Room temperature test of the Continuous Spontaneous Localization model using a levitated micro-oscillator

TL;DR

This study demonstrates a room-temperature levitated micro-oscillator experiment that sets new upper bounds on the CSL collapse rate, showing promise for future collapse model tests.

Contribution

First proof-of-principle experiment using a magneto-gravitational levitated micro-oscillator to test CSL at room temperature, improving bounds significantly.

Findings

Set two orders of magnitude better upper bounds on CSL collapse rate.

Achieved partial sensitivity to the enhanced collapse rate proposed by Adler.

Showed advantages of room-temperature operation over low-temperature experiments.

Abstract

The Continuous Spontaneous Localization (CSL) model predicts a tiny break of energy conservation via a weak stochastic force acting on physical systems, which triggers the collapse of the wave function. Mechanical oscillators are a natural way to test such a force; in particular levitated micro-mechanical oscillator has been recently proposed to be an ideal system. We report a proof-of-principle experiment with a micro-oscillator generated by a micro-sphere diamagnetically levitated in a magneto-gravitational trap under high vacuum. Due to the ultra-low mechanical dissipation, the oscillator provides a new upper bound on the CSL collapse rate, which gives an improvement of two orders of magnitude over the previous bounds in the same frequency range, and partially reaches the enhanced collapse rate suggested by Adler. Although being performed at room temperature, our experiment has already exhibits advantages over those operating at low temperatures previously reported. Our results experimentally show the potential of magneto-gravitational levitated mechanical oscillator as a promising method for testing collapse model. Further improvements in cryogenic experiments are discussed.

Tables

Table 1. Table 1: Upper bounds on the CSL collapse rate λ 𝜆 \lambda . δ ​ T 𝛿 𝑇 \delta T is defined as the temperature increase δ ​ T = T eff − T env 𝛿 𝑇 subscript 𝑇 eff subscript 𝑇 env \delta T=T_{\rm eff}-T_{\rm env} , with T eff subscript 𝑇 eff T_{\rm eff} and T env subscript 𝑇 env T_{\rm env} the effective temperature of the oscillator measured at high vacuum and the environment temperature, respectively. δ ​ S total 𝛿 subscript 𝑆 total \sqrt{\delta S_{\rm total}} is the measured additional force noise beyond the thermal force. σ δ ​ T subscript 𝜎 𝛿 𝑇 \sigma_{\delta T} and σ δ ​ S total subscript 𝜎 𝛿 subscript 𝑆 total \sigma_{\sqrt{\delta S_{\rm total}}} are the corresponding standard deviations at 95 % percent 95 95\% confidence level. Finally, the upper bounds on λ 𝜆 \lambda at 95 % percent 95 95\% confidence level are the calculated for r C = 10 − 7 subscript 𝑟 C superscript 10 7 r_{\text{\tiny C}}=10^{-7}\, m and r C = 10 − 6 subscript 𝑟 C superscript 10 6 r_{\text{\tiny C}}=10^{-6}\, m.

Excess temperature		Excess noise		Upper bound on the collapse rate
$\delta T$	${\sigma }_{\delta T}$	$\sqrt{\delta S_{\mathrm{total}}}$	${\sigma }_{\sqrt{\delta S_{\mathrm{total}}}}$	$\lambda$ ($r_{\text{C}}={10}^{-7}$m)	$\lambda$ ($r_{\text{C}}={10}^{-6}$m)
$6.5K$	40 K	$1.3\times {10}^{-20}N/\sqrt{\mathrm{Hz}}$	$3.3\times {10}^{-20}N/\sqrt{\mathrm{Hz}}$	${10}^{-6.4}{s}^{-1}$	${10}^{-7.4}{s}^{-1}$

Table 2. Table 2: Comparison of the damping rates. input γ 𝛾 \gamma is the input value of damping rate used in the simulation, fitted γ 𝛾 \gamma (nonlinear) is the result of the simulation with nonlinearity added into the equation of motion, fitted γ 𝛾 \gamma (linear) that from the result of simulation without nonlinearity. The first row corresponds to medium vacuum, while the second row to high vacuum.

	input $\gamma$	fitted $\gamma$ (nonlinear)	fitted $\gamma$ (linear)
$P_{\text{MV}}$	0.4 Hz	0.39 Hz	0.38 Hz
$P_{\text{HV}}$	0.0004 Hz	0.00037 Hz	0.00038 Hz

Table 3. Table 3: Measured effective temperature of the oscillator. T eff HV superscript subscript 𝑇 eff HV T_{\rm eff}^{\rm HV} is the effective temperature of the oscillator measured at high vacuum and T eff M ​ V superscript subscript 𝑇 eff 𝑀 𝑉 T_{\rm eff}^{MV} that at medium vacuum. T eff M ​ V = T env superscript subscript 𝑇 eff 𝑀 𝑉 subscript 𝑇 env T_{\rm eff}^{MV}=T_{\rm env} is used for the estimation of the environment temperature. δ ​ T 𝛿 𝑇 \delta T is defined as the temperature increase δ ​ T = T eff H ​ V − T env 𝛿 𝑇 superscript subscript 𝑇 eff 𝐻 𝑉 subscript 𝑇 env \delta T=T_{\rm eff}^{HV}-T_{\rm env} . σ T eff H ​ V subscript 𝜎 superscript subscript 𝑇 eff 𝐻 𝑉 \sigma_{T_{\rm eff}^{HV}} , σ T eff M ​ V subscript 𝜎 superscript subscript 𝑇 eff 𝑀 𝑉 \sigma_{T_{\rm eff}^{MV}} and σ δ ​ T subscript 𝜎 𝛿 𝑇 \sigma_{\delta T} are the corresponding upper bounds at 95 % percent 95 95\% confidence level.

High Vacuum		Medium Vacuum		Difference
$T_{\mathrm{eff}}^{\mathrm{HV}}$	${\sigma }_{T_{\mathrm{eff}}^{\mathrm{HV}}}$	$T_{\mathrm{eff}}^{\mathrm{MV}}$	${\sigma }_{T_{\mathrm{eff}}^{\mathrm{MV}}}$	$\delta T$	${\sigma }_{\delta T}$
297.9 K	16.2 K	291.4 K	4.1 K	6.5 K	40 K

Equations

$${\mathcal{L}}_{\mathrm{CSL}}[{\hat{\rho }}_s(t)]=-{\eta }_i[{\hat{x}}_i,[{\hat{x}}_i,{\hat{\rho }}_s(t)]],$$

$${\eta }_i=\frac{r_{\text{C}}^3\lambda }{{\pi }^{\frac{3}{2}}{m}_0^2}\int d^3\mathbf{k}|{\tilde{\mu }}_s(\mathbf{k}){|}^2{k}_i^2{e}^{-k^2{r}_{\text{C}}^2}$$

$$\eta =\frac{6r_{\text{C}}^4\lambda m^2}{m_0^2{R}^6}\left[e^{-R^2/r_{\text{C}}^2}-1+\frac{R^2}{2r_{\text{C}}^2}(e^{-R^2/r_{\text{C}}^2}+1)\right].$$

$$m\ddot{x}+m\gamma \dot{x}+kx=f_{\mathrm{th}}(t)+f_{\text{CSL}}(t),$$

$$m\ddot{\mathbf{x}}+m\mathbf{\Gamma }\dot{\mathbf{x}}+\mathbf{Kx}+\text{o}({\mathbf{x}}^3)={\mathbf{f}}_{\mathrm{th}}(t)+{\mathbf{f}}_{\mathrm{CSL}}(t)+{\mathbf{f}}_{\mathrm{add}}(t),$$

$$m\ddot{x}+m\gamma \dot{x}+m{\omega }_0^{2}x+m{\omega }_0^2\zeta (t)x=f_{\mathrm{total}}(t),$$

$$\begin{array}{rl}& dx=\frac{p}{m}dt,\\ & dp=(-m{\omega }_0^{2}x-\gamma p)dt+\\ & \sqrt{2m\gamma k_B{T}_{\mathrm{eff}}}dX+m{\omega }_0^2\varsigma xdY,\end{array}$$

$$\frac{\partial P(\epsilon ,t)}{\partial t}=$$

$$+\frac{{\partial }^2}{\partial {\epsilon }^2}[(\gamma k_B{T}_{\mathrm{eff}}\epsilon +\frac{1}{4}{\omega }_0^2{\varsigma }^2{\epsilon }^2)P(\epsilon ,t)].$$

$$P(x^2)=\frac{\gamma +\frac{1}{2}{\omega }_0^2{\varsigma }^2}{\gamma k_B{T}_{\mathrm{eff}}}{\left(1+\frac{m{\omega }_0^4{\varsigma }^2}{2\gamma k_B{T}_{\mathrm{eff}}}{x}^2\right)}^{-\frac{2(\gamma +{\omega }_0^2{\varsigma }^2)}{{\omega }_0^2{\varsigma }^2}}.$$

$$U(\mathbf{x})=-\frac{\chi }{2{\mu }_0}V{\left|\mathbf{B}(\mathbf{x})\right|}^2+mgz+\frac{{\alpha }^{\prime }}{4}IV\xi (\mathbf{x})+U_0.$$

$$\text{F}({\mathbf{x}}_0)=-\nabla U({\mathbf{x}}_0)=0,$$

$$\nabla \cdot \text{F}({\mathbf{x}}_0)<0,$$

$$U(\mathbf{x}+{\mathbf{x}}_0)\approx \sum _i\frac{1}{2}{k}_i{x}_i^2,$$

$$m\ddot{x}+m\gamma \dot{x}+kx+\alpha x^3=f_{\mathrm{total}}(t),$$

$$dx$$

$$dp$$

$$d\epsilon =\frac{\partial \epsilon }{\partial x}dx+\frac{\partial \epsilon }{\partial p}dp=-\frac{\gamma p^2}{m}dt.$$

$$x(t)=X(t)\left[\cos (\omega t)+\frac{\kappa X^2(t)}{12}\cos (3\omega t)+o({\kappa }^2{X}^4(t))\right].$$

$$\omega ={\omega }_0(1+\kappa X^2(t)).$$

$$E_K=\frac{1}{\tau }{\int }_0^{\tau }\frac{1}{2}m{\dot{x}}^2(t)dt,$$

$$V=\frac{1}{\tau }{\int }_0^{\tau }[\frac{1}{2}m{\omega }_0^2{x}^2(t)+\frac{1}{4}\alpha x^4(t)]dt,$$

$$\frac{d{X}^2(t)}{dt}=-\gamma X^2(t)\left[1-\kappa X^2(t)+O({\kappa }^2{X}^4(t))\right],$$

$$X^2(t)\left[\kappa -\left(\frac{\kappa X^2(0)-1}{X^2(0)}\right)e^{\gamma t}\right]=1,$$

$$X^2(t)=X^2(0)e^{-\gamma t}\left[1-\kappa X^2(0)(e^{-\gamma t}-1)+...\right].$$

$$G_{X^2}(t)=⟨X^2(t)X^2(0)⟩,$$

$$G_{X^2}(t)=⟨X^4(0)⟩e^{-\gamma t}\left[1-\kappa X^2(0)(e^{-\gamma t}-1)+...\right].$$

$$R_{X^2}(t)=\frac{⟨X^2(t)X^2(0)⟩}{⟨X^4(0)⟩},$$

$$m{\ddot{x}}_1+m\gamma {\dot{x}}_1+m{\omega }_1^2{x}_1+{\alpha }_1{x}_1^3+\beta x_2^2{x}_1=f_1(t)$$

$$m{\ddot{x}}_2+m\gamma {\dot{x}}_2+m{\omega }_2^2{x}_2+{\alpha }_2{x}_2^3+\beta x_1^2{x}_2=f_2(t).$$

$$\frac{{\sigma }_{T_{\mathrm{eff}}}(t_{\mathrm{mea}})}{T_{\mathrm{eff}}}=\sqrt{\frac{2}{\gamma t_{\mathrm{mea}}}},$$

Full text

I Introduction

The perceived absence of macroscopic quantum superposition attracts the physicists’ interests since the birth of quantum mechanics. Different interpretations and reformulations of quantum mechanics Bohm1 ; Bohm2 ; Durr ; manyworlds1 ; manyworlds2 ; wallace ; dariano have been proposed to comprehensively handle such an issue, however, most of them do not provide direct experimental testability.

A phenomenological and experimentally verifiable Pearle1989 ; GRW ; Ghirardi1990 ; Bassi2003 ; Bassi2013 ; Diosi2014 ; Nimmrichter2014 ; bahrami2014 approach is proposed by collapse models Leggett . They introduce nonlinear and stochastic terms in the Schrödinger equation, which induce a spontaneous collapse of the wave function. Such a collapse is the stronger the larger the system. The origin of the noise remains an open question, and often in the literature it has been linked to gravity Diosi1989 ; Penrose1996 ; Penrose . In this paper we focus on the Continuous Spontaneous Localization (CSL) model, one of the most studied in the literature.

CSL is characterized by two phenomenological parameters: the collapse rate $\lambda$ and its correlation length $r_{\text{\tiny C}}$. The latter can be also understood as the minimum superposition separation necessary to trigger a collapse. The theoretically suggested values for these parameters are $\lambda\simeq 10^{-16}\,$s*-1* and $r_{\text{\tiny C}}=10^{-7}\,$m by Ghirardi et al. GRW ; Ghirardi1990 ; while larger value are considered by Adler: $\lambda\simeq 10^{-8\pm 2}\,$s*-1* at $r_{\text{\tiny C}}=10^{-7}\,$m and $\lambda\simeq 10^{-6\pm 2}\,$s*-1* at $r_{\text{\tiny C}}=10^{-7}\,$m Adler2007 .

The CSL modifications to standard quantum mechanics cause primarily a loss of coherence, which has been studied using cold atoms Kovachy2015 , molecular interferometers Coldmolecular1999 ; Coldmolecular2011 ; Coldmolecular2012 ; Eibenberger2013 ; Interf1 ; Interf2 , and phonon motion in diamonds Diamondsexp ; Diamonds . These compose the first class of experiments, namely the interferometric ones. Collapse models also predict a stochastic force noise acting on the system, whether it is in a quantum or classical state. This opens the way to non-interferometric experiments Bassi2003 ; Bassi2013 ; Diosi2014 , which has already constrained significantly the parameter space of the CSL model. To this class belong experiments with ultra-cold atoms ultra-cold atom ; ultra-cold atom2 , bulk solid matter residual_heating ; Adler2018 ; Phonos2 , planetary temperature observation Planetary ; Planets ; Planets2 , spontaneous X-ray emission x-raycase ; X-ray ; X-ray2 , and optomechanical systems AURIGA ; LIGO1 ; LISA1 ; LISA2 ; gravitation results1 ; gravitation results2 ; Vinante2016 ; Vinante2017 ; Pontin2019 . Many more proposals were analyzed Collett2003 ; Rotational1 ; levitation3 ; MAQRO ; McMillen2017 ; rotschrinski ; multilayer ; mishra2018 .

Non-interferometric experiments test the collapse mechanism at different frequencies, ranging from mHz LISA1 ; LISA2 to $10^{19}$ Hz X-ray . Since the CSL noise is originally assumed to be white, the bound on the collapse parameters is independent from the frequency at which the collapse mechanism is probed. However, this is not the case for colored extensions of the model, where the noise is not white anymore and is typically characterized by a frequency cut-off Adler2007 ; colored1 ; colored2 ; colored3 .

Recently, several studies Interf1 ; ultra-cold atom2 ; Colored-opto ; Adler2018 were performed in this direction, indicating the urgency of testing the CSL noise at different frequencies to probe its spectrum. Opto-mechanics provides the optimal platform for this scope, since frequencies range from sub-mHz to kHz or even higher opto-rev . Among them, the gravitational wave detectors AURIGA, Advanced LIGO and LISA Pathfinder, due to their large test mass, succeeded in setting strong bounds on collapse parameters gravitation results1 ; gravitation results2 ; Rotational1 ; gravitation results3 at frequencies less than $1$ kHz AURIGA , tens of Hz LIGO1 , and sub-Hz LISA1 ; LISA2 respectively. Among them, LISA Pathfinder gives the most strongest upper bound on $\lambda$ LISA2 ; Rotational1 . Also micro-scale solid-state force sensors such as nano-cantilevers provided precise testing of the collapse noise Vinante2016 ; Vinante2017 at frequencies above kHz. In this case, the relative large damping rates are balanced by operating at millikelvin temperatures.

Levitated micro- or nano-mechanical oscillators are ideal for potentially testing collapse models due to their low damping rates. Although they recently attracted considerable theoretical interest levitation1 ; levitation2 ; levitation3 ; levitation4 ; gravitation results3 , an experimental demonstration of their ability for such a purpose has not been performed yet.

Here, we report a first proof-of-priciple test of CSL based on a magnetically levitated micro-mechanical oscillator at room temperature. The levitation is realized with a specially designed magneto-gravitational trap where a test particle of mass of 4.7 pg ($\sim 2.8\times 10^{12}$ amu) is stably levitated for some days in high vacuum. We observed a damping rate $\gamma/2\pi$ of the order of $30\,\mu$Hz at a resonant frequency of the order of 10 Hz. This underlines the noiseless character of magneto-gravitational traps, that can actually provide a sensitive instrument for collapse model testing. As we will discuss in the following paragraphs, for $r_{\text{\tiny C}}=10^{-7}\,$m, we estimate the upper bound $\lambda=10^{-6.4}\,$s*-1* on the collapse rate at $95\%$ confidence level, excluding part of the range of values of the CSL parameters suggested by Adler Adler2007 . This is a significant improvement with respect to the bound obtained from the gravitational-wave detector Advanced LIGO which operates at the same frequency range LIGO1 ; gravitation results1 and proves that magneto-gravitational levitation is a strong competitive platform for testing the limits of quantum mechanics.

II Theoretical model

According to the mass-proportional CSL model Bassi2003 , the collapse of the wave-function leads to a spontaneous diffusion process, which is described by the Lindblad term Nimmrichter2014 ; bahrami2014 ; Diosi2014 :

[TABLE]

where $\hat{\rho}_{s}(t)$ is the density operator describing the center of mass motion, $i=x,y,z$ labels the direction of motion, and

[TABLE]

is the CSL diffusion constant, which depends on the geometry of the object. Here, $m_{0}$ is the atomic mass unit, $k=|\mathbf{k}|$ with $\mathbf{k}=(k_{x},k_{y},k_{z})$, and $\tilde{\mu}_{s}(\mathbf{k})$ is the Fourier transform of the mass density $\mu_{s}(\mathbf{r})$, i.e. $\tilde{\mu}_{s}(\mathbf{k})=\int d\mathbf{r}^{3}\,e^{i\mathbf{k}\cdot\mathbf{r}}\mu_{s}(\mathbf{r})$. In this experiment, the system is a sphere of radius $R$ and mass $m$, for which we have $\tilde{\mu}_{s}(\mathbf{k})={3m[{\sin}(kR)-kR{\cos}(kR)]}/{(kR)^{3}}.$ By substituting $\tilde{\mu}_{s}(\mathbf{k})$, we obtain a single diffusion constant independently from the direction of motion:

[TABLE]

The CSL-induced center of mass diffusion can be effectively described by a stochastic force $f_{\text{\tiny CSL}}(t)$ with zero mean and correlation $\langle f_{\text{\tiny CSL}}(t)f_{\text{\tiny CSL}}(s)\rangle=S_{\text{\tiny CSL}}\,\delta(t-s)$, where $S_{\text{\tiny CSL}}=\hbar^{2}\eta$ is the CSL force spectral density.

We describe the dynamics of our mechanical system as a damped harmonic oscillator subject to environmental and, potentially, CSL noises. Dropping the label $i$, the motion in one direction of the center of mass of our system reads

[TABLE]

with $\gamma/2\pi$ the damping rate, and $k=m\omega_{0}^{2}$ the spring constant. The first term in the right hand side represents the thermal Brownian force noise, whose correlations read $\langle f_{\rm th}(t)f_{\rm th}(s)\rangle=S_{\text{th}}\delta(t-s)$, where $S_{\rm th}=2\gamma mk_{\rm B}T_{\rm env}$ the corresponding power spectral density, which is fully characterized by the environmental temperature $T_{\text{env}}$ rugar1995 . For a system in thermal equilibrium, the additional presence of the collapse force $f_{\text{\tiny CSL}}(t)$ leads to an increase of the temperature of the system Diosi2014 . The effective temperature is thus $T_{\rm eff}=T_{\rm env}+T_{\rm csl}$, where $T_{\rm csl}$, the CSL induced temperature contribution, satisfies the relation $2\gamma mk_{\rm B}T_{\rm csl}=\hbar^{2}\eta$. Here, one assumes that $f_{\rm th}(t)$ and $f_{\rm csl}(t)$ are independent. Fig. 1 (a) shows an intuitive picture of the thermal equilibrium dynamics of the magneto-gravitationally levitated micro-sphere used in this experiment.

It is clear that any other source of noise, such as for example that due to the measurement back-action, also contributes to the total noise Clerk2010 . In Appendix A we discuss different possible noise processes involved in the experiment; however we take a conservative approach and consider all non-thermal noises as potential CSL noise in setting the upper bound.

The total power spectral density is defined as $S_{\rm total}=2\gamma mk_{\rm B}T_{\rm eff}$, which is calculated from the measured $T_{\rm eff}$. By subtracting the thermal Brownian contribution $S_{\rm th}$, we obtain the power spectral density of all additional force noises $\delta S_{\rm total}=S_{\rm total}-S_{\rm th}$. Therefore, $\delta S_{\rm total}=2\gamma mk_{\rm B}\delta T$ provides the estimation of the upper bound of the CSL force as $\hbar^{2}\eta\leq\delta S_{\rm total}$, with $\delta T=T_{\rm eff}-T_{\rm env}$ denoting the rise of the effective temperature. Note that, apart from a barely tunable material density $\sim m/R^{3}$ [cf. Eq. (3)], the ability to test CSL is limited only by the accuracy in determining the thermal Brownian noise. Different methods for such noise signal sensing have been developed for similar purposes Clerk2010 ; rugar2004 ; Kippenberg2012 ; huangpu1 .

III EXPERIMENT DESCRIPTION AND RESULTS

The levitation of diamagnetic systems using magneto-gravitational forces has been already performed with either superconductor Geim1999 or permanent magnets Slezak2018 . The magneto-gravitational trap used in our experiment was generated by a set of micro-machined NdFeB magnets with octagonal bilayer geometry as shown in Fig. 1 (b). In Appendix B, we report details about the trap design. The oscillator is a micro-sphere of polyethylene glycol, whose magnetic susceptibility is $-9.1\times 10^{-6}$ and its density is $1.1\times 10^{3}~{}$kg/m3. The micro-sphere is generated using a home-built nebulizer. A 633 nm laser is focused on the droplet with a power less than 50$\,\mu$W, and the scattered light from the micro-sphere is collected with an objective. The position of the micro-sphere is tracked with a CCD camera, and its motion is recorded in time domain with a photon detector. To isolate the trap from external vibrations, the trap is mounted on a heavy copper frame, which is suspended in a vacuum chamber by means of springs. Because the environmental temperature fluctuations contribute to the measurement uncertainty of the effective temperature of the oscillator, a double layer vacuum chamber and a PID temperature controller are used to maintain the environmental temperature stable. In this way, we achieved fluctuations smaller than 100 mK with an environmental temperature set to 298 K over the whole duration of the experiment ($\sim 10^{5}-10^{6}$ s).

We observed that for electrically charged particles, the dissipation in the experiment is higher than with no charge. This can lead to strong instability of the particles’ motion, which makes them eventually escape from the trap. To avoid this, the charge on the micro-sphere is eliminated by using ultra-violet light. Subsequently, the charge state is checked via a micro-electrode made of a gold wire of 40 $\mu$m in diameter placed near the trapping centre. By applying a voltage $\sim 50$ V, micro-spheres with radius less than 2 $\mu$m can be easily pulled out of the trap if the net charge is nonzero (see Appendix C for further details). Even after having removed the charges, at room temperature micro-spheres with radius smaller than 500 nm are found to escape the trap due to the thermal fluctuation and the limited depth of the trap (see Appendix B for details). The remaining particles were left in the trap in high vacuum for several days. If the particle did not evaporate during this time, it eventually reached the equilibrium thermal distribution, which was observed to be stable within the measurement error.

For micro-spheres of radius close to 1 $\mu$m, direct optical image failed to provide a reliable estimation of the size of the system, and we determined it through the following two methods. In the first method, we made use of the relation between the micro-sphere oscillation damping rate due to the background gas collisions and the pressure, which reads $\gamma=(16/\pi)(P/\nu R\rho)$ (holding for high pressures)optical_levitaion1 , with $P$ and $\nu$ the pressure and the mean speed of the background gas respectively. In this part of the experiment, the pressure was set to $P\sim 10^{-3}$ mbar, so that the damping was fully dominated by the background gas; $\nu$ can be inferred from the environmental temperature, and by measuring $\gamma$ one obtains an estimate for $R$.

The second method simply relied on the equipartition theorem. We measured the oscillators displacement distribution, which follows a Gaussian distribution $P(x)\propto\exp\left(-x^{2}/2\sigma^{2}\right)$, thus determining its standard deviation $\sigma$. The latter is related to the size of the particle through the energy equipartition theorem: $4\pi\sigma^{2}\mu R^{3}\omega_{0}^{2}=3k_{\rm B}T_{\rm env}$, with $\omega_{0}$ the resonant frequency of the oscillator. The results from the two methods are compatible. The micro-sphere used in the experiments, whose results are described below, has radius $R=1.0\,\mu$m, corresponding to $4.7\,$pg and with corresponding potential energy which is thousands of times large than the thermal energy $k_{\text{B}}T$.

After successfully capturing the micro-sphere and eliminating its charge, we proceeded to the measurement of the effective temperature $T_{\rm eff}$ associated to the center of mass motion of the particle. As a first step, we set a medium vacuum ($P_{\rm MV}\sim 10^{-4}$ mbar), and measured the position distribution of the micro-sphere in the $x$-$y$ plane. A typical example of the measured data in a run of few minutes is plotted in Fig. 2. The distribution has an elliptical shape due to the asymmetry of the trap. The distribution can be fitted with a two-dimensional Gaussian distribution, whose long (axis 1) and short (axis 2) standard deviations are denoted by $\sigma_{1}$ and $\sigma_{2}$, respectively. The energy equipartition theorem, which implies $\sigma_{1}/\sigma_{2}=\omega_{1}/\omega_{2}$, is well satisfied within the measurement error, where $\omega_{1}=12.9$ Hz and $\omega_{2}=9.3\,$Hz are the corresponding resonance frequencies (see Appendix D for details on the displacement power spectral density). The effective temperature is then calculated as $T_{\rm eff}=m\omega_{1}^{2}\sigma_{1}^{2}/k_{\text{B}}$ (equally, $m\omega_{2}^{2}\sigma_{2}^{2}/k_{\text{B}}$). Since at a medium vacuum, the thermal Brownian noise from the background gas fully dominates the other noises, we assume that $S_{\rm total}=S_{\rm th}$ and we use this relation to calibrate the environmental temperature as $T_{\rm env}=T_{\rm eff}$. Without loss of generality, we make use of mode 1 ($\omega_{1}/2\pi\approx 12.9$ Hz) for the subsequent measurement.

Next, we determine the dissipation constant $\gamma$, which is another key factor for determining the thermal Bronwnian force noise strength. First, we note that, for $P<10^{-5}$ mbar, the measured power spectral density $S_{x}(\omega)$ shows a strong asymmetric character, deviating substantially from a Lorentzian shape, and is considerably broadened compared to that estimated from the background gas. Such a feature is known Dykman1992 and is due to the nonlinearity of the trap. Therefore, to estimate $\gamma$ we follow the prescription of rugar2001 and make use of the energy autocorrelation defined as $\langle X^{2}(t)X^{2}(0)\rangle$, with $X(t)$ the amplitude of the oscillation. This method is insensitive to the nonlinearity of the trap (see Appendix D for details). The measured autocorrelation curve is then fitted to the exponentially decaying function $\exp(-t/\tau)$, from which we obtain the damping rate $\gamma=1/\tau$. Fig. 3 (a) shows the measured energy autocorrelation for different values of the pressure. In particular, at the highest vacuum $P_{\rm HV}\approx 4\times 10^{-7}\rm mbar$, the measured decay time $\tau\approx 4700$ s corresponds to a damping rate $\gamma/2\pi\approx 34\,\rm\mu Hz$. We also find that the damping rate decreases linearly as the pressure decreases, which shows that the background gas remains the dominant dissipative channel in the experiment, as shown in Fig. 3 (b). Combining the measured effective damping rate $\gamma/2\pi$ and temperature $T_{\rm eff}$, we estimate a force sensitivity of the oscillator in high vacuum as $\sqrt{S_{\rm total}}=9.6\rm\times 10^{-20}~{}N/\sqrt{Hz}$. This value is comparable to that obtained from optical trapping optical_levitaion3 .

By comparing the power spectral densities at medium and high vacuum we find the upper bound on the collapse rate $\lambda$; the main results are summarized in Table 1. For medium vacuum, the background gas is coupled to the system, thus maintaining the temperature of the system at equilibrium with the environmental one. On the contrary, in high vacuum condition, the gas decouples, and thus any potential CSL contribution is not dissipated, thus imposing an effective temperature which is higher than $T_{\text{env}}$. To bound the CSL effect we proceed as follows. The power spectral density of non-thermal forces is estimated via $\delta S_{\rm total}=2m\gamma k_{\rm B}\delta T$, where $\gamma$ is measured at high vacuum, $\delta T=T_{\rm eff}-T_{\rm env}$, with $T_{\rm eff}$ being calculated from the standard deviation of the position distribution $\sigma$ at high vacuum and with $T_{\rm env}$ measured at medium vacuum. We obtain the upper bound $\sqrt{\delta S_{\rm total}}<3.3\times 10^{-20}\rm N/\sqrt{Hz}$ at the 95% confidence level (see Appendix E for details about measurements of $T_{\rm eff}$ and error estimation). Accordingly, the bound on $\lambda$ is calculated through Eq. (3).

Fig. 4(a) compares the excluded values of $\lambda$ at different frequencies for $r_{\text{\tiny C}}=10^{-7}\,$m. In particular, our experiment improves by more than two orders of magnitudes the bound posed by Advanced LIGO LIGO1 ; gravitation results1 at the same frequency.

The upper bound provided by this experiment also partially excludes the range of values of collapse rate suggested by Adler for $r_{\text{\tiny C}}=10^{-7}\,$m Adler2007 , and almost entirely excludes it for $r_{\text{\tiny C}}=10^{-6}\,$m. We also estimated the performances of this experiment by using parameters that are more favourable for CSL testing and which are potentially achievable with our experiment by working at cryogenic condition and for 10*-8* mbar pressure: $R=0.3\,\mu$m, $\gamma/2\pi=10^{-6}\,$Hz and $\delta T=10$ mK. A negative result would imply $\lambda\leq 10^{-11.9}\,\rm s^{-1}$ for $r_{\text{\tiny C}}=10^{-7}\,$m, which would fully rule out Adler’s suggestion. The comparison of our experimental upper bound and the hypothetical upper bound with the strongest bounds reported in the literature are shown in Fig. 4(b), together with the theoretical values for the collapse parameters.

IV SUMMARY AND DISCUSSION

Levitated oscillators have been recently proposed as suitable systems for the collapse model testing levitation1 ; levitation2 ; levitation3 ; levitation4 ; gravitation results3 . Here, we demonstrated that an experiment based on a magneto-gravitational levitated miro-oscillator can place important bounds on the collapse parameters although operating at room temperature. We obtained a new upper bound, which is a significant improvement over previous results in the same frequency range and it partially probes Adler’s theoretical proposal. The system reported here shows a great potential, which would be fully expressed at cryogenic temperatures, where an improvement of several orders of magnitude in bounding the collapse noise is expected.

The performance of the current experiment at room temperature is mainly limited by three factors, which eventually could be improved significantly at lower temperatures. First, the effective temperature measurement precision is worse than tens Kelvins but is expected to reach mK under cryogenic conditions. Differently from other kinds of levitated micro-oscillators, such as electrical electric_levitation , optical optical_levitaion1 ; optical_levitaion2 ; optical_levitaion3 and magnetical levitation Magnet205 ; Magnet206 ; Pino , our magneto-gravitational trap is fully passive with no energy inputs. Thus, it is naturally suitable for low temperature condition. (In principle laser generates an addition force noise. However, the laser intensity is weak at room temperature. Its impact at cryogenic temperature is still to be evaluated.) Second, the minimum radius of the micro-sphere in this experiment is currently limited by the thermal energy, thus, at low temperature, much smaller micro-sphere could be stably trapped and lead to higher precisions in detection. The third potential of improvement is dissipation, which is observed to be constrained by the pressure. Room-temperature experiments show that higher vacuum does not lead to significantly improvement in dissipation Slezak2018 , since eventually other dissipative channels will contribute at lower pressures. However, it is yet to be explored whether dissipation can dicrease at a much lower temperature environment. This work opens a new door for the precise study of collapse models and may provide promising avenues towards breakthrough discoveries in the future.

Acknowledgements.

We thank L. Diósi for helpful comments. We thank Xuan He, Zhangqi Yin, Zhujing Xu, Jingwei Zhou, Fei Xue and Xing Rong for helpful discussion. This work was supported by the National Key R&D Program of China (Grant No. 2018YFA0306600), the National Natural Science Foundation of China (Grant No. 61635012, No. 11675163, No. 11890702, No. 81788101, No. 11761131011, and No. 11722544), the CAS (Grant No. QYZDY-SSW-SLH004, Grant No. GJJSTD20170001), the Fundamental Research Funds for the Central Universities (Grant No. 021314380149), and the Anhui Initiative in Quantum Information Technologies (Grant No. AHY050000). MC and AB acknowledge financial support from the H2020 FET Project TEQ (Grant No. 766900). AB acknowledges financial support from the COST Action QTSpace (CA15220), INFN, and the University of Trieste.

Note Added: After this work was completed, we became aware of similar independent work by Pontin et al. Pontin2019 .

Appendix A Calculation of fluctuation dynamics

The system was modeled by a classical mechanical oscillator with the motion described by the Langevin equations, which, in vectorial form, read:

[TABLE]

where $\mathbf{x}=(x,y,z)$, $m$ is mass of the oscillator, $\mathbf{\Gamma}$ is the damping rate diagonal matrix with elements $\gamma_{ii}$ ($i$ corresponding to x, y and z). When the background gas damping dominates, $\gamma_{ii}$ are isotropic: $\gamma_{ii}=\gamma$. Similarly, $\mathbf{K}$ is the diagonal matrix of the effective spring constants with element $k_{i}=m\omega_{i}^{2}$ and $\omega_{i}$ is the resonance frequency of the oscillator along the $i$-th axis. $\textit{o}(\mathbf{x}^{3})$ includes the higher order terms beyond the linear oscillator, such as Duffing nonlinearity $\alpha_{i}x_{i}^{3}$ and nonlinear couplings between different motions as $\beta_{i,j}x_{i}x_{j}^{2}$ etcNonlinear2009 . The right side of the equation is a sum of force noises. They include the thermal fluctuations ${f}_{\rm th}(t)$, possibly the CSL induced stochastic force ${f}_{\rm CSL}(t)$, and all the additional contributions, e.g. those due to the optical measurements, mechanical vibrations ecc.

Considering the motion in a single direction and dropping the direction label $i$, we estimate the three contributions to the noise in the system. The first one, the thermal force noise, was estimated by using the fluctuation dissipation theorem which gives the relation $\langle f_{\rm th}(t)f_{\rm th}(0)\rangle=2m\gamma k_{B}T_{\rm env}\delta(t)$, where $T_{\rm env}$ is environmental temperature. Equivalently, its strength can be described by the power spectral density $S_{th}(\omega)=2m\gamma k_{B}T_{\rm env}$. The second contribution, $f_{\text{\tiny CSL}}$ has been described in the main text. Within the third contribution, $f_{\text{\tiny add}}$, the optical force noise is the dominant one. It can be written as $f_{\rm opt}(t)=f_{\rm int}(t)+f_{\rm sc}(t)$. The first term $f_{\rm int}(t)$ is the classical optical force due to intensity fluctuations, including both those from the intensity fluctuation $\delta I(t)$ and the position fluctuation of the light position $\textbf{x}_{\rm opt}(t)$ relative to the center of the magneto-gravitational trap. The illumination light intensity fluctuation induced force can be expressed as $-\alpha^{\prime}\nabla\xi[\textbf{x}_{0}]\delta I(t)/4$ and the light spot position fluctuation induced force as $-\alpha^{\prime}I\nabla(\nabla\xi(\textbf{x}_{0})\cdot\delta\textbf{x}_{\rm opt}(t))$, with $\textbf{x}_{0}$ the trapping position, $\xi(\textbf{x}_{0})$ the normalized light field distribution function, and $I$ the average intensity of the illumination light. The second term $f_{\rm sc}(t)$ is the stochastic force due to photon scattering. An additional contribution to $f_{\text{\tiny add}}$ is the parametric noise that is generated from the illumination light intensity, which leads to a fluctuation of the spring constant $k$ via optical force, and is proportional to $\delta I(t)$ parametric cool .

We solved the Fokker-Planck equation for the probability density to obtain the statistical behavior of the system. To this end, the Langevin equations of motion in a single direction are written as:

[TABLE]

where the parametric fluctuation $\zeta(t)$ was approximately taken as a white noise satisfying $\langle\zeta(t)\zeta(0)\rangle=\varsigma\delta(t)$. $f_{\rm total}(t)=f_{\rm th}(t)+f_{\text{\tiny CSL}}(t)+f_{\rm opt}(t)$ is the total force noise and it was also assumed to be white: $\langle f_{\rm total}(t)f_{\rm total}(0)\rangle=2m\gamma k_{B}T_{\rm eff}\delta(t)$, with $T_{\rm eff}$ the effective temperature. It is noted that $f_{\rm total}(t)$ and $\zeta(t)$ are not strictly independent, because both contain the contribution from the illumination light intensity fluctuation $\delta I(t)$. However, such a contribution in $f_{\rm total}(t)$ is small, and so we took the total force noise and the parametric noise as approximately independent. Given this, we write the Langevin equations of motion as follows Kampen1976 :

[TABLE]

with $\operatorname{d}\!X$ and $\operatorname{d}\!Y$ two independent random variables with Gaussian distribution. Setting the energy of the oscillator as $\varepsilon=p^{2}/2m+kx^{2}/2$, with high quality factor $Q=\omega_{0}/\gamma$, the Langevin equations of motion leads to the Fokker-Planck equation for the probability density $P(\varepsilon,t)$ reads

[TABLE]

For a stationary probability distribution $\partial P(\varepsilon,t)/\partial t=0$, and Eq. (8) is:

[TABLE]

This distribution was measured experimentally. It is noted that, for the limiting case $\varsigma\rightarrow 0$, the expression in Eq. (9) approaches the Gaussian distribution.

Appendix B Design of the magneto-gravitational trap

The potential energy density of a small diamagnetic micro-sphere in a magneto-gravitational trap under an illumination light field can be written as Berry1997 :

[TABLE]

Here, the first term is the diamagnetic potential, with $\chi$ and $v$ the magnetic susceptibility and volume of the micro-sphere; the second term is the gravitational potential, with $m$ the mass of the micro-sphere and $z$ is taken opposite to the direction of gravity; the third term is the optical gradient force, with $\alpha^{\prime}$ the real component of the polarisability, $I$ the light field intensity proportional to the light power, and $\xi(x)$ the normalized light field distribution function. The conditions that a diamagnetic micro-sphere can be stably trapped in the equilibrium position $\mathbf{x_{0}}$ are:

[TABLE]

with $\textbf{F}(\mathbf{x})$ the total force of the potential. Near the equilibrium position $\mathbf{x_{0}}$, the potential can be approximately expressed in quadratic form with respect to the displacement $\mathbf{x}$ from $\mathbf{x_{0}}$ as $U(\mathbf{x}+\mathbf{x_{0}})\approx\Sigma_{i,j}\frac{\partial^{2}U(\mathbf{x_{0}})}{\partial x_{i}\partial x_{j}}x_{i}x_{j}$ ($i,j=x,y,z$), which can be put into a diagonalised form as the sum of three independent harmonic oscillators,

[TABLE]

where $k_{i}$ with $i=x,y,z$, are the effective spring constants, leading to the characteristic frequencies of the oscillators $\omega_{i}=\sqrt{k_{i}/m}$. The constant term $U_{0}$ is dropped for convenience. The optical field will also generate an effective potential via the optical force, however such an effect is much smaller than $U(\mathbf{x})$ and can be neglected. Hence, in the trap design, only the magnetic and the gravitational energies were taken into account. The potential function was calculated using a finite element simulation and the result is plotted in Fig. 5.

Appendix C Experimental setup and micro-sphere generation

The experimental apparatus is shown in Fig. 6(a): the magneto-gravitational trap is held in a vacuum system by specially designed springs, with temperature of the inner chamber monitored and controlled to be slightly above the room temperature, and the pressure controlled by a turbomolecular pump of tunable rotation speed. A CCD camera was used to detect the position of the micro-sphere, the magnification $\mathcal{M}$ of the detection optics being calibrated by a standard micro-structure so that the displacement in the $x$-$y$ plane of the micro-sphere is $\mathbf{x}=\mathbf{x^{\prime}}/\mathcal{M}$, where $\mathbf{x^{\prime}}$ is the displacement of the micro-sphere image read out by the CCD camera. In this way, the thermal distribution was obtained.

A photodetector was used to detect the position-dependent scattering light intensity $I_{\rm sc}$, which is proportional to the illumination light $I$ as $I_{\rm sc}\propto I\xi(\textbf{x}$+x0). Since the thermal motion is much bigger than the wavelength, such detection scheme is efficient. The power spectral density in the position $S_{x}(\omega)$ is then calculated from the output photon detector voltage: $S_{x}(\omega)\propto S_{V}(\omega)$, with $S_{V}(\omega)$ the power spectral density of the output voltage. For high quality factor oscillators, the detection nonlinearity does not influence the results.

In order to eliminate the influence of the external vibration, the whole experimental setup is first mounted on an optical table with air legs, and a two-stage spring-mass based suspension is used to further isolate the vibrations, as shown in Fig. 6(b). The resonance frequency of the first stage (the second stage) in $x$-$y$ plane is about 1.5 Hz (4 Hz), and the mass of the first stage is designed to be much heavier than the mass of the second one. We used a very thin wire with diameter about 40 $\mu$m to apply the electric field which was used to pull the micro-sphere, and the wire was mechanically bounded on the first and then second stage before going to the trap, so that vibrations transmitted through the wire to the trap were effectively suppressed.

The micro-sphere used in our experiment is a small polyethylene glycol 400 droplet. To generate such a droplet with desirable diameter, we first mixed polyethylene glycol 400 with dibutyl sebacate (DBS) and ethanol in the proportion of $1:27:1000$ (volume ratio). Subsequently, droplets of the suspension were sprayed into the trap using a home built piezo-atomizer at atmospheric pressure. Ethanol rapidly evaporated after some seconds and a droplet with typical diameter of 3 to 7 $\mu\rm m$ was obtained. Next, a moderated voltage of about a few tens of Volts was applied while the displacement of the droplet was monitored, and an ionizing radiation source (Americium-241) was brought near the droplet. After exposing the droplet to the radiation for a few seconds, the charge on it changed randomly. Once a positively charged droplet was obtained, the pressure was gradually decreased to $10^{\rm-6}\,\rm mbar$ for one day, and then DBS fully evaporated and the diameter of the micro-sphere did not change anymore. Next, a ultra-violet light was used to slowly eliminate the positive charge until the droplet became fully neutralized. This was determined as follows: for a micro-sphere with only a few electron charges, jumps in voltage-displacement response became clear, and eventually the responses dropped to zero when the net charge went to zero by applying a voltage larger than $50$ V. We also observed that the charge state was stable in vacuum ($P<10^{\rm-4}\,\rm mbar$) for a very long time (tens of days or even longer).

Appendix D Influence of nonlinearity on measurement results

The nonlinear term in Eq. (5) becomes important for a motion with large amplitude. For simplicity, we consider the term $\alpha x^{3}$ but temporarily we omit the coupling terms $\beta_{i,j}x_{i}^{2}x_{j}$; the oscillator then becomes a Duffing oscillator with the following equation of motion Nonlinear2009 :

[TABLE]

where $\alpha$ is the Duffing constant. One of the important effects of nonlinearity is a frequency shift and broadening that are proportional to the thermal fluctuation $\alpha k_{B}T_{\rm eff}$ nonlinear1980 . When such a nonlinear thermal broadening becomes larger than the damping rate $\gamma$, the power spectral density shows a non-Lorentz character Dykman1992 . Hence, in the thermal nonlinear regime, the damping rate $\gamma/2\pi$ cannot be obtained by measuring the full width at half maximum based on the power spectral density, which is commonly used with an harmonic oscillator. Instead, we notice that the change of energy over time is still the same as that of a harmonic oscillator. This is because the reduction of energy in the damping process results from the dissipation via the kinetic energy $p^{2}/2m$, while the nonlinearity only modifies the potential energy and preserves energy conservation rugar2001 . Therefore, we extract $\gamma$ from the energy autocorrelation as described below.

From Eq. (14), we first write equations of motion for position and momentum, without the fluctuation $f_{\rm total}(t)$, i.e.

[TABLE]

Then the change of the total energy of the oscillator follows

[TABLE]

Next, we consider a short period during which the dissipation is negligible so the motion of the system can be written as

[TABLE]

Here, $X(t)$ is a vibrational amplitude that is slowly varying, $\kappa=3\alpha/8m\omega_{0}^{2}$ and $\omega$ is an amplitude dependent oscillation frequency which shifts from the resonance frequency $\omega_{0}$ as:

[TABLE]

As $X(t)$ goes to zero, we have $x(t)\approx X(t)\rm{cos}(\omega t)$, as expected.

Then, we define the average kinetic energy $E_{K}$ and average potential energy $V$ as

[TABLE]

[TABLE]

with $\tau$ much shorter than $1/\gamma$ but much longer than $1/\omega$, which can be satisfied for a system with a large quality factor $Q=\omega/\gamma$. By averaging Eq. (16) as $\operatorname{d}\!\varepsilon/\operatorname{d}\!t=-2\gamma E_{K}$, we obtain the differential equation for $X^{2}(t)$:

[TABLE]

Dropping terms of order $\kappa^{2}X^{4}(t)$ or higher, we obtain the solution of Eq. (21)

[TABLE]

Asymptotically, $X(t)$ decays and Eq. (22) can be expanded as follows:

[TABLE]

Next, we define the autocorrelation function of $X^{2}(t)$ as:

[TABLE]

which according to Eq. (D) becomes

[TABLE]

In the experiment, $X^{2}(t)$ is directly measured from the power spectral density $S_{x}(\omega)$ by following standard procedures Poggio2007 ; Huang2016 , as $X^{2}(t)=S_{x}(\omega)b$, where $b$ is the sampling bandwidth satisfying $\gamma\ll b\ll\omega_{0}$. We also define the following normalized autocorrelation:

[TABLE]

which is used to estimate the damping rate $\gamma/2\pi$.

In our system, nonlinearities come not only from the term $\alpha x^{3}$, but also from the coupling of the motion along different axes, as $\beta_{i,j}x_{i}x_{j}^{2}$. We calculated numerically the effects based on two-mode coupling from the equations of motion:

[TABLE]

Here modes 1 and 2 correspond to the motions in the $x$-$y$ plane, while the motion along $z$ axis is neglected, $f_{1}(t)$ and $f_{2}(t)$ are independent white noise with power spectral density $S_{1,2}$ equal to that of the thermal Brownian noise measured experimentally. The values $m$, $\gamma$ and $\omega_{1,2}$ are directly obtained from the experiment. The nonlinearity coefficients $\alpha_{1,2}$ and $\beta$ are tuned so that the full width at half maximum and the shape of the power spectral density $S_{x}(\omega)$ obtained from the numerical simulation and from the experiments agree with each other, as shown in Fig. 7. The corresponding values are $\alpha_{1}=-6.4\,\rm kg/m^{2}s^{2}$, $\alpha_{2}=-2.1\,\rm kg/m^{2}s^{2}$ and $\beta=6.4\,\rm kg/m^{2}s^{2}$. $R_{X^{2}}(t)$ is numerically calculated for medium and high vacuum and the results are shown in Fig. 8. The data are fitted to the exponential decay $\exp(-t/\tau)$, producing the damping rates $\gamma=1/\tau$, which agree well with the values used in numerical simulations (see Table 2).

Appendix E Error estimation

In order to estimate the error on the effective temperature $T_{\rm eff}$ from the measured position distribution, the displacement distributions of the oscillation mode 1 (12.9 Hz) under high vacuum (HV) and medium vacuum (MV) were recorded [Fig. 9 (a) and (b)]. The results are fitted to a Gaussian distribution to give the standard deviation $\sigma^{\rm HV}$($\sigma^{\rm MV}$), from which the effective temperatures $T_{\rm eff}^{\rm HV}$($T_{\rm eff}^{\rm MV}$) are obtained. For a given measurement time $t_{\rm mea}$, the standard deviation $\sigma_{T_{\rm eff}}$ of the measured effective temperature can be derived by following the procedure in Refs. Ranjit2016 and Hebestreit2018 . The results for medium vacuum (MV) and high vacuum (HV) are plotted in Fig. 9 (c) and (d), respectively, as functions of $t_{\rm mea}$. Theoretically, the relative standard deviation of the effective temperature as a function of the measurement time $t_{\rm mea}$ satisfies the relation Hebestreit2018

[TABLE]

and is plotted in Fig. 9 (c) and (d) as straight lines. The measured data agree very well with the theory. Finally, the uncertainty $\sigma_{T_{\rm eff}}$ of the effective temperature is estimated using Eq. (28) by taking $t_{\rm mea}=t_{\rm total}$, the total measurement time. In particular, the total data acquisition time at high vacuum is $9.5\times 10^{5}$ s (about 11 days), which can be further extended to reduce the uncertainty, but this was not done for practical reasons. The effective temperature measured in a medium vacuum is taken as the environmental temperature $T_{\rm eff}^{\rm MV}=T_{\rm env}$ and the temperature difference is $\delta T=T_{\rm eff}^{\rm HV}-T_{\rm eff}^{\rm MV}$. To estimate the upper bound on $\delta T$ with the standard methods Feldman1998 , $T_{\rm eff}^{\rm HV}$ and $T_{\rm eff}^{\rm MV}$ are treated as independent and both following Gaussian distributions, with their corresponding standard deviations $\sigma_{T_{\rm eff}}$ obtained from the measured data [Fig. 9 (c) and (d)]. The threshold $\sigma_{\delta T}$ defined by the $95\%$ confidence level ($\delta T<\sigma_{\delta T}$) is given in Table 3. We note that the measured effective temperature does not coincide with the temperature (298 K) measured by the thermometer in the vacuum chamber. While such a bias is due to the uncertainty in measuring the absolute displacement of the oscillator, there is an uncertainty of less than a few percent in determining the magnification $\mathcal{M}$ of the detection optics, so is in the micro-sphere’s absolute displacement is given by $\mathbf{x}=\mathbf{x^{\prime}}/\mathcal{M}$. This uncertainty is constant during the whole experimental process and only brings about a small error (a few percent) on the final result. Since the power spectral density of additional force noises is defined as $\delta S_{\rm total}=2m\gamma k_{\rm B}\delta T$, we estimate its upper bound as $\delta S_{\rm total}<2m\gamma k_{\rm B}\sigma_{\delta T}$. Finally, we obtain the upper bounds on the CSL collapse rate $\lambda$ from Eq. (3) by using the upper bound on the CSL collapse strength $\eta$ given by $\hbar^{2}\eta<2m\gamma k_{\rm B}\sigma_{\delta T}$.