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Table of contents
1 Introduction
1.1 Quantum information processing with superconducting circuits
1.2 Superconducting circuits quantization
1.2.1 Circuit elements and notations
1.2.2 Circuit quantization
1.2.3 Useful Hamiltonian transformations
1.3 Hamiltonian engineering with parametric pumping
1.3.1 Principles of parametric pumping
1.3.2 Example of the Josephson Parametric Amplifier
1.4 Dissipation engineering
1.5 Using a cavity as a logical qubit: cat qubits
1.5.1 Two-photon cat qubits encoding
1.5.2 Stabilization of the cat-states manifold with dissipation engineering
1.5.3 Four-photon pumping extension
1.6 Physical realization of cat-pumping interaction with a single Josephson junction
1.7 Plan of the manuscript
2 Structural instability of driven transmon circuit
2.1 Strongly driven transmon coupled to a cavity
2.2 Change of frame for numerical simulations
2.3 Simulations in the Floquet-Markov framework for weak dissipation
2.3.1 Floquet-Markov framework for weak dissipation
2.3.2 Numerical simulations of the ac Stark shifts in absence of offset charge
2.3.3 Influence of the offset charge
2.4 Comparison with experimental data
3 Inductively shunted transmon: a solution to dynamical instability
3.1 Model of the driven shunted transmon circuit
3.2 Simulations in the Floquet-Markov framework for weak dissipation
3.3 Rotating-wave approximation results and comparison with numerical simulations
3.4 Choice of parameters for the shunted transmon circuit
4 Floquet-Markov simulations
4.1 Steady-state computation framework
4.1.1 Encoding the circuit Hamiltonian
4.1.2 Floquet code
4.1.3 Running simulations
4.2 Analysis code
4.2.1 Resonant frequencies of the system
4.2.2 Induced Kerr strength
5 Asymmetric Josephson Ring Modulator
5.1 Josephson Ring Modulator
5.1.1 Unshunted Josephson Ring Modulator
5.1.2 Josephson Ring Modulator with shunt inductances
5.2 Asymmetric Josephson Ring Modulator
5.2.1 Circuit Hamiltonian
5.2.2 Quantization of the AJRM circuit embedded in a microwave cavity
5.3 AJRM for the two-photon and four-photon pumping schemes
6 Conclusions and perspectives
Bibliography
