Physics:Continuous spontaneous localization model

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Short description: Quantum mechanical theory of spontaneous collapse

The continuous spontaneous localization (CSL) model is a spontaneous collapse model in quantum mechanics, proposed in 1989 by Philip Pearle.[1] and finalized in 1990 Gian Carlo Ghirardi, Philip Pearle and Alberto Rimini.[2]

Introduction

The most widely studied among the dynamical reduction (also known as collapse) models is the CSL model.[1][2][3] Building on the Ghirardi-Rimini-Weber model,[4] 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:[3]

  • 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: [math]\displaystyle{ \lambda }[/math] and [math]\displaystyle{ r_C }[/math], which are respectively the collapse rate and the correlation length of the model.

Dynamical equation

The CSL dynamical equation for the wave function is stochastic and non-linear:[math]\displaystyle{ \operatorname{d}\!|\psi_t\rangle=\left[ -\frac i\hbar \hat H\operatorname{d}\! t+\frac{\sqrt{\lambda}}{m_0}\int \operatorname{d}\! {\bf x}\,\hat N_t({\bf x})\operatorname{d}\! W_t({\bf x})\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. }[/math]Here [math]\displaystyle{ \hat H }[/math] is the Hamiltonian describing the quantum mechanical dynamics, [math]\displaystyle{ m_0 }[/math] is a reference mass taken equal to that of a nucleon, [math]\displaystyle{ g({\bf x}-{\bf y})=e^{-{({\bf x}-{\bf y})^2}/{4r_C^2}} }[/math], and the noise field [math]\displaystyle{ w_t({\bf x})=\operatorname{d}\! W_t({\bf x})/\operatorname{d}\! t }[/math] has zero average and correlation equal to[math]\displaystyle{ \mathbb E[w_t({\bf x}) w_s({\bf y})]=g({\bf x}-{\bf y})\delta(t-s), }[/math]where [math]\displaystyle{ \mathbb E [\ \cdot\ ] }[/math] denotes the stochastic average over the noise. Finally, we write[math]\displaystyle{ \hat N_t({\bf x})=\hat M({\bf x})-\langle\psi_t|\hat M({\bf x})|\psi_t\rangle, }[/math]where [math]\displaystyle{ \hat M({\bf x}) }[/math] is the mass density operator, which reads[math]\displaystyle{ \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]\displaystyle{ \hat a^\dagger_j({\bf y},s) }[/math] and [math]\displaystyle{ \hat a_j({\bf y},s) }[/math] are, respectively, the second quantized creation and annihilation operators of a particle of type [math]\displaystyle{ j }[/math] with spin [math]\displaystyle{ s }[/math] at the point [math]\displaystyle{ {\bf y} }[/math] of mass [math]\displaystyle{ 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]\displaystyle{ \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]\displaystyle{ \lambda }[/math] and [math]\displaystyle{ r_C }[/math]. Originally, the Ghirardi-Rimini-Weber model[4] proposed [math]\displaystyle{ \lambda=10^{-17}\, }[/math]s[math]\displaystyle{ ^{-1} }[/math] at [math]\displaystyle{ r_C=10^{-7}\, }[/math]m, while later Adler considered larger values:[5] [math]\displaystyle{ \lambda=10^{-8\pm2}\, }[/math]s[math]\displaystyle{ ^{-1} }[/math] for [math]\displaystyle{ r_C=10^{-7}\, }[/math]m, and [math]\displaystyle{ \lambda=10^{-6\pm2}\, }[/math]s[math]\displaystyle{ ^{-1} }[/math] for [math]\displaystyle{ r_C=10^{-6}\, }[/math]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 [math]\displaystyle{ \hat \rho_t }[/math]:[math]\displaystyle{ \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]. }[/math]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 [math]\displaystyle{ m }[/math], it reads[math]\displaystyle{ \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, }[/math]where the off-diagonal terms, which have [math]\displaystyle{ {\bf x}\neq{\bf y} }[/math], decay exponentially. Conversely, the diagonal terms, characterized by [math]\displaystyle{ {\bf x}={\bf y} }[/math], are preserved. For a composite system, the single-particle collapse rate [math]\displaystyle{ \lambda }[/math] should be replaced with that of the composite system[math]\displaystyle{ \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}, }[/math]where [math]\displaystyle{ \tilde \mu(k) }[/math] is the Fourier transform of the mass density of the system.

Experimental tests

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

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[6][math]\displaystyle{ \rho(x,x',t) = \frac{1}{2\pi\hbar} \int_{-\infty}^{+ \infty} \operatorname{d}\! k \int_{-\infty}^{+ \infty} \operatorname{d}\! w\, e^{-ikw/\hbar} F_{{ CSL}}(k, x-x',t) \rho^{{ QM}}(x+w, x'+w, t), }[/math]where [math]\displaystyle{ \rho^{{QM}} }[/math] denotes the statistical operator described by quantum mechanics, and we define[math]\displaystyle{ F_{{ CSL}}(k,q,t)= \exp\bigg[-\lambda \frac{m^2}{m_0^2} t \left(1-\frac{1}{t}\int_0^t \operatorname{d}\!\tau \,e^{-{(q-\frac{k\tau}{m})^2}/{4r_C^2}} \right) \bigg]. }[/math]Experiments testing such a reduction of the interference contrast are carried out with cold-atoms,[7] molecules[6][8][9][10] and entangled diamonds.[11][12]

Similarly, one can also quantify the minimum collapse strength to solve the measurement problem at the macroscopic level. Specifically, an estimate[6] can be obtained by requiring that a superposition of a single-layered graphene disk of radius [math]\displaystyle{ \simeq 10^{-5} }[/math]m collapses in less than [math]\displaystyle{ \simeq 10^{-2} }[/math]s.

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:[13]

  • 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]\displaystyle{ \omega }[/math] for a particle of charge [math]\displaystyle{ Q }[/math] is given by:[14][15][16][17]

[math]\displaystyle{ \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]\displaystyle{ \epsilon_0 }[/math] is the vacuum dielectric constant and [math]\displaystyle{ c }[/math] is the light speed. This prediction of CSL can be tested[18][19][20][21] 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 [math]\displaystyle{ E }[/math] of a free particle of mass [math]\displaystyle{ m }[/math] in three dimensions grows linearly in time according to[3] [math]\displaystyle{ E(t)=E(0)+\frac{3m\lambda\hbar^{2}}{4m_{0}^{2}r_C^{2}}t, }[/math]where [math]\displaystyle{ 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]\displaystyle{ \simeq 10^{-14} }[/math] K per year considering the values [math]\displaystyle{ \lambda=10^{-16} }[/math] s[math]\displaystyle{ ^{-1} }[/math] and [math]\displaystyle{ r_C=10^{-7} }[/math]m. Although small, such an energy increase can be tested by monitoring cold atoms.[22][23] and bulk materials, as Bravais lattices,[24] low temperature experiments,[25] neutron stars[26][27] and planets[26]
  • 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[28][math]\displaystyle{ \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]\displaystyle{ \langle{\hat x^2}\rangle_t^{ (QM)} }[/math] is the free quantum mechanical spread and [math]\displaystyle{ \eta }[/math] is the CSL diffusion constant, defined as[29][30][31][math]\displaystyle{ \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]\displaystyle{ x }[/math] axis; [math]\displaystyle{ \tilde \mu({\bf k}) }[/math] is the Fourier transform of the mass density [math]\displaystyle{ \mu({\bf r}) }[/math]. In experiments, such an increase is limited by the dissipation rate [math]\displaystyle{ \gamma }[/math]. Assuming that the experiment is performed at temperature [math]\displaystyle{ T }[/math], a particle of mass [math]\displaystyle{ m }[/math], harmonically trapped at frequency [math]\displaystyle{ \omega_0 }[/math], at equilibrium reaches a spread in position given by[32][33][math]\displaystyle{ \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]\displaystyle{ k_B }[/math] is the Boltzmann constant. Several experiments can test such a spread. They range from cold atom free expansion,[22][23] nano-cantilevers cooled to millikelvin temperatures,[32][34][35][36] gravitational wave detectors,[37][38] levitated optomechanics,[33][39][40][41] torsion pendula.[42]

Dissipative and colored extensions

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.[3] The dissipative extension of the CSL model[43] gives a remedy. One associates to the collapse noise a finite temperature [math]\displaystyle{ T_{ CSL} }[/math] at which the system eventually thermalizes.[clarification needed] Thus, for a free point-like particle of mass [math]\displaystyle{ m }[/math] in three dimensions, the energy evolution is described by[math]\displaystyle{ E(t)=e^{-\beta t}(E(0)-E_{ as})+E_{ as}, }[/math]where [math]\displaystyle{ E_{ as}=\tfrac32 k_B T_{ CSL} }[/math], [math]\displaystyle{ \beta=4 \chi \lambda /(1+\chi)^5 }[/math] and [math]\displaystyle{ \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]\displaystyle{ T_{ CSL}=1 }[/math] K, although only experiments can indicate a definite value. Several interferometric[6][9] and non-interferometric[23][40][44] tests bound the CSL parameter space for different choices of [math]\displaystyle{ T_{CSL} }[/math].
  • 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]\displaystyle{ w_t({\bf x}) }[/math], 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 [math]\displaystyle{ f(t) }[/math]. The effect can be quantified by a rescaling of [math]\displaystyle{ F_{{ CSL}}(k,q,t) }[/math], which becomes[math]\displaystyle{ 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]\displaystyle{ \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[45] [math]\displaystyle{ f(t)=\tfrac12\Omega_{ C}e^{-\Omega_{ C}|t|} }[/math]. In such a way, one introduces a frequency cutoff [math]\displaystyle{ \Omega_{C} }[/math], whose inverse describes the time scale of the noise correlations. The parameter [math]\displaystyle{ \Omega_{ C} }[/math] works now as the third parameter of the colored CSL model together with [math]\displaystyle{ \lambda }[/math] and [math]\displaystyle{ r_C }[/math]. Assuming a cosmological origin of the noise, a reasonable guess is[46] [math]\displaystyle{ \Omega_{ C}=10^{12}\, }[/math]Hz. As for the dissipative extension, experimental bounds were obtained for different values of [math]\displaystyle{ \Omega_{ C} }[/math]: they include interferometric[6][9] and non-interferometric[23][45] tests.

References

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  2. ↑ 2.0 2.1 Ghirardi, Gian Carlo; Pearle, Philip; Rimini, Alberto (1990-07-01). "Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles". Physical Review A 42 (1): 78–89. doi:10.1103/PhysRevA.42.78. PMID 9903779. Bibcode: 1990PhRvA..42...78G. 
  3. ↑ 3.0 3.1 3.2 3.3 Bassi, Angelo; Ghirardi, GianCarlo (2003-06-01). "Dynamical reduction models" (in en). Physics Reports 379 (5): 257–426. doi:10.1016/S0370-1573(03)00103-0. ISSN 0370-1573. Bibcode: 2003PhR...379..257B. http://www.sciencedirect.com/science/article/pii/S0370157303001030. 
  4. ↑ 4.0 4.1 Ghirardi, G. C.; Rimini, A.; Weber, T. (1986-07-15). "Unified dynamics for microscopic and macroscopic systems". Physical Review D 34 (2): 470–491. doi:10.1103/PhysRevD.34.470. PMID 9957165. Bibcode: 1986PhRvD..34..470G. 
  5. ↑ Adler, Stephen L (2007-10-16). "Lower and upper bounds on CSL parameters from latent image formation and IGM~heating" (in en). Journal of Physics A: Mathematical and Theoretical 40 (44): 13501. doi:10.1088/1751-8121/40/44/c01. ISSN 1751-8113. 
  6. ↑ 6.0 6.1 6.2 6.3 6.4 Toroš, Marko; Gasbarri, Giulio; Bassi, Angelo (2017-12-20). "Colored and dissipative continuous spontaneous localization model and bounds from matter-wave interferometry" (in en). Physics Letters A 381 (47): 3921–3927. doi:10.1016/j.physleta.2017.10.002. ISSN 0375-9601. Bibcode: 2017PhLA..381.3921T. http://www.sciencedirect.com/science/article/pii/S0375960117309465. 
  7. ↑ Kovachy, T.; Asenbaum, P.; Overstreet, C.; Donnelly, C. A.; Dickerson, S. M.; Sugarbaker, A.; Hogan, J. M.; Kasevich, M. A. (2015). "Quantum superposition at the half-metre scale" (in en). Nature 528 (7583): 530–533. doi:10.1038/nature16155. ISSN 1476-4687. PMID 26701053. Bibcode: 2015Natur.528..530K. https://www.nature.com/articles/nature16155. 
  8. ↑ Eibenberger, Sandra; Gerlich, Stefan; Arndt, Markus; Mayor, Marcel; Tüxen, Jens (2013-08-14). "Matter–wave interference of particles selected from a molecular library with masses exceeding 10 000 amu" (in en). Physical Chemistry Chemical Physics 15 (35): 14696–14700. doi:10.1039/C3CP51500A. ISSN 1463-9084. PMID 23900710. Bibcode: 2013PCCP...1514696E. 
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  10. ↑ Fein, Yaakov Y.; Geyer, Philipp; Zwick, Patrick; Kiałka, Filip; Pedalino, Sebastian; Mayor, Marcel; Gerlich, Stefan; Arndt, Markus (2019). "Quantum superposition of molecules beyond 25 kDa" (in en). Nature Physics 15 (12): 1242–1245. doi:10.1038/s41567-019-0663-9. ISSN 1745-2481. Bibcode: 2019NatPh..15.1242F. https://www.nature.com/articles/s41567-019-0663-9. 
  11. ↑ Lee, K. C.; Sprague, M. R.; Sussman, B. J.; Nunn, J.; Langford, N. K.; Jin, X.-M.; Champion, T.; Michelberger, P. et al. (2011-12-02). "Entangling Macroscopic Diamonds at Room Temperature" (in en). Science 334 (6060): 1253–1256. doi:10.1126/science.1211914. ISSN 0036-8075. PMID 22144620. Bibcode: 2011Sci...334.1253L. https://www.science.org/doi/10.1126/science.1211914. 
  12. ↑ Belli, Sebastiano; Bonsignori, Riccarda; D'Auria, Giuseppe; Fant, Lorenzo; Martini, Mirco; Peirone, Simone; Donadi, Sandro; Bassi, Angelo (2016-07-12). "Entangling macroscopic diamonds at room temperature: Bounds on the continuous-spontaneous-localization parameters". Physical Review A 94 (1): 012108. doi:10.1103/PhysRevA.94.012108. Bibcode: 2016PhRvA..94a2108B. 
  13. ↑ Carlesso, Matteo; Donadi, Sandro; Ferialdi, Luca; Paternostro, Mauro; Ulbricht, Hendrik; Bassi, Angelo (February 2022). "Present status and future challenges of non-interferometric tests of collapse models" (in en). Nature Physics 18 (3): 243–250. doi:10.1038/s41567-021-01489-5. ISSN 1745-2481. Bibcode: 2022NatPh..18..243C. https://www.nature.com/articles/s41567-021-01489-5. 
  14. ↑ Adler, Stephen L; Ramazanoğlu, Fethi M (2007-10-16). "Photon-emission rate from atomic systems in the CSL model". Journal of Physics A: Mathematical and Theoretical 40 (44): 13395–13406. doi:10.1088/1751-8113/40/44/017. ISSN 1751-8113. Bibcode: 2007JPhA...4013395A. 
  15. ↑ Bassi, Angelo; Ferialdi, Luca (2009-07-31). "Non-Markovian dynamics for a free quantum particle subject to spontaneous collapse in space: General solution and main properties". Physical Review A 80 (1): 012116. doi:10.1103/PhysRevA.80.012116. Bibcode: 2009PhRvA..80a2116B. 
  16. ↑ Adler, Stephen L; Bassi, Angelo; Donadi, Sandro (2013-06-03). "On spontaneous photon emission in collapse models". Journal of Physics A: Mathematical and Theoretical 46 (24): 245304. doi:10.1088/1751-8113/46/24/245304. ISSN 1751-8113. Bibcode: 2013JPhA...46x5304A. 
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  18. ↑ Fu, Qijia (1997-09-01). "Spontaneous radiation of free electrons in a nonrelativistic collapse model". Physical Review A 56 (3): 1806–1811. doi:10.1103/PhysRevA.56.1806. Bibcode: 1997PhRvA..56.1806F. 
  19. ↑ Morales, A.; Aalseth, C. E.; Avignone, F. T.; Brodzinski, R. L.; Cebrián, S.; Garcı́a, E.; Irastorza, I. G.; Kirpichnikov, I. V. et al. (2002-04-18). "Improved constraints on wimps from the international germanium experiment IGEX" (in en). Physics Letters B 532 (1): 8–14. doi:10.1016/S0370-2693(02)01545-9. ISSN 0370-2693. Bibcode: 2002PhLB..532....8M. 
  20. ↑ Curceanu, C.; Bartalucci, S.; Bassi, A.; Bazzi, M.; Bertolucci, S.; Berucci, C.; Bragadireanu, A. M.; Cargnelli, M. et al. (2016-03-01). "Spontaneously Emitted X-rays: An Experimental Signature of the Dynamical Reduction Models" (in en). Foundations of Physics 46 (3): 263–268. doi:10.1007/s10701-015-9923-4. ISSN 1572-9516. Bibcode: 2016FoPh...46..263C. 
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  23. ↑ 23.0 23.1 23.2 23.3 Bilardello, Marco; Donadi, Sandro; Vinante, Andrea; Bassi, Angelo (2016-11-15). "Bounds on collapse models from cold-atom experiments" (in en). Physica A: Statistical Mechanics and Its Applications 462: 764–782. doi:10.1016/j.physa.2016.06.134. ISSN 0378-4371. Bibcode: 2016PhyA..462..764B. http://www.sciencedirect.com/science/article/pii/S0378437116304095. 
  24. ↑ Bahrami, M. (2018-05-18). "Testing collapse models by a thermometer". Physical Review A 97 (5): 052118. doi:10.1103/PhysRevA.97.052118. Bibcode: 2018PhRvA..97e2118B. 
  25. ↑ Adler, Stephen L.; Vinante, Andrea (2018-05-18). "Bulk heating effects as tests for collapse models". Physical Review A 97 (5): 052119. doi:10.1103/PhysRevA.97.052119. Bibcode: 2018PhRvA..97e2119A. 
  26. ↑ 26.0 26.1 Adler, Stephen L.; Bassi, Angelo; Carlesso, Matteo; Vinante, Andrea (2019-05-10). "Testing continuous spontaneous localization with Fermi liquids". Physical Review D 99 (10): 103001. doi:10.1103/PhysRevD.99.103001. Bibcode: 2019PhRvD..99j3001A. 
  27. ↑ Tilloy, Antoine; Stace, Thomas M. (2019-08-21). "Neutron Star Heating Constraints on Wave-Function Collapse Models". Physical Review Letters 123 (8): 080402. doi:10.1103/PhysRevLett.123.080402. PMID 31491197. Bibcode: 2019PhRvL.123h0402T. 
  28. ↑ Romero-Isart, Oriol (2011-11-28). "Quantum superposition of massive objects and collapse models". Physical Review A 84 (5): 052121. doi:10.1103/PhysRevA.84.052121. Bibcode: 2011PhRvA..84e2121R. 
  29. ↑ Bahrami, M.; Paternostro, M.; Bassi, A.; Ulbricht, H. (2014-05-29). "Proposal for a Noninterferometric Test of Collapse Models in Optomechanical Systems". Physical Review Letters 112 (21): 210404. doi:10.1103/PhysRevLett.112.210404. Bibcode: 2014PhRvL.112u0404B. 
  30. ↑ Nimmrichter, Stefan; Hornberger, Klaus; Hammerer, Klemens (2014-07-10). "Optomechanical Sensing of Spontaneous Wave-Function Collapse". Physical Review Letters 113 (2): 020405. doi:10.1103/PhysRevLett.113.020405. PMID 25062146. Bibcode: 2014PhRvL.113b0405N. 
  31. ↑ Diósi, Lajos (2015-02-04). "Testing Spontaneous Wave-Function Collapse Models on Classical Mechanical Oscillators". Physical Review Letters 114 (5): 050403. doi:10.1103/PhysRevLett.114.050403. PMID 25699424. Bibcode: 2015PhRvL.114e0403D. 
  32. ↑ 32.0 32.1 Vinante, A.; Bahrami, M.; Bassi, A.; Usenko, O.; Wijts, G.; Oosterkamp, T. H. (2016-03-02). "Upper Bounds on Spontaneous Wave-Function Collapse Models Using Millikelvin-Cooled Nanocantilevers". Physical Review Letters 116 (9): 090402. doi:10.1103/PhysRevLett.116.090402. PMID 26991158. Bibcode: 2016PhRvL.116i0402V. 
  33. ↑ 33.0 33.1 Carlesso, Matteo; Paternostro, Mauro; Ulbricht, Hendrik; Vinante, Andrea; Bassi, Angelo (2018-08-17). "Non-interferometric test of the continuous spontaneous localization model based on rotational optomechanics" (in en). New Journal of Physics 20 (8): 083022. doi:10.1088/1367-2630/aad863. ISSN 1367-2630. Bibcode: 2018NJPh...20h3022C. 
  34. ↑ Vinante, A.; Mezzena, R.; Falferi, P.; Carlesso, M.; Bassi, A. (2017-09-12). "Improved Noninterferometric Test of Collapse Models Using Ultracold Cantilevers". Physical Review Letters 119 (11): 110401. doi:10.1103/PhysRevLett.119.110401. PMID 28949215. Bibcode: 2017PhRvL.119k0401V. 
  35. ↑ Carlesso, Matteo; Vinante, Andrea; Bassi, Angelo (2018-08-17). "Multilayer test masses to enhance the collapse noise". Physical Review A 98 (2): 022122. doi:10.1103/PhysRevA.98.022122. Bibcode: 2018PhRvA..98b2122C. 
  36. ↑ Vinante, A.; Carlesso, M.; Bassi, A.; Chiasera, A.; Varas, S.; Falferi, P.; Margesin, B.; Mezzena, R. et al. (2020-09-03). "Narrowing the Parameter Space of Collapse Models with Ultracold Layered Force Sensors". Physical Review Letters 125 (10): 100404. doi:10.1103/PhysRevLett.125.100404. PMID 32955323. Bibcode: 2020PhRvL.125j0404V. https://link.aps.org/doi/10.1103/PhysRevLett.125.100404. 
  37. ↑ Carlesso, Matteo; Bassi, Angelo; Falferi, Paolo; Vinante, Andrea (2016-12-23). "Experimental bounds on collapse models from gravitational wave detectors". Physical Review D 94 (12): 124036. doi:10.1103/PhysRevD.94.124036. Bibcode: 2016PhRvD..94l4036C. 
  38. ↑ Helou, Bassam; Slagmolen, B. J. J.; McClelland, David E.; Chen, Yanbei (2017-04-28). "LISA pathfinder appreciably constrains collapse models". Physical Review D 95 (8): 084054. doi:10.1103/PhysRevD.95.084054. Bibcode: 2017PhRvD..95h4054H. 
  39. ↑ Zheng, Di; Leng, Yingchun; Kong, Xi; Li, Rui; Wang, Zizhe; Luo, Xiaohui; Zhao, Jie; Duan, Chang-Kui et al. (2020-01-17). "Room temperature test of the continuous spontaneous localization model using a levitated micro-oscillator". Physical Review Research 2 (1): 013057. doi:10.1103/PhysRevResearch.2.013057. Bibcode: 2020PhRvR...2a3057Z. 
  40. ↑ 40.0 40.1 Pontin, A.; Bullier, N. P.; Toroš, M.; Barker, P. F. (2020). "An ultra-narrow line width levitated nano-oscillator for testing dissipative wavefunction collapse". Physical Review Research 2 (2): 023349. doi:10.1103/PhysRevResearch.2.023349. Bibcode: 2020PhRvR...2b3349P. 
  41. ↑ Vinante, A.; Pontin, A.; Rashid, M.; Toroš, M.; Barker, P. F.; Ulbricht, H. (2019-07-16). "Testing collapse models with levitated nanoparticles: Detection challenge". Physical Review A 100 (1): 012119. doi:10.1103/PhysRevA.100.012119. Bibcode: 2019PhRvA.100a2119V. 
  42. ↑ Komori, Kentaro; Enomoto, Yutaro; Ooi, Ching Pin; Miyazaki, Yuki; Matsumoto, Nobuyuki; Sudhir, Vivishek; Michimura, Yuta; Ando, Masaki (2020-01-17). "Attonewton-meter torque sensing with a macroscopic optomechanical torsion pendulum". Physical Review A 101 (1): 011802. doi:10.1103/PhysRevA.101.011802. Bibcode: 2020PhRvA.101a1802K. 
  43. ↑ Smirne, Andrea; Bassi, Angelo (2015-08-05). "Dissipative Continuous Spontaneous Localization (CSL) model" (in en). Scientific Reports 5 (1): 12518. doi:10.1038/srep12518. ISSN 2045-2322. PMID 26243034. Bibcode: 2015NatSR...512518S. 
  44. ↑ Nobakht, J.; Carlesso, M.; Donadi, S.; Paternostro, M.; Bassi, A. (2018-10-08). "Unitary unraveling for the dissipative continuous spontaneous localization model: Application to optomechanical experiments". Physical Review A 98 (4): 042109. doi:10.1103/PhysRevA.98.042109. Bibcode: 2018PhRvA..98d2109N. 
  45. ↑ 45.0 45.1 Carlesso, Matteo; Ferialdi, Luca; Bassi, Angelo (2018-09-18). "Colored collapse models from the non-interferometric perspective" (in en). The European Physical Journal D 72 (9): 159. doi:10.1140/epjd/e2018-90248-x. ISSN 1434-6079. Bibcode: 2018EPJD...72..159C. 
  46. ↑ Bassi, A.; Deckert, D.-A.; Ferialdi, L. (2010-12-01). "Breaking quantum linearity: Constraints from human perception and cosmological implications" (in en). EPL (Europhysics Letters) 92 (5): 50006. doi:10.1209/0295-5075/92/50006. ISSN 0295-5075. Bibcode: 2010EL.....9250006B. 



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