I. Kovacic and M. J. Brennan, The Duffing Equation: Nonlinear Oscillators and their Behaviour, ). [2] G. Duffing, Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz (F. Vieweg & Sohn, 1918.
DOI : 10.1002/9780470977859

J. Bourassa, F. Beaudoin, J. M. Gambetta, and A. Blais, Josephson-junction-embedded transmission-line resonators: From Kerr medium to in-line transmon, Physical Review A, vol.86, issue.1, p.13814, 2012.
DOI : 10.1103/PhysRevA.86.013814

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, Physical Review Letters, vol.106, issue.24, p.243601, 2011.
DOI : 10.1103/PhysRevLett.106.243601

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado et al., Dispersive Photon Blockade in a Superconducting Circuit, Physical Review Letters, vol.107, issue.5, p.53602, 2011.
DOI : 10.1103/PhysRevLett.107.053602

G. Kirchmair, B. Vlastakis, Z. Leghtas, S. E. Nigg, H. Paik et al., Observation of quantum state collapse and revival due to the single-photon Kerr effect, Nature, vol.495, issue.7440, p.205, 2013.
DOI : 10.1103/PhysRevLett.77.4281

URL : https://hal.archives-ouvertes.fr/hal-00845264

I. Siddiqi, R. Vijay, F. Pierre, C. M. Wilson, M. Metcalfe et al., RF-Driven Josephson Bifurcation Amplifier for Quantum Measurement, Physical Review Letters, vol.93, issue.20, p.207002, 2004.
DOI : 10.1103/PhysRevLett.93.207002

M. A. Castellanos-beltran and K. W. Lehnert, Widely tunable parametric amplifier based on a superconducting quantum interference device array resonator, Applied Physics Letters, vol.91, issue.8, p.83509, 2007.
DOI : 10.1063/1.2773988

A. Blais, R. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation, Physical Review A, vol.69, issue.6, p.62320, 2004.
DOI : 10.1103/PhysRevA.69.062320

R. Vijay, D. H. Slichter, and I. Siddiqi, Observation of Quantum Jumps in a Superconducting Artificial Atom, Physical Review Letters, vol.106, issue.11, p.110502, 2011.
DOI : 10.1103/PhysRevLett.106.110502

D. Ristè, J. G. Van-leeuwen, H. Ku, K. W. Lehnert, and L. Dicarlo, Initialization by Measurement of a Superconducting Quantum Bit Circuit, Physical Review Letters, vol.109, issue.5, p.50507, 2012.
DOI : 10.1103/PhysRevLett.109.050507

A. Lupa¸sculupa¸scu, S. Saito, T. Picot, P. C. De-groot, C. J. Harmans et al., Quantum non-demolition measurement of a superconducting two-level system, Nature Physics, vol.307, issue.2, p.119, 2007.
DOI : 10.1126/science.1126475

]. F. Mallet, F. R. Ong, A. Palacios-laloy, F. Nguyen, P. Bertet et al., Single-shot qubit readout in circuit quantum electrodynamics, Nature Physics, vol.296, issue.11, p.791, 2009.
DOI : 10.1038/nphys1400

URL : https://hal.archives-ouvertes.fr/hal-00440277

F. R. Ong, M. Boissonneault, F. Mallet, A. C. Doherty, A. Blais et al., Quantum Heating of a Nonlinear Resonator Probed by a Superconducting Qubit, Physical Review Letters, vol.110, issue.4, p.47001, 2013.
DOI : 10.1103/PhysRevLett.110.047001

URL : https://hal.archives-ouvertes.fr/cea-01477717

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, Physical Review Letters, vol.106, issue.16, p.167002, 2011.
DOI : 10.1103/PhysRevLett.106.167002

M. Boissonneault, A. C. Doherty, F. R. Ong, P. Bertet, D. Vion et al., Back-action of a driven nonlinear resonator on a superconducting qubit, Marthaler and M. I. Dykman, pp.22305-725, 1980.
DOI : 10.1103/PhysRevA.85.022305

M. I. Dykman, M. Marthaler, and V. Peano, Quantum heating of a parametrically modulated oscillator: Spectral signatures, Physical Review A, vol.83, issue.5, p.52115, 2011.
DOI : 10.1103/PhysRevA.83.052115

S. André, L. Guo, V. Peano, M. Marthaler, and G. Schön, Emission spectrum of the driven nonlinear oscillator, Physical Review A, vol.85, issue.5, p.53825, 2012.
DOI : 10.1103/PhysRevA.85.053825

F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, Laser Cooling to the Zero-Point Energy of Motion, Physical Review Letters, vol.62, issue.4, p.403, 1989.
DOI : 10.1103/PhysRevLett.62.403

Q. A. Turchette, B. E. Kielpinski, D. King, D. M. Leibfried, C. J. Meekhof et al., Heating of trapped ions from the quantum ground state, Physical Review A, vol.61, issue.6, p.63418, 2000.
DOI : 10.1103/PhysRevA.61.063418

B. Yurke, L. R. Corruccini, P. G. Kaminsky, L. W. Rupp, A. D. Smith et al., Observation of parametric amplification and deamplification in a Josephson parametric amplifier, Physical Review A, vol.39, issue.5, p.2519, 1989.
DOI : 10.1103/PhysRevA.39.2519

M. A. Castellanos-beltran, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, Amplification and squeezing of quantum noise with a tunable Josephson metamaterial, Nature Physics, vol.4, issue.12, p.929, 2008.
DOI : 10.1109/77.403046

M. I. Dykman and M. A. Krivoglaz, Fluctuations in nonlinear systems near bifurcations corresponding to the appearance of new stable states, Physica A: Statistical Mechanics and its Applications, vol.104, issue.3, p.480, 1980.
DOI : 10.1016/0378-4371(80)90010-2

I. Serban, M. I. Dykman, and F. K. Wilhelm, Relaxation of a qubit measured by a driven Duffing oscillator, Physical Review A, vol.81, issue.2, p.22305, 2010.
DOI : 10.1103/PhysRevA.81.022305

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, Physical Review A, vol.76, issue.4, p.42319, 2007.
DOI : 10.1103/PhysRevA.76.042319

C. W. Gardiner and P. Zoller, Quantum Noise : A Handbook of Markovian and Non-Markovian Qu, p.449, 2000.

I. Siddiqi, R. Vijay, M. Metcalfe, E. Boaknin, L. Frunzio et al., Dispersive measurements of superconducting qubit coherence with a fast latching readout, Physical Review B, vol.73, issue.5, p.54510, 2006.
DOI : 10.1103/PhysRevB.73.054510

M. Steffen, S. Kumar, D. P. Divincenzo, J. R. Rozen, G. A. Keefe et al., High-Coherence Hybrid Superconducting Qubit, Physical Review Letters, vol.105, issue.10, p.100502, 2010.
DOI : 10.1103/PhysRevLett.105.100502

J. Gambetta, A. Blais, M. Boissonneault, A. A. Houck, D. I. Schuster et al., Quantum trajectory approach to circuit QED: Quantum jumps and the Zeno effect, Physical Review A, vol.77, issue.1, p.12112, 2008.
DOI : 10.1103/PhysRevA.77.012112

M. Boissonneault, J. M. Gambetta, and A. Blais, Dispersive regime of circuit QED: Photon-dependent qubit dephasing and relaxation rates, Physical Review A, vol.79, issue.1, p.13819, 2009.
DOI : 10.1103/PhysRevA.79.013819

M. Boissonneault, J. M. Gambetta, and A. Blais, Nonlinear dispersive regime of cavity QED: The dressed dephasing model, Physical Review A, vol.77, issue.6, p.60305, 2008.
DOI : 10.1103/PhysRevA.77.060305

J. Gambetta, A. Blais, D. I. Schuster, A. Wallraff, L. Frunzio et al., Qubit-photon interactions in a cavity: Measurement-induced dephasing and number splitting, Physical Review A, vol.74, issue.4, p.42318, 2006.
DOI : 10.1103/PhysRevA.74.042318

R. Zwanzig, Ensemble Method in the Theory of Irreversibility, The Journal of Chemical Physics, vol.33, issue.5, p.1338, 1960.
DOI : 10.1063/1.1731409

V. Peano and M. Dykman, ArXiv e-prints (2013), arXiv:1307