Full counting statistics in interferometers : PROBE models and fluctuation relations

Thesis

Reference

Full counting statistics in interferometers : PROBE models and fluctuation relations

FORSTER, Heidi

Abstract

This work deals with current fluctuations in mesoscopic conductors. The current fluctuations are investigated in the framework of the full counting statistics, and as an example of experimental relevance we chose the electronic Mach-Zehnder interferometer. A powerful method of introducing decoherence into a coherent conductor is the model of voltage and dephasing probes. We develope a formalism for the full counting statistics of the electronic transport through conductors coupled to voltage and dephasing probes. We compare these models to alternative procedures like phase averaging and find that there is perfect agreement between the models for the case of one probe with a single transport channel. The probe gives additional information on internal properties of the conductor via the fluctuations at the probe. We present the joint counting statistics of current at contacts and voltage at voltage probes coupled to the conductor, as well as the joint counting statistics of current at contacts and occupation number at dephasing probes. In linear transport, the Onsager relations and the fluctuation-dissipation theorem are [...]

FORSTER, Heidi. Full counting statistics in interferometers : PROBE models and fluctuation relations. Thèse de doctorat : Univ. Genève, 2008, no. Sc. 3983

URN : urn:nbn:ch:unige-6277

DOI : 10.13097/archive-ouverte/unige:627

Available at:

http://archive-ouverte.unige.ch/unige:627

Disclaimer: layout of this document may differ from the published version.

UNIVERSIT´E DE GEN`EVE FACULT´E DES SCIENCES D´epartement de physique th´eorique Professeur M. B¨uttiker

Full counting statistics in interferometers :

Probe models and fluctuation relations

TH` ESE

pr´esent´ee `a la Facult´e des sciences de l’Universit´e de Gen`eve pour obtenir le grade de Docteur `es sciences, mention physique

par

Heidi F¨ orster

de

Goldkronach, Allemagne

Th`ese No 3983

GEN`EVE

Atelier de reproduction de la Section de physique 2008

Remerciements

J’ai eu la chance de pouvoir faire cette th`ese sous la direction du professeur Markus B¨uttiker. Je voudrais le remercier de m’avoir laiss´e une grande autonomie,

´etant toutefois toujours disponible pour me soutenir. J’ai tout particuli`erement profit´e de sa compr´ehension profonde de la physique, de ses id´ees remarquables et de sa vue d’ensemble de la physique m´esoscopique. J’appr´ecie beaucoup d’avoir eu la possibilit´e de pr´esenter mon travail `a de nombreuses conf´erences et d’avoir pu participer `a des ´ecoles d’´et´e.

Je suis tr`es reconnaissante aux professeurs Wolfgang Belzig, Thomas Ihn, Patrice Roche et David S´anchez d’avoir accept´e le rˆole de jur´e de cette th`ese.

Je tiens `a remercier Sebastian Pilgram et Peter Samuelsson qui m’ont ac- compagn´e dans mes premiers pas en physique m´esoscopique. Ils ont contribu´e d’une mani`ere tr`es importante `a mon travail, notamment pour les chapitres 2 et 3, et c’est grˆace `a eux que j’ai appris les int´egrales de chemins stochastiques et l’´ecriture scientifique.

Un grand merci aussi `a tout mes coll`egues et collaborateurs, Vanessa Chung, Marlies Goorden, Tammy Humphrey, Andrew Jordan, Philippe Jacquet, Philippe Jacquod, Mathias Lunde, Rosa Lopez, Kirsten Martens, Misha Moskalets, Simon Nigg, Cyril Petitjean, Misha Polianski, Valentin Rychkov, David S´anchez, Ja- nine Splettst¨oßer, Eugene Sukhorukov et Rob Whitney du groupe de la physique m´esoscopique pour pleins des discussions int´eressantes et instructives.

Enfin je remercie ma famille et tous mes amis partout dans le monde avec lesquels j’ai pass´e des moments tr`es m´emorables.

Heidi F¨orster

Contents

1 Introduction and summary 1

1.1 Introduction . . . 1

1.1.1 Full counting statistics and scattering approach . . . 4

1.1.2 Incoherent scattering: voltage and dephasing probes . . . . 8

1.1.3 Mach-Zehnder interferometer . . . 11

1.2 Summary . . . 16

1.2.1 Full counting statistics of voltage and dephasing probes . . 16

1.2.2 Current-voltage correlations . . . 18

1.2.3 Fluctuation relations without micro-reversibility in nonlin- ear transport . . . 20

2 Full counting statistics of voltage and dephasing probes 23 2.1 Introduction . . . 23

2.2 Voltage and dephasing probes . . . 25

2.3 Full counting statistics: probabilistic interpretation . . . 27

2.4 Mach-Zehnder interferometer . . . 29

2.5 Stochastic path integral . . . 32

2.5.1 Dephasing probe . . . 33

2.5.2 Voltage probe . . . 35

2.6 The Mach-Zehnder interferometer with the probe . . . 36

2.7 Phase averaging . . . 38

2.8 Classical exclusion models . . . 41

2.9 Two-channel structure . . . 42

2.9.1 Voltage probe . . . 44

2.9.2 Dephasing probe . . . 45

2.10 Comparison to other models . . . 45

2.11 Conclusion . . . 47

2.A Derivation of the stochastic path integral . . . 49

2.B Saddle point corrections . . . 51

2.C Phase averaging and fluctuating phases . . . 52

2.D Proof of equivalence of one-channel dephasing and phase averaging 53 2.E Two probes . . . 55

3 Current-voltage correlations in interferometers 57

3.1 Introduction . . . 57

3.2 Correlations: Langevin approach . . . 59

3.2.1 Voltage probe . . . 60

3.2.2 Dephasing probe . . . 61

3.2.3 Correlations and interference . . . 62

3.3 Full counting statistics of the probe: the stochastic path integral 68 3.3.1 Voltage probe . . . 70

3.3.2 Dephasing probe . . . 72

3.4 Examples . . . 73

3.4.1 Beam splitter. . . 74

3.4.2 Mach-Zehnder interferometer . . . 76

3.4.3 Double barrier . . . 76

3.5 Conclusions . . . 78

3.A Stochastic path integral . . . 80

3.B Higher order correlations . . . 81

4 Fluctuation relations for nonlinear transport 83 4.1 Introduction . . . 83

4.2 Full counting statistics and fluctuation relations . . . 84

4.3 Interaction-induced breakdown of the fluctuation relations . . . . 87

4.3.1 Gauge invariance of the generating function . . . 88

4.3.2 Lack of reversibility in the Mach-Zehnder interferometer . 88 4.4 Fluctuation relations for correlation functions . . . 90

4.4.1 Derivation of the fluctuation relations . . . 91

4.4.2 Comparison of response coefficients . . . 94

4.4.3 Mach-Zehnder interferometer . . . 95

4.5 Two-terminal conductor . . . 96

4.6 Conclusion . . . 101

4.A Appendix A: Derivation of the symmetry relations . . . 102

4.A.1 Symmetry relation . . . 102

4.A.2 Symmetry point of the generating function I . . . 104

4.A.3 Symmetry point of the generating function II . . . 104 4.B Appendix B: Lack of reversibility induced by boundary conditions 106

List of publications 121

Introduction en fran¸ cais

Cette th`ese est consacr´ee `a l’´etude du transport ´electronique dans des conduc- teurs m´esoscopiques. Les syst`emes m´esoscopiques sont des conducteurs d’une taille assez petite pour que les effets quantiques deviennent importants. Par exemple, on peut observer des manifestations des interf´erences dues au com- portement onduleux des particules en mesurant la conductance d’un ´echantillon.

Non seulement la conductance, mais aussi les fluctuations dans un tel conduc- teur ont de l’importance. Les fluctuations proviennent de l’incertitude statistique et de l’incertitude quantique intrins`eque. Elles contiennent des processus `a plu- sieurs particules et apportent des informations qui ne sont pas accessibles par la conductance seule.

Grˆace aux d´eveloppements techniques, il est possible aujourd’hui de fabriquer des ´echantillons d’une tr`es haute qualit´e, qui peuvent ˆetre bien contrˆol´es lors d’une exp´erience. Un exemple ´eminent est l’interf´erom`etre ´electronique de Mach- Zehnder. Celui-ci repr´esente l’analogue ´electronique dans l’´etat solide d’un in- terf´erom`etre optique. Cet interf´erom`etre est d’une grande simplicit´e conceptuelle, exhibitant toutefois une physique tr`es riche. Une des questions principales que les exp´eriences cherchent `a clarifier est de distinguer les sources de d´ecoherence dans un tel syst`eme.

Au niveau th´eorique, il existe de nombreux mod`eles pour expliquer la d´ecoh´erence. Tr`es performant est le mod`ele d’une voltage probe ou d’une depha- sing probe. Ces concepts repr´esentent des m´ethodes pour introduire la diffusion incoh´erente et in´elastique ou la diffusion incoh´erente et semi-´elastique respecti- vement.

Un des enjeux principaux de cette th`ese est de fournir une compr´ehension accrue de ces mod`eles. Pour ceci, nous ne consid´erons pas seulement la conduc- tance, mais ´egalement les fluctuations du transport dans un conducteur. Plus pr´ecisement, on d´etermine la statistique des charges transmises `a travers l’´echantillon. Cette th´eorie est connue sous le nom de full counting statistics.

Nous d´eveloppons un formalisme pour obtenir la full counting statistics dans des conducteurs coh´erents coupl´es `a une voltage probe ou dephasing probe.

de moyennage de phase. Deuxi`emement, nous d´erivons quelles informations suppl´ementaires peuvent ˆetre obtenues en mesurant non seulement le courant

`a travers des conducteurs, mais aussi les fluctuations de tension `a la probe et les corr´elations de courant et tension.

La troisi`eme partie de cette th`ese traite les relations de fluctuations. Pour le transport lin´eaire, les relations de fluctuations sont ´etablies depuis plus d’un si`ecle d´ej`a. En regime de transport non-lin´eaire, des relations de fluctuations pour la full counting statistics ont aussi ´et´e d´evelopp´ees. Ces relations-l`a sont fond´ees sur le principe de la micro-r´eversibilit´e. Dans cette th`ese, nous d´emontrons que dans les conducteurs en pr´esence simultan´e d’un champ magn´etique et d’inter- actions, la micro-r´eversibilit´e peut ˆetre bris´ee. Cet effet se manifeste dans la magneto-asym´etrie de la conductance nonlin´eaire qui a ´et´e mesur´ee dans de nombreuses exp´eriences. Pour ces cas, nous d´erivons de nouvelles relations de fluctuations qui sont valables mˆeme en l’absence de micro-r´eversibilit´e. Ces nou- velles relations peuvent ˆetre verifi´ees exp´erimentalement, par exemple en utilisant l’interf´erom`etre de Mach-Zehnder.

CHAPTER 1

Introduction and summary

1.1 Introduction

In the world of quantum mechanics, particles behave like waves, a cat can be both dead and alive at the same time, and a system feels what is happening to its very distant counterpart. For people who are fascinated by these astonishing effects, mesoscopic physics is an ideal playground. Manifestation of quantum effects can be observed, when the coherence length of the particles, for example electrons in an electrical conductor, exceeds the size of the sample. This can be achieved for example by miniaturization of the conductor under very controlled conditions.

Since the early days of mesoscopic physics, a huge technical development has taken place. Nowadays sample fabrication and state-of-the-art experiments have amazing results, and still, the capacity of controlling structures on micro- to nano-scale and the resourcefulness of experimentalists are increasing. A number of very particular features characterizes the electrical transport in mesoscopic conductors:

• Interference effects: The electrons show wave-like behavior and are diffracted by impurities and by the edge of the sample etc. This results in interference effects that make the physical properties very sensitive to variations in the circuit geometry and other parameters. Interference is thus a useful tool for probing the coherence length in a conductor. Inter- ference effects were at the origin of many discoveries in the beginnings of mesoscopic physics, examples are weak localization [1, 2, 3], persistent cur- rents [4] in normal metals, Aharonov-Bohm oscillations in rings [5, 6, 7, 8]

and universal conductance fluctuations in diffusive conductors [9, 10, 11].

Only recently, in ballistic systems of very high quality, an electrical Mach-

Figure 1.1: An electron micrograph image of a quantum dot, embedded in an Aharonov-Bohm structure (picture taken from Ref. [20]). A two-dimensional elec- tron gas (in the dark grey area) is located at a GaAs/AlGaAs interface and confines the electrons vertically. In addition, metallic gates (light grey) are deposited on top of the structure and confine electrons in the 2DEG below laterally. In this setup, the Aharonov-Bohm ring is formed by two branches enclosing the central metallic island.

In the left branch, a quantum dot is embedded, which is controlled by the gateP. Zehnder interferometer was realized [12], and two-particle Aharonov-Bohm interferences were measured [13].

• Interactions: The charge of the electrons results in repulsive Coulomb interaction between all the electrons in the circuit. In general, this leads to complicated many-particle problems. In experiments, interactions often represent a source of decoherence. On the other hand, interactions manifest themselves in interesting quantization effects like the Coulomb blockade.

There, transport through a quantum dot with small capacitance is blocked by the large charging energy it costs an electron to enter the dot [14, 15, 16, 17]. A quantum dot can be controlled in such a way that it only has a few possible states, e.g. it hosts an excess charge or not. Systems like this can even be maintained in a coherent superpositioned state [18, 19] for a certain time span.

• Reduced dimensionality: In order to observe quantum effects like in- terference or charging effects, it is necessary to confine particles spatially and to reduce the degrees of freedom of their motion. In a two-dimensional electron gas (2DEG), realized for example with the help of semiconductor heterostructures, the free propagation of the carriers is limited to two direc-

1.1 Introduction

tions. Various techniques like electron-beam lithography allow to confine the electrons further to one-dimensional quantum wires or zero-dimensional quantum dots. An example of such a mesoscopic conductor is shown in Fig. 1.1, it is a quantum dot embedded in a ring-shaped structure [20]. The conducting structure can be designed in the desired shape, and can be con- trolled during experiments by applying gate voltages. The most elementary building block of a conductor is a quantum point contact (QPC), a constric- tion in the 2DEG achieved by metallic gates. Within this constriction, the transverse part of the wave function and its associated energies are quan- tized. This is manifest in conductance quantization [21, 22]. Another way of achieving effective one-dimensional motion is to operate the 2DEG in the quantum Hall regime by applying a strong magnetic field. On the Hall plateaus, charge is transported by chiral edge states [23, 24], and transport is insensitive to elastic scattering.

• Sample specific properties: Mesoscopic conductors are so small that transport depends on specific microscopic conditions like the configuration of impurities, as well as on boundary conditions. Therefore, in contrast to macroscopic objects which are described by average values, the fluctu- ations become important. One consequence are the universal conductance fluctuations [9, 10, 11]. By controlling the shape of a quantum dot, one can produce either regular deterministic billiards or chaotic dynamics [25].

• Dependence on the measurement setup: In quantum mechanics, the measurement process is of particular importance. The measurement ap- paratus can have a strong influence on the system investigated, and also back-action can play an important role. For example, in a four-probe mea- surement, the dephasing rate depends strongly on the probe-configuration [26, 27], and the voltage-dependence of the third current cumulant of a tun- nel junction proved to be a consequence of the electromagnetic environment caused by the measurement circuit [28, 29].

Most of the described effects are observed with electrical transport measure- ments. In experiments, the average current through a conductor is analyzed as a function of different parameters, like applied voltage, interaction strength, temperature etc. Most of the principal interference effects mentioned above are visible in the conductance. Since a few years, a lot of effort is equally put into measuring noise and fluctuations [30, 31, 32, 33]. The origin of noise in mesoscopic conductors is two-fold: both the statistical uncertainty and the intrinsic quantum mechanical uncertainty are sources of fluctuations. Random noise like thermal fluctuations obscure the measurement signal and do not contain further useful information. This is expressed by the fluctuation-dissipation theorem which re- lates the equilibrium fluctuations to an average quantity, the linear conductance coefficients. But out of equilibrium, the fluctuations allow additional insight into the properties of the system, corresponding to Landauer’s statement ”the noise is the signal”. In the shot noise regime, fluctuations and correlations originate

from two-particle processes. Therefore, statistical effects like (anti-) bunching of fermions [34, 35, 36] and Hanbury Brown-Twiss effects [13] are only visible when measuring correlations. Also entanglement as a two- (or more) particle property manifests itself in current cross-correlations [37]. Another example where noise allows to characterize transport properties is the measurement of the effective charge in the fractional quantum Hall regime [38, 39, 40].

Current fluctuations are not completely determined by the first and second moment, the mean current and the spectral function. The success of noise mea- surements motivated the experimental investigation of even higher order correla- tion functions [28, 41, 42]. The information on all higher order current cumulants is contained in the full probability distribution of charge traversing the conductor.

This is known as the full counting statistics (FCS). Particularly appealing within this theory is the fact that it gives insight on individual scattering processes con- tributing to transport. First experiments on the statistics of charge transferred through a quantum dot have emerged [43, 44].

In the following, we will give a brief introduction into the theoretical concepts and techniques used in this thesis. In section 1.1.1, we present the full counting statistics within the scattering approach, which is especially adapted for coherent problems. Incoherent processes can be incorporated into the scattering formalism with the help of voltage and dephasing probes, as explained in section 1.1.2. An important example of experimental interest which will appear in every chapter of this thesis, is the Mach-Zehnder interferometer. We introduce it in section 1.1.3 and give a quick overview over recent experimental developments.

1.1.1 Full counting statistics and scattering approach

Transport in mesoscopic systems can be characterized not only via conductance and noise but also with all higher order current correlations. This corresponds to the knowledge of the distribution function of charge transferred through a conductor during a measurement time τ. The distribution function can be ex- pressed via Fourier transform by a generating function. To construct the gen- erating function, it is useful to imagine that electrons passing the sample in a given time are counted, and the approach is thus known as full counting statis- tics [45, 46, 47]. Beyond current and noise, first experiments measured the third cumulant in tunnel junctions [28, 41, 42] and have given additional impetus to this field of research. A number of novel detection schemes [48, 49, 50, 51] were proposed, and recently, remarkable experiments for counting of single electrons appeared [44, 52, 53, 54, 55], even the distribution of charge hopping through a quantum dot was measured [43, 56, 57].

Let us consider a mesoscopic conductor withM terminals as shown in Fig. 1.2 (a). The distribution function is denoted byP(Q) where the vector quantityQ= (Q1, Q2, . . . , QM) describes the charge transfered into each of the M terminals

1.1 Introduction

1 2

2

1

1

1 1

2

1 12

P( )

−

V V

Q

Q

Q

<Q > Q

<Q > <Q >

Figure 1.2: (a) A mesoscopic conductor withM contacts and a scattering region. The reservoirs are characterized by the applied voltages and temperature. The full counting statistics is the distribution of charges Q₁, Q₂, . . . transferred during a measurement through the different leads. Here, infalling particles are counted positively. (b) A sketch of the distribution function of charge transferred in contact 1, indicated are the mean valuehQ1i and the second cumulant

Q²₁

− hQ1i², which describes the width of the distribution.

within the measurement time τ. P(Q) is related to the the cumulant generating function F(Λ) by means of a Fourier transformation

P(Q) = Z

dΛe^{F(Λ)−iΛ·Q} (1.1)

F(Λ) = lnX

Q

P(Q)e^iΛ·Q. (1.2)

The vector Λ = (λ1, λ2, . . . λM) contains the counting variables of the different terminals, they are conjugate variables to Q. The sum and integrals run over all elements of the vector: R

dΛ = (2π)^−MR

dλ1. . . dλM and P

Q = P

Q1...QM. Probability conservation leads to the normalization of the generating function F(0) = lnP

QP(Q) = 0.

All irreducible moments of the distribution function are obtained by taking derivatives of the cumulant generating function with respect to the counting variables and evaluated atΛ= 0. For a long measurement timeτ the transmitted charge into a terminal α is proportional toτ, i.e. one can writeQα =τ Iα. Then, the zero frequency cumulants for the current -the physical observables- can be expressed in terms of the generating function of the charge, Eq. (1.2). Written out explicitely, the first three cumulants are the average current

hIαi= e iτ

∂F

∂λ_α Λ=0

, (1.3)

the auto- or cross-correlations

Sαβ = e² i²τ

∂²F

∂λα∂λβ

Λ=0

, (1.4)

1 2

2

1

= S

(a) (b)

V V

I I

Figure 1.3: (a) The scattering region (shaded) is connected by leads to a number of reservoirs. I₁, I₂ etc. denote the current in the leads or contacts. Coherent single- particle scattering processes are described by the scattering matrix (b), that relates incoming and outgoing current amplitudes.

and the third cumulant (skewness) C_αβγ = e³

i³τ

∂³F

∂λα∂λβ∂λγ

Λ=0

. (1.5)

With the above definitions in hand, it is clear that for characterizing the zero- frequency transport properties, it is sufficient to determine the generating func- tion. While initial derivations were based on the scattering approach to electri- cal transport, subsequently a number of different methods have been developed.

These include an approach based on Keldysh Green’s functions [58, 59, 60, 61], the non-linear sigma model [62], an approach based on a cascade of Boltzmann- Langevin equations [63] and a formulation in terms of a stochastic path integral [64, 65, 66, 67, 68, 69, 70, 71, 72, 73]. These later methods are principally useful when semi-classical transport is dominant. In this work, we concentrate on the zero-frequency counting statistics, but also time- or frequency dependent prob- lems can be treated [74, 75, 76].

Scattering approach

For coherent systems where Coulomb-interaction can be neglected, transport can be described using the scattering approach [77, 78, 79]. Here we only mention the most important aspects of the scattering formalism, without deriving explicitly the formulas. For details the reader is directed to Ref. [33]. The region of interest, the sample, is usually represented as a scattering region connected by ideal leads to a certain number of reservoirs. This is shown in Fig. 1.3, elastic scattering is happening only inside the shaded scattering area. In this representation, the measurement circuit which can be arbitrarily complicated is merged into the contacts. The reservoirs form a free electron gas in three dimensions and are described by equilibrium Fermi distributions. The leads, connecting the contacts to the scattering region are considered to be ideal, no interactions or scattering of impurities take place. Often, the motion of electrons in the leads is free in

1.1 Introduction

only one direction and confined in the transverse directions. Depending on the energy of the carriers, a certain number of transport channels corresponding to the energy levels in the transverse direction are open.

Within the scattering region in Fig. 1.3, the scattering processes can be de- scribed by a scattering matrixS. The scattering matrix relates outgoing current amplitudes to incoming amplitudes as shown in Fig. 1.3 (b). This can be formu- lated by defining creation and annihilation operators ˆa^†,a, for electrons injectedˆ from contacts and ˆb^†,ˆbfor electrons emitted into contacts. For clarity, we restrict the formulae here for the case of only one transport channel per lead, the gener- alization to the multi-channel case is straightforward and can be seen explicitely in the literature [33]. The operator ˆbα describes a particle entering terminal α and is connected to operators for injected particles in all contacts,

ˆbα =X

β

S^αβˆaβ. (1.6)

The current operator in lead α contains the difference of outgoing and incoming particles. Using that processes contributing to electrical transport are situated close the the Fermi energy, it can be written as

Iˆα(t) = e h

Z

dEdE⁰e^i(E−E0)t/h(ˆa^†_α(E)ˆaα(E⁰)−ˆb^†_α(E)ˆbα(E⁰)) = (1.7)

= e

h X

βγ

Z

dEdE⁰e^i(E−E0)t/h[δαβδαγ − Sαβ^† (E)S^αγ(E⁰)]ˆa^†_β(E)ˆaγ(E⁰).

The average current can be easily calculated, since the quantum mechanical aver- age values of combinations of ˆa^†ˆaare given by Fermi functions. To linear response, the average current is expressed in terms of a conductance matrix Gαβ,

hIαi = X

β

GαβVβ, with (1.8)

Gαβ = e² h

Z dE

−∂f(E)

∂E

(δαβ− Sαβ^† (E)S^αβ(E)) (1.9) where V_β is the voltage applied to contact β and f(E) is the equilibrium Fermi function. Current conservation is represented by the unitarity of the scattering matrix,S^†S = 1. Also the noise spectrum Sαβ can be evaluated with the current operator using the definition

S_αβ(ω) = 1 2

Z dtD

∆ ˆI_α(t)∆ ˆI_β(0) + ∆ ˆI_β(0)∆ ˆI_α(t)E

e^iωt (1.10) with ∆ ˆIα(t) = ˆIα(t)− hIαi. In principle, this can be continued in the same way for correlations of higher order like the third moment. However, it is more con- venient to directly calculate the full counting statistics. For a general mesoscopic

conductor described by a scattering matrixS, the generating functionF(Λ) was derived by Levitov and Lesovik [45]:

F(Λ) = 1 h

Z τ 0

dt Z

dE H(Λ) with (1.11)

H(Λ) = ln deth

1 + ˜n

λ˜^†S^†˜λS −1i

. (1.12)

For a conductor with single mode contacts to the M terminals, the scatter- ing matrix has the dimensions M × M. The matrix of occupation numbers

˜

n -here, these are the Fermi functions of the different terminals- is given by

˜

n = diag(n₁, n₂, . . . , n_M), and the matrix ˜λ introduces the counting fields,

˜λ = diag(e^−iλ1, e^−iλ2, . . . , e^−iλM). The effect of the product ˜λ^†S^†λ˜ in Eq. (1.12) is to multiply each element of the scattering matrix by counting fields, Sαβ^† → Sαβ^† e^{iλβ−iλα}. The factor with the counting fields works as a marker which ac- counts for each scattering process. The generalization to many transport modes is straightforward: the dimension of all matrices in Eq. (1.12) grows according to the number of transport channels, but importantly all channels in the same terminal have the same occupation function and counting field, since experiments are not channel specific.

An example for the scattering matrix and the generating function will be given in section 1.1.3 for the Mach-Zehnder interferometer.

1.1.2 Incoherent scattering: voltage and dephasing probes

The scattering approach is a quantum theory for transport through mesoscopic conductors within the single-particle picture. In general however, the particles are subject to the influence of an environment or to interactions with other parti- cles. These are processes that lead to decoherence and exchange of energy. A very powerful concept to introduce such inelastic and incoherent processes on a phe- nomenological level is the model of the voltage [80, 81] or dephasing [82] probe.

An additional terminal -either a voltage or a dephasing probe- is connected to a coherent mesoscopic conductor, as shown in Fig. 1.4. Particles entering the probe are later incoherently reemitted into the conductor. Scattering via the probe, a particle thus looses its phase coherence and in the case of a voltage probe, it also changes its energy.

A voltage probe is a real, physical component used in many mesoscopic experiments [83, 84, 85, 86] and consists of a large metallic contact. The contact is left floating or is connected to a voltmeter, i.e. ideally there is no current drawn at the probe. In response to the injected charge on the probe, the potential Vp = Vp(t) of the floating probe develops fluctuations on the timescale τd = RC, as sketched in Fig. 1.5. Here, R is the total charge relaxation resistance from the probe into the M terminals, where 1/R = PM

α=1Gαp with Gαp the conductance from the probe to terminal α, and C is the total capacitance of the probe. The origin of these fluctuations has a very natural explanation: Injected charges raise the potential Vp which leads to an increase in the outgoing current.

1.1 Introduction

2

1 2

1

p

= S

V

V I I

f (E)

Figure 1.4: A voltage or dephasing probe is an additional ficticious contact attached to the scattering region (compare to Fig. 1.3). This enlarges the scattering matrix by processes into and out of the probe. The probe is a special terminal in the sense that it does not absorbe carriers, but every particle entering the probe is later incoherently reinjected into the conductor.

This consequently reduces the charge on the probe and with this Vp decreases etc. The timescale for the continuous charging and discharging of the probe is just as in classical circuit theory the RC-time. This picture clarifies, why current and current fluctuations at the probe at low frequencies ω <1/τd are completely conserved,

hIpi= 0, ∆Ip = 0. (1.13)

Put differently, for a measurement during a time much longer than τd there is no charge accumulation in the probe.

We assume that the thermalization in the voltage probe is efficient, i.e. charges injected into the probe scatter inelastically on a timescale much shorter than τ_d. The electron occupation is thus described by an equilibrium Fermi-Dirac distribution fp(Vp, T). For a large thermal conductivity between the probe and the surrounding lattice, the temperature T of the probe is given by the lattice temperature. In this case the voltage probe is dissipative and does not conserve heat current. In the opposite case, for a small thermal conductivity, the probe is non-dissipative and the temperature T is instead determined from the additional condition of a conserved heat current at the probe. Here we only consider the former case, however, we emphasize that the latter case could be treated as well within our theoretical scheme. We also note that the problem of removing the heat from an electronic reservoir is important in mesoscopic experiments [87].

Contrary to the voltage probe, the dephasing probe is a conceptual tool used to model quasi-elastic dephasing [82]. A particle injected in an energy in- terval E to E +dE is incoherently emitted after spending an average time τd

inside the probe, in the same energy interval. This allows an energy change of dE much smaller than the applied voltage eV or the temperature kBT. The distribution function np(E) in the energy interval is proportional to the number of carriers in the interval. As scattering in each energy interval is independent,

τ

τ

t V

Figure 1.5: Time-dependent potential Vp(t) at a voltage probe. The current injected from the conductor into the probe gives rise to fast fluctuations on the time scale τ_s = h/eV. The response of the probe gives rise to modulations on a much longer timescale τ_d =RC.

np(E) is -contrary to the distribution function of the voltage probe- a strongly non-equilibrium distribution function. Moreover, the fluctuations of the distri- bution function n_p(E) =n_p(E, t) in each energy interval are independent. Their origin is qualitatively the same as those of the potential fluctuations in the voltage probe: the more charges injected into the probe, the larger is np(E). This leads to an increase in the outgoing current and thus to a reduction of n_p(E). The fluctuations ofnp(E) occur on a timescaleτd, the delay time of the probe. Again there is thus no charge accumulation of the probe on timescales longer than τd, and the current per energy into the probe is conserved up to the frequency 1/τ_d,

hip(E)i= 0, ∆ip(E) = 0. (1.14)

On a conceptual level, the main difference between the models of voltage and dephasing probe lies in the electron distribution functions inside the probe, shown in Fig. 1.6. For the voltage probe it is a equilibrium Fermi function, while the dephasing probe has a non-equilibrium occupation.

Both probe models enlarge the scattering problem by introducing additional processes in and out of the probe (compare Fig. 1.4). The condition that no

f

n

1

a) b)

p

1

0 0 eV eV

p

0 0

eV

n

Figure 1.6: a) The Fermi distribution function of the voltage probe takes on only the values zero or one (at zero temperature). The position of the step is given by the potential at the probe. b) The distribution function of the dephasing probe is a two-step function and has a value between zero and one in the interval [0, eV].

1.1 Introduction

T_A

D D

ϕ

S

A

B

B 4 3

2 1

A

R_A V

Φ

(a) (b)

Figure 1.7: (a) A sketch of the optical Mach-Zehnder interferometer. A beam of monochromatic light emitted by the source S is split with the help of a half-silvered mirrorA. The two partial beams obtain a phase difference ϕbetween their respective paths, before they are recombined at the second half-silvered mirror B and interfere.

Detectors measure the intensity of the light which depends on the phase difference.

(b) A schematic sketch of the electronic Mach-Zehnder interferometer: the four termi- nals act as source and detector, and the electrons move along unidirectional transport channels (indicated by arrows). The interferometer is threaded by a magnetic flux ΦAB which corresponds to the phase differenceϕin the optical case. The transmission (reflection) probability of the beam splitters is T_A and T_B (R_A and R_B) respectively.

Processes from contacts 3 and 4 to 1 and 2 do not interfere and are omitted in this sketch.

charge is absorbed or accumulated inside the probe is formulated in Eqs. (1.13) and (1.14), and reduces the extended problem to a conduction problem with only the true voltage and current contacts. On the level of conductance and noise the reduction from the large coherent problem to the incoherent fewer contact problem is straight forward. The implementation of the fact that no charge accumulates on the probes is more challenging, if we want to capture the entire transport statistics. This is formulated in chapter 2 and expanded in chapter 3.

1.1.3 Mach-Zehnder interferometer

The Mach-Zehnder interferometer occurs in each chapter of this thesis. It is of great experimental relevance and of conceptual simplicity which makes it an ideal example to apply our theories to. In this section, we introduce the Mach-Zehnder interferometer and give a brief overview over the experimental developments and the related literature.

The historical Mach-Zehnder interferometer [88, 89] is an optical interferom- eter as sketched in Fig. 1.7 (a). A monochromatic light beam is split into two partial beams which can aquire a phase difference before they recombine and interfere. For the optical setup, half-silvered mirrors function as beam splitters, and fully reflecting mirrors can be used to guide and redirect the light beams.

The experimental setup has two outputs; in principle, also two different light sources as inputs are possible.

The electronic Mach-Zehnder interferometer is an analog solid state device that allows for electron waves to interfere. It is a four-terminal conductor, and the scattering region consists of two beam splitters and two interferometer arms threaded by a magnetic flux as shown schematically in Fig. 1.7 (b). The four terminals are electron reservoirs, characterized by Fermi-distributions, they act as both emitters and detectors. Importantly, in analogy to the half-silvered mirrors in the optical case, back scattering at the beam splitters is excluded. At beam splitter A, a particle can either choose the upper or lower arm, and at beam splitter B, it goes into either terminal 3 or 4. That means no closed orbits exist, and that left- and right-moving processes are separated. When a particle moves through the device, interference happens due to the Aharonov-Bohm (AB) effect [90]: the different vector potentials in the two arms lead to a phase difference between the arms, just like in the optical setup, Fig. 1.7 (a). The phase difference creates a characteristic interference pattern periodic in the enclosed magnetic flux ΦAB. The interference pattern is manifest, for example, in the electrical conductance measured in terminal 3 or 4.

For the experimental realization of the electronic MZI, a two-dimensional elec- tron gas (2DEG) is constricted via lithographic methods to form an interferome- ter. An image of such a sample is shown in Fig. 1.8 on the left. The interferometer is operated in the quantum Hall regime. Then the electronic transport can be described by chiral edge states which are quasi-one-dimensional and carry charge only in one direction along the edge of the sample. In Fig. 1.8, this is depicted by colored paths and arrows. Note that a difference to the optical setup arises due to the electronic nature of the carriers: current conservation requires a back flow from terminal 3 and 4 to contacts 1 and 2. These processes are indicated in the edge state picture of Fig. 1.8 on the right but omitted in the schematic sketch of Fig. 1.7 (b). The number of transport channels (edge states) is given by the filling factor of the quantum Hall effect. The back-scattering free beam splitters are realized using quantum point contacts (QPC). A QPC is formed by metallic gates, controlled by gate voltages, that form a potential barrier in the Hall bar. Carriers in the edge state can either tunnel through the barrier and continue their path along the same edge, or they are reflected and move along the opposite edge, see Fig. 1.8.

The experimental realization of the electronic MZI is very challenging, a pi- oneering experiment was performed by Ji et al. [12]. They measured the AB- oscillations in the conductance with a visibility of 60%. The visibility for a conductance measurement in contact 3 is defined as (I₃^max−I₃^min)/(I₃^max+I₃^min).

Their data is shown in Fig. 1.9, the curves with blue circles and red points repre- sent the current as a function of magnetic field and of a side-gate voltage which changes the enclosed magnetic flux by changing the effective area inside the in- terferometer arms. Since this first experiment, the field became very active, and presently, experiments from four different research groups are available, namely

1.1 Introduction

Φ

T_A T_B

2 4

1 3

Figure 1.8: Left: A scanning electron microscopy image of a Mach-Zehnder interfer- ometer for filling factor 2, taken from Ref. [94]. The edge channels are indicated, here, the inner channel (full line) is split at the QPCs. Right: A schematic sketch of the MZI in the edge channel picture at filling factor 1. The transmission probabilities of the QPCs areT_AandT_B respectively. Note that in the experimental realization, terminals 2 and 4 are merged.

by Neder et al. [91, 92, 93], Litvin et al. [94, 95], Roulleau et al. [96, 97, 98] and Bieri et al. [99]. The experiments are concerned with investigating the origin of decoherence in the interferometer by measuring the visibility of the interference pattern as a function of different parameters.

All experiments agree in an exponential decay of the visibility with tempera- ture. Using MZI-samples of three different sizes, Roulleau et al.[97] managed to determine a decoherence length of edge states in the integer quantum Hall regime.

They were able to exclude thermal smearing as the decohering mechanism and emphasize that their findings of a decoherence length inversely proportional to temperature is compatible with thermal noise of a fluctuating potential [100] as dephasing source.

A method to control dephasing in the interferometer is to inject carriers into an additional edge state (at filling factor 2) in vicinity to the interfering edge. This edge state acts as a gate onto the interferometer and the noise in the edge will randomize the interference phase. This procedure has been ap- plied in Refs. [92, 93, 98]. In these experiments, the visibility decreases as a function of voltage applied to the additional edge state, in particular, Ref. [98]

finds an exponential decay. Neder et al. [92, 93] deduce from their measurements non-Gaussian noise in the additional edge state as the origin of the decoherence.

Roulleau et al. [98] confirm the assumption of a Gaussian approximation which points towards fluctuations of the electrostatic potentials in the interferometer arms as the source of decoherence.

The visibility as a function of source-drain voltage (the voltage difference V1−V3) shows a particular lobe-structure with distinct maxima and minima that define a new energy scale. Refs. [91, 96, 95, 99] investigate and report the lobe- behavior, while in Refs. [12, 94] an exponential decay with voltage is observed, in agreement with theoretical predictions based on voltage probe models [101].

Figure 1.9: Data of the first measurement of the AB-oscillation in a Mach-Zehnder interferometer (taken from Ref. [12]). The curves show the current as a function of magnetic field (blue circles) and an additional gate voltage (red points) which both change the magnetic flux.

Roulleau et al. [96] explain their data by a model of Gaussian phase average, while Neder et al. [93] put forward an explanation using non-Gaussian noise.

Bieri et al. [99] discovered an asymmetric dependence of the visibility on the transmission of the first beam splitter. This also affects strongly the lobe-pattern and can lead to an increase of coherence with source-drain voltage for certain values of transmission.

A thorough investigation of the visibility as a function of magnetic field, and of the filling factor was pursued by Litvin et al. [95]. They find a maximum of the visibility at filling factor 1.5, when backscattering within the bulk of the 2DEG is maximal; and a vanishing visibility on the Hall plateaus. Although only one edge state actually enters the interferometer, the interference is very sensitive to the building of the second edge. This is in agreement with previous data [91, 96].

On the theoretical side, the Mach-Zehnder interferometer has equally at- tracted a great deal of attention, due to the experimental stimulus. Dephas- ing in one-channel ballistic interferometers was discussed employing models with classical fluctuating potentials both for the dephasing rate in the conductance [27, 100, 102], as well as for noise [103, 104] and higher order fluctuations [105]. Corrections due to a quantum bath were found [106, 107]. Further- more, voltage and dephasing probe models were successfully applied to the MZI [101, 103, 104, 108, 109]. More exotic proposals surfaced, like an interferome- ter in the fractional quantum Hall regime [111], a spin-dependent interferometer [112], and a quantum pump with the MZI [113]. In addition, a good num- ber of theoretical works concentrate on theories on the lobe structure behavior [114, 115, 116, 117], which is by now not fully understood.

1.1 Introduction

The experimental data allows to define a transport regime where a single- particle approach is justified. In the following, we concentrate on case of a co- herent interferometer, neglecting interactions for the moment.

Scattering matrix of the MZI

We describe transport through the Mach-Zehnder interferometer by a scattering matric. The scattering matrix has a simple structure, due to the separation of left- and right-movers: only the elements in the off-diagonal 2×2 blocks of the total 4×4 scattering matrix are non-vanishing [100]. As can be seen in Fig. 1.8 on the right, the interference happens in scattering processes from terminals 1 and 2 towards the terminals 3 and 4. They are contained in lower off-diagonal block of the scattering matrix,

S31 = −p

R_AR_Be^iΦ+p T_AT_B S⁴² = p

TATBe^iΦ−p RARB

S³² = ip

RATB+ip

TARBe^iΦ S⁴¹ = ip

RATBe^iΦ+ip

TARB. (1.15)

Here, we consider equal length of the arms, leading to energy-independent scat- tering amplitudes. The reflection probability of the two beam splitters is denoted by RA = 1−TA and RB = 1−TB. An Aharonov-Bohm flux ΦAB threads the interferometer, and the phase difference between the two arms is Φ = ^2πe_h ΦAB+δ, where δ can represent constant scattering phases. The second off-diagonal block of the scattering matrix is diagonal and describes independent processes from ter- minals 3 and 4 to 1 and 2 which have transmission probability 1 as can be seen in the sketch of Fig. 1.8. Due to current-conservation and the exclusion of back- scattering at the beam splitters, only the transmission probability T31=|S³¹|² is of importance for transport, since T41=T32 = 1−T31 and T42 =T31.

With the above scattering matrix in hand, one obtains with Eqs. (1.11) and (1.12) the full counting statistics for the MZI in the non-interacting coherent case. This will be discussed in detail in later sections, and also decoherence and interactions will be taken into account. For introductory purposes, it is instructive to look at the full counting statistics in the simplest case of zero temperature, and a voltage V applied at terminal 1, as shown in Fig. 1.7 (b). Then, in the energy interval 0≤E ≤eV, the electron occupation function is unity in reservoir 1 and zero in reservoirs 2−4. All charges are incident from contact 1, and only one-particle processes take place. Each infalling particle has exactly two final states, characterized by transmission into either contact 3 or 4. Therefore, the statistics of transmitted charge has a binomial distribution. With this reasoning one obtains the distribution function of the number of charges transmitted into contact 3 without calculation as

P(Q3) = N

Q3

T₃₁^Q3(1−T31)^N−Q3. (1.16)

T_A eiϕ

T_A

B 4 3

2 1

A

R_A ε

Φ p

AB =

B 4 3

2 1

A

R_A

ΦAB

ε (b) φ

(a)

Figure 1.10: (a): A voltage (dephasing) probep is connected to the upper arm of the Mach-Zehnder interferometer with coupling strength ε. The transmission (reflection) probability of the beam splitters isT_A and T_B (R_A and R_B) respectively, and Φ_AB is the magnetic flux through the structure. For a single-channel probe, the full counting statistics of transferred charge does not depend on the nature of the probe (voltage or dephasing probe). In addition, it is identical to the full counting statistics obtained by phase averaging. (b) Phase averaging is achieved by attaching a single-mode scatterer described by a phase distribution to the upper arm.

The distribution depends on the applied voltage V and the measurement timeτ via N =eV τ /h. Note that the transmission probability T₃₁ =R_AR_B+T_AT_B− 2√

RARBTATBcos Φ contains the Aharonov-Bohm oscillations. The correspond- ing generating function can be obtained by Fourier transform, it reads

F(λ₃) =Nln[1 +T₃₁(e^iλ3 −1)]. (1.17) After this brief introduction into specific techniques and the Mach-Zehnder interferometer, we will summarize the contents discussed in the main sections.

Each of the later chapters is independent and recurring subjects like the full counting statistics and the Mach-Zehnder interferometer are briefly repeated for convenience. The bibliography is given at the very end of the thesis.

1.2 Summary

1.2.1 Full counting statistics of voltage and dephasing probes

Voltage and dephasing probes introduce incoherent inelastic and incoherent quasi- elastic scattering into a coherent mesoscopic conductor. They represent addi- tional contacts attached to a conductor where no current is drawn. The origin of decoherence in this model is the escape of a carrier out of the coherent volume, which is, after a delay time, re-injected into the conductor.

The full counting statistics gives insight on individual scattering processes contributing to transport. Therefore, it is of great interest to investigate the effect of voltage and dephasing probes on the transport statistics, and how the scattering processes in and out of the probe appear. Chapter 2 is based on two publications, Refs. [108, 109]. We develop a formalism to implement the

1.2 Summary

1

2

3

QPC1 QPC3

p

T

T

Figure 1.11: A voltage (dephasing) probe is perfectly coupled to a two-channel beam splitter structure. In this case, voltage and dephasing probes have very different effects on the transport: A voltage probe can generate positive current-current correlations in terminals 2 and 3, while for a dephasing probe they are negative.

concepts of voltage and dephasing probes as introduced in section 1.1.2 into the full counting statistics. We formulate a stochastic path integral approach for the full counting statistics which treats the escape and injection process for voltage and dephasing probes. This approach uses separation of timescales in an essential manner: the delay time τd of the probe has to be much larger than the inverse average attempt frequency of scattering wave packetsh/eV, but shorter than the measurement time τ.

We demonstrate that the generating function for an arbitrary conductor con- nected to only one single channel voltage probe is identical with the generating function of the conductor connected to a dephasing probe. A central result of our discussion is that the generating function of the one-channel probe models can also be obtained by phase averaging the generating function of the completely coherent conductor. Phase averaging is performed by connecting the conductor with the same strength as the voltage or dephasing probe to an external single mode scatterer characterized by a single phase with a given distribution. In- terestingly, the appropriate phase distribution which leads to a phase averaged result identical to that found from the voltage and dephasing probes is simply a uniform distribution. A mesoscopic scatterer which provides such a uniform distribution is a chaotic cavity with a long dwell time. We illustrate these results with the help of the Mach-Zehnder interferometer, see Fig. 1.10.

For multi-channel or for multi-probe systems, the transport statistics of volt- age and dephasing probes differs and the equivalence with phase averaging is lost.

As illustrated with the example of a beam splitter structure shown in Fig. 1.11, the differences between dephasing and voltage probes arising in the case of two transport channels can be essential: while in the presence of a dephasing probe

0.2 0.4 0.6 0.8 1

-0.175 -0.15 -0.125 -0.1 -0.075 -0.05 -0.025

0.2 0.4 0.6 0.8 1

-0.175 -0.15 -0.125 -0.1 -0.075 -0.05 -0.025

N ln P

T =T =0.7_{A B}

T =T =0.7_{A B}

=0.1 ε=0.5 ε

Vp

n_p

=0.1 ε=0.5 ε ln P dN_E

T_A

B 4 3

2 1

A

R_A ε

Φ p

AB

(a)

(b)

(c)

Figure 1.12: (a): A voltage (dephasing) probe, described by the potential V_p (the occupation functionn_p) is connected to the Mach-Zehnder interferometer with coupling strengthε. The distribution function of time-averaged voltage ¯V_p at the voltage probe (b) and of time-averaged occupation number ¯n_pat the dephasing probe (c) do not show any interference effects. The distribution function per energy interval PE(¯np) for the dephasing probe is broader thanP( ¯V_p) for the voltage probe. The plots are normalized with respect toN =eV τ /hand dN =dEτ /h.

current-current correlations are negative, a voltage probe can generate positive current-current correlations. Furthermore, a conductor connected to two (or more) voltage probes will in general exhibit a generating function that differs from the one obtained with two dephasing probes. We discuss in detail why there are differences between phase averaging and dephasing as soon as there are two or more probes.

1.2.2 Current-voltage correlations

Usually, the full counting statistics gives the distribution function of charge trans- ferred through a conductor. But also internal properties can reveal characteristics of the transport fluctuations. Quantities of interest are for example charge fluc- tuations inside the conductor which are connected to electrostatic fluctuations of the internal potential landscape in a conductor.

Chapter 3 is based on Ref. [110]. We use the voltage and dephasing probe model to investigate in addition to current fluctuations at the true contacts, also the respective voltage and occupation number fluctuations at the probe. This opens the possibility for obtaining correlations between current and voltage or current and the occupation number. The voltage or dephasing probe is attached directly to the phase coherent volume of the conductor and introduces incoherent events. The condition, that no charge is accumulated on the probe on times scales longer then the delay time τd, is maintained by a fluctuating potential Vp(t) at the voltage probe and a fluctuating occupation numbernp(E, t) at the dephasing

1.2 Summary

2 4

6 0.2

0.4 0.6

0.8 -1

-0.75 -0.5 -0.25 0

2 4

6

N

Φ

1 2

p

2 1

ε

Φ

ln P

(a) (b)

Vp d

T T

Figure 1.13: (a) A voltage probe connected with coupling strength ε to a double barrier. Transmission through the two quantum point contacts defining the double barrier is denoted by T₁ and T₂. (b) In contrast to the Mach-Zehnder interferometer (see Fig. 1.12), here, the distribution function of the time-averaged voltage at the probe depends on the interference phase Φ_d of the double barrier.

probe. It is important to distinguish between the fluctuations on the scale τd

and the fluctuations between results of different measurements. The outcome of a measurement is the voltage at the voltage probe ¯Vp and the occupation number at a dephasing probe ¯np, averaged over the measurement time τ. Only for the quantities ¯Vp and ¯np can we determine the fluctuations and their statistics.

We use a quantum Langevin approach for the average quantities and their fluctuations. For higher moments we develop a stochastic path integral approach and determine the joint probability distribution between the voltage fluctuations at the probe and the charge transferred into contacts of the conductor. A com- parison between the voltage probe and the dephasing probe model reveals that the distribution function for voltage differs from the distribution function of the occupation number even for a single-channel conductor. This is surprising, since in chapter 2 we found that in conductors coupled to a single-channel voltage or dephasing probe, the distribution of transferred charge is identical and does not depend on the nature of the probe. Fig. 1.12 shows the statistics of the voltage at a voltage probe and the occupation number at a dephasing probe coupled to the Mach-Zehnder interferometer. Visibly, the distribution function of the occu- pation number ¯np is broader than the distribution function of the voltage ¯Vp at the voltage probe, the differences are less pronounced for weak coupling.

Of particular interest is the probe model for interferometers, we discuss a Mach-Zehnder interferometer, a double barrier and a triple barrier. The probe acts as a phase breaking element, and a particle entering the probe cannot con- tribute to certain interference processes. This effect is sensitive to the particular setup of the interfering structure, and to which part the probe is coupled to. For the MZI, the absence of flux-dependence in the statistics of the probe parameters (see Fig. 1.12) results from the fact that no particle entering or leaving the probe has the possibility to interfere. In the double barrier, though, the distribution

_+ _+ _+

_ +

_

+ _+ _+ _+

=0 (+)

{

=0 (−)

=0 (+)

{

=0 (+)

_+ _+

= +

=

= _ +

_ {

Figure 1.14: A graphical representation of the relations between response theorems.

The number of circles stands for the order of the cumulant, and the number of vertical lines mean the order of the derivative of the cumulant with respect to voltage. The dif- ferently shaded lines (different colors) represent derivatives with respect to quantities of different terminals. The summation in the last line goes over all possible permutations.

functions of voltage ¯Vp and ¯np depend on the phase, see Fig. 1.13. This is ex- pected, because the probability for a process where a particle enters the probe oscillates with the interference phase Φd and is maximal for Φd =π. Clearly, the more particles enter the probe, the better the mean value is determined, and the narrower is the distribution function, as can be seen in Fig. 1.13.

Phase breaking and which path detection are closely related phenomena. This motivates the investigation of the correlations between the fluctuations at the probe as the which path detectors and the transmitted current that can manifest interference oscillations.

1.2.3 Fluctuation relations without micro-reversibility in non- linear transport

The fluctuation-dissipation theorem, as well as the Onsager relations are corner stones of linear irreversible transport theory. Naturally, the question arises if there exist similar relations which extend beyond the linear regime into truly nonlinear steady state transport. Corresponding efforts in this direction were made simultaneously in both the statistical mechanics community and in the theory of transport fluctuations in mesoscopic systems. This leads to fluctuation relations for the full counting statistics in the presence of a magnetic field:

P(Q, B)

P(−Q,−B) =e^AQ. (1.18)

Here, P(Q, B) is the distribution function of Eq. (1.1) for a conductor subject to a perpendicular magnetic field B. The vector A contains the affinities Aα

defined by Aα = eVα/kBT in each terminal where Vα is the voltage applied to terminal α and T the temperature, here equal in all contacts. This fluctuation relation relies fundamentally on Onsager’s principle of microscopic reversibility away from equilibrium. For linear transport, it leads to the fluctuation dissipation

1.2 Summary

2 1

2 2

1

L R

Φ

1

(a) (b)

Γ C

C C

C

V U V

Γ

T_A

T_B

2 4 1

3

U_o U_i

Figure 1.15: Internal potentials inside a conductor lead to magnetic field asymmetry in nonlinear transport: (a) the Mach-Zehnder interferometer and (b) a quantum Hall bar with a resonant impurity. For both setups, the fluctuation relations, Eq. (1.21) can be explicitly verified.

theorem that relates the equilibrium Nyquist noiseS_αβ⁽⁰⁾ to the linear conductance coefficient G⁽¹⁾_β,α, and to the Onsager relations,

S_αβ⁽⁰⁾=kBT(G⁽¹⁾_α,β+G⁽¹⁾_β,α), and G⁽¹⁾_β,α(B) = G⁽¹⁾_α,β(−B). (1.19) In nonlinear electrical transport however, the deviation from the Onsager reci- procity relations have been of considerable theoretical and experimental interest.

Coulomb interactions in a mesoscopic conductor produce an internal potential landscape which is not necessarily symmetric in magnetic field. This effect leads to magnetic field asymmetry in the nonlinear conductance coefficient. The exper- imental evidence confirm the breakdown of micro-reversibility in the nonlinear transport regime.

Chapter 4 is based on Ref. [118]. We demonstrate that away from linear transport, the usual fluctuation relations for the full counting statistics are not valid in the presence of magnetic field asymmetry. Indeed, this is a consequence of gauge invariance of the full counting statistics which imposes that the trans- mission probability Tαβ between contacts αandβ in a conductor depends via the internal potential on the externally applied voltages Vγ. The internal potential landscape is determined self-consistently and leads to a lack of micro-reversibility:

T_αβ(B,{V_γ})6=T_βα(−B,{V_γ}). (1.20) The lack of reversibility leads to a breakdown of the fluctuation relations for the full counting statistics, Eq. (1.18) away from equilibrium. Surprisingly, we never- theless find novel fluctuation relations for current correlations functions without invoking the principle of microscopic reversibility. These relations are valid for systems with arbitrary interactions, as long as the total energy is conserved in the scattering processes. Importantly, our general findings are independent of particular models describing interactions.

Beyond the linear transport regime, the next order fluctuation relation con- nects the third cumulant at equilibrium C_αβγ⁽⁰⁾ with combinations of the noise