Probing spontaneous wave-function collapse with entangled levitating nanospheres

Wave-function collapse models are considered to be the modified theories of standard quantum mechanics at the macroscopic level. By introducing nonlinear stochastic terms in the Schrödinger equation, these models (different from standard quantum mechanics) predict that it is fundamentally impossible to prepare macroscopic systems in macroscopic superpositions. The validity of these models can only be examined by experiments, and hence efficient protocols for these kinds of experiments are greatly needed. Here we provide a protocol that is able to probe the postulated collapse effect by means of the entanglement of the center-of-mass motion of two nanospheres optically trapped in a Fabry-Pérot cavity. We show that the collapse noise results in a large reduction of the steady-state entanglement, and the entanglement, with and without the collapse effect, shows distinguishable scalings with certain system parameters, which can be used to determine unambiguously the effect of these models.

Figure 1

Sketch of the proposed scheme for testing the CSL theory with entangled nanospheres. Two identical nanospheres are optically trapped with different frequencies ${\omega }_1$ and ${\omega }_2$ in a Fabry-Pérot cavity. The cavity mode is bichromatically driven at the two frequencies ${\omega }_0+{\omega }_1$ and ${\omega }_0-{\omega }_2$. Large and robust entanglement between the two spheres can be generated in the steady state. The output field of the cavity is fed back into the input port through highly reflective mirrors (HRMs) and a controllable beamsplitter (CBS) with a tunable reflection coefficient $r_B$, which is used for reducing the cavity loss and improving the entanglement.

Figure 2

(a) Diffusion rates for the scattering of trapping light $D_t^1$ (blue), cavity light $D_c^1$ (red), air molecules $D_a^1$ (gray), and the collapse rate ${\lambda }_{\mathrm{sph}}^1$ (green) vs the trapping frequency ${\omega }_1$. Note that the curves of $D_c^1$ and $D_a^1$ are very close to the ${\omega }_1$ axis and are no longer visible. (b) The solid line denotes the sum of the diffusion rates $D_t^1$, $D_c^1$, and $D_a^1$ in (a), while the dashed line includes also the collapse rate ${\lambda }_{\mathrm{sph}}^1$ on the basis of the solid line. (c) Steady-state entanglement $E_N$ between the two nanospheres vs the trapping frequency ${\omega }_1$. The blue (red) line refers to the case with (without) the CSL effect. The parameters are $R=0.15r_c$, $r_B=0.99$, $G_2=1.2{\kappa }_{\mathrm{eff}}$, $G_1=0.72G_2$, ${\omega }_2=2{\omega }_1$, $L=4$ cm, $\kappa =20$ kHz (corresponding to $\mathcal{F}=5.9\times {10}^5$ and ${\kappa }_{\mathrm{eff}}=200$ Hz), $R_c=L/1.5$, ${\lambda }_c=1064$ nm, $\mathcal{N}=0.8$, $T=10$ mK, $P_a={10}^{-12}$ Torr, $\lambda ={10}^{-8}$ Hz, $r_c=100$ nm, and we use diamond nanospheres with ${\rho }_0=3.5{\mathrm{g}/\mathrm{cm}}^3$ and $\epsilon =5.76$.

Figure 3

(a)–(c) Total diffusion rates with (dashed line, i.e., $D_t^1+D_c^1+D_a^1+{\lambda }_{\mathrm{sph}}^1$) and without (solid line, i.e., $D_t^1+D_c^1+D_a^1$) the CSL effect vs the trapping frequency ${\omega }_1$. (d)–(f) Steady-state entanglement $E_N$ of the two nanospheres vs the trapping frequency ${\omega }_1$. The blue (red) line corresponds to the case with (without) the CSL effect. The parameters are as follows: (a),(d) $\lambda ={10}^{-9}$ Hz, $R=0.15r_c$, $r_B=0.996$, $G_2=1.2{\kappa }_{\mathrm{eff}}$, $G_1=0.77G_2$. (b),(e) $\lambda ={10}^{-10}$ Hz, $R=0.18r_c$, $r_B=0.999$, $G_2=2.2{\kappa }_{\mathrm{eff}}$, $G_1=0.79G_2$. (c),(f) $\lambda ={10}^{-11}$ Hz, $R=0.22r_c$, $r_B=0.999$, $G_2=2{\kappa }_{\mathrm{eff}}$, $G_1=0.79G_2$. The other parameters are as in Fig. 2.