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arXiv:1712.01071

Minimum Interior Temperature for Solid Objects Implied by Collapse Models

Stephen L. Adler

Published Dec 4, 2017• 1 citation
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Abstract

Heating induced by the noise postulated in wave function collapse models leads to a lower bound to the temperature of solid objects. For the noise parameter values $λ={\rm coupling~strength}\sim 10^{-8} {\rm s}^{-1}$ and $r_C ={\rm correlation~length} \sim 10^{-5} {\rm cm}$, which were suggested \cite{adler1} to make latent image formation an indicator of wave function collapse and which are consistent with the recent experiment of Vinante et al. \cite{vin}, the effect may be observable. For metals, where the heat conductivity is proportional to the temperature at low temperatures, the lower bound (specifically for RRR=30 copper) is $\sim 5\times 10^{-11} (L/r_C) $K, with L the size of the object. For the thermal insulator Torlon 4203, the comparable lower bound is $\sim 3 \times 10^{-6} (L/r_c)^{0.63}$ K. We first give a rough estimate for a cubical metal solid, and then give an exact solution of the heat transfer problem for a sphere.

DOI
10.48550/arxiv.1712.01071

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•Apr 12, 2017•2 citations
Heating induced by the noise postulated in wave function collapse models leads to a lower bound to the temperature of solid objects. For the noise parameter values λ=coupling strength∼10−8s−1λ={\rm coupling~strength}\sim 10^{-8} {\rm s}^{-1}λ=coupling strength∼10−8s−1 and rC=correlation length∼10−5cmr_C ={\rm correlation~length} \sim 10^{-5} {\rm cm}rC​=correlation length∼10−5cm, which were suggested \cite{adler1} to make latent image formation an indicator of wave function collapse and which are consistent with the recent experiment of Vinante et al. \cite{vin}, the effect may be observable. For metals, where the heat conductivity is proportional to the temperature at low temperatures, the lower bound (specifically for RRR=30 copper) is ∼5×10−11(L/rC)\sim 5\times 10^{-11} (L/r_C) ∼5×10−11(L/rC​)K, with L the size of the object. For the thermal insulator Torlon 4203, the comparable lower bound is ∼3×10−6(L/rc)0.63\sim 3 \times 10^{-6} (L/r_c)^{0.63}∼3×10−6(L/rc​)0.63 K. We first give a rough estimate for a cubical metal solid, and then give an exact solution of the heat transfer problem for a sphere.
We discuss limits on the noise strength parameter in mass-proportional-coupled wave-function collapse models implied by bulk heating effects and examine the role of the noise power spectrum in comparing experiments of different types. This comparison utilizes a calculation of the rate of heating through phonon excitation implied by a general noise power spectrum $\ensuremath{\lambda}(\ensuremath{\omega})$. We find that, in the standard heating formula, the reduction rate $\ensuremath{\lambda}$ is replaced by ${\ensuremath{\lambda}}_{\mathrm{eff}}=\frac{2}{3{\ensuremath{\pi}}^{3/2}}\ensuremath{\int}{d}^{3}w{e}^{\ensuremath{-}{\stackrel{P\vec}{w}}^{2}}{\stackrel{P\vec}{w}}^{2}\ensuremath{\lambda}({\ensuremath{\omega}}_{L}(\stackrel{P\vec}{w}/{r}_{c}))$, with ${\ensuremath{\omega}}_{L}(\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}})$ being the longitudinal acoustic-phonon frequency as a function of wave number $\stackrel{P\vec}{q}$, and with rC{r}_{C}rC​ being the noise correlation length. Hence if the noise power spectrum is cut off below ${\ensuremath{\omega}}_{L}(|\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}}|\ensuremath{\sim}{r}_{c}^{\ensuremath{-}1})$, the bulk heating rate is sharply reduced, allowing compatibility with current experimental results. We suggest possible new bulk heating experiments that can be performed subject to limits placed by natural heating from radioactivity and cosmic rays. The proposed experiments exploit the vanishing of thermal transport in the low-temperature limit.
Collapse models postulate that space is filled with a collapse noise field, inducing quantum Brownian motions, which are dominant during the measurement, thus causing collapse of the wave function. An important manifestation of the collapse noise field, if any, is thermal energy generation, thus disturbing the temperature profile of a system. The experimental investigation of a collapse-driven heating effect has provided, so far, the most promising test of collapse models against standard quantum theory. In this paper, we calculate the collapse-driven heat generation for a three-dimensional multi-atomic Bravais lattice by solving stochastic Heisenberg equations. We perform our calculation for the mass-proportional continuous spontaneous localization collapse model with nonwhite noise. We obtain the temperature distribution of a sphere under stationary-state and insulated surface conditions. However, the exact quantification of the collapse-driven heat-generation effect highly depends on the actual value of cutoff in the collapse noise spectrum.
Models of spontaneous wave function collapse predict a small heating rate for a bulk solid, as a result of coupling to the noise field that causes collapse. This rate is small enough that ambient radioactivity and cosmic ray flux on the surface of the earth can mask the heating due to spontaneous collapse. In this paper we estimate the background noise due to $\ensuremath{\gamma}$ radiation and cosmic ray muon flux, at different depths. We demonstrate that a low-temperature underground experiment at a depth of about 6.5 kilometer water equivalent would have a low enough background to allow detection of bulk heating for a collapse rate $\ensuremath{\lambda}$ of ${10}^{\ensuremath{-}16}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$ using presently available technology.
Spontaneous wave-function collapse models, like continuous spontaneous localization, are designed to suppress macroscopic superpositions while preserving microscopic quantum phenomena. An observable consequence of collapse models is spontaneous heating of massive objects. We calculate the collapse-induced heating rate of astrophysical objects, and the corresponding equilibrium temperature. We apply these results to neutron stars, the densest phase of baryonic matter in the Universe. Stronger collapse model parameters imply greater heating, allowing us to derive competitive bounds on model parameters using neutron star observational data, and to propose speculative bounds based on the capabilities of current and future astronomical surveys.
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} m, and partially excludes the enhanced collapse rate suggested by Adler. We discuss future improvements.
Collapse models solve the measurement problem by adding non-linear and stochastic terms to the usual Schroedinger dynamics. The resulting dynamics localizes in position macroscopic systems, while preserves quantum mechanical predictions for microscopic systems. \nTwo of the main consequences of this resulting dynamics are: the loss of any initial spatial coherence, with a decoherence rate which increases with the dimensions of the system (amplification mechanism); and a heating effect induced on the system that, for long times, leads either to a divergent or to a convergent value of the energy, depending on the particular collapse model considered. \nAn interesting class of system to experimentally test collapse models are cold atomic gases. Besides their very low temperatures (T = 10^(-11) - 10^(-7)K), cold atomic gases show mesoscopic quantum properties, i.e. involving a number of atoms of order of N = 10^(3)– 10^(11). These are all necessary ingredients to experimentally test collapse models. In this thesis, we use cold bosonic atomic gas to test one of the most studied and complete among collapse models, i.e. CSL, and also its non-Markovian and dissipative extensions. In particular, in this thesis we put bounds on CSL free-parameters by studying the diffusion effect induced by collapse noise on a free expanding gas, and we compare the theoretical results with experimental data known in literature. The bounds we put are among the strongest ever found non only for CSL model, but especially for its non-Markovian extension. \nIn this thesis we also specialize the CSL dynamics in Bose-Josephson junctions, and we focus on the localization mechanism induced by the noise on initially entangled states. In particular, we focus on decoherence induced by CSL on typical entangled states as atomic coherent states and NOON states. We find that in the latter ones, CSL effects become significant when the NOON state is composed by a number of atoms of 10^(4) or greater. Moreover, we compare CSL decoherence with environmental decoherence induced by typical realistic sources, as an external thermal cloud, imperfections in the trapping laser, and three body effects. We then find which conditions experiments must fulfill in order to test CSL in a Bose-Josephson junction in conditions never