Dynamical Casimir effect in superconducting circuits: a numerical approach

We present a numerical analysis of the particle creation for a quantum field in the presence of time dependent boundary conditions. Having in mind recent experiments involving superconducting circuits, we consider their description in terms of a scalar field in a one dimensional cavity satisfying generalized boundary conditions that involve a time-dependent linear combination of the field and its spatial and time derivatives. We evaluate numerically the Bogoliubov transformation between {\it in} and {\it out}-states and find that the rate of particle production strongly depends on whether the spectrum of the unperturbed cavity is equidistant or not, and also on the amplitude of the temporal oscillations of the boundary conditions. We provide analytic justifications for the different regimes found numerically.