Lindbladian quantizations of nonlinear nonconservative systems

Nonlinear dynamical phenomena such as bifurcations, chaos and synchronization have led to important applications across many branches of science and engineering. Often, such systems rely on inherent nonlinear dissipation and pumping, which are in general difficult to model quantum mechanically. In this work, we present a comprehensive quantization scheme which can quantize a large class of nonlinear dissipative oscillators using Lindblad master equations. We then propose and prove the existence theorem for the Lindblad representation of nonlinear dynamical systems. This provides a systematic technique to quantize dynamical systems with arbitrary degree of nonlinearity. We also propose a heuristic approach called 'minimal-noise' quantization, which minimizes the number of Lindblad operators used. This reduces the relative amount of quantum noise which allows nonlinear effects to manifest more prominently. To highlight the utility of our scheme, we demonstrate various quantum bifurcations by engineering nonlinear dissipations, and quantize a large class of Liénard systems which exhibit quantum limit cycles. Our work opens up the prospect to harness the rich field of nonlinear dynamics for novel quantum technologies.