Master equations in transport statistics

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Master equations in transport statistics

Success and failure of Non-Markovian and higher order corrections

vorgelegt von Diplom-Physiker

Philipp Zedler

aus Darmstadt

Von der Fakult¨at II - Mathematik und Naturwissenschaften der Technischen Universit¨at Berlin

zur Erlangung des akademischen Grades Doktor der Naturwissenschaften

Dr. rer. nat. genehmigte Dissertation

Promotionsausschuss:

Vorsitzende: Prof. Dr. rer. nat. Birgit Kanngießer 1. Gutachter: Prof. Dr. rer. nat. Tobias Brandes 2. Gutachter: Prof. Dr. rer. nat. Andreas Wacker Tag der wissenschaftlichen Aussprache: 4. Juli 2011

Berlin 2011 D 83

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Zusammenfassung

Wie lässt sich kohärentes Tunneln von Elektronen gut mit Mastergleichungen beschrei-ben? Diese Frage stellt sich, weil Mastergleichungen ursprünglich für sequentielles, also klassisches Tunneln konzipiert wurden. Die große Stärke von Mastergleichungen liegt darin, dass sich mit ihnen sehr gut und einfach Mehrteilchen-Wechselwirkungen inner-halb eines mesoskopischen Systems beschreiben lassen. Diese Stärke beruht allerdings darauf, dass die Kopplung zur Außenwelt nur näherungsweise berücksichtigt wird, so dass eine perfekte Beschreibung von kohärenten Tunnelprozessen in das System hinein oder aus dem System heraus im Allgemeinen ausgeschlossen ist. Allerdings wurden Methoden entwickelt, mit denen sich systematisch Korrekturen zur gewöhnlichen Mas-tergleichung berechnen lassen, etwa die Realzeit-Diagrammatik. Normalerweise sollten kohärente Tunnelprozesse deutlich besser beschrieben werden, wenn höhere Korrekturen berücksichtigt werden.

In dieser Arbeit beschäftigen wir uns mit einer bestimmten Methode, um solche Ko-rrekturen selbstkonsistent auszurechnen. Sie wurde kürzlich von Jonas Pedersen und Andreas Wacker entwickelt. Um die Qualität der Korrekturen beurteilen zu können, vergleichen wir alle Ergebnisse mit der exakten Lösung eines einfachen Modells in nicht-trivialen Realisierungen. Um Hinweise zu erhalten, ob auch Korrelationen zwischen Elektronen gut beschrieben werden, untersuchen wir Größen aus dem Gebiet der vollen Zählstatistik (Full Counting Statistics).

Eine weitere Möglichkeit, Mastergleichungen zu verbessern, besteht darin, Ged¨ achnis-effekte (auch: nicht markoffsche Effekte) zu berücksichtigen. Diese werden bei der Her-leitung der Mastergleichung erzeugt und häufig hinterher weggelassen. Wir verwenden neue Formalismen, die die Berechnung nicht markoffscher Korrekturen vereinfachen und testen die Ergebnisse anhand der exakten Lösung.

Außer den Näherungen stellen wir auch mehrere exakte Lösungen unseres Testmodells vor (ein resonantes Niveau mit Zu- und Abfluss von Elektronen). Mit diesen berech-nen wir verschiedene physikalische Größen bis hin zu frequenzabhängigen Kumulanten, bei denen sich besonders gut der Übergang von rein sequentiellem zu rein kohärentem Tunneln veranschaulichen lässt. Außerdem geben wir ein Beispiel, wie sich Wechsel-wirkungseffekte in die exakten Lösungen einbauen lassen.

Die Arbeit schließt mit einer Beurteilung der untersuchten Lösungen und Näherungen, insbesondere mit der Empfehlung, bei kleinen Spannungen alle behandelten Korrekturen zu berücksichtigen, aber bei ungewöhnlichen spektralen Eigenschaften generell mit Mas-tergleichungen vorsichtig zu sein.

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Abstract

How can we gain a good description of coherent tunneling of electrons? This question arises as originally the master equation was made to describe sequential tunneling which is classical. The strength of master equations lies in the fact that they give an easy and good description of many particle effects within a mesoscopic system. However, this stength emerges because the coupling to the outside world is only approximately incorporated. Thus, a perfect description of tunneling processes into and out of the system is in general excluded. But methods have been developed that give systematic corrections to usual master equations, for example real time diagrammatics. Usually, when higher order corrections are incorporated this should lead to a remarkably better description of coherent tunneling processes.

In this thesis we are dealing with a certain method to evaluate such corrections in a self-consistent way which was recently developed by Jonas Pedersen and Andreas Wacker. To be able to make statements about its accuracy we compare all results to the exact solution of a simple model in non-trivial realizations. To get a hint if also correlations will be described well, we investigate quantities form the field of Full Counting Statistics. A further possibility to improve master equations consists of including memory effects (also: Non-Markovian effects). They are created in the derivation of the master equation and often neglected afterwards. We will use a new formulation which simplifies the evaluation of Non-Markovian corrections and we test the results using the exact solution. Apart from the approximations we also present several exact solutions of our test model (a resonant level with a source and a drain of electrons). We use them to eval-uate various physical quantities up to the frequency-dependent cumulants where it is especially easy to visualize the transition form purely sequential to purely coherent tun-neling. We furthermore give an example of how interaction effects can be built into our exact solutions.

The work concludes with statements about the solutions and approximations. Partic-ularly, we suggest that one should incorporate all corrections considered here for systems with small bias, but that if the spectral properties are unusual one should be careful with master equations.

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Acknowledgments

I would like to take the opportunity and say thank you to a few people who have con-tributed their ideas to this work.

My supervisor Tobias Brandes has created a paradisian environment for research, so I could develop and follow my own ideas and discuss them at large with him and many collegues, visitors and collaborators.

Tom´aˇs Novotn´y had the idea that I should generalize the second order von-Neumann approach to the Full Counting Statistics. He always had clear suggestions and opinions and I enjoyed the collaboration with him greatly, as well.

Gernot Schaller, Gerold Kießlich and Clive Emary have collaborated with me on Non-Markovian effects and Dynamical Coarse Graining and always had ideas when I was stuck during my reseach.

Robert Hussein and Anja Metelmann have collaborated with me on Green’s functions and phonons and have helped me in many intense discussions to better understand our common problems.

Jin-Jun Liang, Marten Richter and Carsten Weber have shared their office with me for years and helped me in various ways with many everyday problems that appeared in my calculations.

With many other members of our institute I had inspiring discussions, especially with Christina P¨oltl, Carlos L´opez-Mon´ıs, David Marcos, Gerardo Paz, Martin Kliesch, Malte Vogl, Victor Bastidas, Mathias Hayn, Mario Schoth, Yin-Tsan Tang, Georg Engelhardt, and Emely Wiegand.

I spent a week in Lund where people had a lot of time and interest to discuss my and their work, especially Andreas Wacker who is my co-supervisor, Olov Karlstr¨om, and Peter Samuelson.

I also had fruitful discussions with our numberless visitors especially with Rafael Sanchez, Ramon Aguado, Shmuel Gurvitz, Christian Flindt, John. H. Reina, Federica Haupt and with Daniel Urban.

I should not forget to mention Peter Kopietz, Peyman Pirooznia, Francesca Sauli, Mat-hias Geueke, Friedrich Heß, Jan Lamprecht, Claudia Bl¨oser and Kirsten Vogeler who are the most important collaborators and friends whom I have met during my studies in Frankfurt am Main. Crucial for my decision to become a physicist were my two most important physics teachers Mrs. Frank and Mr. Spengler as well as discussions with my parents.

It is very strange, but the music by Clueso always helped me to proceed in my calcula-tions.

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1. Introduction

This work is about two different topics. The original purpose was to make statements about the accuracy of approximations that were recently suggested as improvements to the usual master equation [1, 2]. In order to obtain reliable results the approximations had to be contrasted with an exact solution. The author took pleasure in the seek for exact solutions, so the second topic of this work became his test model which is the single resonant level model in non-trivial realizations. Various solutions and approximations to this model are one important golden thread in this work. They are numbered asten approaches _{(five approximations and five exact solutions).}

Master equations are an approved and successful tool to describe open quantum sys-tems [3]. They were first developed in quantum optics [4] and later adopted to mesoscopic transport [5]. They are made for systems where a tiny region requires a quantum me-chanical treatment while its coupling to the outside world can be described classically – electrons will enter and exit the tiny region sequentially, not in a coherent superposition. A complementary method is scattering theory [6]. It is made for systems where our tiny region acts on crossing electrons as an external potential and interactions between particles are not important. For the intermediate regime where transport is coherent and interacting at the same time there is no general reliable standard approach (yet). It is possible to extend the scattering-like approaches in a way that interactions can be included. This works especially well for electron-phonon interactions [7]. Here we choose the other possible way where one extends master equations such that coherent effects will be taken into account.

Master equations can be derived with a perturbative treatment of the coupling between the system and its environment. It is often sufficient to incorporate this coupling only to lowest order. Of course, there are systems where more accuracy is necessary and this fact can be shown in experiments [8, 9, 10]. To explain why more accuracy is needed one often constructs pictures where electrons perform cotunneling which means that they do not tunnel one by one but instead assist each other during the tunneling process. A theory for a systematic improvement of the master equation is given by the formalism of real time diagrams [11] which can alternatively be formulated as the recently developed Liouvillian Perturbation Theory [12, 13]. In this work we are going to investigate another technique to evaluate higher order tunneling processes. This is the 2nd order von-Neumann approach [1] which can be related to the other two formalisms. A widely different way of improving master equations is the inclusion of Non-Markovian effects which result from a memory in the system. That master equations are often for-mulated without memory has two good reasons: Firstly, a calculation with simple rates is much easier and often covers everything. Secondly, the results of Non-Markovian cal-culations can exhibit strange features like negative probabilities while after a Markovian

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approximation (and an additional secular approximation) one can feel certain that all results will be physical to the effect that the probability is conserved [14]. However, the shape of a Non-Markovian master equation is much more flexible that that of a Marko-vian master equation and the derivation contains less approximations. So we can expect that Non-Markovian corrections will often improve the results. It is not easy to observe Non-Markovian corrections, because they have no effect on the stationary expectation values of the occupations or the current. That is why recently there was developed a method to systematically evaluate Non-Markovian corrections for quantities where they matter: to stationary cumulants in the Full Counting Statistics [2].

The way how approximations are derived gives strong indications of where they should be valid and where not. Even so, one should make tests to check whether the approx-imations perform as it was expected. The single resonant level model [15, 16] is ideal for such tests. It has a whole series of exact solutions among which one can choose to one’s own taste. The Hamiltonian itself is simple, but we can evaluate complicated quantities like the Full Counting Statistics and we can choose parameters which make the solutions non-trivial. Here we choose two different non-trivial realizations. Either we incorporate the full dependency on the electronic occupations in the source and the drain or we choose structured tunneling rates which are a function of energy. The most natural approach to electronic transport in this model is scattering theory. But here we are interested in what happens when we choose the “wrong” approach - the master equation, which could easily handle interactions within the system but can be insufficient for coherent tunneling processes.

When we apply the “right” approach to the single resonant level model we obtain a starting point for the perturbative extension to interacting models which is typically done with Keldysh Green’s functions [17]. This is why at the end of this work we shift our attention to the formulation of more complicated quantities in terms of Keldysh Green’s functions. For the actual inclusion of interactions we give a little example of how we can treat electron-phonon interactions by using linear response theory.

This work is organized as follows: In chapter 2 we introduce the single resonant level model and exact solutions with a few methods: with the retarded and advanced Green’s function, the time-dependent Fermion operators, an exact master equation and with scattering theory. In parallel we introduce interesting quantities like occupation, current and zero frequency noise. In section 2.5 we show how the model can be extended by including electron-phonon interactions in the framework of linear response theory. In chapter 3 we introduce the concept of the Full Counting Statistics and investigate Non-Markovian corrections in first order master equations. In chapter 4 we generalize the second order von-Neumann approach to the Full Counting Statistics and investigate higher order corrections. In chapter 5 we introduce the formalism of Keldysh Green’s functions and calculate the Frequency-dependent noise and skewness.

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2. The single resonant level model and

exact solutions

2.1. Introduction to the single resonant level model

We give a brief historical background before introducing the Hamiltonian and the thermodynamic concept. The retarded and advanced Green’s func-tion are evaluated as an established standard tool which can easily be obtained for the considered model. They give access to the propagators and make the self-energy appear in a natural way. The tunneling rates are introduced in the context of the self-energy and the thermodynamic limit. We argue that when the tunneling rates are smooth functions of the energy this implies the treatment of the leads in the thermodynamic limit.

2.1.1. The Hamiltonian

In 1961 Philip W. Anderson was seeking for conditions required for the stability of localized magnetic moments like those of iron, cobalt or nickel dissolved in non-magnetic metals [15]. He found that a proper treatment of Coulomb interactions on the d-shells of the localized atoms was essential. Hence he did not use scattering at the potential of the impurities, but instead introduced a separate impurity’s d-orbital wave function, which enabled him to take special care of the Coulomb interactions at the impurity. This extra level is only easy to handle if one assumes that it is orthogonal to the Wannier functions of the free electrons. Anderson does not see a big problem in this assumption, for example because the plane waves needed to construct the localized level are in general far from the Fermi level and can therefore be neglected in the calculation. Nevertheless he points out that there are also ways to make the impurity wave function orthogonal to all Wannier functions [18]. Anderson chooses the “physically unrealistic case of a single nondegenerate level”, because this allows him to show the principles and can easily be generalized to more levels. His Hamiltonian, which he formulates in a compact second quantisation notation, is now known as the Anderson model.

At the same time Ugo Fano worked on the effect of autoionization on excitation spectra. Therefore he coupled a discrete level to one or several continua which can be regarded as a generalized Anderson model without Coulomb interaction [16]. This model is exactly solvable and Fano provided the solution with a method similar to what we will use here to evaluate the retarded and advanced Green’s functions. Fano generalized

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Anderson’s model in two ways. On the one hand he had not only one but several discrete levels and on the other he added several different continua. An Anderson model with several different continua, but without Coulomb interaction is sometimes referred to as the Fano Anderson model. We will call it the single resonant level model.

In this work we deal with a localized level connected to two leads that are again connected to a source of a constant voltage, which we treat as two particle reservoirs that create differing chemical potentials in the two leads. We distinguish the leads by saying that one is on the left and the other one is on the right side. The Hamiltonian is

H = ǫdd†d+ X kα ǫkαc†kαckα+ X kα tkαd†ckα+t∗kαc†kαd . (2.1)

The first part associates the occupation of the single resonant level with an energy ǫd. The letter d is choosen because the level can be experimentally realized as a quantum dot, so we will refer to the corresponding state as the state of the dot and use the symbolsd†, dfor the Fermion operators that create and anihilate electrons on the dot. The second part in Eq. (2.1) describes a free gas of electrons with momentumk and a dispersionǫkα. Electrons are created and anihilated with the operatorsc†kαandckα. The indexα distinguishes the left lead (α=L) from the right lead (α=R). The third part in Eq. (2.1) describes hopping between the dot and the leads with transition amplitudes tkα.

We initialize the system by coupling each lead to a thermodynamic bath with the inverse temperatureβα and chemical potential µα. At time t0 we release this coupling in order to avoide using thermodynamic concepts in a non-equilibrium system which we create by switching on a coupling between the leads and the dot state as described in the Hamiltonian. A sketch of this is given in Fig. 2.1.

2.1.2. Approach One: The retarded and advanced Green’s functions

A way to find the Eigenenergies of a Hamiltonian is to determine its retarded and advanced Green’s functions. They are functions of energy and their poles are located at the Eigenenergies of the Hamiltonian. For several purposes (like easier usage of tools from complex analysis or the connection with the propagators) we would like to shift the poles away from the real axis. Therefore we introduce a tiny shiftδ and define the retarded (R) and advancedA Green’s function as the resolvent

GR/A(E) := lim

δ→0(E− H ±iδ)

−1_. _(2.2)

We are going to evaluate the single particle Green’s functions for the single resonant level model. The same calculation for the same model can also be found in Ref. [19]. We consider the Hamiltonian as a huge matrix acting on the single particle states|kαi

with an electron of momentumkin leadα and on the single particle state _|d_iwhere the electron is on the dot. The single particle Green’s functions must obey the equation

lim

δ→0(E− H ±iδ)G

R/A₍_E_{) =}

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2.1. Introduction to the single resonant level model ǫd ǫd ǫkL ǫkL ǫkR ǫkR βL, µL βL, µL βR, µR βR, µR tkL tkR t < t0 t > t0

left heat bath left lead center right lead right heat bath

Figure 2.1.: Initialization of the system for times t < t0 and internal dynamics for times t > t0. Before t =t0 the system is in equilibrium such that we can safely describe it with results from Thermodynamics (with inverse temperature βα and chemical potentials µα). After t=t0 we consider the internal non-equilibrium dynamics of an isolated system and deny ourselves the usage of thermodynamic concepts in non-equilibrium.

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We will use a common convention and omit writing down the limit explicitly while replacingδ by 0+. It is important that 0+>0 wherever this shifts poles away from the real axis. The matrix elements will be denoted with indices as

Gµ,νR/A(E) =hµ|GR/A(E)|νi with|µi,|νi ∈ |di,|kαi. (2.4) We evaluate them by using the action of the creation and anihilation operators. Taking the element_hd_{| · · · |}d_i, we obtain

E₋ǫd±i0+ Gd,dR/A(E)− X k,α tkαGkα,d(E) = 1. (2.5) The matrix elementshkα| · · · |di lead to

(E−ǫkα±i0+)Gkα,dR/A(E)−t∗kαG R/A

d,d (E) = 0, (2.6) and with this and Eq. (2.5) we find

Gd,dR/A(E) =

1

E₋ǫd−ΣR/A(E)

(2.7) where we have introduced a self-energy that describes the difference between the full and the free Green’s function by shifting the energyǫd:

ΣR/A(E) :=X k,α

|tkα|2 E−ǫkα±i0+

. (2.8)

By inserting Eq. (2.7) into Eq. (2.6) we can write

Gkα,dR/A(E) = t∗_kα E−ǫkα±i0+ 1 E₋ǫd−ΣR/A(E) . (2.9)

We may reverse Eq. (2.3), and thus write GR/A₍_E₎₍_E_{− H ±}_i₀+_{) =}

1. The matrix element_hd_|..._|kα_i produces Gd,kαR/A(E) = tkα E₋ǫkα±i0+ 1 E₋ǫd−ΣR/A(E) (2.10) and with matrix element_hkα_|..._|k′α′_i we obtain

Gkα,kR/A′_α′(E−ǫk′_α′ ±i0+)−GR/A kα,dtk′α′ = δk,k′δα,α′, (2.11) and thus GR/Akα,k′_α′ = δkk′δ_α,α′ E−ǫk′_α′±i0+ + t∗_kα E−ǫkα±i0+ tk′_α′ E−ǫk′_α′±i0+ 1 E₋ǫd−ΣR/A(ω) . (2.12) In the textbook [20] there is an attempt to obtain these Green’s functions without using an imaginary part and with a diagonalization. However, the formalism shown there is

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2.1. Introduction to the single resonant level model

fragile and can easily lead to contradictions. So here we have chosen the more standard and clean way and have includes imaginary parts.

As the Hamiltonian is constant, the propagator for positive times can be simply written as

U(t) =e−iHtθ(t). (2.13) Its Fourier transform is

Z

dteiEt_U(t) = Z ∞

0

dtei(E−H)t. (2.14) This integral can only be performed, if the integrand vanishes for large values of t. To assure ourselves of this, we put the termi0+ _{into the exponent such that}

Z

dteiEtU(t) = Z ∞

0

dtei(E−H+i0+)t = −1

i (E− H+i0

+₎−1 ₌ _i_GR₍_E₎_. _(2.15) That means that by reversing the Fourier transform we can obtain the propagators from the Green’s functions by using the relations

U(t) = i Z _dE 2πe −iEt_GR₍_E₎_, U†(t) = −i Z dE 2πe −iEt_GA₍_E₎_. _(2.16) The retarded and advanced Green’s functions can be evaluated in a very similar way for several levels in series. For two levels this is done in Ref. [21] and [22, 23] and in App. A.1. An expression for an arbitrary number of levels can be found in Ref. [24]. We can make use of the retarded and advanced Green’s functions when we want to construct propagators for time-dependent quantities like in section 3.4. We can also use them to evaluate the spectral function. However, as we deal with a non-equilibrium system the applications of the spectral function are limited. Only with special theorems like the Meir-Wingreen formula [25] one can use it to evaluate quantities like the current. 2.1.3. Tunneling rates

We define the tunneling rate Γα(E) as the probability according to Fermi’s golden rule, that a particle performs a transition between the central level and the lead at side α, with energy E:

Γα(E) := 2π X

k

|tkα|2δ(E−ǫkα), α∈ {L,R}. (2.17) It is handy to introduce the abbreviation

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The tunneling rate is twice the imaginary part of the advanced self energy (Eq. (2.8)): Γ(E) = 2 ImΣA(E)=iΣR(E)₋ΣA(E). (2.19) This can be seen by using the identity lim

δ→0 1

x+iδ = P1x −πiδ(x) where P denotes the principal value. We can use the identity to decompose the self-energy ΣR/A(E) into its real and imaginary part:

ΣR/A(E) =X k,α P |tkα| 2 E₋ǫ_kα ∓iπ X k,α |tkα|2δ(E−ǫkα). (2.20) We know that the self energy’s imaginary part correspondence to the lifetime while its real part induces a level shift [26, 17]. The real part is usually given the name

Λ(E) = ReΣA(E). (2.21) We give separate definitions of the two parts of this level shift function which depend on the two distinct reservoirs as

Λα(E) = X k P |tkα| 2 E₋ǫkα (2.22) such that Λ(E) = ΛL(E) + ΛR(E). As the self-energy should be an analytic function that vanishes for large modulus ofE, it is already determined as soon as either real or

imaginary part are given. This can be expressed by the Kramers-Kronig relation [20]: Λα(E) = 1 2 Z ∞ −∞ dE′ π P Γα(E) E−E′, Γα(E) = −2 Z ∞ −∞ dE′ π P Λα(E) E₋E′. (2.23)

For the sake of simplicity one usually assumes constant tunneling rates (confer the next section). When the tunneling rates are constant, the level shift function vanishes, because the integrand in equation (2.23) becomes an odd function.

In this work we are not interested in leads of finite size where the tunneling rate would consist of a few delta peaks, but we prefer to describe the reservoirs in the thermodynamic limit. In this limit for a system of volume V and dimension D we can perform the replacement _V1 P_k _→R _(2π)dkD (confer the book [27]) and thus express the tunneling rate

as Γα(E) = V Z _dk (2π)D−1|tkα| 2_δ₍_E₋_ǫ kα). (2.24) This expression makes it visible that to obtain finite values of Γα(E) we need to perform the thermodynamic limit V _{→ ∞} in a way that _|t_kα_|2 _∼ 1

V. That particularly means that the probability that the single state |kαi will become occupied due to a tunneling event becomes exactly zero. So the occupations in the leads will never change (only coherences will).

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2.2. Approach Two: _{Time-dependent Fermion operators}

2.2. Approach Two: Time-dependent Fermion operators

To determine the time dependence of the creation and anihilation operators with Heisenberg’s equation of motion is probably that solution of the sin-gle resonant level model that needs the least pre-knowledge. With constant tunneling rates the differential equations can easily be solved with a Laplace transform. We use the results to evaluate the dot occupation and the cur-rent. The method is simple and flexible for the given model with constant tunneling rates, but there is no approved way how to apply it to models with interactions.

2.2.1. Solution with constant tunneling rates

Without loss of generality we can choose the energy scale such that ǫd = 0, for the creators this results in the following system of differential equations:

d dtd † ₌ _i_[ H, d†] = iX k,α t∗_kαc†_kα (2.25) d dtc † kα = i[H, c † kα] = iǫkαc † kα+itkαd†. (2.26) We Laplace-transform these two equations and gain

zdˆ†(z)₋d†(0) = iX k,α

t∗_kαˆc†_kα(z), (2.27) zˆc†_kα(z)₋c†_kα(0) = iǫkαcˆ†_kα(z) +itkαdˆ†(z). (2.28) We resolve Eq. (2.28) to ˆc†_kα(z) and insert the result in Eq. (2.27). This gives

 z+X k,α |tkα|2 z−iǫkα  _dˆ†₍_z_{) =} _d†_{(0) +}X k,α it∗_kα z−iǫkα c†_kα(0). (2.29)

Now we restrict ourselves to the case of constant tunneling rates. This limit is used in nearly all calculations with master equations. It is very successful, also in describing a lot of today’s experiments, for example Ref. [28] and Ref. [29]. The limit is justified if the energy scale responsible for the strength of the tunneling is much larger than the energy scale for structures in the tunneling rates. The assumption of constant tunneling rates leads to an extreme simplification. The Laplace space version of the self energy

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becomes X k,α |tkα|2 z₋iǫkα = Z dEX k,α δ(E₋ǫkα) | tkα|2 z₋iE = Γ Z _dE 2π 1 (z₋iE) = Γ Z _dE 2π z (z−iE)(z+iE) + Γ Z _dE 2π iE (z−iE)(z+iE)

⇓ The second integral vanishes,

⇓ because the integrand is an odd function.

= Γ/2 (2.30)

From this together with equation (2.26) and (2.28) we obtain the dot operator ˆ d†(z) = 1 z+ Γ/2d †_{(0) +}X k,α it∗_kα 1 z₋iǫkα 1 z+ Γ/2c † kα(0). (2.31) We insert this into Eq. (2.28) to obtain the lead operator

ˆ c†_kα(z) = 1 z₋iǫkα c†_kα(0) + itkα (z₋iǫkα)(z+ Γ/2) d†(0) −tkα X k′_,α′ t∗_k′_α′ (z−iǫkα)(z−iǫk′_α′)(z+ Γ/2) c†_k′_α′(0). (2.32)

For the transformation back to time-dependent quantities we have to collect all the corresponding residues. For the dot creator this is straight forward and yields

d†(t) = e−Γ/2td†(0) +X k,α t∗_kαe iǫkαt₋_e−Γ/2t ǫkα−iΓ/2 c†_kα(0). (2.33) When we perform the Laplace transform backwards on Eq. (2.32), we must be careful, because the term under the sum, which usually has three poles of first order, will get a second order pole, whenǫka=ǫk′_b. Thus we have to perform a seperate calculation for

this case. Briefly the result: c†_kα(t) = tkα eiǫkαt₋_e−Γ/2t ǫkα−iΓ/2 d†(0) + ( eiǫkαt 1 +it |tkα| 2 ǫkα−iΓ/2 − |tkα|2 eiǫkαt₋_e−Γ/2t (ǫkα−iΓ/2)2 ) c†_kα(0) +tkα X k′_,α′ ǫk′α′6=ǫkα t∗_k′_α′ " eiǫkαt (ǫkα−iΓ/2)(ǫkα−ǫk′_α′) + e iǫk′α′t (ǫk′_α′−iΓ/2)(ǫ_k′_α′ −ǫ_kα) + e −Γ/2t (ǫ_k′_α′−iΓ/2)(ǫ_kα−iΓ/2) # c†_k′_α′(0) (2.34)

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For the anihilation operators we can simply take the adjoints of the equations (2.33) and (2.34). In App A.1 there is a sketch how to determine the dot operators for two resonant levels. In Ref. [30] the author and co-workers have evaluated the Fermion operators for the single resonant level with two Lorentzian tunneling rates.

2.2.2. Occupation of the quantum dot

We start with the simplest task and evaluate the stationary occupation. For large times all terms with e−Γ/2t will have vanished as Γ>0. This simplifies the dot operator to

lim t→∞d †₍_t_{) =} X k,α t∗_kα e iǫkαt ǫkα−iΓ/2 c†_kα(0). (2.35)

We said in section 2.1.1 that we initialize our system with Fermi distributions in the leads, so_hc†_kαckαi=fkα and for different quantum numberskα6=k′α′ we have hc†_kαck′_α′i= 0.

With that pre-condition we can glue together the time-dependent Fermion operators and get the stationary occupation

¯ nd := lim t→∞hd †₍_t₎_d₍_t₎_i = X kα |tkα|2 ǫ2 kα+ Γ2/4 fkα = Z dE 2π ΓLfL(E) + ΓRfR(E) E2_{+ Γ}2_/₄ . (2.36) At infinite bias this is simply

¯

nd = ΓL/Γ. (2.37)

At finite bias and zero temperature it becomes ¯ n_d = 1 2 + ΓL πΓarctan µL Γ/2 + ΓR πΓarctan µR Γ/2. (2.38) At finite temperature we can use the result from App. A.6.2 where we show that the Fermi function can be expressed with the Digamma functionψand how we can use that to solve the integral. We find

¯ nd = 1 2 + X α Γα Γ 1 2πi ψ 1 2+βα iΓ/2−µα 2πi +ψ 1 2 −βα −iΓ/2−µα 2πi . (2.39) In figure 2.2.a) we show how the occupation increases with the chemical potential in the right lead. In figure 2.2.b) we show that high temperatures diminish the influence of the chemical potentials.

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a) Asymetric bias changes b) Temperature changes 0.2 0.4 0.6 0.8 -10 -5 0 5 10 V /ΓL ¯ nd 0.3 0.4 0.5 0.1 1 10 100 1000 T /ΓL ¯ nd

c) Start with empty dot d) Start in “steady state”

0 0.1 0.2 0.3 0.4 0.5 0 1 2 3 4 5 t∗ΓL nd ( t ) 0.43 0.44 0.45 0.46 0.47 0 1 2 3 4 5 t_∗ΓL nd ( t )

Figure 2.2.: The stationary dot occupation ¯ndas a function of an asymmetric voltageV and of temperatureT, and the time-dependent dot occupationnd(t) starting with an empty dot and with its stationary value. Parameters: _{a) ΓL}_{= ΓR,}

µL=ǫd,V =µL−µRb) ΓL= ΓR,βL=βR= 1/T,µL=ǫdµR=ǫd−10ΓL c) and d) ΓL = ΓR = µL−ǫd, µR = ǫd−2ΓL. It is clear that the dot occupation will increase with the right chemical potential when the left chemical potential is fixed like in a). At high temperatures the effect of the voltage becomes unimportant like in b). We find transient behaviour in the dot occupation in c) and even in d) where the initial occupation is the same as the stationary one. This shows that for a complete description of a stationary state the knowledge about coherences between dot and lead is essential.

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The time-dependent behaviour is governed by the initial conditions. We choose Fermi-distributed electrons in both leads, an initial occupation nd(0) of the dot and no coher-ences. This leads to

nd(t) = hd†(t)d(t)i = e−Γtnd(0) +X k,α |tkα|2 1 +e−Γt−e−Γ/2 t(e−ǫkαt₊_e−iǫkαt₎ ǫ2 kα+ Γ2/4 fkα = e−Γtnd(0) + (1 +e−Γt)¯nd −e−Γ/2 tX α Γα Z dE 2π (eiEt+e−iEt) E2_{+ Γ}2_/₄ fα(E). (2.40) At infinite bias that is

nd(t) = e−Γtnd(0) + ΓL

Γ 1−e

−Γt _(2.41)

and at finite bias and zero temperature the integral can be solved using App. A.5 with the result nd(t) = e−Γtnd(0) + 1 +e−Γt ¯ nd− X α iΓα 2πΓ{E1[(−iµα+ Γ/2)t]−E1[(iµα+ Γ/2)t]} +ie −Γt_Γα

2πΓ {E1[(iµα−Γ/2)t]−E1[(−iµα−Γ/2)t]}

−e −Γt_Γ α Γ/2 θ(µα) . (2.42)

In figure 2.2.c) we show how the dot occupation approaches its stationary value starting with an empty dot. In figure 2.2.d) we choose the stationary value of the dot’s occupation as initial condition. Despite that the time-dependent dot occupation swings a bit before returning to the initial value. Why?

That happens because the initial condition is not properly stationary. It does not contain coherences between states on the dot and states in the leads. At the beginning of the time evolution in figure 2.2.d) these coherences must be built. Only after that is done the quantum-mechanical time evolution leads the system to the proper stationary state. Of course, we could choose this as the initial state for a very boring plot where the time-dependent occupation is just a straight line.

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2.2.3. The current

The current is the change of the number of particles in leadα, which is described by the operator

Nα := X

k

c†_kαckα. (2.43) It’s time-derivative can easily be obtained by applying Heisenberg’s equation of motion:

Iα := d dtNα = −i[H,Nα] = iX k t∗_kαc†_kαd−tkαd†ckα . (2.44)

To evaluate the current we simply use the expressions for the creation and anihilation operators from Eq. (2.33) and (2.34). After calculating a bit, we gain

hIa(t)i = −2Im Z dω 2πΓa ( 1−e(iω−Γ/2)t ω+ iΓ/2 fa(ω)− e−Γt−e(iω−Γ/2)t ω₋iΓ/2 nd(0) +X b Z dω′ 2π Γb "

e(iω−iω′)t−e(iω−Γ/2)t−1 +e(iω′−Γ/2)t

|ω′₊_i_Γ_/₂_|2₍_ω₋_ω′₎

+e

(−iω′₋_Γ/2)t

−e−Γt₋e(iω−iω′)t+e(iω−Γ/2)t

|ω′₊_i_Γ_/₂_|2₍_ω₋_i_Γ_/₂₎ # fb(ω′) ) . (2.45) In the long-time limit this reduces to

lim t→∞hIR(t)i = −tlim→∞IL(t) = Z dE 2π ΓLΓR E2_{+ Γ}2_/₄[fL(E)−fR(E)]. (2.46) This is a well-known result that can also be derived with a Landauer formula [26, 31] or with Keldysh Green’s functions [17]. In figure 2.3 a few elementary properties of the current are shown. a) We cannot reach arbitrarily large values just by increasing the tunneling rates. The size of the transport window (i.e. µL−µR) limits the magnitute of the current. b) When we increase only one of the tunneling rates the current will vanish for very large values as it is suppressed by the accuring asymmetry. c) A current will only flow if the dot level is inside or at least close to the transport window, i.e. if µR.ǫd.µL. d) With sufficiently small tunneling rates the current will be suppressed if the temperature T becomes remarkably larger than the bias V = µL−µR, because this diminishes the difference of the left and the right Fermi function.

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2.2. Approach Two: _{Time-dependent Fermion operators} I /β ΓL/β = ΓR/β a) ΓL= ΓR increases 0 0.1 0.2 0.3 0.01 0.1 1 10 100 1000 I /β ΓR/β b) ΓR increases 0 0.05 0.1 0.15 0.2 0.01 0.1 1 10 100 1000 I /β ǫd/β c) Dot level moves

0 0.1 0.2 0.3 0.4 0.5 -30 -20 -10 0 10 20 30 I / ΓL kBT /ΓL d) Temperature increases 0 0.1 0.2 0.3 0.01 0.1 1 10 100

Figure 2.3.: The current with constant tunneling rates and finite bias as a function of both tunneling rates, the right tunneling rate, the level position and of temperature. Parameters: _a) _µ_L ₌ ₋_µ_R ₌ _β _{b) Γ}_L ₌ _µ_L ₌ ₋_µ_R ₌ _β

c) ΓL = ΓR = β, µL = −µR = 10β d) ΓL = ΓR = µL = −µR When we increase both tunneling rates the current first increases and then saturates like in a). Asymmetry suppresses the current like in b). A current flows only if the dot level is within or near the transport window as shown in c). High temperature suppresses a directed current which can be seen in d).

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2.3. Approach Three: An exact master equation

The application of Heisenberg’s equation of motion to products of two Fermion operators leads to differential equations that are much more difficult to solve than differential equations dealing with only one Fermion operator like in the previous section. However, this procedure leads to a master equa-tion. This is very desirable, especially as we will compare exact results with approximated master equations later. We solve the differential equations by guessing pole structures.

2.3.1. The equations of motion

In order to avoid differential equations we are going to use the Laplace transform. For expectation values we introduce the new symbol

d

h. . .i(z) := Z ∞

0

dte−zth. . .i(t). (2.47) As initial condition we choose that

hd†(0)d(0)_i = n0

hc†_kα(0)ckα(0)i = fkα (2.48) and all coherences shall be zero. We apply Heisenberg’s equation of motion in Laplace space to the dot occupation:

zh[d†_d_i₍_z₎_{− h}_d†₍₀₎_d₍₀₎_i ₌ _i_h_[_{H, d}\†_d_]_i₍_z₎ = iX kα t∗_kα_h\c†_kαd_i(z)₋tkαh\d†ckαi(z) . (2.49)

The dot occupation couples to the coherences. So for them as well we formulate the equations of motion: z_h\c†_kαd_i(z) = i_h[H, c†_kαd]_i = iǫkα \ hc†_kαdi(z) +itkαh[d†di(z)−i X k′_α′ tk′_α′ \ hc†_kαck′_α′i(z) (2.50) zhd\†_d kαi(z) = −iǫkαh\d†ckαi(z)−it∗kαh[d†di(z) +i X k′_α′ t∗_k′_α′ \ hc†_k′_α′ckαi(z) (2.51)

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2.3. Approach Three: _{An exact master equation}

We see that we still need a fourth quantity: a product of two lead operators: zhc†_kα\ck′_α′i(z)−δ_kα,k′_α′f_kα = i \ h[H, c†_kαck′_α′]i(z) = i(ǫkα−ǫk′_α′) \ hc†_kαck′_α′i(z) +it_kαhd\†c_k′_α′i(z) −it∗_k′_α′ \ hc†_kαdi(z) (2.52) We insert Eq. (2.52) into Eq. (2.50) and (2.51), introduce the self energy in Laplace space ˆ ΣR/A(E, z) := X kα |tkα|2 E₋ǫkα±iz (2.53) and obtain h iz+ǫkα−ΣˆR(ǫkα, z) i_\ hc†_kαdi(z) = tkαfkα−tkαh[d†di(z) +tkα X k′_α′ tk′_α′hd\†c_k′_α′i(z) ǫk′_α′ −ǫ_kα−iz, h −iz+ǫkα−ΣˆA(ǫkα, z) i _\ hd†_c kαi(z) = t∗kαfkα−t∗kα [ hd†_d_i₍_z₎ +t∗_kαX k′_α′ t∗_k′_α′ \ hc†_k′_α′di(z) ǫk′_α′ −ǫ_kα+iz. (2.54) This together with Eq. (2.49) determines the time-dependence of _hd†d_i. To enable ourselves to use methods from analysis we switch from the k-sum to an integration by defining B_αR(E, z) := 2πX k t∗_kαδ(E₋ǫkα) \ hc†_kαd_i(z), B_αA(E, z) := 2πX k tkαδ(E−ǫkα)h\d†ckαi(z). (2.55) where the letters R and A indicate that the two quantities should have retarded and advanced nature. Such a statement is usually made for the retarded and advanced Green’s functions where it results from tiny imaginary parts in the definition, like here in Eq. (2.2). However, in the present context the retarded/advanced nature automatically emerges form the Laplace transform that gives us imaginary parts iz.

With this definition we can reformulate the equations (2.49) and (2.54) as z_h[d†_d_i₍_z₎_{− h}_d†_d_i_{(0) =} _iX α Z _dE 2π B_αR(E, z)₋B_αA(E, z), (2.56)

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E_±iz+ ΣR/A(E, z)BR/A(E, z) = Γα(E)fα(E)−Γα(E)hd†di+ Γα(E) X α′ Z _dE′ 2π BA/R₍_E′_{, z}₎ E′₋_E_∓_iz. (2.57)

In general it is very difficult to solve equation (2.57). But the solution becomes easy for the trivial case with constant tunneling rates and infinite bias. For constant tunneling rates we can evaluate the self-energy by omitting the antisymmetric part in the following integral: ΣR/A(E, z) = X α Z dE′ 2π Γα E₋E′_±_iz = ₋Γ Z dE′ 2π E′+E_∓iz (E′₎2_{+ (}_E_±_iz₎2 = ₋Γ(E_∓iz) Z dE′ 2π 1 (E′₋_E_∓_iz₎₍_E′₊_E_±_iz₎ = ∓iΓ/2. (2.58)

Then the integral on the right side of equation (2.57) vanishes and B_LR/A(E, z) = ΓL 1₋_h[d†_d_i₍_z₎ E±iΓ/2±iz, B_RR/A(E, z) = −ΓR [ hd†_d_i₍_z₎ E±iΓ/2±iz. (2.59) By putting these results into (2.57) one can check that the integrand on the right side is really zero, because all poles are in the same half plane and the integrand vanishes with 1/E2.

A master equation is a linear (differential) equation for the system’s reduced density matrix (confer chapter 3). For our model the reduced density matrix has two entries: the probability fo find the dot empty, ˆρ00(z) := 1−ρˆ11(z), and the probability to find the dot occupied, ˆρ11(z) :=h[d†di(z). As a vector we write

ˆ ρS(z) := ˆ ρ00(z) ˆ ρ11(z) . (2.60)

Combining equation (2.56) with the solution (2.59) gives the equation of motion for ˆ ρS(z): zρˆS(z)−ρS(0) = −ΓL ΓR ΓL −ΓR | {z } =:L ˆ ρ00(z) ˆ ρ11(z) (2.61)

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2.3.2. Finite bias and constant tunneling rates

At finite bias we can solve equation (2.57) using an Ansatz. We guess that BR/Aα (E, z) should have a retarded/advanced character of its pole structure. That means that all poles should be in the lower/upper complex half plane except those which are inherited from the Fermi function. This motivates the Ansatz

B_αR/A(E, z) = fα(E)bR/A_α (E, z) (2.62) where bR/Aα (E, z) is a function that has all poles in the lower/upper half plane and vanishes with 1/E or quicker. This Ansatz enables us to solve the integral on the right hand side of equation (2.57) by closing the contour in that half plane wherebR/Aα (E, z) has no poles and by using that the Fermi function produces the residues₋1/βα at

E_αnR/A := µα±(2n+ 1)iπ/βα, n= 0,1,2,3, . . . (2.63) The integral becomes

Z _dE′ 2π BαR/A(E′, z) E′₋_E_±_iz = Z _dE′ 2π bR/Aα (E′, z) E′₋_E_±_izfα(E) = ±i ∞ X n=0 bR/Aα (EαnR/A, z) EαnR/A−E±iz ·−1 βα = ± i βα ∞ X n=0 bR/Aα (EαnR/A, z) E₋EαnR/A∓iz . (2.64)

With that result we can check the Ansatz. Only for the integral in equation (2.57) it is not obvious that the Anstz fits. But equation (2.64) shows that the integral over BαR/A(E, z), which appears in the expression for BαA/R(E, z), yields only poles in the lower/upper half plane as desired.

With the Ansatz we obtain the following solution: B_αR/A(E, z) = 1 E_±iΓ/2_±iz " Γαfα(E)−Γαρˆ11(z)± Γα βα ∞ X n=0 bA/Rα (EA/Rαn , z) E₋EαnA/R±iz # . (2.65) It is not necessary to compute the coefficientsbA/Rα (EαnA/R, z) as we do not need them. We are not interested in the quantity BαR/A(E, z) itself, but only in the occupation ˆρ11(z). When we put the formal solution (2.65) into equation (2.56) the terms containing the unknown coefficiens do not contribute to the result: they appear in a term that vanishes with 1/E2 and has both poles in the same complex half plane. We can evaluate the remaining integral by using the Digamma functionψand App. A.6.2. With the definition

Fα(z) := 1 2 + 1 2πiψ 1 2 +βα iΓ/2 +iz−µα 2πi −₂1_πiψ 1 2−βα −iΓ/2₋iz₋µα 2πi (2.66)

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a) Right tunneling rate ΓR(E) E/Ω δR= Ω δR = 2 Ω δR = 4 Ω ΓR ( E ) / Ω 0 0.5 1 1.5 2 -4 -2 0 2 4 6 b) Spectral functionS(E) 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3 -4 -2 0 2 4 6 E/Ω δR= 1000 Ω δR= Ω δR= Ω/1000 S ( E ) / Ω

Figure 2.4.: Right tunneling rate and the spectral function for various widths. Param-eters:_{/ a)} _ǫ_R_{= Ω b)} _ǫ_R_{= Ω = Γ}_L_{When the peak of the tunneling rate is}

very sharp (δR= Ω/1000) the central peak of the spectral function is shifted towards ₋ǫR and a strong side peak occurs.

we can formulate the master equation zρˆ(z)−ρ(0) = X α −ΓαFα(z) Γα[1−Fα(z)] ΓαFα(z) −Γα[1−Fα(z)] | {z } =:W(z) ˆ ρ(z) (2.67)

where we have defined the kernel [3]_W(z). The difference between the kernel and the Liouvillian is that the kernel depends on the Laplace-variable z or on time while the Liouvillian is always constant. Master equations with a z-dependent kernel are called Non-Markovian contrasting with Markovian ones that have a constant Liouvillian. That the exact master equation is Non-Markovian can be seen as a motivation for the interest in Non-Markovian master equation that is rising at the moment [2, 32, 33, 34, 35, 36, 37] and motivates our chapter 3.

2.3.3. A Lorentzian tunneling rate

In this work we use a very simple model to compare evaluation schemes and approxima-tions. In order to obtain interesting results we need non-trivial versions of the model. We have already discussed the case where the Fermi functions cause non-trivial behaviour. Here we make another choice and use non-trivial spectral properties. While staying in the infinite bias limit and leaving the left tunneling rate constant we choose a right tun-neling rate that is peaked around a certain energy and hence creates side peaks in the spectral function.

Due to its simple analytic properties we use a Lorentzian function to model the peak. The same choice has been made for the single resonant level model in the references

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2.3. Approach Three: _{An exact master equation} Re(E) Im(E) ǫR+iδR Re(E) Im(E) ǫR+iδR

Figure 2.5.: Left: The poles ofBR_{, right: The poles of} _BA_{, marked with crosses.} [38, 39, 40, 41]. There are many ways to normalize the tunneling rate. We have chosen the representation

ΓR(E) = 2|Ω|2

δR (E−ǫR)2+δ2R

(2.68) where the parameter |Ω|2 _{controls the integral over the tunneling rate as}

Z ∞ −∞

ΓR(E)dE = 2π|Ω|2. (2.69) In figure 2.4 the tunneling rate and the spectral function [26] S(E) = _2πi1 GA(E) ₋ GR₍_E₎_{are depicted. We can see that when the structure of the tunneling rate becomes} relevant (i.e. if the widthδRis sufficiently small) the central peak of the spectral function is shifted towards ₋ǫRand a side peak occurs. We evaluate the right self-energy:

ΣR/A_R (E, z) := Z dE′ 2π Γ(E′) E₋E′_±_iz = −2|Ω|2δR Z dE′ 2π 1 (E′₋_E_∓_iz₎₍_E′₋_ǫ R−iδR)(E′−ǫR+iδR) = |Ω| 2 E₋ǫR±iδR±iz . (2.70)

We know fro Eq. (2.58) that for the constant tunneling rate the self energy is

ΣR/A_L (E, z) = ∓iΓL/2 (2.71) We evaluate the quantity BαR/A(E, z) in four steps:

1. We make an Ansatz for the pole structure. On the one hand we said that BαR/A(E, z) has a retarded/advanced nature such that the poles should be located in the lower/upper complex half plane. On the other hand,BαR/A(E, z) should inherit the pole

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structure of the right tunneling rate which has its poles located atE =ǫR±iδR. So we have good reasons to assume that there exist functionsbR/Aα (E, z) which have poles only in the lower/upper half complex plane such that

B_RR/A(E, z) = b R/A R (E, z) E₋ǫR∓iδR , B_LR/A(E, z) = bR/A_L (E, z). (2.72) Fig. 2.5 shows a possible realization of such a pole structure.

2. With that assumption it becomes very easy toperform the integralsin Eq. (2.57): Z _dE′ 2π B_RR/A(E′, z) E′ ₋_E_±_iz = ∓i bR/A_R (ǫR±iδR, z) E₋ǫR∓iδR∓iz , Z _dE′ 2π B_LR/A(E′, z) E′ ₋_E_±_iz = 0. (2.73)

We thus can solve Eq. (2.57) immediately with the result

B_LR/A(E, z) = ΓL ˆ w00(z)±ib A/R R (ǫR∓iδR,z) E−ǫR±iδR±iz E_±iz₋ΣR/A₍_{E, z}₎ B_RR/A(E, z) = ΓR(E) −w11(z)±ib A/R R (ǫR∓iδR,z) E−ǫR±iδR±iz E±iz−ΣR/A₍_{E, z}₎ . (2.74)

3. We still need to determine the coefficients bA/R_R (ǫR∓iδR, z). To do so we can take the expressions forBαR/A(E) determined by Eq. (2.74), insert them into

bR/A_R (ǫR±iδR, z) = lim E→ǫR±iδR

[E−(ǫR±iδR)]BRR/A(E, z) (2.75) which follows directly from definition (2.72), and solve the resulting system of linear equations. A longer algebraic calculation yields

bR/A_R (ǫR±iδR, z) = | Ω|2₍_±_iǫ R+δR+ ΓL/2 +z) ǫ2 R+ (δR+ ΓL/2 +z) δR+ ΓL/2 +z+ 2 |Ω| 2 2δR+z wˆ11(z). (2.76)

4. Now we cancheck our Ansatz. In Eq. (2.74) all poles ofB as a function ofE have usual retarded/advanced nature except the one atE=ǫR∓iδR that appears in ΓR(E). This shows that our assumptions about the pole structure ofBL andBR were correct.

To extract a master equation from Eq. (2.56) we still have to perform integrals over BL and BR each. As BL has poles in only one half plane, one could assume that the integral over BL is zero. However, the first term vanishes only with 1/E, so we cannot close the integration contour. Only the second term (containing bR/A_R ) is zero. For the

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ǫR/Ω

¯

nd

a) Peak positionǫR changes

0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 -10 -5 0 5 10 δR/Ω

b) Level width δR changes

¯ nd 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.001 0.01 0.1 1 10 100 1000

Figure 2.6.: The occupation of the quantum dot with a Lorentzian right tunneling rate.

Parameters: _a) _δ_R_{= Ω = ΓL. b)} _ǫ_R_{= Ω = ΓL.}

first term we neglect the quickly vanishing part and obtain Z _dE 2πB R/A L (E, z) = Z _dE 2π ΓLwˆ00(z) E_±iz₋ΣˆR/A₍_{E, z}₎ = ΓLwˆ00(z) Z _dE 2π E (E±iz±iΓL/2)(E−ǫR±iδR±iz)− |Ω|2 = _∓iΓL/2 ˆw00(z). (2.77) B_RR/A(E, z) vanishes fast enough, so we can simply close the contour in the half plane with only one pole, so

Z dE 2πB R/A R (E, z) = ±i_E_→lim_ǫ_R±_iδ R (E−ǫR∓iδR)BRR/A(E, z) = _±ibR/A_R (ǫR±iδR, z). (2.78) Inserting the integrals obtained in Eq. (2.77) and (2.78) into Eq. (2.56) we can formulate the master equation

zρˆ(z)−ρ(0) = ˆW(z)ˆρ(z) (2.79) with ˆ W(z) =      −ΓL/2 2|Ω| 2_(δ R+ΓL/2+z) ǫ2 R+(δR+ΓL/2+z) δR+ΓL/2+z+2 |Ω| 2 2_δR+z ΓL/2 − 2|Ω| 2_(δ R+ΓL/2+z) ǫ2 R+(δR+ΓL/2+z) δR+ΓL/2+z+2 |Ω| 2 2_δR+z     . (2.80) We can solve Eq. (2.79) and obtain the stationary solution

ˆ

ρ(z) = 1₋_Wˆ(0)−1ρ(0). (2.81) In figure 2.6 we see that the occupation is smallest when the peak of the right tunneling rate is in resonance with the dot level or when the width of the right tunneling rate takes a value similar to Ω. Both makes the right barrier more transparent so that the electrons can leave the dot more quickly.

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2.4. Approach Four: Scattering Theory

Scattering theory is the best tool to evaluate the transport statistics of sys-tems where interactions are absent or weak. Current, noise (and also all higher cumulants) can be evaluated with only one integration each [42, 43]. The central role of the transmission coefficient enables us to state in a rather general way that a single dot coupled with a Lorentzian tunneling rate is equivalent to two dots in series with flat tunneling rates.

2.4.1. Application to the single resonant level

When we talk about scattering theory in mesoscopic transport we mean ways to express transport quantities in terms of the transmission coefficientT(E) which is the probability that a particle approaching the junction with energyE will pass through it. For simple Hamiltonians that describe particles in an external potential, but no interactions involv-ing two or more particles, scatterinvolv-ing theory will produce exact results for the steady state. As the singel resonant level model has this kind of Hamiltonian, in this thesis we use scattering theory as a convenient and established tool to produce exact solutions. For the transmission coefficient of the single resonant level we refer to the paper [44] or the book [17] or the appendix A.2 where it is shown that

T(E) = ΓL(E)ΓR(E)

|E₋ǫd−ΣR(E)|2

. (2.82)

For the currentI we have the Landauer formula [31, 26] I := _hIi =

Z dE

2πT(E) [fL(E)−fR(E)]. (2.83) For the noise_hhI(2)_ii(the 2nd cumulant of the current operator) we have the Landauer-B¨uttiker formula [6, 26] hhI(2)_ii := _h(_{I − hIi})2_i = Z dE 2π n T(E)fL(E) 1−fL(E) +fR(E) 1−fR(E) +T(E)1−T(E)[fL(E)−fR(E)]2o. (2.84) Also the characteristic function which we will introduce in the next chapter in Eq. (3.2) can be expressed with the transmission coefficent [45, 46, 47]. From the characteristic function we can evaluate all cumulants. It is given by

g(χ, t) = t Z dE 2π ln 1 +T(E)(eiχ−1)fL(E) 1−fR(E) +(e−iχ−1)fR(E) 1−fL(E) . (2.85)

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2.4. Approach Four: _{Scattering Theory} F ΓL/ΓR 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0.001 0.01 0.1 1 10 100 1000

Figure 2.7.: The Fano factor at infinite bias and constant tunneling rates. We see that symmetry reduces the noise.

The most simple realization of our model has infinite bias and constant tunneling rates. There the transmission coefficient is

T(E) = ΓLΓR

E2_{+ Γ}2_/₄, (2.86) where we have omitted ǫd because it has no effect on anything. So we get the current

I = ΓLΓR

Γ (2.87)

and the noise

hhI(2)_ii = ΓLΓR Γ

Γ2_L+ Γ2_R

Γ2 . (2.88)

This result suggests to introduce the Fano factor

F := _hhI(2)_ii/I. (2.89) With the Fano factor we measure the deviation from Schottky’s noise [48] which would be created by a Poissonian process. For a Poissonian process the Fano factor (and also all corresponing quantities for higher correlations) is F = 1. When the Fano factor is larger that means that the current is noisier because the electrons come in bunches. A Fano factor smaller than 1, by contrast, indicates anti-bunching which means that the electrons pass the system at regular intervals. In figure 2.7 we see the Fano factor for the very simple case we are considering. In presence of the high symmery ΓL = ΓR it exhibits a minimum with value 1/2. With growing asymmetry it moves asymptotocally towards the value 1.

All approximations used in this work will give the exact results for this special case. Therefore we have two non-trivial cases to challenge the approximations that we will introduce later. For the first case we keep the tunneling rates constant while allowing finite bias with the two chemical potentials µL and µR and the two inverse tempera-tures βL and βR. This will not change the transmission coefficient, but the transport quantities. For the current we reproduce the results from the section about equations of

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-0.4 0 0.4 -40 -20 0 20 40 -0.4 0 0.4 -0.4 0 0.4 0.4 0.6 0.8 1 1.2 1.4 0 5 10 15 20 25 30 35 J /k B T , hh I (2 ) ii /k B T F V /kBT V /kBT

a) Current J and noiseS b) Fano factor F

J hhI(2)_ii J hhI(2)_ii J hhI(2)ii ǫ d = k B T ǫ d = 5_k B T ǫ_d = 10_k B T ǫd=kBt ǫd= 5kBT ǫd= 10kBT

Figure 2.8.: Noise with constant tunneling rates and finite bias for three different level positionsǫd. Parameters: ΓL= ΓR=kBT and µL=−µR=V /2.

motion which we give here explicitely using the Digamma function from App. A.6.2:

I = ΓLΓR 2πΓ " ψ 1 2+ βL(iΓ/2−µL) 2πi +ψ 1 2− βL(−iΓ/2−µL) 2πi −ψ 1 2 + βR(iΓ/2−µR) 2πi −ψ 1 2 − βR(−iΓ/2−µR) 2πi # . (2.90)

For the noise the situation is not that simple as to our knowledge in products of Fermi functions we cannot easily separate the residues in the upper half plane from those in the lower half plane such that the integral cannot be easily solved. So we must either perform a standard numerical integration or we can take account of the character of the integrand by formulating a numerical solution as a truncated sum over residues. In figure 2.8 there is a sketch about how current, noise and Fano factor behave as a function of the voltage. We regard cases with three levels of asymmetry characterized by the value ofǫd. For very small voltage the Fano factor diverges because the current (by which one must devide) becomes zero quicker than the noise.

As the second case we choose inifinite bias, a constant left tunneling rate and a right tunneling rate which has Lorentzian shape:

ΓR(E) = 2|Ω|2

δR (E−ǫR)2+δR2

. (2.91)

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2.4. Approach Four: _{Scattering Theory} 0.2 0.4 0.6 0.8 1 0.0001 0.01 1 100 0.0001 0.01 1 100 0.0001 0.01 1 100 0.2 0.4 0.6 0.8 1 F F δR/Ω δR/Ω δR/Ω

a) in resonance b) off-resonant c) very off-resonant

Figure 2.9.: The Fano factorF with a Lorentzian tunneling rate with the dot levelǫdand the level of the peak of the tunneling rateǫR a) in resonance (ǫd−ǫR= 0), b) off-resonant (ǫd−ǫR = 1.5Ω) and c) very off-resonant (ǫd−ǫR = 10 Ω).

Parameters: _ΓL _{= Ω. We see that a second minimum occurs when the}

detuning ǫd−ǫR is strong. and noise: I = 4ΓLδRΓ¯ |Ω| 2 (ΓLδR+ 2|Ω|2)¯Γ2+ 4ǫ2ΓLδR , hhI(2)_ii = I 1₋4ΓLδR|Ω| 2 ¯ Γ × 4ǫ2(Γ3_L+ 8δ3_R) + ¯Γ3(¯Γ2+ 2ΓLδR+ 4|Ω|2) (ΓLδR+ 2|Ω|2)¯Γ2+ 4ǫ2ΓLδR2 ! . (2.92) with the abbreviation ¯Γ := ΓL+ 2δRand withǫ:=ǫR−ǫd. In figure 2.9 we see the Fano factor as a function of the level width δR. This dependence will be shown a few times in this thesis because it is a very mean test for all approximations. Here we only introduce one main feature: For very small or very large width δR the Fano factor asymptotically approaches the value 1. If the energy levels of dot and tunneling rate are in resonance there will be only one minimum. As the levels move away from each other a second local minimum emerges which becomes the dominant one for large level detuning.

2.4.2. A Lorentzian tunneling rate and a system with two levels

One reason for our choice of the Lorentzian tunneling rate was not mentioned yet: it enables us to make a mapping to a system with two levels in series [50] (confer App. A.1) described by the Hamiltonian

¯ H = ¯ǫLd†_LdL+ ¯ǫRd†_RdR+ ¯TCd†_LdR+ ¯TC∗d † RdL +X k,a ¯ ǫkac†kacka+ X k,a (¯tkad†acka+ h.c.) (2.93)

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with a left and a right dot state, dL and dR, and a coupling ¯TC between them. Each lead couples only to its adjacent dot. The correspondence does not apply to all features of the two models. For some properties we will hardly find a correspondence, e.g. what in the single level model will correspond to the right dot’s occupation in the two-level model? For other properties there might be correspondences we are not aware of here. The statement made here restricts to the transmission coefficient. In reference [51] we find that for the two level system (with constant tunneling rates) we have

¯ T(ω) = Γ¯LΓ¯R|T¯C| 2 (ω₋¯ǫL+iΓ¯L/2)(ω−¯ǫR+iΓ¯R/2)− |T¯C|2 2. (2.94) Comparison to Eq. (2.82), with appropriate ΓR(ω) and Λ(ω), shows that ¯T(ω) =T(ω) by the correspondence ¯ ΓL ↔ ΓL, |T¯C|2 ↔ |Ω|2, ¯ ΓR ↔ 2δR, ¯ ǫL ↔ ǫd, ¯ ǫR ↔ ǫR. (2.95)

Since the characteristic function can be expressed solely in terms of the transmission probabilities[52, 45, 53], not only the first two cumulants [54], but all steady state current cumulants, of the two models coincide.

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2.5. Including interactions

In Ref. [21] the author and co-workers have derived and used a Langevin equation for a bosonic mode that is coupled to either a single resonant level or to two resonant levels in series. There this was done by expanding the Feynman-Vernon influence functional around the classical path in the electron-phonon interaction. A connection with linear response theory was stated but not explained in detail. So here we will take the opportunity to derive the linear response results.

In the chapters 3 and 4 we will deal with master equations that can be derived using a systematic expansion in the strength of the coupling between a little system and its environment. By contrast, “traditional” many body perturbation theory [55, 26] needs a non-interacting Hamiltonian as a starting point and formulates a systematic expan-sion in the interaction between the particles. So the detailed knowledge gained in this chapter about the single resonant level model can be used in many ways to approximate interacting systems in a way that is complementary to the usual master equation ap-proach. The textbooks [26] and [17] give an overview about how this complementary approach can be used in electronic transport. It is problematic to treat Coulomb interac-tions with an expansion in the interaction as the perturbative series might not converge [26, 56, 57]. This is especially true at low temperatures (around the Kondo temperature) where a complementary method to the usual master equation aproach is mostly needed. However, “traditional” perturbation theory is often a good tool to approximate coupling between electrons and phonons.

As an example we extend the single resonant level model by coupling the occupation of the dot to the x coordinate of an harmonic oscillator. Therefore we supplement the original Hamiltonian from Eq. (2.1) by an oscillator part such that

H = ǫdd†d+ X k,α ǫkαc†_kαckα+ X k,α tkαd†ckα+t∗kαc†kαd +pˆ 2 2m + 1 2mω 2 0xˆ2−λd†dxˆ (2.96) where ˆx and ˆp are the position and the momentum operator. This system is some-times called the Anderson Holstein model [58]. The Anderson Holstein model has an exactly solution for the case where only one single electron is present [59, 60]. However, when we assume that the two leads are Fermi occupied at different chemical potentials the problem becomes a non-equilibrium many body problem which has not been solved (yet). The mesoscopic approach would be to treat the coupling to the leads perturba-tively. The remaining system can either be fomulated with a differential equation for the coordinates ˆxand ˆp[61] or can be written in the Energy basis of the oscillator. This procedure makes it necessary to truncate the Fock space of the oscillator, like in Ref. [62, 63]. The complementary approach that uses “traditional” many particle perturba-tion theory can be realized by various means. To focus on the current one can use the

Master equations in transport statistics