Dynamic mesoscopic conductors: single electron sources, full counting statistics, and thermal machines

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Thesis

Reference

Dynamic mesoscopic conductors: single electron sources, full counting statistics, and thermal machines

HOFER, Patrick

Abstract

We theoretically investigate questions regarding the controlled emission and entanglement of individual electrons in mesoscopic circuits, the statistics of current fluctuations and electron waiting times for phase-coherent quantum transport, and thermal machines such as heat engines and refrigerators at the nano-scale. Chapter 2 focuses on dynamic single-electron sources, specifically on the generation of ac spin currents in topological insulators, Mach-Zehnder interferometry with periodic voltage pulses, and the on-demand entanglement of few-electron excitations. In Chapter 3, we present a novel theory for joint electron waiting times and we connect the occurence of negative values in the full counting statistics to a peculiar interference effect. Chapter 4 discusses a heat engine based on the interference in a Mach-Zehnder interferometer as well as implementations for the arguably smallest thermal machines making use of a Josephson junction coupled to harmonic oscillators.

HOFER, Patrick. Dynamic mesoscopic conductors: single electron sources, full counting statistics, and thermal machines . Thèse de doctorat : Univ. Genève, 2016, no.

Sc. 5006

URN : urn:nbn:ch:unige-889778

DOI : 10.13097/archive-ouverte/unige:88977

Available at:

http://archive-ouverte.unige.ch/unige:88977

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

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UNIVERSIT É DE GEN ÈVE Département de physique théorique

FACULT ´E DES SCIENCES Professeur M. B¨uttiker Professeur E. V. Sukhorukov Professeur C. Flindt (Aalto)

Dynamic Mesoscopic Conductors

Single Electron Sources, Full Counting Statistics, and Thermal Machines

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

Patrick P. Hofer

de Meilen (ZH)

Th` ese N

^o

5006

GEN` EVE

Atelier de reproduction ReproMail

2016

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Acknowledgments

Completing this thesis was only possible thanks to the guidance, collaboration, and support of many people.

I start by acknowledging the people who supervised me. I started my PhD under the supervision of Markus B¨uttiker who introduced me to the field of mesoscopic physics. He sadly passed away on October 4, 2013.

Although he was supervising me for less than a year, I learned a lot from this brilliant physicist, and his advice and opinions defined the topics and style of the research presented in this thesis. I am very grateful to all the people who helped during this difficult time in October 2013, especially Andrew Jordan and G´eraldine Haack who by that time were former group members. I would also like to point out Ruth Durrer who right away expressed the full support of the department and Eugene Sukhorukov who took over the formal responsibilities of supervising the PhD students of our group. The scientific supervision of the group was taken over by Christian Flindt. This was a tremendous effort and it is thanks to him that the group was able to continue to produce scientific results and to look ahead. I would like to thank Christian for all his advice and motivation as a supervisor as well as for the good times I had with him as a friend. After another year in Geneva, I went to Montreal for one year where I was supervised by Aashish Clerk. I am very grateful to Aashish for giving me the opportunity to profit from a different perspective on mesoscopic physics and learn many new techniques and concepts. I was impressed by his ability of ensuring a high quality output by critically analyzing the work of himself and his collaborators and focusing on the essential physics.

I would also like to thank all my previous supervisors and teachers, notably Christoph Bruder who taught me quantum mechanics, as well as many other theory classes, and guided me through most of my masters at the University of Basel.

In addition to the supervisors who guided me and had many of the ideas that I worked on, I profited from a number of great collaborators. My thanks to Hugo Aramberri, Joe Bowles, David Dasenbrook, Mart´ı Perarnau-Llobet, Pierre Delplace, Marcus Huber, Christoph Schenke, Ralph Silva, Björn Sothmann, Jean- René Souquet, Nicolas Brunner, Ivar Martin, and Alberto Morpurgo. It was a pleasure to collaborate with all of you. I especially want to point out David Dasenbrook who started his PhD shortly before me and in the same group. In addition to inspiring discussions about physics, we spend a lot of time together playing and listening to music and getting to know Geneva. Special thanks also to Jean-René Souquet who motivated me to play hockey in Montreal and with whom I had many interesting discussions about physics as well as other topics.

I also want to express my thanks to the many friends and colleagues who made the time during my PhD as enjoyable as it was. I want to thank my office mates in Geneva, Dania Kambly, Jian Li, Giulia Cusin, and Davide Racco for the friendly and relaxed working atmosphere. I also want to thank all the people who were members of the Büttiker group while I did my PhD, David Dasenbrook, Dania Kambly, Konrad Thomas, Pierre Delplace, Géraldine Haack, Jian Li, Johan Ott, Christoph Schenke, Björn Sothmann, and Christian Flindt. The discussions during our group meetings and coffee breaks were a healthy mix of being productive and entertaining. Special thanks to Johan for his sense of humor and for organizing many nice evenings, to Pierre for welcoming me to the Büttiker group with his joyful jokes, and to Björn for all the fun facts. Thanks also to Francine Gennai-Nicole and Cécile Jaggi-Chevalley for competently handling all the bureaucracy and thanks to Andreas Malaspinas for the IT support. I also want to thank the members of the band le Fuzzball, David Dasenbrook, Irwin Law, Andrew Terrett, Amiyna Farouque, Joe Bowles, Margot Hill Clarvis, and Mona El Isa. Our rehearsals and concerts are a very welcome distraction and source of inspiration. I would also like to thank Michelle Büttiker who often joined us for the coffee breaks and came to some of our concerts. Thanks also to Noé Cuneo for the language tandems and for proofreading the French part of this thesis and thanks to Andrea Agazzi for his adventurous nature.

I further want to thank the people who made my stay in Montreal unforgettable. Thanks to my colleagues Jean-Ren´e Souquet, Alexandre Baksic, Nicolas Didier, F´elix Beaudoin, and Martin Houde, for teaching me French during lots of funny conversations in the bars of Montreal. Thanks to Ann-Laurien Haag, Zeno

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Schumacher, Andreas Spielhofer, Gord Margison, and Aaron Mascaro for all the nice hikes, sushi dinners, hockey games, and parties. Thanks to my entertaining room-mates Kubila¨ı Iksel, Jean de La Thibauderie, Reda Berrada, Jordan Stoker, Nicolas Corizzi, and Schopenhauer the cat.

I am grateful to Nicolas Brunner, Carlo Beenakker, Aashish Clerk, Christian Flindt, and Eugene Sukho- rukov for agreeing to be on the committee for my thesis defense.

Last but not least I want to thank my parents, Gertrud and Peter Hofer, my sister Karina Gisiger and her family, and my girlfriend Heidi Potts for all the support and love during these four years. Thank you.

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Introduction

“...the imagination of nature is far, far greater than the imagination of man.”

Richard Feynman

This thesis is about driven systems which behave according to the laws of quantum mechanics in a way that can not be explained by classical theories. Although this thesis is theoretical in nature, the systems under investigation are meant to be implementable experimentally (either now or in the near future) in condensed matter or quantum optics laboratories. This defines a minimum size for the devices under consideration which are typically made out of more than 10¹⁰ atoms (this is roughly how many atoms are in one cubic micrometer of crystalline silicon). This thesis constitutes thus a contribution to mesoscopic physics, the branch of physics that investigates quantum effects in systems that are in between the very microscopic, like single atoms or molecules, and the macroscopic, like everyday life objects which consist of the order of 10²³ particles (number of atoms in a cubic centimeter of crystalline silicon).

Quantum mechanics has been immensely successful in explaining the microscopic world. For instance, it provided the answers to important questions such as “Why does the electron in a hydrogen atom not fall into the nucleus?” and led to a whole new understanding of chemistry. At the same time, through its counter- intuitive predictions, it challenged fundamental beliefs, such as the principle of locality which states that an object is only affected by its immediate surroundings. Albeit its immense success and the vast amount of past and ongoing research on quantum mechanics, its counter-intuitive nature keeps puzzling physicists and there is no general agreement on the answer to the very fundamental question “What happens in a measurement?”. It is extremely surprising that a theory which leaves open such fundamental questions can be one of the most successful theories in predicting physical outcomes. In light of that, one could be inclined to believe that the investigation of physical systems and observables is the easier route to understanding quantum mechanics than to directly tackle the fundamental questions which defied a satisfactory explanation for so long.

In addition to obtaining a better understanding of quantum mechanics, a major motivation to investigate mesoscopic systems is to develop technologies which exploit quantum effects such as coherence and entanglement. There are to date a number of technological applications that rely on quantum mechanics, such as the laser, which relies on the wave-particle duality of light, or sensitive magnetometers that rely on superconducting quantum interference devices (SQUIDs). Many of these applications, especially those that rely on fragile quantum properties such as coherence and entanglement, are used for fundamental research only and are not yet available in commercial products. The principal reason for this is that quantum effects are mainly observed in systems that are isolated from their surroundings and cooled to very low temperatures (typically in the millikelvin range for the systems considered here). However, there is an increasing body of literature on applications that make use of quantum coherence and entanglement and it has been speculated that quantum technologies will be responsible for many of the technological advances in the 21st century [Dowling03].

The goal of the present thesis is to contribute towards the development of such technologies that harness quantum coherence and entanglement. Mesoscopic physics is tailored for such investigations as it considers quantum effects in systems that are big enough to be manipulated in proof-of-principle experiments. The choice of the physical systems discussed in this thesis is mainly determined by recent experimental progress.

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Most of the thesis focuses on driven electronic systems, where the controlled emission and interference of single electrons has recently been reported [F`eve07; Dubois13b; Bocquillon13]. Such an impressive level of control constitutes a promising avenue towards quantum technologies. The rest of the thesis discusses microwave resonators which are coupled to a Josephson junction. Since microwaves have wavelengths of the order of centimeters, these resonators are relatively big objects, which facilitates their fabrication. However, they nevertheless behave quantum mechanically and offer a versatile test-bed to probe quantum behavior.

The body of this thesis is grouped into three chapters. In Chap.2, we discuss our proposals that make use of single-electron sources. We show how these sources can be used to create entangled electrons and a pure ac-spin current in a quantum spin Hall material. Furthermore, we discuss interferometry of single-electron excitations and show how they can be used to create and detect entanglement using a single electron. Before discussing these proposals, we give an introduction to the concepts and theories that are used to study these systems.

Chapter3is dedicated to full counting statistics (FCS) and waiting time distributions (WTD). These are tools to describe and characterize electronic transport. The WTD in particular is a promising and still rather unexplored candidate to characterize the above mentioned single-electron emitters. Our contribution to this theory as well as illustrations are discussed in Sec. 3.1. In addition to characterizing electronic transport, the FCS can be used to describe a variety of dynamical processes. Interestingly, this quantity can fail to be positive and is therefore a quasi-probability distribution. In Sec. 3.2, we describe how negative FCS can be linked to an unusual interference phenomenon and thus directly indicates non-classical behavior much like negativities in the widely used Wigner function.

In Chap.4, we discuss thermal machines that are based on quantum effects. In addition to a heat engine that relies on the interference of single electrons, we discuss a heat engine as well as an absorption refrigerator based on microwave resonators coupled to a Josephson junction. Coupling the resonators to different thermal baths, together with the peculiar coupling provided by the Josephson junction, allows for the construction of very efficient, small, and autonomous thermal machines. Such machines are of considerable interest in the quantum thermodynamics community as they allow us to study energy conversion in quantum systems with only a few degrees of freedom. Our proposals may pave the way for experimental studies of such thermal machines.

Most of the work presented in this thesis is published in scientific journals. While an effort has been undertaken in unifying the notation and making the thesis self-consistent, many text-passages have been copied verbatim from the published papers. Throughout the thesis, we set~= 1 and e <0 is the charge of an electron.

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Chapter 2

Single-electron sources

This chapter discusses nanoscopic electronic devices which are able to emit single electrons above the Fermi sea with the use of time-dependent voltages. Allowing for a time-dependent control on the very smallest of electronic scales, such devices have motivated a number of theoretical as well as experimental works investigating the coherent propagation of single and few electron states through mesoscopic conductors.

Many of the considered setups are motivated from the field of quantum optics which is why this field has become known as electron quantum optics [Bocquillon14].

Figure 2.1: Single-electron sources. (a) Mesoscopic capacitor. An ac voltage applied to a gate capacitively coupled to a quantum dot leads to the emission of single electrons that have a Lorentzian profile in energy and an exponential profile in space. Reprinted from [Gabelli06]. (b) Levitons. Applying Lorentzian voltage pulses to an Ohmic contact leads to the emission of single electrons with a Lorentzian profile in space and an exponential profile in energy. Reprinted from [Dubois13b].

The two single-electron sources discussed in this thesis are illustrated in Fig.2.1. The first one is provided by a mesoscopic capacitor, the quantum analogue of a classicalRC-circuit. There, a top gate is used to “push out” a single electron of a quantum dot. By applying an ac voltage, an alternating stream of electrons and holes can be emitted. Due to the Lorentzian shape of the energy levels in the quantum dot, these particles have a Lorentzian profile in energy or, equivalently, an exponential profile in space. Experimentally, the mesoscopic capacitor has first been used as a single-electron source by Fèveet al. [Fève07], followed by a theoretical discussion by Moskalets, Samuelsson, and Büttiker [Moskalets08].

The second method of producing single-electrons is provided by applying Lorentzian voltage pulses to an Ohmic contact. This yields electrons with a Lorentzian profile in space and an exponential profile in energy. In terms of the wavefunction the two sources can thus be said to be dual to each other. This method of creating single-electron excitations has first been discussed by Levitov and Lesovik [Levitov96], was further studied by Keeling, Klich, and Levitov [Keeling06], and experimentally implemented by Dubois et al. [Dubois13b] who coined these excitationslevitons.

In our works we investigate the mesoscopic capacitor as a means to inject single-electrons into the edge states of two-dimensional topological insulators (see Sec. 2.3). We find that this can lead to the emission of time-bin entangled particles (Sec. 2.3.3) and to the creation of a pure ac-spin current (Sec.2.3.4). Addi-

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tionally, we investigate single- and two-particle interferometry with levitons (see Sec.2.4). A Mach-Zehnder interferometer reveals signatures of time-dependent coherent transport (Sec. 2.4.1) and a Hanbury-Brown and Twiss interferometer can be used to demonstrate mode-entanglement of a single leviton (Sec.2.4.2). Be- fore discussing these works in detail, we introduce the Floquet scattering theory that we employ to describe these systems and we give a brief introduction to on-demand entanglement generated by single-electron sources.

2.1 Floquet scattering theory

The Floquet scattering theory was developed by Moskalets and B¨uttiker in a series of papers. Here we closely follow the book written by Moskalets [Moskalets12] where all the relevant references can be found. At the heart of the Floquet scattering theory lies the Floquet theorem [Floquet83] which states that the solutions to the Schr¨odinger equation for a periodically time-dependent Hamiltonian ˆH(t+T) = ˆH(t) can be written as

|Ψ(t)i=e^−iEt|Φ(t)i, with |Φ(t+T)i=|Φ(t)i. (2.1) Note that this theorem (as well as its proof) is analogous to the Bloch theorem. Expanding the periodic part of the state |Φ(t)iin a Fourier series yields the following form for the solutions of the Schr¨odinger equation

|Ψ(t)i=e^−iEt

∞

X

q=−∞

e^−iqΩt|ψ_qi, (2.2)

with Ω = 2π/T. From the last equation, we can infer that energy is not uniquely defined in a system described by a time-dependent Hamiltonian. Indeed, shifting E → E+pΩ and relabeling |ψqi → |ψq+pi leaves the state invariant. Thus the quasi-energy or Floquet-energy E is only defined up to the energy quantum Ω and (in analogy to Bloch waves) it suffices to consider the energy rangeE∈[0,Ω].

Considering non-interacting electrons, one can calculate a scattering matrix which connects incoming states with outgoing states analogous to time-independent scattering theory [Blanter00]. To this end, we consider a mesoscopic conductor which connects metallic contacts through a scattering region. As usual, we neglect backscattering at the contact interface. Furthermore, we assume that the Hamiltonian is time- independent in the contacts. In this case, far away from the scattering region, the states look like plane waves (or, in a crystal, Bloch waves) and can be characterized by their energy, their direction of propagation, an index for the contact, and an additional channel index which characterizes the transversal part of the wavefunction. Although these quantum numbers are motivated by the local Hamiltonian in contactα, they uniquely determine the wavefunction in every part of the conductor (together with the Schr¨odinger equation).

We denote these scattering-states as Ψ^in/outα,n (E), where α denotes the contact, n the channel, and E the energy. The superscript reads infor states propagating toward the scatterer andoutfor states propagating toward contactα. Using Eq. (2.1), the scattering states forxclose to contactβ can be expressed as

Ψⁱⁿ_α,n(E) =δβ,α

e^−ik^α,nⁱⁿ ^(E)x q|v_α,nⁱⁿ (E)|

ψ_α,nⁱⁿ (E)e^−iEt+X

m

∞

X

q=−∞

[S_βα^mn(Eq, E)]^∗ eîk^β,môut^(E^q^)x q|vôut_β,m(Eq)|

ψ^out_β,m(Eq)e^−iE^q^t,

Ψ^out_α,n(E) =δβ,α

eîkôut^α,n^(E)x q|v_α,nôut(E)|

ψ_α,n^out(E)e^−iEt+X

m

∞

X

q=−∞

S_αβ^nm(E, Eq)e^−ik^β,mⁱⁿ ^(E^q^)x q|vⁱⁿ_β,m(Eq)|

ψ_β,mⁱⁿ (Eq)e^−iE^q^t.

(2.3)

Herexgrows towards contactβ,Eq =E+qΩ,kα,nîn/out(E) is the absolute value of thekvector corresponding to channelnin contactαwith propagation direction specified by the superscript,vα,nîn/out(E) is the corresponding group velocity, and ψîn/outα,n (E) includes the transversal part of the wavefunction (and, for Bloch waves, a function periodic in x). Note that Ψîn/outα,n (E) is an eigenstate of the local Hamiltonian close to contactα.

The coefficients S_αβ^nm(E⁰, E) can be found by wavefunction matching and define the Floquet scattering matrix. They relate current amplitudes in the incoming channel defined by the quantum numbersβ,m, E to current amplitudes in the outgoing channel defined by the quantum numbers α,n,E⁰.

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Conservation of current implies that the scattering matrix is unitary, i.e. it fulfills the conditions X

q,α

S^∗_αβ(E_q, E_l)S_αγ(E_q, E_p) =δ_l,pδ_β,γ, X

q,α

S^∗_γα(El, Eq)Sβα(Ep, Eq) =δl,pδβ,γ.

(2.4)

Here and below we included the channel index in the indexαwhich now denotes the contact and the channel.

We note that both the incoming as well as the outgoing scattering states form a complete orthogonal basis (up to bound states which do not contribute to transport). This is known as the completeness theorem in scattering theory [Farina73]. The incoming and the outgoing scattering states are simply related by the scattering matrix [cf. Eqs. (2.3) and (2.4)]

Ψ^out_α (E) =X

β,q

S_αβ(E, E_q)Ψⁱⁿ_β(E_q), Ψⁱⁿ_α(E) =X

β,q

S_βα^∗ (E_q, E)Ψôut_β (E_q). (2.5) Finally, we introduce the second quantized operators â_α(E) which annihilate the incoming states Ψⁱⁿ_α(E) and the operators ˆbα(E) which annihilate the outgoing states Ψôut_α (E). These operators fulfill the canonical anti-commutation relations

{â_α(E),â^†(E⁰)}=δ_α,βδ(E−E⁰), {â^(†)_α (E),â^(†)(E⁰)}= 0, {ˆbα(E),ˆb^†(E⁰)}=δα,βδ(E−E⁰), {ˆb^(†)_α (E),ˆb^(†)(E⁰)}= 0,

(2.6) and inherit the relation given in Eq. (2.5)

ˆb_α(E) =X

β,q

S_αβ(E, E_q)ˆa_β(E_q), ˆa_α(E) =X

β,q

S_βα^∗ (E_q, E)ˆb_β(E_q). (2.7) With the help of these operators, any operator of interest can be expressed. For transport quantities, the basic operator is the current operator. Under the assumption that the relevant energies are close to the Fermi energy such that the group velocities can be treated as energy-independent, the current operator takes on the simple form [B¨uttiker92]

Iˆα(t) = e 2π

X

n

Z

dEdE⁰e^i(E−E⁰^)th

ˆb^†_α,n(E)ˆbα,n(E⁰)−ˆa^†_α,n(E)ˆaα,n(E⁰)i

, (2.8)

where we reinstated the channel index such that ˆIαdenotes the total current flowing through contactα. In the frequency domain, the current operator is given by the Fourier transform of the last expression

Iˆα(ω)≡ Z ∞

−∞

dte^iωtIˆα(t) =eX

n

Z dEh

ˆb^†_α,n(E)ˆbα,n(E+ω)−ˆa^†_α,n(E)ˆaα,n(E+ω)i

, (2.9)

In order to calculate the average values for the current and current correlators, one needs to know the occupation of the scattering states. Because the incoming states originate from a single contact, we can assume

hâ^†_α(E)âβ(E⁰)i=δα,βδ(E−E⁰)fα(E), hâ^†_α(E)âβ(E⁰)i=δα,βδ(E−E⁰)[1−fα(E)], (2.10) where fα(E) denotes the Fermi distribution which describes the local equilibrium distribution in contactα

f_α(E) = 1 1 +e^E−µα^{kB Tα}

, (2.11)

where µ_α denotes the chemical potential and T_α the temperature in contact αand k_B is the Boltzmann constant.

Employing Eqs. (2.7) and (2.10), we find that the outgoing states are not distributed according to equilibrium distributions. Indeed outgoing states in different channels and at different energies can be correlated

hˆb^†_α(E)ˆbβ(E⁰)i= X

p,q,γ

δ(E⁰−Ep)S_αγ^∗ (E, Eq)Sβγ(Ep, Eq)fγ(Eq). (2.12)

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We note that a scatterer with periodicity T can only correlate states at energies that differ by an integer multiple of Ω.

We are now in the position to calculate averages involving arbitrary many current operators. To this end, we express the quantity of interest in terms of averages of ˆa operators using the scattering matrix [cf. Eq. (2.7)]. Since we are dealing with non-interacting electrons, the average of an arbitrary number of ˆ

a operators can be broken down into averages of pairs using Wick’s theorem [Wick50] which can then be evaluated using Eq. (2.10). In the following we present some expressions which are obtained in this manner without their explicit derivation.

Expressions for observables

In the frequency domain, the average current can be written as I_α(ω)≡ hIˆ_α(ω)i=

∞

X

l=−∞

2πδ(ω−lΩ)I_α,l, (2.13)

with the Fourier coefficients I_α,l = e

2π X

q,β

Z

dES_αβ^∗ (E, E_q)S_αβ(E_l, E_q) [f_β(E_q)−f_α(E)]. (2.14) In terms of these coefficients, the time dependent current reads

I_α(t)≡ hIˆ_α(t)i=

∞

X

l=−∞

e^−ilΩtI_α,l. (2.15)

Often we will be interested in the zero frequency current Iα≡Iα,0=

Z δω

−δω

dω

2πhIˆα(ω)i= 1 T

Z T 0

dthIˆα(t)i

=X

q,β

Z

dE|Sαβ(E, Eq)|²[fβ(Eq)−fα(E)],

(2.16)

where δωdenotes a bandwidth much smaller than Ω.

As expressed in Landauer’s famous quote: “The noise is the signal” [Landauer98], much information can be gained by going beyond the average current and investigating current correlators. In the frequency domain, the noise correlator is defined as

1

2h{∆ ˆIα(ω),∆ ˆIβ(ω⁰)}i=

∞

X

l=−∞

2πδ(ω+ω⁰−lΩ)P^αβ,l(ω), (2.17) where the current noise operator reads ∆ ˆIα(ω) = ˆIα(ω)− hIˆα(ω)iand {·,·} denotes the anti-commutator.

The quantity of interestP^αβ,l(ω) is called the noise spectral density. It relates to the noise correlator in the time domain as

P^αβ(t, t⁰) = 1

2h{∆ ˆI_α(t),∆ ˆI_β(t⁰)}i=

∞

X

l=−∞

e^−ilΩt⁰ Z ∞

−∞

dω

2πe^−iω(t−t⁰⁾P^αβ,l(ω). (2.18) As for the average current, we will often be interested in the zero frequency noise which can be written as

P^αβ≡P^αβ,0(0) = 1 2

Z T 0

dτ Z ∞

−∞

dth{∆ ˆIα(t),∆ ˆIβ(t+τ)}i

=e² 2π

Z dE

δαβ

fα(E) [1−fα(E)] +X

γ,n

|Sαγ(En, E)|²fγ(E) [1−fγ(E)]

−X

n

|S_βα(E_n, E)|²f_α(E) [1−f_α(E)]− |S_αβ(E_n, E)|²f_β(E) [1−f_β(E)]

+X

γ,δ

X

n,m,p

[f_γ(E_n)−f_δ(E_m)]²

2 S_αγ^∗ (E, E_n)S_αδ(E, E_m)S_βδ^∗ (E_p, E_m)S_βγ(E_p, E_n)

.

(2.19)

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Finally, we note that the expressions for the time independent case can be recovered usingS(E_n, E_m) = δ_n,mS(E), whereS(E) is the scattering matrix for the time independent problem.

Adiabatic limit

We will now discuss a particularly simple limit, called the adiabatic limit, which is correct for Ω →0. In this limit, the scatterer changes over timescales much longer than the time a single electron stays in the scatterer. The scatterer can then be described by the instantaneous scattering matrix S(E, t) which is the scattering matrix that describes the static system obtained by “freezing” the dynamic system at timet. In the adiabatic limit, scattering of an electron has the effect of multiplying the wavefunction by S(E, t). By Fourier expanding the instantaneous scattering matrix

S(E, t) =

∞

X

n=−∞

e^−inΩtS_n(E), (2.20)

we identify Sn(E) as the amplitudes with which a scattered electron absorbsnphotons which make up the Floquet scattering matrix

Sn(E) =S(En, E) =S(E, E_−n) = 1 T

Z T 0

e^inΩtS(E, t). (2.21) We note that in the adiabatic limit, the Floquet scattering matrix does not change when changing its energy arguments by nΩ (since the adiabatic limit corresponds to Ω →0). In practice, processes involving many photons can often be neglected and n is limited by some value n_max. One can then check the validity of the adiabatic approximation by verifying thatS(E_n_max, E)≈S(E, E_−n_max) for the relevant energies. In the adiabatic limit, the unitarity of the Floquet scattering matrix is reduced to the unitarity of the instantaneous scattering matrix

S^†(E, t)S(E, t) =1. (2.22)

The time-dependent current induced by a dynamic scatterer in the adiabatic limit reads Iα(t) =−ie

2π Z

dE

−∂f(E)

∂E S(E, t)∂S^†(t, E)

∂t

αα

. (2.23)

Herefα(E) =f(E)∀αwhich is in principle only valid if all the contacts are at the same chemical potential.

As we show below, (possibly time-dependent) voltages at the contacts can be incorporated into the scattering matrix.

Mixed energy time representation

In the non-adiabatic case, it can still be useful to introduce the Fourier transforms of the Floquet scattering matrix

Sin(t, E) =

∞

X

n=−∞

e^−inΩtS(En, E), Sout(E, t) =

∞

X

n=−∞

e^inΩtS(E, En). (2.24) It turns out that these matrices have a physical interpretation. Sin(t, E) describes scattering of particles incident with energyEwhich leave the scatterer at timetwhileSout(E, t) describes particles which enter the scatterer at time t and leave it with energyE [Moskalets12]. In the adiabatic limit, both of these matrices reduce to the frozen scattering matrix. Due to this interpretation, it is often easier to obtain the Floquet scattering matrix via Fourier transform of the last equation than by a direct calculation.

Time-dependent voltages

At first sight, it seems as if time-dependent voltages can not be described by Floquet scattering theory because the contacts are no longer time-independent. However, using a gauge transformation, the time-dependent voltages can be treated as time-dependent scattering phases that are picked up when electrons leave or enter the contacts. Then the contacts can still be treated as time-independent and in local equilibrium. To illustrate this, we consider a one-dimensional conductor where the region x <0 defines the contact which is

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driven with a periodic voltageV(t) with frequency Ω. Linearizing the spectrum, the right-moving electrons (the ones leaving the contact) are governed by the single-particle Schr¨odinger equation

i∂tΨ(x, t) = ˆHΨ(x, t) = [−ivF∂x+eV(t)Θ(−x)] Ψ(x, t), (2.25) where vF denotes the Fermi energy and Θ(x) the Heaviside step function. Using a time dependent gauge transformation, the voltage can be mapped onto a Dirac delta potential

Hˆ⁰=e^{iϕ(t)Θ(−x)}Heˆ −iϕ(t)Θ(−x)

−∂_tϕ(t)Θ(−x) =−iv_F∂_x+δ(x)ϕ(t), (2.26) where the phaseϕ(t) reads

ϕ(t) = Z t

t₀

dt⁰eV(t⁰), (2.27)

and t0 is a time in the distant past where the voltage was switched on.

The solutions of the Schr¨odinger equation defined by the Hamiltonian in Eq. (2.26) read

Ψ⁰(x, t) = Ψ₀(x, t)e^{−iϕ(t−x/v}^F^)Θ(x), (2.28) where Ψ₀(x, t) is a solution to the Schr¨odinger equation in the absence of the voltage. We thus see that the time-dependent voltage can be treated as a scattering phase which is obtained when the electrons leave the contact (at x= 0). Because this phase is energy independent, the adiabatic approximation holds and the Fourier coefficients of the scattering phase define the Floquet scattering amplitudes

Sn= 1 T

Z T 0

dte^inΩte^−iϕ(t). (2.29)

Another way to see that the adiabatic approximation is valid is by noting that the scattering region is infinitesimal (the phase is picked up at a single point in space). Therefore the time an electron spends in the scattering region goes to zero.

In this treatment, the electrons that leave the contactαare described by the operators ˆ

a⁰_α(E) =X

n

Snˆaα(E_−n), (2.30)

where the ˆa(E) operators are in local equilibrium as described by Eq. (2.10).

In addition to time dependent voltages, constant voltages can be treated in the same manner. Indeed a voltage equal to eV = Ω is described by Sn =δn,1. A constant voltage thus increases the energy of all the electrons by eV. For a linear spectrum, this is equivalent to filling all states up to an energyeV above the equilibrium chemical potential (i.e. to shifting the Fermi distributions by eV).

Finally, we note that electrons entering a driven contact pick up the phase −ϕ(t). This can become important for preserving current conservation. It is also important to note that the assumption of a linear spectrum is crucial in the above derivation. In practice, this restricts the applicability of this method to voltages much smaller than the Fermi energy.

2.2 On-demand entanglement from single-electron sources

This section is closely based on Ref. [Hofer16c].

An important and recurring theme of the work presented in this thesis is entanglement in electronic structures. In particular, on-demand entanglement that is provided by single-electron sources. We thus give a brief introduction to this topic before introducing the sources that are used for the entanglement creation.

As we will now show, the on-demand entanglement obtained from single-electron sources can often be traced back to the entanglement of a single electron that is in a superposition of spatially separated modes [Hofer16c; Dasenbrook16a]. Using a quantum point contact (QPC), a single electron can be partitioned between two locally separated parties Alice and Bob, creating the state

|Ψ1i= 1

√2 ˆc^†_A+ ˆc^†_B

|0i= 1

√2 |1A,0Bi+|0A,1Bi

, (2.31)

where ˆc^†_α creates a particle localized with party α and |0i = |0_A,0_Bi denotes the state with no (excess) particles with either party. As discussed in Sec. 2.4.2(see also Ref. [Friis16b]), this state can be considered

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entangled. The reason that this is not obvious lies in the fact that the local operations Alice and Bob can perform are constrained by parity and charge superselection rules. This constrains measurements to be in the Fock basis, excluding measurements of superpositions (such as |0i+|1i) which are required to violate a Bell inequality with the state given in Eq. (2.31).

The entanglement in Eq. (2.31) can nevertheless be accessed using a second copy of the same state [Bartlett07;Dasenbrook16a] as we will now illustrate. Since we are dealing with Fermions, we can not have two particles in the same state. We thus introduce an additional degree of freedom that we denote by the subscript ±. The state given by two copies of Eq. (2.31) then reads

|Ψ2i= 1

2 ˆc^†_A,++ ˆc^†_B,+

ˆ

c^†_A,−+ ˆc^†_B,−

|0i. (2.32)

Here±could denote any additional degree of freedom such as the orbital degree of freedom [Dasenbrook16a], the time the particles traverse the QPC [Splettstoesser09; Hofer13], the nature of the particle (electron vs.

hole) [Dasenbrook15], or the spin degree of freedom [Lebedev04].

In order to discuss the structure of the last state, it is helpful to write it in first-quantized form like the right hand side of Eq. (2.31). The additional degree of freedom ± allows for different first-quantized representations of the last state which correspond to different bi-partitions of the Hilbert space H. To illustrate this, we consider two different bi-partitions. The first bi-partition is determined by the parties Alice and Bob

H=H^A⊗ H^B. (2.33)

The basis states of subspace α=A, B are given by

|0αi, |+αi= ˆc^†_α,+|0αi, |−^αi= ˆc^†_α,−|0αi, |2αi= ˆc^†_α,+ˆc^†_α,−|0αi. (2.34) We will call this bi-partition the AB-partition. Since Alice and Bob are assumed to be localized in space but separated from each other, we will be mostly interested in entanglement with respect to this, local bi-partition. With respect to the AB-partition, the state in Eq. (2.32) reads

|Ψ2i= 1

2 |2A,0Bi+|0A,2Bi+|+A,−^Bi+|−^A,+Bi

, (2.35)

which can clearly not be written as a product state. Furthermore, as discussed in Sec. 2.4.2, this state can be used to violate a Bell inequality without the use of measurements that are forbidden by superselection rules.

A different, but in principle equally valid bi-partition of the Hilbert space is given by

H=H⁺⊗ H⁻, (2.36)

and will be denoted as the ±-partition. Here the basis states of the subspacej=±are given by

|0ji, |Aji= ˆc^†_A,j|0ji, |Bji= ˆc^†_B,j|0ji, |2ji= ˆc^†_A,jcˆ^†_B,j|0ji. (2.37) Noting that the vacuum is the same in both bi-partitions |0i=|0A,0Bi=|0+,0₋iwe find that the state in Eq. (2.32) can be written as

|Ψ2i=1

2 |A+i+|B+i

⊗ |A₋i+|B₋i

, (2.38)

which is a product state. Thus we find that |Ψ2iis entangled with respect to the AB-partition, but it is not entangled with respect to the±partition. This underlines the fact that entanglement is defined with respect to a bi-partition (or, more generally, a multi-partition) of the Hilbert space [Zanardi04]. Because the degree of freedom ±labels the particles, a lack of entanglement in the±-partition reflects the fact that there is no entanglement between the two particles. This is expected because the state is constructed by taking two uncorrelated copies of the single-particle state in Eq. (2.31).

Now we are in a position to close our argument which unambiguously shows that Eq. (2.31) should be considered entangled. If the state in Eq. (2.31) was separable, then a product of two copies of this state, given in Eq. (2.38), should also be separable. A violation of a Bell inequality using |Ψ₂i (as discussed in Sec.2.4.2) therefore demonstrates the entanglement present in|Ψ1i.

Having established that the entanglement in|Ψ2iconstitutes single-particle entanglement, we note that Alice and Bob can use the state|Ψ2iin order to entangle the two particles, i.e. create entanglement in the

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±-partition. Using a detector which only measures the number of particles but not their internal degree of freedom ±, Alice and Bob can project onto the subspace where each party receives a single particle. This yields the Bell state

|Φi= 1

√2 |+_A,−^Bi+|−^A,+_Bi

, (2.39)

which is fully entangled with respect to both the AB as well as the ±-partition. In the ±-partition, it is thus the projection (a local operation with respect to theAB-partition) which created the entanglement (see also Refs. [Wiseman03;Vaccaro03;Lebedev04]). Since this projection constitutes a non-local operation with respect to the ±-partition, it is not surprising that it can create entanglement between the two particles.

The above procedure is reminiscent of the entanglement distillation discussed in Ref. [Popescu95].

We conclude that the entanglement between the parties Alice and Bob originates from the single-particle entanglement of the state in Eq. (2.31), whereas entanglement between the two particles can be created by the projection onto the subspace containing a single particle at each location. These concepts are illustrated with examples in Secs.2.3.3and2.4.2.

2.3 Mesoscopic Capacitors in quantum spin Hall insulators

In this section, we discuss our works on the mesoscopic capacitor in two-dimensional quantum spin Hall insulators. In these topological insulators, current is carried by counter-propagating states with opposite spin polarization localized at the edges of the material. In Sec. 2.3.1, we give a brief introduction to edge states of topological insulators. For comparison and further reference, we then introduce the mesoscopic capacitor in quantum Hall insulators in Sec.2.3.2before presenting our own work in Secs.2.3.3and2.3.4.

2.3.1 Edge states of topological insulators

Topological insulators are materials which are described by a topological number characterizing their phase.

Within the boundaries of the topological insulator, this integer number is constant and thus constitutes a global property of the insulator. This is in stark contrast to more conventional materials (e.g. a magnet) which can be described by a local order parameter (e.g. the magnetization) which might vary within the material.

Topological insulators are protected by the energy gap in their spectrum. This means that the gap has to close for the topological number to change. At the boundaries of topological insulators, the phase changes from topologically non-trivial in the insulator to topologically trivial in the vacuum. As a consequence, the gap has to close leading to edge states localized at the boundary of a topological insulator. We now briefly introduce two kinds of two-dimensional topological insulators which exhibit one-dimensional, current- carrying edge states. For further information on the quickly expanding field of topological insulators, we refer the reader to the book by Bernevig and Hughes [Bernevig13] and the reviews by Qi and Zhang [Qi11]

and by Hasan and Kane [Hasan10]. For more information on quantum Hall effects, we recommend the review by Goerbig [Goerbig09].

Quantum Hall insulators

The most prominent two-dimensional topological insulator is provided by a two-dimensional electron gas brought into the quantum Hall regime by a strong perpendicular magnetic field. We illustrate this by considering free electrons in two-dimensions. The single-particle Hamiltonian in real space governing such a system reads

Hˆ = 1

2m(−i∇ −eA)²+eV(y) +gµ_BBˆσ_z, (2.40) where∇= (∂_x, ∂_y) andA= (A_x, A_y),V(y) is a confinement potential,B =∂_xA_y−∂_yA_xthe magnetic field which points in the z-direction, g is the g-factor, µ_B the Bohr magneton, and the Pauli matrix ˆσ_z acts on the spin degree of freedom. In the Landau gauge (Ax=−ByandAy= 0) the Hamiltonian is translationally invariant in the x-direction implying that the momentum in the x-direction is a good quantum number. In the absence of a confinement potential, the eigenstates of the last Hamiltonian can be expressed in terms of the Hermite polynomialsHn(x)

Ψn,k_x,σ(x, y) =e^ik^x^xHn

y−y0

lB

e^−(y−y⁰^)/2l²^B|σi, (2.41)

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with the eigenenergies

E_n,σ=nω_C+σgµ_BB (2.42)

Here we introduced the magnetic length lB = 1/p

|eB|, y0 =kxl²_B, the cyclotron frequency ωC =|eB|/m and σ= (−)1 for spin up (down) electrons.

We note that the eigenstates are localized aroundy0which is proportional tokx, while their eigenenergies are independent of kx. The spectrum of a two-dimensional electron gas in the quantum Hall regime thus consists of flat bands which are called Landau levels [see Fig. 2.2 (c)]. Whenever the chemical potential is located in between two Landau levels, the system is insulating. Furthermore, it is described by a topological invariant [Thouless82], the so-called Chern number denotedν. For the simple example of free electrons, the Chern number is given by the number of filled Landau levels below the chemical potential and can thus take on any positive integer number.

At the boundary of a quantum Hall insulator, edge states appear due to the topological nature of the system. To see this, we introduce a confinement potential which limits the size of the system. We consider the strip geometry sketched in Fig. 2.2(a) and thus take the potential to be independent of x. Assuming furthermore that V(y) varies slowly as a function of y compared to the magnetic length, we can replace the argument of the potential by y0 due to the localization of the wavefunction [cf. Eq. (2.41)]. Then the only effect of the potential is to add the term eV(y0) to the energy [Goerbig09]. The Landau levels thus bend upwards for states that are localized at the edges following the confinement potential. Because the localization in they-direction depends linearly onkx, this affects the spectrum as illustrated in Fig.2.2(c).

We note that the effect of a quickly varying potential is similar [Goerbig09]. Due to this upward bending, all Landau levels which are below the chemical potential in the bulk cross the chemical potential somewhere close to the edge. Therefore the Chern number indicates how many edge states there are. Furthermore, all states at a given edge propagate in the same direction [cf. Fig.2.2(a) and (c)] while the states at the other edge propagate in the other direction motivating the namechiral edge states. This large spatial separation between counter propagating states suppresses backscattering and results in the quantized Hall resistance which can be measured with a relative uncertainty as small as 3·10⁻¹¹[Schopfer13] and currently serves as the resistance standard

Finally, we note that the behavior discussed above can be interpreted in a semi-classical picture where the electrons in the bulk are confined to closed cyclotron orbits due to the Lorentz force [Goerbig09]. The electrons close to the edge can not complete their orbits but get scattered of the edge. This leads to propagating “skipping orbits”, as illustrated in Fig. 2.2 (a), which constitute the current carrying edge states.

Quantum spin Hall insulator

The second type of topological insulators we discuss are two-dimensional quantum spin Hall insulators. The basic idea is to combine a quantum Hall effect for spin up electrons with a quantum Hall effect for spin down electrons corresponding to an inverted magnetic field. The resulting system exhibits counter propagating edge states with opposite spin polarization.

Here we focus on systems that can be described by the Hamiltonian proposed by Bernevig, Hughes, and Zhang (BHZ) [Bernevig06] which includes quantum wells of HgTe/CdTe [Bernevig06], InAs/GaSb/AlSb [Liu08] and Ge/GaAs [Zhang13]. The Hamiltonian reads

Hˆ =

ˆh 0 0 ˆh_{T R}

, ˆh=

C+M −(B+D) ˆp² Apˆ+

Apˆ₋ C − M+ (B − D) ˆp²

, (2.43)

where A, B, C, D, and M are material specific parameters, ˆp² = ˆp²_x+ ˆp²_y, and ˆp_± = ˆp_x±iˆp_y and ˆh_{T R} is obtained from by ˆh by time reversal (i.e. complex conjugation and ˆpj → −pˆj). Here ˆh describes states with positive spin (or total angular momentum) polarization while ˆhT R describes the time-reversed states with negative spin (or total angular momentum) polarization. The full Hamiltonian is thus invariant under time-reversal. We follow the common trend in the literature in denoting the states described by ˆh (ˆh_{T R}) as spin up (down) states even when these states do not correspond to eigenstates of the spin operator. In Sec.2.3.4, we discuss the basis states of Eq. (2.43) and their spin polarization in more detail.

It can be shown that the above Hamiltonian supports counter-propagating, spin-polarized edge states forM/B>0 with their propagation direction determined by the sign ofA/B[Qi11]. Since the propagation direction is coupled to the spin of the particles, these states are called helical edge states. Figure 2.2 (a)

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illustrates these edge states while Fig.2.2(d) shows the spectrum of a quantum spin Hall insulator in a strip geometry.

x

Figure 2.2: Edge states of topological insulators and their corresponding spectra. (a) Illustration of a quantum Hall insulator in a strip geometry. The insulator is characterized by the Chern number ν = 1 while the surrounding vacuum is topologically trivial (ν = 0). In a semiclassical picture, the bulk states are confined to closed cyclotron orbits while at the edge, skipping orbits provide chiral one-dimensional edge states as illustrated at the lower edge. A top gate (gray shading) can be used to guide the edge states. (b) Illustration of a quantum spin Hall insulator in a strip geometry. The insulator is characterized by the Z2

topological numberZ2 = 1 and exhibits helical edge states. Due to time-reversal symmetry, backscattering within a pair of edge states is prohibited. A top gate (gray shading) only changes the penetration depth and can not be used to guide the edge states. (c) Spectrum of a quantum Hall insulator. The momentum in x-direction is directly related to the localization iny-direction. Within the bulk, the spectrum exhibits flat bands called Landau levels which are spin split by the Zeeman energy ∆EZ = 2gµBB. The Chern number is given by the number of Landau levels that lie below the chemical potential µ. At the edges, the bands bend upwards due to the confinement potential. The same effect can be exploited to guide the edge states using a top gate (gray shading). (d) Spectrum of a quantum spin Hall insulator. The band gap of the bulk states is crossed by the spin polarized edge states. Here only the edge states of one edge are shown (adding the other edge states would result in spin degeneracy at each k_x). The crossing atk_x = 0 is protected by time-reversal symmetry guaranteeing that the edge states cross the chemical potentialµ.

In Eq. (2.43), there are no terms coupling the two spin blocks and thus backscattering within a pair of helical edge states is obviously absent. Interestingly, coupling the two spin blocks does not introduce backscattering as long as the Hamiltonian remains time-reversal invariant. This can be motivated from the spectrum shown in Fig.2.2(d). Time-reversal invariance dictates that the spectrum remains invariant when invertingkx→ −kxand exchanging spin up and down. This implies that there exists a degeneracy of the edge states atkx= 0. Since this degeneracy can only be lifted by breaking time-reversal invariance or by closing the bulk gap (then the edge states can hybridize with the bulk bands), the edge states necessarily cross the chemical potential (which lies within the band gap) even if the two spin blocks are coupled [Hasan10;Qi11].

Unlike in the quantum Hall regime, where multiple topologically protected edge states can coexist, only

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a single pair of edge states is protected in quantum spin Hall insulators. Indeed if there is an even number of edge states, they can hybridize and a gap can open even for couplings which respect time-reversal symmetry.

Only if there is an odd number of edge states is one pair guaranteed to cross the full gap. Thus, quantum spin Hall insulators are characterized by aZ² topological invariant denotedZ2 equal to 0 (1) if there is an even (odd) number of edge states [Hasan10;Qi11].

Top gates

For many setups investigating transport through edge channels, it is essential to control the location of the edge states. For instance, a quantum-point contact (QPC) is formed by bringing the edge states of opposite edges in close proximity. In this section, we briefly discuss how this is possible in quantum Hall insulators using top gates and why the same method does not work in quantum spin Hall insulators.

A local top gate [as illustrated in Figs.2.2(a), (b) and (c)] adds the termeV_tg(x, y) to the Hamiltonian.

In principle, one would have to solve the Schr¨odinger equation in the presence of this potential. Here we only give a handwaving motivation for the qualitative influence of the top gate.

For a quantum Hall insulator, the effect of the top gate is equivalent to the effect of the confinement potential. For a given x0, the bands will be shifted by the amount eVtg(x0, y0 =kxl²_B). As illustrated in Figs. 2.2(a) and (c), this can change the Chern number underneath the top gate. If this happens, the edge state will propagate around the top gate (at the interface of two distinct topological phases) allowing for a controlled manipulation of the edge state localization.

In quantum spin Hall insulators, there is no correspondence betweenkxand the localization iny-direction.

Furthermore, there is no insulating state with a different topological number available by shifting the chemical potential. Therefore, as long as the chemical potential remains away from the bulk bands, the top gate only locally shifts the bands corresponding to the edge states. This might change their penetration depth [Krueckl11] but it does not move the states away from the edge of the sample. As a consequence, controlling the localization of helical edge states is extremely challenging and to date no QPCs have been experimentally implemented.

2.3.2 Mesoscopic capacitors in quantum Hall insulators

The mesoscopic capacitor, first investigated by Büttiker, Prêtre and Thomas [Büttiker93] is sketched in Fig.2.1(a). It consists of an electron gas connected to a metallic contact and capacitively coupled to a top gate. A QPC pinches off part of the electron gas forming a quantum dot underneath the top gate. The plates of the capacitor are provided by the top gate and the electron gas underneath it. With the QPC being the quantum analogue of a resistor, the mesoscopic capacitor constitutes the quantum analogue of an RC-circuit.

A charged capacitor in a classicalRC-circuit discharges over a timescale given by τ_RC=RC. At small frequencies, the mesoscopic capacitor still behaves like anRC circuit described by the dwell timeτ_D=R_qC_q [Moskalets08]. However, the values ofRq andCq depend on the specifications of the driving voltage. In the linear response regime,Rq =h/2e² is given by half the resistance quantum and becomes independent of the QPC transmission [B¨uttiker93]. This quantized value has been called the B¨uttiker resistance [Glattli14].

Here we focus on the mesoscopic capacitor as a source of single electrons. More information on the mesoscopic capacitor as an RC-circuit can be found in [Gabelli12] and references therein. Operating the capacitor as a single-electron source is made possible by the peaked density of states in the quantum dot.

Whenever the time-dependent top gate moves a filled energy level above the Fermi energy, an electron is emitted from the dot. Moving an empty level below the Fermi energy leads to refilling of the level which is equivalent to the emission of a hole. For a quantitative analysis, we resort to Floquet scattering theory.

Here we focus on an implementation of the mesoscopic capacitor in the quantum Hall regime at filling factorν = 1. The electrons are then confined to one dimensional edge states as sketched in Fig.2.3. We note that in this case the electrons can originate from a different contact than the one they end up in. Due to the chirality of the edge states, all electrons incident on the capacitor will eventually leave the capacitor in the same direction (to the right in Fig.2.3). The scattering matrix is thus a single number. The mesoscopic capacitor in the quantum Hall regime as a single-electron source has been discussed in numerous publications including the textbook by Moskalets [Moskalets12]. Here we loosely follow Ref. [Hofer16b]. Because this setup makes use of a chiral edge state, we also denote it as the chiral capacitor.

In order to calculate the Floquet scattering matrix, we first consider a time-independent top gate potential

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Figure 2.3: Mesoscopic capacitor in the quantum Hall regime. An elongated chiral edge state is coupled via a QPC to a quantum dot. Within the quantum dot, the closed motion leads to a spiked density of states.

Moving energy levels of the dot above and below the Fermi energy using a top gate potential U(t) leads to the periodic emission of electrons and holes.

U(t) =U0. The time-independent scattering matrix then reads S(E) =r−d²

∞

X

p=1

r^p−1e^ip(E−eU⁰^)τ⁰ = r−e^i(E−eU⁰^)τ⁰

1−re^i(E−eU⁰^)τ⁰, (2.44) where r and dare the reflection and transmission amplitudes of the QPC (here chosen to be real without loss of generality) andτ₀is the time it takes an electron to complete one round-trip within the quantum dot.

In this case, no (additional) current is generated by the capacitor. To see how a time-dependent top gate potential does generate a current, it is instructive to inspect the density of states of the capacitor [B¨uttiker94]

ρ() = 1

2πiS^∗(E)dS(E) dE = 1

π

∞

X

j=−∞

1 2γ

(E−eU0−j∆)²+ ¹₂γ², (2.45) which consists of a series of Lorentzian peaks with spacing ∆ = 2π/τ₀ and widths

γ=−2 τ0

lnp 1−d²

' d² τ0

= 1 τD

, d²1. (2.46)

We now consider the simplest case which is given by the adiabatic limit where the top gate changes on timescales much slower than the dwell time τD. The frozen scattering matrixS(E, t) is then given by Eq. (2.44) after substitutingU₀ withU(t). Furthermore, we only consider a single quantum dot level which formally corresponds to the limit d²→0. Considering a top gate potential that varies linearly in time, we can write

E−eU(t) =−t−t^E 2ΓτD

, (2.47)

where t^E corresponds to the time at which the considered resonance level is located at energy E and Γ parametrizes how fast the top gate potential is varied. The adiabatic approximation is then valid as long as ΓτD. Expanding the scattering matrix in Eq. (2.44) using the last expression, we find [Moskalets12]

S(E, t) =t−t^E+iΓ t−t^E−iΓ = exp

iπ−2iarctan

t−t^E Γ

. (2.48)

While the resonance crosses the energy E, the phase of the frozen scattering matrix winds from 2π to 0. This phase winding eventually gives rise to the emission of exactly one particle (an electron or a hole).

The current that is emitted by moving one energy level above the Fermi energy can be calculated using Eq. (2.23). At zero temperature, we find

I(t) = e π

Γ

(t−t^E^F)²+ Γ², (2.49)

which corresponds to a Lorentzian current pulse that carries a single elementary charge. Moving a filled level above the Fermi energy allows the electron to leave the capacitor. Since this is done slowly, the Lorentzian shape of the energy level determines the temporal shape of the current pulse.

The adiabatic limit can also be used to approximate periodic processes. To this end we assume that every level crossing is described by the frozen scattering matrix given in Eq. (2.48). Figure2.4(b) illustrates

Dynamic mesoscopic conductors: single electron sources, full counting statistics, and thermal machines

Thesis

Reference

Dynamic mesoscopic conductors: single electron sources, full counting statistics, and thermal machines

HOFER, Patrick

HOFER, Patrick. Dynamic mesoscopic conductors: single electron sources, full counting statistics, and thermal machines . Thèse de doctorat : Univ. Genève, 2016, no.

Sc. 5006

URN : urn:nbn:ch:unige-889778

DOI : 10.13097/archive-ouverte/unige:88977

Dynamic Mesoscopic Conductors

Single Electron Sources, Full Counting Statistics, and Thermal Machines

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

Patrick P. Hofer

de Meilen (ZH)

Th` ese N

5006

GEN` EVE

Atelier de reproduction ReproMail

2016

Acknowledgments

Contents

Chapter 1

Introduction

Chapter 2

Single-electron sources

2.1 Floquet scattering theory

2.2 On-demand entanglement from single-electron sources

2.3 Mesoscopic Capacitors in quantum spin Hall insulators

2.3.1 Edge states of topological insulators

x

2.3.2 Mesoscopic capacitors in quantum Hall insulators