We recall theoretical studies on transient transport through interacting mesoscopic systems. It is shown that a generalized master equation (GME) written and solved in terms of many-body states provides the suitable formal framework to capture both the effects of the Coulomb interaction and electron--photon coupling due to a surrounding single-mode cavity. We outline the derivation of this equation within the Nakajima-Zwanzig formalism and point out technical problems related to its numerical implementation for more realistic systems which can neither be described by non-interacting two-level models nor by a steady-state Markov-Lindblad equation. We first solve the GME for a lattice model and discuss the dynamics of many-body states in a two-dimensional nanowire, the dynamical onset of the current-current correlations in electrostatically coupled parallel quantum dots and transient thermoelectric properties. Secondly, we rely on a continuous model to get the Rabi oscillations of the photocurrent through a double-dot etched in a nanowire and embedded in a quantum cavity. A~many-body Markovian version of the GME for cavity-coupled systems is also presented.
In this review article, we present a non-equilibrium quantum transport theory for transient electron dynamics in nanodevices based on exact master equation derived with the path integral method in the fermion coherent-state representation. Applying the exact master equation to nanodevices, we also establish the connection of the reduced density matrix and the transient quantum transport current with the Keldysh nonequilibrium Green functions. The theory enables us to study transient quantum transport in nanostructures with back-reaction effects from the contacts, with non-Markovian dissipation…
Electrons in the active region of a nanostructure constitute an open many-body quantum system, interacting with contacts, phonons, and photons. We review the basic premises of the open system theory, focusing on the common approximations that lead to Markovian and non-Markovian master equations for the reduced statistical operator. We highlight recent progress on the use of master equations in quantum transport, and discuss the limitations and potential new directions of this approach.
Within the hierarchical quantum master equation (HQME) framework, an approach is presented, which allows a numerically exact description of nonequilibrium charge transport in nanosystems with strong electronic-vibrational coupling. The method is applied to a generic model of vibrationally coupled transport considering a broad spectrum of parameters ranging from the nonadiabatic to the adiabatic regime and including both resonant and off-resonant transport. We show that nonequilibrium effects are important in all these regimes. In particular in the off-resonant transport regime, the inelastic c…
In this paper, the exact transient quantum transport of non-interacting nanostructures is investigated in the presence of initial system-lead correlations and initial lead-lead correlations for a device system coupled to general electronic leads. The exact master equation incorporating with initial correlations is derived through the extended quantum Langevin equation. The effects of the initial correlations are manifested through the time-dependent fluctuations contained explicitly in the exact master equation. The transient transport current incorporating with initial correlations is obtaine…
We present a hierarchical quantum master equation (HQME) approach, which allows the numerically exact evaluation of higher-order current cumulants in the framework of full counting statistics for nonequilibrium charge transport in nanosystems. The novel methodology is exemplarily applied to a model of vibrationally coupled electron transport in a molecular nanojunction. We investigate the influence of cotunneling on avalanche-like transport, in particular in the nonresonant transport regime, where we find that inelastic cotunneling acts as trigger process for resonant avalanches. In this regim…
In this work a practical scheme is developed for the first-principles study of time-dependent quantum transport. The basic idea is to combine the transport master-equation with the well-known time-dependent density functional theory. The key ingredients of this paper include: (i) the partitioning-free initial condition and the consideration of the time-dependent bias voltages which base our treatment on the Runge-Gross existence theorem; (ii) the non-Markovian master equation for the reduced (many-body) central system (i.e. the device); and (iii) the construction of Kohn-Sham master equation f…
We investigate to which extent a many-body Bloch-Redfield master equation description of quantum transport is consistent with the exact generalized equilibrium conditions known as exchange fluctuation theorems. Thereby we identify a class of master equations for which this is the case. Beyond this class, we find deviations which exhibit characteristic scaling laws as functions of the dot-lead tunneling, the inter-dot tunneling, and the temperature. These deviations are accompanied by an increase of lead energy fluctuations inherent in the Bloch-Redfield equation beyond rotating-wave approximat…
Time-dependent currents in molecular junctions can be caused by structural fluctuations or interaction with external fields. In this publication, we demonstrate how the hierarchical quantum master equation approach can be used to study time-dependent transport in a molecular junction. This reduced density matrix methodology provides a numerically exact solution to the transport problem including time-dependent energy levels, molecule-lead coupling strengths and transitions between electronic states of the molecular bridge. Based on a representative model, the influence of a time-dependent mole…
Time-resolved electron transport in nano-devices is described by means of a time-nonlocal quantum master equation for the reduced density operator. Our formulation allows for arbitrary time dependences of any device or contact parameter. The quantum master equation and the related expression for the electron current through the device are derived in fourth order of the coupling to the contacts. It is shown that a consistent sum up to infinite orders induces level broadening in the device. To facilitate a numerical propagation of the equations we propose to use auxiliary density operators. An e…
We construct a particle-number(n)-resolved master equation (ME) approach under the self-consistent Born approximation (SCBA) for quantum transport through mesoscopic systems. The formulation is essentially non-Markovian and incorporates the interlay of the multi-tunneling processes and many-body correlations. The proposed n-SCBA-ME goes completely beyond the scope of the Born-Markov master equation, being applicable to transport under small bias voltage, in non-Markovian regime and with strong Coulomb correlations. For steady state, it can recover not only the exact result of noninteracting tr…
Based on an exact formulation, we present a master equation approach to transport through Majorana zero modes (MZMs). Within the master equation treatment, the occupation dynamics of the regular fermion associated with the MZMs holds a quite different picture from the BdG S-matrix scattering process, in which the "positive" and "negative" energy states are employed, while the master equation treatment does not involve them at all. Via careful analysis for the structure of the rates and the rate processes governed by the master equation, we reveal the intrinsic connection between both approache…
For quantum transport through mesoscopic system, a quantum master equation approach is developed in terms of compact expressions for the transport current and the reduced density matrix of the system. The present work is an extension of Gurvitz's approach for quantum transport and quantum measurement, namely, to finite temperature and arbitrary bias voltage. Our derivation starts from a second-order cummulant expansion of the tunneling Hamiltonian, then follows conditional average over the electrode reservoir states. As a consequence, in the usual weak tunneling regime, the established formali…
The authors apply the generalized master equation to analyze time-dependent transport through a finite quantum wire with an embedded subsystem. The parabolic quantum wire and the leads with several subbands are described by a continuous model. We use an approach originally developed for a tight-binding description selecting the relevant states for transport around the bias-window defined around the values of the chemical potential in the left and right leads in order to capture the effects of the nontrivial geometry of the system in the transport. We observe a partial current reflection as a m…
For an open quantum system, we investigate the pumped current induced by a slow modulation of control parameters on the basis of the quantum master equation and full counting statistics. We find that the average and the cumulant generating function of the pumped quantity are characterized by the geometrical Berry-phase-like quantities in the parameter space, which is associated with the generator of the master equation. From our formulation, we can discuss the geometrical pumping under the control of the chemical potentials and temperatures of reservoirs. We demonstrate the formulation by spin…
We demonstrate that with a stepwise introduction of complexity to a model of an electron system embedded in a photonic cavity and a carefully controlled stepwise truncation of the ensuing many-body space it is possible to describe the time-dependent transport of electrons through the system with a non-Markovian generalized quantum master equation. We show how this approach retains effects of an external magnetic field and the geometry of an anisotropic electronic system. The Coulomb interaction between the electrons and the full electromagnetic coupling between the electrons and the photons ar…
We discuss the use of super-fermion formalism to represent and solve quantum kinetic equations for the electron transport problem. Starting with the Lindblad master equation for the molecule connected to two metal electrodes, we convert the problem of finding the nonequilibrium steady state to the many-body problem with non-Hermitian Liouvillian in super-Fock space. We transform the Liouvillian to the normal ordered form, introduce nonequilibrium quasiparticles by a set of canonical nonunitary transformations and develop general many-body theory for the electron transport through the interacti…
We use a generalized Master equation (GME) formalism to describe the non-equilibrium time-dependent transport through a short quantum wire connected to semi-infinite biased leads. The contact strength between the leads and the wire are modulated by out-of-phase time-dependent functions which simulate a turnstile device. One lead is fixed at one end of the sample whereas the other lead has a variable placement. The system is described by a lattice model. We find that the currents in both leads depend on the placement of the second lead. In the rather small bias regime we obtain transient curren…
We present a theoretical approach to solve Markovian master equation for quantum transport with stochastic telegraph noise. Considering probabilities as functionals of a random telegraph process we use the Novikov's functional method to convert stochastic master equation to a set of deterministic differential equations. The equations are then solved in the Laplace space and the expression for the probability vector averaged over ensemble of realisations of stochastic process is obtained. We apply the theory to study manifestations of telegraph noise in transport properties of molecular junctio…
The interplay between interference effects and electron-electron interactions in electron transport through an interacting double quantum dot system is investigated using a hierarchical quantum master equation approach which becomes exact if carried to infinite order and converges well if the temperature is not too low. Decoherence due to electron-electron interactions is found to give rise to pronounced negative differential resistance, enhanced broadening of structures in current-voltage characteristics and an inversion of the electronic population. Dependence on gate voltage is shown to be …
This paper presents a comprehensive review of the wave-function approach for derivation of the number-resolved Master equations, used for description of transport and measurement in mesoscopic systems. The review contains important amendments, clarifying subtle points in derivation of the Master equations and their validity. This completes the earlier works on the subject. It is demonstrated that the derivation does not assume weak coupling with the environment and reservoirs, but needs only high bias condition. This condition is very essential for validity of the Markovian Master equations, w…
We report an observation of charge conservation breaking in a model study of electronic current noise of transport through a dissipative double quantum dot within generalized master equation formalism. We study the current noise through a double quantum dot coupled to two electronic leads in the high bias limit and a dissipative heat bath in the weak coupling limit. Our calculations are based on the solution of a Markovian generalized master equation. Zero-frequency component of the current noise calculated within the system, i.e., between the two dots, via the quantum regression theorem exhib…
A simple scheme capable of treating transport in molecular junctions in the language of many-body states is presented. An ansatz in Liouville space similar to generalized Kadanoff-Baym approximation is introduced in order to reduce exact equation-of-motion for Hubbard operator to quantum master equation (QME)-like expression. A dressing with effective Liouville space propagation similar to standard diagrammatic dressing approach is proposed. The scheme is compared to standard QME approach, and its applicability to transport calculations is discussed within numerical examples.
We derive a Markovian master equation for the single-electron density matrix, applicable to quantum cascade lasers (QCLs). The equation conserves the positivity of the density matrix, includes off-diagonal elements (coherences) as well as in-plane dynamics, and accounts for electron scattering with phonons and impurities. We use the model to simulate a terahertz-frequency QCL, and compare the results with both experiment and simulation via nonequilibrium Green's functions (NEGF). We obtain very good agreement with both experiment and NEGF when the QCL is biased for optimal lasing. For the cons…
We present a novel hierarchical quantum master equation (HQME) approach which provides a numerically exact description of nonequilibrium charge transport in nanosystems with electronic-vibrational coupling. In contrast to previous work [Phys. Rev. B $\bf{94}$, 201407 (2016)], the active vibrational degrees of freedom are treated in the reservoir subspace and are integrated out. This facilitates applications to systems with very high excitation levels, for example due to current-induced heating, while properties of the vibrational degrees of freedom, such as the excitation level and other momen…
Beyond the second-order Born approximation, we develop an improved master equation approach to quantum transport by virtue of a self-consistent Born approximation. The basic idea is replacing the free Green's function in the tunneling self-energy by an effective reduced propagator under the Born approximation. We found that the effect of this simple improvement is remarkable, for instance, it can not only recover the exact result of noninteracting transport under arbitrary voltages, but also predict the challenging nonequilibrium Kondo effect. In addition to having an elegant structure, the ap…
We give a detailed comparison of the hierarchical quantum master equation (HQME) method to a continuous-time quantum Monte Carlo (CT-QMC) approach, assessing the usability of these numerically exact schemes as impurity solvers in practical nonequilibrium calculations. We review the main characteristics of the methods and discuss the scaling of the associated numerical effort. We substantiate our discussion with explicit numerical results for the nonequilibrium transport properties of a single-site Anderson impurity. The numerical effort of the HQME scheme scales linearly with the simulation ti…
Transport of electrons through two-dimensional semiconductor structures on the nanoscale in the presence of perpendicular magnetic field depends on the interplay of geometry of the system, the leads, and the magnetic length. We use a generalized master equation (GME) formalism to describe the transport through the system without resorting to the Markov approximation. Coupling to the leads results in elastic and inelastic processes in the system that are described to a high order by the integro-differential equation of the GME formalism. Geometrical details of systems and leads leave their fing…
Master equations are increasingly popular for the simulation of time-dependent electronic transport in nanoscale devices. Several recent Markovian approaches use "extended reservoirs" - explicit degrees of freedom associated with the electrodes - distinguishing them from many previous classes of master equations. Starting from a Lindblad equation, we develop a common foundation for these approaches. Due to the incorporation of explicit electrode states, these methods do not require a large bias or even "true Markovianity" of the reservoirs. Nonetheless, their predictions are only physically re…
Based on our recent work on quantum transport [Li et al., Phys. Rev. B 71, 205304 (2005)], where the calculation of transport current by means of quantum master equation was presented, in this paper we show how an efficient calculation can be performed for the transport noise spectrum. Compared to the longstanding classical rate equation or the recently proposed quantum trajectory method, the approach presented in this paper combines their respective advantages, i.e., it enables us to tackle both the many-body Coulomb interactionand quantum coherence on equal footing and under a wide range of …
We propose a self-consistent generalized quantum master equation (GQME) to describe electron transport through molecular junctions. In a previous study [M.Esposito and M.Galperin. Phys. Rev. B 79, 205303 (2009)], we derived a time-nonlocal GQME to cure the lack of broadening effects in Redfield theory. To do so, the free evolution used in the Born-Markov approximation to close the Redfield equation was replaced by a standard Redfield evolution. In the present paper, we propose a backward Redfield evolution leading to a time-local GQME which allows for a self-consistent procedure of the GQME ge…
A quantum master equation (QME) is derived for the many-body density matrix of an open current-carrying system weakly coupled to two metal leads. The dynamics and the steady-state properties of the system for arbitrary bias are studied using projection operator techniques, which keep track of number of electrons in the system. We show that coherences between system states with different number of electrons, n, (Fock space coherences) do not contribute to the transport to second order in system-lead coupling. However, coherences between states with the same n may effect transport properties whe…
We study shot noise in tunneling current through a double quantum dot connected to two electric leads. We derive two master equations in the occupation-state basis and the eigenstate basis to describe the electron dynamics. The approach based on the occupation-state basis, despite widely used in many previous studies, is valid only when the interdot coupling strength is much smaller than the energy difference between the two dots. In contrast, the calculations using the eigenstate basis are valid for an arbitrary interdot coupling. We show that the master equation in the occupation-state basis…
The superconducting reservoir effect on the current carrying transport of a double quantum dot in Markovian regime is investigated. For this purpose, a quantum master equation (QME) at finite temperature is derived for the many-body density matrix of an open quantum system. The dynamics and the steady-state properties of the double quantum dot system for arbitrary bias are studied. We will show that how the populations and coherencies of the system states are affected by superconducting leads. The energy parameter of system contains essentially four contributions due to dots system-electrodes …
Based on our recently developed quantum transport theory in term of an exact master equation, the corresponding particle-number resolved ($n$-resolved) master equation and the related shot noise spectrum formalism covering the full frequency range are constructed. We demonstrate that the noise spectra of transport current through single quantum dot and double quantum dots show characteristic steps and/or peak-dips in different tunneling frequency regimes through tuning the applied bias voltage and/or gate voltage at low temperatures. The peak-dips crossing the tunneling resonant frequencies is…
The consistency with Onsager's theorem is examined for commonly used perturbative approaches, such as the Redfield and second-order von Neumann master equations, for thermoelectric transport through nanostructures. We study a double quantum dot, which requires coherences between states for a correct description, and we find that these perturbative approaches violate Onsager's theorem. We show that the deviations from the theorem scale with the lead-coupling strength in an order beyond the one considered systematically in the respective approach.
We study the transport properties of the Fano-Anderson model with a Lorentzian-shaped density of states in one of the electronic reservoirs. We explicitly show that the energy dependence of the density of states can cause non-Markovian effects and that the non-Markovian master equation may fail if these effects are strong. We evaluate the stationary current, the zero frequency current noise and the occupation dynamics of the resonant level by means of a quantum master equation approach within different approximation schemes and compare the results to the exact solution obtained by scattering t…
An important class of approaches to the description of electronic transport through molecules and quantum dots is based on the master equation. We discuss various formalisms for deriving a master equation and their interrelations. It is shown that the master equations derived by Wangsness, Bloch, and Redfield and by Koenig et al. are equivalent. The roles of the large-reservoir and Markov approximations are clarified. The Markov approximation is traced back to nonzero bias voltage and temperature, whereas interactions and the corresponding rapid relaxation in the leads are shown to be irreleva…
We discuss the description of full counting statistics in quantum transport with a nonMarkovian master equation. We focus on differences arising from whether charge is considered as a classical or a quantum degree of freedom. These differences manifest themselves in the inhomogeneous term of the master equation which describes initial correlations. We describe the influence on current and in particular, the finite-frequency shotnoise. We illustrate these ideas by studying transport through a quantum dot and give results that include both sequential and cotunneling processes. Importantly, the n…
Solution of master equations by fermionic-duality: Time-dependent charge and heat currents through an interacting quantum dot proximized by a superconductor
We study qubit-mediated energy transfer between two electron reservoirs by adopting a numerically-exact influence functional path-integral method. This non-perturbative technique allows us to study the system's dynamics beyond the weak coupling limit. Our simulations for the energy current indicate that perturbative-Markovian Master equation predictions significantly deviate from exact numerical results already at intermediate coupling, $πρα_{j,j'}\gtrsim 0.4$, where $ρ$ is the metal (Fermi sea) density of states, taken as a constant, and $α_{j,j'}$ is the scattering potential energy of electr…
We construct a number($n$)-resolved master equation (ME) approach under self-consistent Born approximation (SCBA) for noise spectrum calculation. The formulation is essentially non-Markovian and incorporates properly the interlay of the multi-tunneling processes and many-body correlations. We apply this approach to the challenging nonequilibrium Kondo system and predict a profound nonequilibrium Kondo signature in the shot noise spectrum. The proposed $n$-SCBA-ME scheme goes completely beyond the scope of the Born-Markovian master equation approach, in the sense of being applicable to the shot…
Thermal transport through a Coulomb-blockade quantum dot (QD) coupled to two metallic leads is studied using five different approaches to the master equation in which sequential and coutuneling terms are taken into account. In the presence of intradot Coulomb interactions, a plateau in the thermo-particle, the heat, and the energy currents is seen. The current plateau diminishes at a high thermal bias between the leads. It is shown that the Pauli, the Redfield, the Lindblad-type equation with first order tunneling rates, and first-order von-Neumann master equations give very similar thermal tr…
We study coherent transport through a double quantum dot. Its two electronic leads induce electronic matter and energy transport and a phonon reservoir contributes further energy exchanges. By treating the system-lead couplings perturbatively, whereas the coupling to vibrations is treated non-perturbatively in a polaron-transformed frame, we derive a thermodynamic consistent low-dimensional master equation. When the number of phonon modes is finite, a Markovian description is only possible when these couple symmetrically to both quantum dots. For a continuum of phonon modes however, also asymm…
QmeQ is an open-source Python package for numerical modeling of transport through quantum dot devices with strong electron-electron interactions using various approximate master equation approaches. The package provides a framework for calculating stationary particle or energy currents driven by differences in chemical potentials or temperatures between the leads which are tunnel coupled to the quantum dots. The electronic structures of the quantum dots are described by their single-particle states and the Coulomb matrix elements between the states. When transport is treated perturbatively to …
The hierarchical quantum master equation (HQME) approach is an accurate method to describe quantum transport in interacting nanosystems. It generalizes perturbative master equation approaches by including higher-order contributions as well as non-Markovian memory and allows for the systematic convergence to the numerically exact result. As the HQME method relies on a decomposition of the bath correlation function in terms of exponentials, however, its application to systems at low temperatures coupled to baths with complexer band structures has been a challenge. In this publication, we outline…
We consider electronic transport through a single-molecule junction where the molecule has a degenerate spectrum. Unlike previous transport models, and theories a rate-equations description is no longer possible, and the quantum coherences between degenerate states have to be taken into account. We present the derivation and application of a master equation that describes the system in the weak-coupling limit and give an in-depth discussion of the parameter regimes and the new phenomena due to coherent on-site dynamics.
An ab initio based theoretical approach to describe nonequilibrium many-body effects in molecular transport is developed. We introduce a basis of localized molecular orbitals and formulate the many-body model in this basis. In particular, the Hubbard-Anderson Hamiltonian is derived for single-molecule junctions with intermediate coupling to the leads. As an example we consider a benzenedithiol junction with gold electrodes. An effective few-level model is obtained, from which spectral and transport properties are computed and analyzed. Electron-electron interaction crucially affects transport …
We present a method which uses density functional theory (DFT) to treat transport through a single molecule connected to two conducting leads for the weak and intermediate coupling. This case is not accessible to standard non-equilibrium Green's function (NEGF) calculations. Our method is based on a mapping of the Hamiltonian on the molecule to a limited set of many-body eigenstates. This generates a many-body Hamiltonian with parameters obtained from ground state L(S)DA-DFT calculations. We then calculate the transport using many-body Green's function theory. We compare our results with exist…
We formulate a quantum theory of electron waiting time distributions for charge transport in nano-structures described by non-Markovian generalized master equations. We illustrate our method by calculating the waiting time distribution of electron transport through a dissipative double quantum dot, where memory effects are present due to a strongly coupled heat bath. We consider the influence of non-Markovian dephasing on the distribution of electron waiting times and discuss how spectral properties of the heat bath may be detected through measurements of the electron waiting time.
We investigate the zero and finite temperature transport properties of a quasi-two-dimensional electron gas in a GaAs/InGaAs/GaAs quantum well under a magnetic field, taking into account many-body effects via a local-field correction. We consider the surface roughness, roughness-induced piezoelectric, remote charged impurity and homogeneous background charged impurity scattering. The effects of the quantum well width, carrier density, temperature and local-field correction on resistance ratio are investigated. We also consider the dependence of the total mobility on the multiple scattering eff…
In this paper, we develop a quantum transport theory to describe photonic transport in photonic networks. The photonic networks concerned in the paper consist of all-optical circuits incorporating photonic bandgap waveguides and driven resonators. The photonic transport flowing through waveguides are entirely determined from the exact master equation of the driven resonators. The master equation of the driven resonators is obtained by explicitly eliminating all the waveguide degrees of freedom while the back-reactions between resonators and waveguides are fully taken into account. The relation…
We present a formalism for calculating the non-symmetrized quantum current noise within the Born-Markov approximation for the master equation. The formalism is particularly well suited to obtaining the current noise for quantum transport in mesoscopic devices such as a superconducting single electron transistor (SSET). As an example of the method, we obtain explicit results for the double Josephson-quasiparticle (DJQP) resonance in a SSET. Our calculations reveal the asymmetries that develop in the current noise as well as clarifying the behavior at high frequencies. Our findings are consisten…
The simulation of quantum transport in a realistic, many-particle system is a nontrivial problem with no quantitatively satisfactory solution. While real-time propagation has the potential to overcome the shortcomings of conventional transport methods, this approach is prone to finite size effects that are associated with modeling an open system on a closed spatial domain. Using a master equation framework, we exploit an equivalence between the superoperators coupling an open system to external particle reservoirs and non-Hermitian terms defined at the periphery of a quantum device. By taking …
Quantum master equations are common tools to describe the dynamics of many-body systems open to an environment. Due to the interaction with the latter, even for the case of non-interacting electrons, the computational cost to solve these equations increases exponentially with the particle number. We propose a simple scheme, that allows to study the dynamics of $N$ non-interacting electrons taking into account both dissipation effects and Fermi statistics, with a computational cost that scales linearly with $N$. Our method is based on a mapping of the many-body system to a specific set of effec…
