Upper Bounds on Spontaneous Wave-Function Collapse Models Using Millikelvin-Cooled Nanocantilevers

Collapse models predict a tiny violation of energy conservation, as a consequence of the spontaneous collapse of the wave function. This property allows us to set experimental bounds on their parameters. We consider an ultrasoft magnetically tipped nanocantilever cooled to millikelvin temperature. The thermal noise of the cantilever fundamental mode has been accurately estimated in the range 0.03–1 K, and any other excess noise is found to be negligible within the experimental uncertainty. From the measured data and the cantilever geometry, we estimate the upper bound on the continuous spontaneous localization collapse rate in a wide range of the correlation length $r_C$. Our upper bound improves significantly previous constraints for $r_C>10^{-6}\mathrm{m}$, and partially excludes the enhanced collapse rate suggested by Adler. We discuss future improvements.

Figure 1

(a) Scheme of the mechanical resonator. We focus on the fundamental bending mode of a high aspect ratio nanocantilever with length $L$, width $w$, and thickness $d$. A ferromagnetic microsphere with radius $R$ is attached to the free end of the cantilever. (b) Calculated CSL-induced heating $\mathrm{\Delta }{T}_{\mathrm{CSL}}$ of the cantilever fundamental mode as function of $r_C$. The total effect (solid line) includes two terms associated, respectively, to the microsphere (dashed line)and to the cantilever (dotted line), as well as a correlation term. Because of higher density, the contribution of the sphere is largely dominant for $r_C>10^{-7}\mathrm{m}$.

Figure 2

Main panel: cantilever mode temperature $T_m=E/k_B$ as a function of the bath temperature $T$. Data in the saturation region $T<25\mathrm{mK}$ are excluded by the data analysis. The straight line represents the best fit with $T_m=\alpha T+T_0$. Inset: examples of acquired averaged spectra at two representative temperatures $T=11\mathrm{mK}$ and $T=1.01\mathrm{K}$, with the respective best fit with a Lorentzian curve. Figures reproduced from Ref. [27].

Figure 3

Exclusion plot in the $\lambda -r_C$ plane based on our experimental data, compared with the best experimental upper bounds reported so far and with the proposed theoretical lower bounds. Continuous (red) curve: upper limit on the CSL collapse rate $\lambda$, as function of the characteristic length $r_C$. The region above the curve is excluded at 95% confidence level. Dashed (blue) curve: upper limit on $\lambda$ from spontaneous x-ray emission [32]. Dotted (black) line: foreseen upper limit from the proposed future upgraded setup (see text). Hollow (purple) circle: best upper limit on $\lambda$ from matter-wave interferometry at $r_C=10^{-7}\mathrm{m}$ [20, 21, 22]. Filled (violet) circle: conservative lower bound on CSL parameters according to Ghirardi et al. [2]. Green bars: optimistic lower bounds on $\lambda$ at $r_C=10^{-7}\mathrm{m}$ and $r_C=10^{-6}\mathrm{m}$ as suggested by Adler [5].