Thesis
Quantum Single-Particle Generator andShot Noise in Mesoscopic Systems
Eiki Iyoda
Department of Physics, University of Tokyo
December, 2011

Abstract
In this thesis, we study quantum nature of mobile quantum objects
through the setup of the Hong-Ou-Mandel-type experiment, which reﬂects statistical properties of the objects. We ﬁrst study eﬀect of environmental noise in solid state devices for single-photon generation. We
derive density matrix of emitted photons in a fully analytic form, andcalculate survival probabilities, spectra, and purities of emitted photons
under pure dephasing induced by energy level ﬂuctuation. We discuss
what extent quality of quantum coherence of emitted photons is aﬀectedby noise, and how it is improved by time ﬁltering. We also study effect of environmental noise and Coulomb interaction on single-electron
generation using a chiral edge state in integer quantum Hall eﬀect. We
discuss how noise and the Fermi-edge singularity of edge states aﬀect
quantum coherence of emitted electrons. These results will provide a
useful aspect on future application on quantum information technology
using mobile quantum objects. Finally, we calculate ﬁnite-temperaturenoise due to fractional charge excitations in order to access fractional statistical properties of quasiparticles. We propose an extended shot noise
calculated from experimentally available data, and demonstrate that it
is useful to determine the statistical angle of quasiparticles.
This is a version, which is uploaded on the Web. Figures taken fromprevious studies are omitted.

Acknowledgments
I am sincerely grateful to Prof. Takeo Kato for continuous discussions
and tolerance encouragement throughout my graduate course. I would
also like to thank Prof. Kazuki Koshino for stimulating discussions. I
wish to acknowledge to Tatsuya Fujii for powerful discussions. I am
grateful to Prof. Yasuhiro Utsumi and Prof. Kensuke Kobayashi who
give me many chance to think about mesoscopic physics. I thanks to Prof.
Takao Aoki and Prof. Keiichi Edamatsu for a joint research. I am deeply
grateful to Ken-Ichiro Imura and Hideaki Maebashi for their hearty encouragement. I thank to all the people who spend invaluable time together at physics meetings, workshops, and summer schools. I would
like to thank my colleagues of Institute for Solid State Physics, University of Tokyo. Especially, Shunsuke Furuya, Shintaro Takayoshi, Yasuo
Ohta, Kouhei Ohnishi, Shun Maruyama, Ryo Tamura and the memberof Kato group, Kazuyoshi Yoshimi, Hiroshi Shinaoka, Yuji Hamamoto,
Itofumi Takeuchi, Noritaka Tamura, Hiroyuki Abe, and Shigeru Sumi
for daily discussion and conversation. I am indebted to Hiroko Eguchi
who enabled me to live a good laboratory life.
Finally, I acknowledge my family for heartful encouragement andﬁnancial supports. I also acknowledge the ﬁnancial support by GCOE
for Phys. Sci. Frontier, MEXT.

Contents
1 Introduction
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Purpose of this thesis . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Single photon generator (Chap. 3) . . . . . . . . . .
1.2.2 Single electron generator (Chap. 4) . . . . . . . . . .
1.2.3 Shot noise in FQH edge states (Chap. 5) . . . . . . .

7
7
12
12
12
13

2 Review
2.1 Single Photon Generation . . . . . . . . . . . . . . . . . . .
2.1.1 Cavity QED systems . . . . . . . . . . . . . . . . . .
2.1.2 Cavity QED systems in solid state devices . . . . .
2.1.3 Dephasing eﬀect in cavity QED systems . . . . . . .
2.2 Single Electron Generation . . . . . . . . . . . . . . . . . . .
2.2.1 Analogy between optics and electronics . . . . . . .
2.3 Fractional quantum Hall systems . . . . . . . . . . . . . . .

14
14
14
17
18
19
19
21

3 Single-Photon Generator
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Heisenberg equations . . . . . . . . . . . . . . . . .
3.3.2 State vector . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Decay of dot excitation . . . . . . . . . . . . . . . . .
3.3.4 Density matrix of emitted photon . . . . . . . . . .
3.3.5 Pulse shape . . . . . . . . . . . . . . . . . . . . . . .
3.3.6 Frequency Spectrum . . . . . . . . . . . . . . . . . .

24
24
26
28
28
29
31
32
34
34

4

CONTENTS

5

3.3.7 Purity . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Zeno and Anti-Zeno eﬀects . . . . . . . . . . . . . .
3.4.2 Pulse shape and spectrum . . . . . . . . . . . . . . .
3.4.3 Purity . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.4 Time ﬁltering . . . . . . . . . . . . . . . . . . . . . .
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix 3.A Relation between coincidence probability andpurity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix 3.B Proof of Eqs. (3.50) and (3.51) . . . . . . . . . . .

35
37
37
38
41
43
45

4 Single-Electron Generator
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Pure dephasing due to environment noise . . . . . . . . . .
4.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Result . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Eﬀect of Coulomb Interaction . . . . . . . . . . . . . . . . .
4.3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49
49
50
50
51
52
52
53
58

5 Shot Noise and Fractional Statistics
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Model and method . . . . . . . . . . . . . . . . . . . . . . .
5.3 Non-equilibrium Kubo formula . . . . . . . . . . . . . . . .
5.3.1 Landauer formula . . . . . . . . . . . . . . . . . . .
5.3.2 Interacting case . . . . . . . . . . . . . . . . . . . . .
5.3.3 Non-equilibrium Kubo formula . . . . . . . . . . .
5.4 Results and discussions . . . . . . . . . . . . . . . . . . . .
5.4.1 Current and Current Noise . . . . . . . . . . . . . .
5.4.2 Fano factor and peak structure . . . . . . . . . . . .
5.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix 5.A Keldysh Green’s function . . . . . . . . . . . . .

60
60
61
63
64
64
64
66
66
67
72
75
75

46
47

CONTENTS

6

Appendix 5.B Calcultion of Current and Current Noise Power 76
Appendix 5.C Calculation of Shot Noise from Eq.(5.9) . . . . . 78
6 Summary82

Chapter 1
Introduction
In this chapter, we overview the background of the present research,and present the purpose of this thesis. A detailed review of previous
experimental and theoretical studies related to this thesis is given in
Chap. 2.

1.1 Overview
Coherent control of quantum states in physical systems is now of much
importance in various research ﬁelds. Quantum manipulation of quantum states not only provides appealing demonstration on the foundationof quantum mechanics such as examination of breaking of Bell’s inequality, but also gives a vital building block of quantum information technology (quantum communication, quantum computation, etc.) [1]. In anumber of early studies, quantum manipulation was tried to spatiallyﬁxed objects such as ions trapped by laser, cavity-QED systems, Josephson devices, quantum dots, and so on. Recently, mobile quantum objects
have been attracting much interest, because they are expected to play animportant role in quantum information transfer between spatially-ﬁxed
quantum objects. Mobile quantum objects are also useful in experimental demonstration of ‘non-locality’ in quantum mechanics as nonlocal
correlation between spatially-separate quantum objects is easily generated.
7

8

CHAPTER 1. INTRODUCTION

Realization of mobile quantum objects has been considered so far
in two kinds of particles, bosons and fermions. Let us ﬁrst discuss theﬁrst type of particles, i.e., bosons. The most famous object of this kind
is a photon. Quantum nature of photons has been of a central topic
for a few decades in the research ﬁeld called ‘quantum optics’ [2, 3].
Technology of quantum optics is being developed even now, and has
enable us to manipulate a quantum state of a single photon, to entangle
a few photons, and to utilize photons for quantum communication. In
spite of these successes, there are a disadvantage in use of photons as
mobile quantum objects; interaction between photons is usually weak.
To increase interaction between photons, one may consider use of excellent nonlinear optical materials. Another promising way is to employ a
combined system of a single atom and a cavity, which is called a cavity
quantum electrodynamics (cavity QED) system [4, 5]. This system cannaturally introduce strong interaction between photons through nonlinearity of the system prohibiting an atom to be excited twice. Further
recent progress in quantum optics enables us to fabricate a novel type ofthe cavity QED system composed by an optical cavity and a quantum
dot, both of which is made in semiconductors [6]. The semiconductor
cavity QED has an advantage that an atomic system (a quantum dot) is
spatially ﬁxed and can couple cavity photons continuously, in contrast
with the original cavity QED system where atoms stay in a cavity witha ﬁnite dwell time. The semiconductor cavity QED is now considered
to be a promising candidate for quantum information processing withphotons in solid state devices. The cavity QED system in solid state
devices provides a novel feature originating from strong inﬂuence of theenvironment. The atomic level is no longer isolated, and couples to thebackground ﬂuctuation. This environment eﬀect induces new eﬀects as
discussed in Sec. 2.1.
The second type of mobile quantum objects are fermions. The most
famous object following fermi statistics is, of course, an electron. In
contrast with photons, quantum control of mobile electrons started only
several years ago. For generation of nonlocal quantum correlation, onedimensional electron systems have frequently been considered theoreti-

1.1. OVERVIEW

9cally. A chiral edge state in the integer quantum Hall (IQH) states [7] is agood candidate for a quantum propagation of electrons, since backward
scattering by impurities and potential inhomogeneity is well suppressed.
Actually in 2007, a single-electron injection from a quantum dot into anedge state has been realized experimentally by using a rapid temporalchange of a gate voltage [79]. This result reminds us of a fermionic analogy of experiments in quantum optics if an edge state and a tunneling
junction of edge states are considered to be a light-propagating channel
and a half-mirror. This experiment opens up a possibility of a novel
quantum device using mobile quantum objects. For actual application,
we should note that electrons in chiral-edge channels system may suffer strong perturbation from an (electronic) environment because of the
Coulomb interaction. The coupling between electrons and an environment induces pure dephasing, and aﬀects quality of quantum information carried by propagating electrons in edge channels. (Detailed review
is given in Sec. 2.2.)
The most remarkable diﬀerence between photons and electrons is thechange of the wave function in exchanging indistinguishable particles;
the sign of the wave function is invariant in exchanging bosons (likephotons), whereas it is changed in exchanging fermions (like electrons).
This diﬀerence is strikingly observed in the so-called Hong-Ou-Mandel
experiment [9]), which is a variant of the Hanbury Brown-Twiss experiment [10, 11, 12, 13, 14, 15]. We show a schematic setup of the
Hong-Ou-Mandel experiment in Fig. 1.1 (a), where two input particles
simultaneously enter a half mirror M from a channel 1 and 2, and scatterout into a channel 3 or 4. If the particles are bosons, both of two particles
always scatter out into the same channel, i.e., both go into the channel 3
with a probability 1/2 and into the channel 4 with a probability 1/2. On
the other hand, if the particles are fermions, each of them always scattersout respectively into the diﬀerent channels, i.e., one of two goes into thechannel 3 and the other into the channel 4. In other words, the coincidence probability P34 that one of two particles goes out into the channel
3 and the other into the channel 4 is 0 for bosons, and 1 for fermions.
The fermionic analogy of the Hong-Ou-Mandel experiment is actually

CHAPTER 1. INTRODUCTION

10
(a)

1

(b)

4

IQH state

1

4
R

half
mirror MT3

2

junction M

2

3
(c)

FQH state

1

4
e＊

3

2

Figure 1.1: Schematic illustrations of systems treated in this thesis. (a)Hong-Ou-Mandel interference experiment (b) two chiral edge channels
in IQH, coupled to each other at a junction M (c) two chiral edge channels
in FQH. quasi-particle with charge e∗ tunnels at a junction M

realized in the IQH edge states as shown in Fig. 1.1 (b). Here, two chiraledge channels are coupled to each other at a junction M. We assume thatwhen one electron in channel 1 enters the junction M, it go through thejunction into the channel 3 with a probability T, and tunnels(reﬂects)
into the channel 4 with a probability R. Similarly, we assume that if one
electron in channel 2 enters the junction M, it go through the junction
into the channel 4 with a probability T, or tunnels into the channel 3 witha probability R. For the half-mirror condition T = R = 1/2, the system
shown in Fig. 1.1 (b) provides an analogy of the fermionic Hong-OuMandel experiment. Although several experimental groups have now
been trying to realize this type of experiments by using a single-electron
generator in semiconductors, its experimental demonstration has notbeen reported so far.

1.1. OVERVIEW

11

In this thesis, we point out that the Hong-Ou-Mandel experiment canbe utilized for evaluation of quality of quantum information carried bymobile quantum objects. Actually, as shown in Sec. ??, the coincidence
probability P34 is related to the purity of one incident particle state as
P34 = 1 ∓ P where −(+) is for bosons(fermions). Here, the purity P =Trρ2 is a measure of how the wave function of the incident particle
retains quantum coherence, and takes 1 and 0 for a pure state and a fully
mixed state, respectively. Thus, the coincide probability in the HongOu-Mandel experiment gives a direct measure of quantum coherenceof a single particle. One of the main subjects in this thesis is to study
how dephasing process in a single particle generator made in solid state
devices aﬀects coherence of quantum mobile objects by evaluating thecoincide probability in the Hong-Ou-Mandel experiment setup.
In condensed matter physics, it is known that there exists the third
type of mobile quantum objects called an anyon. Anyons are describedas elementally quasi-particle excitations in the fractional quantum Hall
(FQH) eﬀect [16, 17]. Although precise theoretical description on statistics in exchanging two anyons is diﬃcult in general, simple argument
is possible for some special cases. We show one example in Fig. 1.1 (c),where the setup is almost the same as Fig. 1.1 (b) except for replacing theregion between two edge channels from the IQH state to the FQH state.
When we take the reﬂection probability R as small (R = 1 − T ≪ 1), thefundamental charge-transfer process at the junction is dominated by one
quasi-particle excitation in FQH states with a fractional charge e∗ The
fractional charge e∗ is equal to νe in Laughlin state, where ν is a ﬁlling
factor. Then, the eﬀective charge e∗ can be observed by measurement ofshot noise, i.e., partition noise at the junction [18, 19, 20, 21]. Statistics
of anyons, however, is not characterized by an eﬀective charge e∗ , but
by a statistical angle θ deﬁned by the extended commutation relation
ci c†j − eiθ c†j ci = δij . Whereas the statistical angle of anyons in FQH statesis given simply by θ = νπ for the Laughlin states ν = 1/(2n + 1), it
takes more complex values in general hierarchical FQH states, and isin general independent of the eﬀective charges e∗ . The statistical angle
has been studied experimentally by the Aharonov-Bohm oscillation in

12

CHAPTER 1. INTRODUCTIONa ring fabricated by FQH edge states [22]. However, direct observation of anyons statistics in two-particle exchange process such as the
Hong-Ou-Mandel-type experiment still remains a challenge.

1.2 Purpose of this thesis
In this thesis, we theoretically study the following three subjects, all ofwhich are closely related to statistics of mobile quantum objects.

1.2.1 Single photon generator (Chap. 3)
We study a quantum nature of a single photon generated from a cavity
QED system in semiconductors [23]. We consider dephasing eﬀect due
to white-noise energy-level ﬂuctuation in a quantum dot by introducing
an additional one-dimensional bosonic port. We show that all the quantities of interest on a generated photon can be calculated analytically inthis simpliﬁed model. We show that novel features on quantum coherence of a generated photon in solid state devices are obtained throughcalculation of a survival probability, a spectrum, and a purity despite
simpleness of the model.

1.2.2 Single electron generator (Chap. 4)
We study a quantum nature of a single electron generated from a quantum dot into an IQH edge channel. As source of dephasing on electrons,
we consider (1) white-noise energy level ﬂuctuation in a quantum dot
and (2) (screened) Coulomb interaction between an electron in a dot
and electrons in a edge channel. We calculate a survival probability,a spectrum, and a purity in this system. In particular, we show that
Coulomb interaction induces the so-called ‘fermi-edge singularity’ effect, and strongly aﬀects coherence of a generated electron.

1.2. PURPOSE OF THIS THESIS

13

1.2.3 Shot noise in FQH edge states (Chap. 5)
We study current noise at ﬁnite temperatures under weak reﬂection between two FQH edge states [24]. We deﬁne an (extended) Fano factor at
ﬁnite temperatures, which can be obtained from measurement of noise
power and nonlinear diﬀerential conductance. We show that this extended Fano factor is useful to experimentally detect a statistical angle.
We demonstrate that the extended Fano factors show diﬀerent behaviors
for ν = 1/5 and ν = 2/5 even though the fractional charge is given bye∗ /5 for both FQH states.

Chapter 2
Review
In this chapter, we present a detailed review on single-particle generation. We describe previous experimental and theoretical works on individual topics, single-photon generation (Sec. 2.1), single-electron generation (Sec. 2.2), fractional charge excitation and shot noise in FQH edge
states (Sec. 2.3).

2.1 Single Photon Generation
2.1.1 Cavity QED systems
As mentioned in Chap. 1, a key factor in future development of quantum
optics is control of strong nonlinear coupling between photons. One
hopeful candidate is a cavity QED system [4]. Fig. 2.1 shows a schematic
ﬁgure of a typical cavity QED setup. Two mirrors located in an extreme
high vacuum constitute a high-Q cavity for a photon frequency ωc . One
of two mirrors has weak photon leakage, whose rate is denoted with κ,
into an output light channel connected to a photon counter. One atom
ﬂows slowly through the cavity, and stays in the cavity in a ﬁnite dwelltime. Atoms in the cavity can interact with the photon ﬁeld througha dipole interaction. For simplicity, an atom is assumed to have only
two quantum levels with an energy diﬀerence ωa , and its coupling to
the cavity ﬁelds to be a constant g independent of the photon frequency
14

2.1. SINGLE PHOTON GENERATION

g

γ

15

κ

T

Figure 2.1: Characteristic setup of the cavity QED system, composed of
an atom, a cavity, and an environment (an output channel).

ω around the cavity frequency ωc . Then, the system Hamiltonian is
described within the rotating wave approximation as
H0 = ωa σ† σ + ωc a† a + g(σ† a + a† σ),(2.1)

where a(a† ) is an annihilation(a creation) operator of the cavity photon
ﬁeld, and σ is a transition operator of the atom. This model is known as
the Jaynes-Cummings model [25, 26]. There are several methods to add
leakage eﬀect into an output channel. The simplest method to describe
it in a fully quantum mechanical way is to introduce a one-dimensionalbosonic port [27, 28, 29, 30], whose Hamiltonian is given as
r
Z
Z
κ
†
Hleak = kbk bk dk +dk(b†k a + a† bk ).
(2.2)
2π
The cavity QED system has an important feature useful for quantum
manipulation of photons. Let us ﬁrst consider the Jaynes-Cummings
Hamiltonian H0 for the nearly tuned case (ωa ∼ ωc ) in the space spannedby two state bases; one is a state of an excited atom plus no cavity photon,and the other is of a ground-state atom plus one cavity photon. These twostates are mixed by the atom-cavity coupling g, and the eigenenergy of H0p
is given as E = ω̄± (δω)2 + g2 , where ω̄ = (ωa +ωc )/2 and δω = (ωc −ωa ).
For the perfectly tuned case (δω = 0), the energy splitting is 2g, which iscalled the vacuum Rabi coupling. By a similar way, the energy splitting

16

CHAPTER 2. REVIEW

Figure 2.2: Three types of optical cavities in semicondoctors. (a) A
photonic-crystal-slab nanocavity, (b) a micropillar, and (c) a microdisk.
(Cited from [6].)between high-energy states, a N-photon plus excited-atom state and a√
N + 1-photon plus ground-state-atom state is calculated as 2g N + 1.
Clearly, the photon ﬁelds yields nonlinearity with respect to the cavity
photon intensity N by the presence of atoms. This nonlinearity of photon
ﬁelds is useful for nonlinear optics such as laser physics [6]. Even in
linear optics, the cavity QED system is useful in quantum manipulationof photons and atomic states. For actual use of the cavity QED, it isrequired to increase the vacuum Rabi coupling g. Although the couplingg was usually smaller than κ in early studies of the cavity QED, it has
been increased in recent experiments close to (or in some case largerthan) κ. We call the former condition g < κ as weak coupling and thelatter g > κ as strong coupling. In the strong coupling condition g > κ,an appropriately prepared initial state can oscillate in the Hilbert spacespanned by two states (N photon plus excited atom and N + 1 photon√
plus ground-state atom) with a frequency 2g N + 1. This is called thevacuum Rabi oscillation.

For single photon generation, we may naively expect that large coupling strength g is favorable since rapid emission of photons into thecavity can be realized. The best condition for single photon generationis, however, nontrivial. One reason is that for large values of g photon emission into an output channel is bottlenecked by leakage rate κ.
Another reason is that environment eﬀect such as unintended emission
into vacuum. For example, single photon generation by using π-pulse
excitation of an atom has been studied theoretically [31, 32].

2.1. SINGLE PHOTON GENERATION

17

2.1.2 Cavity QED systems in solid state devices

Rapid development of nanoscale fabrication technique has now enabled
to realize a cavity QED system in solid state devices. Weak Rabi-vacuum
coupling between a semiconductor quantum dot and a cavity photon
ﬁeld as well as its application to single photon generation has alreadybeen realized a decade ago [33, 34, 35, 36, 37]. Realization and control
of strong Rabi-vacuum coupling are now one of hot topics in quantum
optics in semiconductors [6], since they are useful for creating compact
optical devices. There are several ways for fabrication of cavities to realize a large vacuum Rabi splitting. In Fig. 2.2, we summarized three
types of optical cavities, a photonic-crystal-slab nanocavity [38], a micropiller [39], and a microdisk [40]. Though there are advantages anddisadvantages for each of these types of cavities, the cavity frequency,
the leakage frequency, and Rabi vacuum coupling in these systems areroughly given as ωc ≃ 3 − 5 × 105 GHz, κ ≃ 40GHz, and g ≃ 40 − 200GHz,
respectively [39, 38, 40]. This strong-coupling cavity-dot system has beenutilized to fabricate a single-dot laser [41] and to generate nonclassicallight [42, 43].
Another important physical system for realization in cavity QED in
solid state devices is a superconductoring device [44]. After the ﬁrst
experimental realization [45], there have appeared a number of remarkable experimental results including generation of single microwave photons [46], generation of Fock states [47], and tomography of arbitrarystates between a qubit and a cavity ﬁeld [48]. We, however, will not discuss experimental relevance of this thesis to the superconducting cavity
QED, because environment noise in this system is governed by low frequency 1/ f noise [49], which is diﬃcult to be treated in the method
adopted in this thesis based on white noise. Although we should keep
in mind importance of dephasing problem in the superconducting cavity
QED, we will left it as a future problem here.

18

CHAPTER 2. REVIEW

Figure 2.3: (a) A photoluminescence spectra of a photonic-crystal-slab
cavity coupled to a quantum dot. Each curve is obtained by changing
Xe pressure, which condensate on a semiconductor surface. (b) Changeof peak positions. (Cited from [50].)

2.1.3 Dephasing eﬀect in cavity QED systems
Single photon generator made by the cavity QED system in solid state devices has several advantages on scalability and easiness of experimentswith no need of an extreme vacuum as in the original cavity QED system. There, however, are a disadvantage that the system suﬀers strong
environment noise. Environment noise is considered to strongly aﬀect
the ﬂuorescence spectra of solid-state, and make its behavior qualitatively diﬀerent from the atomic cavity QED. In Fig. 2.3, we show an
example of a spectrum of an emitted photon from the cavity in applying a pump light whose frequency is tuned to excite a ground state ofa quantum dot to an target excited state [50]. In usual experiments,
the transition energy of a quantum dot is almost constant, whereas thecavity energy can be varied by changing the temperature, the intensityof pump light, and other experimental conditions. In the experimentshown in Fig. 2.3, quantity of condensate Xe on the surface is controlled
to change the cavity energy. The peak changing its position dependingon the Xe concentration corresponds to the cavity energy, and the othersmall invariant peaks correspond to the transition energies of a single
quantum dot. One can clearly see a Rabi vacuum splitting when two
peaks marge. Then, there is one mystery of this data; the cavity peak is
not expected to appear if the detuning energy (the diﬀerence of the cavity
and transition energy) is much larger than the linewidth of two peaks.
The same feature has been observed in other experiments [39, 38, 40].
This feature is inherent in the cavity QED in solid state devices, and is
not observed in the atomic cavity QED. Further experimental studies
have characterized this peak more throughly [51, 52, 53, 54, 55, 56, 57]

2.2. SINGLE ELECTRON GENERATION

19

and now the appearance of ﬂuorescence at the cavity frequency in theoﬀ-resonant condition is considered to be explained by dephasing eﬀect
due to the environment noise, which is unavoidable more or less in solidstate devices.
In parallel to experimental studies, there have appeared several theoretical studies to account for the above experimental results on thephotoluminescence spectra. Simple analyses based on the stochastic
Schrödinger equation [58, 59] or the Master equation [60, 61, 62] has
successfully explained appearance of the peak at the cavity frequency
for the detuned case. These analyses implicitly assume white-noise typeﬂuctuation as the environment noise, and are insuﬃcient to explain details of experiments such as temperature dependences. More realistic
model of the environment noise based on phonons [63, 64, 65, 66, 67]
and charge ﬂuctuation of background carriers [68] has also been studied. Whereas the understanding of photoluminescence spectra has thus
been proceeding, dephasing eﬀect on the indistinguishability of photons,
which is crucial to the Hong-Ou-Mandel experiment on emitted photons
from two single-photon generators, has not studied so far.

2.2 Single Electron Generation
2.2.1 Analogy between optics and electronics
In a long history of mesoscopic physics, which is a research area tostudy quantum nature of electrons in solids, analogy with optics has
played an important role. For example, the term Fabry-Perot interferometer, which is realized in quantum wires with scatterers, was clearly
imported from optics. In a level of one-particle interference experiments
such as Young’s interference experiments [69] (or the Aharonov-Bohm
interference experiments [70]), there appears no diﬀerence between photons and electrons. On the other hand, important information of statistics
of quantum particles is obtained from two-particle correlation. Therefore, the analogy with quantum optics has been utilized actively in studyof shot noise in mesoscopic conductors [71, 72]. Two-particle correlation

20

CHAPTER 2. REVIEW

Figure 2.4: Schematic of single-electron(hole) injection into an edge state.
(a) Temporal change of a gate voltage applied to a quantum dot, (b) aschematic viewgraph of the experimental setup, and (c) energy diagramof electron(hole) injection. (Cited from Ref. [79]).

Figure 2.5: Experimental results of single electron generation. Temporalchange of the gate voltage (the red lines), the averaged current (the blacklines), and the ﬁtted curve with exponential functions (the blue lines).
(Cited from Ref. [79]).

has been measured in various types of experiments in mesoscopic systems such as shot noise reduction in point contacts [73], fermionic Hanbury Brown-Twiss experiments [74, 75], and two-particle Mach-Zenderinterferometer utilizing shot noise measurement [76, 77]. Now, there
are a number of experimental supports on fermion statistics of electrons
through quantum-optics-like experiments. We note that nonlocal quantum correlation breaking the Bell’s inequality can be produced even fromthermal source of electrons in, e.g., two-channel edge states [78].
Two-particle correlation measurement explained so far is for nonequilibrium steady state. Recently, there was one breakthrough in experiments on mesoscopic systems in temporal domain. In 2007, French experimental group has realized single electron sources by using a quantum dot and a time-controlled gate in order of sub-nanosecond [79].
Fig. 2.4 shows schematic of their experiment. They have applied thetime-dependent gate voltage shown in Fig. 2.4 (a) to a quantum dot
coupled to a chiral edge channel through a point contact (Fig. 2.4 (b)).
Then, one electron and one hole are injected into a chiral edge channel
in this period (Fig. 2.4 (c)) with appropriate tuning of the voltage-pulse
amplitude. The average current induced in this temporal change of thegate voltage is shown in Fig. 2.5. At a rapid change of the gate-voltage

2.3. FRACTIONAL QUANTUM HALL SYSTEMS

21(shown by the red lines), average additional currents are induced. The
current shows exponential decay with ﬁnite decay rates, which are given
as τ = 0.9ns, 3.6ns, and 10ns in the three graphs given in Fig. 2.5. The
decay rates agree with theoretical values derived from the transmission
probability D at a point contact, which has been estimated from the linear ac response. Moreover, the integral of the averaged current for a halfof the period is shown to coincide with a charge e when the decay rate is
suﬃciently large.
Realization of dynamical control of electron(hole) injection opens up
a novel type of experiments. The most important experiment in thiscategory is the Hong-Ou-Mandel experiment explained in Chap. 1 (seeFig. 1.1), though it has not succeeded so far. With expectation of thisnovel type of quantum-optical experiments, several theoretical proposals
such as two-particle collider shot noise [80] and nonlocal AharonovBohm eﬀect [81] have been presented. This experiment also provides
a possibility of application to quantum information processing using
moving electrons in edge channels.
After the experiment of Ref. [79], theoretical calculation based on
noninteracting electron models [85, 87] and Master equations [86] hasbeen performed to explain detailed feature of experiments of singleelectron generation [88, 83, 84]. Although dephasing eﬀect on injected
single electrons has also been studied by several authors [88, 89], it isbased on a phenomenological model using an additional bosonic port(see Sec. 2.1.1). In our knowledge, there is no theoretical trial so far todeal with Coulomb interaction eﬀect, which is expected to be strong insingle-electron injection process. We will study dephasing eﬀect due toenvironment noise and Coulomb interaction in Chap. 4.

2.3 Fractional quantum Hall systems
Incompressive two-dimensional ﬂuid in FQH eﬀect has provided a number of interesting phenomena after its discovery [16]. This state emerges
by drastic eﬀect of Coulomb interaction between electrons in clean samples of two-dimensional electron gases under strong perpendicular mag-

22

CHAPTER 2. REVIEW

Figure 2.6: Longitudinal and transverse resistivity of two-dimensional
electron gases showing various hierarchical states of the FQH eﬀect.
(Cited from Ref. [94]).

Figure 2.7: Shot noise measurement for the ν = 1/3 Laughlin state. (Cited
from Ref. [19]).

netic ﬁeld, and is described by Laughlin’s wave functions [17] for a simple ﬁlling factor ν = 1/(1 + 2n). The hierarchical states of FQH has been
considered in the picture of composite particles composed of electronsand some amount of ﬂux, and have succeeded in explaining of experimental observation. We show an example of experimental observation
on hierarchical structures of FQH states. We show an example of such
hierarchical structures in FQH in Fig. 2.6.
One of the most remarkable properties in FQH states is existence ofquasi-particle excitations with a fractional charge e∗ . Direct measurementof fractional charge is possible in measurement of shot noise. We show
one example of shot noise measurement for determination of fractionalcharge e∗ at the ν = 1/3 Laughlin state [19] in Fig. 2.7. Comparison withthe theoretical prediction of shot noise power indicates that the fractionalcharge for the ν = 1/3 Laughlin state seems to be not e but close to e/3.
The quasi-particle with fractional charges obeys fractional statistics [123], characterized by a statistical angle θ upon an adiabatic ex-

Figure 2.8: (a) Setup of the Aharonov Bohm for quasiparticles with fractional charges. (Cited from Ref. [22]). (b) Experimental setup for measurement of current cross-correlation to detect statistical angles. (Cited
from Ref. [95]).

2.3. FRACTIONAL QUANTUM HALL SYSTEMS

23

change process of two quasi-particles,Ψ(r1 , r2 ) = eiθ Ψ(r2 , r1 ).

(2.3)

The issue to experimentally observe θ has also become an intriguingquestion. It has been proposed to use a kind of the Aharanov-Bohm (AB)eﬀect in equilibrium [96, 97, 98, 99, 100] (Fig. 2.8 (a)), and has performed
successfully in recent experiments [22, 101]. In order to further detect
two-particle correlation between two fractional-charge excitations, a proposal to use cross-correlation of currents has been presented [95], though
it has not been performed experimentally so far.
In Chap. 5, we study ﬁnite-temperature noise in FQH edge states. We
propose another method to detect a statistical angle in experiments, byextending deﬁnition of shot noise into the ﬁnite-temperature regime.

Chapter 3
Single-Photon Generator
3.1 Introduction
As reviewed in the previous chapter, solid-state cavity QED systems
composed of semiconductor quantum dots and cavities have recently
been attracting much attention [6]. Strong coupling between a single
dot and a cavity has been conﬁrmed through a large vacuum Rabi splitting [39, 38, 40], and excellent performances have been reported in generating indistinguishable photons [90, 91] and entangled photon pairs [92],both of which are useful for quantum information processing [93].
In contrast with real atoms, semiconductor quantum dots are strongly
inﬂuenced by environmental noise sources. The ﬂuorescence spectra ofsolid-state and atomic cavity QED are qualitatively diﬀerent. Whena solid-state system is excited by a pump light of the dot frequency,
a spectral peak appears at the cavity frequency in spite of the large
detuning between them [39, 38, 40]. Further experimental studies havecharacterized this peak more throughly [51, 52, 53, 54, 55, 56, 57] and haverevealed that the ﬂuorescence at the cavity frequency is due to radiative
decay of the dot. Subsequent theoretical studies accounted for puredephasing of the dot through the stochastic Schrödinger equation [58, 59]
or the Master equation [60, 61, 62] and successfully explained the peakat the cavity frequency. The inﬂuence of pure dephasing on radiative
decay of the dot has also been understood in terms of the quantum Zeno24

3.1. INTRODUCTION

25

and anti-Zeno eﬀects [102, 60].
Therefore, when designing a single-photon source using solid-statecavity QED systems, it is crucial to quantitatively consider pure dephasing of the dot. The performance of such photon sources should be
evaluated from two aspects. One is the collection eﬃciency; namely, theprobability that the emitted photon is transferred to the intended spatial
mode (i.e., the radiation pattern of the cavity). This has been discussed
in several studies in terms of the ratio of radiative decay rates [58, 61, 62].
The other one is the indistinguishability of generated photons, which canbe measured by two-photon interference experiments and is evaluatedby the purity. Single photons with high purity are required for quantum
information processing, particularly for constructing scalable quantum
circuits.
In this chapter, we investigate the properties of a single photon emitted by a solid-state cavity QED system and quantitatively observe theeﬀects of pure dephasing. We treat the ﬁve elements of the overall system(the dot, the cavity, radiation leaking from the cavity, non-cavity radiation modes, and the environment causing pure dephasing of the dot) as
active quantum-mechanical degrees of freedom, and analytically derive
the density matrix of the emitted photon in the real-space representation.
This density matrix contains full information about the emitted photon,
including its pulse proﬁle, the frequency spectrum, and the purity. These
quantities are observed as functions of the pure dephasing rate of thedot. We reveal the optimum condition for maximizing the purity of theemitted photon.
This chapter is organized as follows. In Sec.3.2, we introduce ourmodel. In Sec.3.3, we analytically derive the state vector and the density
matrix of the emitted photon in the real-space representation. This density matrix contains full information about the emitted photon, including
its pulse proﬁle, the frequency spectrum, and the purity. In Sec.3.4, we
visualize analytical results for speciﬁc parameters. We reveal the optimum condition for maximizing the purity of the emitted photon. Finallyin Sec.3.5 we summarize this chapter.

26

CHAPTER 3. SINGLE-PHOTON GENERATOR

Figure 3.1: Schematic illustration of the solid-state cavity QED system
considered. It consists of a quantum dot, a cavity, photon leakage fromthe cavity (b ﬁeld), non-cavity radiation modes (c ﬁeld), and a reservoir
ﬁeld, which causes pure dephasing of the dot (d ﬁeld).

3.2 Model

We investigate radiative decay of an excited quantum dot placed insidea cavity, as illustrated in Fig. 3.1. This solid-state cavity QED system
consists of the following ﬁve components: (i) a quantum dot, (ii) a cavity, (iii) a photon ﬁeld leaking from the cavity (referred to as the b ﬁeld
hereafter), (iv) non-cavity radiation modes (c ﬁeld), and (v) a reservoir
ﬁeld, which causes pure dephasing of the dot (d ﬁeld) [104]. The annihilation operators corresponding to these components are respectively
denoted by σ, a, bk , ck , and dk , where k is a one-dimensional wavenumber.
Note that σ is a Pauli operator, whereas the other operators are bosonic.

3.2. MODEL

27

Putting ℏ = c = 1, the Hamiltonian of the overall system is given by
H = H0 + H1 + H2 + H3 ,(3.1)

H0 = ωd σ† σ + ωc a† a + g(σ† a + a† σ),ZhipH1 = dk kb†k bk + κ/(2π)(a† bk + b†k a) ,
Z
h
i
p
†
†
†
H2 = dk kck ck + γ/(2π)(σ ck + ck σ) ,Zqhi
H3 = dk kd†k dk + γp /πσ† σ(d†k + dk ) .

(3.2)
(3.3)
(3.4)
(3.5)

The parameters are deﬁned as follows (see Fig. 3.1). ωd and ωc respectively denote the resonance frequencies of the dot and the cavity, g represents the coupling between them, κ is the escape rate of cavity photons, γ
is the radiative decay rate of the dot into non-cavity modes, and γp is thepure dephasing rate of the dot. H0 describes the Rabi oscillation between
the dot and the cavity (Jaynes–Cummings Hamiltonian), H1 describes
leakage of a cavity photon to its radiation pattern, H2 describes the radiative decay of the dot into unintended directions, and H3 describes pureRRdephasing of the dot. We can conﬁrm that N ≡ σ† σ+a† a+ dreb†r e
br + drec†r ecrcommutes with the Hamiltonian. Therefore, the number of excitations
is conserved in the dot, cavity, b and c ﬁelds.
We assume that the dot is initially (t = 0) in the excited state while
the other ﬁelds are in their vacuum states. Then, denoting the overall
vacuum state by |0⟩, the initial state vector is given by|ψi ⟩ = σ† |0⟩.

(3.6)

The Hamiltonian of Eq. (3.1) and the initial state vector of Eq. (3.6) form
the basis of our analysis.
For later convenience, we introduce the real-space representation of
the b ﬁeld (cavity leakage). It is deﬁned byZ−1/2e
br = (2π)
dkeikr bk .
(3.7)
The r < 0 (r > 0) region represents the incoming (outgoing) ﬁeld. e
cr ande
dr can be formally deﬁned in a similar manner. Our main concern lies in
the properties of single photon emitted in the b ﬁeld.

CHAPTER 3. SINGLE-PHOTON GENERATOR

28

To model the pure dephasing of the dot, the d ﬁeld interacts with thedot so as to conserve the dot excitation. Using Eqs. (3.2) and (3.5), the dotpHamiltonian can be rewritten as [ωd + f (t)]σ† σ, where f (t) = 2γp [de0 (t)+de†0 (t)] is the ﬂuctuation of the dot resonance frequency induced by thed ﬁeld. Using Eqs. (3.6) and (3.10), we can conﬁrm that ⟨ f (t) f (t′ )⟩i =2γp δ(t − t′ ), where ⟨· · ·⟩i = ⟨ψi | · · · |ψi ⟩. Therefore, the present model
assumes a white noise spectrum for ﬂuctuation of the dot resonance.

3.3 Analysis
In this section, we present analytical results that are rigorously derivablefrom the model of Sec. 3.2. We solve the time evolution of the overallsystem and derive several formulae to characterize the emitted single
photon (density matrix, pulse shape, spectrum, and purity). These analytical results are visualized in the next section for speciﬁc parameters.

3.3.1 Heisenberg equations
Here, we present the Heisenberg equations for the system (σ, a) and ﬁeld
(bk , ck , dk ) operators. Deriving the raw Heisenberg equations for the ﬁeld
operators from Eq. (3.1) and transforming them into real-space representations, we obtain the following relations that connect the incoming(r < 0) and outgoing (r > 0) ﬁelds:√ebr (t) = e
br−t (0) − i κθ(r)θ(t − r)a(t − r),√ecr (t) = e
cr−t (0) − i γθ(r)θ(t − r)σ(t − r),qeedr (t) = dr−t (0) − i 2γp θ(r)θ(t − r)σ† (t − r)σ(t − r),

(3.8)
(3.9)
(3.10)

where θ(x) is the Heaviside step function. From the raw Heisenberg
equations for the system operators and the above input–output relations,3.3. ANALYSIS

29

the Heisenberg equations for system operators are given byded σ − ig(1 − 2σ† σ)aσ = −iωdt− i(1 − 2σ† σ)Nc (t) − i Nd† (t)σ + σ† Nd (t) ,dec a − igσ − iNb (t),a = −iωdt(3.11)(3.12)

ed = ωd −i(γ/2+γp ) and ω
ec = ωc −iκ/2 are respectively the complex
where ωfrequencies of the dot and the cavity, and the noise operators are deﬁnedp√√b−t (0), Nc (t) = γeby Nb (t) = κec−t (0), and Nd (t) =2γp de−t (0). Notethat the noise operators are the initial-time operators and consequently
N j (t)|0⟩ = 0 (j = b, c, d).

3.3.2 State vector
The state vector of the overall system at an arbitrary time t is determinedby |ψ(t)⟩ = e−iHt |ψ(0)⟩. Since the initial dot excitation is conserved in thedot, cavity, b, and c ﬁelds, the state vector can be written as


Z
Z
†
†
†
†
e
ψ(t) = α0 (t)σ + β0 (t)a + drγ0 (r, t)br + drδ0 (r, t)e
cr |0⟩
+
Z

∞ Z
X


dm x αm (x, t)σ† + βm (x, t)a† +m=1drγm (r, x, t)e
b†r +

Z


drδm (r, x, t)ec†rde†x1 · · · de†xm |0⟩,

(3.13)

R
where m denotes the number of excitations in the d ﬁeld and dm x
denotes a multi-dimensional integral with respect to x = (x1 , x2 , · · · , xm ).
We can set x1 ≤ · · · ≤ xm without loss of generality. As we show later,these coeﬃcients are nonzero only when 0 ≤ r ≤ x1 ≤ · · · ≤ xm ≤ t.
First, we discuss α0 and β0 . From Eq. (3.13), we can conﬁrm thatα0 (t) = ⟨σ(t)σ† (0)⟩ and β0 (t) = ⟨c(t)σ† (0)⟩, where ⟨· · ·⟩ = ⟨0| · · · |0⟩. From
Eqs. (3.11) and (3.12), their equations of motion are given by!
!
!
ed −ig−iω
α0 (t)
d α0 (t)=,(3.14)
dt β0 (t)ec−ig −iωβ0 (t)

30

CHAPTER 3. SINGLE-PHOTON GENERATOR

ed and λ2 = −iωecFigure 3.2: λ1 and λ2 in the complex plane. λ1 = −iωwhen g = 0. Dotted lines show their traces as g increases.

with the initial conditions α0 (0) = 1 and β0 (0) = 0. The solutions are
given byα0 (t) = A1 eλ1 t + A2 eλ2 t ,(3.15)

β0 (t) = B1 eλ1 t + B2 eλ2 t ,(3.16)

where λ1 and λ2 are the two eigenvalues of the 2 × 2 matrix in Eq. (3.14)
ec )/(λ1 − λ2 ), A2 = (λ2 + iωec )/(λ2 − λ1 ), and(see Fig. 3.2), A1 = (λ1 + iω
B1 = −B2 = −ig/(λ1 − λ2 ). The real parts of λ1 and λ2 are always negativeand consequently α0 and β0 vanish as t → ∞. Equation (3.13) also implies
that γ0 (r, t) = ⟨ebr (t)σ† (0)⟩ and δ0 (r, t) = ⟨ecr (t)σ† (0)⟩. From Eqs. (3.8) and(3.9), we have√
γ0 (r, t) = −i κβ0 (t − r),(3.17)√
δ0 (r, t) = −i γα0 (t − r).
(3.18)
Rigorously, θ(r)θ(t − r) should appear on the right-hand sides of theseequations. However, below, we implicitly assume 0 ≤ r ≤ x1 ≤ · · · ≤xm ≤ t and omit the Heaviside functions.
Next, we proceed to investigate higher-order quantities. We consider α1 and β1 as examples. Applying the same reasoning as thatpfor γ0 and δ0 , we have α1 (x, t) = −i 2γp ⟨σ(t)σ† (t − x)σ(t − x)σ† (0)⟩

3.3. ANALYSIS

31pand β1 (x, t) = −i 2γp ⟨a(t)σ† (t − x)σ(t − x)σ† (0)⟩. Thus, we need toevaluate the two-time correlation functions, ⟨σ(t2 )σ† (t1 )σ(t1 )σ† (0)⟩ and⟨a(t2 )σ† (t1 )σ(t1 )σ† (0)⟩ with t2 > t1 > 0. Their equations of motion with respect to t2 have the same form as Eq. (3.14) and their initial values (t2 → t1 )are α0 (t1 ) and 0, respectively. This implies that the two-time correlation
functions can be factorized as ⟨σ(t2 )σ† (t1 )σ(t1 )σ† (0)⟩ = α0 (t2 − t1 )α0 (t1 ) and⟨a(t2 )σ† (t1 )σ(t1 )σ† (0)⟩ = β0 (t2 − t1 )α0 (t1 ). Thus, we haveq
α1 (x, t) = −i 2γp α0 (t − x)α0 (x),qβ1 (x, t) = −i 2γp α0 (t − x)β0 (x).

(3.19)(3.20)

Repeating the same arguments, all coeﬃcients can be written as productsof α0 and β0 , as follows: q mαm (x, t) = −i 2γp α0 (t − xm )α0 (xm − xm−1 ) · · · α0 (x2 − x1 )α0 (x1 ),q mβm (x, t) = −i 2γp α0 (t − xm )α0 (xm − xm−1 ) · · · α0 (x2 − x1 )β0 (x1 ),(3.21)(3.22)qm√ γm (r, x, t) = −i κ −i 2γp α0 (t − xm )α0 (xm − xm−1 ) · · · α0 (x2 − x1 )β0 (x1 − r),(3.23)qm√δm (r, x, t) = −i γ −i 2γp α0 (t − xm )α0 (xm − xm−1 ) · · · α0 (x2 − x1 )α0 (x1 − r).
(3.24)

3.3.3 Decay of dot excitation
The state vector of Eq. (3.13) fully describes the dynamics of the overallsystem, including both its transient and asymptotic behaviors. In thissection, as an example of a transient phenomenon, we analyze the decayof the dot excitation. The survival probability of the dot excitation isdeﬁned as
P(t) = ⟨ψ(t)|σ† σ|ψ(t)⟩.

(3.25)

CHAPTER 3. SINGLE-PHOTON GENERATOR

32

From Eqs. (3.13) and (3.21), we haveZ
2
P(t) = |α0 (t)| + 2γp dx|α0 (t − x)|2 |α0 (x)|2 + · · · .

(3.26)We here introduce the Laplace transform of |α0 |2 , which is deﬁned byR∞
L|α0 |2 (z) = 0 dte−zt |α0 (t)|2 . It is given by
L|α0 |2 (z) =X

Am A∗n
,
∗
z
−
λ
−
λ
m
n
m,n=1,2(3.27)

where λ1,2 and A1,2 are deﬁned in Sec. 3.3.2. The Laplace transform ofP(t) is then given by
LP (z) =L|α0 |2 (z).
1 − 2γp L|α0 |2 (z)(3.28)

P(t) is obtained by analyzing the poles of this function in the z-plane. We
denote the four roots of the equation 1−2γp L|α0 |2 (z) = 0 by µ j ( j = 1, · · · , 4)(see Fig. 3.3). P(t) is then given by
P(t) =

4X

E j eµ j t ,(3.29)

j=1

Q
Ej =

∗
m′ ,n′ =1,2 (µ j − λm′ − λn′ )Qi(,j) (µ j − µi )X

Am A∗n
.
∗
µ
−
λ
−
λ
j
m
n
m,n=1,2(3.30)

Note that the real parts of µ j are always negative and that the survival
probability P(t) vanishes in the t → ∞ limit, as expected.

3.3.4 Density matrix of emitted photon
In the t → ∞ limit, the initial dot excitation is completely transformed
into a photon propagating in the intended mode (b ﬁeld) or in unintended
directions (c ﬁeld). In the following subsections, we analyze the photon
emitted in the b-ﬁeld. It is fully characterized by its density matrix ρ̂(t).
In the real space representation, the matrix element ρ(r, r′ , t) is given byρ(r, r′ , t) = ⟨ψ(t)|eb†r′ ebr |ψ(t)⟩.

(3.31)

3.3. ANALYSIS

33

Figure 3.3: µ j ( j = 1, · · · , 4) in the complex plane. When γp is absent,µ1 = λ1 + λ∗1 , µ2 = λ1 + λ∗2 , µ3 = λ2 + λ∗1 , and µ4 = λ2 + λ∗2 . Dotted lines
indicate their traces as γp increases. Real parts of µ j are always negative
for any γp .

We make the following three comments regarding this quantity: (i) ρ̂(t)is Hermitian, namely ρ(r′ , r, t) = ρ∗ (r, r′ , t). Therefore, we need consider
only the r < r′ region in this subsection. (ii) As we will see later, ρ(r, r′ , t) =ρ(r − t, r′ − t) in the t → ∞ limit. This reﬂects translational motion of theRemitted photon. (iii) Trρ̂(t) = drρ(r, r, t) represents the probability ofﬁnding the emitted photon in the b ﬁeld. This is unity when γ = 0.
Using Eqs. (3.13) and (3.23), the matrix element can be rewritten asZ′∗′
ρ(r, r , t) = κβ0 (t − r)β0 (t − r ) + 2γp κ dxβ0 (x − r)β∗0 (x − r′ )|α0 (t − x)|2 + · · · .
(3.32)We here introduce the Laplace transform of β0 β∗0 , which is deﬁned byR∞
Lβ0 β∗0 (r, r′ , z) = 0 dte−zt β0 (t − r)β∗0 (t − r′ ). It is given by
Lβ0 β∗0 (r, r′ , z) =X

Bm B∗n′′eλm (r −r)−r z ,∗z − λm − λnm,n=1,2(3.33)

where λ1,2 and B1,2 are deﬁned in Sec. 3.3.2. The Laplace transform ofCHAPTER 3. SINGLE-PHOTON GENERATOR

34

ρ(r, r′ , t) is then given byLρ (r, r′ , z) =κLβ0 β∗0 (r, r′ , z).
1 − 2γp L|α0 |2 (z)(3.34)

By analyzing the poles of this function in the z-plane, ρ(r, r′ , t) is obtained
as follows:′ρ(r, r , t) =

4 X
2
X

′

′

ρ jm eλm (r −r)−µ j (t−r ) ,(3.35)

j=1 m=1

Q

2∗ X
m′ ,n′ =1,2 (µ j − λm′ − λn′ )Qρ jm =

i(,j) (µ j − µi )Bm B∗n,µ j − λm − λ∗nn=1(3.36)

where 0 < r < r′ < t and µ j ( j = 1, · · · , 4) are deﬁned in Sec. 3.3.3. We cancheck that this quantity depends on only two variables, r − t and r′ − t.

3.3.5 Pulse shape
The pulse shape of the emitted photon is characterized by the intensity
distributionf (r, t) = ⟨ψ(t)|eb†r e
br |ψ(t)⟩ = ρ(r, r, t),(3.37)

which is the diagonal element of the density matrix. This quantity is realand positive in 0 < r < t. By setting r′ = r in Eq. (3.35), we havef (r, t) =4Xf j e−µ j (t−r) ,(3.38)

j=1

Q
fj =

2
∗
X
m′ ,n′ =1,2 (µ j − λm′ − λn′ )Qi,j (µ j − µi )Bm B∗n
.
∗
µ
−
λ
−
λ
j
m
n
m,n=1

(3.39)

3.3.6 Frequency Spectrum
The frequency spectrum of the emitted photon is deﬁned by
S(k, t) = ⟨ψ(t)|b†k bk |ψ(t)⟩.

(3.40)

3.3. ANALYSIS

35

Apparently, S(k, t) becomes independent of t in the t → ∞ limit, and weare interested in S(k) = limt→∞ S(k, t). By deﬁnition, S(k, t) is the Fouriertransform of the density matrix element:
ZZ
1′
S(k, t) =drdr′ eik(r −r) ρ(r, r′ , t).
(3.41)
2π
R∞
We consider the Laplace transform of S(k, t) deﬁned by LS (k, z) = 0 dte−zt S(k, t).
Using Eqs. (3.34) and (3.41), it is given by
L′β0 β∗0 (k, z)κLS (k, z) =,2π 1 − 2γp L|α0 |2 (z)whereL′β0 β∗0 (k, z) =
=

Z ∞

ZZ
dt

0

Z ∞
0

(3.42)′

drdr′ eik(r −r)−zt β∗0 (t − r′ )β∗0 (t − r)
2
X

2

Bm
(e−ikt − eλm t ) .
dte−zt
λ
+
ik
m=1 m

(3.43)

Note that L′β0 β∗0 has a pole at z = 0. Since our interest lies in the t → ∞limit of S(k, t), we need to investigate the pole of LS (k, z) at z = 0 only.
Therefore, S(k) = (κ/2π)[1 − 2γp L|α0 |2 (0)]−1 Resz=0 [L′β0 β∗0 (k, z)]. After some
calculations, we obtain
S(k) =N,ed )(k − ωec ) − g2 |2|(k − ω(3.44)

where N = (κg2 /2π)[1 − 2γp L|α0 |2 (0)]−1 is a factor that is independentof k. This spectral shape was predicted by Glauber [103], and it was
conﬁrmed in recent theoretical studies based on the quantum Langevinequations [58, 59] and the Master equations [60, 61, 62].

3.3.7 Purity
Quantum information processing requires high indistinguishability between single photons. A popular measure of indistinguishability ofphotons is the coincidence probability Pco in two-photon interference
experiments (see Fig. 3.4). When two indistinguishable photons are simultaneously input to a beamsplitter, they always appear at the same

CHAPTER 3. SINGLE-PHOTON GENERATOR

36

beam
splitteremittedphotondetectorsourceFigure 3.4: Schematic illustration of two-photon interference experiment.
The coincidence probability vanishes when the two input photons arecompletely indistinguishable.

output port (Hong–Ou–Mandel interference), namely, Pco = 0. However,pure dephasing generates entanglement between an emitted photon andthe environment of its source, and it makes two-photon interference imperfect (Pco > 0). The coincidence probability is related to the purity P
of a photon by Pco = (1 − P)/2 (see Appendix 3.A for the derivation).
The purity is deﬁned in terms of the density matrix ρ̂(t) byP≡

Tr[ρ̂2 (t)].
Tr[ρ̂(t)]2(3.45)As expected, this quantity becomes independent of t when t is suﬃciently
large. Using the real-space matrix element, the purity can be rewritten
as
R
P=

R
drdr′ |ρ(r, r′ , t)|2drdr′ ρ(r, r′ , t)ρ(r′ , r, t)
= R
R
2
2 .
drρ(r, r, t)drρ(r, r, t)(3.46)

3.4. NUMERICAL RESULTS

37

Using Eq. (3.36), we have
P=
Pn =
Pd =

Pn
,
Pd
4
2
X
X

2ρ jm ρ∗j′ m′j, j′ =1 m,m′ =1(µ j + µ∗j′ )(λm + λ∗m′ )(3.47)

4 X
2
X
ρ jm

−µ j
j=1 m=1.

,(3.48)(3.49)

Pd is an eﬃciency of two photons to go into the intended modes and
Pd = 1 if γ = 0.

3.4 Numerical results
The analytical results derived in the previous section are rigorous andapplicable to any set of parameters, (ωd −ωc , g, κ, γ, γp ). In this section, we
visualize these results by employing speciﬁc parameters. Throughout
this section, we assume for simplicity that radiative decay to unintended
modes does not occur (γ = 0).

3.4.1 Zeno and Anti-Zeno eﬀects
First, we observe the eﬀects of pure dephasing on decay of the dot
excitation. We focus on the weak-coupling regime (κ = 6g) in thissubsection, where the dot decays monotonically without revival andobeys the exponential decay law to a high accuracy. The decay rate ofthe dot is well deﬁned in this case and is given by Γ = limt→∞ [− log P(t)/t].
This reduces to min j |Re µ j |, where µ j is deﬁned in Sec. 3.3.3.
Figure 3.5 shows the temporal behavior of the survival probability
P(t). In Fig. 3.5(a), where the dot is in resonance with the cavity (ωd = ωc ),the decay becomes slower as pure dephasing increases. In contrast, inFig. 3.5(b), where the dot is detuned from the cavity (ωd − ωc = 600µeV), the decay becomes faster under small pure dephasing (γp = 200µeV), whereas the decay becomes slower under larger pure dephasing

CHAPTER 3. SINGLE-PHOTON GENERATOR

38
1.0

(b)

(a)

0.8
0.6
0.4
0.2
0.0
0

2

4

6

80

10

20

30

40

50

60

Figure 3.5: Survival probability S(t) for (a) resonant (ωd = ωc ) and (b) detuned (ωd − ωc = 600 µeV) cases. g = 25 µeV and κ = 150 µeV. The
values of γp are indicated in the ﬁgures.

(γp = 12800 µeV). Such suppression and enhancement of decay canbe interpreted as the quantum Zeno and anti-Zeno eﬀects, since puredephasing plays the same role as repeated measurements in destroying quantum coherence [102, 60]. Previous analyses predict that theanti-Zeno eﬀect can be observed only when the dot–cavity detuning islarge and pure dephasing is small [102]. This agrees with the present
numerical results.
Figure 3.6 shows the dependence of the radiative decay rate Γ on
the pure-dephasing rate γp . To clearly observe the Zeno and anti-Zenoeﬀects, Γ(γp ) is normalized by the free decay rate of Γ(0): Γ(γp )/Γ(0) < 1
indicates the Zeno eﬀect, whereas Γ(γp )/Γ(0) > 1 exhibits the anti-Zenoeﬀect. Non-monotonous behavior of Γ(γp ) is clearly observed for large
detuning.

3.4.2 Pulse shape and spectrum
In this subsection, we examine the pulse shape and the frequency spectrum of the emitted photon using the same parameters as those used in
the previous subsection. First, we observe the results for the resonant(ωd = ωc ) case. The pulse shapes f (r, t) of the emitted photon are shownin Fig. 3.7(a) for three pure dephasing rates γp . Each pulse shape is nor-

3.4. NUMERICAL RESULTS

1.2
1.0
0.8
0.6
0.4
0.2
0.00

39

Anti-Zeno
Zeno

125

250

375

Figure 3.6: Dependence of the radiative decay rate Γ̃ on the puredephasing rate γp . The radiative decay rate is normalized by the freedecay rate (i.e., the rate for γp = 0). g = 25 µeV and κ = 150 µeV.
ωd − ωc = 0 (solid), 66 µeV (dotted) and 150 µeV (dashed).

Rtmalized [ −∞ dr f (r, t) = 1] since γ = 0 is assumed here and a single photon
is necessarily generated in the b ﬁeld. The pulse becomes longer as puredephasing increases. This is consistent with the quantum Zeno eﬀect
discussed in the previous subsection: In the resonant case, the decay ofthe dot becomes monotonously slower with increasing pure dephasing.
Figure 3.7(b) shows the spectra S(k) of the photon for the same parameters as Fig. 3.7(a). These spectra have a single peak at k = ωd (= ωc ) andR∞
are normalized [ −∞ dkS(k) = 1]. The spectrum broadens with increasing
pure dephasing. This is conﬁrmed by Fig. 3.7(c) in which the spectralRwidth (deﬁned as ∆ = [ dk(k − ωd )2 S(k)]1/2 ) is plotted as a function of
the pure dephasing rate. Thus, the pulse broadens in both real and frequency spaces with increasing γp and it is thus not Fourier-limited. Thisimplies the mixedness of the emitted photon when γp , 0. The purity ofthe photon is discussed later in Sec. 3.4.3.
Next, we observe the results for the detuned case. Figure 3.8 (a)shows the pulse shapes of the photon. The pulse shape is approximately exponential except for the oscillatory behavior at the very initial

CHAPTER 3. SINGLE-PHOTON GENERATOR

40

0.4

1.0

(a)

0.3

0.6

0.2

0.4

0.1

0.2

0.0-8

(b)

0.8

-6

-4

0 0.0
-100

-2

150

0

-50

50

100

(c)

100
50
00

125

250

375

Figure 3.7: (a) Pulse shape f (r, t) and (b) spectrum S(k) of emitted photon.
g = 25 µeV, κ = 150 µeV and ωd − ωc = 0. γp = 0 (solid), 50 µeV (dotted),and 200 µeV (dashed). (c) Spectral width as a function of γp .

stage. The pulse length is inversely proportional to the decay rate ofthe dot. Figure 3.8(b) shows the photon spectra for the same parameters
as Fig. 3.8(a). A notable diﬀerence from the resonant cases is that the
spectra are doubly peaked with peaks at both the dot frequency ωd andthe cavity frequency ωc . The widths of these peaks are determined by
ed | = γ/2+γp and |Imω
ec | = κ/2. Therefore, the width of the peak at ωd|Imω
is sensitive to pure dephasing. When pure dephasing is weak (γp ≪ κ/2)
as in atomic cavity QED systems, the dominant peak of S(k) appears at ωd .
In contrast, when pure dephasing is strong (γp ≫ κ/2) as in solid-statesystems, the dominant peak of S(k) appears at ωc . This partly explains
the detuned peaks observed in the resonance ﬂuorescence spectrum insolid-state cavity QED systems.
R
The mean energy of the emitted photon is evaluated by Ep = dk kS(k).
Due to energy conservation, the mean photon energy is expected to al-3.4. NUMERICAL RESULTS

41

0.03 (a)10 (b)

0.02

0.1

0.01

0.001

0.00

-40

-30

-20

-10

0

-500

-250

0

250

500

Figure 3.8: (a) Pulse shape f (r, t) and (b) spectrum S(k) of emitted photon.
g = 25 µeV, κ = 150 µeV and ωd − ωc = 600 µeV. γp = 0 (solid), 50 µeV(dotted), and 200 µeV (dashed).

ways be identical to the dot frequency (i.e., Ep = ωd ). However, Fig. 3.8(b)clearly shows that the mean photon energy is sensitive to pure dephasing
and may deviate from the dot frequency when pure dephasing is present.
This discrepancy can be resolved by considering the energy released toRthe environment during decay, which is given by Ee = dk k⟨d†k dk ⟩. It canbe shown that (see Appendix 3.B for derivation)
2γp
κ
ωd +
ωc ,
κ + 2γpκ + 2γp2γp(ωd − ωc ).
Ee =κ + 2γpEp =(3.50)(3.51)

Thus, energy conservation is satisﬁed when the environmental energy
is included (Ep + Ee = ωd ). An important insight here is that while puredephasing coupling never induces a transition in the dot, this coupling
enables energy exchange between the dot and the environment. Whenthe cavity frequency exceeds the dot frequency, Eq. (3.51) indicates thatthe dot may absorb environmental energy during decay [104].

3.4.3 Purity
The pulse shape and spectrum presented in Figs. 3.7(a) and (b), respectively, indicate that the emitted photon is not Fourier limited when puredephasing is present and it is thus in a mixed state. Here, we observe

CHAPTER 3. SINGLE-PHOTON GENERATOR

42
1.0

(b)

(a)

0.8
0.6
0.4
0.2
0.0
-4
2.5 10

0.025

2.5

250

0.025

2.5

250

Figure 3.9: Dependences of the purity on γp . In (a), g and κ are ﬁxed
(g = 25 µeV, κ = 50 µeV) and the detuning is varied: ωd − ωc = 0(solid) and 200 µeV (dotted). In (b), g and ωd − ωc are ﬁxed (g = 25 µeV,
ωd − ωc = 0) and κ is varied: κ = 50 µeV (solid), 200 µeV (dotted) and 2.5µeV (dashed).

how the purity of the emitted photon depends on the system parameters.
The purity is approximately determined by the product of τ (dot excitation lifetime before it leaves the cavity) and γp (pure dephasing rate):
P ≃ 1 when γp τ ≪ 1, whereas P ≃ 0 when γp τ ≫ 1. In Fig. 3.9(a), thepurity is plotted as a function of γp by ﬁxing the escape rate of the cavity
photon (κ = 50 µeV) and varying the detuning ωd − ωc . As expected,
the purity is unity in the γp → 0 limit and decreases with increasing γp .
Detuning between the dot and cavity reduces the purity of the emitted
photon because the excitation lifetime τ increases with increasing detuning. In Fig. 3.9(b), the purity is plotted as a function of γp by assuming
there is no detuning (ωd = ωc ) and varying κ. The purity is maximized
when κ = 50 µeV, where is comparable to g. This can be understood as
follows: When κ ≪ g, the excitation lifetime τ increases and the purity
is degraded due to the long lasting Rabi oscillation between the dot andthe cavity. In contrast, when κ ≫ g, τ increases due to overdamping.
The optimal value of κ is more clearly seen in Fig. 3.10, where the purity is plotted as a function of κ for several values of γp . The purity is
maximized when κ is comparable to g, regardless of γp .

3.4. NUMERICAL RESULTS

43

1.0
0.8
0.6
0.4
0.2
0.0
0.25

2.5

25

250

Figure 3.10: Dependences of the purity on κ. g and ωd − ωc are ﬁxed
(g = 25 µeV, ωd − ωc = 0). γp = 2.5 µeV (solid), 25 µeV (dotted), and 250µeV (dashed).

3.4.4 Time ﬁltering
In order to increase the purity of emitted photons, one eﬀective way is
only to use photons emitted into an output channel in an early stage,
because such a photon has no time to be aﬀected by dephasing due tothe environment. The purity of photons ﬁltered in the temporal regime
0 ≤ t ≤ T is written as
P(T) =Pn (T) =Pn (T),
Pd (T)"∗
2ρ jm ρ∗j′ m′ 1 − e(µ j +µ j′ )t(3.52)#∗(µ +µ∗ )te(λm +λm′ )t − e j j′+,
λm + λ∗m′ − µ j − µ∗j′

(λm + λ∗m′ )µ j + µ∗j′

2
4 X
2
X
ρ jm
Pd (T) = (1 − eµ j t ) .
−µj
j=1 m=1(3.53)(3.54)The denominator Pd (T) is a square of a probability that a photon isemitted during 0 ≤ t ≤ T, and equals to a probability that both of two
photon sources produce one photon. We call Pd (T) a eﬃciency of timeﬁltering. Although the numerator Pn (T) is always smaller than Pn (∞),
the total purity P(T) is improved from Pd (∞) as the denominator Pd (T)becomes smaller than 1.

CHAPTER 3. SINGLE-PHOTON GENERATOR

44
1.0
0.8

1

(a)

0.6

0.01

0.4

0.001

0.2

10-4

0.0

(b)

0.1

10-5
0.025

2.5

250

0.025

2.5

250

Figure 3.11: (a) Purities P(T) and (b) eﬃciencies Pd (T) as a function of
γp . The detuning is taken as ωd − ωc = 0 (solid) and 200 µeV (dotted) byﬁxing g and κ (g = 25 µeV, κ = 50 µeV). In the ﬁgure (a), thin and thicklines show the purity without ﬁltering and with time ﬁltering (T = 2/g),
respectively.

Fig. 3.11(a) shows the improved purity when the ﬁltering time is taken
as T = 2/g, and other parameters are taken as the same as Fig. 3.9 (a).
Both in the on-resonant and oﬀ-resonant case, the purity is improvedby time ﬁltering and converged to a ﬁnite value in the limit of γp →∞. Fig. 3.11(b) shows the eﬃciency Pd (T). The eﬃciency decreases
exponentially in the limit of γp → ∞. This can be understood by increaseof the decay time due to the Zeno eﬀect.
In actual experiments, time ﬁltering with too small T is not realistic,
because the eﬃciency Pd (T) gets small down to an unacceptable level.
Fig. 3.12 shows the improved purity with ﬁxed eﬃciency (Pd (T) = 1/2)
as a function of γp . Maximum magniﬁcation of the purity is expectednaively to be an inverse of the eﬃciency. This maximum is achieved
when γp is large, however, in that case T is also large. Too large ﬁltering time T is also unrealistic, because in real experiment there is aspontaneous emission into a vacuum with a ﬁnite rate γ, neglected in
the present calculation. Probably, the optimum value of T is of order of
1/g. If we need photons whose purity P > 0.6, the pure dephasing rate
must be less than 0.03g without ﬁltering, however, with ﬁltering, puredephasing rate can be large within < 0.08g.

3.5. SUMMARY

0.8

500 (b)(a)Filtering Time T

1.0

45

100
50

0.6
0.4
0.2
0.0

0.25

25

2500

10
5
1

0.25

25

2500

Figure 3.12: (a) Dependences of the purity on γp . g, κ and ωd − ωc are
ﬁxed (g = 25 µeV, κ = 50 µeV ωd − ωc = 0). The solid and dottedline are the purity with time ﬁltering (T = 2/g and the ﬁxed eﬃciency
Pd = 0.5) and without time ﬁltering, respectively. (b) Dependences of
the corresponding ﬁltering time T on γp .

3.5 Summary
We analyzed radiative decay of an excited dot in a solid-state cavity
QED system and investigated the quantum-mechanical properties of theemitted photon, stressing the eﬀects of pure dephasing. Our analysis
is based on a model in which all the elements of the system (including
environmental ones) are treated as active quantum-mechanical degrees
of freedom. We rigorously solved the time evolution of the overallsystem, and derived analytical expressions for the density matrix, pulse
shape, spectrum, and purity of the emitted photon. These analyticalresults were visualized under realistic parameters. The main results are
summarized as follows: (i) Changes in the dot decay rate due to puredephasing can be explained in terms of quantum Zeno and anti-Zenoeﬀects. The emitted photon pulse length is approximately given by the
inverse of the dot decay rate. (ii) The emitted photon spectrum agrees
with the Glauber formula. The mean energy of an emitted photon is
not necessarily identical to that of the dot and energy conservation isseemingly broken. However, the present analysis revealed that the dot
can exchange energy with the environment through pure dephasing
coupling. Energy conservation holds when the environmental energy isCHAPTER 3. SINGLE-PHOTON GENERATOR

46

included. (iii) The purity of the emitted photon is calculated as a measureof indistinguishability. The purity is approximately determined by the
product of the pure dephasing rate and excitation lifetime before leaving
the cavity. When the pure dephasing rate γp is ﬁxed, the optimum
condition for maximizing the purity is ωd = ωc and κ ∼ g. (iv) Eﬀects
of time ﬁltering are discussed. The purity is improved by ﬁltering out
photons emitted later. In actual experiments, the optimum ﬁltering time
will depend on to what extent we can permit a lowered eﬃciency.

Appendix 3.A

Relation between coincidence probability and purity

Here we prove the relation Pco = (1 − P)/2 between the coincidence
probability Pco and the purity P. We consider the following situation(see Fig. 3.4). Two solid-state emitters (S1 and S2 ) respectively emit
single photons simultaneously and deterministically (assuming γ = 0
for simplicity) into two input ports (P1 and P2 ) of a beamsplitter. These
two photons are mixed by the beamsplitter and are output into the ports(P3 and P4 ). We denote the photon ﬁeld operators for the port P j by eb(j)r
( j = 1, · · · , 4), and the pure-dephasing reservoir operators for the emitterS j by de(j)r ( j = 1, 2). Assuming t → ∞ and γ = 0 (and therefore δm = 0) in
Eq. (3.13), the state vector of P1 photon is given by|ψ1 ⟩ =∞ Z
X

drdm xγm (t, r, x)eb†(1)r |x⟩,(3.55)

m=0

where |x⟩ = de†(1)x1 · · · de†(1)xm |0⟩. Thus, the emitted photon is entangled withthe environment of its source. The input state vector including both P1
and P2 photons is then given by|ψ12 ⟩ =∞ Z
Xdrdr′ dm xdn x′ γm (t, r, x)γn (t, r′ , x′ )eb†(1)reb†(2)r′ |x; x′ ⟩,m,n=0

where |x; x′ ⟩ = de†(1)x1 · · · de†(1)xm de†(2)x′ · · · de†(2)x′n |0⟩.
1

(3.56)

3.B. PROOF OF EQS. (3.50) AND (3.51)

47√
The beamsplitter mixes two photons as eb†(1)r → [eb†(3)r + eb†(4)r ]/ 2 and√eb†(2)r → [eb†(3)r − eb†(4)r ]/ 2, but obviously does not aﬀect the environmental
degrees of freedom. Then, the output state vector is given by |ψout ⟩ =|ψ33 ⟩ + |ψ44 ⟩ + |ψ34 ⟩, where
∞ Z
′
′
X
′ m
n ′ γm (t, r, x)γn (t, r , x ) e† e†|ψ33 ⟩ =b(3)r b(3)r′ |x; x′ ⟩,(3.57)
drdr d xd x
2
m,n=0
∞ Z
X
γm (t, r, x)γn (t, r′ , x′ ) e† e†|ψ44 ⟩ = −drdr′ dm xdn x′
b(4)r b(4)r′ |x; x′ ⟩, (3.58)
2
m,n=0
∞ Z
X
|ψ34 ⟩ =drdr′ dm xdn x′m,n=0γm (t, r, x)γn (t, r′ , x′ ) − γm (t, r′ , x)γn (t, r, x′ ) e† e†b(3)r b(4)r′ |x; x′ ⟩. (3.59)
2
It is readily conﬁrmed that ⟨ψ33 |ψ33 ⟩ + ⟨ψ44 |ψ44 ⟩ + ⟨ψ34 |ψ34 ⟩ = 1 and⟨ψ33 |ψ33 ⟩ = ⟨ψ44 |ψ44 ⟩. The coincidence probability Pco , namely, the probability to ﬁnd single photons in both ports 3 and 4, is given by Pco =⟨ψ34 |ψ34 ⟩. Since the density matrix element is given by∞ ZX′
ρ(r, r , t) =dm xγ∗m (r′ , x, t)γm (r, x, t),(3.60)
m=0

Pco is recast into the following form:
Z
1 1
1−P
P11 = −
drdr′ ρ(r, r′ , t)ρ(r′ , r, t) =.
(3.61)
2 2
2
When pure dephasing is present, the purity of the emitted photon becomes less than unity and the coincidence probability becomes nonzero.

Appendix 3.B

Proof of Eqs. (3.50) and (3.51)

R
R
Here we analytically evaluate Ep = dk k⟨b†k (t)bk (t)⟩i and Ee = dk k⟨d†k (t)dk (t)⟩iin the t → ∞ limit, where ⟨· · ·⟩i = ⟨ψi | · · · |ψi ⟩. Switching to the real-space
representations, we have
Z
D
E
i
Ep =
dr (∂reb†r )ebr − eb†r (∂reb†r ) ,
(3.62)
i
2Z
D
E
i
Ee =
dr (∂r de†r )der − de†r (∂r de†r ) .
(3.63)i2

CHAPTER 3. SINGLE-PHOTON GENERATOR

48

d †We can conﬁrm from Eq. (3.8) that ⟨(∂r b†r )br ⟩i = −κθ(t − τ)θ(τ)⟨( dτa )a⟩i ,where τ = t − r. Using similar equations, we have
Z
iκ ∞
d †
d
dτ ( dτEp = −a )a − a† ( dτa) i ,(3.64)
2 0
Z ∞
d †
d †
dτ ( dτσ σ)σ† σ − σ† σ( dτσ σ) i .
(3.65)
Ee = −iγp0Using Eqs. (3.11), (3.12) and their conjugates, we have
Z ∞
Z

gκ ∞
†
Ep = κωcdτ⟨a a⟩ +dτ ⟨a† σ⟩ + ⟨σ† a⟩ ,
2 0
Z 0∞

dτ ⟨a† σ⟩ + ⟨σ† a⟩ .
Ee = gγp(3.66)(3.67)

0R∞
R∞
Thus, we need to evaluate I1 = 0 dτ⟨σ† σ⟩, I2 = 0 dτ⟨a† a⟩, and I3 =R∞†
0 dτ⟨σ a⟩.
The equations of motion for ⟨σ† σ⟩, ⟨a† a⟩, and ⟨σ† a⟩ are given byd †⟨σ σ⟩ = −γ⟨σ† σ⟩ − ig(⟨σ† a⟩ − c.c.),dtd †⟨a a⟩ = −κ⟨a† a⟩ + ig(⟨σ† a⟩ − c.c.),dtd †e∗d − ωec )⟨σ† a⟩ + ig(⟨a† a⟩ − ⟨σ† σ⟩).
⟨σ a⟩ = i(ωdt(3.68)(3.69)(3.70)

Integrating these equations with respect to τ, we have 1 = γI1 + ig(I3 − I3∗ ),e∗d − ωec )I3 +ig(I2 −I1 ). When γ = 0, these equations
0 = κI2 −ig(I3 −I3∗ ), 0 = i(ωe∗c − ωed )/(ωec + ω
ed − ω
e∗c − ωe∗d ).
are solved to yield I2 = 1/κ and I3 = (i/g) × (ωSince Ep = κωc I2 +(gκ/2)(I3 +I3∗ ) and Ee = gγp (I3 +I3∗ ), we obtain Eqs. (3.50)
and (3.51).

Chapter 4
Single-Electron Generator
4.1 Introduction
On-demand coherent single electron source is a promising candidate fortransport of quantum information with mobile electrons [79]. For actual
application, however, we have to evaluate eﬀect of strong Coulombinteraction between a mobile electron and background (environment)
electrons, and to make eﬀorts for reducing its dephaing eﬀect. Whereasdephasing eﬀect due to environment noise has been studied by a simple
phenomenological model so far [88, 89], there is no theoretical study toclarify eﬀect of Coulomb interaction between electrons.
In this chapter, we consider dephasing eﬀect in single-electron generation caused by environment noise to a quantum dot, and by Coulombinteraction between electrons. The former calculation is almost the sameas the one for single-photon generation, and is helpful to understanddephasing process on electrons. For the latter calculation, we employ
a powerful diagrammatic method of the Keldysh type [105, 106]. Weshow that the present problem is closely related to the so-called Fermiedge singularity, which originates from anomalous edge eﬀect in X-ray
absorption [107, 108, 109, 110, 111]. We note that dephasing eﬀect toquantum states of a localized electron in a quantum dot has been studied
to explain strange nonequilibrium eﬀect in the Mach-Zehnder interferometer made by edge channels of the IQH state [112, 113, 114].
49

CHAPTER 4. SINGLE-ELECTRON GENERATOR

50

quantum dotedge state
Figure 4.1: A model of single electron generator.

This chapter is organized as follows. We investigate quantum natureof emitted electrons under dephasing eﬀect. We ﬁrst calculate survival
probabilities, spectra, and purities of injected electrons under environment noise in Sec. 4.2. Next, we calculate them under dephasing caused
by Coulomb interaction in Sec. 4.3. We ﬁnally summarize the results in
Sec. 4.4.

4.2 Pure dephasing due to environment noise
4.2.1 Model
Our model consists of a quantum dot connected to a chiral edge modein IQH states illustrated in Fig. 4.1, and is described by the Hamiltonian
H0 = Hdot + Hedge + Ht ,Hdot = ϵd d† d,Z
Hedge = dkka†k ak ,Ht = g(ã†0 d + d† ã0 ),

(4.1)
(4.2)
(4.3)
(4.4)

where Hdot , Hedge , and Ht express a quantum dot, a one-dimensional
chiral edge channel, and tunneling between a dot and a edge channel atthe origin. Here, ϵd is an energy level of a quantum dot, g is a tunneling
amplitude between the quantum dot and the one-dimensional chiraledge state, the wavenumber k of the edge state is measured from the4.2. PURE DEPHASING DUE TO ENVIRONMENT NOISE

51

Fermi wavenember, and the Fermi velocity is set to be unity. The level
broadening is given as Γ = g2 in this unit.
We consider the Hamiltonian H = H0 + Henv , where the latter denotes eﬀect of environment noise. Here, we employ a simple model forenvironment noise by using an additional one-dimensional bosonic port
as
Z
q
i
h
†
†
†
(4.5)
Henv = dk kck ck + γp /πd d(ck + ck ) ,where γp is a pure dephasing rate.

4.2.2 Result
Under the condition Γ ≪ ϵd , we can calculate analytically the survival probability P1 (t) = ⟨ψ(t)|d†1 d1 |ψ(t)⟩, the density matrix of emitted
electrons ρ(r, r′ , t) = ⟨ψ(t)|b†r′ br |ψ(t)⟩, the pulse proﬁle f (r, t) = ρ(r, r, t),
R R′
the spectrum S(k, t) = dr dr′ e−ik(r−r ) ρ(r, r′ , t), and the purity P(T) =R R ′R
dr dr ρ(r, r′ , t)ρ(r′ , r, T)/( drρ(r, r, T))2 (including to time-ﬁltering effect by looking at electrons emitted during 0 < t < T) in the same way ofChap. 3. The results are given as
P1 (t) = e−Γt ,ρ(r, r′ , t) = Γe−(iϵd +Γ/2+γp )(r′ −r)−Γ(t−r′ )

(4.6)
,

(r < r′ ),f (r, t) = Γe−Γ(t−r) ,Γ/2 + γp1,
π (k − ϵd )2 + (Γ/2 + γp )2


2
2Γ2
1 − e−2ΓT e−ΓT−2γp T − e−2ΓT
P(T) =−/ 1 − e−ΓT .
Γ + 2γp2Γ
Γ − 2γpS(k, t) =

(4.7)
(4.8)
(4.9)
(4.10)

We note that ρ(r, r′, t) for (r > r′ ) is obtained by the symmetric relation
ρ(r, r′ , t) = ρ∗ (r′ , r, t). We stress that the pulse proﬁle, which has beenmeasured in experiment of single electron generation by average current,
is not aﬀected by pure dephasing rate γp . In other words, we cannot get
information of pure dephasing from the pulse proﬁle. The spectrum
becomes a Lorenzian form with a peak at k = ϵd . The peak width is given
as Γ + 2γp , and is increased by introduction of pure dephasing. It is clear

52

CHAPTER 4. SINGLE-ELECTRON GENERATOR

that the emitted electron is not Fourier-limited in the presence the puredephasing as observed in calculation of Chap. 3.
For no time ﬁltering (T → ∞), the purity, which is a direct measureof coherence in two-particle interferometer, is given as
P ≡ P(T → ∞) =1.
1 + 2γp /Γ

(4.11)The purity monotonically varies from 1 to 0 as γp increases from 0 to ∞.
This is a main result for the case of environment noise.
Eﬀect of time ﬁltering is discussed in parallel to the one for single
photon generation. Eﬃciency of the ﬁltering are independent of puredephasing rate, and ﬁltering time T for a ﬁxed eﬃciency η is explicitly
derived as−1√ 
log 1 − η .
(4.12)T=Γ
The purity is improved as Eq. (4.10) by time ﬁltering paying a cost of
reduction of eﬃciency.

4.3 Eﬀect of Coulomb Interaction
4.3.1 Model
Next, we investigate eﬀect of pure dephasing caused by Coulomb interaction. The Hamiltonian for Coulomb interaction is taken as
HC = Ud† dã†0 ã0 ,(4.13)

where U is a local (screened) Coulomb interaction between an electron
in a dot and electrons in a edge channel at the origin.
If we focus on electrons in the edge channel, its Hamiltonian changes
depending on whether an electron in the dot is emitted into a edge
channel or not. We deﬁne the Hamiltonian of the edge channel for theempty dot (nd = 0) and the ﬁlled dot (nd = 1) asZ†
h0 = ϵd d d + dkka†k ak ,(4.14)
Z†(4.15)
h1 = ϵd d d + dkka†k ak + Uã†0 ã0 .

4.3. EFFECT OF COULOMB INTERACTION
(a)

(b)

53
(c)

t4
t8 t 7

t6 t5

t`2

0

t3 t2

t1

t`1

t

(d)

(e)
...
0

t

0

t

Figure 4.2: (a) Examples of diagrams (n = 8, n′ = 2); (b)(c) the diagrams
representing high energy process neglected in the present calculation;(d) a diagram of the self-energy Σ(t); (e) Dyson equation for the survival
probability.

Since we assumed local interaction, the eﬀect of the Coulomb interaction
can be expressed by the phase shift of electrons in the edge channel. We
assume that electrons gain a phase −δ if the dot is ﬁlled (nd = 1), whereδ > 0 for repulsive interaction (U > 0).

4.3.2 Method
In order to treat Coulomb interaction, we develop a ﬁeld-theoreticalmethod. The essence of our method is to express a arbitrary observablein the following form
Z t
dt1 · · · dtn dt′1 · · · dt′n′⟨O(t)⟩ =t1 >···>tn ,t′1 >···>tn′

ψ(0) V(t′n′ ) · · · V(t′1 )OV(t1 ) · · · V(tn ) ψ(0)(4.16)

as in a usual procedure deriving Keldysh Green’s functions [105, 106].
Here, V is a interaction representation of the tunneling Hamiltonian
Ht . Fig. 4.2 (a) shows an example of diagrams. The horizontal two

54

CHAPTER 4. SINGLE-ELECTRON GENERATOR

straight lines represent a state of a dot, and the other lines present aelectron propagating in the edge channel. In the present calculation, weonly treat the leading term for small tunneling amplitude g, and neglect
diagrams such as Fig.4.2 (b) and (c) because they describe high energy
processes [106].
The survival probability P1 (t) and the decay probability P0 (t) = 1 −P1 (t) are formulated by the sum of diagrams under a constraint thatboth of the dot states in the upper and lower horizontal line ends thestate nd = 1 and nd = 0, respectively. When we deﬁne the lowest-orderself-energy Σ(t) as shown in Fig. 4.2 (d), the survival probability P0 (t) is
calculated from the Dyson equation shown in Fig. 4.2 (e). It is convenient
to employ the Laplace transform to sum up diagrams appearing theDyson equation. The ﬁnal result of the Laplace transformation of P1 (t)and P0 (t) are given asZ ∞1,(4.17)P1 (λ) ≡e−λt P1 (t) =λ − Σ(λ)
0Z ∞−Σ(λ)/λ.
(4.18)P0 (λ) ≡e−λt P0 (t) =λ − Σ(λ)
0
In a similar way, the density matrix ρ(k, k̄, t) is formulated by the sum of
diagrams under a constraint that an electron with a wavelength k and ahole with a wavelength k̄ ends at a time t.
First, we consider the survival probability P1 (t). In the leading contribution for small g, the self-energy Σ(t) is calculated asΣ(t) = 2Re (−ig)2 eiϵd t eih1 t ã†0 e−ih0 t ã0 0 ,(4.19)


where ⟨· · · ⟩0 = Z1 ρ0 · · · , Z = trρ0 , and ρ0 = e−βϵ is a density matrix at
initial state which we here take as a product state of a ﬁlled dot (nd = 1)
and a Fermi sea at zero temperature. To calculate this average, we adopt
the method in Refs. [115, 116, 117]. We ﬁrst deﬁne an operator ŵ in the
single-particle Hilbert space by!
X
1
ρ0 eih1 t e−ih0 t = exp
wk,k′ a†k ak .
(4.20)
Z′k,k

4.3. EFFECT OF COULOMB INTERACTION

55

By using a relation ak e−ih1 t = e−ih1 t ak e−ikt and the determinant formuladerived recently [115], the average in the self-energy is calculated asZΣ(t) = −2g2 Re e−i(k−ϵd )t dkdk̄ det(1̂ + eŵ )(1̂ + e−ŵ )−1(4.21)kk̄ ,The determinant and the inverse (1̂+e−ŵ )kk̄ can be calculated by RiemannHilbert factorization in energy domain [117]. Performing the integral
respect to k̄, we obtain the self energy asΣ(t) = −2g2 Re eiϵd t L(t)D(t) ,(4.22)

2a
Z
dk
−ξ0
L(t) =(1 − n(k))
e−ikt ,
(4.23)
2π
k + iη
D(t) = (−iξ0 t)−a ,2(4.24)

where ξ0 is high energy cutoﬀ and a is a normalized phase shift a = δ/π,
η is a low-energy cutoﬀ appearing in the Riemann-Hilbert factorization,and n(k) is the Fermi distribution function at zero temperature. We notethat L and D represent open and closed diagrams in the work by Nozieres
and de Dominicis, respectively [108]. The Laplace transform of Σ(t) isthen given byZ
Σ(λ) ≡ dte−λt Σ(t)(4.25)"
a2 −2a #g2ξ= − Γ(1 − 2a)Γ(2a − a2 )Im e2iπa,(4.26)π
ϵd + iλwhere Γ(x) is the Gamma function.
Next, we consider density matrix of emitted electrons. To calculatedensity matrix ρ(k, k̄, t), the self-energies deﬁned by the diagrams in
Fig.4.3 (a) are needed. For example, Σ(1A)
is given bykk̄ZZdk′ dk′′ t(1A)Σkk̄ =dτ(−ig)(ig)eiϵd τ eih1 τ ak′′ eih0 (t−τ) a†k̄ ak e−ih0 t a†k′ 0(4.27)
2(2π) 0ZZ
dk′ dk′′ t′′(4.28)=
dτg2 eiϵd τ−ik τ−i(k−k̄)t eŵ ak′′ a†k̄ ak a†k′ 02(2π) 0ZZ
dk′ dk′′ t
2 iϵd τ−ik′′ τ−i(k−k̄)t
ŵ −1
=
dτg
e
det(1̂ + eŵ )(1̂ + eŵ )−1
k′′ k̄ (1̂ + e )kk′
(2π)2 0(4.29)

CHAPTER 4. SINGLE-ELECTRON GENERATOR

56

(a)

(b)

Figure 4.3: diagrams of Σkk̄

In a similar way,
Z
Σ(1B)
=
kk̄

dk′ dk′′
(2π)2

Z t
0

′′

ŵ −1
dτg2 eiϵd τ−ik τ−i(k−k̄)t det(1̂ + eŵ )(1̂ + e−ŵ )−1
k′′ k̄ (1̂ + e )kk′

(4.30)
Using a relation (1̂ + eŵ )−1+ (1̂ + e−ŵ )−1= δk′′ k̄ , we obtaink′′ k̄
k′′ k̄Σ(1)≡ Σ(1A)+ Σ(1B)
kk̄
kk̄
kk̄
Z
Z t
′
dk
=
dτg2 eiϵd τ−ik̄τ−i(k−k̄)t det(1̂ + eŵ )(1̂ + eŵ )−1
kk′(2π)2 0(4.31)(4.32)The Laplace transform of Σ(1)is derived as
kk̄
Z
(λ) =
Σ(1)
kk̄

(t)
dte−λt Σ(1)kk̄

g2
= i Γ(1 − a2 )ξ0



(4.33)
ξ0k + iη

2a1
λ + i(k − k̄)ξ0ϵd − k + iλ

1−a2
(4.34)

Σ(2)is obtained by complex conjugate of Σ(2)
.
kk̄
k̄k
We mention that hole excitation in edge state (k < 0) does not contribute the density matrix at all. This is because that the sign of the4.3. EFFECT OF COULOMB INTERACTION

57

phase shift is opposite to electrons due to attractive interaction. For theattractive scattering, singular behavior due to the Fermi surface eﬀect
is much more weak, and can be shown to be small in comparison with
contribution from electrons.
The density matrix of emitted electron is obtained by
ρ(k, k̄, λ) =Σkk̄ (λ)λ − Σ(λ)(4.35)In order to asymptotic behavior of the spectrum (t → ∞), we multiply λand take a limit λ → 0 and k → k̄. The spectrum can be written as
S(k) = 2Re[Λkk ]/(−Σ(0))

"

#
2

(4.36)

2a 
1−a
ξ0
ξ0
, (4.37)
k + iη
iη − (k − ϵd )a  1−a2 2
2g2
ξ0
ξ0
2
Γ(1 − a ) sin ϕ 2, (4.38)=(−Σ(0))ξ0k + η2(ϵd − k)2 + η2
η
η
ϕ = 2a tan−1 + (1 − a2 ) tan−1.
(4.39)kϵd − k−2g2
Γ(1 − a2 )Im=(−Σ(0))ξ0

where Λkk̄ ≡ limλ→0 λΣkk̄ and −Σ(0) behaves as a normalization factor of
the spectrum. We can check the following sum rule:Zdk−Σ(λ)/λΣkk (λ) =.
(4.40)2π
λ − Σ(λ)
This relation guarantee that the sum of the spectrum of emitted electrons
is equal to the decay probability P0 (t)(= 1 − P1 (t)).
Fig. 4.4 shows the spectrum of emitted electrons. If the phase shiftis small enough, the spectrum is Lorentzian which peak is at ϵd . With
increasing the phase shift, a singular peak near the Fermi sea grows
up, and the main peak at ϵd change its form into asymmetric one. We
note that the small peak near the Fermi sea is caused by closed loop
contribution in Nozieres’s work, whereas the growth of the peak near
the Fermi sea indicate degrade of the purity of the emitted electrons.
Finally, we evaluate the purity. We deﬁne ρ̃(λ) = ρ(λ + i(k̄ − k)) andΣ̃(λ) = Σ(λ + i(k̄ − k)) to take the time dependence for suﬃciently large t.

CHAPTER 4. SINGLE-ELECTRON GENERATOR
30 (b)

10.00 (a)

20

1.00

10

0.10
0.0

Spectrum S(k)Spectrum S(k)

58

0.5

1.0 1.5 2.0
k [units of g2]

0
0.0

0.5

1.0

1.5

2.0

2

k [units of g ]Figure 4.4: (a) A linear plot and (b) a log plot of spectrum S(k). Parameters
are taken as ϵd = g2 , η = 0.01g2 , and the phase shift is taken as a = δ/π =
0.1 (solid), 0.2 (dotted), and 0.3 (dashed).

We obtain ρ̃(λ) as

"

1−a2
2a 
ξ0
ξ0
ρ̃(λ) =
k + iη
ϵd − k̄ + iλ−iλ + k̄ − k + iΣ̃
2a 
1−a2 #ξ0
ξ0−,
ϵd − k − iλk̄ − iηgΓ(1 − a2 )/λξ0(4.41)

Asymptotic behavior of ρ̃ in t → ∞ is derived by limλ→0 λρ̃(λ), which
has singular points at k = k̄ and k̄ − k + iΣ(0) = 0. If we can neglect latter
term, the purity is ﬁnally written as
Z
dkdk̄
P=
ρ(k, k̄)ρ(k̄, k)(4.42)
(2π)2
Z
1
Σ(0)2
=
dxd
x̄
×
(4πB(1 − 2a, 2a − a2 ) sin(2aπ))2(x − x̄)2 + Σ(0)2
1
1
2 −
2a
1−a
2a
(x + iη) (1 − x̄ + iη)(x̄ − iη) (1 − x − iη)1−a2

2(4.43)

This is a main result for dephasing caused by Coulomb interaction.

4.4 Summary
We have discussed dephasing eﬀect on quantum coherence of single electrons injected from a quantum dot. We have shown that environment

4.4. SUMMARY

59

noise does not change the spatial proﬁle of emitted electrons, whereas
the spectrum and the purity is aﬀected by pure dephasing due to theenvironment. For dephasing caused by Coulomb interaction, we haveobtained a singular behavior in the spectrum of emitted electrons. This
singular behavior originates from the Fermi surface eﬀect, which is famous as a name of the Fermi edge singularity. We expect that thissingular change of quantum nature in emitted electrons strongly aﬀect
the degrade of the purity.

Chapter 5
Shot Noise and Fractional
Statistics
This chapter is organized as follows. After breif introduction in Sec. 5.1,
let us start with our model in Sec.5.2. An operator of quasi-particles inthe edge states is constructed by bosonization techniques of Hamiltonian
representing incompressible liquids. In Sec.5.3, non-equilibrium Kubo
formula in mesoscopic systems is introduced to deﬁne shot noise at ﬁnite
temperatures. In Sec.5.4, we discuss Fano factor (normalized shot noise)and propose a way to indirectly obtain statistical angle in hierarchicalFQH states.

5.1 Introduction
The fractional quantum Hall (FQH) eﬀect occurs in the two dimensional
electron system subject to a strong magnetic ﬁeld. A conﬁning geometry in low disorder samples makes the edges gapless modes, which
carries fractionally charged quasi-particles. The quasi-particles are characterized by exotic features of fractional charge and fractional statistics.
The fractional statistics is determined by a phase gained by adiabatic
exchange process of quasi-particles. The direct observation of theseproperties has been a stimulating problem. The fractional charge has
been observed both well by many of shot noise measurements[18, 19, 20,60

5.2. MODEL AND METHOD

61

120, 121, 122], which is now the well-established scheme.
Fractional statistics is also appealing and mutual statistics has beenmeasured. Other attempts were made in Ref.[124], and Refs.[125, 126]
where cross-correlation in three edge states was studied for the Laughlin
states at zero temperature. Kim extended their works into hierarchical FQH states at ﬁnite frequency and temperature, and then discussed
statistics for ν = 1/5, 2/5 [95]. It is well known that quasi-particles inν = 1/5, 2/5 FQH states have the same charge, but obey diﬀerent statistics. Shot noise measurement in the low-temperature limit succeeded
to conﬁrm the former [120], but failed to do the latter. Ref.[95] gave an
idea to the problem, but not realized in experiments. We revert back tothe standard set up, and discuss shot noise in two edge states instead ofthree ones. What is most important point is that ﬁnite temperature eﬀects
are considered on the basis of a recent work [127]. It is shown that theapproach enables us to detect the diﬀerence of statistics in ν = 1/5, 2/5
FQH states. Finally we discuss a method to determine statistical angle
itself in hierarchical FQH states.
This chapter is organized as follows. Let us start with our model in
Sec.5.2. An operator of quasi-particles in the edge states is constructedby bosonization techniques of Hamiltonian representing incompressible
liquids. In Sec.5.3, non-equilibrium Kubo formula in mesoscopic systems
is introduced to deﬁne shot noise at ﬁnite temperatures. In Sec.5.4,
we discuss Fano factor (normalized shot noise) and propose a way toindirectly obtain statistical angle in hierarchical FQH states.

5.2 Model and method
Fractional quantum hall liquid has energy gap in bulk states due to
Coulomb interaction and it is incompressible. In a system with conﬁning
geometry, edge states exist to satisfy gauge invariance. Canonical quantization of Hamiltonian representing density excitation at edges leads
that the density operator obeys Kac-Moody algebra. By making appropriate bosonic operators from the density operators, it is shown that theedge states can be written as chiral Tomonaga-Luttinger liquid. Oper-

62

CHAPTER 5. SHOT NOISE AND FRACTIONAL STATISTICS

Figure 5.1: Schematic illustration of setup to detect shot noise.

ators for electrons and quasi-particles can be composed and condition
from electrons’ anti-commutation relation shows that the quasi-particles
are of IQHE or Laughlin’s FQHE states.
Let us consider the quasi-particle tunneling through the QPC set at
x = 0 between edge states[18, 131, 72]. The gapless edge states are described by chiral Tomonaga-Luttinger liquids. We begin with Laughlin
states with ﬁlling fractions ν = 1/(2n + 1) for simplicity. The Hamiltonianis given by right/left going edge modes HR , HL and the tunneling part
HB :H = HR + HL + HB2Z
∂ϕR,L (x)vF ∞HR,L =
dx
,
π −∞
∂x

(5.1)

HB = tB ψ†R (0)ψL (0) + h.c.,

(5.3)

(5.2)

where we put Planck constant ℏ one, vF is the Fermi velocity, tB is thetunneling amplitude of quasi-particles, ϕR,L (x) are chiral boson ﬁelds.
The density operator obeys the U(1) Kac-Moody algebra. The operator
for electron can be composed by imposing that the electron obeys theanti-commutation relation. However, the same procedure cannot givethe operator for quasi-particle because it obeys a diﬀerent statistics. Wen

5.3. NON-EQUILIBRIUM KUBO FORMULA

63

introduced a duality transformation: ν → 1/ν, and thus ψR,L (x) represent
the vertex operators in c = 1 CFT as√
1
ψR,L (x) = √ e±ikF x : e∓i νϕR,L (x) : .
2π

(5.4)

This construction is also available for hierarchical Laughlin’s stateswith K-matrix. K-matrix allows us to write the condition for a ﬁlling
factor to make hierarchical states. In such a ﬁlling factor, edge states
have multi modes naively. A mode with the maximum exponent carries
charge current and other modes are charge neutral. If we neglect charge
neutral modes, we can construct an operator of quasi-particles by replaceν to an exponent α. In this study, we neglect charge neutral modes,
however, there is a theoretical work which states the charge neutralmodes becomes relevant at especially low temperature. We discuss laterin this relation.
The quasi-particle hopping with an unit charge e∗ generates backscattering current:ÎB ≡ ie∗ eiω0 t tB ψ†R (0)ψL (0) − e−iω0 t t∗B ψ†L (0)ψR (0) .

(5.5)

Following the discussion of a gauge transformation [72], source-drain
bias voltage V is incorporated into phase factor ω0 = e∗ V. The backscattering current and current noise are obtained as
Z
IB = ÎB (t) ,

S=

dt′



δÎB (t), δÎB (t′ ).

These quantities can be calculated on the basis of Schwinger-Keldysh
formalism(Appendix 5.B).

5.3 Non-equilibrium Kubo formula
In this section, let us introduce shot noise at ﬁnite temperatures, proposedin the context of a generalization from the Kubo formula: non-equlibrium
Kubo Formula.

64

CHAPTER 5. SHOT NOISE AND FRACTIONAL STATISTICS

5.3.1 Landauer formulaFirst, we start with a non-interacting case such as ν = 1. In this case,Landauer formula is correct and charge current IB0 and its noise S0 can be
written asZdω0T0 ( fL − fR ),(5.6)
IB = e
π
Z

dω 
0
2
S =e
T0 fL (1 − fL ) + fR (1 − fR )
π
Z

dω 
= e2
T0 ( fL − fR )2 ,(5.7)
π
where fL and fR are Fermi distribution functions of left and right reservoirs and transmission probability T0 = |tB |2 /v2F . At zero temperature,
thermal noise vanishes in the ﬁrst line of Eq. 5.7. The second line of
Eq. 5.7 clearly originates from non-equilibrium nature, called as shotnoise in Landauer formula. At ﬁnite temperatures, shot noise separates
from thermal noise. Thus it enables us to investigate shot noise in viewof thermal ﬂuctuation.

5.3.2 Interacting case
Next, we turn to the discussion about interacting case. It is well known
that for correlated electrons Landauer formula is not satisﬁed in principle. At zero temperature, thermal noise must vanish and non-equilibrium
shot noise can be deﬁned without any problems. However, correlation eﬀects generally merge thermal noise and shot noise at ﬁnite
temperatures??(Eq.5.48). interactions mix bias voltage dependence ofthermal noise and non-equilibrium noise at ﬁnite temperature. Thus,
there is arbitrariness about deﬁnition of shot noise. It is a problem whata kind of thermal noise at ﬁnite temperature should be subtracted fromtotal noise and shot noise.

5.3.3 Non-equilibrium Kubo formula
It is well known that the standard Kubo formula determines linear conductance. In contrast, a relation to diﬀerential conductance G under5.3. NON-EQUILIBRIUM KUBO FORMULA

65

ﬁnite bias voltages were derived [127]. Thus this formula was called as
the nonequilibrium Kubo formula in mesoscopic systems. This formula
is valid for any quantum transport systems with two reservoirs whose
Hamiltonian is described by
H = HL + HR + Hc + HcL + HcR .

(5.8)

HL and HR describes left/right reservoirs connected to ﬁnite conductor
Hc HcL and HcR describes couplings between the left/right reservoirs andthe conductor. The formula can be applied to various systems, including
quantum dots, quantum point contacts, and chiral Luttinger liquids on
the quantum Hall edge. Then, it was also proposed to deﬁne shot noiseat any temperature Sh as the following formula:Sh ≡ − ⟨{δI, e(δNL − δNR }⟩ ,(5.9)

where δA ≡ A − ⟨A⟩. I and NL,R are the charge current and number ofparticles in left/right reservoirs. It was proved that Sh in eq.(5.9) has
several aspects expected as shot noise. (i) In a non-interacting system, Sh
directly gives the Landauer-type shot noise at ﬁnite temperatures. (ii) At
zero temperature, Sh is agreement with the standard shot noise: current
noise S at T = 0. (iii) In the linear response regime, Sh = 0 and eq.(5.10)
reproduces the Nyquist-Johnson relation. As a result Sh is qualiﬁed as
shot noise at ﬁnite temperatures. Actually using Sh it was successful tostudy shot noise of the Kondo eﬀect in a quantum dot [128, 129, 130].
In this study, we apply the formula to the system represented by 5.1,
replacing I and e Eq. 5.9 to IB and e∗ respectively.
The nonequilibrium Kubo formula also satisﬁes the relation:
Sh = S − 4kB TG,(5.10)

where S is current noise, G = ∂V IB is diﬀerential conductance, kB is
Boltzmann constant, and T is temperature in reservoirs. This relationshows us what variations of thermal noise should be subtracted fromcurrent noise to deﬁne shot noise at ﬁnite temperatures in experiments.
Our framework gives a prospective way to study shot noise at ﬁnite

66

CHAPTER 5. SHOT NOISE AND FRACTIONAL STATISTICS

temperatures: Sh in eq.(5.9) is directly calculated, and its prediction
is examined through S − 4kB TG in experiments. In the following theapproach is applied to edge states.

5.4

Results and discussions

In this section, we discuss shot noise and fano factor at ﬁnite temperature.
When we show the graphs for speciﬁc parameters, we are taking an unitof |tB |2 /v2F = 1.

5.4.1

Current and Current Noise

Conventionally information on a fractional charge is extracted from Poisson noise. Thus, when the quasi-particle weakly transmits through the
QPC, it is suﬃcient to calculate these transport quantities up to the lowest
order O(t2B ). These expressions can be rewritten into Landauer forms:Zdω∗
IB = eT(ω)( fL − fR ),(5.11)
2πZdω
S = e∗ 2
T(ω)( fL (1 − fL ) + fR (1 − fR ))
π
Z
dω
∗2
+e
T(ω)( fL − fR )2 .
(5.12)π
The transmission probability is characterized by the t-matrix t(ω) ≡πtB ρ(ω):
T (ω) ≡ t∗ (ω − ω0 /2) t (ω + ω0 /2) ,tB
ρ(ω) =
cosh
vF



βω
2



2π
βvF

ν−1 |Γβων
+ i 2π
2

πΓ(ν)



|2

.

(5.13)

Here ρ(ω) ≡ −ImGr (ω) /π is the density of states (DOS) at x = 0 and Gr (ω)
is retarded Green’s function at x = 0. Figure 5.2 shows the ω-dependenceof transmission probability for ν = 1, 1/3, 1/5 for V = 1, 2. In the ν = 1
IQH state or non-interacting edge states, T(ω) in eq.(5.13) leads to a
constant |tB |2 /v2F and remains unchanged when a bias voltage is tuned.

5.4. RESULTS AND DISCUSSIONS

67

Τ(ω)

ν=1,ω =1,20
1.0
0.8 ν=1/3
ω =1
0.6 0 2
0.4
0.2
0.0
-2
-1

0

1

2

ω
Figure 5.2: Energy dependence of density of states ρ(ω) for ν = 1, 1/3, 1/5
and β = 1.

When electrons are non-interacting (ν = 1), the density of states areconstant. However, when Coulomb interactions are turned on (ν = 1/3),
T(ω) has double peaks at chemical potentials: ω = ±ω0 /2 = ±e∗ V/2.
In the ν = 1 IQH state or non-interacting edge states, T(ω) in eq.(5.13)
leads to a constant |tB |2 . Within our approximation, only the transmission
probability is renormalized by Coulomb interaction through the DOS. In
contrast the Fermi distribution function is unrenormalized as

fL,R ≡

1.
1 + exp(β(ω ± ω0 /2))

(5.14)

5.4.2 Fano factor and peak structure
Seemingly, the second line in eq.(5.12) might be interpreted as shot noisein view of a Landauer-formula sense. However, shot noise formula ineq.(5.9) modiﬁes the naive prediction. Following the same approxima-

68

CHAPTER 5. SHOT NOISE AND FRACTIONAL STATISTICS

tion, eq.(5.9) can be calculated, and rewritten into a Landauer-like form:
Sh = SL + δSL ,Z
dω∗2
SL = eT(ω)( fL − fR )2 ,πZdω
δSL = e∗ 2T(ω)( fL − fR )(yL − yR ),π(5.15)(5.16)

where SL represents a Landauer-type shot noise and δSL does the correction term. Here,
 
 

βω
βω
1
ν
1
y(ω) ≡ tanh
− Im Ψ
+i
,
2
2
π
2
2π
yL,R ≡ y (ω ± ω0 /2) ,(5.17)

where Ψ(z) is the digamma function. In case of ν = 1, because δSL = 0
and T(ω) = |tB |2 /v2F , it is exempliﬁed that Sh is equivalent to the Landauertype shot noise SL for ν = 1 at ﬁnite temperatures. Thus the correction
term δSL plays an essential role in FQH states.
Let us discuss the feature in view of the nonequilibrium Kubo formula. G = ∂V IB is calculated using eq.(5.11), and then the resulting G and
S are substituted into S − 4kB TG. Therefore, we conﬁrm that the result is
identical with Sh in eq.(5.15), so that the nonequilibrium Kubo formulaeq.(5.10) is satisﬁed. In the context it is found that δSL corresponds tothe V-derivative of T(ω). As shown in Fig.5.2, T(ω) depends on V.
To proceed a further discussion, we introduce the following Fanofactor:Fν ≡Sh.
2e∗ IB(5.18)The diﬀerent point compared to a standard Fano factor is to be normalized by an unit charge e∗ . In the low-temperature/high-bias limit, Sh /2IBconverges to e∗ , as discussed later. With the normalization factor at zerotemperature, it enables us to focus on thermal ﬂuctuation of shot noise.
Here in Fig.5.3 Fν for ν = 1/3 at a ﬁxed inverse temperature β = 1 iscompared to FLν and δFLν deﬁned by5.4. RESULTS AND DISCUSSIONS

69

F
Fν

1.2
1.0
0.8
0.6
0.4
0.2

FLν
δFLν
20

40

V
60

80

100

Figure 5.3: Bias dependence of Fν (solid line), FLν (dashed line) and δFLν(dashed-dotted line) for ν = 1/3 at β = 1SL,2e∗ IBδSLδFLν ≡ ∗ .
2e IB
FLν ≡

(5.19)(5.20)

FLν monotonously changes, on the other hand δFLν exhibits non-monotonous
behavior with increasing bias voltage V. The total Fano factor Fν which isthe sum of them converges to 1 in the high-bias limit. The fact represents
that the charge of the quasiparticle is e∗ because Fν is normalized by theunit charge. In contrast, we ﬁnd the peak structure at a ﬁnite bias which
originates from the correction noise δFLν .
If we regard the Fano factor as an eﬀective charge function, it turns outthat the enhancement from the unit charge occurs. Transmitted quasiparticles are seemed to tend to come together due to thermal ﬂuctuation.
The peak structure is a sign for carried charges to bunch induced bythermal ﬂuctuation. Therefore we call the eﬀect ”thermal bunching”.
Note that this ”thermal bunching” is diﬀerent from the bunching which
originates from statistics of quasi-particles [95].
Fig.5.4(a) shows Fν at a ﬁxed ν = 1/3 for several inverse temperatures
β. As lowering temperature, a peak position moves to a lower bias. What

70

CHAPTER 5. SHOT NOISE AND FRACTIONAL STATISTICS

(a)

(b)

Fν
1.2
1.0
0.8
0.6
0.4
0.2

Fν

ν=1
ν=1/3
ν=1/5
ν=1/7

2.0
1.5

β=0.5
β=1.0
β=2.0

5

10

15

20

1.0
25

V V 0.5
30

1

2

3

4

V

Figure 5.4: (a) Fν for β = 0.5, 1.0, 2.0 at ν = 1/3 (solid line, dashed line,
dash-dotted line); (b) Fν for ν = 1, 1/3, 1/5, 1/7 at β = 1 (solid line, dashedline, dash-dotted line, dotted line)

is universal nature is that the peak height is unchanged. On the other
hand in Fig.5.4(b) Fν at a ﬁxed β = 1 is drawn for some parameters ν.
The peak structure does not appear in the ν = 1 IQH state. The peakheight becomes larger for smaller ν, namely, stronger magnetic ﬁeld or
Coulomb interaction. Figure 5.5(a) shows ﬁlling factor dependence ofthe peak height. We ﬁnd that the peak height diverges in the limit ofν → 0 as,lim Fν |max →

ν→+0

1
,
πν

(5.21)and the right hand side approximates well even with ﬁnite ﬁllings. Fig1 e∗ V
ure 5.5(b) shows ﬁlling factor dependence of the peak position y ≡ 2π.
kB T
As easily seen, the peak position is equal to the ﬁlling factor in the limitof ν → 0.
Up to now, our discussion has been restricted to Laughlin stateswith ν = 1/(2n + 1). As said in the introduction, we would like toconsider statistics for ν = 1/5, 2/5. Here thus we extend our discussion
into hierarchical FQH states (ex.ν = 2/5, 3/7, 2/9, · · · ). Those states arecharacterized by ﬁlling fraction, unit charge and statistical angle:ν=2n(p − 1) + 1
p1e∗ = eθ=π2np + 1
2np + 1
2np + 1

(5.22)

5.4. RESULTS AND DISCUSSIONS
(b)
12
10
8
6
4
2
0
0.0

0.5

!
!
!
!
!
!
!
!
!
!
!
!
!
!

Peak position y

Peak height h

(a)

71

!
!
!
!
!

0.1

0.2

0.3

0.4

0.5

!

0.4
0.3
!

0.2
!

0.1

!
!
!!
!!!
!
!
!
!
!
!
!

0.0
0.0

!

0.1

fillingNufactor !

0.2

0.3

0.4

0.5

filling Nu
factor !

Figure 5.5: (a) Dots shows dependence of the peak height h of the Fanofactor Fν on the ﬁlling factor ν = 1/(1 + 2n) and the solid line shows
1 e∗ Vh = 1/πν. (b) Dots shows dependence of the peak position y = 2πofkB Tthe Fano factor Fν on the ﬁlling factor ν = 1/(1 + 2n) and the solid line
shows y = ν.

which are originally deﬁned through the K-matrix [131, 132] and n andp are positive integer. Our formalism developed above is described by
the quasi-particle Green function at x = 0. This treatment can be also
justiﬁed when multiple tunneling processes can be neglected (discusslater). Thus the extension changes the exponent ν to α in eq.(5.29):α=1.
p(2np + 1)

(5.23)

Furthermore, the current eq.(5.11), current noise eq.(5.12) and shotnoise eq.(5.15) have been expressed as a frequency-integral form, in
accordance with the concept of Landauer formula. Concerning the current and current noise, integrated results were derived for Laughlin
states[72]. According to the same idea, shot noise can be also calculated.
The result is straightforwardly extended into hierarchical FQH states,
and thus the Fano factor is governed by a scaling function:
 


2
1 e∗ VFα = Im Ψ α + i.
(5.24)π
2π kB T
This function is characterized by exponent α.

72

CHAPTER 5. SHOT NOISE AND FRACTIONAL STATISTICS

2.0

1.5

y

0.0
1.0

−0.5
0.5

0.5 −1.0

0.0
0.0

0.2

1.0

0.4

0.6

0.8

1.0

α
Figure 5.6: Contour plot of diﬀerential Fano factor ∂V Fα =-1.0,0.5,0,0.5,1.0. The thick line shows ∂V Fα = 0.

Taking advantage of the scaling form, let us reexamine the peakstructure of the Fano factor. We plot ∂V Fα taking α as a continuous
parameter(Fig.5.6). It is found that α < 1/2 is the suﬃcient condition for
emergence of peak. It is easily shown that α = 1/p(2np + 1) is less than
1/2, and thus the peak structure develops in all type of hierarchical FQH
states represented by Eq.(5.22).

5.4.3 Discussion
As an experimentally relevant case, let us discuss statistics in FQH stateswith ν = 1/5, 2/5. The quantities listed in eq.(5.22) are speciﬁcallyobtained for these states:ν = 1/5
ν = 2/5e∗1/5 = e/5e∗2/5 = e/5

θ1/5 = π/5(n, p) = (2, 1)

θ2/5 = 3π/5(n, p) = (1, 2)

As mentioned in the introduction, e∗1/5 = e∗2/5 = e/5 has been conﬁrmed
through shot noise measurement in the low-temperature limit[120]. How-

5.4. RESULTS AND DISCUSSIONS

73

ever, there still remain the problem on statistics.
To address the issue we begin with generic relations among e∗ ,θ,νand α in eqs.(5.22) and (5.23). Each of e∗ ,θ,ν and α is determined by two
integer parameters: n and p. Thus the independent quantities become
two of them, and others are given by them. Therefore the statistical angle
discussed here is represented in all type of hierarchical FQH states asihe(5.25)
θ=π 1−α ∗ −1 .
e
The result yields one-on-one relations between α and θ using e∗1/5 = e∗2/5 =
e/5:




1
θ1/5
1
θ2/5
α1/5 =
1−, α2/5 =1−.
(5.26)4π
4π
In conclusion, in order to see the diﬀerence of statistical angles there is a
way to discuss exponents α.
We substitute α1/5 , α2/5 in eq.(5.26) into eq.(5.24) for ν = 1/5, 2/5
respectively, and show the Fano factors for ν = 1/5, 2/5 in Fig.(5.7). In the
low-temperature/high-bias limit, both Fano factors converge to 1. This isa generic feature of the Fano factor normalized by unit charge e∗ . In thepresent case for ν = 1/5, 2/5, even if the normalization by e is considered,
the limiting values are equal: e∗1/5 /e = e∗2/5 /e = 1/5. It turns out that shotnoise in the low-temperature/high-bias limit cannot distinguish statistics.
What is striking is that the Fano factors of ν = 1/5, 2/5 have diﬀerence
at ﬁnite temperature/bias. The Fano factor is determined by observable
quantities thruogh eq.(5.10) and (5.18). Analyzing its Fano factor, it ispossible to distinguish statistics of θ1/5 and θ2/5 in experiments.
The estimation of exponent α has been already reported by analyzing the power-law dependence of tunneling current: I ∝ V α [21]. Our
approach makes it possible to obtain both α and e∗ in the shot noise measurement with the scaling function which does not contain non-universal
parameters eq.(5.24) .
In this study, we discuss Fano factor deﬁned by shot noise. It isdiﬃcult to subtract thermal noise at ﬁnite temperature appropriately.
Then, there is a naive question, ”Why we need non-equilibrium noise?

74

CHAPTER 5. SHOT NOISE AND FRACTIONAL STATISTICS

Fα
3.0
2.0
1.0
0.2

0.4

0.6

0.8

1.0

Figure 5.7: Fano factors of ν = 1/5 (Laughlin state) and ν = 2/5 (hierarchical FQH state). Both quasi-particles have the same fractional charge
e/5.

Isn’t the current noise power S suﬃcient?” To clarify this, a Fano factor
deﬁned as a ratio between total current noise S and backscattering current
IB is,

S= coth(ω0 β/2),2e∗ IB(5.27)and it does not include α. A subtle additional problem with this ratio isdiverging at V → 0 limit. Our shot noise does not diverge for any bias
voltages.
Finally we comment on a closely-related work by Ferarro et al.[133].
They pointed out that the tunneling is dominated by multiple particles
at T < T∗ , in contrast the single particle at T > T∗ . The dynamics ofneutral edge mode determines the crossover temperature T∗ , evaluatedas 50mK in the experiment[134]. Thus our treatment, which has focusedon the single-particle process, still stands in the region of T > T∗ .

5.5. SUMMARY

75

5.5 Summary
In summary the ﬁnite-temperature shot noise at FQH edge states has
been studied on the basis of the nonequilibrium Kubo formula. The
peak structure of Fano factor has been found at a bias voltage. We havenamed the phenomena ”thermal bunching” because this is a sign for
quasi-particles to weakly glue, mediated by thermal ﬂuctuation. The
phenomena has been determined by a scaling function characterized by
an exponent of quasi-particle Green function. In ν = 1/5, 2/5 FQH states,
exponents have been given by only statistical angles. Detecting thediscrepancy of Fano factors, one can measure the diﬀerence of statistics.
Finally we have proposed an indirect way to determine a statistical angle
from exponent ﬁtted to the scaling function and unit charge estimatedat suﬃciently low temperature within T > T∗

Appendix 5.A

Keldysh Green’s function

The quasi-particle Green’s function is deﬁned by
η η

1 2
GR,L
(x, t) ≡ −i TK ψR,L (x, tη1 )ψ†R,L (0, 0η2 ) .

(5.28)

Under the Hamiltonian(Eq. 5.1), this correlations can be calculated. The
greater and lesser part of Green function are
!ν
π
i
±i
βvF
,
(5.29)G±∓
R,L (x, t) =π
2π sinh βvF (−x + tvF ± iϵ)
where ϵ is an inﬁnitesimal positive number where η1 , η2 = ± represents±∓
a branch of the Keldysh contour. We also deﬁne G±∓
R,L (t) ≡ GR,L (0, t) tosimplify.
These Green’s functions satisfy symmetry-like relations,
η,η̄

η̄,η

Gr (−x, −t) = −Gr (x, t),
η ,η̄

η,η̄

Gr 1 (−x, −t) = G−r (x, t),
η,η̄

η,η̄

Gr (x, t) = G−r (x, −t).

(5.30)(5.31)(5.32)

76

CHAPTER 5. SHOT NOISE AND FRACTIONAL STATISTICS

These relations are alike to ”particle-hole symmetry”, ”parity symmetry”
and ”time reversal” respectively.
We also deﬁne Fourier transformationZ
G(ω) = dtG(t)eiωt .
(5.33)

Appendix 5.B Calcultion of Current and Current Noise PowerIn this appendix, we represent backscattering current IB and current
noise power S by Green’s functions. These quantity can be written by
 
 Z

1X
η1
IB =
TK ÎB (t ) exp −i dτ1 HB (τ1 ),2 ηK(5.34)

1

S(t, t1 ) = ÎB (t)ÎB (t1 ) + ÎB (t1 )ÎB (t) − 2 ÎB (t) ÎB (t1 ) ,
 Z

X 
η̄
=
TK ÎB (tη1 )ÎB (t11 ) exp −i dτ1 HB (τ1 )(5.35)(5.36)K

η1

S = S(ω = 0),(5.37)

where η̄ ≡ −η. We expand these quantity in tB to second order. The
backscattering current is
Z
D n
oE
X
  † η2IB
η2 ϵ1iϵω0 t+iϵ1 ω0 t1
† η1
η1 ϵ
=
ϵη2 dt1 e
TK ψR (t )ψL (t ) ψR (t1 )ψL (t1 ),e∗ |tB |2 /2 η η ϵϵ1 2

1(5.38)
where η1 , η2 are indices of Keldysh contour(±), ϵ, ϵ1 = ± and † η+ψR (t 1 )ψL (tη1 ) ≡ ψ†R (tη1 )ψL (tη1 ) † η−ψR (t 1 )ψL (tη1 ) ≡ ψ†L (tη1 )ψR (tη1 ).

(5.39)(5.40)

Using Wick’s theorem, we take apart these correlations into productsof Green’s functions. Terms such as ψL ψL and ψ†L ψ†L vanishes, so
ϵ = −ϵ1 . Resulting Green’s functions only depend on t − t1 , so we shall5.B. CALCULTION OF CURRENT AND CURRENT NOISE POWER 77
rewrite t − t1 into t.

X Z
 iωt η2 η1
IB
η η
η η
η η
dt
e GR (−t)GL1 2 (t) − e−iωt GR1 2 (t)GL2 1 (−t) ,
=
η
2
∗
2
e |tB | /2 η η
1 2

=

XZ

(5.41)η ηη ηdt(η2 − η1 )eiω0 t GR2 1 (−t)GL1 2 (t).

(5.42)

η1 η2

In the last line, we change t to −t and replace η1 and η2 of the last term.
With help of Eq.5.31, we obtainXZ η¯1 η1
2
IB
iω0 t
=
dte
η
G
(−t)
.
(5.43)
1
Re∗ |tB |2
η1

Similarly, the current noise power S can be represented by Green’s functions asX η̄1 η1
2
S(t)
=
−
2
cos(ω
t)
G
(−t)
,
(5.44)
0Re∗ 2 |tB |2
η
1

S
S(ω)
=
,
e |tB |2e∗ 2 |tB |2 ω→0XZ η̄ η
2= −2
dt cos(ω0 t) GR1 1 (−t) .
∗2(5.45)(5.46)

η1

These integrals can be performed by replacing t′ = t ∓ iϵ ± β/2 [72].
The results are 2ν−1

 
2
ω0 β
ω0 β
IB
1
2π
=
sinh
Γ ν+i
(5.47)
e∗ |tB |2 π2 v2νβ22πF 2ν−1

 
2
ω0 β
ω0 β
S(ω)
2
2π
=
cosh
Γ ν+i
.
(5.48)e∗ 2 |tB |2 π2 v2νβ2
2πFFourier transformation of the lasser and greater Green’s functions are
obtained by the same path-integral and we obtain a relationGηη̄ (ω) = ηi f η (ω)ρ(ω).

(5.49)

Using this relation, we can rewrite Eqs.5.43 and 5.46 into Landauer-likeform in Eqs.5.11 and 5.12 by straightforward calculation.

CHAPTER 5. SHOT NOISE AND FRACTIONAL STATISTICS

78

Appendix 5.C

Calculation of Shot Noise fromEq.(5.9)

Shot noise deﬁned by non-equilibrium Kubo formula can be calculated
from Sh = S − 4kB TG. However, if we calculate Sh from Eq.(5.9), weshould care a treatment for δN(t). Shot noise is deﬁned by Eq.5.9 andcan be written by
 Z

X 
η1
η̄1
IB = −2
TK δÎB (t )δQ(t ) exp −i dτ1 HB (τ1 ), (5.50)
Z

K

η1


1
dx N[ψ†R (x, t)ψR (x, t) − ψ†L (x, t)ψL (x, t)] ,(5.51)
2
where N means normal order. We represent Sh by Green’s function,
however, scrupulous care should be taken in handling δQ because of thenormalization. When applying Wick’s theorem, correlations which includes normal ordered terms should be normalized by operator productexpansion such as,
N(t) =

ηN[ψ†R (x, tη̄1 )ψR (x, tη̄1 )]ψ†R (0, tη1 )ψR (t, t12 )−1
ηTK ∂x ϕR (x, tη̄1 )ψ†R (0, tη1 )ψR (t, t12 )
=
2π
η̄ η
η̄ η
η η
= −(GR01 2 (x, t − t1 ) − GR01 1 (x, 0))GR02 1 (0, t1 − t),(5.52)(5.53)(5.54)

where Gr0 is non-interacting Green’s function and in the last line weapply operator product expansion. Thus Sh can be written as
X ηη ZSh=iτz 2 2 dxdt2 cos(ω0 t)Gη1 η2 (t)Gη2 η1 (−t)
e∗ 2 |tB |2
η1 η2
X η̄ ηη̄ η×Gr01 2 (x, t) − Gr01 1 (x, 0) .
(5.55)rη̄ ηη̄ ηIntegration of Gr01 2 (x, t) − Gr01 1 (x, 0) respect to x works as a kernel for

integral respect to t. When performing integral respect to t, singularities
are relevant, however, above expression is not suitable for extract the
singularities. In order to treat the singularities properly, we replace thex depending terms as
η̄ η

η̄ η

η̄ η

η η̄

η η

Gr01 2 (x, t) − Gr 1 1 (x, 0) → −Gr01 1 (x, 0)Gr02 1 (−x, −t)Gr 2 1 (0, −t),(5.56)

5.C. CALCULATION OF SHOT NOISE FROM EQ.(5.9)

79where Gr is an pointer of singularities and equal to inverse of Gr0 near
the singularities. To proceed calculation, we use next relations (as shown
later),
Z
 vF t G−+ (0, −t) (r = R)
R0
η̄η
ηη̄
β
(5.57)
dxGr0 (x, 0)Gr0 (−x, t) =v
t
 F G+− (0, −t) (r = L)
β

Z

η̄η

η̄η̄

dxGr0 (x, 0)Gr0 (−x, t) =R0 vF t G−+ (0, −t) − θ(t)iηGη̄η (0, −t)

R0
R0
β
v
t
+−
F
 G (0, −t) − θ(−t)iηGη̄η (0, −t)R0R0β(r = R)(r = L)(5.58)First, we consider a case of η1 = η2 = η. Using Eqs.(5.57,5.57),
XZ
η̄η
ηη̄
ηη
dxGr0 (x, 0)Gr0 (−x, −t)Gr0 (0, −t)rvF t −+(GR0 (0, −t) + G+−
R0 (0, −t))×βππ(θ(−ηt) sinh (t + iηϵ) + θ(ηt) sinh (t − iηϵ)),
β
β





t + iηϵ t + iηϵ
t − iηϵ t − iηϵvF t=
θ(−ηt)++ θ(ηt)+βt + iϵt − iϵt + iϵt − iϵ2vF t(5.59)=β=In these transformations, we picked up singular parts near τ = 0 whichsurvive when we integrate respect to t and used a relation,x + iη2 ϵη2 − η1→1+2πixδ(x).
x + iη1 ϵ
2(5.60)

Next, we see a case of −η1 = η2 = η. In the same way in Eq.(5.59), using
Eqs.(5.58), we obtain
XZ
2vF t
η̄η̄
ηη
η̄η
− iη
(5.61)
dxGr0 (x, 0)Gr0 (−x, −t)Gr0 (0, −t) =βr
Eq.(5.55) can be calculated by using Eqs.(5.56,5.59) and (5.61). As easily
seen, the part of η1 = η2 = η (Eq.5.59) vanishes, because the integrand is

CHAPTER 5. SHOT NOISE AND FRACTIONAL STATISTICS80

odd in τ. Only the part of −η1 = η2 = η (Eq.) survives and we obtain
Z
Z
X
2
2
Sh
−2t X
ηη̄
ηη̄
=
i
dt2
cos
ω
t
η
G
(0,
t)
−
dt2
cos
ω
t
η
G
(0,
t)
0
0
e∗ 2 |tB |2
β η
η
=

S
e∗ 2 |tB |2−

4 ∂ IB,βe∗ ∂V e∗ |tB |2(5.62)

here we used∂∂sin ω0 t = ∗sin ω0 t.
(5.63)∂ω0e ∂VEq. (5.62) reproduces non-equilibrium Kubo formula Sh = S − 4kB TG.
We prove Eqs.(5.57,5.58). First, we prove of Eq.(5.57). We show theprove only for r = R. The left hand side of Eq.(5.57) is
Z
Z
−1 η
η̄η
ηη̄
g (x)gη (x − vF t),(5.64)
dxGR0 (x, 0)GR0 (−x, −t) = dx2(2π)
where gη (x) ist cos ω0 t =η

g (x) ≡πβsinh πβ (x + iηϵ).

We deﬁne the Fourier transformation of g asZ1
ηg (k) =
dxeikx gη (x)2π= −ηi f (ηk),
1f (k) =.
1 + eβk
We substitute Eq.(5.66) to Eq.(5.64).
Z
η̄η
ηη̄
dxGR0 (x, 0)GR0 (−x, −t)Zdk=f (ηk) f (−ηk)e−ikvF t
2πZ−1
dk f ′ (k)e−ikvF t
=
2πβ


Z

−1 
=
f (k)e−ikvF t − dk f (k)(−ivF t)e−ikvF t
2πβ
Z
vF t
dk
=
(−i f (k))e−ikvF t
β
2πvF t −+
G (0. − t).
=β R(5.65)(5.66)(5.67)(5.68)(5.69)(5.70)(5.71)(5.72)(5.73)

5.C. CALCULATION OF SHOT NOISE FROM EQ.(5.9)
In Eq.(5.70), we use f (ηk) f (−ηk) = − f ′ (k)/β.
Next, we prove Eq.(5.58). The left hand side of Eq.(5.58) is
Z
Z
−1 η
η̄η
η̄η̄
dxGR0 (x, 0)GR0 (−x, −t) = θ(−τ) dx
g (x)gη (x − vF t)
2
(2π)
Z
−1 −η
g (x)g−η (x − vF t).
+ θ(τ) dx
2
(2π)

81

(5.74)

The ﬁrst term of the right hand side is equal to θ(−t)×Eq.(5.57) and thesecond term isZdk− θ(t)f (ηk)2 e−ikvF t(5.75)
2π
Z
Z
dk
θ(t)
−ikvF t= −ηiθ(t)(−ηi f (ηk))e−
dk f ′ (k)e−ikvF t(5.76)
2π
2πβ
vF t
η̄η
(0, −t)(5.77)= −ηiθ(t)GR (0, −t) + θ(t) G−+β R
In Eq.(5.76), we use a relation f (ηk)2 = f (ηk)+ f ′ (k)/β. Thus we substitutethe Eq.(5.77) to Eq.(5.74) and obtainZ
vF t −+
η̄η
η̄η
η̄η̄
G (0, −t) − ηiθ(t)GR (0, −t).
(5.78)
dxGR0 (x, 0)GR0 (−x, −t) =βChapter 6
Summary
In this thesis, we have theoretically investigated three subjects related to
the Hong-Ou-Mandel-type experiment to characterize statistics of mobile particles. We summarize the results obtained in this thesis as wellas problems left for a future study.
In Chap. 3, we have studied dephasing eﬀect on single photon generator made in the semiconductor cavity QED [23]. By employing a simple
model with one-dimensional bosonic channel, we have obtained analytical results for quantum nature of photons emitted from single-photon
generator. For realistic experimental parameters, we have evaluated survival probabilities, spectra, and purities as a function of various parameters, and have discussed optimal condition for generation of coherent
single photons. Preliminary results on improvement of purities by timeﬁltering have also shown. Detailed discussion on eﬀect of various ﬁltering technique, which is useful for realization of excellent single-photon
emission for quantum information, is an important future problem. The
theoretical treatment of the non-Markovian noise such as 1/ f noise is also
a important problem left for future study to describe superconducting
cavity QED systems.
In Chap. 4, we have studied a fermionic analogy of single-photon
generation, i.e., single-electron generation into an integer quantum Halledge state. We have considered dephasing due to level ﬂuctuation andCoulomb interaction between an electron in a quantum dot and elec82

83
trons in a edge state. In particular for the latter eﬀect, we have derived
ﬁeld-theoretical method to approach this problem including nontrivial Fermi-surface eﬀects such as Anderson orthogonality theorem andthe Fermi-edge singularity. We have calculated survival probabilities,
spectra, and purities, and have discussed how environment noise andCoulomb interaction with electrons in an edge channel degrades quantum coherence of electrons. For further description in actual single
electron generation, ﬁnite-temperature eﬀect, coexistence of noise andCoulomb interaction, and dephasing eﬀect on several theoretical proposals for quantum information processing should be studied in moredetail, and are left for a future problem.
In Chap. 5, we have studied current and current noise at ﬁnite temperatures under weak reﬂection between two fractional quantum Halledge states [24]. We have deﬁned an (extended) Fano factor at ﬁnite temperatures, and have calculated it for dominant channel of backscatteringchannel at moderate temperatures for which neutral modes are irrelevant. We have demonstrate that this extended Fano factor is useful todetermine a statistical angle, which characterizes the exchange relation
between two fractional charge excitations. Extension of this discussion
toward non-Abelian statistics followed by quasi-particles of ν = 5/2
fractional quantum Hall states is an important future problem.

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