continuous spontaneous localization model

The most widely studied among the dynamical reduction (also known as collapse) models is the CSL model.{{Cite journal|last1=Bassi|first1=Angelo|last2=Ghirardi|first2=GianCarlo|date=2003-06-01|title=Dynamical reduction models|url=http://www.sciencedirect.com/science/article/pii/S0370157303001030|journal=Physics Reports|language=en|volume=379|issue=5|pages=257–426|doi=10.1016/S0370-1573(03)00103-0|issn=0370-1573|arxiv=quant-ph/0302164|bibcode=2003PhR...379..257B|s2cid=119076099}} Building on the Ghirardi-Rimini-Weber model,{{Cite journal|last1=Ghirardi|first1=G. C.|last2=Rimini|first2=A.|last3=Weber|first3=T.|date=1986-07-15|title=Unified dynamics for microscopic and macroscopic systems|journal=Physical Review D|volume=34|issue=2|pages=470–491|doi=10.1103/PhysRevD.34.470|pmid=9957165|bibcode=1986PhRvD..34..470G}} the CSL model describes the collapse of the wave function as occurring continuously in time, in contrast to the Ghirardi-Rimini-Weber model.

Some of the key features of the model are:

• The localization takes place in position, which is the preferred basis in this model.
• The model does not significantly alter the dynamics of microscopic systems, while it becomes strong for macroscopic objects: the amplification mechanism ensures this scaling.
• It preserves the symmetry properties of identical particles.
• It is characterized by two parameters: $\backslash lambda$ and $r\_C$, which are respectively the collapse rate and the correlation length of the model.

The CSL dynamical equation for the wave function is stochastic and non-linear:$$\backslash operatorname\{d\}\backslash !|\backslash psi\_t\backslash rangle=\backslash left[\; -\backslash frac\; i\backslash hbar\; \backslash hat\; H\backslash operatorname\{d\}\backslash !\; t+\backslash frac\{\backslash sqrt\{\backslash lambda\}\}\{m\_0\}\backslash int\; \backslash operatorname\{d\}\backslash !\; \{\backslash bf\; x\}\backslash ,\backslash hat\; N\_t(\{\backslash bf\; x\})\backslash operatorname\{d\}\backslash !\; W\_t(\{\backslash bf\; x\})\backslash right.$$

\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.Here $\backslash hat\; H$ is the Hamiltonian describing the quantum mechanical dynamics, $m\_0$ is a reference mass taken equal to that of a nucleon, $g(\{\backslash bf\; x\}-\{\backslash bf\; y\})=e^\{-\{(\{\backslash bf\; x\}-\{\backslash bf\; y\})^2\}/\{4r\_C^2\}\}$, and the noise field $w\_t(\{\backslash bf\; x\})=\backslash operatorname\{d\}\backslash !\; W\_t(\{\backslash bf\; x\})/\backslash operatorname\{d\}\backslash !\; t$ has zero average and correlation equal to$$$$

\mathbb E[w_t({\bf x}) w_s({\bf y})]=g({\bf x}-{\bf y})\delta(t-s),

where $\backslash mathbb\; E\; [\backslash \; \backslash cdot\backslash \; ]$ denotes the stochastic average over the noise. Finally, we write$$\backslash hat\; N\_t(\{\backslash bf\; x\})=\backslash hat\; M(\{\backslash bf\; x\})-\backslash langle\backslash psi\_t|\backslash hat\; M(\{\backslash bf\; x\})|\backslash psi\_t\backslash rangle,$$where $\backslash hat\; M(\{\backslash bf\; x\})$ is the mass density operator, which reads$$$$

\hat M({\bf x})=\sum_j m_j\sum_s\hat a^\dagger_j({\bf x},s)\hat a_j({\bf x},s),

where $\backslash hat\; a^\backslash dagger\_j(\{\backslash bf\; y\},s)$ and $\backslash hat\; a\_j(\{\backslash bf\; y\},s)$ are, respectively, the second quantized creation and annihilation operators of a particle of type $j$ with spin $s$ at the point $\{\backslash bf\; y\}$ of mass $m\_j$. 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 $\backslash hat\; M(\{\backslash bf\; x\})$ ensures the collapse in the position basis.

The action of the CSL model is quantified by the values of the two phenomenological parameters $\backslash lambda$ and $r\_C$. Originally, the Ghirardi-Rimini-Weber model proposed $\backslash lambda=10^\{-17\}\backslash ,$s$^\{-1\}$ at $r\_C=10^\{-7\}\backslash ,$m, while later Adler considered larger values:{{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|journal=Journal of Physics A: Mathematical and Theoretical|language=en|volume=40|issue=44|pages=13501|doi=10.1088/1751-8121/40/44/c01|issn=1751-8113|arxiv=quant-ph/0605072|s2cid=250685315 }} $\backslash lambda=10^\{-8\backslash pm2\}\backslash ,$s$^\{-1\}$ for $r\_C=10^\{-7\}\backslash ,$m, and $\backslash lambda=10^\{-6\backslash pm2\}\backslash ,$s$^\{-1\}$ for $r\_C=10^\{-6\}\backslash ,$m. Eventually, these values have to be bounded by experiments.

From the dynamics of the wave function one can obtain the corresponding master equation for the statistical operator $\backslash hat\; \backslash rho\_t$:$$$$

\frac{\operatorname{d}\! \hat\rho_t}{\operatorname{d}\! t}

=-\frac{i}{\hbar}\left[{\hat H},{\hat \rho_t}\right]

-\frac{\lambda}{2m_0^2}\int\operatorname{d}\!{\bf x}\int\operatorname{d}\!{\bf y}\,g({\bf x}-{\bf y})

\left[{\hat M({\bf x})},\left[{{\hat M({\bf y})},{\hat \rho_t}}\right]\right].

Once the master equation is represented in the position basis, it becomes clear that its direct action is to diagonalize the density matrix in position. For a single point-like particle of mass $m$, it reads$$$$

\frac{\partial \langle{{\bf x}|\hat \rho_t|{\bf y}}\rangle}{\partial t}=-\frac{i}{\hbar}\langle{{\bf x}|\left[{\hat H},{\hat \rho_t}\right]|{\bf y}}\rangle-\lambda\frac{m^2}{m_0^2}\left(1-e^{-\tfrac{({\bf x}-{\bf y})^2}{4r_C^2}}\right)\langle{{\bf x}|\hat \rho_t|{\bf y}}\rangle,

where the off-diagonal terms, which have $\{\backslash bf\; x\}\backslash neq\{\backslash bf\; y\}$, decay exponentially. Conversely, the diagonal terms, characterized by $\{\backslash bf\; x\}=\{\backslash bf\; y\}$, are preserved. For a composite system, the single-particle collapse rate $\backslash lambda$ should be replaced with that of the composite system$$$$

\lambda\frac{m^2}{m_0^2}\to\lambda\frac{r_C^3}{\pi^{3/2}m_0^2}\int\operatorname{d}\!{\bf k}|\tilde\mu({\bf k})|^2e^{-k^2r_C^2},

where $\backslash tilde\; \backslash mu(k)$ is the Fourier transform of the mass density of the system.

Contrary to most other proposed solutions of the measurement problem, collapse models are experimentally testable. Experiments testing the CSL model can be divided in two classes: interferometric and non-interferometric experiments, which respectively probe direct and indirect effects of the collapse mechanism.

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{{Cite journal|last1=Toroš|first1=Marko|last2=Gasbarri|first2=Giulio|last3=Bassi|first3=Angelo|date=2017-12-20|title=Colored and dissipative continuous spontaneous localization model and bounds from matter-wave interferometry|url=http://www.sciencedirect.com/science/article/pii/S0375960117309465|journal=Physics Letters A|language=en|volume=381|issue=47|pages=3921–3927|doi=10.1016/j.physleta.2017.10.002|issn=0375-9601|arxiv=1601.03672|bibcode=2017PhLA..381.3921T|s2cid=119208947}}$$\backslash rho(x,x\text{'},t)\; =\; \backslash frac\{1\}\{2\backslash pi\backslash hbar\}\; \backslash int\_\{-\backslash infty\}^\{+\; \backslash infty\}\; \backslash operatorname\{d\}\backslash !\; k\; \backslash int\_\{-\backslash infty\}^\{+\; \backslash infty\}\; \backslash operatorname\{d\}\backslash !\; w\backslash ,\; e^\{-ikw/\backslash hbar\}\; F\_\{\{\; CSL\}\}(k,\; x-x\text{'},t)\; \backslash rho^\{\{\; QM\}\}(x+w,\; x\text{'}+w,\; t),$$where $\backslash rho^\{\{QM\}\}$ denotes the statistical operator described by quantum mechanics, and we define$$F\_\{\{\; CSL\}\}(k,q,t)=\; \backslash exp\backslash bigg[-\backslash lambda\; \backslash frac\{m^2\}\{m\_0^2\}\; t\; \backslash left(1-\backslash frac\{1\}\{t\}\backslash int\_0^t\; \backslash operatorname\{d\}\backslash !\backslash tau\; \backslash ,e^\{-\{(q-\backslash frac\{k\backslash tau\}\{m\})^2\}/\{4r\_C^2\}\}\; \backslash right)\; \backslash bigg].$$Experiments testing such a reduction of the interference contrast are carried out with cold-atoms,{{Cite journal|last1=Kovachy|first1=T.|last2=Asenbaum|first2=P.|last3=Overstreet|first3=C.|last4=Donnelly|first4=C. A.|last5=Dickerson|first5=S. M.|last6=Sugarbaker|first6=A.|last7=Hogan|first7=J. M.|last8=Kasevich|first8=M. A.|date=2015|title=Quantum superposition at the half-metre scale|url=https://www.nature.com/articles/nature16155|journal=Nature|language=en|volume=528|issue=7583|pages=530–533|doi=10.1038/nature16155|pmid=26701053|bibcode=2015Natur.528..530K|s2cid=205246746|issn=1476-4687}} molecules{{Cite journal|last1=Eibenberger|first1=Sandra|last2=Gerlich|first2=Stefan|last3=Arndt|first3=Markus|last4=Mayor|first4=Marcel|last5=Tüxen|first5=Jens|date=2013-08-14|title=Matter–wave interference of particles selected from a molecular library with masses exceeding 10 000 amu|journal=Physical Chemistry Chemical Physics|language=en|volume=15|issue=35|pages=14696–14700|doi=10.1039/C3CP51500A|pmid=23900710|arxiv=1310.8343|bibcode=2013PCCP...1514696E|issn=1463-9084|doi-access=free}}{{Cite journal|last1=Toroš|first1=Marko|last2=Bassi|first2=Angelo|date=2018-02-15|title=Bounds on quantum collapse models from matter-wave interferometry: calculational details|journal=Journal of Physics A: Mathematical and Theoretical|volume=51|issue=11|pages=115302|doi=10.1088/1751-8121/aaabc6|issn=1751-8113|arxiv=1601.02931|bibcode=2018JPhA...51k5302T|s2cid=118707096}}{{Cite journal|last1=Fein|first1=Yaakov Y.|last2=Geyer|first2=Philipp|last3=Zwick|first3=Patrick|last4=Kiałka|first4=Filip|last5=Pedalino|first5=Sebastian|last6=Mayor|first6=Marcel|last7=Gerlich|first7=Stefan|last8=Arndt|first8=Markus|date=2019|title=Quantum superposition of molecules beyond 25 kDa|url=https://www.nature.com/articles/s41567-019-0663-9|journal=Nature Physics|language=en|volume=15|issue=12|pages=1242–1245|doi=10.1038/s41567-019-0663-9|bibcode=2019NatPh..15.1242F|s2cid=203638258|issn=1745-2481}}, entangled diamonds{{Cite journal|last1=Lee|first1=K. C.|last2=Sprague|first2=M. R.|last3=Sussman|first3=B. J.|last4=Nunn|first4=J.|last5=Langford|first5=N. K.|last6=Jin|first6=X.-M.|last7=Champion|first7=T.|last8=Michelberger|first8=P.|last9=Reim|first9=K. F.|last10=England|first10=D.|last11=Jaksch|first11=D.|date=2011-12-02|title=Entangling Macroscopic Diamonds at Room Temperature|url=https://www.science.org/doi/10.1126/science.1211914|journal=Science|language=en|volume=334|issue=6060|pages=1253–1256|doi=10.1126/science.1211914|issn=0036-8075|pmid=22144620|bibcode=2011Sci...334.1253L|s2cid=206536690}}{{Cite journal|last1=Belli|first1=Sebastiano|last2=Bonsignori|first2=Riccarda|last3=D'Auria|first3=Giuseppe|last4=Fant|first4=Lorenzo|last5=Martini|first5=Mirco|last6=Peirone|first6=Simone|last7=Donadi|first7=Sandro|last8=Bassi|first8=Angelo|date=2016-07-12|title=Entangling macroscopic diamonds at room temperature: Bounds on the continuous-spontaneous-localization parameters|journal=Physical Review A|volume=94|issue=1|pages=012108|doi=10.1103/PhysRevA.94.012108|arxiv=1601.07927|bibcode=2016PhRvA..94a2108B|hdl=1887/135561|s2cid=118344117|hdl-access=free}} and mechanical oscillators {{Cite journal |last1=Schrinski |first1=Björn |last2=Yang |first2=Yu |last3=von Lüpke |first3=Uwe |last4=Bild |first4=Marius |last5=Chu |first5=Yiwen |last6=Hornberger |first6=Klaus |last7=Nimmrichter |first7=Stefan |last8=Fadel |first8=Matteo |title=Macroscopic Quantum Test with Bulk Acoustic Wave Resonators |journal=Physical Review Letters |date=29 March 2023 |volume=130 |issue=13 |pages=133604 |doi=10.1103/PhysRevLett.130.133604|arxiv=2209.06635}}.

Similarly, one can also quantify the minimum collapse strength to solve the measurement problem at the macroscopic level. Specifically, an estimate can be obtained by requiring that a superposition of a single-layered graphene disk of radius $\backslash simeq\; 10^\{-5\}$m collapses in less than $\backslash simeq\; 10^\{-2\}$s.

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:{{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}}

• 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 $\backslash omega$ for a particle of charge $Q$ is given by:{{Cite journal|last1=Adler|first1=Stephen L|last2=Ramazanoğlu|first2=Fethi M|date=2007-10-16|title=Photon-emission rate from atomic systems in the CSL model|journal=Journal of Physics A: Mathematical and Theoretical|volume=40|issue=44|pages=13395–13406|doi=10.1088/1751-8113/40/44/017|arxiv=0707.3134|bibcode=2007JPhA...4013395A|s2cid=14772616 |issn=1751-8113}}{{Cite journal|last1=Bassi|first1=Angelo|last2=Ferialdi|first2=Luca|date=2009-07-31|title=Non-Markovian dynamics for a free quantum particle subject to spontaneous collapse in space: General solution and main properties|journal=Physical Review A|volume=80|issue=1|pages=012116|doi=10.1103/PhysRevA.80.012116|arxiv=0901.1254|bibcode=2009PhRvA..80a2116B|s2cid=119297164}}{{Cite journal|last1=Adler|first1=Stephen L|last2=Bassi|first2=Angelo|last3=Donadi|first3=Sandro|date=2013-06-03|title=On spontaneous photon emission in collapse models|journal=Journal of Physics A: Mathematical and Theoretical|volume=46|issue=24|pages=245304|doi=10.1088/1751-8113/46/24/245304|issn=1751-8113|arxiv=1011.3941|bibcode=2013JPhA...46x5304A|s2cid=119307432}}{{Cite journal|last1=Bassi|first1=A.|last2=Donadi|first2=S.|date=2014-02-14|title=Spontaneous photon emission from a non-relativistic free charged particle in collapse models: A case study|url=http://www.sciencedirect.com/science/article/pii/S0375960114000073|journal=Physics Letters A|language=en|volume=378|issue=10|pages=761–765|doi=10.1016/j.physleta.2014.01.002|issn=0375-9601|arxiv=1307.0560|bibcode=2014PhLA..378..761B|s2cid=118405901}}

\frac{\operatorname{d}\! \Gamma(\omega)}{\operatorname{d}\! \omega}=\frac{\hbar Q^2\lambda}{2\pi^2\epsilon_0c^3m_0^2r_C^2\omega},

where $\backslash epsilon\_0$ is the vacuum dielectric constant and $c$ is the light speed. This prediction of CSL can be tested{{Cite journal|last=Fu|first=Qijia|date=1997-09-01|title=Spontaneous radiation of free electrons in a nonrelativistic collapse model|journal=Physical Review A|volume=56|issue=3|pages=1806–1811|doi=10.1103/PhysRevA.56.1806|bibcode=1997PhRvA..56.1806F}}{{Cite journal|last1=Morales|first1=A.|last2=Aalseth|first2=C. E.|last3=Avignone|first3=F. T.|last4=Brodzinski|first4=R. L.|last5=Cebrián|first5=S.|last6=Garcı́a|first6=E.|last7=Irastorza|first7=I. G.|last8=Kirpichnikov|first8=I. V.|last9=Klimenko|first9=A. A.|last10=Miley|first10=H. S.|last11=Morales|first11=J.|date=2002-04-18|title=Improved constraints on wimps from the international germanium experiment IGEX|journal=Physics Letters B|language=en|volume=532|issue=1|pages=8–14|doi=10.1016/S0370-2693(02)01545-9|arxiv=hep-ex/0110061|bibcode=2002PhLB..532....8M|issn=0370-2693|doi-access=free}}{{Cite journal|last1=Curceanu|first1=C.|last2=Bartalucci|first2=S.|last3=Bassi|first3=A.|last4=Bazzi|first4=M.|last5=Bertolucci|first5=S.|last6=Berucci|first6=C.|last7=Bragadireanu|first7=A. M.|last8=Cargnelli|first8=M.|last9=Clozza|first9=A.|last10=De Paolis|first10=L.|last11=Di Matteo|first11=S.|date=2016-03-01|title=Spontaneously Emitted X-rays: An Experimental Signature of the Dynamical Reduction Models|journal=Foundations of Physics|language=en|volume=46|issue=3|pages=263–268|doi=10.1007/s10701-015-9923-4|issn=1572-9516|arxiv=1601.06617|bibcode=2016FoPh...46..263C|s2cid=53403588}}{{Cite journal|last1=Piscicchia|first1=Kristian|last2=Bassi|first2=Angelo|last3=Curceanu|first3=Catalina|last4=Grande|first4=Raffaele Del|last5=Donadi|first5=Sandro|last6=Hiesmayr|first6=Beatrix C.|author6-link=Beatrix Hiesmayr|last7=Pichler|first7=Andreas|date=2017|title=CSL Collapse Model Mapped with the Spontaneous Radiation|journal=Entropy|language=en|volume=19|issue=7|pages=319|doi=10.3390/e19070319|arxiv=1710.01973|bibcode=2017Entrp..19..319P|doi-access=free}} by analyzing the X-ray emission spectrum from a bulk Germanium test mass.

• Heating in bulk materials. A prediction of CSL is the increase of the total energy of a system. For example, the total energy $E$ of a free particle of mass $m$ in three dimensions grows linearly in time according to $$$$

E(t)=E(0)+\frac{3m\lambda\hbar^{2}}{4m_{0}^{2}r_C^{2}}t,

where $E(0)$ is the initial energy of the system. This increase is effectively small; for example, the temperature of a hydrogen atom increases by $\backslash simeq\; 10^\{-14\}$ K per year considering the values $\backslash lambda=10^\{-16\}$ s$^\{-1\}$ and $r\_C=10^\{-7\}$m. Although small, such an energy increase can be tested by monitoring cold atoms.{{Cite journal|last1=Kovachy|first1=Tim|last2=Hogan|first2=Jason M.|last3=Sugarbaker|first3=Alex|last4=Dickerson|first4=Susannah M.|last5=Donnelly|first5=Christine A.|last6=Overstreet|first6=Chris|last7=Kasevich|first7=Mark A.|date=2015-04-08|title=Matter Wave Lensing to Picokelvin Temperatures|journal=Physical Review Letters|volume=114|issue=14|pages=143004|doi=10.1103/PhysRevLett.114.143004|pmid=25910118|arxiv=1407.6995|bibcode=2015PhRvL.114n3004K|doi-access=free}}{{Cite journal|last1=Bilardello|first1=Marco|last2=Donadi|first2=Sandro|last3=Vinante|first3=Andrea|last4=Bassi|first4=Angelo|date=2016-11-15|title=Bounds on collapse models from cold-atom experiments|url=http://www.sciencedirect.com/science/article/pii/S0378437116304095|journal=Physica A: Statistical Mechanics and Its Applications|language=en|volume=462|pages=764–782|doi=10.1016/j.physa.2016.06.134|issn=0378-4371|arxiv=1605.01891|bibcode=2016PhyA..462..764B|s2cid=55562244}} and bulk materials, as Bravais lattices,{{Cite journal|last=Bahrami|first=M.|date=2018-05-18|title=Testing collapse models by a thermometer|journal=Physical Review A|volume=97|issue=5|pages=052118|doi=10.1103/PhysRevA.97.052118|arxiv=1801.03636|bibcode=2018PhRvA..97e2118B}} low temperature experiments,{{Cite journal|last1=Adler|first1=Stephen L.|last2=Vinante|first2=Andrea|date=2018-05-18|title=Bulk heating effects as tests for collapse models|journal=Physical Review A|volume=97|issue=5|pages=052119|doi=10.1103/PhysRevA.97.052119|arxiv=1801.06857|bibcode=2018PhRvA..97e2119A|s2cid=51687442}} neutron stars{{Cite journal|last1=Adler|first1=Stephen L.|last2=Bassi|first2=Angelo|last3=Carlesso|first3=Matteo|last4=Vinante|first4=Andrea|date=2019-05-10|title=Testing continuous spontaneous localization with Fermi liquids|journal=Physical Review D|volume=99|issue=10|pages=103001|doi=10.1103/PhysRevD.99.103001|arxiv=1901.10963|bibcode=2019PhRvD..99j3001A|doi-access=free}}{{Cite journal|last1=Tilloy|first1=Antoine|last2=Stace|first2=Thomas M.|date=2019-08-21|title=Neutron Star Heating Constraints on Wave-Function Collapse Models|journal=Physical Review Letters|volume=123|issue=8|pages=080402|doi=10.1103/PhysRevLett.123.080402|pmid=31491197|arxiv=1901.05477|bibcode=2019PhRvL.123h0402T|s2cid=119272121}} and planets

• 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{{Cite journal|last=Romero-Isart|first=Oriol|date=2011-11-28|title=Quantum superposition of massive objects and collapse models|journal=Physical Review A|volume=84|issue=5|pages=052121|doi=10.1103/PhysRevA.84.052121|arxiv=1110.4495|bibcode=2011PhRvA..84e2121R|s2cid=118401637}}$$$$

\langle{\hat x^2}\rangle_t=\langle{\hat x^2}\rangle_t^{ (QM)}+\frac{\hbar^2\eta t^3}{3 m^2},

where $\backslash langle\{\backslash hat\; x^2\}\backslash rangle\_t^\{\; (QM)\}$ is the free quantum mechanical spread and $\backslash eta$ is the CSL diffusion constant, defined as{{Cite journal|last1=Bahrami|first1=M.|last2=Paternostro|first2=M.|last3=Bassi|first3=A.|last4=Ulbricht|first4=H.|date=2014-05-29|title=Proposal for a Noninterferometric Test of Collapse Models in Optomechanical Systems|journal=Physical Review Letters|volume=112|issue=21|pages=210404|doi=10.1103/PhysRevLett.112.210404|arxiv=1402.5421|bibcode=2014PhRvL.112u0404B|s2cid=53337065}}{{Cite journal|last1=Nimmrichter|first1=Stefan|last2=Hornberger|first2=Klaus|last3=Hammerer|first3=Klemens|date=2014-07-10|title=Optomechanical Sensing of Spontaneous Wave-Function Collapse|journal=Physical Review Letters|volume=113|issue=2|pages=020405|doi=10.1103/PhysRevLett.113.020405|pmid=25062146|arxiv=1405.2868|bibcode=2014PhRvL.113b0405N|hdl=11858/00-001M-0000-0024-7705-F|s2cid=13151177|hdl-access=free}}{{Cite journal|last=Diósi|first=Lajos|date=2015-02-04|title=Testing Spontaneous Wave-Function Collapse Models on Classical Mechanical Oscillators|journal=Physical Review Letters|volume=114|issue=5|pages=050403|doi=10.1103/PhysRevLett.114.050403|pmid=25699424|arxiv=1411.4341|bibcode=2015PhRvL.114e0403D|s2cid=14609818}}$$$$

\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,

where the motion is assumed to occur along the $x$ axis; $\backslash tilde\; \backslash mu(\{\backslash bf\; k\})$ is the Fourier transform of the mass density $\backslash mu(\{\backslash bf\; r\})$. In experiments, such an increase is limited by the dissipation rate $\backslash gamma$. Assuming that the experiment is performed at temperature $T$, a particle of mass $m$, harmonically trapped at frequency $\backslash omega\_0$, at equilibrium reaches a spread in position given by{{Cite journal|last1=Vinante|first1=A.|last2=Bahrami|first2=M.|last3=Bassi|first3=A.|last4=Usenko|first4=O.|last5=Wijts|first5=G.|last6=Oosterkamp|first6=T. H.|date=2016-03-02|title=Upper Bounds on Spontaneous Wave-Function Collapse Models Using Millikelvin-Cooled Nanocantilevers|journal=Physical Review Letters|volume=116|issue=9|pages=090402|doi=10.1103/PhysRevLett.116.090402|pmid=26991158|arxiv=1510.05791|bibcode=2016PhRvL.116i0402V|hdl=1887/46827|s2cid=10215308|hdl-access=free}}{{Cite journal|last1=Carlesso|first1=Matteo|last2=Paternostro|first2=Mauro|last3=Ulbricht|first3=Hendrik|last4=Vinante|first4=Andrea|last5=Bassi|first5=Angelo|date=2018-08-17|title=Non-interferometric test of the continuous spontaneous localization model based on rotational optomechanics|journal=New Journal of Physics|language=en|volume=20|issue=8|pages=083022|doi=10.1088/1367-2630/aad863|arxiv=1708.04812|bibcode=2018NJPh...20h3022C|issn=1367-2630|doi-access=free}}$$$$

\langle{\hat x^2}\rangle_{ eq}=\frac{k_B T}{m\omega_0^2}+\frac{ \hbar^2 \eta}{2m^2 \omega_0^2 \gamma },

where $$

k_B

is the Boltzmann constant. Several experiments can test such a spread. They range from cold atom free expansion, nano-cantilevers cooled to millikelvin temperatures,{{Cite journal|last1=Vinante|first1=A.|last2=Mezzena|first2=R.|last3=Falferi|first3=P.|last4=Carlesso|first4=M.|last5=Bassi|first5=A.|date=2017-09-12|title=Improved Noninterferometric Test of Collapse Models Using Ultracold Cantilevers|journal=Physical Review Letters|volume=119|issue=11|pages=110401|doi=10.1103/PhysRevLett.119.110401|pmid=28949215|arxiv=1611.09776|bibcode=2017PhRvL.119k0401V|hdl=11368/2910142|s2cid=40171091|hdl-access=free}}{{Cite journal|last1=Carlesso|first1=Matteo|last2=Vinante|first2=Andrea|last3=Bassi|first3=Angelo|date=2018-08-17|title=Multilayer test masses to enhance the collapse noise|journal=Physical Review A|volume=98|issue=2|pages=022122|doi=10.1103/PhysRevA.98.022122|arxiv=1805.11037|bibcode=2018PhRvA..98b2122C|s2cid=51689393}}{{Cite journal|last1=Vinante|first1=A.|last2=Carlesso|first2=M.|last3=Bassi|first3=A.|last4=Chiasera|first4=A.|last5=Varas|first5=S.|last6=Falferi|first6=P.|last7=Margesin|first7=B.|last8=Mezzena|first8=R.|last9=Ulbricht|first9=H.|date=2020-09-03|title=Narrowing the Parameter Space of Collapse Models with Ultracold Layered Force Sensors|url=https://link.aps.org/doi/10.1103/PhysRevLett.125.100404|journal=Physical Review Letters|volume=125|issue=10|pages=100404|doi=10.1103/PhysRevLett.125.100404|pmid=32955323|arxiv=2002.09782|bibcode=2020PhRvL.125j0404V|s2cid=211258654}} gravitational wave detectors,{{Cite journal|last1=Carlesso|first1=Matteo|last2=Bassi|first2=Angelo|last3=Falferi|first3=Paolo|last4=Vinante|first4=Andrea|date=2016-12-23|title=Experimental bounds on collapse models from gravitational wave detectors|journal=Physical Review D|volume=94|issue=12|pages=124036|doi=10.1103/PhysRevD.94.124036|arxiv=1606.04581|bibcode=2016PhRvD..94l4036C|hdl=11368/2889661|s2cid=73690869|hdl-access=free}}{{Cite journal|last1=Helou|first1=Bassam|last2=Slagmolen|first2=B. J. J.|last3=McClelland|first3=David E.|last4=Chen|first4=Yanbei|date=2017-04-28|title=LISA pathfinder appreciably constrains collapse models|journal=Physical Review D|volume=95|issue=8|pages=084054|doi=10.1103/PhysRevD.95.084054|arxiv=1606.03637|bibcode=2017PhRvD..95h4054H|doi-access=free}} levitated optomechanics,{{Cite journal|last1=Zheng|first1=Di|last2=Leng|first2=Yingchun|last3=Kong|first3=Xi|last4=Li|first4=Rui|last5=Wang|first5=Zizhe|last6=Luo|first6=Xiaohui|last7=Zhao|first7=Jie|last8=Duan|first8=Chang-Kui|last9=Huang|first9=Pu|last10=Du|first10=Jiangfeng|last11=Carlesso|first11=Matteo|date=2020-01-17|title=Room temperature test of the continuous spontaneous localization model using a levitated micro-oscillator|journal=Physical Review Research|volume=2|issue=1|pages=013057|doi=10.1103/PhysRevResearch.2.013057|arxiv=1907.06896|bibcode=2020PhRvR...2a3057Z|doi-access=free}}{{cite journal|last1=Pontin|first1=A.|last2=Bullier|first2=N. P.|last3=Toroš|first3=M.|last4=Barker|first4=P. F.|title=An ultra-narrow line width levitated nano-oscillator for testing dissipative wavefunction collapse|journal=Physical Review Research|year=2020|volume=2|issue=2|page=023349|doi=10.1103/PhysRevResearch.2.023349|arxiv=1907.06046|bibcode=2020PhRvR...2b3349P|s2cid=196623361}}{{Cite journal|last1=Vinante|first1=A.|last2=Pontin|first2=A.|last3=Rashid|first3=M.|last4=Toroš|first4=M.|last5=Barker|first5=P. F.|last6=Ulbricht|first6=H.|date=2019-07-16|title=Testing collapse models with levitated nanoparticles: Detection challenge|journal=Physical Review A|volume=100|issue=1|pages=012119|doi=10.1103/PhysRevA.100.012119|arxiv=1903.08492|bibcode=2019PhRvA.100a2119V|s2cid=84846811}} torsion pendula.{{Cite journal|last1=Komori|first1=Kentaro|last2=Enomoto|first2=Yutaro|last3=Ooi|first3=Ching Pin|last4=Miyazaki|first4=Yuki|last5=Matsumoto|first5=Nobuyuki|last6=Sudhir|first6=Vivishek|last7=Michimura|first7=Yuta|last8=Ando|first8=Masaki|date=2020-01-17|title=Attonewton-meter torque sensing with a macroscopic optomechanical torsion pendulum|journal=Physical Review A|volume=101|issue=1|pages=011802|doi=10.1103/PhysRevA.101.011802|arxiv=1907.13139|bibcode=2020PhRvA.101a1802K|hdl=1721.1/125376|s2cid=214317541 |hdl-access=free}}

The CSL model consistently describes the collapse mechanism as a dynamical process. It has, however, two weak points.

• CSL does not conserve the energy of isolated systems. Although this increase is small, it is an unpleasant feature for a phenomenological model. The dissipative extensions of the CSL model{{Cite journal|last1=Smirne|first1=Andrea|last2=Bassi|first2=Angelo|date=2015-08-05|title=Dissipative Continuous Spontaneous Localization (CSL) model|journal=Scientific Reports|language=en|volume=5|issue=1|page=12518|doi=10.1038/srep12518|pmid=26243034|pmc=4525142|arxiv=1408.6446|bibcode=2015NatSR...512518S|issn=2045-2322|doi-access=free}}{{Cite journal |last1=Di Bartolomeo |first1=Giovanni |last2=Carlesso |first2=Matteo |last3=Piscicchia |first3=Kristian |last4=Curceanu |first4=Catalina |last5=Derakhshani |first5=Maaneli |last6=Diósi |first6=Lajos |date=2023-07-06 |title=Linear-friction many-body equation for dissipative spontaneous wave-function collapse |url=https://link.aps.org/doi/10.1103/PhysRevA.108.012202 |journal=Physical Review A |language=en |volume=108 |issue=1 |page=012202 |doi=10.1103/PhysRevA.108.012202 |issn=2469-9926|arxiv=2301.07661 }} gives a remedy. One associates to the collapse noise a finite temperature $T\_\{\; CSL\}$ at which the system eventually thermalizes.{{clarify|What does this mean?|date=August 2020}} Thus, as an example, for a free point-like particle of mass $m$ in three dimensions, the energy evolution in Ref. is described by$$$$

E(t)=e^{-\beta t}(E(0)-E_{ as})+E_{ as},

where $$

E_{ as}=\tfrac32 k_B T_{ CSL}, $\backslash beta=4\; \backslash chi\; \backslash lambda\; /(1+\backslash chi)^5$ and $\backslash chi=\backslash hbar^2/(8\; m\_0\; k\_B\; T\_\{\; CSL\}r\_C^2)$. Assuming that the CSL noise has a cosmological origin (which is reasonable due to its supposed universality), a plausible value such a temperature is $T\_\{\; CSL\}=1$ K, although only experiments can indicate a definite value. Several interferometric and non-interferometric{{Cite journal|last1=Nobakht|first1=J.|last2=Carlesso|first2=M.|last3=Donadi|first3=S.|last4=Paternostro|first4=M.|last5=Bassi|first5=A.|date=2018-10-08|title=Unitary unraveling for the dissipative continuous spontaneous localization model: Application to optomechanical experiments|journal=Physical Review A|volume=98|issue=4|pages=042109|doi=10.1103/PhysRevA.98.042109|arxiv=1808.01143|bibcode=2018PhRvA..98d2109N|hdl=11368/2929989|s2cid=51959822|hdl-access=free}}{{Cite journal |last1=Di Bartolomeo |first1=Giovanni |last2=Carlesso |first2=Matteo |date=2024-04-01 |title=Experimental bounds on linear-friction dissipative collapse models from levitated optomechanics |url=https://iopscience.iop.org/article/10.1088/1367-2630/ad3842 |journal=New Journal of Physics |volume=26 |issue=4 |pages=043006 |doi=10.1088/1367-2630/ad3842 |issn=1367-2630|arxiv=2401.04665 }} tests bound the CSL parameter space for different choices of $T\_\{CSL\}$.

• 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 $w\_t(\{\backslash bf\; x\})$, whose correlation is proportional to a Dirac delta in time, a non-white noise is considered, which is characterized by a non-trivial temporal correlation function $f(t)$. The effect can be quantified by a rescaling of $F\_\{\{\; CSL\}\}(k,q,t)$, which becomes$$$$

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],

where $\backslash bar\backslash tau=\backslash int\_0^t\backslash operatorname\{d\}\backslash !\; s\backslash ,f(s)$. As an example, one can consider an exponentially decaying noise, whose time correlation function can be of the form{{Cite journal|last1=Carlesso|first1=Matteo|last2=Ferialdi|first2=Luca|last3=Bassi|first3=Angelo|date=2018-09-18|title=Colored collapse models from the non-interferometric perspective|journal=The European Physical Journal D|language=en|volume=72|issue=9|pages=159|doi=10.1140/epjd/e2018-90248-x|arxiv=1805.10100|bibcode=2018EPJD...72..159C|issn=1434-6079|doi-access=free}} $f(t)=\backslash tfrac12\backslash Omega\_\{\; C\}e^\{-\backslash Omega\_\{\; C\}|t|\}$. In such a way, one introduces a frequency cutoff $\backslash Omega\_\{C\}$, whose inverse describes the time scale of the noise correlations. The parameter $\backslash Omega\_\{\; C\}$ works now as the third parameter of the colored CSL model together with $\backslash lambda$ and $r\_C$. Assuming a cosmological origin of the noise, a reasonable guess is{{Cite journal|last1=Bassi|first1=A.|last2=Deckert|first2=D.-A.|last3=Ferialdi|first3=L.|date=2010-12-01|title=Breaking quantum linearity: Constraints from human perception and cosmological implications|journal=EPL (Europhysics Letters)|language=en|volume=92|issue=5|pages=50006|doi=10.1209/0295-5075/92/50006|issn=0295-5075|arxiv=1011.3767|bibcode=2010EL.....9250006B|s2cid=119186239}} $\backslash Omega\_\{\; C\}=10^\{12\}\backslash ,$Hz. As for the dissipative extension, experimental bounds were obtained for different values of $\backslash Omega\_\{\; C\}$: they include interferometric and non-interferometric tests.

{{Reflist}}

Category:Interpretations of quantum mechanics

Category:Quantum measurement