Arbitrary length XX spin chains boundary-driven by non-Markovian environments

G. Mouloudakis1,2,∗ T. Ilias1,† and P. Lambropoulos1,2

1Department of Physics, University of Crete, P.O. Box 2208, GR-71003 Heraklion, Crete, Greece2Institute of Electronic Structure and Laser, FORTH, P.O.Box 1527, GR-71110 Heraklion, Greece

(Dated: November 19, 2021)

In this work we provide a recursive method of calculating the wavefunction of a XX spin chaincoupled at both ends to non-Markovian reservoirs with arbitrary spectral density. The methodis based on the appropriate handling of the time-dependent Schrodinger’s equations of motion inLaplace space and leads to closed form solutions of the transformed amplitudes, for arbitrary chainlengths as well as arbitrary initial conditions, within the single-excitation subspace. Results on thedynamical as well as state transfer properties of the system for various combinations of parametersare also presented. In particular, detailed quantitative comparisons for Lorentzian and Ohmicreservoirs are illustrated.

I. INTRODUCTION

One dimensional many-body systems such as quantumspin chains arise in many contexts throughout quantuminformation theory as well as condensed matter physics,due to their versatility as basic resources for the imple-mentation of solid-state devices for quantum computingand quantum communication tasks [1]. Among thesetasks, faithful quantum state transfer [2–9] and long-distance entanglement [10–15] have been investigated forvarious spin chain configurations in great detail through-out the last 20 years or so, with the research on thesefields being still active.

In recent years, much interest has arisen in the studyof properties of open quantum systems interacting withexternal environments, since dissipation and decoherenceare inevitably present in any experimental realization ofsuch systems. Such systems may consist of just a pairof qubits, with focus on the effects of the environmentaldissipation on the generated bipartite entanglement [16–20], up to a whole quantum spin chain with focus on statetransfer [21–24] or short- and long-distance correlations[25–28]. Extensions that account for the non-Markoviancharacter of the surrounding environment have been alsomade [29–35], revealing interesting effects associated withthe memory character of the reservoir, which enable theexchange of information between the system and the en-vironment within finite times.

At the same time, attention has been drawn to a spe-cial class of open quantum systems, i.e. the so-calledboundary-driven open quantum systems [36]. These arequantum systems coupled to external environments attheir edges and are usually studied for their transport[37–40] and thermodynamic properties [41–43]. For 1Dboundary-driven open quantum systems, the differenttemperature or chemical potential between the two edgebaths will cause a current flow from one bath to theother, mediated by the quantum system which can reach

∗ gmouloudakis@physics.uoc.gr† Current Affiliation: Institut fur Theoretische Physik, Albert-

Einstein-Allee 11, Universitat Ulm, 89069 Ulm, Germany

a non-equilibrium steady-state (NESS). The study of thetransport properties of such systems, focusing on ways ofcontrolling the associated current flows efficiently, seemstherefore promising owing to potential applications insolid-state devices such as diodes or transistors [44]. Onthe other hand, the dynamical and quantum state trans-fer properties of such systems [45] are also essential froma practical perspective for the implementation of quan-tum information tasks.

Among the tools available for the study of open quan-tum systems, tensor network numerical methods basedon matrix product states have proven quite efficient forthe simulation of their dynamical behaviour [46, 47].The method of matrix product states can in many casesyield analytical expressions for the expectation valuesof observables in the NESS of open quantum systemsboundary-driven by Lindblad operators for Markovianreservoirs [48–51]. Perturbative approaches [52] as well asapproaches based on quantum trajectories using numeri-cal Monte Carlo techniques [53] have also been exploredin great detail.

In this paper, we extend the study of the dynam-ics of an arbitrary length XX chain boundary-driven bytwo non-Markovian reservoirs. Our theory is cast interms of the Schrodinger’s equations of motion withinthe single-excitation subspace, which can be solved recur-sively in Laplace space, yielding closed form solutions forthe transformed amplitudes for a broad class of spectraldensities of the boundary reservoirs, as well as arbitraryinitial conditions. Results on the quantum state transferproperties of the system for various parameters of the tworeservoirs are also presented. Our work extends the ana-lytical techniques used for dealing with boundary-drivenopen quantum systems, by providing a systematic wayof calculating the whole wavefunction of the spin chainand therefore enabling the study of numerous propertiesof the system at hand.

In much of the extensive literature with methods forthe handling of non-Markovian reservoirs, the Lorentzianor Ohmic spectral densities are used for illustration. Yet,the analytical forms of those spectral densities are fairlydifferent. In order to explore the impact of those formson the behaviour of realistic physical systems, we have

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in addition performed comparative calculations for thosetwo reservoirs. Our results, discussed in section III seemto reveal what one might call differences in the Marko-vianity of the two reservoirs.

The article is organized as follows: In section II we pro-vide the theoretical description of our problem in termsof the time-dependent Schrodinger equation (TDSE) andobtain closed form solutions for the transformed ampli-tudes for arbitrary number of sites, arbitrary spectraldensities of the boundary reservoirs and arbitrary ini-tial conditions within the single-excitation subspace. Insection III we illustrate the potential of our approachthrough the application to two types of non-Markovianboundary reservoirs, namely Lorentzian and Ohmic, bystudying the dynamical or state transfer properties of thesystem, while in section IV we provide some concludingremarks as well as future directions related to our work.

II. THEORY

Our system consists of an N qubit (spin) chain in-teracting with two environments E1 and E2 through itsfirst and last qubit, with coupling strengths g1 and g2,respectively. The interaction between each pair of neigh-boring qubits in the chain is denoted by J . A schematicpresentation of our system is depicted in Fig. 1. TheHamiltonian of our system H = HS + HE + HI , con-sists of three parts; namely, the Hamiltonian of the XXchain HS , the Hamiltonian of the two environments HEand the chain-environments interaction Hamiltonian HI ,given by the relations (~ = 1):

HS =ωe

N∑i=1

|e〉i i〈e|+ ωg

N∑i=1

|g〉i i〈g|

+

N−1∑i=1

J2

(σ+i σ−i+1 + σ−i σ

+i+1

) (1a)

HE =∑λ

ω1λaE1

λ

†aE1

λ +∑λ

ω2λaE2

λ

†aE2

λ (1b)

HI =∑λ

g1(ω1λ)(aE1

λ σ+1 + aE1

λ

†σ−1

)+∑λ

g2(ω2λ)(aE2

λ σ+N + aE2

λ

†σ−N

) (1c)

where ωg and ωe are, respectively, the energies of theground and excited state of each spin (all spins are as-sumed to be identical), ωiλ, i = 1, 2 is the energy of

the λ-mode photon of each environment, aEjλ and a

Ejλ

†,

j = 1, 2, are the annihilation and creation operators ofeach environment, respectively, and σ+

i = |e〉i i〈g| and

σ−i = |g〉i i〈e|, i = 1, . . . , N are the qubit raising andlowering operators, respectively.

g1

1 2 3 N-1 N

E1 E2𝒥 𝒥 𝒥 g2

FIG. 1. Schematic presentation of the system at study. AHeisenberg XX spin chain of arbitrary length is coupled tonon-Markovian reservoirs at its boundaries.

The wavefunction of the compound system (spin chain+ environments) in the single-excitation space is ex-pressed as:

|Ψ(t)〉 =

N∑i=1

ci(t) |ψi〉+∑λ

cE1

λ (t)∣∣∣ψE1

λ

⟩+∑λ

cE2

λ (t)∣∣∣ψE2

λ

⟩ (2)

where,

|ψi〉 = |g〉1 |g〉2 . . . |g〉i−1 |e〉i |g〉i+1 . . . |g〉N |0〉E1|0〉E2

(3a)

∣∣ψ1λ

⟩= |g〉1 |g〉2 . . . |g〉N |00 . . . 01λ0 . . . 00〉E1

|0〉E2(3b)

∣∣ψ2λ

⟩= |g〉1 |g〉2 . . . |g〉N |0〉E1

|00 . . . 01λ0 . . . 00〉E2(3c)

For our derivation, it is useful to adopt the transfor-mations ci(t) = e−i[ωe+(N−1)ωg]tci(t) , i = 1, . . . N and

cEjλ (t) = e−i(Nωg+ωjλ)tc

Ejλ (t), j = 1, 2 for the qubits’ and

environments’ amplitudes. The equations of motion ofthe transformed amplitudes, resulting from the TDSE,are:

idc1(t)

dt=J2c2(t) +

∑λ

g1(ω1λ)e−i∆

1λtcE1

λ (t) (4a)

idci(t)

dt=J2

[ci−1(t) + ci+1(t)] , i = 2, . . . , N − 1 (4b)

idcN (t)

dt=J2cN−1(t) +

∑λ

g2(ω2λ)e−i∆

2λtcE2

λ (t) (4c)

idcE1

λ (t)

dt= g1(ω1

λ)ei∆1λtc1(t) (4d)

idcE2

λ (t)

dt= g2(ω2

λ)ei∆2λtc2(t) (4e)

3

where ∆jλ ≡ ω

jλ − (ωe − ωg) ≡ ωjλ − ωeg, j = 1, 2.

Formal integration of Eqns. (4d) and (4e), under the

assumption that cE1

λ (0) = cE2

λ (0) = 0, and substitutionback to Eqns. (4a) and (4c), yields:

dc1(t)

dt=− iJ

2c2(t)

−∫ t

0

∑λ

e−i∆1λ(t−t′) [g1(ω1

λ)]2c1(t′)dt′

(5a)

dcN (t)

dt=− iJ

2cN−1(t)

−∫ t

0

∑λ

e−i∆2λ(t−t′) [g2(ω2

λ)]2cN (t′)dt′

(5b)

The summation over all possible modes of each en-vironment can be replaced by an integration that re-quires the specification of the environment’s spectral den-

sity Jj(ωjλ), according to

∑λ

[gj(ω

jλ)]2→∫dωjλJj(ω

jλ),

j = 1, 2. In what follows, we keep the spectral density ofboth environments arbitrary. The resulting set of differ-ential equations then is:

dc1(t)

dt= −iJ

2c2(t)−

∫ t

0

R1(t− t′)c1(t′)dt′ (6a)

dci(t)

dt= −iJ

2[ci−1(t) + ci+1(t)] , i = 2, . . . , N − 1

(6b)

dcN (t)

dt= −iJ

2cN−1(t)−

∫ t

0

R2(t− t′)cN (t′)dt′ (6c)

where we have introduced the definition

Rj(t) ≡∫ ∞

0

Jj(ωjλ)e−i∆

jλtdωjλ, j = 1, 2 (7)

Taking now the Laplace transform of the above set ofdifferential equations, and using the property of the con-

volution transform, L[∫ t

0f(t′)g(t− t′)dt′

]= F (s)G(s),

where F (s) and G(s) are the Laplace transforms of thefunctions f(t) and g(t), respectively, we obtain:

sF1(s) = c1(0)− iJ2F2(s)−B1(s)F1(s) (8a)

sFi(s) = ci(0)−iJ2

[Fi−1(s) + Fi+1(s)] , i = 2, . . . , N−1

(8b)

sFN (s) = cN (0)− iJ2FN−1(s)−B2(s)FN (s) (8c)

where we have used the fact that ci(0) = ci(0), sinceci(t) = e−i[ωe+(N−1)ωg ]tci(t). In the above system ofalgebraic equations, Fi(s) is the Laplace transform ofci(t), while Bj(s) is the Laplace transform of the functionRj(t), j = 1, 2.

As detailed in the Appendix, the set of Eqns. (8), canbe organized in such a way that each Fi(s), i = 2, . . . , Nis expressed in terms of F1(s) as:

Fi(s) = Ai(s)F1(s) +Ai−1(s)(ik)B1(s)F1(s)

−(ik)

i−1∑n=1

Ai−n(s)cn(0), i = 2, . . . , N(9)

where k ≡ 2J and Ai(s) obeys the relation:

Am+2(s) = (iks)Am+1(s)−Am(s), m = 1, . . . , N(10)

with A1(s) = 1 and A2(s) = iks. Given these conditions,one can easily verify that the mth term of this sequenceis given by:

Am(s) =

[(iks) + i

√k2s2 + 4

]m − [(iks)− i√k2s2 + 4]m

2mi√k2s2 + 4

, m = 1, . . . , N (11)

Taking equation (9) for i = N − 1 and i = N , we obtain:

FN−1(s) = AN−1(s)F1(s) +AN−2(s)(ik)B1(s)F1(s)− (ik)

N−2∑m=1

AN−1−m(s)cm(0) (12)

FN (s) = AN (s)F1(s) +AN−1(s)(ik)B1(s)F1(s)− (ik)

N−1∑m=1

AN−m(s)cm(0) (13)

4

Substituting Eqns. (12) and (13) back into Eqn. (8c) and using the relation∑N−1m=1 AN−m(s)cm(0) =∑N−2

m=1 AN−m(s)cm(0) +A1(s)cN−1(0), we can, after some straightforward algebraic manipulations, find that F1(s) isgiven by:

F1(s) =ikcN (0)− k2 [s+B2(s)]A1(s)cN−1(0) + (ik)

∑N−2m=1

{ik [s+B2(s)]AN−m(s)−AN−1−m(s)

}cm(0)

ik [s+B2(s)]AN (s)−{

1 + k2B1(s) [s+B2(s)]}AN−1(s)− ikB1(s)AN−2(s)

(14)

The inverse Laplace transform of Eqn. (14) providesthe time evolution of the amplitude c1(t). The rest of theamplitudes are given by the inverse Laplace transformsof Fi(s), i = 2, . . . , N which are recursively related toF1(s) through Eqn. (9).

The analytical method we have developed above allowsus to obtain the time evolution of the amplitude of anyspin, for an arbitrary number of sites as well as for arbi-trary initial conditions, within the single-excitation sub-space, by simply calculating the inversion integral of thecorresponding Laplace transform. Through our method,the number of sites N has become an arbitrary and eas-ily modifiable parameter entering expression (14). Theinversion integral can be readily calculated numerically(or even analytically in some cases), after specification ofthe spectral density of each environment, necessary forthe calculation of the Laplace transforms Bj(s) of thefunctions Rj(t).

III. RESULTS & DISCUSSION

A. Lorentzian spectral density

As a first example, we consider the case of an XX chainboundary-driven by two non-Markovian reservoirs withLorentzian spectral densities, given by:

Jj(ωjλ) =

g2j

π

γj2(

ωjλ − ωjc

)2

+ (γj2 )2

, j = 1, 2 (15)

where gj are the coupling strengths between the edgespins and the boundary reservoirs in units of frequency,while γj and ωjc , j = 1, 2 are the widths and the peakfrequencies of each distribution, respectively.

For analytical simplification of Eqn. (7), we extendthe lower limit of the integration over frequency from 0to −∞. Note that such extension is not in general validfor any DOS; it is however well justified in the case of aLorentzian spectral density with positive peak frequencyand width such that the distribution has practically neg-ligible extension to negative frequencies. The necessarycondition for this is γj << ωjc . In this case the frequencyintegrals can be calculated analytically, yielding:

0 10 20 30 400.00.20.40.60.81.0

t

Pop

ulat

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t

t0.0

0.2

0.4

0.6

0.8

1.0

Sum

of

Pop

ulat

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0.0

0.2

0.4

0.6

0.8

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Sum

of

Pop

ulat

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t

P1(t) PN(t)PCh(t)

(a) (b)

(c) (d)

g1=g2=0.3 g1=g2=1.5

0 10 20 30 400.00.20.40.60.81.0

Pop

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ions

0 200 400 600 800 1000 0 200 400 600 800 1000

FIG. 2. Dynamics of the populations of the first qubit (blackline), N th qubit (teal line) and channel qubits (orange line)for c1(0) = 1, N = 5, J = 1, γ1 = γ2 = 0.02, ∆1

c = ∆2c=0

and (a) g1 = g2 = 0.3, (b) g1 = g2 = 1.5. In panels (c)and (d) we show the long-time dynamics of the sum of allqubit populations in the spin chain for g1 = g2 = 0.3 andg1 = g2 = 1.5, respectively.

Rj(t) =g2j

π

γj2

∫ +∞

−∞

e−i(ωjλ−ωeg)t(

ωjλ − ωjc

)2

+(γj

2

)2 dωjλ= g2

j e−i∆j

cte−γj2 t, j = 1, 2

(16)

where ∆jc ≡ ωjc − ωeg, j = 1, 2. The Laplace transforms

of Rj(t) are then given by the expressions:

Bj(s) =g2j

s+γj2 + i∆j

c

, j = 1, 2 (17)

Using these expressions of Bj(s), j = 1, 2, we can eas-ily obtain the time evolution of the amplitudes ci(t),i, . . . , N , using Eqns. (9), (11) and (14). Note that inall of our calculations, we adopt the value J = 1 for thestrength of the interaction between neighboring qubits,as a reference value for the comparison with the othercouplings entering the formalism.

In Fig. 2(a) we study the dynamics of the popula-tions of the edge as well as the intermediate spins in theg1, g2 < J case for c1(0) = 1. For the intermediatequbits, we adopt the term ”channel qubits” , which is

5

0 200 400 600 8000.00.20.40.60.81.0

t

Populations

t t

t t t0 200 400 600 800

0.00.20.40.60.81.0

SumofPopulations

0 200 400 600 8000.00.20.40.60.81.0

Populations

0 200 400 600 8000.00.20.40.60.81.0

SumofPopulations

0 200 400 600 8000.00.20.40.60.81.0

Populations

0 200 400 600 8000.00.20.40.60.81.0

SumofPopulations

PCh(t)

00.0

t

0.1

0.2

400 800 00.0

t

0.1

0.2

400 800 00.0

t

0.1

0.2

400 800

N=3

P1(t)PN(t)

(a) (b) (c)

(d) (e) (f)

N=4 N=5

N=3 N=4 N=5

FIG. 3. Long-time dynamics of the populations P1(t), PN (t), PCh(t) (inset) as well as their sum, for spin chains of small lengthand c1(0) = 1, J = 1, γ1 = γ2 = 0.01, g1 = g2 = 1.8, ∆1

c = ∆2c=0. Panels (a) and (d): N = 3, panels (b) and (e): N = 4,

panels (c) and (f): N = 5.

frequently used in quantum information protocols andrepresents the sum of the populations of all except the 2edge qubits. The corresponding occupation probability

is PCh(t) =∑N−1i=2 |ci(t)|2 =

∑N−1i=2 |ci(t)|2. Since the

coupling between the qubits is larger than the couplingsbetween the edge spins and the reservoirs, we observeoscillations in the edge and channel qubit populations,associated with the spreading of the initial excitationthroughout the whole chain. Eventually, the populationwill be lost due to the environmental dissipation. Thiscan be seen through the long-time dynamical picture ofthe sum of all qubit populations, as in Fig. 2(c). Onthe the other hand, in the case where g1, g2 > J , asexamined in Fig. 2(b), since the initial excitation is onthe first qubit, the population is essentially ”trapped”,albeit not permanently, between the first qubit and en-vironment 1, with only a small portion of the populationleaking to the channel. In this case the exact value of g2

is of little relevance since the population of the N th siteat any time t is practically negligible. As seen in Fig.2(d), the strong coupling between the first qubit and thereservoir causes the total population to be lost throughthe reservoir much faster than in the previous case.

Although, as discussed above, the strong coupling be-tween the environment and the edge qubits may resultto population trapping (Fig. 2b), if the chain length issmall (up to 5 qubits or so), the long-time dynamics ofthe chain reveal an oscillatory population transfer be-tween the two edges of the chain (Fig. 3) along with thedissipation to the reservoirs. In that case, the populationof the N − 2 channel qubits remains low for any time t(see inset of Figs. 3a, 3b and 3c). This effect ceases tooccur for spin chains of longer length since the excita-tion cannot be easily transferred between the two edges

of the chain. As seen in Fig. 3, the frequency of these os-cillation is highly sensitive to N , decreasing as the latterincreases.

0 500 1000 1500 20000.0

0.2

0.4

0.6

0.8

1.0

Sum

of

Pop

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Sum

of

Pop

ulat

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t

t0 500 1000 1500 2000

0.0

0.2

0.4

0.6

0.8

1.0

N=5

N=11

(a)

(b)

g1=g2=3

g1=g2=2g1=g2=1

FIG. 4. Long-time dynamics of the sum of all qubit pop-ulations in the spin chain with the initial excitation in thecentral qubit c(N+1)/2(0) = 1 and J = 1, γ1 = γ2 = 0.02,

∆1c = ∆2

c=0, (a) N = 5 and (b) N = 11. Black line:g1 = g2 = 1, teal line: g1 = g2 = 2 and orange line:g1 = g2 = 3.

6

In Fig. 4 we examine the long-time dynamics of thetotal population for chains with odd number of sitesand initial excitation on the centre of the chain, i.e.c(N+1)/2(0) = 1. In Figs. 2(c) and 2(d) we observedthat, if the initial excitation is on the first qubit, the totalpopulation of the chain decays much faster for increasingvalues of g1. This of course is also true for cN (0) = 1 andincreasing values of g2, which is also obvious from symme-try arguments. However, as seen in Fig. 4(a), if the ini-tial excitation is on the centre of the chain, increasing themagnitude of the couplings between the edge spins andthe corresponding environment leads to the completelyopposite effect, namely a much slower decay of the totalpopulation in the long-time dynamics picture. Such aneffect occurs only as long as the boundary couplings arelarger than the coupling between the qubits J . In theopposite regime, the increase of the boundary couplings(to values still less than J ) results to faster decay of thetotal population to the reservoirs. These results indicatethat, for large boundary couplings, the evolution of theedge spins tends to freeze, alluding to a Quantum Zenoeffect, leading thus to an effective hindering of the to-tal decay to the reservoirs trough them. This hinderingof the decay becomes even more pronounced for increas-ing chain lengths (Fig. 4(b)), which physically can beattributed to the increased number of pathways for theevolution of the initial excitation throughout the wholechain and therefore the smaller likelihood for the popula-tion to reach the edge spins and be lost to the reservoirs.

It may be useful at this point to note that the dissipa-tion of the total population to the reservoirs can also beslowed down by choosing the detunings ∆1

c and ∆2c larger

than the boundary coupling constants. In that regime ofparameters, the dominant modes of the Lorentzian spec-tral densities of the boundary reservoirs are off-resonantwith the qubit frequency and do therefore damp the sys-tem less. That might be argued to be obvious. Be thatas it may, the statement is valid only in the context ofnon-Markovian reservoirs. Because the DOS of a non-Markovian reservoir must at some frequency exhibit amore or less pronounced peak, as it must depart from theslowly varying, smooth DOS which is necessary for theMarkovian approximation. It could therefore be viewedas yet another feature of non-Markovian behavior.

It is also important to note that the effect in which theincreased coupling constants between the edge spins andthe environments act like a barrier and protect the totalpopulation of the chain from decaying to the latter, donot necessary require an initial excitation in the centre ofthe chain. This is just one among the choices of possibleinitial conditions where this phenomenon occurs. Thegeneral rule is that the initial excitation should be any-where but the edge spins, in order to avoid the trappingof the population between the edge spins and their cor-responding reservoirs, as was the case in Fig. 2(b). Forexample, the increase of g1 and g2 for a chain system ini-

tially prepared in the state 1√N−2

∑N−1i=2 |ψi〉, will result

to the same phenomenon occurring in Fig. 4.

0 10 20 30 400.50.60.70.80.91.0

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Fid

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yM

ax A

v. F

idel

ity

0.6

0.7

0.8

0.9

1.0

0 10 20 30 40N

(a)g1=g2=0.0g1=g2=0.1g1=g2=0.3

N=10

(b)

FIG. 5. (a) Dynamics of the average-state fidelity of statetransfer between the two edges of the spin chain for N =10 and (b) Maximum Average Fidelity as a function of thenumber of spin sites N . In both panels J = 1, γ1 = γ2 = 0.5,∆1c = ∆2

c=0 and black line/dots: g1 = g2 = 0, teal line/dots:g1 = g2 = 0.1 and orange line/dots: g1 = g2 = 0.3.

In Fig. 5, we study the state transfer properties be-tween the two edge spins of the chain, in terms of theaverage-state fidelity [54]

F(t) =1

2+|cN (t)|2

6+|cN (t)|

3(18)

which involves an average over all possible sender-qubitstates. In Fig. 5(a) we show the dynamics of the average-state fidelity for a spin chain with N = 10 sites, for dif-ferent values of the boundary couplings g1 and g2. As be-comes evident, the first peak of the average-state fidelityis very sensitive to the boundary couplings, decreasing astheir values increase. At the same time subsequent peaksin time tend to increase their maximum value. The po-sition of the first peak, indicating the transfer time isslightly decreased by increasing the boundary couplings,however larger values of the latter lead to non-faithfulstate transfer in general. In Fig. 5(b) we examine thebehaviour of the maximum value of the average-state fi-delity as a function of the number of spin sites for chainlength up to 40 sites. As N increases, the maximumaverage-state fidelity tends towards lower values that de-pend also on the values of the boundary couplings ofthe edge spins with the reservoirs. Our results qualita-tively agree with a similar study by Feng-Hua Ren et al.[45], where they study the state transfer properties of the

7

same system using the dephasing and dissipation modelsin terms of the quantum state diffusion approach.

B. Ohmic spectral density

As a second example, we consider the case of boundaryreservoirs characterized by Ohmic spectral densities [55,56], according to the relation:

Jj(ωjλ) = Njg2

jωjc

(ωjλωjc

)Sjexp

(−ωjλωjc

), j = 1, 2

(19)where gj are qubit-environment coupling constants inunits of frequency, ωjc are the so-called Ohmic cut-offfrequencies and Sj , j = 1, 2, are the Ohmic parameters,characterizing whether the spectrum of the reservoirs issub-Ohmic (S < 1), Ohmic (S = 1) or super-Ohmic(S > 1). Nj is a normalization constant given by therelation Nj = 1

(ωjc)2Γ(1+Sj)

, where Γ(z) is the gamma

function.Using Eqn. (19) back to Eqn. (7), we perform the

integration over frequency for arbitrary Sj and find thatRj(t) is given by the expression:

Rj(t) = Njg2jω

jc

∫ ∞0

(ωjλωjc

)Sje−ωjλ

ωjc e−i(ω

jλ−ωeg)tdωjλ

= g2j eiωegt

(iωjct+ 1

)−1−Sj, j = 1, 2

(20)

Note that, unlike much of the literature, in our use ofOhmic spectral densities we found it necessary to includethe normalization factor Nj which renders meaningfulthe comparison with the Lorentzian spectral density, dis-cussed in subsection IIIC.

The Laplace transform of Eqn. (20) can also be calcu-lated analytically, yielding:

Bj(s) = −g2j

i1−Sj

ωjce−iKj(s) [Kj(s)]

SjΓ (−Sj ,−iKj(s))

, j = 1, 2

(21)

where Kj(s) ≡ (s− iωeg) /ωjc , j = 1, 2 and Γ(a, z) is theincomplete gamma function. In the special case whereSj are integers, it is useful to use the incomplete gammafunction property

Γ(−n, z) =1

n!

[e−z

zn

n−1∑k=0

(−1)k(n− k − 1)!zk

+(−1)nΓ (0, z)

] (22)

0 2 4 6 80.00

ωλ

Spe

ctra

lDen

sity ωc g=0.3

0.01

0.02

0.03

0.04

0.05

0 10 20 30 40 500.0

0.2

0.4

0.6

0.8

1.0

t

Sum

ofP

opul

atio

ns

t0 10 20 30 40 50

0.0

0.2

0.4

0.6

0.8

1.0

Sum

ofP

opul

atio

ns

ω 2ω 1c =

c = 0.5 ω2ω1c = c = 1.0

ω2ω1c =

c = 1.5

S1=S2=1

S1=S2=2

S1=S2=3

(a)

(b)S =1S =2S =3

FIG. 6. Dynamics of the sum of all qubit populations in aN = 6 qubit chain, boundary driven by reservoirs with Ohmicspectral densities. The parameters used in panel (a) are ω1

c =ω2c = 1, black line: S1 = S2 = 1, teal line: S1 = S2 = 2,

orange line: S1 = S2 = 3, while in panel (b), S1 = S2 = 1,black line: ω1

c = ω2c = 0.5, teal line: ω1

c = ω2c = 1.0 and orange

line: ω1c = ω2

c = 1.5. In both panels, J = 1, g1 = g2 = 0.3and ωeg = 1 and c1(0) = 1. The inset in panel (b) shows theOhmic spectral density as a function of ωλ for various Ohmicparameters S (S = 1: solid line, S = 2: dotted line andS = 3: dashed line) and g = 0.3 (coupling strength), ωc = 1.

which holds for integer n, to express Bj(s) in termsof Γ(0,−iKj(s)). Using Eqn. (21), as in the previouscase, we can easily calculate the Laplace inversion nu-merically and find the time evolution of the amplitudesci(t), i, . . . , N .

In Fig. 6 we study the dynamics of the sum of allqubit populations in the chain for various parameters ofthe Ohmic reservoirs. The dynamics of this quantity pro-vide us straightforward information of how fast the ini-tial excitation gets lost in the two environments. As Fig.6(a) indicates, by increasing the Ohmic parameters Sj ,j = 1, 2 of the reservoirs, the sum of populations decaysmore slowly, indicating a slower decay of the single ex-citation to the environments. Physically, this effect isattributed to the fact that by increasing Sj , the spec-tral density distribution of the environments is mainlypeaked in the vicinity of modes whose frequency is largerthan the qubit frequency ωeg = 1 (see inset of Fig. 6(b)),and therefore by being off-resonant with the qubit fre-quency, they damp the system less. A similar behaviour

8

0 10 20 30 40 500.0

0.2

0.4

0.6

0.8

1.0

t

Populations

0 10 20 30 40 500.0

0.2

0.4

0.6

0.8

1.0

t

Populations

(a) (b)

(c) (d)

P1(t)

PN(t)

PCh(t)

g1=g2=0.3g1=g2=1.4

0 10 20 30 40 500.0

0.2

0.4

0.6

0.8

1.0

t

PopulationP1(t)

S1=S2=1.0S1=S2=1.0

0 10 20 30 40 500.0

0.2

0.4

0.6

0.8

1.0

PopulationP1(t)

N=4N=8

N=12

S1=S2=0.5S1=S2=1.5

S1=S2=3.0

t

g1=g2=1.4

500.0

t

PopulationP1(t)

100 150 200

0.2

0.4

0.0

0.2

0.4

0.0

0.2

0.4

50 100 150t

200PopulationP1(t)

100 200 300PopulationP1(t)

t

g1=g2=1.4S1=S2=2.0

FIG. 7. (a) Dynamics of the populations of the first qubit (black line), N th qubit (teal line) and channel qubits (orange line) forc1(0) = 1, N = 6, J = 1, ωeg = 1, ω1

c = ω2c = 1, S1 = S2 = 1 and g1 = g2 = 0.3. (b) Same as in panel (a) but for g1 = g2 = 1.4.

(c) Dynamics of the population P1(t) for various values of the environmental Ohmic parameters and c1(0) = 1, N = 6, J = 1,ωeg = 1, ω1

c = ω2c = 1, g1 = g2 = 1.4. Black line: S1 = S2 = 0.5, teal line: S1 = S2 = 1.5 and orange line: S1 = S2 = 3.0. (d)

Dynamics of the population P1(t) for various values of the number of qubit sites and c1(0) = 1, J = 1, ωeg = 1, ω1c = ω2

c = 1,S1 = S2 = 2.0, g1 = g2 = 1.4. Black line: N = 4, teal line: N = 8, orange line: N = 12. In the insets of panels (b), (c) and (d)we show the long-time dynamics of the population P1(t).

is observed in Fig. 6(b), where the dynamics are shownas function of the cut-off frequencies, for Ohmic type ofenvironments (Sj=1). Since for Ohmic distributions thepeak of the distribution coincides with the cut-off fre-quency ωjc , j = 1, 2, increase of the cut-off frequenciesresults to distributions whose modes extend mainly be-yond ωeg, resulting to a slower decay to the boundaryenvironments. Note that in general, for arbitrary Ohmicparameters, the Ohmic distributions are always peakedat the frequencies ωjλ = Sjωjc , j = 1, 2. The character-istic oscillations observed in both panels of Fig. 7 areindicative of the non-Markovian character of the Ohmicboundary reservoirs, enabling the exchange of populationbetween them and the spin chain within finite times.

In Fig. 7(a) we examine the dynamics of the popu-lations of the edge as well as the channel qubits of thechain for Ohmic boundary reservoirs (Sj=1, j = 1, 2) inthe weak boundary couplings regime g1, g2 < J . The ini-tial single-excitation is on the first qubit (c1(0) = 1). Thedynamics indicate oscillations between the populations ofthe chain damped by the two Ohmic environments. Thesituation however may change drastically in the strongboundary couplings regime g1, g2 > J , as shown in Fig.7(b). In this case the dynamics of the population P1(t)indicate a long-time stability behaviour around a finitevalue for times much larger than any other time scale of

the system. Note that further increase of the boundarycoupling constants will not result to the increase of thislong-time stability value. Still, as seen in Fig. 7(c), thisvalue is highly affected by the Ohmic parameters of theenvironments (more specifically by the Ohmic parameterof the first environment which is the one that commu-nicates with the first qubit). In general, increase of theOhmic parameters results to more complex behaviour asindicated by the orange line of Fig. 7(c). In the long-time dynamics picture, the population of the first qubitexhibits an oscillatory behaviour with a frequency thatdepends on the Ohmic parameter of its boundary reser-voir. These non-coherent oscillations become faster forsuper-Ohmic reservoirs and are essentially an effect asso-ciated with the non-Markovianity of the boundary reser-voirs that drive the chain, since they indicate popula-tion exchanges between the chain and the latter for verylong times. Another parameter that affects considerablythis oscillatory behaviour is the number of qubit sites.In Fig. 7(d) we plot the dynamics of P1(t) for a chainof various number of qubit sites N , boundary-driven bysuper-Ohmic environments with Sj=2, j = 1, 2. For asmall number of qubit sites, the long-time dynamics inthe inset of Fig. 7(d) indicate a long-living coherent os-cillatory behaviour (black line) while for chains that con-sist of larger number of qubits, the population of the

9

g1=g2=1 g1=g2=2g1=g2=3

(b)

(a)

0 20 40 60 80 1000.00.20.40.60.81.0

t

Sum

ofP

opul

atio

ns

0.0

ω

0.20.40.60.81.01.2

0 2 4 6 8 10

Spe

ctra

l Den

sity g=1 Lorentzian

ωc=3 (peak freq.)γ=0.65

Ohmicωc=1 (cut-off)S=3

FIG. 8. (a) Spectral density of a Lorentzian distribution withpeak frequency ωc = 3, width γ = 0.65 and g = 1 (solidline) compared with an Ohmic distribution with Ohmic pa-rameter S = 3, cut-off frequency ωc = 1 and g = 1 (dashedline). (b) Dynamics of the sum of all qubit populationsin a N=7 chain and initial excitation in the center of thechain (c(N+1)/2(0) = 1), for various coupling strengths be-tween the boundary reservoirs and the edge qubits. Blackline: g1 = g2 = 1, teal line: g1 = g2 = 2 and orange line:g1 = g2 = 3. The solid lines corresponds to Lorentzian bound-ary reservoirs, while the dashed lines correspond to Ohmic.The rest of the parameters (except the coupling strengths)correspond to the parameters of panel (a). The qubit fre-quency is chosen to be resonant with the peak frequency ofthe two distributions, namely ωeg = 3.

first qubit exhibit small non-coherent oscillations arounda stabilized value (orange line). Our results are just afirst glimpse on the richness of the effects one may ex-pect from the boundary-driving of chains with Ohmictype of reservoirs.

C. Comparing Lorentzian to Ohmic spectraldensities

So far we have separately studied the cases ofboundary-driven chains by reservoirs with Lorentzianand Ohmic spectral densities. In Fig. 8 we provide a casefor comparison between the two. As seen in Fig. 8(a) wehave chosen both distributions to be peaked exactly onresonance with the qubit frequency ωeg = 3. Althoughthe freedom in the choice of the parameters of each distri-bution, results to many possible lineshapes of their spec-tral densities, the general tendency for typical parameters

of the two distributions is that Ohmic distributions areusually wider than Lorentzian distributions. In Fig. 8(b)we use the distributions of Fig. 8(a) as spectral densitiesof the boundary reservoirs and study the dynamics of thesum of all qubit populations in a N = 7 chain for variousvalues of the boundary couplings gj , j = 1, 2 and ini-tial excitation in the center of the chain. The solid linescorrespond to Lorentzian boundary reservoirs while thedashed lines correspond to Ohmic. As also argued in thediscussion of Fig. 4, since the initial excitation is not onthe edge qubits of the chain and the boundary couplingsare larger than the qubit-qubit coupling J , increasingthe values of gj , j = 1, 2, results to slower dissipationof the total excitation in the environments. Compari-son between the solid (Lorentzian) and dashed (Ohmic)lines, reveals two major differences: First, the dynamicsof the total population in the boundary-driven chain byLorentzian reservoirs is more sensitive to the increase ofthe boundary couplings compared to the Ohmic-drivenchain. Moreover, for the particular combination of pa-rameters chosen, the total excitation in the chain tendsto live longer in Lorentzian-driven chains compared toOhmic-driven ones, which can be attributed to the muchbroader profile of the Ohmic spectral density comparedto the Lorentzian one. Second, it is evident that thedynamics of the sum of all qubit populations exhibitmore vivid oscillations in the Lorentzian case comparedto the Ohmic one. Such oscillations are indicative ofthe non-Markovian character of the boundary-reservoirsand therefore the exchange of populations between thechain and the reservoirs within finite times. Even if bothtypes of reservoirs are non-Markovian, the Ohmic spec-tral density of Fig. 8(a) tends to be substantially flatterthan the Lorentzian spectral density, resembling thus aMarkovian environment for which no oscillations are ex-pected between the chain and its boundaries.

Finally, before concluding, we should note that the dy-namics of the average-state fidelity of the state transferbetween the edges of a chain boundary-driven by Ohmicreservoirs display qualitatively more or less similar be-haviour as the corresponding average-state fidelity dy-namics of a boundary-driven chain by Lorentzian reser-voirs, as in Fig. 5.

IV. CONCLUDING REMARKS

In this paper, we present a formalism for the systematicway of calculating the wavefunction of an arbitrary lengthXX spin chain, boundary-driven by non-Markovian envi-ronments of arbitrary spectral density. The theory is castin terms of Schrodinger’s equations of motion, within thesingle-excitation subspace, and leads to closed form solu-tions for the Laplace transforms of the amplitudes for ar-bitrary initial conditions, using a recursive method. Theinversion integrals can be easily calculated numerically(or even analytically in some cases), upon specificationof the spectral density of each environment. The calcu-

10

lation of the wavefunction of the chain within the single-excitation subspace enables the study of a number of in-teresting properties of the system. As an illustration ofthe potential of the approach, we consider Lorentzian andOhmic spectral densities for the boundary environmentsand study aspects of the dynamical, as well as of the statetransfer properties of the system. Our results demon-strate the memory effects of the non-Markovian environ-ments. The concomitant information exchange betweenthe latter and the spin chain, give rise to a plethora ofinteresting effects such as population trapping, oscilla-tions between the populations of the edge spins or theprotection of the spin populations against dissipation forincreased values of the boundary couplings.

One aspect of our results that turns out to be revealingand somewhat surprising has to do with the comparativeanalysis of the dynamics of the chain under two quitedifferent reservoirs; namely Lorentzian and Ohmic. Weadopt the general notion that any reservoir, with spectraldensity not smoothly and slowly varying as a function ofenergy (frequency), in an extended range around the res-onance with the qubit, is non-Markovian. This entailssome degree of memory, which in turn implies that theloss into the reservoir would not be monotonic. In muchof the work and approaches on non-Markovian reservoirs,it is the Lorentzian that has served as a typical illustra-tion. Physically, that behavior should be expected for aLorentzian spectral density as it exhibits a well-definedpeak. But so does an Ohmic as well as a superohmicspectral density. However, our direct quantitative com-parison of the effect of a Lorentzian DOS versus thatof a superohmic in section IIIC, reveals effects of an al-most qualitative difference between the two; the effect ofthe superohmic resembles an almost Markovian reservoir,which implies a much lower degree of non-Markovianity.

This concrete example complements earlier general for-mulations of measures of non-Markovian behavior [57].

Our work was motivated by the rapidly growing inter-est in the field of boundary-driven open quantum sys-tems and in particular the need for the development ofthe necessary analytical or numerical tools for the studyof their properties. The results provide the backgroundfor a number of extensions, such as accounting for dif-ferent spin chain configurations, as well as the theoret-ical description of the system, within our formal devel-opment, beyond the single-excitation. It would in addi-tion be interesting to investigate whether the behaviorfor long times of the edge qubits, which communicate di-rectly with the boundary reservoirs (Fig. 7), could beuseful in the survival of long-distance quantum correla-tions, for chains consisting of large number of qubit sites,boundary-driven by Ohmic reservoirs.

ACKNOWLEDGMENTS

GM would like to acknowledge the Institute of Elec-tronic Structure and Laser (IESL), FORTH, for finan-cially supporting this work through its Research and De-velopment Program Π∆E00710. We are also thankfulto Prof. Martin B. Plenio for his critical reading of themanuscript.

APPENDIX A

The system of Eqns. (8) is a set of N algebraic equa-tions that can be solved recursively as follows: First wesolve Eqn. (8a) for F2(s) and Eqn. (8b) for Fi+1(s) tofind

F2(s) = (iks)F1(s) + (ik)B1(s)F1(s)− (ik)c1(0) (A.1a)

Fi+1(s) = −Fi−1(s) + (iks)Fi(s)− (ik)cj(0), i = 2, . . . , N − 1 (A.1b)

where k ≡ 2/J . Using these two relations, we can express each Fi(s) in terms of F1(s). The first few terms up toF6(s) can be organized as follows:

F2(s) = (iks)F1(s) + (ik)B1(s)F1(s)− (ik)c1(0) (A.2a)

F3(s) =[(iks)2 − 1

]F1(s) + (iks)(ik)B1(s)F1(s)− (ik) [(iks)c1(0) + c2(0)] (A.2b)

F4(s) =[(iks)3 − 2(iks)

]F1(s) +

[(iks)2 − 1

](ik)B1(s)F1(s)− (ik)

{ [(iks)2 − 1

]c1(0) + (iks)c2(0)− c3(0)

}(A.2c)

F5(s) =[(iks)4 − 3(iks)2 + 1

]F1(s) +

[(iks)3 − 2(iks)

](ik)B1(s)F1(s)

− (ik){ [

(iks)3 − 2(iks)]c1(0) +

[(iks)2 − 1

]c2(0) + (iks)c3(0) + c4(0)

} (A.2d)

11

F6(s) =[(iks)5 − 4(iks)3 + 3(iks)

]F1(s) +

[(iks)4 − 3(iks)2 + 1

]B1(s)F1(s)

− (ik){ [

(iks)4 − 3(iks)2 + 1]c1(0) +

[(iks)3 − 2(iks)

]c2(0) +

[(iks)2 − 1

]c3(0) + (iks)c4(0) + c5(0)

}(A.2e)

Careful inspection of the above system of equations reveals that Fi(s), follows a pattern of the form:

Fi(s) = Ai(s)F1(s) +Ai−1(s)(ik)B1(s)F1(s)− (ik)

i−1∑n=1

Ai−n(s)cn(0), i = 2, . . . , N (A.3)

where Ai(s) obeys the relation:

Am+2(s) = (iks)Am+1(s)−Am(s), m = 1, . . . , N (A.4)

with A1(s) = 1 and A2(s) = iks. The mth term of this sequence is given by:

Am(s) =

[(iks) + i

√k2s2 + 4

]m − [(iks)− i√k2s2 + 4]m

2mi√k2s2 + 4

, m = 1, . . . , N (A.5)

[1] Simone Paganelli, Salvatore Lorenzo, Tony J. G. Apollaro,Francesco Plastina, and Gian Luca Giorgi, Phys. Rev. A 87,062309 (2013).

[2] J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, Phys.Rev. Lett. 78, 3221 (1997).

[3] Matthias Christandl, Nilanjana Datta, Artur Ekert, andAndrew J. Landahl, Phys. Rev. Lett. 92, 187902, (2004).

[4] C. Di Franco, M. Paternostro, and M. S. Kim, Phys. Rev.Lett. 101, 230502 (2008).

[5] David Petrosyan, Georgios M. Nikolopoulos, and P.Lambropoulos, Phys. Rev. A 81, 042307 (2010).

[6] N. Y. Yao, L. Jiang, A. V. Gorshkov, Z.-X. Gong, A. Zhai,L.-M. Duan, and M. D. Lukin, Phys. Rev. Lett. 106, 040505(2011).

[7] A. K. Pavlis, G. M. Nikolopoulos, and P. Lambropoulos,Quantum Inf. Process. 15, 2553–2568 (2016).

[8] A. Gratsea, G. M. Nikolopoulos, and P. Lambropoulos,Phys. Rev. A 98, 012304 (2018).

[9] M. B. Plenio, J. Hartley and J. Eisert, New. J. Phys. 6, 36(2004).

[10] Sougato Bose, Phys. Rev. Lett. 91, 207901 (2003).[11] L. Campos Venuti, S. M. Giampaolo, F. Illuminati, and P.

Zanardi, Phys. Rev. A 76, 052328 (2007).[12] Salvatore M Giampaolo and Fabrizio Illuminati, New J.

Phys. 12, 025019 (2010).[13] S. Sahling, G. Remenyi, C. Paulsen, P. Monceau, V.

Saligrama, C. Marin, A. Revcolevschi, L. P. Regnault, S.Raymond, and J. E. Lorenzo, Nature Physics 11, 255–260(2015).

[14] Marta P. Estarellas, Irene D’Amico, and Timothy P. Spiller,Phys. Rev. A 95, 042335 (2017).

[15] Francesco Plastina and Tony J. G. Apollaro, Phys. Rev.Lett. 99, 177210 (2007).

[16] T. Yu and J. H. Eberly, Phys. Rev. Lett. 93, 140404 (2004);97, 140403 (2006); J. H. Eberly and T. Yu, Science 316, 555(2007).

[17] M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P.

Walborn, P. H. Souto Ribeiro and L. Davidovich, Science316, 579 (2007).

[18] J. Laurat, K. S. Choi, H. Deng, C. W. Chou, and H. J.Kimble, Phys. Rev. Lett. 99, 180504 (2007).

[19] P. Marek, J. Lee and M. S. Kim, Phys. Rev. A 77, 032302(2008).

[20] A. Al-Qasimi and D. F. V. James, Phys. Rev. A 77, 012117(2008).

[21] M. B. Plenio and S. F. Huelga, New J. Phys. 10, 113019(2008).

[22] Zhou Lan, Lu Jing, and Shi Tao, Commun. Theor. Phys. 52,226 (2009).

[23] Zhao-Ming Wang, Feng-Hua Ren, Da-Wei Luo, Zhan-YuanYan, and Lian-Ao Wu, J. Phys. A: Math. Theor. 54, 155303(2021).

[24] S. R. Clark, J. Prior, M. J. Hartmann, D. Jaksch and M. B.Plenio, New J. Phys. 12, 025005 (2010).

[25] Mohammad Reza Pourkarimi, Majid Rahnama, and HosseinRooholamini, Int. J. Theo. Phys. 54, 1085–1097 (2015).

[26] Zhao-Yu Sun, Kai-Lun Yao, Wei Yao, De-Hua Zhang, andZu-Li Liu, Phys. Rev. B 77, 014416 (2008).

[27] Gehad Sadiek and Samaher Almalki, Phys. Rev. A 94,012341 (2016).

[28] Morteza Rafiee, Eur. Phys. J. D 72, 152 (2018).[29] B. Bellomo, R. Lo Franco and G. Compagno, Phys. Rev.

Lett. 99, 160502 (2007).[30] R. Lo Franco, B. Bellomo, S. Maniscalco, and G. Compagno,

Int. J. Mod. Phys. B 27, 1345053 (2013).[31] Tomas Ramos, Benoıt Vermersch, Philipp Hauke, Hannes

Pichler, and Peter Zoller, Phys. Rev. A 93, 062104 (2016).[32] Fujing Liu, Xingxiang Zhou, and Zheng-Wei Zhou, Phys.

Rev. A 99, 052119 (2019); Phys. Rev. A 100, 019901(E)(2019).

[33] Zhao-Ming Wang, Feng-Hua Ren, Da-Wei Luo, Zhan-YuanYan, and Lian-Ao Wu, Phys. Rev. A 102, 042406 (2020).

[34] G. Mouloudakis and P. Lambropoulos, Quantum Inf.Process. 20, 331 (2021)

12

[35] Heinz-Peter Breuer, Elsi-Mari Laine, Jyrki Piilo, andBassano Vacchini, Rev. Mod. Phys. 88, 021002 (2016).

[36] G. T. Landi, D. Poletti, and G. Schaller, arXiv:2104.14350(2021).

[37] K. Yamanaka and T. Sasamoto, arXiv:2104.11479 (2021).[38] E. Mascarenhas, G. Giudice, and V. Savona, Quantum 1, 40

(2017).[39] Abhishek Dhar, Keiji Saito, and Peter Hanggi, Phys. Rev. E

85, 011126 (2012).[40] Marko Znidaric, Phys. Rev. Lett. 112, 040602 (2014)[41] F. Barra, Nature Scientific Reports 5, 14873 (2015).[42] Amikam Levy and Ronnie Kosloff, EPL 107, 20004 (2014).[43] Gabriele De Chiara, Gabriel Landi, Adam Hewgill, Brendan

Reid, Alessandro Ferraro, Augusto J. Roncaglia, and MauroAntezza, New J. Phys. 20, 113024 (2018).

[44] Nianbei Li, Jie Ren, Lei Wang, Gang Zhang, Peter Hanggi,and Baowen Li, Rev. Mod. Phys. 84, 1045 (2012).

[45] Feng-Hua Ren, Zhao-Ming Wang, and Yong-Jian Gu, Q. Inf.Process. 18, 193 (2019).

[46] D. Jaschke, M. L. Wall, and L. D. Carr, Comp. Phys.Comm. 225, 59-91 (2018).

[47] Regina Finsterholzl, Manuel Katzer, Andreas Knorr, and

Alexander Carmele, Entropy 22, 984 (2020).[48] M. Znidaric, J. Phys. A: Math. Theor. 43, 415004 (2010).[49] D. Karevski, V. Popkov, and G. M. Schutz, Phys. Rev. Lett.

110, 047201 (2013).[50] Tomaz Prosen, Phys. Rev. Lett. 106, 217206 (2011).[51] Tomaz Prosen, Phys. Rev. Lett. 107, 137201 (2011).[52] M. Michel, J. Gemmer, and G. Mahler, Eur. Phys. J. B 42,

555–559 (2004).[53] Hannu Wichterich, Markus J. Henrich, Heinz-Peter Breuer,

Jochen Gemmer, and Mathias Michel, Phys. Rev. E 76,031115 (2007).

[54] G. M. Nikolopoulos and I. Jex, Quantum State Transfer andNetwork Engineering, Quantum Science and Technology(Springer-Verlag, Berlin Heidelberg, 2014).

[55] A. J. Leggett, S. Chakravarty, A. T. Dorsey, Matthew P. A.Fisher, Anupam Garg, and W. Zwerger, Rev. Mod. Phys.59, 1 (1987); Rev. Mod. Phys. 67, 725 (1995).

[56] U. Weiss, Quantum Dissipative Systems (World Scientific,Singapore, 2001).

[57] Heinz-Peter Breuer, Elsi-Mari Laine and Jyrki Piilo, Phys.Rev. Lett. 103, 210401 (2009).