  Axion-like particles (ALPs) are promising dark matter candidates. They are
typically described by a classical field, motivated by large phase space
occupation numbers. Here we show that such a description is accompanied by a
quantum effect: squeezing due to gravitational self-interactions. For a typical
QCD axion today, the onset of squeezing is reached on $\mathrm{\mu s}$-scales
and grows over millennia. Thus within the usual models based on the classical
Schr\"odinger-Poisson equation, a type of Gross-Pitaevskii equation, any viable
ALP is nonclassical. We also show that squeezing may be relevant on the scales
of other self-gravitating systems such as galactic haloes, or solitonic cores.
Conversely, our results highlight the incompleteness and limitations of the
classical single field description of ALPs.
