Dephasing and Decoherence in Open Quantum Systems: A ......An additional very special thanks is due...
130
Dephasing and Decoherence in Open Quantum Systems: A Dyson's Equation Approach Item Type text; Electronic Dissertation Authors Cardamone, David Michael Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 27/01/2021 02:01:49 Link to Item http://hdl.handle.net/10150/195386
Transcript of Dephasing and Decoherence in Open Quantum Systems: A ......An additional very special thanks is due...
Dephasing and Decoherence in Open QuantumSystems: A Dyson's Equation Approach
Item Type text; Electronic Dissertation
Authors Cardamone, David Michael
Publisher The University of Arizona.
Rights Copyright © is held by the author. Digital access to this materialis made possible by the University Libraries, University of Arizona.Further transmission, reproduction or presentation (such aspublic display or performance) of protected items is prohibitedexcept with permission of the author.
Download date 27/01/2021 02:01:49
Link to Item http://hdl.handle.net/10150/195386
Dephasing and Decoherence in Open Quantum Systems: A
Dyson’s Equation Approach
by
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: 08/04/05Bruce R. Barrett
Date: 08/04/05Charles A. Stafford
Date: 08/04/05Sumitendra Mazumdar
Date: 08/04/05Michael A. Shupe
Date: 08/04/05Koen 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: 08/04/05Dissertation Director: Bruce R. Barrett
Date: 08/04/05Dissertation Director: Charles A. Stafford
4
STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfillment of requirements for anadvanced degree at The University of Arizona and is deposited in the University Libraryto be made available to borrowers under rules of the Library.
Brief quotations from this dissertation are allowable without special permission, pro-vided that accurate acknowledgment of source is made. Requests for permission forextended quotation from or reproduction of this manuscript in whole or in part may begranted by the head of the major department or the Dean of the Graduate College whenin 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 hasinspired 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 theirconfidence. 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 havehelped me in numerous ways over the years. I was fortunate to travel professionally quitea bit during my grad student years, and James Little, Geoff Schmidt, and Jeremy Jonesfacilitated this in all those important ways that friends do. I also thank Tucson KendoKai for helping me find the determination, courage, and character necessary to a gradstudent’s lifestyle.
On the professional side, I could not have been more honored or fortunate to workunder the tutelage and supervision of Profs. Bruce Barrett and Charles Stafford. They,too, gave me their confidence. Much more useful than teaching me physics (althoughthey did that as well), they showed me how to learn physics. I shall never forget theirtireless efforts to guide me on the long journey from inexperienced student to practicingphysicist.
An additional very special thanks is due to Prof. Sumit Mazumdar, with whom I havealso had the privilege of collaborating the last two years. Although Sumit had manyanswers, equally valuable in this collaboration were his questions. He took the time togive excellent and thoughtful career advice, which was key in getting me where I amtoday.
Indeed, the entire community of the University of Arizona Department of Physicshave been welcoming and helpful to me during my time here. Over the years, my thesiscommittee, 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 toohave 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 allthe helpful staff, Mike Eklund and Phil Goisman always went above and beyond the callof 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 dueto 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 withsuch 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-Buttiker 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 junc-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 intro-
duction 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 − V characteristic of a Quantum Interference Effect Transistor . . . . . 107
12
LIST OF TABLES
3.1 Free-electron densities of states for nanoscale systems of different dimen-
sionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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, deco-
herence 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 superdeforma-
tion. 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 under-
stood 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]:
Dψ(x) = −ρ(x), (1.1)
where D is a general linear differential operator in x, which is an ordered set of coordinates.
The Green function is defined by the related equation
D1G(x2, x1) = −δ(x1 − x2), (1.2)
where the subscript i on D indicates to which set of coordinates xi it applies, and δ is the
Dirac delta.
Using G, a solution to Eq. (1.1) can be constructed, given a set of boundary conditions.
The secular equation for D is
Dφn(x) = λnφn(x), (1.3)
where φn and λn are D’s eigenfunctions and eigenvalues, respectively. The eigenfunctions
are a complete, orthonormal set of states:
∞∑
n=−∞φn(x1)φ
∗n(x2) = δ(x1 − x2),
∫
dτφn(x)φ∗m(x) = δnm, (1.4)
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:
G(x2, x1) =∞∑
n=0
an(x1)φn(x2). (1.5)
17
Substituting this result into Eq. (1.2) yields
D1
∞∑
n=0
an(x1)φn(x2) =
∞∑
n=0
an(x1)λnφn(x2) = −δ(x1 − x2). (1.6)
It follows that
∫
dτ2
∞∑
n=0
an(x1)λnφn(x2)φ∗m(x2) = −
∫
dτ2δ(x1 − x2)φ∗m(x2) = −φ∗m(x1) = am(x1)λm,
(1.7)
where dτi is the volume element corresponding to the basis xi. The Green function is thus
seen to be [12]
G(x2, x1) = −∞∑
n=0
φ∗n(x1)φn(x2)
λn. (1.8)
This expansion for G(x2, x1) is closely related to the charge fluctuation resonance expan-
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 G(x2, x1) and integrating, we find
∫
dτ1
∞∑
n=0
φ∗n(x1)φn(x2)
λnD1ψ(x1) =
∫
dτ1G(x2, x1)ρ(x1). (1.9)
Writing the solution ψ(x) as an expansion in the eigenstates and eigenvalues of D,
ψ(x) =
∞∑
n=0
bnφn(x), (1.10)
we arrive at [12]
∫
dτ1G(x2, x1)ρ(x1) =
∫
dτ1
∞∑
n=0m=0
φ∗n(x1)φn(x2)
λnλmbmφm(x1) =
∞∑
n=0
bnφn(x2) = ψ(x2).
(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
∇2 1
|r1 − r2|= −4πδ(r1 − r2), (1.13)
it is apparent that the Green function corresponding to D = ∇2 is [19]
G(r2, r1) =1
4π|r1 − r2|+ F(r2, r1), (1.14)
where F(r2, r1) can be any function which satisfies the Laplace equation,
∇2F(r2, r1) = 0. (1.15)
The choice of F(r2, r1) depends on the boundary conditions of the system. Green
derived a simple theorem for any two fields A and B [11],∫
VdτA∇2B +
∮
dσAdB
dn=
∫
VdτB∇2A+
∮
dσBdA
dn, (1.16)
which follows from Gauss’s theorem. The surface integrals are taken over the boundary
of the volume V , and ddn represents differentiation with respect to a unit vector normal
to the boundary. Performing the integration of Eq. (1.11), we find
ψ(r2) =
∫
dτ1
[
1
4π|r1 − r2|+ F(r2, r1)
]
ρ(r1). (1.17)
By Eqs. (1.15) and (1.12), Equation (1.16) implies∫
Vdτ1F(r2, r1)ρ(r1) =
∮
dσ1
[
F(r2, r1)dψ(r1)
dn1− dF(r2, r1)
dn1ψ(r1)
]
, (1.18)
where the subscripts on dσ1 and n1 indicates they correspond to r1. If V includes the
entire region containing charge, we find
ψ(r2) =
∫
Vdτ1
ρ(r1)
4π|r1 − r2|+
∮
dσ1
[
F(r2, r1)dψ(r1)
dn1− dF(r2, r1)
dn1ψ(r1)
]
. (1.19)
19
Since it appears only in the surface integral with ψ(r1), it is manifest that F(r2, r1) is
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 Schrodinger
picture, state kets time evolve while the operators remain stationary. For the moment
assuming time-translational symmetry [20], a state ket is given by
|ψ(t)〉 = e−iHt/~|ψ0〉, (1.20)
where |ψ0〉 is the ket at time t = 0. Equation (1.20) follows from the defining property
of the Hamiltonian H; it is the operator that generates translations in time [21].
H has real eigenvalues; thus it is Hermitian. The time-evolution operator e−iHt/~ is
consequently unitary:(
e−iHt/~
)−1=(
e−iHt/~
)†= eiHt/~. (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)〉 = |ψ0〉. (1.22)
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 quan-
tum mechanical understanding of operators in place of integrals. The Green function
(operator) is thus
G(t) = −e−iHt/~. (1.23)
In the Green function approach, it is almost always easier to consider the energy
domain. We construct the Fourier transform:
G(E) = −∫ ∞
−∞
dt
~ei(E−H)t/~. (1.24)
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],
G(E) ≡ iG(t≥0)(E + i0+) =−i~
∫ ∞
0dt ei(E−H+i0+)t/~ =
1
E −H + i0+, (1.25)
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 superde-
formed 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 envi-
ronment 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 sta-
tistical 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 [H0,HI ] 6= 0. It is assumed that
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 ob-
servables to the known eigenstates of H0. The secular equation of H0 is
H0|φn〉 = En|φn〉, (2.2)
24
which pertains to an exact, possibly many-body, solution. It is assumed that in the
absence of the perturbation HI , the system lies in the ground state |φ0〉, and that each
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 H0:
H(t) = H0 + limη→0
HIe−η|t|, (2.3)
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]:
|ψ(t+ dt)〉 =
[
1 − iH(t)dt
~
]
|ψ(t)〉, (2.4)
which implies the time-dependant Schodinger equation. It follows that, in the general
case [39],
|ψ(t)〉 = T e− i
~
R tt0
dt′H(t′)|ψ(t0)〉 = −G(t, t0)|ψ(t0)〉. (2.5)
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
|ψI(t)〉 = T e− i
~
R tt0
dt′HI(t′)|ψI(t0)〉 = GI(t, t0)|ψI(t0)〉. (2.6)
Expectation values are maintained by defining operators as
OI(t) = ei~H0tOe−
i~H0t, (2.7)
where O is the Schrodinger-picture operator. The interaction picture is neither the rest
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 O(t) is
〈O(t)〉 =〈ψI(t)|OI(t)|ψI(t)〉
〈ψI(t)|ψI(t)〉=
〈φ0|GI(−∞, t)OI(t)GI(t,−∞)|φ0〉〈φ0|GI(−∞, t)GI(t,−∞)|φ0〉
. (2.8)
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
GI(−∞, t) = GI(∞, t). (2.9)
We can thus rewrite Eq. (2.8) to read
〈O(t)〉 =〈φ0|GI(∞, t)OI(t)GI(t,−∞)|φ0〉
〈φ0|GI(∞, t)GI(t,−∞)|φ0〉. (2.10)
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 |φo〉. Then, slowly turn
on HI , and evolve the state to time t. Operate on this state with the relevant operator,
OI(t). Now, to get back to the state |φ0〉, time evolve the state forward further to t = ∞,
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]:
〈O(t)〉 =〈φo|T [OI(t)S] |φo〉
〈φ0|S|φo〉, (2.11)
where the S-matrix is defined as
S ≡ GI(∞,−∞) = T e−i~
R ∞−∞ dt′HI (t′). (2.12)
We can write the exponential expansions explicitly:
T [OI(t)S] =
∞∑
n=0
(
− i
~
)n 1
n!
∫ ∞
−∞dt1 · · ·
∫ ∞
−∞dtnT [OI(t)HI(t1) · · ·HI(tn)]
S =∞∑
n=0
(
− i
~
)n 1
n!
∫ ∞
−∞dt1 · · ·
∫ ∞
−∞dtnT [HI(t1) · · ·HI(tn)]
(2.13)
The picture we have is thus of all possible numbers of operations of the perturbation
HI , taking place both before and after evaluation of the operator OI(t). Note that time
evolution by H0, from t = −∞ to t = ∞, is included in the definition (2.7) of the
interaction-picture operator OI(t), due again to time-reversal symmetry.
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 contractions
N(all products with n contractions). (2.14)
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) ≡ C(AB) ≡ T(AB) − N(AB). (2.15)
The two operators must be brought next to each other before the contraction is evalu-
ated. After evaluation, the result is to be removed from further ordering operators. Of
particular interest are contractions of the form
C[
an(t2)a†m(t1)
]
= Θ(t2 − t1)[
an(t2), a†m(t1)
]
±, (2.16)
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 |φ0〉. It is easily verified that the expectation value of any normal ordering in
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]
Θ(t2 − t1)
⟨
φ0
∣
∣
∣
∣
[
an(t2), a†m(t1)
]
±
∣
∣
∣
∣
φ0
⟩
= iG0nm(t2, t1), (2.17)
where G0nm(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. HI and O must be written in terms of second quantization operators,
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 OI and
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 “out-
going” 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 re-tarded self-energy Σ?. The proper self-energy Σ is the sum of the minimal set of self-energypieces from which Σ? can be built.
our consideration to diagrams which consist, topologically, of only one piece, we auto-
matically include all diagrams of the numerator and denominator.
In general then, after this cancellation, all diagrams have a single incoming and out-
going 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]
S = G000(∞,−∞) +
∑
kk′
∫ ∞
−∞dt1
∫ ∞
−∞dt2G
00k′(∞, t2)Σ
?k′k(t2, t1)G
0k0(t1,−∞), (2.18)
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 |φ0〉. While Eq. (2.18)
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 |φ0〉. This allows an iterative expansion for Σ? to be built up [39], as in Fig. 2.1.
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
G−1(E) =[
G0(E)]−1 − Σ(E), (2.23)
which is also known as Dyson’s equation.
Since G0(E) is simply given by
G0(E) =1
E −H0 + i0+, (2.24)
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 proper-
ties 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:
H0 = ε0|0〉〈0|. (2.25)
The retarded Green function of energy arising from this Hamiltonian is
G0(E) =1
E −H0 + i0+=
|0〉〈0|E − ε0 + i0+
. (2.26)
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
G0(t) =
∫ ∞
−∞
dE
2π
e−iEt/~|0〉〈0|E − ε0 + i0+
= e−iε0t/~|0〉〈0|, (2.27)
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
HI = ε1|1〉〈1| + V |1〉〈0| + V ∗|0〉〈1|, (2.28)
which depends on the energy ε1 of the new state |1〉, as well as the tunneling matrix
element V , which takes the system from state |0〉 to the new state.
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
within the |1〉 state, and finally returns with interaction V ∗. All diagrams in the full
self-energy are simply iterations of this process for this simple system.
The proper retarded self-energy is thus
Σ00 =V V ∗
E − ε1 + i0+=
|V |2E − ε1 + i0+
. (2.29)
Dyson’s equation (2.23) now yields the full Green function:
G−100 (E) = E − ε0 + i0+ − |V |2
E − ε1 + i0+=
(E − ε0 + i0+) (E − ε1 + i0+) − |V |2(E − ε0 + i0+) (E − ε1 + i0+)
; (2.30)
G00(E) =(E − ε0 + i0+) (E − ε1 + i0+)
(E − ε0 + i0+) (E − ε1 + i0+) − |V |2 . (2.31)
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,
ε± =ε0 + ε1
2± 1
2
√
(ε1 − ε0)2 + 4|V |2, (2.32)
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
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 = H0 +∑
k
εk|k〉〈k| + Vk|k〉〈0| + V ∗k |0〉〈k|, (2.33)
where k labels the states of the continuum.
The levels of the continuum only communicate through the original |0〉 state, so proper
self-energy terms are merely variations on Eq. (2.29), one for each of the states in the
continuum. The result is
Σ00(E) =∑
k
|Vk|2E − εk + i0+
. (2.34)
To shed light on this result, we appeal to the Dirac identity
limη→0+
1
x+ iη≡ P
(
1
x
)
− iπδ(x), (2.35)
where P(x) denotes the Cauchy principle value of x, to arrive at
Σ00(E) = −iπ∑
k
|Vk|2δ(E − εk) = − i
2Γ(E), (2.36)
where
Γ(E) = 2π∑
k
|Vk|2δ(E − εk). (2.37)
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
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):
G00(E) =1
E − ε0 + iΓ/2. (2.38)
The density operator is defined [21] by summing over the Hamiltonian’s diagonal basis
|`〉:ρ(E) =
∑
`
δ(E − ε`)|`〉〈`|, (2.39)
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) =∑
`
|`〉〈`|E − ε` + i0+
=∑
`
−iπδ(E − ε`)|`〉〈`|. (2.40)
It follows that
ρ(E) = − 1
πIm[G(E)], (2.41)
where the density of states is the trace of this quantity. In the current system, this equals
ρ0(E) =1
π
Γ/2
(E − ε0)2 + (Γ/2)2, (2.42)
a Lorentzian. Due to its contact with the continuum, the state |0〉 has been broadened
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
is filled with non-interacting particles.
33
2.3 Summary
The Dyson’s equation approach derived in this chapter has been of fundamental impor-
tance 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 pertur-
bation 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
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 superde-
formed 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 quan-
tum 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 dy-
namics 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 theircommon 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 3directions, respectively. Θ is the Heaviside step function.
d Common Name Free-electron density of states
3 bulkm∗
π2~3
√2m∗E
2 quantum wellm∗
π~2
∑
i
Θ(E −Ei)
1 quantum wire1
π
√
m∗
2~2
∑
ij
Θ(E −Eij)√
E −Eij
0 quantum dot∑
ijk
δ (E −Eijk)
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) Anarray of vertical quantum dots. The horizontal bars are 5µm. The main picture is fromRef. [84], while the inset diagram of a single dot was added in Ref. [28]. (b) A lateraldouble 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 is280×280nm2.
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 im-
plications 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 quan-
tized 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 param-
eter 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 resultdemonstrates the discrete nature of the density of states, as well as Coulomb blockadeoscillations. (b) Electron addition energies for two dots of diameterD. The peaks indicateespecially stable electron numbers, implying shell closures. The magic numbers of thesedots 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 exci-
tations between levels. Thanks to remarkable efforts by experimentalists, quantum dot
systems usually have effective electron temperatures of around 10–15mK [26]. This cor-
responds 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 dotsare coupled to each other, and each to its own macroscopic lead. This situation is alsofrequently known as a triple-barrier system. In the two-level model, the two dots arecompletely described by their energy levels ε1 and ε2, which are defined in the absence ofboth leads and the tunneling between the two dots V . Γ1/~ and Γ2/~ give the rates forelectrons 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
Hdots = ε1d†1d1 + ε2d
†2d2 − V d†1d2 − V ∗d†2d1. (3.2)
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
Hleads =
2∑
α=1
∑
k∈α
εkc†kck, (3.3)
41
where εk 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 =∑
〈nα〉
∑
k∈α
(
Vnkd†nck + H.c.
)
. (3.4)
Here Vnk parameterizes the tunneling between dot n and state k of lead α. The notation
〈nα〉 reminds us that Vnk 6= 0 only if it would connect a dot to its own lead.
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 appropri-
ate juncture to draw analogy to a very well studied problem, that of the spin-boson
Hamiltonian [105]:
HSB = −1
2~∆SBσx +
1
2εSBσz +
∑
ν
(
1
2mνωνx
2ν +
p2α
2mν
)
+1
2q0σz
∑
ν
Cνxν . (3.5)
Here σx and σz are the Pauli matrices for spin- 12 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
ε1 − ε2, respectively.
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 oscilla-
tors, 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 surround-
ing 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]:
J(ω) =π
2
∑
α
C2α
mαωαδ(ω − ωα). (3.6)
The spirit of this result should seem quite familiar: we are interested only in the oscilla-
tors’ 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, how-
ever, we take the opportunity to re-examine the textbook problem of the closed system’s
43
ε1
ε+
ε−
ε2ω
(a) (b)
Figure 3.4: Hybridization of energy levels in the double-dot system without leads. (a) Theenergy levels in the absence of V , ε1 and ε2, are combined by their tunneling matrixelement to give two new levels, ε+ and ε−, as given by Eq. (3.7). ω =
√∆2 + 4V 2 is
the detuning of the two new orbitals. (b) The graph of ε±−εV (solid lines) vs. ∆/V shows
the anticrossing which results when two coupled levels approach each other. The dashedlines give
ε1,2−εV . 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:
ε± = ε±
√
(
∆
2
)2
+ V 2 (3.7)
where ε ≡ (ε1 + ε2)/2 is the mean of the two isolated energy levels, and ∆ ≡ ε2 − ε1
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:
ψ± =1
√
1 +(
ε±−ε1
V
)2
[
ψ1 +ε± − ε1V
ψ2
]
, (3.8)
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
Gdots(E) =(
E −Hdots + i0+)−1
=
E − ε1 + i0+ V
V E − ε2 + i0+
−1
=1
(E − ε1 + i0+)(E − ε2 + i0+) − V 2
E − ε2 + i0+ −V−V E − ε1 + i0+
. (3.9)
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
Gdots(E)ψ± =1
E − ε± + i0+ψ±, (3.10)
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
(Schrodinger 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
Gdots(t) =
∫ ∞
−∞
dE
2πGdots(E)e−iEt. (3.11)
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]
P1(t) = |G11(t)|2 =1
ω2
(
∆2 sin2 ωt
2+ ω2 cos2 ωt
2
)
, (3.12)
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 oscilla-tions continue forever.
while
P2(t) = |G21(t)|2 = 1 − P1(t) =4V 2
ω2sin2 ωt
2. (3.13)
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)
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 subsec-
tion: 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
Σnm(E) = − i
2δnmΓn(E), (3.15)
where the tunneling rate into and out of dot n is given by the Fermi’s Golden Rule result
(2.37)
Γn(E) = 2π∑
〈nα〉
∑
k∈α
|Vnk|2δ(E − ε′k). (3.16)
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
G(E) =[
G−1dots(E) − Σ(E)
]−1=
E − ε1 + i2Γ1(E) −V
−V E − ε2 + i2Γ2(E)
−1
. (3.17)
Inverting this result we find [37]
G(E) =
{[
E − ε1 +i
2Γ1(E)
] [
E − ε2 +i
2Γ2(E)
]
− V 2
}−1
×
E − ε2 + i2Γ1(E) V
V E − ε1 + i2Γ2(E)
, (3.18)
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 thefull system of dots and leads, as given by Eqs. (3.19) and (3.20). In both, ∆ = V . (a) Anexample of the underdamped case, where decoherence from the leads is relatively weak.Γ = 3Γ′ = .3V . (b) An example of the overdamped case. Γ = 3Γ′ = 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]
P1(t) =V 2
|ω|2 e−Γt/~
(
ωi + Γ′
Γ′ − ωieωit +
Γ′ − ωi
ωi + Γ′ e−ωit +
iωr + Γ′
iωr − Γ′ eiωrt +
iωr − Γ′
iωr + Γ′ e−iωrt
)
(3.19)
and
P2(t) =2V 2
|ω|2 e−Γt/~ (coshωit− cosωrt) . (3.20)
Here
Γ ≡ Γ1 + Γ2
2, Γ′ ≡ Γ2 − Γ1
2, (3.21)
and ωr and ωi are the real and imaginary parts of the complex Rabi frequency
ω ≡ ωr + iωi ≡√
4V 2 + (∆ − iΓ′)2/
~. (3.22)
Figure 3.6 shows two examples of behaviors which can result from this solution.
Taking the limits Γ1,Γ2 → 0+ of Eqs (3.19) and (3.20) yields Eqs. (3.12) and (3.13).
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
falls as e−Γt/~ 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± =ε1 + ε2
2± ~ω
2, (3.23)
where it is to be understood that εi = εi − iΓi/2 are the poles of the Green function
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):
T12(E) = Γ1Γ2|G12(E)|2. (3.24)
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
R=
2e2
hT12(εF ), (3.25)
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)
This quantity is either purely real or purely imaginary, depending on the relative values
of 2|V | and |Γ′|.In the case that 2|V | > |Γ′|, the Rabi frequency is purely real. Working with the
algebraically simpler probability P2(t), we find [37]
P2(t) =4V 2
ω2e−Γt/~ sin2 ωt
2. (3.27)
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
the presence of the exponential envelope function e−Γt/~, 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],
P2(t) =4V 2
|ω|2 e−Γt/~ sinh2 |ω|t2. (3.28)
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
element of interdot tunneling, while 2Γ1+Γ2
~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 Γn’s that is to be compared with 4|V |. We explore this surprising result further by
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 energies 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 mea-
surement 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]
to G(t) = e−Γt/~Gdots(t), so the effect of the leads is merely to contribute the decay term.
As Γ′ 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 2Γ′ 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 inter-
disciplinary 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 en-
countered 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
En =−13.6eV
n2, (4.1)
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 wavefunc-
tions of the Hamiltonian into a spherical basis:
ψk(r) =∑
nlm
C(k)nlmRn(r)Y m
l (θ, φ), (4.2)
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 approximatelyproportional to the nuclear radius. Nuclei near shell closures in one or both fermionsystems 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 nucle-ons and α-particles from stable nuclei. The small oscillations with period 2 are due tothe 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
appropriate to the central potential, and Y ml (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
Pk =
∫
d3r
∣
∣
∣
∣
∣
∑
nlm
C(k)nlmRn(r)Y m
l (θ, φ)
∣
∣
∣
∣
∣
2
=
∫
d3r∑
nlm
∣
∣
∣C
(k)nlm
∣
∣
∣
2|Rn(r)|2 |Y m
l (θ, φ)|2 , (4.3)
where the final step makes use of the fact that the Y ml (θ, φ) are orthogonal. For a mean
field axially symmetric about the z axis, the angular momentum projection on that axis
Lz commutes with the Hamiltonian, and thus the eigenstates ψk(r) do not mix different
m values. Thus, the only m-dependence of the C(k)nlm can be to determine whether a state
is filled or not:
C(k)nlm =
0, state unfilled
C(k)nl , state filled
. (4.4)
An important property of the spherical harmonics is [2]
l∑
m=−l
|Y ml (θ, φ)|2 =
2l + 1
4π. (4.5)
It is clear now that in the case of a filled shell,
Pk =∑
nl
(2l + 1)
∫ ∞
0dr∣
∣
∣C(k)nl
∣
∣
∣
2|Rn(r)|2 . (4.6)
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 statenormal 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 sphericalnuclei do not begin mid-shell. (b) Portion of the table of nuclei, showing calculatedvalue of the lowest angular momentum quantum number for which SD states occur, fromRef. [111]. In contrast with (a), SD states occur for every nuclide, but never as theground state. The letter denoted on the chart gives the mixture of standard quadrupolesuperdeformation 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
HEHO =~
2
2m+m
2
[
ω23x
23 + ω2
⊥(
x21 + x2
2
)]
, (4.7)
where m is the effective mass, xi is the ith body-fixed coordinate, and ωi is the frequency
for oscillation in the ith axis. The symmetry implies ω1 = ω2 = ω⊥. The eigenenergies of
HEHO are
EEHOn3n⊥ =
(
n3 +1
2
)
~ω3 + (n⊥ + 1) ~ω⊥, (4.8)
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 afunction of a deformation parameter δosc = ω⊥−ω3
ω , where the average oscillator frequencyω = 1
3 (2ω⊥ + ω3). Small-integer values of axis ratios are noted. In particular, note thenew 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 closuresappears. 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]:
Q =
√
16π
5
∫
d3r Y 0∗
2 (θ, φ)r2ρ(r) =
∫
d3r(
3x23 − r2
)
ρ(r). (4.9)
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
ρ(r) =
Ze4
3πR2
⊥R3
, r⊥ ≤ R⊥
√
1 − x23
R3
0, r⊥ > R⊥
√
1 − x23
R3
, (4.10)
where R⊥ and R3 are the maximum extent of the nucleus along the two symmetric axes
and the third axis, respectively, and r⊥ =√
x21 + x2
2 is the distance of the point r from
the x3-axis. The result for the quadrupole moment is
Q =2
5Ze(
R23 −R2
⊥)
. (4.11)
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 L 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]:
ErotorI =
I(I + 1)~2
2I+Erotor
0 , (4.12)
where Erotor0 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
∆E(I) =I(I + 1) − (I − 2)(I − 1)~2
2I=
2(2I − 1)~2
2I. (4.13)
The peak spacing observed in a decay experiment, or “double difference”, is thus
δE ≡ ∆E(I) − ∆E(I − 2) =2(2I − 1) − 2[2(I − 2) − 1]~2
2I=
4~2
I, (4.14)
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 experi-
mental [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]’sstudy of 152Dy. Angular momentum values I are noted in units of ~. Also noteworthy isthe 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
6 10 14 18 22spin of initial level
0
50
100
in−
band
SD
inte
nsity
(%
)
190Hg
192Hg
194Hg
192Pb
194Pb
196Pb
(a) (b)
Figure 4.6: (a) Decay profiles of several SD bands in the A ≈ 190 mass region. Note howsuddenly each decay occurs. (b) The profiles of (a), but shifted in angular momentum sothat the leftmost point, the last point in which the SD band is experimentally observed toretain any strength, are aligned. The profiles exhibit a universal behavior. Both graphsare 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
SD
ND
Ene
rgy
Angular Momentum
E2
E1
Figure 4.7: Schematic diagram of the SD decay process. The nucleus enters an SDband at a very high angular momentum, for which the state is yrast. It then decays viaE2 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 statisticaltransitions, to the lower-lying ND band. Finally, the nucleus continues to decay downthe 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
64Po
tent
ial
Deformation
SD
ND
ND
Pote
ntia
l
Deformation
SD
(a) (b)
Figure 4.8: Schematic examples of potentials historically used to model SD decay, as afunction of deformation. The potentials change with angular momentum, as well. (a)Double-well potential, which accurately reflects the physical process [175, 176]. The statesof each well, SD or ND, are broadened according to their coupling with the electromag-netic field. (b) Potentials appropriate to describe nuclear fission, but not the SD decayprocess. 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 arestrongly broadened by the ND continuum [190]. (b) can be read as the result of taking acontinuum (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
ND
SΓε ΓN V SD
εSN
Figure 4.9: Schematic diagram of SD decay at a particular angular momentum, under thetwo-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 = εSc†ScS + εN c
†N cN + V
(
c†ScN + c†NcS)
. (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 elec-
tromagnetic field and its coupling with the nucleus, respectively, are the sources of deco-
herence. The electromagnetic field is clearly a continuum of photonic states, similar to
the spin-boson Hamiltonian (3.5). We may thus denote it by
HEM =∑
n=S,N
∑
v
ε(n)ν a(n)†
ν a(n)ν , (4.17)
where ε(n)ν is the energy of the νth oscillator state, and a
(n)ν 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
Hc =∑
ν
[
V (S′)ν c†S′cSa
(S′)†ν + H.c.
]
+∑
νN ′
[
V (N ′)ν c†N ′cNa
(N ′)†ν + H.c.
]
, (4.18)
where S′ denotes the lower-angular-momentum SD state to which decay can occur, and
similarly N ′ runs over all lower-angular-momentum ND states to which the nucleus can
decay. V(n)ν 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 approx-
imation 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]
ρ(U) =
√π
48a1/4U−5/4e2
√aU , (4.19)
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
〈β|M|α〉2 =
(
~2e2
2m
2
πΓGDR
NZ
A
)
EγΓ2GDR
(
E2γ −E2
0
)2+E2
γΓ2GDR
1
ρ (Uβ), (4.20)
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]:
ΓE1(U) ≈ 4!4
3π
e2
~c
1
mc2ΓGDR
E40
NZ
A
(
U
a
)5/2
, (4.21)
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
ΓN ≈ ΓE1. (4.22)
U is generally renormalized to U−2∆p, where 2∆p, known as the backshift parameter,
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
Table 4.1: Experimental inputs and results of the two-level model, for decaying SD states, with the angular momentumquantum number I given in parentheses. Where different models or experiments have generated differing inputs, all areshown to give the fullest possible picture of the two-level model. References given in the right-most column correspond toexperimental measurements of FN and ΓS, as well as measurements and estimations related to determination of ΓN and DN ,as discussed in Sec. 4.3.2. The tunneling width Γ↓ is then determined by Eq. (4.39), either directly, or in the cases of 152Dy(26)and 192Pb(10), statistically, as explained in Sec. 4.4.2. The Gaussian orthogonal ensemble average of the tunneling matrixelement, 〈V 〉, is determined from ΓS , ΓN , DN , and Γ↓, as given by Eq. (4.61)
nucleus(I) FN ΓS ΓN DN Γ↓ 〈V 〉 〈V 〉/DN Refs.(meV) (meV) (eV) (meV) (eV)
152Dy(28) 0.40 10.0 17. 220. 11. 35. 0.16 [126]152Dy(26) 0.81 7.0 17. 194. 140.† 120. 0.62 [126]192Hg(12) 0.26 0.128 0.613 135. 0.049 8.7 0.064 [154, 189]192Hg(10) 0.92 0.050 0.733 89. 0.37 15. 0.17 [154, 189]192Pb(16) <0.01 0.487 0.192 1,362. <0.0050 <29. < 0.021 [157, 172]192Pb(14) 0.02 0.266 0.201 1,258. 0.0056 34. 0.027 [157, 172]192Pb(12) 0.34 0.132 0.200 1,272. 0.10 170. 0.13 [157, 172]192Pb(10) 0.88 0.048 0.188 1,410. 1.9† 1000. 0.71 [157, 172]192Pb(8) >0.75 0.016 0.169 1,681. >0.067 >250. >0.15 [157, 172]194Hg(12) 0.42 0.097 4.8 16.3 0.071 0.49 0.030 [138, 146, 148, 183]194Hg(10) >0.91 0.039 4.1 26.2 >0.44 >2.1 >0.080 [138, 146, 148, 183]194Hg(12) 0.40 0.108 21. 344. 0.072 5.0 0.015 [126]194Hg(10) 0.97 0.046 20. 493. 1.6 35. 0.071 [126]194Hg(12) 0.40 0.086 1.345 19. 0.060 0.97 0.051 [148, 189]194Hg(10) ≥0.95 0.033 1.487 14. ≥1.1 ≥3.0 ≥0.21 [148, 189]194Hg(15) 0.10 0.230 4.0 26.5 0.026 0.52 0.020 [148, 183]194Hg(13) 0.16 0.110 4.5 19.9 0.021 0.34 0.017 [148, 183]194Hg(11) >0.93 0.048 6.4 7.2 >0.71 >0.60 >0.083 [148, 183]194Pb(10) 0.10 0.045 0.08 21,700. 0.0053 1100. 0.051 [140, 149, 183, 193]194Pb(8) 0.38 0.014 0.50 2,200. 0.0087 72. 0.031 [140, 149, 183, 193]194Pb(6) >0.91 0.003 0.65 1,400. >0.032 >77. >0.055 [140, 149, 183, 193]194Pb(12) <0.01 0.125 0.476 236. <0.0013 <2.7 <0.011 [172, 183]194Pb(10) 0.10 0.045 0.470 244. 0.0051 6.1 0.025 [172, 183]194Pb(8) 0.35 0.014 0.445 273. 0.0077 8.8 0.032 [172, 183]194Pb(6) >0.96 0.003 0.405 333. >0.088 >39. >0.12 [172, 183]
†Calculated statistically, as explained in Sec. 4.4.2.
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
Γn2 :
G0(E) =
1
E−εS+iΓS2
0
0 1
E−εN+iΓN2
. (4.23)
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
V =
0 V
V 0
. (4.24)
Dyson’s equation (2.22) then allows us to include its effects to all orders [106]:
G(E) =[
G−10 − V
]−1=
E − εS + iΓS2 −V
−V E − εN + iΓN2
−1
=1
(
E − εS + iΓS2
)(
E − εN + iΓN2
)
− V 2
E − εN + iΓN2 V
V E − εS + iΓS2
, (4.25)
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
dPdecayn (t) =
Γn
~Pn(t)dt. (4.26)
Thus, the branching ratios are [106]
Fn =
∫ ∞
0dPdecay
n (t) =Γn
~
∫ ∞
0dtPn(t). (4.27)
In the open system, it is the identity∑
n
Fn ≡ 1 that replaces∑
n
Pn(t) as a reflection of
particle conservation.
Since the probabilities Pn(t) are not readily accessible by experiment, clarity is en-
hanced by eliminating them from our formulas. This can be readily accomplished via
Parseval’s Theorem,∫ ∞
−∞dt|F (t)|2 ≡
∫ ∞
−∞
dE
2π|F (E)|2, (4.28)
which relates the continuous function F (t) to its Fourier transform F (E). The branching
ratios are thus shown to be [106]
Fn =Γn
~
∫ ∞
−∞dt|GnS(t)|2 =
Γn
2π~
∫ ∞
−∞dE|GnS(E)|2, (4.29)
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]
FN =(1 + ΓN/ΓS)V 2
∆2 + Γ2(1 + 4V 2/ΓNΓS)
(4.30)
for the ND branching ratio, and
FS = 1 − FN =∆2 + Γ
2 (1 + 4V 2/ΓNΓS
)
− (1 + ΓN/ΓS) V 2
∆2 + Γ2(1 + 4V 2/ΓNΓS)
(4.31)
for the fraction of nuclei which continue down the SD band. In analogy with Chapter 3,
∆ is here defined as εN − εS , and Γ ≡ ΓN +ΓS2 .
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:
Γ↓ = 2π
∫ ∞
−∞dEρS(E)V 2ρN (E), (4.32)
where
ρn(E) =Γn/2π
(E − εn)2 + (Γn/2)2(4.33)
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]
Γ↓ =2ΓV 2
∆2 + Γ2 . (4.34)
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]:
FN =ΓNΓ↓/
(
ΓN + Γ↓)
ΓS + ΓNΓ↓/ (ΓN + Γ↓). (4.35)
Likewise, Equation (4.31) becomes
FS =ΓS
ΓS + ΓNΓ↓/ (ΓN + Γ↓). (4.36)
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
Γout =
(
1
Γ↓ +1
ΓN
)−1
=ΓNΓ↓
ΓN + Γ↓ , (4.37)
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:
FN
FS=
Γout
ΓS=
ΓNΓ↓
ΓN + Γ↓
/
ΓS. (4.38)
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 V appears to order Ns, and a numerator, in which V can be at
most order Ns−1. As we saw in Sec. 3.3.2, the role of the denominator of G(E) is solely to
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 O(
V 2)
is sufficient for an exact result for Γ↓, although a derivation of the branching
ratios themselves 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]
Γ↓ =FNΓNΓS
ΓN − FN (ΓN + ΓS). (4.39)
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]
FN <ΓN
ΓS + ΓN, (4.40)
or, equivalently,
ΓN > ΓSFN
FS. (4.41)
Experimental results for which these two inequalities break down indicate either a depar-
ture 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]:
P0(ΓN ) =1
Γ0N
√2π
e−
„
ΓN−Γ0N√
2Γ0N
«2
, (4.42)
where the Γ0N 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 P0(ΓN ):
P0(Γ↓) = P0(ΓN )
∣
∣
∣
∣
dΓN
dΓ↓
∣
∣
∣
∣
=
(
ΓN/Γ↓)2
Γ0N
√2π
e−
„
ΓN−Γ0N√
2Γ0N
«2
. (4.43)
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↓:
Γ↓ > ΓSFN
FS, (4.44)
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,
P(Γ↓) =
0, Γ↓ ≤ ΓSFNFS
A(
ΓN
Γ↓
)2e−
„
ΓN−Γ0N√
2Γ0N
«2
Γ↓ > ΓSFNFS
, (4.45)
where A is the new normalization constant, determined as usual by
∫ ∞
−∞dΓ↓P(Γ↓) = 1. (4.46)
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 experimen-
tally 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 orig-
inal 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,
such as 152Dy(26) and 192Pb(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 � DN , for which we recover the result (4.34) for Γ↓. This
reasoning is similar to that of Sec. 3.2.1.
76
In the opposite limit, V � DN , the result for Γ↓ is
limV/DN→∞
Γ↓ = 2π〈V 2〉DN
, (4.47)
where 〈· · · 〉 represents an energy average. This is the limit in which an infinite number
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
DN � ΓN (4.48)
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]
DN �√
〈V 2〉, (4.49)
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], ran-
dom 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 infor-
mation, 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]:
P(s) =π
2se−πs2/4. (4.50)
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:
Ps (∆) =1
sDNΘ
(
s
2− |∆|DN
)
. (4.51)
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 decayingSD level within the GOE, given by Eqs. (4.52) and (4.64). The average detuning of thenearest level is 〈|∆|〉 ≡ 〈|∆1|〉 = DN/4, and the average detuning of the next-nearest levelis 〈|∆2|〉 = 3DN/4. From Ref. [187]
The Total Probability Theorem now yields a distribution for ∆ [187]:
P (∆) =
∫ ∞
0dsPs (∆)P (s) =
π
2DNerfc
(√π|∆|DN
)
, (4.52)
where erfc (x) is the complementary error function of x. This distribution is illustrated
in Fig. 4.10. The average detuning is
〈|∆|〉 =
∫ ∞
−∞d∆|∆|P (∆) =
DN
4. (4.53)
The typical deviation of |∆| from its average, for cases of real decays, is characterized by
σ|∆| =
√
2
∫ ∞
0d∆
(
∆ − DN
4
)2
P(∆) = DN
√
1
3π− 1
16≈ .8353〈|∆|〉. (4.54)
The fact σ|∆| . 〈|∆|〉 reassures us that 〈|∆|〉 is a meaningful measure of the detunings
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,
ΓS ,ΓN � V,∆, (4.55)
79
where we assume, as implied by the fact σ∆ ∼ 〈∆〉, that ∆ will not stray too far from
〈∆〉. Thus, it is to be expected that only two parameters significantly affect the decay-out
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
P (V ) = 2P (∆)
∣
∣
∣
∣
d∆
dV
∣
∣
∣
∣
, (4.56)
where 2∣
∣
d∆dV
∣
∣ is the Jacobian associated with the change of variables. Solving the definition
of Γ↓ (4.34) for |∆| yields
|∆| =
√
2Γ
Γ↓
(
V 2 − Γ↓Γ2
)
. (4.57)
The requirement that |∆| be real yields a lower bound on V :
V ≥ Vmin =
√
1
2Γ↓Γ. (4.58)
Taking the derivative of Eq. (4.57) gives
∣
∣
∣
∣
d∆
dV
∣
∣
∣
∣
=
√
√
√
√
2ΓΓ↓
1 − Γ↓Γ2V 2
. (4.59)
Within the assumption of ND states obeying the GOE, then, V is drawn from the prob-
ability distribution [187]
P(V ) =
0, V < Vmin
2ΓV πΓ↓|∆|DN
erfc(√
π |∆|DN
)
, V ≥ Vmin
, (4.60)
where |∆| is evaluated according to Eq. (4.57). The average value of V is thus
〈V 〉 =
∫ ∞
−∞dV V P(V ) =
√
Γ↓
2Γ
[
DN
4+ O
(
Γ2
DN
)]
. (4.61)
The width of the distribution (4.60) is characterized by
σV =
√
∫ ∞
Vmin
dV (V − 〈V 〉)2 P(V ) =
√
D2N
(
13π3/2 − Γ↓
32Γ
)
+√
π4 Γ
2+ O
(√
Γ3Γ↓)
. 〈V 〉.(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 inves-
tigations 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, ∆2 ≡ εN2 − εS . In this
section, ∆ ≡ ∆1 ≡ εN1 − εS , to make the notation uniform for clarity. Encouraged by
the successes of a statistical approach previously, we again turn to the GOE to determine
a probability distribution for ∆2 in analogy to P(∆1) given by Eq. (4.52).
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 P(∆2), we would be including two different “regimes” of behavior in
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 condi-
tional probability distribution for ∆2:
Ps(∆2) =1
sDNΘ
( |∆2|DN
− s
2
)
Θ
(
s− |∆2|DN
)
. (4.63)
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]:
P(∆2) =
∫ ∞
0dsPs(∆2)P(s) =
π
2DN
[
erf(√π|∆2|
)
− erf
(√π
2|∆2|
)]
, (4.64)
where erf(x) is the error function of x. This distribution is shown in Fig. 4.10. Its average
detuning is
〈|∆2|〉 =
∫ ∞
−∞d∆2|∆2|P(∆2) =
3DN
4, (4.65)
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 = εSc†ScS + εN1c
†N1cN1 + εN2c
†N2cN2 + V1
(
c†ScN1 + c†N1cS
)
+ V2
(
c†ScN2 + c†N2cS
)
.
(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
G0(E) =
1
E−εS+iΓS2
0 0
0 1
E−εN1+iΓN1
2
0
0 0 1
E−εN2+iΓN2
2
, (4.67)
while the self-energy for tunneling is given by
V =
0 V1 V2
V1 0 0
V2 0 0
. (4.68)
Once again including the effects of V exactly to all orders via Dyson’s Equation (2.22),
we compute the full inverse Green function of energy,
G−1(E) =
E − εS + iΓS2 −V1 −V2
−V1 E − εN1 + iΓN1
2 0
−V2 0 E − εN2 + iΓN2
2
, (4.69)
and hence the Green function itself:
G(E)=h“
E−εS+iΓS2
”“
E−εN1+iΓN1
2
”“
E−εN2+iΓN2
2
”
+V 21
“
E−εN2+iΓN2
2
”
+V 22
“
E−εN1+iΓN1
2
”i−1
×
„
E−εN1+iΓN1
2
«„
E−εN2+iΓN2
2
«
V1
„
E−εN2+iΓN2
2
«
V2
„
E−εN1+iΓN1
2
«
V1
„
E−εN2+iΓN2
2
« „
E−εS+iΓS2
«„
E−εN2+iΓN2
2
«
−V 22
V1V2
V2
„
E−εN1+iΓN1
2
«
V1V2
„
E−εS+iΓS2
«„
E−εN1+iΓN1
2
«
−V 21
.
(4.70)
83
(a) ΓS = 10−3ΓN
(b) ΓS = ΓN
Figure 4.11: Numerical results comparing the two- and three-state total ND branchingratios. Superscripts indicate the number of levels included in the model. Since theydecay via the same statistical process, the widths of the ND states in the three-levelmodel are set equal, i.e. ΓN1 = ΓN2 = ΓN . In (a), ΓS/ΓN = 10−3, while (b) shows thecase ΓS = ΓN . In both, Γ/DN ≡ (ΓN + ΓS)/(2DN ) = 10−4. The levels are taken attheir mean detunings (4.53) and (4.65) within the GOE, under the assumption that theylie 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 com-
binations 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/〈∆1〉, for cases of both large and small ΓS/ΓN .
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 ΓS/ΓN ∼ 1 is noticeably different: there is a distinct maximum in the
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 〉/DN , and the two-level model shares in the common
prediction that V increases with decreasing I. Comparing the values of 〈V 〉/DN with the
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
noted that, since the ND levels are uncoupled, each contributes separately to the self-
85
energy due to tunneling:
VSS =∑
ν
|Vν |2E − εν + iΓNν
2
, (4.72)
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.
ΓNν is set equal for each ND state. Finally, |V |2 is approximated with its energy average,
〈V 2〉. These approximations yield the result:
VSS ≈∞∑
ν=−∞
〈V 2〉E − νDN − ∆ + iΓN
2
=π〈V 2〉DN
cot
(
πE − ∆ + iΓN
2
DN
)
. (4.74)
Dyson’s equation (2.23) includes V to all orders [186]:
GSS(E) ={
[G0(E)]SS − VSS
}−1=
1
E − εS + iΓS2 − Γ↓
2 cot
(
πE−∆+i
ΓN2
DN
) , (4.75)
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
0
FN
1.0
0.5
0.40.2√
〈V 2〉/∆ 0
FN
1.0
0.5
0.40.2√
〈V 2〉/∆
(a) (b)
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 itstypical value in the GOE, 〈∆〉 = DN
4 , while (b) shows the result for ∆ = DN2 , its maximum
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 V1 and
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 espe-
cially 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 stud-
ied 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 con-
tacts 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 ide-
alization 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 contactwith three macroscopic leads. The colored spheres represent individual carbon (green),hydrogen (purple), and sulfur (yellow) atoms, while the three gold structures are metalliccontacts. 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 to-
day’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 mea-
sure 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 ter-
minal 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 atom-
ically 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 con-ductance measurements on small molecules. (a), from Ref. [211], shows a gold nanograinbeing used to contact a small number of molecules in a SAM. (b), from Ref. [212], showsthe method of using an auxiliary SAM to support a single molecule for STM conductancemeasurement.
(a)
(b)
Figure 5.3: MCBJ technique. (a) Scanning electron micrograph of a lithographicallyconstructed 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 isformed 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]
Hmol =∑
nσ
εnd†nσdnσ −
∑
〈nm〉σtnm
(
d†nσdnσ + H.c.)
+∑
nm
Unm
2QnQm, (5.2)
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]
Unm =11.13eV
√
1 + 0.6117(
Rnm/A)2, (5.3)
where Rnm is the distance between sites n and m.
Qn =∑
σ
d†nσdnσ −∑
α
CnαVα
e− 1 (5.4)
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 Vα is the
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],
C = U−1/e2, (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 neg-
ligible. 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 Vα, along with
their Fermi energies and temperatures. Each lead possesses a continuum of states, so
that their Hamiltonian is
Hleads =∑
α
∑
k∈ασ
εkc†kσckσ, (5.6)
where εk is the energy of the single-particle level k in lead α, and the ckσ are the annihi-
lation operators for the states in the leads.
Tunneling between molecule and leads is provided by the final term of the Hamilto-
nian,
Htun =∑
〈αa〉
∑
k∈ασ
(
Vakd†aσckσ + H.c.
)
. (5.7)
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
Σnm(E) = − i
2δnm
∑
〈aα〉Γα(E)δna, (5.8)
by analogy to Eq. (2.36). Here the notation 〈aα〉 refers to sites a that tunnel with lead
α. The tunneling rate is given by Eq. (2.37):
Γα(E) = 2π∑
k∈ασ
|Vnk|2δ(
E − ε′k)
. (5.9)
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†nσdnσd†mρdmρ
∼=⟨
d†mρdmρ
⟩
d†nσdnσ −⟨
d†nρdnσ
⟩
d†nσdmρδσρ. (5.10)
This result can be viewed as a consequence of Wick’s theorem (2.14), although not rig-
orous. 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],
HHFmol =
∑
nσ
(
εn −∑
mα
UnmCnαVα
e
)
d†nσdnσ +∑
〈nm〉σtnm
(
d†nσdmσ + H.c.)
+∑
nσmρ
Unm
(⟨
d†mρdmρ
⟩
d†nσdnσ −⟨
d†mρdnσ
⟩
d†nσdmρδσρ
)
. (5.11)
In this approximation, the retarded molecular Green function is
Gmol(E) =1
E −HHFmol + i0+
. (5.12)
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 sym-
metry 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 O(t) might be evaluated on either
the outgoing or return trip.
If we wish to be able to calculate 〈O(t)〉 for all experimentally accessible times, then,
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
ds =
dt, first half of contour
−dt, second half of contour, (5.13)
replaces t, and a contour-ordering operator Ts replaces T. In particular, Equation (2.13)
becomes
Ts [OI(s)S] =
∞∑
n=0
(
− i
~
)n 1
n!
∫ ∞
−∞ds1 · · ·
∫ ∞
−∞dsnTs [OI(s)HI(s1) · · ·HI(sn)]
S =∞∑
n=0
(
− i
~
)n 1
n!
∫ ∞
−∞ds1 · · ·
∫ ∞
−∞dsnTs [HI(s1) · · ·HI(sn)]
.
(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 func-
tions [9]:
Gtnm(t1, t2) = −i
⟨
T[
an(t2)a†m(t1)
]⟩
, Gtnm(t1, t2) = −i
⟨
T[
an(t2)a†m(t1)
]⟩
, (5.15)
97
t
sFigure 5.4: The Keldysh contour, used to map a non-time-reversal-symmetric problemonto a symmetric one in a larger space. Adiabatic switching on and off still occurs att = ±∞. The contour consists of two branches, one which time evolves forward fromt = −∞ to t = ∞, and one which evolves backward from ∞ to −∞. When the twobranches are taken in sequence, contour-reversal symmetry holds. The Green functionon this contour, which obeys Dyson’s equation, consists of parts which contour evolvewithin 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
Gsnm(s1, s2) = −i
⟨
Ts
[
an(s2)a†m(s1)
]⟩
(5.16)
obeys Dyson’s equation.
Experiments, of course, are not done on time-loop contours, and we must make con-
tact 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 indepen-
dent 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]:
G =
Gt −G<
G> −Gt
, (5.17)
where, in the time domain,
G<nm(t1, t2) ≡ i
⟨
a†m(t1)an(t2)⟩
, G>nm(t1, t2) ≡ −i
⟨
an(t2)a†m(t1)
⟩
. (5.18)
98
We label the elements of the self-energy correspondingly
Σ =
Σt −Σ<
Σ> −Σt
. (5.19)
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]:
G<(E) = [1 +G(E)Σ(E)]G0<(E)[
1 + Σ†(E)G†(E)]
+G(E)Σ<(E)G†(E). (5.21)
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 volt-
age. 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⟨
d†nσdmρ
⟩
. Through the Keldysh approach, we now compute an
inverse expression for the correlation 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:
G<nσ,mρ(t1, t2) ≡ i
⟨
d†mσ(t1)dnρ(t2)⟩
=
∫ ∞
−∞
dE
2πG<
nσ,mρ(E)e−iω(t2−t1)/~ (5.22)
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]:
Σ<ab(E) = iδab
∑
〈αa〉Γα(E)fα(E) (5.23)
99
where the Fermi distribution function for electrons in lead α is
fα(E) =1
1 + e(E−µα)/kBT, (5.24)
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]
⟨
d†mσdnρ
⟩
=∑
〈αa〉
∫ ∞
−∞
dE
2πGnσ,aσ′ (E)G∗
aσ′ ,mρ(E)Γα(E)fα(E). (5.25)
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):
G−1(E) = G−1mol(E) − Σ(E), (5.26)
which completes the self-consistent loop.
5.2.5 Landauer-Buttiker 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 Buttiker [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,
Iα(t) ≡ −e⟨
dNα
dt
⟩
= − ie~〈[H,Nα]〉 , (5.27)
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:
Iα(t) =2e
~
∑
〈αa〉
∑
k∈ασ
Re(
iV ∗ak
⟨
c†kσ(t)daσ(t)⟩)
. (5.28)
100
i⟨
c†kσ(t)daσ(t)⟩
is an equal-time “<” Green function connecting leads and molecule.
When applied to this quantity, Dyson’s equation yields
i⟨
c†kσ(t)daσ(t)⟩
=
∫ ∞
−∞
dE
2πe−iEt/~
∑
b
G<aσ,bσ(E)Vbkg
†kσ,kσ(E), (5.29)
where g(E) is the retarded Green function within the leads. Continuing to work in the
energy domain, the current is thus
Iα =2e
h
∑
σ
∫ ∞
−∞dE Tr
[
G<(E)Σ†(E)]
. (5.30)
Applying Eq. (5.21), this result becomes
Iα = −2e
h
∑
β
∫ ∞
−∞dE{
Tr[
Γ(α)(E)G(E)Γ(β)(E)G†(E)]
fβ(E) + Fαβ(E)}
, (5.31)
where the second term is due to the equilibrium term of the Keldysh result. Γ(α) is defined
by
Γ(α) = −2Im[
Σ(α)]
, (5.32)
where Σ(α) is the retarded self-energy due only to lead α. The transmission function
is [4, 235]
Tαβ = Tr[
Γ(α)GΓ(β)G†]
, (5.33)
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
Fαβ(E) = Fαβ(E)fβ(E). (5.34)
Continuity of charge requires
∑
α
Iα = 0 =∑
αβ
2e
h
∫ ∞
−∞dE [Fαβ(E)fβ(E) − Tαβ(E)fα(E)] . (5.35)
We conclude
Fαβ(E) = Tαβ(E), (5.36)
and so
Iα =2e
h
∑
β
∫ ∞
−∞dE Tαβ(E)[fβ(E) − fα(E)]. (5.37)
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 semicon-
ductor 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
Γ1
Γ2
2
Σ1
∼(3)(x)3 Γ3A1
Γ1
Γ2
2
1 . . .AN 3
(a) (b)
Figure 5.5: Schematic diagrams of QuIETs [227]. In each, the voltage on lead 3 modulatesthe coherent suppression of current between leads 1 and 2. (a) shows a generic QuIETbased on benzene. The retarded self-energy of lead 3, Σ(3), is determined by a controlvariable x. The real part of Σ(3) provides phase relaxation, while the imaginary partprovides 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 further away from resonance with charge carriers in benzene. N numbers the allylradicals, 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 T21 = 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
Σ(3)nm = − i
2Γ3δnmδnA (5.38)
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
(b)(a)
1
2
1
2
1
2
3
Figure 5.6: Cancellation of paths in a QuIET, and lifting of that effect by the introductionof a third lead. (a) Two-lead experiment to measure the conductance of benzene whenthe leads are in the meta configuration. Shown are the two most direct paths between thetwo leads, which cancel exactly, as do all other paths with the same endpoints in a similarpairwise fashion. (b) Example of a new path allowed when a third lead is included. Suchpaths are not canceled, and so contribute to the total current.
(a) (b) (c)
Figure 5.7: Effective tranmission probability T21 of the device shown in Fig. 5.5a, cal-culated in the linear response via the self-consistent Hartree-Fock method outlined inSec. 5.2.4. Here, Γ1 = 1.2eV and Γ2 = .48eV, but the choice of energy width does notaffect the qualitative results. εF is the Fermi energy of the leads, chosen so that thedevice is charge neutral. (a) is the result in the absence of Σ(3). (b) is the result when
Σ(3)AA = −iΓ3/2, where A is the site on the ring to which lead 3 connects, and Γ3 = 0
in the lowest curve, increasing by .24eV with each successive one. (c) is the case of Eq.(5.40), i.e. purely real Σ(3). 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]
T21 = T21 +T23T31
T23 + T31. (5.39)
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].
Finally, Figure 5.7c shows the effect of a real Σ(3)AA, 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
Σ(3)nm = δnmδnA
∑
ν
|tν |2E − εν + i0+
, (5.40)
where tν and εν are the hoppings and energies of the ν th orbital of the isolated additional
molecular complex, respectively. In general, the effect of a side-molecular orbital on reso-
nance 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 assump-
tion 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 fourarrangements 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 de-
structive 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 decoher-
ence 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
(b) (c)(a)
Figure 5.9: The three different arrangements for the third lead on benzene when thefirst two leads, bold, are in the meta configuration. (a) is the choice which couplesmost strongly to the conducting orbitals of benzene. (b) is a second case which allowsfor QuIET operation. The third possibility, (c), decouples entirely from the conductingmolecular orbitals by symmetry. A third lead in this configuration cannot break thecoherent current suppression at all, and so the device does not function as a QuIET.
carriers.
A typical I − V diagram for a QuIET is shown in Fig. 5.10, demonstrating that the
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 electron-
ics. Similar quantities are often estimated to be . .5eV [200, 239] within a scattering
107
Figure 5.10: I − V characteristic of the QuIET shown in Fig. 5.5b, with N = 1. Thecalculation is done at room temperature. The current in lead 1 is graphed, and Vαβ ≡Vα − Vβ. Here, Γ1 = Γ2 = 1eV. Γ3 is taken as .0024eV, which allows the slight flow ofcurrent in the base necessary for a BJT-style device. A field-effect-transistor-style devicewith almost identical I −V can be achieved by taking Γ3 = 0. The curve for I3 is for thecase 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 broad-
enings 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 V3 . .75V, the side-molecular orbital is strongly off resonance, and
the scale of the leakage current through the molecule is set by the sequential tunneling
rate Γ/~ = 1~
Γ1Γ2
Γ1+Γ2, a result expected for systems, like the QuIET, that are far from
any charge fluctuation resonance [16]. For higher V3, the side molecule begins to play
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 molec-
ular 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-Buttiker
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 molec-
ular transistor. The Quantum Interference Effect Transistor, which operates via a tunable
breaking of an otherwise exact coherence effect, is motivated by the detailed understand-
ing 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 quan-
tum 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 classi-
cal 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 con-
sidering 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 intu-
ition 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 sys-
tems. 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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