Analysis of various factors affecting the non-Markovian dynamics associated with a hierarchical environment based on collision model

Abstract

We propose a quantum collision model in which the environment is abstractively divided into two hierarchies including “environment-bus” that has direct interactions with the system and “environment-stations” that has not. Based on the model, we investigate the effects of initial system–environment correlations, initial states of environment, and various interactions on the dynamics of open quantum systems associated genuinely with such a hierarchical environment. We illustrate that the initial quantum correlation between the system and environment leads to a transition from Markovian to non-Markovian dynamics, while for initial classical correlation the transition can only be confirmed to happen when the couplings rather than the correlations in environment are present. In addition, we investigate the degree of non-Markovianity varying with environment initial states and reveal that the interaction strength between two environmental hierarchies plays an important role in it. In particular, we show that in such a hierarchically structured environment the degree of non-Markovianity is not equivalent to memory effects of the environment-stations as a reservoir due to the presence of the environment-bus.

Appendices: The methods of non-Markovianity witness with or without initial correlation

1.1 Appendix 1: Non-Markovianity witness without initial correlations

We introduce the usual degree of non-Markovianity (N) proposed in Refs. [47,48,49,50,51],

$$\begin{aligned} \textit{N}=\max _{\{\rho ^{N}_s(0), \rho _s^{N\perp }(0)\}}\int _{\Omega _+^N}\partial _t D(\rho ^{N}_s(t), \rho _s^{N\perp }(t))\mathrm{d}t, \end{aligned}$$

(42)

where $D(\rho ^{N}_s(t), \rho _s^{N\perp }(t))=\frac{1}{2}\Vert \rho ^{N}_s(t)-\rho _s^{N\perp }(t)\Vert _1$ with $\Vert .\Vert _1$ being the trace norm, is the trace distance between two evolved states and $\Omega _+^N=\bigcup _i(a_i^N,b_i^N)$ is the union of all the time intervals $(a_i^N,b_i^N)$ in our observation window within which $\partial _t D(\rho ^{N}_s(t), \rho _s^{N\perp }(t))>0$. $\rho ^{N}_s(0)$ and $\rho _s^{N\perp }(0)$ are two initial orthogonal states of the system, and their corresponding time-evolved states are $\rho ^{N}_s(t)$ and $\rho _s^{N\perp }(t)$. The maximization is performed over all possible orthogonal pairs of initial system states. As the evolution under scrutiny in this paper proceeds in discrete temporal steps, we will employ the discretized version of Eq. (42), which is obtained as [37, 52, 53]

$$\begin{aligned} \textit{N}= & {} \max _{\{\rho ^{n1}_s(0), \rho _s^{n1\perp }(0)\}}\sum _{n\in \Omega _+}[D(\rho ^{n1}_s(n+1), \rho _s^{n1\perp }(n+1))\nonumber \\&\quad -\,D(\rho ^{n1}_s(n), \rho _s^{n1\perp }(n))] \end{aligned}$$

(43)

with the time-evolved states of the system $\rho ^{n1}_s(n)$ and $\rho _s^{n1\perp }(n)$ obtained starting from a pair of the initial orthogonal states $\rho ^{n1}_s(0)$ and $\rho _s^{n1\perp }(0)$ after n steps of our protocol. $\Omega _+=\bigcup _n(n,n+1)$ is the union of all the collision step intervals $(n,n+1)$ in our observation window within which $D(\rho ^{n1}_s(n+1), \rho _s^{n1\perp }(n+1))-D(\rho ^{n1}_s(n), \rho _s^{n1\perp }(n))>0$.

1.2 Appendix 2: Non-Markovianity witness with initial quantum correlation

We consider the method which can be used to witness the non-Markovian dynamics when the initial correlation between the system and environment is present. Generally, the measure for non-Markovian dynamics is defined as namely the degree of non-Markovianity based on trace distance between different system states or the concurrence between system and reference system [47, 48, 54]. However, the two ways defining the degree of non-Markovianity both require that all the system states over the Bloch sphere have to be taken, and the state of environment must be the same, which indicates that the definition of the degree of non-Markovianity only is suitable when the system initially has no correlation with environment. So far, how to measure the non-Markovian dynamics with initial system–environment correlations is still an open question. In view of this, inspired by the previous definition of the degree of non-Markovianity, we make use of the non-monotonicity of the trace distance D[$\rho ^{q}_{s}(n),\widetilde{\rho }^{q}_{s}(n)$] between two states of system to witness the non-Markovian dynamics of an open quantum system with initial quantum correlation between the system and environment, and define

$$\begin{aligned} \mathrm{NE}^q=\sum _{n\in \Omega _+}\Delta D(n) , \end{aligned}$$

(44)

as the non-Markovian effect (NE) in this paper. Here $\Delta D(n)=D[\rho ^{q}_{s}(n+1),\widetilde{\rho }^{q}_{s}(n+1)] -D[\rho ^{q}_{s}(n),\widetilde{\rho }^{q}_{s}(n)]$ with $\widetilde{\rho }^{q}_{s}(n)$ being the state of system after 2n collisions with E-S corresponding to the initial state $\widetilde{\rho }^{q}_{s,b}(0)=\mathrm{Tr}_b(\rho ^{q}_{s,b}(0))\otimes \mathrm{Tr}_s(\rho ^{q}_{s,b}(0))$ and here the definition of $\Omega _+$ is the same as that in Eq. (43), in which $\Delta D(n)>0$.

1.3 Appendix 3: Non-Markovianity witness with initial classical correlation

We introduce the method which makes use of the non-monotonicity of the trace distance between two states of system to witness the effect of initial classical correlation on the non-Markovian dynamics of the system. However, the quantification of the NE is expressed by an alternative method differently from Eq. (44), which can be written as

$$\begin{aligned} \mathrm{NE}^{ci}= & {} \max _{\{\delta \in [0,\pi ],\varphi \in [0,2\pi ]\}}\sum _{n\in \Omega _+}\Delta D(n)\end{aligned}$$

(45a)

$$\begin{aligned} \Delta D(n)= & {} D\left[ \rho ^{ci}_{s}(n+1),\widetilde{\rho }^{ci}_{s}(n+1)\right] -D\left[ \rho ^{ci}_{s}(n),\widetilde{\rho }^{ci}_{s}(n)\right] , \end{aligned}$$

(45b)

where $i=1,2$ correspond respectively to collision model case I and collision model case II, $\widetilde{\rho }^{ci}_{s}(n)$ is the system state after 2n collisions with E-S corresponding to the initial state $\widetilde{\rho }^{ci}_{s,b}(0)=\widetilde{\rho }^{ci}_{s}(0)\otimes \mathrm{Tr}_s(\rho ^{ci}_{s,b}(0))$ with the initial system state $\widetilde{\rho }^{ci}_{s}(0)=\cos \frac{\delta }{2}|2\rangle +\sin \frac{\delta }{2}e^{i\varphi }|0\rangle , \delta \in [0,\pi ],\varphi \in [0,2\pi ]$. The maximization is performed by taking all possible system states $\widetilde{\rho }^{ci}_{s}(0)$ over the Bloch sphere. It is noted that the above definition is a little different from Eq. (44) where for the initial state $\widetilde{\rho }^{q}_{s,b}(0)=\mathrm{Tr}_b(\rho ^{q}_{s,b}(0))\otimes \mathrm{Tr}_s(\rho ^{q}_{s,b}(0))$ the initial states of both the system and E-B are fixed, while here only the initial state of E-B is fixed and the same as Eq. (44), but the initial states of the system is taken over the whole Bloch sphere.