Time-dependent transport in open systems based on quantum master equations
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Time-dependent transport in open systems based on quantum master equations
I. Knezevic ∗ Department of Electrical and Computer Engineering, University of Wisconsin - Madison, Madison, WI 53706, USA
B. Novakovic
Department of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, USA
Electrons in the active region of a nanostructure constitute an open many-body quantum system, interactingwith contacts, phonons, and photons. We review the basic premises of the open system theory, focusing on thecommon approximations that lead to Markovian and non-Markovian master equations for the reduced statisticaloperator. We highlight recent progress on the use of master equations in quantum transport, and discuss thelimitations and potential new directions of this approach.
I. INTRODUCTION
The term master equation traditionally refers to differen-tial equations that describe the time evolution of the probabil-ity that a given physical system will occupy a set of allowedstates. These equations are typically first order in time andoften, but not necessarily, linear in the probabilities. Nowa-days, the term master equation is used more broadly: in thetheory of few-level open systems, it refers to equations thatdescribe the evolution of the open system’s statistical operator(usually called the reduced statistical operator or the reduceddensity matrix ) in the presence of coupling with an environ-ment that has a large, perhaps infinite, number of degrees offreedom and is responsible for the irreversibility in the opensystem’s evolution [1, 2]. Master equations are sometimes ob-tained as an “educated guess” (i.e. phenomenologically), butcan often be derived from the general framework of the opensystem evolution and a few reasonable assumptions. Masterequations can be divided into
Markovian master equations ,in which temporal evolution of the reduced statistical oper-ator does not depend on its past but only its current state,and non-Markovian equations , in which the so-called memoryeffects play an important role and involve information aboutthe evolution of the environment. How to best quantify non-Markovian effects in open quantum systems [3–6] and howto experimentally control the information flow between thesystem and the environment, potentially driving the systembetween the Markovian to the non-Markovian regime [7], arecurrently very active areas of inquiry.Electronic systems in semiconductor nanostructures areopen quantum systems, exchanging particles and informationwith the rapidly dephasing reservoirs of charge, often referredto as contacts, and possibly interacting with phonons or pho-tons as well [8]. In the open system theory, environments arecommonly considered to be bosonic, which is fine for the in-teraction of electrons with light or lattice vibrations. How-ever, electronic transport in the presence of contacts is a caseof a fermionic open system coupled to fermionic reservoirs,which is a largely unexplored problem [9, 10]. When refer-ring to electronic transport calculations, the use of the term ∗ [email protected] master equation falls in two camps: on the one hand, wehave few-level models (e.g. the resonant-level models usedfor quantum dots [11]) for which master equations continueto refer to the dynamics of the reduced statistical operator.On the other hand, if we strive to account for the generallycontinuous single-particle energy spectrum of an electron in ananostructure (e.g. when capturing current in structures with-out resonances) and the fact that many electrons are availableto populate them, then calculating the full many-body reducedstatistical operator becomes both intractable and unnecessary,as a great deal of information can be obtained from the single-particle quantities. In this case, master equation can refer tothe equations for the time evolution of the single-particle den-sity matrix (e.g. Redfield-type equations [12]) or just its diag-onal terms (e.g. the Pauli master equation [13]).In this paper, we review the basic premises and recentprogress on the use of Markovian and non-Markovian masterequations in the description of quantum electronic transport.In Sec. II, we present the basics of the open system formalism,including the concept of complete positivity of a dynamicalmap. We discuss microscopic derivations of Markovian mas-ter equations in Sec. III, focusing on the weak-coupling limit,and follow with examples from quantum transport in Sec. IV.In Sec. V, we overview general features of non-Markovianmaster equations and present the Nakajima-Zwanzig projec-tion operator technique. Examples of non-Markovian masterequations for time-dependent quantum transport are given inSec. VI. A summary and outlook conclude this paper in Sec.VII. II. GENERAL OPEN SYSTEM FORMALISM
Consider a quantum-mechanical system S interacting withan environment E . This composite S + E system is gener-ally described by the full statistical operator that, like otheroperators, lives in the Liouville space, which is (in the caseof finite systems) isomorphic to the square of the compositeHilbert space, H = ( H S ⊗ H E ) . Here, H S and H E are theHilbert spaces while H S and H E are the Liouville spaces ofthe system and environment, respectively. Operators acting onthe Liouville space are often called superoperators . If S + E is closed, the dynamics of its statistical operator ρ is given bythe Liouville equation (in the units of ~ = 1 ) dρdt = − i L ρ = − i [ H, ρ ] . (1) L is the Liouville superoperator and H the total S + E Hamil-tonian, generally of the form H = H S ⊗ I E + I S ⊗ H E + H int .The integral form of the Liouville equation is ρ ( t ) = U ( t, t ) ρ ( t ) U † ( t, t ) , (2a) U ( t, t ) = T exp (cid:18)Z tt − iH ( t ) dt (cid:19) , (2b)with T denoting time ordering. In the case of a time-indepen-dent H , U ( t, t ) = exp ( − iH ( t − t )) .Assume that we are interested only in the evolution of S .Its statistical operator can be obtained by tracing out the E degrees of freedom, i.e., ρ S = Tr E ( ρ ) . (3) S is often referred to as the reduced system and ρ S as the re-duced statistical operator or the reduced density matrix . Wewill use the term reduced statistical operator, because the termdensity matrix is usually reserved for the single-particle quan-tity in quantum transport studies.While the dynamics of S + E , a closed system, is unitary,the dynamics of S is not. If the environment has a large num-ber of degrees of freedom, on the timescales accessible in ex-periment the evolution of the reduced system S effectivelybecomes irreversible. Quite generally, the dynamics of thereduced statistical operator is given by ρ S ( t ) = Tr E (cid:0) U ( t, t ) ρ ( t ) U † ( t, t ) (cid:1) (4a) dρ S dt = − i Tr E L ρ = − i Tr E [ H, ρ ] . (4b)The central goal of the open system theory is to obtain the evo-lution of the (relatively small) reduced system S while mini-mizing the information that has to be gathered about the (rel-atively large) environment E . This quest is understandablyvery difficult, and approximations must be employed to yieldtractable equations.A first major simplification is assuming that, at some pointin the S + E evolution, the S and E were decoupled. It is as-sumed that, up until a certain point in time, usually designatedas t = 0 , S and E were mutually isolated, non-interacting,and therefore the initial ρ is of the uncoupled, tensor-productform, ρ (0) = ρ S (0) ⊗ ρ E (0) . (5)Thereafter, the interaction is turned on, presumably adiabati-cally. (A reader interested in the field of nanoelectronics canimmediately ask if this assumption is ever satisfied in elec-tronic systems, and the answer is “sometimes.” For instance,it can be considered true when we have high tunnel barriersbetween the active region ( S ) and contacts ( E ) in a nanos-tructure, and have let the active region and environment eachrelax on its own, with minimal tunneling between them.) The assumption of an uncorrelated initial state is a very ap-pealing one to adopt, because it guarantees [14] the existenceof a subdynamics (also known as reduced dynamics ), i.e. itguarantees that the evolution of the reduced statistical operatorcan in principle be fully described within H S . In other words,the existence of a subdynamics means there exists a generallynonunitary evolution superoperator W ( t, , such that ρ S ( t ) = W ( t, ρ S (0) . (6) W ( t, is often referred to as a dynamical map . In general,there exists a non-Hermitian generator of the dynamical map , K ( t ) , which satisfies W ( t,
0) = T Z t K ( t ′ ) dt ′ ; K ( t ) = d W dt . (7)The differential equation of motion for a subdynamics can bewritten as dρ S dt = K ( t ) ρ S ( t ) . (8)Clearly, it is practically impossible to obtain K and W fromfirst principles, and approximations are commonly made tothe structure of K . Generator K should be such that the result-ing evolution does not violate the unit trace or the positivity of ρ S . While the former is quite easily satisfied (any generatorthat acts as a commutator or a sum of commutators will pre-serve the trace of ρ S ), the latter is generally a tall order and is,in fact, not fulfilled by many common approximations. Com-plete positivity of a dynamical map [15] is a stronger criterionthan positivity (i.e. requiring that the map preserve the posi-tivity of the statistical operator). Namely, if you have two sys-tems whose evolution is such that the density matrix of eachremains positive (i.e. the evolution operators W for each sub-system are positive maps), it is not guaranteed that the com-posite map (their tensor product) will be a positive map, i.e.it is not guaranteed that the composite statistical operator willremain positive throughout evolution. If, however, each oneof the evolution maps is completely positive, then the tensorproduct is completely positive. In essence, complete positivityof the evolution map is a stricter criterion than positivity andnecessary for a successful description of composite systems[1].A time-independent generator K corresponds to Markovianapproximations. Evolution operators W ( t, t ′ ) generated by atime-independent K form a semigroup, with K then referredto as the semigroup generator. It has been shown by Lindblad[16] that the most general case of a generator of a completelypositive Markovian evolution must be of the form (given inthe Schr¨odinger picture): K ρ S = − i [ H S , ρ S ] + X k γ k (cid:16) [ A k , ρ S A † k ] + [ A k ρ S , A † k ] (cid:17) , (9)where γ k are nonnegative coefficients. The last term on theright-hand side is often referred to as the dissipator . The dy-namical map W ( t, t ′ ) generated by K from Eq. (9) is a com-pletely positive Markovian map. Lindblad’s form of K is veryuseful because it enables development of physically reason-able approximate forms of semigroup generators. III. MICROSCOPIC DERIVATIONS OF MARKOVIANMASTER EQUATIONS
In the most general terms, completely positive Markovianequations for the subsystem dynamics can be obtained in theweak-coupling limit [17–20], singular coupling limit [2], andby coarse graining over time [21] (discussed in more detail inSec. VI A). In electronic systems, there is also the large biaslimit with contacts that have a constant density of states [11](also referred to as the wide-band limit [22, 23]), which wewill discuss separately. The weak-coupling limit is of partic-ular importance, being applicable to electron-phonon interac-tion and electron-contact coupling in the case of tunnel barri-ers, so we discuss it in more detail.
A. The weak-coupling limit
The total dynamics in the interaction picture can be writtenas (in differential and integral forms) ddt ρ ( t ) = − i [ H I ( t ) , ρ ( t )] , (10a) ρ ( t ) = ρ (0) − i Z t ds [ H I ( s ) , ρ ( s )] , (10b)where H I ( t ) is the interaction Hamiltonian in the interactionpicture. Putting the integral form in the right-hand side ofthe differential form results in dρ ( t ) dt = − i [ H I ( t ) , ρ (0)] − R t ds [ H I ( t ) , [ H I ( s ) , ρ ( s )] . Tracing out this equation over theenvironment degrees of freedom, we obtain dρ S ( t ) dt = − i Tr E [ H I ( t ) , ρ (0)] (11) − Z t ds Tr E [ H I ( t ) , [ H I ( s ) , ρ ( s )]] . It is commonly assumed that Tr E [ H I ( t ) , ρ (0)] = 0 . Thisassumption is often satisfied: for instance, if the initial sta-tistical operators of the system and environment are grand-canonical or canonical equilibrium ensembles, they will con-tain pairs of the creation and annihilation operators associatedwith the single-particle spectra for S and E . The interactionHamiltonian is usually linear in these operators, i.e. it is com-monly assumed to be of the form (in the Schr¨odinger picture) P α A α ⊗ B α , where A α are the system and B α the envi-ronment operators. Tracing out the product of this interactionHamiltonian with the environment statistical operator over en-vironmental states gives zero.In electronic systems, the above approximation is satisfiedfor the electron-phonon interaction (the interaction Hamilto-nian is linear in phonon creation and annihilation operators)as well as for typical model Hamiltonians that describe theinteraction of the device with the contacts (Hamiltonian lin-ear in the contact and device creation/annihilation operators)if the evolution starts from equilibrium. However, if contact-active region electron-electron interaction is deemed impor-tant and is part of the interaction Hamiltonian, then the term Tr E [ H I ( t ) , ρ (0)] would survive. So the equation we are focusing on, which is still exact pro-vided approximation Tr E [ H I ( t ) , ρ (0)] = 0 holds, is: dρ S ( t ) dt = − Z t ds Tr E [ H I ( t ) , [ H I ( s ) , ρ ( s )]] . (12)The Born approximation assumes that the interaction isweak, so that the environment is negligibly affected by it andno considerable S - E correlations arise due to it over timeon the timescales relevant to S . As a result, we can write ρ ( t ) ≈ ρ S ( t ) ⊗ ρ E (0) and, consequently, dρ S ( t ) dt = − Z t ds Tr E [ H I ( t ) , [ H I ( s ) , ρ S ( s ) ⊗ ρ E ]] (13)Equation (13) has memory. The Markov approximation as-sumes that the interaction magnitude is such that the evolu-tion will depend only on the present state of the system, notits prior evolution, so ρ S ( s ) is replaced by ρ S ( t ) . This as-sumption is valid on timescales coarser than the decay timeof environmental correlations [2]. We can switch from s to t − s , with s now denoting the temporal distance from t , andintegrate over all values of s , because we expect the integrandto be negligible for large values of s (i.e. environmental cor-relations decay rapidly with increasing s ), finally arriving at dρ S dt = − Z ∞ ds Tr E [ H I ( t ) , [ H I ( t − s ) , ρ S ( t ) ⊗ ρ E ]] . (14)Equation (14) is the Redfield equation [24] and it still hasmemory.For an interaction Hamiltonian of the form P α A α ⊗ B α ,we can define the Fourier transforms of A α and B α based onthe system and environment spectra, A α ( ω ) = X ε P ( ε ) A α P ( ε + ω ) , (15)where P ( ε ) projects onto the eigenspace of H S correspond-ing to eigenvalue ε . As a result, A α ( t ) = e iH S t A α ( ω ) e − iH S t = e − iωt A α ( ω ) . In that case, the interaction Hamil-tonian in the interaction picture becomes H I ( t ) = P α,ω e − iωt A α ( ω ) ⊗ B α ( t ) , where B α ( t ) = e iH E t B α e − iH E t .Finally, the evolution of the reduced statistical operator be-comes ddt ρ S ( t ) = X ω,ω ′ X α,β e i ( ω − ω ′ ) t Γ αβ ( ω ) (cid:0) A β ( ω ) ρ S ( t ) A † α ( ω ′ ) − A † α ( ω ′ ) A β ( ω ) ρ S ( t ) (cid:1) + h . c . , (16)where Γ αβ ( ω ) = R ∞ ds e iωs Tr E (cid:0) B † α ( t ) B β ( t − s ) (cid:1) . Whenthe typical timescales for the system evolution, proportionalto | ω − ω ′ | − , are much shorter than the expected relaxationtimescales for the system, the so-called secular approximation (also known as the rotating-wave approximation or RWA) canbe applied: all terms with ω − ω ′ = 0 are considered as varyingtoo fast, so that their average contribution on the timescalesrelevant to S can be neglected. As a result, we obtain theweak-limit Markovian equation of motion ddt ρ S ( t ) = X ω X α,β Γ αβ ( ω ) (cid:0) A β ( ω ) ρ S ( t ) A † α ( ω ) − A † α ( ω ) A β ( ω ) ρ S ( t ) (cid:1) + h . c . . (17)Let us define χ αβ ( ω ) = 12 i (cid:0) Γ αβ ( ω ) − Γ ∗ βα ( ω ) (cid:1) (18a) γ αβ ( ω ) = 12 (cid:0) Γ αβ ( ω ) + Γ ∗ βα ( ω ) (cid:1) . (18b) χ corresponds to the so-called Lamb shift , an effective correc-tion to the system Hamiltonian of the form H LS = X ω,α,β χ αβ ( ω ) A † α ( ω ) A β ( ω ) . (19) H LS commutes with H S , so it shares the eigenvectors with H S and simply corrects the H S energy levels, and is thereforenot a true dissipative term. γ defines the coefficient of the truedissipator, D ( ρ S ) = X ω X α,β γ αβ ( ω ) (cid:0) [ A β ( ω ) ρ S ( t ) , A † α ( ω )]+ [ A β ( ω ) , ρ S ( t ) A † α ( ω )] (cid:1) . (20)This dissipator is of Lindblad form, which can be shown afterproving that γ αβ is positive definite and diagonalizing it.If the system Hamiltonian is diagonalized in a basis | n i as H S = P n ε n | n ih n | , then we can derive an equationof motion for the populations of the eigenstates ρ S ( n, t ) = h n | ρ S ( t ) | n i as dρ S ( n, t ) dt = X n ′ S ( n, n ′ ) ρ S ( n, t ) − S ( n ′ , n ) ρ S ( n ′ , t ) , (21) S ( n, n ′ ) = P αβ γ α ( ε ′ n − ε n ) h n ′ | A α | n ih n | A β | n ′ i being thetransition rates obtained from Fermi’s golden rule [2]. Equa-tion (21) is known as the Pauli master equation . IV. EXAMPLES OF MARKOVIAN MASTER EQUATIONSIN QUANTUM TRANSPORTA. Pauli master equation for electron-phonon interaction
An example of the Pauli master equation in the treatment ofelectron-phonon interaction in devices is the work of Fischetti[13, 25]. He has shown that, in the Born-Markov approxi-mation and the van Hove limit (time tends to infinity whilethe coupling strength tends to zero, so that interaction squaredtimes time remains constant and nonzero during the limitingprocedure)[17], the master equation for the fermionic activeregion will include the exclusion principle, thus generally be-coming non-linear for high population of states. Scatteringstates | µ i that diagonalize the single-electron Hamiltonian in the active region can be obtained from the solution of the cou-pled Schr¨odinger and Poisson equations with open boundaryconditions. In order to accurately compute spatially resolvedquantities, such as charge density and potential, in the numer-ical implementation, an appropriately dense set of scatteringstates is obtained through a mapping onto standing-wave-typesolutions (details can be found in [25]).In the work, the active region – contact interaction is treatedthrough a boundary injection/collection term that acts as asource to the equation. The Pauli master equation in the basisof scattering states reads ∂ρ S ( µ,t ) ∂t = P λ S ( µ, λ ) ρ S ( µ, t )[1 − ρ S ( λ, t )] (22) − S ( λ, µ ) ρ S ( λ, t )[1 − ρ S ( µ, t )] + (cid:16) ∂ρ S ( µ,t ) ∂t (cid:17) con . , where the source term for contact j is given by ∂ρ ( j ) S ( µ, t ) ∂t ! con . ∼ υ ⊥ ( ~k µ,j )[ f ( j ) ( ~k µ,j ) − ρ ( j ) S ( µ, t )] . (23)Here, f ( j ) ( ~k µ,j ) is the distribution function in contact j and υ ⊥ ( ~k µ,j ) is the perpendicular component of velocity associ-ated with state µ and normal to the active region/contact j boundary. Figure 1 shows a comparison between the Paulimaster equation and ensemble Monte Carlo simulation of asilicon nin diode.There is a concern that the Pauli master equation does notconserve current outside of the steady state. It has been shownthat current is conserved as long as coupling to the contacts islocal [26]. B. Markovian equations for system-environment coupling
One of the early contributions aimed specifically at thetreatment of transport in electronic systems via master equa-tions was the paper by Gurvitz and Prager [11]. In their work,the approximation of high bias has enabled the Markov ap-proximation. They discuss resonant transfer in mesoscopicdevices, focusing on resonant states as the only relevant eigen-states of the electronic Hamiltonian in the systems of interest.The resonant level model is commonly adopted [22, 27, 28].The open system has two terminals and is coupled only to theleft and right reservoirs, such that the resonant levels are com-fortably inside the transport window (the range of energiesbetween the Fermi levels of the two contacts) and the densityof contact states is constant throughout. Markovian evolu-tion can be obtained in the form of the density matrix in thebasis of the resonant states, with off-diagonal terms makingit different from the phenomenological rate equations [29].Around the same time, Stoof and Nazarov [30] investigatedtime-dependent resonant tunneling via two discrete states inthe presence of resonant-frequency irradiation based on a phe-nomenological Markovian master equation for the full statis-tical operator of this two-level system.
FIG. 1. Calculated potential energy, electron charge density (toppanel), average drift velocity and average kinetic energy (bottompanel) for an nin silicon diode at 77 K biased to 0.25 V. The solidlines refer to results calculated using the master equation, the dashedlines to results obtained using a semiclassical full-band Monte Carlosimulation employing identical parameters. Reprinted with permis-sion from Ref. [13], M. V. Fischetti, Phys. Rev. B 59, 4901 (1999).(c) 1999 The American Physical Society
An enhancement to the work of Gurvitz and Prager [11]was put forth by Li et al. [31] for a system with multiple reso-nances, such as a quantum dot, that is connected to the reser-voirs via barriers through which tunneling is relatively weak.Starting from the Born approximation and working with con-ditional density matrices that correspond to a fixed number ofelectrons getting onto the dot at a given time, the authors areable to derive a Markovian equation of the Lindblad form thatdoes not require the wide-band limit [11].The work of Harbola, Esposito, and Mukamel [32] usesprojection operators (see more in Sec. V A below) to derive ahierarchy of quantum master equations for the many-body sta-tistical operators representing the system with a given numberof electrons. They show that Fock-space coherences betweenstates with different populations do not contribute to transportto second order in system-environment coupling, but coher-ences between different many-body states with the same n areappreciable.Espostio and Galperin [33] derived a time-local Markovianmaster equation for molecular transport based on the Redfieldequation, which is nonlocal in time, and supplanting it with akind of time-reversed Redfield evolution that enables a self-consistent procedure for deriving the generator.Pedersen and Wacker [34] worked in the basis of the FIG. 2. The time-dependent current calculated with the method ofPedersen and Wacker [34] (solid line) and with the time-dependentGreen function method [37] (dashed line) in response to steplikemodulation of the bias, with step height µ L . The coupling to theleft and right contacts are Γ L = Γ R = Γ / , the temperature is k B T = 0 . , and the half-width of the band is W = 30Γ .Reprinted with permission from Ref. [34], J. N. Pedersen and A.Wacker, Phys. Rev. B 72, 195330 (2005). (c) 2005 The AmericanPhysical Society full many-body Hamiltonian and derived Markovian mas-ter equations for few-level systems coupled to a continuumof lead states. The long-time evolution coincides with thenon-Markovian description based on time-dependent Green’sfunctions (see Fig. 2). The evolution they describe is nu-merically tractable and contains considerably more informa-tion than the rate equations. The approach is referred to asthe second-order von Neumann approach (2vN for short), in-dicating that the correlations between two tunneling eventsare included [35]. Based on a diagrammatic expansion, Karl-str¨om et al. recently showed the equivalence between the 2vNapproach and the resonant tunneling approximation, and dis-cussed the limitations of the technique in the calculation ofhigher order cumulants [36]. V. MICOSCOPIC DERIVATIONS OF NON-MARKOVIANEQUATIONS
It is known that the general, completely positive, non-Mar-kovian evolution of an open system that started in an uncorre-lated state (5) can be written as ρ S ( t ) = X i R i ( t ) ρ S (0) R † i ( t ) . (24)This form is usually referred to as the operator-sum represen-tation or Kraus representation [38], where R i ( t ) are the Krausoperators. Approximate Kraus maps based on physically rea-sonable assumptions have been constructed [39]. However, incontrast to Markovian evolution, where the Lindblad equation(9) specifies the required form for a generator of a completelypositive dynamical map, there are no similarly compact cri-teria to determine if an approximate non-Markovian map iscompletely positive or not. A. Nakajima-Zwanzig and time-convolutionless (TCL)projection operator techniques
A general and widely applied technique for the derivationof non-Markovian master equations up to a given order in the S − E interaction is the Nakajima-Zwanzig projection oper-ator technique [40, 41]. Commonly, terms up to the secondor fourth order in the interaction are retained, but completepositivity of the resulting master equations is generally notguaranteed.In the S + E Liouville space H , any environment density ρ E matrix generates a projection operator P whose action isgiven by P x = Tr E ( x ) ⊗ ρ E , x ∈ H . P is a projector,meaning that P = P . The range (space of images) of P isisomorphic to the system Liouville space H S . The comple-mentary projector is Q = 1 − P .By projecting the Liouville equation (1) onto the ranges of P and Q , we obtain two equations of motion i ∂∂t P ρ = PLP ρ + PLQ ρ, (25a) i ∂∂t Q ρ = QLP ρ + QLQ ρ. (25b)If the interaction Hamiltonian, in the interaction picture, is ofthe form ǫH I ( t ) , where ǫ is a unitless number characterizingthe smallness of the interaction, we can formally solve theequation for Qρ as Q ρ S ( t ) = G ( t, t ) Q ρ S ( t ) + ǫ Z tt ds G ( t, s ) QL ( s ) P ρ S ( s ) , (26)where G ( t, s ) = T exp h ǫ R ts ds ′ QL ( s ′ ) i . Substituting thisequation into (25a) above, we obtain the Nakajima-Zwanzigequation : ddt P ρ S = ǫ PL ( t ) G ( t, t ) Q ρ ( t ) + ǫ PL ( t ) P ρ S ( t )+ ǫ Z tt ds PL ( t ) G ( t, s ) QL ( s ) P ρ S ( s ) . (27)Commonly, in the case of an uncorrelated initial state (5), theinitial environment density matrix is chosen to generate theprojection operator P , which means that P ρ (0) = ρ (0) and Q ρ (0) = 0 . Alternatively, the projector may be chosen soas to annul the odd moments of the interaction Hamiltonian.The choice of P generally depends on the application in mind,and P is often assumed to be associated with the equilibriumcanonical or grand canonical statistical operator for the envi-ronment.The time-convolutionless (TCL) projection operator tech-nique , originally due to Shibata et al. [42], writes the Na-kajima-Zwanzig equation in a form that depends only on theinstantaneous ρ S ( t ) , and all the memory effects are relegatedto certain evolution operators, which opens doors to system-atic approximations, even if the operators are still quite un-wieldy and a partial trace does technically need to be taken over the equation after everything. (Partial-trace-free time-convolutionless equations of motion and the related conceptof memory dressing have been proposed in [43, 44].) Here,we quote the TCL equation in the form without the inhomo-geneity, i.e. for Q ρ (0) = 0 . ddt P ρ ( t ) = K ( t ) P ρ ( t ) , (28)where K ( t ) = ǫ PL ( t )[1 − Σ( t )] − P , (29a) Σ( t ) = ǫ Z tt ds G ( t, s ) QL ( s ) PU ( t, s ) , (29b) U ( t, s ) = T exp (cid:20) − ǫ Z ts ds ′ L ( s ) ′ (cid:21) . (29c)Obviously, there is an assumption that − Σ( t ) is invertible[43]. Upon performing a Taylor expansion of − Σ( t ) interms of ǫ , we can get a series K ( t ) = P n ǫ n K n ( t ) , where K = 0 , K = 0 , K ( t ) = R tt dt ′ PL ( t ) L ( t ′ ) P , and K ( t ) = R tt dt R tt dt R tt dt PL ( t ) L ( t ) L ( t ) L ( t ) P .If K ≈ K , the TCL equation yields the Redfield equation(14). A number of examples of TCL equations with secondand fourth order coupling can be found in [2]. Timm [45]discusses a diagrammatic expansion of time-convolutionlessequations.It is also worth noting that a projection operator techniquecan be used to derive the well-known semiconductor Blochequations [46]. VI. EXAMPLES OF NON-MARKOVIAN TRANSPORTEQUATIONS IN QUANTUM TRANSPORT
One of the early non-Markovian approaches to electrontransport in nanostructures was put forth by Bruder andSchoeller [47]. Time-dependence was introduced either byperiodic modulation of the Fermi energy or by time-dependentperturbations to the quantum states in the dot. The authors fo-cused on the effects of the Coulomb interaction in the limit oflow tunneling rates but finite level spacing.Vaz and Kyriakidis [48–50] calculated the full Redfield ten-sor in Fock space for a two-level system (Fig. 3). The authorsfind that Fock-space coherences between states with differentparticle numbers are robust and may be preserved even in thepresence of tunneling into and out of the dot. The authors alsonote that, while Redfield dynamics could potentially violatepositivity of the statistical operator, they have not observed itin practice [48].Recently, Gudmundsson and co-authors [51] used a non-Markovian transport equation to analyze time-dependenttransport in a few-mode nanowire containing a localized re-gion and focused on the effect of nontrivial geometry. Theauthors pay attention to the fact that arbitrary decisions wherethe active region ends and contacts begin lead to inconsisten-cies, and that an effective overlap between the S and E wavefunctions will yield effective interaction Hamiltonian matrix FIG. 3. Markovian and non-Markovian time evolution of populationprobabilities in a quantum dot with two transport channels and fourstates. The plots are for symmetric source and drain tunnel barriers,and varying orbital asymmetry. A 6 meV bias symmetric about theFermi energy is assumed. The two transport channels have energies ± elements. This important issue was discussed in detail byRossi [52].Zedler et al. [53] present an interesting analysis of non-Markovian versus Markovian equations in the weak couplinglimit on the example of a quantum dot coupled to contactswith a Lorentzian density of states (i.e. contacts with a finiteelectron lifetime), thereby going beyond the high bias limit,and conclude that one must be careful with non-Markovianmaster equations as they do not necessarily perform betterthan their Markovian counterparts when non-Markovian ef-fects are strong, and are not in general guaranteed to conservepositivity. The authors compare the exact solution for a sin-gle level system with dynamical coarse graining [54], non-Markovian master equation, and the Markovian master equa-tion limit [53]. FIG. 4. Time-dependent occupation probability of the single boundstate in a dot coupled to contacts with a Lorentzian density of stateswith width ǫ R . Calculation is presented for the exact solution,dynamical coarse graining (DCG), non-Markovian master equation(NMM), and Markovian master equation (MMM). Approximationsparameters are ε d = Γ R, ε d − Γ R, = Γ L = 0 . R, , where ε d , Γ R, , and Γ L are the dot energy level and the rates of tunneling intothe right and left contacts, respectively. Reprinted with permissionfrom [53], P. Zedler et al. , Phys. Rev. B 80, 045309 (2009). (c) 2009The American Physical Society A. Coarse graining over contact relaxation time
As many nanostrucures have no resonances, the work byNovakovic and Knezevic [55, 56] emphasizes the continu-ous spectrum in the open active region, with forward- andbackward-propagating scattering states, whose asymptoticforms are plane waves (a combination of injected and reflectedwaves in the incoming contact, transmitted wave in the outgo-ing one). The model interaction Hamiltonian couples eachscattering state only with the plane wave with the same wavenumber k from the injecting contact, i.e. H int = X k> ∆ k d † k c k,L + ∆ − k d †− k c − k ′ ,R + h.c. (30) c † k,L ( c k,L ) and c †− k ′ ,R ( c − k ′ ,R ) create (destroy) an electronwith a wavevector k in the left and − k ′ in the right contact, re-spectively, d k and d † k do the same for active-region states, and k ′ − k = 2 m ∗ eV / ~ ( k and k ′ are the wave numbers cor-responding to the same energy in the two contacts separatedby bias V ). The hopping coefficients ∆ k and ∆ − k are pro-portional to the current I k carried by each mode, ∆ k = I k e T k ,where T k is the transmission coefficient of mode k [56].To obtain a tractable theoretical approach, the full dynam-ics is coarse grained over the momentum-relaxation time ofthe contacts. Contact relaxation occurs on timescales of or-der − femtoseconds [57, 58], owing to fast electron-electron scattering that results in a drifted Fermi-Dirac distri-bution [59].To coarse grain, we partition the time axis into intervalsof length τ , t n = nτ , so the environment interacts with thesystem in approximately the same way during each interval [ t n , t n +1 ] [21], dρ S dt ≈ ρ S,n +1 − ρ S,n τ = K τ ρ S,n , (31)where K τ = R τ K ( t ′ ) dt ′ τ = R tn +1 tn K ( t ′ ) dt ′ τ is the averaged valueof the map’s generator over any interval [ t n , t n +1 ] ( K is resetat each t n ). If the coarse-graining time τ is short enough,then the short-time expansion of K can be used to perform thecoarse-graining [55].Each term in the short-time expansion of K turns out to bea sum of independent contributions over single-particle states,so in reality we have a multitude of two-level problems, onefor each | k i , where the two levels are a particle being in | k i (”+”) and a particle being absent from | k i (”-”). Each such2-level problem is cast on its own 4-dimensional Liouvillespace, with ρ k = (cid:0) ρ ++ k , ρ + − k , ρ − + k , ρ −− k (cid:1) T being the reducedstatistical operator that describes the occupation of | k i andevolves according to a master equation dρ k dt = K τ,k ρ k . (32)The equations for f ± k = ρ ++ ± k become df k dt = − τ ∆ k f k + τ ∆ k f Lk ( k d ( t )) , (33a) df − k dt = − τ ∆ − k f − k + τ ∆ − k f R − k ′ ( k d ( t )) . (33b)The above equations describe non-Markovian evolution, be-cause drifted Fermi-Dirac distribution functions in the con-tacts depend on time through the drift wave vector (relatedto current). As the transient progresses, the current and thecharge density in the structure change, which in turn changesthe potential profile, the scattering states available to elec-trons, the transmission coefficients, and, to a small degree,the interaction matrix elements ∆ ± k , as well as the aforemen-tioned contact distribution functions. Moreover, there may bewell-like confined states that cannot be populated by tunnelingbut only by scattering in the active region. These considera-tions have been addressed in detail in [56, 60].Figure 5 depicts the potential, charge density, and currentdensity for a single ellipsoidal valley in an nin silicon diodeat room temperature. The left and right contacts are doped to cm − , whereas the middle region is intrinsic (undoped).In the three main panels, the momentum relaxation time inthe contacts is taken to be τ =
120 fs, based on the textbookmobility values for the above doping density. The charac-teristic response time of the current is of order hundreds ofpicoseconds, three orders of magnitude greater than τ . Thetransient duration can be thought of as the inverse of a typical ∆ k τ among the k ’s participating in the current flow; shorter τ means weaker coupling and a slower transient (inset). x [nm] V [ m e V ] t=0 pst=0.48 pst=105 pst=585 pst=1185 ps x [nm] n [ m − ] t=0 pst=0.48 pst=105 pst=585 pst=1185 ps t [ps] J d e v [ A / m ] t [ps] J d e v [ a / m ] τ =120 fs τ =360 fs τ =40 fs (a)(b)(c) FIG. 5. Potential (a), charge density (b), and current density (c) inthe nin diode as a function of time upon the application of -25 mV tothe left contact. The n -type regions are doped to cm − and con-tact momentum relaxation time is τ =120 fs, as calculated from thetextbook mobility value corresponding to the contact doping density.Inset to panel (c): effect of different contact momentum relaxationtimes τ (equal to the coarse-graining times for the active-region dy-namics) on the duration of the transient. VII. CONCLUSION
Electrons in the active region of nanostructures constitutean open many-body quantum system, coupled with reservoirsof charge, as well as interacting with phonons and photons.We overviewed the basics of the open system theory, withspecial focus on the approximations that lead to Markovianand non-Markovian master equations for the reduced statisti-cal operator, and highlighted some recent applications of bothtypes of master equations in quantum transport theory andsimulation.It should be noted that this review did not discuss otherwidely applied techniques for time-dependent quantum trans-port, such as the Wigner function simulation [61, 62], non-equilibrium Green’s functions [28], time-dependent densityfunctional theory [37], Bohmian trajectories [63], or full quan-tum statistics [64], which will receive due attention in otherreviews in this special issue. Also, we did not discuss semi-conductor Bloch equations [46, 65], which are often employ-ed to address ultrafast optics in semiconductors, and whichdeserve much more space than available here.We conclude with some thoughts on the limitations of themaster equation framework, as well as potential avenues forfurther developments.
Active region/contact partitioning.
An obvious question iswhere the active region ends and the contacts begin; there isno a good answer to this question, especially for structuresthat have no resonances. In large and complex physical sys-tems it is impossible to treat all degrees of freedom quantum-mechanically, so a boundary between the quantum and theclassical (rapidly dephasing) parts has to be adopted, but aboundary should be moved until convergence is reached andthe physics no longer varies with its position [66, 67]. Rossi[52] has argued that, in the Wigner function simulations, thisseemingly arbitrary introduction of the contact/active regionboundary results in artifacts that have conceptual, rather thancomputational origin.A related issue is that the reduced statistical operator for-malism requires that we be able to write the total many-bodyFock space as a tensor product of the Fock spaces of the sys-tem and environment, and that we write down an interactionHamiltonian between the two. With S and E containing elec-trons, we can try to split the total S + E single-particle Hilbertspace into S and E subspaces spanned by specific eigenvec-tors of the position operator, then construct Fock spaces basedon these spatially separated single-particle spaces, and finallyform a tensor product of said Fock spaces. Unfortunately, thisframework artificially makes the interaction local and is nota good choice for capturing current flow that the full S + E Fock space can describe. Rossi [52] shows that consistencyrequires that the effective interaction depend on the overlapbetween contact states and active region states, where bothcontact and active region states in principle extend throughoutthe whole coordinate space.
Validity of the RWA approximation.
The usual secular orRWA approximation – assuming that the system energy lev-els are so large that the spacing between them is much greater than the system relaxation rate – works well for optical sys-tems and is amply applied in the derivations of master equa-tions for electronic transport, but may not necessarily hold. Infact, in nanostructures with a continuum of states, the spacingbetween relevantly coupled levels is small and easily smallerthan the expected system relaxation rate, especially in the caseof strong coupling with the contacts. Therefore, the oppositelimit, that of quantum Brownian motion [2, 68] may be moreapplicable in electronic systems with densely spaced systemstates strongly coupled to the environment. This is a directionin which quantum master equations may have a lot to offer toquantum transport studies [69].
Uncorrelated initial state.
Considering that, in reality, thecontacts and active region share a Fock space, once we parti-tion it into spatially-determined subspaces and if there are notunnel barriers, it is not easy to justify the approximation of anuncorrelated initial state. Taking a close look into correlatedinitial states [70] can be a very fruitful direction of research,one where a tight coupling between approaches that do notadopt contact/active region partitioning, such as TDDFT, withmaster equations would likely be necessary.
High-frequency transport.
Another direction in which themaster equation approaches can grow is to look into systemswith continua of states and realistic fermionic reservoirs, witha more complete account of intra-reservoir dephasing. Thiswork has opportunities to interface with modern experimentalwork on GHz-frequency response of nanostructures [71, 72].
Deriving single-particle techniques from statistical opera-tor nonunitary dynamics.
Capturing the entire statistical op-erator is feasible only in very small systems. With the sta-tistical operator being the “parent” concept from which sin-gle particle quantities such as the density matrix and Green’sfunctions can be derived, it is reasonable to expect that agood non-Markovian approximation for the many-body sta-tistical operator of the electronic system would come first,and from its non-unitary evolution one can further derivesingle-particle techniques [73]. An open direction of re-search is to look at single-particle kinetic approaches that orig-inate from non-Markovian approximations for the evolutionof the reduced statistical operator. Time-convolutionless non-Markovian equations, thus far underutilized in quantum trans-port theory, could enable systematic development of single-particle non-Markovian formalisms that are of a fixed order inthe interaction.
VIII. ACKNOWLEDGEMENT
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