, PHASE TRANSITIONS IN DRIVEN-DISSIPATIVE SYSTEMS

. , 43 3.3.1 Analogies and differences with respect to the equilibrium case, p.46

. , 49 3.4.1 First-order phase transition

. .. Conclusions,

. .. The-model, 99 6.2 Emergence of a second-order phase transition

. .. Conclusions,

. .. Conclusions, 115 first part of Eq. (2.61), keeping only the highest order in ?

F. Minganti, A. Biella, N. Bartolo, and C. Ciuti, Spectral theory of Liouvillians for dissipative phase transitions, Phys. Rev. A, vol.98, p.42118, 2018.

R. Rota, F. Minganti, A. Biella, and C. Ciuti, Dynamical properties of dissipative XYZ Heisenberg lattices, New Journal of Physics, vol.20, p.45003, 2018.

F. Vicentini, F. Minganti, R. Rota, G. Orso, and C. Ciuti, Critical slowing down in driven-dissipative Bose-Hubbard lattices, Phys. Rev. A, vol.97, p.13853, 2018.

N. Bartolo, F. Minganti, J. Lolli, and C. Ciuti, Homodyne versus photon-counting quantum trajectories for dissipative Kerr resonators with two-photon driving, The European Physical Journal Special Topics, vol.226, p.2705, 2017.

N. Bartolo, F. Minganti, W. Casteels, and C. Ciuti, Exact steady state of a Kerr resonator with one-and two-photon driving and dissipation: Controllable Wigner-function multimodality and dissipative phase transitions, Phys. Rev. A, vol.94, p.33841, 2016.

F. Minganti, N. Bartolo, J. Lolli, W. Casteels, and C. Ciuti, Exact results for Schrödinger cats in driven-dissipative systems and their feedback control, Scientific Reports, vol.6, p.26987, 2016.

M. Planck, Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum, Verhandlungen der Deutschen Physikalischen Gesellschaft, vol.2, p.237, 1900.

E. Einstein, Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt, p.132, 1905.

C. Cohen-tannoudji, B. Diu, and F. Laloe, Quantum Mechanics Volume, vol.1, 1991.

L. Landau and E. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, Course of Theoretical Physics, vol.3, 1981.

L. Pítajevskíj and S. Stringari, Bose-Einstein Condensation, International Series of Monographs on Physics, 2003.

L. D. Landau and E. M. Lifshitz, Statistical Physics, Course of Theoretical Physics, vol.5, 2013.

C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, 2008.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys, vol.71, p.463, 1999.

I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys, vol.80, p.885, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00195515

N. Ashcroft and N. Mermin, Solid State Physics, 1976.

C. Kittel, Introduction to Solid State Physics, 2004.

R. P. Feynman, Rev. Mod. Phys, vol.29, p.205, 1957.

M. Tinkham, Introduction to Superconductivity, 2004.

A. J. Leggett, Superfluidity, Rev. Mod. Phys, vol.71, p.318, 1999.

P. Rossi, The Birth of Modern Science, 2001.

Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, 2011.

H. Breuer and F. Petruccione, The Theory of Open Quantum Systems, 2007.

C. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, 2004.

S. Haroche and J. M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons, 2006.

H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations

I. Carusotto and C. Ciuti, Quantum fluids of light, vol.85, p.299
URL : https://hal.archives-ouvertes.fr/hal-00913374

R. J. Schoelkopf and S. M. Girvin, Wiring up quantum systems, Nature, vol.451, p.664, 2008.

J. Q. You and F. Nori, Atomic physics and quantum optics using superconducting circuits, Nature, vol.474, p.589, 2011.

B. Deveaud, The Physics of Semiconductor Microcavities: From Fundamentals to Nanoscale Devices, 2007.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity optomechanics, vol.86, p.1391, 2014.

J. J. Hopfield, Theory of the Contribution of Excitons to the Complex Dielectric Constant of Crystals, Phys. Rev, vol.112, p.1555, 1958.

C. Ciuti, P. Schwendimann, and A. Quattropani, Theory of polariton parametric interactions in semiconductor microcavities, Semiconductor Science and Technology, vol.18, p.279, 2003.

A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Colloquium: Nonequilibrium dynamics of closed interacting quantum systems, Rev. Mod. Phys, vol.83, p.863, 2011.

J. Eisert, M. Friesdorf, and C. Gogolin, Quantum many-body systems out of equilibrium, review Article, vol.11, p.124, 2015.

S. Sachdev, Quantum Phase Transitions, 2001.

M. P. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Boson localization and the superfluid-insulator transition, Phys. Rev. B, vol.40, p.546, 1989.

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms, Nature, vol.415, p.39, 2002.

T. E. Lee, S. Gopalakrishnan, and M. D. Lukin, Unconventional Magnetism via Optical Pumping of Interacting Spin Systems, Phys. Rev. Lett, vol.110, p.257204, 2013.

J. Jin, A. Biella, O. Viyuela, L. Mazza, J. Keeling et al., Cluster Mean-Field Approach to the Steady-State Phase Diagram of Dissipative Spin Systems, Phys. Rev. X, vol.6, p.31011, 2016.

S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. Büchler et al., Quantum states and phases in driven open quantum systems with cold atoms, Nat. Phys, vol.4, p.878, 2008.

F. Verstraete, M. M. Wolf, and J. I. Cirac, Quantum computation and quantum-state engineering driven by dissipation, Nat. Phys, vol.5, p.633, 2009.

E. M. Kessler, G. Giedke, A. , S. F. Yelin, M. D. Lukin et al., Dissipative phase transition in a central spin system, Phys. Rev. A, vol.86, p.12116, 2012.

A. A. Houck, H. E. Tureci, and J. Koch, On-chip quantum simulation with superconducting circuits, Nat Phys, vol.8, p.292, 2012.

M. Fitzpatrick, N. M. Sundaresan, A. C. Li, J. Koch, and A. A. Houck, Observation of a Dissipative Phase Transition in a One-Dimensional Circuit QED Lattice, Phys. Rev. X, vol.7, p.11016, 2017.

M. Müller, S. Diehl, G. Pupillo, and P. Zoller, Engineered Open Systems and Quantum Simulations with Atoms and Ions, Adv. At. Mol. Opt. Phys, vol.61, p.1, 2012.

H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran et al., Probing many-body dynamics on a 51-atom quantum simulator, Nature, vol.551, p.579, 2017.

E. Gil-santos, M. Labousse, C. Baker, A. Goetschy, W. Hease et al., Light-Mediated Cascaded Locking of Multiple NanoOptomechanical Oscillators, Phys. Rev. Lett, vol.118, p.63605, 2017.

J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun et al., Bose-Einstein condensation of exciton polaritons, Nature, vol.443, p.409, 2006.

J. M. Fink, A. Dombi, A. Vukics, A. Wallraff, and P. Domokos, Observation of the Photon-Blockade Breakdown Phase Transition, Phys. Rev. X, vol.7, p.11012, 2017.

S. R. Rodriguez, W. Casteels, F. Storme, N. Carlon-zambon, I. Sagnes et al., Probing a Dissipative Phase Transition via Dynamical Optical Hysteresis, Phys. Rev. Lett, vol.118, p.247402, 2017.

T. Fink, A. Schade, S. Höfling, C. Schneider, and A. Imamoglu, Signatures of a dissipative phase transition in photon correlation measurements, Nature Physics, vol.14, p.365, 2018.

H. J. Carmichael, Breakdown of Photon Blockade: A Dissipative Quantum Phase Transition in Zero Dimensions, Phys. Rev. X, vol.5, p.31028, 2015.

H. Weimer, Variational Principle for Steady States of Dissipative Quantum Many-Body Systems, Phys. Rev. Lett, vol.114, p.40402, 2015.

M. Benito, C. Sánchez-muñoz, and C. Navarrete-benlloch, Degenerate parametric oscillation in quantum membrane optomechanics, Phys. Rev. A, vol.93, p.23846, 2016.

J. J. Mendoza-arenas, S. R. Clark, S. Felicetti, G. Romero, E. Solano et al., Beyond mean-field bistability in driven-dissipative lattices: Bunchingantibunching transition and quantum simulation, Phys. Rev. A, vol.93, p.23821, 2016.

W. Casteels, F. Storme, A. L. Boité, and C. Ciuti, Power laws in the dynamic hysteresis of quantum nonlinear photonic resonators, Phys. Rev. A, vol.93, p.33824, 2016.

W. Casteels and C. Ciuti, Quantum entanglement in the spatial-symmetry-breaking phase transition of a driven-dissipative Bose-Hubbard dimer, Phys. Rev. A, vol.95, p.13812, 2017.

W. Casteels, R. Fazio, and C. Ciuti, Critical dynamical properties of a first-order dissipative phase transition, Phys. Rev. A, vol.95, p.12128, 2017.

M. Foss-feig, P. Niroula, J. T. Young, M. Hafezi, A. V. Gorshkov et al., Emergent equilibrium in many-body optical bistability, Phys. Rev. A, vol.95, p.43826, 2017.

M. Biondi, G. Blatter, H. E. Türeci, and S. Schmidt, Nonequilibrium gas-liquid transition in the driven-dissipative photonic lattice, Phys. Rev. A, vol.96, p.43809, 2017.

A. Biella, F. Storme, J. Lebreuilly, D. Rossini, R. Fazio et al., Phase diagram of incoherently driven strongly correlated photonic lattices, Phys. Rev. A, vol.96, p.23839, 2017.

V. Savona, Spontaneous symmetry breaking in a quadratically driven nonlinear photonic lattice, Phys. Rev. A, vol.96, p.33826, 2017.

C. Sánchez-muñoz, A. Lara, J. Puebla, and F. Nori, Hybrid systems for the dissipative preparation of mechanical Schrödinger cats, 2018.

L. M. Sieberer, S. D. Huber, E. Altman, and S. Diehl, Dynamical Critical Phenomena in Driven-Dissipative Systems, Phys. Rev. Lett, vol.110, p.195301, 2013.

L. M. Sieberer, S. D. Huber, E. Altman, and S. Diehl, Nonequilibrium functional renormalization for driven-dissipative Bose-Einstein condensation, Phys. Rev. B, vol.89, p.134310, 2014.

E. Altman, L. M. Sieberer, L. Chen, S. Diehl, and J. Toner, Two-Dimensional Superfluidity of Exciton Polaritons Requires Strong Anisotropy, Phys. Rev. X, vol.5, p.11017, 2015.

T. E. Lee, H. Häffner, and M. C. Cross, Antiferromagnetic phase transition in a nonequilibrium lattice of Rydberg atoms, Phys. Rev. A, vol.84, p.31402, 2011.

C. Chan, T. E. Lee, and S. Gopalakrishnan, Limit-cycle phase in driven-dissipative spin systems, Phys. Rev. A, vol.91, p.51601, 2015.

M. F. Maghrebi and A. V. Gorshkov, Nonequilibrium many-body steady states via Keldysh formalism, Phys. Rev. B, vol.93, p.14307, 2016.

R. Rota, F. Storme, N. Bartolo, R. Fazio, and C. Ciuti, Critical behavior of dissipative two-dimensional spin lattices, Phys. Rev. B, vol.95, p.134431, 2017.

V. R. Overbeck, M. F. Maghrebi, A. V. Gorshkov, and H. Weimer, Multicritical behavior in dissipative Ising models, Phys. Rev. A, vol.95, p.42133, 2017.

D. Roscher, S. Diehl, and M. Buchhold, Phenomenology of a First Order Dark State Phase Transition, 2018.

E. G. Dalla-torre, E. Demler, T. Giamarchi, and E. Altman, Dynamics and universality in noise-driven dissipative systems, Phys. Rev. B, vol.85, p.184302, 2012.

J. Marino and S. Diehl, Driven Markovian Quantum Criticality, Phys. Rev. Lett, vol.116, p.70407, 2016.

W. H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Rev. Mod. Phys, vol.75, p.715, 2003.

J. F. Poyatos, J. I. Cirac, and P. Zoller, Quantum Reservoir Engineering with Laser Cooled Trapped Ions, Phys. Rev. Lett, vol.77, p.4728, 1996.

H. Tan, G. Li, and P. Meystre, Dissipation-driven two-mode mechanical squeezed states in optomechanical systems, Phys. Rev. A, vol.87, p.33829, 2013.

C. Arenz, C. Cormick, D. Vitali, and G. Morigi, Generation of two-mode entangled states by quantum reservoir engineering, Journal of Physics B: Atomic, Molecular and Optical Physics, vol.46, p.224001, 2013.

M. Asjad and D. Vitali, Reservoir engineering of a mechanical resonator: generating a macroscopic superposition state and monitoring its decoherence, Journal of Physics B: Atomic, Molecular and Optical Physics, vol.47, p.45502, 2014.

A. Roy, Z. Leghtas, A. D. Stone, M. Devoret, and M. Mirrahimi, Continuous generation and stabilization of mesoscopic field superposition states in a quantum circuit, Phys. Rev. A, vol.91, p.13810, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01266483

Z. Leghtas, S. Touzard, I. M. Pop, A. Kou, B. Vlastakis et al., Confining the state of light to a quantum manifold by engineered two-photon loss, Science, vol.347, p.853, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01240210

J. Kerckhoff, M. A. Armen, and H. Mabuchi, Remnants of semiclassical bistability in the few-photon regime of cavity QED, Opt. Express, vol.19, p.24468, 2011.

A. Gilchrist, K. Nemoto, W. J. Munro, T. C. Ralph, S. Glancy et al., Schrödinger cats and their power for quantum information processing, Journal of Optics B: Quantum and Semiclassical Optics, vol.6, p.828, 2004.

A. Ourjoumtsev, R. Tualle-brouri, J. Laurat, and P. Grangier, Generating Optical Schrödinger Kittens for Quantum Information Processing, Science, vol.312, p.83, 2006.

M. Mirrahimi, M. Leghtas, V. Albert, S. Touzard, R. Schoelkopf et al., Dynamically protected cat-qubits: a new paradigm for universal quantum computation, New Journal of Physics, vol.16, p.45014, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01089514

H. Goto, Universal quantum computation with a nonlinear oscillator network, Phys. Rev. A, vol.93, p.50301, 2016.

S. Puri, S. Boutin, and A. Blais, Engineering the quantum states of light in a Kerrnonlinear resonator by two-photon driving, npj Quantum Information, vol.3, p.18, 2017.

L. Landau and E. Lifshitz, Mechanics, Course of Theoretical Physics, vol.1, 1982.

H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 2002.

M. Maggiore, A Modern Introduction to Quantum Field Theory, 2005.

M. Hillery, An introduction to the quantum theory of nonlinear optics, Acta Physica Slovaca. Reviews and Tutorials, vol.59, p.1, 2010.

R. W. Boyd and D. Prato, Nonlinear Optics

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaître et al., SpinOrbit Coupling for Photons and Polaritons in Microstructures, Phys. Rev. X, vol.5, p.11034, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01275245

A. Verger, C. Ciuti, and I. Carusotto, Polariton quantum blockade in a photonic dot, Phys. Rev. B, vol.73, p.193306, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00107401

A. L. Boité, Strongly correlated photons in arrays of nonlinear cavities, Theses, Université Paris Diderot-Paris, vol.7, 2015.

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin et al., Bosonic Condensation and Disorder-Induced Localization in a Flat Band, Phys. Rev. Lett, vol.116, p.66402, 2016.

F. Baboux, E. Levy, A. Lemaître, C. Gómez, E. Galopin et al., Measuring topological invariants from generalized edge states in polaritonic quasicrystals, Phys. Rev. B, vol.95, p.161114, 2017.

M. Mili?evi?, T. Ozawa, G. Montambaux, I. Carusotto, E. Galopin et al., Orbital Edge States in a Photonic Honeycomb Lattice, Phys. Rev. Lett, vol.118, p.107403, 2017.

U. Vool and M. Devoret, Introduction to quantum electromagnetic circuits, International Journal of Circuit Theory and Applications, vol.45, p.897

B. D. Josephson, The discovery of tunnelling supercurrents, Rev. Mod. Phys, vol.46, p.251, 1974.

F. R. Ong, M. Boissonneault, F. Mallet, A. Palacios-laloy, A. Dewes et al., Circuit QED with a Nonlinear Resonator: ac-Stark Shift and Dephasing, Phys. Rev. Lett, vol.106, p.167002, 2011.

S. Schmidt and J. Koch, Circuit QED lattices: Towards quantum simulation with superconducting circuits, Annalen der Physik, vol.525, p.395, 2013.

Y. Makhlin, G. Schön, and A. Shnirman, Quantum-state engineering with Josephsonjunction devices, Rev. Mod. Phys, vol.73, p.357, 2001.

J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster et al., Charge-insensitive qubit design derived from the Cooper pair box, Phys. Rev. A, vol.76, p.42319, 2007.

J. A. Schreier, A. A. Houck, J. Koch, D. I. Schuster, B. R. Johnson et al., Suppressing charge noise decoherence in superconducting charge qubits, Phys. Rev. B, vol.77, p.180502, 2008.

E. T. Jaynes and F. W. Cummings, Comparison of quantum and semiclassical radiation theories with application to the beam maser, Proceedings of the IEEE, vol.51, p.89, 1963.

V. Gorini, A. Kossakowski, and E. C. Sudarshan, Completely positive dynamical semigroups of N-level systems, Journal of Mathematical Physics, vol.17, p.821, 1976.

G. Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics, vol.48, p.119, 1976.

H. Carmichael, Statistical Methods in Quantum Optics 2: Non-Classical Fields, 2007.

H. Wiseman and G. Milburn, Quantum Measurement and Control, 2010.

M. G. Paris, The modern tools of quantum mechanics, The European Physical Journal Special Topics, vol.203, p.61, 2012.

S. Barnett, Quantum Information, 2009.

D. F. Walls and G. J. Milburn, Quantum Optics

K. Mølmer, Y. Castin, and J. Dalibard, Monte Carlo wave-function method in quantum optics, J. Opt. Soc. Am. B, vol.10, p.524, 1993.

A. J. Daley, Quantum trajectories and open many-body quantum systems, Advances in Physics, vol.63, p.77, 2014.

C. Gardiner, Stochastic Methods: A Handbook for the Natural and Social Sciences

G. S. Agarwal and E. Wolf, Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. I. Mapping Theorems and Ordering of Functions of Noncommuting Operators, Phys. Rev. D, vol.2, p.2161, 1970.

G. S. Agarwal and E. Wolf, Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. II. Quantum Mechanics in Phase Space, Phys. Rev. D, vol.2, p.2187, 1970.

K. Vogel and H. Risken, Quasiprobability distributions in dispersive optical bistability, vol.39, p.4675

A. Serafini, Quantum Continuous Variables: A Primer of Theoretical Methods, 2017.

W. Greiner, D. Rischke, L. Neise, and H. Stöcker, Thermodynamics and Statistical Mechanics, 2012.

T. Kato, Perturbation theory for linear operators, Classics in Mathematics, 1995.

K. Macieszczak, M. Gut?, I. Lesanovsky, and J. P. Garrahan, Towards a Theory of Metastability in Open Quantum Dynamics, Phys. Rev. Lett, vol.116, p.240404, 2016.

B. Horstmann, J. I. Cirac, and G. Giedke, Noise-driven dynamics and phase transitions in fermionic systems, Phys. Rev. A, vol.87, p.12108, 2013.

B. Baumgartner and H. N. , Analysis of quantum semigroups with GKS-Lindblad generators: II. General, Journal of Physics A: Mathematical and Theoretical, vol.41, p.395303, 2008.

A. Nagy and V. Savona, Driven-dissipative quantum Monte Carlo method for open quantum systems, Phys. Rev. A, vol.97, p.52129, 2018.

A. Biella, J. Jin, O. Viyuela, C. Ciuti, R. Fazio et al., Linked cluster expansions for open quantum systems on a lattice, Phys. Rev. B, vol.97, p.35103, 2018.

M. Biondi, S. Lienhard, G. Blatter, H. E. Türeci, and S. Schmidt, Spatial correlations in driven-dissipative photonic lattices, vol.19, p.125016, 2017.

S. Finazzi, A. L. Boité, F. Storme, A. Baksic, and C. Ciuti, Corner-Space Renormalization Method for Driven-Dissipative Two-Dimensional Correlated Systems, Phys. Rev. Lett, vol.115, p.80604, 2015.

A. Kshetrimayum, H. Weimer, and R. Orús, A simple tensor network algorithm for two-dimensional steady states, Nature Communications, vol.8, p.1291, 2017.

G. Vidal, Classical Simulation of Infinite-Size Quantum Lattice Systems in One Spatial Dimension, Phys. Rev. Lett, vol.98, p.70201, 2007.

M. Zwolak and G. Vidal, Mixed-State Dynamics in One-Dimensional Quantum Lattice Systems: A Time-Dependent Superoperator Renormalization Algorithm, Phys. Rev. Lett, vol.93, p.207205, 2004.

J. Cui, J. I. Cirac, and M. C. Bañuls, Variational Matrix Product Operators for the Steady State of Dissipative Quantum Systems, Phys. Rev. Lett, vol.114, p.220601, 2015.

P. D. Drummond and D. F. Walls, Quantum theory of optical bistability. I. Nonlinear polarisability model, vol.13, p.725

S. Diehl, A. Tomadin, A. Micheli, R. Fazio, and P. Zoller, Dynamical Phase Transitions and Instabilities in Open Atomic Many-Body Systems, Phys. Rev. Lett, vol.105, p.15702, 2010.

A. Tomadin, S. Diehl, and P. Zoller, Nonequilibrium phase diagram of a driven and dissipative many-body system, Phys. Rev. A, vol.83, p.13611, 2011.

A. L. Boité, G. Orso, and C. Ciuti, Steady-State Phases and Tunneling-Induced Instabilities in the Driven Dissipative Bose-Hubbard Model, Phys. Rev. Lett, vol.110, p.233601, 2013.

A. L. Boité, G. Orso, C. Ciuti, . Bose-hubbard, and . Model, Relation between drivendissipative steady states and equilibrium quantum phases, Phys. Rev. A, vol.90, p.63821, 2014.

J. Jin, D. Rossini, R. Fazio, M. Leib, and M. J. Hartmann, Photon Solid Phases in Driven Arrays of Nonlinearly Coupled Cavities, Phys. Rev. Lett, vol.110, p.163605, 2013.

J. Jin, D. Rossini, M. Leib, M. J. Hartmann, and R. Fazio, Steady-state phase diagram of a driven QED-cavity array with cross-Kerr nonlinearities, Phys. Rev. A, vol.90, p.23827, 2014.

R. M. Wilson, K. W. Mahmud, A. Hu, A. V. Gorshkov, M. Hafezi et al., Collective phases of strongly interacting cavity photons, Phys. Rev. A, vol.94, p.33801, 2016.

P. D. Drummond and C. W. Gardiner, Generalised P-representations in quantum optics, vol.13, p.2353

W. Bailey, Cambridge tracts in mathematics and mathematical physics, 1972.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, 1964.

T. Macrobert, Functions of a Complex Variable, 1954.

G. Kryuchkyan and K. Kheruntsyan, Exact quantum theory of a parametrically driven dissipative anharmonic oscillator, Optics Communications, vol.127, p.230, 1996.

C. H. Meaney, H. Nha, T. Duty, and G. J. Milburn, Quantum and classical nonlinear dynamics in a microwave cavity, EPJ Quantum Technology, vol.1, p.1, 2014.

M. Elliott and E. Ginossar, Applications of the Fokker-Planck equation in circuit quantum electrodynamics, Phys. Rev. A, vol.94, p.43840, 2016.

H. M. Gibbs, S. L. Mccall, and T. N. Venkatesan, Differential Gain and Bistability Using a Sodium-Filled Fabry-Perot Interferometer, Phys. Rev. Lett, vol.36, p.1135, 1976.

A. Dorsel, J. D. Mccullen, P. Meystre, E. Vignes, and H. Walther, Optical Bistability and Mirror Confinement Induced by Radiation Pressure, Phys. Rev. Lett, vol.51, p.1550, 1983.

G. Rempe, R. J. Thompson, R. J. Brecha, W. D. Lee, and H. J. Kimble, Optical bistability and photon statistics in cavity quantum electrodynamics, Phys. Rev. Lett, vol.67, p.1727, 1991.

V. R. Almeida and M. Lipson, Optical bistability on a silicon chip, Opt. Lett, vol.29, p.2387, 2004.

A. Baas, J. P. Karr, H. Eleuch, and E. Giacobino, Optical bistability in semiconductor microcavities, Phys. Rev. A, vol.69, p.23809, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00012338

T. Boulier, M. Bamba, A. Amo, C. Adrados, A. Lemaitre et al., Polariton-generated intensity squeezing in semiconductor micropillars, Nature Communications, vol.5, p.3260, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01092955

I. Carusotto and C. Ciuti, Spontaneous microcavity-polariton coherence across the parametric threshold: Quantum Monte Carlo studies, Phys. Rev. B, vol.72, p.125335, 2005.
DOI : 10.1103/physrevb.72.125335
URL : https://hal.archives-ouvertes.fr/hal-00004775

P. D. Drummond and M. Hillery, The Quantum Theory of Nonlinear Optics

R. Labouvie, B. Santra, S. Heun, and H. Ott, Bistability in a Driven-Dissipative Superfluid, Phys. Rev. Lett, vol.116, p.235302, 2016.
DOI : 10.1103/physrevlett.116.235302
URL : http://arxiv.org/pdf/1507.05007

S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg et al., Observation of a Localized Flat-Band State in a Photonic Lieb Lattice, Phys. Rev. Lett, vol.114, p.245504, 2015.

W. Casteels, R. Rota, F. Storme, and C. Ciuti, Probing photon correlations in the dark sites of geometrically frustrated cavity lattices, Phys. Rev. A, vol.93, p.43833, 2016.

M. Biondi, E. P. Van-nieuwenburg, G. Blatter, S. D. Huber, and S. Schmidt, Incompressible Polaritons in a Flat Band, Phys. Rev. Lett, vol.115, p.143601, 2015.
DOI : 10.1103/physrevlett.115.143601
URL : https://link.aps.org/accepted/10.1103/PhysRevLett.115.143601

A. Amo and J. Bloch, Exciton-polaritons in lattices: A non-linear photonic simulator, Comptes Rendus Physique, vol.17, 2016.
DOI : 10.1016/j.crhy.2016.08.007
URL : https://doi.org/10.1016/j.crhy.2016.08.007

D. Tanese, E. Gurevich, F. Baboux, T. Jacqmin, A. Lemaître et al., Fractal Energy Spectrum of a Polariton Gas in a Fibonacci Quasiperiodic Potential, Phys. Rev. Lett, vol.112, p.146404, 2014.

H. J. Carmichael, Photon Antibunching and Squeezing for a Single Atom in a Resonant Cavity, Phys. Rev. Lett, vol.55, p.2790, 1985.
DOI : 10.1103/physrevlett.56.539.2

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, Strongly Interacting Photons in a Nonlinear Cavity, Phys. Rev. Lett, vol.79, p.1467, 1997.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup et al., Photon blockade in an optical cavity with one trapped atom, Nature, vol.436, 2005.
DOI : 10.1109/cleo.2006.4628590
URL : https://authors.library.caltech.edu/93397/1/04628590.pdf

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink et al., Observation of Resonant Photon Blockade at Microwave Frequencies Using Correlation Function Measurements, Phys. Rev. Lett, vol.106, p.243601, 2011.
DOI : 10.1103/physrevlett.106.243601
URL : http://arxiv.org/pdf/1102.0461

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. Hollenberg, Quantum phase transitions of light, Nature Physics, vol.2, p.856, 2006.

M. J. Hartmann, F. G. Brandão, and M. B. Plenio, Strongly interacting polaritons in coupled arrays of cavities, Nature Physics, vol.2, p.849, 2006.
DOI : 10.1109/cleoe-iqec.2007.4386794
URL : http://arxiv.org/pdf/quant-ph/0606097

D. G. Angelakis, M. F. Santos, and S. Bose, Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays, Phys. Rev. A, vol.76, p.31805, 2007.
DOI : 10.1103/physreva.76.031805
URL : http://arxiv.org/pdf/quant-ph/0606159

M. J. Hartmann, F. G. Brandão, and M. B. Plenio, Quantum many-body phenomena in coupled cavity arrays, Laser and Photonics Reviews, vol.2, p.527, 2008.
DOI : 10.1002/lpor.200810046
URL : http://arxiv.org/pdf/0808.2557

J. Lebreuilly, A. Biella, F. Storme, D. Rossini, R. Fazio et al., Stabilizing strongly correlated photon fluids with non-Markovian reservoirs, Phys. Rev. A, vol.96, p.33828, 2017.
DOI : 10.1103/physreva.96.033828
URL : https://arpi.unipi.it/bitstream/11568/876443/2/PhysRevA.96.033828_CavityNonMarkovian.pdf

M. J. Hartmann, F. G. Brandão, and M. B. Plenio, Effective Spin Systems in Coupled Microcavities, Phys. Rev. Lett, vol.99, p.160501, 2007.
DOI : 10.1103/physrevlett.99.160501
URL : https://authors.library.caltech.edu/67712/1/PhysRevLett.99.160501.pdf

A. Kay and D. G. Angelakis, Reproducing spin lattice models in strongly coupled atomcavity systems, Europhysics Letters), vol.84, 2008.
DOI : 10.1209/0295-5075/84/20001
URL : http://arxiv.org/pdf/0802.0488

J. Qian, G. Dong, L. Zhou, and W. Zhang, Phase diagram of Rydberg atoms in a nonequilibrium optical lattice, Phys. Rev. A, vol.85, p.65401, 2012.

M. Viteau, P. Huillery, M. G. Bason, N. Malossi, D. Ciampini et al., Cooperative Excitation and Many-Body Interactions in a Cold Rydberg Gas, Phys. Rev. Lett, vol.109, p.53002, 2012.

J. Qian, L. Zhang, J. Zhai, and W. Zhang, Dynamical phases in a one-dimensional chain of heterospecies Rydberg atoms with next-nearest-neighbor interactions, Phys. Rev. A, vol.92, p.63407, 2015.

W. Casteels, R. M. Wilson, and M. Wouters, Gutzwiller Monte Carlo approach for a critical dissipative spin model, Phys. Rev. A, vol.97, p.62107, 2018.

R. Orús and G. Vidal, Infinite time-evolving block decimation algorithm beyond unitary evolution, Phys. Rev. B, vol.78, p.155117, 2008.

R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Annals of Physics, vol.349, p.117, 2014.

F. Verstraete, J. J. García-ripoll, and J. I. Cirac, Matrix Product Density Operators: Simulation of Finite-Temperature and Dissipative Systems, Phys. Rev. Lett, vol.93, p.207204, 2004.

C. Joshi, F. Nissen, and J. Keeling, Quantum correlations in the one-dimensional driven dissipative XY model, Phys. Rev. A, vol.88, p.63835, 2013.

L. Bonnes, D. Charrier, and A. M. Läuchli, Dynamical and steady-state properties of a Bose-Hubbard chain with bond dissipation: A study based on matrix product operators, Phys. Rev. A, vol.90, p.33612, 2014.

A. Biella, L. Mazza, I. Carusotto, D. Rossini, and R. Fazio, Photon transport in a dissipative chain of nonlinear cavities, Phys. Rev. A, vol.91, p.53815, 2015.

G. Vidal, Efficient Classical Simulation of Slightly Entangled Quantum Computations, Phys. Rev. Lett, vol.91, p.147902, 2003.

G. Vidal, Efficient Simulation of One-Dimensional Quantum Many-Body Systems, Phys. Rev. Lett, vol.93, p.40502, 2004.

B. S. Chissom, Interpretation of the Kurtosis Statistic, The American Statistician, vol.24, p.19, 1970.

C. C. Gerry, Non-classical Properties of Even and Odd Coherent States, Journal of Modern Optics, vol.40, p.1053, 1993.

L. Gilles, B. M. Garraway, and P. L. Knight, Generation of nonclassical light by dissipative two-photon processes, Phys. Rev. A, vol.49, p.2785, 1994.

E. Schrödinger, Die gegenwärtige Situation in der Quantenmechanik, Naturwissenschaften, vol.23, p.807, 1935.

M. Wolinsky and H. J. Carmichael, Quantum noise in the parametric oscillator: From squeezed states to coherent-state superpositions, Phys. Rev. Lett, vol.60, p.1836, 1988.

M. Everitt, T. Spiller, G. G. Milburn, R. D. Wilson, and A. M. Zagoskin, Engineering Dissipative Channels for Realising Schrödinger Cats in SQUIDs, Frontiers in ICT, vol.1, 2014.

L. Krippner, W. J. Munro, and M. D. Reid, Transient macroscopic quantum superposition states in degenerate parametric oscillation: Calculations in the large-quantumnoise limit using the positive P representation, Phys. Rev. A, vol.50, p.4330, 1994.

L. Sun, A. Petrenko, Z. Leghtas, B. Vlastakis, G. Kirchmair et al., Tracking photon jumps with repeated quantum non-demolition parity measurements, Nature, vol.511, p.444, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01089512

D. Vitali, S. Zippilli, P. Tombesi, and J. Raimond, Decoherence control with fully quantum feedback schemes, Journal of Modern Optics, vol.51, p.799, 2004.

S. Zippilli, D. Vitali, P. Tombesi, and J. M. Raimond, Scheme for decoherence control in microwave cavities, Phys. Rev. A, vol.67, p.52101, 2003.

C. Sayrin, I. Dotsenko, X. Zhou, B. Peaudecerf, T. Rybarczyk et al., Real-time quantum feedback prepares and stabilizes photon number states, Nature, vol.477, p.73, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00637734

X. Zhou, I. Dotsenko, B. Peaudecerf, T. Rybarczyk, C. Sayrin et al., Field Locked to a Fock State by Quantum Feedback with Single Photon Corrections, Phys. Rev. Lett, vol.108, p.243602, 2012.
URL : https://hal.archives-ouvertes.fr/tel-00737657

P. Campagne-ibarcq, E. Flurin, N. Roch, D. Darson, P. Morfin et al., Persistent Control of a Superconducting Qubit by Stroboscopic Measurement Feedback, Phys. Rev. X, vol.3, p.21008, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00910214

P. Goetsch, P. Tombesi, and D. Vitali, Effect of feedback on the decoherence of a Schrödinger-cat state: A quantum trajectory description, Phys. Rev. A, vol.54, p.4519, 1996.

M. Brune, S. Haroche, J. M. Raimond, L. Davidovich, and N. Zagury, Manipulation of photons in a cavity by dispersive atom-field coupling: Quantum-nondemolition measurements and generation of "Schrödinger cat" states, Phys. Rev. A, vol.45, p.5193, 1992.

T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, Quantum computation with optical coherent states, Phys. Rev. A, vol.68, p.42319, 2003.

M. S. Sarandy and D. A. Lidar, Adiabatic approximation in open quantum systems, Phys. Rev. A, vol.71, p.12331, 2005.