Multilevel quantum description of decoherence in superconducting qubits

We present a multilevel quantum theory of decoherence for a general circuit realization of a superconducting qubit. Using electrical network graph theory, we derive a Hamiltonian for the circuit. The dissipative circuit elements (external impedances, shunt resistors) are described using the Caldeira-Leggett model. The master equation for the superconducting phases in the Born-Markov approximation is derived and brought into the Bloch-Redfield form in order to describe multilevel dissipative quantum dynamics of the circuit. The model takes into account leakage effects, i.e., transitions from the allowed qubit states to higher excited states of the system. As a special case, we truncate the Hilbert space and derive a two-level (Bloch) theory with characteristic relaxation ${(T}_1)$ and decoherence ${(T}_2)$ times. We apply our theory to the class of superconducting flux qubits; however, the formalism can be applied for both superconducting flux and charge qubits.