Strictly nonclassical behavior of a mesoscopic system

We experimentally demonstrate the strictly nonclassical behavior in a many-atom system using a recently derived criterion [E. Kot et al., Phys. Rev. Lett. 108, 233601 (2012)] that explicitly does not make use of quantum mechanics. We thereby show that the magnetic-moment distribution measured by McConnell et al. [Nature (London) 519, 439 (2015)] in a system with a total mass of $2.6\times {10}^5$ atomic mass units is inconsistent with classical physics. Notably, the strictly nonclassical behavior affects an area in phase space ${10}^3$ times larger than the Planck quantum $\hslash$.

Figure 1

(a) Experimental setup. $N=3\times {10}^3 {}^{87}\mathrm{Rb}$ atoms are trapped in an optical cavity by a far-detuned dipole trap, and prepared in the state $5S_{1/2}, F=1$. The magnetic moment of the atomic ensemble $M=N{\mu }_B$ is initialized along the $\widehat{x}$ axis, perpendicular to the cavity axis. A vertically polarized weak incident light pulse experiences a weak Faraday polarization rotation due to the atomic magnetization such that a horizontally polarized photon is sometimes detected. If detector $D$ measures such a photon, the magnetization of the Rb gas is measured along $\widehat{y}, \widehat{z}$, or $\frac{1}{\sqrt{2}}(\widehat{y}\pm \widehat{z})$, yielding magnetization distributions schematically indicated in (b). In classical physics, the measured distributions should arise from an underlying probability distribution $\rho (M_y,M_z)$. Here, using a variation of the criterion in Ref. [38], we show that there is no classical probability distribution $\rho (M_y,M_z)$ consistent with the measured result, thereby ruling out any classical description of the magnetic moment.

Figure 2

Reconstructing the rotationally averaged marginal atomic magnetic-moment distribution $\overline{F}\left(M^2\right)$ from the measured photon number distribution $g\left(n\right)$. (a) shows $g\left(n\right)$ (red solid circles) for the distribution of interest after registering a heralding click on detector $D$. The blue squares show the distribution for the initial state with the atomic magnetic moments prepared along the $\widehat{x}$ axis for reference. For the red solid line, we first apply Eq. (3) solving for the magnetic-moment distribution $\overline{F}\left(M^2\right)$, and then convert the distribution back into expected photon counts. This tests the numerical reliability of our data processing method. The inset shows a log-linear scale plot of the same data, which displays the large-photon-number behavior more clearly. (b) The reconstructed magnetic-moment distribution $\overline{F}\left(M^2\right)$ using our method. The blue line is for the reference state. The inset shows for illustration purposes the rotationally averaged distribution $P\left(M\right)$ assuming $\langle M\rangle =0$. All the error bars and error bands represent one standard deviation.

Figure 3

The mean value $\langle \mathcal{F}\rangle$ of the trial function vs the cutoff order $N_c$. The blue squares correspond to the reference state with all magnetic moments aligned, which has $\langle \mathcal{F}\rangle >0$, and does not violate classical physics. The analytic asymptotic limit for the reference state (blue squares) is $2/(N_c+2)$. The solid line here only joins the points in the plot. The red circles correspond to the state of interest. When $N_c$ is larger than 10, $\langle \mathcal{F}\rangle$ becomes negative, which is forbidden by classical physics. $\langle \mathcal{F}\rangle$ is monotonically decreasing and approaching $-0.024$ when $N_c$ is increasing. Here, all error bars represent one standard deviation.