Effects and detection of quantum noise

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Effects and Detection of Quantum Noise

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof.dr.ir. J.T. Fokkema, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 30 januari 2006 om 13.00 uur door

Jens TOBISKA

Filosofie Magisterexamen med fysik som huvud¨amne Ume˚a Universiteit, Zweden

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Samenstelling van de promotiecommissie: Rector Magnificus, voorzitter

Prof. dr. Yu. V. Nazarov Technische Universiteit Delft, promotor Prof. dr. ir. G. E. W. Bauer, Technische Universiteit Delft

Prof. dr. C. W. J. Beenakker, Universiteit Leiden, Nederland Prof. dr. W. Belzig, Universiteit Konstanz, Duitsland Prof. dr. ir. J. E. Mooij, Technische Universiteit Delft Prof. dr. A. Shelankov, Universiteit Ume˚a, Zweden

Het onderzoek beschreven in dit proefschrift is financieel ondersteund door de stichting voor Fundamenteel Onderzoek der Materie (FOM).

Published by: Jens Tobiska

Printed by: Sieca Repro B.V., Delft Cover design: Just noise?

An electronic version of this thesis is available at: http://www.library.tudelft.nl/dissertations/ Casimir Ph.D. Series, Delft-Leiden, 2006-01

ISBN: 90-8593-008-1

All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilised in any form or by any means, electronic or me-chanical, including photocopying, recording or by any information storage and retrieval system, without permission from the publisher.

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PREFACE

Four years in the theory group in Delft—a truly unique experience. During this time I got to know quite a few people (voluntarily or inevitably) without whom this thesis would not exist as is. Through them I not only learned a lot about physics but also about human personalities. This is the place to thank you for the stimulating-crazy-enjoyable-eventful-entertaining environment. If anybody feels that he/she should (not?) belong to this list or feels wrongly acknowledged, don’t panic! Be assured that this is not intentional and that this preface represents my own humble and subjective point of view (and not my employer’s!). Other people’s opinion may differ. Let’s start the praising. . .

This thesis is the result of a long history of fortunate events so let me start by thanking the people from math.-nat.-techn., TUBS and the physics department of Ume˚a University, particularly Andrei Shelankov and Jørgen Rammer who introduced me to the field of quantum transport. After a short interlude—doing something good—I ended up in Delft as an OIO (a palindrome!). Ensuing years were spend among fellow theoreticians. So thanks go to all former and present members of the Delft theory group for their input/output and sharing good times. After this all-in-one acknowledgement let me still point out some of the highlights among you.

First of all there is Yuli Nazarov my supervisor. Many words and rumours have been spread about him. After such a long time working with him I feel authorised to add my own. . . Looking back, I feel privileged having had Yuli as a promotor. Your deep insight into physics (even of the experimental kind) still impresses me and was highly stimulating for research. Through you I learned a lot about physics and—equally important—the correct way to pursue physics. There was always time for discussion, something which can not be taken for granted from a supervisor. Thanks for all that and much more.

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your practical tips on anything Japanese. Unforgotten are also Yaroslav Blanter (for having me learn about advanced mail-setups), Milena Grifoni for her con-tinuous good mood and efforts to make theoreticians run and Jos Thijssen for teaching me how to solve undergraduate homework. It was hardstikke mooi to have cheerful Miriam Blaauboer in the group as well as Henk Bl¨ote who beauti-fied F327 with Chinese roses—poor roses! A major factor in keeping the whole group operational was Yvonne Zwang and her delicious coffee. That has always (well, often) been a good reason for getting up before 1030am.

Most time however I spend with you fellow Masters, PhDs and postdocs comprising the footfolk. Some funny guys unfortunately left for one reason or another; others managed to stay with me (almost) till the end: Jo¨el, Dani, Siggi, Dima I—the physicist, Freek, Gabriele, Xuhui, Fabian, Henri, Moosa, Antonio, Hayk. . . Thanks a lot for comments, suggestions, good times, bad (?) times, fruitful discussions and the tiramisu.

I had the pleasure of sharing the room with two “extremes”. Wouter, thanks a lot for discussions, translations and the “sollicitatie”; maybe one day I will speak Dutch. Oleg, master of the borschtsch, always energetic and bubbling with ideas. Hope that something good comes out of our project and that your omen comes true. Spasibo, Vika for taming him once in a while. I am indebted to Jeroen and Izak for struggling through all those terms without getting lost. I learned a lot from both of you. Special thanks to Jeroen for the flamethrower! Then there is Omar, able to comment on anything for as long as one wishes to listen. Your craziness has surely been an enrichment to the group. Thanks a lot for discussing NEMS and not FIFA with me. Alex. . . , his quotes could tell more about him than anything else (don’t worry I am not going to release them yet). Without his philosophies life at theory would have been soooo much more boring. I don’t know what Sijmen’s secret is for staying sane in this environment. Anyway thanks a lot for convincing me of the beauty of Dutch culture and for the beachvolley. Only shortly I had the pleasure of getting to know Dima II—the bald—who introduced me to the power of the diffusion equation and did not tire of telling about different fighting techniques. Later he and Babak even joined the F327 volleyball-club (and Wouter was not amused)! Thanks (and sorry) for all that.

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Saito, Hugues Pothier, In`es Safi, Alessandro Romito, Markus Kindermann, Udo Hartmann, . . . , and others who I forgot but should be mentioned here.

I am deeply grateful to NTT Basic Research Labs for organising the excellent autumn school and for the hospitality at my second visit. I am indebted to Yasuhiro Tokura for accepting me as a postdoc and am looking forward to a fruitful and interesting time at NTT.

Let me not forget my experimental friends and colleagues. First and fore-most the people formerly known as NF: Anna, Monica, Saverio. Many thanks for bringing me back to (experimental) earth and for the great times we had. Grazie for many interesting discussions about physics and detailing experimen-tal stuff in simple words to me. I am also grateful to the guys from downstairs notably Lieven Vandersypen, Eugen Onac and Adrian Lupascu for enlightening discussions and Samir Etaki for advice/tips. Thanks also for letting me join in group meetings and the infamous Thursday soccer matches.

Not to be forgotten are also the nameless construction workers who in the summer of 2005 provided a whole range of noise samples right at work. Fur-thermore I am especially grateful to all the people who provided a connection to the real world and a counterweight to nerd-life. First of all thanks to my family for continued support, understanding, proofreading and bearing with my — far-away work-place choices. Ren´e, despite of having changed to the evil side, thanks for discussing REAL problems of hula-hooping and for making this thesis a little more digestible for mathematician. Tack, Doreen for letting him do so and the great company on our trip to the North pole.

A large part of my free time in the last two years was spend at the best club ever: Punch, hup blauw!. Thanks a lot to all of you for letting me oldie relive some student life. And H12, hope you are going to win more often without me—Baaaaaaaaaaaaanaaaaaaaaaaaaan! I am very happy that many of my old friends did not lose contact despite the distance. Kudos to the DOKO-guys and Erasmus people (especially Flo who risked being a paranimf). Let me end this preface by saying that—believe it or not—I might miss you. Hope to see you one day somewhere, maybe over a chocolate fondue?

Jens Tobiska

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1. INTRODUCTION

Noise a nuisance? Many people and most parents would probably agree. Traffic, neighbours and modern music have all lead to this (mis?)-conception. Noise is usually perceived as something bad and if possible avoided. However there are also some good aspects about it which are not immediately apparent during the first exposure to noise. Apart from providing the basis for this thesis—a very mundane motivation indeed—a scientific study of noise is not a nuisance at all. One aspect that is worthwhile studying is its effect on the environment. Traffic leads to the construction of noise barriers, inconsiderate neighbours may cause knocking and counter-knocking on walls or even court trials, while some music facilitates apparently random motion of certain groups of people. Noise emitted from children is a rather complicated object to study as it can result in a variety of reactions of stressed parents. Specifically for this parent-child system there exists certainly a back-action from the parental reactions to the children expressed for instance by feeding, which—if everything goes well—leads to a lowering of the noise level. Obviously this is of major interest to everybody and is one of the motivations for studying the effects of noise on the environment.

But this is not all there is to noise. If we choose a different point of view, we can ask how noise is actually perceived. Generally there is a source (e.g. cars, neighbours, children) and a detector (e.g. ear, brain). The auditory system of humans is fortunately fairly well developed and allows us to distinguish between noise coming from different sources. Furthermore an experienced parent may distinguish between the request for food and the demand of playing expressed by a babbling baby. Such a “detector” therefore even allows to discriminate different states of a single source. In the interest of the hungry child this dis-crimination is of vital importance. We also note that a silent (noiseless) child does not provide any (audible) information about her state.

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influence of noise or, and more interestingly, in utilising the noise, enabling processes to happen in the environment that would not exist otherwise, (ii) the detection of noise can provide us with important information about the source of noise, that we would not obtain in the absence of it. We will elaborate on both of these points in this thesis, hopefully improving the bad reputation of noise.

To avoid misunderstanding, the noise in physics which we study is somewhat different to what is commonly referred to as noise and we will define it in the following. Specifically we are not interested in sound, like in the examples above but in other physical quantities. In a physical context, what is noise?

Physical quantities are generally functions of time X(t). This explicit time dependence can be of several different kinds. The position of the earth towards the sun is a periodic function of time, the world population is a monotonous function of time and many other examples can be imagined. However there are also some quantities whose values stay more or less constant over time. Think of the outside air temperature or atmospheric pressure in a month, population dynamics of some country or exchange rates between currencies (figure 1.1). Note also that the first two examples can be reduced to this case by subtraction of the periodic motion and monotonous part respectively. All of these processes are fairly well described by a single number which is the average value

hXi ≡ _T1 Z T /2

−T /2

dt X(t). (1.1)

The choice of the averaging interval T understandably depends on the considered system and interest. Temperature data in metrology are presented as averages over days, months, years and even 1000s of years when looking at glacial periods in the history of the earth for instance. Physicists sometimes write T → ∞ meaning an average over an interval larger than all other time scales relevant to the system. Averaging with a restricted time interval whose size is usually determined by an inherent time scale of the system is done on a regular basis. If we asked for the world population we might be satisfied with an interval of days.

Clearly the operation in equation (1.1) discards a huge amount of informa-tion, namely the value of X at any moment in time. And this information is important indeed. A careless traveller to Ume˚a might only check the average winter temperature of approximately −5and be surprised by the substantial

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be satisfied with a little less information and still catch some of the features of the full function X(t).

Expectedly the values of X in figure 1.1 deviate from the average value in some arbitrary way. This observation leads to the following definition. Noise are stochastic (random) fluctuations in time of a physical quantity from a mean value. The reasons for the existence of such fluctuations are manifold and very much system dependent. Still on a general level we can classify them as extrinsic and intrinsic noise sources. The former are due to external parameters and can be influenced from the outside by isolating the system from its environment. Intrinsic noise on the other hand is a property of the system itself and always exists. We can not get rid of it by any means. Exchange-rates (figure 1.1(b)) are an example of a system where big fluctuations are not desirable and can be reduced from the outside. Despite all external efforts small fluctuations usually persist.

While the study of extrinsic noise can provide us with information about the environment and its coupling to the system, intrinsic noise, as we will see, enables to look into the microscopic processes going on in the system, some-thing that often can not or only with great difficulties be accessed by other means. This separation into extrinsic and intrinsic sources has been observed by Schottky [3], when studying the current fluctuations in vacuum tubes.

To quantify noise we need the value of X at different times. The simplest such quantity is the autocorrelation function [4]

CX(t) = h∆X(t + t′)∆X(t′)i ≡ 1

T Z T /2

−T /2

dt′∆X(t + t′)∆X(t′), (1.2) where ∆X(t) ≡ X(t) − hXi is the deviation from the mean value. By virtue of its construction the autocorrelation function contains the memory of the system: “How much does the system characterised by X remember about its state after a time t?”. This function provides a measure for the “randomness” of a signal. If the correlations are decaying over some time, the system is stochastic. A constant or periodic autocorrelation function on the other hand tells us that there are recurring events in the signal. Based on the autocorrelation function one can define its Fourier transform, the noise power or power spectral density

SX(ω) =

Z ∞

−∞

dt eiωth∆X(t)∆X(0)i. (1.3)

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-30 -25 -20 -15 -10 -5 0 5 10 15/02 01/02 15/01 01/01 15/12 01/12 temperature [ oC] (a) 0.8 0.9 1 1.1 1.2 1.3 1.4 1999 2000 2001 2002 2003 2004 2005 2006

exchange rate Euro-Dollar

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1.1. Thesis layout 5

Colours of Noise Depending on the shape of the power spectral density one distinguishes several limiting cases. The classification is usually done by assum-ing a frequency dependence of 1/fβ with some exponent β for the density and f = ω/2π. If the spectrum is flat (S(ω) = const), the noise is termed white since different frequencies contribute with equal power to the signal. Having equal power at all frequencies the total power of such a system would be infinite. In practice one is limited to certain frequency bands. A white spectrum is then one which is constant over some band. This restriction to a band is assumed for the definition of other coloured noises as well. From equations (1.2) and (1.3) it follows that the signal is delta-correlated (CX(t) ∝ δ(t)). Thus the quantity

X does not know about its value at previous times expressing the fact that the system has no memory.

Another relevant noise is pink (or 1/f or flicker) noise. Its power spectral density is proportional to the reciprocal frequency. It is encountered in the systems we study and will be further discussed below.

There is also brown noise, proportional to 1/f2_{. Its name stems from}

Brow-nian motion as it can be generated by adding a random offset to a sample to obtain the next.

An example for the usefulness and application of noise is blue noise with a density proportional to frequency. Interestingly this is good noise for adding to a quantised signal with the purpose of reducing quantisation error. This mech-anism has developed long ago in retinal cells of mammals [5] and is nowadays used in processing digital audio and video data where it is called dithering.

Not associated with a colour but still widely known is the noise originating from AC-devices. It is emitted at multiples of the AC frequency (50/60 Hz) and is normally an unwanted signal.

1.1 Thesis layout

This thesis is organised as follows. We start with an initial discussion of general concepts related to noise and its detection. This part includes also a discus-sion of theoretical techniques employed in this thesis. Subsequent chapters are concerned with specific systems.

In chapter 3 we investigate a typical situation encountered in actual experi-ments: a quantum coherent conductor embedded in an electromagnetic environ-ment with impedance Zω ≪ G−1Q . We show that the environment changes the

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correction to the generating functional, that is to all moments of the current. This correction is studied analytically and numerically. An important result in that chapter is a universal relation for cumulants.

The subsequent chapters are concerned with the actual detection of noise. By means of a realistic setup the detection of noise emitted by a quantum point contact using a quantum tunnelling detector is studied in chapter 4. It is shown that the detector signal can be related to two distinct processes, a two photon-process and a process due to interacting electrons in the point contact. The detection of noise thus provides information about microscopic processes, notably electron-electron interactions, happening in the source.

Another example of a detection scheme which allows to extract information about the source is presented in chapter 5. There, a Josephson junction is employed in order to measure the full distribution of current fluctuations Pτ(I)

or—equivalently—all moments of it. This is of prime experimental interest since it is at the moment rather difficult to access anything but the second moment.

References

[1] European Central Bank, www.ecb.int, 11/10/2005.

[2] Koninklijk Nederlands Meteorologisch Instituut (KNMI) Afdeling Seismolo-gie, www.knmi.nl, 21/10/2005.

[3] W. Schottky, ¨Uber spontane Stromschwankungen in verschiedenen Elek-trizit¨atsleitern, Annalen der Physik 57 (1918), 541.

[4] N. G. van Kampen, Stochastic processes in physics and chemistry, rev. and enl. ed., Elsevier Science Publishers B.V., Amsterdam, 1992.

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2. FLUCTUATIONS AND NOISE

2.1 Statistical interpretation

Going back to the definition of noise as a stochastic process, it is natural to ask: “What is the probability to find a certain value of X during a time interval τ ?” Denoting this probability distribution by Pτ(X), we can assign in the usual

way [1] expectation values

hXi = X

X

XPτ(X), and average fluctuations (2.1)

hX2i = X

X

X2Pτ(X). (2.2)

Here we defined it for discrete valued X. The generalisation to continuous X is straightforward. Clearly the first quantity represents the average of X and the second gives a measure for the size of the deviation from this average—the noise. Both of these and any similar property will be referred to as moments or correlators of the distribution function. Equivalently the fluctuations can be characterised by irreducible moments or cumulants. These can be defined from the correlators defined above and will be denoted by hh. . . ii. For the first two we have

hhXii ≡ hXi (2.3)

hhX2ii ≡ h(∆X)2i (2.4)

In many cases and in the systems we are studying, Pτ(X) is a single peaked

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X hhX2_ii

hXi hhX3ii Pτ(X)

Fig. 2.1: _{A typical probability distribution. Average hXi, average deviation} hhX2_{ii and third moment (asymmetry, hhX}3_{ii) are indicated.}

moments. Besides the definition given above the second order correlator (equa-tion 2.2) alone is also referred to as noise. This is due to historical reasons as the second order correlator was the first quantity related to fluctuations being studied. Higher moments which still represent noise in the definition sense are a more recent concept, at least in experimental studies.

As we will see below, it is often much more convenient to work with a generating functional F (χ) instead of the distribution. Defining it as the Fourier transform

F (χ) =X

X

eiχXPτ(X), (2.5)

we can easily generate all moments of the distribution function by differentiation hXn_{i =} ∂n ∂(iχ)nF (χ) χ=0 . (2.6)

Cumulants are generated in the same way by taking derivatives of ln F (χ). This generalises the definition given by equation (2.4) to higher order cumulants.

If the events leading to the probability distribution Pτ(N ) are independent,

the characteristic function is a product of functions of each type of event, F (χ) = Q

iFi. In the limit of long measurement times, τ → ∞, the distribution goes

to a normal distribution as a consequence of the central limit theorem. A large part of this thesis is concerned with the calculation of the generating functional for concrete systems.

Summarising, there are two equivalent statistical formulations of noise. It can be fully characterised by either Pτ(X) respectively its Fourier transform

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2.2. Quantum transport 9

2.2 Quantum transport

As we have seen the statements made so far apply to a huge class of systems, from everyday life to tiny fields of physics and biology and length scales ranging from light-years to atomic distances. This thesis is concerned with systems belonging to the field of mesoscopic solid state physics [2]. Lying in-between the microscopic and the macroscopic world, these systems inherit some interesting features of quantum physics while still being experimentally manageable. The relevant length scale in this context is the correlation length, that is the size over which the system behaves as a quantum system. In solid state this mesoscopic size lies typically between nano- and micrometre which has resulted in the term nanoscience as a recent replacement for mesoscopic physics.

The huge theoretical and experimental efforts undertaken in this field are mainly motivated by the prospect of fabricating a quantum computer [3]. In contrast to conventional computers which use classical quantities to achieve their goal and operate with 0s and 1s, a quantum computer makes full use of the quantum mechanical superposition principle using 0 and 1 “at the same time”. At least theoretically this allows for a huge increase of speed over its classical counterpart. In analogy the basic unit of such a machine has been termed qubit. Different realizations of quantum mechanical two-level systems representing a qubit have been proposed and experimentally realized. The most promising ones employ persistent currents in a superconducting flux-qubit or make use of charge or spin degrees of freedom.

So far quantum computation has been demonstrated in a molecule using the spin of nuclei [4]. Single and coupled qubits have been operated in solid state as well [5, 6]. It is expected that further advances will lead to a successful implementation. Easy scalability in solid state and experience with similar systems should help in this respect. However there are still several obstacles to be overcome. A major one results from the small size of such systems. Operating with single electrons or spins makes them vulnerable to external disturbances or noise. On the positive side, understanding the complex mechanisms in these systems we may channel the noise to good use. As explained in the previous subsection, measuring noise we can learn something about microscopic processes occurring in such systems. Such knowledge may be used in designing better or completely new systems. In any case a deeper understanding of noise in nano-structures is required and provides the main motivation for this work.

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semiconduc-tors, carbon nanotubes, molecules, DNA, even single atoms can be combined and shaped into hetero-structures to perform a specific task.

These nano-structures have in common that by restricting the electron mo-tion they channel currents, charges and voltages much like on a larger scale is done in conventional electronics. Correspondingly these are also the fluctuat-ing quantities we study. Due to their small size quantum effects are expected to be relevant for the transport of electrons through these systems requiring a more sophisticated description than Kirchhoff rules. The appropriate quantum theory will be presented below.

2.2.1 Scattering theory

Quantum electrons are characterised by a wave function ψ(r, t), whose squared absolute value |ψ(r, t)|2 _{gives the probability to find the electron at point r at}

time t. In the nano-structures we study, electrons are confined by the design elements and influenced by impurities and defects present in the system. Col-lecting these effects into a time independent potential U (r), the dynamics is described by the time independent Schr¨odinger equation

Eψ(r) = −~ 2 2m∇ 2_{+ U (r, t)} ψ(r) (2.7)

and the time dependence is given by the energy ψ(r, t) = exp(−iEt/~)ψ(r). It is clear that for sufficiently complicated potentials the Schr¨odinger equation is not readily solved. Fortunately for the systems of interest a lot can be learned from simple forms of U (r) which allow for an analytic solution of (2.7). One such textbook example is an infinite one-dimensional waveguide with rectangular cross-section and a barrier of length d (figure 2.3(a)).

The potential takes the form

U (r) =    ∞ |y| > a/2, |z| > b/2, 0 |y| < a/2, |z| < b/2, |x| > d/2, U0 |y| < a/2, |z| < b/2, |x| < d/2. (2.8)

Away from the barrier the solution of (2.7) is found from the ansatz

ψ(r) = ψkx(x)ψn(y, z), (2.9)

where ψn(y, z) are standing waves corresponding to the quantisation in the

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(a) (b)

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x a, b 0 d U 0 (a)

0

0.5

1

0

1

2

3

T (E ) E/U0 (b)

Fig. 2.3: _{(a) Waveguide with rectangular cross section a × b and a potential} bar-rier of length d and height U0. (b) Transmission through a barrier. If

the energy is less than the barrier height E < U0, tunneling is

expo-nentially suppressed. At large energies E ≫ U0transmission is perfect

T → 1. (d = 4.5 ~/√2mU0for this plot).

wave function ψkx = exp(±ikxx). The energy spectrum reads

En,kx = (~kx)2 2m + π2_~2 2m n2 y a2 + n2 z b2 ! . (2.10)

States with a definite n corresponding to different modes of the transverse quan-tisation are referred to as transport channels. Note that a given maximum en-ergy restricts the number of channels. Equation (2.10) also provides an estimate for the typical size of systems in which the effects we discuss become relevant. If the dimensions a, b are sufficiently small, such that the corresponding energies are larger or comparable to other energy scales in the system, quantum effects should be noticeable. For typical voltages applied this gives length scales in the range of micro- to nanometre.

Any transport property will be determined by ψkx(x). A wave sent in from the left (exp(ikxx)) will be partially reflected (r exp(−ikxx)) and transmitted

(t exp(ikxx)). Within the barrier a similar quantisation to (2.10) holds with

a wave vector κ = pk2

x− 2mU0/~2. Matching the wave functions and their

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a(x), b(x) x

Fig. 2.4: _{Adiabatic waveguide with variable rectangular cross section a(x)×b(x).}

determines the amount of the transmitted wave

T (E) ≡ |t|2= 4k

2_κ2

(k2_{− κ}2₎2_sin2_{(κd) + 4k}2_κ2. (2.11)

A classical particle with energy E < U0would always be reflected T (E < U0) =

0, while T (E > U0) = 1. Quantum mechanics changes this simple picture,

giving electrons the chance to tunnel through the barrier even if their energy is below the barrier height. On the other hand an electron which in the classical case would be transmitted with probability one acquires a non-zero probability for being reflected. Far away from the barrier edge, the transmission approaches the classical values 0, 1 (figure 2.3(b)).

This discussion can be extended to a more realistic model of an adiabatic barrier, that is a waveguide whose transverse dimensions vary slowly on the scale of the transverse dimensions themselves (figure 2.4). In this case the wavefunctions can locally be approximated by those of the ideal wave-guide studied above. Transport is in the same way found from an x-dependent ψ(x) satisfying a one dimensional Schr¨odinger equation. The role of the potential is played by a channel dependent energy En(x) which takes the same form as in

(2.10) with coordinate-dependent widths a(x), b(x). Thus for any channel there is an effective potential barrier at the narrowest part of the waveguide. The larger the channel indices ny,zthe higher this barrier becomes. We can employ

the same reasoning that lead to the transmission coefficient (2.11). For a given energy there will be a channel-dependent transmission coefficient Tn(E) which

is mainly determined by the height of the barrier. Unless E is close to one of the maxima of En(x), the transmission will be either perfect or zero for that

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Scattering matrix

In the previous subsection we studied scattering by means of a concrete example that is a constriction in a wave guide. Surprisingly this provides a good model for a quantum point contact (QPC), that is a constriction in a two dimensional electron gas. The proof by comparison with experiment we postpone to the next subsection. Here we would like to generalise this simple model to a wider class of nano-structures. At this point it is necessary to precede the discussion with a comment. Even though the systems of figure 2.2 are man-made in a controlled way, the shape of the potential to be used in a theoretical description is not known due to the presence of defects and impurities in the samples. Furthermore for sufficiently complicated systems the potential by design might be too complicated to be treated exactly. The electrons will feel all of these potential variations and scatter, thereby influencing the transport properties.

Fortunately for sufficiently low temperatures and voltages this scattering takes place elastically. It can then be described by a scattering matrix. To make contact to quantum transport let us calculate the current in such a setup. For this it is important to realize that the nano-structure considered is part of a larger electric circuit, in the end it will be connected to macroscopic pads each kept at a constant voltage. These pads contain a large number of electrons at thermal equilibrium with some electro-chemical potential µ and can thus be described by a Fermi distribution f (E) = 1/{exp[(E − µ)/kBT ] + 1}. For

simplicity let us also assume that there are two reservoirs only with a scattering region in-between and that electrons are transferred through a wave guide to the scatterer. The last condition is not important for the following discussion but simplifies the expressions by restricting the spatial dimension to one.

Within the wave guides the wave function is a superposition of left and right going plane waves in the channel n

ψ(x) =X n 1 √ 2π~vn h aαne±ik (n) x x+ b αne∓ik (n) x x i , (2.12)

where α = L, R in the respective wave guide (x ≶ 0) and the upper/lower sign corresponds to the left/right region. In the scattering region the wave function is not known and may have a very complicated form. The coefficients aαnbαn in equation (2.12) are not independent. The amplitude for an outgoing wave depends linearly on the amplitudes of all incoming waves

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The coefficients constitute the scattering matrix s = sLL sLR sRL sRR = r t′ t r′ . (2.14)

The elements of the reflection and transmission sub-matrices r, r′_{, t, t}′ _{are the}

amplitudes for scattering between certain channels. As an example |rnn′|2is the probability for an electron incident from the left in channel n to be reflected to the left into channel n′_{. In general these matrices are symmetric with inversion}

of the magnetic field rT_{(B) = r(−B), . . . . Any scattering matrix has to be}

unitary (s−1_{= s}

) as a result of flux conservation—an electron is either reflected or transmitted.

2.2.2 Landauer formula

So far the discussion proceeded using terminology like wave functions and trans-mission or reflection probability. Much more relevant are observable quantities. To calculate current we note that due to charge conservation it is uniform and we evaluate it in the left region. There are three contributions to the current: i) electrons from the left reservoir with velocity vx to the right, ii) electrons

that originated from the left reservoir and were reflected, now moving to the left with −vx and iii) electrons from the right reservoir that were transmitted

now moving to the left with −vx. Summing all those contributions up and

summing/integrating over channels and wave vectors/energy gives the following expression for the current

I = 2e 2π X n Z ∞ 0

dkxvx(kx)(1 − Rn(E))(fL(E) − fR(E)). (2.15)

The voltage is incorporated into the Fermi functions fL(E − V ) = fR(E). Here

we introduced the probability of being reflected to the channel n from all possible starting channels, Rn(E) = Pn′|rnn′|2 and a factor 2 for spin. The latter could as well be incorporated into the scattering matrix. Changing integration variables from kxto energy removes the dependence on velocity since vx= 1~

∂E ∂kx. Making use of the unitarity of the scattering matrix we can replace 1 − Rn =

P

n′|tn′_n|2 = (t

t)nn = Tr(t

t) = P

nTn with 0 ≤ Tn ≤ 1 the transmission

eigenvalues. The current then takes on a suggestive form I =2e

h Z

dE Tr(t

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In general Tn should depend on energy. Assuming that they don’t change

much on the scale set by the remaining integrand, that is voltage, they can be replaced by their value at the Fermi energy. The remaining integral evaluates to eV which leads to the well known Landauer expression for the conductance (GQ = 2e2/h, conductance quantum)

G = GQ

X

n

Tn(EF). (2.17)

It is indeed surprising that despite of the complicated structure of the scatterer the conductance can be expressed in such a simple form. Before continuing let us make some notes on the model used. In reality there is nothing like a wave guide connecting scattering regions and reservoirs. A typical quantum point contact is created by reducing a 2-dimensional (quasi-infinite) system in a narrow region whose length is short. One would therefore have to solve the Schr¨odinger equation without the boundary conditions set by the wave guide. However we note that the infinite limit can be reached by widening the wave guide, the spectrum becoming continuous and the channel number infinite in this process in accordance with equation (2.10). Still the transport would be determined by the small scattering region which is characterised by a set of transmissions which are typically close to zero or one.

This striking relation for the conductance (2.17) has first been observed in a GaAs-AlGaAs semiconductor heterostructure [10]. By applying a gate voltage, a constriction with variable width could be formed in a two-dimensional electron gas. Since the barrier height depends on the width of the constriction through equation (2.10) new transport channels will open whenever the top crosses through the Fermi energy of the leads. In a conductance measurement this can be seen as a staircase with steps of height GQ (figure 2.5).

2.2.3 Sources of Noise

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Fig. 2.5: Quantised conductance in a quantum point contact. (from [10])

Thermal Noise

The motion of charge carriers in a conductor does not occur along straight lines. Instead they are randomly wiggling around, their motion being due to temperature. By electromagnetic interaction, this random motion will induce a fluctuating voltage at the contact which is termed thermal or Johnson-Nyquist noise. Thermal noise is the dominant source at temperatures kBT ≫ eV . It has

a white spectrum and is the only relevant noise source at equilibrium (V = 0). The noise power is related to the system conductance by a special form of the fluctuation-dissipation theorem [11]

SI = 4kBT G (2.18)

As can be seen from this expression, the detection of thermal noise does not provide more information than a conductance measurement. On the other hand it can be used for temperature measurements given a certain conduc-tance [12] though measurements employing shot noise are experimentally ad-vantageous [13].

Shot Noise

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Fig. 2.6: Partition/shot noise. A noiseless current of equally spaced pulses (solid lines) is transferred into a noisy current (dashed lines) with a stochastic distribution of puls separations by random transmission/reflection on a barrier.

Reducing the current and temperature, eventually we will enter a regime where carriers can be observed as individual current pulses. Due to random transmissions and reflections in the conductor they arrive stochastically at the contacts. The average over a long time gives the average current. For the simplest system—a single barrier—this randomisation is depicted in figure 2.6. These fluctuations being due to the discreteness of the charge carriers are called shot noise [14]. They have been first observed in vacuum diodes by Schottky. He could also distinguish such fluctuations—which he called Schroteffekt—and those related to temperature—W¨armeeffekt. Shot noise is relevant at large voltages, eV ≫ kBT . The power spectrum is white and in the case of electrons

incident on a barrier with transmission probability T , is proportional to the average current ¯I

SI = 2e ¯I(1 − T ). (2.19)

As a function of the applied voltage the noise in such a conductor crosses over from a constant value given by thermal noise (2.18) at small voltages to a linearly increasing shot noise at eV ∼ kBT (see figure 2.7). In the limit of low

transmission T ≪ 1 expression (2.19) reduces to the Schottky formula

SI = 2e ¯I. (2.20)

In this limit the electrons are rarely transmitted and consequently uncorrelated. Their distribution is given by a Poissonian (in time, equation (2.36)). Based on this formula it is custom to introduce the Fano factor F = SI/(2e ¯I) as the

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2.2. Quantum transport 19 0 0.5 1 1.5 2 2.5 3 -4 -2 0 2 4 shotnoise thermal noise eV /kBT S / (4 kB T G )

Fig. 2.7: _{Crossover from thermal to shot noise at eV ∼ k}BT (F = 1).

(a) (b)

Fig. 2.8: (a) Observation of fractional charge e∗_{= e/3 in the fractional quantum}

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anti−bunching bunching

+

Fig. 2.9: A crosscorrelation measurement on a beam of bunched and anti-bunched particles. Positive crosscorrelations are indicated by “+”.

The expression (2.20) is strictly true for carriers with charge e. In super-conductors or fractional quantum Hall systems however this charge has to be replaced by an effective charge e → q, which represents the actual charge of carriers (q = 2e in superconductors). The shotnoise amplitude in such systems provides information about the nature of the carriers, notably their charge, something that is not found from a dc conductance measurement (figure 2.8).

Up to now it was implicitly assumed that the fluctuations are measured at a certain probe. Multiterminal conductors provide additional possibilities for such a measurement. Due to their design it is possible to correlate current fluc-tuations obtained at different probes of the conductor. Such correlations can be viewed as the analog of the famous Hanbury Brown and Twiss (HBT) experi-ment in astrophysics [18]. From the intensity correlations between two detectors measuring the light emitted from a star they could infer its diameter. It is long known that intensity-intensity correlations of a thermal source of bosons are positive, as a result of their statistics. Particles obeying Bose-Einstein statistics tend to cluster or bunch (figure 2.9).

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negative cross-correlations. Such behaviour—the fermionic equivalent of the HBT-experiment—has been demonstrated in edge channels of a semiconductor Hall bar [19], with the help of a fermionic beam-splitter fabricated in a two-dimensional GaAs electron gas [20] and in vacuum [21]. For low frequencies and constant potentials in the terminals, current-current correlations in nano-electronic systems are always negative [22].

These assumptions are clearly a simplification as electrons in typical nano-structures are interacting (see chapter 4) and voltages are usually fluctuating. It has been demonstrated in a variety of setups that such interaction effects can lead to positive correlation effects. Even though the often heard statement: “fermions = negative cross-correlation” is obviously not true, still multi-probe measurements can provide additional information about statistics and interac-tion effects of particles.

Another idea along the same lines consists in splitting Cooper pairs of a superconductor into two normal arms in a Y-shaped geometry. Such system has been proposed as a way of observing entangled electrons in a solid state setup [23]—the dream of many experimental physicists. If the electrons origi-nating from a Cooper pair end up in different arms, it should result in positive cross correlations. Note that this is different to the positive sign observed for anti-bunched particles; the statistics of particles plays no role here. Rather it is the simultaneous emission of two electrons into the probe arms that causes such correlations.

Many theoretical works were devoted to this setup including a calculation of higher moments of noise [24]. A splitting has recently been observed experi-mentally [25] and cross correlation measurements which would provide further evidence for a successful splitting are planned [26].

Clearly shotnoise is the quantity one should study in order to learn about microscopic processes in the source. It can provide information about inter-actions between carriers, their charge and statistics and possibly many other properties that can not be obtained, by a pure conductance measurement.

Quantum Noise

Even at zero temperature there is still one source of noise which we did not cover so far. It is due to the quantum nature of the system and is represented by the zero-point fluctuations. This source is relevant at ~ω ≫ kBT . Evidently, since it

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quantum equivalent, such that equation (1.2) reads CI(t) = h∆ ˆI(t + t′)∆ ˆI(t′)i ≡

X

i

ρiihi|∆ ˆI(t)∆ ˆI(0)|ii, (2.21)

where ˆH is the Hamiltonian of the system and ρii the diagonal elements of the

density matrix. The power density is found in the same way as for the classical case by Fourier transforming CI. An important difference to the classical

quan-tities follows directly from the observation that in general current operators at different times do not commute. Consequently the correlator is not symmetric with time-reversal. Instead it satisfies CI(t) = CI(−t)∗and the noise spectrum

acquires an antisymmetric component S(ω) 6= S(−ω). By itself this does not mean much. However it has been shown that this asymmetry can actually be accessed by coupling the system to a detector [27]. This will be discussed in chapter 4 of this thesis. The result is that the positive and negative branch cor-respond to the emission and absorption spectrum of the source. In the absence of both voltage and temperature, the emission is zero. However the zero-point fluctuations can still absorb energy such that S(ω) 6= 0.

An equilibrium system (V = 0) satisfies the detailed balance condition

S(ω) = e~ω/kBT_S(−ω), _(2.22)

which goes to the classical limit S(ω) = S(−ω) if ~ω ≪ kBT .

For the noise power of a voltage biased conductor with transmission eigenval-ues Tnthere exists a general formula for the non-symmetrised spectrum valid for

arbitrary voltage and temperature [14]. The limiting expressions for thermal-, shot- and quantum noise can be obtained in the limits of large temperature, volt-age and frequency respectively. Figure 2.7 follows as the low frequency/classical limit ω → 0. S(ω) = GQ ( 2~ω coth _~ ω 2kBT X n Tn2+ (~ω + eV ) coth ~ω + eV 2kBT + (~ω − eV ) coth ~ω − eV 2kBT X n Tn(1 − Tn) ) (2.23) 1/f Noise

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2.3. Noise detection 23

appearance in rather different systems, there has been no uniform theory of pro-cesses leading to such distinct noise. In the systems we study their appearance has been related to the random motion of impurities between two positions or, equivalently, the random occupation of charging centres. A single such centre causes a Lorentzian contribution to the spectrum. However in sufficiently large conductors we can expect to find many of them differing in their typical time scales τifor occupation/de-charging. Assuming that these times are distributed

over a large interval τ ∈ [τ1, τ2] following a distribution P (τ ) ∝ 1/τ,

averag-ing with respect to τ results in the typical 1/f -behaviour for the noise power (ω ∈ [1/τ2, 1/τ1]) S(ω) = Z dτ P (τ ) τ 1 + ω2_τ2 ∝ 1 ω. (2.24)

The occurrence of 1/f -noise is a separate field of research and is not covered in this thesis which concentrates on the more general other sources.

2.3 Noise detection

In the previous section we have seen that noise is not only an unwanted signal. Rather it contains a wealth of information not present in averaged observable quantities. The most important aspect of studying noise has not been covered so far. How do we actually detect the noise emitted by a source? In princi-ple one could measure time-resolved, that is continuously obtain values of the fluctuating quantity and later do statistics on the data, calculating for instance the noise power. In practice however this might be difficult to achieve if the typical time of fluctuations is very small requiring an extremely fast detector. That is one of the reasons why experiments on noise in quantum transport have to this day remained a tricky subject. Though direct time-resolved measure-ments following the definitions in the previous section exist [29], noise is usually measured in a different way.

The following discussion of the measurement of classical noise follows Gavish et al. [30] (see figure 2.10). In a typical situation the fluctuating current from a sample is send through a filter. This can be for instance an RLC circuit having a resonance frequency Ω = (LC)−1

2 and a certain bandwidth ∆_f ≪ Ω. The resulting current can then be decomposed into slowly varying Ic,s(t) and a fast

oscillation with frequency Ω [31]

If(t) = Ic(t) cos(Ωt) + Is(t) sin(Ωt). (2.25)

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Ω

SM(ΩΩΩΩ)

V

DC Square Law Detector optionally preceded by a Homodyne or Heterodyne Detector

Sample RLC Filter Amplifier Detector

DC Voltage Resonance Frequency ΩΩΩΩ, Width ∆∆∆∆f E.g., FET, Degenerate Parametric Amplifier, Non Degenerate Parametric Amplifier Display Sample Filter Amplifier Detector

Display J(t) If(t)

SM(ΩΩΩΩ)= Ia2(t) /∆∆∆∆f or Ia,c2(t) /∆∆∆∆f

Ia(t)

Fig. 2.10: A typical setup of a noise measurement,(from [30]).

resulting squared currents are sampled at many (N ≫ 1) different times and the following quantity is calculated and displayed.

SM(Ω) = 1 N ∆f N X n=1 Ia2(tn) (2.26)

It is important at this point that the sampling times tn originate from a large

time interval T0which is larger than all other times in the system (~/eV , ~/kBT ,

1/∆f). At first glance there is no relation to the noise power as defined in

equation (1.3) since this quantity depends only on averages of squared currents taken at the same time.

The justification for using S(Ω) defined by equation (2.26) instead of the originally defined power density is closely related to the usage of a filter in the measurement setup. Since the biggest time in the system is T0, all time

integrations in the defining integrals can be restricted to this size. The power spectral density then takes the following simple form

S(ω) = |I(ω)|

2

T0 . (2.27)

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2.4. Theoretical techniques 25

the input signal J(ω) If(ω) =

γJ(Ω), if |ω − Ω| . ∆f

0, if |ω − Ω| & ∆f (2.28)

with some constant γ. Thus, according to (2.27), a similar relation between the noise powers of original and filtered/measured signal holds (with proportionality γ2_{). Assuming that the system is stationary, i.e., I}2_{(t) does not depend on time,}

and noting that

I2_{(t) =} 1

π Z

dω SJ(ω), (2.29)

we obtain the important relation 1 ∆f

I2

f(t) = γ2SJ(Ω), (2.30)

which tells us that the noise power can be accessed from the procedure expressed by equation (2.26).

The discussion as presented is of course idealised but nevertheless contains the essential ideas of noise detection. In a real experiment, the measurement apparatus would introduce additional noise notably thermal noise. Importantly this derivation also assumed classical noise. Different detection schemes for classical and quantum noise and concrete setups are presented in later chapters. Quantum noise detection is discussed in chapter 4.

2.4 Theoretical techniques

2.4.1 Full Counting Statistics

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Such a theory was pioneered by Levitov and Lesovik [32, 33, 34] and called Full Counting Statistics (FCS) reflecting that it describes the statistics of dis-crete objects (electrons) beyond the second moment (“full”). In agreement with general practice we will adopt FCS to denote the theory as well as the generating functional whose calculation is its aim. Their idea was to introduce in analogy to classical physics an operator of transferred charge during time τ through the operator of current, ˆI(t) as

ˆ Q =

Z τ

0

dt ˆI(t). (2.31)

Based on this operator they applied the textbook measurement procedure, that is, the probability of an outcome (tranferred charge) is given by the square of the projection of the system state on an eigenstate of ˆQ. However there is a problem with such a procedure which is due to the artificial description of the measurement process. They found that the transferred charge in this scheme was not quantised in units of e.

Later they improved their model to a more realistic charge detector by in-troducing a spin-1/2 precessing in the magnetic field which is created by the current [33, 34]. Its angle of precession is then proportional to the transferred charge. The concrete realisation of the detector is not important and differ-ent schemes lead to the same statistics. Applying the projection paradigm to the detector they were able to obtain the distribution of transferred quantized charge Pτ(n). There are alternative ways of defining a charge transfer statistics

that do not require a detector. A counting statistics can be obtained as an intrinsic property of the system as has been demonstrated by Shelankov and Rammer [35].

An important result is obtained for electrons tunnelling through a barrier with transmission T at zero temperature and finite voltage V between the reser-voirs (figure 2.11). In this limit the reservoir states are either filled or empty and electron transfer can only occur in one direction in a strip of size eV . Clearly the only source of noise is shot noise due to the random transfer of charges. The probability to transfer n electrons during the time τ is then

Pτ(n) =

N n

Tn_{(1 − T )}N −n_. _(2.32)

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2.4. Theoretical techniques 27 f_r f_l E eV x T

Fig. 2.11: _{Electrons tunneling through a barrier with transmission T between} two reservoirs.

distribution. For the considered setup it results in the following plausible inter-pretation: During time τ there are N independent attempts to pass the barrier. These attempts are successful with probability T and rejected with 1 − T . The probability to transfer n particles is given by Pτ(n). Thus, equation (2.32) can

be seen as a result of electrons gambling. If the chance to win (transfer) in one game is T and the total number of games N, the probability to win n times is Pτ(n).

For completeness and comparison with later formulae we also list the cu-mulant generating function (cf. equation (2.5)) and the first few cucu-mulants of transferred charge

ln F (χ) = N ln_{1 + T (e}iχ_{− 1) ,} _(2.33)

hhn2ii = NT (1 − T ), (2.34)

hhn3ii = NT (1 − T )(1 − 2T ). (2.35) Thus any noise in this zero-temperature example is due to the probabilistic nature of the charge transfer, that is to deviations of T from zero or one.

In the limit of small T ≪ 1, the transmissions become rare and consequently the Pauli principle does not influence the electron transfer. The distribution assumes Poissonian character

P (n) = (N T )

n

n! e

−N T _(2.36)

and the corresponding generating function reads

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All cumulants are equal to N T .

At finite temperature the situation is more complicated. In addition to par-tition noise resulting from random transmission/reflection, temperature intro-duces additionally fluctuations in the occupation numbers of the leads. Further-more electrons can be transferred in both directions as states become available. Still it is possible to write down a generating functional for fluctuations. Es-sentially, reflection/transmission probabilities have to be modified accounting for the level occupation. For left-right transfer the transmission probability is proportional to T fL(1 − fR) where fL,Rdenote the Fermi functions of the leads.

The general expression valid for arbitrary temperature as found by Levitov and Lesovik reads [34] ln F (χ) = τ h Z dE ln1 + T (eieχ − 1)fL(1 − fR) +(e−ieχ− 1)fR(1 − fL) . (2.38)

Equation (2.33) (and (2.37)) follow from this expression in the limit V → 0 (T ≪ 1). For simplicity all expressions in this section were written for a single channel conductor with transmission T . The generalisation of single channel F to many channels with transmissions Tn is obtained by summation over

chan-nels, ln F (χ) = P

nln Fn(χ) = ln (QnFn(χ)), where Fn(χ) are the presented

generating functionals with T replaced by Tn. This simple rule—summation

over channels—expresses the fact that electrons in different channels are not correlated. Thereby every channel contributes separately to the statistics and the generating functional factorises into those of individual events or channels. Note that the transmission coefficients in (2.38) can depend on energy. In the same way as for the derivation of the Landauer formula above, this energy dependence can be neglected if it is sufficiently slow and—in this limit—the integration over energy can be performed for arbitrary voltage and tempera-ture. The result is a rather complicated expression and presented in [34]. More instructive it is to look at the second cumulant, that is current noise. Differen-tiation of (2.38) twice with respect to iχ and putting χ = 0 yields the second cumulant of transmitted charge

hhQ2ii = τGQ X n Tn Z dE{fL(1−fL)+fR(1−fR)+(1−Tn)(fL−fR)2]}, (2.39)

which even in equilibrium (V = 0) does not vanish

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If we note that for large detection times and small frequency, the cumulant of transferred charge is related to the zero-frequency current noise hhQ2_{ii =}

τ SI(0)/2, we recover the Johnson-Nyquist result SI(0) = 4kBT G.

In the opposite limit of large voltage eV ≫ kBT , we find from (2.39) that

hhQ2_{ii =} τ

2SI(0) = τ eV GQ X

n

Tn(1 − Tn). (2.41)

For completeness we note that the Fano factor defined as ratio of noise to current for a multi-channel conductor with transmission eigenvalues {Tn} reads

F = P nTn(1 − Tn) P nTn . (2.42)

Equation (2.41) is the generalisation of expression (2.19) for the shotnoise to the multi-channel case. It is an important result since it shows that transport properties are not only characterised by the sum of transmission eigenvalues as in the Landauer expression for the conductance. In fact the full set of Tn is

necessary to determine these properties completely, conversely the set of Tn—

sometimes called the PIN-code of the contact—can be accessed from transport measurements [36]. Theoretically any zero-frequency cumulant measurement provides another equation for determining {Tn}. However access to higher order

cumulants is experimentally difficult and it might be more realistic to study the parameter dependence of lower order cumulants as is done in chapter 3.

Ensemble variations and the transmission distribution

On a microscopic basis it is clear that there are many factors which determine this set—defects, impurities and uncertainties in the design process will all lead to variations of {Tn} between different samples. Instead of discussing a

con-crete realization of {Tn} for a given sample it is more meaningful to get rid of

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The previous results have shown that many interesting transport quantities are expressed as a sum over a function of transmission eigenvaluesP

nf (Tn). If

the average conductance of a nano-structure is large hGiensemble≫ GQ, there are

many channels present. The transmissions are then densely distributed on the interval [0, 1] and the sum can be replaced by averaging with the transmission distribution * X n f (Tn) + ensemble = Z 1 0 dT f(T )P (T ). (2.44) Transport properties are therefore not dependent on a concrete realization of the nano-structure. On the other hand sample-to-sample variations can become significant if hGiensemble ≃ GQ and (2.44) is not applicable. To get a better

understanding let us look at transmission distributions for concrete systems. The simple exercise in chapter 2.2 demonstrated that transport in a quantum point contact is carried by a number NQPCof almost open channels with Tn. 1

and an infinite set of closed channels. For transport relevant are only the open channels. Application of equation (2.43) then gives

PQPC(T ) = NQPCδ(1 − T ). (2.45)

In practice there will always be a little disorder present in the point contact which introduces some probability for reflection thereby spreading the distribu-tion to values close to 1 (figure 2.12(a)). Increasing the disorder more we will enter a new regime where electrons start being scattered mainly on the impurity potential. The crossover happens if the resistance due to disorder is on the order of the resistance of the contact RQPC.

Complementarily the transmission distribution of a tunnel junction with Tn≪ 1 should have a maximum at small T . Introducing disorder close to the

junction will surprisingly enhance the transmission. The simplest process which clearly leads to an enhancement is an electron which was rejected from the tunnel contact being reflected back from the disorder thereby playing another “game” for transmission. The limit on the amount of disorder for this description to be valid is in the same way as for the QPC given by comparing resistances of the disorder and the tunnel junction. For more disorder the electron starts to randomly move around between impurities—a process called diffusion (assuming G ≫ GQ). The contact plays no role as transport properties are governed by the

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2.4. Theoretical techniques 31 T ≪ 1 T ≈ 1 (a) 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 T 2 GQ Pd iff (T )/ G (b)

Fig. 2.12: (a) Two processes causing a smearing of the transmission distribution due to disorder. (b) Transmission distribution of a diffusive conduc-tor.

on the design details [40]

Pdiff(T ) = hGi

2GQ

1

T√1 − T. (2.46)

Though one might expect a diffusive conductor to have all kinds of transmis-sions, this surprising result shows that most channels are either almost closed or open (figure 2.12(b)).

2.4.2 Keldysh technique

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Feynman and Vernon [43] and Kadanoff and Baym [44]. Some often cited refer-ence works include Rammer and Smith [45], Danielewicz [46], Kleinert [47] and Mahan [48].

Besides being the theory of choice of many people for describing systems that are not in thermal equilibrium, the Keldysh formalism allows the convenient for-mulation of full counting statistics (FCS), which describes the complete statis-tics of measurement outcomes of a quantum mechanical variable. After some general remarks on the technique, the FCS in Keldysh space is introduced.

The theory is usually formulated starting from an interacting Hamiltonian H = H0+ HI(t) and assumes that the system starts from a ground-state of the

non-interacting Hamiltonian H0 at some point in the distant past (t0→ −∞).

It is assumed that the solution to the Schr¨odinger equation with H0are known.

The initial density matrix of the system is taken to be diagonal ˆ ρ =X n ρn|nihn| (2.47) with eigenvalues ρn = e−En/kBT. (2.48)

The switching on of interactions is assumed to happen adiabatically. Typically one is interested in the time evolution of averages of (Heisenberg) operators φH(x, t) representing physical quantities. They are related to those in the

inter-action picture by the free Hamiltonian in the usual way [49] φ = eiH0(t−t0)/~_φ

He−iH0(t−t0)/~. (2.49)

The interaction Hamiltonian HI is transformed in the same way. Using the

transformed interaction Hamiltonian one can define a time evolution operator U (t′, t) ≡ T+exp " i ~ Z t′ t dτ HI(τ ) # , (2.50)

where T+orders the (generally not-commuting) operators by time (forward time

ordering).

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t t+

t−

Fig. 2.13: The Keldysh contour

A useful reformulation can be obtained by inserting an identity, essentially mov-ing (in time) to infinity and back, into equation (2.51) and other operator aver-ages

ˆ

U (−∞, t) ˆφ Û(t, −∞) = Û (−∞, t) Û (t, ∞)T+[ Û (∞, t) ˆφ Û(t, −∞)].

Changing the ordering of operators in the time-ordered product, this expression can be rewritten as

TK[ ˆU

(∞, −∞) ˆU(∞, −∞) ˆφ].

The ordering of operators is done along the Keldysh contour (figure 2.13) extending along the real time axis from −∞ to ∞ and back. Naturally we can define two different times corresponding to the two branches, t± and will use

the notation x± ≡ x(t±) to refer to the resulting coordinates of the system.

One might be tempted to replace ˆUˆ

U → 1. However this is forbidden by the special time-ordering procedure; we will use the short-hand notation

TKexp −_~i Z K dt HI(t) ≡ TK[ ˆU (∞, −∞) ˆU(∞, −∞)]. (2.53) Instead of working with averages of products of operators it is much more convenient to define a generating functional Z[j] for arbitrary products. This mathematical trick is no different to that adopted in the (Feynman) perturbation expansion where a source term is added to the Hamiltonian. Here it requires some care with the special time-ordering along the contour. It is easily checked using equations (2.51), (2.52), (2.53) that functional derivatives of

Z[j] = TKexp −_~i Z K dt HI(t) + i Z K dx j(x)φ(x) (2.54) generate the relevant operator averages.

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interest (X, φ) and an integral/sum over the fluctuations with some kernel. It would be desirable to extract in the same way a function “Pτ(φ)” describing the

statistics of fluctuations. However it has been realized [50] that the measure-ment outcomes in some systems can not be interpreted in terms of a positive probability distribution. Therefore, equation (2.54) provides the generalisation to a larger class of systems.

In the following subsection the correspondence between probabilities and the generating functional will be discussed by means of a detector coupled to the quantity of interest. The artificially introduced counting field will be naturally replaced by the detector coordinate (cf. equation 2.5).

The concepts presented in this subsection lie at the heart of the Keldysh technique. Nothing has been said so far about the actual evaluation of the path integrals and it is not clear that any simplification has been achieved. It is hoped that the examples presented in this thesis will provide some justification for such manipulations.

2.4.3 FCS with a detector

Recently it has been realized [51] that the FCS of current can be obtained by a modification of the Keldysh formalism, introducing a fictitious counting field as a source term. When studying the FCS of superconductors [50] a problem appeared. An interpretation in terms of attempts and probabilities failed, as the desired probability density could take on negative values. This conflict was finally solved by Nazarov and Kindermann [52] and conditions for the inter-pretability in terms of probabilities were given. For the case of superconductors the problem was traced down to a violation of gauge invariance. However that did not mean the breakdown of the theory. In the same paper a refined formu-lation in terms of the time evolution of the density matrix was given. Since it is relevant for understanding subsequent chapters we summarise the results of that paper here.

Though the FCS has been defined for a general variable ˆA, we shall use the suggestive notation ˆI to mean current fluctuations. The basic idea of [52] was to refine the measurement process of Levitov Lesovik. Instead of the definition of a charge transfer operator as in equation (2.31) they introduced a detector with coordinate ˆx and conjugate momentum ˆq ([ˆx, ˆq] = i~). The coupling between the system and the detector was chosen such that during the measurement time τ the Heisenberg equation of motion resembles equation (2.31)

(45)

The measurement paradigm was then applied to the detector momentum. Fol-lowing Feynman and Vernon the model can be seen as a system coupled to an environment (detector), the time evolution of the system can thus be reduced to an integral over the system degrees only. The disturbance due to the detector can be reduced to an influence functional.

For further reference we review the main steps of the derivation and refer for details to references [43, 52]. The starting point is the Hamiltonian

ˆ

H(t) = ˆHsys− ατ(t)ˆx ˆI + qˆ 2

2m (2.56)

where the linear coupling term is assumed to be present mainly during the measurement time τ and is adiabatically switched on and off outside of that interval by ατ(t). The aim is then to calculate the dynamics, that is the time

evolution of the density matrix. In the usual way it can be obtained by mul-tiplication with the time-ordered and anti-time-ordered (conjugate) evolution operator ˆU = T±exp{∓iR dt ˆH(t)/~}. Inserting complete sets of states, the

operators in the Hamiltonian can be transfered to their eigenvalues. Assum-ing that the initial density matrix factorises into detector and system ρsysρin

and doing the integration over system and momentum the time evolved density matrix takes the form

ρf(x+, x−) =

Z

dx+1dx−1K(x+, x−; x+1, x−1, τ )ρin(x+1, x−1) (2.57)

where the Kernel includes the influence functional and the detector action (for definitions see [52]). Clearly the measurement process as described presents a disturbance to the system. In order to reduce the (classical) back-action we would like to have a static detector. This is achieved by letting the mass m → ∞, a similar reasoning to ideal ammeter/voltmeter having zero/infinite resistance. In this limit equation (2.57) takes a suggestive form in Wigner representation

ρf(x, q) = Z dq1Pτ(x, q − q1)ρin(x, q1), (2.58) Pτ(x+, x−) = TrsysT+e− i ~R dt[ ˆHsys−ατ(t)x +_I]_ˆ ρsysT−e i ~R dt[ ˆHsys−ατ(t)x −_I]_ˆ , (2.59) Pτ(x, q) = 1 2π Z dze−iqzPτ x +z 2, x − z 2 (2.60) and ρ(x, q) is found in the same way from ρ(x+_{, x}−_{). This set of equations}

Effects and detection of quantum noise