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Interactions between photons (nonlinearities) enable a powerful form ofcontrol over the state of light. This control has enabled technologies such aslight sources at new wavelengths, ultra-short optical pulses, frequency-combmetrology systems, even quantum light sources. Common to a wide variety ofnonlinear optical technologies is an equilibrium between an energy source, suchas an external laser, and dissipation, such as radiation loss or absorption. Inthe vast majority of these systems, the coupling between the system and theoutside world (which leads to loss) is well-described as ``Markovian,'' meaningthat the outside world has no memory of its past state. In this work, weintroduce a class of driven-dissipative systems in which a nonlinear cavityexperiences non-Markovian coupling to the outside world. In the classicalregime, we show that these non-Markovian cavities can have extremely lowthresholds for nonlinear effects, as well as self-pulsing instabilities at THzrates, and rich phase diagrams with alternating regions of stability andinstability. In the quantum regime, we show how these system, when implementedon state-of-the-art platforms, can enable generation of strongly squeezedcavity states with intensity fluctuations that can be more than 15 dB below theclassical limit, in contrast to the Markovian driven-dissipative cavity, inwhich the limit is 3 dB. In the regime of few-photon nonlinearity, suchnon-Markovian cavities can enable a deterministic protocol to generate Fockstates of high order, which are long-desired, but still elusive at opticalfrequencies. We expect that exploiting non-Markovian couplings in nonlinearoptics should in the future lead to even richer possibilities than thosediscussed here for both classical and quantum light manipulations.

Driven-dissipative phases and dynamics in non-Markovian nonlinear photonics