Thesis
Reference
Mesoscopic noise effects in weakly and strongly interacting systems
CHERNII, Iurii
Abstract
Mesoscopic physics considers systems of appropriate size and at time scales of the order of decoherence time, which allows us to analyse the transformation of information from a quantum state to a classical signal, occurring in a process of quantum measurement. We concentrate on the limiting cases of weak and strong coupling between a mesoscopic source of noise and a subsystem functioning as a detector. In the ``weak coupling'' part we elaborate on the idea of cross-correlation measurements for quantum noise detection and develop a set of original methods to tackle the perturbation theory up to fourth order. In the ``strong coupling'' part we provide an exact solution for one-dimensional system out of equilibrium, strongly interacting with a localized state and predict some remarkable experimentally verifiable features, such as Fermi-edge singularity in the noise induced transition rates and unusually pronounced manifestation of non-Gaussian noise effects.
CHERNII, Iurii. Mesoscopic noise effects in weakly and strongly interacting systems . Thèse de doctorat : Univ. Genève, 2014, no. Sc. 4666
URN : urn:nbn:ch:unige-458739
DOI : 10.13097/archive-ouverte/unige:45873
Available at:
http://archive-ouverte.unige.ch/unige:45873
Disclaimer: layout of this document may differ from the published version.
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UNIVERSITÉ DE GENÈVE FACULTÉ DES SCIENCES
Département de physique théorique Docteur E. Sukhorukov
Mesoscopic noise effects in weakly and strongly interacting systems
THÈSE
présentée à la Faculté des sciences de l’Université de Genève pour obtenir le grade de Docteur ès sciences, mention physique
par
Iurii Chernii
de Kyiv (Ukraine)
Thèse No 4666
GENÈVE
Atelier d’impression ReproMail 2014
Remerciements
Je tiens à remercier tout d’abord le directeur de la thèse, Prof. Eugene Sukhorukov. A part de sa fonction impeccable concernée directement de la supervision de mon travail scientifique, il a fournit également le soutien moral énorme. Je vais ainsi reconnaître sa patience et sa com- préhension remarquables, sans lesquelles il aurait été problématique pour moi de procéder pendant quelques temps les plus difficiles qui peuvent avoir lieu dans la vie de chaque sans exception étudiant de doctorat.
Notre petite groupe de recherche se composait seulement, pour la plupart de temps, de moi et de Ivan Levkivskyi, à qui je suis reconnaissant pour les innombrables discussions scientifiques fructueuses et l’encouragements continues, ainsi que pour nos aventures alpines partagées qui m’ont aidé à maintenir un esprit sain et un bonne état physique. Les travaux pré- cédents fait par Ivan, Eugene, et leurs autres collègues sur le développement de la technique de bosonisation hors-équilibre, a fourni le fondement solide indispensable pour le deuxième projet inclus dans cette thèse. Egalement comme les calculs numériques de la statistique de comptage complètes (FCS) des excitations après un point de contact quantique, menées par Ivan, ont permis d’étendre la gamme de paramètres couverts dans cette partie du travail. Le soutien des membres ultérieures de notre groupe : Artur Slobodeniuk et Edwin Idrisov était également utile, en particulier pendant la rédaction de ce manuscrit.
Je suis d’ailleurs reconnaissant à l’administration du Département de Physique Théorique, notamment les secrétaires Cécile Jaggi-Chevalley et Francine Gennai-Nicole pour toute as- sistance. Ainsi que à la directrice, Prof. Ruth Durrer, pour la possibilité de poursuivre mon travail au-delà des délais habituels. Et après tout, aux membres de la commission de thèse : Prof. Thomas lhn, Prof. Goran Johansson, Dr. Patrice Roche, et Dr. Christian Flindt qui ont mis leur apport à la pointe culminante, la soutenance de la thèse.
J’ai aussi beaucoup apprécié l’ambiance conviviale et joyeuse crée et toujours maintenue par tous les membres anciens et actuels de notre département, surtout ceux qui résidaient au deuxième étage du bâtiment des Sciences I, où j’ai eu le plaisir de travailler au cours de ces cinq longues années.
Finalement, ma gratitude aux membres de ma famille, de leur soutien indubitablement indispensable, malgré la distance géographique qui nous séparait ces derniers ans.
ii Remerciements
Merci tout le monde !
Les effets de bruit mésoscopique dans les systèmes à faible et forte interaction.
Résumée de thèse en langue française
Le présent travail traite des concepts relatifs à la détection de bruit quantique dans les systèmes mésoscopiques. C’est-à-dire aux systèmes à l’échelle intermédiaire entre les di- mensions dominées par des effets quantiques, et ceux permettant déjà une description par la physique classique. Nous considérons les deux types des systèmes très différents : le premier caractérisé par l’interaction faible entre la source du bruit et le détecteur, et l’autre par l’inter- action forte. L’attention particulière a été payée aux effets de bruit non-Gaussien et de bruit hors équilibre. Ces deux concepts sont au présent parmi les phénomènes les plus concernés dans le domaine de la physique mésoscopique.
Dans la première partie nous avons étudié les possibilités de détection de la composante quantique de bruit généré dans un circuit mésoscopique au moyen de la détection ´Ssur puce’
de la corrélation croisée à l’aide d’une paire des détecteurs à deux niveaux. Les avantages de cette approche sont inhérents à ceux des détecteurs sur puce, permettant d’éviter la perte de cohérence dans les fluctuations sur le chemin vers le détecteur ; et aussi des mesures de corrélation croisée qui assure une protection contre le composant classique de bruit, ainsi que les sources de bruit locales non corrélées. Une installation intelligemment conçu sur puce permet donc la de séparer l’information utile sans la nécessité d’une amplification spéciale à haute fréquence et à très faible température.
En même temps, le couplage faible est le facteur déterminant de notre méthode qui permet la séparation des échelles de temps. Les fluctuations rapides de courant dans le circuit induit des transitions rares dans les détecteurs de points quantiques, qui peuvent alors être décrites par une ´Séquation maîtresse’. Les informations pertinentes sur le bruit sont codées dans les taux de transition, qui sont déjà un processus d’un niveau plus lent. Enfin, la corrélation croisée des signaux des deux détecteurs peut être obtenue facilement et est la quantité finale observée.
Nous avons analysé les conditions physiques et avons trouvé une gamme de paramètres, où la composante Gaussienne quantique peut être la contribution dominante de la corréla- tion croisée. Alors que la contribution non-Gaussienne malheureusement est supprimée par les ordres supplémentaires de la faiblesse de couplage. Outre les résultats scientifiques, nous
iv
allons également noter les méthodes originales qui ont été développés dans le travail. Il s’agit notamment du calcul direct de l’évolution temporelle dans le quatrième ordre de la théorie de perturbation, qui a permis d’éviter les divergences qui apparaissent dans les ordres supérieurs de la théorie des perturbations dans la représentation énergétique. La matrice de transition pourrait être trouvée par conséquent grâce à la séparation des échelles de temps, par extrac- tion de la contribution irréductible sur l’échelle de temps intermédiaire entre le temps de corrélation, et le temps de l’évolution des occupations de détection.
La deuxième partie et consacrée aux systèmes unidimensionnels à couplage fort. Nous avons étudié les effets de couplage fort d’une charge localisée à des canaux électroniques unidimensionnels. Des systèmes similaires ont déjà été pris en compte dans l’équilibre ther- mique, mais nous nous attendons à la physique la plus intéressante lorsque les transitions à l’impureté sont induites par le bruit de partitionnement dans le canal hors équilibre. Ce bruit hors équilibre est réalisé par un séparateur de faisceau QPC d’un interféromètre de Mach- Zehnder, dont le canal sert de l’un des manches. Le couplage fort se reflète dans la phase de dispersion, qu’un électron du canal s’échange avec la charge localisée en passant par le point quantique. Un tel système peut être considéré comme un détecteur de l’état du point quantique, également comme détecteur de ´Squel chemin’ de passage de l’électron dans. Mais puisque le couplage est fort, il est plus approprié pour traiter le système dans son ensemble.
L’effet du bruit hors équilibre et le backaction des perturbations de la mer Fermi lors d’une transition d’effet tunnel à l’état localisé, doivent donc être prisent en compte non perturbative- ment. Cependant, grâce à une transformation unitaire, l’Hamiltonien peut être partiellement diagonalisé, et le couplage fort peut être représenté comme une modification de l’Hamilto- nien tunnel. Cela nous permet de trouver les taux des transitions de la ´nrègle d’Or˙z de Fermi en second ordre en couplage tunnel. En utilisant la technique de bosonisation non-équilibre, le problème est alors réduit au calcul des statistiques de comptage complet des fluctuations hors équilibre après le partitionnement QPC.
Nous résolvons donc le problème de manière analytique dans la limite de Markov pour des désaccords d’énergie faible et de petite transmission ou réflexion dans le partitionnement QPC. Pour les transparents intermédiaires et les énergies grandes, nous faisons appel à des calculs numériques exacts. La conclusion principale est que les taux de transition ont la dé- pendance énergétique caractéristique, qui indique que le couplage fort conduit à un effet de backaction non-perturbative et la catastrophe d’orthogonalité se manifestant dans la singula- rité de Fermi dans les taux de transition. Nous constatons également une asymétrie de la chute de la visibilité et de la dépendance non-négligeable de la position d’immersion sur la transpa- rence du QPC de partitionnement. Cette asymétrie est due à la composante non-Gaussienne du bruit qui cause la brisure de symétrie de signe de charge. Il décale la position de la réso- nance à effet tunnel en fonction du signe du courant dissipatif et de caractère des fluctuations hors équilibre dues à la transparence faible ou grande de la QPC.
Contents
1 Introduction 1
1.0.1 Quantum measurements . . . 2
1.1 Noise . . . 2
1.1.1 Full counting statistics . . . 3
1.1.2 Non-Gaussianity . . . 4
1.2 Mesoscopic physics . . . 5
1.2.1 Mesoscopic structures . . . 5
1.3 Role of the coupling strength in mesoscopic devices . . . 8
1.3.1 Weak coupling . . . 8
1.3.2 Strong coupling . . . 8
2 Direct access to quantum fluctuations through cross-correlation measurements 11 2.1 On-chip noise detection . . . 12
2.2 Two-level detectors . . . 16
2.2.1 Pair of two-level detectors . . . 17
2.2.2 Time evolution . . . 18
2.3 Full counting statistics of the QPC currents . . . 20
2.3.1 Cross-correlator . . . 21
2.3.2 Perturbative approach . . . 23
2.4 Cumulant expansion . . . 25
2.5 Current and phase correlation functions . . . 26
2.6 The cross-correlator: Results . . . 28
2.6.1 Classical noise . . . 28 v
vi Contents
2.6.2 Quantum noise . . . 30
2.6.3 Analysis of physical conditions for the quantum noise detection. . . 32
2.7 Conclusions . . . 33
3 Fermi edge singularity in one-dimensional systems far from equilibrium 37 3.1 Experiment and background . . . 38
3.2 The model . . . 41
3.3 Transition rates and the nonequilibrium bosonization . . . 43
3.4 Analytic solution in the Markovian limit . . . 44
3.5 Exact numerical calculations . . . 45
3.6 Visibility of the AB oscillations . . . 49
3.7 Conclusions . . . 53
4 Summary of results 55 4.1 Weak coupling and cross-correlation measurements . . . 55
4.2 Strong coupling in one-dimensional systems . . . 56
Chapter 1
Introduction
Quantum Mechanics has been developed in the early 20th century to embrace a growing array of experimental observations that could not be explained with the help of classical physics.
The theory is based on a set of concepts and principles that, unlike in the case of classical physics, to the most part, do not represent the intuitive categories in which the surrounding world is perceived. These are the concepts such as: wave functions, particles indistinguisha- bility, quantum superposition, non-commutativity of the observables’ operators, etc.
A range of more advanced theories have since been developed, further elaborating on the same foundations, such as quantum field theory, or string theory. However, even the basic implications of QM never cease to excite and bewilder the curious minds. A particular chal- lenge is to build experiments that would demonstrate the manifestations of quantum physics possibly unambiguously and in the most well controllable systems. Such experiments range from the various verifications of the EPR ‘paradox’ and the Bell theorem [1, 2] to the most complex many body effects in ultracold atom systems [3]
The reason for this is that the quantum mechanical effects are only revealed under certain conditions, that allow for the so-called quantum coherence to be preserved. Depending on the system and context, these conditions commonly include: small number of particles involved;
low ambient temperature; fast observation; and small length scales. While in the everyday life – at normal conditions, when none of the above conditions are satisfied – we normally observe all the phenomena in the classical limit. The relation between the quantum nature of the reality on a microscopic level and the macroscopic observations is a complex and insuf- ficiently explored field. Investigation of an example of such a transition will be an integral part of the first project presented in this thesis. It is particularly related to the subject of quan- tum measurements, that investigates how a certain information about quantum states can be processed by a measurement apparatus and eventually displayed in a classical, macroscopic signal.
2 Introduction
1.0.1 Quantum measurements
An observable quantity is represented in the quantum mechanics formalism by a Hermitian operator acting in a Hilbert space [4]. The possible values of the quantity correspond then to the eigenvalues of this operators.
Historically, the simplest kind of measurements had been considered - the so called pro- jective measurements. It assumes a classical system the state of which can be known, con- sidered as a detector, interacting with the given quantum system. This interaction assumes a collapse of the state of the quantum system into one of the possible eigenstates of the respec- tive observable operator, corresponding to the eigenvalue indicated by the resulting classical state of the detector. The probability of a particular result is related to the corresponding
’proportion’ of the final state in the composition of the initial quantum state, called the prob- ability amplitude. Hence the state is seen as projected onto one of it’s components during the process of measurement. This view of a measurement, set as a textbook quantum mechanics postulate, is of course oversimplified. However, it highlights the probabilistic nature of the outcome of an act of measurement.
In the general case a measurement is a much more complex process, related to interaction and the information flow between a series of open quantum systems. The concept of the on-chip detectors, that we use in the first part, is to illustrate how a number of consecutive stages of this flow can be identified, where the information from the relevant quantum system is converted step by step into the final classical signal.
The experimental advances have made it most relevant to be concerned with the funda- mental limitations on the precision and efficiency of the measurement processes, and renewed interest in noise and specifically its quantum-mechanical characteristics.
1.1 Noise
In many everyday applications noise is commonly considered as an unwanted phenomenon, as it - almost by definition - makes it more difficult to distinguish what is meant as the useful signal. It can be, however, of great academic interest [12], since noise can be the only output of a studied system, which carries the information about its properties. Moreover, as the size of systems grow, the relative power of the noise tends to decrease due to the interference of independent sources, typically as one over the square root of the system size. A spectral function
S(ω) =
∫
dteiωt⟨ˆj(t)ˆj(0)⟩ (1.1) is one of the characteristic properties of a noisy quantity, such as a current operatorˆj(t).
And the zero-frequency is the noise powerS(0) is a common measure of the strength of fluctuations.
Note that the spectral function S(ω)is always a symmetric function of frequency for a classical quantity, sincej(t)andj(0)can be swapped around. This allows one to point out one important distinction in the context of the present work, between quantum and clas-
1.1 Noise 3
sical signals, indicated by the asymmetry of S(ω). The asymmetry is caused by the non- commutativity of the operators that represent different observables, or the same operator in different moments of time, and is one of the most distinct quantum features. Normally, we accompany with the epithet “quantum” the effects that do not have a direct classical analogy, and thus are the signatures of the quantum mechanical nature of the processes.
Although the spectral function may not be a complete characteristic, since it does not take into account the higher-than-second order correlators, there is, however, a wide class of systems for which it is. These are the systems where the respective quantities have a Gaussian probability distribution. In that case, all the higher order averages of such quantities can be expressed in terms of the first and second order moments. Or in terms of the second moment alone, if only central moments are considered, which is often the case. A perfect illustration is the well known example of Wick’s theorem, applicable to Gaussian systems:
⟨j1j2j3j4⟩=⟨j1j2⟩⟨j3j4⟩+⟨j1j3⟩⟨j2j4⟩+⟨j1j4⟩⟨j2j3⟩, (1.2) wherejk can be the operators of currentj(t)taken at moments of timet=tk. In terms of this simple kind of averages, such as the left side of (1.2), it is not always easy to tell whether a process is, indeed, Gaussian. But special combinations of this averages are known to be more practical.
1.1.1 Full counting statistics
For a simple process, such as for example electron tunneling where an outcome can be quan- tified with a single natural number, ifPT(n)is the probability thatnelectrons pass, during the observation timeT, one can define the moment generating function [18]
g(λ, T) =∑
n
eλnPT(n) (1.3)
whereλis the counting variable. It turns out thatg(λ, T)has a great practical use in the description of the statistics of fluctuating quantities, such as the number of passing electrons.
Indeed, we first note, that in the limit of measurement timeT ≫ tc much longer than the correlation time, the logarithmlogg(λ, T)becomes linear in timeT, because of the multi- plicative property of the joint probability of independent, uncorrelated processes. This allows one to define thecumulant generating function
H(λ) = lim
t→∞
1
t logg(λ, T); (1.4)
and, correspondingly, the irreducible moments, or so called cumulants,cm=∂mH(λ)/∂λm|λ→0. For a more detailed overview please refer to the section [2.3].
The logarithm in the definition ofH(λ), analogously to its role in the definition of the free energy and other similar objects, allows to cancel out the reducible contributions, such as in the formula (1.2). What remains – the cumulants – is a convenient measure of the non-trivial correlations of the corresponding order. Particularly, if all the cumulants of a certain quantity
4 Introduction
starting from third order and higher, are zero, then this means identically that this quantity has a Gaussian distribution.
It should be noted, however, that the definition of such irreducible moments for more complex quantities in realistic systems is not entirely universal, and they are constructed individually for a particular sort of system, see section [2.4]. Moreover, there may be many different variants, due to the various possibilities of the operator ordering. In fact, it is often a non-trivial question as to which particular quantity of the said order is being measured in an experiment, therefore a very detailed consideration of the measurement process itself is required from every theoretical consideration.
1.1.2 Non-Gaussianity
Since the observed signal is usually averaged, or integrated over time due to the slow response of the amplification circuits, this signal is a result of the contribution of many independent processes. This has an important implication for the statistics of such a signal due to the Central Limit Theorem (CLT). The latter states that under certain conditions, such as a fi- nite variance, the distribution of a sum of a big number of equivalent independent random variables will converge to the Gaussian distribution.
The quantity of interest often enters to the measurable signal with a certain multiplica- tive prefactor, such as a coupling constant. This means that every consecutive order moment enters with one more power of the prefactor. Thus for the small prefactors the relative contri- bution of the higher moments will be increasingly suppressed and harder to detect.
Non-Gaussianity is, therefore, an important property of a certain quantity or a phenom- ena, a deviation from the constrictions imposed by the CLT. Such a deviation indicates that some of the requirements of the theorem are not satisfied. This can be for a number of rea- sons: the elementary processes can be correlated; it can be an indication of many-particle entanglement [13]; or the response time of the amplifier can turn out to be sufficiently short to resolve few elementary processes.
It also presents a distinctive feature – the sensitivity to the sign of the respective quantity, which is not available on the Gaussian level for apparent reason. This can be the sign of the charge carriers in the transport systems, or the sign of the coupling parameter, and is often linked to non-equilibrium effects.
This explains the particular interest that non-Gaussian effects attract in various fields, also outside of condensed matter physics, encouraged by the universal challenges of their experimental detection [14]. In condensed matter systems, one of the approaches is to make a detector as small and as close to the system as possible. This often assumes realizing the detector as a part of the same semiconductor sample, or on the same chip. The notion of on-chip detectors will be elaborated in the section 2.1.
It is now suitable to note that we are particularly interested in the quantum aspects of noise. For this the system has to be small enough, so the quantum effects are pronounced, but still produce a classical, usable for further processing, signal in the output of the detector. The actual size of a sample to correspond to these requirements is determined by the characteristic
1.2 Mesoscopic physics 5
length scales of the system.
1.2 Mesoscopic physics
The pure basic quantum mechanics formalism can be successfully applied to the systems with few degrees of freedom, where the Schrodinger equation can be solved directly, in one form or another, to answer various questions about the properties of the system such as the energy spectrum and probability density distributions of the eigenstates. These are, for example, the classical exactly solvable problems of simple atoms, or of a single particle scattering on a potential. There are also examples of integrable systems with arbitrarily many degrees of freedom, such as harmonic oscillator chains and arrays, etc., however, these are rather exceptional model systems that rarely have macroscopic realizations since the integrability is easily destroyed by any perturbations. Moreover, the macroscopic systems are well know to behave in accordance with the classical concepts, and the many particle systems are well described by the statistical mechanics and thermodynamics.
The question arises, what are the length scales, where each of these descriptions are applicable, and where it is substituted by the other. The broad field ofmesoscopic physics, with its distinct methods is determined by the intermediate length scales, where the transition takes place from the behavior governed by quantum mechanics, to that of classical physics.
These scales are characterized by the dephasing, or the phase relaxation length. In electronic transport this is normally bounded roughly between the Fermi wave-length as the principal quantum length scale, and the inelastic scattering length, at which the electrons lose quantum coherence and are thermalized together with the sample.
1.2.1 Mesoscopic structures
The advances of the experimental techniques [5] made it possible to artificially build struc- tures that manifest outstanding novel properties. Besides contraction of the linear size of a system, the behavior can be drastically changed and the many interesting effects can be made relevant by reducing the dimensionality.
When two bulk materials with different Fermi energies are brought into contact, move- ment of the electrons between the two sides allows for the leveling of the Fermi energy. The transversal potential well that appears due to the corresponding tilting of the potential around the interface, makes possible the existence of bound states in the direction perpendicular to the boundary. In the case when only one, the lowest, bound state falls below the Fermi level, it is occupied by electrons that are, however, free in the plane parallel to the interface. Hence a two-dimensional electron gas (2DEG) can be formed [6].
The 2DEG typically is a very clean medium, since by the fabrication technique, any impurities are distanced from its plane. This leads to a remarkable increase of the mean free path by up to thousand times – from the order of 100nm in typical metals, to 1− 100µmin 2DEG. This dramatic extension of the appropriate distances range is decisive for the experimental accessibility of the mesoscopic physics effects.
6 Introduction
Figure 1.1: a) Layer of the two-dimensional electron gas (blue) in a semiconductor heterostructure interface between GaAs and AlGaAs, b) the band structure of the two materials before they are brought into contact, and c) formation of the transverse quantum well due to the redistribution of the charge carriers near the interface
Local application of an external electric field, can shift the electrons occupation function, emptying the transversely localized states. This can be done by locating electrostatic gates with appropriate gate voltage bias, therefore depleting some regions of the 2DEG. Various structures can be formed then, forming two-dimensional ‘printed circuit board’-like electrical circuits.
Further, with the help of two side gates, a narrow junction between two regions can lead to quantization and bound states in one of the in-plane directions, thus leaving the electrons free only in one dimension. This is how a quantum point contact (QPC) can be made. The number of transverse modes, or bound states, is determined by the gates voltage bias, and therefore so is the number of the transmission channels. Since the conductivity of a single one-dimensional channel is given by the conductance quantum
Gq = 2e2
h , (1.5)
the conductivity of a QPC is represented by a staircase-like function of the gate voltage [7].
Transitions between the steps of this staircase are particularly important, since a QPC tuned
1.2 Mesoscopic physics 7
Figure 1.2:The quantum Hall effect: classical cyclotron orbits and the skipping trajectories along the edges in a strong magnetic field, that form an isolator in the bulk and the chiral edge states, respectively.
to work at one of these transitions, can be used as a highly sensitive electrometer. Similarly, extended linear structures – quantum wires – can be also realized.
Figure 1.3:A quantum point contact is formed by applying the gate voltageVGto the gate electrodes placed on top of the heterostructure containing the two-dimensional electron gas, depleting it locally, and resulting in a junction between the two regions separated by the gate.
A very different approach to formation of a one-dimensional system is the application of a strong magnetic field perpendicular to the 2DEG [8]. This leads to the quantum Hall (QH) effect, which forms edge channels along the boundaries of the 2DEG, corresponding to the classical skipping orbits. The physics of the QH edge channels is incredibly rich, but in many cases it can be dealt with in fairly simple terms, thanks to the effective theories describing the propagation of the edge excitations [69]. Such is the bosonization technique, that is the foundation of chapter 3.
8 Introduction
Various fabrication techniques are also available to produce the effectively zero-dimensional objects - quantum dots [9]. The latter, containing as few localized states as two, are a realiza- tion of the simplest quantum model, a two-level system.
1.3 Role of the coupling strength in mesoscopic de- vices
The mesoscopic structures listed above, provide quite a rich arsenal for designing experimen- tal setups with a wide choice of possible operating regimes and parameters. In the previously considered context of measurements, we particularly distinguish how the source of noise, or thesystem part, and thedetector part of the setup are interacting with each other. We announce here the two limiting cases of weak and strong coupling, that reflect the essential difference between the systems studied in the two following chapters.
1.3.1 Weak coupling
The opposite limiting case to a projective measurement is a perfect non-invasive measure- ment, when the interaction with the detector does not produce any disturbance of the system’s state [10]. This is of course impossible, since there is a fundamental relation between the gain of information from a quantum system and the rate of the loss of coherence of this system [11]. But unlike in the case of an idealized instance of a projective measurement, the amount of the extracted information is continuously controllable by the strength of the weak coupling between the quantum system and the measurer, and by the duration of this interaction. Par- ticularly, weak coupling allows for a separation of the time scales of the different relevant processes in the system.
An example of such a system is considered in chapter 2, where multiple photons from an electric circuit, weakly interacting with a two-level system, cause rare stochastic transitions.
An important feature of this kind of processes is that they assume, or automatically provide a small parameter. This, in turn, opens the doors to the vast arsenal of mathematical methods based on series expansions, the perturbation theory. This also defines the class of systems where such a description can be productive - where the system is composed of nearly isolated subsystems.
1.3.2 Strong coupling
As the strength of interaction increases, a meaningful separation of the whole system into subsystems of a detector and a measured system, as well as a definition of the boundary between them, becomes less apparent. Generally, a description of each example of a sys- tem with strong interaction requires an individual approach. Neither perturbative expansions are possible in the absence of a small parameter, nor a measurement can be considered as projective, since the measured subsystem can not remain in a pure state due to interaction.
1.3 Role of the coupling strength in mesoscopic devices 9
Nevertheless, for certain systems the problem can be solved exactly. This is illustrated in chapter 3, where either an electronic interferometer or a quantum dot (see Fig. 3.1) can be seen as a detector of the state of the other. In both cases, backaction of what is chosen as a detector onto the other part is not small. The two parts together are found in an entangled state, and the two-way influence determines the behavior of the whole system. However, the effective theory [70] describing a one dimensional system such as a quantum Hall edge channel, allowed us to solve the model analytically, even in presence of strong Coulomb interaction and away from equilibrium.
Chapter 2
Direct access to quantum fluctuations through
cross-correlation measurements
Detection of the quantum fluctuations by conventional methods meets certain obstacles, since it requires high-frequency measurements. Moreover, quantum fluctuations are normally dom- inated by classical noise, and are usually further obstructed by various accompanying effects such as a detector backaction. In this chapter we demonstrate that these difficulties can be bypassed by performing cross-correlation measurements. We propose to use a pair of two- level detectors, weakly coupled to a collective mode of an electric circuit. Fluctuations of the current source accumulated in the collective mode induce stochastic transitions in the de- tectors. These transitions are then read off by s quantum point contact (QPC) electrometers and translated into two telegraph processes in the QPC currents. Since both detectors interact with the same collective mode, this leads to a certain fraction of correlated transitions. These correlated transitions are fingerprinted in the cross-correlations of the telegraph processes, which can be detected at zero frequency, i.e., with a long time measurement. Concerning the dependance of the cross-correlator on the detectors’ energy splittingsε1andε2, the most interesting region is at the degeneracy pointsε1 =±ε2, where it exhibits a sharp non-local resonance, that stems from higher-order processes. We find that at certain conditions the main contribution to this resonance comes from the quantum noise. Namely, while the reso- nance line shape is weakly broadened by the classical noise, the height of the peak is directly proportional to the square of the quantum component of the noise spectral function [15].
12 Direct access to quantum fluctuations through cross-correlation measurements
2.1 On-chip noise detection
It has been recently understood that the noise phenomena are not necessarily something detri- mental in physical experiments, but instead, they may carry useful new information about the underlaying processes [16, 17]. This information may be difficult to extract from the mea- surements of average quantities, therefore properties of the fluctuations become themselves a valuable subject of research. Particularly, one may study higher irreducible moments (or cumulants) of physical quantities [18]. It is especially interesting to detect the essentially quantum properties of noise, such as non-symmetrized correlators, reflecting the quantum non-commutativity [12].
To motivate this interest, it is useful to introduce an example of a quantum mechanical op- eratorˆj(t), representing a fluctuating electric current. Suppose this current is measured by a classical ammeter of certain bandwidth, and the raw measurement data is stored as a sequence of numbersj(tk). The classical properties related to the symmetrized correlators, such as the noise powerSsym(ω) = ∫
dτexp(iωτ)⟨{δˆj(t), δˆj(t+τ)}⟩, or higher order symmetrized correlators, can be extracted from the raw data in postprocessing (hereδˆj(t) = ˆj(t)−¯jis the deviation from the average current¯j). In contrast, the quantum (anti-symmetrized) parts of the correlators cannot be found in principle from postprocessing of such raw data, since this information is lost in the measurement of the average values⟨j(tk)⟩. Instead, these quantum properties may be inferred from the measurement of more complex quantities and systems, occurring naturally, or engineered on purpose. However, in this kind of measurements, it may not always be immediately clear which particular quantity is measured. Specifically, there is certain ambiguity concerning the questions of the operator ordering in the complex quantities, and of the process of their reduction to the classical values. Thus, a careful approach requires the knowledge of the detailed model of the detector and of the process of measurement.
Hence one arrives at the notion of a mesoscopic on-chip detector [23, 19, 22, 20, 21].
It is a part of the measurement apparatus, which interacts directly with the system, and is responsible for transforming the quantum information into a classical signal. As an exam- ple, the two-level detectors have been studied theoretically and implemented in experiments [25, 24, 28, 29, 27, 26], and their operation is now quite well understood. Such a detector
Figure 2.1:A schematic representation of the two-level detector as a double well structure with the level splittingε. Stochastic transitions in the detector are induced by the tunneling coupling∆and assisted by photons coming from the collective mode. The state of the detector is monitored by the QPC electrometer, located nearby one of the wells.
2.1 On-chip noise detection 13
consists of a double quantum dot, or a similar structure, which has two energetically relevant quantum states (see Fig. 2.1), that have different spatial charge distributions. Then a non- invasive QPC electrometer [30, 31, 32, 33, 34, 35, 36, 37, 38] may be used to read out the state of the two-level detector, and the resulting signal can then be amplified by conventional means. In a properly adjusted operating regime, the two-level detector is weakly coupled to the mesoscopic system, and at short times they evolve together quantum mechanically. Due to the weakness of coupling, fluctuations in the system induce rare stochastic (nonadiabatic) transitions in the detector. Because of the noisy nature of these fluctuations, the state of the detector becomes decohered to a statistical mixture. [31, 39, 40, 41] In this case, the QPC electrometer effectively senses the already classical state of the two-level detector, and thus, it actually satisfies our definition of an on-chip detector.
While the on-chip detector approach should at least clarify as what exactly is measured, the extraction of the information about the quantum fluctuations may still be challenging for some general reasons. The quantum effects often appear as small corrections to classical con- tributions, thus a high relative accuracy may be needed. Another source of complication is the fact that the system of interest is subjected to the perturbations induced by the measuring device itself,[42] and other extrinsic sources of noise. Therefore, some advanced techniques have to be employed to carefully extract the useful information. One such technique, that demonstrated certain success at isolating the properties of the measured system is a so called cross-correlation technique [43, 44]. The main idea is that two (or possibly more) detectors are used to measure a certain fluctuating signal from the same system. Then, since the detec- tors are meant to be independent, any local processes at one of the detectors should not lead to cross-correlations in the fluctuations measured by different detectors. Therefore, the mea- surement of the cross-correlations of the detectors’ outputs, gives certain level of protection from the local unwanted sources of noise, and from the detectors’ backaction to the system, and thus may enhance the accuracy of the experiments [46, 47, 45, 49, 50, 48].
In this chapter we consider using the cross-correlation technique to gain access to quan- tum fluctuations of current. We propose the following measurement setup: a mesoscopic system, source of current noise, is incorporated into an electric circuit, so the fluctuations of its current are accumulated on a capacitor (see Fig. 2.2). An electric charge on the capacitor
Figure 2.2:The equivalent scheme of the measurement electric circuit coupling the noise sourcej(t) to detectors via the collective modeQ.
14 Direct access to quantum fluctuations through cross-correlation measurements
plays the role of a so-called collective mode, to which a pair of two-level detectors are cou- pled. We assume the weak coupling regime, when the evolution of the detectors’ state can be described by the master equation for the average state occupations. Then fluctuations of the collective mode induce rare stochastic transitions in the detectors. These transitions generate two telegraph processes in the outputs of corresponding QPC electrometers, (see Fig. 2.4) and the cross-correlator of these outputs has to be measured in the markovian (long time) limit.
Figure 2.3: A diagram of the hierarchy of the time scales in the measurement process: the noise correlation timeτc, the decoherence timeτd, switching time of the detectorsτs, and and the proposed measurement timeτm.
Throughout the chapter we rely on the concept of time scale separation. In our model there are several independent small parameters (such as the coupling constant α, and the tunneling amplitude∆) that produce the following hierarchy of time scales (see Fig. 2.3).
The smallest scale is the noise correlation timeτc, at which the coherent quantum-mechanical evolution of the joint system of the noise source, the circuit, and the detector takes place.
However, starting from the decoherence timeτd ≫ τc, the noise source plays the role of a heat bath for the detector. Then the evolution of the average occupations of the detector states is described by a master equation. These occupations vary on the characteristic timeτs≫τd of the order of the inverse transition rates. Finally, the longest time scaleτm ≫ τsis the markovian limit of the telegraph processes, where the cross-correlator should be measured.
To describe the quantum mechanical evolution of the joint system of the collective mode ϕand the two-level detectors on short time scalest < τd, it is rather more adequate to think of the two detectors in the space of four states|11⟩,|12⟩,|21⟩,|22⟩. A calculation of the reduced density matrix for the detectors involves averaging over the fluctuations of the collective mode ϕ. To thenth order of the perturbation expansion with respect to weak tunneling∆, this requires finding averages of the Keldysh ordered products of the corresponding number of exponential phase operatorseiαϕ (so called vertex operators). In the weak coupling regime α ≪ 1, we use the cumulant expansion, since every next cumulant enters with one extra power of the small coupling constant, and we limit ourselves to the third cumulant. The main contribution comes from the times as long as the decoherence timeτd, where the cumulants can be expressed in terms of the zero-frequency expansion of the noise spectral functions.
Quantum corrections to the classical long time asymptotic of the cumulants are small, and thus, they can be taken into account perturbatively in the coupling constantα.
On the time scalest > τd, we find the transition rates perturbatively in ∆. We show that the most interesting effects do not appear on the level of the standard P(E)theory
2.1 On-chip noise detection 15
[51, 52, 53], which accounts for the tunneling to the lowest (second) order, only. Notice, that some of the transitions between these states, such as|11⟩ |22⟩or|12⟩|21⟩, that arise in the 4th order in∆, directly correspond to simultaneous, correlated switching of the detectors and lead to cross-correlations at long times (see Fig. 2.4). However, computation of the transition rates in the energy representation encounters well known divergences in higher orders of the perturbation theory. We avoid those divergences by considering directly the time evolution of the detector states. Namely, at times longer then the decoherence timet > τd, but much shorter than the switching timet ≪ τs, the master equation description suggests linear in time drift of the occupations probabilities from the initial distribution. On the other hand, in the fourth order we find quadratic in time terms. We identify them with the reducible contribution generated in the perturbation expansion and, accordingly, find the transition rates by extracting the irreducible part from the long time asymptotic of the occupation probabil- ities. As typical for the perturbation theory in higher orders, immense numbers of terms are generated. Nevertheless, we managed to find all our results analytically [54].
Note, that the transition rates may be measured from the time-resolved observation of the telegraph processes [32, 33, 34, 35, 36, 37, 38]. Then, by fitting their dependence on the controllable parameters of the system, such as energy splittings and coupling constants, one may try to infer some of the noise properties. The drawback of this approach is that the large amounts of real-time data need to be recorded and analyzed. This also limits the possible measurement pace, since the real-time switching resolution is required to extract the transition rates. Instead, we expect that by measuring directly the cross-correlator of the two telegraph processes on the time scaleτmmuch longer than the switching timeτs, one can considerably simplify the implementation of experiments. To evaluate the resulting cross-correlator of the telegraph processes, we generalize for the case of two detectors the approach that has been proposed in Refs. [55] and [38] to study the statistics of bistable systems. We present an exact general result, which is convenient to use if the transition matrix is symmetric, i.e., for the classical noise. For the case of quantum noise we also develop a perturbative in tunneling calculation which is better suitable for analytical computations.
Figure 2.4:An example of the two telegraph processes generated by (partially) correlated switching of the two detectors. One of the mechanisms of cross-correlation is illustrated by correlated transitions
|11⟩ |22⟩(dashed lines), and anti-correlated|12⟩ |21⟩(dotted lines), both governed by the anti-diagonal elements of the transition matrix. The net cross-correlator may be estimated simply by counting the number of dashed lines, minus the number of dotted lines.
16 Direct access to quantum fluctuations through cross-correlation measurements
Finally, we analyze the physically different contributions to the cross-correlator from classical and quantum noise. Note, that although the two detectors do not interact directly, an effective non-local interaction emerges between the detectors due to their coupling to the collective mode [see Eq. (2.8)]. The effective interaction does not depend on the properties of the noise, but it leads to the trivial second order contribution to the cross-correlator. This contribution is not of much interest, since it can be activated by subjecting both, or even any one of the detectors to local classical noise. However, since it is proportional to the strength of the effective interaction, it can be minimized by tuning the parameters of the circuit. In contrast, the more interesting kind of contributions arise as a consequence of the correlated response of the detectors to the fluctuations of the collective mode in the higher (fourth) order processes. The distinctive feature of these contributions is a sharp peak of the non-local resonance at the degeneracy point, where the detectors’ level splittings satisfy ε1±ε2 = 0. We note, that the non-local resonance contains both classical and quantum parts of noise. Remarkably though, the classical part in the 4th order is also proportional to the effective interaction strength and thus may be rendered small, while the purely quantum contribution to the cross-correlator survives if the effective interaction is “switched off”. The corresponding conditions (see Sec. 2.6.3) can be achieved, basically, if the noise temperature is sufficiently high, and the coupling constants are made small enough.
In the rest of the chapter we present a model of two-level detectors in the next section, then, in Sec. 2.3 we give a more detailed overview of the counting statistics approach, gen- eralize it for calculation of the cross-correlations, and present two methods of evaluating the cross-correlator, each more practical for classical and quantum noise respectively; in Sec. 2.4 and 2.5 we show how the cumulant expansion is applied to calculate the averages needed to find the transition rates; finally, in Sec. 2.6 we present the results for the classical and quantum noise cases, and analyze the conditions needed to access the required measurement regime.
2.2 Two-level detectors
An electric circuit may be modeled[51] by a set of bosonic fields,ϕk, and their conjugated
“charges”qk. We wish to single out one of these fields, a so called collective modeQand its phaseϕ, which are linearly coupled to the rest of the fields and enter quadratically to the corresponding Hamiltonian of the circuit. One can show that any such Hamiltonian may be transformed to the form, where all the couplings are carried by the phaseϕonly:
Hc = Q2
2C +Hn(ϕ, qk, ϕk). (2.1) This model is sufficiently general to describe any circuit with the usual linear elements, such as resistors, capacitors and inductances, and can also include such mesoscopic elements as tunnel junctions by adding non-quadratic potentials to the Hamiltonian.
Let us assume that the collective modeQis linearly coupled, with a dimensionless cou- pling strengthα, to a two-level detector represented by the Hamiltonian
H0= ε
2σz+ ∆σx+ α
2CσzQ, (2.2)
2.2 Two-level detectors 17
whereεis the energy level splitting,∆ ≪ εis a weak level mixing and we use the units e = ~ = 1 throughout the paper. The constant ∆ can also be understood as the tun- neling amplitude between the levels, or as the quantum level broadening. This type of quantum detectors have been considered in a number of works and have been implemented experimentally.[25, 24, 28, 29, 27, 26]
In what follows, we treat the term∆σxas a smallest perturbation. We, therefore, render the total HamiltonianH=H0+Hcin a more convenient form by performing the following transformation:
H′ =eiα2σzϕHe−iα2σzϕ. (2.3) This transformation affects only those terms that do not commute with σz or ϕ. Since [ϕ, Q] =i, it shifts the chargeQby−ασz/2, canceling the linear coupling termασzQ/2C and bringing interactions in the form of operatorse±iαϕ,
H′ =ε 2σz+ ∆
( 0 eiαϕ e−iαϕ 0
)
− α2
8C +Hc, (2.4)
where the energy is also shifted by a constant−α2/8C. We switch to the interaction picture with the time dependent tunneling Hamiltonian
HI(t) = ∆
( 0 ei[αϕ(t)+εt]
e−i[αϕ(t)+εt] 0 )
. (2.5)
This suggests for a perturbative expansion in powers of∆, which will be justified by accept- ing that the tunneling constant∆is the smallest energy scale in the system. Particularly, it must be smaller than the level broadening introduced by noise.
2.2.1 Pair of two-level detectors
Now, let us consider two such detectors, both coupled to the same circuit via the chargeQ, H0=∑
j=1,2
H(j)0 =∑
j=1,2
{εj
2σz(j)+ ∆jσ(j)x + αj 2Cσ(j)z Q
}
, (2.6)
where we denote quantities belonging to the different detectors with an additional index j= 1,2. In this case, the transformation analogous to Eq. (2.3),
H′=e2i∑jαjσ(j)z ϕHe−2i∑jαjσ(j)z ϕ (2.7) leads to
H′= ∑
j=1,2
{εj
2 σz(j)+ ∆j
( 0 eiαjϕ e−iαjϕ 0
)
− αj2 8C
}
+Hc+ Ec
2 σ(1)z σ(2)z . (2.8) Thus, besides changing the energy by a constant, it also generates the cross-term with
Ec=α1α2/2C. (2.9)
18 Direct access to quantum fluctuations through cross-correlation measurements
The cross-term represents the effective non-local interaction between the detectors mediated by the circuit collective mode. At the degeneracy pointsε1 = ±ε2this interaction leads to the quantum avoided-crossing level splitting of the value
∆ε= 4∆1∆2Ec/ε21. (2.10)
Finally, after switching to the interaction picture, the tunneling Hamiltonian for two de- tectors takes the form
HI =
0 ∆2eiα2ϕ+i(ε2−Ec)t ∆1eiα1ϕ+i(ε1−Ec)t 0
∆2e−iα2ϕ−i(ε2−Ec)t 0 0 ∆1eiα1ϕ+i(ε1+Ec)t
∆1e−iα1ϕ−i(ε1−Ec)t 0 0 ∆2eiα2ϕ+i(ε2+Ec)t 0 ∆1e−iα1ϕ−i(ε1+Ec)t ∆2e−iα2ϕ−i(ε2+Ec)t 0
.
(2.11) Note that ifEc = 0the tunneling Hamiltonian (2.11) reduces to a tensor sumHI =H(1)I ⊗ E+E⊗ H(2)I . We will see that the presence ofEc leads to a trivial mechanism of cross- correlations even in the presence of only local noise. However, these cross-correlations vanish with smallEc, and thus may become dominated by some more interesting phenomena, such as effects of quantum fluctuations in higher order processes.
2.2.2 Time evolution
To study cross correlations in the detectors’ output, we wish to consider the long-time limit, where evolution of the detectors’ states can be described by the master equation
P = ˆ˙ MP (2.12)
for the occupation probabilitiesP = (p1, . . . , p4). Such description is valid if the decoher- ence timeτd = (α2R2S)−1 (see Sec. 2.5) is much shorter,τd ≪ ∆−1, than the quantum time scale associated with the quantum level repulsion of a single detector, orτd≪(∆ε)−1 in the degeneracy point. These conditions are equivalent to a requirement that the classical level broadening due to the noise is always stronger then the quantum avoided-crossing level repulsion. Or, in other words, one may say that the stochastic switching timeτs= 1/(∆2τd) is longer than the dephasing time,τs≫τd.
The time dependence of the reduced density matrix of the detectors is ρ(t) = Trc
[Uρ(0)U˜ †]
, (2.13)
where
U = ˆT exp {
−i
∫ t 0
dt′HI(t′) }
(2.14)
2.2 Two-level detectors 19
is the interaction picture evolution operator, the initial condition is represented by the full density matrix of the system of detectors together with the circuitρ(0) =˜ ρ(0)×ρc(0), and Trcmeans averaging over the circuit degrees of freedom.
As soon asρ(t)is found, one observes that the off-diagonal elements ofρdecay expo- nentially over the timeτd. Therefore we can concentrate our attention on the probabilities
pk(t) =ρkk(t) = Tr [
ρ(0)×ρc(0)U†|k⟩⟨k|U ]
, (2.15)
where the trace is over all the degrees of freedom of the system. Equation (2.15) may be recast in the form
P(t) =[
E+ ˆM(t)]
P(0), (2.16)
where the elements of the time dependent transition probability matrixMˆ(t)read {Mˆ(t)
}
kl = Trc
[
ρc(0)⟨k|U†|l⟩⟨l|U|k⟩]
−δkl. (2.17)
They may be found from (2.15) as the perturbative expansion Mˆ(t) = ∑
nMˆn(t)with respect to the tunneling Hamiltonian (2.11). Comparing the expression (2.16) with a solution of equation (2.12), P(t) = exp{M tˆ }P(0), one finds that the transition matrix Mˆ can be obtained from the long time asymptotics of the irreducible part ofMˆ(t)as
Mˆ = lim
t→∞
1 t log[
E+ ˆM(t)]
. (2.18)
Figure 2.5: Typical quadratic in time dependence (dashed line) of the elements of the bare matrix Mˆ4(t), and linear in time (solid line) for the corrected matrixMˆ4(t)−Mˆ22(t)/2. Time scale in the units of the decoherence timeτd.
Note, that due to the specific structure of (2.11), only even powers of∆j are present in the diagonal elements of the perturbation series for the density matrix (2.13). Therefore, to the fourth order in∆j, one can writeMˆ(t) = ˆM2(t) + ˆM4(t). The time dependence of an arbitrary element of theMˆ2(t)appears to be quite simple:
{Mˆ2(t)}
kl=C0(t) +C1t, (2.19)
20 Direct access to quantum fluctuations through cross-correlation measurements
whereC1is a constant, and the first termC0(t)is a decaying oscillating function of time. This decaying term only represents the artefact of the chosen decomposition of the initial condition
˜
ρ=ρ×ρc, which, due to the detector-circuit interaction, does not describe a stationary state.
However, the stationary, linear in time behavior is restored after the decoherence timeτd, as illustrated by the decaying oscillations in the Fig. 2.5.
The elements of the next term,Mˆ4(t), may have a more complex structure:
{Mˆ4(t)}
kl=C0′(t) +[
C1′ +D1′(t)]
t+C2′t2, (2.20) where the functionsC0′(t)andD′1(t)are analogous toC0(t), the constantC1′ is analogous to C1, and there is a quadratic in time termC2′t2. This last term is a reducible part of theMˆ4(t), which is exactly eliminated in the logarithm in Eq. (2.18). Particularly, to the 4th order one has:
Mˆ = lim
t→∞
1 t
[Mˆ2(t) + ˆM4(t)−1 2
Mˆ22(t) ]
. (2.21)
This expression provides a conclusive point in the calculation of the transition matrixMˆ from the explicit time dependence of the occupationsP(t).
2.3 Full counting statistics of the QPC currents
In this section we present the method for calculating the statistics of a QPC current using the generating functions approach. We first recall the simpler case of a single detector [55, 38]
and later generalize it to the case of two detectors. Consider a QPC as a charge detector tuned to detect the state of the two-level system (see Fig. 2.1). The statistics of the current passing through the corresponding conductance levelsk= 1,2of the QPC is described at long times by the moment generating function
gk(λ, t) =∑
n
exp(λn)fk(n) =eHk(λ)t, (2.22) wherefk(n)is the probability thatnelectrons are transferred while the levelskis occupied,t is the total time of the measurement, andHk(λ)is the function generating current cumulants:
⟨⟨Ikm⟩⟩= ∂mHk
∂λm
λ→0
. (2.23)
One can see thatgk(λ, t)satisfy the equations
˙
gk(λ, t) =Hk(λ)gk(λ, t). (2.24) This equations may be modified in order to take into account mixing of the current channels induced by switching of the detector. IntroducingG(λ, t) ≡(g1, g2), the extended master equation reads:
G(λ, t) = ˆ˙ WG(λ, t), (2.25)
