  Quantum walks function as essential means to implement quantum simulators,
allowing one to study complex and often directly inaccessible quantum processes
in controllable systems. In this contribution, the notion of a driven Gaussian
quantum walk is introduced. In contrast to typically considered quantum walks
in optical settings, we describe the operation of the walk in terms of a
nonlinear map rather than a unitary operation, e.g., by replacing a
beam-splitter-type coin with a two-mode squeezer, being a process that is
controlled and driven by a pump field. This opens previously unattainable
possibilities for quantum walks that include nonlinear elements as core
components of their operation, vastly extending their range of applications. A
full framework for driven Gaussian quantum walks is developed, including
methods to dynamically characterize nonlinear, quantum, and quantum-nonlinear
effects. Moreover, driven Gaussian quantum walks are compared with their
classically interfering and linear counterparts, which are based on classical
coherence of light rather than quantum superpositions. In particular, the
generation and boost of highly multimode entanglement, squeezing, and other
quantum effects are studied over the duration of the nonlinear walk.
Importantly, we prove the quantumness of the evolution itself, regardless of
the input state. A scheme for an experimental realization is proposed.
Furthermore, nonlinear properties of driven Gaussian quantum walks are
explored, such as amplification that leads to an ever increasing number of
correlated quantum particles, constituting a source of new walkers during the
walk. Therefore, a concept for quantum walks is proposed that leads to -- and
even produces -- directly accessible quantum phenomena, and that renders the
quantum simulation of nonlinear processes possible.
