Emulating time interfaces via smooth temporal transitions

The introduction of time as a new degree of freedom to control wave-matter interactions have opened new avenues to fully control wave propagation in four dimensions (x,y,z,t). Time interfaces (rapid changes of the constitutive relations of the medium where a wave propagates) have recently become popular as they are the temporal analogue of spatial interfaces. While recent groundbreaking experimental work has demonstrated time interfaces from water waves, microwaves and the optical regime, rapidly changing, for instance, the permittivity of the medium requires carefully engineered structures. Here, we study the possibility of implementing smooth temporal transitions of the refractive index of the medium in order to mimic the response of time interfaces via adiabatic modulations. It is shown that indeed, as long as the signal is inside the structure during the modulation time, there are some values of rising/falling time of the adiabatic modulation that enables a full approximate emulation of a time interface. These results may open further avenues to explore by relaxing the speed at which the time-modulation should be introduced when designing four-dimensional media.
Comment: 24 pages. 5 figures, 79 references