PHYSICAL REVIEW A 88, 032115 (2013)

Embedding non-Markovian quantum collisional models into bipartite Markovian dynamics

Adrian A. BudiniConsejo Nacional de Investigaciones Cientıficas y Tecnicas (CONICET), Centro Atomico Bariloche,

Avenida E. Bustillo Km 9.5, (8400) Bariloche, Argentinaand Universidad Tecnologica Nacional (UTN-FRBA), Fanny Newbery 111, (8400) Bariloche, Argentina

(Received 1 August 2013; published 25 September 2013)

A wide class of non-Markovian completely positive master equations can be formulated on the basis ofquantum collisional models. In this phenomenological approach the dynamics of an open quantum system ismodeled through an ensemble of stochastic realizations that consist in the application at random times of a(collisional) completely positive transformation over the system state. In this paper, we demonstrate that thesekinds of models can be embedded in bipartite Markovian-Lindblad dynamics consisting of the system of interestand an auxiliary one. In contrast with phenomenological formulations, here the stochastic ensemble dynamicsan the interevent time interval statistics are obtained from a quantum measurement theory after assuming thatthe auxiliary system is continuously monitored in time. Models where the system intercollisional dynamics isnon-Markovian [B. Vacchini, Phys. Rev. A 87, 030101(R) (2013)] are also obtained from the present approach.The formalism is exemplified through bipartite dynamics that lead to non-Markovian system effects such as anenvironment-to-system backflow of information.

DOI: 10.1103/PhysRevA.88.032115 PACS number(s): 03.65.Yz, 42.50.Lc, 03.65.Ta, 02.50.Ga

I. INTRODUCTION

The description of open quantum systems through local-in-time Markovian evolutions is well understood from bothmathematical [1] and physical [2] points of view. As iswell known, under a completely positive condition, Lindbladequations provide the more general evolution structure of thesystem density matrix [1,2]. On the other hand, in recent yearsan ever-increasing interest has been paid to establishing a non-Markovian generalization of the open quantum system theoryformulated in terms of nonlocal-in-time evolutions [3]. Thereexist diverse formalisms for describing memory effects. Oneleading program consists in generalizing Lindblad equationsby replacing the rates of each dissipative channel by a time-convoluted kernel function. A wide class of both phenomeno-logical [4–8] and theoretical approaches [9–22] were formu-lated for building and characterizing master equations of thatkind, which in turn lead to a completely positive solution map.

In the category of phenomenological approaches, quantumcollisional models (QCMs) provided a fundamental tool forestablishing a non-Markovian generalization of Lindbladequations [4,5]. In this formalism, the evolution of an openquantum system follows from an average performed over anensemble of stochastic realizations of the system state. Eachrealization consists in the application, at random times, of acompletely positive transformation. The events can be read asa “collision” or interaction with the environment. Dependingon the statistics of the collision times and the system intereventdynamics, different non-Markovian master equations wereestablished [5–8]. Over that basis, the emergence of non-Markovian effects such as a system-to-environment backflowof information [23,24] were also analyzed in the recentliterature [25,26].

The collisional superoperator, the interevent system dynam-ics, and the collision time statistics are the main ingredientsof the approach. They must be defined, in an arbitraryway, from the beginning. Therefore, besides its usefulness,the QCM model does not have an associated microscopic

description, and neither it is completely understood whichkind of underlying mechanism may induce the structure of thestochastic dynamics. The main goal of this paper is to providea rigorous physical framework to answer these issues.

The basic idea consists in embedding the non-Markoviansystem evolution in a Markovian bipartite dynamics. It isdefined by the system of interest and an auxiliary (ancilla)system. We demonstrate that there exist bipartite Markovianinteractions that induce the same system non-Markoviandynamics. In this way, “microscopic interactions” that leadto the master equations associated with the QCM are found.On the other hand, by assuming that the auxiliary system iscontinuously monitored in time, over the basis of a(Markovian) quantum jump approach [27–29], we find thatthe realizations of the QCM can be put in one-to-onecorrespondence with the realizations of the measurementapparatus. In this way, the stochastic dynamics of the QCM isestablished from a quantum measurement theory. In addition,this modeling allows to describe the interevent statistics fromthe Markovian-Lindblad description.

In Ref. [8] Vacchini introduced a generalized QCM where,in contrast to previous approaches [5–7], the system intereventdynamics is defined by a non-Markovian propagator. On thebasis of an underlying tripartite Markovian dynamics, we showthat this generalization can also be described within the presentframework. Even when the stochastic realizations consist ofsuccessive collisional events with a non-Markovian intereventdynamics [8], they cannot be read as the result of a continuousmeasurement action performed over the system of interest.In fact, in contrast with the results of Ref. [22], here wedemonstrate that QCMs can consistently be recovered whenmeasuring the auxiliary ancilla system. The non-Markovianquantum jump approach developed in [22] relies on moregeneral bipartite interactions. Additionally, the monitoringaction is performed over the system of interest.

It is interesting to note that collisional models were alsoproposed as a phenomenological tool for deriving Markovian

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ADRIAN A. BUDINI PHYSICAL REVIEW A 88, 032115 (2013)

irreversible dynamics [1,30]. Furthermore, from a quantuminformation perspective [31], similar approaches were intro-duced by considering collisions with a string of auxiliaryqubits systems [32–34]. When the system-string interaction isdefined by partial swap and controlled-NOT qubits operations,specific Markovian master equations describe the system dy-namics [33]. Generalization of these ideas to non-Markoviandynamics were considered recently in Refs. [35–37]. Thestretched relation of these results with the present formalismis also investigated.

The paper is outlined as follows. In Sec. II we presentthe Markovian embedding, where the system density matrixis obtained by using projector techniques [3]. In Sec. III,from a standard quantum measurement theory, we obtain thestochastic ensemble dynamic after assuming that the auxiliarysystem is subjected to a measurement process. These resultsrely on the standard quantum jump approach [27–29] appliedto bipartite dynamics. In Sec. IV, we analyze some examplesthat exhibit the main features of the present approach. Abackflow of information from the system to the environmentis explicitly shown. In Sec. V, some generalizations of thestandard collisional approach are provided. The dynamicspresented in Ref. [8] is recovered from a tripartite Markoviandynamics. The formalisms of Refs. [35–37] are analyzed inthis context. In Sec. VI we present the conclusions.

II. MARKOVIAN EMBEDDING

In this section, it is demonstrated that non-MarkovianQCMs can be obtained by tracing out a bipartite Markoviandynamics. We deal with the case of stationary renewalstatistics.

A. Phenomenological renewal collisional models

The superoperator Es that defines each collisional event iswritten as

Es[ρ] =∑

α

VαρV †α ,

∑α

V †αVα = Is , (1)

where the set of operators {Vα} act on the system Hilbert space.Is is the identity matrix. Between collision events the systemdynamics is defined by an arbitrary Lindblad generatorLs (uni-tary plus dissipative contributions). Thus, given that the lastevent happened at time t ′, the interevent evolution follows fromthe propagator exp[(t − t ′)Ls]. By assuming that the collisiontimes define a renewal process, with waiting time distributionw(t) [5], it is possible to demonstrate that the average systemdensity matrix ρs

t is governed by the equation [6]:

d

dtρs

t = Ls

[ρs

t

] +∫ t

0dt ′k(t − t ′) Cs

{exp[(t − t ′)Ls]ρ

st ′}. (2)

The superoperator Cs and the kernel function read

Cs = Es − Is , k(u) = uw(u)

1 − w(u), (3)

where u is a Laplace variable [f (u) ≡ ∫ ∞0 dte−utf (t)].

Notice that here, due to the assumed (stationary) renewalproperty, the kernel does not depend separately on the timevariables t and t ′. On the other hand, if [Cs ,Ls] = 0, in an

interaction representation with respect to Ls , Eq. (2) (underthe replacement Ls → 0) recovers the evolution introducedin Ref. [5].

B. Bipartite Markovian dynamics

We introduce a bipartite arrangement defined by the systemof interest S and an auxiliary (ancilla) system A. Their jointdensity matrix is ρsa

t . Therefore their marginal density matricesfollow from a partial trace:

ρst = Tra

[ρsa

t

], ρa

t = Trs[ρsa

t

]. (4)

The bipartite dynamics is defined by a Markovian Lindbladequation,

d

dtρsa

t = Lρsat = (Ls + La + Csa)ρsa

t , (5)

where the (arbitrary) Lindblad generators Ls and La define thesystem and ancilla dynamics, respectively. The contributionCsa introduces their mutual interaction.

Now we ask about the possibility of finding specific system-ancilla interactions such that the marginal system densitymatrix ρs

t [Eq. (4)] fulfills the evolution (2). With this goalin mind, the superoperator Csa is defined as

Csa[ρ] = 1

2

∑α,l

γl([Vαl,ρV†αl] + [Vαlρ,V

†αl]), (6)

where γl are dissipative rates and the operator Vαl is

Vαl = Vα ⊗ |al〉 〈a0| . (7)

The set of operators {Vα} are the same as those in Eq. (1).The states {|a0〉 , |al〉}, l = 1,2, . . . , dim{Ha} − 1, form acomplete orthogonal normalized basis in the Hilbert spaceHa of the ancilla system. Hence, except for the state |a0〉 ,

the index l runs over all available states. Notice that operators(7) introduce irreversible ancilla transitions between the state|a0〉 and any of the remaining possible states |al〉 , that is,|a0〉 � |al〉 .

1. Ancilla dynamics

With the previous choice of operators [Eq. (7)], it is simpleto write down a closed Markovian evolution for the ancillastate ρa

t . From Eqs. (5) and (6) we get

d

dtρa

t = Laρat = (La + Ca)ρa

t . (8)

The extra Lindblad term reads

Caρat = 1

2

∑l

γl

([Al,ρ

at A

†l

] + [Alρ

at ,A

†l

]), (9)

where Al = |al〉 〈a0| . Straightforwardly, this superoperatorcan be rewritten as

Caρat = − 1

2γ{|a0〉〈a0|,ρa

t

}+ + γ 〈a0| ρa

t |a0〉 ρa. (10)

Here, {· · · }+ denotes an anticommutation operation, and theancilla state ρa is

ρa =∑

l

γl

γ|al〉 〈al| , γ =

∑l

γl, (11)

which in fact satisfies Tra [ρa] = 1.

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2. Non-Markovian system dynamics

In contrast to Eq. (8), the evolution of the system state ρst is

non-Markovian. Its calculation is a little more involved, whichhere is obtained by using a projector technique [2,3]. I nowintroduce the projectors P and Q:

Pρsat = Tra

[ρsa

t

] ⊗ ρa, P + Q = Isa, (12)

where Isa is the identity matrix in the bipartite system-ancillaHilbert space, and ρa is the ancilla state (11). The election ofthis projector definition will become clear in the next section.

The bipartite evolution (5) can be projected in relevant andirrelevant contributions [3]

d

dtPρsa

t = PL(P + Q)ρsat , (13)

d

dtQρsa

t = QL(P + Q)ρsat . (14)

On the other hand, in consistency with the projector definition(12), a separable state defines the bipartite initial condition,

ρsa0 = ρs

0 ⊗ ρa, (15)

where ρs0 is an arbitrary system state. With this initial state, it

follows that Qρsa0 = 0. Therefore, Eq. (14) can be integrated

[3] as Qρsat = ∫ t

0 dt ′ exp[QL(t − t ′)]QLPρsat ′ , which in turn,

after replacing in Eq. (13), leads to the convoluted evolution

d

dtPρsa

t = PLPρsat + PL

∫ t

0dt ′ exp[QL(t − t ′)]QLPρsa

t ′ .

(16)

The superoperator L is defined by Eq. (5). From Eqs. (6) and(7), it can be rewritten as

L[•] = (Ls + La)[•] − 12γ {|a0〉〈a0|,•}+

+ γ Es[〈a0| • |a0〉] ⊗ ρa, (17)

where the collision superoperator Es and the ancilla state ρa

are defined by Eqs. (1) and (11), respectively. Equations (12)and (17) lead to

PL[•] = {Ls(Tra [•]) + γ Cs(〈a0| • |a0〉)} ⊗ ρa, (18)

where Cs follows from Eq. (3). With these last two expressionsit is possible to evaluate all contributions in Eq. (16). Byusing that 〈a0| ρa |a0〉 = 0, we get PLPρsa

t = Ls[ρst ] ⊗ ρa,

and QLPρsat ′ = ρs

t ′ ⊗ La[ρa], where the ancilla superoperatorLa follows from Eq. (8). We have also used that Ca[ρa] = 0[see Eqs. (10) and (11)]. Similarly, it is possible to demonstratethat QL(ρs

t ′ ⊗ La[ρa]) = (Ls + La)(ρst ′ ⊗ La[ρa]), which by

induction implies the expression

exp[QLt]QLPρsat ′ = exp[(Ls + La)t]

(ρs

t ′ ⊗ La[ρa]). (19)

By introducing the previous results in Eq. (16), using thatTra[La(•)] = 0, straightforwardly we recover the convolutedevolution (2) with the kernel function

k(t) = γ 〈a0| exp (tLa)La[ρa] |a0〉 , (20a)

= γd

dt〈a0| exp (tLa)[ρa] |a0〉 . (20b)

This is the main result of this section. It demonstrates thatthe non-Markovian evolution (2) also arises as the marginal

dynamics of a Markovian bipartite dynamics. In addition, herethe kernel function is not arbitrary. In fact, it is completelydetermined from the ancilla dynamics [see Eqs. (8) and(20)]. Notice that the solution map ρs

0 → ρst associated with

the evolution (2) with the kernel (20) is, by construction,completely positive.

III. QUANTUM MEASUREMENT THEORY

In the previous section we have found an underlyingbipartite Markovian dynamics that leads to the non-Markoviansystem dynamics. Here, over the same basis we find a clearphysical interpretation to the ensemble of realizations [5,6]associated with the master equation (2).

A. Quantum jumps in the bipartite dynamics

The realizations of the collision model do not rely ona quantum measurement theory. Nevertheless, this link canbe established by studying the bipartite dynamics when ameasurement process is performed over the ancilla system.Specifically, we assume that the apparatus is sensitive to allancilla transitions |a0〉 � |al〉. As the bipartite dynamics isMarkovian, from a standard quantum jump approach [27,28]it is possible to associate each realization of the monitoringprocess with a realization in the system-ancilla Hilbert spacesuch that

ρsat = ρst

sa(t). (21)

Here, ρstsa(t) is a stochastic density matrix and the overbar

denotes an ensemble average. The time evolution of ρsat is

defined by Eq. (5). As usual, the stochastic dynamics ofρst

sa(t) consists of disruptive transformations associated to eachrecording event, while in the intermediate time intervals it issmooth and nonunitary [27,28].

Consistently with a quantum measurement theory [2], ineach detection event the bipartite state suffers the (measure-ment) transformation

ρ → Mρ = J ρ

Trsa[J ρ], (22)

where the superoperator J takes into account all possibletransitions |a0〉 � |al〉 that lead to a detection event. Assumingthat La does not induce these kinds of transitions, from Eq. (6)we write

Mρ =∑

αl γlVαlρV†αl

{Trsa[∑

αl γlV†αlVαlρ]}

, (23a)

= Es 〈a0| ρ |a0〉Trs[〈a0| ρ |a0〉] ⊗ ρa. (23b)

This last expression follows from the definition of the operatorsVαl [Eq. (7)]. On the other hand, the conditional evolution ofρst

sa(t) between detection events is given by the normalizedpropagator [27,28]

Tc(t)ρ = T (t)ρ

Trsa[T (t)ρ], (24)

where the un-normalized propagator T (t) is

T (t)ρ = exp[tD]ρ. (25)

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Here, the exponential superoperator is defined by the generatorD, which is the complement ofJ , that is,L = D + J . Hence,from Eq. (6) it reads

Dρ = (Ls + La)ρ − γ

2{|a0〉 〈a0| ,ρ}+. (26)

The measurement transformation M and the propagatorTc(t) completely define the structure of the realizations ofρst

sa(t). It only remains to define an algorithm that allowsthe random detection times to be obtained. Here they arecharacterized through a survival probability function P0(t |ρ)[27]. Given that at time τ the bipartite system state is ρ, theprobability of not having any detection up to time t is [29]

P0(t − τ |ρ) = Trsa[T (t − τ )ρ] = Trsa[etDρ]. (27)

With this function the realizations can be obtained as follows:Given the initial state ρsa

0 , the time t1 of the first detection eventfollows by solving the equation P0(t1 − 0|ρsa

0 ) = r, where r

is a random number in the interval (0,1). The dynamic ofρst

sa(t) in the interval (0,t1) is defined by Eq. (24). At t = t1the disruptive transformation [Eq. (23)] ρst

s (t1) → Mρstsa(t1) is

applied. The subsequent dynamics is the same. In fact, afterthe nth measurement event at time tn, ρ

stsa(tn) → Mρst

sa(tn), thetime tn+1 for the next detection event follows from P0(tn+1 −tn|Mρst

sa(tn)) = r, where again r is a random number in theinterval (0,1). The dynamic in the interval (tn,tn+1) is definedby the conditional propagator (24). The realizations generatedwith this algorithm fulfill Eq. (21) (see, for example, AppendixA of Ref. [22]).

B. Stochastic realizations

The standard quantum jump approach allows the realiza-tions of ρst

sa(t) to be defined. Straightforwardly from this itis possible to obtain the partial stochastic dynamics of eachsystem,

ρsts (t) = Tra

[ρst

sa(t)], ρst

a (t) = Trs[ρst

sa(t)]. (28)

Furthermore, from Eq. (21), the relations ρst = ρst

s (t) andρa

t = ρsta (t) are also valid. Given the separable initial condition

(15), from Eqs. (23) and (26) it is simple to realize that ρstsa(t)

becomes separable at all times:

ρstsa(t) = ρst

s (t) ⊗ ρsta (t). (29)

In fact, given the absence of initial correlations, the conditionaldynamic (24) remains separable [see Eq. (26)]. Furthermore,in each detection event, given a separable input, the postmea-surement state also becomes separable. Nevertheless, noticethat ρst

s (t) and ρsta (t) are statistically correlated. Below, we

describe their dynamics.

1. Ancilla realizations

After taking a partial trace over Eq. (23), from Eq. (29) wededuce that in each measurement event the ancilla state suffersthe transformation

ρsta (t) → Trs

[Mρst

sa(t)] = Jaρ

sta (t)

Tra[Jaρst

a (t)] = ρa, (30)

where the ancilla superoperator Ja is

Ja[ρ] = γ 〈a0| ρ |a0〉 ρa. (31)

Hence, the collapsed ancilla state is always the same [Eq. (11)].Similarly, from Eqs. (25) and (26), we deduce that betweendetection events the conditional ancilla dynamics is definedby the (un-normalized) superoperator Ta(t)ρ = exp[tDa]ρ,

where

Daρ = Laρ − γ

2{|a0〉 〈a0| ,ρ}+. (32)

For separable initial conditions, this propagator also applies atthe initial time. This simplification explains the chosen initialstate (15) and the projectors (12).

From Eqs. (26) and (27), we notice that the survivalprobability can be rewritten as [P0(t − τ |ρ) → P0(t − τ )]:

P0(t − τ ) = Tra {exp[Da(t − τ )]ρa} . (33)

In fact, the ancilla state is always the same after a detectionevent. Consequently, the measurement statistics correspondto a renewal process, that is, the interevent probabilitydistribution is always the same. On the other hand, it issimple to realize that the measurement transformation (30),the conditional ancilla dynamics defined by Eq. (32), andthe survival probability (33) also arise by formulating thequantum jump approach over the basis of Eq. (8). In fact,La = Da + Ja.

2. System realizations

Given the separability property (29), from Eq. (23) itfollows that in each detection event (ancilla measurementapparatus) the system suffers the transformation

ρsts (t) → Tra

[Mρst

sa(t)] = Es

[ρst

s (t)], (34)

that is, the transformation associated to a collision event. Onthe other hand, given that a measurement event happened attime τ, from Eqs. (24) and (26) we deduce that the posteriorsystem conditional evolution is given by

ρsts (t) = Tra

[Tc(t − τ )ρst

sa(τ )] = exp[Ls(t − τ )]ρst

s (τ ). (35)

This interevent evolution also corresponds to the dynamicsof the QCM. Therefore, by assuming that the measurementprocess is performed over the ancilla system, the realizationsof the system of interest have the same structure than in thephenomenological QCM. This is the main result of this section.Notice that each system collisional event happens when themeasurement apparatus detects an ancilla transition.

The renewal property of the realizations was provenpreviously. In fact, from the survival probability (33) we definethe waiting time distribution w(t) = −(d/dt)P0(t), whichdelivers

w(t) = −Tra{Da exp[tDa]ρa}, (36a)

= γ 〈a0| exp(tDa)[ρa] |a0〉 . (36b)

In deriving this expression we used Eq. (32) and thatTra[Laρ] = 0. Hence, in the present modeling the quantumjump approach allows the waiting time distribution to bewritten in terms of the ancilla dynamics. Indeed, fromEqs. (34) and (35), we deduce that the ancilla dynamics mainlydetermine the statistics of the system realizations.

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C. Consistence between master equation and ensemble ofrealizations

For showing the consistency of the developed results, itremains to demonstrate that the waiting time distribution (36),which determines the realizations statistics, and the kernel(20), which determines the density matrix evolution, fulfill inthe Laplace domain the relation (3).

The Laplace transform of Eq. (36) reads

w(u) = γ 〈a0| 1

u − Da

[ρa] |a0〉 , (37)

while from Eq. (20) we obtain

k(u)

u= γ 〈a0| 1

u − La

[ρa] |a0〉 . (38)

In deriving this expression we used that 〈a0| ρa |a0〉 = 0. Onthe other hand, using that La = Da + Ja , it follows that

1

u − La

=∞∑

n=0

[1

u − Da

Ja

]n 1

u − Da

. (39)

By introducing this expression in Eq. (38) and by using thedefinition (31) we get

k(u)

u=

∞∑n=1

w(u) = w(u)

1 − w(u), (40)

which recovers the relation (3) associated to the phenomeno-logical approach.

IV. EXAMPLE

In this section, we study the dynamics of a two-levelsystem, which in turn may be read, for example, as a qubitunit. In quantum information arrangements it is expected thatdecoherence and dissipation are “mediated” by interactionswith extra quantum subunits. Therefore, as ancilla we consideranother system whose dynamics is able to develop quantumcoherent effects. For simplicity it is also taken as a two-levelsystem.

In the approach developed in the previous sections, the col-lision statistics is completely defined by the ancilla dynamics.Hence, in the next example, its structure depends on underlyingquantum coherent effects. We remark that this feature is foreignin phenomenological formulations, where the waiting timedistribution is usually defined by a linear combination ofdecaying exponential functions [5–8]. We demonstrate thatthese kinds of distributions arise when the ancilla dynamics iscompletely incoherent. This property motivates the dynamicsstudied below. Both dephasing and dissipative channels areformulated.

A. Dephasing channel

As system we consider a two-level system whose Hamilto-nian reads Hs = hωsσz/2, where ωs is the transition frequencybetween its eigenstates, denoted as |±〉 , while σz is the z-Paulimatrix. The ancilla system is also a two-level system. In aninteraction representation with respect to Hs , the evolution of

the bipartite state ρsat reads

dρsat

dt= −i�

2

[Is ⊗ σx,ρ

sat

] + γ

2

([V,ρsa

t V †] + [Vρsa

t ,V †]).(41)

The first unitary contribution defines the ancilla Hamiltonian.It is given by the x-Pauli matrix σx , written in the basis of σz

eigenstates: |±〉. The Lindblad contribution is written in termsof the operator [see Eq. (7)]

V = σz ⊗ σ. (42)

Here, σ = |−〉〈+| is the lowering operator acting on the ancillastates |±〉. Hence, V leads to a dissipative coupling betweenboth systems. The initial bipartite state [see Eq. (15)] is takenas

ρsa0 = ρs

0 ⊗ |−〉〈−|, (43)

where ρs0 is an arbitrary system state. The ancilla begins in its

lower state.Performing the partial trace ρa

t = Trs[ρsat ], the bipartite

evolution (41) leads to

dρat

dt= −i�

2

[σx,ρ

at

] + γ

2

([σ,ρa

t σ †] + [σρa

t ,σ †]). (44)

This marginal ancilla dynamics corresponds to a quantumfluorescent system [2,28], where γ defines its natural decayrate while � is the Rabi frequency. On the other hand, theinteraction defined by Eq. (42) leads to a decoherence systemchannel [33]. Hence, only the system coherences are affectedby the undesirable interaction.

1. System stochastic realizations

The measurement apparatus records the ancilla transitions|+〉 � |−〉 . Therefore, from Eqs. (23) and (42) we deduce thatin each measurement event the ancilla collapses to its groundstate ρa = |−〉 〈−| , while the system suffers the completelypositive transformation

Es[ρ] = σzρσz. (45)

As is well known, this superoperator leads to a change ofsign in the system coherences [5]. On the other hand, duringthe successive measurement events the system dynamics isfrozen, that is, it does not evolve. This conclusion followsfrom Eqs. (35) and (41).

The statistics of the time interval between successivedetection events define a renewal process. Its probabilitydistribution is given by Eq. (36). Under the associations |a0〉 →|+〉 , andDa[ρ] = −(i�/2)[σx,ρ] − (1/2)γ {σ †σ,ρ}+, we getthe waiting time distribution

w(t) = 4γ�2e−γ t/2

{sinh[(t/4)

√γ 2 − 4�2]√

γ 2 − 4�2

}2

. (46)

Notice that Eqs. (45) and (46) completely define the systemrealizations.

In Fig. 1 we show a realization of the system coherence〈+| ρst

s (t) |−〉 . In order to show the consistence of thedeveloped approach, it was obtained from the realizations of

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FIG. 1. Realizations of matrix elements of the stochastic densitymatrix ρsa

st (t) and ρsst(t): (a) 〈++| ρst

sa(t) |−+〉, (b) 〈+−| ρstsa(t) |−−〉,

and (c) 〈+| ρsts (t) |−〉 . The characteristic parameters of the bipartite

evolution (41) satisfy �/γ = 6.

the underlying bipartite dynamics,

〈+|ρsts (t)|−〉 = 〈++| ρst

sa(t)|−+〉 + 〈+−|ρstsa(t)|−−〉,

(47)

that is, from the partial trace of ρstsa(t). The states {|ij 〉},

i,j = +, − , provide a complete basis of the bipartite Hilbertspace. The realizations of ρst

sa(t) follow from a “standardMarkovian quantum jump approach” formulated on the basisof Eq. (41). We have taken the initial condition ρst

sa(0) =|x+〉 〈x+| ⊗ |−〉 〈−| , where |x+〉 = (1/

√2)(|+〉 + |−〉) is an

eigenstate of σx. In Fig. 1(a), we see that in each recordingevent the bipartite coherence 〈++| ρst

sa(t) |−+〉 collapses tozero:

〈++|Mρstsa(t) |−+〉 = 0. (48)

This result follows from the action of the operator (42),which induces the ancilla transitions |+〉 � |−〉 . On the otherhand, the bipartite coherence 〈+−| ρst

sa(t) |−−〉 suffers thedisruptive changes 〈+−| ρst

sa(t) |−−〉 → −〈+−| ρstsa(0) |−−〉

[Fig. 1(b)]. By calculating the measurement transformation(23), from Eq. (42) we get

〈+−|Mρstsa |−−〉 = − 〈++| ρst

sa |−+〉〈++| ρst

sa |++〉 + 〈−+| ρstsa |−+〉 .

By an explicit calculation of the conditional evolution definedby the operator D [Eq. (26)], it follows that the quotient ofthe previous bipartite matrix elements is an invariant of the

conditional evolution, delivering the observed property

〈+−|Mρstsa(t) |−−〉 = − 〈+| ρst

s (0) |−〉 , (49)

where we have used that 〈+−| ρstsa(0) |−−〉 = 〈+| ρst

s (0) |−〉[Eq. (43)]. Therefore, in each measurement event the coher-ence 〈+−| ρst

sa(t) |−−〉 , besides a change of sign, recovers itsinitial value.

In Fig. 1(c) we plot the realization of 〈+| ρsts (t) |−〉

obtained from Eq. (47), that is, by adding the two bipartitecoherences. As both coherences 〈++|Mρst

sa(t) |−+〉 and〈+−|Mρst

sa(t) |−−〉 always oscillate in a complementary way,during the interevent time intervals 〈+| ρst

s (t) |−〉 is constant,while in the measurement events it changes sign. In thisway, we explicitly show that the underlying quantum jumpapproach led to the realizations of the phenomenologicalcollision model. In fact, the action of the superoperator (45)only introduces a change of sign in the system coherences. In asimilar way, it is possible to show that the system populationsare not affected by the dynamics, that is, 〈±| ρst

s (t) |±〉 =〈±| ρst

s (0) |±〉 .

2. Density matrix evolution

In Fig. 2 we show the average coherence behavior obtainedfrom the ensemble of realizations shown in Fig. 1 (noisycurve). Furthermore, we present the exact solution of thecoherence that follows from the master equation (2) (black fullline). Taking into account the underlying Lindblad equation(41), it can be written as

d

dtρs

t =∫ t

0dt ′k(t − t ′)Cs

[ρs

t ′]. (50)

The superoperator Cs = (Es − Is), from Eq. (45), reads

Cs[•] = 12 ([σz, • σz] + [σz • ,σz]). (51)

FIG. 2. System coherence. Full line, exact solution Eq. (53).Dotted (noisy) line, average coherence 〈+| ρst

s (t) |−〉 obtained byaveraging 103 realizations. Gray line, relative entropy E(ρs

t ||ρs∞),

Eq. (55). The parameters are the same as in Fig. 1, �/γ = 6.

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On the other hand, the kernel is determined by the generalexpression (20). From Eq. (44) it follows that

k(t) = 2γ�2e−(3/4)γ t

{sinh[(t/4)

√γ 2 − 16�2]√

γ 2 − 16�2

}. (52)

This kernel and the waiting time distribution (46) fulfill theLaplace relation (3).

Consistently with the system stochastic realizations,Eq. (50) does not modify the populations, 〈±| ρs

t |±〉 =〈±| ρs

0 |±〉 . On the other hand, working in a Laplace domain,the coherences c±

t ≡ 〈±| ρst |∓〉 read

c±t = c±

0

{e−γ t 2�2

γ 2 + 2�2+ e−γ t/4

[γ 2

γ 2 + 2�2cosh(ϕt)

+ γ (γ 2 + 8�2)

4(γ 2 + 2�2)

sinh(ϕt)

ϕ

]}, (53)

where for shortening the expression we introduced the “fre-quency,”

ϕ =√

(γ /4)2 − �2. (54)

The same expression follows from the alternative solutionc±t = 〈±| ρs

t |∓〉 = 〈±±| ρsat |∓±〉 + 〈±∓| ρsa

t |∓∓〉 , whereρsa

t is the solution of the bipartite evolution (41). Notice thatin Eq. (53), besides a monotonic decaying contribution, thetwo remaining terms may develop an oscillatory behavior. Asshown in Fig. 2, Eq. (53) correctly fits the average ensemblebehavior.

3. Environment-to-system backflow of information

The analysis of Refs. [25,26] demonstrate that QCMsmay lead to non-Markovian effects such as an environment-to-system backflow of information [23]. This property orphenomenon can be defined on the basis of “any measure”that in the Markovian case presents a monotonic time decaybehavior [2]. One well-known example is the relative entropybetween two states [24,25]. As we are not interested inquantifying the non-Markovian effects, for simplicity here weconsider the relative entropy with respect to the stationarystate,

E(ρs

t

∣∣∣∣ρs∞

) = Trs[ρs

t

(ln2 ρs

t − ln2 ρ∞s

)], (55)

where ρ∞s = limt→∞ ρs

t . Hence, the backflow of informationarises if there exist times t2 > t1 such that E(ρs

t2||ρs

∞) >

E(ρst1||ρs

∞). Below we show that this feature arises in thedynamics described previously.

In Fig. 2 we also plotted E(ρst ||ρs

∞) (gray full line)where ρs

t is the solution of Eq. (50). The stationary stateis the diagonal matrix ρs

∞ = diag{〈+| ρs0 |+〉 , 〈−| ρs

0 |−〉}.Clearly the time behavior is nonmonotonous, indicating abackflow of information. Furthermore, the oscillatory behaviorof E(ρs

t ||ρs∞) is correlated with the oscillatory behavior of the

coherences, which arise whenever ϕ is a complex quantity,that is, from Eq. (54), � > (γ /4).

4. Incoherent ancilla dynamics

For the dynamics (41), the ancilla dynamics developquantum coherent effects [Eq. (44)], which in turn determinethe waiting time distribution [Eq. (46)]. Here, we introduce

an alternative ancilla dynamics which only induces incoherenttransitions. Instead of Eq. (41), for the same system S, we takethe bipartite evolution as

dρsat

dt= γ

2

([V,ρsa

t V †] + [Vρsa

t ,V †])+ β

2

([A,ρsa

t A†] + [Aρsa

t ,A†]), (56)

with initial condition ρsa0 = ρs

0 ⊗ |−〉〈−|, while

V = σz ⊗ σ, A = Is ⊗ σ †. (57)

Hence, the ancilla dynamics [Eq. (8)] only lead to the

incoherent (classical) transitions |+〉 γ� |−〉 and |−〉 β� |+〉 .

The corresponding statistical behavior is defined by a (two-level) classical rate master equation.

We assume that the recording apparatus is only sensitiveto the ancilla transition |+〉 � |−〉 , that is, the transitioninduced by the operator V. In this situation, from Eqs. (23)and (57), we deduce that the collisional superoperator againreads Es[ρ] = σzρσz [Eq. (45)]. Thus, the system evolutionis given by Eq. (50). Nevertheless, the kernel follows fromEq. (3), where the waiting time distribution can be calculatedfrom Eq. (36). We get

w(u) =(

γ

u + γ

)(β

u + β

). (58)

In the time domain w(t) is the convolution of two exponentialfunctions. The system coherences become c±

u = c±0 (u + γ +

β)/[2u2 + 2u(γ + β) + γβ], which can also be written asa lineal combination of decaying exponential functions.Independently of the initial conditions, in this case thedynamics does not present an environment-to-system backflowof information, suggesting that underlying coherent effectsmay be necessary for the development of this phenomenon.

Taking an ancilla system with a higher number of states, allof them coupled via incoherent transitions, the waiting timedistribution becomes defined by more complex expressionswhich in the time domain are linear combinations of decayingexponential functions. For example, taking a unidirectionalcoupling |a0〉 � |a1〉 � · · · |am〉 � |a0〉 , all of them with rateγ, the waiting time distribution becomes w(u) = [γ /(u +γ )]m+1. These kinds of distributions, which rely on incoherentancilla dynamics, were considered, for example, in Ref. [8].

B. Dissipative channels

In the previous example, Eqs. (50) and (51) define anon-Markovian decoherence channel. One may also considerinteractions that lead to dissipative channels. For example,by maintaining the ancilla dynamics (44), a depolarizing [31]non-Markovian channel arises by introducing two bipartiteLindblad terms [α = x,y in Eq. (6)] defined by the operatorsVx = √

pσx ⊗ σ, and Vy = √1 − pσy ⊗ σ, where the param-

eter p satisfies 0 < p < 1. With the same collision statistics[Eq. (46)], in this case the stationary system state becomesρs

∞ = (1/2) Is . A thermal stationary state can be obtained byconsidering a generalized amplitude damping superoperator[31].

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V. GENERALIZED COLLISIONAL MODELS

In the previous sections we associated the basic masterequation of the collision model [Eq. (2)] with an underlyingMarkovian microscopic dynamics [Eq. (5)]. Furthermore, therealizations of the model, given that the ancilla system iscontinuously monitored in time, were established on the basisof the quantum jump approach. In this section, we show thatthese results also apply in different possible generalizations ofthe basic approach.

A. Nonstationary renewal collision dynamics

The basic ingredients of the present approach remain validwhen the evolution of the ancilla system, in the bipartite Lind-blad dynamics (5), depends explicitly on time, La → La(t).Under this situation, the main change is the measurementstatistics. While it remains a renewal process, the waiting timedistribution explicitly depends on the observation time. Thiscase can be worked out with the elements introduced in theprevious sections.

B. Nonrenewal collision statistics

With the same system realizations, the formalism maybecome nonrenewal when the measurement process is non-renewal. Basically this situation occurs whenever the ancillaresetting state is not always the same. This case arises, forexample, when the operators (7) are generalized as

Vαlk = Vα ⊗ |al〉⟨ak

0

∣∣ . (59)

Hence, instead of a unique state |a0〉 , here many of them playthe same role. Assuming that the measurement apparatus issensitive to “all transitions”

∣∣ak0

⟩� |al〉 , the stochastic ancilla

becomes nonrenewal. This case may correspond, for example,to optical cascade systems [29].

While the structure of the system realizations remains thesame, the statistics of the interevent time intervals can only bedetermined by knowing the ancilla state at all times. Therefore,for generating the system realizations, unavoidably one mustalso generate the ancilla realizations.

C. Non-Markovian intercollision dynamics

Maintaining the renewal property, in Ref. [8] Vacchini intro-duced an interesting generalization that consists in assumingthat the interevent dynamics is non-Markovian. This situationnaturally arises when considering a system interacting succes-sively with a string of qubits systems [35–37].

Instead of the Markovian evolution defined by Eq. (35), itis taken as

ρsts (t) = G(t − τ )

[ρst

s (τ )], (60)

where G(t) is an arbitrary (trace preserving), completelypositive propagator that cannot be written as a semigroup,G(t) �= exp[tLs] [8]. Here, we demonstrate that this case canalso be covered with the present formalism.

The generalized QCM can be embedded in a tripartiteunderlying Lindblad equation. Hence, besides the system ofinterest S, the ancilla system A, we consider an extra auxiliarysystem B. The evolution of their joint density matrix ρsab

t is

written as

d

dtρsab

t = Lρsabt = (Lsb + La + Csab)ρsab

t . (61)

The first superoperator reads

Lsb = Ls + Lb + Csb. (62)

Here, Ls and Lb are arbitrary Lindblad equations for thesystems S and B, respectively. Csb is an extra Lindbladcontribution that introduces an arbitrary interaction (unitaryand dissipative) between them. As before, La defines thedynamics of the ancilla system A. The contribution Csab

introduces a dissipative interaction between the three systems,

Csab[ρ] = 1

2

∑α,l,m

γl([Vαlm,ρV†αlm] + [Vαlmρ,V

†αlm]), (63)

where γl are the dissipative rates and the operators are

Vαlm = Vα ⊗ |al〉 〈a0| ⊗ |b0〉 〈bm| . (64)

The system operators Vα and the states |al〉 are the same as inEq. (7), where the index l = 1,2, . . . , dim{Ha} − 1 does notinclude the single state |a0〉 . On the other hand, the states |bm〉 ,

m = 0,1, . . . dimHb − 1 form a complete orthonormal basisin the Hilbert space Hb of B. Notice that here the state |b0〉must be included in the summation index m. For simplicity,the tripartite initial state is chosen to be separable,

ρsab0 = ρs

0 ⊗ ρa ⊗ |b0〉 〈b0| , (65)

where ρs0 is an arbitrary system state and ρa follow from

Eq. (11).We determine the system realizations over the basis of

a standard quantum jump approach formulated on the basisof Eq. (61). As before, the measurement apparatus is onlysensitive to transitions of the auxiliary system A. Therefore thetransformation associated with each detection event, instead ofEq. (23), here reads

Mρ = Es[∑

m 〈a0bm| ρ |a0bm〉]Trs[

∑m 〈a0bm| ρ |a0bm〉] ⊗ ρa ⊗ |b0〉 〈b0| . (66)

The collisional superoperator Es is given by Eq. (1). On theother hand, the (tripartite) conditional dynamics can be writtenas in Eqs. (24) and (25). Nevertheless, here the superoperatorD reads

Dρ = (Lsb + La)ρ − γ

2{|a0〉 〈a0| ,ρ}+. (67)

In deriving this result we used that∑

α V †αVα = Is , and∑dimHb−1

b=0 |bm〉 〈bm| = Ib. With the previous definition of D,

the expression for the survival probability [Eq. (27)] remainsalmost the same, P0(t − τ |ρ) = Trsab[etDρ].

Over the basis of the previous two equations and the initialcondition (65), it is simple to conclude that the tripartitestochastic state ρst

sab(t) [ρstsab(t) = ρsab

t ] can be written at alltimes as

ρstsab(t) = ρst

sb(t) ⊗ ρsta (t). (68)

The dynamics for the ancilla state ρsta (t) remains the same as

before, that is, Eqs. (30)–(32) are not modified by the introduc-tion of system B. Consequently, the measurement statistics,

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defined by the survival probability (33), or equivalently, thewaiting time distribution (36), are also the same.

The induced stochastic system dynamics follows fromρst

s (t) = Trab[ρstsab(t)]. Hence, in each recording event the state

suffers the disruptive transformation

ρsts (t) → Trab

[Mρst

sab(t)] = Es

[ρst

s (t)]. (69)

This expression follows straightforwardly from Eq. (66), afterusing Eq. (68) and noting that

∑m 〈bm| • |bm〉 = Trb[•]. On

the other hand, the intercollision dynamic [Eq. (35)] hereis ρst

s (t) = Trab[Tc(t − τ )ρstsab(τ )]. Given that Tc(t) follows

from Eqs. (24) and (25), the operator D [Eq. (67)] and theseparability property defined by Eqs. (66) and (68) lead to

ρsts (t) = Trb{exp[(t − τ )Lsb] |b0〉 〈b0|}ρst

s (τ ). (70)

This conditional dynamics recovers the phenomenologicalproposal Eq. (60). Hence, the non-Markovian propagator G(t)reads

G(t) = Trb[exp (tLsb) |b0〉 〈b0|]. (71)

This is the main result of this section. It implies that thegeneralized phenomenological approach of Ref. [8] can bedescribed over the basis of a tripartite Markovian evolution. IfLsb = Ls + Lb, that is, when the system S and the ancillaB do not interact, the formalism of the previous section,G(t) = exp(tLs), is recovered. Hence, given the structure ofthe operators (64), it becomes clear that the main role of systemB is to modify the intercollision system dynamics.

The realizations defined by the measurement transforma-tion (66) and the interevent dynamics (71) are similar to thosefound in Ref. [22], where a non-Markovian generalizationof the quantum jump approach was defined over a similarbasis by assuming that the system of interest is submitted toa measurement process. Nevertheless, the present treatmentexplicitly demonstrates that collisional dynamics can only belinked with a quantum measurement theory if the monitoringaction is performed over the auxiliary ancilla system.

The nonlocal character of the propagator G(t) can be shownby writing Eq. (71) in the Laplace domain as G(u) = Trb[(u −Lsb)−1|b0〉〈b0|]. This expression can be rewritten as G(u) ={Tra[(u − Lsb)−1(u − Lsb)|b0〉〈b0|]}−1{[G(u)]−1}−1. Using inthe curly brackets that X−1Y−1 = (YX)−1, where X and Y arearbitrary matrices, it follows that G(u) = {[G(u)]−1(uTra((u −Lsb)−1|b0〉〈b0|] − Tra[(u − Lsb)−1Lsb|b0〉〈b0|)]}−1, which inturn leads to

G(u) = 1

u + K(u), (72)

where the system superoperator K(u) is

K(u)={

Trb

[1

u − Lsb

|b0〉〈b0|]}−1

Trb

[1

u − Lsb

Lsb|b0〉〈b0|].

Hence, in the time domain we get

d

dtG(t) =

∫ t

0dt ′K(t − t ′)G(t ′), (73)

where K(t − t ′) is defined by its Laplace transform K(u).The evolution of ρs

t can be obtained from Eq. (61) byusing projector techniques. A simpler way is to calculate theaverage behavior of the ensemble of stochastic realizations

(see Ref. [8]). On the other hand, the QCM introduced byCiccarello, Palma, and Giovannetti in Ref. [36], which relieson interaction with a qubits string, can also be recovered fromthe present approach. In fact, as demonstrated in Ref. [8], itarises by taking Es → Is . Hence, each collision only resets theevolution induced by G(t). The results presented by Rybaret al. in Ref. [35] rely on a similar approach. All non-Markovian effects arise because the ancilla string start ina correlated state [37]. Nevertheless, in our approach thatformalism seems to be equivalent to a system-ancilla dynamicscoupled via a unitary evolution, which in turn leads to arandomlike superposition of Hamiltonian system propagators.Therefore, extra analyses are necessary for establishing a fullmapping between both approaches.

In what follows we analyze how different underlyingdynamics lead to dephasing and dissipative intercollisiondynamics [8,36].

1. Dephasing intercollision dynamics

In this example, both the system and the ancillas are two-level systems. Their tripartite Markovian evolution is given byEq. (61). In an interaction representation with respect to thesystem Hamiltonian, we write

Lsb[ρ] = −i

h[Hsb,ρ] = −iλ

2[σz ⊗ Ia ⊗ σx,ρ], (74)

where σj , j = x,y,z, are the Pauli matrices defined in eachHilbert space. Hence, the system of interest S and the auxiliarysystem B are coupled via a Hamiltonian interaction. Theisolated dynamics of ancilla A is unitary,

La[ρ] = −i

h[Ha,ρ] = −i�

2[Is ⊗ σx ⊗ Ib,ρ]. (75)

The dissipative tripartite interaction [Eq. (63)] reads

Csab[ρ] = γ

2

∑m=0,1

([Vm,ρV †m] + [Vmρ,V †

m]). (76)

The index m = 0,1, runs over the basis {|b0〉 , |b1〉} of systemB. The two operators Vm are

Vm = σx ⊗ σ ⊗ |b0〉 〈bm| , (77)

where, as before, σ is the lowering operator, here defined inthe Hilbert space of system A. Consistently with Eq. (65), theinitial tripartite state is

ρsab0 = ρs

0 ⊗ |−〉 〈−| ⊗ |b0〉 〈b0| . (78)

From the previous definitions, Eqs. (66) and (69) lead tothe collision system superoperator

Es[ρ] = σxρσx. (79)

Notice that σx arises from the first (system) operator contribu-tion in Eq. (77). On the other hand, the dynamics of system A

again is defined by Eq. (44). Consequently, the waiting timedistribution is given by Eq. (46). The intercollision dynamicfollows from Eqs. (71) and (74). By an explicit calculation,we get the completely positive (non-Markovian) dephasingsuperoperator

G(t)ρ = 12 [1 + d(t)]ρ + 1

2 [1 − d(t)]σzρσz, (80)

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ADRIAN A. BUDINI PHYSICAL REVIEW A 88, 032115 (2013)

which in turn can be rewritten as

G(t)ρ =(

〈+| ρ |+〉 d(t) 〈+| ρ |−〉d(t) 〈−| ρ |+〉 〈−| ρ |−〉

). (81)

The function d(t) defines the system coherence behavior. Itreads d(t) = cos(λt).

In the first example worked out in Ref. [8], the superoperatoris given by Eq. (79), while the propagator G(t) is givenby Eq. (80) (see Supplemental Material of [8]). Hence, ourresults provide a clear microscopic description for that phe-nomenological model. The waiting time distribution, insteadof Eq. (46), is a classical one like Eq. (58). That case can berecovered, replacing the ancilla dynamics (75) by

La[ρ] = β

2

([A,ρsa

t A†] + [Aρsa

t ,A†]) (82)

with the operator

A = Is ⊗ σ † ⊗ Ib. (83)

As explained previously, diverse “underlying classical” wait-ing time distributions can be obtained by adding extra ancillastates, all of then coupled by incoherent transitions.

2. Dissipative intercollision dynamics

Instead of the dephasing evolution (80), the intercollisiondynamics may also lead to dissipative effects. This property isdefined by the superoperator Lsb [Eq. (74) in the previousexample]. For example, Lsb may correspond to a Jaynes-Cumming interaction, which couples the system to a set ofbosonic field modes initially in the vacuum state [2,24]. Thiscase, which has been studied in Refs. [8,36], can be analyzedover the basis developed previously.

VI. SUMMARY AND CONCLUSIONS

Phenomenological QCMs provided an important theo-retical tool for establishing and describing non-Markoviancompletely positive dynamics. In this paper we have developeda solid physics basis for understanding this approach. It relieson a Markovian embedding of the non-Markovian systemdensity matrix evolution, which in turn allows derivation of thephenomenological trajectories from a quantum measurementtheory.

First, we focused our analysis on the leading case in whichthe collision statistics is defined by a renewal process, whilethe interevent dynamics is defined by a Markovian quantumsemigroup. By using projector techniques we demonstratedthat the non-Markovian density matrix evolution [Eq. (2)]can be obtained, without involving any approximation, from abipartite Markovian dynamics, where the system of interest in-teracts with an auxiliary ancilla system [Eq. (5)]. The memorykernel that determines the system evolution becomes definedby the ancilla dynamics [Eq. (20)]. The proposed Markovian

embedding allows association of a clear microscopic dynamicsto the QCM. In fact, Lindblad equations are linked withwell-defined microscopic dynamics.

In a second step, we assumed that the ancilla systemis continuously monitored in time. Hence, over the basisof the quantum jump approach formulated for the bipartitedynamics, we find that the realizations of the QCM arerecovered from the marginal conditional stochastic systemdynamics [Eq. (21)]. In fact, each recording event of the ancillameasurement apparatus leads to the collisional transformationsof the phenomenological approach [Eq. (34)]. The intercolli-sion system dynamics follows from the conditional bipartitedynamics between detection events [Eq. (35)]. The waitingtime distribution of the interevent time interval also becomesdefined by the ancilla dynamics [Eq. (36)]. In this way, thephenomenological realizations of the collisional approachwere derived from a quantum measurement theory.

The Markovian embedding and the link with the quantumjump approach were explicitly shown through an examplewhere the dynamics of both the system of interest andthe auxiliary one develop in two-dimensional Hilbert spaces(Figs. 1 and 2). In contrast to phenomenological formulations,here the collision statistics arise from quantum coherenteffects developing in the ancilla Hilbert space. A system-to-environment backflow of information characterizes the dy-namics. In opposition, when the ancilla dynamic is completelyincoherent, this feature is absent.

The previous finding provides a solid basis for propos-ing different generalizations of the QCM. For example,nonstationary renewal collision dynamics can be obtainedby introducing an explicit time dependence in the ancilladynamics. Nonrenewal collision statistics can be related to anonrenewal ancilla measurement process. On the other hand,we showed that by introducing a second auxiliary system, theintercollision dynamics becomes defined by a non-Markovianpropagator, Eq. (71). This finding allowed us to recover a recentproposed generalization of the QCM [8], which in fact can alsobe embedded in a Markovian evolution, and their realizationsderived from a quantum measurement theory. From thisresult, we also concluded that some non-Markovian collisionalmodels formulated in terms of qubit logical operations [36] canalso be recovered from our formalism.

The present analysis allows us to read the phenomenologi-cal QCMs from a different perspective. Besides a solid physicalbasis of the corresponding non-Markovian dynamics, thedeveloped approach provides an alternative powerful tool fordescribing non-Markovian memory effects in open quantumsystems.

ACKNOWLEDGMENT

This work was supported by CONICET, Argentina, underGrant No. PIP 11420090100211.

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