  A Superconducting QUantum Interference Device (SQUID) modulated by a fast
oscillating magnetic flux can be used as a parametric amplifier, providing gain
with very little added noise. Here, we develop linearized models to describe
the parametrically flux-pumped SQUID in terms of an impedance. An unpumped
SQUID acts as an inductance, the Josephson inductance, whereas a flux-pumped
SQUID develops an additional, parallel element which we have coined the
``pumpistor.'' Parametric gain can be understood as a result of a negative
resistance of the pumpistor. In the degenerate case, the gain is sensitive to
the relative phase between the pump and signal. In the nondegenerate case, gain
is independent of this phase.
  We develop our models first for degenerate parametric pumping in the
three-wave and four-wave cases, where the pump frequency is either twice or
equal to the signal frequency, respectively. We then derive expressions for the
nondegenerate case where the pump frequency is not a multiple of the signal
frequency, where it becomes necessary to consider idler tones which develop.
For the nondegenerate three-wave case, we present an intuitive picture for a
parametric amplifier containing a flux-pumped SQUID where current at the signal
frequency depends upon the load impedance at an idler frequency. This
understanding provides insight and readily testable predictions of circuits
containing flux-pumped SQUIDs.
