  Causality - the principle stating that the output of a system cannot
temporally precede the input - is a universal property of nature. Here, we show
that analogous input-output relations can also be realized in the spectral
domain by leveraging the peculiar properties of time-modulated non-Hermitian
photonic systems. Specifically, we uncover the existence of a broad class of
complex time-modulated metamaterials which obey the time-domain equivalent of
the well-established frequency-domain Kramers-Kronig relations (a direct
consequence of causality). We find that, in the scattering response of such
time-modulated systems, the output frequencies are inherently prohibited from
spectrally preceding the input frequencies, hence we refer to these systems as
'spectrally causal'. We explore the consequences of this newly introduced
concept for several relevant applications, including broadband perfect
absorption, temporal cloaking of an 'event', and truly unidirectional
propagation along a synthetic dimension. By emulating the concept of causality
in the spectral domain and providing new tools to extend the field of
temporally modulated metamaterials ("chrono-metamaterials") into the complex
realm, our findings may open unexplored opportunities and enable relevant
technological advances in various areas of photonics and, more broadly, of wave
physics and engineering.
