Testing collapse models with levitated nanoparticles: Detection challenge

We consider a nanoparticle levitated in a Paul trap in ultrahigh cryogenic vacuum, and look for the conditions which allow for a stringent noninterferometric test of spontaneous collapse models. In particular we compare different possible techniques to detect the particle motion. Key conditions which need to be achieved are extremely low residual pressure and the ability to detect the particle at ultralow power. We compare three different detection approaches based, respectively, on an optical cavity, an optical tweezer, and an electrical readout, and for each one we assess advantages, drawbacks, and technical challenges.

Figure 1

Current upper limits on the CSL model collapse rate from different noninterferometric experiments. Solid lines refer to bounds from mechanical or cold atom experiments, and dashed lines refer to experimental bounds probing the CSL field at very high frequency, specifically, x-ray spontaneous emission (dashed blue) [12], bulk heating in solid matter (dashed dark green) [25], cold atoms (light green) [26], the cantilever 2016 experiment (orange) [19], the cantilever 2017 experiment (red) [20], LISA Pathfinder early data (gray) [21, 22], and final data (black) [23]. The theoretical parameters suggested by Adler [9] and GRW [1] are also shown.

Figure 2

Internal temperature of the nanosphere as a function of the gas pressure and the absorbed power. The particle is a silica nanosphere with $R=200\mathrm{nm}$, the residual gas is helium at $T_g=300\mathrm{mK}$, the thermal accommodation factor has been set to $\alpha =0.4$, and the emissivity has been set to ${\epsilon }_{\mathrm{abs}}=0.1$.

Figure 3

Thermal force noise as a function of the residual gas pressure. It includes the effects of both gas collisions and blackbody radiation. Thick solid curves refer from top to bottom to an absorbed power of ${10}^{-10}, {10}^{-14}, {10}^{-18}$, and ${10}^{-22}\mathrm{W}$. Blackbody radiation is always negligible except for the flattening of the ${10}^{-10}\mathrm{W}$ curve at low pressure. As in Fig. 2, the particle is a silica nanosphere with $R=200\mathrm{nm}$, the residual gas is helium at $T_g=300\mathrm{mK}$, the thermal accommodation factor has been set to $\alpha =0.4$, and the blackbody emissivity has been set to ${\epsilon }_{\mathrm{abs}}=0.1$. As a reference we also plot the estimated contribution from the Paul trap bias noise (middle horizontal dashed blue line, see text for details), the CSL force noise for $r_c={10}^{-7}\mathrm{m}$ and $\lambda ={10}^{-10}\mathrm{Hz}$ (top dot-dashed black line), and GRW values $r_c={10}^{-7}\mathrm{m}$ and $\lambda =1\times {10}^{-16}\mathrm{Hz}$ (bottom dot-dashed green line).

Figure 4

Simplified scheme of the experiment. A Nd:Yag laser at $1064\mathrm{nm}$ is locked on the fundamental mode of an optical cavity by means of a standard Pound-Drever-Hall technique. A silica bead is held at the center of the cavity field by a linear Paul trap. Measurement of the particle dynamic is obtained by homodyning the transmitted cavity field.

Figure 5

Cavity output homodyne spectrum normalized to shot noise. Black is the total noise, blue (long-dashed) is the quantum noise, red dot-dashed is the thermal noise, and red dashed is the photon recoil. Finally, the (dashed) horizontal green line represents the typical frequency noise that could hinder the particle detection.

Figure 6

Imprecision noise and measurement bandwidth as a function of the collapse rate $\lambda$ that would provide a signal to noise ratio of 1 given the estimated total force noise acting on the particle and assuming $r_c={10}^{-7}\mathrm{m}$. Continuous lines refer to a frequency noise limited detection while dashed lines refer to a shot-noise limited one. See main text for more details.

Figure 7

(a) Detection using a paraboloidal mirror and (b) detection using a dual counterpropagating beam setup.

Figure 8

Optical tweezer homodyne spectrum normalized to shot noise. Black is the total noise, blue (long-dashed) is the quantum noise (photon recoil and backaction), and red (dot-dashed) is the thermal noise. Finally, the (dashed) green horizontal line represents the noise floor estimated using NEP to quantify the detector's dark noise which is expected to significantly hinder the detection.

Figure 9

Imprecision noise and measurement bandwidth as a function of the collapse rate that would provide a signal to noise ratio of 1 given the estimated total force noise acting on the particle and assuming $r_C={10}^{-7}\mathrm{m}$. Continuous lines refer to a NEP noise limited detection while dashed lines refer to a shot-noise limited one.

Figure 10

Scheme of a SQUID-based detection of the levitated particle. The motion of the charge $q$ induces a current $I$ in the electrodes of capacitance $C_{\text{el}}$. The current is enhanced by a low-loss superconducting transformer with mutual inductance $M$ and high inductance ratio ($L_p\gg L_s$) before being measured by a SQUID of input inductance $L_i$ ($L_i\simeq L_s$). A further way to boost the current and improve impedance matching is to add a capacitor $C$ ($C\gg C_{\text{el}}$) in order to tune the $\mathit{LC}$ resonance to the mechanical resonance of the trapped particle.

Figure 11

Spectrum of a levitated particle measured by a SQUID through a superconducting transformer. The particle is a silica nanosphere with $R=200\mathrm{nm}$, charge $q={10}^{3}e$, the resonance frequency $f_0=1\mathrm{kHz}$, the residual gas is helium at $T_g=300\mathrm{mK}$, and $P_g={10}^{-12}\mathrm{mbar}$. The SQUID noise is $N_{\hslash }=10$ and the transformer has $L_p=10\mathrm{H}$, coupling $k=0.8$, and $Q=1\times {10}^6$. The various noise components are total noise (solid black), gas collision noise (dot-dashed red), electrical noise (dashed green), SQUID backaction (blue long-dashed line), and SQUID imprecision (dashed horizontal gray line). The total force noise around resonance corresponds to a CSL collapse rate $\lambda \simeq 6\times {10}^{-14}\mathrm{Hz}$.

Figure 12

Displacement noise and measurement bandwidth, expressed as a function of the minimum measurable collapse rate $\lambda$, evaluated at the standard value $r_C={10}^{-7}\mathrm{m}$, for the untuned SQUID readout. The parameters are the same as in Fig. 11, except for the gas pressure, which is variable, in order to vary the force noise and thus the detectable $\lambda$.

Figure 13

Spectrum of a levitated particle measured by a SQUID through a superconducting transformer with tuned $\mathit{LC}$ resonance. Parameters are exactly the same as in Fig. 11, except that a capacitor $C$ is added in parallel to the primary inductor to tune the electrical and mechanical frequencies. The spectrum shown here is expressed as current at the input of the SQUID. The color and line type coding for the different contributions is the same as in Fig. 11. Note that the electrical noise (dashed green) is hardly visible as it almost coincides with the total noise (solid black) everywhere except in the narrow antiresonance between the two peaks.

Figure 14

Spectrum in the same situation as Fig. 13, but now expressed as force noise on the levitated particle and zoomed in a narrow range around the deep antiresonance visible in Fig. 13.

Figure 15

CSL exclusion plot, where we have reproduced the current bounds already discussed in Fig. 1, together with two representative hypothetical bounds from a levitated nanoparticle. The curve labeled by 1 represents the best scenario considered in this paper based on an optical readout, corresponding to a recoil-limited force noise of $4\times {10}^{-46}{\mathrm{N}}^2/\mathrm{Hz}$ and a bandwidth of the order of $0.5\left(0.1\right)\mathrm{Hz}$ for a quantum limited cavity (tweezer) detection. The curve labeled by 2 represents an extreme scenario where the dominating effect is the thermal noise at $300\mathrm{mK}$ and $P={10}^{-13}\mathrm{mbar}$. This might be in principle achieved with a SQUID detection and ultranarrow bandwidth $\approx {10}^{-5}\mathrm{Hz}$, and provided that electronics noise is suppressed to negligible values.