Continuous spontaneous localization model

The Continuous Spontaneous Localization (CSL) model is a spontaneous collapse model in quantum mechanics, proposed in 1989 by Philip Pearle and finalized in 1990 Gian Carlo Ghirardi, Philip Pearle and Alberto Rimini.

Introduction

The most widely studied among the dynamical reduction (also known as collapse) models is the Continuous Spontaneous Localization (CSL) model. The model works as a paradigm of collapse models, accounting for all the desired features and overcoming the difficulties faced by the Ghirardi-Rimini-Weber (GRW) model. The CSL model describes the collapse as occurring continuously in time, in contrast to GRW.

The main features of the model are:

• The localization takes place in position, which is the preferred basis.
• The model does not alter the dynamics of the microscopic system, while it acts firmly on macroscopic objects: the amplification mechanism ensures this scaling.
• It preserves the symmetry properties of identical particles, in distinction with the GRW model.
• It is characterized by two parameters: $\lambda$ and $r_{C}$, which are respectively the collapse rate and the correlation length of the model.

Dynamical equation

The CSL dynamical equation for the wavefunction is stochastic and non-linear in the wavefunction; it reads$$\operatorname {d} \!|\psi _{t}\rangle =\left[-{\frac {i}{\hbar }}{\hat {H}}\operatorname {d} \!t+{\frac {\sqrt {\lambda }}{m_{0}}}\int \operatorname {d} \!{\bf {x}}\,{\hat {N}}_{t}({\bf {x}})\operatorname {d} \!W_{t}({\bf {x}})\right.\left.-{\frac {\lambda }{2m_{0}^{2}}}\int \operatorname {d} \!{\bf {x}}\int \operatorname {d} \!{\bf {y}}\,g({\bf {x}}-{\bf {y}}){\hat {N}}_{t}({\bf {x}}){\hat {N}}_{t}({\bf {y}})\operatorname {d} \!t\right]|\psi _{t}\rangle ,$$where ${\hat {H}}$ is the Hamiltonian describing the quantum mechanical dynamics, $m_{0}$ is a reference mass taken equal to that of a nucleon, $g({\bf {x}}-{\bf {y}})=e^{-{({\bf {x}}-{\bf {y}})^{2}}/{4r_{C}^{2}}}$, and the noise field $w_{t}({\bf {x}})=\operatorname {d} \!W_{t}({\bf {x}})/\operatorname {d} \!t$ 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 $\mathbb {E} [\ \cdot \ ]$ denotes the stochastic average over the noise. Finally, we introduced$${\hat {N}}_{t}({\bf {x}})={\hat {M}}({\bf {x}})-\langle \psi _{t}|{\hat {M}}({\bf {x}})|\psi _{t}\rangle ,$$where ${\hat {M}}({\bf {x}})$ is the mass density operator, which reads$${\hat {M}}({\bf {x}})=\sum _{j}m_{j}\sum _{s}{\hat {a}}_{j}^{\dagger }({\bf {x}},s){\hat {a}}_{j}({\bf {x}},s),$$where ${\hat {a}}_{j}^{\dagger }({\bf {y}},s)$ and ${\hat {a}}_{j}({\bf {y}},s)$ are, respectively, the second quantized creation and annihilation operators of a particle of type $j$ with spin $s$ at the point ${\bf {y}}$ of mass $m_{j}$. The use of these operators satisfies the conservation of the symmetrical properties of identical particles. Moreover, the mass proportionality implements autometically the amplification mechanism. The choice of the form of ${\hat {M}}({\bf {x}})$ ensures the collapse in the position basis.

The action of the CSL model is quantified by the values of the two phenomenological parameters $\lambda$ and $r_{C}$. Originally, GRW proposed $\lambda =10^{-17}\,$s$^{-1}$ at $r_{C}=10^{-7}\,$m, while later Adler considered larger values: $\lambda =10^{-8\pm 2}\,$s$^{-1}$ for $r_{C}=10^{-7}\,$m, and $\lambda =10^{-6\pm 2}\,$s$^{-1}$ for $r_{C}=10^{-6}\,$m. Eventually these values are bounded by experiments.

From the dynamics of the wavefunction one can obtain the corresponding master equation for the statistical operator ${\hat {\rho }}_{t}$:$${\frac {\operatorname {d} \!{\hat {\rho }}_{t}}{\operatorname {d} \!t}}=-{\frac {i}{\hbar }}\left[{\hat {H}},{{\hat {\rho }}_{t}}\right]-{\frac {\lambda }{2m_{0}^{2}}}\int \operatorname {d} \!{\bf {x}}\int \operatorname {d} \!{\bf {y}}\,g({\bf {x}}-{\bf {y}})\left[{{\hat {M}}({\bf {x}})},\left[{{{\hat {M}}({\bf {y}})},{{\hat {\rho }}_{t}}}\right]\right].$$Once the master equation is represented in the position basis, it becomes clear that its direct action is that of diagonalizing the state in position. In the case of 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 ${\bf {x}}\neq {\bf {y}}$, decay exponentially. Conversely, the diagonal terms, characterized by ${\bf {x}}={\bf {y}}$, are preserved. For a composite system, the single-particle collapse rate $\lambda$ should be substitute with that of the composite system$$\lambda {\frac {m^{2}}{m_{0}^{2}}}\to \lambda {\frac {r_{C}^{3}}{\pi ^{3/2}m_{0}^{2}}}\int \operatorname {d} \!{\bf {k}}|{\tilde {\mu }}({\bf {k}})|^{2}e^{-k^{2}r_{C}^{2}},$$where ${\tilde {\mu }}(k)$ is the Fourier transform of the mass density of the system.

Experimental tests

Contrary to other solutions of the measurement problem, collapse models are experimentally verifiable. We divide the experiments testing the CSL model 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 collapse models, which is the collapse of the wavefunction. They include all the 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$$\rho (x,x',t)={\frac {1}{2\pi \hbar }}\int _{-\infty }^{+\infty }\operatorname {d} \!k\int _{-\infty }^{+\infty }\operatorname {d} \!w\,e^{-ikw/\hbar }F_{CSL}(k,x-x',t)\rho ^{QM}(x+w,x'+w,t),$$where ${\textstyle \rho ^{QM}}$ denotes the statistical operator described by quantum mechanics, and we define$$F_{CSL}(k,q,t)=\exp {\bigg [}-\lambda {\frac {m^{2}}{m_{0}^{2}}}t\left(1-{\frac {1}{t}}\int _{0}^{t}\operatorname {d} \!\tau \,e^{-{(q-{\frac {k\tau }{m}})^{2}}/{4r_{C}^{2}}}\right){\bigg ]}.$$Among the experiments testing such a reduction of the interference contrast, we count those with cold-atoms, with molecules and with entangled diamonds.

Similarly, one can also quantify the minimum collapse strength to actually 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 $\simeq 10^{-5}$m collapses in less than $\simeq 10^{-2}$s.

Non-interferometric experiments

Non-interferometric experiments consist in all the collapse models testings which are not based on the preparation of a superposition. They exploit an indirect effect of collapse models, which consists in a Brownian motion induced by the interaction with the collapse noise. The effect of this noise is an effective stochastic force acting on the system, and several experiments can be design to quantify such a force. Among them we can count:

• Radiation emission from charged particles:

If the particle is electrically charged, the action of the coupling with the collapse noise will induce a radiation emission. This result is in net contrast with the predictions of quantum mechanics, where no radiation is expected. The predicted CSL-induced emission rate at frequency $\omega$ for a particle of charge $Q$ is given by:$${\frac {\operatorname {d} \!\Gamma (\omega )}{\operatorname {d} \!\omega }}={\frac {\hbar Q^{2}\lambda }{2\pi ^{2}\epsilon _{0}c^{3}m_{0}^{2}r_{C}^{2}\omega }},$$where $\epsilon _{0}$ is the vacuum dielectric constant and $c$ is the light speed. The CSL predictions are tested analyzing the X-ray emission spectrum from a bulk Germanium test mass.

• Heating in bulk materials

One of the predictions 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 actually small; for example, an hydrogen atom is heated by $\simeq 10^{-14}$K per year considering the values $\lambda =10^{-16}$s$^{-1}$ and $r_{C}=10^{-7}$m. Although small, such a potential energy increase can be tested considering the heating effect in cold atoms and bulk materials, as Bravais lattices, low temperature experiments, neutron stars and planets.

• Diffusive effects

Another prediction of the CSL model is the increase of the center-of-mass position spread of a system. For a free particle, the position spread in one dimension reads$$\langle {{\hat {x}}^{2}}\rangle _{t}=\langle {{\hat {x}}^{2}}\rangle _{t}^{(QM)}+{\frac {\hbar ^{2}\eta t^{3}}{3m^{2}}},$$where $\langle {{\hat {x}}^{2}}\rangle _{t}^{(QM)}$ is the free quantum mechanical spread and $\eta$ is the CSL diffusion constant, defined as$$\eta ={\frac {\lambda r_{C}^{3}}{2\pi ^{3/2}m_{0}^{2}}}\int \operatorname {d} \!{\bf {k}}\,e^{-{\bf {k}}^{2}r_{C}^{2}}k_{x}^{2}|{\tilde {\mu }}({\bf {k}})|^{2},$$where we considered the motion along the $x$ axis and denoted with ${\tilde {\mu }}({\bf {k}})$ the Fourier transform of the mass density $\mu ({\bf {r}})$. In an experiment, such an increase is limited by the dissipation rate $\gamma$. Assuming that the experiment is performed at temperature $T$, a particle, which has mass $m$ and it is harmonically trapped at frequency $\omega _{0}$, at the equilibrium reaches a position spread given by$$\langle {{\hat {x}}^{2}}\rangle _{eq}={\frac {k_{B}T}{m\omega _{0}^{2}}}+{\frac {\hbar ^{2}\eta }{2m^{2}\omega _{0}^{2}\gamma }},$$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, gravitational wave detectors, levitated optomechanics, torsion pendulum.

Dissipative and colored extensions

The CSL model describes coherently the collapse mechanism. It has, however, two week points.

• CSL does not conserve energy at long time-scales

Although this increase is small, it is an unexpected feature also for a phenomenological model. The dissipative extension of the CSL model gives a remedy. One associates to the collapse noise a finite temperature $T_{CSL}$ at which the system will eventually termalize. Thus, for a free point-like particle of mass $m$ in three dimensions, the energy evolution is described by$$E(t)=e^{-\beta t}(E(0)-E_{as})+E_{as},$$where $E_{as}={\tfrac {3}{2}}k_{B}T_{CSL}$, $\beta =4\chi \lambda /(1+\chi )^{5}$ and $\chi =\hbar ^{2}/(8m_{0}k_{B}T_{CSL}r_{C}^{2})$. 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 tests bounded the CSL parameter space for different choices of $T_{CSL}$.

• The CSL noise spectrum is white

If one assumes a physical origin to the CSL noise, then its spectrum cannot be white but colored. In particular, in place of the white noise $w_{t}({\bf {x}})$, whose correlation is proportional to a Dirac delta in time, we now have a non-white noise characterized by a non-trivial temporal correlation function $f(t)$ of the noise. The effect can be quantified by a rescaling of $F_{CSL}(k,q,t)$, which becomes$$F_{cCSL}(k,q,t)=F_{CSL}(k,q,t)\exp \left[{\frac {\lambda {\bar {\tau }}}{2}}\left(e^{-(q-kt/m)^{2}/4r_{C}^{2}}-e^{-q^{2}/4r_{C}^{2}}\right)\right],$$where ${\bar {\tau }}=\int _{0}^{t}\operatorname {d} \!s\,f(s)$. As an example, one can consider an exponentially decaying noise, whose time correlation function can be of the form $f(t)={\tfrac {1}{2}}\Omega _{C}e^{-\Omega _{C}|t|}$. In such a way, one introduces a frequency cutoff $\Omega _{C}$, whose inverse describes the time scale of the noise correlations. The parameter $\Omega _{C}$ works now as the third parameter of the colored CSL model together with $\lambda$ and $r_{C}$. Assuming a cosmological origin of the noise, a reasonable guess is $\Omega _{C}=10^{12}\,$Hz. As for the dissipative extension, experimental bounds were obtained for different values of $\Omega _{C}$: they comprehend interferometric and non-interferometric tests.

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Category:Interpretations of quantum mechanics Category:Quantum measurement