Collision model for non-Markovian quantum dynamics

We study the applicability of collisional models for non-Markovian dynamics of open quantum systems. By allowing interactions between the separate environmental degrees of freedom in between collisions we are able to construct a collision model that allows us to study quantum memory effects in open system dynamics. We also discuss the possibility to embed non-Markovian collision model dynamics into Markovian collision model dynamics in an extended state space. As a concrete example we show how, using the proposed class of collision models, we can discretely model non-Markovian amplitude damping of a qubit. In the time-continuous limit, we obtain the well-known results for spontaneous decay of a two-level system into a structured zero-temperature reservoir.

Figure 1

First two steps of a collision model where the open system, initialized in state $|\phi \rangle$, interacts with a new and uncorrelated subenvironment at each collision. All subenvironments are initially at state $|0\rangle$.

Figure 2

First two steps of a collision model where correlations are propagated by the interaction between the different subenvironments. The initial state of the system is $|\phi \rangle$, and all environmental degrees of freedom are initially at state $|0\rangle$.

Figure 3

Comparison of ${\eta }_{\tau }$ obtained from continuous and collision models. Case $\tilde{\lambda }=1/3$ corresponds to Markovian evolution, and $\tilde{\lambda }=5$ corresponds to non-Markovian evolution of the open system. Other parameters are $\delta \tau =0.02{\tilde{\lambda }}^{-1}$ and $\tilde{\mathrm{\Omega }}=0$. Collision model parameters $G,g$, and $\varphi$ are obtained using Eqs. (13, 14, 15).

Figure 4

Non-Markovianity of the collision model after $n=30$ steps and using the optimal initial-state pair $|{\phi }_{\pm }\rangle =\frac{1}{\sqrt{2}}\left(|0\rangle \pm |1\rangle \right)$. We chose uniformly 14 400 values $g,G\in S_{\epsilon }$, where $\epsilon =0.01$. We have chosen $\epsilon \ne 0$ in order to avoid divergences. The gray line corresponds to the boundary between divisible and indivisible dynamics of the continuous-in-time model, $\tilde{\lambda }=1/2$.

Figure 5

Data from Fig. 4 are mapped to $(\delta \tau ,\tilde{\lambda })$ coordinates using Eqs. (17) and (18). The solid line is the boundary $\tilde{\lambda }=1/2$ between divisible and indivisible dynamics. The area confined by dark gray (nonsolid) lines corresponds to the image of $S_{\epsilon }$, where $\epsilon =0.01$ under the same mapping. Light gray (nonsolid) lines indicate the image of $S_{{\epsilon }^{\prime }}$ with ${\epsilon }^{\prime }=0.005$. The dotted lines correspond to the image of the boundary $g=\epsilon$ of the set $S_{\epsilon }$. We thus see how the time-continuous limit $\delta \tau \to 0$ is obtained for fixed $\tilde{\lambda }$ by letting $g\to 0$.

Figure 6

Qubit+ancilla embedding of the non-Markovian collision model, first three steps. We have left out single-qubit unitaries $U_0$ for the system qubit for simplicity.