On Time-Dependent Dephasing and Quantum Transport
OOn Time-Dependent Dephasing and Quantum Transport
Saulo V. Moreira, Breno Marques, and Fernando L. Semi˜ao Centro de Ciˆencias Naturais e Humanas, Universidade Federal do ABC - UFABC, Santo Andr´e, Brazil
The investigation of the phenomenon of dephasing assisted quantum transport, which happenswhen the presence of dephasing benefits the efficiency of this process, has been mainly focusedon Markovian scenarios associated with constant and positive dephasing rates in their respectiveLindblad master equations. What happens if we consider a more general framework, where time-dependent dephasing rates are allowed, thereby permitting the possibility of non-Markovian sce-narios? Does dephasing assisted transport still manifest for non-Markovian dephasing? Here, weaddress these open questions in a setup of coupled two-level systems. Our results show that themanifestation of non-Markovian dephasing assisted transport depends on the way in which theincoherent energy sources are locally coupled to the chain. This is illustrated with two differentconfigurations, namely non-symmetric and symmetric. Specifically, we verify that non-Markoviandephasing assisted transport manifested only in the non-symmetric configuration. This allows us todraw a parallel with the conditions in which time-independent Markovian dephasing assisted trans-port manifests. Finally, we find similar results by considering a controllable and experimentallyimplementable system, which highlights the significance of our findings for quantum technologies.
I. INTRODUCTION
Dephasing assisted transport means energy currentsenhanced by dephasing [1, 2]. This implies that theopen system dynamics may surpass the correspondentunitary evolution in terms of transport efficiency. Onthe one hand, this defied the notion that, in general,the presence of noise tends to jeopardize the efficiencyof tasks performed by quantum systems [3]. On theother hand, it helped us to understand energy trans-port behavior in quantum systems subject to heavilynoisy conditions in harsh natural environments, whichnonetheless shows an outstanding ability to effectivelytransfer energy. A paradigmatic example is the widelystudied Fenna-Mathew-Olson (FMO) complex, a struc-ture present in green sulphur bacteria which channelsthe energy captured from solar light to a reaction centre[4–9]. As well as this, the comprehension of dephasingassisted transport is of central importance for quantumtechnologies. Indeed, the possibility of exploiting it toachieve improved efficiencies is very appealing from thepoint of view of practical implementations, principallyfor quantum technology applications including controlledquantum systems [10–14].The theoretical studies of dephasing assisted transporthave been mainly focused on time-independent interac-tion between the system and environment [1, 15, 16].Therefore, investigations of time-dependent dephasing ina transport scenario that includes more general Marko-vian as well as non-Markovian evolutions have the poten-tial to drive new applications in the context of quantumtechnologies [17]. Furthermore, in the last years therehas been a great interest in the fundamental and prac-tical aspects of non-Markovianity [12, 18–25]. With thetools resulting from these studies and the experimentaladvances that have been reported, it is natural to en-visage new possibilities to exploit such systems in thecontext of quantum transport. In this work, we willstudy how the presence of time-dependent dephasing in a chain of coupled two-level systems affects quantum trans-port efficiency in Markovian and non-Markovian scenar-ios. In doing so, we tackle a relevant question in the fieldof open quantum systems, which is the impact of time-dependent scenarios on quantum transport. We will beusing the fact that important examples of non-Markovianevolutions can be characterized by Lindblad-like masterequations for which the time dependent decoherence rateachieves negative values [17, 26–29].This paper is organized as follows. First, we reviewthe canonical form of the Lindblad-like master equationsand the characterization of non-Markovianity via masterequations in section II. Then, we describe the transportmodel in section III. In section IV, we present our resultsand analyse transport efficiency in some time-dependentdephasing scenarios, and extend the analysis for resultsobtained in the context of a controlled quantum systemin section V. In section VI, we present our conclusions.
II. CHARACTERIZING TIME-DEPENDENTNON-MARKOVIAN EVOLUTIONS
Time-local master equations [30, 31] can be expressedin a Lindblad-like form as˙ ρ ( t ) = − i [ H ( t ) , ρ ]+ d − (cid:88) k γ k ( t ) (cid:18) ˆ L k ( t ) ρ ˆ L † k ( t ) − { ˆ L † k ( t ) ˆ L k ( t ) , ρ } (cid:19) , (1)with a unique set of functions γ k ( t ), not necessarily pos-itive for all times [29]. Here, d is dimension of the statespace, H ( t ) is a Hermitian operator, and ˆ L k ( t ) consti-tutes an orthonormal basis of traceless operators, i.e.,Tr[ ˆ L k ( t )] = 0 , Tr[ ˆ L † j ( t ) ˆ L k ( t )] = δ jk . (2)Since any time-local master equation can be written inthis canonical form, in which each γ k ( t ) is uniquely de- a r X i v : . [ qu a n t - ph ] F e b termined, it turns out that Eq.(1) may be used to char-acterize non-Markovianity [29]. In fact, γ k ( t ) ≥ γ k ( t ), for some k and at any instant of time t , indicates non-Markovianity.The fact that each γ k ( t ) is unique in Eq.(1) motivatedthe use of f k ( t ) ≡ max[0 , − γ k ( t )] ≥ , (3)as a indicator of non-Markoviany in the channel k , andits integration in time F k ( t, t (cid:48) ) = (cid:90) t (cid:48) t dsf k ( s ) , (4)as a quantifier of the total amount of non-Markovianityof a given channel k in an interval of time from t to t (cid:48) [29].In general, γ k ( t ) must satisfy certain constraints for acompletely positive evolution. For instance, consider amaster equation for a two-level system given by˙ ρ ( t ) = − i [ H ( t ) , ρ ( t )] + 12 (cid:88) k γ k ( t )( σ k ρ ( t ) σ k − ρ ( t )) , (5)where σ i are Pauli matrices ( σ = σ x , σ = σ y , σ = σ z ),and H ( t ) is Hermitian. Complete positivity of the map,in the interval from 0 to t , is garanteed if the followingset of conditions are fulfilled [33]Γ j + Γ k ≤ l , (6)for all permutations j, k, l of 1 , , j ≡ exp( − (cid:82) t ds [ γ k ( s ) + γ l ( s )]).Let us illustrate it with a simple case where γ ( t ) = γ ( t ) = 0 and γ ( t ) = γ ( t ), i.e.,˙ ρ ( t ) = − i [ H ( t ) , ρ ( t )] + 12 γ ( t )( σ z ρ ( t ) σ z − ρ ( t )) . (7)It is straightforward to show that (cid:82) t γ ( s ) ds ≥ γ ( t ) = sin( νt ), where ν is integer and γ ≥ t . III. THE MODEL
We consider a linear chain of N two-level systems ina first-neighbor coupling model, whose Hamiltonian isgiven by ( (cid:126) = 1) H = N (cid:88) i =1 ω i σ zi + N − (cid:88) i =1 λ i ( σ + i σ − i +1 + σ + i +1 σ − i ) , (8) where σ + i is the operator causing transition from groundto excited state in site i , σ − i = ( σ + i ) † , σ zi and ω i arethe Pauli z operator and the energy associated with i thsite, respectively, and λ i is the coupling constant betweensites i and i + 1. This model has been extensively usedto describe quantum transport, and this kind of interac-tion can be implemented, for instance, in the context oftrapped ions and circuit QED [25, 35].In turn, the chain is considered to be locally coupledto incoherent energy sources, responsible for incoherentinjection and extraction of energy. More specifically, weconsider energy injection at site 1 and extraction at site k , where 2 ≤ k ≤ N . This situation is described by thefollowing terms, to be added to the master equation L inj ρ = 12 κ inj (2 σ +1 ρσ − − σ − σ +1 ρ − ρσ − σ +1 ) , L ext ρ = 12 κ ext (2 σ − k ρσ + k − σ + k σ − k ρ − ρσ + k σ − k ) , (9)where κ inj ( κ ext ) describes the rate of injection (extrac-tion) of energy into (out) the chain. In order to simplifythe notation, we omitted the time-dependence of ρ ( t ) inEq.(9). From now on, we will adopt this simplified nota-tion.Finally, we consider that each site is also subjectedto local dephasing. This assumption of local couplingto the environment is reasonable for weak intercouplingstrength between the sites of the chain when compared tothe local frequencies [36–41]. For the sake of simplicity,we will assume that each site is subjected to equivalentdephasing environments. Therefore, the total dephasingto which the chain is subjected is given by L deph ρ = N (cid:88) i γ ( t )( σ zi ρσ zi − ρ ) . (10)Then, non-Markovianity is the result of γ ( t ) assumingnegative values. Finally, the total master equation rep-resenting the evolution of the system will read˙ ρ = − i [ H, ρ ] + L deph ρ + L inj ρ + L ext ρ. (11)For the investigation of transport efficiency, we willconsider the stationary value of the rate of variation ofthe total number operator ˆ N = (cid:80) i σ + i σ − i . In a broadsense, it can be called exciton current. In the stationarystate, one finds Tr[ ˆ N ˙ ρ ] = Tr[ ˆ N L in ρ ] + Tr[ ˆ N L ext ρ ] = 0.Consequently, Tr[ ˆ N L in ρ ] = − Tr[ ˆ N L ext ρ ]. As a fig-ure of merit for transport efficiency, we then consider J ˆ N = | Tr[ ˆ N L ext ρ ] | = κ e xt p ext ( ∞ ), where p ext ( ∞ ) isthe asymptotic population of the extraction site. To bemore specific, we will be evaluating the rescaled current˜ J ˆ N ≡ ( κ ext N ) − J ˆ N . IV. TIME-DEPENDENT DEPHASINGASSISTED TRANSPORT
We start our analysis by considering time-dependentdephasing models with sinoidal time dependence. Wefocus on two specific situations, called symmetric and non-symmetric configurations, having in mind the case N = 7 as a benchmark for the Markovian case [15]. Inthe non-symmetric configuration, the 5th site is the ex-traction site, what breaks inversion symmetry. In thesymmetric configuration, the extraction site is the 7thsite. Chains of different sizes can also be studied andthey present similar behaviors to the ones presented here.In both cases, the chain has uniform frequencies ω i = ω and inter-site couplings λ i = λ , and the injection siteis always the first site. In the symmetric configuration,the extraction site is on the other tip of the chain, i.e.the last site. All other choices for extraction site willlead to non-symmetric configurations. As shown in Ref.[15], Markovian dephasing-assisted transport manifestsonly in the non-symmetric configuration. We investigatewhat happens when non-Markovian dephasing shows upin these configurations. To construct a scenario, we set ω i = ω , λ i = λ = 0 . ω , and κ ext = κ inj = 0 . ω for allsimulations, i.e. all frequencies and couplings are set inunits of ω . A. Non-symmetric configuration
In Fig. 1, we plot the current ˜ J ˆ N as a function of γ ≥ γ ( t ) = γ sin( νt ),and different values of ν in the non-symmetric configu-ration. We also consider the average of these three sinefunctions, for which ν = 0 . , ,
4. This model happensto be non-Markovian for any finite value of the posi-tive constant γ . The Markovian case corresponding to γ ( t ) = γ is also plotted as a benchmark. The first thingto be noticed is that dephasing-assisted transport mani-fests in both cases: Markovian and non-Markovian. Thiscorresponds to the first portion of the curves where thecurrent increases with γ . Regardless of being Marko-vian or not, there is always an optimal value of γ abovewhich dephasing becomes detrimental. Notwithstanding,we see that the non-Markovian cases are more efficientthan their Markovian counterpart for higher dephasingmagnitudes γ .In Fig. 2, we consider the current ˜ J ˆ N for anothermodel, for which γ ( t ) = γ + γ sin( t ) in the non-symmetricconfiguration, with γ = 1. The resulting dynamics isnon-Markovian for 0 < γ <
1. One can also see the time-independent Markovian benchmark in the same plot. Asa glimpse of how rich the transport scenario is in thepresence of time-dependent dephasing, this model doesnot present efficiency enhancement by dephasing. Com-pared to the Markovian case for γ = 0, i.e. closed systemdynamics, the case with γ ( t ) = γ + sin( t ) is always lessefficient. This is in clear contrast to the model consideredbefore. However, the present model shows an interestingnon-monotonic behavior with γ , and it also turns out tobe more efficient than the time-independent Markoviancounterpart for higher values of γ . FIG. 1: Non-symmetric configuration – current ˜ J ˆ N as a func-tion of γ/ω for γ ( t ) = γ sin( νt ), where ν = 0 . , ,
4, and thenormalized sum of these functions. The Markovian case cor-responding to γ ( t ) = γ is also plotted.FIG. 2: Non-symmetric configuration – current ˜ J ˆ N as a func-tion of γ/ω for γ ( t ) = γ + γ sin( t ). The dotted red verticalline corresponds to γ = 1. The system is decreasingly non-Markovian in the interval 0 < γ <
1. For γ ≥
1, the systemis Markovian since we have γ ( t ) ≥ t . The gray curvecorresponds to γ ( t ) = γ . B. Symmetric configuration
Now, we focus on the symmetric configuration, andonce again plot the current ˜ J ˆ N as a function of γ for γ ( t ) = γ sin( νt ), where ν = 0 . , ,
4. As before, wealso plot the case with the average of the aforementionedsine functions, what guarantees a fair comparison withthe other cases. The Markovian case corresponding to γ ( t ) = γ is also plotted and shows a monotonic behavioras γ is increased. In other words, there is no Marko-vian dephasing-assisted transport in the symmetric case,in agreement with Ref. [15]. For each non-Markoviancurve shown in Fig. 3, we also have that non-Markoviandephasing assisted transport does not manifest, as themaximum current is reached for γ = 0. Nevertheless, itis remarkable to see that the non-Markovian cases be-comes once again more efficient than their Markoviancounterpart as γ is increased.In Fig. 4, we consider once again the model givenby γ ( t ) = γ + γ sin( νt ), with γ = 1 in the symmetricconfiguration. As we can see, a non-monotonic behaviour FIG. 3: Symmetric configuration – current ˜ J ˆ N as a functionof γ/ω for γ ( t ) = γ sin( νt ), where ν = 0 . , ,
4, and the nor-malized sum of these functions. The Markovian case corre-sponding to γ ( t ) = γ is also plotted. is also observed in this case, and, by comparing it to thethe benchmark, we see that it can also help efficiencyregardless of being Markovian ( γ <
1) or not ( γ ≥ FIG. 4: Symmetric configuration – current ˜ J ˆ N as a function of γ/ω for γ ( t ) = γ + γ sin( t ). The dotted red vertical line cor-responds to γ = 1. The system is decreasingly non-Markovianin the interval 0 < γ <
1. For γ ≥
1, the system is Marko-vian since we have γ ( t ) ≥
0. The gray curve corresponds to γ ( t ) = γ . C. Spread of occupations and efficiency
Next, we seek to analyse how the spread of occupa-tions correlates with the current maximum in the time-dependent dephasing scenarios presented above. We con-sider the spread of occupations ∆ n [15], with n i = p i ( ∞ ),∆ n = 1 − (cid:32) N (cid:88) i n i − n k (cid:33) , (12)where n k is the population of the extraction site k , n k = p k ( ∞ ). The maximum of this quantity is associatedwith a minimum spread of the occupations. A correlationbetween the maximal of ∆ n and the maximum of the cur-rent is verified in Ref. [15] for several time-independent Markovian cases. Here, we certify that, for the time-dependent non-Markovian cases studied above, the sametendency is verified: ∆ n is maximum when the currentis maximum, as shown in Figs. 5 and 6, which showsplots of ∆ n and the current as a function of γ . These re-sults suggest that this quantity is an indicator of optimaltransport scenarios in the more general time-dependentand non-Markovian dephasing picture. V. EXAMPLE: CONTROLLED QUANTUMSYSTEM
In the scope of controlled quantum systems, time-dependent dephasing, including non-Markovian evolu-tions, can be introduced and externally controlled [12,18–21]. Usually one can achieve it through controlledauxiliary systems. Specifically, a model in the context ofnuclear magnetic resonance (NMR) experiments, wherea Ising-like interaction takes place between two spin 1/2systems, is studied in Ref. [18]. One of these two-levelsystems is considered to be the system of interest, andthe other is is seen as part of the environment, providinga structured bath. The strength of the coupling betweenthe system and the environment is given by a parameter J , and θ is a parameter which gives the state in whichthe environment is initialized, before the interaction. Itturns out that the parameters J and θ are controllablein the NMR experimental realization. In particular, thefollowing superoperator can be engineered for any site i of the chain, such that the master equation describingthe system’s state ρ is given by [18] L i ρ = 12 γ i ( t )( σ zi ρσ zi − ρ ) − is i ( t )[ σ zi , ρ ] , (13)where γ i ( t ) = γ i + πJ sin (2 θ ) sin(2 πJt )3 + 2 cos(4 θ ) sin ( πJt ) + cos(2 πJt ) , (14)is a time-dependent dephasing rate, and s i ( t ) = 2 πJ cos(2 θ )3 + 2 cos(4 θ ) sin ( πJt ) + cos(2 πJt ) , (15)is an environment-induced time-dependent energy shift.As mentioned before, in Eqs.(14) and (15), J and θ arefully controlled parameters.First of all, it is worth studying the behaviour of thefunction γ i ( t ) in Eq.(14), which is an odd and periodicfunction satisfying the condition for a completely pos-itive evolution for any value of γ i ≥ J = 1. In Fig. 7, we plot γ i ( t ) for several values ofthe parameter θ while keeping γ i = 0. We can see that,for γ i = 0 the system is always non-Markovian, and itsnon-Markovianity increases as θ increases in the interval[0 , π/ γ i = γ so that (a) γ ( t ) = γ sin(0 . t ) (b) γ ( t ) = γ sin( t ) (c) γ ( t ) = γ sin(4 t ) (d) γ ( t ) = ( γ ( t ) + γ ( t ) + γ ( t ))(e) γ ( t ) = γ sin(0 . t ) (f) γ ( t ) = γ sin( t ) (g) γ ( t ) = γ sin(4 t ) (h) γ ( t ) = ( γ ( t ) + γ ( t ) + γ ( t )) FIG. 5: ∆ n (dashed purple line) and the current ˜ J ˆ N (solid line) as a function of γ/ω for each non-Markovian evolution,corresponding to the γ ( t ) in the caption of each figure. The plots in Figs. (a)-(d) correspond to the non-symmetric configuration,while the plots in Figs. (e)-(h) correspond to the symmetric configuration. (a) Non-symmetric config. (b) Symmetric config. FIG. 6: ∆ n (dashed purple line) and the current ˜ J ˆ N (green/coral solid lines) as a function of γ/ω , for γ ( t ) = γ + sin( t ) forthe non-symmetric and symmetric configurations.FIG. 7: γ ( t ) in Eq. (14) with γ = 0 and J = 1 and differentvalues of θ . all γ i ( t ) are the same. Thus, the whole chain will be sub-jected to the following total master equation, which alsotakes into account the coupling to the incoherent energysources responsible for injection and extraction of energy, as described before,˙ ρ = − i [ H, ρ ] + N (cid:88) i γ i ( t )( σ zi ρσ zi − ρ ) − is i ( t )[ σ zi , ρ ]+ L inj ρ + L ext ρ. (16) A. Non-symmetric configuration
We now focus on the model with time-dependent de-phasing as described by Eqs. (14), (15) and (16). First,we consider γ = 0 in Eq. (14). In Fig. 8, we have the cur-rent ˜ J ˆ N plotted as a function of θ , see the solid green line.We see that non-Markovian dephasing assisted transporthappens, as the maximum of the current is associatedwith a value θ (cid:54) = 0. This is similar to the behavior dis-cussed before and illustrated in Fig. 1, but here θ is theparameter controlling the non-Markovian dephasing. Inother words, by increasing θ , one increases the presenceof non-Markovian dephasing in the system’s evolution.The effect of increasing the positive contribution γ isshown in Fig. 9 for fixed J and θ ( J = 1 and θ = 0 . γ ( t ) is plotted in Fig. 7), where the solidgreen line is the plot of the current as a function γ inthe non-symmetric configuration. The system is initiallynon-Markovian and becomes Markovian for the value of γ ≈ .
17 correspondent to the dotted red line. For γ (cid:38) .
17, the system is Markovian. We see that theincrease in the Markovian contribution, γ , cannot leadto an increase in the current, ˜ J ˆ N . We note that the sameeffect – the decrease of ˜ J ˆ N as γ increases – is observedfor other values of J and θ . Therefore, the decrease inthe non-Markovianity of the system by increasing γ jeop-ardizes transport efficiency. FIG. 8: Current ˜ J ˆ N as a function of θ (rad) for γ = 0. Thesolid green line represents the non-symmetric case, while thedashed coral line represents the symmetric one.FIG. 9: Current ˜ J ˆ N as a function of γ/ω for J = 1 and θ =0 .
52. The dashed red vertical line corresponds to γ = 1 . <γ (cid:46) .
17. For γ (cid:38) .
17, the system is Markovian since we have γ ( t ) >
0. The green line represents the non-symmetric case,while the dashed coral line represents the symmetric one.
B. Symmetric configuration
We consider the case γ = 0 in Eq.(14) in the sym-metric configuration. In Fig. 8, we have the current˜ J ˆ N plotted as a function of θ , see the dashed coral line.We see that non-Markovian dephasing assisted transportdoes not happen, as we have a monotonic behavior of the current with θ . This behavior is similar to the cases dis-cussed before for Fig. 3, where we also have a monotonicbehavior for the current in all the cases.Next, we study what happens when the positive contri-bution γ is increased, by taking J = 1 and θ = 0 .
52 onceagain, but now in the symmetric configuration. In Fig.9, the dashed coral line shows the current as function of γ . As in the non-symmetric case, we see a monotonicdecreasing behavior of the current ˜ J ˆ N as γ decreases. Asbefore, the non-Markovian system becomes Markovianfor γ ≈ .
17, indicated by the dotted red line in Fig. 9.As in the non-symmetric case, therefore, non-Markovianscenarios are shown to be associated with a greater trans-port efficiency.
VI. CONCLUSION
The investigation of the influence of time-dependentdephasing rates γ ( t ) on the efficiency of quantum trans-port is a very relevant open problem in the contextof open quantum systems research, notably in non-Markovian scenarios for which γ ( t ) reaches strictly neg-ative values. Here, we provided a systematic investiga-tion of this phenomenon for a chain of coupled two-levelsystems, which is, in turn, locally coupled to incoherentsources of energy, a scenario which accounts for energyinjection and extraction in and out of the chain. An ex-citon current is then established and evaluated when thesystem is in the stationary state. Specifically, the mod-els treated here are characterized by complete positivemaps associated with time-dependent dephasing ratesdescribed by linear combinations of sine functions andconstants in the canonical representation of the corre-sponding master equation. Based on that, we studiedthe behavior of the exciton current in the non-symmetricand symmetric scenarios. We find that the phenomenonof non-Markovian dephasing-assisted transport occurs inthe non-symmetric cases, thereby establishing a parallelwith the time independent Markovian cases investigatedelsewhere [15].As a final remark, one can also investigate theso-called “maximally non-Markovian evolution” [42],to find that different degrees of non-Markovianity inthat model do not change the efficiency of quantumtransport. This indicates that the generalization ofnon-Markovian dephasing-assisted transport, beyondthe models studied here, is not straightforward, andcertainly deserves to be further investigated. We hopeour work will serve as a motivation for further advancesrelated to this interesting problem. Acknowledgements – S.V.M. acknowledges supportfrom the Brazilian agency CAPES. B. M. and F. L.S. acknowledge partial support from the BrazilianNational Institute of Science and Technology of Quan-tum Information (CNPq-INCT-IQ 465469/2014-0) andCAPES/PrInt Process No. 88881.310346/2018-01.F.L.S. also acknowledges partial support from CNPq (Grant No. 305723/2020-0). [1] M. B. Plenio, and S. F. Huelga, New J. Phys. ,065802 (2005).[4] G. S. Engel, T. R. Calhoun, E. L. Read, T. -K. Ahn,T. Mancal, Y. -C. Cheng, R. E. Blankenship, and G. R.Fleming, Nature
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