Dissipative extension of the Ghirardi-Rimini-Weber model

In this paper, we present an extension of the Ghirardi-Rimini-Weber model for the spontaneous collapse of the wave function. Through the inclusion of dissipation, we avoid the divergence of the energy on the long-time scale, which affects the original model. In particular, we define jump operators, which depend on the momentum of the system and lead to an exponential relaxation of the energy to a finite value. The finite asymptotic energy is naturally associated to a collapse noise with a finite temperature, which is a basic realistic feature of our extended model. Remarkably, even in the presence of a low-temperature noise, the collapse model is effective. The action of the jump operators still localizes the wave function and the relevance of the localization increases with the size of the system, according to the so-called amplification mechanism, which guarantees a unified description of the evolution of microscopic and macroscopic systems. We study in detail the features of our model, at the level of both the trajectories in the Hilbert space and the master equation for the average state of the system. In addition, we show that the dissipative Ghirardi-Rimini-Weber model, as well as the original one, can be fully characterized in a compact way by means of a proper stochastic differential equation.

Figure 1

Effect of the localization mechanism on (a) a Gaussian wave function $|\psi \rangle =|{\varphi }^{\alpha ,\beta ,\gamma }\rangle$ [see Eq. (5)] and (b) a superposition of Gaussian wave functions $|\psi \rangle =\frac{1}{\sqrt{2}}\left(\right|{\varphi }^{\alpha ,0,\gamma }\rangle +|{\varphi }^{-\alpha ,0,\gamma }\rangle )$ [see Eq. (9)]. The position probability distribution $\mathcal{P}\left(x\right)={\left|\langle x|\psi \rangle \right|}^2$ before the localization is given by the blue dashed line, the probability distribution after the localization ${\mathcal{P}}_y\left(x\right)={\left|\langle x|{\psi }_y\rangle \right|}^2$ is given by the black dotted dashed line, while the red line represents the normalized Gaussian distribution associated with the jump operator [see Eq. (2)]. The states after the localization are given by Eq. (6) (a) and by Eq. (10) with $y=\alpha$ (b).

Figure 2

Expected value of the position variance [see Eq. (41)] as a function of time (blue dots), compared with the deterministic unitary evolution (red line); the statistical mean is ob-tained over a sample of ${10}^5$ trajectories. (a), (b) Microscopic system: $M={10}^{-27}$ kg and jump rate given by $\lambda ={10}^{-16}{\text{s}}^{-1}$, with $\gamma =r_c^2$ (a) and $\gamma ={10}^{-6}{r}_c^2$ (b). (c), (d) Macroscopic system: $M={10}^{-3}$ kg and jump rate given by ${\lambda }_{\text{macro}}=N\lambda ={10}^7{\text{s}}^{-1}$, with $\gamma ={10}^6{r}_c^2$ (c) and $\gamma ={10}^{12}{r}_c^2$ (d). The initial variance is ${\left({\Delta }_{{\varphi }_0}X\right)}^2=\gamma /2$.