Quantum non-Markovian piecewise dynamics from collision models

TL;DR

This paper introduces a quantum collision model with memory that explicitly reproduces a broad class of non-Markovian piecewise dynamics in open quantum systems, linking microscopic interactions to observed non-Markovian behavior.

Contribution

It provides the first explicit microscopic collision model realization of non-Markovian piecewise quantum dynamics with memory effects.

Findings

The collision model reproduces the non-Markovian dynamics in the continuous-time limit.

It links microscopic collision processes to macroscopic non-Markovian behavior.

The model offers a new way to simulate and understand quantum memory effects.

Abstract

Recently, a large class of quantum non-Markovian piecewise dynamics for an open quantum system obeying closed evolution equations has been introduced [B. Vacchini, Phys. Rev. Lett. 117, 230401 (2016)]. These dynamics have been defined in terms of a waiting-time distribution between quantum jumps, along with quantum maps describing the effect of jumps and the system's evolution between them. Here, we present a quantum collision model with memory, whose reduced dynamics in the continuous-time limit reproduces the above class of non-Markovian piecewise dynamics, thus providing an explicit microscopic realization.

Equations

$$\dot{\rho }=-i[\hat{H},\rho ]+\sum _k{\gamma }_k\left({\hat{L}}_k\rho {\hat{L}}_k^{†}-\frac{1}{2}\{{\hat{L}}_k^{†}{\hat{L}}_k,\rho \}\right),$$

$${\mathcal{R}}_t[\rho ]=e^{\hat{R}t}\rho e^{{\hat{R}}^{†}t},\mathcal{J}[\rho ]=\sum _k{\gamma }_k{\hat{L}}_k\rho {\hat{L}}_k^{†},$$

$${\rho }_t={\mathcal{R}}_t[{\rho }_0]+\sum _{j=1}^{\infty }{\int }_0^{t}d{t}_j\dots {\int }_0^{t_2}d{t}_1\dots {\mathcal{R}}_{t-t_j}\mathcal{J}\dots \mathcal{J}{\mathcal{R}}_{t_2-t_1}\mathcal{J}{\mathcal{R}}_{t_1}[{\rho }_0]$$

$${\rho }_t$$

$$g(t)=\exp \left[-{\int }_0^t\mathrm{d}s\varphi (s)\right],f(t)=\varphi (t)\exp \left[-{\int }_0^t\mathrm{d}s\varphi (s)\right],$$

$$\varphi (t)$$

$$\dot{\rho }={\int }_0^t\mathrm{d}{t}^{\prime }\mathcal{W}(t-t^{\prime })[\rho (t^{\prime })]+\mathcal{I}(t)[{\rho }_0],$$

$$\mathcal{W}(t)=\frac{\mathrm{d}}{\mathrm{d}t}\left[f(t){\mathcal{E}}_t\right]\mathcal{Z}+\delta (t)f(0){\mathcal{E}}_0\mathcal{Z},\mathcal{I}(t)=\frac{\mathrm{d}}{\mathrm{d}t}\left[g(t){\bar{\mathcal{E}}}_t\right].$$

$${\mathcal{U}}_{\tau }[\sigma ]=e^{-i{\hat{H}}_{SM}\tau }\sigma e^{i{\hat{H}}_{SM}\tau },$$

$${\mathcal{S}}_n[\sigma ]=p\sigma +(1-p){\hat{S}}_{Mn}\sigma {\hat{S}}_{Mn},$$

$${\sigma }_0=\left({\rho }_0\otimes {\bar{\eta }}_M\right)\otimes {\eta }_1\otimes {\eta }_2\otimes \cdot \cdot \cdot$$

$${\sigma }_n={\mathcal{S}}_n{\mathcal{U}}_{\tau }{\mathcal{S}}_{n-1}{\mathcal{U}}_{\tau }\dots {\mathcal{S}}_{{}_2}{\mathcal{U}}_{\tau }{\mathcal{S}}_{{}_1}{\mathcal{U}}_{\tau }[{\sigma }_0],$$

$$p=e^{-\Gamma \tau }$$

$$p^k=(p^{\frac{1}{\tau }}{)}^{k\tau }\to e^{-\Gamma t^{\prime }}.$$

$${\rho }_t=e^{-\Gamma t}{\bar{\mathcal{E}}}_t[{\rho }_0]+\sum _{j=1}^{\infty }{\Gamma }^j{e}^{-\Gamma t}{\int }_0^{t}d{t}_j\dots {\int }_0^{t_2}d{t}_1{\mathcal{E}}_{t-t_j}\dots {\mathcal{E}}_{t_2-t_1}{\bar{\mathcal{E}}}_{t_1}[{\rho }_0].$$

$${\bar{\mathcal{E}}}_t\rho ={\mathrm{Tr}}_M\left\{{\mathcal{U}}_t[\rho {\bar{\eta }}_M]\right\},{\mathcal{E}}_t\rho ={\mathrm{Tr}}_M\left\{{\mathcal{U}}_t[\rho {\eta }_M]\right\},\mathcal{Z}=\mathbb{I}$$

$${\sigma }_0=\left({\rho }_0\otimes {\bar{\eta }}_M\right)\otimes \left({\xi }_1\otimes {\eta }_1\right)\otimes \left({\xi }_2\otimes {\eta }_2\right)\otimes \dots ,$$

$${\mathcal{S}}_n[\sigma ]$$

$$\tilde{\mathcal{Z}}[{\rho }_{{}_{SM}}]$$

$$\mathcal{Z}[\rho ]$$

$${\mathrm{Tr}}_{n_1{n}_2}\left\{{\mathcal{S}}_n({\rho }_{{}_{SM}}{\xi }_{n_1}{\eta }_{n_2})\right\}$$

$${\rho }_{SM}^{(1)}={\mathcal{U}}_{\tau }\left[{\rho }_{SM}^{(0)}\right].$$

$${\rho }_{SM}^{(2)}$$

$${\rho }_{SM}^{(3)}$$

$${\rho }_{SM}^{(4)}$$

$${\rho }_{SM}^{(n)}$$

$$\pi (k_j,\dots ,k_1)=\left(\prod _{\ell =k_j+1}^{n-1}{p}_{\ell }\right)q_{k_j}\left(\prod _{\ell =k_{j-1}+1}^{k_j-1}{p}_{\ell }\right)q_{k_{j-1}}\dots q_{k_2}\left(\prod _{\ell =k_1+1}^{k_2-1}{p}_{\ell }\right)q_{k_1}\left(\prod _{\ell =1}^{k_1-1}{p}_{\ell }\right).$$

$$p(t_k-t_{k-1})={\mathrm{e}}^{-{\int }_0^{t_k-t_{k-1}}\mathrm{d}s\varphi (s)},$$

$$\left(\prod _{\ell =k+1}^j{p}_{\ell }\right)q_k$$

Full text

1 Introduction

Prompted by the growing impact of quantum technologies, the study of non-Markovian (NM) quantum dynamics is currently a topical field [1, 2, 3, 4]. Besides the goal of defining, witnessing and even quantifying on a rigorous basis the degree of quantum “non-Markovianity” of an open system dynamics, efforts are under way to advance the longstanding quest for the non-Markovian counterpart of the celebrated Gorini-Kossakowski-Lindblad-Sudarshan master equation (ME) [5, 6]. As a pivotal requisite, which may be easily violated [7, 8], a well-defined NM ME must entail a completely positive and trace-preserving (CPT) dynamics for an arbitrary initial state and for suitably large classes of operators and parameters appearing in its expression. While the set of known NM dynamics described by well-behaved MEs (in the above sense) is still relatively small, remarkable progress was made in the last few years.

A relatively new approach to quantum NM dynamics is based on quantum collision models (CMs) [9, 10, 11, 12, 13, 14, 15, 16, 17]. A CM is a simple microscopic framework for describing the open dynamics of a system $S$ in contact with a bath, where the latter is assumed to consist of a large number of elementary subsystems, the “ancillas”. The open dynamics of $S$ results from its successive pairwise collisions with the bath ancillas, each collision being typically described by a bipartite unitary on $S$ and the involved ancilla.

In the continuous-time limit, a CM leads to a Lindblad ME with no need to resort to the Born-Markov approximation [18]. Such appealing property prompted NM generalizations of the simplest memoryless CM, whose continuous-time-limit dynamics is ensured by construction to be CPT. A significant instance is the CM in Refs. [12, 19], recently extended in Ref. [15], which produced a new NM memory-kernel ME. The peculiar structure of this memory-kernel ME and corresponding dynamical map inspired further investigations [20, 21, 22, 23, 24] from different viewpoints, which allowed to further enlarge the class of known NM dynamics governed by well-defined MEs.

One of these viewpoints builds on the well-known quantum-jumps picture of the Lindblad ME [25, 26, 27, 28] to devise a far larger, NM class of piecewise dynamics characterized by a waiting time distribution, a CPT map describing the effect of jumps and a collection of CPT maps accounting for the evolution between jumps Ref. [20, 21, 23]. This class of piecewise dynamics obeys a memory-kernel ME [23]. Given that this general ME encompasses the reduced ME of the CM in Ref. [15] only as a special case, it is natural to wonder whether a generalized CM can be constructed giving rise to the piecewise-dynamics ME with no restrictions. In this work, we prove that such a CM indeed exists and show that it can be defined as a non-trivial generalization of Ref. [15] where collisions occur in the form of probabilistic SWAP operations. Among its major distinctive features are the doubling of each ancilla into a pair of subancillas, which allows to introduce the jump map that was fully absent in Ref. [15], and the introduction of time-step-dependent swap probabilities, which allows to reproduce waiting time distributions of arbitrary shape unlike Ref. [15] that was restricted to exponential ones. This extension is of particular importance to comply with possible experimental implementations as well as encompass all the different features of the interaction dynamics that might give rise to non-Markovianity.

This paper is organized as follows. In Section 2, we review the class of NM quantum dynamics introduced in Refs. [20, 21, 23]. As anticipated, the main purpose of this work is demonstrating the existence of a quantum CM with memory, which in the continuous-time limit reproduces the above class of NM dynamics. Since this CM is an extension of the one in Ref. [15], the latter is reviewed in Section 3 and a brief introduction to quantum CMs is provided. These introductory sections, in particular, allow us to introduce most of the notation and formalism that we use later in Section 4, where the main results of this work are presented. Owing to its central importance, Section 4 is structured in a number of subsections so as to better highlight the different essential aspects of the proposed CM: the initial state, the way system-ancilla collisions are modelled, the discrete dynamics, its continuous-time limit and, at last, the reduced dynamics of the open system. Our conclusions along with some comments and outlook are given in Section 5 Some technical proofs are presented in Appendix A.

2 Review of non-Markovian piecewise quantum dynamics

The prototypical Markovian dynamics of an open quantum system $S$ is described by the Gorini-Kossakowski-Lindblad-Sudarshan ME [5, 6], which reads

[TABLE]

where $\rho(t)$ is the $S$ density operator, $\{...,...\}$ stands for the anticommutator, $\hat{H}$ is a Hermitian operator, $\{\gamma_{k}\}$ are positive rates, and where $\{\hat{L}_{k}\}$ are jump operators. By introducing the maps

[TABLE]

where we defined the non-Hermitian operator $\hat{R}=-i\hat{H}-\frac{1}{2}\sum_{k}\gamma_{k}\hat{L}^{\dagger}_{k}\hat{L}_{k}$, the solution of the Lindblad ME (1) can be written as the Dyson series [26]

[TABLE]

with $0\leq t_{1}\leq t_{2}\leq...\leq t$. Eq. (3) shows that the time evolution of $S$ can be viewed as an underlying dynamics described by the evolution map $\mathcal{R}_{t}$ interrupted by jumps each transforming the system state according to the jump map $\mathcal{J}$. Index $j$ in Eq. (3) indeed represents the number of jumps occurred up to time $t$ at instants $\{t_{1},t_{2},...,t_{j}\}$ such that $0\leq t_{1}\leq t_{2}\leq...\leq t_{j}\leq t$. Note that the maps (2) are not trace-preserving.

Both the Lindblad ME Eq. (1) and the representation Eq. (3) for its exact solution have been taken as a starting point for possible generalizations leading to well-defined dynamics to be described by means of memory kernel MEs, which can describe memory effects in the time evolution. Starting from the seminal work in Ref. [29], different approaches have been devised along this line [30, 31, 32, 33, 34]. One of us recently extended these results investigating a NM generalization of Eq. (3) [20, 21, 23], which in its most general form can be expressed as [23]

[TABLE]

Compared to Eq. (3), the jump map $\cal J$ [see Eq. (2)] is turned into the CPT map $\cal Z$, while ${\mathcal{R}}_{t}$ is replaced by the CPT evolution map $\bar{\mathcal{E}}_{t}$ before any jump has occurred and by the CPT evolution map $\mathcal{E}_{t}$ after the first jump (if any) has taken place. Maps $\cal Z$, $\bar{\mathcal{E}}_{t}$ and $\mathcal{E}_{t}$ are fully unspecified, but for the requirement of being CPT. Importantly, while in Eq. (3) the statistical weight of each possible trajectory is determined by the non-trace-preserving maps $\mathcal{R}_{t}$, $\mathcal{J}$ and the initial state [21], in Eq. (LABEL:dysonNM) these statistical weights are assigned independently of the maps $\bar{\mathcal{E}}_{t}$, ${\mathcal{E}}_{t}$, $\cal Z$ and the initial state. Indeed, the functions $f(t)$ and $g(t)$, appearing in Eq. (LABEL:dysonNM), stand for an arbitrarily chosen waiting time distribution, namely the probability density for the distribution in time of the jumps, and its associated survival probability $g(t){=}1{-}\int_{0}^{t}{\rm d}t^{\prime}f(t^{\prime})$, that is the probability that no jump has taken place up to time $t$. The waiting time distribution and the associated survival probability can always be expressed in the form

[TABLE]

where the positive function

[TABLE]

is known as hazard rate function or simply hazard function [35]. The meaning of this is that $\phi(t)\operatorname{dt}$ provides the probability for a jump to take place in the time interval $(t,t+\operatorname{dt}]$, given that no jump has taken place up to time $t$. Accordingly, the time-dependent coefficient $f(t\!-\!t_{j})\!\ldots\!f(t_{2}\!-\!t_{1})g(t_{1})$ in Eq. (LABEL:dysonNM) gives the probability density that $j$ jumps take place at times $\{t_{1},t_{2},...,t_{j}\}$, while the pre-factor of the first term on the rhs is the probability that no jumps occurred up to time $t$ (this pre-factor indeed multiplies the jump-free evolution map $\bar{\mathcal{E}}_{t}$). The jumps are thus distributed in time according to a renewal process, which in particular entails that after each jump the process starts anew. Note that, in fact by construction, the dynamical map defined by Eq. (LABEL:dysonNM) is ensured to be CPT. Importantly, it can be shown [23] that it obeys the memory-kernel ME

[TABLE]

where

[TABLE]

The corresponding open dynamics, at variance with the Lindbladian case [see Eqs. (1) and (3)], is in general NM [20].

3 Collision models with memory

A quantum CM [36, 37, 38] is a simple microscopic model for describing the open dynamics of a system $S$ in contact with a bath $B$. In its prototypical version, a CM assumes that $B$ comprises a huge number of elementary, identical and non-interacting ancillas all initialized in the same state $\eta$. The $S$-$B$ interaction process occurs via successive pairwise “collisions” between $S$ and the ancillas, each of these collisions being described by a bipartite unitary operation $\hat{U}_{n}$. By hypothesis, $S$ can collide with each ancilla only once. After $n$ collisions, the state of $S$ is given by $\rho_{n}{=}\Phi^{n}[\rho_{0}]$, where the CPT map $\Phi$ is defined as $\Phi[\rho]{=}{\rm Tr}_{n}\{\hat{U}_{n}(\rho{\otimes}\eta_{n})\hat{U}_{n}^{\dagger}\}$. Note that, despite the apparent dependance on $n$ (see e.g. the partial trace over the $n$th ancilla), the map $\Phi$ does not depend on $n$ since the bath initial state and system-ancilla interaction Hamiltonian are fully homogeneous. It can be shown [18] that in the continuous-time limit the dynamics of such a simple CM is described by a Lindblad ME of the form (1), a result which can be expected based on the discrete semigroup property enjoyed by the collision map, $\Phi^{n+m}=\Phi^{n}\Phi^{m}$. The open dynamics of $S$ corresponding to such a paradigmatic CM is thereby fully Markovian.

There are several ways to endow the basic CM just described with memory so as to give rise to a NM dynamics [9, 12, 13, 14, 16, 17]. The one of concern to us, given the goals of the present paper, is the CM with memory of Ref. [15], which can be regarded as a generalization of a model first put forward in Refs. [12, 19]. The general structure of the CM in Ref. [15] is in many respects analogous to the basic memoryless CM described in the previous paragraph except that the system undergoing collisions with the bath ancillas is now bipartite, comprising the very open system under study $S$ plus an auxiliary system $M$, the “memory”, whose Hilbert space dimension is the same as each ancilla’s one. A sketch of the CM is given in Fig. 1(a). Systems $S$ and $M$ interact all the time according to the pairwise unitary evolution map

[TABLE]

where $\hat{H}_{S\!M}$ is the joint $S$-$M$ Hamiltonian. Here and throughout this paper, $\sigma$ stands for a joint state of the $S$-$M$ system and all the bath ancillas.

By hypothesis, only $M$ is in direct contact with the bath [see Fig. 1(a)]. This interaction takes place through successive collisions, each being described by the pairwise non-unitary quantum map

[TABLE]

where $\hat{S}_{Mn}$ is the swap unitary operator exchanging the states of $M$ and the $n$th ancilla. The CPT map (2), which depends parametrically on the probability $p$, can be interpreted as a probabilistic partial SWAP gate: the memory and ancilla states are either swapped or left unchanged with probability $p$.

The initial state of the overall system ($S$, $M$ and the bath ancillas) is assumed to be

[TABLE]

where in particular $\rho_{0}$ ($\bar{\eta}_{M}$) is the initial state of $S$ ($M$). Throughout this paper, tensor product symbols will be omitted whenever possible to avoid using too cumbersome notation.

By calling $\sigma_{n}$ the overall state at step $n$, the dynamics proceeds according to

[TABLE]

namely an $S$-$M$ unitary dynamics goes on all the time, being interrupted at each fixed time step $\tau$ by a collision between $M$ and a “fresh” ancilla (i.e. one still in the initial state $\eta$) described by the non-unitary map (2). Equivalently, one can view each $\mathcal{U}_{\tau}$ itself as embodying the effect of a unitary collision that is however internal to the joint $S$-$M$ system in such a way that the overall CM dynamics results from subsequent $M$-ancilla collisions interspersed with internal ones that involve $S$ and $M$ only [15].

Like for any CM, the dynamics just defined is discrete. One can however define a continuous-time limit by assuming that the duration of each time step $\tau$ becomes very small while the step number $n$ gets very large in such a way that $n\tau\!\rightarrow t$, where $t$ is a continuous time variable. The assumption of a very large number of steps demands an additional prescription for the continuous-time limit of the probability $p$ entering Eq. (2) since Eq. (4) clearly features $p$’s powers $\{p^{k}\}$ for all positive integers $k{\leq}n$. This task is carried out by first defining a rate $\Gamma$ that allows to express $p$ as

[TABLE]

(which is always possible) and assuming next that $\Gamma\tau\!\ll\!1$ in such a way that $p\simeq 1$. This ensures that $p^{k}$, for any $k$ smaller than $n$ and yet large enough so that $k\tau\rightarrow t^{\prime}<t$ is finite, be not washed out in the continuous-time limit. Indeed, this yields

[TABLE]

By finally noting that, consistently with the hypothesis $\Gamma\tau\ll 1$, $1{-}p\!=\!1{-}e^{-\Gamma\tau}{\simeq}\Gamma\tau$ [cf. Eq. (2)] and that since the CM is well defined for any choice of $\eta$, $\bar{\eta}$ and $\mathcal{U}_{t}$ it is possible to describe the reduced evolution of $S$ by the following CPT map

[TABLE]

which is a special case of (LABEL:dysonNM) for

[TABLE]

In fact it can be shown [15] that the map (7) obeys a memory-kernel ME of the form (7).

Yet, the CM in fact lacks the jump map $\mathcal{Z}$ and is, in addition, apparently constrained to a purely exponential waiting time distribution $f(t){=}\Gamma e^{-\Gamma t}$ [the corresponding hazard function $\phi(t)$ being thus constant].

In the next section, we show how to construct a CM with memory whose continuous-time limit yields ME (7) in the most general case, including in particular an arbitrary jump map $\mathcal{Z}$ and an arbitrary waiting time distribution $f(t)$.

4 A generalized collision model with memory

The CM to be defined here is a non-trivial generalization of the CM of Ref. [15] reviewed in the last section. Just like in Ref. [15], the system undergoing collisions with the bath ancillas comprises $S$ and a memory $M$ that are subject to a coherent mutual coupling giving rise to the unitary evolution map (1). At variance with Ref. [15], however, now each bath ancilla is bipartite as well, consisting of a pair of “subancillas”: one subancilla has the same Hilbert space dimension as $S$, while the other subancilla has the same dimension as $M$. A sketch of this generalized CM with memory is displayed in Fig. 1(b).

4.1 Initial state

The initial joint state reads

[TABLE]

where $\xi$ ($\eta$) is the initial state of the subancilla having the same dimension as $S$ ($M$). In full analogy with Eq. (3), $\rho_{0}$ ($\bar{\eta}_{M}$) is the initial state of $S$ ($M$).

4.2 System-ancilla collisions

A further distinctive feature of the generalized CM with memory is that the collisions with the ancillas now involve $S$ as well. By definition, the collision between $S$-$M$ and the $n$th bipartite ancilla is described by the non-unitary four-partite CPT map

[TABLE]

where $n_{1}$ and $n_{2}$ are the two $n$’s subancillas that are isodimensional to $S$ and $M$, respectively, while $\hat{V}_{Sn_{1}}$ is a unitary operator acting on $S$ and subancilla $n_{1}$. Map (10) therefore swaps the states of $M$ and $n_{2}$ and, at the same time, applies the unitary $\hat{V}_{Sn_{1}}$ on $S$ and $n_{1}$, or leaves unchanged the state of $S$, $M$, $n_{1}$ and $n_{2}$ with probability $p_{n}$. Note that, unlike the CM of the previous section [cf. Eq. (2)], now we allow the probability $p_{n}$ to be in general step-dependent. The reason for this will become clear later on.

Based on Eq. (10) and the ancilla’s initial state [cf. Eq. (9)], it is convenient to define a bipartite CPT map on $S$ and $M$ as

[TABLE]

where $\mathcal{Z}$ is the CPT map on $S$ defined by

[TABLE]

The proof of the last step in Eq. (11) is given in Appendix A.

Eqs. (11) and (12) entail that the collision with the $n$th ancilla [see Eq. (10)] changes the reduced state of $S$ and $M$, $\rho_{{}_{\!S\!M}}$, according to

[TABLE]

The essential effect of the collision, thereby, is to either leave with probability $p_{n}$ the $S$-$M$ state unchanged or, with probability $1{-}p_{n}$, to apply the CPT map $\mathcal{Z}$ on $S$ by simultaneously resetting the $M$’s state to $\eta$.

4.3 Discrete dynamics

Similarly to the CM in Ref. [15] (see previous section), the initial state (9) evolves through an underlying $S$-$M$ unitary dynamics that is interrupted at each fixed time step $\tau$ by a collision described by Eq. (10) involving a fresh bipartite ancilla that is still in state $\xi\otimes\eta$. Accordingly, the overall state at the $n$th step is given by $\sigma_{n}=\mathcal{S}_{n}\,\mathcal{\,}U_{\tau}\,\mathcal{S}_{n-1}\,\mathcal{U}_{\tau}\,\ldots\mathcal{S}_{2}\,\mathcal{U}_{\tau}\,\mathcal{S}_{1}\,\mathcal{U}_{\tau}[\sigma_{0}]$.

Starting from $\rho_{S\!M}^{(0)}{=}\rho_{0}\otimes\bar{\eta}_{M}$ [see Eq. (9)], at the end of the first step the reduced $S$-$M$ state is turned into

[TABLE]

Next, the collision with ancilla 1 described by map $\mathcal{S}_{1}$ [see Eq. (10)] takes place followed by another application of the $S{-}M$ unitary. At the end of the second step, the $S$-$M$ state thus reads

[TABLE]

where the trace is taken over the $n$th ancilla for $n=1$ and to simplify the notation we set $q_{n}=1{-}p_{n}$. By replacing in the last identity the state at the end of the first step (14), Eq. (15) can be expressed as a function of the initial $S$-$M$ state only as

[TABLE]

Since the elapsed time of the process is an integer multiple of the time step $\tau$ and given that a jump (if any) occurs at the end of each time step $\tau$, at the second step either 0 or 1 jumps have taken place. The former and latter cases correspond to the terms featuring zero or one $\widetilde{\mathcal{Z}}$ in Eq. (16) as highlighted by the captions. At the end of the 3rd step, after the application of maps $\mathcal{S}_{2}$ and $\mathcal{U}_{\tau}$, an analogous calculation leads to

[TABLE]

showing that, as expected, 0, 1 or 2 jumps are possible in this case corresponding to as many applications of the map $\widetilde{\mathcal{Z}}$. In a similar fashion, at the 4th step we get

[TABLE]

In order to write down the $n$th-step state in a compact form, having in mind the structure of Eq. (LABEL:dysonNM), we first note that based on Eq. (1) any $k$th power of the map $\mathcal{U}_{\tau}$ is given by $\mathcal{U}_{\tau}^{k}=\mathcal{U}_{k\tau}$ (in the following we will further set $\mathcal{U}_{k}\equiv\mathcal{U}_{k\tau}$ to simplify the notation).

By induction, the $n$th-step state for arbitrary $n{\geq}2$ is given by

[TABLE]

where $\pi(k_{j},\ldots,k_{1})$ stands for the probability to perform exactly $j$ jumps at specific steps $\{k_{j},\ldots,k_{1}\}$ and reads

[TABLE]

4.4 Continuous-time limit

In order to perform the continuous-time limit, in analogy with Eq. (5) we introduce the quantities

[TABLE]

corresponding respectively to the probability of no jump or one jump to take place in each small time interval $t_{k}-t_{k-1}$, which in the case of constant hazard function $\phi(s)$ reduces to a Poisson distribution for the jumps. According to the definition of a renewal process, the jump probabilities thereby depend only on the elapsed time. In this representation, the various contributions appearing in Eq. (20), in the limit of a large number of short steps such that the time intervals between steps become increasingly small, can be written as

[TABLE]

This shows that the function $\phi(t)$ has indeed the role of hazard rate function [cf. Eq. (5], which determines the renewal process describing the time distribution of jumps. Hence, we can thus finally identify

[TABLE]

The first term in (LABEL:formula) accordingly becomes

[TABLE]

Therefore, Eq. (LABEL:formula) in the continuous-time limit reads

[TABLE]

4.5 Reduced dynamics

So far, we have focused on the bipartite system $S$-$M$, working out its evolution. We now consider the resulting reduced dynamics for the system $S$, which embodies the degrees of freedom of the open quantum system of interest. We first recall that $\rho_{SM}(0)=\rho_{0}\otimes\bar{\eta}_{M}$ [see Eq. (9)], which ensures the existence of the reduced dynamical map of $S$. When this expression is replaced in Eq. (23) upon taking the trace over $M$ we get

[TABLE]

By next introducing, according to Eq. (8), the CPT maps $\mathcal{E}_{t}$ and $\bar{\mathcal{E}}_{t}$, whose definition is thus identical to the model in Ref. [15], and recalling Eqs. (11) and (12), we get

[TABLE]

The argument of the partial trace can be expressed by iteration according to

[TABLE]

which finally leads to the expression

[TABLE]

where for the sake of simplicity we have removed the nested square brackets in the last expression. When this result is replaced in Eq. (25), we end up with Eq. (LABEL:dysonNM). Accordingly, the reduced dynamics of $S$ in the continuous-time limit necessarily obeys ME (7) with no restrictions.

We can therefore conclude that the generalized collision model with memory constructed here is indeed able to reproduce altogether the piecewise NM dynamics with jumps considered in Ref. [23].

5 Conclusions and outlook

The Gorini-Kossakowski-Lindblad-Sudarshan ME has been for over 40 years the workhorse of open quantum systems theory. It embodies the basic reference for open dynamics that lack memory effects. Clearly, though, in the case of strong coupling and/or structured reservoirs a memoryless Markovian description fails to faithfully capture the relevant features of the dynamics. Many non-trivial challenges follow, in particular the need for more general evolution equations that ensure a well-defined (i.e., CPT) dynamics and, at the same time, effectively describe memory effects. On top of this, it is highly desirable that these theoretical descriptions be associated with corresponding environmental models thus providing an underlying microscopic interpretation and, possibly, a controlled implementation of such non-Markovian dynamics.

Both the above aspects were the focus of this paper. Starting from a recently proposed family of memory-kernel MEs corresponding to a large class of generally non-Markovian time evolutions, we showed that any such ME admits a microscopic CM from which it can be obtained as the equation governing its continuous-time-limit reduced dynamics.

Specifically, the considered time evolutions consist of piecewise dynamics in which a continuous, generally non-Markovian, time evolution is interrupted at random times, distributed according to a general waiting time distribution, by a quantum jump described by a general CPT transformation. These dynamics obey a closed memory-kernel ME. In this work, we showed that one such ME can be obtained as the continuous-time limit of a CM where memory effects are due to auxiliary degrees of freedom (which we indeed called memory) mediating the action of the environment on the system. As a distinctive feature of the CM, each bath ancilla is bipartite comprising a pair of subancillas. Each collision occurs in the form of a map that with some probability swaps the state of the memory and one subancilla, while a unitary is at the same time applied on the system under study and the other subancilla. As a further hallmark of the considered CM, the probability for such swap-and-unitary operation can depend at will on the time step.

As remarked in the main text, the ancillas’ doubling along with the step-dependance of the aforementioned probability are the crucial features marking the difference between the CM in Ref. [15] and the one addressed here (which can thus be viewed as a non-trivial generalization of the former). They allow to introduce a jump map as well as a waiting time distribution of arbitrary shape.

It is interesting to note that the term “collisional model” was at times used in the literature (see e.g. [29]) to denote a quantum dynamics that is interrupted at random times by “collisions” – that is jumps in fact – just like in the framework addressed in Ref. [23]. In this respect, our work provides a connection between this definition of CM, based on random-time collisions, and the one used throughout the paper, where instead collisions occur at fixed times.

We finally point out that ME (7) was obtained in Ref. [23] within a general framework based on the quantization of a family of classical stochastic dynamics. Since this quantization involves non-commuting operators, ME (7) arises only as one of two possible cases corresponding to different operator orderings. The question whether or not a class of underlying CMs can be devised even for the ME arising in the other case [23] – which is qualitatively different from ME (7) – is under ongoing investigations.

Appendix A

We here provide the proof of the last identity in Eq. (11). Let us first recall the starting point, namely the definition of the map $\widetilde{\mathcal{Z}}$ given in the first line of Eq. (11), omitting the tensor product symbol to simplify the notation

[TABLE]

and consider two orthonormal bases $\{|\mu\rangle_{M}\}$ and $\{|\nu\rangle_{n_{2}}\}$ in the Hilbert spaces of $M$ and $n_{2}$, respectively. In terms of these vectors, the swap operator $\hat{S}_{Mn_{2}}$ is expressed as

[TABLE]

Using this expression in Eq. (A.1) the rhs explicitly reads

[TABLE]

so that taking the partial trace over $n_{2}$ we end up with

[TABLE]

By recalling that the state $\rho$ of the reduced system is just the marginal of $\rho_{SM}$ and by noting that the expression in square brackets swaps $\eta_{n_{2}}$ and $\eta_{M}$, we finally get

[TABLE]

which according to the definition (12) of the map $\mathcal{Z}$ concludes the proof.

Acknowledgments

We acknowledge support from the EU Project QuPRoCs (Grant Agreement 641277) and the Fulbright Commission.