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

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