  We consider Schwinger pair production in 1+1 dimensional de Sitter space,
filled with a constant electric field $E$. This can be thought of as a model
for describing false vacuum decay beyond the semiclassical approximation, where
pairs of a quantum field $\phi$ of mass $m$ and charge $e$ play the role of
vacuum bubbles. We find that the adiabatic "in" vacuum associated with the flat
chart develops a space-like expectation value for the current $J$, which
manifestly breaks the de Sitter invariance of the background fields. We derive
a simple expression for $J(E)$, showing that both "upward" and "downward"
tunneling contribute to the build-up of the current. For heavy fields, with
$m^2\gg eE,H^2$, the current is exponentially suppressed, in agreement with the
results of semiclassical instanton methods. Here $H$ is the inverse de Sitter
radius. On the other hand, light fields with $ m \ll H$ lead to a phenomenon of
infrared hyperconductivity, where a very small electric field $mH \lesssim eE
\ll H^2$ leads to a very large current $J \sim H^3 /E$. We also show that all
Hadamard states for $\phi$ necessarily break de Sitter invariance. Finally, we
comment on the role of initial conditions, and "persistence of memory" effects.
