Fig. 3: Transmission-line models for passive and active temporal boundaries with Drude dispersion.

a top: An increase in plasma frequency can be modeled as the shorting of a series inductor in the shunt branch of a transmission-line. In this scenario, the current \(J\) in the branch remains unchanged, along with the flux linkage on \({L}_{a}\), whereas the flux on \({L}_{b}\) is lost. Mechanically, this scenario is analogous to particles leaving the system (e.g. droplets leaving a water bucket) with their instantaneous velocity, carrying their momentum away. The analogy is corroborated by panel (a, bottom), which shows how the total energy density (continuous blue line) in the system is reduced, along with the energy in the forward wave (dashed line), and the electromagnetic momentum \(P\) (dashed red line) is reduced. By contrast, (b, top) reducing the plasma frequency by opening the short can only preserve the current in the branch if the additional inductor \({L}_{b}\) is prepared in such a state that its current at the switching instant matches the one of the original inductor, (b, bottom) increasing the total energy and net momentum density. This is analogous to additional particles entering a mechanical system with its identical instantaneous velocity, thereby increasing its total momentum without affecting its velocity. Instead, in a passive system, (d, top) voltage spikes on the respective inductors resulting from the TI drive a finite current through the additional inductor \({L}_{b}\), while abruptly reducing the current on \({L}_{a}\) within an infinitesimal time interval, such that the total flux linkage is redistributed between the two inductors, reducing the current \({J}_{c}\) across the branch. Mechanically, this is analogous to adding particles with no initial velocity into the system, and letting the system redistribute its momentum among them. As a result, (c, bottom) the total electromagnetic momentum is conserved, whereas the total energy is reduced. (d, top) In a dual fashion, conserving flux linkage upon an increase in plasma frequency would require a gain mechanism capable of replacing the flux lost by shorting the inductor \({L}_{b}\) in the branch by increasing the current on \({L}_{a}\). Expectedly, (d, bottom) this increases the energy density in the system, while conserving its momentum