  Interactions between photons (nonlinearities) enable a powerful form of
control over the state of light. This control has enabled technologies such as
light sources at new wavelengths, ultra-short optical pulses, frequency-comb
metrology systems, even quantum light sources. Common to a wide variety of
nonlinear optical technologies is an equilibrium between an energy source, such
as an external laser, and dissipation, such as radiation loss or absorption. In
the vast majority of these systems, the coupling between the system and the
outside world (which leads to loss) is well-described as ``Markovian,'' meaning
that the outside world has no memory of its past state. In this work, we
introduce a class of driven-dissipative systems in which a nonlinear cavity
experiences non-Markovian coupling to the outside world. In the classical
regime, we show that these non-Markovian cavities can have extremely low
thresholds for nonlinear effects, as well as self-pulsing instabilities at THz
rates, and rich phase diagrams with alternating regions of stability and
instability. In the quantum regime, we show how these system, when implemented
on state-of-the-art platforms, can enable generation of strongly squeezed
cavity states with intensity fluctuations that can be more than 15 dB below the
classical limit, in contrast to the Markovian driven-dissipative cavity, in
which the limit is 3 dB. In the regime of few-photon nonlinearity, such
non-Markovian cavities can enable a deterministic protocol to generate Fock
states of high order, which are long-desired, but still elusive at optical
frequencies. We expect that exploiting non-Markovian couplings in nonlinear
optics should in the future lead to even richer possibilities than those
discussed here for both classical and quantum light manipulations.
