  It is universally accepted that noise may bring order to complex
nonequilibrium systems. Most strikingly, entirely new states not seen in the
noiseless system can be induced purely by including multiplicative noise -- an
effect known as pure noise-induced transitions. It was first observed in
superfluids in the 1980s. Recent results in complex nonequilibrium systems have
also shown how new collective states emerge from such pure noise-induced
transitions, such as the foraging behavior of insect colonies, and schooling in
fish. Here we report such effects of noise in a quantum-mechanical system
without a classical limit. We use a minimal model of a nonlinearly damped
oscillator in a fluctuating environment that is analytically tractable, and
whose microscopic physics can be understood. When multiplicative environmental
noise is included, the system is seen to transition to a limit-cycle state. The
noise-induced quantum limit cycle also exhibits other genuinely nonclassical
traits, such as Wigner negativity and number-parity sensitive circulation in
phase space. Such quantum limit cycles are also conservative. These properties
are in stark contrast to those of a widely used limit cycle in the literature,
which is dissipative and loses all Wigner negativity. Our results establish the
existence of a pure noise-induced transition that is nonclassical and unique to
open quantum systems. They illustrate a fundamental difference between quantum
and classical noise.
