Proper treatment of energy and momentum in time-modulated plasmas

Time-varying media enable electromagnetic nonreciprocity, frequency conversion, and the production of momentum space bandgaps in time crystals. However, most theoretical descriptions contain fundamental omissions in their treatment of how the medium properties are modulated, notably in the creation and removal of polarizable constituents. Neglecting the consequences of the momentum that these particles contribute to the medium plays a significant role in the outcome, and in some cases gives rise to nonphysical amplification of the waves. This is a problem traceable to the methods by which the medium's properties are modulated, and in this article, we give a specific example for the case in which the polarizable constituents are free-electron carriers injected and removed by ionization and recombination. We show that when momentum conservation is accounted for consistently in the modulation process, one avoids spurious postcollapse field growth commonly interpreted as amplification, or gain. This analysis applies broadly to other polarizable constituents and points to the importance of addressing how these constituents of the medium are introduced or altered; otherwise one encounters questions about how the surroundings perform work on the medium to account for this predicted gain.