  We propose that periodic driving can stabilize a new type of order,
"period-doubled superconductivity", in which a superconducting order parameter
oscillates at half the frequency of the drive. Despite having a zero
time-averaged order parameter, the ordered state exhibits perfect conductivity
and a Meissner effect. Our theory predicts that this phase may be realized as
the steady state of materials shown to exhibit photo-induced superconductivity.
We propose to detect the period-doubled oscillation of the order parameter by
utilizing a Josephson junction between a photo-induced superconductor and a
conventional superconductor. Our theory can also be realized via parametric
driving of cold bosonic atoms in an optical lattice.
