Opto-mechanical tests of collapse models

TL;DR

This paper reviews opto-mechanical methods for testing collapse models, emphasizing how system complexity affects the divergence from quantum mechanics and exploring non-interferometric approaches to validate collapse theories.

Contribution

It introduces a comprehensive review of non-interferometric opto-mechanical tests for collapse models, highlighting their potential to probe the models' parameter space.

Findings

Non-interferometric opto-mechanical tests can effectively distinguish collapse models from quantum mechanics.

The divergence between collapse models and quantum predictions increases with system mass and size.

Opto-mechanical platforms offer scalable and sensitive means to test collapse theories.

Abstract

The gap between the predictions of collapse models and those of standard quantum mechanics widens with the complexity of the involved systems. Addressing the way such gap scales with the mass or size of the system being investigated paves the way to testing the validity of the collapse theory and identify the values of the parameters that characterize it. Here, we review the recently proposed non-interferometric approach to the testing of collapse models, focusing on the opto-mechanical platform.

Equations

$$⟨{\hat{\mathbf{x}}}^2(t)⟩={⟨{\hat{\mathbf{x}}}^2(t)⟩}_{\text{QM}}+\frac{\lambda {\hslash }^2{t}^3}{2m_0^2{r}_{\text{C}}^2},$$

$$\frac{d{\hat{x}}_t}{dt}=\frac{{\hat{p}}_t}{m},\text{and}\frac{d{\hat{p}}_t}{dt}=-m{\omega }_{\text{m}}^2{\hat{x}}_t+\hslash \chi {\hat{a}}_t^{†}{\hat{a}}_t-{\gamma }_{\text{m}}{\hat{p}}_t+{\hat{\xi }}_t+{\hat{F}}_t^{\text{CSL}},$$

$${\hat{V}}_{\text{CSL}}=-\hslash \sum _j\frac{m_j}{m_0}\int d\mathbf{x}{\hat{\Psi }}_j^{†}(\mathbf{x},t){\hat{\Psi }}_j(\mathbf{x},t)N(\mathbf{x},t),$$

$$\mathbb{E}[N(\mathbf{x},t)]=0,\text{and}\mathbb{E}[N(\mathbf{x},t)N(\mathbf{y},s)]=\lambda \delta (t-s)G(\mathbf{x}-\mathbf{y}),$$

$${\mathcal{S}}_{xx}(\omega )=\int \frac{d\Omega }{4\pi }\mathbb{E}\left[⟨\{\tilde{x}(\omega ),\tilde{x}(\Omega )\}⟩\right],$$

$${\mathcal{S}}_{xx}(\omega )$$

$$+\frac{\hslash m{\gamma }_{\text{m}}\omega \coth \left(\frac{\hslash \omega }{2k_{\text{B}}T}\right)+{\mathcal{S}}_{\text{CSL}}}{m^2|d(\omega ){|}^2},$$

$$|d(\omega ){|}^2=({\omega }_{\text{m,eff}}^2(\omega )-{\omega }^2{)}^2+{\gamma }_{\text{m,eff}}^2(\omega ){\omega }^2.$$

$${\mathcal{S}}_{\text{CSL}}=\frac{{\hslash }^2\lambda r_{\text{C}}^3}{{\pi }^{3/2}{m}_0^2}\int d\mathbf{k}{\left|\tilde{\mu }\left(\mathbf{k}\right)\right|}^2{e}^{-{\mathbf{k}}^2{r}_{\text{C}}^2}{k}_x^2,$$

$$\hslash m{\gamma }_{\text{m}}\omega \coth \left(\frac{\hslash \omega }{2k_{\text{B}}T}\right)+{\mathcal{S}}_{\text{CSL}}\to 2m{\gamma }_{\text{m}}{k}_{\text{B}}(T+\Delta T_{\text{CSL}})$$

$$\Delta T_{\text{CSL}}=\frac{{\mathcal{S}}_{\text{CSL}}}{2m{\gamma }_{\text{m}}{k}_{\text{B}}}.$$

$${\mathcal{S}}_{\text{CSL}}=\frac{{\hslash }^2\lambda r_{\text{C}}^3}{2{\pi }^{3/2}{m}_0^2}\int d\mathbf{k}{\left|\tilde{\mu }\left(\mathbf{k}\right)\right|}^2{e}^{-{\mathbf{k}}^2{r}_{\text{C}}^2}{k}_x^2(1-e^{ia{k}_x}),$$

$${\mathcal{S}}_{\text{CSL}}^{\text{rot}}=\frac{{\hslash }^2\lambda r_{\text{C}}^3}{{\pi }^{3/2}{m}_0^2}\int d\mathbf{k}{\left|k_y{\partial }_{k_z}\tilde{\mu }(\mathbf{k})-k_z{\partial }_{k_y}\tilde{\mu }(\mathbf{k})\right|}^2{e}^{-r_{\text{C}}^2{\mathbf{k}}^2}.$$

Full text

Opto-mechanical tests of collapse models

Matteo Carlesso

matteo.carlesso@ts.infn.it

Department of Physics, University of Trieste, Strada Costiera 11, 34151 Trieste, Italy

Istituto Nazionale di Fisica Nucleare, Trieste Section, Via Valerio 2, 34127 Trieste, Italy

  

Mauro Paternostro

m.paternostro@qub.ac.uk

Centre for Theoretical Atomic, Molecular and Optical Physics, School of Mathematics and Physics, Queen’s University, Belfast BT7 1NN, United Kingdom

The gap between the predictions of collapse models and those of standard quantum mechanics widens with the complexity of the involved systems. Addressing the way such gap scales with the mass or size of the system being investigated paves the way to testing the validity of the collapse theory and identifying the values of the parameters that characterize it.

Despite increasing sensitivities which are taking experiments closer to regimes where the potential differences between collapse-based formulations and standard quantum theory should become apparent, the task of finding the precise value of the parameters of a given collapse model is nevertheless difficult. In fact, environmental decoherence – which at the statistical level has the same signature as collapse models – could mask any collapse-induced effects, thus biasing the interpretation of related experimental observations.

The current efforts aimed at the test of collapse models can be notionally split into two broad classes: interferometric and non-interferometric tests. The former, which aim at directly probing the validity of the quantum superposition principle, provide a natural test for any collapse model. They rely on the creation of a spatial superposition and, after a suitable time of free evolution – necessary for the propagation of the collapse effects – on the subsequent measurement of its interference contrast. The comparison of such contrast, which is weakened by the environmental and collapse noises, with the predictions of quantum mechanics provides experimental upper bounds to the collapse parameters. The most successful experiments in this context have been performed using matter-wave interferometry and are extensively discussed elsewhere Toroš et al. (2017); Toroš and Bassi (2018). Here we focus on the second class of experimental assessments, namely the non-interferometric ones, with the declared goal of illustrating their potential for the successful falsification of collapse models in close-to-state-of-the-art platforms.

The remainder of this Chapter is organised as follows: In Sec. I we review the recently proposed non-interferometric approach to the testing of collapse models. Sec. II specialises our assessment to the opto-mechanical platform. In particular, we focus on the description of two recent thought experiments, which have paved the way to the design of experimental routes to the falsification of collapse mechanisms. In Sec. III we assess quantitative bounds provided by a set of experiments that broadly fall into the category of non-interferometric settings. Finally, Secs. IV and V address the open questions linked to plausible extensions of standard and nearly canonical formulations of collapse theories and the use of rotational degrees of freedom of mechanical rotors as ultra-sensitive tools for the inference of the minuscule effects of collapse models.

I Non-interferometric experiments: a new perspective in collapse model testing

Differently from interferometric tests, where a superposition needs to be created, sustained and finally measured, non-interferometric assessments tests do not rely on the availability of high-quality non-classical resource states. A plethora of different experiments fall in this class, from those involving the x-ray radiation spontaneously emitted from Germanium Adler et al. (2013); Bassi and Donadi (2014); Donadi et al. (2014) to those focussing on the change of the internal energy of matter-like systems Adler and Vinante (2018); Bahrami (2018); Adler et al. (2019), from the monitoring of the free expansion of cold atoms Bilardello et al. (2016) to experiments based on the dynamics of opto-mechanical systems, which are currently considered to be one of the most promising platforms for the delicate discrimination between collapse-based models and standard quantum mechanics.

Here, we review the proposals put forward in Ref. Bahrami et al. (2014); Nimmrichter et al. (2014); Diósi (2015), which have planted the seeds for the opto-mechanical exploration of collapse models via non-interferometric approaches. For concreteness, we will focus on the Continuous Spontaneous Localization (CSL) model Ghirardi et al. (1990); Pearle (1989); Bassi and Ghirardi (2003), which is characterized by parameters $\lambda$ and $r_{\text{\tiny C}}$: the first is the collapse rate, while the second is the correlation distance.

To illustrate the effects induced by the CSL model, we consider a confined system of mass $m$ whose dimensions are, for the sake of simplicity, point-like. The system is initially in thermal equilibrium at temperature $T$, which we shall assume to be small so as to make thermal fluctuations irrelevant. The confining mechanism is then switched off and the system is allowed to freely evolve for a time $t$, when measurement of the position of the system is performed. During the free evolution, the effect of the CSL mechanism can be read out in the spread of the position, which reads

[TABLE]

where $m_{0}$ is the mass of a nucleon, $\braket{\hat{\bf x}^{2}(t)}_{\text{\tiny QM}}$ gives the contribution due to quantum mechanics, and the last term is due to the CSL effect. There is a qualitative difference between the evolution of the spread due to quantum mechanics (which is $\sim t^{2}$) and the contribution arising from the collapse mechanism ($\sim t^{3}$). The diffusion induced by the environment has a behaviour similar to the one due the collapse mechanism Romero-Isart (2011). On this basis, a way to extrapolate the parameters of CSL would pass through the observation of the diffusive Brownian process and the consequent establishment of bounds on the collapse parameters. This idea was put forward in Ref. Goldwater et al. (2016), which considered a levitated charged nanosphere in a Paul trap supported by an optical cavity [the latter being needed for passive cooling of the system, cf. Fig. 1 (a)]. Clearly, the standard decoherence sources, such as thermal photon emission, absorption and scattering as well as the collision with the residual gas particles, would also contribute to the diffusive motion of the system. The analysis performed in Goldwater et al. (2016); McMillen et al. (2017) is, in this context, particularly useful as reporting a comparison between possible diffusive contributions from collapse models and analogous terms resulting from standard decoherence mechanisms. By following ideas akin to those pursued in Ref. Goldwater et al. (2016), quantitative bounds on the CSL parameters were derived from a cold atom experiment Bilardello et al. (2016), where the free expansion of the gas cloud was characterized and compared with the collapse-induced diffusion. The corresponding upper bounds are reported in Fig. 2.

II Opto-mechanical system as a probe of the collapse mechanism

Let us now turn to the role played by opto-mechanical experiments in the assessment of collapse models. They focus on an indirect effect provided by the collapse mechanism, which is an extra Brownian-like motion of the center of mass of the mechanical component of an opto-mechanical system. Such motion leads to an extra diffusion mechanism that can be detected through standard experimental techniques and, under suitable conditions, provide information on the undergoing collapse mechanism. To give a concrete example, we assume a single-sided Fabry-Perot cavity endowed with an end-cavity mechanical oscillator and driven by an external laser, which also provides the mechanism for the measurement of the mechanical motion [cf. Fig. 1 (b)]. The latter is influenced by a phononic environment (at non-zero temperature) and, allegedly, the CSL-like collapse noise. The action of the latter can be added to the Langevin equations governing the opto-mechanical motion, which read Bahrami et al. (2014)

[TABLE]

where $\omega_{\text{\tiny m}}$ and $\gamma_{\text{\tiny m}}$ are the harmonic frequency of the mirror and its damping constant, $\chi$ denotes the coupling of the mechanical oscillator with the cavity field, whose creation and annihilation operators are $\hat{a}^{\dagger}$ and $\hat{a}$ respectively. Here, $\hat{\xi}_{t}$ and $\hat{F}_{t}^{\text{\tiny CSL}}$ denote the stochastic forces due to the environment and the collapse mechanism, respectively. Indeed, the collapse action can be mimicked by adding to the Schrödinger equation a stochastic potential $\hat{V}_{\text{\tiny CSL}}$, whose corresponding force is given by $\hat{F}_{\text{\tiny CSL}}(t)=\tfrac{i}{\hbar}[\hat{V}_{\text{\tiny CSL}}(t),\hat{p}]$. In the case of CSL we have Adler et al. (2013)

[TABLE]

where $\hat{\Psi}_{j}^{\dagger}({\bf x},t)$ and $\hat{\Psi}_{j}({\bf x},t)$ are respectively the creation and annihilation operators of a $j$-type particle of mass $m_{j}$, and $N({\bf x},t)$ is a stochastic noise inducing the collapse, whose mean and correlator are

[TABLE]

with $\mathbb{E}$ the stochastic average over the noise and $G({\bf x})={e^{-{\bf x}^{2}/4r_{\text{\tiny C}}^{2}}}$. Eq. (4) gives a clear interpretation of $\lambda$ and $r_{\text{\tiny C}}$ as the collapse rate and the noise correlation distance respectively.

The signatures of the collapses of the mechanical motion can be tracked through the density noise spectrum (DNS), whose definition reads

[TABLE]

where $\tilde{x}(\omega)$ is the Fourier transform of the fluctuations of $\hat{x}_{t}$. Following the derivation in Ref. Paternostro et al. (2006), one finds

[TABLE]

where $|\alpha|^{2}$ denotes the intensity of the intra-cavity laser, $\Delta$ is the laser-cavity detuning, $T$ is the environmental temperature, and $\kappa$ is the cavity dissipation rate. Moreover we have introduced the susceptibility function $1/|d(\omega)|^{2}$ with

[TABLE]

Here, $\omega_{\text{\tiny m,eff}}(\omega)$ and $\gamma_{\text{\tiny m,eff}}(\omega)$ denote the effective mechanical frequency and damping rate, respectively. Finally, $\mathcal{S}_{\text{\tiny CSL}}$ quantifies the action of CSL noise, which can be obtained from $\mathbb{E}[\braket{\hat{F}_{\text{\tiny CSL}}(t)\hat{F}_{\text{\tiny CSL}}(t^{\prime})}]=\mathcal{S}_{\text{\tiny CSL}}\delta(t-t^{\prime})$ with Carlesso et al. (2018a)

[TABLE]

where $\tilde{\mu}({\bf k})$ is the Fourier transform of the mass density. Here, due to the presence of the mass density, two aspects can be considered. First, $\mathcal{S}_{\text{\tiny CSL}}$ is proportional to the square of the mass $m$ of the system. Thus, heavier masses can provide a stronger signature of the collapse mechanism. Second, Eq. (8) strongly depends on the geometry of the system and in particular on the ratio between its size $L$ and $r_{\text{\tiny C}}$. Indeed, in the limit of $r_{\text{\tiny C}}\ll L$ the collapse noise will act incoherently on parts of the system which are more distant than $r_{\text{\tiny C}}$, while for $r_{\text{\tiny C}}\sim L$ such action will be coherent. Finally, for $r_{\text{\tiny C}}\gg L$, the collapse action will be still coherent but unfocused on the system, thus effectively loosing strength. The dependence of $\mathcal{S}_{\text{\tiny CSL}}$ on the geometry of the system is clearly visible in the shape of the corresponding upper bounds on the collapse parameters. Indeed, as it is shown in Fig. 1, once the dimensions $L$ of the system are fixed, one has the strongest bound on $\lambda$ for the value of $r_{\text{\tiny C}}\sim L$. This reflects in the characteristic $V$-shaped form of the bounds of the CSL parameters.

Eq. (6) gives insight in the collapse action on the mechanical oscillator. This is the change of the equilibrium temperature of the system from the environmental one $T$ to an enhanced effective one. Indeed, in the limit for high temperatures of the environment this reads Carlesso et al. (2018a)

[TABLE]

with

[TABLE]

One should notice that, here, another parameter of the opto-mechanical setup plays an important role, namely the damping rate $\gamma_{\text{\tiny m}}$ that quantifies mechanical dissipation. Clearly, the more the system dissipates, the faster the thermalization process to the environmental temperature, and the smaller the collapse contribution. On the contrary, in the limit of no dissipation (i.e. for $\gamma_{\text{\tiny m}}\to 0$), $\Delta T_{\text{\tiny CSL}}$ diverges: this is exactly what should be expected from the model, whose collapse noise can be associated to an infinite-temperature bath. In passing, we remark that generalizations of collapse models have been proposed Smirne and Bassi (2015); Nobakht et al. (2018); Smirne et al. (2014) where the noise inducing collapse is associated with a finite temperature $T_{\text{\tiny CSL}}$ and an ensuing dissipative process. We refer to Sec. IV for details on such models.

As underlined in Ref. Nimmrichter et al. (2014), the thermal noise, proportional to $\coth(\tfrac{\hbar\omega}{2k_{\text{\tiny B}}T})$, is not the only limitation in detecting the collapse-induced diffusion. Indeed, the measurement process also contributes to enhancing the noise in the readout signal, thus screening the signal from the collapse mechanism. Clearly, a precise characterization of the thermal effects and the measurement backaction would provide stronger upper bounds to the collapse parameters.

III Experimental bounds

The first application that we consider is the one reported in Ref. Carlesso et al. (2016), where three experiments – LIGO, AURIGA and LISA Pathfinder – have been considered. The first two are gravitational wave detectors, while the last one is only a prototype of a future gravitational wave detector. In all such experiments, a mechanical resonator is monitored through optical techniques. Due to the mass of the systems ($\sim 2$ kg for LISA Pathfinder, $\sim 40\,$kg for LIGO and $\sim 2300\,$kg for AURIGA), the back-action of the optics can be neglected, and one considers only the last term in Eq. (6), which depends explicitly on the experiment considered. The single arm of LIGO and LISA Pathfinder consists of two masses, modelled as harmonic oscillators, whose relative distance is monitored. Conversely, AURIGA is a resonant bar whose elongation is measured. For the latter, one can model the system as two half-mass harmonic oscillators oscillating in counterphase. Thus, the modelling is the same for all three experiments. Eq. (8) is consequently modified to read

[TABLE]

where $a$ is the distance between the two masses. Such systems are well outside the quantum realm due to their masses, which also prevent their use in interferometric experiments. However, they set important bounds on the collapse parameters, which are here reported in Fig. 1.

The second application that we aim at covering is that reported in Refs. Vinante et al. (2016, 2017), where a heavy micrometrical sphere is attached to a silicon cantilever, which acts as a mechanical resonator. As the sphere is ferromagnetic, in place of the optics, a low noise SQUID can be employed to monitor the mechanical motion of the cantilever. The system is placed in high vacuum and low temperature to minimize the thermal action of the environment. Moreover, in order to better characterize the thermal component of the noise, different measurements of the DNS of the system were performed at different temperatures of the environment, ranging from $11$ mK to $\sim 1\,$K. Thus, by exploiting Eq. (10), one can determine upper bounds on the collapse parameters $\lambda$ and $r_{\text{\tiny C}}$, which are reported in Fig. 1.

IV Testing of the dissipative and colored CSL models

The CSL model have two weaknesses Bassi and Ghirardi (2003). The first is the steady increase in the energy of any (free) system in time, e.g. an hydrogen atom is heated by $\simeq 10^{-14}$ K per year taking the values $\lambda=10^{-16}$ s*-1* and $r_{\text{\tiny C}}=10^{-7}$ m. Although the increment is small, it is not realistic feature even for a phenomenological model. On the other hand, one expects that, through a dissipative mechanisms, the system will eventually termalize to the finite temperature of the collapse noise. Although there are theoretical arguments suggesting the value of such a temperature to be $T_{\text{\tiny CSL}}\simeq 1\,$K Smirne and Bassi (2015); Smirne et al. (2014), one needs to validate them. While an interferometric investigation was performed in Ref. Toroš and Bassi (2018); Toroš et al. (2017), and a non-interferometric measurement of the free expansion of a cold-atom cloud was studied in Ref. Bilardello et al. (2016), the theoretical setting for an opto-mechanical test of the dissipative extension of the CSL model was proposed in Ref. Nobakht et al. (2018). Fig. 3 shows how the experimental bounds change when the dissipation is explicitly considered in the collapse mechanism for two values of the $T_{\text{\tiny CSL}}$.

The second weakness of the CSL model is that its noise has a white spectrum. This is clearly an approximation as no physical noise can be perfectly white. Conversely, one expects the existence of a cutoff frequency $\Omega_{\text{\tiny C}}$ above which the collapse mechanism is negligible. Theoretical arguments suggest $\Omega_{\text{\tiny C}}\sim 10^{12}\,$Hz Adler and Bassi (2007, 2008). The introduction of the cutoff changes the predictions of the model: the correlations of the noise in Eq. (4) are modified in $\mathbb{E}[N({\bf x},t)N({\bf y},s)]=\lambda f(t-s)G({\bf x}-{\bf y})$, where $f(t)$ describes the time correlations of the collapse noise. Correspondingly, the DNS in an opto-mechanical system becomes $\mathcal{S}_{\text{\tiny cCSL}}(\omega)=\mathcal{S}_{\text{\tiny CSL}}\times\tilde{f}(\omega)$, where $\tilde{f}(\omega)$ is the Fourier transform of $f(t)$ Carlesso et al. (2018b). Bounds on the CSL parameters for colored noise were studied in detail in Ref. Bilardello et al. (2016); Toroš et al. (2017); Carlesso et al. (2018b). In particular, upper bounds from high frequencies experiments (or involving small time scales) are weakened when moving to small value of $\Omega_{\text{\tiny C}}$. Fig. 3 shows the upper bounds to the colored CSL extension for two values of $\Omega_{\text{\tiny C}}$.

V Proposals for future testing

Opto-mechanical proposals have been put forward aimed at strengthening the current upper bounds on the collapse parameters. One consists in the modification of the cantilever experiment in Ref. Vinante et al. (2017), where the homogeneous mass is substituted with one made of several layers of two different materials Carlesso et al. (2018c). This will increment the effect of the CSL noise for the values of $r_{\text{\tiny C}}$ of the order of the thickness of the layers. The hypothetical upper bounds that can be inferred from such scheme are shown in Fig. 4.

A second possible test focuses on the rotational degrees of freedom in place of the vibrational ones Schrinski et al. (2017); Carlesso et al. (2018a). The former can quantify the CSL action in a form similar to that in Eq. (10), where the collapse-induced contribution to the temperature is related to the rotational degrees of freedom and reads $\Delta T_{\text{\tiny CSL}}^{\text{\tiny rot}}={\mathcal{S}_{\text{\tiny CSL}}^{\text{\tiny rot}}}/{2k_{\text{\tiny B}}D_{\phi}}$, where $D_{\phi}$ is the rotational damping rate and

[TABLE]

Eq. (12) quantifies the stochastic torque induced on the system by the collapse noise. When such a scheme is applied to macroscopic systems, it can provide a sensible improvement of the bounds on the collapse parameters, cf. Fig. 4. A direct application was considered in Carlesso et al. (2018a), where the bound from LISA Pathfinder Carlesso et al. (2016) can be significantly improved by also considering the rotational degrees of freedom.

The above are only two of several proposals Collett and Pearle (2003); Goldwater et al. (2016); Kaltenbaek et al. (2016); McMillen et al. (2017); Mishra et al. (2018) suggested over the past few years aimed to push further the exploration of the CSL parameter space.

Acknowledgements.

The authors acknowledge the invaluable collaboration with A. Bassi, H. Ulbricht, S. Donadi, A. Vinante on the topics of this work. They are grateful acknowledge support from the H2020 Collaborative project TEQ (grant agreement 766900) and COST Action CA15220. MC acknowledges support from INFN. MP acknowledges support from the DfE-SFI Investigator Programme (grant 15/IA/2864), the Royal Society Wolfson Research Fellowship (RSWF\R3\183013) and the Leverhulme Trust Research Project Grant (grant nr. RGP-2018-266).