Nonclassicality of axionlike dark matter through gravitational self-interactions

Axionlike 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 $\mu \mathrm{s}$ scales and grows over millennia. Thus within the usual models based on the classical Schrödinger-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.

Figure 1

Squeezing generated through unitary time evolution of a dark matter coherent state $|\alpha ⟩$ in the Hartree approximation visualized by the Wigner function $W(z,t)=(\pi \hslash {)}^{-1}⟨\alpha |\widehat{D}(z)(-1{)}^{{\widehat{a}}^{†}\widehat{a}}{\widehat{D}}^{†}(z)|\alpha ⟩$, see [47], where $\widehat{D}(z)=e^{z{\widehat{a}}^{†}-z^{*}\widehat{a}}$. Darker hues of blue correspond to larger values of $W$. Top: initial coherent state. Middle: onset of squeezing. Bottom: maximal achievable squeezing for $\sqrt{N}=2000$. The initial time $t=0$ is an arbitrary moment in the late universe (i.e., close to today). The red (large) dot indicates the classical solution $a_{\mathrm{cl}}(t)=\alpha =2000$, and the black (small) dot the quantum mean field $⟨\widehat{a}(t)⟩$. The classical solution is time independent since (3f) holds for the ALP.

Figure 2

Squeezing timescales $t_{\mathrm{sqz}}$ and $t_{\max }$ (upper panel) and maximal squeezing (lower panel), and their scaling with $N$, for the Kerr model. Full lines are exact, dashed lines approximations used in the main text.

Figure 3

$\mathrm{Var}(\widehat{n})$ for a squeezed coherent state $|\beta ,\rho e^{2i\varphi }⟩$ with $\beta =\sqrt{N}={10}^6$ and $\rho =r(t)$ with $0\le r(t)\le r_{\max }$ of the Kerr oscillator (b3). Inserting $\varphi =0$ corresponds to amplitude squeezing, accompanied by $\widehat{n}$-squeezing, whereas inserting $\varphi ={\theta }_{-}(t)$ of the Kerr evolution prevents $\widehat{n}$-squeezing. The dashed line shows $\mathrm{Var}(\widehat{n})$ for a reference Kerr state.