EEnergy transport in the presence of entanglement
A. A. Cifuentes and F. L. Semi˜ao Centro de Ciˆencias Naturais e Humanas, Universidade Federal do ABC, 09210-170, Santo Andr´e, S˜ao Paulo, Brazil
In this work, we investigate how the presence of initial entanglement affects energy transport in a network.The network have sites dedicated to incoherent input or output of energy and intermediate control sites whereinitial entanglement can be established. For short times, we found that initial entanglement in the control sitesprovides a robust efficiency enhancer for energy transport. For longer times, dephasing considerably dampsthe quantum correlations, and the advantage of having initial entanglement tends to disappear in favor of thewell known mechanism of noise-assisted transport. We careful study these two mechanisms in this work, andwe believe our findings may be usuful for a better understanding of the relation between nonclassicality andtransport, a topic of potential interest for quantum technologies.
I. INTRODUCTION
Quantum control [1] and quantum transport (QT) [2] playa prominent role in modern applications of quantum dynam-ics. In particular, quantum networks (QN) are extensivelyused to investigate the phenomenon of energy propagationand its relation to quantum coherence. Paradigmatic examplesare that of light harvesting complexes [3–7] and nano devices[8]. This kind of investigation led to the discovery of noise-assisted transport (NAT) [3, 9], where the interaction with theenvironment helps to enhance transport efficiency when thesystem parameters are appropriately tuned. In the context oflight harvesting complexes, there are investigations about thecapability of the dynamics in promoting the manifestation ofquantum correlations [10–12]. Unfortunately, complete con-trol over system state preparation and its evolution is not yetpossible in real photosynthetic systems, let alone the use ofcharacterization tools from quantum information science.A natural step is then to ask how especially arranged QNof controlled quantum systems can be used to critically assessthe role of quantum coherences and correlations in the phe-nomenon of energy transport [13, 14]. In this work, we areinterested in this kind of investigation. In other words, we areinterested in the active role of state preparations and externalcontrol over transport efficiency. These situate our work inthe context of quantum technologies where, instead of hav-ing a naturally occurring network, such as a photosyntheticcomplex, one can deliberately engineer a system, prepare itsstates, and drive it to study its response. This is precisely thecase of setups such as trapped ions, cavity or circuit quan-tum electrodynamic systems, just to name a few examples. Inthese systems, control is usually achieved by means of inter-action with external fields. In particular, it has been recentlyshown that cleverly chosen external time dependent drivingscan assist quantum tunneling between nodes in a QN [15–17].Here, we employ this ideia to engineer a QN suitable to ourpurpose of studying how quantum correlations actively influ-ence energy transport.This article is organized as follows. In Section II, wepresent the network model used in this work. Our resultsare presented in Section III, where we carefully investigatethe role of quantum correlations and environment in the effi-ciency of energy transport through the network. In Section IV,we conclude our findings and present our final remarks.
II. NETWORK AND EQUATION OF MOTION
The basic two-dimensional network element we are inter-ested in is depicted in Fig.1 (top), where one can see the spa-tial distribution of the sites or nodes. Γ s Γ d γ γ γ γ ω +(Θ +Θ )∆ωω +Θ ∆ωω xy i =1 i =2 i =3 i =4 (0,1) (0,0) (1,1) (1,0) ω +Θ ∆ω Figure 1. (Color Online) The network and its basic elements. Top:Four two-level systems labeled by i occupy site i = ( i , i ) formingthe network. The lines connecting the sites represent their coherentinteraction. The site-dependent energies area also indicated. Bot-tom: Different dynamical elements of the effective network. It isrepresented the energy pump ( Γ s ) in site , the energy drain ( Γ d ) insite , the decoherence γ k , and the effective energy hopping betweensites (two-ended dashed arrows). Also, the possibility of having en-tangled states initially prepared for the sites and is represented bythe shadowed area delimited by a dashed contour. We say that site i ( i = 1 , , . . . ) is located at i = ( i , i ) ,where i and i are natural numbers (zero included). Byclosely following the driving mechanism in [15, 16], we con-sider the system Hamiltonian to be ˆ H ( t ) = ˆ H ( t ) + ˆ H c , (1)with the total on-site energies subjected to external driving a r X i v : . [ qu a n t - ph ] J a n described by ˆ H ( t ) = (cid:126) (cid:88) j [ ω j + ∆ ω j + η d,j ω d,j cos ( ω d,j t + φ j )] ˆ σ + j ˆ σ − j , (2)and the coherent hopping given by ˆ H c = (cid:126) (cid:88) ij ; k>j c jk (cid:0) ˆ σ + j ˆ σ − k + ˆ σ − j ˆ σ + k (cid:1) , (3)where ∆ ω j = ∆ ω (Θ j + Θ j ) , (4)with Θ , Θ positive integers [18], ˆ σ + j and ˆ σ − j two-level rais-ing and lowering operators at site j , respectively, η d,j and ω d,j are basically the amplitude and angular frequency of thedriving field acting on site j , and c jk the coherent transferrate between the sites j and k . More precisely, the product η d,j ω d,j gives the strength of the interaction external drive-site j and it depends monotonically on the field amplitude andfrequency. Also, the external drive frequency is chosen to besite-dependent with the phases φ i = i φ x + i φ y . (5)All these driving parameters and the energy ladder ∆ ω j arecontrolled externally and provide a variable tool to design ef-fective interactions in the network [15, 16].For the sake of simplicity, we will now choose the drivingfield controlled parameters such that ω d,j = ω d and η d,j = η d ,for each site j . For the network depicted in Fig.1 (top), wechose initially resonant sites ω j = ω and ∆ ω = rω d , with r a positive integer, which forms an energy ladder implementedvia the time-independent part of the external field. In par-ticular, we depicted the case r = 1 in Fig.1 (top). FromEq.(4), it follows that ∆ ω = 0 , ∆ ω = Θ ∆ ω, ∆ ω =(Θ + Θ )∆ ω , and ∆ ω = Θ ∆ ω . Also, by using Eq.(5),one finds that φ = 0 ,φ = φ x ,φ = φ x + φ y ,φ = φ y . (6)The hopping Hamiltonian (3) naturally appears in varied sce-narios. It may represent, for instance, dipole-dipole interac-tion among two-level atoms or molecules in free space [19].Another well known situation where Hamiltonian (3) appearsis the dispersive interaction of two-level systems with a com-mon bosonic mode [20]. The role of the driving in Eq.(2) isto suppress the coupling between particular pairs of sites. Ourgoal is to carefully choose the phases and amplitudes of theexternal driving field to design an effective regime where thetransitions between sites ↔ and ↔ are suppressed.With this, we can study how energy injected in site arrive atsite through indirect pathways ↔ ↔ and ↔ ↔ ,as depicted in Fig.1 (bottom). This is a situation found, for in-stance, in the description of the conduction of potassium ionsin the KcsA channel [21–23]. This suppression is achieved asfollows. By transforming the system Hamiltonian (1) to an interac-tion picture with respect to ˆ H , and taking into account thecondition ∆ ω = rω d ( r a positive integer), a rotating waveapproximation (RWA) can be performed to obtain ˆ H I = (cid:126) (cid:88) ij ; k>j τ jk (cid:0) ˆ σ + j ˆ σ − k + ˆ σ − j ˆ σ + k (cid:1) , (7)with τ jk ≡ c jk F f ( r,j,k ) ( η d , ∆ φ j,k ) e − i f ( r,j,k )2 ( φ j + φ k ) , (8)where f ( r, j, k ) ≡ r [(Θ j + Θ j ) − (Θ k + Θ k )] , (9) ∆ φ j,k ≡ φ j − φ k , (10)and F χ ( ξ, ζ, θ ) ≡ ∞ (cid:88) s = −∞ J s ( ξ ) J s + χ ( ζ ) e i ( s + χ ) θ , (11)with J s being the Bessel function of first kind and order s .The rotating wave approximation used to obtain (7) is validonly for c j, k (cid:28) ω , ω d [14].Now, we carefully look into the content of Eq. (7). For thesake of simplicity, let us consider once again r = 1 . Thedynamics of energy migration between sites i and j is ruledby F f (1 ,j,k ) (see Eq.(8)). The magnitude of this quantity forthe pair of diagonal sites - and - is plotted in Fig. 2 as afunction of the driving parameters. Figure 2. (Color Online) Hopping amplitude between the sites j and k [Eq. (11)] as a function of the driving amplitude η d and phase ∆ φ jk . In particular, it is considered |F | ≡ (cid:12)(cid:12) F f (1 ,j,k ) (cid:12)(cid:12) where r = 1 .Left: Sites 1 and 3. Right: Sites 4 and 2. The parameters used in thisplots were Θ = 1 and Θ = 0 . Direct inspection of plots in Fig 2 and the use of Eq.(10)thus reveal that the choice φ x = φ y = π leads to the soughtsuppression of hopping along those diagonal sites. This is trueregardless of the driving amplitude η d for the range of parame-ters considered. It is this choice of φ x and φ y that will be usedfrom now on. We must now emphasize, however, that we willstill keep using the full original (non RWA) time-dependentHamiltonian (1) in the simulations. The RWA argument wasjust used to understand how the external driving can effec-tively suppress hopping between particular pair of sites.The dynamics of the QN depicted in Fig.1 (bottom), whichis our system of interest, is ruled by the following masterequation [24] d ˆ ρdt = − i (cid:126) (cid:104) ˆ H ( t ) , ˆ ρ (cid:105) + L s (ˆ ρ ) + L d (ˆ ρ ) + L deph (ˆ ρ ) , (12)where the source superoperator L s (ˆ ρ ) = Γ s (cid:0) − (cid:8) ˆ σ − ˆ σ +1 , ˆ ρ (cid:9) + 2ˆ σ +1 ˆ ρ ˆ σ − (cid:1) (13)accounts for the incoherent input of energy into the systemthrough site at a pump rate Γ s , the drain superoperator L d (ˆ ρ ) = Γ d (cid:0) − (cid:8) ˆ σ +3 ˆ σ − , ˆ ρ (cid:9) + 2ˆ σ − ˆ ρ ˆ σ +3 (cid:1) (14)represents an incoherent loss of energy through site at a rate Γ d , and the dephasing superoperator L deph (ˆ ρ ) = N (cid:88) k =1 γ k (cid:0) − (cid:8) ˆ σ + k ˆ σ − k , ˆ ρ (cid:9) + 2ˆ σ + k ˆ σ − k ˆ ρ ˆ σ + k ˆ σ − k (cid:1) , (15)destroys quantum coherence in the network, where γ k is a site-dependent dephasing rate. In all these, { (cid:63), ρ } denotes the an-ticommutator { (cid:63), ρ } ≡ (cid:63) ρ + ρ (cid:63) . One of our goals is to studythe contribution of each of these terms in the master equationto the dynamics of energy propagation in the QN depicted inFig.1 (bottom). Moreover, we want to investigate it takinginto account the presence of initial quantum correlations inthe control sites and .An important figure of merit is how efficiently energyleaves the system through site which works as a drain (rate Γ d ). This is quantified by the integrated population of site P = (cid:90) t p ( t (cid:48) ) dt (cid:48) , (16)where p ( t (cid:48) ) is the occupation of site at instant t (cid:48) > . Thisquantity is proportional to the transport efficiency η eff through P = η eff / d [9, 25].In this work, we use the entanglement of formation (EoF)to quantify bipartite entanglement between pairs of sites [26].In addition to that, we also include other form of quantumcorrelation in our study, the so called Quantum Discord (QD)[27]. The former is interesting because it spots quantumnessfor a set of states which does not necessarily contains entan-glement. In this sense, the QD adds generality to our study. III. RESULTS
The just presented formalism allows one to explore site-dependent dephasing scenarios. However, from now on wewill be adopting the same decoherence rates for all sites γ k = γ , a feasible choice for quantum technologies. Nev-ertheless, if one aims at studying transport in natural systemssuch as photosynthetic complexes, site-dependent dephasingsshould necessarily be taken into account in accordance withexperimental observations and computational simulations. A. Efficiency enhancers - entanglement versus dephasing
We would like to start our investigations by considering twoinitial states, with same mean energy (one excitation), but withdifferent types of correlations. The first state is ρ ent = | ψ (cid:105)(cid:104) ψ | , (17)where | ψ (cid:105) = ( | gegg (cid:105) + | ggge (cid:105) ) / √ . For this initial prepara-tion, sites and start in their ground state and they are notcorrelated with sites and , which share a bipartite maxi-mally entangled state that contains one quantum of excitation.Experimentally, bipartite entangled states such as the one con-sidered here has been generated in a variety of setups rangingfrom photons [28] to massive particles [29]. In the scenariodefined by Eq.(17), sites and are then quantum correlated.The second initial state to be considered in this subsection is ρ mix = ( | gegg (cid:105) (cid:104) gegg | + | ggge (cid:105) (cid:104) ggge | ) / , (18)where once again the sites and are initially in their groundstate, but now sites and are just classically correlated ina maximally mixed state. One can consider Eq.(18) as thelimit of Eq.(17) when previous decoherence ( t < on thedecoupled model ( c = 0) had fully acted and completely de-stroyed the coherences i.e., the nondiagonal terms in the basis {| gegg (cid:105) , | ggge (cid:105)} .In Fig. 3, we present the efficiency quantifier (16) for anobservation time ct = 10 (interval of integration) and a de-phasing rate γ = c/ what means that the dephasing timesare around one order of magnitude longer than the hoppingtimes (coherent dynamics). Within this time interval, whichcan be called a short interaction time, quantum effects havea chance to manifest or to bring some influence on transport.In the long-time regime, to be briefly discussed next, quantumeffects usually become irrelevant since dephasing generallykills the coherences. P γ -mix ( ) γ -mix ( ) γ -ent ( ) γ -ent ( ) η d ω d / c Figure 3. (Color Online) Integrated population of site [Eq.(16)] asa function of the rescaled driving strength η d ω d /c , for the choices: ω j = ω ∀ j , ∆ ω = ω / . φ x = π , φ y = π . Θ = 1 , Θ = 0 (energy ladder along x direction). c j,k = c = ω / ∀ j, k , ω d = ω / . γ k = γ ∀ k , γ (0) = γ = 0 , γ (1) = γ = c/ . Γ d = c/ , Γ s = 2 Γ d . -ent : Initial state is Eq. (17). -mix : Initial state is Eq.(18). In this scenario, by comparing the plots in Fig. 3, it is clearthat the initial presence of entanglement helped transport, i.e.,resulted in values of P that surpassed those obtained with theinitially mixed (non-entangled) situation. In other words, en-tanglement worked as a resource for QT. Another interestingfeature of Fig. 3 is the presence of NAT for the minima whenentanglement was originally present in sites e . By increas-ing the dephasing, the efficiency increased in those regions.This is not true for the maxima where dephasing tends to bedestructive. This is expected because dephasing destroys en-tanglement in our model, and entanglement is precisely theingredient for the pronounced maxima. For the initial mixedstate, the phenomenon of NAT is practically absent in Fig. 3.All these interesting features are confirmed by Fig. 4 wherewe provide a more general picture of the problem throughvariation of dephasing over a broad range. It is interesting tosee that for the initial preparation with entanglement (plot onthe left), increasing the dephasing γ is generally beneficial forthe minima, and that this feature is practically not manifestedfor the non-entangled initial situation (plot on the right). Onthe contrary, dephasing acted as a hinderer when no entangle-ment was initially present. Figure 4. (Color Online) Efficiency indicator P as a function ofthe rescaled driving strength η d ω d /c , and the dephasing γ/c . Theparameters for the plot are as in Fig. 3. Left: Considering the initialstate Eq. (17). Right: Considering the initial state Eq. (18). We now present the long-time behavior of the population p of the last site, fixing two values of η d ω d /c : the first min-imum and the first maximum of P with initial entanglementin Fig. 3. The results are shown in Fig. 5, where one can seethat initial entanglement, as expected, gradually looses its ca-pacity to boost transport. The only enhancer left is dephasingthrough NAT. This mechanism is clearly manifest in Fig. 5giving the fact that, for times longer than ct ≈ , the curveswith non null dephasing γ (1) are above the ones with null de-phasing γ (0) .Before finishing this section, we would like to go a littledeeper in our investigation about the entanglement-assistedtransport phenomenon already observed in previous plots. Wewill do this in two directions. First, we would like to see howFig. 3 changes when the input of energy is changed. Thisis shown in Fig. 6. The plots are very illustrative becausethey once again show the competition between entanglement-assisted transport and noise-assisted transport, now for a dif- ct γ -mix ( ) γ -ent ( ) γ -mix ( ) γ -ent ( ) γ -mix ( ) γ -ent ( ) γ -mix ( ) γ -ent ( ) ctp p Figure 5. (Color Online) Temporal dependence of the population ofsite , p , for the first minimum ( η d ω d /c = 18 . ) and maximum( η d ω d /c = 38 . ) values in Fig. 3. -ent : Initial state is Eq. (17). -mix : Initial state is Eq. (18). Top: First maximum. Bottom: Firstminimum. The other parameters for the plot are as in Fig. 3. ferent scenario where the increase of noise comes from othersources than pure dephasing. For Γ (1) , the beneficial effectcoming from initial entanglement is still quite clear, almostlike in Fig. 3. To see this, compare, for instance, the maximaarising from the situation with initial maximal entanglementwith the situation with and maximal mixedness. When the in-coherent input of energy is increased to Γ (2) , the initial max-imally mixed state and the initial maximally entangled stateare practically equivalent in terms of efficiency. The reasonwhy initial entanglement starts loosing importance when theenergy input rate increases is that adding more energy alsoadds more noise. This is so because the energy input and out-put are both incoherent process corresponding to non-unitaryterms in the network master equation, see Eqs.(13) and (14).The second direction we want to explore is the variationof the initial entanglement. Up to now, we worked only withmaximal entanglement versus non-entanglement at all, bothstates with only one excitation shared between sites and .We now keep considering the one excitation sector, but withthe state | ψ (cid:105) = cos θ | eg (cid:105) + sin θ | ge (cid:105) for sites and . Theother two sites are still considered to be initially in the groundstate. The variation of θ makes the entanglement of formationvary from zero ( θ = 0) to one ( θ = π/ . In Fig. 7, we onceagain consider the efficiency indicator P in the first maxi-mum η d ω d /c = 38 . [see Fig. 3]. However, we now have it Γ -ent ( ) Γ -ent ( ) Γ -mix ( ) Γ -mix ( ) η d ω d / cP Figure 6. (Color Online) Integrated population of site , for the two drain rates Γ (1) = c/ and Γ (2) = c/ . The chosen dephasingrate is γ = c/ . -ent : Initial state is Eq. (17). -mix : Initial state isEq. (18). The other parameters for the plot are as in Fig. 3.Figure 7. (Color Online) Efficiency indicator P , as a function ofdephasing and entanglement in the initial state, for the first maximum( η d ω d /c = 38 . ) in Fig. 3. The drain rate is Γ d = c/ . The otherparameters are as in Fig. 4. as a function of dephasing and the entanglement in the initialstate | ψ (cid:105) . One can see that states with more initial entangle-ment lead to higher values of P . This confirms the robustnessof the entanglement-assisted transport mechanism for a wholeclass of states (all pure states with one excitation shared bycontrol sites and ). Finally, one can see once again that, forinitial pure states and the maxima of P in Fig. 3, the increaseof dephasing γ is a hindrance to transport. B. Quantum correlations survival
Since the initial presence of entanglement was seen to bebeneficial for transport in the short-time behavior, it would beinteresting to have a close look at its dynamics. This couldhelp us to better understand the conclusions previously pre-sented about the effect of quantum correlations (QC) overtransport for our system of interest. As said before, we will ct D ( )( ) EoF D ct D ( )( ) EoF D Figure 8. (Color Online) Temporal dependence of the QC betweenthe sites and . Insets: Zoom intended to highlight the existence ofQC apart from entanglement for some times. The initial state is Eq.(17). γ k = 0 ∀ k and Γ d = c/ . Top: First maximum. Bottom:First minimum. The other parameters for the plot are as in Fig. 3. be employing quantifiers of entanglement and quantum dis-cord QD. For the latter, it is important to distinguish betweenthe situation where projective measurements are thought to acton one subsystem or another. In our case, the subsystems aresites and , and we will then denote the case where projec-tive measurements are intended to act on by D (2) , and whenintended on by D (4) . The dynamical behavior of quantumcorrelations for the same parameters considered in Fig. 5 isshown in Fig. 8. As expected, at long times the oscillatory be-havior of the quantum correlations is completely damped ren-dering the system state to be essentially classicality correlated.It is interesting to see that there are times where quantum dis-cord remains finite in spite of the fact that entanglement goesto zero. In Fig. 9, we present the average correlations overthe same time spam consider to evaluate P in Fig. 3, for aninitially maximally entangled situation. From this plot, it isclear that also on average quantum correlations remain finiteduring the transport for the whole range of driving strengths η d ω d /c considered in Fig. 3. From Fig. 9 it is also possible tosee that entanglement is a bit more sensitive to the choice ofthe driving strength than discord in the sense that the formeroscillates more stronger than the latter as the driving strengthis varied. IV. CONCLUSIONS
In this work, we studied the relation between transport ef-ficiency and initial presence of entanglement in a network.The network used in this work is the simplest one needed toassess the effect of having entanglement in the intermediatesites, i.e., between two sites not directly connected to energysources or sinks. In fact, we found that initial entanglementprovides a robust enhancer of transport efficiency. For shorttimes, entanglement-assisted transport showed up for all pos-sible initial pure states with only one excitation in the network.In this time domain, we showed that quantum correlations sur-vive dynamically and on average thus rendering the transportto be quantum in essence. On the other hand, for longer times,these correlations progressively vanish and it comes to a pointwhere only noise-assisted transport is available as a transportenhancer. We also showed that, for short-times, noise-assistedtransport is imaterial for the maximally mixed initial situation.
ACKNOWLEDGMENTS
A.A.C. acknowledges to “Coordenac¸ ˜ao de Aperfeic¸oamen-to de Pessoal de N´ıvel Superior” (CAPES). FLS acknowl-edges partial support from the Brazilian National Institute of Science and Technology of Quantum Information (INCT-IQ)and CNPq under Grant No. 307774/2014-7. We would alsolike to thank Ms. Marcela Herrera and Prof. Roberto Serra forgranting access to their computing facilities. D ( )( ) EoF D η d ω d / c Figure 9. (Color Online) Temporal average of QC between the sites and for the situation where the system is initially in an entangledstate. A large dephasing situation is considered, with γ k = c ∀ k and Γ d = c/ , for the initial state Eq. (17). The other parameters forthe plot are as in Fig. 3.[1] H. Rabitz, Focus on Quantum Control , New J. Phys. , 105030(2009).[2] M. Mohseni, Y. Omar, G. Engel and M. B. Plenio, Quantumeffects in Biology , Cambridge University Press, Cambridge,2014).[3] M. B. Plenio and S. F. Huelga,
Dephasing-assisted transport:quantum networks and biomolecules , New J. Phys. , 113019(2008).[4] F. Caruso, A. W. Chin, A. Datta, S. F. Huelga and M. B. Plenio, Highly efficient energy excitation transfer in light-harvestingcomplexes: The fundamental role of noise-assisted transport ,J. Chem. Phys. , 105106 (2009).[5] A. W. Chin, A. Datta, F. Caruso, S.F. Huelga and M. B. Plenio,
Noise-assisted energy transfer in quantum networks and light-harvesting complexes , New J. Phys. , 65002 (2010).[6] A. W. Chin, S. F. Huelga and M. B. Plenio, Coherence anddecoherence in biological systems: principles of noise-assistedtransport and the origin of long-lived coherences , Phil. Trans.R. Soc. A , 3638 (2012).[7] Bao-quan Ai and Shi-Liang Zhu,
Complex quantum networkmodel of energy transfer in photosynthetic complexes , Phys.Rev. E , 061917 (2012).[8] F. L. Semi˜ao, K. Furuya and G. J. Milburn, Vibration-enhancedquantum transport , New J. Phys. , 083033 (2010).[9] M. Mohseni, P. Rebentrost, S. Lloyd and A. Aspuru-Guzik, Enviroment-Assisted quantum walks in photosynthetic energytransfer , J. Chem. Phys. , 174106 (2008).[10] F. Caruso, A. W. Chin, A. Datta, S. F. Huelga, and Martin B.Plenio,
Entanglement and entangling power of the dynamics inlight-harvesting complexes , Phys. Rev. A , 062346 (2010).[11] M. Sarovar, A. Ishizaki, G. R. Fleming, and K. B. Whaley, Quantum entanglement in photosynthetic light-harvesting com- plexes , Nat. Phys. , 462 (2010).[12] K. Bradler, M. M. Wilde, S. Vinjanampathy, and D. B. Uskov, Identifying the quantum correlations in light-harvesting com-plexes , Phys. Rev. A , 062310 (2010).[13] F. Nicacio and F. L. Semiao, Transport of correlations in a har-monic chain , Phys. Rev. A , 012327 (2016).[14] F. Nicacio and F. L. Semiao, Coupled harmonic systems asquantum buses in thermal environments , J. Phys. A: Math.Theor. Synthetic Gauge Fieldsfor Vibrational Excitations of Trapped Ions , Phys. Rev. Lett. , 150501 (2011).[16] A. Bermudez, T. Schaetz and D. Porras,
Photon-assisted-tunneling toolbox for quantum simulations in ion traps , NewJ. Phys. , 053049 (2012).[17] M. Grifoni and P. H¨anggi, Driven quantum tunneling , Phys.Rep. , 229 (1998).[18] M. Gl¨uck, A. R. Kolovskya and H. C. Korsch,
Wannier-Starkresonances in optical and semiconductor superlattices , Phys.Rep. , 103 (2002).[19] Z. Ficek and S. Swain,
Quantum Interference and Coherence:Theory and Experiments (Springer, New York, NY, 2005).[20] P. P. Munhoz and F. L. Semi˜ao,
Multipartite entangled stateswith two bosonic modes and qubits , Eur. Phys. J. D , 509(2010).[21] J. Morais-Cabral, Y. Zhou, and R. MacKinnon, Energetic op-timization of ion conduction rate by the K + selectivity filter ,Nature,
37 (2001).[22] S. Bern`eche and B. Roux,
Energetics of ion conduction throughthe K + channel , Nature,
73 (2001).[23] A. A. Cifuentes and F. L. Semi˜ao,
Quantum model for a peri-odically driven selectivity filter in a K + ion channel , J. Phys. B: At. Mol. Opt. Phys. , 225503 (2014).[24] H. J. Carmichael, Statistical Methods in Quantum Optics 1:Master Equations and Fokker-Planck Equations (Springer-Verlag, Berlin-Heidelberg, 2002).[25] P. Rebentrost, M. Mohseni, I. Kassal, S. Lloyd and A. Aspuru-Guzik,
Environment-Assisted Quantum Transport , New J. Phys. , 033003 (2009).[26] W. K. Wootters, Entanglement of formation of an arbitrarystate of two qubits , Phys. Rev. Lett , 2245 (1998).[27] H. Ollivier and W. H. Zurek, Quantum discord: A measure ofthe quantumness of correlations , Phys. Rev. Lett , 017901(2001).[28] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A.Zeilinger, Violation of Bell’s Inequality under Strict Einstein Locality Conditions , Phys. Rev. Lett. , 5039 (1998).[29] M. Steffen, M. Ansmann, R. C. Bialczak, N. Katz, E. Lucero,R. McDermott, M. Neeley, E. M. Weig, A. N. Cleland, and JohnM. Martinis, Measurement of the Entanglement of Two Super-conducting Qubits via State Tomography , Science , 1423(2006).[30] A. Vaziri and Martin B. Plenio,
Quantum coherence in ionchannels: resonances, transport and verification , New J. Phys. , 085001 (2010).[31] G. N. Watson, A Treatise on the Theory of Bessel Functions (2 ndnd