciuti, C. & Carusotto, I. Quantum vacuum radiation spectra from a semiconductor microcavity with a time-modulated vacuum Rabi frequency. Phys. Rev. Lett. 98, 103602 (2007).

    ADS  Google Scholar 

  89. Auer, A. & Burkard, G. Entangled photons from the polariton vacuum in a switchable optical cavity. Phys. Rev. B 85, 235140 (2012).

    ADS  Google Scholar 

  90. Todorov, Y. et al. Ultrastrong light-matter coupling regime with polariton dots. Phys. Rev. Lett. 105, 196402 (2010).

    ADS  Google Scholar 

  91. Jouy, P. et al. Transition from strong to ultrastrong coupling regime in mid-infrared metal-dielectric-metal cavities. Appl. Phys. Lett. 98, 231114 (2011).

    ADS  Google Scholar 

  92. Geiser, M. et al. Ultrastrong coupling regime and plasmon polaritons in parabolic semiconductor quantum wells. Phys. Rev. Lett. 108, 106402 (2012).

    ADS  Google Scholar 

  93. Delteil, A. et al. Charge-induced coherence between intersubband plasmons in a quantum structure. Phys. Rev. Lett. 109, 246808 (2012).

    ADS  Google Scholar 

  94. Forn-Díaz, P., Romero, G., Harmans, C. J. P. M., Solano, E. & Mooij, J. E. Broken selection rule in the quantum Rabi model. Sci. Rep. 6, 26720 (2016).

    ADS  Google Scholar 

  95. Baust, A. et al. Ultrastrong coupling in two-resonator circuit QED. Phys. Rev. B 93, 214501 (2016).

    ADS  Google Scholar 

  96. Chen, Z. et al. Single-photon-driven high-order sideband transitions in an ultrastrongly coupled circuit-quantum-electrodynamics system. Phys. Rev. A 96, 012325 (2017).

    ADS  Google Scholar 

  97. Yoshihara, F. et al. Characteristic spectra of circuit quantum electrodynamics systems from the ultrastrong- to the deep-strong-coupling regime. Phys. Rev. A 95, 053824 (2017).

    ADS  Google Scholar 

  98. Yoshihara, F. et al. Inversion of qubit energy levels in qubit-oscillator circuits in the deep-strong-coupling regime. Phys. Rev. Lett. 120, 183601 (2018).

    ADS  Google Scholar 

  99. Forn-Díaz, P. et al. Ultrastrong coupling of a single artificial atom to an electromagnetic continuum in the nonperturbative regime. Nat. Phys. 13, 39 (2017). The first experiment to demonstrate ultrastrong coupling between a qubit and a continuum of light modes in an open waveguide.

    Google Scholar 

  100. Magazzù, L. et al. Probing the strongly driven spin-boson model in a superconducting quantum circuit. Nat. Commun. 9, 1403 (2018).

    ADS  Google Scholar 

  101. Puertas Martinez, J. et al. A tunable Josephson platform to explore many-body quantum optics in circuit-QED. Preprint at http://arxiv.org/abs/1802.00633 (2018).

  102. Langford, N. K. et al. Experimentally simulating the dynamics of quantum light and matter at deep-strong coupling. Nat. Commun. 8, 1715 (2017). Experimental quantum simulation of the quantum Rabi model, demonstrating photonic Schrödinger’s cat states in the ground state of that model.

    ADS  Google Scholar 

  103. Braumüller, J. et al. Analog quantum simulation of the Rabi model in the ultrastrong coupling regime. Nat. Commun. 8, 779 (2017).

    ADS  Google Scholar 

  104. Scalari, G. et al. Ultrastrong coupling of the cyclotron transition of a 2D electron gas to a THz metamaterial. Science 335, 1323 (2012).

    ADS  Google Scholar 

  105. Scalari, G. et al. Ultrastrong light-matter coupling at terahertz frequencies with split ring resonators and inter-Landau level transitions. J. Appl. Phys. 113, 136510 (2013).

    ADS  Google Scholar 

  106. Maissen, C. et al. Ultrastrong coupling in the near field of complementary split-ring resonators. Phys. Rev. B 90, 205309 (2014).

    ADS  Google Scholar 

  107. Keller, J. et al. Critical softening of cavity cyclotron polariton modes in strained germanium 2D hole gas in the ultra-strong coupling regime. Preprint at http://arxiv.org/abs/1708.07773 (2017).

  108. Muravev, V. M., Andreev, I. V., Kukushkin, I. V., Schmult, S. & Dietsche, W. Observation of hybrid plasmon-photon modes in microwave transmission of coplanar microresonators. Phys. Rev. B 83, 075309 (2011).

    ADS  Google Scholar 

  109. Paravicini-Bagliani, G. L. et al. Tomography of an ultrastrongly coupled polariton state using magneto-transport in the quantum regime. Nat. Phys. https://doi.org/10.1038/s41567-018-0346-y (2018).

  110. Todisco, F. et al. Ultrastrong plasmon-exciton coupling by dynamic molecular aggregation. ACS Photonics 5, 143 (2018).

    Google Scholar 

  111. Schwartz, T., Hutchison, J. A., Genet, C. & Ebbesen, T. W. Reversible switching of ultrastrong light-molecule coupling. Phys. Rev. Lett. 106, 196405 (2011).

    ADS  Google Scholar 

  112. Kéna-Cohen, S., Maier, S. A. & Bradley, D. D. C. Ultrastrongly coupled exciton-polaritons in metal-clad organic semiconductor microcavities. Adv. Opt. Mater. 1, 827 (2013).

    Google Scholar 

  113. Gubbin, C. R., Maier, S. A. & Kéna-Cohen, S. Low-voltage polariton electroluminescence from an ultrastrongly coupled organic light-emitting diode. Appl. Phys. Lett. 104, 233302 (2014).

    ADS  Google Scholar 

  114. Mazzeo, M. et al. Ultrastrong light-matter coupling in electrically doped microcavity organic light emitting diodes. Appl. Phys. Lett. 104, 233303 (2014).

    ADS  Google Scholar 

  115. Barachati, F. et al. Tunable third-harmonic generation from polaritons in the ultrastrong coupling regime. ACS Photonics 5, 119 (2018).

    Google Scholar 

  116. Eizner, E., Brodeur, J., Barachati, F., Sridharan, A. & Kéna-Cohen, S. Organic photodiodes with an extended responsivity using ultrastrong light-matter coupling. ACS Photonics 5, 2921 (2018).

    Google Scholar 

  117. Benz, F. et al. Single-molecule optomechanics in “picocavities”. Science 354, 726 (2016).

    ADS  Google Scholar 

  118. Pirkkalainen, J.-M. et al. Cavity optomechanics mediated by a quantum two-level system. Nat. Commun. 6, 6981 (2015).

    Google Scholar 

  119. Beaudoin, F., Gambetta, J. M. & Blais, A. Dissipation and ultrastrong coupling in circuit QED. Phys. Rev. A 84, 043832 (2011).

    ADS  Google Scholar 

  120. Ridolfo, A., Leib, M., Savasta, S. & Hartmann, M. J. Photon blockade in the ultrastrong coupling regime. Phys. Rev. Lett. 109, 193602 (2012).

    ADS  Google Scholar 

  121. Stassi, R., Savasta, S., Garziano, L., Spagnolo, B. & Nori, F. Output field-quadrature measurements and squeezing in ultrastrong cavity-QED. New J. Phys. 18, 123005 (2016).

    ADS  Google Scholar 

  122. Ridolfo, A., Savasta, S. & Hartmann, M. J. Nonclassical radiation from thermal cavities in the ultrastrong coupling regime. Phys. Rev. Lett. 110, 163601 (2013).

    ADS  Google Scholar 

  123. Garziano, L., Ridolfo, A., De Liberato, S. & Savasta, S. Cavity QED in the ultrastrong coupling regime: photon bunching from the emission of individual dressed qubits. ACS Photonics 4, 2345 (2017).

    Google Scholar 

  124. Ciuti, C. & Carusotto, I. Input-output theory of cavities in the ultrastrong coupling regime: the case of time-independent cavity parameters. Phys. Rev. A 74, 033811 (2006).

    ADS  Google Scholar 

  125. Savasta, S. & Girlanda, R. Quantum description of the input and output electromagnetic fields in a polarizable confined system. Phys. Rev. A 53, 2716 (1996).

    ADS  Google Scholar 

  126. Di Stefano, O., Kockum, A. F., Ridolfo, A., Savasta, S. & Nori, F. Photodetection probability in quantum systems with arbitrarily strong light-matter interaction. Sci. Rep. 8, 17825 (2018).

  127. Lolli, J., Baksic, A., Nagy, D., Manucharyan, V. E. & Ciuti, C. Ancillary qubit spectroscopy of vacua in cavity and circuit quantum electrodynamics. Phys. Rev. Lett. 114, 183601 (2015).

    ADS  Google Scholar 

  128. Cirio, M., Debnath, K., Lambert, N. & Nori, F. Amplified optomechanical transduction of virtual radiation pressure. Phys. Rev. Lett. 119, 053601 (2017).

    ADS  Google Scholar 

  129. De Liberato, S., Gerace, D., Carusotto, I. & Ciuti, C. Extracavity quantum vacuum radiation from a single qubit. Phys. Rev. A 80, 053810 (2009).

    ADS  Google Scholar 

  130. Takashima, K., Hatakenaka, N., Kurihara, S. & Zeilinger, A. Nonstationary boundary effect for a quantum flux in superconducting nanocircuits. J. Phys. A Math. Theor. 41, 164036 (2008).

    ADS  MathSciNet  MATH  Google Scholar 

  131. Werlang, T., Dodonov, A. V., Duzzioni, E. I. & Villas-Bôas, C. J. Rabi model beyond the rotating-wave approximation: generation of photons from vacuum through decoherence. Phys. Rev. A 78, 053805 (2008).

    ADS  Google Scholar 

  132. Dodonov, A. V., Celeri, L. C., Pascoal, F., Lukin, M. D. & Yelin, S. F. Photon generation from vacuum in non-stationary circuit QED. Preprint at http://arxiv.org/abs/0806.4035 (2008).

  133. Carusotto, I., De Liberato, S., Gerace, D. & Ciuti, C. Back-reaction effects of quantum vacuum in cavity quantum electrodynamics. Phys. Rev. A 85, 023805 (2012).

    ADS  Google Scholar 

  134. Garziano, L., Ridolfo, A., Stassi, R., Di Stefano, O. & Savasta, S. Switching on and off of ultrastrong light-matter interaction: photon statistics of quantum vacuum radiation. Phys. Rev. A 88, 063829 (2013).

    ADS  Google Scholar 

  135. Shapiro, D. S., Zhukov, A. A., Pogosov, W. V. & Lozovik, Y. E. Dynamical Lamb effect in a tunable superconducting qubit-cavity system. Phys. Rev. A 91, 063814 (2015).

    ADS  Google Scholar 

  136. Moore, G. T. Quantum theory of the electromagnetic field in a variable-length one-dimensional cavity. J. Math. Phys. 11, 2679 (1970).

    ADS  Google Scholar 

  137. Johansson, J. R., Johansson, G., Wilson, C. M. & Nori, F. Dynamical Casimir effect in a superconducting coplanar waveguide. Phys. Rev. Lett. 103, 147003 (2009).

    ADS  Google Scholar 

  138. Wilson, C. M. et al. Observation of the dynamical Casimir effect in a superconducting circuit. Nature 479, 376 (2011).

    ADS  Google Scholar 

  139. Nation, P. D., Johansson, J. R., Blencowe, M. P. & Nori, F. Colloquium: Stimulating uncertainty: amplifying the quantum vacuum with superconducting circuits. Rev. Mod. Phys. 84, 1 (2012).

    ADS  Google Scholar 

  140. Macrì, V. et al. Nonperturbative dynamical Casimir effect in optomechanical systems: vacuum Casimir-Rabi splittings. Phys. Rev. X 8, 011031 (2018).

    Google Scholar 

  141. Ridolfo, A., Vilardi, R., Di Stefano, O., Portolan, S. & Savasta, S. All optical switch of vacuum Rabi oscillations: the ultrafast quantum eraser. Phys. Rev. Lett. 106, 013601 (2011).

    ADS  Google Scholar 

  142. Huang, J.-F. & Law, C. K. Photon emission via vacuum-dressed intermediate states under ultrastrong coupling. Phys. Rev. A 89, 033827 (2014).

    ADS  Google Scholar 

  143. Di Stefano, O. et al. Feynman-diagrams approach to the quantum Rabi model for ultrastrong cavity QED: stimulated emission and reabsorption of virtual particles dressing a physical excitation. New J. Phys. 19, 053010 (2017).

    Google Scholar 

  144. Buluta, I. & Nori, F. Quantum simulators. Science 326, 108 (2009).

    ADS  Google Scholar 

  145. Georgescu, I. M., Ashhab, S. & Nori, F. Quantum simulation. Rev. Mod. Phys. 86, 153 (2014).

    ADS  Google Scholar 

  146. Baumann, K., Guerlin, C., Brennecke, F. & Esslinger, T. Dicke quantum phase transition with a superfluid gas in an optical cavity. Nature 464, 1301 (2010).

    ADS  Google Scholar 

  147. Longhi, S. Jaynes-Cummings photonic superlattices. Opt. Lett. 36, 3407 (2011).

    ADS  Google Scholar 

  148. Crespi, A., Longhi, S. & Osellame, R. Photonic realization of the quantum Rabi model. Phys. Rev. Lett. 108, 163601 (2012).

    ADS  Google Scholar 

  149. Dimer, F., Estienne, B., Parkins, A. S. & Carmichael, H. J. Proposed realization of the Dicke-model quantum phase transition in an optical cavity QED system. Phys. Rev. A 75, 013804 (2007).

    ADS  Google Scholar 

  150. Ballester, D., Romero, G., García-Ripoll, J. J., Deppe, F. & Solano, E. Quantum simulation of the ultrastrong-coupling dynamics in circuit quantum electrodynamics. Phys. Rev. X 2, 021007 (2012).

    Google Scholar 

  151. Grimsmo, A. L. & Parkins, S. Cavity-QED simulation of qubit-oscillator dynamics in the ultrastrong-coupling regime. Phys. Rev. A 87, 033814 (2013).

    ADS  Google Scholar 

  152. Pedernales, J. S. et al. Quantum Rabi model with trapped ions. Sci. Rep. 5, 15472 (2015).

    ADS  Google Scholar 

  153. Felicetti, S. et al. Spectral collapse via two-phonon interactions in trapped ions. Phys. Rev. A 92, 033817 (2015).

    ADS  Google Scholar 

  154. Puebla, R., Hwang, M.-J., Casanova, J. & Plenio, M. B. Probing the dynamics of a superradiant quantum phase transition with a single trapped ion. Phys. Rev. Lett. 118, 073001 (2017).

    ADS  Google Scholar 

  155. Fedortchenko, S. et al. Quantum simulation of ultrastrongly coupled bosonic modes using superconducting circuits. Phys. Rev. A 95, 042313 (2017).

    ADS  Google Scholar 

  156. Aedo, I. & Lamata, L. Analog quantum simulation of generalized Dicke models in trapped ions. Phys. Rev. A 97, 042317 (2018).

    ADS  Google Scholar 

  157. Felicetti, S., Romero, G., Solano, E. & Sabín, C. Quantum Rabi model in a superfluid Bose-Einstein condensate. Phys. Rev. A 96, 033839 (2017).

    ADS  Google Scholar 

  158. Felicetti, S. et al. Quantum Rabi model in the Brillouin zone with ultracold atoms. Phys. Rev. A 95, 013827 (2017).

    ADS  Google Scholar 

  159. Lv, D. et al. Quantum Simulation of the quantum Rabi model in a trapped ion. Phys. Rev. X 8, 021027 (2018).

    Google Scholar 

  160. MarkoviĆ, D. et al. Demonstration of an effective ultrastrong coupling between two oscillators. Phys. Rev. Lett. 121, 040505 (2018).

    ADS  Google Scholar 

  161. Mezzacapo, A. et al. Digital quantum Rabi and Dicke models in superconducting circuits. Sci. Rep. 4, 7482 (2014).

    Google Scholar 

  162. Lamata, L. Digital-analog quantum simulation of generalized Dicke models with superconducting circuits. Sci. Rep. 7, 43768 (2017).

    ADS  Google Scholar 

  163. Leggett, A. J. et al. Dynamics of the dissipative two-state system. Rev. Mod. Phys. 59, 1 (1987).

    ADS  Google Scholar 

  164. Weiss, U. Quantum Dissipative Systems, 4th edn. (World Scientific, 2012).

  165. Bourassa, J. et al. Ultrastrong coupling regime of cavity QED with phase-biased flux qubits. Phys. Rev. A 80, 032109 (2009).

    ADS  Google Scholar 

  166. Le Hur, K. Kondo resonance of a microwave photon. Phys. Rev. B 85, 140506 (2012).

    Google Scholar 

  167. Peropadre, B., Zueco, D., Porras, D. & García-Ripoll, J. J. Nonequilibrium and nonperturbative dynamics of ultrastrong coupling in open lines. Phys. Rev. Lett. 111, 243602 (2013).

    ADS  Google Scholar 

  168. Leppäkangas, J. et al. Quantum simulation of the spin-boson model with a microwave circuit. Phys. Rev. A 97, 052321 (2018).

    ADS  Google Scholar 

  169. Snyman, I. & Florens, S. Robust Josephson-Kondo screening cloud in circuit quantum electrodynamics. Phys. Rev. B 92, 085131 (2015).

    ADS  Google Scholar 

  170. Sanchez-Burillo, E., Zueco, D., Garcia-Ripoll, J. J. & Martin-Moreno, L. Scattering in the ultrastrong regime: nonlinear optics with one photon. Phys. Rev. Lett. 113, 263604 (2014).

    ADS  Google Scholar 

  171. Díaz-Camacho, G., Bermudez, A. & García-Ripoll, J. J. Dynamical polaron Ansatz: a theoretical tool for the ultrastrong-coupling regime of circuit QED. Phys. Rev. A 93, 043843 (2016).

    ADS  Google Scholar 

  172. Hoi, I.-C. et al. Demonstration of a single-photon router in the microwave regime. Phys. Rev. Lett. 107, 073601 (2011).

    ADS  Google Scholar 

  173. Gheeraert, N., Bera, S. & Florens, S. Spontaneous emission of Schrödinger cats in a waveguide at ultrastrong coupling. New J. Phys. 19, 023036 (2017).

    ADS  Google Scholar 

  174. Gheeraert, N. et al. Particle production in ultrastrong-coupling waveguide QED. Phys. Rev. A 98, 043816 (2018).

  175. Goldstein, M., Devoret, M. H., Houzet, M. & Glazman, L. I. Inelastic microwave photon scattering off a quantum impurity in a Josephson-junction array. Phys. Rev. Lett. 110, 017002 (2013).

    ADS  Google Scholar 

  176. Meaney, C. P., Duty, T., McKenzie, R. H. & Milburn, G. J. Jahn-Teller instability in dissipative quantum systems. Phys. Rev. A 81, 043805 (2010).

    ADS  Google Scholar 

  177. Larson, J. Jahn-Teller systems from a cavity QED perspective. Phys. Rev. A 78, 033833 (2008).

    ADS  Google Scholar 

  178. Dereli, T., Gül, Y., Forn-Díaz, P. & Müstecaplioglu, Ö. E. Two-frequency Jahn-Teller systems in circuit QED. Phys. Rev. A 85, 053841 (2012).

    ADS  Google Scholar 

  179. Levine, G. & Muthukumar, V. N. Entanglement of a qubit with a single oscillator mode. Phys. Rev. B 69, 113203 (2004).

    ADS  Google Scholar 

  180. Hirokawa, M. The Rabi model gives off a flavor of spontaneous SUSY breaking. Quantum Stud. Math. Found. 2, 379 (2015).

    MathSciNet  MATH  Google Scholar 

  181. Garziano, L., Stassi, R., Ridolfo, A., Di Stefano, O. & Savasta, S. Vacuum-induced symmetry breaking in a superconducting quantum circuit. Phys. Rev. A 90, 043817 (2014).

    ADS  Google Scholar 

  182. Gong, Z., Hamazaki, R. & Ueda, M. Discrete time-crystalline order in cavity and circuit QED systems. Phys. Rev. Lett. 120, 040404 (2018).

    ADS  Google Scholar 

  183. Ruggenthaler, M., Tancogne-Dejean, N., Flick, J., Appel, H. & Rubio, A. From a quantum-electrodynamical light-matter description to novel spectroscopies. Nat. Rev. Chem. 2, 0118 (2018).

    Google Scholar 

  184. Romero, G., Ballester, D., Wang, Y. M., Scarani, V. & Solano, E. Ultrafast quantum gates in circuit QED. Phys. Rev. Lett. 108, 120501 (2012).

    ADS  Google Scholar 

  185. Wang, Y., Guo, C., Zhang, G.-Q., Wang, G. & Wu, C. Ultrafast quantum computation in ultrastrongly coupled circuit QED systems. Sci. Rep. 7, 44251 (2017).

    ADS  Google Scholar 

  186. Stassi, R. et al. Quantum nonlinear optics without photons. Phys. Rev. A 96, 023818 (2017).

    ADS  Google Scholar 

  187. Stassi, R. & Nori, F. Long-lasting quantum memories: extending the coherence time of superconducting artificial atoms in the ultrastrong-coupling regime. Phys. Rev. A 97, 033823 (2018).

    ADS  Google Scholar 

  188. Kyaw, T. H., Felicetti, S., Romero, G., Solano, E. & Kwek, L.-C. Scalable quantum memory in the ultrastrong coupling regime. Sci. Rep. 5, 8621 (2015).

    ADS  Google Scholar 

  189. Nataf, P. & Ciuti, C. Protected quantum computation with multiple resonators in ultrastrong coupling circuit QED. Phys. Rev. Lett. 107, 190402 (2011).

    ADS  Google Scholar 

  190. Wang, Y., Zhang, J., Wu, C., You, J. Q. & Romero, G. Holonomic quantum computation in the ultrastrong-coupling regime of circuit QED. Phys. Rev. A 94, 012328 (2016).

    ADS  Google Scholar 

  191. Cao, X., You, J. Q., Zheng, H. & Nori, F. A qubit strongly coupled to a resonant cavity: asymmetry of the spontaneous emission spectrum beyond the rotating wave approximation. New J. Phys. 13, 073002 (2011).

    ADS  Google Scholar 

  192. Lizuain, I., Casanova, J., García-Ripoll, J. J., Muga, J. G. & Solano, E. Zeno physics in ultrastrong-coupling circuit QED. Phys. Rev. A 81, 062131 (2010).

    ADS  Google Scholar 

  193. Cao, X., Ai, Q., Sun, C.-P. & Nori, F. The transition from quantum Zeno to anti-Zeno effects for a qubit in a cavity by varying the cavity frequency. Phys. Lett. A 376, 349 (2012).

    ADS  MATH  Google Scholar 

  194. Seah, S., Nimmrichter, S. & Scarani, V. Refrigeration beyond weak internal coupling. Phys. Rev. E 98, 012131 (2018).

    ADS  Google Scholar 

  195. Felicetti, S., Romero, G., Rossini, D., Fazio, R. & Solano, E. Photon transfer in ultrastrongly coupled three-cavity arrays. Phys. Rev. A 89, 013853 (2014).

    ADS  Google Scholar 

  196. Lindner, N. H., Refael, G. & Galitski, V. Floquet topological insulator in semiconductor quantum wells. Nat. Phys. 7, 490 (2011).

    Google Scholar 

  197. Claassen, M., Jiang, H.-C., Moritz, B. & Devereaux, T. P. Dynamical time-reversal symmetry breaking and photo-induced chiral spin liquids in frustrated Mott insulators. Nat. Commun. 8, 1192 (2017).

    ADS  Google Scholar 

  198. Hübener, H., Sentef, M. A., De Giovannini, U., Kemper, A. F. & Rubio, A. Creating stable Floquet-Weyl semimetals by laser-driving of 3D Dirac materials. Nat. Commun. 8, 13940 (2017).

    ADS  Google Scholar 

  199. Tame, M. S. et al. Quantum plasmonics. Nat. Phys. 9, 329 (2013).

    Google Scholar 

  200. Sentef, M. A., Ruggenthaler, M. & Rubio, A. Cavity quantum-electrodynamical polaritonically enhanced electron-phonon coupling and its influence on superconductivity. Sci. Adv. 4, eaau6969 (2018).

  201. Schlawin, F., Cavalleri, A. & Jaksch, D. Cavity-mediated electron-photon superconductivity. Preprint at http://arxiv.org/abs/1804.07142 (2018).

  202. Kockum, A. F., Macrí, V., Garziano, L., Savasta, S. & Nori, F. Frequency conversion in ultrastrong cavity QED. Sci. Rep. 7, 5313 (2017).

    ADS  Google Scholar 

  203. Garziano, L., Stassi, R., Macrí, V., Savasta, S. & Di Stefano, O. Single-step arbitrary control of mechanical quantum states in ultrastrong optomechanics. Phys. Rev. A 91, 023809 (2015).

    ADS  Google Scholar 

  204. Garziano, L. et al. Multiphoton quantum Rabi oscillations in ultrastrong cavity QED. Phys. Rev. A 92, 063830 (2015).

    ADS  Google Scholar 

  205. Ma, K. K. W. & Law, C. K. Three-photon resonance and adiabatic passage in the large-detuning Rabi model. Phys. Rev. A. 92, 023842 (2015).

    ADS  Google Scholar 

  206. Garziano, L. et al. One photon can simultaneously excite two or more atoms. Phys. Rev. Lett. 117, 043601 (2016).

    ADS  Google Scholar 

  207. Ebbesen, T. W. Hybrid light-matter states in a molecular and material science perspective. Acc. Chem. Res. 49, 2403 (2016).

    Google Scholar 

  208. Bennett, K., Kowalewski, M. & Mukamel, S. Novel photochemistry of molecular polaritons in optical cavities. Faraday Discuss. 194, 259 (2016).

    ADS  Google Scholar 

  209. Kowalewski, M., Bennett, K. & Mukamel, S. Cavity femtochemistry: manipulating nonadiabatic dynamics at avoided crossings. J. Phys. Chem. Lett. 7, 2050 (2016).

    Google Scholar 

  210. Martínez-Martínez, L. A., Ribeiro, R. F., Campos-González-Angulo, J. & Yuen-Zhou, J. Can ultrastrong coupling change ground-state chemical reactions? ACS Photonics 5, 167 (2018).

    Google Scholar 

  211. Ruggenthaler, M. et al. Quantum-electrodynamical density-functional theory: bridging quantum optics and electronic-structure theory. Phys. Rev. A 90, 012508 (2014).

    ADS  Google Scholar 

  212. Schäfer, C., Ruggenthaler, M. & Rubio, A. Ab initio nonrelativistic quantum electrodynamics: bridging quantum chemistry and quantum optics from weak to strong coupling. Phys. Rev. A 98, 043801 (2018).

  213. Chikkaraddy, R. et al. Single-molecule strong coupling at room temperature in plasmonic nanocavities. Nature 535, 127 (2016).

    ADS  Google Scholar 

  214. Ćwik, J. A., Kirton, P., De Liberato, S. & Keeling, J. Excitonic spectral features in strongly coupled organic polaritons. Phys. Rev. A 93, 033840 (2016).

    ADS  Google Scholar 

  215. Johansson, J. R., Nation, P. D. & Nori, F. QuTiP 2: a Python framework for the dynamics of open quantum systems. Comput. Phys. Commun. 184, 1234 (2013).

    ADS  Google Scholar 

  216. Chang, W.-H. et al. Efficient single-photon sources based on low-density quantum dots in photonic-crystal nanocavities. Phys. Rev. Lett. 96, 117401 (2006).

    ADS  Google Scholar 

  217. Raimond, J. M., Brune, M. & Haroche, S. Manipulating quantum entanglement with atoms and photons in a cavity. Rev. Mod. Phys. 73, 565 (2001).

    ADS  MathSciNet  MATH  Google Scholar 

  218. Nataf, P. & Ciuti, C. Vacuum degeneracy of a circuit QED system in the ultrastrong coupling regime. Phys. Rev. Lett. 104, 023601 (2010).

    ADS  Google Scholar 

  219. Rossatto, D. Z., Villas-Bôas, C. J., Sanz, M. & Solano, E. Spectral classification of coupling regimes in the quantum Rabi model. Phys. Rev. A 96, 013849 (2017).

    ADS  Google Scholar 

  220. Rabi, I. I. Space quantization in a gyrating magnetic field. Phys. Rev. 51, 652 (1937).

    ADS  MATH  Google Scholar 

  221. Xie, Q.-T., Cui, S., Cao, J.-P., Amico, L. & Fan, H. Anisotropic Rabi model. Phys. Rev. X 4, 021046 (2014).

    Google Scholar 

  222. Hopfield, J. J. Theory of the contribution of excitons to the complex dielectric constant of crystals. Phys. Rev. 112, 1555 (1958).

    ADS  MATH  Google Scholar 

  223. Gardiner, C. W. & Zoller, P. Quantum Noise, 3rd edn. (Springer, 2004).

  224. Bamba, M. & Ogawa, T. Dissipation and detection of polaritons in the ultrastrong-coupling regime. Phys. Rev. A 86, 063831 (2012).

    ADS  Google Scholar 

  225. Bamba, M. & Ogawa, T. Recipe for the Hamiltonian of system-environment coupling applicable to the ultrastrong-light-matter-interaction regime. Phys. Rev. A 89, 023817 (2014).

    ADS  Google Scholar 

  226. Bamba, M., Inomata, K. & Nakamura, Y. Superradiant phase transition in a superconducting circuit in thermal equilibrium. Phys. Rev. Lett. 117, 173601 (2016).

    ADS  Google Scholar 

  227. Bamba, M. & Ogawa, T. System-environment coupling derived by Maxwell's boundary conditions from the weak to the ultrastrong light-matter-coupling regime. Phys. Rev. A 88, 013814 (2013).

    ADS  Google Scholar 

  228. De Liberato, S. Comment on “System-environment coupling derived by Maxwell’s boundary conditions from the weak to the ultrastrong light-matter-coupling regime”. Phys. Rev. A 89, 017801 (2014).

    ADS  Google Scholar 

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Acknowledgements

Numerical simulations were performed using QuTiP215. The authors thank Y.-X. Liu, X. Gu, O. Di Stefano and A. Ridolfo for useful discussions. The authors also thank A. Settineri, M. Cirio and S. Ahmed for technical assistance with some of the figures. A.F.K. acknowledges partial support from a JSPS Postdoctoral Fellowship for Overseas Researchers (P15750). A.M. and F.N. acknowledge support from the Sir John Templeton Foundation. S.D.L. acknowledges support from a Royal Society research fellowship and thanks F.N. for his hospitality at RIKEN during the course of this work. F.N. also acknowledges support from the MURI Center for Dynamic Magneto-Optics via the Air Force Office of Scientific Research (AFOSR) award No. FA9550-14-1-0040, the Army Research Office (ARO) under grant No. W911NF-18-1-0358, the Asian Office of Aerospace Research and Development (AOARD) grant No. FA2386-18-1-4045, the Japan Science and Technology Agency (JST) through the Q-LEAP program, the ImPACT program, and CREST grant No. JPMJCR1676, the Japan Society for the Promotion of Science (JSPS) through the JSPS-RFBR grant No. 17-52-50023 and the JSPS-FWO Grant No. VS.059.18N, and the RIKEN-AIST Challenge Research Fund.

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Authors and Affiliations

  1. Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama, Japan

    Anton Frisk Kockum, Adam Miranowicz, Salvatore Savasta & Franco Nori

  2. Wallenberg Centre for Quantum Technology, Department of Microtechnology and Nanoscience, Chalmers University of Technology, Gothenburg, Sweden

    Anton Frisk Kockum

  3. Faculty of Physics, Adam Mickiewicz University, Poznań, Poland

    Adam Miranowicz

  4. School of Physics and Astronomy, University of Southampton, Southampton, UK

    Simone De Liberato

  5. Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Università di Messina, Messina, Italy

    Salvatore Savasta

  6. Physics Department, University of Michigan, Ann Arbor, MI, USA

    Franco Nori

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Frisk Kockum, A., Miranowicz, A., De Liberato, S. et al. Ultrastrong coupling between light and matter. Nat Rev Phys 1, 19–40 (2019). https://doi.org/10.1038/s42254-018-0006-2

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