Collapse-Model Tests

Collapse-model tests compare measured coherence or noise with a quantitative stochastic modification of quantum dynamics. They constrain parameters of a specified theory, not the word “collapse” in every interpretation. Standard conditional state update is not itself a source of spontaneous heating. This page develops stable inference methods; changing best bounds require a dated assessment of the chosen model and data.

Helpful background. Objective Collapse Theories explains the dynamical motivation.

A model must predict more than lost visibility

Localization theories typically introduce a rate and a length scale. Their stochastic pure-state evolution is arranged so that the ensemble density operator has a consistent evolution. An illustrative single-particle localization kernel is

$$\left.\partial_t\rho(x,x')\right|_{\rm loc} =-\lambda\left[1-e^{-(x-x')^2/(4r_C^2)}\right]\rho(x,x').$$

Here $\lambda$ is the saturated coherence-decay rate for this example and $r_C$ is its localization length. This is not the complete mass-density CSL master equation. Many-particle amplification depends on mass distribution, correlations, and the definition of the rate parameter.

For a maintained separation $d$, the additional visibility factor is

$$\frac{V}{V_0} =\exp\!\left[-\lambda t(1-e^{-d^2/(4r_C^2)})\right].$$

For $d\ll r_C$, the exponent scales as $-\lambda t d^2/(4r_C^2)$; for $d\gg r_C$, it saturates at $-\lambda t$. An experiment probes a curve in the $(\lambda,r_C)$ plane, not one universal collapse number.

Interferometric and noninterferometric channels

Interferometry searches for additional loss of branch coherence. Mechanical heating and force-noise measurements search for momentum diffusion caused by localization noise. Radiation experiments can constrain fluctuations coupled to charged constituents. These channels test related parameters only after specifying the same noise spectrum, spatial kernel, mass coupling, and ultraviolet behavior.

A bound derived for white noise cannot automatically exclude a colored-noise model with a high-frequency cutoff. A dissipative model may change heating predictions. Likewise, astrophysical estimates depend on thermal history, composition, and background modeling. These dependencies are reasons to state assumptions, not reasons to ignore the experiments.

A defensible exclusion calculation

Let measured data be $D$, collapse parameters $\theta=(\lambda,r_C)$, and nuisance parameters $\eta$. Build a likelihood $L(D\mid\theta,\eta)$ that includes known decoherence, calibration, and detector effects. Report how $\eta$ is constrained or eliminated and which statistical construction defines the exclusion.

As an elementary illustration, suppose a calibrated experiment bounds extra coherence loss by $V/V_0>0.95$ during $0.1$ s at $d\gg r_C$. The example kernel gives $\lambda<-\ln(0.95)/0.1\simeq0.513\ {\rm s}^{-1}$. At $d=0.1r_C$, the same visibility sensitivity allows a rate roughly $400$ times larger. Geometry can dominate the translation from contrast to a model bound.

An observed discrepancy is not automatically a collapse detection. A positive case requires characteristic parameter dependence and exclusion of ordinary mechanisms; a null result excludes only the tested parameter region. The experimental roadmap places these tests alongside other foundational questions.

Exercise

Expand the kernel at small separation and identify the effective $D_{pp}$ in $\dot\rho=-D_{pp}[x,[x,\rho]]/\hbar^2$.

Solution

The leading kernel is $-\lambda(x-x')^2/(4r_C^2)$. Therefore $D_{pp}=\lambda\hbar^2/(4r_C^2)$ in this single-particle model. This relates coherence loss to heating, but the full CSL geometry factor must be restored before using it for a macroscopic resonator.

References

• Bassi, A., and Ulbricht, H., “Collapse models: from theoretical foundations to experimental verifications,” Journal of Physics: Conference Series 504, 012023 (2014), arXiv:1401.6314 — overview of inference strategies.
• Bassi, A., Lochan, K., Satin, S., Singh, T. P., and Ulbricht, H., “Models of wave-function collapse, underlying theories, and experimental tests,” Reviews of Modern Physics 85, 471–527 (2013), arXiv:1204.4325 — model definitions and experimental channels.