Environmentally induced shift of the quantum arrival time

Using a simple model potential, we study the effects of weak Markovian dissipation on the quantum arrival time. The interaction with the environment is incorporated into the dynamics through a Markovian master equation of Lindblad type, which allows us to compare time-of-arrival distributions and approximate crossing probabilities for different dissipation strengths and temperatures. We also establish a connection to an earlier study where quantum tunneling with dissipation was investigated, which leads us to some conclusions concerning the formulation of the continuity equation in the Lindblad theory.

Figure 1

(Color online) Upper panels: TOA distribution defined in terms of the probability density current for low (left) and high (right) temperatures $\left[ϵ=\hslash \omega /\left(k_{B}T\right)\right]$, shown for three different environmental coupling strengths $\left(\xi =\lambda /\omega \right)$. Lower panels: approximate crossing probabilities in the case of low (left) and high (right) temperatures. The following values were used in the calculations: $\omega =3.0$, $m=1.0$, $x_0=4.0$, $p_0=-15.0$, and ${\Delta }_x=1.0$. All quantities are given in atomic units.

Figure 2

(Color online) Comparison of the probabilities from Eqs. (17, 21) for the case of low temperatures with a weak dissipation (left panel) and high temperatures with a stronger dissipation (right panel). Note that the two curves overlap in the former case. The values used in the calculations are as stated in the caption of Fig. 1, with $ϵ=\hslash \omega /\left(k_{B}T\right)$ and $\xi =\lambda /\omega$. All quantities are given in atomic units.