Dephasing and Decoherence in Open Quantum Systems: a Dyson's Equation Approach

Dephasing and Decoherence in Open Quantum Systems: A Dyson's Equation Approach

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Authors Cardamone, David Michael

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Link to Item http://hdl.handle.net/10150/195386 Dephasing and Decoherence in Open Quantum Systems: A Dyson’s Equation Approach

David Michael Cardamone

A Dissertation Submitted to the Faculty of the DEPARTMENT OF PHYSICS In Partial Fulﬁllment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY In the Graduate College THE UNIVERSITY OF ARIZONA

2 0 0 5 3

THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE

As members of the Dissertation Committee, we certify that we have read the dissertation prepared by David M. Cardamone entitled Dephasing and Decoherence in Open Quantum Systems: A Dyson’s Equation Approach and recommend that it be accepted as fulﬁlling the dissertation requirement for the Degree of Doctor of Philosophy Date: 08/04/05 Bruce R. Barrett

Date: 08/04/05 Charles A. Staﬀord

Date: 08/04/05 Sumitendra Mazumdar

Date: 08/04/05 Michael A. Shupe

Date: 08/04/05 Koen Visscher

Final approval and acceptance of this dissertation is contingent upon the candidate’s submission of the ﬁnal copies of the dissertation to the Graduate College.

We hereby certify that we have read this dissertation prepared under our direction and recommend that it be accepted as fulﬁlling the dissertation requirement.

Date: 08/04/05 Dissertation Director: Bruce R. Barrett

Date: 08/04/05 Dissertation Director: Charles A. Staﬀord 4

STATEMENT BY AUTHOR

This dissertation has been submitted in partial fulﬁllment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

SIGNED: David Michael Cardamone 5

ACKNOWLEDGEMENTS

From the bottom of my heart, the greatest thanks I have must go to my wife, Martha. Through the years, she has been an unparalleled source of love, hope, advice, support, and friendship. Her wisdom has informed each choice I have made, and her example has inspired me. Thank you, Martha. I must also thank my parents and grandparents, who provided me with a safe child- hood, allowed me the luxury of exploring my own interests, and, above all, gave me their confidence. By their example, they taught me a sense of personal responsibility, ethics, and morality, which shaped the person I have become. On a final personal note, I do not want to forget my friends in Tucson who have helped me in numerous ways over the years. I was fortunate to travel professionally quite a bit during my grad student years, and James Little, Geoff Schmidt, and Jeremy Jones facilitated this in all those important ways that friends do. I also thank Tucson Kendo Kai for helping me find the determination, courage, and character necessary to a grad student’s lifestyle. On the professional side, I could not have been more honored or fortunate to work under the tutelage and supervision of Profs. Bruce Barrett and Charles Stafford. They, too, gave me their confidence. Much more useful than teaching me physics (although they did that as well), they showed me how to learn physics. I shall never forget their tireless efforts to guide me on the long journey from inexperienced student to practicing physicist. An additional very special thanks is due to Prof. Sumit Mazumdar, with whom I have also had the privilege of collaborating the last two years. Although Sumit had many answers, equally valuable in this collaboration were his questions. He took the time to give excellent and thoughtful career advice, which was key in getting me where I am today. Indeed, the entire community of the University of Arizona Department of Physics have been welcoming and helpful to me during my time here. Over the years, my thesis committee, including those mentioned above as well as Profs. Mike Shupe, Koen Visscher, and Srin Manne, have always found time to help me with advice or encouragement. So too have the other faculty of the department, including especially Keith Dienes, Fulvio Melia, Jan Refelski, Bob Thews, Bira van Kolck, J. D. Garcia, and Carlos Bertulani. Among all the helpful staff, Mike Eklund and Phil Goisman always went above and beyond the call of duty without complaint, for which I wish to express my appreciation and admiration. My understanding of the scientific issues discussed in this work has benefitted enor- mously from numerous engaging discussions over the years. In particular, thanks are due to Chang-hua Zhang, Jerome Burki,¨ Jeremie Korta, Ned Wingreen, Peter von Brentano, Micah Johnson, Ryoji Okamoto, Dan Stein, Anna Wilson, Paul Davidson, Mahir Hussein, Adam Sargeant, and George Kirczenow. The pleasure of discussion and collaboration with such outstanding physicists is one I hope will continue for many years. 6

For Martha, meae vitae. 7

TABLE OF CONTENTS

LIST OF FIGURES ...... 10

LIST OF TABLES ...... 12

ABSTRACT ...... 13

CHAPTER 1: INTRODUCTION ...... 14 1.1 Green Functions ...... 15 1.1.1 General theory of Green functions ...... 16 1.1.2 Electrostatic Green functions ...... 18 1.1.3 Quantum mechanical Green functions ...... 19 1.2 Physical Systems ...... 20 1.2.1 Coupled quantum dots ...... 21 1.2.2 Decay of superdeformed nuclei ...... 21 1.2.3 Molecular electronics ...... 22

CHAPTER 2: DYSON’S EQUATION ...... 23 2.1 Derivation ...... 23 2.1.1 S-matrix expansion ...... 23 2.1.2 Diagrammatic approach ...... 26 2.2 Self-Energies ...... 29 2.2.1 Hybridization: adding a second level ...... 30 2.2.2 Decoherence: a single continuum ...... 31 2.3 Summary ...... 33

CHAPTER 3: COUPLED QUANTUM DOTS ...... 34 3.1 Quantum Dots ...... 35 3.1.1 History and fabrication ...... 36 3.1.2 Experimental studies ...... 37 3.2 Two-Level Model ...... 38 3.2.1 Realm of Applicability to Quantum Dots ...... 39 8

TABLE OF CONTENTS –Continued

3.2.2 Hamiltonian of the coupled dot system ...... 39 3.2.3 Spin-boson analogy ...... 41

3.3 Green Function Treatment ...... 42 3.3.1 Without leads ...... 43 3.3.2 Including the leads ...... 45 3.4 Limiting Cases ...... 48 3.4.1 Identical dots ...... 49 3.4.2 Identical lead couplings ...... 50 3.5 Summary ...... 50

CHAPTER 4: DECAY OF SUPERDEFORMED NUCLEI ...... 52 4.1 Nuclear Deformation ...... 52 4.1.1 Normal deformation ...... 53 4.1.2 Superdeformation ...... 56 4.1.3 Experimental signatures of deformation ...... 57 4.2 Decay Process ...... 60 4.2.1 Experiments ...... 60 4.2.2 Double-well paradigm ...... 63 4.3 Two-State Model ...... 65 4.3.1 Two-state Hamiltonian ...... 66 4.3.2 Energy broadenings ...... 67 4.3.3 Green function treatment ...... 70 4.3.4 Branching ratios ...... 70 4.4 Tunneling Width ...... 72 4.4.1 Relation between branching ratios and tunneling width ...... 72 4.4.2 Measurement of the tunneling width ...... 73 4.4.3 Limits of the tunneling width ...... 75 4.5 Statistical Theory of Tunneling ...... 76 4.5.1 Gaussian orthogonal ensemble ...... 76 4.5.2 Implications for tunneling ...... 77 9

TABLE OF CONTENTS –Continued

4.6 Adding More Levels ...... 80 4.6.1 Three-state model ...... 80 4.6.2 Inﬁnite-band approximation ...... 84 4.7 Summary ...... 86

CHAPTER 5: MOLECULAR ELECTRONICS ...... 88 5.1 Fabrication of Single-Molecular Systems ...... 88 5.1.1 Scanning-tunneling microscopic techniques ...... 89 5.1.2 Mechanically controllable break junction ...... 90 5.1.3 Other techniques ...... 92 5.2 Modeling Molecular Electronics ...... 92 5.2.1 Hamiltonian ...... 93 5.2.2 Hartree-Fock approximation ...... 95 5.2.3 Non-equilibrium Green function theory ...... 95 5.2.4 Equal-time correlation functions ...... 98 5.2.5 Landauer-Buttik¨ er formalism ...... 99 5.3 Quantum Interference Eﬀect Transistor ...... 101 5.3.1 Tunable conductance suppression ...... 101 5.3.2 Finite voltage ...... 105 5.4 Summary ...... 108

CHAPTER 6: DISCUSSION ...... 109

REFERENCES ...... 111 10

LIST OF FIGURES

2.1 Dyson’s equation expansion of the full retarded self-energy Σ? ...... 28

3.1 Electron micrograph images of experimental quantum dots ...... 36 3.2 Experimental spectra and addition energies of quantum dots ...... 38 3.3 Schematic diagram of the double quantum dot system ...... 40 3.4 Hybridization of energy levels in the double-dot system without leads . . 43 3.5 Coherent Rabi oscillations in the double-dot system without leads . . . . 45 3.6 Mixture of coherent and incoherent behavior in the full system ...... 47

4.1 Evidence for shell closures in the ﬁrst excited state of even-even nuclei . . 54 4.2 Evidence for shell closures in nuclear separation energies ...... 54 4.3 Normal deformation and superdeformation on the table of nuclides . . . . 56 4.4 Superdeformation from a harmonic oscillator potential ...... 58 4.5 Decay spectrum of superdeformed 152Dy ...... 61 4.6 Universality in the decay of superdeformed nuclei of A 190 ...... 62 ≈ 4.7 Diagram of the superdeformed decay process ...... 63 4.8 Types of potentials historically used to model superdeformed decay . . . . 64 4.9 Two-level model of superdeformed decay ...... 65 4.10 Gaussian orthogonal ensemble probability distributions for the energies of the two levels on either side of the decaying superdeformed level . . . . . 78 4.11 Branching ratios in the two- and three-level model ...... 83 4.12 Branching ratios in the two- and iniﬁte-level models ...... 86

5.1 Artist’s conception of the Quantum Interference Eﬀect Transistor, a single- molecular device ...... 89 5.2 Scanning tunneling microscope approach to creating single-molecule junctions ...... 91 5.3 Mechanically controllable break junction approach to creating single-molecule junctions ...... 91 5.4 The Keldysh contour ...... 97 5.5 Schematic diagrams of Quantum Interference Eﬀect Transistors ...... 102 11

LIST OF FIGURES –Continued

5.6 Cancellation of paths in a QuIET, and lifting of that effect by the introduction of a third lead ...... 103 5.7 Transmission probabilities for various Quantum Interference Effect Tran- sistors ...... 103 5.8 Lead configurations for a Quantum Interference Effect Transistor based on [18]-annulene ...... 105 5.9 Possible placements for a third led in the benzene QuIET ...... 106 5.10 I characteristic of a Quantum Interference Effect Transistor . . . . . 107 − V 12

LIST OF TABLES

3.1 Free-electron densities of states for nanoscale systems of diﬀerent dimensionality ...... 35

4.1 Inputs and results of the two-level model for speciﬁc superdeformed decays 69 13

ABSTRACT

In this work, the Dyson’s equation formalism is outlined and applied to several open quantum systems. These systems are composed of a core, quantum-mechanical set of discrete states and several continua, representing macroscopic systems. The macroscopic systems introduce decoherence, as well as allowing the total particle number in the system to change. Dyson’s equation, an expansion in terms of proper self-energy terms, is derived. The hybridization of two quantum levels is reproduced in this formalism, and it is shown that decoherence follows naturally when one of the levels is replaced by a continuum. The work considers three physical systems in detail. The first, quantum dots coupled in series with two leads, is presented in a realistic two-level model. Dyson’s equation is used to account for the leads exactly to all orders in perturbation theory, and the time dynamics of a single electron in the dots is calculated. It is shown that decoherence from the leads damps the coherent Rabi oscillations of the electron. Several regimes of physical interest are considered, and it is shown that the difference in couplings of the two leads plays a central role in the decoherence processes. The second system relates to the decay-out of superdeformed nuclei. In this case, decoherence is provided by coupling to the electromagnetic field. Two, three, and infinite-level models are considered within the discrete system. It is shown that the two-level model is usually sufficient to describe decay-out for the classic regions of nuclear superdeformation. Furthermore, a statistical model for the normal-deformed states allows extraction of parameters of interest to nuclear structure from the two-level model. An explanation for the universality of decay profiles is also given in that model. The final system is a proposed small molecular transistor. The Quantum Interference Effect Transistor is based on a single monocyclic aromatic annulene molecule, with two leads arranged in the meta configuration. This device is shown to be completely opaque to charge carriers, due to destructive interference. This coherence effect can be tunably broken by introducing new paths with a real or imaginary self-energy, and an excellent molecular transistor is the result. 14

CHAPTER 1

INTRODUCTION

In the last century, the study of physics experienced an unprecedented Renaissance. With the advent of quantum mechanics and relativity, a Kuhnian paradigm shift [1] occurred in the discipline. These theories allowed an understanding of experimental systems far beyond the realm of everyday experience, but the advance came at a price: the body of knowledge became compartmentalized, and it was now necessary to categorize physical systems according to their length scales, energy scales, or even the forces involved. It is not always immediately clear what should happen in systems that lie at the interface of these different theoretical regimes. This problem is to be distinguished from that of limits, which were in all cases understood at the founding of the new theories. The present work, for example, is concerned with the interface between quantum-mechanical and classical systems. The appropriate limit, of course, is well known as the Bohr correspondence principle [2], which states that a classical system is obtained if a quantum system is taken in the limit ~ 0, → where h = 2π~ is Planck’s constant. The question of interface between the classical and quantum-mechanical regimes, by contrast, asks an entirely different question: what happens to a system that is neither wholly quantum-mechanical, nor wholly classical? Especially in the case of quantum mechanics, an understanding of interfacial systems is of paramount importance, for the simple reason that no system of physical interest is purely quantum-mechanical. This is not mere pedantry: a system of physical interest is, by definition, limited to the set of those upon which measurements are, or will be, performed. As such, these systems always contain non-quantum components. When the interplay between the classical and quantum regimes is significant, the system is referred to as mesoscopic [3]. In general, mesoscopic systems have coherence lengths comparable to their size, so that they are primarily quantum, but non-quantum effects are also important in them. Such effects invariably arise from interaction with a classical, macroscopic system [4]. The additional system can generate dephasing, in which particles accrue additional phase individually, as well as decoherence, in which phase coherent effects are utterly destroyed [5–7]. A mesoscopic system can be of any 15 length scale on which quantum coherence can exist: this thesis deals with nanoscale, molecular, and femtoscale (nuclear) mesoscopic systems. This is not to say that quantum mechanics is useless without reference to an external theory, for we emphasize that an investigation of the interface between two theoretical frameworks need only make reference to the more general one. In this case, the Bohr correspondence principle tells us that quantum mechanics contains all the information of classical mechanics. Our task is therefore not to seek “new physics”, for the quantum theory is, in this sense, complete. Rather, we wish to understand more fully what we already have. As we study each mesoscopic system, the classical components will be built up from discrete and fully quantum-mechanical sets of states. The purpose of this thesis is to explore the role of dephasing and decoherence in three open quantum systems, which particles are free to enter and leave via macroscopic continua. To study these, we make use of the Dyson’s equation formalism [8], based on equilibrium and non-equilibrium Green functions [4, 9]. The goals will be to determine the effect of phase-breaking processes in these systems, and to develop a rigorous and intuitive approach, which will be applicable to the study of future such systems, as well.

1.1 Green Functions

Classical phenomena are characterized by continua of states, in which a variable may take on any value within a particular range, whereas quantum-mechanical systems generally have discrete sets of allowable values for certain variables. Indeed, the name of the theory was determined by this remarkable and wholly new phenomenon: we refer to variables which are restricted to a discrete set of values as quantized. We can expect, then, that while the quantum mechanical parts of the systems under study are characterized by discrete states, the classical components consist of continua. A formalism is required, therefore, which deals well with such large numbers of states in a rigorous, quantum-mechanically exact manner. Such a formalism is found in the Dyson’s equation approach to many-body quantum theory, which centers on treatment of the quantity known as the Green function [10]. It is remarkable that many-body theory provides such an apt solution to the problem, as we are not primarily interested in many-body eﬀects. In fact, throughout this work, 16 interactions between the quantum system and classical environment are treated as entirely single-particle eﬀects. The reason the Green function formalism performs so well in such circumstances is that many-body theory is, at its essence, a theory of many states, which is precisely the item of current interest.

1.1.1 General theory of Green functions

The Green function approach was first put forward by its namesake, George Green, as a method to find the electrostatic potential of a charge configuration with an arbitrary set of boundary conditions [11]. Green’s solution to this problem is, in fact, quite general, not only to the case of the Poisson equation so described, but to any linear differential equation [12]: ψ(x) = ρ(x), (1.1) D − where is a general linear differential operator in x, which is an ordered set of coördinates. D The Green function is defined by the related equation

(x , x ) = δ(x x ), (1.2) D1G 2 1 − 1 − 2 where the subscript i on indicates to which set of co¨ordinates x it applies, and δ is the D i Dirac delta. Using , a solution to Eq. (1.1) can be constructed, given a set of boundary conditions. G The secular equation for is D φ (x) = λ φ (x), (1.3) D n n n where φ and λ are ’s eigenfunctions and eigenvalues, respectively. The eigenfunctions n n D are a complete, orthonormal set of states:

∞ φ (x )φ∗ (x ) = δ(x x ), dτφ (x)φ∗ (x) = δ , (1.4) n 1 n 2 1 − 2 n m nm n= X−∞ Z where dτ is the volume element corresponding to the basis x, δnm is the Kronecker delta, and the integral is to be taken over the entire space of x. This property allows an expansion of the Green function in terms of the eigenstates:

∞ (x , x ) = a (x )φ (x ). (1.5) G 2 1 n 1 n 2 n=0 X 17

Substituting this result into Eq. (1.2) yields

∞ ∞ a (x )φ (x ) = a (x )λ φ (x ) = δ(x x ). (1.6) D1 n 1 n 2 n 1 n n 2 − 1 − 2 n=0 n=0 X X It follows that

∞ dτ a (x )λ φ (x )φ∗ (x ) = dτ δ(x x )φ∗ (x ) = φ∗ (x ) = a (x )λ , 2 n 1 n n 2 m 2 − 2 1 − 2 m 2 − m 1 m 1 m Z n=0 Z X (1.7) where dτi is the volume element corresponding to the basis xi. The Green function is thus seen to be [12] ∞ φ (x )φ (x ) (x , x ) = n∗ 1 n 2 . (1.8) G 2 1 − λ n=0 n X This expansion for (x , x ) is closely related to the charge ﬂuctuation resonance expan- G 2 1 sion, which has seen a great deal of success in the nanoscopic literature [13–18]. The paramount utility of the Green function follows directly from the eigenfunction expansion (1.8). Multiplying Eq. (1.1) by (x , x ) and integrating, we ﬁnd G 2 1

∞ φn∗ (x1)φn(x2) dτ1 1ψ(x1) = dτ1 (x2, x1)ρ(x1). (1.9) λn D G Z nX=0 Z Writing the solution ψ(x) as an expansion in the eigenstates and eigenvalues of , D

∞ ψ(x) = bnφn(x), (1.10) n=0 X we arrive at [12]

∞ φ (x )φ (x ) ∞ dτ (x , x )ρ(x ) = dτ n∗ 1 n 2 λ b φ (x ) = b φ (x ) = ψ(x ). 1G 2 1 1 1 λ m m m 1 n n 2 2 Z Z n=0 n n=0 mX=0 X (1.11) This is an extraordinary result, and the central conclusion that underlies all work using Green functions. Equation (1.11) says that any linear diﬀerential equation may be solved by applying an integral transform to the inhomogeneous term ρ(x). The kernel of this transform is the Green function, deﬁned by Eq. (1.2), solution of which is generally a much simpler task than the original equation (1.1). In essence, Eq. (1.11) gives the inverse of . D 18

1.1.2 Electrostatic Green functions

Green introduced this method of treating linear differential equations to solve the Poisson equation [11] 2ψ(r) = ρ(r), (1.12) ∇ − a problem of central importance at the time due to its application in electrostatics, where ρ/4π is the charge density and ψ(r) the electrostatic potential. Since it is also, nearly invariably, the first use of Green functions encountered in a physicist’s education [19], it may be of some value to briefly review the solution. From the identity 1 2 = 4πδ(r r ), (1.13) ∇ r r − 1 − 2 | 1 − 2| it is apparent that the Green function corresponding to = 2 is [19] D ∇ 1 (r , r ) = + (r , r ), (1.14) G 2 1 4π r r F 2 1 | 1 − 2| where (r , r ) can be any function which satisfies the Laplace equation, F 2 1 2 (r , r ) = 0. (1.15) ∇ F 2 1 The choice of (r , r ) depends on the boundary conditions of the system. Green F 2 1 derived a simple theorem for any two fields A and B [11], dB dA dτA 2B + dσA = dτB 2A + dσB , (1.16) ∇ dn ∇ dn ZV I ZV I which follows from Gauss’s theorem. The surface integrals are taken over the boundary d of the volume V , and dn represents differentiation with respect to a unit vector normal to the boundary. Performing the integration of Eq. (1.11), we find 1 ψ(r ) = dτ + (r , r ) ρ(r ). (1.17) 2 1 4π r r F 2 1 1 Z | 1 − 2| By Eqs. (1.15) and (1.12), Equation (1.16) implies dψ(r ) d (r , r ) dτ (r , r )ρ(r ) = dσ (r , r ) 1 F 2 1 ψ(r ) , (1.18) 1F 2 1 1 1 F 2 1 dn − dn 1 ZV I 1 1 where the subscripts on dσ1 and n1 indicates they correspond to r1. If V includes the entire region containing charge, we find ρ(r ) dψ(r ) d (r , r ) ψ(r ) = dτ 1 + dσ (r , r ) 1 F 2 1 ψ(r ) . (1.19) 2 1 4π r r 1 F 2 1 dn − dn 1 ZV | 1 − 2| I 1 1 19

Since it appears only in the surface integral with ψ(r ), it is manifest that (r , r ) is 1 F 2 1 ﬁxed by the boundary conditions of a particular physical system.

1.1.3 Quantum mechanical Green functions

Next, we consider a Green function approach to the quantum mechanical process of time evolution. The formalism which results forms the foundation of the current work, allowing as it does the complete solution of all time dynamics for a particular quantum system. We begin with the quantum-mechanical initial-value problem. In the Schr¨odinger picture, state kets time evolve while the operators remain stationary. For the moment assuming time-translational symmetry [20], a state ket is given by

iHt/~ ψ(t) = e− ψ , (1.20) | i | 0i where ψ is the ket at time t = 0. Equation (1.20) follows from the deﬁning property | 0i of the Hamiltonian H; it is the operator that generates translations in time [21]. iHt/~ H has real eigenvalues; thus it is Hermitian. The time-evolution operator e− is consequently unitary: 1 iHt/~ − iHt/~ † iHt/~ e− = e− = e . (1.21) This operator time evolves backwards by an amount t. Acting from the left on Eq. (1.20), then, we arrive at the linear diﬀerential equation

~ eiHt/ ψ(t) = ψ . (1.22) | i | 0i

This is the equation to be solved by the Green function approach. Equation (1.22) is included for completeness, but, of course, we already know the solution: Equation (1.20) corresponds exactly to Equation (1.11), with the usual quantum mechanical understanding of operators in place of integrals. The Green function (operator) is thus iHt/~ (t) = e− . (1.23) G − In the Green function approach, it is almost always easier to consider the energy domain. We construct the Fourier transform:

∞ dt i(E H)t/~ (E) = e − . (1.24) G − ~ Z−∞ 20

The result is poorly deﬁned, but this should not trouble us, since we have not speciﬁed boundary conditions. For example, if we wish to solve an initial-value problem, we should construct the retarded Green function [9],

(t 0) + i ∞ i(E H+i0+)t/~ 1 G(E) i ≥ (E + i0 ) = − dt e − = , (1.25) ≡ G ~ E H + i0+ Z0 − where the superscript (t 0) denotes that the integral of Eq. (1.24) should be evaluated ≥ only for nonnegative t. The notation 0+ is used throughout this work to denote a quantity which is to be taken to zero from the right when the calculation is finished. The restriction to nonnegative times is the crucial characteristic of the retarded Green function. The i0+ is added to make integrals over positive time converge, and the overall factor of i is an unobservable notational convenience, equivalent to giving the same phase rotation to all states. The final result (1.25) is of central importance, and clearly highlights the inverse relationship between Green function and Hamiltonian. In addition to the retarded Green function (1.25), we shall occasionally make use of the advanced Green function G† as well. This operator, the Hermitian adjoint of G, corresponds to boundary conditions for which the final state of the system is known.

1.2 Physical Systems

In the next chapter, we develop the formalism of Dyson’s equation [8], which allows extrapolation from a system with understood dynamics to a new one. By partitioning the Hamiltonian into solved and unsolved terms, Dyson’s equation develops a controlled expansion in terms of a quantity Σ, called the retarded proper self-energy. In general, there is no reason for this expansion to converge quickly. If the self- energy is complicated, an arbitrary number of terms may be necessary to ensure even a qualitatively accurate description of the physics [9]. In this work, however, we take the tactic of choosing physical systems that lend themselves to reasonable approximations which reduce Dyson’s equation to an exactly soluble form. Then, the sum can be done exactly to all orders in the perturbation Σ. While we restrict ourselves to the study of three systems in detail, this is by no means the limit of the approach. The broadly general nature of the formalism allows for many other applications, as well. 21

1.2.1 Coupled quantum dots

Examination of particular physical systems begins in Chapter 3 with the problem of two coupled quantum dots. Since quantum dots, nanoscale electron systems, are often described as “artifical atoms” [22–24], this system can be thought of, in some sense, as an artifical diatomic molecule [25, 26]. We shall explore all coupling regimes, corresponding to both covalent and ionic bonding. Coupled quantum dot systems have generated a plethora of interesting experimental and theoretical results [26–28]. It has further been suggested that such a system could be used as a logic gate [29, 30] or a qubit, the building block of a quantum computer [31–34]. Naturally, questions of environmental effects are central to such possibilities, especially the latter. A purely isolated quantum system is of no use as a qubit, and yet coherence effects must be well preserved if the device is to be useful. As the first physical system we shall investigate, a simple approach is taken to the physics of the coupled quantum dot. The system is well approximated by a two-level model, with each dot coupled to its own macroscopic lead [15, 35–37]. An electron is injected into one dot, and its time behavior is calculated exactly via Dyson’s Equation. The results are contrasted with those for the same system neglecting lead effects. The interplay between classical leads and quantum mechanical dots is seen to damp the Rabi oscillations characteristic to the isolated two-state system, opening a new expanse of interesting physical regimes for experimental exploration.

1.2.2 Decay of superdeformed nuclei

In Chapter 4, the two-level model is expanded to treat the decay-out process of superdeformed bands in nuclear physics. These bands consist of axially symmetric, ellipsoidal states with major-to-minor axis ratios of about 2:1, an important prediction of the shell model [38]. Intraband decay occurs as the nucleus loses angular momentum to the environment through coupling to the electromagnetic continuum. Superdeformed bands abruptly lose their strength to less deformed bands through a statistical decay process. We demonstrate conclusively that this phenomenon is well understood within a simple two-level model, exactly solvable in the Dyson’s Equation approach. The decoherent effects of electromagnetic decay processes are included on the 22 same footing as the coherent effects, which mix levels of different deformation. Via a statistical model, parameters of direct consequence to nuclear structure theory are extracted from experimental results. Furthermore, the consistency of the two-level approximation is verified via the addition of first one, then an infinite number of extra levels.

1.2.3 Molecular electronics

Chapter 5 continues the investigations by moving to a treatment of molecular electronic systems. These experimentally realized systems consist of small molcules, whose electron dynamics are governed by a discrete set of states, in contact with macroscopic metallic leads. Theoretical modeling of such a system at finite voltages presents several new challenges. The model of previous chapters is expanded to non-equilibrium, interacting systems. A comprehensive picture for modeling molecular electronics is shown to follow directly from the techniques of this work. Moreover, the intuitive understanding granted by our study of the previous systems motivates proposal of a small molecular transistor based on the interplay of coherent and decoherent effects. The Quantum Interference Effect Transistor, as it is called, operates based on tunable breaking of a coherent current suppression caused by perfect destructive interference of paths through the molecule. The results, which mimic the current-voltage characteristics of classical transistors in all important regards, are valid for a large class of molecules and lead arrangements. 23

CHAPTER 2

DYSON’S EQUATION

In this chapter, we shall derive and begin to use the most central result of Green function theory, Dyson’s equation. This formalism provides a perturbative expansion, whereby a Hamiltonian is partitioned into solved and unsolved parts, and the eﬀect of the the unsolved part is included systematically. In this work, we shall focus on methods of summing the series to all orders, so that the results of Dyson’s equation remain exact. In the latter part of this chapter, we begin this process with two simple examples. A single quantum level is placed in contact with both another single level, and an inﬁnite continuum of states representing a classical system. In both these cases, Dyson’s equation can be summed exactly to all orders.

2.1 Derivation

Derivation of Dyson’s equation [8, 39] centers on the development of a diagrammatic expansion of the quantity known as the S-matrix, closely related to the Green function. The essential problem is to describe the time dynamics of a system whose Hamiltonian

H = H0 + HI (2.1)

consists of a part H0 whose eigenvalues and eigenstates are known, and an additional part HI . It is not immediately clear how the addition of HI impacts the behavior of the system for the general case, in which the commutator [H , H ] = 0. It is assumed that 0 I 6 H0 and HI can be written in a second-quantized form [39], that is, in terms of creation and annihilation operators.

2.1.1 S-matrix expansion

The ﬁrst step in constructing Dyson’s equation is to link the expectation values of observables to the known eigenstates of H0. The secular equation of H0 is

H φ = E φ , (2.2) 0| ni n| ni 24 which pertains to an exact, possibly many-body, solution. It is assumed that in the absence of the perturbation H , the system lies in the ground state φ , and that each I | 0i piece is separately Hermitian. Since we are interested in finite times only, it is permissible to rewrite the full Hamil- tonian (2.1) so that for infinite times t , it reduces to H : → ∞ 0 η t H(t) = H0 + lim HI e− | |, (2.3) η 0 → an approach which is known as adiabatic “switching on” [39]. We note that the derivation of Sec. 1.1.3 no longer strictly applies, as the Hamiltonian is now time-dependant. The defining property of H is that it generates instantaneous time translations [21]: iH(t)dt ψ(t + dt) = 1 ψ(t) , (2.4) | i − ~ | i which implies the time-dependant Schödinger equation. It follows that, in the general case [39], i t dt H(t ) ψ(t) = T e− ~ t0 0 0 ψ(t ) = (t, t ) ψ(t ) . (2.5) | i R | 0 i −G 0 | 0 i The exponential function is to be interpreted, as usual, as a Taylor expansion. The symbol T represents a time-ordering of each term in the series, so that operators evaluated at earlier times are on the right.

To diﬀerentiate the eﬀects of H0 from those of HI, it is customary to work within the interaction picture of quantum mechanics [2, 21, 39]. Kets time-evolve as

i t dt H (t ) ψ (t) = T e− ~ t0 0 I 0 ψ (t ) = (t, t ) ψ (t ) . (2.6) | I i R | I 0 i GI 0 | I 0 i Expectation values are maintained by deﬁning operators as

i i H0t H0t (t) = e ~ e− ~ , (2.7) OI O where is the Schr¨odinger-picture operator. The interaction picture is neither the rest O frame of state kets, nor of operators. We note, therefore, that HI = HI (t) itself, which governs the time development of kets, rotates with the frequency H0/~, as described by Eq. (2.7). The advantage of Eq. (2.3) is that it allows us to make contact between expectation values and the unperturbed ground state. The expectation value of (t) is O ψ (t) (t) ψ (t) φ ( , t) (t) (t, ) φ (t) = h I |OI | I i = h 0|GI −∞ OI GI −∞ | 0i. (2.8) hO i ψ (t) ψ (t) φ ( , t) (t, ) φ h I | I i h 0|GI −∞ GI −∞ | 0i 25

In this and the next two chapters, we shall deal with physical systems strictly in equilibrium. These systems possess time reversal symmetry, so that

( , t) = ( , t). (2.9) GI −∞ GI ∞

We can thus rewrite Eq. (2.8) to read

φ ( , t) (t) (t, ) φ (t) = h 0|GI ∞ OI GI −∞ | 0i. (2.10) hO i φ ( , t) (t, ) φ h 0|GI ∞ GI −∞ | 0i The numerator of this equation can be read as a prescription for evaluating an expectation value at time t. First, begin at the unperturbed ground state φ . Then, slowly turn | oi on HI, and evolve the state to time t. Operate on this state with the relevant operator, (t). Now, to get back to the state φ , time evolve the state forward further to t = , OI | 0i ∞ switching oﬀ HI again. Since the system is time-reversal symmetric, you are guaranteed to end up right back where you started. Making use of the time-ordering operator T , we can write Eq. (2.10) in a more compact form [39]: φ T [ (t)S] φ (t) = h o| OI | oi, (2.11) hO i φ S φ h 0| | oi where the S-matrix is deﬁned as

i ∞ dt H (t ) S ( , ) = T e− ~ 0 I 0 . (2.12) ≡ GI ∞ −∞ R−∞

We can write the exponential expansions explicitly:

n ∞ i 1 T [ (t)S] = ∞ dt ∞ dt T [ (t)H (t ) H (t )] OI −~ n! 1 · · · n OI I 1 · · · I n n=0 Z−∞ Z−∞ X n (2.13) ∞ i 1 S = ∞ dt ∞ dt T [H (t ) H (t )] −~ n! 1 · · · n I 1 · · · I n n=0 X Z−∞ Z−∞ The picture we have is thus of all possible numbers of operations of the perturbation H , taking place both before and after evaluation of the operator (t). Note that time I OI evolution by H , from t = to t = , is included in the deﬁnition (2.7) of the 0 −∞ ∞ interaction-picture operator (t), due again to time-reversal symmetry. OI 26

2.1.2 Diagrammatic approach

The S-matrix expansion detailed in the previous section allows time evolution for the full Hamiltonian (2.1) to be computed without knowing its eigenstates, but instead using the ground state of H0. Obviously, a formalism developed from this result has the potential to be of great use in the study of a number of systems. The ﬁrst important step toward this goal is to build a diagrammatic expansion, known as Feynman-Dyson perturbation theory [8, 39–42]. To bridge the gap between equations and diagrams, we turn to the result known as Wick’s theorem [39, 43]. The theorem states that time-ordered products of creation and annihilation operators, such as those found in the numerator and denominator of Eq. (2.11), can be written in terms of two new operations, called contractions and normal- ordering:

T(ABC Z) = N(all products with n contractions). (2.14) · · · n conXtractions That is, the time ordering of operators is equal to the normal ordering of those operators, plus all normal orderings with one contraction, plus those with two, and so on. The normal ordering N is a permuting operator similar to T, but before applying it, the operators of its argument must be broken down into their constituent creation and annihilation operators. Then, the normal-ordering operator places all creation operators to the left of the annihilation operators, accruing the usual factor of 1 each time a − fermion operator passes through another. The diﬀerence between the time ordering and normal ordering of two operators is called a contraction:

contraction(AB) (AB) T(AB) N(AB). (2.15) ≡ C ≡ −

The two operators must be brought next to each other before the contraction is evaluated. After evaluation, the result is to be removed from further ordering operators. Of particular interest are contractions of the form

a (t )a† (t ) = Θ (t t ) a (t ), a† (t ) , (2.16) C n 2 m 1 2 − 1 n 2 m 1 h i h i since all other types of contractions in the S-matrix expansion (2.13) vanish. [ ] indi- · · · cates the anticommutator or commutator, as appropriate to the algebra of the operators. 27

As is customary, we choose the vacuum, defined as the state which vanishes when acted upon by any annihilation operator, to be the ground state of the unperturbed system φ . It is easily verified that the expectation value of any normal ordering in | 0i such a state vanishes, unless all operators to be ordered are contracted. This is a very important result, as it simplifies use of Wick’s theorem (2.14) considerably. The only expectation values which remain in Eq. (2.11), after application of Wick’s theorem, are those which look like [39]

0 Θ(t t ) φ a (t ), a† (t ) φ = iG (t , t ), (2.17) 2 − 1 0 n 2 m 1 0 nm 2 1 h i

0 where Gnm(t2, t1) is the retarded Green function of time [9], as deﬁned in Sec. 1.1.3. In this case, having deﬁned a vacuum state, we have explicitly considered both particle and hole excitations, which is the role of the (anti)commutator. The superscript 0 denotes that this Green function is related to H0, which is contained within the interaction-picture operators, rather than the full Hamiltonian. A complete picture of the diagrammatic approach to the S-matrix expansion (2.11) now emerges. H and must be written in terms of second quantization operators, I O and Wick’s theorem applied, after which two types of functions remain. Retarded Green functions (2.17), resulting from the movement of the interaction-picture operators and OI HI , are commonly represented by single-line arrows [39]. The other type of term which remains is the c-number part of the interaction, which may act on a single state and time, or may act on several. Such interactions are represented in the diagrammatic expansion by wavy or dashed lines, depending on their physical origin.

For term n in the expansions (2.13), each diagram has nNop/2 vertices, where Nop is the number of creation and annihilation operators which compose HI . The nature of the vertices themselves depends on the character of HI. Two-body forces, for example, create vertices of four Green function lines. All topological possibilities, with appropriate prefactors, must be summed to calculate a particular term of the expansions.

The presence of OI (t) in the numerator gives each diagram an “incoming” and “outgoing” line, which stretch to t = , while the denominator has no such Green functions. ∞ The denominator can be eliminated from consideration, therefore, by factoring all such “disconnected” pieces from the numerator, and canceling [39, 44]. That is, by restricting 28 Σ Σ Σ? = Σ + + Σ + Σ · · · Σ

Figure 2.1: Schematic diagram showing the Dyson’s equation expansion of the full retarded self-energy Σ?. The proper self-energy Σ is the sum of the minimal set of self-energy pieces from which Σ? can be built. our consideration to diagrams which consist, topologically, of only one piece, we automatically include all diagrams of the numerator and denominator. In general then, after this cancellation, all diagrams have a single incoming and outgoing line, in between which lie n vertices, connected to each other and the exterior Green functions by Green function lines and interactions; that is, each diagram with n > 0 can be represented schematically by Fig. 2.1. Clearly, the full S-matrix is given by [39, 40]

0 ∞ ∞ 0 ? 0 S = G ( , ) + dt1 dt2G ( , t2)Σ (t2, t1)G (t1, ), (2.18) 00 ∞ −∞ 0k0 ∞ k0k k0 −∞ Xkk0 Z−∞ Z−∞ where the quantity Σ?, known as the full retarded self-energy, is simply the sum of all things that can happen to a particle that starts and ﬁnishes in φ . While Eq. (2.18) | 0i may almost seem to be a tautology, it is extremely important. It tells us that if we can understand the full self-energy, we know the S-matrix, and, by extension, the full time dynamics of the system’s observables. The key observation here is that certain terms of Σ? can be singled out as “proper” self-energy contributions. This subset of self-energy contributions can be deﬁned as the minimal set from which all others can be built. That is, at no point in a proper self- energy diagram, except, of course, for the beginning and the end, does the system return to φ . This allows an iterative expansion for Σ? to be built up [39], as in Fig. 2.1. | 0i 29

Mathematically, the iterative equation is simpliﬁed by working in the energy domain

G(E) = G0(E) + G0(E)Σ(E)G0(E) + G0(E)Σ(E)G0(E)Σ(E)G0(E) + (2.19) · · · = G0(E) 1 + Σ(E)G0(E) + Σ(E)G0(E)Σ(E)G0(E) + (2.20) · · · = G0(E) [1 + Σ(E)G(E)] (2.21)

= G0(E) + G0(E)Σ(E)G(E) = G0(E) + G(E)Σ(E)G0(E). (2.22)

The same is true in the time domain, but intermediate times must be integrated over. Furthermore, a direct algebraic consequence of Eq. (2.22) is

1 0 1 G− (E) = G (E) − Σ(E), (2.23) − which is also known as Dyson’s equation. Since G0(E) is simply given by

1 G0(E) = , (2.24) E H + i0+ − 0 a physical problem is entirely solved if the proper retarded self-energy Σ is known. For many problems, this remains an intractable requirement [9]. In studying the basic properties of coherence and decoherence, however, we shall require only the simplest of one-body forces in HI . As such, Equation (2.23) provides a full, exact solution.

2.2 Self-Energies

In this work, two types of self-energy will be key. The ﬁrst is the most basic possible, that from a single level. The second is the self-energy due to a full, inﬁnite continuum of states. As a classical system, the continuum gives rise to decoherence. In each case, we begin with a single, isolated level, which forms the solved part of the Hamiltonian: H = ε 0 0 . (2.25) 0 0| ih | The retarded Green function of energy arising from this Hamiltonian is

1 0 0 G0(E) = = | ih | . (2.26) E H + i0+ E ε + i0+ − 0 − 0 30

As implied by Eq. (1.25), taking the real part of the pole of G(E) yields the single energy in the system. Time evolution is given by

iEt/~ 0 ∞ dE e− 0 0 iε0t/~ G (t) = | ih +| = e− 0 0 , (2.27) 2π E ε0 + i0 | ih | Z−∞ − as is to be expected. Our goal for the remainder of this chapter is to gain intuition about how addition of diﬀerent types of terms to this Hamiltonian changes the Green function.

2.2.1 Hybridization: adding a second level

H0 generates time evolution within a single quantum level. We can ask what eﬀect the inclusion of a second level has on the time dynamics. We consider an addition to the Hamiltonian

H = ε 1 1 + V 1 0 + V ∗ 0 1 , (2.28) I 1| ih | | ih | | ih | which depends on the energy ε of the new state 1 , as well as the tunneling matrix 1 | i element V , which takes the system from state 0 to the new state. | i We must now consider the proper retarded self-energy due to addition of the new level. It consists of only one diagram: the system leaves 0 with the interaction V , propagates | i within the 1 state, and ﬁnally returns with interaction V . All diagrams in the full | i ∗ self-energy are simply iterations of this process for this simple system. The proper retarded self-energy is thus

V V V 2 Σ = ∗ = | | . (2.29) 00 E ε + i0+ E ε + i0+ − 1 − 1 Dyson’s equation (2.23) now yields the full Green function:

2 + + 2 1 + V (E ε0 + i0 ) (E ε1 + i0 ) V G− (E) = E ε + i0 | | = − − − | | ; (2.30) 00 − 0 − E ε + i0+ (E ε + i0+) (E ε + i0+) − 1 − 0 − 1 (E ε + i0+) (E ε + i0+) G (E) = − 0 − 1 . (2.31) 00 (E ε + i0+) (E ε + i0+) V 2 − 0 − 1 − | | It should be noted that this is an exact result, which includes the proper self-energy to all orders. This is the power of Dyson’s equation (2.22). In passing, we note that the Green function now has two poles,

ε0 + ε1 1 2 2 ε = (ε1 ε0) + 4 V , (2.32) 2 2 − | | p 31 which give the two hybridized energy levels of the problem. It is a general consequence of Eq. (1.25) that the real part of the poles of the green function are always the eﬀective energy levels in a system. The eﬀect of the second level on the coherent time dynamics of the system is clear. Due to the presence of the level 1 , the system can leave 0 , propagate for an arbitrary | i | i amount of time, and then return. During this time, it accrues a phase based, not only on the time spent, but on the energy and tunneling matrix parameters introduced to the system with the extra level. Equation (2.31) provides an exact, mathematical description of the full time dynamics which result when this process is included.

2.2.2 Decoherence: a single continuum

Now that all the building blocks are in place, the next logical topic to consider is the most basic system which exists at the interface of quantum mechanics and the classical world. Instead of for a single level, we construct the self-energy due to a full continuum of states. This schematic problem will serve as the prototype for consideration of more complicated open quantum systems. The Hamiltonian is

H = H + ε k k + V k 0 + V ∗ 0 k , (2.33) 0 k| ih | k| ih | k | ih | Xk where k labels the states of the continuum. The levels of the continuum only communicate through the original 0 state, so proper | i self-energy terms are merely variations on Eq. (2.29), one for each of the states in the continuum. The result is 2 Vk Σ00(E) = | | + . (2.34) E εk + i0 Xk − To shed light on this result, we appeal to the Dirac identity 1 1 lim iπδ(x), (2.35) η 0+ x + iη ≡ P x − → where (x) denotes the Cauchy principle value of x, to arrive at P i Σ (E) = iπ V 2δ(E ε ) = Γ(E), (2.36) 00 − | k| − k −2 Xk where Γ(E) = 2π V 2δ(E ε ). (2.37) | k| − k Xk 32

Equation (2.37) happens also to be the result of Fermi’s Golden Rule in this system. This is simply due to the fact that the proper self-energy Σ is second-order in Vk and diagonal. In such a case, Dyson’s equation provides a prescription for iterating the processes of second-order perturbation theory to all orders exactly. For a more detailed discussion of the applicability of Fermi’s Golden Rule to exact results, the reader is excouraged to examine Sec. 4.4.1. Equation (2.37) is an exact result, even in cases without an ideal continuum. It is easy to see, however, that in the limit of an ideal continuum, whose states all couple equally to 0 , it produces a constant function of energy. Taking this limit, as we shall often do in | i this work, is analogous to making use of the correspondence principle to bring a quantum system to its classical limit. Now that we know the self-energy, we can sum its eﬀects to all orders exactly using Dyson’s equation (2.23): 1 G (E) = . (2.38) 00 E ε + iΓ/2 − 0 The density operator is deﬁned [21] by summing over the Hamiltonian’s diagonal basis ` : | i ρ(E) = δ(E ε ) ` ` , (2.39) − ` | ih | X` where ε` are the eigenenergies. ρ(E) is thus directly related to the imaginary part of the Green function of energy by the Dirac identity:

` ` G(E) = | ih | + = iπδ(E ε`) ` ` . (2.40) E ε` + i0 − − | ih | X` − X` It follows that 1 ρ(E) = Im[G(E)], (2.41) −π where the density of states is the trace of this quantity. In the current system, this equals

1 Γ/2 ρ (E) = , (2.42) 0 π (E ε )2 + (Γ/2)2 − 0 a Lorentzian. Due to its contact with the continuum, the state 0 has been broadened | i into a Breit-Wigner resonance. Its rate of decay is Γ/~. By particle-hole symmetry, this rate is also the rate of ﬁlling the level in the event that 0 is empty and the continuum | i is ﬁlled with non-interacting particles. 33

2.3 Summary

The Dyson’s equation approach derived in this chapter has been of fundamental importance to the theoretical solution of countless systems. Perhaps just as importantly, the simple form of the result (2.23) often provides an intuitive insight into the physics of a system, which might be lost in an unnecessarily complex treatment. In this work, we specifically explore the use of Dyson’s equation in situations where the self-energy is known, and the summation (2.22) can be carried out exactly to all orders in the perturbation of Σ. The results of Sec. 2.2 suggest that a quantum system can be affected in two ways by the addition of a single-particle self-energy, such as those important to the study of open quantum systems. First, if the state 0 becomes vacant, a particle from an external | i state may refill it. Since the new particle’s phase would then be completely uncorrelated to the old electron’s, the net result is a total loss of phase coherence, or decoherence. The other possibility is that the same particle may propagate for a time in the external system, and then re-enter the original set of levels. Such a process contributes an arbitrary phase to the process, and so is called dephasing. It is worth noting that, in the case of an infinite set of levels, as for a macroscopic continuum, dephasing and decoherence are indistinguishable. In the next chapter, we begin to apply the results found thus far to the study of a nanoscale system, two coupled quantum dots. The two-state model used is a synthesis of the results of Secs. 2.2.1 and 2.2.2, in that both states are coupled to a macroscopic continuum. A similar model is used in Chapter 4 to examine the decay-out of superdeformed nuclear bands. Thus, even the most basic results of the Green function approach are seen to generate solutions in a striking variety of problems. 34

CHAPTER 3

COUPLED QUANTUM DOTS

We now turn our attention to a particular system: a series arrangement of two quantum dots, coupled to each other, and each to a macroscopic lead. An important goal of this analysis is to lay a firm pedagogical foundation for the more complex treatments of later chapters, and as such, we shall endeavor to maintain as simple and straightforward an approach as can be implemented, and yet arrive at interesting physics. The focus will remain on equilibrium systems, and the treatment is largely restricted to a non- interacting, two-state model of time dynamics. Fortunately, the nature of the physical system is such that these assumptions retain a great deal of physical merit over large regions of experimental interest. Over the last two decades, quantum dots have been found to be a rich and varied physical system. Part of their appeal comes from parallels to natural systems [28, 45]: quantum dots can be seen as “artificial atoms”, nanoscale analogues to nuclei, and test systems for theories of quantum chaos, to name only a few. Whereas natural systems are generally limited to specific examples, however, the quantum dot is a remarkably tunable structure, so that an entire range of parameters can be explored. Quantum dots are thus often the ultimate proving ground for theories of mesoscopic fermion systems. It was perhaps natural that, once the community had begun to come to terms with the existence of the “artificial atom”, study of multiple-dot systems began to thrive. Theorists and experimentalists alike have found lattices of quantum dots a subject of great interest [18, 46–53], as well as smaller “artificial molecules” involving just a few dots [15, 17, 18, 25, 26, 28, 36, 37, 52, 54–80]. Besides their inherent interest, many important technological applications of coupled quantum dot systems have been proposed [29–34, 72, 81]. Great strides which have been made toward understanding the internal electron dynamics of double-dot systems, but it is a tendency in the field to consider coupling to the macroscopic leads as, at best, a source of experimental error to be minimized, or a generic source of uniform level broadening [26]. Nevertheless, experimental systems run the gamut from weak to very strong lead coupling [45], and it seems appropriate to 35

Table 3.1: Dimensionalities d available to nanoscale quantum systems, along with their common names and free-electron densities of states. m∗ is the electron’s eﬀective mass. Ei, Eij, and Eijk are the discrete energies which results from conﬁnement in 1, 2, or 3 directions, respectively. Θ is the Heaviside step function.

d Common Name Free-electron density of states m∗ √ 3 bulk π2~3 2m∗E m ∗ Θ (E E ) 2 quantum well π~2 − i Xi 1 m∗ Θ (E Eij) 1 quantum wire − π 2~2 E E r ij ij X − p 0 quantum dot δ (E Eijk) − Xijk consider the decoherent systems in the leads as sources of intriguing physics in their own right. Furthermore, the intuitive Dyson’s equation approach presented in the previous chapters provides an ideal tool for the study, since it automatically treats both coherence and decoherence on an equal footing, without the need for prejudicial assumption.

3.1 Quantum Dots

Quantum dots are zero-dimensional nanoscale systems of confined electrons. In them, we find an experimental realization of a textbook quantum mechanics problem, i. e. the particle in a box. More than that, however, the quantum dot represents a rich and varied “sandbox” of mesoscopic and many-body physics. The experimental and theoretical mastery of these systems has opened up countless frontiers full of engrossing problems for the physicist to study. From a theoretical perspective, a quantum dot is simply the logical continuation in a progression of mesoscopic devices defined by the dimensionality of their confinement (see Table 3.1). The exact meaning of “confinement” can be made rigorous from microscopic considerations: essentially the electrons should be restricted to a length scale ` which is much less than their spacing [82, 83]. For our purposes, however, it is sufficient to require that a particular experimental system possess a discrete density of states, the unmistakable hallmark of full three-dimensional confinement. 36

Figure 3.1: Electron micrograph images of experimental quantum dot systems. (a) An array of vertical quantum dots. The horizontal bars are 5µm. The main picture is from Ref. [84], while the inset diagram of a single dot was added in Ref. [28]. (b) A lateral double quantum dot system from Ref. [26]. The darker grey is the quantum well structure, and the lighter is the electrodes, which deﬁne the two quantum dots in the center tunably. The area available to the left dot is a 320 320nm2, while the region of the right dot is × 280 280nm2. ×

3.1.1 History and fabrication

The first quantum dots [84–86] were an outgrowth of earlier experiments into fabrication of quantum wells and developments in the technology of lithography. These “vertical” quantum dots (see Fig. 3.1a) began life as semiconductor herterostructures, usually GaAs- AlGaAs or GaAs-InGaAs. With proper doping, mobile electrons are trapped in a two- dimensional region near the interface of the heterostructure, forming a quantum well system. To form a quantum dot, the surrounding heterostructure material was removed by lithography, thus providing the additional two dimensions of electron confinement. Vertical quantum dots have been constructed in the series arrangement we discuss in this work. Another, more common, possibility for this type of experiment, however, is the lateral quantum dot (see Fig. 3.1b). This style of device also makes use of a semiconductor heterostructure to form a two-dimensional system. The difference lies in the method used to provide lateral confinement. Instead of etching, metallic electrodes are deposited on the surface of the quantum well device [87–89]. These are gated to tunably control all 37 matrix elements of the system’s Hamiltonian. Lateral quantum dots have become even more common in recent years, as they take full advantage of the tunability of quantum dot systems. Semiconductor quantum dots are not the only nanoscale systems that have been demonstrated to posses discrete electronic energy levels. Metallic nanograins [90, 91], self- assembling islands [92, 93], quantum “corrals” made via scanning tunneling microscope [94], and macromolecular systems [95, 96] have all been fabricated with physics similar to the original heterostructure systems. While the series arrangement we shall discuss here may be more difficult to contrive in some of these systems than in others, the theory we develop is general enough to apply to any.

3.1.2 Experimental studies

There exists a large variety of experimental techniques available to the study of quantum dot systems. Rather than attempt a review of the experimental literature, we choose to focus on the techniques and results that are the most relevant for the case of quantum dots coupled in series. For a comprehensive introduction to the varied types of experiments on quantum dots, and their implications, see Ref. [27] Experiments characterizing the electronic spectra of dots (see Fig. 3.2a) are of the most fundamental interest. The observation of discrete energy levels [84–86], with its implications for electronic confinement, has already been mentioned as a result of paramount importance. Vindicating analogies to atomic and nuclear systems, a further result of great import was the observation of shell structure in quantum dots [97]. The presence of shell closures and “magic numbers” (see Fig. 3.2b) has significant benefit to the applicability of the two-level model we shall use. A second, related class of experiments center on electron transport through and among dots. Charging and discharging of such a small nanostructure is dominated by the quantized nature of the electric charge. This Coulomb blockade effect leads to oscillations in the conductance of quantum dots [14]: only when points of charge degeneracy in parameter space are neared is current allowed to flow (see Fig. 3.2a). This, in turn, allows one to count how many electrons are on a dot by emptying it, and then slowly refilling while counting current peaks [98]. By this method, an experimental system can be “tailored” 38

(a)

(b)

Figure 3.2: Results of an experimental study of the spectra of vertical quantum dots, from Ref. [97]. (a) Current vs. gate voltage in a dot of diameter .5µm. This result demonstrates the discrete nature of the density of states, as well as Coulomb blockade oscillations. (b) Electron addition energies for two dots of diameter D. The peaks indicate especially stable electron numbers, implying shell closures. The magic numbers of these dots are 2, 6, and 12. The inset shows a diagram of the dots used in the study. to have whatever number of particles is desired. Studies of devices such as single-electron boxes [99], turnstiles [100], pumps [101], and transistors [102] have demonstrated the viability of this idea. Many other types of experiments fall outside the scope of this work. Measurements of the optical, magnetic, and even phonon properties of quantum dots have been made [27]. One prominent regime of quantum dot physics that will be intentionally absent from the present study is the Kondo eﬀect in quantum dots [103, 104]. For the purposes of this work, we assume that the system is far away from the regime for which such lead-lead or lead-dot correlations are relevant.

3.2 Two-Level Model

We assume that electrons in each dot are limited to only one energy level. As long as the dots remain within their ground states, this is known to be a rather good approximation [26]. Previous theoretical studies applying this model to coupled quantum dots have met with much success [15, 35–37]. For a discussion of how to move beyond the two-level 39 model, the reader is encouraged to examine Sec. 4.6 of the present work.

3.2.1 Realm of Applicability to Quantum Dots

The two-level approximation amounts to an assumption that interactions with those electrons of less energy in each dot can be treated as constant, and that states with higher energy can be neglected. In a non-interacting picture, this is permissible so long as any tunneling matrix elements connecting our two states to other states are much less than the energy difference with those states [37]. When this is true, such extra states will play very little role in the dynamics of the system. These conditions are especially likely to be fulfilled if the dots are filled exactly to shell closure. Then, the extra electron we inject is forced to begin a new shell, and interacts mainly in a mean-field sense with the lower electrons. This situation is reminiscent of the “core” approach to the shell model studies of nuclear physics. Due to the artificial and tunable nature of quantum dot systems, and especially the ability to controllably add and remove particles from systems, such a situation is quite realizable in practice. A further consideration is that the temperature of the system must not cause excitations between levels. Thanks to remarkable efforts by experimentalists, quantum dot systems usually have effective electron temperatures of around 10–15mK [26]. This corresponds to a level spacing of .86–1.29µeV, comfortably below those common in quantum dots. Of course, the two-level model will be most accurate when we need not ignore electrons of lower energy levels, because there is only one electron on the dot. Such single-electron devices have indeed been fabricated [99–102]. They provide the best opportunity for experimental verification of theoretical studies, such as ours, that rely solely on single- particle effects.

3.2.2 Hamiltonian of the coupled dot system

For the rest of this discussion, we assume the conditions of the preceeding argument are met, so that we are justiﬁed in working within the two-level model. Each dot is coupled to its own lead: this series arrangements of dots and leads is shown in Fig. 3.3. 40

Figure 3.3: Schematic diagram of the double quantum dot system. Two quantum dots are coupled to each other, and each to its own macroscopic lead. This situation is also frequently known as a triple-barrier system. In the two-level model, the two dots are completely described by their energy levels ε1 and ε2, which are deﬁned in the absence of both leads and the tunneling between the two dots V . Γ1/~ and Γ2/~ give the rates for electrons to enter and leave the double-dot system through each lead. From Ref. [37].

The Hamiltonian of the system can be written as the sum of three terms [37]:

H = Hdots + Hleads + Htun, (3.1) where Hdots generates the time evolution of the two-level system in the absence of the leads, Hleads is likewise the Hamiltonian of the isolated leads, and Htun gives the coupling between them. The isolated dot Hamiltonian in the two-level model is

H = ε d† d + ε d† d V d† d V ∗d† d . (3.2) dots 1 1 1 2 2 2 − 1 2 − 2 1

Here εn is the isolated energy level of dot n, V parameterizes the hopping from dot 2 to dot 1, and dn is the operator which annihilates an electron in the state of dot n. Gauge invariance allows us to choose exactly one phase in the problem: we use this freedom to set V real and positive. Equation (3.2) neglects interdot interactions. We shall extend a similar model to fully include the Coulomb interaction in Chapter 5. In general, we assume the leads are ideal Fermi gasses, devoid of the complications of many-body correlations such as Kondo physics, superconductivity, etc. They posses a diagonal representation, and in that basis their Hamiltonian can be written 2

Hleads = kck† ck, (3.3) α=1 k α X X∈ 41

where k is the energy of a particular state k in lead α, and ckσ annihilates an electron in state k. Coupling between the leads and dots is provided by the third term of the Hamiltonian:

Htun = Vnkdn† ck + H.c. . (3.4) nα k α hXi X∈

Here Vnk parameterizes the tunneling between dot n and state k of lead α. The notation nα reminds us that V = 0 only if it would connect a dot to its own lead. h i nk 6 The source of decoherence, dephasing, and dissipation in this system is clear from Eq. (3.4). Through it, electrons can leave the system. New electrons can enter the system as well, and, having random phase, they can only contribute to the dynamics in incoherent ways. A combination of these two results is also possible: an old electron may be replaced by a different one from the leads, thus causing decoherence. Similarly, an electron may tunnel into a lead, propagate via Eq. (3.3) for arbitrary time, and itself return later with a new phase. Equation (3.2) describes a well understood quantum-mechanical problem. The new and interesting physics is introduced by the leads in the terms (3.3) and (3.4). These contributions require an infinite-dimensional Fock space. It is important to realize that one does not really care what happens to electrons in the leads, only in the dots. The beauty of the Dyson’s equation approach is that it allows us to incorporate the effects of the leads exactly, as they appear from inside the dots.

3.2.3 Spin-boson analogy

Having constructed the two-state Hamiltonian of our system, we arrive at an appropriate juncture to draw analogy to a very well studied problem, that of the spin-boson Hamiltonian [105]:

1 1 1 p2 1 H = ~∆ σ + ε σ + m ω x2 + α + q σ C x . (3.5) SB −2 SB x 2 SB z 2 ν ν ν 2m 2 0 z ν ν ν ν ν X X 1 Here σx and σz are the Pauli matrices for spin- 2 systems. This Hamiltonian is often used to describe two-state systems, especially spin or isospin systems, in contact with a bath of harmonic oscillators. The oscillators usually represent bosonic states, such as phonons or photons. 42

The first two terms are to be taken in analogy to our Hdots; that is, they define the closed, well understood two-level system. The Pauli matrices play the role of the quadratic operators in Eq. (3.2). ~∆SB and εSB should be taken in analogy to our V and ε ε , respectively. 1 − 2 The harmonic oscillators, labeled by the index ν, usually represent a bath of bosonic states, such as phonons or photons, which provide dissipation to the spin states. The internal Hamiltonian of the oscillators themselves is given by the third term in Eq. (3.5), comparable to bosonic excitations of the lead system described by Eq. (3.3). The oscillators, with position and momentum operators xν and pν, respectively, are defined by their mass and frequency parameters mν and ων.

The ﬁnal term in the spin-boson Hamiltonian (3.5) plays a similar role to Htun of Eq. (3.4); its purpose it to provide the coupling between the spin and bosonic systems.

Although physical meaning may be ascribed in certain systems, in the general case q0Cν simply parameterizes the coupling of each oscillator to the two-level system [105]. The spin-boson formalism is itself a powerful tool, and many parallels can be drawn with the Dyson’s equation approach. The most central conclusion of the theory surrounding its use is that the eﬀect of the boson system on the spin part is entirely contained within the spectral function of the coupling [105]:

π C2 J(ω) = α δ(ω ω ). (3.6) 2 m ω − α α α α X The spirit of this result should seem quite familiar: we are interested only in the oscillators’ eﬀect on the spin system, and so we “trace out” the extra degrees of freedom, and replace them with J(ω). Equation (3.6) should be compared with the functions Γ(E), for example Eq. (2.37), which arise in the Dyson’s equation approach. Although Γ(E) is not, strictly speaking, a spectral function, its derivation from the retarded self-energy Σ is reminiscent of the method used to extract a spectral function from a Green function [9].

3.3 Green Function Treatment

Having constructed the Hamiltonian of the complete system in Sec. 3.2.2, we are ready to attempt a solution of the problem using Dyson’s equation. Before doing that, however, we take the opportunity to re-examine the textbook problem of the closed system’s 43

(a) ε+ (b)

ε1 ω ε2

ε−

Figure 3.4: Hybridization of energy levels in the double-dot system without leads. (a) The energy levels in the absence of V , ε1 and ε2, are combined by their tunneling matrix 2 2 element to give two new levels, ε+ and ε , as given by Eq. (3.7). ω = √∆ + 4V is − ε ε the detuning of the two new orbitals. (b) The graph of V− (solid lines) vs. ∆/V shows the anticrossing which results when two coupled levels approach each other. The dashed ε1,2 ε lines give V− . After Ref. [26].

Hamiltonian, given by Eq. (3.2). This will allow us to compare and contrast with the results of the full problem, the better to determine the eﬀect of the leads.

3.3.1 Without leads

The ﬁrst step in understanding the full problem will be to remind ourselves of some results for the two-level system without coupling to an environment. Diagonalizing the Hamiltonian is a simple problem in elementary quantum mechanics. The eigenenergies are simply the results for two hybridized levels:

∆ 2 ε = ε + V 2 (3.7) 2 s where ε (ε + ε )/2 is the mean of the two isolated energy levels, and ∆ ε ε ≡ 1 2 ≡ 2 − 1 is their detuning. The physical meaning of Eq. (3.7) is demonstrated in Fig. 3.4. The eigenstates themselves are the well known antibonding and bonding solutions to the two- state problem: 1 ε ε1 ψ = ψ1 + − ψ2 , (3.8) 2 V ε ε1 1 + V− r where ψn are the wavevectors of each isolated dot. 44

As a simple exercise, let us see if we can arrive at this result within the Green function formalism. We work within the basis of the individual dots. The retarded Green function of the system is given by

1 + − + 1 E ε1 + i0 V Gdots(E) = E Hdots + i0 − = − + −  V E ε2 + i0  −  +  1 E ε2 + i0 V − − = + + 2 . (3.9) (E ε1 + i0 )(E ε2 + i0 ) V  V E ε + i0+ − − − − − 1   As expected from Eq. (1.25), the real part of the poles of the Green function gives us the energies of the eigenstates. Solving for the poles recovers Eq. (3.7).

The retarded Green function Gdots(E) is characterized by the same eigenfunctions as the Hamiltonian Hdots. The secular equation is

1 G (E)ψ = ψ, (3.10) dots E ε + i0+ − where the prefactor on the right is the eigenvalues of the retarded Green function. Equa- tion (3.8) follows directly from Eq. (3.10).

It is, of course, natural and necessary that the eigenvalues and eigenstates of Hdots, the generator of time translations for the system, should be contained within Gdots(E), since that quantity determines the time dynamics of the problem. We raise the point here not only to serve as an instructional example, but also to emphasize a key element of the theory: the Green function and Hamiltonian formulations of quantum mechanics are completely interchangeable. The correspondence is exactly that between a differential (Schrödinger equation) and integral (Green function) approach, rigorous within linear inverse theory [12]. For a generic state, the time evolution is most easily seen from the time-domain Green function ∞ dE iEt G (t) = G (E)e− . (3.11) dots 2π dots Z−∞ If a single electron is localized in dot 1 at time 0, the probability of finding it in the same dot at a later time t is [26, 37]

1 ωt ωt P (t) = G (t) 2 = ∆2 sin2 + ω2 cos2 , (3.12) 1 | 11 | ω2 2 2 45

Figure 3.5: Example of the coherent Rabi oscillations described by Eqs. (3.12) and (3.13). In this graph, ∆ = V . In the absence of dissipation from the leads, the coherent oscillations continue forever. while 4V 2 ωt P (t) = G (t) 2 = 1 P (t) = sin2 . (3.13) 2 | 21 | − 1 ω2 2 Equations (3.12) and (3.13) describe the coherent Rabi oscillations of a two-level system [21]. The frequency of oscillation is given by the detuning between the bonding and antibonding orbitals: ω = ∆2 + 4V 2 ~. (3.14) p . An example of these coherent oscillations is shown in Fig. 3.5.

3.3.2 Including the leads

We proceed now to include the leads in our description of the system. The motivation for this procedure is evident from the results (3.12) and (3.13) of the previous subsection: since the Rabi oscillations continue forever, without decoherence, dephasing, or dissipation, such an approach clearly neglects an important part of the physics. Since there are no processes which allow an electron to tunnel from dot 1 into a lead, propagate within the lead, and then return by tunneling into dot 2, or visa versa, the tunneling rates of leads 1 and 2 are uncorrelated. This is an essential point, as it tells us 46 the retarded self-energy is diagonal, and that we are dealing with two separate broadening phenomena. At this stage, we may refer ourselves to the solution of the single level in contact with a macroscopic continuum of states (Sec. 2.2.2), and simply write down the solution. The retarded self-energy has the form

i Σ (E) = δ Γ (E), (3.15) nm −2 nm n where the tunneling rate into and out of dot n is given by the Fermi’s Golden Rule result (2.37) 2 Γ (E) = 2π V δ(E ε0 ). (3.16) n | nk| − k nα k α hXi X∈ Equation (3.15) contains all time evolution within the leads. Dyson’s equation (2.23) allows us to use it to solve for the eﬀect of the leads exactly to all orders in perturbation theory. The full Green function is thus

1 i − 1 1 E ε1 + 2 Γ1(E) V G(E) = G− (E) Σ(E) − = − − . (3.17) dots −  i  V E ε2 + 2 Γ2(E) − −   Inverting this result we ﬁnd [37]

1 i i − G(E) = E ε + Γ (E) E ε + Γ (E) V 2 − 1 2 1 − 2 2 2 − i E ε2 + Γ1(E) V − 2 , (3.18) ×  V E ε + i Γ (E) − 1 2 2   which should be compared with Eq. (3.9). We now make an assumption, known as the broad-band approximation and often utilized in the literature, that the energies of the dots lie well within continua of the leads. If this is the case, the densities of states in the leads can be eﬀectively replaced with a constant function, and the tunneling matrix elements Vnk are uniform. The Γn(E) may then be replaced by constant parameters. If the leads are indeed good metallic conductors, this approximation is very well justiﬁed. Further, since the focus of this work is the interplay between quantum and macroscopic systems, the case of uniform continua in the leads is of primary interest here. 47

(a) (b)

Figure 3.6: Examples of the mixture of coherent and incoherent behavior found in the full system of dots and leads, as given by Eqs. (3.19) and (3.20). In both, ∆ = V . (a) An example of the underdamped case, where decoherence from the leads is relatively weak. Γ = 3Γ0 = .3V . (b) An example of the overdamped case. Γ = 3Γ0 = 3V .

Let us again consider the case that our single electron is localized in dot 1 at time t = 0. After performing the Fourier transform of Eq. (3.18) by contour integration, we are able to compute the probabilities [37]

2 V Γt/~ ωi + Γ0 ωit Γ0 ωi ωit iωr + Γ0 iωrt iωr Γ0 iωrt P1(t) = 2 e− e + − e− + e + − e− ω Γ0 ωi ωi + Γ0 iωr Γ0 iωr + Γ0 | | − − (3.19) and 2 2V Γt/~ P (t) = e− (cosh ω t cos ω t) . (3.20) 2 ω 2 i − r | | Here Γ1 + Γ2 Γ2 Γ1 Γ , Γ0 − , (3.21) ≡ 2 ≡ 2 and ωr and ωi are the real and imaginary parts of the complex Rabi frequency

ω ω + iω 4V 2 + (∆ iΓ )2 ~. (3.22) ≡ r i ≡ − 0 q Figure 3.6 shows two examples of behaviors which can result from this solution. Taking the limits Γ , Γ 0+ of Eqs (3.19) and (3.20) yields Eqs. (3.12) and (3.13). 1 2 → We note, however, that, except in this limit, the identity P1(t)+P2(t) = 1 no longer holds: 48 the open nature of the system allows particles to both enter and leave the system. The incoherent physics introduced by the leads is reﬂected in the real exponential (hyperbolic) functions in Eqs. (3.19) and (3.20). Whereas ωr governs the coherent Rabi oscillations of the system, ωi describes the decoherence processes due to the leads. For long times, the decoherence dominates, as we should expect. As t , the probability in both dots → ∞ Γt/~ falls as e− due to the inﬁnite bath of states into which electrons can escape. We can understand our general result more by comparing Eq. (3.22) with its coun- terpart Eq. (3.7) in the isolated system. Whereas in the isolated case, we considered the energies of the Hamiltonian, which are observable and hence real, in the lead-coupled situation we generalize the discussion to consideration of the complex poles of the Green function ε˜ + ε˜ ~ω ε˜ = 1 2 , (3.23) 2 2 where it is to be understood that ε˜ = ε iΓ /2 are the poles of the Green function i i − i for dots isolated from each other, i.e. in the limit V 0. Equation (3.23) shows that → ~ω remains the displacement in the complex plane between the poles of the full Green function. Since it determines time dynamics of the system, the retarded Green function, together with the lead couplings Γi, naturally contains all linear-response conductance information. The transmission probability between leads 1 and 2 for charge carriers of a given energy is given by the multi-terminal current formula (see Sec. 5.2.5):

T (E) = Γ Γ G (E) 2. (3.24) 12 1 2| 12 |

In the linear response, all charge carriers have energy equal to the Fermi energy εF of the leads. Thus, the conductance of the double-dot device is equal to [4]

1 2e2 = T (ε ), (3.25) R h 12 F where the prefactor is simply the conductance quantum.

3.4 Limiting Cases

Eqs. (3.19) and (3.20) represent a complete solution of our model, in as much as they contain all information regarding the time evolution of the experimental observables in 49 the problem. Greater understanding of the physical meaning of this result can be gained through examination of the limiting cases of identical dots and identical leads.

3.4.1 Identical dots

We first consider the case of identical dots: ε1 = ε2. When this is true, the complex Rabi frequency (3.22) simplifies to ω = 4V 2 Γ 2 ~. (3.26) − 0 p . This quantity is either purely real or purely imaginary, depending on the relative values of 2 V and Γ . | | | 0| In the case that 2 V > Γ , the Rabi frequency is purely real. Working with the | | | 0| algebraically simpler probability P2(t), we find [37]

2 4V Γt/~ 2 ωt P (t) = e− sin . (3.27) 2 ω2 2

In this limit, incoherent interference has vanished entirely, and the solution exhibits only the coherent behavior characteristic of Rabi oscillations. In fact, Equation (3.27) is nearly identical to the result (3.13) for an isolated double-dot system. It diﬀers only in Γt/~ the presence of the exponential envelope function e− , which appears equally in both

P1(t) and P2(t), signifying the statistical escape of electrons from the discrete system to the inﬁnite-dimensional Fock space of the leads. In the case of identical dots but 2 V < Γ , we ﬁnd a purely imaginary ω. Then [37], | | | 0| 2 4V Γt/~ 2 ω t P (t) = e− sinh | | . (3.28) 2 ω 2 2 | | The lack of circular functions denotes an absence of coherent Rabi oscillations in this regime. In this limit, transport between the two dots is totally incoherent. These results possess a particularly intriguing characteristic. V is the tunneling matrix

Γ1+Γ2 element of interdot tunneling, while 2 ~ is approximately the frequency of tunneling events coupling the dot system to the macroscopic world. One might have supposed that a competition between these two quantities would determine the extent to which the system exhibits coherent or incoherent behavior. Instead, we ﬁnd it is the diﬀerence of Γ ’s that is to be compared with 4 V . We explore this surprising result further by n | | examining the case of identical leads, next. 50

3.4.2 Identical lead couplings

Motivated by the results of the previous subsection, we turn now to a consideration of identical lead couplings, as opposed to identical dots. Rather than require ε1 = ε2, we set Γ1 = Γ2. In this case [37], ω = 4V 2 + ∆2 ~ (3.29) . is again purely real, regardless of the energiesp of the individual dots. Fully coherent Rabi oscillation is thus recovered, and equation (3.27) once again describes the time dynamics. We can understand this surprising result, and the results of Sec. 3.4.1, in terms of measurement theory. Incoherent transport between the two dots corresponds to measurement of the two-dot system by the macroscopic bath of states. If Γ1 = Γ2, the presence of an electron in either dot 1 or dot 2 is seen as the same by the macroscopic environment, and so no measurement can take place regarding where the electron is. In this case, therefore, transport can only be coherent. Moreover, the Green function of time factorizes [15, 106] Γt/~ to G(t) = e− Gdots(t), so the eﬀect of the leads is merely to contribute the decay term.

As Γ0 is increased, the distinction the environment makes between the dots comes into competition with V , which tends to mix the two dot states.

3.5 Summary

Using the Green function formalism, we have solved the two-level model for the case of quantum dots in equilibrium with a macroscopic reservoir. Dyson’s equation (2.22) allowed us to exactly sum all diagrams which included interaction with the leads. Unlike most studies present in the literature, this treatment lets us consider regimes where decoherence may play a central role. Indeed, the result is that a competition between the hopping matrix element V and the difference in coupling of the two dots to the environment 2Γ0 determines the mixture of coherent and incoherent behavior, a result well understood in the context of measurement theory. We have deliberately kept the treatment simple so as not to obscure the important physics of coherence and decoherence in this system. The most significant ways we might improve the model of this chapter are the inclusion of other levels in the dots and the inclusion of interdot interactions. Although the treatment is for different physical systems, 51 the reader interested in such further steps is encouraged to examine the later chapters of this work, where both such issues are addressed more fully. Presently, we turn our attention to a natural system which, while physically quite different, nevertheless exhibits much of the same physics of the two-dot model of this chapter. The wide scope of problems that can be treated with the Dyson’s equation approach is a central aspect of its utility and power. Chapter 4 deals with the decay-out process of superdeformed nuclear bands. 52

CHAPTER 4

DECAY OF SUPERDEFORMED NUCLEI

Our next topic is the decay of superdeformed nuclei. While, on the surface, the interdisciplinary leap from the time-dynamics of quantum dots may seem great, it will become clear that the two problems are, in reality, not at all dissimilar in their underlying physics. This, in itself, illustrates one of the central reasons for choosing a Green function approach: it illuminates, rather than obscures, the underlying physical principles of a problem, and so theoretical insights gained for one system are not lost when we move to the next. Superdeformed nuclei are a striking example of a counterintuitive phenomenon encountered in the study of a many-body, quantum mechanical system. If one pictures the nucleus as a rotating drop of ﬂuid, some deformation, i.e. departure from a spherical shape to an ellipsoidal one, may be expected on purely classical grounds. The strength of the internal forces of the nucleus, however, indicate that one should expect this result to be very slight, as indeed it usually is. For certain highly excited, high angular momentum nuclei, however, a set of states is observed with approximately 2:1 major-to-minor-axis ratios. These superdeformed (SD) states, like their normally deformed (ND) cousins, form rotational bands, which are observed to persist over many states, and then suddenly decay into ND bands of lower energy. An understanding of this process becomes our current goal.

4.1 Nuclear Deformation

The departure of nuclei from a spherical shape is of enduring interest in nuclear structure theory. As a topic, it lies at the intersection of two important, extraordinarily successful, pictures of nuclear behavior, the shell model of Mayer and Jensen [107] and Bohr and Mottelson’s collective motion model [108]. Moreover, the phenomenon cannot be properly understood without taking a first step beyond these essentially single-particle models, to a picture which includes the pairing force, a “residual” many-body interaction [109]. The ground states of most nuclei are found experimentally to be somewhat ellipsoidal, simply because many-body effects make such a configuration of nucleons energetically favorable. The phenomenon can be thought of as small admixtures of spherical harmonics 53 beyond the monopole entering into the mean-field potential for nucleons, and creating a lower-energy state. This is to be distinguished from the larger deformations that fall under the heading of superdeformation: in these cases, the entire shell structure of the nucleus is different. Ground states and low-angular-momentum states are never superdeformed.

4.1.1 Normal deformation

Many nuclei, known as normally deformed, are found experimentally to possess a major- to-minor axis of about 1.3:1. In general, a nucleus with filled neutron and proton shells is spherical. As a nucleon shell begins to be filled, nuclei tend to become prolate, with one axis larger than the other two. Once the point of shell half-filling is passed, deformation is generally oblate, with one axis smaller than the other two. The starting point to understanding deformation of nuclei is the shell model, which, like the atomic shell model, centers on a mean-field picture of stability, in which the single-particle spectrum tends to clump into close-lying groups of levels called shells [107]. Nuclear shells exist for both the nucleon and proton systems, and those nuclei for which both species of shell are closed are the most stable. Shell closures occur at the “magic numbers”: for neutrons, these are 2, 8, 20, 28, 50, 82, and 126; and for protons, 2, 8, 20, 40, and 82. Experimental signals of a major shell closure include large excitation energies (see Fig. 4.1) and large proton and neutron separation energies (see Fig. 4.2) [38]. We recall from study of a quantum mechanical particle in a central, spherical potential that shells arise from the degeneracy of levels of the same principal quantum number n. Neglecting magnetic interactions, for example, the energies of the electron in a hydrogen atom are 13.6eV E = − , (4.1) n n2 while the orbital quantum number l, azimuthal quantum number m, and spin s are left undetermined by a generic measurement of energy. This leads to a shell of 2n2 levels, degenerate until lifted by fine and hyperfine splitting. In order to examine deformation, it is advisable to decompose the spatial wavefunctions of the Hamiltonian into a spherical basis:

r (k) m ψk( ) = CnlmRn(r)Yl (θ, φ), (4.2) Xnlm 54

Figure 4.1: Energies of the ﬁrst excited state of nuclei with even proton number (Z) and neutron number (N), multiplied by A1/3 = (Z + N)1/3, which is approximately proportional to the nuclear radius. Nuclei near shell closures in one or both fermion systems have higher excitation energies, since the ground state is so stable. From Ref. [38], based on data from Ref. [110].

Figure 4.2: Separation energies, calculated from experimental binding energies, of nucleons and α-particles from stable nuclei. The small oscillations with period 2 are due to the pairing interactions, and the larger steps are due to shell closures. From Ref. [38] 55

where the label k identiﬁes a particular wavefunction, Rn(r) is the radial wavefunction m appropriate to the central potential, and Yl (θ, φ) are the spherical harmonics. In the case that the mean-ﬁeld is spherically symmetric, only one term in Eq. (4.2) will play a role, in analogy to the hydrogen atom. The occupation probability of a state k is then

2 2 3 (k) m 3 (k) 2 m 2 Pk = d r CnlmRn(r)Yl (θ, φ) = d r Cnlm Rn(r) Yl (θ, φ) , (4.3) | | | | Z nlm Z nlm X X m where the ﬁnal step makes use of the fact that the Yl (θ, φ) are orthogonal. For a mean ﬁeld axially symmetric about the z axis, the angular momentum projection on that axis

Lˆz commutes with the Hamiltonian, and thus the eigenstates ψk(r) do not mix diﬀerent (k) m values. Thus, the only m-dependence of the Cnlm can be to determine whether a state is ﬁlled or not:

(k) 0, state unﬁlled Cnlm = . (4.4)  C(k), state ﬁlled  nl An important property of the spherical harmonics is [2]

l 2l + 1 Y m(θ, φ) 2 = . (4.5) | l | 4π m= l X− It is clear now that in the case of a ﬁlled shell,

2 ∞ (k) 2 Pk = (2l + 1) dr Cnl Rn(r) . (4.6) 0 | | nl Z X The integral over angular coordinates has vanished, and so we may read Eq. (4.6) as follows: knowledge of how to mix the various spherical harmonics is not required to compute the probability of finding a nucleon in any single-particle eigenstate. Since the spherical harmonics are the generators of a general deformation in spherical coordinates, the state k cannot be found to be deformed by any measurement. Clearly, the spherical symmetry of Eq. (4.6) holds for filled shells, in which the second line of Eq. (4.4) is always used. If the intra-shell ordering of levels is such that nucleons fill up closed subshells of a given orbital angular momentum l before full shell closure is reached, it will result in further examples of spherical nuclei. If, however, the ordering of levels is such that groups of states with the same l do not fill easily, spherical nuclei will 56

(a) (b)

Figure 4.3: (a) Table of nuclides showing experimentally observed regions of ground state normal deformation, from Ref. [38]. The large shaded area shows the stable nuclides, while the smaller islands are the regions of deformation. Note that regions of spherical nuclei do not begin mid-shell. (b) Portion of the table of nuclei, showing calculated value of the lowest angular momentum quantum number for which SD states occur, from Ref. [111]. In contrast with (a), SD states occur for every nuclide, but never as the ground state. The letter denoted on the chart gives the mixture of standard quadrupole superdeformation ε with a cross-axial γ mode for the lowest-energy SD state. be exclusively found near shell closures. The presence or absence of normal deformation is thus seen to be closely related to the fine structure of nuclear shells. In fact, the latter is the case, due to a many-body effect called the pairing interaction [109]. Since the shell model is a mean-field model, corrections, called residual interactions, must be included to compensate for the absence of many-body effects. The pairing interaction is one such: a strong, short-range, attractive force, so that two nucleons starting a new shell find it energetically favorable to overlap their wavefunctions as much as possible. This leads them to form pairs of azimuthal angular momentum m , with | | m as large as possible. m = 0 states, necessary to close any group of specific orbital | | angular momentum l, thus tend to fill last in any shell. As Fig. 4.3a shows, the conclusion that spherical nuclei occur only near major shell closures agrees well with experimental evidence.

4.1.2 Superdeformation

The phenomenon of superdeformation, like that of normal deformation, has its roots in the shell structure of nuclear ground states. In this case, it is related to new sets of magic 57 numbers and shell closures, which form as deformation is applied to the shell model. In fact, sets of shell closures called hyperdeformed states are posited to exist for even greater deformations than are currently available to experiment [112]. As an example, we consider the elliptical harmonic oscillator, which can be taken as a lowest-order approximation to an axially symmetric mean field. The Hamiltonian of such a system is ~2 m 2 2 2 2 2 HEHO = + ω3x3 + ω x1 + x2 , (4.7) 2m 2 ⊥ th where m is the effective mass, xi is the i body-fixed coördinate, and ωi is the frequency th for oscillation in the i axis. The symmetry implies ω1 = ω2 = ω . The eigenenergies of ⊥ HEHO are EHO 1 ~ ~ En3n = n3 + ω3 + (n + 1) ω , (4.8) ⊥ 2 ⊥ ⊥ where the two quantum numbers n3 and n are nonnegative integers. Figure 4.4 shows ⊥ how these levels move as a function of deformation. New sets of shell closure appear at axes which are in the ratios of small integers, and are most strong when one of those integers is 1. Superdeformation, in particular, corresponds to the shell closures at axis ratios 2:1 and 1:2. In fact, superdeformation is a general prediction, not only of nuclear shell models, but of shell models in general. New sets of shell closures and magic numbers corresponding to large elliptical deformations are known in quantum dot systems [28], and they have have been predicted for systems of cylindrical symmetry, such as nanowires [113], as well. SD states have been observed across a wide variety of nuclei. They were first observed in 152Dy [114], and the mass region near A 150 has seen considerable experimental ≈ interest [114–126]. Even more study has been dedicated to the A 190 region, for ≈ example Refs. [117, 127–160]. Since the discovery of these regions, lighter regions of SD nuclei have also been identified [161–169]. Figure 4.3b shows the results of calculations for the onset of superdeformation across the table of nuclides.

4.1.3 Experimental signatures of deformation

The typical SD decay experiment involves collision of two heavy ions [170]. At high angular momenta, an SD state is often yrast, meaning it is the lowest-energy state for that angular momentum. When large amounts of angular momentum stay within the system, 58

Figure 4.4: Eigenenergies of the elliptical harmonic oscillator Hamiltonian (4.7), as a ω ω3 function of a deformation parameter δosc = ⊥ω− , where the average oscillator frequency 1 ω = 3 (2ω + ω3). Small-integer values of axis ratios are noted. In particular, note the new sets of⊥ shell closures and magic numbers which occur for prolate (2:1) and oblate (1:2) superdeformation. At the axis ratio 3:1, another set of hyperdeformed shell closures appears. From Ref. [108].

SD nuclei are thus often produced from such reactions. In general, two experimental signatures are usable to detect these nuclei. An intrinsic method of detecting an SD state is related to its quadrupole electric moment. We recall, from electrostatics, that in the multipole expansion of the electric ﬁeld due to a charge distribution, the quadrupole moment about the body-ﬁxed 3-axis is [19]:

16π 3 0 2 3 2 2 Q = d r Y ∗(θ, φ)r ρ(r) = d r 3x r ρ(r). (4.9) 5 2 3 − r Z Z This quantity provides a measure of a nucleus’s ellipsoidal deformation from a spherical charge distribution. Making an assumption that the nucleus is an ellipsoid of revolution, with charge inside 59 it distributed evenly, the charge density is

2 Ze x3 4 2 , r R 1 R3 r 3 πR R3 ⊥ ≤ ⊥ − ρ( ) = ⊥ , (4.10)  q x2 0, r > R 1 3  ⊥ ⊥ − R3 q where R and R3 are the maximum extent of the nucleus along the two symmetric axes ⊥ 2 2 r and the third axis, respectively, and r = x1 + x2 is the distance of the point from ⊥ the x3-axis. The result for the quadrupole momenp t is

2 2 2 Q = Ze R3 R . (4.11) 5 − ⊥ Note that, since the x3-axis is defined as the body-fixed axis that is not equal to the other two in length, prolate nuclei have a positive Q, and oblate nuclei a negative one. According to Eq. (4.11), the quadrupole moment of an SD nucleus is roughly three times that of an ND state. Experiments have succeeded in using this property to identify SD bands [124, 128, 130, 134, 146–148, 152, 155, 156, 168]. It is not always convenient to measure the electric quadrupole moment of a decaying nucleus. Fortunately, superdeformed nuclei possess another experimental signature: their rotational spectrum. When a state breaks spherical symmetry, as the SD and ND states do, the orbital angular momentum operator Lˆ no longer commutes with the Hamiltonian. Thus, time evolution causes transitions between states of differing angular momentum in the body-fixed frame. These occur via electromagnetic decay, so the transition energies can be measured. The energy levels of a quantum mechanical rotor are well known [2]:

I(I + 1)~2 Erotor = + Erotor, (4.12) I 2I 0

rotor where E0 is the energy of the zero-orbital-angular-momentum bandhead state, I is the orbital angular momentum quantum number about the body-ﬁxed axis of rotation, and I is the moment of inertia about the axis of rotation. In general, I may be a function of angular momentum, so that the spectrum is distorted from the constant-I “rigid rotor” case. ND nuclei, for example, experience a phenomenon called “centrifugal stretching”, which causes the moment of inertia increases with I, so that the spectrum is compressed relative to the rigid rotor [108]. By contrast, the SD states are much more rigid against this eﬀect [111, 171]. In this case, I may be approximated by a constant. 60

Because of their strong quadrupole deformation, SD nuclear decay is dominated by coupling to the quadrupole electric ﬁeld, called E2 decay. The symmetry of E2 matrix elements is such that two units of angular of momentum are lost with each event. This, together with the nearly constant I , provides a unique character to SD rotational bands. The total energy of the photons emitted in an E2 decay from a state with angular momentum ~I is

I(I + 1) (I 2)(I 1)~2 2(2I 1)~2 ∆E(I) = − − − = − . (4.13) 2I 2I

The peak spacing observed in a decay experiment, or “double diﬀerence”, is thus

2(2I 1) 2[2(I 2) 1]~2 4~2 δE ∆E(I) ∆E(I 2) = − − − − = , (4.14) ≡ − − 2I I a constant. In addition to an exceptionally large quadrupole moment, then, a second experimental signature of SD nuclei is their nearly uniform “picket fence” rotational decay spectrum (see Fig. 4.5).

4.2 Decay Process

We turn our attention now to the decay processes of SD nuclei, a subject of much experimental [118, 119, 126, 133, 134, 138, 140, 143, 146, 147, 149–151, 153–155, 157, 159, 160, 172] and theoretical interest [106, 111, 169, 170, 173–189]. While the process of decay is intriguing in itself, a thorough understanding of this phenomenon is also regarded as a promising avenue to explore aspects of microscopic nuclear structure [111].

4.2.1 Experiments

Superdeformed bands are typically populated in the laboratory by heavy-ion reactions. A collision, for example of 48Ca and 108Pd [114], is used to produce highly excited, high- angular-momentum nuclei, which then decay to lower energies by shedding photons and nucleons. If little angular momentum is lost in the decay processes, the metastable state so resulting is likely to be superdeformed, if possible, since at high angular momenta SD states are yrast. From there, the nuclei decay down the SD rotational band, losing two units of angular momentum by E2 decay at each step [171]. These decay events are readily observed, and, 61

Figure 4.5: Example of the “picket fence” decay spectrum of SD bands, from Ref. [170]’s study of 152Dy. Angular momentum values I are noted in units of ~. Also noteworthy is the extremely sharp loss of strength from the band, another hallmark of SD decay. with proper data analysis, the unique “picket fence” decay spectrum can be extracted. The strength of this signal is proportional to the number of nuclei in the experimental ensemble which have remained in the SD band. Nuclei may exit the band through decays into other SD bands, or into ND states which exist at the same angular momentum. In practice, SD nuclei generally travel quite easily down a rotational band for some time. These bands are observed to retain their strength for many states, with negligible losses to others. Quite suddenly, however, the band decays (see Fig. 4.6a): sometime after it has ceased to be yrast, it loses almost all of its strength over just one or two states. After a series of statistical decays through unrelated states, the nuclei then continue via decays dominated by E1 (electric dipole) transitions down the ND rotational band [171]. This whole process is outlined by Fig. 4.7 Since decay-out occurs while the nucleus is still well above the SD bandhead, it is 62

100 (a) (b)

190Hg 192Hg 50 194 Hg 192Pb 194Pb 196

in−band SD intensity (%) Pb

0 6 10 14 18 22 spin of initial level Figure 4.6: (a) Decay profiles of several SD bands in the A 190 mass region. Note how ≈ suddenly each decay occurs. (b) The profiles of (a), but shifted in angular momentum so that the leftmost point, the last point in which the SD band is experimentally observed to retain any strength, are aligned. The profiles exhibit a universal behavior. Both graphs are from Ref. [189].

a subject of great interest. The SD state still has very high energy when it leaves the rotational band, so the density of states to which it decays is nearly constant. In short, there seems to be no reasonable explanation, a priori, why SD rotational bands lose their strength so quickly.

The decay-out process becomes even more shocking when the results from different SD bands within the same mass region are compared. When corrected for the different angular momenta at which decay occurs, many decay profiles overlap almost perfectly (see Fig. 4.6b) [189]. This universal behavior suggests a strikingly elegant and simple physical processes lies behind the decay of SD bands. 63 Energy

Angular Momentum

Figure 4.7: Schematic diagram of the SD decay process. The nucleus enters an SD band at a very high angular momentum, for which the state is yrast. It then decays via E2 transitions beyond the angular momentum for which ND states lie lower in energy. Suddenly, over the course of only a few states, the SD band loses its strength, via statistical transitions, to the lower-lying ND band. Finally, the nucleus continues to decay down the ND rotational band via E1 transitions.

4.2.2 Double-well paradigm

Without exception, theoretical efforts to describe the decay-out of superdeformed nuclei model the transition via a potential function of both the deformation ε and angular momentum quantum number I [189]. It was noted early in the theoretical study of the decay processes [175, 176] that a double-well potential models the phenomenon much more accurately than a traditional fission-style decay, in which the state of a single well decays through a barrier into a continuum of states (see Fig. 4.8). The most appropriate picture for SD decay is thus found to be two sets of discrete states, separated by a potential barrier, and each state broadened by a different coupling to the electromagnetic 64

(a) (b) Potential Potential

ND SD SD

ND Deformation Deformation Figure 4.8: Schematic examples of potentials historically used to model SD decay, as a function of deformation. The potentials change with angular momentum, as well. (a) Double-well potential, which accurately reflects the physical process [175, 176]. The states of each well, SD or ND, are broadened according to their coupling with the electromagnetic field. (b) Potentials appropriate to describe nuclear fission, but not the SD decay process. The ND states are represented by a continuum. In the case of the double-humped (dashed) barrier, the bound states of the secondary well may play a perturbative role, with their importance depending on the energy of the decaying SD state, but they are strongly broadened by the ND continuum [190]. (b) can be read as the result of taking a continuum (infinite ND well depth) approximation of (a).

field. The continuum of the electromagnetic spectrum allows each state to irreversibly decay to lower-energy states. In addition to explaining the specifics of the decay process, a central goal in modeling SD decay is the extraction of the shape of the barrier. Of special interest is the behavior of the barrier as a function of angular momentum. A phenomenological understanding of how, or indeed if, the barrier height changes with I would yield significant insight into the underlying, microscopic roots of collective nuclear phenomena [178]. The double-well picture, then, as contrasted with a single well plus a continuum of ND states, has been shown to be essential to understanding how SD nuclei decay [175, 176]. Unfortunately, this insight is often trivialized due to an inclination in the community to draw analogies to incoherent fission processes. Many attempts have been made to consider SD decay as an analogue to a single- or double-humped fission barrier, failing to consider the important role coherence effects, such as the Rabi oscillations, play in the decay process. This framework has motivated continuum and many-level 65

Γ εS ε S N Γ N V SD ND

Figure 4.9: Schematic diagram of SD decay at a particular angular momentum, under the two-state approximation. V is the tunneling matrix element connecting the two states. Electromagnetic decay to lower-lying states gives each state its finite width, ΓN or ΓS. εN and εS are the energies of the two levels in the absence of V . From Ref. [187]. approximations for the ND well, which are algebraically identical to the single-well decay problem already known to be insufficient. Furthermore, such approximations are not justified experimentally. One strength of the Dyson’s equation approach is that it can include both coherent and incoherent effects on the same footing, and, in fact, determines automatically the regimes in which either, or both, are important. Thus, we are motivated to attempt a Green function solution to the two-well decay of SD nuclei. Once the general solution is reached, it will become clear that, in cases of experimental interest, the coherent effects play a significant role.

4.3 Two-State Model

The two-state model of superdeformed nuclear decay [106] was first put forward by Stafford and Barrett in 1999. The basic assumption is that only one state of the ND well, the nearest in energy to the SD state, plays an important role in the decay process. Other states are neglected. We are thus left with the system shown in Fig. 4.9 This assumption is similar to the two-level approximation made in Sec. 3.2 of this work. If the tunneling matrix elements which connect the SD state to all other ND are much less than the energy difference of those states from the SD level, the approximation is good. In this case, however, we deal with a collective mode of the nucleus, and so 66 interactions and shell closures are already included.

4.3.1 Two-state Hamiltonian

We again divide the full Hamiltonian into three separate parts:

H = Hnuc + HEM + Hc, (4.15) where the ﬁrst term gives the dynamics of the bare nucleus without considering the electromagnetic ﬁeld [106]:

Hnuc = εScS† cS + εN cN† cN + V cS† cN + cN† cS . (4.16) Here, V is the tunneling matrix element which takes the nucleus through the barrier from the SD state to the single ND one, εS and εN are the energies of the nucleus in the isolated

(i.e., in the absence of V ) SD and ND wells, respectively, and cS and cN annhilate the nucleus from the appropriate state. As in Chapter 3, we take V to be real and positive without loss of generality.

In this system, HEM and Hc, which are the Hamiltonian of the environmental electromagnetic ﬁeld and its coupling with the nucleus, respectively, are the sources of decoherence. The electromagnetic ﬁeld is clearly a continuum of photonic states, similar to the spin-boson Hamiltonian (3.5). We may thus denote it by

(n) (n) (n) HEM = εν aν †aν , (4.17) v n=XS,N X (n) th (n) where εν is the energy of the ν oscillator state, and aν annihilates a photon in that state. The superscript n keeps track of the state of the nucleus: even after the nucleus has left the two-well system by electromagnetic decay, it has a deﬁnite deformation, which aﬀects the orthogonal modes of the dressed elctromagnetic vacuum. The coupling term is similarly

(S0) (S0) (N 0) (N 0) Hc = V c† cSa † + H.c. + V c† cN a † + H.c. , (4.18) ν S0 ν ν N 0 ν ν X h i XνN 0 h i where S0 denotes the lower-angular-momentum SD state to which decay can occur, and similarly N 0 runs over all lower-angular-momentum ND states to which the nucleus can (n) decay. Vν are the matrix elements characterizing each process. 67

4.3.2 Energy broadenings

Analogous to the approach of Chapter 3, we shall use Dyson’s equation to exactly include the eﬀects of HEM and Hc. Section 3.3.2 demonstrated how broadening functions are linked to the Hamiltonian, and there is little to be gained by reiteration in this case. Instead, we note that SD decay is a nearly perfect idealization of the broad-band approximation introduced in that section: the inﬁnite, degenerate nature of photonic excitations provides almost exactly constant functions Γn(E) in this case.

In fact, the decay rates of the SD and ND levels are ΓS/~ and ΓN /~, respectively, and, as such, these quantities can be extracted from experiment. ΓS, in particular, is well determined by current results. When they are measured, it can be obtained directly from the quadrupole moments for the E2 transition down the SD band. In other cases, methods such as recoil-distance and Doppler-shift attenuation have been used to extract the lifetime of SD states from SD rotational band decay spectra [172]. The experimental uncertainty in ΓS is typically of the order of 10% [187].

ΓN is less well known, usually only to within a factor of 2 or more [187]. The standard method of its extraction is to assume a Fermi-gas, cranking model density of levels [191]

√π 5/4 2√aU ρ(U) = U − e , (4.19) 48a1/4 where a is a parameter to be ﬁt to experiment, and U is the excitation energy above yrast. The use of a density of levels of this form is the main source of the high uncertainty in estimates of ΓN , since real nuclear spectra are often much more complicated. Parenthetically, we note that 1/ρ(U), evaluated at the energy of the ND state, is the average level spacing in the isolated ND well, DN , which enters as a parameter into many statistical eﬀorts to model SD decay. Like ΓN , this quantity should be treated as a theoretical estimate, albeit based on experimental data. The choice of density of levels gives a strong model-dependence to values of both ΓN and DN . Statistical E1 decay, which dominates decay from the ND well, is characterized by the well known giant dipole resonance (GDR) [38]. The mean-square transition matrix element for absorption of the GDR is

~2 2 2 2 e 2 NZ EγΓGDR 1 β α = 2 , (4.20) h |M| i 2m πΓGDR A E2 E2 + E2Γ2 ρ (Uβ) γ − 0 γ GDR 68

where the width ΓGDR and centroid E0 are ﬁxed by nuclear data, Uβ(α) is the energy of the destination (initial) state β(α), and Eγ is the incident photon energy. Based on the results (4.19) and (4.20), Døssing and Vigezzi derived the width due to the inverse, decay process [192]: 4 e2 1 Γ NZ U 5/2 Γ (U) 4! GDR , (4.21) E1 ≈ 3π ~c mc2 E4 A a 0 which is then evaluated at the estimated excitation energy of the ND state. Finally, the decay width of an ND state is estimated by the simple association

Γ Γ . (4.22) N ≈ E1

U is generally renormalized to U 2∆ , where 2∆ , known as the backshift parameter, − p p takes the pairing interaction into account. Recently, however, detailed analyses [153, 172] have suggested that the magnitude of ∆p should be much smaller than usually used, or neglected entirely. This, then, is an additional factor which adds uncertainty to current knowledge of ΓN .

Table 4.1 gives experimental results for ΓS, as well as the Fermi-gas estimates of

DN and ΓN . For a particular decay, only a single number is generally estimated in the literature for either characteristics of the ND well, ΓN or DN . Since the two-level model considers only a single level, well defined in energy, in the ND well, its use is especially appropriate. 69 , to N are D tum all ond matrix Dy(26) and 152 N momen of ] ] ] ] ] Γ corresp inputs, of 183 183 193 193 193 tunneling ] ] ] ] ] ] ] ] ] ] ] ] , ] ] ] , , ] , , cases ] ] ] ] angular 189 183 183 189 189 189 183 172 183 183 172 183 148 172 172 183 148 183 172 183 183 the the column , , , , , , , , , , , , , , , , , , , , , [126 [126 [126 [126 Refs. differing of the in 146 146 149 149 149 [154 [148 [148 [154 [148 [148 [172 [157 [148 [172 [157 [172 [157 [157 [172 [157 or , , , , , , t-most with determination erage [138 [138 [140 [140 [140 v a to righ generated N directly ble e states, the v /D i 0.16 0.071 0.62 0.015 0.17 0.030 0.020 0.017 0.051 0.032 0.064 0.083 0.025 0.031 0.21 0.080 0.13 0.12 0.027 0.71 0.055 0.011 0.051 0.15 0.021 ha related in V > ≥ > > > < > SD h < either ensem ts en giv ying 49 0 52 34 97 8 7 1 i 0.60 2.1 3.0 2.7 0. 5. 0. 0. 0. 8. 8. 6. (4.39), V 35. 35. 15. 72. 39. 77. 34. erimen 29. h > > ≥ < (eV) 120. 170. 250. deca > > < 1000. 1100. estimations > Eq. exp (4.61) orthogonal for y b or and Eq. References † ↓ ts del, y † Γ b dels 0.071 0.072 1.6 0.026 0.021 0.71 0.37 0.060 0.0077 0.44 0.049 0.0087 0.0051 1.1 0.088 0.032 0.0013 0.0056 0.10 1.9 0.0053 0.067 0.0050 mo 11. (meV) del. > > ≥ > > < > < 140. Gaussian en mo el mo t giv 3 5 9 2 2 determined The el 7. as o-lev N 16. 26. 19. 26. 89. 19. 14. , 194. 220. 344. 493. 273. 135. 244. 333. 236. w measuremen ↓ D t (eV) then 2,200. 1,400. 1,258. 1,272. 1,410. 1,681. 1,362. o-lev Γ differen as 21,700. w is 4.4.2. t the ↓ ell and Γ w , of the N Sec. N Where 4.4.2. Γ as 0.50 0.445 0.613 0.733 0.201 0.200 0.188 4.8 1.345 4.0 4.5 0.08 0.470 0.169 0.65 0.405 0.192 4.1 1.487 6.4 0.476 of D , in (meV) 17. 17. 21. 20. , S width Sec. Γ N results Γ in , S 014 014 0 128 050 266 132 048 097 108 046 086 230 110 045 045 016 003 003 0 487 039 033 048 125 theses. S and Γ picture 0. 0. 7. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. Γ and (meV) 10. N explained F tunneling paren explained as of from 38 35 40 81 26 92 02 34 88 42 40 97 40 10 16 10 10 N ossible , in as 0.75 0.91 0.96 0.01 0.91 0.95 0.93 0.01 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. inputs F p The ts , > > > < > ≥ > < en tal giv 4.3.2. fullest I determined statistically erimen statistically er Pb(8) Pb(8) Pb(6) Pb(8) Pb(6) Sec. the Hg(12) Hg(10) Pb(16) Pb(14) Pb(12) Pb(10) Hg(12) Hg(10) Hg(12) Hg(10) Hg(12) Hg(10) Hg(15) Hg(13) Hg(11) Pb(10) Pb(12) Pb(10) Dy(28) Dy(26) b is measuremen ucleus(I) e , 192 194 194 194 194 in 192 192 192 192 192 192 194 194 194 194 194 194 194 194 194 194 194 194 152 152 n i Exp um tal giv V n h Pb(10), to t, 4.1: Calculated † tum 192 erimen wn discussed able T elemen quan sho exp and as 70

4.3.3 Green function treatment

Our approach to Dyson’s Equation diﬀers slightly from the case of quantum dots (Sec.

3.3.2), for the sake clarity. Since ΓS and ΓN are a priori constants, it is quite natural to think of the isolated SD and ND wells as the components of H0, each with its own bath of bosonic states [188]. The energy levels εS and εN are broadened, as is usual with states of ﬁnite lifetime, into Lorentzian Breit-Wigner resonances. In the Green function formalism, this is equivalent to a shifting of the poles below the real axis by an amount

Γn 2 : 1 ΓS 0 E εS+i 2 G0(E) = − . (4.23)  1  0 Γ E ε +i N  − N 2    Note that, in this approach, the two wells are completely unmixed by H0. Each state in Eq. (4.23) propagates and decays independently, without interaction with the other. Of the Hamiltonian (4.15), the only physics left unincluded in Eq. (4.23) is the third term of Eq. (4.16), that which allows tunneling through the barrier. It can be described by the simple retarded self-energy

0 V Vˆ = . (4.24) V 0    Dyson’s equation (2.22) then allows us to include its eﬀects to all orders [106]:

1 − 1 E ε + i ΓS V 1 ˆ − S 2 G(E) = G0− V = − − −  V E ε + i ΓN  h i − − N 2  ΓN 1 E εN + i V = − 2 , (4.25) ΓS ΓN 2 ΓS E ε + i E ε + i V  V E εS + i  − S 2 − N 2 − − 2   a result which should be compared with Eq. (3.18).

4.3.4 Branching ratios

Rather than the probabilities Pn(t), which are as expected from Eqs. (3.19) and (3.20), the quantities measured by experimental studies are the fractions of nuclei which decay down either band, called the branching ratios. Table 4.1 gives the ND branching ratios 71

FN measuerd in decay experiments. The probability of a decay from a speciﬁc well n in a time dt is Γ d decay(t) = n P (t)dt. (4.26) Pn ~ n Thus, the branching ratios are [106]

Γ F = ∞ d decay(t) = n ∞ dtP (t). (4.27) n Pn ~ n Z0 Z0

In the open system, it is the identity F 1 that replaces P (t) as a reﬂection of n ≡ n n n particle conservation. X X

Since the probabilities Pn(t) are not readily accessible by experiment, clarity is enhanced by eliminating them from our formulas. This can be readily accomplished via Parseval’s Theorem, dE ∞ dt F (t) 2 ∞ F˜(E) 2, (4.28) | | ≡ 2π | | Z−∞ Z−∞ which relates the continuous function F (t) to its Fourier transform F˜(E). The branching ratios are thus shown to be [106]

Γ Γ F = n ∞ dt G (t) 2 = n ∞ dE G (E) 2, (4.29) n ~ | nS | 2π~ | nS | Z−∞ Z−∞ where we have made use of the experimental fact that the nucleus is initially localized in the SD well. Either Equation (4.27) or (4.29) may be used to compute the branching ratios of the two-state model. The integral can be done analytically by the usual contour approach. The results are [106] 2 (1 + ΓN /ΓS) V FN = (4.30) 2 2 2 ∆ + Γ (1 + 4V /ΓN ΓS) for the ND branching ratio, and

2 2 2 2 ∆ + Γ 1 + 4V /ΓN ΓS (1 + ΓN /ΓS) V FS = 1 FN = − (4.31) − 2 2 2 ∆ + Γ (1 + 4V /ΓN ΓS) for the fraction of nuclei which continue down the SD band. In analogy with Chapter 3, ∆ is here deﬁned as ε ε , and Γ ΓN +ΓS . N − S ≡ 2 72

4.4 Tunneling Width

We now introduce the concept of the tunneling width, Γ↓, which gives the net rate at which nuclei tunnel between the two wells. Fermi’s Golden Rule can be used to calculate this quantity: ∞ 2 Γ↓ = 2π dEρS(E)V ρN (E), (4.32) Z−∞ where Γ /2π ρ (E) = n (4.33) n (E ε )2 + (Γ /2)2 − n n is the density of states in well n. In this case, ρn are simply the usual Breit-Wigner resonances associated with width-broadened discrete energy levels. The result for Γ↓ is [106] 2ΓV 2 Γ↓ = 2 . (4.34) ∆2 + Γ Despite its derivation from Fermi’s Golden Rule, Equation (4.34) has meaning in an exact sense, as we shall shortly see. For this reason, we can simply take it as the deﬁnition of

Γ↓.

4.4.1 Relation between branching ratios and tunneling width

With the deﬁnition (4.34) of Γ↓, Equation (4.30) can be rewritten into a more transparent form [187]: ΓN Γ↓/ ΓN + Γ↓ FN = . (4.35) ΓS + ΓN Γ↓/ (ΓN +Γ↓) Likewise, Equation (4.31) becomes

ΓS FS = . (4.36) ΓS + ΓN Γ↓/ (ΓN + Γ↓)

Equations (4.35) and (4.36) have a clear physical interpretation. They are precisely the results we expect to ﬁnd in the case of a two-step decay process: in order to leave the SD well, the nucleus must ﬁrst tunnel out of the SD well, and then decay down the ND band. The problem, therefore, is, in its essence, one of series-added rates. The rate to leave the SD band is thus

1 1 1 − Γ Γ Γ = + = N ↓ , (4.37) out Γ Γ Γ + Γ ↓ N N ↓ 73 which should be compared with the rate associated with remaining within the SD band,

ΓS. Together with the identity FS + FN = 1, the ratio of these two rates define the branching ratios: F Γ Γ Γ N = out = N ↓ Γ . (4.38) F Γ Γ + Γ S S S N ↓ Equations (4.35) and (4.36) follow directly. The reader may find this result surprising. The grouping of quantities defined by Eq.

(4.34) to be Γ↓ plays exactly the role of a tunneling rate in the two-stage analysis of the problem. Fermi’s Golden Rule, in general accurate only to second order in V , apparently provides an exact solution to this problem. The reason for this serendipitous success of Fermi’s Golden Rule is the simplicity of the two-level model. In the general problem of Ns states, the Green function consists of a denominator, in which Vˆ appears to order Ns, and a numerator, in which Vˆ can be at most order N 1. As we saw in Sec. 3.3.2, the role of the denominator of G(E) is solely to s − determine the complex Rabi frequency of the problem in the two-level case. In problems with Ns > 2, it similarly only aﬀects the problem in ways that can be absorbed into the poles of the Green function, and thus does not determine the relevance of Fermi’s Golden Rule. In this sense, the character of the time dynamics is governed by the numerator. In the two-level problem, therefore, the Golden Rule’s inclusion of perturbation theory to V 2 is suﬃcient for an exact result for Γ , although a derivation of the branching O ↓ ratiosthemselv es requires use of the information in the denominator of G(E).

4.4.2 Measurement of the tunneling width

The relation (4.35) between Γ↓ and FN can be inverted to ﬁnd [187]

FN ΓN ΓS Γ↓ = . (4.39) Γ F (Γ + Γ ) N − N N S

This is a central result, as it clearly demonstrates that Γ↓ is a measurable quantity, in the sense that a typical SD-decay experiment determines all of the parameters necessary for its knowledge. Table 4.1 gives values of Γ↓, as extracted from the experimental data.

Equation (4.39) demonstrates the measurable nature of Γ↓, but its form may trouble the reader. As an energy width, Γ↓ is inherently positive deﬁnite, and thus a negative 74 value is unphysical. We must thus infer the inequality [187]

ΓN FN < , (4.40) ΓS + ΓN or, equivalently, FN ΓN > ΓS . (4.41) FS Experimental results for which these two inequalities break down indicate either a departure form the validity of the two-level approximation, or that one or more of the three experimental inputs is poorly known. In fact, Table 4.1 includes two SD states, 152Dy(26) and 192Pb(10), for which the inequalities (4.40) and (4.41) are violated. As mentioned in Sec. 4.3.2, however, the standard error in ΓN is expected to be at least a factor of 2. In both cases, the only two currently known, this factor is more than suﬃcient to explain the negative result of Eq. (4.39). The conclusion that a third state plays an important role in the dynamics of these decays is therefore premature, for a major factor in the appearance of an anomalous sign is likely the use of Eq. (4.19) in place of the true, unknown density of states. In cases where the positivity of Eq. (4.39) is violated, then, it behooves us to remember that, like all experimentally-based results, the value quoted for ΓN is really the centroid of a probability distribution. For want of better information, we can assume a Gaussian shape of this distribution, and take the half-width to be the estimated error [188]:

2 Γ Γ0 N − N 1 √2Γ0 (Γ ) = e−„ N « , (4.42) 0 N 0 √ P ΓN 2π

0 where the ΓN is the quoted experimental value. The quoted experimental values of ΓS and FN could, of course, be treated similarly, but since their uncertainties are so much less than that of ΓN , we forego this process and treat them as known quantities. The probability distribution for Γ follows from (Γ ): ↓ P0 N 2 Γ Γ0 2 N N Γ /Γ −0 dΓN N ↓ −„ √2Γ « (Γ↓) = (Γ ) = e N . (4.43) P0 P0 N dΓ 0 √ ↓ ΓN 2π

Here, ΓN is given in terms of Γ↓ by Eq. (4.39).

A physical value for Γ↓ can be extracted by explicitly demanding the validity of the two-level model. Requiring that inequality (4.41) be satisﬁed imposes a similar restriction 75

on Γ↓: FN Γ↓ > ΓS , (4.44) FS due to the dual relationship between the two quantities. This result can be “forced” onto the probability density function (4.43) by truncating it at the minimum value,

0, Γ Γ FN ↓ S FS 2 ≤ (Γ↓) = Γ Γ0 , (4.45)  2 N − N P Γ √2Γ0 F N −„ N « N  e Γ↓ > ΓS F A Γ↓ S where is the new normalization constant, determined as usual by A ∞ dΓ↓ (Γ↓) = 1. (4.46) P Z−∞

A typical value for Γ↓, as extracted from the experimental evidence, then, is the median of the distribution (4.45). That result is given, in the cases of the two anomalous decays, in Table 4.1. The preceding analysis is not intended for the ideal case, in which the three experimentally determined quantities, ΓS, ΓN , and FN , are well known. In such cases, violations of the inequalities (4.40) and (4.41), should they occur, would indicate that the original two-level approximation is invalid. With the current state of aﬀairs, however, the method outlined above provides a statistical approach to continuing our analysis in cases, 152 192 such as Dy(26) and Pb(10), where our current poor understanding of ΓN impedes conclusions. It is to be hoped that this approach will eventually be outmoded by an improved understanding of the spectrum to which the ND level decays, or a more direct measurement of ΓN .

4.4.3 Limits of the tunneling width

One can view the two-level approximation as one of two possible limits of the true many- level ND spectrum. In the general case, an approximate tunneling width into the ND well Γ↓ may still be calculated via Eq. (4.32), except that we should take ρN (E) as the many-peaked density of states of the ND well, and V 2 is now a function of energy. For an average peak spacing in the ND well DN , we observe that the two-level model is equivalent to the limit V D , for which we recover the result (4.34) for Γ . This N ↓ reasoning is similar to that of Sec. 3.2.1. 76

In the opposite limit, V D , the result for Γ is N ↓ V 2 lim Γ↓ = 2π h i, (4.47) V/D N →∞ DN where represents an energy average. This is the limit in which an inﬁnite number h· · · i of ND levels participate signiﬁcantly in the decay, often called the continuum limit. It is worth noting (see Table 4.1), that, experimentally, this is never an appropriate limit, as

D Γ (4.48) N N always holds, indicating that the correct picture is of a discrete, well separated spectrum of levels in the ND well. Furthermore, many-level models consistently ﬁnd [189]

D V 2 , (4.49) N h i p which is the central condition for the validity of a two- or few-level model.

4.5 Statistical Theory of Tunneling

While it was shown in the previous section that Γ↓ is an experimentally ﬁxed quantity, a method of determining V , and thus the shape of the potential barrier, is not a priori clear. Solving Eq. (4.34) for V requires knowledge of ∆, but the experimental spectrum of the ND rotational states is not generally known. Even if it were, only an estimate of ∆ would be possible, since the quantity is a theoretical construct, representing as it does the detuning between levels in the isolated wells, not the eigenenergies of the full Hamiltonian that are available from experiment. Our answer to this conundrum will be to construct an estimate based on theoretical considerations. We make use of a common tool of random-matrix theory, the Gaussian Orthogonal Ensemble (GOE). This statistical approach allows us to ﬁnd a probability distribution for ∆, from which, in turn, follows a probability distribution for V . With the current types of SD-decay experiments, this is the most successful we can hope to be at determining the tunneling matrix element.

4.5.1 Gaussian orthogonal ensemble

Developed to deal with the problem of an unknown nuclear Hamiltonian [194–197], random matrix theory has in time found use in a host of other disciplines. Its wide utility 77

[45, 198] is due to an unorthodox, yet often successful, approach to a common problem: how can one best determine the eigenvalues of a Hamiltonian about which one knows little, aside from its space-time symmetries? The answer provided by random-matrix theory is simply to ignore any other information, and focus on the statistical properties of an ensemble of Hamiltonians with the appropriate symmetries. We posit a Hamiltonian HND, the potential VND(ε) of which is simply the isolated ND well of the SD-decay problem. We are not really sure what shape the well has, or even, perhaps, if it has a uniquely determined shape. All we need know is that HND possesses both time-reversal symmetry and either rotational symmetry or integral angular momentum [45]. In other words, the spectrum corresponding to HND’s real eigenvalues is solely governed by level repulsion. The ensemble of such matrices is known as the Gaussian Orthogonal Ensemble. A probability density function for its energy levels is given by the Wigner “surmise” [199]:

π πs2/4 (s) = se− . (4.50) P 2

Here s is the spacing between two adjacent levels, in units of the average spacing DN . The average level spacing, and the space-time symmetries of the problem, are the only inputs into a generic random-matrix theory problem. We shall use Eq. (4.50) to build a statistical theory of the tunneling matrix element V .

4.5.2 Implications for tunneling

The ﬁrst step in building a statistical theory of V is to consider the behavior of ∆ under the assumption of GOE-distributed states in the ND well [187]. First, consider only the two ND levels which lie just above and below the SD state in energy, one of which must be the nearest-energy ND level of interest to the two-level model. The two ND states are separated by a spacing sDN , whose probability distribution is given by the

Wigner surmise (4.50). Since the ND and SD states are isolated from each other, εS is uncorrelated with the energy of either state, and the probability distribution for the detuning of the SD level with either of the two ND levels must be rectangular: 1 s ∆ (∆) = Θ | | . (4.51) Ps sD 2 − D N N The Heaviside step function chooses the nearer of the two ND states. 78

Figure 4.10: Probability distributions for the two ND levels on either side of the decaying SD level within the GOE, given by Eqs. (4.52) and (4.64). The average detuning of the nearest level is ∆ ∆ = D /4, and the average detuning of the next-nearest level h| |i ≡ h| 1|i N is ∆ = 3D /4. From Ref. [187] h| 2|i N

The Total Probability Theorem now yields a distribution for ∆ [187]:

∞ π ∆ (∆) = ds (∆) (s) = erfc √π | | , (4.52) P Ps P 2D D Z0 N N where erfc (x) is the complementary error function of x. This distribution is illustrated in Fig. 4.10. The average detuning is D ∆ = ∞ d∆ ∆ (∆) = N . (4.53) h| |i | |P 4 Z−∞ The typical deviation of ∆ from its average, for cases of real decays, is characterized by | | 2 ∞ DN 1 1 σ ∆ = 2 d∆ ∆ (∆) = DN .8353 ∆ . (4.54) | | s − 4 P 3π − 16 ≈ h| |i Z0 r The fact σ ∆ . ∆ reassures us that ∆ is a meaningful measure of the detunings | | h| |i h| |i typical to experiment. At this point, we note that the branching ratios (4.30) and (4.31) depend on only four parameters, ∆, V , ΓS, and ΓS. The data in Table 4.1 and the results (4.53) and (4.54) indicate that, among these four values, a separation of scales exists. In general,

Γ , Γ V, ∆, (4.55) S N 79 where we assume, as implied by the fact σ ∆ , that ∆ will not stray too far from ∆ ∼ h i ∆ . Thus, it is to be expected that only two parameters signiﬁcantly aﬀect the decay-out h i process, ΓS/ΓN and V/∆. Other combinations of the four parameters are expected to have little importance in practice. The pieces are now in place to construct a probability density function for V . In general, such a function is given by d∆ (V ) = 2 (∆) , (4.56) P P dV

d∆ where 2 dV is the Jacobian associated with the change of variables. Solving the deﬁnition of Γ↓ (4.34) for ∆ yields | | 2Γ Γ Γ ∆ = V 2 ↓ . (4.57) | | Γ − 2 s ↓ The requirement that ∆ be real yields a lower bound on V : | | 1 V Vmin = Γ↓Γ. (4.58) ≥ r2 Taking the derivative of Eq. (4.57) gives

d∆ 2ΓΓ = ↓ . (4.59) dV v Γ↓Γ u1 2V 2 u − t Within the assumption of ND states ob eying the GOE, then, V is drawn from the probability distribution [187]

0, V < Vmin (V ) = , (4.60) 2ΓV π ∆ P  erfc √π | | , V V Γ ∆ D DN min  ↓| | N ≥ where ∆ is evaluated according to Eq. (4.57). The average value of V is thus | | 2 Γ D Γ V = ∞ dV V (V ) = ↓ N + . (4.61) h i P s2Γ 4 O DN Z−∞ " !# The width of the distribution (4.60) is characterized by

∞ 2 √π 2 3 σ = dV (V V ) (V ) = D2 1 Γ↓ + Γ + Γ Γ V N 3π3/2 32Γ 4 ↓ s Vmin − h i P − O Z r p . V . h i (4.62) 80

Equation (4.61) thus provides a typical value for V in the two-level model. Equation (4.60) is a central result, since it represents the maximal information we can have about V without prior knowledge of the shape of the potential. Earlier attempts to consider a statistical theory of SD decay [175, 176, 180, 184, 185] focused on average values of Γ↓ and FN . The ﬂuctuations in FN are much larger than the mean, indicating that the average value has little meaning for comparison to experiment. Given the experimentally measured branching ratio FN , on the other hand, the approach outlined here allows the essential parameters ∆ and V to be determined, in the sense of “sharp” probability distributions whose typical values are comparable to their means.

4.6 Adding More Levels

Thus far we have treated only a two-level model of the SD decay process, and it is reasonable to ask what eﬀect the inclusion of further ND levels has. To that end, we include discussion of the results of two investigations into similar models with more ND levels. Throughout this work, an emphasis has been the utility of adopting the minimally complex approach, while still describing realistic physics. In this light, we present investigations into three-level and inﬁnite-level models as checks on the accuracy and limits of applicability for the two-level model.

4.6.1 Three-state model

As a first step toward determining the applicability of the two-level model, Cardamone, Stafford, and Barrett considered the addition of a third level, the energy level of the ND well next-nearest in energy to the SD level [187]. To analyze the situation, they made use of the GOE to determine likely placement of the second level, ∆ ε ε . In this 2 ≡ N2 − S section, ∆ ∆ ε ε , to make the notation uniform for clarity. Encouraged by ≡ 1 ≡ N1 − S the successes of a statistical approach previously, we again turn to the GOE to determine a probability distribution for ∆ in analogy to (∆ ) given by Eq. (4.52). 2 P 1 There are two cases of physical interest: the ND states may lie on the same side, or on opposite sides, of the SD level. Due to level repulsion, the first case is the less likely. In addition, level repulsion causes the the next-nearest ND level to be further from the SD level, on average, than it would be in the second case, that of bracketing ND levels, 81 and thus the effect of the third level on tunneling dynamics is appreciably suppressed in the same-side case relative to the bracketing case. While the Wigner surmise (4.50) does allow us to give each case its proper weight when constructing (∆ ), we would be including two different “regimes” of behavior in P 2 the same probability distribution. No real decay is represented by average values of a statistical distribution. Rather, we hope to create a distribution whose typical values reflect experiment. For this reason, it is undesirable to include both regimes in the same distribution; we opt instead to consider mainly the bracketing case, taking the role of “devil’s advocate” for the two-level model, and content ourselves with the knowledge that those rare decays for which the two nearest ND states fall on the same side of the SD level conform even better to the predictions of the two-level model than those cases studied here. Based on the assumption of bracketing ND levels, then, we can write down a conditional probability distribution for ∆2: 1 ∆ s ∆ (∆ ) = Θ | 2| Θ s | 2| . (4.63) Ps 2 sD D − 2 − D N N N As in Eq. (4.51), the distribution is rectangular, since all of the ND levels’ placements are uncorrelated with each other. The two step functions specify that the level be the next-nearest neighbor. Proceeding, we arrive at the total probability distribution [187]: π √π (∆ ) = ∞ ds (∆ ) (s) = erf √π ∆ erf ∆ , (4.64) P 2 Ps 2 P 2D | 2| − 2 | 2| Z0 N where erf(x) is the error function of x. This distribution is shown in Fig. 4.10. Its average detuning is 3D ∆ = ∞ d∆ ∆ (∆ ) = N , (4.65) h| 2|i 2| 2|P 2 4 Z−∞ as one might expect from Eq. (4.53). Now that we have some idea where the next-nearest ND level lies, we move to the Green function treatment of the problem. The Hamiltonian of the three-level system can again be divided into three parts, as in Eq. (4.15). The first part, corresponding to the closed, nuclear system, is

Hnuc = εScS† cS + εN1cN† 1cN1 + εN2cN† 2cN2 + V1 cS† cN1 + cN† 1cS + V2 cS† cN2 + cN† 2cS . (4.66) 82

Note that, since no observables depend on the phase differences of the states in the discrete system, we are free to choose both tunneling matrix elements V1 and V2 to be real and positive, without loss of generality. This is a consequence of the fact that there is no direct tunneling between the two ND states. Also without loss of generality, we assume for notational convenience that the state N1 is the ND state nearer in energy, and N2 is the further. The other parts of the Hamiltonian, which describe the environmental electromagnetic field and its coupling to the discrete two-well system, are essentially the same as Eq. (4.17) and (4.18), with the modification that the sum over discrete states must include both N1 and N2. Thus, the Green function of energy in the absence of inter-well tunneling is now 1 Γ 0 0 E ε +i S − S 2  1  G0(E) = 0 ΓN1 0 , (4.67) E εN1+i  − 2   0 0 1   ΓN2   E εN2+i 2   −  while the self-energy for tunneling is given by

0 V1 V2 ˆ V = V1 0 0  . (4.68) V 0 0   2    Once again including the eﬀects of Vˆ exactly to all orders via Dyson’s Equation (2.22), we compute the full inverse Green function of energy,

E ε + i ΓS V V − S 2 − 1 − 2 1 Γ G− (E) =  V E ε + i N1 0  , (4.69) − 1 − N1 2  V 0 E ε + i ΓN2   − 2 − N2 2    and hence the Green function itself:

1 ΓS ΓN1 ΓN2 2 ΓN2 2 ΓN1 − G(E)= E εS+i 2 E εN1+i 2 E εN2+i 2 +V1 E εN2+i 2 +V2 E εN1+i 2 h“ − ”“ − ”“ − ” “ − ” “ − ”i ×

ΓN1 ΓN2 ΓN2 ΓN1 E ε +i E ε +i V1 E ε +i V2 E ε +i „ − N1 2 «„ − N2 2 « „ − N2 2 « „ − N1 2 «

ΓN2 ΓS ΓN2 2 V1 E ε +i E ε +i E ε +i V V1V2 .  „ − N2 2 « „ − S 2 «„ − N2 2 «− 2 

ΓN1 ΓS ΓN1 2  V2 E ε 1+i V1V2 E ε +i E ε 1+i V   „ − N 2 « „ − S 2 «„ − N 2 «− 1    (4.70) 83

3 (a) ΓS = 10− ΓN

(b) ΓS = ΓN

Figure 4.11: Numerical results comparing the two- and three-state total ND branching ratios. Superscripts indicate the number of levels included in the model. Since they decay via the same statistical process, the widths of the ND states in the three-level 3 model are set equal, i.e. ΓN1 = ΓN2 = ΓN . In (a), ΓS/ΓN = 10− , while (b) shows the case Γ = Γ . In both, Γ/D (Γ + Γ )/(2D ) = 10 4. The levels are taken at S N N ≡ N S N − their mean detunings (4.53) and (4.65) within the GOE, under the assumption that they lie on opposite sides of the SD state, and the tunneling matrix elements are taken equal: V1 = V2 = V . From Ref. [187]

The Parseval’s Theorem result, Eq. (4.29), once again yields the branching ratios. When a nucleus decays through the ND well, a measurable event, in the form of photon emission, occurs, and so decays through each of the ND states add classically. The full ND branching ratio in the three-level model is thus

FN = FN1 + FN2. (4.71)

While the quantities GnS(E) are integrable analytically (e.g., by contour integration), it is more illuminating to simply plot the results. As noted in Sec. 4.5.2, only the combinations of parameters ΓS/ΓN and V/∆ are expected to be relevant to describing a particular decay. Figure 4.11 explores the parameter space numerically by plotting FN vs. 4V/ ∆ , for cases of both large and small Γ /Γ . h 1i S N One conclusion of the numerical results is that the two-level model is, in general, well justified. For the more usual case that ΓS/ΓN is small, very little effect is found from 84 the addition of a third level. The conclusion for this regime is thus that adding levels beyond the nearest neighbors is largely superfluous to understanding the dynamics of SD decay-out [187]. The situation for Γ /Γ 1 is noticeably different: there is a distinct maximum in the S N ∼ three-level result. It is a consequence of Eq. (4.30) that as the SD width approaches the ND width, much larger tunneling matrix elements V are required to cause decay out. In analogy to Eq. (3.22), then, the separate Rabi oscillations involving N1 and N2 are faster. Interference between paths involving N1 and N2 is thus of greater significance, and the possibility for appreciable constructive interference exists. As V becomes very large, the two phases are essentially randomized with respect to each other, and interference effects become less important again. Obviously, this sort of physics is completely neglected in a two-level approach, and so some departure from that model’s results are to be expected. All things considered, however, the agreement between three- and two-level models is seen to remain quite good. A further important understanding that can be drawn from Fig. 4.11 is the physics behind the puzzling near-universality of SD decays in the A 190 region (Fig. 4.6). Table ≈ 4.1 includes a calculation of V /D , and the two-level model shares in the common h i N prediction that V increases with decreasing I. Comparing the values of V /D with the h i N graphs of Fig. 4.11, the universal character of many decays is shown to be a consequence of the steep character of FN vs. V in the relevant domain. As V increases, FN increases so quickly that the nuclei must decay over only a few states. After only one or two SD states of appreciable decay, at the usual rate of increasing V , the few-level models predict the branching ratios have become so high that essentially all strength is gone from the SD band.

4.6.2 Inﬁnite-band approximation

Dzyublik and Utyuzh further investigated the applicability of the two-level approximation by examining the limit of many ND levels [186]. They considered an evenly spaced ND band, with each ND state coupling to the SD level with the same tunneling matrix element, V 2 . Considering, as we have done, the tunneling as a perturbation, they h i noted that,p since the ND levels are uncoupled, each contributes separately to the self- 85 energy due to tunneling: 2 Vν VˆSS = | | , (4.72) ΓNν ν E εν + i 2 X − in analogy to Eq. (2.34). Here, Vν gives the coupling between the SD level and the ND state labeled by ν. After several reasonable approximations, the sum in Eq. (4.72) can be computed analytically [186]. An equally-spaced ND band is assumed, so that

εν = νDN + ∆, (4.73) where ν runs over a set of consecutive integers. Further, the ground state of the ND well lies far below the SD level, so that the sum can be approximated by a sum on all integers. Γ is set equal for each ND state. Finally, V 2 is approximated with its energy average, Nν | | V 2 . These approximations yield the result: h i

2 2 ΓN ∞ V π V E ∆ + i 2 VˆSS h i = h i cot π − . (4.74) ≈ ΓN D D ν= E νDN ∆ + i 2 N N ! X−∞ − − Dyson’s equation (2.23) includes Vˆ to all orders [186]:

1 − 1 GSS(E) = [G0(E)] VˆSS = , (4.75) SS ΓN − ΓS Γ E ∆+i 2 E εS + i ↓ cot π − n o − 2 − 2 DN where G0(E) is simply the Green function of a single width-broadened level given by Eq. (2.38). The branching ratios can be computed from Eq. (4.29).

Results comparing FN in the inﬁnite-band approximation and two-level approximation are shown in Fig. 4.12. As we have seen, the approximations of the inﬁnite-band model tend to overestimate FN , while that of the two-level model tends to underestimate it. The two cases can thus be viewed as approximate upper and lower bounds on the true branching ratio. The proximity of the two curves, even in the unlikely, worst-case scenario shown in Fig. 4.12b, speaks to the determinative importance of the nearest-neighbor state in the decay-out process. Even when all of the states neglected by the two-level model are included, to all orders in perturbation theory, their contribution to the main experimentally observed quantity is slight. 86

1.0 1.0

FN FN

0.5 0.5

(a) (b) 2 0 0.2 0.4 √ V /∆ 2 h i 0 0.2 0.4 √ V /∆ h i Figure 4.12: Comparison of results of the inﬁnite-level model (solid curves) of Ref. [186] and the two-level model (dotted curves). (a) shows the result when ∆ is taken at its typical value in the GOE, ∆ = DN , while (b) shows the result for ∆ = DN , its maximum h i 4 2 value consistent with the model of Ref. [186]. In both graphs, adapted from Ref. [186], DN = 100eV, ΓS = 0.1meV, and ΓN = 10mev.

4.7 Summary

As this chapter has shown, the Green function formalism can be employed to solve the double-well model of decay-out of SD bands in several diﬀerent approximations. The most important of these is the two-level model, in which only one level in each well participates meaningfully in the dynamics. This model yields a very clear picture of the decay process, as well as several important results. It explains why the decay proﬁle of A 190 nuclei ≈ are all so similar, and why the decay-out happens so quickly. Furthermore, the tunneling width Γ↓ takes on a rigorous, experimentally measurable meaning in the two-level model.

The rate Γ↓/~ is the net rate for tunneling through the barrier between SD and ND wells. Because of its simplicity, a statistical theory of tunneling between the two levels can be extracted from the two-level model. Once a statistical assumption for the probability distribution of level spacings in the ND well is made, for example via the Wigner surmise of random matrix theory, a probability distribution for the tunneling matrix element V can be calculated. This represents the maximal knowledge extractable for the current style of SD-decay experiments, and thus a great success. Knowledge of V can be used to 87 determine the shape of the barrier between SD and ND wells, which in turn may speak volumes about nuclear structure. The Green function formalism, considered as a controlled expansion in the number of levels, further predicts the accuracy of the two-level model. Including a third level is a numerically and analytically simple way to do this; the result is that the two-level approximation is excellent for small ΓS/ΓN , and remains quite good over the full range of that ratio which occurs in nature. When ΓS/ΓN approaches the maximum of its known range, as is the case in the A 150 region of SD nuclei, the stronger couplings V and ≈ 1 V2 can lead to important interference effects, which cause some modification to FN . The solution of an infinite-ND-band model via the Green function approach has also been presented. With various reasonable assumptions, this model is also solvable exactly. It, in essence, predicts an approximate upper bound on experimental results, whereas the two-level model gives an approximate lower bound, which is expected to be especially accurate in the 190 mass region, where interference effects are almost absent. That the two models hardly differ in their calculated branching ratios is a great victory for the extraordinarily transparent and simple two-level model. This and the previous chapter have shown how powerful the Green function approach can be. The use of Dyson’s equation to take perturbations into account exactly is especially useful when appropriate approximations are available to simplify a problem to its bare essentials. The next chapter will take an important step beyond what we have studied so far, to expand the equilibrium theory described in Chapter 2 to cases where time- reversal symmetry is violated. Such a theory is essential to describing non-equilibrium processes, such as charge transport at finite voltage. 88

CHAPTER 5

MOLECULAR ELECTRONICS

As a final example of the utility, and versatility, of the Dyson’s equation approach to mesoscopic systems, we shall study the physics of small, single-molecular devices. Within the last decade, experimentalists have grown increasingly adept at making electrical contacts to single molecules, and at fashioning such devices with specific properties. It has not, of course, escaped the notice of the community, or the world at large, that such devices may well play an important role in the technology of the near future. Moreover, the advent of such techniques has opened the gates to a host of systems of basic scientific interest. Molecular electronic systems are an elegant physical example of the archetype at the heart of this work: the coherent quantum system which interfaces with the classical environment. While the molecules themselves are, generally speaking, good quantum mechanical systems, the leads introduce decoherence and dephasing into the system. As we shall see, not only are these effects often non-negligible, but they can give rise to interesting, and useful, physical phenomena in their own right. For this reason, the method explored in previous chapters is ideal to apply to molecular electronic systems. Treating exactly, as it does, the decoherent effects of external systems on the coherent dynamics of electrons in the molecule, Dyson’s equation obviates the need for a priori assumption of a system’s character. Thus, if interesting physics is to be found from coherent or incoherent behavior, or from a combination of these effects, the approach can reveal it.

5.1 Fabrication of Single-Molecular Systems

The conductance theory of molecular electronic systems is, in many ways, simply an idealization of that of semiconductor nanostructures [4, 200]. Atomic site orbitals, of known number and character, take the place of energy levels in each dot, and the parameters of the molecular system are generally well known through chemistry and chemical physics. From a fabrication point of view, however, the problems are entirely diﬀerent. A single molecule must be suspended between at least two electrically active surfaces, which serve as the macroscopic leads (see Fig. 5.1). Although simply described in words, this system 89

Figure 5.1: (color) Artist’s conception of the molecular device proposed in this chapter, the Quantum Interference Eﬀect Transistor. The small molecular component is in contact with three macroscopic leads. The colored spheres represent individual carbon (green), hydrogen (purple), and sulfur (yellow) atoms, while the three gold structures are metallic contacts. Image by Helen Giesel. is obviously a major challenge to create in practice. Throughout the last decade, however, it has been done, and with increasing control and precision.

5.1.1 Scanning-tunneling microscopic techniques

The ﬁrst successful investigations of the conductance properties of single molecules were made with a scanning-tunneling microscope (STM). The device, a central tool in today’s experimental condensed matter world, consists of two electrically active surfaces: a monocrystalline substrate, above which a metallic, atomic-scale tip can be positioned with a precision of fractions of an angstrom in three dimensions [201]. The relative bias of the two surfaces is controllable, so that when they are brought near to each other, a tunnel junction is formed. 90

Although the device is famous for its remarkable topographic images, it is also ideal for measurements of molecular electronic properties. The tip is positioned over a molecule deposited on the lower surface, and biased relative to the molecule and substrate. A two- terminal current measurement can thus be made across an individual molecule, and the differential conductance found. Monatomic molecules were the first investigated using this method [202–205]. While STMs are remarkably capable of positioning single atoms, however, difficulty in expand- ing the technique arose from their inability to manipulate larger molecules. In particular, their was no clear way to make a single molecule “stand up”, so that the STM would measure conductance through the whole molecule. For this reason, the first successful STM measurements of conductance through larger molecules were on the highly symmetric and stable C60 system [206, 207]. A further development came with the use of the self-assembled monolayer (SAM), a mechanically stable single layer of molecular conductors which automatically construct themselves in a parallel arrangement [208, 209]. Many earlier measurements had been made of the conductance of entire monolayers, but use of an STM tip as the second terminal allowed construction of circuits containing only a few, parallel molecular elements. The first such experiments, shown schematically in Fig. 5.2a, made use of a metallic nanograin contact between tip and molecules [210, 211]. As facility with the technique grew, it became possible to contact single molecules of the SAM directly with an atomically sharp tip [212]. Originally, this was achieved through use of a secondary, shorter SAM, which was used to “prop up” the conducter, as in Fig. 5.2b.

5.1.2 Mechanically controllable break junction

A second method of making conductance measurements on single molecules, making use of the experimental technique called the mechanically controllable break junction (MCBJ) and shown in Fig. 5.3, has also seen much success. The MCBJ is simply a well controlled method of fracturing a thin ( .1mm) metallic wire [213–215]. It produces two clean, ∼ atomically sharp contacts, whose spacing can be adjusted on the atomic scale. If a SAM is allowed to grow in the system before breaking, a single-molecular junction can be fabricated [216–218]. 91

(a) (b)

Figure 5.2: Schematic diagrams of methods by which an STM has been used to make conductance measurements on small molecules. (a), from Ref. [211], shows a gold nanograin being used to contact a small number of molecules in a SAM. (b), from Ref. [212], shows the method of using an auxiliary SAM to support a single molecule for STM conductance measurement.

(b)

(a)

Figure 5.3: MCBJ technique. (a) Scanning electron micrograph of a lithographically constructed gold MCBJ, before breaking. The scale bar is 1µm. From [219]. (b) ∼ Schematic of MCBJ device used in a single-molecule conductance experiment. The piezo “e” is used to bend the bar “a” and fracture the notched gold wire “c”. The SAM is formed from the solution “f”. From [216]. 92

5.1.3 Other techniques

A few other techniques have seen success in producing single-molecular junctions, but they are of less general use than the STM and MCBJ methods. Conventional lithographic contacts have been used with much success to measure the electrical properties of carbon nanotubes [220–222]. This approach has the advantage that multiple-lead measurements are straightforward, and four-terminal experiments have been successful [221]. Further, an atomic force microscope can be used to simultaneously take topographic and electrical data [220]. Nevertheless, simple lithography is obviously not appropriate for generic molecules, which are much smaller than nanotubes. As of yet, no general technique exists for connecting more than two leads to a small molecule. Recent success has been reported, however, in combining the STM technique with a nearby single-atom probe, whose state was shown to aﬀect the molecule suspended in the STM tunnel junction [223]. Although this technique currently only provides an electrostatic interaction, it is a promising avenue of exploration toward eventual multiple- lead conﬁgurations. Another is to combine the STM and MCBJ techniques.

5.2 Modeling Molecular Electronics

The success of the Dyson’s equation approach at modeling the systems of the previous chapters motivates its application to the issues of molecular electronics. Like the others, these systems possess both a discrete, fully quantum-mechanical component, as well as the classical leads. One important change in our model at this stage, however, will be in the addition of a higher-order term to the discrete system’s Hamiltonian, representing the Coulomb force between electrons. In this work, a self-consistent mean-ﬁeld picture will be used to treat this physics. Another diﬀerence from previous chapters is the essential non-equilibrium nature of a molecular electronic system modeled at nonzero voltage. Since time-reversal symmetry, an assumption of the derivations of Sec. 2.1, is not valid in such a system, the retarded Green function and self-energy do not provide a full picture of the system’s time dynamics. Later in this section, we shall turn to the Keldysh non-equilibrium Green function to move beyond this limitation. 93

5.2.1 Hamiltonian

We write the Hamiltonian of this system, in the usual way, as the sum of three terms:

H = Hmol + Hleads + Htun. (5.1)

The ﬁrst is the extended Hubbard model molecular Hamiltonian [224]

Unm H = ε d† d t d† d + H.c. + Q Q , (5.2) mol n nσ nσ − nm nσ nσ 2 n m nσ nm σ nm X hXi X where dnσ annihilates an electron on atomic site n with spin σ, εn are the atomic site energies, and tnm are the tunneling matrix elements. The ﬁnal term of Eq. (5.2) contains intersite and same-site Coulomb interactions, as well as the electrostatic eﬀects of the leads. The interaction energies are modeled according to the Ohno parameterization [225, 226]

11.13eV Unm = , (5.3) 2 1 + 0.6117 Rnm/A˚ q where Rnm is the distance between sites n and m.

Cnα α Q = d† d V 1 (5.4) n nσ nσ − e − σ α X X is the eﬀective charge operator for atomic site n, where the second term represents the polarization charge on site n due to capacitive coupling with lead α. Here is the Vα voltage on lead α, and Cnα is the capacitance between site n and α, chosen to correspond with the interaction energies of Eq. (5.3). That is [18],

1 2 C = U − /e , (5.5) where C and U are the full capacitance and interaction matrices, respectively, each of which includes the leads as well as the atomic sites. This amounts to an approximation that the presence of a macroscopic lead does not alter the internal electrostatics of the molecule or other leads too strongly, which is consistent with the general point of view taken in this work that correlations between the continua and discrete systems are negligible. The ﬁnal degrees of freedom in the lead site-capacitances Cnα are determined by the locations of the leads [227]. 94

With lead-site interactions treated at the level of capacitances, the electronic situation of the leads is completely determined by the externally controlled voltages , along with Vα their Fermi energies and temperatures. Each lead possesses a continuum of states, so that their Hamiltonian is

Hleads = kck†σckσ, (5.6) α k α X X∈σ where k is the energy of the single-particle level k in lead α, and the ckσ are the annihilation operators for the states in the leads. Tunneling between molecule and leads is provided by the ﬁnal term of the Hamilto- nian,

Htun = Vakdaσ† ckσ + H.c. . (5.7) αa k α hXi X∈σ

Vak are the tunneling matrix elements for moving from a level k within lead α to the nearby site a. Coupling of the leads to the molecule via inert molecular chains, as may be desirable for fabrication purposes, can be included in the eﬀective Vnk, as can the eﬀect of any substituents used to bond the leads to the molecule [227]. In equilibrium, this system is directly analogous to those of the previous chapters. If no lead couples to more than one site, the self-energy due to the leads is

i Σ (E) = δ Γ (E)δ , (5.8) nm −2 nm α na aα hXi by analogy to Eq. (2.36). Here the notation aα refers to sites a that tunnel with lead h i α. The tunneling rate is given by Eq. (2.37):

2 Γ (E) = 2π V δ E ε0 . (5.9) α | nk| − k k α X∈σ

As usual, we shall take the broad-band limit in the leads, and approximate Γα(E) with a constant parameter characterizing the lead-site coupling, which shifts the poles of the Green function into the complex plane. It is important to note that through this process the density of states (2.41) becomes a continuous, width-broadened function. Due to the open nature of the system, electrons can occupy all energies [227]. 95

5.2.2 Hartree-Fock approximation

Since we wish to calculate the response of molecular electronic systems to finite bias, self-consistency requires a treatment of electron-electron interactions, represented by the third term of Eq. (5.2). Assuming that many-body correlations do not play a significant role, a qualitatively accurate picture of the physics is attainable through a mean-field approach. To treat Coulomb interactions, we choose the well known Hartree-Fock approximation [227],

d† d d† d = d† d d† d d† d d† d δ . (5.10) nσ nσ mρ mρ ∼ mρ mρ nσ nσ − nρ nσ nσ mρ σρ D E D E This result can be viewed as a consequence of Wick’s theorem (2.14), although not rigorous. Nevertheless, the large body of previous work indicates that, in general, this approximation is justiﬁed at the qualitative level important to our investigation.

Application of Eq. (5.10) to Hmol yields its Hartree-Fock approximation [227],

HF Cnα α H = ε U V d† d + t d† d + H.c. mol n − nm e nσ nσ nm nσ mσ nσ mα ! nm σ X X hXi

+ Unm dmρ† dmρ dnσ† dnσ dmρ† dnσ dnσ† dmρδσρ . (5.11) nσ − Xmρ D E D E

In this approximation, the retarded molecular Green function is

1 Gmol(E) = . (5.12) E HHF + i0+ − mol 5.2.3 Non-equilibrium Green function theory

Realistic modeling of a molecular device requires a consideration of the molecule’s current response to finite voltages. Such a system clearly does not posses the time-reversal symmetry exploited in Sec. 2.1.1, and so the conclusions of that analysis must be revisited. In Sec. 2.1, we considered the effects of adding an additional piece to the Hamiltonian by adiabatically turning it on at t = , time-evolving forward to t = , and adia- −∞ ∞ batically switching it off again. If the Hamiltonian possesses time-reversal symmetry, the ending state is the same as the starting one, and so this is a prescription for taking an expectation value. In the more general case, there is no guarantee that the two states are 96 the same, so it is inappropriate to use them both in the calculation of expectation values. The analysis of Sec. 2.1.1 breaks down from the point of Eq. (2.9). The breakthrough of Keldysh [228] was to construct a rigorous mapping of the non- equilibrium problem to an equilibrium one of larger Hilbert space, and construct the Green function in the new space. The question of whether the initial and final state are the same, after all, is merely a question of boundary values, so we should expect the general approach of Green function theory to remain valid. The problem is to evaluate the matrix elements of Eq. (2.8) without appealing to time-reversal symmetry. Clearly, a simple way to do this is to move forward from time to t, and then turn around and go back to again. In fact, we could even go on −∞ −∞ past t on the first leg of the journey, and turn around some time later, as long as we are careful to consider the possibility that the operator (t) might be evaluated on either O the outgoing or return trip. If we wish to be able to calculate (t) for all experimentally accessible times, then, hO i it would behoove us to construct a theory in which we wait until t to turn around. → ∞ This is the root of the time-loop contour which lies at the heart of Keldysh theory. On this contour, shown in Fig. 5.4, the complete discussion of Sec. 2.1 holds, where the parameter s, defined by dt, first half of contour ds = , (5.13)  dt, second half of contour  − replaces t, and a contour-ordering operator Ts replaces T. In particular, Equation (2.13) becomes

n ∞ i 1 T [ (s)S] = ∞ ds ∞ ds T [ (s)H (s ) H (s )] s OI −~ n! 1 · · · n s OI I 1 · · · I n n=0 Z Z X n −∞ −∞ . ∞ i 1 S = ∞ ds ∞ ds T [H (s ) H (s )] −~ n! 1 · · · n s I 1 · · · I n n=0 Z−∞ Z−∞ X (5.14) Although we focused on the retarded quantities, the derivation of Dyson’s equation in Sec. 2.1.2 applies equally well to the time-ordered and anti-time-ordered Green functions [9]:

t t G (t , t ) = i T a (t )a† (t ) , G (t , t ) = i T a (t )a† (t ) , (5.15) nm 1 2 − n 2 m 1 nm 1 2 − n 2 m 1 D h iE D h iE 97 t s Figure 5.4: The Keldysh contour, used to map a non-time-reversal-symmetric problem onto a symmetric one in a larger space. Adiabatic switching on and oﬀ still occurs at t = . The contour consists of two branches, one which time evolves forward from ∞ t = to t = , and one which evolves backward from to . When the two −∞ ∞ ∞ −∞ branches are taken in sequence, contour-reversal symmetry holds. The Green function on this contour, which obeys Dyson’s equation, consists of parts which contour evolve within each branch, as well as parts which take the system between the two. so long as the appropriate self-energies are used. Here T is an ordering operator, like T and N, that places later operators to the right. Thus, in the non-equilibrium case, the contour-ordered Green function

s G (s , s ) = i T a (s )a† (s ) (5.16) nm 1 2 − s n 2 m 1 D h iE obeys Dyson’s equation. Experiments, of course, are not done on time-loop contours, and we must make contact with real notions of time and energy. We note that, where solution of an equilibrium problem required only one Green function, it is clear from Eq. (5.16) that three independent Green functions are required in the non-equilibrium case: Gs contains pieces which propagate forward in time along the first branch of the contour, pieces which go backward in time along the second branch, and pieces which move the system from the first branch to the second branch. It is often convenient to choose a 2 2 matrix representation, in × which one of the elements is not independent: Green functions of different representations are then connected by similarity transforms [229]. The representation most useful to us is due to Craig [230]:

Gt G< G = − , (5.17) G> Gt  −   where, in the time domain,

< > G (t , t ) i a† (t )a (t ) , G (t , t ) i a (t )a† (t ) . (5.18) nm 1 2 ≡ m 1 n 2 nm 1 2 ≡ − n 2 m 1 D E D E 98

We label the elements of the self-energy correspondingly

Σt Σ< Σ = − . (5.19) Σ> Σt  −   In the event that we can consider only stationary, non-transient behavior in the system, it remains meaningful to work in the energy domain. Dyson’s equation then reads

G(E) = G0(E) + G0(E)Σ(˜ E)G(E) = G0(E) + G(E)Σ˜ (E)G0(E), (5.20) which represents four coupled equations. Of the diﬀerent elements, we shall be most interested in G<, since we note that it is, in fact, precisely the two-time correlation function. Returning to the deﬁnitions (2.17), (5.15), and (5.18) of Green functions allows us to solve Eq. (5.20) and write the answer in terms of retarded functions of energy [231]:

< 0< < G (E) = [1 + G(E)Σ(E)] G (E) 1 + Σ†(E)G†(E) + G(E)Σ (E)G†(E). (5.21) h i This result is known as the Keldysh equation. Typically, the self-energy is used to treat the source of non-equilibrium behavior in the system, for example leads at a finite voltage. When this is the case, the first term of Eq. (5.21) consists of entirely equilibrium quantities. As such, it can be fixed by comparison with a related equilibrium problem.

5.2.4 Equal-time correlation functions

Equation (5.11) gives an approximate molecular Hamiltonian in terms of the equal-time correlation functions dnσ† dmρ . Through the Keldysh approach, we now compute an inverse expression forDthe correlationE functions in terms of the Hamiltonian. This self- consistent loop can then be evaluated numerically. As noted in the previous section, correlation functions arise naturally in the Keldysh formalism:

< ∞ dE < iω(t2 t1)/~ G (t , t ) i d† (t )d (t ) = G (E)e− − (5.22) nσ,mρ 1 2 ≡ mσ 1 nρ 2 2π nσ,mρ D E Z−∞ G<(E) is given by the Keldysh result (5.21). iΣ< gives the rate for electrons to enter − the molecule from each lead [4, 9]:

< Σab(E) = iδab Γα(E)fα(E) (5.23) αa hXi 99 where the Fermi distribution function for electrons in lead α is 1 fα(E) = , (5.24) (E µα)/k T 1 + e − B with Boltzmann’s constant kB, lead temperature T , and electrochemical potential µα in lead α. The ﬁrst term of Eq. (5.21) is ﬁxed by the equilibrium conditions of the system. In terms of the correlation functions, then, it corresponds to a simple gate voltage applied to the entire molecule. It can be thus be absorbed into the atomic site energies in the molecule. The remaining term of the Keldysh result, inserted into Eq. (5.22), yields [227]

∞ dE d† dnρ = Gnσ,aσ (E)G∗ (E)Γα(E)fα(E). (5.25) mσ 2π 0 aσ0 ,mρ αa Z D E hXi −∞ Although the retarded Green function G(E) alone no longer completely determines the time-dynamics of the system, Dyson’s equation remains true for it. Thus, it is given by Eq. (2.23): 1 1 G− (E) = G− (E) Σ(E), (5.26) mol − which completes the self-consistent loop.

5.2.5 Landauer-Buttik¨ er formalism

The observables of primary interest in molecular electronic systems are the currents in each lead. A formalism to treat this topic was originally developed as part of scattering matrix theory by Landauer [232, 233] and Buttik¨ er [234] for nanoscale systems, and later shown to be consistent with the Keldysh approach [235]. This formalism has seen much success in the molecular electronics literature, as well [200]. To derive the multi-terminal current formula from the Keldysh formalism, we begin with the deﬁnition of current in lead α, dN ie I (t) e α = [H, N ] , (5.27) α ≡ − dt − ~ h α i where Nα is the number of electrons in lead α. Our task is therefore to evaluate this expectation value. Htun is the only part of the Hamiltonian which does not commute with Nα. Using the fermion anticommutator relations, the commutator is evaluated: 2e Iα(t) = ~ Re iVak∗ ck†σ(t)daσ (t) . (5.28) αa k α hXi X∈σ D E 100

i ck†σ(t)daσ(t) is an equal-time “<” Green function connecting leads and molecule. WhenD applied Eto this quantity, Dyson’s equation yields

∞ dE iEt/~ < i c† (t)d (t) = e− G (E)V g† (E), (5.29) kσ aσ 2π aσ,bσ bk kσ,kσ D E Z−∞ Xb where g(E) is the retarded Green function within the leads. Continuing to work in the energy domain, the current is thus

2e ∞ < I = dE Tr G (E)Σ†(E) . (5.30) α h σ X Z−∞ h i Applying Eq. (5.21), this result becomes

2e ∞ (α) (β) I = dE Tr Γˆ (E)G(E)Γˆ (E)G†(E) f (E) + (E) , (5.31) α − h β Fαβ Xβ Z−∞ n h i o where the second term is due to the equilibrium term of the Keldysh result. Γˆ(α) is deﬁned by Γˆ(α) = 2Im Σ(α) , (5.32) − h i where Σ(α) is the retarded self-energy due only to lead α. The transmission function is [4, 235] (α) (β) Tαβ = Tr Γˆ GΓˆ G† , (5.33) h i which is the probability for an electron injected from lead β to coherently travel directly to lead α. The equilibrium term’s contribution can at most depend on a single Fermi function. Therefore, we can rewrite it

(E) = F (E)f (E). (5.34) Fαβ αβ β Continuity of charge requires 2e I = 0 = ∞ dE [F (E)f (E) T (E)f (E)] . (5.35) α h αβ β − αβ α α X Xαβ Z−∞ We conclude

Fαβ(E) = Tαβ(E), (5.36) and so 2e I = ∞ dE T (E)[f (E) f (E)]. (5.37) α h αβ β − α Xβ Z−∞ This result is the multi-terminal current formula [234]. 101

5.3 Quantum Interference Eﬀect Transistor

Although there has been considerable commercial and scientific interest, a small molecular transistor has yet to be discovered. Needless to say, such a device would solve significant problems, transferring existing technology to smaller length scales. The current semiconductor technology, like all commercial transistors, relies on raising and lowering an energy barrier of order kBT or greater to achieve its switching characteristic, a mechanism which faces significant power consumption and cooling challenges at the approaching nanometer scale [236]. Furthermore, lithographic techniques are nearing an inherent optical limit. Finally, transistors are now so small that much further reduction may lead to undesired electron coherence effects. From the point of view of this work, we are inspired to study the subject by the intriguing interplay of coherent and decoherent effects found in the systems of previous chapters. In the proposed Quantum Interference Effect Transistor (QuIET), the effects of self-energies similar to those we have explored in the earlier chapters of this work provide a mechanism whereby a coherence effect prohibiting charge flow is tunably broken, thus creating transistor behavior.

5.3.1 Tunable conductance suppression

The structure of the QuIET is shown schematically in Fig. 5.5. The first two leads, which play roles similar to the emitter and collector terminals of a bipolar junction transistor (BJT), are attached to a monocyclic aromatic annulene, such as benzene, one third of the way around the ring from each other. As we shall see, destructive interference makes this device completely opaque to charge flow. This effect can be broken by either the decoherence-inducing effect of a third lead (Fig. 5.5a), or simply by bringing an additional level into resonance with the charge carriers, thus introducing extra phase (Fig. 5.5b) [227]. Our first goal is to understand the coherent current suppression of the two-lead device. Charge carriers can take all possible paths between leads 1 and 2, but in the absence of Σ˜ (3), the self-energy due to a third lead or side-complex, these paths all lie within the benzene ring. We operate the QuIET in the regime where there is little charge transfer between it and the leads. In the limits of linear response and no charge transfer, each 102

(a) (b) 2 Γ 2 2 Γ ∼ 2 (3) Γ Γ Σ (x) 1 1 1 . . . Γ 1 3 A1 3 AN 3

Figure 5.5: Schematic diagrams of QuIETs [227]. In each, the voltage on lead 3 modulates the coherent suppression of current between leads 1 and 2. (a) shows a generic QuIET based on benzene. The retarded self-energy of lead 3, Σ˜ (3), is determined by a control variable x. The real part of Σ˜ (3) provides phase relaxation, while the imaginary part provides decoherence, as discussed in Sec. 2.2. (b) shows a specific example of a tunable Re Σ˜ (3) . The electrostatic effect of lead 3 can bring an orbital of the allyl chain closer or furtherh i away from resonance with charge carriers in benzene. N numbers the allyl radicals, and may run from 1 to . ∞ injected carrier has momentum equal to the Fermi momentum of the ring kF = π/2a, where a = 1.397A˚ is the intersite spacing of benzene. It is clear that the phase difference between the two most direct paths through the ring, shown in Fig. 5.6a, is π, and they cancel each other exactly. Similarly, all paths through the ring exactly cancel in a pairwise fashion. Transport of carriers is completely forbidden, unless lifted by the addition of new paths involving a third lead, as in Fig. 5.6b [227]. It is a consequence of Luttinger’s theorem [237] that this coherent suppression of current persists into the interacting regime, as demonstrated in Fig. 5.7a. The effective transmission T˜21 = T21 is calculated according to the self-consistent Hartree-Fock model outlined in Sec. 5.2.4. In this plot Σ˜ (3) = 0, and so the transmission at the Fermi energy is wholly suppressed by coherence. Figure 5.7b demonstrates the effect of introducing an imaginary self-energy i Σ˜ (3) = Γ δ δ (5.38) nm −2 3 nm nA to the system, where A is the site across the molecule from one of the leads. As we saw in Sec. 2.2.2, this corresponds to allowing electrons to tunnel to and from a macroscopic continuum of states. Physically, this might be realized by bringing an STM tip very close to the site. With a third lead present, paths outside the molecular system are available to the 103

(a) (b) 2 2 1

3 1 1

Figure 5.6: Cancellation of paths in a QuIET, and lifting of that eﬀect by the introduction of a third lead. (a) Two-lead experiment to measure the conductance of benzene when the leads are in the meta conﬁguration. Shown are the two most direct paths between the two leads, which cancel exactly, as do all other paths with the same endpoints in a similar pairwise fashion. (b) Example of a new path allowed when a third lead is included. Such paths are not canceled, and so contribute to the total current.

(a) (b) (c)

Figure 5.7: Effective tranmission probability T˜21 of the device shown in Fig. 5.5a, calculated in the linear response via the self-consistent Hartree-Fock method outlined in Sec. 5.2.4. Here, Γ1 = 1.2eV and Γ2 = .48eV, but the choice of energy width does not affect the qualitative results. εF is the Fermi energy of the leads, chosen so that the device is charge neutral. (a) is the result in the absence of Σ˜ (3). (b) is the result when Σ˜ (3) = iΓ /2, where A is the site on the ring to which lead 3 connects, and Γ = 0 AA − 3 3 in the lowest curve, increasing by .24eV with each successive one. (c) is the case of Eq. (3) (5.40), i.e. purely real Σ˜ . One resonance was used with εν = εF and tν = 1eV. 104 charge carriers. The effective transmission function, taking into account not only paths within the molecule, but those outside as well, is [4]

T23T31 T˜21 = T21 + . (5.39) T23 + T31

This remains true out of equilibrium, as well, if the third lead is taken to be an ideal, inﬁnite-impedance voltage probe, for which the total steady-state current is zero. As

Fig. 5.7b shows, the transmission function at the Fermi energy increases from zero as Γ3 is increased. Decoherence from the third lead lifts the coherent current suppression [227]. ˜ (3) Finally, Figure 5.7c shows the eﬀect of a real ΣAA, as is the case for a second molecule bonded to the hydrocarbon ring (e.g., Fig. 5.5b). This was the case considered abstractly in Sec. 2.2.1. The retarded self-energy has the form

t 2 Σ˜ (3) = δ δ | ν| , (5.40) nm nm nA E ε + i0+ ν ν X − th where tν and εν are the hoppings and energies of the ν orbital of the isolated additional molecular complex, respectively. In general, the eﬀect of a side-molecular orbital on resonance with the carriers from the leads is to contribute a Fano antiresonance, which blocks current ﬂow through only one arm of the annulene, thus destroying the coherent current suppression outlined above. Alternatively, one can imagine that the second molecule’s orbitals hybridize with those of the annulene, and a state that connects leads 1 and 2 is created in the gap [227]. In Fig. 5.7c, we have taken only a single orbital for the purposes of illustration. Clearly, transistor behavior based on this mechanism is requisite upon an assumption that the device be operated well within the gap of benzene. Numerical simulations indicate that the most transistor-like response is found for bias . 1 2V. Another, − related, consideration is that in equilibrium, charge transfer between the molecule and leads should not play an important role. For this to be the case, the work function of the metallic leads must be comparable to the chemical potential of the benzene [227]. Fortunately, this is the case with many bulk metals, among them palladium, iridium, platinum, and gold [238]. While we focus mainly on benzene, the QuIET mechanism applies to any monocyclic aromatic annulene with leads 1 and 2 positioned so the two most direct paths have a phase 105

(a) (b) (c) (d)

Figure 5.8: All possible configurations for leads 1 and 2 in a QuIET based on [18]- annulene. The bold lines represent the positioning of the two leads. Each of the four arrangements has a different phase difference associated with it: (a) π, (b) 3π, (c) 5π, and (d) 7π. difference of π. Furthermore, larger molecules have other possible lead configurations, based on phase differences of 3π, 5π, etc. Figure 5.8 shows the lead configurations for a QuIET based on [18]-annulene. Of course, benzene provides the smallest possible QuIET. The position of the third lead or molecular complex affects the degree to which destructive interference is suppressed. For benzene, the most effective location for site A is shown in Fig. 5.9a. It may also be the site immediately between the other two leads, as shown in Fig. 5.9b. The QuIET operates in this configuration as well, although since the third lead’s coupling to the current carriers is less, the transistor effect is somewhat lessened. In the third, three-fold symmetric configurations of leads, Fig. 5.9c, completely decouples site A from the charge carriers within benzene. Because of this, the decoherence or dephasing necessary to QuIET operation is not provided in this configuration. For each monocyclic aromatic annulene, exactly one three-fold symmetric lead configuration exists and yields no transistor behavior.

5.3.2 Finite voltage

In order to create a transistor using coherent current suppression, it is necessary to tunably break the eﬀect. In the case of decoherence from an imaginary Σ˜ (3), we can imagine bringing an STM tip or other metallic lead closer or farther away from the annulene, or interposing a molecular complex of tunable transparency between it and the third lead. In the case of a real Σ˜ (3), an arrangement such as that of Fig. 5.5b is necessary. The electrostatic eﬀect of positive (negative) voltage applied to the third lead is then to bring the anti-bonding (bonding) orbital(s) of the side molecule into resonance with the 106

(a) (b) (c)

Figure 5.9: The three different arrangements for the third lead on benzene when the first two leads, bold, are in the meta configuration. (a) is the choice which couples most strongly to the conducting orbitals of benzene. (b) is a second case which allows for QuIET operation. The third possibility, (c), decouples entirely from the conducting molecular orbitals by symmetry. A third lead in this configuration cannot break the coherent current suppression at all, and so the device does not function as a QuIET. carriers. A typical I diagram for a QuIET is shown in Fig. 5.10, demonstrating that the − V device is quite reminiscent in operation to a classical transistor. The currents in leads 1 and 2 exhibit a broad resonance as the third lead’s voltage is increased. Furthermore, for nonzero Γ3, the device amplifies current in the third lead, providing emulation of the classical BJT, whereas for Γ3 = 0, it acts like a field effect transistor (FET). For the calculation shown in Fig. 5.10, the QuIET of Figs. 5.1 and 5.5b was used, with the number of allyl linkages N = 1. The transistor behavior is interpreted as due to the tunable coherence mechanism introduced in Sec. 5.3.1. If hopping between the benzene ring and the base complex is set to zero, as a check, full current suppression is restored and almost no current flows between leads 1 and 2. Furthermore, the transistor effect persists for arbitrarily small Γ3, which is consistent with our conclusion that transport through the non-canceling paths is enhanced by the electrostatic effect of the third lead. Finally, the fact that the three-fold symmetric lead arrangement yields no appreciable current resonance strongly supports our conclusion [227].

Estimation of the Γn’s remains an open question in the ﬁeld of molecular electronics. Similar quantities are often estimated to be . .5eV [200, 239] within a scattering 107

Figure 5.10: I characteristic of the QuIET shown in Fig. 5.5b, with N = 1. The − V calculation is done at room temperature. The current in lead 1 is graphed, and Vαβ ≡ . Here, Γ = Γ = 1eV. Γ is taken as .0024eV, which allows the slight flow of Vα − Vβ 1 2 3 current in the base necessary for a BJT-style device. A field-effect-transistor-style device with almost identical I can be achieved by taking Γ = 0. The curve for I is for the − V 3 3 case of 1.00V bias voltage; I3 for other biases looks similar. 108 formalism [240], whereas values as high as 1eV have been suggested [241]. For the broadenings of leads 1 and 2, this value has been used to compute Fig. 5.10, but nothing in our arguments depends strongly on these quantities. Furthermore, at . .75V, the side-molecular orbital is strongly off resonance, and V3 the scale of the leakage current through the molecule is set by the sequential tunneling rate Γ/~ = 1 Γ1Γ2 , a result expected for systems, like the QuIET, that are far from ~ Γ1+Γ2 any charge fluctuation resonance [16]. For higher , the side molecule begins to play V3 an important role. As described by Ref. [200], Γ alone then no longer determines the scale of the current. Instead, the rate-limiting process is travel within the side molecule, and varying Γ1 and Γ2 has little effect. Thus, smaller energy widths actually enhance the QuIET’s current contrast dramatically, by reducing the leakage current while maintaining the peak current.

5.4 Summary

In this chapter, we have further developed the Dyson’s equation formalism to treat molecular electronic systems. This included introducing the Keldysh nonequilibrium Green function formalism to treat systems at finite voltage, computing the Hartree-Fock mean- field approximation to the molecular Hamiltonian, and deriving the Landauer-Buttik¨ er approach to coherent transport through the system. Together with the original Dyson’s equation approach, these adaptations form a comprehensive model for treating molecular electronic systems, which is widely and commonly applied in the literature. Furthermore, this model was successfully applied to a novel proposal for a small molecular transistor. The Quantum Interference Effect Transistor, which operates via a tunable breaking of an otherwise exact coherence effect, is motivated by the detailed understanding of open quantum systems we have gained in our tour of the subject. The success of this intuition in the molecular electronic system demonstrates the value, not only of the elegant method of Dyson’s equation, but also of the far-reaching, interdisciplinary outlook it fosters. 109

CHAPTER 6 DISCUSSION

In this work, we have developed a systematic and powerful method of treating quantum systems in interface with a larger environment. This formalism, the Dyson’s equation approach, can exactly include a wide variety of realistic models for effects from both classical continua of states, as well as discrete quantum-mechanical contributions to the system. In either case, particle number within the original subsystem is no longer conserved, and so the system is termed open. While the addition of either a single discrete state or an infinite number are two different limits of the same physics, physicists often think of them in different terms. This arises naturally in the Dyson’s equation formalism, as the single state contributes a real self-energy, and thus gives rise to a simple hybridization of levels. The continuum, on the other hand, as is appropriate for interaction with a truly classical system, yields a purely imaginary self-energy. This leads to the phenomenon known as decoherence, a consideration of which is necessary to a complete picture of any experimental quantum system. We examined three systems in detail, coupled quantum dots in equilibrium, decaying superdeformed nuclei, and a proposed molecular electronic system known as the Quantum Interference Effect Transistor. In the first case, the coupled quantum dots, we learned how important the effects of the classical leads can be on such a system. Without considering them, Rabi oscillations of charge between the two dots can continue forever, a conclusion which rankles our sense of realistic systems. Indeed, a more realistic treatment of the systems, we found, will include the leads, and the result agrees well with our intuition from classical vibrational systems. The leads contribute a damping of the coherent oscillations, which can be either negligible, in the case of extreme underdamping, or, in the case of moderate underdamping or overdamping, can contribute significantly to the time dynamics of the system. Consideration of the leads is especially significant, in light of the many coherent technological applications which have been proposed for quantum dots. The Dyson’s equation approach was also shown to be remarkably successful in the case of decaying superdeformed nuclei. It automatically treats the electromagnetic physics of 110 decay on the same footing as the coherent physics of shape change, an absolute necessity if the decay process is to be truly understood. Careful consideration of the number of levels necessary to accurately model the physical systems finds that the essential physics is encompassed in only one or two normally deformed levels. The few-level models are particularly successful, as far as furthering our understanding of the experimental systems. They explain the arresting universality present in superdeformed decay profiles, and the two-level model allows extraction of a probability distribution for the Hamil- tonian’s tunneling matrix element, a quantity of great interest to microscopic nuclear structure studies. The final system studied was the Quantum Interference Effect Transistor. Recent experiments having indicated that three-lead small molecular devices are just around the corner, this theoretical proposal has great potential for technological impact, especially since it so well mimics all types of classical transistor of use to industry. The system is a very interesting physical study in its own right, as well, combining as it does all the concepts of decoherence and dephasing of central interest to this work. Rather than struggling with such effects as undesirable properties to be minimized, the study of the Quantum Interference Effect Transistor demonstrated that, with the exact and intuitive understanding granted by the Green function formalism, they can be at the roots of intriguing, interesting, and useful phenomena. This year marks the internationally recognized World Year of Physics, chosen for the centennial anniversary of Einstein’s annus mirabilis. It is inspiring to think that one hundred years later, the theories developing at that time still bear interesting fruit. Even more so, we have seen how the techniques introduced by Green in the early nineteenth century remain not only valid, but extraordinarily relevant. They prompt us to seek the connections and commonalities among subfields, from which arise a basic and intuitive understanding of the discipline as a whole. 111

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