Fig. 2: Transmission line models for passive and active temporal boundaries.

a, b Assuming that the bound charge Q is conserved, a permittivity increase \({\varepsilon }_{1}\to {\varepsilon }_{2}\) (a), represented in a transmission-line as the on-switching of a parallel lumped capacitor \({C}_{1}\), results in charge redistribution, which reduces the voltage \(V\) (electric field). In turn, this redistribution reduces (a, bottom) the total energy density \(U\) (continuous, dashed and dotted lines denote respectively total, forward and backward energy densities in logarithmic scale, normalized to the initial value \({U}_{0}\)), while the momentum flux density (red dot-dashed line) is conserved (\({P}_{0}\) denotes the initial momentum). By contrast, a (b) permittivity reduction can be achieved while conserving bound charges by transferring the charges before off-switching the capacitance \({C}_{1}\), which requires gain (realized, e.g., via a current source), thus (b, bottom) increasing the energy in the system. c, d Voltage (electric field) continuity instead is observed in a passive system upon a permittivity drop, whereby the bound charges in \({C}_{1}\) (depicted in red) are switched out of the system, (c, bottom) reducing the total energy, as well as the momentum density. Finally, imposing electric field continuity upon (d) a permittivity increase is equivalent to assuming that the additional capacitance enters the system with a dynamically varying, nonzero bound charge, produced, e.g., by a current source, in order to preserve the voltage, resulting in (d, bottom) an increase in total energy and momentum density