Non-Markovian dynamics for a free quantum particle subject to spontaneous collapse in space: General solution and main properties

We analyze the non-Markovian dynamics of a quantum system subject to spontaneous collapse in space. After having proved, under suitable conditions, the separation of the center-of-mass and relative motions, we focus our analysis on the time evolution of the center of mass of an isolated system (free-particle case). We compute the explicit expression of the Green’s function, for a generic Gaussian noise, and analyze in detail the case of an exponential correlation function. We study the time evolution of average quantities, such as the mean position, momentum, and energy. We eventually specialize to the case of Gaussian wave functions and prove that all basic facts about collapse models, which are known to be true in the white-noise case, hold also in the more general case of non-Markovian dynamics.

Figure 1

Linear-linear graph showing the time evolution of $\sigma \left(t\right)$ for a 1 kg particle, for different values of $\gamma$, for the same initial condition $\sigma \left(0\right)=1$, and with ${\lambda }_0=5.00\times {10}^{-3}{\text{m}}^{-2}{\text{s}}^{-1}$. The curve with $\gamma =\infty$ corresponds to the white-noise case. Time is measured in seconds and distances in meters.

Figure 2

Log-linear graph, showing the large time behavior of $\sigma \left(t\right)$ for a 1 kg particle, for different values of $\gamma$, for the same initial condition $\sigma \left(0\right)=1$, and with ${\lambda }_0=5.00\times {10}^{-3}{\text{m}}^{-2}{\text{s}}^{-1}$. The white-noise case would appear in the graph as a straight line with value $1.27\times {10}^{-15}\text{m}$. Time is measured in seconds and distances in meters.

Figure 3

Linear-linear graph, showing to which value an initial spread $\sigma \left(0\right)=1\text{m}$ decreases after ${10}^{-3}\text{s}$, as a function of $\gamma$. The mass of the particle has been set equal to $m=1.01\times {10}^{-3}\text{kg}$, corresponding to the total mass of a system containing Avogadro’s number of nucleons. The coupling constant ${\lambda }_0$ has been given the GRW value $5.00\times {10}^{-3}{\text{m}}^{-2}{\text{s}}^{-1}$. Larger values of $\gamma$ imply a faster collapse in space. Time is measured in seconds and distances in meters.

Figure 4

Linear-linear graph, showing to which value an initial spread $\sigma \left(0\right)=1\text{m}$ decreases after $3.33\times {10}^{-2}\text{s}$ as a function of $\gamma$. The mass of the particle has been set equal to $m=1.06\times {10}^{-18}\text{kg}$, which is the mass of a system of $n^{2}N\left(n=5640,N=20\right)$ nucleons, with $n$ and $N$ taken from Eq. (8) in [13]. The coupling constant ${\lambda }_0$ has been given Adler’s middle value $1.12\times {10}^6{\text{m}}^{-2}{\text{s}}^{-1}$. The dashed line corresponds to the white-noise case. Time is measured in seconds and distances in meters.