  We develop a photonic description of short, one-dimensional electromagnetic
pulses, specifically in the language of electrical transmission lines. Current
practice in quantum technology, using arbitrary waveform generators, can
readily produce very short, few-cycle pulses in microwave TEM guided structures
(coaxial cables or coplanar waveguides) in a very low noise, low temperature
setting. We argue that these systems attain the limit of producing pure
coherent quantum states, in which the vacuum has been displaced for a short
time, and therefore short spatial extent. When the pulse is bipolar, that is,
the integrated voltage of the pulse is zero, then the state can be described by
the finite displacement of a single mode. Therefore there is a definite mean
number of photons, but which have neither a well defined frequency nor
position. Due to the Paley-Wiener theorem, the two-component photon
'wavefunction' of this mode is not strictly bounded in space even if the vacuum
displacement that defines it is bounded. This wavefunction's components are,
for the case of pulses moving in a specific direction, complex valued, with the
real and imaginary parts related by a Hilbert transform. They are thus akin to
the 'analytic signals' of communication theory. When the pulse is unipolar no
photonic description is possible -- the photon number can be considered to be
divergent. We consider properties that photon counters and quantum
non-demolition detectors must have to optimally convert and detect the photons
in several example pulses, and we discuss some consequence of this optimization
for the application of very short pulses in quantum cryptography.
