Role of coherence in quantum-dot-based nanomachines within the Coulomb blockade regime
RRole of coherence in quantum-dot-based nanomachines within the Coulomb blockade regime
Federico D. Ribetto,
1, 2
Ra´ul A. Bustos-Mar´un,
1, 3, ∗ and Hern´an L. Calvo
1, 2 Instituto de F´ısica Enrique Gaviola (CONICET) and FaMAF, Universidad Nacional de C´ordoba, Argentina Departamento de F´ısica, Universidad Nacional de R´ıo Cuarto, Ruta 36, Km 601, 5800 R´ıo Cuarto, Argentina Facultad de Ciencias Qu´ımicas, Universidad Nacional de C´ordoba, Argentina
During the last decades, quantum dots within the Coulomb blockade regime of transport have been proposedas essential building blocks for a wide variety of nanomachines. This includes thermoelectric devices, quantumshuttles, quantum pumps, and even quantum motors. However, in this regime, the role of quantum mechanicsis commonly limited to provide energy quantization while the working principle of the devices is ultimately thesame as their classic counterparts. Here, we study quantum-dot-based nanomachines in the Coulomb blockaderegime, but in a configuration that resembles the quantum mechanics’ paradigmatic experiment: the double-slit. We show that the coherent superposition of states appearing in this configuration can be used as the basisfor different forms of “true” quantum machines. We analyze the efficiency of these machines against differentnon-equilibrium sources (bias voltage, temperature gradient, and external driving) and the factors that limit it,including decoherence and the role of the different orders appearing in the adiabatic expansion of the charge/heatcurrents.
I. INTRODUCTION
The high degree of control and the discrete energy spec-trum of coupled quantum dots (QDs), sometimes referred toas quantum dot molecules, make them especially suitable forthe manipulation of charge and energy fluxes in nanoscale.This is crucial for nanoscopic heat and charge management,the development of new quantum information technologies,and the design of different forms of quantum machines.
Inthis regard, experimental and theoretical studies have shownthat quantum-dot-based designs may provide remarkable per-formances in thermoelectric devices that exchange electricaland thermal energies.
The pumping of charge and heaton quantum-dot-based driven systems have been extensivelystudied.
In recent years, the reverse process in whichheat or charge currents are used to propel a mechanical devicehas also gained considerable attention.
In all the above systems, it is assumed that the typicalsize of the device is smaller than the characteristic coherencelength of electrons. It is clear then that quantum mechanicsbecomes crucial for the description of these forms of nanode-vices, which can be put together under the generic name ofquantum machines. Depending on the type of energy conver-sion involved they are usually referenced as (adiabatic) quan-tum motors, (adiabatic) quantum pumps (or generators), quan-tum heat engines, or quantum heat pumps.
In a quantummotor, a dc electric current is transformed into mechanicalwork while in a quantum pump, an ac electrical or mechanicaldriving is turned into a dc electric current. Quantum heat en-gines and heat pumps are very similar systems but the powersource involve temperature gradients instead of bias voltages,and the focus is shifted from charge currents to heat currents.The role of quantum mechanics on quantum-dot-based ma-chines strongly depends on the system’s conditions. Here,we focus on the adiabatic regime, where the modulation ofthe system’s parameters is slow as compared with the typicaltime spent by the electrons inside it. However, even withinthis condition there are different transport regimes that shouldbe distinguished. For example, in the ballistic regime, de-scribed by a mean-field approximation of the electron repul-
Figure 1. (a) Example of the type of studied systems: a DQD coupledto some mechanical degree of freedom. Here, the dots are weaklycoupled to each other and to source/drain leads. The dots’ ener-gies are modulated by the gate voltages generated from the capac-itive coupling to a charged rotor. Panels (b) and (c) are simplifiedschemes for the DQD system in parallel and in series, respectively.Lead-dot tunneling events are characterized by four tunneling rates( Γ S1 , Γ S2 , Γ D1 and Γ D2 ) while the interdot coupling is describedby t c . If the two drain leads in (a) are kept at the same voltage, weobtain the parallel configuration. However, by establishing a biasvoltage between the two drain leads while removing the source leadin (a), we obtain the series configuration. Throughout this work, theparallel DQD without interdot coupling will be dubbed the double-slit configuration. sion, the working principle of adiabatic quantum pumps andmotors can be attributed to interference effects of the electronspassing through the modulation region. On the otherhand, in the Coulomb blockade regime, quantum effects aretypically restricted to energy quantization so that the internalpumping mechanism, beyond the quantization of the trans-ported charge, resembles that of a classical pump. In this case,some form of rate equation relating the occupation probabili-ties of the quantum-dot states is typically used to describe thesystem, while the coherence between them can be disregardedin a first approximation. Quantum pumping, shuttle transport, and even adiabatic quantum motors have been studied by using this approach. Other strategies, a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b I THEORETICAL FRAMEWORK like the nonequilibrium Green’s function formalism, have alsobeen used in the past to study quantum pumping within theCoulomb blockade regime.
However, the working princi-ple of the device can also be explained by relying on a classi-cal analogous.Based on the above, it is fair to wonder, once in the adi-abatic and Coulomb blockade regimes, how “quantum” ananomachine based on quantum dots can be. In this con-text, the weak interdot coupling regime provides a useful plat-form to test the role of quantum coherences between the dots’states. This is so because the degeneracy of dots’ states bringstogether both occupations and coherences on the same timescale. As a consequence of that, coherences survive evenat the steady-state of the system. In particular, in Ref. [49]charge pumping was studied for a double quantum dot (DQD)coupled in series. They found that the coupling between co-herences and occupations is responsible for charge pumping.However, given that both coherence and electron transportrely entirely on the interdot coupling, taking the coherenceto zero trivially sets the current to zero. Thus, although the“quantumness” of the pumping mechanism is clearly present,its effect is somewhat hidden. On the other hand, in Ref. [50]the authors analyzed charge pumping in an Aharonov-Bohminterferometer configuration of the dots. As in this case thereis no explicit interdot coupling, the role of quantum superpo-sition becomes more clear.In this work, we exploit the weak interdot coupling regimein a DQD to analyze the role of coherence in a broad class ofquantum machines such as charge/heat pumps and nanomo-tors driven by bias voltages or temperature gradients. Forthis to be achieved, we focus on a particular configuration ofthe quantum dots, which resembles the double-slit experimentand has no classical counterpart. The configuration consistsof a parallel DQD within the Coulomb blockade regime withno direct interdot coupling and where the system is coupledto classical degrees of freedom. The latter provides the dots’energy level modulation. Fig. 1 shows an example of our pro-posal. Through this model, we show that the above mentionedregime dominated by coherences also applies to quantum mo-tors fueled by a finite bias voltage. In addition, we includein our description an external force acting on the (classical)mechanical component of the system. Such a force allows usto bring together the two operation modes (pump and motor)of the device on a same basis. These ideas are also extendedto the case where the leads are subjected to different tempera-tures, giving rise to coherence induced quantum heat enginesand refrigerators. We analyze the performance of these ma-chines and the factors that limit it, including decoherence andthe role of the different orders of the adiabatic expansion.The paper is organized as follows. In Sec. II, we presentthe theoretical framework, including a brief overview of thereal-time diagrammatic approach, the expressions for the ob-servables, the definitions of the efficiencies, and the used de-coherence model. In Sec. III, we apply the formalism to theparticular example of a DQD weakly coupled to two exter-nal leads and capacitively coupled to a rotor. In Sec. IV, westudy the performance of the different operational regimes ofthe double-slit configuration of the DQD. Finally, in Sec. V, we summarize the main results.
II. THEORETICAL FRAMEWORKA. Hamiltonian model
We consider a system composed of QDs in which mechani-cal and electronic degrees of freedom are present and coupledto each other. From now on we call this system the local sys-tem , and we model it by the following Hamiltonian ˆ H local = ˆ H el ( ˆ X ) + ˆ P m + U ( ˆ X , t ) , (1)where ˆ X = ( ˆ X , ..., ˆ X N ) is the vector (operator) of mechani-cal coordinates while ˆ P = ( ˆ P , ..., ˆ P N ) represents their asso-ciated momenta, m is the effective mass related to ˆ X , and U denotes some external mechanical potential that may be actingon the local system. We use an explicit time dependence in U to denote that an external and nonconservative force might beacting on the mechanical subsystem (see below). The Hamil-tonian ˆ H el includes the electronic degrees of freedom of thesystem, that are participating in the transport, as well as theircoupling to the mechanical ones through ˆ H el ( ˆ X ) = (cid:88) i E i ( ˆ X ) | i (cid:105)(cid:104) i | , (2)where the sum runs over all possible electronic many-bodyeigenstates | i (cid:105) . The system is then weakly coupled to externalleads so the total Hamiltonian reads ˆ H total = ˆ H local + (cid:88) r ˆ H r + ˆ H tun . (3)The leads are described as non-interacting electrons reservoirsthrough the Hamiltonian ˆ H r = (cid:88) kσ (cid:15) rk ˆ c † rkσ ˆ c rkσ , (4)where ˆ c † rkσ ( ˆ c rkσ ) is the creation (annihilation) operator foran electron with state-index k and spin projection σ = {↑ , ↓} in the lead r , which we typically take as source and drainreservoirs, i.e., r = { S , D } . These reservoirs are assumedto be always in equilibrium, characterized by a temperature T r and an electrochemical potential µ r . Finally, the tunnelcoupling between the local system and the leads is given bythe tunnel Hamiltonian ˆ H tun = (cid:88) rkσ(cid:96) ( t r(cid:96) ˆ d † (cid:96)σ ˆ c rkσ + H . c . ) , (5)where t r(cid:96) denotes the tunneling amplitude, which we assumeto be k and spin independent for simplicity. The fermionicoperator ˆ d † (cid:96)σ ( ˆ d (cid:96)σ ) creates (annihilates) an electron with spin σ in the quantum dot (cid:96) composing the local system. Thetunnel-coupling strengths, defined as Γ r(cid:96) = 2 πρ r | t r(cid:96) | , quan-titatively describe the rate at which electrons enter (leave) the2 Stationary state regime II THEORETICAL FRAMEWORK quantum dot (cid:96) from (to) the r -reservoir. We also define thetotal tunnel rate Γ as Γ = (cid:80) r(cid:96) Γ r(cid:96) . The reservoirs are takento be in the wideband limit where their densities of states ρ r are assumed to be energy independent. Throughout this paper,we set e = 1 for the absolute value of the electron charge and (cid:126) = 1 . B. Stationary state regime
We suppose that the dynamics of the electronic and me-chanical degrees of freedom are well separated from eachother, and therefore we can treat them through the Born-Oppenheimer approximation. Under this approximation, themechanical coordinates can be treated as classical variablesobeying the following Langevin-like equation m ¨ X + F ext = F + ξ , (6)where F = − (cid:104)∇ ˆ H el (cid:105) = i (cid:104) [ ∇ ˆ H el , ˆ P ] (cid:105) is the mean value ofthe current-induced forces (CIFs) while ξ stands for its fluc-tuation. Later on we will see that a friction component arisesfrom expanding F in terms of the velocity of the mechani-cal coordinates. The term F ext represents an externalforce applied to the mechanical part of the local system andis related to the potential U in Eq. (1). This force will be, ingeneral, opposed to the bias induced direction of the CIF, sowe define it with a minus sign for convenience. As we shallsee later, in our model such a quantity appears as the key toolto set up the different operation modes of the electromechan-ical device. If we manage to calculate the expectation valueof the CIF (see Sec. II D) then we can use Eq. (6) to integratethe classical equations of motion and derive the effective dy-namics of the local system, including both electronic and me-chanical degrees of freedom. In realistic systems, friction andstochastic forces may have different origins, such as the cou-pling to other phononic degrees of freedom. Here, however,we are only interested in the quantum effects of CIFs. Thus,we will only take into account friction and stochastic forcesthat arise from the coupling to the electronic degrees of free-dom.Before continuing, some comments about the system arein order. First, we will focus on systems whose mechanicalpart is capable of reaching a stationary regime characterizedby a steady cyclic motion (with some frequency Ω ∝ ˙ X ) andwhose dynamics can be described by an angular Langevinequation. If we assume that this rotor follows a circular tra-jectory then only one parameter, the angle θ , is needed for thestudy of its dynamics. In this case we can project Eq. (6) on the angular direction ˆ θ to obtain the following angularform ¨ θ = 1 I ( F − F ext + ξ θ ) , (7)where I is the moment of inertia of the mechanical subsystem, F is the current-induced torque, F ext is the torque associatedto the external force, and ξ θ is the stochastic torque whichcomes from the angular projection of the CIF’s fluctuation.Second, in addition to the supposition of cyclic mechanicalmotion, we consider that the terminal velocity reached by thesystem is constant, i.e. ˙ θ = Ω , during the whole cycle. This isalso justified for large values of I , where the variation of theangular velocity (together with its fluctuations) along the cy-cle becomes negligible. Both numerical and analyticalprocedures for the calculation of ˙ θ , before and after reachingstationarity, have been carried out in Refs. 31 and 6.We are now in position to derive a relation between thework related to the torques F and F ext . This is done by in-tegrating Eq. (7) over a whole period of the system at the sta-tionary state, yielding W = (cid:90) τ F ˙ θ d t = (cid:90) τ F ext ˙ θ d t = W ext . (8)The equation implies that, once the cycle is completed, thework related to the CIF is balanced by the work done by theexternal mechanical force. This equality is fundamental in thesense that it defines the stationary state condition mentionedbefore and allows us to extract the value of ˙ θ = Ω . C. Generalized master equation
In this section we introduce the formalism that describes thedynamics of the electronic part of the system. This will allowus to calculate the expectation value of the CIF, together withother relevant observables like charge and heat currents, whileexactly taking into account the strong Coulomb interaction inthe local system. We assume that, before certain initializationtime t , the leads and the local system are decoupled, suchthat the total density matrix is factorized as ˆ ρ = ˆ p res ⊗ ˆ p .Here ˆ p res describes the leads’ density matrix, while ˆ p repre-sents the reduced density matrix of the local system. Whenboth subsystems are coupled together, the relevant informa-tion of the local system dynamics at times t > t is encodedin ˆ p ( t ) = Tr res [ˆ ρ ( t )] , where Tr res is the trace over the reser-voirs’ degrees of freedom. The time evolution of the matrixelements is governed by the generalized master equation dd t p βα ( t ) = − i (cid:88) β (cid:48) ,α (cid:48) L β,β (cid:48) α,α (cid:48) ( t ) p β (cid:48) α (cid:48) ( t ) + (cid:88) α (cid:48) ,β (cid:48) (cid:90) t −∞ d t (cid:48) W β,β (cid:48) α,α (cid:48) ( t, t (cid:48) ) p β (cid:48) α (cid:48) ( t (cid:48) ) , (9)where p βα ( t ) = (cid:104) α | ˆ p ( t ) | β (cid:105) and we have taken the limit t → −∞ , in order to neglect any transient effect. The first3 Generalized master equation II THEORETICAL FRAMEWORK term in the r.h.s. of this equation takes into account the inter-nal dynamics of the QDs through the Liouvillian superopera-tor L • ≡ [ ˆ H el , • ] , while the second term describes state tran-sitions due to electron tunneling processes between the leadsand the local system. This is quantified by the kernel super-operator W , which represents all irreducible diagrams in theKeldysh double contour, and whose matrix elements W α,α (cid:48) β,β (cid:48) describes the transition between states α (cid:48) and β (cid:48) at time t (cid:48) ,and states α and β at time t , due to tunnel processes.To simplify the notation, we gather the diagonal and off-diagonal elements of the reduced density matrix into a vector, ˆ p → p ≡ ( p d , p n ) T , yielding a matrix representation for both W and L superoperators, i.e. W → W and L → L . Here,the diagonal and off-diagonal elements of the reduced densitymatrix are contained in p d and p n , respectively. Thus we canthink of W and L as composed by the following block matri-ces W = (cid:18) W dd W dn W nd W nn (cid:19) , L = (cid:18) L dd L dn L nd L nn (cid:19) . (10)As we already mentioned, Eq. (2) tells us that the dots’ energylevels are affected by the cyclic mechanical motion, charac-terized by a frequency Ω proportional to the mechanical ve-locities ˙ X . If we assume that the dwell time of the elec-trons in the local system is much shorter than the mechani-cal period τ = 2 π/ Ω , then it is possible to perform a fre-quency expansion on p ( t ) . Strictly speaking, this adiabatic approximation holds if the adiabaticity condition Ω / Γ (cid:28) k B T /δ(cid:15) is satisfied, where δ(cid:15) stands for the energyamplitude of the QDs’ energy levels. This allows us to ex-pand the reduced density matrix as p ( t ) = (cid:80) k ≥ p ( k ) ( t ) with p ( k ) ∼ (Ω / Γ) k . The first term, p (0) ( t ) , represents the steady-state solution at which the electronic part of the system arriveswhen the mechanical coordinates are frozen at time t . In otherwords, this order corresponds to the adiabatic electronic re-sponse to the mechanical motion. Note that here we are re-ferring to the steady state of the electronic part of the system,which should not be confused with the steady-state regime ofthe mechanical degrees of freedom mentioned in the previoussection. From now on, every time we talk about stationarity,it will be referred to as the mechanical part of the local sys-tem. Higher orders terms ( k > ) represent nonadiabatic cor-rections due to retardation effects in the electronic responsementioned earlier.On top of this adiabatic expansion for small Ω , we performa perturbative expansion in the tunnel coupling strengths, tak-ing only terms up to first order in Γ (which is reasonable inthe weak tunnel coupling limit considered here). Higher-orderprocesses, like cotunneling, are therefore ignored throughoutthis paper. This double expansion gives rise to the followinghierarchy of equations W eff p (0) = , W eff p ( k ) = dd t p ( k − , (11)where we have defined the effective kernel W eff as the zero-frequency Laplace transform of W − i L , with both matrices evaluated up to first order in Γ . We omit the frequency or-der superscript in the effective kernel since at this level ofapproximation it is always O (Ω ) . The above set of equa-tions, combined with the normalization condition on the re-duced density matrix, e T p ( k ) = δ k , allows us to iterativelycalculate p (0) and any nonadiabatic correction p ( k ) . The vec-tor e T ≡ (1 , . . . , , , . . . , T is a representation of the localsystem’s trace operator, where the number of ones equals thedimension of the reduced Hilbert space. In light of this, thenonadiabatic corrections can be written as p ( k ) = (cid:18) ˜ W − dd t (cid:19) k p (0) . (12)Here ˜ W − represents the pseudo inverse kernel, defined as ˜ W ij ≡ W eff ij − W eff ii , in order to exclude the zero eigenvaluethrough the normalization condition. Since the effective ker-nel is linear in Γ , the k -term of the reduced density matrix, p ( k ) , is proportional to (Ω / Γ) k . This forces us to assume Ω < Γ , in order to avoid any divergence. Once we get p (0) and any required nonadiabatic correction p ( k ) , we can pro-ceed with the calculation of all observables related to the per-formance of adiabatic quantum machines. In the next sectionwe discuss the procedure used to achieve this task. D. Observables
Now we are going to make use of the formalism describedin the previous section to determine the expectation values ofa set of observables. First, we consider the charge current I r ( t ) ≡ (cid:104) ˆ I r ( t ) (cid:105) and the heat current J r ( t ) ≡ (cid:104) ˆ J r ( t ) (cid:105) , bothassociated with the r -lead. For these quantities we take thesign convention that in each lead the particle and heat currentsare positive when particles and heat are flowing towards thelead, thus we can write the currents in the lead r as I r ( t ) = dd t Tr[ ˆ N r ˆ ρ ( t )] , (13) J r ( t ) = dd t Tr[( ˆ H r − ˆ N r µ r )ˆ ρ ( t )] , (14)where ˆ N r is the number operator for the electrons in the reser-voir r . We also address the CIF which, unlike the previous ob-servables, constitutes a local quantity. As we showed before,Eq. (2) tells us that the mechanical part of the system onlyinteracts with the local parameters of the dots via their many-body eigenenergies. This implies that the CIF only consists offermionic dot operators, and therefore we can write its expec-tation value as F ( t ) = − Tr local [ ∇ ˆ H el ˆ p ( t )] , (15)where the gradient is taken with respect to the mechanical co-ordinates X .The adiabatic expansion developed in Sec. II C can also beperformed over any observable R of interest ( I , J , and F inour case), R ( t ) = (cid:88) k ≥ R ( k ) ( t ) . (16)4 Observables II THEORETICAL FRAMEWORK
To lowest order in Γ , the R ( k ) terms can be written as R ( k ) = e T W R p ( k ) , (17)where W R stands for the kernel associated to the observable R . The charge and heat currents flowing from the lead r intothe device are represented by the following kernels [ W I r ] ij = − n i [ W eff r ] ij , (18) [ W J r ] ij = − ( E i − µ r n i )[ W eff r ] ij , (19)where n i and E i are the number of particles and energy as-sociated with the local system’s eigenstate | i (cid:105) , respectively,and W eff r is the r -lead evolution kernel such that W eff = (cid:80) r W eff r . Regarding the ν component of the CIF, we candirectly construct a diagonal matrix kernel from: [ W F ν ] ij = − ∂E i ∂X ν δ ij , (20)where again we make use of its local condition. As in the case of p (0) , the zeroth-order terms I (0) r ( t ) and J (0) r ( t ) describe the steady-state currents flowing through thesystem in a stationary situation where all time-dependent pa-rameters are kept constant at time t . The only way for theseterms to be nonzero is when the system is subject to a biasvoltage or a temperature gradient since, in this case, the timevariation of the mechanical parameters has no effective rolein the observables. Higher-order terms represent additionalcontributions to the steady-state currents due to the delayedresponse of the system to the mechanical motion.A similar analysis applies to the CIF, where we take con-tributions up to first order in the mechanical velocity Ω , i.e. F ( t ) = F (0) + F (1) . The lowest order term can be split into( i ) an equilibrium contribution, which is conservative and itcan be interpreted as the Helmholtz’s free energy of the localsystem, and ( ii ) a nonequilibrium term, which appears as aconsequence of temperature gradients or bias voltages amongthe leads. The first adiabatic correction to the CIF, propor-tional to Ω , gives the frictional force that dissipates energyfrom the mechanical part of the local system toward the elec-tronic reservoirs. For systems with multiple mechanical de-grees of freedom, it also contributes to the energy exchangebetween modes and, for finite voltages, it can even allow theflux of energy from the leads towards the mechanical degreesof freedom.
If we now perform an adiabatic expansion of the torque F ,integrate it over a cycle, and use Eq. (8), we get the relation W F = s (cid:88) k (cid:18)(cid:90) π d θk ! ∂ k F ∂ ˙ θ k (cid:12)(cid:12)(cid:12)(cid:12) ˙ θ =0 (cid:19) ˙ θ k = s (cid:88) k C ( k ) F ˙ θ k , (21)where s is the sign of ˙ θ and gives the direction in which thetrajectory is traversed. Here, we define the force coefficients C ( k ) F which are independent of the direction of motion of thesystem, not obvious a priori . If we take terms up to k = 1 ,the angular velocity can be obtained from Eq. (21) as follows ˙ θ = Ω = C ext − C (0) F C (1) F , (22) where we defined W ext = s C ext to keep track of every term’ssign. Note that, for the mechanical subsystem to achieve a sta-tionary regime in the present simple model, the stability con-dition C (1) F < should be fulfilled, which implies a positive“friction coefficient”. E. Efficiency
Previously we stated that the mechanical subsystem per-forms a cyclic motion along a circular trajectory while affect-ing the dots’ energy levels. If we define a closed trajectory C for the system’s parameters that are being modulated, thenthe work W (0) F done by the zeroth-order contribution of theCIF can be calculated by performing a line integral of F (0) along this trajectory or, for two parameters and with the aid ofStokes’ theorem, we can calculate it in the following way W (0) F = (cid:120) S ∇ × F (0) · d S ≡ (cid:120) S B F · d S . (23)This means that the work associated with the zeroth-order CIFcan be understood as the surface integral of a curvature vector B F = ∇ × F (0) (the curl of the force), which is frequency in-dependent. Analogously, we can define a pumping curvaturefor the charge current flowing from/to reservoir r as Q (1) I r = (cid:73) C ∂I (1) r ∂ ˙ X · d X = (cid:120) S B I r · d S , (24)and the same can be done for the pumped heat Q (1) J r , via thecurvature B J r . These relations (which are only valid to firstorder in Ω ) highlight the geometrical nature of these observ-ables in the sense that they only depend on the chosen trajec-tory C . Equations like (23), (24), and similar provide a geometricalapproach to the study of adiabatic quantum devices, which hasbeen discussed by several authors.
One immediate con-clusion from these equations is that the trajectory followed bythe modulation parameters should enclose a finite area. Thisimplies that there must be at least two out-of-phase parametersmodulating the device.With the help of the geometric curvatures B F , B I r , and B J r , one can set a convenient working point in the parame-ters’ space around which a trajectory C will be defined. Forexample, if the goal is to design a nanomotor, this trajectoryshould enclose regions of large B F . On the other hand, if acharge pump is desired, then we should create a closed trajec-tory over regions where B I r is large. All these quantities, to-gether with their integrals, are not independent but related viaorder-by-order energy conservation, the second law of ther-modynamics, and Onsager’s reciprocal relations. Theorder-by-order energy conservation is given by (cid:88) r (cid:16) Q ( k ) I r δV r + Q ( k ) J r (cid:17) = −W ( k − F . (25)Here, the superscript ( k ) indicates the order in the frequencyexpansion and δV r = δµ r /e where δµ r = µ r − µ ( µ is the5 Efficiency II THEORETICAL FRAMEWORK reference chemical potential). The second law of thermody-namics can be expressed in the following form for the type ofsystems treated here (cid:88) k (cid:34) W ( k ) F + (cid:88) r (cid:18) Q ( k ) I r δV r + Q ( k ) J r δT r T r (cid:19)(cid:35) ≤ , (26)where δT r = T r − T ( T is the reference temperature).Onsager’s reciprocal relations appear in the linear regime oftransport, characterized by low bias voltages, small temper-ature gradients, and low velocities of the moved parame-ters. For example, in a two-lead configuration with r = { S , D } and when the leads are kept at the same temperature(i.e. δT r = 0 ), Onsager’s reciprocal relations imply ˜ Q (1) I ∆ V = −W (0) F , (27)where ˜ Q (1) I ≡ ˜ Q (1) I, S , ∆ V = δV S − δV D , and we use a tilde inthe pumped charge to denote that this quantity is being evalu-ated in the limit of zero bias. Similarly, when no bias voltageis applied between the contacts we have (cid:88) r ˜ Q (1) J r δT r T r = −W (0) F , (28)where, again, the tilde states that the pumped heat is evaluatedat zero thermal gradient between the leads.Eq. (26) allows us to derive bound expressions for the de-vice’s efficiencies which, as usual, are defined as the ratio be-tween the output and input powers per cycle. Before doingthis, we first need to know how to determine the operationalmode of the device, namely, whether the device acts like a mo-tor or a pump. Using Eq. (22), the sign s of the constant ve-locity ˙ θ can be determined and, with it, the sign of W ext . Thelatter determines the direction of the energy flux between thelocal system and the external agent that is acting on it through U , see Eq. (1). If W ext > , the energy current flows from theleads to the dots and, there, it is transformed into mechanicalwork, so the device operates as an electric motor/heat enginedepending on the nonequilibrium source. On the contrary, if W ext < the external agent is performing mechanical workwhich is then dissipated through the dots to the leads, so thedevice operates as a pump.Now considering that only a bias voltage is applied, thegenerated electrical current delivers an input energy Q I ∆ V per cycle, while the output energy is W ext = W (0) F + W (1) F .Thus, the efficiency of this electrical motor is given by η em = − W (0) F + W (1) F ∆ V (cid:16) Q (0) I + Q (1) I + Q (2) I (cid:17) ≤ , (29)in consistence with Eqs. (25) and (26) for a truncation in thefrequency expansion up to first order in the CIF, that impliesa second-order term in the currents. In the opposite casewhere W ext < , now the input and output energies swaproles, so the efficiency of this electrical pump is η ep = − ∆ V (cid:16) Q (0) I + Q (1) I + Q (2) I (cid:17) W ext ≤ . (30) Such a quantity, however, is only well defined in the casewhere the total amount of pumped charge is opposed to thatgiven by the natural direction of the bias current. In theused sign convention for the charge currents this means that Q I ∆ V > .A similar analysis can be done in the case where one re-places the bias voltage by a temperature gradient betweenthe contacts, such that the device can operate either as aheat engine or a refrigerator. By establishing different tem-peratures in the leads, defined as T hot = T + ∆ T / and T cold = T − ∆ T / , a heat current flows through the DQD sys-tem which, in turn, may activate its mechanical component. Inthis scenario where W ext > , the device is driven by the heatcurrent coming from the hot lead, − Q J hot . This means thatthe device operates as a heat engine with efficiency: η he = − W (0) F + W (1) F Q (0) J hot + Q (1) J hot + Q (2) J hot ≤ ∆ TT hot , (31)where Q J hot is defined as the time integral of J hot over a pe-riod given by Ω , while J hot is taken up to second order in thisquantity. On the other hand, when W ext < the heat currentflows against the temperature gradient. Assuming that the to-tal amount of pumped heat to the cold reservoir is negative, Q J cold < , we can define the efficiency (or coefficient of per-formance) of this heat pump or refrigerator by the expression: η hp = Q (0) J cold + Q (1) J cold + Q (2) J cold W ext ≤ T cold ∆ T , (32)where again the heat current J cold is taken up to second orderin Ω . Finally, it is convenient to define normalized efficiencieswith respect to the maximum theoretical value, given byEqs. (31) and (32), i.e. ˜ η he = T hot ∆ T η he , and ˜ η hp = ∆ TT cold η hp . (33) F. Decoherence model
One of the key questions motivating this work is whetherquantum coherence play a role in the operation of QD-basednanodevices such as adiabatic quantum motors and pumps.In this regard, studying the effect of decoherence on the ma-chines’ performance is crucial.Calculating decoherent relaxation times from a microscopictheory would require identifying the dephasing mechanisms,which is beyond the scope of this work. Instead, we choose aphenomenological approach that consists of inserting therelaxation times directly into the master equations. In ourcase, this implies adding to the kernel W eff a decoherent rate Γ φ ≡ /T . The inclusion of Γ φ is only done in the di-agonal elements of the nn block of W eff , i.e. [ W effnn ] ii → [ W effnn ] ii − Γ φ . The phenomenological rate Γ φ describesany decoherent process that may occur in the quantum dots,present even in the absence of a coupling to the leads. This6 Hamiltonian and physical model III DQD IN THE WEAK INTERDOT COUPLING REGIME type of decoherence destroys the information about the rela-tive phase in a superposition of states α and β ( p αβ ) withoutchanging the populations of the states ( p αα and p ββ ). Withouta coupling to the reservoirs, this formally leads to a decay ofthe off-diagonal matrix element p αβ ( t ) . In our case, however,there is also a replenishing mechanism given by the fact thatwhen electrons enter into the system, they do it in a superposi-tion state. Therefore, it is expected that coherences p αβ reach a Γ φ -dependent steady-state at long times. In the following sec-tions, we will take Γ φ as an “external knob” that can be used totest the effect of decoherence on the machines’ performance. III. DQD IN THE WEAK INTERDOT COUPLING REGIME
In this section we will apply the formalism and assumptionsdescribed previously to the particular example of a double dotweakly coupled to two external leads and capacitively coupledto a rotor.
A. Hamiltonian and physical model
The local system we are about to study is a DQD devicecomposed of two single-level spin-degenerate quantum dotscoupled to each other, together with a rotative mechanicalpiece placed in their proximity and capacitively coupled tothem. At the same time, the whole device is weakly coupledto source ( r = S ) and drain ( r = D ) leads, as depicted inFig. 1. By weak coupling we mean that the broadening due totunneling events is much smaller than the temperature broad-ening, i.e., Γ (cid:28) k B T . Notice that, depending on the choiceof the tunnel rates Γ r(cid:96) , it is possible to configure the doublequantum dot arrangement either in series or in parallel [seeFigs. 1(b) and (c)]. The asymmetry between source and drainrates is quantified by the factor λ = (Γ S − Γ D ) / Γ , (34)where Γ r = Γ r + Γ r . In addition, for a specific lead r = { S, D } , we define the lead-dot asymmetry factor as λ r = (Γ r − Γ r ) / Γ r . (35)These factors will be useful later on for setting different sys-tem configurations and for the search of a suitable workingpoint (see Secs. III C and IV). The local system is representedby the electronic Hamiltonian ˆ H el = (cid:88) (cid:96) E (cid:96) ˆ n (cid:96) + U ˆ n ˆ n + U (cid:48) (cid:88) (cid:96) ˆ n (cid:96) (ˆ n (cid:96) − − t c (cid:88) σ ( ˆ d † σ ˆ d σ + H . c . ) , (36)where ˆ n (cid:96) is the (cid:96) -dot particle number operator, defined as ˆ n (cid:96) = (cid:80) σ ˆ d † (cid:96)σ ˆ d (cid:96)σ , while E (cid:96) = E (cid:96) ( X ) represents the onsiteenergy of each dot (cid:96) = { , } , which is locally tuned by itscoupling to the mechanical part of the system. t c denotes the interdot coupling amplitude while U and U (cid:48) represent the interand intradot Coulomb interactions, respectively. For the sakeof simplicity, we will take in the following these parametersto be much larger than all other energy scales in the system( U, U (cid:48) → ∞ ), such that the double-dot device can only besingly occupied or empty. Due to these assumptions, the onlystates relevant for our system are | (cid:105) and | (cid:96)σ (cid:105) , where the for-mer means that both quantum dots are empty and the latterthat there is one electron with spin σ in the dot (cid:96) .Applying a bias voltage and/or a temperature gradient be-tween the leads will cause charge and heat to flow throughthe dots. If the mechanical piece is coupled to the DQD thenan energy exchange between these subsystems is possible. AsFig. 1(a) suggests, the cyclic motion of the rotor (which can bethought of as an electrical dipole) modifies the quantum dotsenergy levels, similarly to the action of externally controlledgate voltages. In agreement with Eqs. (25)-(28), once a biasvoltage or a temperature gradient is applied, the current flow-ing through the dots release part of its energy to the mechan-ical subsystem making it to rotate. The opposite scenario canbe achieved by applying an external force into the mechanicalsystem, such that its motion produces a finite current throughthe electronic device. In App. A we discuss in more detail theexample shown in Fig. 1(a) and how it might be possible tocontrol the coupling between the quantum dots and the rotor.Another possibility could be a dipolar molecule in proximityto the quantum dots such that there exists a capacitive cou-pling between the subsystems. In any case, for the purposeof the present work, it does not really matter the specific de-tails of the mechanical system, but its effects on the electronicHamiltonian. What enables this energy conversion is the de-pendence of the energy levels of the dots on the position of themechanical rotor, which in this case can be characterized byan angle θ . This θ -dependence is related to physical character-istics such as the rotor’s length and its position with respect tothe DQD, and the coupling strength between the rotor and thedots. A strict derivation of this angular dependence requiresan accurate knowledge of the rotor’s details, which can yieldcomplex parameterizations for the dots’ onsite energies. Asthe aim of the work is to unveil the role of coherences on CIFsand not to focus on specific details of a particular device, weassume a simple θ -dependence for the dots’ energies, givenby E ( θ ) = ¯ E + δ E cos( θ ) + δ (cid:15) sin( θ ) ,E ( θ ) = ¯ E + δ E cos( θ ) − δ (cid:15) sin( θ ) , (37)where δ E and δ (cid:15) describe the electromechanical coupling. Ac-cording to the model shown in Fig. 1(a) and discussed inApp. A, they are related to the capacitances acting on theDQD. In the energy space, these equations describe an ellip-tic trajectory of radius δ E and δ (cid:15) around the working point ( ¯ E , ¯ E ) . This trajectory is convenient given the typical shapeof the curvatures for the configuration of interest of the DQD,see Fig. 3-(a) for example. We assume that these energies ¯ E (cid:96) can be externally tuned (for example, by external gate volt-ages) so that the working point can be chosen favorably. Ifwe are thinking in the performance of motors or pumps, thenthis convenience lies on the fact that, to get useful work or7 Regime of parameters III DQD IN THE WEAK INTERDOT COUPLING REGIME pumped charge/heat, we need to find some region in the pa-rameter space where their associated curvatures are non-zero(cf. Sec. II E). Obviously, the above parametric approach alsoapplies to the energy difference (cid:15) = E − E and the meanlevel energy E = ( E + E ) / , such that these can also betreated as tunable parameters through the following equations: E ( θ ) = ¯ E + δ E cos( θ ) , and (cid:15) ( θ ) = ¯ (cid:15) + 2 δ (cid:15) sin( θ ) . (38)Importantly, at the level of approximation used in this work,the energy difference between the dots needs to be taken per-turbatively, i.e. (cid:15) ∼ Γ . As we shall see next, the regions inwhich the curvature associated with the CIF is non-zero liesbelow this constrain, such that we can safely define a trajec-tory enclosing the relevant region of B F with δ (cid:15) on the orderof Γ . B. Regime of parameters
With the purpose of studying the potential role of quantumcoherences on these devices, we will now focus on the weakinterdot coupling regime where t c ∼ Γ . Adiabatic quantummotors, heat engines, and charge/heat pumps, in the strongcoupling regime ( t c (cid:29) Γ ) has already been addressed. There, it was shown that coherences have no important con-tributions to any of the quantities of interest (e.g., charge andheat currents, CIFs, etc.) and can therefore be disregardedto lowest order in Γ . On the contrary, in the weak couplingregime, the role of the coherent superposition among the DQDstates becomes crucial for the operation of electron pumps.This was studied in Ref. [49]. Due to the connection betweenadiabatic quantum motors and pumps [cf. Eq. (25)-(28)], itis expected that coherent effects are also relevant for the per-formance of quantum motors and heat engines in the weakcoupling regime.Whether or not a system is in the weak or in the strongcoupling regimes depends on the comparison between Γ andthe energy difference between the eigenstates of the system.When this difference is much bigger than Γ , coherent effectscan be disregarded, at least to the lowest order in Γ . Herewe are in the opposite case, which occurs when both (cid:15) and t c are of the order of Γ . The assumption implies that single-electron states are almost-degenerate and guarantees the co-herences’ survival, laying the ground for the study of theirpotential effect on autonomous quantum machines like the onestudied here.With respect to the kernel W eff , see Eqs. (10), (11) andApp. B, all kernel blocks depend on the mean level energy E . However, the (cid:15) -dependence only enters in the W effnn block,which contains local information of the system through the Li-ouvillian L . As discussed before in section II E, the geomet-rical nature of the first-order pumped charge and heat, and thezero-order work of CIFs implies that a two-parameter depen-dence is necessary for these quantities to be nonzero. In oursystem, this condition can only be fulfilled if there is a cou-pling between the diagonal and non-diagonal blocks of W eff .The coupling of the kernels’ blocks inevitably leads, in turn,to the coupling of occupations and coherences of the reduced density matrix. Therefore, we can state that occupations andcoherences need to be coupled to have finite pumping/work inDQD-based nanodevices within the weak coupling regime.In the present regime, the single dot states | (cid:96)σ (cid:105) al-ready form the local eigenbasis to compute the densitymatrix. The vector p then adopts the form p = (cid:16) p , p ↑ ↑ , p ↓ ↓ , p ↑ ↑ , p ↓ ↓ , p ↑ ↑ , p ↓ ↓ , p ↑ ↑ , p ↓ ↓ (cid:17) T . Its componentsrepresent the occupation probabilities for the device to be ei-ther in the empty state | (cid:105) ( p ) or in the singly occupied state | (cid:96)σ (cid:105) ( p (cid:96)σ(cid:96)σ ), and coherent superpositions between single parti-cle eigenstates | (cid:96)σ (cid:105) and | (cid:96) (cid:48) σ (cid:105) ( p (cid:96) (cid:48) σ(cid:96)σ ). As we shall see later on,when the DQD is connected in series, such a coupling will beprovided by t c . However, in the parallel configuration we willsee that even in the absence of t c , the coupling between p d and p n still holds. This is due to the fact that, under this con-dition, electrons coming from the leads enter into the DQD ina coherent superposition of states. C. Role of coherences and the double-slit configuration -4-2024-10 -5 0 5 10 15 -1-0.500.51 -1-0.500.51-4 -2 0 2 4
Figure 2. (a) Map of the charge current curvature B I as a func-tion of ¯ E and ¯ (cid:15) for a DQD in series and in the absence of biasvoltages and temperature gradients. The shown curvature is normal-ized to its maximum absolute value within the shown map, B max ∼ . e/ ( k B T ) . (b) A cut of the curvature for ¯ E = 3Γ [see red arrowin (a)] and for several decoherence rates: Γ φ = 0 , . , . , . ,and 1, in units of Γ . The darkest curve corresponds to the case wherethere is no decoherence ( Γ φ = 0 ) while the lightest one denotes thecase of highest decoherence rate ( Γ φ = Γ ). The rest of the curvesare for intermediate values of Γ φ . The other parameters used are Γ = t c = 0 . k B T , λ = 0 . , λ S = 1 . , and λ D = − . . A serially coupled DQD in the weak interdot couplingregime was considered in Ref. [49]. There it was shown thatthe system is capable of pumping charge without an appliedbias voltage if the DQD is asymmetrically coupled to the leads( λ (cid:54) = 0 ). This can be seen in Fig. 2(a) where we show a map ofthe charge current curvature B I as a function of ¯ E and ¯ (cid:15) . Thisquantity allows one to determine those regions in the spaceof parameters over which a closed trajectory can be traced forthe production of a net pumped charge current after one mod-ulation cycle. In the figure, we observe a two-lobe patternwith opposite signs. The shift of sign of the current curvatureis due to a renormalization of energy levels attributed to theCoulomb interaction.
As discussed in section II E, mov-8
Role of coherences and the double-slit configuration III DQD IN THE WEAK INTERDOT COUPLING REGIME ing the parameters E and (cid:15) such as their trajectory encirclesany region of Fig. 2(a), without a change of sign, ensures afinite pumped charge. This alone proves that the system canbe used as an adiabatic quantum pump.The configuration in series of the DQD, see Fig. 1(c), al-lows one to access different operational modes of the device,i.e., adiabatic quantum motors, adiabatic quantum pumps, etc.However, in this case, quantum coherences come entirelyfrom the coupling between the quantum dots, i.e., the super-position states form when electrons hop sequentially from onedot to the other. This is clear also when one analyses the struc-ture of the effective kernel W eff . There, for the series config-uration, the nd and dn blocks of the effective kernel are onlygiven by the local Hamiltonian, such that: W effdn / nd = − i L dn / nd . (39)These blocks are responsible for the arising of coherences andfor the coupling with the W effnn block (which ultimately leadsto finite pumping). In this configuration, all matrix elementsin W effdn / nd are proportional to t c , i.e., there are no contribu-tions from the evolution kernel W . Thus, e.g., taking t c = 0 not only destroys any coherent superposition but also triviallycuts charge/heat transport through the local system. In thissense, the parallel configuration of the DQD, see Fig. 1(b), of-fers a richer example to study the role of coherences. There,coherences do not solely come from the interdot coupling butalso from the particles entering simultaneously to both dots.This comes from the fact that novel tunnel processes are en-abled, since the matrix elements of W nd / dn ∝ t r, t ∗ r, becomenon-zero (see App. B).In contrast to the configuration in series, in the parallel sce-nario it is possible to pump charge or heat even if the dotsare decoupled ( t c = 0 ), see Fig. 3(a). This particular casedoes not have a classical analogous as classical particles en-tering one dot, do not have a way of getting information fromthe other one. Then, a “classical” DQD should behave as twoindependent single parameter systems and, because of that,the total pumped charge, e.g., should be zero. Therefore, thequantum nature of electrons, which allows for a coherent su-perposition of the wave functions, it is the ultimate responsi-ble for the pumping of charge and heat, and the productionof finite work from the CIFs. One can interpret that quan-tum coherences are what allows a particle to get informationfrom both dots making the pumping to depend on two param-eters, E and E (or E and (cid:15) ). In this sense, devices based onthis configuration can be considered as “true” quantum ma-chines. Because of the similarity between the DQD in parallelwith t c = 0 and the paradigmatic double-slit experiment wedubbed this the “double-slit” configuration of the DQD.Before analyzing the effect of decoherence on the seriesand double-slit configuration of the DQDs, we want to ana-lyze an interesting case of the system’s parameters that leadsto zero pumping. As we pointed out before, the coupling be-tween the E -dependent block of the effective kernel ( W effdd )and the (cid:15) -dependent one ( W effnn ), is what provides the twoparameters needed for finite pumping. However, even in thepresence of such a coupling, we found that setting λ S = λ D leads to zero pumping, in the absence of a bias voltage or a -2-1012-10 -5 0 5 10 15 -1-0.500.51 -1-0.500.25-2 -1 0 1 2 Figure 3. (a) Map of the charge current curvature B I as a func-tion of ¯ E and ¯ (cid:15) for a DQD in parallel and in the absence of biasvoltages and temperature gradients. The shown curvature is nor-malized to its maximum absolute value B max ∼ . e/ ( k B T ) whithin the shown map. (b) A cut of the pseudomagnetic field for ¯ E = 3Γ [see red arrow in (a)] and for several decoherence rates: Γ φ = 0 , . , . , . , and 1, in units of Γ . The darkest curvecorresponds to the case where there is no decoherence ( Γ φ = 0 )while the lightest one denotes the case of highest decoherence rate( Γ φ = Γ ). The rest of the curves are for intermediate values of Γ φ .The used parameters are Γ = 0 . k B T , t c = 0 , λ = 0 , λ S = 0 . , λ D = − . . temperature gradient, independently of the choice of λ . Thisoccurs despite the fact that there is no obvious inversion sym-metry [see Eqs. (34) and (35)] and even when parameters E and (cid:15) enclose a finite area in the parameters’ space. How-ever, this particular case can be understood once one realizesthat the pumping currents become proportional to each other: I (1) r = (Γ r / Γ r (cid:48) ) I (1) r (cid:48) , where r, r (cid:48) = { S , D } , see App. C.The proportionality between both currents implies that eachone can be written as the total time derivative of the averagezeroth-order occupation number (cid:104) ˆ n (cid:105) and, thus, they integrateto zero for a whole cycle, see App. C.As a way to test the role coherences on DQD-based quan-tum machines in the series configuration, we show in Fig. 2(b)the current curvature for a fixed mean energy, ¯ E = 3Γ , anddifferent decoherence rates Γ φ . There it can be seen that Γ φ produces an amplitude decay and a widening of the curvaturepeaks which can be attributed to a gradual attenuation in thecoherent coupling between the quantum dots. Interestingly,for intermediate values of Γ φ there are some regions in pa-rameter’s space where decoherence increases the magnitudeof the curvature due to its broadening effect, see for example ¯ (cid:15) ∼ .In Fig. 3(a) we show the current curvature B I but for thedouble-slit configuration. Now, we observe a three-lobe pat-tern but dominated by a single sign (here the absolute valueof the central peak is much greater than that corresponding tothe side peaks). The role of Γ φ on the curvature, see Fig. 3(b),is the same as in the configuration in series, but the fact that t c = 0 in this case, allows us to give a more direct interpreta-tion of its effect for sufficiently large values. In this limitingsituation, the two dots become effectively decoupled since thecharacteristic survival time of the superposition now goes like / Γ φ . Then, each dot is unaware of the other dot’s existenceas all phase information gets lost much faster than the typical9 Role of coherences and the double-slit configurationIV QUANTUM MACHINES BASED ON THE DOUBLE-SLIT DQD time spent by the electron in the DQD system. As mentioned,this leads to a monoparametric scheme (with E the only pa-rameter being modulated), such that B I → and therefore noworking device can be created. IV. QUANTUM MACHINES BASED ON THEDOUBLE-SLIT DQD
In the previous section, we discussed the role of coherencein charge pumping, but the studied device admits other op-erational regimes. Here, we study the effects of applying anexternal driving force together with a bias voltage or a tem-perature gradient to the double-slit configuration. This takesthe system into different operational regimes, namely: electricmotor, charge pump, heat engine, or heat pump. We start bydescribing the effect of a bias voltage and an external force.The shown charge current curvatures in Figs. 2 and 3 areclear indications that the device may operate as a charge pumpin the situation where there is no applied bias voltage. Al-though not shown, the force curvature B F displays a simi-lar pattern thus implying that the device could also work asan electrical motor for the chosen parameters. However, tounderstand in more detail the device’s operational behavior,we need to take into account the effect of the external force F ext . This force will finally determine if the system is opera-tional or not, together with its subsequent working mode, i.e.,motor or pump. For simplicity we assume that the externalforce is constant and points along the tangential direction ˆ θ ,i.e. F ext = F ext ˆ θ , so its associated torque F ext is constantalong the whole trajectory and we can take W ext = 2 πs F ext .One of the interesting aspects of the system analyzed hereis that one can control its operational mode externally, shift-ing from a pump to a motor by moving the bias voltage orthe externally applied torque. For the case of the motor, thesystem will act as such when the work done by the CIFs over-comes the work done by the external agent, W ext . Then, fora fixed voltage, obviously F ext acts as the external knob thatcontrols whether the system is a motor or not. For the case ofthe charge pump, the system will act as such when the pumpedcurrent overcomes the leakage current given by the zero-ordercontribution (proportional to ∆ V at least for small voltages)and the second-order contribution (proportional to Ω ). For afixed voltage, here also F ext acts as the external knob, but fora different reason. In this case, F ext is controlling the com-pensation between the leakage and pump currents. Note thateach order of the current depends differently on Ω which is ul-timately controlled by F ext . One can also take the voltage asthe pump knob. Here, one is mainly controlling the zero-ordercurrent, which at some point will be so large that it cannot becounteracted by the pumped current. All these features arereflected in the cross-shaped efficiency shown in Fig. 4(a).In Fig. 4(a) we show a map of the electric motor/pump ef-ficiencies η em and η ep as a function of the external torque F ext and the bias voltage ∆ V for the double-slit configura-tion. The plot allows us to visualize the regions in whichthe device becomes operational, while giving us a quantita-tive idea on the performance it can reach. The vertical arm of this cross-shaped map corresponds to the motor efficiency[see also panel (b), η em for ∆ V = 0 . k B T ], while the hori-zontal arm depicts the charge pump efficiency [see also panel(c), η ep for ∆ V = 0 . k B T ]. Notice that, in principle, notall regions in the shown map are well defined as the obtainedfrequencies not always fulfill the adiabaticity condition dis-cussed in Sec. II C and upon which our expansion is justified.Concretely, the unshaded regions in the figures correspond to Ω δ(cid:15)/ Γ k B T ≤ . , in which our expansion should be ade-quate.To evaluate the role of the decoherence in the efficiencies,in Fig. 4(b) and (c) we show η em and η ep , respectively, for afixed bias voltage and different values of Γ φ . In general, wecould say that the decoherence rate has an adverse effect overthe device’s performance in the sense that it reduces the de-vice’s maximum efficiency. This result was expected since,as seen before in Sec. III C, Γ φ tends to reduce the peaks inthe curvature B I ( B F ), which is proportional to the pumpedcharge (CIF work). For the electric motor this is quite evidentas all curves progressively fall off below the zero-decoherencecase. Note also that the range within which the device actsas a motor always decreases with Γ φ . This range is deter-mined by the crossing of the curves with the line η = 0 , seeFig. 4(b). For the electric pump, however, the situation is dif-ferent. There, even though there is an overall decrease of theefficiency, decoherence injection makes the system more re-silient to the effects of the external force. In Fig. 4(c) we cansee that for Γ φ = 0 , the device can only bear torques up toapproximately 0.3 k B T / rad , a small value compared to thetorques it can withstand for Γ φ > . Interestingly, this allowsthe system to be “activated” by decoherence in regions where,in principle, it wouldn’t be operational. Such effect has al-ready been discussed in a similar quantum system in Ref. [27],where the performance of an adiabatic quantum motor wasimproved with the aid of decoherence. Notice that, in thatcase, the electromechanical device was described within theLandauer-B¨uttiker formalism, as is usual in systems where theCoulomb interaction between electrons can be taken as a meanfield. Here we see that a similar decoherence-induced activa-tion appears in the pumping regime under strong Coulombinteraction.In the energy range shown in Fig. 4(a) and for the consid-ered set of parameters, the maximum efficiency achieved was η max ≈ . , a small value if one compares it with the stronginterdot coupling regime discussed in Ref. [31], where effi-ciencies up to 75% were obtained. While in the motor oper-ation mode this value can be increased (see below), for theelectric pump, the small value of η seems not so easy to over-come. The reason behind that lies in the interplay between thedifferent orders of the transported charge per cycle, Q ( k ) I , asdepicted in Fig. 5(a). Notice that each order obeys a differentlaw as one moves the external torque. This is so because, first, Ω depends linearly on F ext [see Eq. (22)], and second, eachcontribution to the charge current has a different dependenceon the angular frequency Ω , i.e., Q ( k ) I ∝ Ω k − . Therefore,e.g., in regions where the frequency is extremely small (ascompared with Γ ), the amount of transported charge in a pe-riod is dominated by its zeroth-order contribution, as many10 V QUANTUM MACHINES BASED ON THE DOUBLE-SLIT DQD -0.4-0.200.20.4-0.15 -0.1 -0.05 0 0.05 0.1 0.15 00.250.50.751 00.250.50.751 0 0.005 0.01 0.015 0.02 0 0.1 0.2 0.3 0.4 0.5
Figure 4. (a) Map of the efficiencies η em and η ep as functions of the external torque F ext and the bias voltage ∆ V . The efficiencies havebeen normalized with respect to the maximun value η max = 1 . × − achieved in the shown region. Shaded areas denote the regionswhere the adiabaticity condition is not fulfilled (see main text). (b) Electric motor efficiencies as a function of F ext for ∆ V = 0 . k B T anddifferent decoherence rates in units of Γ : Γ φ = F ext for ∆ V = 0 . k B T and the same values of Γ φ as in (b). The other used parameters are Γ = 0 . k B T , t c = 0 , λ = 0 , λ S = 0 . , and λ D = − . .The chosen trajectory is given by ¯ E = 1 . k B T , ¯ (cid:15) = 0 , δ E = 5 k B T , and δ (cid:15) = 0 . k B T . -0.5-0.2500.250.5-0.5 -0.25 0 0.25 0.5 -0.5 -0.25 0 0.25 0.5 Figure 5. Different order contributions to the transported charge as afunction of the external torque F ext . The sum of all these contribu-tions, denoted by Q I , is shown in solid black. The gray area indicatesthe region where the device is capable of pumping charge and there-fore becomes operational. We considered the same parameters as inFig. 4(c) with Γ φ = 0 (a) and Γ φ = 0 . (b). electrons can flow through the system even though the me-chanical motion is almost frozen. The first-order contribution,which is the relevant quantity for the device’s performance,only surpasses the sum of the other contributions in a smallrange of the external force, see the gray area in Fig. 5(a).In other words, the first-order current only prevails over theleaking effect, given by the zeroth- and second-order currents,in a small range of F ext . The effect of decoherence on thecurrent is complex, as it affects each order in a different way.Figs. 5 (a) and (b) show an example of this. There, one can no-tice that, although all contributions to Q I decrease with an in-crease of decoherence, the second-order term is affected muchdramatically, making it negligible within the shown range ofparameters. As a consequence of that, the range over whichcharge pumping is possible is considerably extended.As can be seen in Fig. 4(a), increasing ∆ V shrinks the re-gion over which charge can be pumped, to the point where thisis no longer possible, i.e., Q (1) I cannot exceed the other contri- butions. This is expected since dots’ energies are very similarin the considered approximation, and hence most of the parti-cles easily flow through the system (along the bias direction)without exchanging its momentum with the mechanical part.Being limited to work only in the low-bias voltage regionsmakes it difficult to increase efficiency. The situation is quitedifferent for the motor regime. There, the working region islimited by F ext , but CIFs can easily be risen to compensatefor that by just increasing ∆ V . Therefore, there is more free-dom to look through the space of parameters. We perform awide numerical exploration and find that adiabatic quantummotors can achieve efficiencies up to 50%, but in a large biasregime ( ∆ V ∼ k B T ).We have also studied other operation regimes of our system.In particular, we explored its role as a heat pump (refrigerator)and as a heat engine (temperature-driven motor), see Fig. 6.The results are similar to those described above for the chargepump and the electric motor regimes. The main differencesare: ( i ) the heat pump is more sensitive to decoherence, seeFig. 6(c) and notice the different Γ φ used values with respectto Fig. 4(c), and ( ii ) due to the way in which the efficienciesare affected by Γ φ , we can conclude that there is no activa-tion by decoherence at least in this regime of the parameters.We observe that efficiencies of the order of 4% were obtainedwhen taking temperature gradients close to the limit of zerotemperature in the cold reservoir [where Carnot’s efficiency is1 and which is out of the range of Fig. 6(a)]. On the otherhand, the quantum refrigerator achieves an efficiency whichis approximately 2% of Carnot’s limit. Again, these valuesare small when compared to the ones reported for the stronginterdot coupling regime, where efficiencies higher than 50%of Carnot’s limit were obtained for both the heat engine andthe refrigerator operational modes. As discussed above, thereason behind these low values lies in the fact that leakage cur-rents are dominant in the considered regime of parameters. InFig. 7 we show the contributions for the transported heat percycle as a function of the external torque for Γ φ = 0 and 0.05 Γ . We can see that both the zeroth and second-order trans-11 I ACKNOWLEDGEMENTS -0.2-0.100.10.2-0.1 -0.05 0 0.05 0.1 00.250.50.751 00.250.50.751 0 0.002 0.004 0.006 0.008 0.01 0 0.05 0.1 0.15 0.2
Figure 6. (a) Map of the normalized efficiencies ˜ η he and ˜ η hp as a function of F ext and ∆ T . These functions have been divided with respect tothe maximum value ˜ η max = 1 . × − achieved in the shown map. As in Fig. 4, the shaded areas denote the regions where the adiabaticitycondition is not satisfied. (b) Plots of ˜ η he vs F ext for ∆ T = 0 . T and for different decoherence rates (in units of Γ ): Γ φ = ˜ η hp vs F ext for ∆ T = 0 . T and for the same values of Γ φ as in (b). The other used parametersare: Γ = 0 . k B T , t c = 0 , λ = 0 , λ S = 0 . , and λ D = − . . The chosen trajectory is given by ¯ E = 5 k B T , ¯ (cid:15) = 0 , δ E = 5 k B T , and δ (cid:15) = 0 . k B T . -0.5-0.2500.250.5-0.05 0 0.05 0.1 0.15 -0.05 0 0.05 0.1 0.15 Figure 7. Different order contributions to the transported head fromthe cold reservoir as a function of the external torque F ext . Thesum of all these contributions, denoted by Q J cold , is shown in solidblack. The gray area indicates the region where the device is capableof pumping heat and therefore becomes operational. We consideredthe same parameters as in Fig. 6(c) with Γ φ = 0 (a) and Γ φ = 0 . (b). ported heat (i.e., those coming from the leakage currents) arealmost not affected by decoherence, while the first order con-tribution clearly decays with Γ φ . Despite that, we remark thatthe goal of the present work is not to obtain efficient pumpsor motors but to study up to what extent coherences may playa role in QD-based quantum machines within the Coulombblockade regime. V. CONCLUSIONS
We studied quantum-dot-based nanomachines in theCoulomb blockade regime in a situation where the coherencescan dominate the transport properties of the device. We fo-cused our analysis to what we called the double-slit config-uration. In this setup, coherences do not come from the in-terdot coupling, which is zero, but from the particles enter-ing/leaving the two dots simultaneously. Therefore, the only way particles entering the system get information from thetwo dots is through a coherent superposition of states. Thismakes the modulation manifold effectively bi-parametric, asrequired in the adiabatic regime. In this sense, the double-slitconfiguration can be used as the basis for different forms of“true” quantum machines, namely: quantum motors, quantumpumps, quantum heat engines, and quantum heat pumps.We analyzed the impact of decoherence on the above ma-chines. As expected, we found that the overall result is todecrease the efficiency of the machines. In the strong deco-herence limit, this can be interpreted as the situation in whichthe quantum superposition is destroyed, so the electrons inthe device can no longer access the two parameters, and theamount of pumped charge/heat or useful work per cycle goesto zero. However, for intermediate values of Γ φ , its effect ismore complex due to two main factors. The first one is that,although decoherence tends to decrease the maximum of thegeometric curvatures (current, heat, and force), it also widensthem. This causes that, under specific parameters, some formsof quantum machines can indeed be activated by decoherence,in the sense that they require a minimum amount of it to oper-ate. The other important factor is that decoherence can affectthe orders of the adiabatic expansions of the observables dif-ferently. This is the reason for the found differences betweencharge and heat pumps regarding the effect of decoherence onthem. VI. ACKNOWLEDGEMENTS
We acknowledge financial support by Consejo Nacionalde Investigaciones Cient´ıficas y T´ecnicas (CONICET); Sec-retar´ıa de Ciencia y Tecnolog´ıa de la Universidad Na-cional de C´ordoba (SECYT-UNC); and Agencia Nacional dePromoci´on Cient´ıfica y Tecnol´ogica (ANPCyT, PICT-2018-03587).12
EFFECTIVE EVOLUTION KERNEL
Appendix A: Trajectory in the parameter space
In Sec. III A we stated that it is convenient to take an ellip-tic trajectory around the origin of the energy space in order totake advantage of the shape of B I and thus increase the effi-ciency of the device. More specifically, this elliptic trajectoryshould be much wider along the E axis than in the (cid:15) axis [cf.Fig. 3(a)]. With this in mind, we now show how the exper-imental setup displayed in Fig. 1 can be configured to allowfor such a trajectory. Let C and C be the capacitances of theside contacts and C the capacitance of the central contact,displayed in the middle of the two dots. For the consideredconfiguration, the dots’ energies can be described by E i ( θ ) = E (0) i + q ( θ ) C + q i ( θ ) C i , i = { , } , (A1)where E (0) i is the energy in the absence of contacts, the q ’sdenote the amount of charge accumulated in each one of thecontacts, as a function of the rotor’s position. For the specificgeometry of the rotor and the used configuration for the con-tacts, we could argue that q ( θ ) = − q ( θ ) , as the charges inthe rotor are assumed to be the same in magnitude, but oppo-site in sign. Besides, we could simplify the above dependenceby stating that C = C . Due to the position of the centralcontact with respect to the C contact, it is reasonable to ex-pect a phase shift of π/ in q , i.e., q ( θ ) = q ( θ + π/ .Accordingly, we replace these assumptions in the above ex-pressions and obtain E , ( θ ) = E (0)1 , + q ( θ ) C ± q ( θ + π/ C . (A2)If we now define E = ( E + E ) / , (cid:15) = E − E , E (0) =( E (0)1 + E (0)2 ) / , (cid:15) (0) = E (0)1 − E (0)2 , C E = C and C (cid:15) = C / , we arrive at the following parametric equations: E ( θ ) = E (0) + q ( θ ) C E , (cid:15) ( θ ) = (cid:15) (0) + q ( θ + π/ C (cid:15) . (A3)Thus, if an elliptic trajectory with E max (cid:29) (cid:15) max is desired,then it is enough to take C (cid:15) (cid:29) C E . Appendix B: Effective evolution kernel
In this appendix we show how the blocks of the effectiveevolution kernel W eff are related to the energy parameters E and (cid:15) . As discussed in Sec. II C, this effective kernel is de-fined as the sum of the evolution kernel W and the Liouvil-lian L , which can be decomposed into two contributions, L dot and L c , by separating the t c -dependent term in the electronicHamiltonian of Eq. (36). To study the energy dependence,we will treat these components individually. For the systemtreated in this work, the matrix elements of the evolution ker-nel W depend on the DQD’s eigenenergies in the followingway W α,α (cid:48) β,β (cid:48) ∝ Γ (cid:88) pηr (cid:88) ij (cid:2) − pφ r ( q η,rij ) − i πf ( pq η,rij ) (cid:3) . (B1) Here, p = ± is an index that distinguishes forward ( + ) frombackward ( − ) time evolutions on a Keldysh double contourdiagram while η = ± is a particle index denoting the annihi-lation/creation of an electron in the r -lead. The indexes i and j run over the DQD’s eigenstates, and f ( x ) = [1 + exp( x )] − is the usual Fermi function. The function φ ( x ) is defined as: φ r ( x ) = − Re ψ (cid:18)
12 + i x π (cid:19) + ln D πk B T r , (B2)where ψ is the digamma function and D denotes the reser-voir’s bandwidth, which we assume to be independent of r forsimplicity. The argument in the above functions correspondsto the energy difference between initial and final eigenstates,with respect to the r -lead electrochemical potential and di-vided by the thermal energy, i.e. q η,rij = E i − E j − ηµ r k B T r . (B3)If we set the energy of the empty state as reference, i.e., E =0 , then all non-vanishing elements of W depend only on theenergies E (cid:96)σ of the singly occupied states. At the same time,as in the approximation mentioned in Sec. III B the effectivekernel W eff must be taken up to first order in Γ , and since allelements in W are multiplied by a prefactor proportional to Γ ,the energy differences entering in q η,rij need to be taken up tozeroth order in the perturbation parameter. This means that thearguments q η,rij can only retain the zeroth order contribution,so all elements in W only depend on the mean level energy E . If we now consider the Liouvillian L c , we can see that [ L c ] α,α (cid:48) β,β (cid:48) = (cid:104) β | ˆ H c | α (cid:105) δ α (cid:48) β (cid:48) − (cid:104) α (cid:48) | ˆ H c | β (cid:48) (cid:105) δ αβ , (B4)where ˆ H c accounts for the interdot coupling Hamiltonian [lastterm in the r.h.s. of Eq. (36)]. Due to the Kronecker deltasand the off-diagonal structure of ˆ H c in the local basis, thisLiouvillian will only contribute to the dn and nd blocks of W eff with terms of the form ± i t c / . Lastly, we study theenergy dependence of the Liouvillian L dot . In this case it canbe shown that [ L dot ] α,α (cid:48) β,β (cid:48) = ( E α − E α (cid:48) ) δ αβ δ α (cid:48) β (cid:48) . (B5)This means that L dot will only contribute to the nn block of W eff . Since we are working in the local basis, this impliesthat L dot is diagonal, whose elements are ± (cid:15) . The explicitmatrix representation of the complete Liouvillian L = L dot + L c is thus given by L dd = , L dn = t c − − − − , (B6)together with L nd = L Tdn and L nn = (cid:15) diag(1 , , − , − .Regarding the decoherence rates, all the blocks of the deco-herence matrix Γ φ are zero, except for the nn block, which issimply Γ φ times the 4 × SYMMETRIC COUPLINGS TO THE LEADS
To summarize this analysis, we conclude that all blocks of W eff are E -dependent but only its nn block depends on theenergy difference (cid:15) . With regard to the interdot coupling, wecan see that for the configuration in series, the elements in L c are the only ones connecting the dd and nn blocks of the effec-tive kernel, such that coherences are completely determinedby this parameter. However, in the configuration in parallel,additional matrix elements proportional to t r, t ∗ r, [cf. Eq. (5)]contribute in the dn and nd blocks of W , such that coherencesmay even survive without any interdot coupling. Appendix C: Symmetric couplings to the leads
Here we go into detail about the recovery of the (cid:15) ↔ (cid:15) symmetry by taking the same asymmetry factors when cou-pling the DQD system with the leads, i.e., λ S = λ D . Underthis condition, the tunneling rates satisfy Γ r,i = Γ r Γ r (cid:48) Γ r (cid:48) ,i , (C1)where r, r (cid:48) = { S , D } , and i = { , } . In the absence of anybias voltage or temperature gradient, as the asymmetry factorsonly enter in W eff through the tunneling rates, this results in similar relations when decomposing the effective kernel in its r -lead components, such that the same can be applied for thecharge currents [cf. Eq. (18)], W eff r = Γ r Γ r (cid:48) W eff r (cid:48) ⇒ I ( k ) r = Γ r Γ r (cid:48) I ( k ) r (cid:48) . (C2)On the other hand, charge conservation on the first order cur-rents gives rise to the following relation (cid:88) r I (1) r = − dd t (cid:104) ˆ n (cid:105) (0) , (C3)where ˆ n is the DQD occupation number operator. Hence, ifwe make use of Eq. 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