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{{Short description|Quantum mechanical theory of spontaneous collapse}}
The '''continuous spontaneous localization''' ('''CSL''') model is a [[Objective-collapse theory|spontaneous collapse model]] in [[quantum mechanics]], proposed in 1989 by Philip Pearle.<ref name=":0">{{Cite journal|last=Pearle|first=Philip|date=1989-03-01|title=Combining stochastic dynamical state-vector reduction with spontaneous localization
== Introduction ==
The most widely studied among the [[Objective-collapse theory|dynamical reduction]] (also known as collapse) models is the CSL model.<ref name=":0" /><ref name=":1" /><ref name=":2">{{Cite journal|
* The localization takes place in position, which is the preferred basis in this model.
* The model does not significantly alter the dynamics of
* It preserves the symmetry properties of identical particles.
* It is characterized by two parameters: <math>\lambda</math> and <math>r_C</math>, which are respectively the collapse rate and the correlation length of the model.
== Dynamical equation ==
The CSL dynamical equation for the
\left.-\frac\lambda{2m_0^2}\int\operatorname{d}\!{\bf x}\int\operatorname{d}\!{\bf y}\,g({\bf x}-{\bf y})\hat N_t({\bf x})\hat N_t({\bf y})\operatorname{d}\! t \right]|\psi_t\rangle
\mathbb E[w_t({\bf x}) w_s({\bf y})]=g({\bf x}-{\bf y})\delta(t-s),
</math>where <math>\mathbb E [\ \cdot\ ]</math> denotes the stochastic average over the noise. Finally, we
\hat M({\bf x})=\sum_j m_j\sum_s\hat a^\dagger_j({\bf x},s)\hat a_j({\bf x},s),
</math>where <math>\hat a^\dagger_j({\bf y},s)</math> and <math>\hat a_j({\bf y},s)</math> are, respectively, the second quantized creation and annihilation operators of a particle of type <math>j</math> with spin <math>s</math> at the point <math>{\bf y}</math> of mass <math>m_j</math>. The use of these operators satisfies the conservation of the symmetry properties of identical particles. Moreover, the mass proportionality implements automatically the amplification mechanism. The choice of the form of <math>\hat M({\bf x})</math> ensures the collapse in the position basis.
The action of the CSL model is quantified by the values of the two phenomenological parameters <math>\lambda</math> and <math>r_C</math>. Originally, the Ghirardi-Rimini-Weber model<ref name=":3" /> proposed <math>\lambda=10^{-17}\,</math>s<math>^{-1}</math> at <math>r_C=10^{-7}\,</math>m, while later Adler considered larger values:<ref>{{Cite journal|last=Adler|first=Stephen L|date=2007-10-16|title=Lower and upper bounds on CSL parameters from latent image formation and IGM~heating
From the dynamics of the
\frac{\operatorname{d}\! \hat\rho_t}{\operatorname{d}\! t}
=-\frac{i}{\hbar}\left[{\hat H},{\hat \rho_t}\right]
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== Experimental tests ==
Contrary to most other proposed solutions of the measurement problem, collapse models are experimentally testable.
=== Interferometric experiments ===
[[Interferometry|Interferometric]] experiments can detect the direct action of the collapse, which is to localize the wavefunction in space. They include all experiments where a superposition is generated and, after some time, its interference pattern is probed. The action of CSL is a reduction of the interference contrast, which is quantified by the reduction of the off-diagonal terms of the statistical operator<ref name=":4">{{Cite journal|
Similarly, one can also quantify the minimum collapse strength to
=== Non-interferometric experiments ===
Non-interferometric experiments consist in CSL tests, which are not based on the preparation of a superposition. They exploit an indirect effect of the collapse, which consists in a Brownian motion induced by the interaction with the collapse noise. The effect of this noise amounts to an effective stochastic force acting on the system, and several experiments can be designed to quantify such a force. They include:<ref>{{Cite journal |last1=Carlesso |first1=Matteo |last2=Donadi |first2=Sandro |last3=Ferialdi |first3=Luca |last4=Paternostro |first4=Mauro |last5=Ulbricht |first5=Hendrik |last6=Bassi |first6=Angelo |date=February 2022 |title=Present status and future challenges of non-interferometric tests of collapse models |url=https://www.nature.com/articles/s41567-021-01489-5 |journal=Nature Physics |language=en |volume=18 |issue=3 |pages=243–250 |doi=10.1038/s41567-021-01489-5 |arxiv=2203.04231 |bibcode=2022NatPh..18..243C |s2cid=246949254 |issn=1745-2481}}</ref>
* ''Radiation emission from charged particles''. If a particle is electrically charged, the action of the coupling with the collapse noise will induce the emission of radiation. This result is in net contrast with the predictions of quantum mechanics, where no radiation is expected from a free particle. The predicted CSL-induced emission rate at frequency <math>\omega</math> for a particle of charge <math>Q</math> is given by:<ref>{{Cite journal|
<math display="block">
\frac{\operatorname{d}\! \Gamma(\omega)}{\operatorname{d}\! \omega}=\frac{\hbar Q^2\lambda}{2\pi^2\epsilon_0c^3m_0^2r_C^2\omega},
</math>where <math>\epsilon_0</math> is the vacuum dielectric constant and <math>c</math> is the light speed. This prediction of CSL can be tested<ref>{{Cite journal|last=Fu|first=Qijia|date=1997-09-01|title=Spontaneous radiation of free electrons in a nonrelativistic collapse model
* ''Heating in bulk materials''. A prediction of CSL is the increase of the total energy of a system. For example, the total energy <math>E</math> of a free particle of mass <math>m</math> in three dimensions grows linearly in time according to<ref name=":2" /> <math display="block">
E(t)=E(0)+\frac{3m\lambda\hbar^{2}}{4m_{0}^{2}r_C^{2}}t,
</math>where <math>E(0)</math> is the initial energy of the system. This increase is effectively small; for example, the temperature of a hydrogen atom increases by <math>\simeq 10^{-14}</math> K per year considering the values <math>\lambda=10^{-16}</math> s<math>^{-1}</math> and <math>r_C=10^{-7}</math>m. Although small, such an energy increase can be tested by monitoring cold atoms.<ref name=":6">{{Cite journal|
* ''Diffusive effects''. Another prediction of the CSL model is the increase of the spread in position of center-of-mass of a system. For a free particle, the position spread in one dimension reads<ref>{{Cite journal|last=Romero-Isart|first=Oriol|date=2011-11-28|title=Quantum superposition of massive objects and collapse models
\langle{\hat x^2}\rangle_t=\langle{\hat x^2}\rangle_t^{ (QM)}+\frac{\hbar^2\eta t^3}{3 m^2},
</math>where <math>\langle{\hat x^2}\rangle_t^{ (QM)}</math> is the free quantum mechanical spread and <math>\eta</math> is the CSL diffusion constant, defined as<ref>{{Cite journal|
\eta=\frac{\lambda r_C^3}{2\pi^{3/2}m_0^2}\int\operatorname{d}\!{\bf k}\,e^{-{\bf k}^2r_C^2}k_x^2|\tilde \mu({\bf k})|^2,
</math>where the motion is assumed to occur along the <math>x</math> axis; <math>\tilde \mu({\bf k})</math> is the Fourier transform of the mass density <math>\mu({\bf r})</math>. In experiments, such an increase is limited by the dissipation rate <math>\gamma</math>. Assuming that the experiment is performed at temperature <math>T</math>, a particle of mass <math>m</math>, harmonically trapped at frequency <math>\omega_0</math>, at equilibrium reaches a spread in position given by<ref name=":8">{{Cite journal|
\langle{\hat x^2}\rangle_{ eq}=\frac{k_B T}{m\omega_0^2}+\frac{ \hbar^2 \eta}{2m^2 \omega_0^2 \gamma },
</math>where <math>
k_B
</math> is the Boltzmann constant. Several experiments can test such a spread. They range from cold atom free expansion,<ref name=":6" /><ref name=":7" /> nano-cantilevers cooled to millikelvin temperatures,<ref name=":8" /><ref>{{Cite journal|
== Dissipative and colored extensions ==
The CSL model
* ''CSL does not conserve the energy of isolated systems''. Although this increase is small, it is an
E(t)=e^{-\beta t}(E(0)-E_{ as})+E_{ as},
</math>where <math>
E_{ as}=\tfrac32 k_B T_{ CSL}</math>, <math>\beta=4 \chi \lambda /(1+\chi)^5</math> and <math>\chi=\hbar^2/(8 m_0 k_B T_{ CSL}r_C^2)</math>. Assuming that the CSL noise has a cosmological origin (which is reasonable due to its supposed universality), a plausible value such a temperature is <math>T_{ CSL}=1</math> K, although only experiments can indicate a definite value. Several interferometric<ref name=":4" /><ref name=":10" /> and non-interferometric<ref name=":7" /><ref name=":11" /><ref>{{Cite journal|
* ''The CSL noise spectrum is white''. If one attributes a physical origin to the CSL noise, then its spectrum cannot be white, but colored. In particular, in place of the [[white noise]] <math>w_t({\bf x})</math>, whose correlation is proportional to a Dirac delta in time, a [[Colors of noise|non-white noise]] is considered, which is characterized by a non-trivial temporal correlation function <math>f(t)</math>. The effect can be quantified by a rescaling of <math>F_{{ CSL}}(k,q,t)</math>, which becomes<math display="block">
F_{{cCSL}}(k,q,t)=F_{{ CSL}}(k,q,t) \exp\left[ \frac{\lambda \bar\tau}{2}\left( e^{-(q-k t/m)^2/4r_C^2}-e^{-q^2/4r_C^2} \right) \right],
</math>where <math> \bar\tau=\int_0^t\operatorname{d}\! s\,f(s)</math>. As an example, one can consider an exponentially decaying noise, whose time correlation function can be of the form<ref name=":12">{{Cite journal|
== References ==
{{Reflist}}
[[Category:Interpretations of quantum mechanics]]
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