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particle current trough a coherent mesoscopic conductor connected at its left and right hand side to reservoirs is usually described in the non - interacting case by a formula , due to landauer , based on the following physical picture : electrons in the left ( right ) reservoir which are fermi distributed with chemical potential @xmath0 ( @xmath1 ) and inverse temperature @xmath2 ( @xmath3 ) can come close to the conductor and feed a scattering state that can transmit it to the right ( left ) reservoir . all possible dissipative processes such as thermalization occur in the reservoirs while the system formed by the conductor and the leads is assumed to be coherent . the probability of being transmitted is a property of the conductor connected to the leads which is treated as a scattering system . in this picture , the probability that an outgoing electron comes back to the conductor before being thermalized is neglected , the contact is said to be reflectionless . this description of the non - equilibrium steady state ( ness ) current through a finite system has been rigorously proved in some particular limiting situations  @xcite , such as infinite reservoirs , but several difficulties prevail the understanding of non - equilibrium states in general and the description of current in more general situations , for instance in the case of interacting particles . two frameworks are usually considered to study these open quantum systems : one deals with the properties of the state of the total ( infinite ) system @xcite where reservoirs are explicitly considered as part of the system . the other is based on the master equation of the reduced density operator , obtained by tracing out the reservoirs degrees of freedom and is better suited to be applied to different systems and to compute explicitly some averaged ness properties , at the price of several approximations such as _ e.g. _ , born - markov ( see e.g. @xcite ) .    in this paper we explore particle current in a model where we mimic the leads that connect the reservoirs with the system , as a finite non - interacting system with a finite number of levels ( which we call meso - reservoir ) . the reservoirs ( called here super - reservoirs ) are modeled by local lindblad operators representing the effect that a markovian macroscopic reservoirs have over the meso - reservoirs . in sec . [ sec.2 ] we introduce the model and briefly review the method we use to solve it . in sec . [ sec.3 ] we analyze the particle current operator and indicate the quantities that should be computed for a full description of the current . in sec . [ landauer ] we briefly present the landauer formula that is expected to apply in some appropriate limits to our model and in sec .  [ numeric ] we analyze the numerical results we obtained with our model and compare them with the current predicted by the landauer formula , validating the applicability of our model but also going beyond by computing the full probability distribution function ( pdf ) of the current . in sec . [ conclu ] we present some conclusions and discuss interesting perspectives of our study . we consider a one - dimensional quantum chain of spinless fermions coupled at its boundaries to meso - reservoirs comprising a finite number of spinless fermions with wave number @xmath4 ( @xmath5 ) . the hamiltonian of the total system can be written as @xmath6 , where @xmath7 is the hamiltonian of the chain with @xmath8 the nearest neighbor hopping , @xmath9 the onsite potential and @xmath10 the annihilation / creation operator for the spinless fermions on the site @xmath11 of the chain ( conductor ) . the chain interacts through the term @xmath12 with the meso - reservoirs @xmath13 . here @xmath14 denotes the left and right meso - reservoir . they share the same spectrum with a constant density of states @xmath15 in the band @xmath16 $ ] described by @xmath17 and @xmath18 are annihilation / creation operator of the left and right meso - reservoirs . the system is coupled to the leads only at the extreme sites of the chain with coupling strength @xmath19 that we choose @xmath4-independent   dependent super - reservoir to meso - reservoir couplings @xmath20 @xmath21 . we assume that the density matrix of the chain - meso - reservoirs system evolves according to the many - body lindblad equation @xmath22   + \sum_{k,\alpha , m } \left ( 2l_{k , \alpha , m } \rho l_{k , \alpha , m}^\dag -\{l_{k , \alpha , m}^\dag l_{k , \alpha , m},\rho\ } \right ) , \label{eq : lindbladmodel}\end{aligned}\ ] ] where @xmath23 and @xmath24 are operators representing the coupling of the meso - reservoirs to the super - reservoirs , @xmath25 are fermi distributions , with inverse temperatures @xmath26 and chemical potentials @xmath27 , and @xmath28 $ ] and @xmath29 denote the commutator and anti - commutator , respectively . the parameter @xmath30 determines the strength of the coupling to the super - reservoirs and to keep the model as simple as possible we take it constant . the form of the lindblad dissipators is such that in the absence of coupling to the chain ( i.e. @xmath31 ) , when the meso - reservoir is only coupled to the super - reservoir , the former is in an equilibrium state described by fermi distribution . @xcite . to analyze our model we use the formalism developed in @xcite . there it is shown that the spectrum of the evolution superoperator is given in terms of the eigenvalues @xmath32 ( so - called rapidities ) of a matrix @xmath33 which in our case is given by @xmath34 where @xmath35 and @xmath36 denote @xmath37 zero matrix and @xmath38 unit matrix , @xmath39 is the pauli matrix , and @xmath40 is a matrix which defines the quadratic form of the hamiltonian , as @xmath41 in terms of fermionic operators @xmath42 . the ness average of a quadratic observable like @xmath43 is given  @xcite in terms of the solution @xmath44 of the lyapunov equation @xmath45 with @xmath46 and @xmath47 as follows : consider the change of variables @xmath48 , the ness average of the quadratic observable @xmath49 is determined by the matrix @xmath44 through the relation @xmath50 . wick s theorem can be used to obtain expectations of higher - order observables and in fact , the full probability distribution for these expectation values in some cases . the operator representing the current flowing from the @xmath4-th level of the meso - reservoir to the chain is given by @xmath51 while the current trough the site @xmath52 of the chain is @xmath53 in the steady state the average current is conserved in this model  @xcite and thus @xmath54 is independent of @xmath52 . moreover , if we define the current from the left meso - reservoir as @xmath55 , we have that @xmath56 . it is not difficult to note that current satisfies @xmath57 with @xmath58 now we are in a position to compute the full non equilibrium current distribution in terms of @xmath59 and @xmath60 . for this we consider the generating function @xmath61 which , using eq.([moments ] ) , gives @xmath62 the probability distribution @xmath63 is the inverse fourier transform of @xmath64 , thus we get @xmath65 which is normalized . we note that normality and positivity of probability lead an interesting inequality : @xmath66 an equivalent result holds for the current from the @xmath4-level of the meso - reservoir to the system : @xmath67    these expressions are expected because @xmath68 and @xmath69 are the possible eigenvalues of the operator @xmath70 ( similarly for @xmath71 ) , but they show that @xmath59 and @xmath60 contains all the information about the current . we will study these quantities numerically , thus we need to solve the above mentioned lyapunov equation and note that in the @xmath72 variables ( we use the primed variables to indicate indices as they appear in variable @xmath73 , i.e. if @xmath11 is a site in the chain , then @xmath74 ) @xmath75 thus @xmath76 and @xmath77 wick s theorem implies @xmath78    to simplify the discussion , in what follows we reduce the number of parameters by assuming constant hopping and onsite energy @xmath79 and @xmath80 . moreover we fix @xmath30 such that @xmath81 . in that case , we showed @xcite that the transport quantities become roughly independent of @xmath30 . in fact , in that case , the coupling super - reservoir  meso - reservoir is stronger than that of the system ( chain )  meso - reservoir . then the meso - reservoir are driven to a near equilibrium state weakly dependent of @xmath30 . we explore now the behavior of the current as a function of @xmath15 , @xmath82 , @xmath83 and contrast the observation with expectations based on the landauer formula . qualitative explanation of the current behavior is also provided . the landauer formula  @xcite provides an almost explicit expression for the ness average current as a function of the parameters of the system . in units where @xmath84 and @xmath85 it reads @xmath86 where @xmath87\ , $ ] is the transmission probability written here in terms of @xmath88 the retarded and advanced green function of the system connected to the leads and of @xmath89 the imaginary part of the self - energy @xmath90 . the self energies @xmath91 have only terms at the boundaries of the chains  @xcite i.e. , @xmath92 , where @xmath93 if @xmath94 and @xmath95 if @xmath96 and @xmath97 recalling that both leads have the same spectrum , we assume a constant density of lead states @xmath98 in the range @xmath16 $ ] , thus @xmath99 is independent of @xmath100 inside the interval and zero otherwise . for the real part of the self - energy we have the principal value integral @xmath101 the landauer formula is expected to hold when the leads have a dense and wide spectrum . therefore , we restrict ourselves to the case that @xmath102 and @xmath103 , the so called wide - band limit , where @xmath104 can be neglected . the transmission coefficient is then @xmath105 and we need to compute the wide - band limit retarded green function @xmath106 note that in the previous expression we have set @xmath107 which sets the energy axis origin and @xmath108 which sets the energy scale . thanks to a recursion relation , this matrix can be inverted  @xcite and one explicitly finds the relevant element of the green function @xmath109^{-1}=\left(\omega + \frac{i \gamma _ l}{2}\right ) \left(\omega + \frac{i \gamma_r } { 2}\right ) \sum     _ { k=0}^{\left\lfloor \frac{n-2}{2}\right\rfloor } ( -1)^k \omega ^{n-2 - 2 k }     \binom{n-2-k}{k}\\    -\left(2 \omega + \frac{1}{2 } i ( \gamma_l + \gamma_r ) \right ) \sum     _ { k=0}^{\left\lfloor \frac{n-3}{2}\right\rfloor } ( -1)^k \omega ^{n-3 - 2 k }     \binom{n-3-k}{k}+\sum _ { k=0}^{\left\lfloor \frac{n-4}{2}\right\rfloor } ( -1)^k     \omega ^{n-4 - 2 k } \binom{n-4-k}{k } , \end{array } \label{g1n}\ ] ] where @xmath110 is the largest integer smaller than @xmath111 . in the next section we compute numerically the integral in eq.([eq.landauer ] ) and compare with the results obtained in our model . , width=264 ]    in fig . [ esquema ] we depict in blue and red two fermi distributions with @xmath112 levels and parameters @xmath113 and @xmath114 respectively . in the middle ( brown ) the spectrum of a chain with @xmath115 sites and @xmath116 the width of the chain energy band is @xmath117 with @xmath115 levels inside and is centered around @xmath107 . from this picture , we expect that decreasing the width @xmath118 of the populated energy interval @xmath119 $ ] is equivalent to increase the width of the conduction band @xmath83 . this is confirmed in the left panel of fig . [ j - t ] where we show that the current is roughly independent of @xmath83 for @xmath120 and decreases with @xmath83 for @xmath121 ( black dots ) , when the conduction band extends beyond the region populated by electrons in the reservoirs . the red dots are obtained for a larger @xmath118 for which the conduction band is always inside the populated region . in the rest of our numerical examples we set @xmath122 and @xmath123 . analogously , in the right panel of fig . [ j - t ] we show that for fixed @xmath83 the current grows linearly with @xmath118 and saturates at @xmath124 .     for @xmath125 , @xmath126 , @xmath127 , @xmath128 and @xmath129 . in the left panel ( black ) @xmath130 and ( red ) @xmath131 . right panel : same parameters except : @xmath132.,title="fig:",width=264 ]   for @xmath125 , @xmath126 , @xmath127 , @xmath128 and @xmath129 . in the left panel ( black ) @xmath130 and ( red ) @xmath131 . right panel : same parameters except : @xmath132.,title="fig:",width=264 ]    what is perhaps more interesting is the scaling of the current with @xmath133 . in fig . [ j - vr ] we plot the current @xmath134 as a function of the coupling to the right lead , showing that the main trend of the current is @xmath135 . in the inset we show that there are deviations to this law .     and @xmath136 . the red dots are obtained by a numerical computation of the landauer formula with @xmath137 , and the continuous line is @xmath138 . the inset is @xmath139 . , width=264 ]    we have analyzed this behavior using the landauer formula , that allows deeper analytical exploration . since temperature is very low we take it exactly zero in both reservoirs , thus the landauer formula is @xmath140.$ ] changing the representation of the green matrix from position basis to the energy basis ( of the isolated chain ) one can see that it is possible to approximate ( isolated resonances approximation ) the integrand as a superposition of @xmath141 lorentzians each located at @xmath142 $ ] and with a width @xmath143/(n+ 1 ) , \ , j=1,\ldots n$ ] along the energy axis . if @xmath144 , the integral can be extended from @xmath145 to @xmath146 because the green function decays exponentially outside the energy band of the chain , thus the result @xmath147 is obtained . now we can also explore fluctuations and compute @xmath148 . in fig.[fluct]a , we computed for the same parameters than in fig.[j - t]a the quantity @xmath149 . we see that as soon as the current decreases because some modes of the chain goes out of the populated energy region , the fluctuation increases . as a function of other parameters like @xmath150 or @xmath141 ( data not shown ) , the average @xmath148 does not have important changes ( see fig.[fluct]b ) .     as function of @xmath83 for @xmath151 ( black dots ) and @xmath152 ( red dots ) as in fig . [ j - t]a . panel b : @xmath149 for @xmath153 as a function of @xmath154 . in both figures the other parameters are : @xmath155 and @xmath136 with @xmath156 in panel a and @xmath157 in panel b. , title="fig:",width=264 ]   as function of @xmath83 for @xmath151 ( black dots ) and @xmath152 ( red dots ) as in fig . [ j - t]a . panel b : @xmath149 for @xmath153 as a function of @xmath154 . in both figures the other parameters are : @xmath155 and @xmath136 with @xmath156 in panel a and @xmath157 in panel b. , title="fig:",width=264 ] we showed that in the wide - band limit , the numerical results found in our model indeed correspond to what is expected on the basis of the landauer formula , a formula which is usually interpreted as if the reservoirs were always in an equilibrium ( gran canonical ) distribution , not perturbed by the presence of the system . the landauer formula emphasizes the role of the fermi distributions of the reservoirs and provides an accurate description of the current if the assumption of reflectionless contacts is justified . in this respect , a very interesting relation was found in  @xcite and proved in  @xcite between the current and the occupation of the meso - reservoir @xmath158 $ ] . this is an exact relation that in the appropriate limit should converge to the landauer formula . note that it implies that the occupation difference with respect to the fermi distribution is @xmath159 . it is a very interesting relation because it links the current , which is the fingerprint of the non equilibrium state , to the difference in distribution to the equilibrium case . something similar has been found in classical systems where the fractal nature of the non equilibrium state is determined by the current  @xcite . moreover in  @xcite we analyzed how onsager reciprocity relation is broken in the system and found that @xmath160 grows with @xmath30 implying that despite the almost @xmath30 independent value of the current , the dissipative mechanisms in the super - reservoir play an important role . a deeper study of these effects , that are beyond the landauer picture , can be studied in the context of the model presented here and deserve further investigation . the authors gratefully acknowledge the hospitality of nordita where part of this work has been done during their stay , within the framework of the nonequilibrium statistical mechanics program , and support from esf . fb thanks fondecyt project 1110144 and sa thanks fondecyt project 3120254 .    99 w aschbacher , v jasi , y pautrat , and c pillet , j. math . phys . * 48 * , 032101 ( 2007 ) . s. ajisaka , f. barra , c.mejia-monasterio and t. prosen arxiv:1204.1321 s. ajisaka , f. barra , c.mejia-monasterio and t. prosen , arxiv:1205.1167 . d. ruelle j. stat . phys . * 98 * , 57 ( 2000 ) ; comm . * 224 * , 3 ( 2001 ) . s. ajisaka , s. tasaki and f. barra , bussei - kenkyu * 97 * 483 ( 2011/2012 ) ; arxiv:1110.6433 h. wichterich , m. henrich , h - p . breuer , j. gemmer , and m. michel , phys . e * 76 * , 31115 ( 2007 ) . t. prosen , n. j. phys . * 10 * , 3026 ( 2008 ) . a. a. dzhioev and d. s. kosov , j. chem . phys . 044121 * 134 * ( 2011 ) . t. prosen j. stat . p077020 ( 2010 ) . s. datta , _ electronic transport in mesoscopic systems _ ( cambridge university press , cambridge , 1995 ) . m. zilly , _ electronic conduction in linear quantum systems : coherent transport and the effects of decoherence _ phd . dissertation universitt duisburg - essen ( http://duepublico.uni-duisburg-essen.de/servlets/derivateservlet/derivate-24279/dissertation_zilly.pdf ) . f. barra and p. gaspard , j. phys . a math and gen . * 32 * 3357 ( 1999 ) . p. gaspard , _ chaos , scattering and statistical mechanics _ ( cambridge university press , cambridge , 1998 ) .
we study particle current in a recently proposed model for coherent quantum transport . in this model a system connected to mesoscopic fermi reservoirs ( meso - reservoir ) is driven out of equilibrium by the action of super reservoirs thermalized to prescribed temperatures and chemical potentials by a simple dissipative mechanism described by the lindblad equation . 
 we compare exact ( numerical ) results with theoretical expectations based on the landauer formula .