An out-of-equilibrium non-Markovian Quantum Heat Engine

Marco Pezzutto1,2, Mauro Paternostro3, and Yasser Omar1,21Instituto de Telecomunicações, Physics of Information and Quantum Technologies Group, Lisbon, Portugal

2 Instituto Superior Técnico, Universidade de Lisboa, Portugal3 Centre for Theoretical Atomic, Molecular and Optical Physics,

School of Mathematics and Physics, Queen’s University Belfast BT7 1NN, United Kingdom

We study the performance of a quantum Otto cycle using a harmonic work medium and un-dergoing collisional dynamics with finite-size reservoirs. We span the dynamical regimes of thework strokes from strongly non-adiabatic to quasi-static conditions, and address the effects thatnon-Markovianity of the open-system dynamics of the work medium can have on the efficiency ofthe thermal machine. While such efficiency never surpasses the classical upper bound valid forfinite-time stochastic engines, the behaviour of the engine shows clear-cut effects induced by boththe finiteness of the evolution time, and the memory-bearing character of the system-environmentevolution.

I. INTRODUCTION

The study of work- and heat-exchanges at the quan-tum scale [1–3] is paving the way to the understandingof how quantum fluctuations influence the energetics ofnon-equilibrium quantum processes. In turn, such fun-damental progress is expected to have significant reper-cussions on the design and functioning of quantum heatmachines [4–9].

Such devices thus play the role of workhorses for the ex-plorations of the potential advantages stemming from theexploitation of quantum resources for thermodynamic ap-plications at the nano-scale [10, 11]. Theoretical modelsof microscopic heat engines based on the use of work-ing fluid comprising two-level systems [12] or quantumharmonic oscillators [13] have been introduced. Such de-signs appear increasingly close to grasp in light of therecent progresses in the experimental management of(so far classical) thermal engines using individual par-ticles [14, 15] or mechanical systems [16–18].

Is it possible to pinpoint genuine signatures of quan-tum behaviour that influence the thermodynamics of asystem in ways that could never be produced by a clas-sical mechanism [19]? How would quantum mechanicsenhance the performance of a quantum thermal enginebeyond anything achievable classically [20–23]? Do co-herences in the energy eigenbasis [24–27] or non-thermalreservoirs [28, 29] represent exploitable (quantum) ther-modynamic resources? In an attempt at providing an-swers to such burning questions, in this paper we studythe finite-time thermodynamics of a heat engine opera-ting an Otto cycle whose work fluid is a quantum har-monic oscillator. Hot and cold environments are model-led via a collections of spin-1/2 particles (Fig. 1). Thework strokes of the cycle are implemented via parametricchanges of the frequency of the harmonic oscillator, whileheat exchanges result from collisional dynamics with theenvironments that may allow for memory effects [30].The significant flexibility and richness of dynamical con-ditions of collisional models is perfectly suited to the ex-ploration of non-Markovian dynamics in a wide range ofconditions [31–38].

The scope of our study is twofold: on the one hand, weinvestigate work transformations of controlled yet varia-ble duration, spanning the whole range from an infinitelyslow (and thus adiabatic) transformations, to the oppo-site extreme of a sudden quench. On the other hand, byincluding intra-environment interactions, we allow for theemergence of memory effects and thus non-Markovianityin the dynamics of the engine. We investigate numeri-cally the behaviour of the engine and its performance inthe two cross-overs from adiabaticity to sudden quench,and from Markovianity to non-Markovianity. We aim atidentifying the optimal trade-off between efficiency andspeed, and the role and impact of memory effects on theengine performance.

Among the results reported in this paper is the demon-stration that the efficiency of the device always decreasesas we approach the sudden-quench regime, and the quan-tification of an optimal time at which the power output ismaximum. Intra-environment interactions, in turn, seemto have no effect on the long-time engine performance.However, they affect the transient of the evolution of theengine by seemingly lowering the efficiency of the heat-transfer process – at least in the case when the both theengine and the environment particles are initialized in athermal state. In no case we observe a performance ex-ceeding the classical bounds, which is in agreement withthe result reported in [19]. We do observe however astrong connection between non-Markovianity and the co-herences in the initial engine state.

The remainder of the article is structured as follows.Sec. II introduces our model for heat engine, describ-ing how the work and heat transformations are real-ized. Sec. III presents the results of our quantitativeanalysis, while in Sec. IV we draw our conclusions. Fi-nally, Appendix A reviews key concepts of quantum non-Markovianity and its characterization used in our quan-titative analysis.

arX

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26

Jun

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2

(a) (b)

1

<H>

ω2

Win

Wout

Qout

Qin

ω1

1

2

3

4

2

3

4

ω

FIG. 1. (a) We study an engine performing an Otto cycle witha quantum harmonic oscillator as the working fluid, which inturn interacts with two environments composed of spin-1/2particles. Work is done on/by the oscillator by changing thefrequency of its potential, while in isolation from the envi-ronments (cf. Sec. II B). Heat is exchanged with the latterthrough collisions with the spin-1/2 particles (cf. Sec. II C).Additional intra-environment interactions allow the environ-ments to keep memory of past interactions with the engine.(b) As its classical version, the cycle is composed by fourstrokes: two isentropic (strokes 1 and 3), where work is per-formed on or by the engine, and two (strokes 2 and 4), duringwhich heat is exchanged with the reservoirs. In our model,the control parameter is the oscillator frequency ω, whosechanges play the role of an effective modification in volume inthe classical version of the engine. Therefore, strokes 2 and4 are analogous to isocoric transformations. On the verticalaxis, we report the average internal energy of the oscillator〈H〉, which quantifies the energy exchanges resulting from thefour strokes.

II. THE ENGINE MODEL

We study a model of heat engine operating accordingto an Otto cycle, whose working fluid is a quantum har-monic oscillator governed by the Hamiltonian

Hs(t) =p2

2m+mω2(t)x2

2. (1)

The subscript "s" stands for "system" as we may regardthe engine as our main system of interest. The Otto cycleconsists of two work strokes and two heat strokes. Thework strokes are implemented by changing the frequencyω of the harmonic potential. The hot and cold environ-ments are modelled as a collection of spin-1/2 particleswith Hamiltonian

H(n)e =

1

2~ωeσze,n, ωe > 0, e = c, h

for the n-th particle. The subscripts "h,c" stands foreither a hot- or a cold-reservoir particle. The engine in-teracts with them through a collisional model, similarto the one employed in Ref. [35]. The details of thesedynamical processes, pictured in Fig. 1, are outlined infollowing Subsections.

A. Details of the cycle operation andthermodynamics of the process

We are now going to outline the protocol throughwhich the Otto cycle is implemented, and its thermo-dynamics. Let us define the thermodynamic quantitiesthat will be central to our analysis.

We start with the internal energy of the engine, whichis defined as

E := Tr[ρsHs]. (2)

The second quantity of relevance is the work done on/bythe engine during a work-producing stroke. As, in anOtto cycle, no heat is exchanged in one of such strokes,the difference between the values of the internal energyof the engine at the initial and final points of the strokequantifies the exchanged work. We thus have

W := E(k)in − E

(k)fin , (3)

where k = 1, 3 identifies the work-producing strokes. Inwhat follows, we use the usual convention that W > 0when work is performed by the engine. This is also inagreement with a definition of the average exchangedwork based on the so-called two-projective-measurementapproach [39].

Similarly to the above considerations, no work is ex-changed during a heat-exchanging stroke, so that the dif-ference between the values of the internal energy of theengine at the initial and final points of the stroke providesan estimate of the exchanged heat Q. Therefore

Q := E(k)fin − E

(k)in , (4)

where Q > 0 if it is absorbed by the work medium, andk = 2, 4 is the label for the heat-producing strokes. Anengine-environment interaction that conserves the totalenergy [such as the one illustrated in Sec. II C], is a physi-cally sound description of a heat transfer process, as itis well suited to describe the heat exchange as a flow ofenergy from one system (engine or environment) to theother. Moreover, it is consistent with a more generaldefinition of the exchanged heat as the difference of theenvironment internal energy.

The environmental particles are assumed to be all pre-pared in a single-particle thermal state,

ρ(n)e =

e−βeH(n)e

Tr [e−βeH(n)e ]

with βe = 1/(κTe) the inverse temperature of the e =c,h environment. We have also assumed the hierarchyof temperatures Tc < Th. The work fluid is assumed tobe initialized in a thermal state at initial temperature Tssuch that Tc < Ts < Th.

With reference to Fig. 1, our Otto cycle is implementedwith the following steps:

3

Stroke 1–Compression: let the initial engine internalenergy be E0. The oscillator frequency is changedfrom ωc to ωh in isolation from any environment.The final energy is E1 and the work done on theengine is Win = E0 − E1 < 0.

Stroke 2a–Contact with hot environment: the en-gine interacts with a hot-environment particle andthe final internal energy is E2. The engine absorbsthe heat Qin = E2 − E1 > 0.

Stroke 2b–Intra-environment interaction: the in-ternal interaction between the environmental par-ticles may propagate some memory of the enginestate and feed it back at a later stage. This stephas no direct effect on the thermodynamics of theengine.

Stroke 3–Expansion: the oscillator’s frequency ischanged from ωh back to ωc in isolation from anyenvironment. The final energy is E3 and the workperformed by the engine is Wout = E3 − E2 > 0.

Stroke 4a–Contact with cold environment: theengine interacts with a cold-environment particleand the final internal energy is E4. The engine hastransferred an amount of heat Qout = E4 −E3 < 0to the environment.

Stroke 4b–Intra-environment interaction: similarto stroke 2b.

The final state of the engine becomes the initial state ofa new cycle and the steps are iterated, involving newenvironmental particles. The dynamics thus proceedsthrough discrete time steps, each of them being a fulliteration of the Otto cycle. At the end of each cycle, wecompute the power output of a cycle, and its efficiency.By denoting with T the total duration of one cycle, thepower output is P := (Win+Wout)/T , while the efficiencyreads η := (Win +Wout)/Qin.

Let us recall the usual expression for the internal ener-gy of a quantum harmonic oscillator E = ~ω(1/2 + 〈n〉)with n the number operator. Let us define as ni = 〈n〉i(i = 0, . . . , 4) the average occupation number at the be-ginning (i = 0) and after step i ≥ 1 of the protocol, suchthat Ei = ~ωe(1/2 + ni), with ωe = ωc at the beginningand after strokes 3 and 4, and ωe = ωh after strokes 1and 2. We finally assume n4 = n0 if the engine performsa stationary cycle. Using Eqs. (2)-(4) we have

η =E2 − E3 + E0 − E1

E2 − E1= 1− ωc(n3 − n0)

ωh(n2 − n1).

If the work transformations are performed adiabatically,the populations remain unchanged, thus n1 = n0 andn2 = n3, leading to the remarkably simple theoreticalefficiency

ηth = 1− ωcωh, (5)

irrespectively of the details of the heat exchanges.

B. Work transformations

The work strokes are implemented through a unitarytransformation on the engine alone, isolated from thecold or hot environment. A theoretical description ofsuch processes was developed in Ref. [40] and further ex-tended in Ref. [39]. In the following, we summarise thekey steps of this treatment, which represent the basis forour implementation of the work strokes.

We wish to find a wave-function ψ(x, t) satisfying theSchrödinger equation

i~∂tψ(x, t) = Hs(t)ψ(x, t) (6)

within the time interval [0, τ ], with ω(0) = ω1 andω(τ) = ω2. In the following, ω1 and ω2 will be either ωcor ωh depending on which work transformation is beingperformed. The Hamiltonian in Eq. (1) can be written,at any fixed time t, in the second quantization formalismas

Hs(t) = ~ω(t)(1/2 + a†(t)a(t)

), (7)

where the operators

a(t) =

√mω(t)

2~x+ i

√1

2m~ω(t)p (8)

and a†(t) = [a(t)]† depend explicitly on time. FromEq. (7), we obtain the instantaneous eigenvalues Etn =~ω(t)(1/2 + n(t)) and the wave-function φtn(x) of itseigenvectors, which are just a slight generalization of thesolutions for the time-independent quantum harmonic os-cillator. Explicitly

φtn(x) =4

√mω(t)

π~1√

2nn!e−

mω(t)2~ x2

Hn

(x

√mω(t)

~

),

(9)where Hn(z) is the n-th Hermite polynomial of argumentz. The superscript t aims at reminding that t here playsjust the role of a label.

It can be seen by direct inspection that Eq. (6) admitssolutions satisfying the Gaussian ansatz

ψ(x, t) = exp[i(A(t)x2 + 2B(t)x+ C(t)

)/2~]. (10)

By inserting this formula into Eq. (6), we obtain a systemof three differential equation for the coefficients A, B, Creading

dA

dt= −A

2

m−mω2(t) (11)

dB

dt= −A

mB (12)

dC

dt= i~

A

m− 1

mB2. (13)

Eq. (11) can be mapped into the equation of motion of aclassical time-dependent oscillator with amplitude X(t),

4

through the substitution A = mX/X. Explicitly

d2X

dt2+ ω2(t)X = 0. (14)

Once a parametrization is chosen for the transformationω(t), solving such equation gives A(t), from which alsoB(t) and C(t) can be found by integrating Eqs. (12)and (13). In [39] it is shown that by choosing theparametrization

ω2(t) = ω22 + t(ω2

1 − ω22)/τ,

an analytic solution to such problem can be found. Werefer to the mentioned reference for the full expression.Another key result, obtained in Ref. [40], is the expressionof the propagator

U(x, τ |x0, 0) =

√m

2πi~X(τ)e

im2~X(τ)

(X(τ)x2−2xx0+Y (τ)x20),

(15)

where now X(t) and Y (t) are two specific solutions ofEq. (14) satisfying the boundary conditions

X(0) = 0, X(0) = 1,

Y (0) = 1, Y (0) = 0.

Having the explicit form of the propagator U(x, τ |x0, 0),we now have all the tools to describe the effect of the worktransformation ω1 → ω2 (for arbitrary values of ω1,2) onthe engine density operator ρ(x, y; t), considered here asdependent on position. We have

ρ(x0, y0; 0) 7→

ρ(x, y; τ) =

∫U(x, τ |x0, 0)ρ(x0, y0; 0)U†(y, τ |y0, 0)dx0dy0

(16)

One further step is required, with the aim of makingthe above transformation amenable to numerical treat-ment: the expansion of both the density matrix ρ andthe operator U on the basis given by the eigenfunctionsin Eq. (9). Let us define

ρmn(t) := 〈φtm|ρ(t)|φtn〉, (17)

Umn := 〈φτm|U(τ, 0)|φ0n〉, (18)

where we omitted the position dependencies since theyare integrated over in the scalar products. It should bestressed that the Umn elements are computed by takingscalar products with two different sets of eigenfunctions,the effect of U(τ, 0) being precisely that of implemen-ting the transformation from one Hamiltonian to another.Eq. (16) then becomes

ρmn(0) 7→ ρkl(τ) =∑mn

Ukmρmn(0)U†nl.

0.1 0.5 1 5 100.0

0.5

1.0

1.5

2.0

τ

Q*

Deviation from adiabaticity

FIG. 2. Deviation from the adiabatic regime, as captured bythe Q∗ factor as a function of the work stroke duration τ , forω1 = 1 and ω2 = 4. The green (lighter) line represents thelimit Q∗ = 1 for τ →∞.

In Ref. [40] an expression for the generating function oftransition probabilities from the initial to the final eigen-states P τm,n = |〈φτm|U(τ, 0)|φ0

n〉|2 is provided:

P (u, v) =∑m,n

umvnP τm,n

=

√2

Q∗(1− u2)(1− v2) + (1 + u2)(1 + v2)− 4uv,

Remarkably, the above expression depends on the detailsof the parametrization ω(t) only through the factor Q∗,whose expression for the most general transformation isprovided in [39]:

Q∗ =ω2

1

(ω2

2X(τ)2 + X(τ)2)

+(ω2

2Y (τ)2 + Y (τ)2)

2ω1ω2(19)

The factor tends to the limiting value Q∗ → 1 for τ →∞,and becomes increasingly greater than 1 as τ becomessmaller, as exemplified in Fig. 2.

The following special cases are of particular interest:

• No transformation is performed, ω2 = ω1. It can beshown that the propagator (15) becomes the iden-tity operator embodied by U(x, τ |x0, 0) = δ(x−x0).The matrix elements (18) are Umn = δmn, since theinitial and final eigenbasis coincide.

• Sudden quench, τ → 0. Also in this caseU(x, τ |x0, 0) → δ(x − x0), because the transfor-mation is so quick that the density operator is leftunchanged. Its matrix elements ρmn, however, un-dergo a unitary change of basis through the matrixUmn = 〈φ(2)

m |φ(1)n 〉, where the superscripts refer to

the frequencies ω1, ω2.

• Adiabatic transformation, τ → ∞. The initialeigenstates are mapped one-to-one to the final ones,

5

infinitely slowly. The propagator can be expressedas U(τ, 0) =

∑n |φτn〉

⟨φ0n

∣∣, which gives Umn = δmn.

From now on, we will denote the duration τ of the worktransformations by τw.

C. Heat exchanges: collisional model

Let us now introduce the model for the environmentsand for how they interact, both with the engine andwithin themselves. The interactions are implementedthrough a collisional model. When the engine interacts,say, with the cold environment, it undergoes an inter-action with one, and only one, environment particle ata time, through a unitary operator acting non-triviallyonly on the Hilbert spaces of the engine and said particle.This is what we refer to as a collision, meaning a "point-like" interaction between the engine and one element ofthe environment. With the thermodynamic limit in sight,it is assumed that the engine never interacts twice withthe same environment particle: at each collision, the en-gine interacts with a new, "fresh" particle.

The unitary Vse through which the interaction happensis generated by a resonant excitation conserving Hamil-tonian

Hse = J(aσ+

e + a†σ−e), (20)

Vse = exp

[− i

~Hseτse

], e = c, h (21)

where the main coupling constant J and the interactiontime τse rule the interaction strength, and are assumedto be the same for both environments. As mentioned inSec. II A, at the moment of interaction the engine fre-quency matches exactly that of the environment parti-cle it is interacting with. In the most basic, completelymemoryless implementation of such a model, only oneparticle for each environment actually stored at any time.Indicating by Hs, Hc and Hh Hilbert spaces of the en-gine, of a cold and a hot particle respectively, the minimaltotal Hilbert space is

H = Hc ⊗Hs ⊗Hh.

With reference to Fig. 1 (a), suppose the engine is inthe state ρs at the beginning of iteration n of the thermo-dynamic cycle, and interacts with the n-th cold particleinitially in the state ρ(n)

c ,

ρ(n)c ⊗ ρs ⊗ ρ(n)

h →

ρcsh = (Vsc ⊗ Ih)(ρ(n)c ⊗ ρs ⊗ ρ(n)

h

)(V †sc ⊗ Ih),

and a hot particle ρ(n)h , while present, is left unaf-

fected [41]. After interaction, we take the local marginalsof both parties ρs = Trc,h[ρcsh] and ρ

(n)c = Trs,h[ρcsh]

and possibly use them to compute the various thermo-dynamic quantities as explained in Sec. II A. The particle

ρ(n)c is then discarded, in practice traced away from the

global density matrix, and a new one ρ(n+1)c is included

in the model in its place, such the global state ready forthe next step is

ρ(n+1)c ⊗ ρs ⊗ ρ(n)

h

We now take a step further and introduce intra-environment collisions, thus allowing the environmentsto carry over memory of past interactions with the en-gine, opening the possibility for the engine dynamics tobe non-Markovian. We wish therefore to consider twoparticles for each environment at any given time. In or-der to do so, we need to extend the Hilbert space we workwith to

H = Hc,b ⊗Hc,a ⊗Hs ⊗Hh,a ⊗Hh,b,

where the additional subscript "a" indicates now the first(hot or cold) environment particle interacting with theengine and "b" indicates the second, that is particlesn and n + 1 in our example. Before is it traced away,the environment particle n undergoes a further collisionwith the particle n+ 1, through a Heisenberg interactionHamiltonian:

Hee = Jee(σxnσ

xn+1 + σynσ

yn+1 + σznσ

zn+1

), (22)

Vee = exp

[− i

~Heeτee

], ee = cc, hh (23)

with coupling constant Jcc(Jhh) and interaction timeτcc(τhh) for the cold (hot) environment. As explainedin [42], [32] and [35], the interaction acts effectively as apartial swap, exchanging the states of the two particleswith probability sin2(2Jeeτee). In particular, a perfectswap is achieved for Jeeτee = π/4.

Continuing in our example, after the subsequent appli-cation of Vsc and Vcc, the engine and particle n+1 will bein general in a correlated state, ρ(n+1)

sc even if they haven’tinteracted directly yet. After tracing away the cold par-ticle n, "shifting" particle n+ 1 from position c, b to c, ain our Hilbert space, and including a new particle n+ 2in position (c, b), the global state is

ρ(n+2)c ⊗ ρ(n+1)

sc ⊗ ρ(n)h ⊗ ρ(n+1)

h .

This completes the description of one full heat stroke.The state is now ready for the next stroke, which will bea work one. The interactions between the engine and thehot environment, and within the hot environment, wouldoccur in exactly the same way. Therefore, at the end ofa full cycle, composed of all the steps of Sec. II A, theglobal state looks like

ρ(n+2)c ⊗ ρ(n+1)

sch ⊗ ρ(n+2)h ,

with the engine correlated to the hot and cold parti-cles (n + 1). More details on this model of system-environment interaction can be found in [35].

6

0 20 40 60 80 1000.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

number of iterations

-Degree Evolution

Jeeτee=0.65π

4

Jeeτee=0.5π

4

Jeeτee=0.35π

4

Jeeτee=0.2π

4

Jeeτee=0.05π

4

FIG. 3. Signatures of non-Markovianity: time evolution ofthe degree of non-Markovianity N , with intra-environmentinteraction Jeeτee increasing from blue to red (bottom to top).Nearly adiabatic work strokes (Jτw = 32) and pure initialstates

∣∣ψ±test⟩. Inset: final N against the intra-environmentinteraction Jeeτee. The dashed line is a guide for the eye.

Finally, the total cycle duration is T = 2(τw + τse),taking into account only the steps in which the engine isdirectly involved and assuming the intra-environment in-teractions to occur at the same time as the work strokes.

III. RESULTS

We present here the results on the engine perfor-mance and the possible influence of non-Markovianityon its operations. First, we present evidence of non-Markovianity in the engine dynamics and its depen-dence on intra-environment interactions. We then in-vestigate the crossover from adiabatic to sudden quenchwork strokes in the purely Markovian regime. Finally weshow how the non-Markovianity affects the engine per-formance.

Few words on the choice of parameters. We work inunits where ~ = 1, κ = 1, and also choose J = Jcc =Jhh = 1 without loss of generality. The temperaturesof the environments are Tc = 0.1 and Th = 10, givinga Carnot efficiency of 0.99 and a Curzon-Ahlborn effi-ciency of 0.9 as theoretical upper bounds. The engine isinitialized in a thermal state at Ts = 0.5 unless otherwisestated. While the initial temperature is not very cru-cial for the thermodynamics, the fact of the initial statebeing diagonal in the energy eigenbasis does impact thebehaviour of the engine.

We chose an interaction between the engine and theenvironments of moderate strength Jτse = 0.3, in or-der for it to be quite weak, but not so weak that theheat exchanged per cycle becomes too small, comparedto the exchanged work. The chosen environment frequen-cies are ωc = 1 and ωh = 4, close enough to each otherthat the work exchanged is not too big, compared tothe exchanged heat, and such that the adiabatic regime(τw → +∞) is approximated well at τw = 16 and verywell at τw = 32. The gap between ωh and ωc is nonthe-

0 20 40 60 80 1000.0

0.5

1.0

1.5

2.0

number of iterations

Evolution of coherence

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.01.2

Initial Coherence

Non-Markovianity Degree

Jeeτee=0.6π

4

Jeeτee=0.45π

4

Jeeτee=0.3π

4

Jeeτee=0.15π

4

Jeeτee=0.π

4

FIG. 4. Time evolution of the coherence C in the enginedensity matrix, with intra-environment interaction Jeeτee in-creasing from blue to red (bottom to top). Pure initial states∣∣ψ±test⟩. The work strokes are nearly adiabatic, Jτw = 32. In-set: degree of non-Markovianity N against initial coherencein the pure initial states

∣∣ψ±α ⟩, with α on the x axis. Thedashed line is a guide for the eye.

less big enough that, in the sudden quench regime, theQ∗ factor is appreciably different from 1, in fact surpas-sing 2 as can be seen in Fig. 2. The theoretical efficiencyin the adiabatic case is ηth = 0.75.

The density matrices of the environment particles arerepresented in the eigenbasis { |0〉 , |1〉 } of the Hamilto-nian ~ωeσze/2.

As the initial temperature Ts is low, the initial pop-ulations decay quite fast, being negligible (below ma-chine precision) above level n = 20. Therefore, in mostsimulations we could safely truncate the Fock space atn = 30, checking that the matrices representing the uni-taries U, Vse, and Vee in the truncated space remain ap-proximately unitary. We performed tests extending theFock space up to n = 50 to confirm that the results werenot appreciably different than those obtained truncatingat n = 30. Since at any time we simulate two particles foreach environment, the dimension of the total dimensionof the density matrix is 24 × 31 = 496.

A. Non-Markovianity of the engine dynamics

As summarized in Appendix A, in order to quantifythe degree of non-Markovianity N according to the BLPmeasure, one has to choose pairs of initial states, makethem evolve in time according to the specific dynamicsin object, and observe the behaviour of the trace dis-tance. It is then required to maximise N over all pos-sible choices of initial pairs. In our case, however, thestate of the engine is represented by a 31 × 31 complexhermitian matrix and the maximization is an extremelydemanding task. We chose therefore heuristically a cou-ple of pure orthogonal states

∣∣ψ±test⟩, guided by analogywith the spin-1/2 particle case in which often the optimal

7

(a) (b)

0.050.10 0.50 1 5 100.0

0.2

0.4

0.6

0.8

1.0

Jτw

η∞

Efficiency

0.050.10 0.50 1 5 100

10

20

30

40

50

60

Jτw

N∞

Approach to stationary cycle

FIG. 5. (a) Stationary cycle efficiency η∞ against the dura-tion of the work stroke τw, for engine-environment couplingconstant J = 1. The dashed line is a guide for the eye, thegreen (solid) line represents the theoretic adiabatic efficiencyηth = 1 − ωc/ωh. (b) Number of iterations N∞ required toreach the stationary cycle against the duration of the workstroke τw, for engine-environment coupling constant J = 1.The dashed line is a guide for the eye, the green (solid) linerepresents the limit τw →∞.

pair is |±〉 = (|0〉 ± |1〉)/√

2 [32, 35]. The choice was

∣∣ψ±test⟩ =|0〉 ± |10〉√

2, (24)

as we found that pure states in the form (|0〉 ± |n〉)/√

2,with a high degree of coherence in the energy eigen-basis, seem to be particularly favourable to boost thenon-Markovianity measure. This particular choice

∣∣ψ±test⟩turned out to be the one yielding the biggest N . Eventhough we do not have evidence that this is the best pairof states maximising the non-Markovianity degree N , weare confident that the results may still provide a valuableinsight on the non-Markovian character of the dynamics.

Fig. 3 presents the behaviour of the non-Markovianitydegree N in its dependence on the intra-environment in-teraction strength and in its time evolution, in the case ofadiabatic work strokes. The non-Markovian behaviour isintrinsically a property of the dynamics during the tran-sient to stationary state. Fig. 4 shows the dynamics ofthe total internal coherence of the engine, quantified by

C :=∑i 6=j

|ρij |.

The coherence in the stationary state settles to a quitesmall value, irrespective of the initial state. Further-more, the more the dynamics is non-Markovian, thelonger the coherences survive. This is most likely a di-rect consequence of the fact that the interaction withnon-Markovian environments slows down the approachto the stationary state (see also Fig. 7). The inset ofFig. 4 shows the relation between non-Markovianity andthe initial coherence present in the engine, when initiali-zed in the state |ψ±α 〉 = (|0〉 ± α |10〉)/Nα, normalized byNα, with α = m× 0.1, m = 0, 1, . . . , 10. The connectionbetween the presence of coherence in the initial states andtheir effectiveness in the detection of non-Markovianityis evident.

0.05 0.10 0.50 1 5 100.00

0.01

0.02

0.03

0.04

0.05

Jτw

P∞

Power

FIG. 6. Power output against the duration of the work strokeτw, for engine-environment coupling constant J = 1. Thedashed line is a guide for the eye. The power vanishes forτw → ∞, as the efficiency approaches the limit ηth while thecycle duration grows as ∼ 2τw. In the sudden quench limit,instead, it approaches a finite value, being η non-zero for τw →0 while the cycle duration is ∼ 2τse.

B. Engine performance

Fig. 5 and 6 summarize the behaviour of the enginein the Markovian regime, with no intra-environment in-teractions, focusing on the crossover from adiabatic tosudden quench work strokes. We can see that the sta-tionary cycle efficiency η∞ reaches the expected limit ηthin the adiabatic case, and decreases as we depart fromadiabaticity. The duration of the work strokes τw alsoaffects the number of iterations N∞ it takes for the en-gine to reach the stationary regime, which grows as weapproach the sudden quench regime. This further indi-cates a drop of the engine performance as we move awayfrom adiabaticity. The power output per single iterationP∞, however, has a maximum around τw = 1, since atthat point the efficiency deviates only slightly from ηth.

Fig. 7 and 8 present the behaviour of the performancein the most general case of the engine operating with non-adiabatic work strokes and non-Markovian environments.Non-Markovianity seem to always affect negatively theperformance, but it does so more pronouncedly as wedeviate from the adiabatic regime. In particular, the effi-ciency in the adiabatic case is mostly independent of thenon-Markovian character of the dynamics, approachingin fact ηth, while for smaller durations of the work strokesit drops more pronouncedly as the intra-environment in-teractions become stronger. The power output, therefore,decreases accordingly.

IV. CONCLUSIONS AND OUTLOOK

In this work we studied the out-of-equilibrium ther-modynamics and performance of a quantum Otto cycle

8

0.0 0.1 0.2 0.3 0.4 0.50

50

100

150

Jeeτee4π

N∞

Approach to stationary cycle

τW=2

τW=4

τW=8

τW=16

τW=32

FIG. 7. Number of iterations N∞ to the stationary cycle,against the intra-environment interaction Jeeτee. The dashedlines are guides for the eye. The duration τw of the workstrokes increases from the red to the blue curve (top to bot-tom).

employing a harmonic oscillator as working fluid, in in-teraction with a finite-size environment through a colli-sional dynamics that may allow for memory effects, andthus for the emergence of non-Markovianity. We ex-plored the crossover from adiabatic to sudden quenchwork strokes and found that, while departing from theadiabatic regime induces a drop in the efficiency, it ispossible to find an optimal duration of the work strokessuch that the power output is maximised. We do notobserve better than classical performance, at least in thecase when the both the engine and the environment par-ticles are initialized in thermal states, which is consis-tent with the what reported in [19] on the thermody-namics of heat machines with quantized energy levels.Non-Markovianity signatures are observed in the enginedynamics, and even though such memory effects do notimpact the stationary engine performance, they do affectthe approach to the asymptotic cycle, slowing it down.Non-Markovianity is however found to be closely con-nected with the presence of initial coherences in the en-ergy eigenbasis of the engine.

ACKNOWLEDGMENTS

M Pezzutto thanks the Centre for Theoretical Atomic,Molecular, and Optical Physics, School of Mathemat-ics and Physics, Queen’s University Belfast for hos-pitality during the development of this work. MPezzutto and Y Omar thank the support from Fun-dação para a Ciência e a Tecnologia (Portugal)through programmes PTDC/POPH/POCH and projectsUID/EEA/50008/2013, IT/QuNet, ProQuNet, partiallyfunded by EU FEDER, from the QuantERA project The-BlinQC, and from the JTF project NQuN (ID 60478).Furthermore, M Pezzutto acknowledges the support fromthe DP-PMI and FCT (Portugal) through scholarshipSFRH/BD/52240/2013. M Paternostro acknowledgessupport from the EU Collaborative project TEQ (grant

0.0 0.1 0.2 0.3 0.4 0.50.0

0.2

0.4

0.6

0.8

Jeeτee4π

η∞

Efficiency

τW=32

τW=16

τW=8

τW=4

τW=2

FIG. 8. Stationary cycle efficiency η∞ against the intra-environment interaction Jeeτee. The dashed lines are guidesfor the eye. The duration τw of the work strokes decreasesfrom the blue to the red curve (top to bottom).

agreement 766900), the DfE-SFI Investigator Programme(grant 15/IA/2864), the Royal Society Newton Mobil-ity Grant NI160057, and COST Action CA15220. Allauthors gratefully acknowledge support from the COSTAction MP1209.

Appendix A: Quantum non-Markovianity

Recently, the issue of non-Markovianity of quan-tum dynamics has received considerable attention aimedat characterizing the phenomenology of non-Markovianopen-system dynamics through general tools of broad ap-plicability. Such efforts are based on the formal assess-ment of the various facets with which non-Markovianityis manifested.

One of such approaches to the definition of Markovian-ity is based on the notion of divisibility [43]: a dynamicalprocess is said to be Markovian if, for any two timest ≤ s there is a completely positive map Γ(s, t) suchthat ρ(s) = Γ(s, t)ρ(t), and such maps form a semigroup.The requirement of complete positivity links very tightlyMarkovianity with the behaviour of the dynamical mapson entangled states, in the sense that a Markovian pro-cess is always expected to degrade entanglement. In orderto assess the (non-)Markovian character of the dynamics,one needs to perform a tomography of the process so asto reconstruct the map Γ.

Another widely employed approach, introduced inRefs. [44, 45], is based on the concept of informationbackflow. The starting point is the fact that any com-pletely positive trace-preserving (CPTP) map is a con-traction for the trace distance [46], defined as

D(ρ1, ρ2) := 12‖ρ1 − ρ2‖ ,

where ‖A‖ = Tr√A†A is the trace-1 norm and ρ1,2 are

two density matrix of the system under scrutiny. Thetrace distance is a metric in the space of density matri-ces, closely related to their distinguishability: a value of

9

D(ρ1, ρ2) = 1 implies perfect distinguishability.The contractive character of D(ρ1, ρ2) under CPTP

maps is the key idea for the quantification of non-Markovianity based on information backflow: Markovianmaps can not increase the distinguishability of any twogiven states. If, however, one can find a pair of initialstates and a time t for which contractivity is violated,resulting in

σ(t) =dD(ρ1(t), ρ2(t))

dt> 0, (A1)

this is held as a signature of non-Markovianity in the

dynamics. Such criterion can be used to build a quanti-tative measure N of non-Markovianity as

N := max{ρ1,ρ2}

∫Σ+

σ(t)dt, (A2)

where Σ+ is the time window where σ(t) > 0, and maxi-mise on the choice of initial states. While finding theoptimal pair of initial states is in general challenging, thetask is often simplified owing to the result reported inRef. [47], where it is proven that the states maximizingN must be orthogonal and belonging to the boundary ofthe state space.

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