Laplace transform approach for the dynamics of N qubits coupled to a resonator

An approach to use the method of Laplace transform for the perturbative solution of the Schr\"{o}dinger equation at any order of the perturbation for a system of $N$ qubits coupled to a cavity with $n$ photons is suggested. We investigate the dynamics of a system of $N$ superconducting qubits coupled to a common resonator with time-dependent coupling. To account for the contribution of the dynamical Lamb effect to the probability of excitation of the qubit, we consider counter-rotating terms in the qubit-photon interaction Hamiltonian. As an example, we illustrate the method for the case of two qubits coupled to a common cavity. The perturbative solutions for the probability of excitation of the qubit show excellent agreement with the numerical calculations.