Experimental investigation of the effect of classical noise on quantum non-Markovian dynamics
Simone Cialdi, Claudia Benedetti, Dario Tamascelli, Stefano Olivares, Matteo G. A. Paris, Bassano Vacchini
EExperimental investigation of the effect of classical noiseon quantum non-Markovian dynamics
Simone Cialdi, ∗ Claudia Benedetti, Dario Tamascelli,Stefano Olivares, Matteo G. A. Paris, and Bassano Vacchini † Quantum Technology Lab, Dipartimento di Fisica “Aldo Pontremoli”,Universit`a degli Studi di Milano, Via Celoria 16, I-20133 Milan, Italy andINFN, Sezione di Milano, Via Celoria 16, I-20133 Milan, Italy
We provide an experimental study of the relationship between the action of different classical noises on thedephasing dynamics of a two-level system and the non-Markovianity of the quantum dynamics. The two-levelsystem is encoded in the photonic polarization degrees of freedom and the action of the noise is obtained via aspatial light modulator, thus allowing for an easy engineering of different random environments. The quantumnon-Markovianity of the dynamics driven by classical Markovian and non-Markovian noise, both Gaussian andnon-Gaussian, is studied by means of the trace distance. Our study clearly shows the different nature of thenotion of non-Markovian classical process and non-Markovian quantum dynamics.
I. INTRODUCTION
The characterization of quantum non-Markovianprocesses has recently attracted a lot of attention:besides its conceptual interest it might open theway to obtain improved performances in quantumthermodynamics [1], higher sensitivities in quantummetrology [2, 3] and techniques for complex quan-tum systems probing [4]. A natural and intriguingquestion is the relationship between the proposednotions of quantum non-Markovian dynamics (see[5–7] for reviews) and the classical notion of memo-ryless or Markovian process. This parallel has beenthe object of different theoretical studies [8–13] andhas indeed provided the motivation for one of thefirst approaches to the problem [14].In this paper we address this question from a newviewpoint, relying on the experimental realizationof quantum dynamics depending on a classical ran-dom processes. In such a way we relate a classicalinput with a quantum output and investigate the fea-tures of the latter with respect to the former. To thisaim we need to generate a wide variety of classi-cal processes with known features. We perform thistask by obtaining such stochastic processes as solu-tion of suitable stochastic differential equations, so ∗ [email protected] † [email protected] that we can obtain both Gaussian and non-Gaussian,Markovian and non-Markovian classical processes.The experimental implementation is based on aquantum optics setup which allows to engineer ina controlled way a dephasing dynamics determinedby a classical stochastic process, which affects thepolarization degrees of freedom of photons. A suit-able configuration allows to address in parallel ahigh number of realizations of the process, and av-erage them automatically in the detection stage.The paper is organized as follows. In Sect. II weintroduce the model of the system and the environ-ment used to relate a classical input with a quantumoutput. In Sect. III we discuss how to generate clas-sical noises with known features. We introduce theexperimental setup in Sect. IV and discuss the ob-tained results in Sect. V. We draw our conclusionsand final remarks in Sect. VI. II. THE MODELA. Evolution map
In order to investigate by means of experimenthow the different hallmarks of classical noise, suchas being Gaussian or Markovian, affect the featuresof a quantum dynamics, especially in view of theproperty of non-Markovianity, we consider the fol-lowing simple but versatile model. The Hamiltoniandescribing the time evolution of the system is given a r X i v : . [ qu a n t - ph ] S e p by H ( t ) = X ( t ) (cid:126) ω σ z , (1)where X ( t ) denotes a classical stochastic processwith independent increments, describing the effectof the environmental noise on the two-level systemof interest, and ω denotes the natural energy split-ting of the two-level system, fixing scale and dimen-sions. The time evolution operator determining theeffect of each single realization of the noise reads U ( t,
0) = e − iω σ z (cid:82) t d τX ( τ ) , so that upon averaging over the environmental influ-ence one obtains for the reduced system dynamics ρ S ( t ) = Λ( t )[ ρ S (0)]= E [ U ( t, ρ S (0) U ( t, † ] , where E [ · ] denotes the expectation value over thesample space of the noise. We are interested in in-vestigating the behavior of ρ S ( t ) in its dependencefrom the noise X ( t ) .To this aim we denote by X ( t ) = (cid:90) t d τ X ( τ ) (2)the integral over time of the stochastic process, lead-ing to U ( t,
0) = diag(e − iω X ( t ) , e + iω X ( t ) ) , so thatthe reduced system dynamics is fully captured bythe transformation of the off-diagonal matrix ele-ment ρ S (0) t −→ ρ S (0) E [exp( − iω X ( t ))] . (3)This simple description allows for a high experi-mental freedom in the implementation of the envi-ronmental noises and a unique characterization ofnon-Markovianity of the ensuing quantum dynam-ics. Indeed the evolution corresponds to a puredephasing dynamics, for which all definitions ofquantum non-Markovianity coincide [5–7], so thata clearcut signature of quantum memory effects canbe provided. In such a way the considered modelwell serves the purpose of exploring the effects ofdifferent classical noises on a quantum dynamics. B. Features of quantum dynamics
The signature of the quantum dynamics that wewant to address, in its dependence on the noise de-scribing the effect of the environment, is its non-Markovianity. In the considered model, once wefix the noise acting in the Hamiltonian, we obtaina quantum dynamics which, while being given bya unitary transformation for each fixed realizationof the noise, provides a time dependent collectionof completely positive trace preserving maps uponaveraging over the noise. This collection of mapsdescribes a stochastic quantum dynamics, whichwould generally arise as a consequence of the in-teraction with a suitable quantum environment. In-deed, as it has been recently considered, any clas-sical average can be seen to arise from a micro-scopic description with ancillary quantum degreesof freedom initialized in a classical state [15]. Inthis respect the obtained dynamics provides a well-defined reduced quantum dynamics, whose featurescan be studied in view of the relationship betweenthe properties of the input noise and the features ofthe output maps. As a figure of merit we will con-sider the non-Markovianity of the ensuing quantumdynamics as characterized by the behavior in timeof the trace distance between two initially distinctsystem states. This viewpoint was first introducedin [16, 17] and connected to a notion of informa-tion exchange between system and environment. Inparticular, it is important to stress that this phys-ical intuition remains confirmed even if the map,as in this case, is obtained upon averaging with re-spect to a classical label. This is an important issue,which has been the object of different investigations[15, 18, 19]. For the present simple model, as al-ready stressed, essentially all proposed definitionsof non-Markovianity coincide [6] since the dynam-ics is captured by the transformation Eq. (3). Thenatural quantifier of non-Markovianity for this classof models is therefore the behavior of the quantity D [ { X ( t ) } ] = | E [exp( − iω X ( t ))] | , (4)which describes the dephasing effect of the environ-mental noise on the system dynamics. In the tracedistance formalism this estimator is obtained con-sidering a pair of states which maximize the pos-sible revivals of the considered quantity. A mono-tonic decrease in time of this quantity correspondsto a Markovian dynamics, while a non-Markoviandynamics is obtained if revivals in time appear. III. CLASSICAL NOISE GENERATION
In considering classical stochastic processes, twoclasses stand out in their relevance for applicationsand theoretical treatments, namely Gaussian andMarkovian processes. In both cases a relatively sim-ple description applies, at variance with the case ofa generic process. As a matter of fact, while inthe general case a description of the process callsfor knowledge of all its correlation functions, fix-ing the probability for given outcomes of the ran-dom variable at given times, in the case of Gaus-sian and Markovian processes a drastic simplifica-tion applies. A Gaussian process is in fact fixed byfirst and second moments only, while a Markovianprocess is determined by initial probability distribu-tion and transition probability [20]. It is thereforeof interest to explore the effect of noise on a quan-tum dynamics classifying the classical noises withrespect to these two distinctive features.To the aim of generating in a simple way thesedifferent type of noises we consider as starting pointtwo Markovian processes whose realizations can beeasily simulated. Our starting tools are thereforeWiener and random telegraph noise, both Marko-vian: Gaussian the former, non-Gaussian the latter.Stochastic processes X ( t ) with different features tobe used in the dynamics given by Eq. (1) will beobtained as solution of stochastic differential equa-tions with different input noises.As Markovian Gaussian process we will considera Ornstein-Uhlenbeck process X OU ( t ) with frictioncoefficient γ and diffusion constant σ , which can beobtained as solution of a linear stochastic differen-tial equation driven by Wiener noise. We will fur-ther denote as X RTN ( t ) a random telegraph noisewith switching rate γ and step one, whose realiza-tions therefore jump from plus to minus one accord-ing to a Poisson process with rate γ . Such a pro-cess is still Markovian, but its probability density isnot in Gaussian form. Using the realization of theseprocesses as input noise we can obtain noises withdifferent features. Let us first consider the equation dY ( t ) = − κY ( t ) dt + dX OU ( t ) , (5)with κ a positive rate. The process Y ( t ) is stillGaussian due to linearity of the equation but non-Markovian, since its determination requires theknowledge of X OU ( t ) up to time t [21, 22]. Usingthe same strategy we can obtain the increments ofa process which is neither Gaussian nor Markovianconsidering the stochastic differential equation dZ ( t ) = − µZ ( t ) dt + dX RTN ( t ) , (6)with µ a positive rate. Using as seeds Wiener andrandom telegraph noise we are thus able to gener-ate, via the stochastic differential equations givenby Eq. (5) and Eq. (6), increments of stochastic pro-cesses which share or lack the distinct features ofGaussianity and Markovianity according to the cor-responding well established classical definitions.In order to estimate the effect of the differentnoises on the dynamics we further need to evaluatethe expectation value of the integral over time of theconsidered noise, defined as in Eq. (3). The analyticevaluation of this quantity is only feasible in spe-cial cases. For the case of the Ornstein-Uhlenbeckprocess we define X OU ( t ) ≡ (cid:90) t d τ X OU ( τ ) , (7)and exploiting Gaussianity one obtains for the quan-tity determining the dephasing of the two-level sys-tem D [ { X OU ( t ) } ] ≡ | E [exp( − iω X OU ( t ))] | = exp (cid:18) − ω σ γ (2 γt − − e − γt + 4 e − γt ) (cid:19) , (8)where γ and σ denote respectively friction and dif-fusion coefficient of the process.For the case of random telegraph noise, definingon the same footing X RTN ( t ) ≡ (cid:90) t d τ X RTN ( τ ) , (9)one can show that the dephasing factor takes theform [23] D [ { X RTN ( t ) } ] ≡ | E [exp( − iω X RTN ( t ))] | = e − γt (cid:104) cosh( νt ) + γν sinh( νt ) (cid:105) , (10)with ν = (cid:112) γ − ω . These explicit expressionsallow to estimate the dephasing factor of Eq. (4)and study its monotonicity properties as a func-tion of time. As we discuss in Sect. V these es-timates are indeed confirmed by the experimentalresults, and validate the theoretical analysis. It ap-pears in particular that while the dephasing due toOrnstein-Uhlenbeck noise D [ { X OU ( t ) } ] is a de-creasing function of time for any value of γ and σ , the contribution corresponding to the random telegraph noise D [ { X RTN ( t ) } ] can also exhibit anoscillating behavior [24]. Note that both pro-cesses are examples of Markovian colored noise andhave an exponentially decaying correlation func-tion, namely [20] E [ X OU ( t ) X OU ( s )] = σ γ exp( − γ | t − s | ) and E [ X RTN ( t ) X RTN ( s )] = exp( − γ | t − s | ) . In order to consider a classical non-Markovian pro-cess, still retaining the property of Gaussianity, werefer to Eq. (5). The relevant quantity is again D [ { Y ( t ) } ] , which can be evaluated exploiting thefact that Y ( t ) is again Gaussian and relying on theproperties of the Ornstein-Uhlenbeck process. Theresult reads D [ { Y ( t ) } ] = | E [exp( − iω Y ( t ))] | (11) = exp (cid:26) − ω σ ( γ − κ ) − ( γe − κt − κe − γt ) + γκ (2 e − ( γ + κ ) t − e − γt − e − κt ) γκ ( γ − κ ) ( γ + κ ) (cid:27) , where according to Eq. (7) and (9) we have de-noted the integrated process as Y ( t ) ≡ (cid:82) t d τ Y ( τ ) .The dephasing factor shows a monotonic decay-ing behavior for all possible values of the con-stants γ and κ , friction coefficient of the Ornstein-Uhlenbeck and rate appearing in the stochastic dif-ferential equation Eq. (5) respectively.The last process that we will consider is the so-lution of Eq. (6), which is neither Gaussian norMarkovian due to the fact that the driving noise iscolored and non-Gaussian. The evaluation of thecorresponding dephasing factor D [ { Z ( t ) } ] = | E [exp( − iω Z ( t ))] | , (12)with Z ( t ) ≡ (cid:82) t d τ Z ( τ ) , calls for a numerical eval-uation since we can no more exploit the importantsimplification in the evaluation of the characteristicfunction warranted for Gaussian processes. In par-ticular, as confirmed by the experiment, it appearsthat depending on the parameter values also in this case an oscillating behavior can show up. It thusappears that in this context non-Markovianity of thequantum dynamics appears when the relevant clas-sical process is non-Gaussian, rather then being re-lated to a lack of the Markov property.In various theoretical papers and experimentalimplementations [25] it has been shown that theappearance of quantum non-Markovianity in situ-ations in which the environmental interaction canbe characterized by a spectral density is typicallyrelated to a non-trivial peak structure of the rele-vant frequency spectrum. In this respect it is naturalto investigate also in the present framework the re-lationship between spectral properties of the noiseand features of the quantum dynamics. For bothOrnstein-Uhlenbeck and random telegraph noise thespectrum has a Lorentzian shape centered in zero,corresponding to the exponential decay of the two-time correlation function. The correlation functionof the process Y ( t ) takes instead the form E [ Y ( t ) Y ( s )] = (cid:18) σγ − ν (cid:19) (cid:110) γ − γ | t − s | − e − γ ( t + s ) ) + ν − ν | t − s | − e − ν ( t + s ) )+ γνγ + ν (e − νt − γs + e − γt − νs − e − ν | t − s | − e − γ | t − s | ) (cid:27) Figure 1. Spectrum of the process Z ( t ) obtained by thenumerical evaluation of the stochastic differential equa-tion Eq. (6) for the case µ = 0 . (red line) and µ = 1 (blu line). The coefficient γ characterizing the RTN is setto 0.5. The process is only asymptotic stationary, with as-sociated power spectrum S ( ω ) = σ π ω ( γ + ω )( κ + ω ) (13)featuring a double-peaked structure and a dip atsmall frequencies. The same feature is shared bythe spectrum of the process Z ( t ) arising as solutionof Eq. (6), which can be evaluated numerically andis shown in Fig. 1.Despite the non trivial structure of the powerspectrum, as follows from Eq. (11) the trace dis-tance still exhibits a monotonically decaying behav-ior, reflecting a Markovian dynamics, for the Y ( t ) process, while oscillations may be present for the Z ( t ) process. It therefore appears that in this con-text the correlation function of the classical processand the associated power spectrum does not embody the relevant information in characterizing the mem-ory properties of the quantum dynamics. IV. EXPERIMENTAL IMPLEMENTATION
The effect of a classical noise on a quantum de-phasing dynamics can be experimentally investi-gated in a quantum optics setup. To this aim weencode the quantum degrees of freedom in the po-larization state of photons and let the noise affectthe phase information. Efficiently generating andaveraging over the different realizations of the noiseprovides the major obstacle in order to experimen-tally study the effect of classical disturbance on aquantum dynamics. This difficulty can be overcomeby exploiting a recently realized all-optical quan-tum simulator [26]. This apparatus allows to obtainmany realizations of the considered stochastic pro-cess in parallel and directly averages over them atthe detection stage. While details of the experimen-tal setup have been given in [4, 26], we will hereprovide the logical scheme of the apparatus, rep-resented in Fig. 2. The core of the apparatus is aspatial light modulator (SLM) placed in the Fourierplane between the two lenses L1 and L2 of the 4Fsystem. The SLM is a 1D liquid crystal mask ( pixels, µ m/pixel) used to introduce a differ-ent phase (externally controlled by the computer) toeach pixel, implementing the simulation of the dy-namical map. This device thus imprints differentphases depending on the position and on the polar-ization state of the incoming photon. In the experi-mental device photons are generated by parametricdown-conversion and a suitable grating provides aspatial separation of the different frequency com-ponents. The SLM acts differently on the differentspectral components, thanks to their spatial separa- Figure 2. Schematic diagram of our apparatus. A coupleof frequency-entangled photons is generated via paramet-ric down-conversion (PDC) through a BBO crystal, us-ing a . nm laser diode as pump. One photon is sentvia a multi-mode fiber (MMF) to the single-photon detec-tor D2. The other is sent through a single-spatial-modeand polarization preserving fiber (SMF) to the 4F system(composed by two diffraction gratings G1-G2 and twolenses L1-L2). The initial state of the photon is preparedby the half-wave plate H1. T is the tomographic appara-tus, made of a quarter-wave plate, a half-wave plate and apolarizer. The photon is then sent through a MMF to thesingle-photon detector D1. Finally, an electronic devicemeasures the coincidence counts (CC) and sends them tothe computer (PC). tion, and thus allows to encode in parallel differentrealizations of the noise. This experimental setupfurther allows to perform the average over the real-izations of the noise by collecting the different spa-tial components through the lens L2 and the gratingG2 into a multi-mode fiber (MMF). The detectionstage is in fact performed after recollecting the sig-nal via the MMF, so that one averages over the spec-tral components and therefore the different realiza-tions of the noise. We observe that the parametricdown-conversion (PDC) spectrum is selected by thelimited width of the H plate mount. For this rea-son we are limited to use n = 100 out of the pixel available on the SLM (which corresponds to realizations in parallel of the noise).As shown in the logical scheme Fig. 3 this simpleexperimental setting nicely reproduces the frame-work considered in [15], namely the description ofthe overall reduced dynamics arising as a mixture of Figure 3. Logical scheme of the experimental setting. Thepreparation stage involves generation of the photons andspatial separation of the different spectral componentsvia a grating. The dynamical stage involves interactionwith different regions of the SLM, imprinting differentphases depending on the realization of the noise associ-ated to the region, corresponding to a Hamiltonian inter-action U X ( ξ ) ( t ) with a fixed noise realization. The detec-tion stage involves recombination of the different spectralcomponents by means of a MMF and a final photon de-tection. Markovian dynamics. In the present case in particu-lar the system dynamics which get mixed are givenby unitary maps U X ( ξ ) ( t ) , each characterized by asingle realization of the stochastic process. In theexperimental realization of the scheme it clearly ap-pears how non-Markovianity arises because of thepresence of degrees of freedom dynamically cou-pled to the observed ones and later averaged over. V. EFFECT OF NOISE ON THE QUANTUMDYNAMICS
We here report about the experimental results forthe realizations of the different kind of noises con-sidered in Sec. III, using the apparatus described inSec. IV. In order to generate the different noises wehave numerically solved the stochastic differentialequations considered in Sec. III. The obtained val-ues have been passed over to the SLM so as to affectthe phase of the photons according to the dynamicsgiven by Eq. (1). (a)(b) � � � � �������������������� [ � �� ( � )] � � � � �������������������� [ � ��� ( � )] � � � � �������������������� [ � ( � )] TheoryExperimental dataSimulated
Figure 4. Behavior of the quantum non-Markovianityquantifier D defined in Eq. (4) for the case of classicalMarkovian processes. In panel (a) we consider the Gaus-sian Ornstein-Uhlenbeck process X OU with γ = 0 . and σ = 0 . ; in panel (b) we consider the non-Gaussian butstill Markovian random telegraph noise process X RTN with γ = 0 . . Blu dots represent the experimental dataand the red line is the analytic solution. The green dashedline is the average of 100 simulated curves, each obtainedwith 100 realizations of the noise. The dashed areas corre-spond to the σ (darker) and σ (lighter) interval aroundthe averaged coherence and σ is the standard deviation ofthe sampled curves. In the first case (a) the quantum dy-namics does exhibit a Markovian behavior, correspondingto a monotonic decrease of coherence, while for the RTN(b) the resulting quantum dynamics is non-Markovian. The values obtained in correspondence to the dif-ferent realizations have been encoded in differentregions of the SLM, thus allowing for an easy im-plementation of the average as depicted in the log-ical scheme Fig. 3. Given that the aim of the workis the comparison between non-Markovianity of thequantum dynamics and the features of the classicalnoise, for each kind of noise we have studied thebehavior of the trace distance as a function of time. (a)(b) � � � � �������������������� [ � ( � )] � � � � �������������������� [ � ( � )] � � � � �������������������� [ � ( � )] (c) � � � � � ����������������� [ � ��� ( � )] TheoryExperimental dataSimulated
Figure 5. Same non-Markovianity quantifier D shown inFig. 4 for the case of classical non-Markovian processes.In panel (a) we consider the Gaussian but non-Markovianprocesses Y ( t ) with k = 1 ; in panel (b) we considerthe non-Gaussian and non-Markovian process Z ( t ) with µ = 1 ; in panel (c) we consider the same process Z ( t ) with µ = 0 . . In the first two panels the quantum dynam-ics does exhibit a Markovian behavior, corresponding toa monotonic decrease of coherence, at variance with theclassical property. In the last panel, instead, one also hasquantum revivals corresponding to a non-Markovian be-havior. As in Fig. 4 the blue dots represent the experi-mental data and the red line the analytic solution, whenit exists. The green dashed line is the averaged non-Markovianity and the shaded areas corresponds to σ and σ regions around the mean value. We keep track of time by encoding in the SLMthe different values of the processes at discretizedtimes with step of order in inverse units of therate appearing in the stochastic differential equa-tion characterizing the given process. As discussedin Sec. II B, we consider the quantity D defined inEq. (4) as quantifier of the non-Markovian featuresof the dynamics, which in particular fixes the behav-ior of the coherences. In Fig. 4 we show the exper-imental data referring to the quantum signature ofnon-Markovianity for two classical Markovian pro-cesses, namely Ornstein-Uhlenbeck D [ { X OU ( t ) } ] and random telegraph noise D [ { X RTN ( t ) } ] . Whilethe former quantity is monotonically decreasing, thelatter clearly shows a damped oscillating behavior,corresponding to a quantum non-Markovian behav-ior. Note that both processes have a power spec-trum of the form Eq. (13). While both processes areclassically Markovian, only Ornstein-Uhlenbeck isGaussian. The theoretical and numerical previsionsare in very good agreement with the experimentaldata (see also the shaded regions in Fig. 5)We further consider two non-standard classi-cal processes obtained as solution of the stochas-tic differential equations (5) and (6) respectively.The process Y ( t ) is Gaussian but classically non-Markovian. Despite these properties and the nontrivial power spectrum given by (13), as shown inFig. 5 the quantity D [ { Y ( t ) } ] is monotonically de-creasing in time. Again the experimental points arein agreement with the analytical estimate (11). Inthe case of D [ { Z ( t ) } ] , there exist values for the pa-rameters that make revivals of the trace distance ap-pear. We highlight again that the structured spec-trum of both the Y ( t ) and Z ( t ) processes cannot bedirectly connected to memory effects. For such non- Gaussian process the experimental points are com-pared to the results obtained via a numerical simu-lation of the process, further allowing to obtain itspower spectrum shown in Fig. 1. Again the classi-cal non-Markovianity of the process is not reflectedin the quantum signature. VI. CONCLUSIONS AND OUTLOOK
We address the quantum non-Markovianity of asingle-qubit dephasing map in terms of the Marko-vianity of the stochastic process generating thenoise. In particular, we considered four random pro-cesses with different Gaussianity and Markovianitytraits. We showed that the Markovianity of the clas-sical stochastic process does not affect the informa-tion backflow to the system, i.e. classical lack ofMarkovianity is not directly related to memory ef-fects. However, we showed evidence that the non-Gaussianity of the noise can be related with oscilla-tions of the trace-distance.
ACKNOWLEDGEMENTS
The author acknowledges support from the JointProject “Quantum Information Processing in Non-Markovian Quantum Complex Systems” funded byFRIAS, University of Freiburg and IAR, NagoyaUniversity, from the FFABR project of MIUR andfrom the Unimi Transition Grant H2020. Bas-sano Vacchini gratefully acknowledges useful dis-cussions with Alberto Barchielli and Matteo Grego-ratti. [1] Felix Binder and Luis A. Correa and ChristianGogolin and Janet Anders and Gerardo Adesso, eds.,
Thermodynamics in the Quantum Regime (Springer,2018)[2] A. W. Chin, S. F. Huelga, and M. B. Plenio, Phys.Rev. Lett. , 233601 (2012)[3] J. F. Haase, A. Smirne, S. F. Huelga, J. Kołodynski,and R. Demkowicz-Dobrzanski, Quantum Meas.Quantum Metrol. , pp. 13 (2018) [4] A. Smirne, S. Cialdi, G. Anelli, M. G. A. Paris, andB. Vacchini, Phys. Rev. A , 012108 (2013)[5] A. Rivas, S. F. Huelga, and M. B. Plenio, Rep. Prog.Phys. , 094001 (2014)[6] H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini,Rev. Mod. Phys. , 021002 (2016)[7] I. de Vega and D. Alonso, Rev. Mod. Phys. ,015001 (2017) [8] B. Vacchini, A. Smirne, E.-M. Laine, J. Piilo, andH.-P. Breuer, New J. Phys. , 093004 (2011)[9] B. Vacchini, J. Phys. B , 154007 (2012)[10] G. Guarnieri, A. Smirne, and B. Vacchini, Phys.Rev. A , 022110 (2014)[11] M. Gessner, M. Ramm, H. Haeffner, A. Buchleitner,and H.-P. Breuer, EPL , 40005 (2014)[12] L. Li, M. J. Hall, and H. M. Wiseman, Physics Re-ports , 1 (2018), ISSN 0370-1573, concepts ofquantum non-Markovianity: A hierarchy[13] A. Smirne, D. Egloff, M. G. D´ıaz, M. B. Plenio, andS. F. Huelga, Quantum Science and Technology ,01LT01 (2019)[14] G. Lindblad, Comm. Math. Phys. , 281 (1979)[15] H.-P. Breuer, G. Amato, and B. Vacchini, New Jour-nal of Physics , 043007 (2018)[16] H.-P. Breuer, E.-M. Laine, and J. Piilo, Phys. Rev.Lett. , 210401 (2009)[17] E.-M. Laine, J. Piilo, and H.-P. Breuer, Phys. Rev. A , 062115 (2010)[18] D. Chru´sci´nski and F. A. Wudarski, Phys. Lett. A , 1425 (2013) [19] N. Megier, D. Chru´sci´nski, J. Piilo, and W. T. Strunz,Scientific Reports , 6379 (2017)[20] C. W. Gardiner, Handbook of stochastic methodsfor physics, chemistry and the natural sciences ,Vol. 13 of
Springer Series in Synergetics , 3rd edn.(Springer-Verlag, Berlin, 2004)[21] R. F. Fox, Journal of Mathematical Physics , 2331(1977)[22] A. Hern´andezMachado and M. San Miguel, J. Math.Phys. , 1066 (1984)[23] B. Abel and F. Marquardt, Phys. Rev. B , 201302(2008)[24] C. Benedetti, M. G. A. Paris, and S. Maniscalco,Phys. Rev. A , 012114 (2014), ISSN 10502947[25] B.-H. Liu, L. Li, Y.-F. Huang, C.-F. Li, G.-C. Guo,E.-M. Laine, H.-P. Breuer, and J. Piilo, NaturePhysics , 931 (2011)[26] S. Cialdi, M. A. C. Rossi, C. Benedetti, B. Vacchini,D. Tamascelli, S. Olivares, and M. G. A. Paris, Ap-plied Physics Letters110