Nonequilibrium plasmons and transport properties of a ... · Keywords: quantum wire, Luttinger liquid, shot noise, full counting statistics. 1. Introduction The recent discovery of

Fizika Nizkikh Temperatur, 2006, v. 32, No. 12, p. 1522–1544

Nonequilibrium plasmons and transport properties of a

double-junction quantum wire

Jaeuk U. Kim1, Mahn-Soo Choi2, Ilya V. Krive3,4, and Jari M. Kinaret3

1Department of Physics, G�teborg University, SE-412 96 G�teborg, Sweden

2Department of Physics, Korea University, Seoul 136-701, Korea

3Department of Applied Physics, Chalmers University of Technology, SE-412 96 G�teborg, Sweden

4B. Verkin Institute for Low Temperature Physics and Engineering, 47 Lenin Ave., Kharkov 61103, Ukraine

Received January 3, 2006, revised January 22, 2006

We study theoretically the current-voltage characteristics, shot noise, and full counting statis-tics of a quantum wire double barrier structure. We model each wire segment by a spinlessLuttinger liquid. Within the sequential tunneling approach, we describe the system’s dynamicsusing a master equation. We show that at finite bias the nonequilibrium distribution of plasmonsin the central wire segment leads to increased average current, enhanced shot noise, and full count-ing statistics corresponding to a super-Poissonian process. These effects are particularly pro-nounced in the strong interaction regime, while in the noninteracting case we recover results ob-tained earlier using detailed balance arguments.

PACS: 71.10.Pm, 72.70.+m, 73.23.Hk, 73.63.–b

Keywords: quantum wire, Luttinger liquid, shot noise, full counting statistics.

1. Introduction

The recent discovery of novel one-dimensional(1D) conductors with non-Fermi liquid behaviors hasinspired extensive research activities both in theoryand experiment. The generic behavior of electrons in1D conductors is well described by the Luttingerliquid (LL) theory, a generalization of the Tomo-naga–Luttinger (TL) model [1–3]. Luttinger liquidsare clearly distinguished from Fermi liquids by manyinteresting characteristics. The most important exam-ples among others would be (a) the bosonic nature ofthe elementary excitations (i.e., the collective densityfluctuations) [4], (b) the power-law behavior of cor-relations with interaction dependent exponents [5–8],and (c) the spin-charge separation [9]. All these cha-racteristics cast direct impacts on transport propertiesof an 1D system of interacting electrons, including theshot noise and the full counting statistics as we discussin this paper.

In this paper we will consider a particular struc-ture, namely, the single-electron transistor (SET),made of 1D quantum wires (QWs). A conventional

SET with noninteracting electrodes itself is one of themost extensively studied devices in recent years invarious contexts [10]. The SET of interacting 1DQWs has attracted renewed interest as a tunable de-vice to test the LL theory and thereby improve the un-derstanding of the 1D interacting systems. In parti-cular, the recent experimental reports on thetemperature dependence of the resonance level widthin a SET of semiconductor QWs [11] and on the tem-perature dependence in a SWNT SET [12] stirred acontroversy, with the former consistent with the con-ventional sequential tunneling picture [13,14] and thelatter supporting the correlated sequential tunnelingpicture [12,15]. The issue motivated the more recenttheoretical works based on a dynamical quantumMonte Carlo method [16] and on a function renor-malization group method [17–20], and still remaincontroversial.

Another interesting issue on the SET structure of1D QWs is the effects of the plasmon modes in thecentral QW [21–24]. It was suggested that theplasmon excitations in the central QW leads to a

© Jaeuk U. Kim, Mahn-Soo Choi, Ilya V. Krive, and Jari M. Kinaret, 2006

power-law behavior of the differential conductancewith sharp peaks at resonances to the plasmon modes[21]. The plasmon excitations were also ascribed tothe shot noise characteristics reflecting the strong LLcorrelations in the device [22]. Note that in these bothworks, they assumed fast relaxation of plasmon exci-tations. We hereafter refer to this approach as «equi-librium plasmons». However, it is not clear, espe-cially, in the presence of strong bias (eV k TB�� ) howwell the assumption of equilibrium plasmons can bejustified. This question is important since the plasmonmodes excited by electron tunneling events can influ-ence the subsequent tunnelings of electrons and henceact as an additional source of fluctuations in current.In other words, while the average conductance maynot be affected significantly, the effects of the«nonequilibrium plasmons» can be substantial on shotnoise characteristics [25–31]. Indeed, a recent worksuggest that the nonlinear distribution of the plasmonexcitations itself is of considerable interest [23] and itcan affect the transport properties through the de-vices, especially the shot noise, significantly [23,24].Moreover, even if the precise mechanism of plasmonrelaxation in nanoscale structures is not well knownand it is difficult to estimate its rate, recent computersimulations on carbon nanotubes indicate that theplasmon life time could be of the order of a picosec-ond, much longer than those in three-dimensionalstructures [32].

It is therefore valuable to investigate systemati-cally the effects of nonequilibrium distribution ofplasmon excitations with finite relaxation rate on thetransport properties through a SET of 1D QWs. Inthis work we will focus on the shot noise (SN) charac-teristics and full counting statistics (FCS), which aremore sensitive to the nonequilibrium plasmon excita-tions than average current-voltage characteristics. Wefound that the SN characteristics and the FCS both

indicate that the fluctuations in the current throughthe devices is highly super-Poissonian. We ascribe thiseffects to the additional conduction channels via exci-tation of the plasmon modes. To demonstrate this rig-orously, we present both analytic expression of simpli-fied approximate models and numerical results for thefull model system. Interestingly, the enhancement ofthe noise and hence the super-Poissonian character ofthe FCS is more severe in the strong interaction limit.Further more, the sensitive dependence of the super-Poissonian shot noise on the nonequilibrium plasmonexcitations and their relaxation may provide a usefultool to investigate the plasmon relaxation phenomenain 1D QWs.

The paper is organized as following: In Sec. 2 weintroduce our model for a 1D QW SET and briefly ex-amine the basic properties of the tunneling rateswithin the golden rule approximation. We also intro-duce the master equation approach to be used through-out the paper, and discuss the possible experimentalrealizations. Before going to the main parts of the pa-per, in Sec. 3, we first review the results of the previ-ous work [23], namely, the nonequilibrium distribu-tion of the plasmons in the central QW in the limit ofvanishing plasmon relaxation. This property will beuseful to understand the results in the subsequent sec-tions. We then proceed to investigate the consequenceof the nonequilibrium plasmons in the context of thetransport properties. We first consider average currentin Sec. 4, and then discuss shot noise in Sec. 5.Finally, we investigate full counting statistics inSec. 6. Section 7 concludes the paper.

2. Formalism

The electric transport of a double barrier structurein the (incoherent) sequential tunneling regime can bedescribed by the master equation [14,33,34]

��

� � � � � � � �tP N n t N n N n P N n t N( , { }, ) [ ( , { } , { }) ( , { }, ) ( , {� � n N n P N n t

nN

� ��

�� } , { }) ( , { }, )]{ }

, (1)

where P N n t( , { }, ) is the probability that at time t thereare N (excess) electrons and { } ( , , ..., , ...)n n n nm� 1 2plasmon excitations (i.e., collective charge excitations),that is, nm plasmons in the mode m on the quantumdot. The transitions occur via single-electron tunnelingthrough the left (L) or right (R) junctions (see Fig. 1).The total transition rates � in master equation (1) aresums of the two transition rates �L and �R where�L/R N n N n( , { } , { })� � � is the transition rate from a

quantum state ( , { })N n� � to another quantum state( , { })N n via electron tunneling through L/R-junction.

Master equation (1) implies that, with known transi-tion rates, the occupation probabilities P N n t( , { }, ) canbe obtained by solving a set of linear first order differen-tial equations with the probability conservation

P N n tN n

( , { }, ),{ }� � 1.

Nonequilibrium plasmons and transport properties of a double-junction quantum wire

Fizika Nizkikh Temperatur, 2006, v. 32, No. 12 1523

In the long time limit the system converges to asteady-state with probability distribution

lim ( , { }, )t

P N n t��

� P N nst ( , { }),

irrespective of the initial preparation of the system.To calculate the transition rates, we start from the

Hamiltonian of the system. The reservoir temperatureis assumed zero (T � 0), unless it is stated explicitly.

2.1. Model and Hamiltonian

The system we consider is a 1D quantum wire SET.Schematic description of the system is that a finitewire segment, which we call a quantum dot, is weaklycoupled to two long wires as depicted in Fig. 1. Thechemical potential of the quantum dot is controlled bythe gate voltage (VG) via a capacitively coupled gateelectrode. In the low-energy regime, physical proper-ties of the metallic conductors are well described bylinearized dispersion relations near the Fermi points,which allows us to adopt the Tomonaga–LuttingerHamiltonian for each wire segment. We model thesystem with two semi-infinite LL leads and a finite LLfor the central segment. The leads are adiabaticallyconnected to reservoirs which keep them in internalequilibria. The chemical potentials of the leads arecontrolled by source-drain voltage (V), and the wiresare weakly coupled so that the single-electron tunnel-ing is the dominant charge transport mechanism, i.e.,we are interested in the sequential tunneling regime.Rigorously speaking, the voltage drop between thetwo leads (V) deviates from the voltage drop betweenthe left and right reservoirs (sayU) if electron trans-port is activated [35,36]. However, as long as the tun-neling amplitudes through the junctions (barriers) areweak so that the Fermi golden rule approach is appro-priate, we estimate V U� .

The total Hamiltonian of the system is then givenby the sum of the bosonized LL Hamiltonian� � � �H H H HL D R0 � accounting for three isolated

wire segments labeled by � � ( , , )L D R , and the tunnel-ing Hamiltonian �HT accounting for single-electronhops through the junctions L and R at XL and XR, re-spectively:

� � � .H H HT� 0 (2)

Using standard bosonization technique, the Hamil-tonian describing each wire segment can be expressedin terms of creation and annihilation operators for col-lective excitations ( �†b and �b) [4,37]. For the semi-in-finite leads, it reads

� � �,†

,H mb b L RM

mm

m��

� �

�

� ��

� �1 1

for = , , (3)

where the index labels the M transport sectors of theconductor and m the wave-like collective excitations oneach transport sector. The effects of the Coulomb inter-action in 1D wire are characterized by the Luttinger pa-rameter g�: g � 1 for noninteracting Fermi gas and0 1� �g for the repulsive interactions (g �� 1 in thestrong interaction limit). Accordingly, the velocities ofthe collective excitations are also renormalized asv v /gF� �� . The energy of an elementary excitation insector is given by � �� v /L where L is the lengthof the wire and � the Planck constant. For instance, ifthe wire has a single transport channel (usually referredto as spinless electrons), e.g., a wire with one transportchannel under a strong magnetic field, the system’s dy-namics is determined by collective charge excitations(plasmons) alone ( �� and M � 1). If, however, thespin degrees of freedom survive, the wire has two trans-port sectors (M � 2); plasmons ( �� ) and spin-waves( �� ) [37]. If the system has two transport channelswith electrons carrying spin (M � 4), as is the case withSWNTs, the transport sectors are total-charge-plasmons( �� ), relative-charge-plasmons ( �� ), total-spin-waves ( �� , and relative-spin-waves ( �� ) [38,39].

For the short central segment, the zero-mode needto be accounted for as well, which yields

� � �,†

,H mb bDm

m m� � ��

�

��

� �1

� ��

��

�

�� 2 2

2 2

MgN N

MgN EG r( � ) � . (4)

In the second line of Eq. (4), which represents thezero-mode energy of the quantum dot, the operator �N�measures the ground-state charge, i.e., with no excita-tions, in the -sector. The zero-mode energy systemati-cally incorporates Coulomb interaction in terms of the

1524 Fizika Nizkikh Temperatur, 2006, v. 32, No. 12

Jaeuk U. Kim, Mahn-Soo Choi, Ilya V. Krive, and Jari M. Kinaret

3 quantum wiresconnected to reservoirs

Fig. 1. Model system. Two long wires are adiabaticallyconnected to reservoirs and a short wire is weakly coupledto the two leads. Tunneling resistances at junction pointsXL and XR are RL/R, and the junction capacitances areconsidered equal C CL R� . Quantum dot is capacitivelycoupled to the gate electrode.

Luttinger parameter g� in the QD. To refer to thezero-mode energy later in this paper, we define the«charging energy» EC as the minimum energy cost toadd an excess electron to the QD in the off-Coulombblockade regime, i.e.,

E E E /gC D D N /G� � ��[ ( , ) ( , )] .2 0 1 0 1 2 �� (5)

Note that this is twice the conventional definition,and includes the effects of finite level spacing. Hence,the charging energy is smallest in the noninteractinglimit (g� � 1) and becomes the governing energy scalein the strong interaction limit (g� �� 1). The origin ofthe charging energy in conventional quantum dots isthe long range nature of the Coulomb interaction. Inthe theory of Luttinger liquid, the long range inter-action can easily be incorporated microscopicallythrough the interaction strength g� . For the effect ofthe finite-range interaction across a tunneling junc-tion, for instance see Refs. 40, 41.

Note that charge and spin are decoupled in Lut-tinger liquids, which implies the electric forces affectthe (total) charge sector only; due to intrinsic e e� in-teractions, g� � 1 but g� �� 1, and the gate voltageshifts the band bottom of the (total) charge sector asseen by the dimensionless gate voltage parameter NGin Eq. (4). As will be shown shortly, the transportproperties of the L/R-leads are determined by the LLinteraction parameter g� and the number of the trans-port sector M. In this work, we consider each wiresegment has the same interaction strength for the (to-tal) charge sector, g g L� ��

( ) g gR D� �( ) ( )� . Accord-

ingly, the energy scales in the quantum are written by� � ��

( )DF Dv /g L� and � � �� 0 � �� v /LF D.

We consider the ground-state energy in the QD isthe same as those in the leads, by choosing the refer-ence energy Er in Eq. (4) equals the minimum valueof the zero-mode energy,

EN

Mg

N

Mg

M

MrG G�

�

��

�

��

�

�

��

min ,( ) ( )� � ��

�

�

�

2 20

2

1

2

1

2, (6)

where min( , )x x� denotes the smaller of x and �x , andthe gate charge NG is in the range N MG � [ , ]0 .

The zero-mode energy in the QD,

EMg

N NM

N EG r02 0 2

2 2� � �

��

��

� �

( ) (7)

yields degenerate ground states for N � 0 and N � 1excess electrons when N M g /G � � [( ) ]1 1 2

2� . Here

we replaced N� by the number of the total excesselectrons N since N N� � , and N� are all either evenor odd integers, simultaneously; in the case of theSWNTs with N N

i s i s� �

, ,excess electrons, where

i � 1 2, is the channel index and s � � �, is the spinindex of conduction electrons (M = 4), N N� � ,

N N Nii

i� � � �( ), , , N N Nss

s�� ( ), ,1 2 ,

N N Ni

ii i�� ( ) ( ), ,1 .

From now on we consider only one spin-polarized(or spinless) channel unless otherwise stated — ourfocus is on the role of Coulomb interactions, and theadditional channels only lead to more complicated ex-citation spectra without any qualitative change in thephysics we address below. A physical realization ofthe single-channel case may be obtained, e.g., by ex-posing the quantum wire to a large magnetic field.

For the system with high tunneling barriers, theelectron transport is determined by the bare electronhops at the tunneling barriers. The tunneling events inthe DB structure are described by the Hamiltonian

� [ � ( ) � ( ) .]†

,

H t X XT DL R

� ��

�

� � h.c , (8)

where � ( )†��

X and � ( )�� X are the electron creationand annihilation operators at the edges of the wiresnear the junctions at XL and XR. As mentioned ear-lier, the electron field operators �� and � †� are relatedto the plasmon creation and annihilation operators �band �†b by the standard bosonization formulae. Differ-ent boundary conditions yield different relations be-tween electron field operators and plasmon operators.Exact solutions for the periodic boundary conditionhave been known for decades [3,37] but the openboundary conditions which are apt for our system ofconsideration has been investigated only recently (see,for example, Refs. 38, 42–44).

The dc bias voltage V V VL R� between L and Rleads is incorporated into the phase factor of thetunneling matrix elements t t ieV t/� � �� | | exp( ) by atime-dependent unitary transformation [10]. HereVL/R � VC/CR/L is voltage drop across the L/R-junction where C C C / C CL R L R� ( ) is the effectivetotal capacitance of the double junction, and the baretunneling matrix amplitudes | |tL/R are assumed to beenergy independent. Experimentally, the tunnelingmatrix amplitude is sensitive to the junction propertieswhile the capacitance is not. For simplicity, the capaci-tances are thus assumed to be symmetric C CL R�throughout this work. By junction asymmetry we meanthe asymmetry in (bare) squared tunneling amplitudes| |tL/R

2. The parameter R t / tL R� | | | |2 2 is used to de-

scribe junction asymmetry; R � 1 for symmetric junc-tions and R �� 1 for highly asymmetric junctions.

It is known that, at low energy scales in thequantum wires with the electron density away from



half-filling, the processes of backward and Umklappscattering, whose processes generate momentum trans-fer across the Fermi sea (� 2kF), can be safely ignoredin the middle of ideal 1D conductors [37], includingarmchair SWNTs [38]. The Hamiltonian (2) does notinclude the backward and Umklapp scattering (exceptat the tunneling barriers) and therefore it is validaway from half-electron-filling.

We find that, in the regime where electron spindoes not play a role, the addition of a transport chan-nel does not change essential physics present in a sin-gle transport channel. Therefore, we primarily focusattention to a QW of single transport channel withspinless electrons and will comment on the effects dueto multiple channel generalization, if needed.

2.2. Electron transition rates

The occupation probability of the quantum statesin the SET system changes via electron tunnelingevents across L/R-junctions. In the single-electrontunneling regime, the bare tunneling amplitudes| |tL/R are small compared to the characteristic energyscales of the system and the electron tunneling is thesource of small perturbation of three isolated LLs. Inthis regime, we calculate transition rates �L/R be-tween eigenstates of the unperturbed Hamiltonian H0to the lowest nonvanishing order in the tunneling am-plitudes | |tL/R . In this golden rule approximation, weintegrate out lead degrees of freedom since the leadsare in internal equilibria, and the transition rates aregiven as a function of the state variables and the ener-gies of the QD only [21,23],

�L/R N n N n( , { } , { })� � � �

� �2 2� � ��

| | ( ) ({ }, { })t W n nL/R L/R D .(9)

In Eq. (9)WL/R is the change in the Gibbs free energyassociated with the tunneling across the L/R-junction,

W E N n E N n N N eVL/R D D L/R� � � � ��( , { }) ( , { }) ( )� .

(10)

Here E N n N n H N nD D( , { }) , { }| � | , { }� ! is the energy ofthe eigenstate | , { }N n ! of the dot and L/R correspondto � / . For the QD with only one transport channelwith spinless electrons only,

E N n mnN N

gED p

mm

G

Dr( , { })

( ),�

�"

#$$

%

&''

��

�

��1

2

2 (11)

where � ��p � accounting that we consider onlycharge plasmons and n n b b nm m m m m� !| � � |

† is the num-ber of plasmons in the mode m. Note that the excita-tion spectrum (11) in the QD consists of chargeexcitations which are formed by changing electron

number N in the zero-mode, and plasmon excitationswhich are neutral excitations. In this work, wesharply distinguish those two different excitations.

The function �( )x in Eq. (9) is responsible for theplasmon excitations on the leads, and given by (see,e.g., Ref. 14)

� ��

�( ) ( , ) ( , )†� ! �

�

�

(12 0� � � � �dt X X ti te � �

��

�

��

�

�

��

�

��

�

��

12

1 2 12 2

1 1

� ��

)* )�

�

�

� �v

g

vi

F

/ g

F

+�

( ) ,

,, ,

,,

2 2

1e ��

*

/

�( ),

(12)

where ) � 1/k TB is the inverse temperature in theleads, + is a short wavelength cutoff, and �( )z is thegamma function. The exponent * � �( )g /M1 1 is acharacteristic power law exponent indicating interac-tion strength of the leads with M transport sectors(hence, in our case, M � 1). At g � 1 (noninteractingcase), the exponent * � 0 and it grows as g - 0(strong interaction). The decrease of the exponent *with increasing M implies that the effective interac-tion decreases due to multi-channel effect, and theLuttinger liquid eventually crosses over to a Fermiliquid in the many transport channel limit [45,46].

For the noninteracting electron gas, the spectral density� �( ) is the TDOS multiplied by the Fermi–Diracdistribution function fFD( )� � [ exp( )]1

1 )� ; � �( ) �� f / vFD F( ) ( )� �� for g � 1. At zero temperature, � �( ) isproportional to a power of energy,

lim ( ) ( )(| | )

( )T Fv

/

��

01

1� � �

��

*

�

.�

�

�, (13)

where � � +�

�v / gF/ g1 1( ) is a high energy cut-off.

At zero temperature � �( ) is the TDOS for the negativeenergies and zero otherwise (as it should be), imposedby the unit step function .( )�� .

The function � D in Eq. (9) accounts for the plasmontransition amplitudes in the QD, and is given by

� /D N N Dn n N n X N n({ }, { }) | , { }|� ( )| , { } |,

†� � � � ! � �12� �

� � !� /N N DN n X N n, | , { }| � ( )| , { } |12� � , (14)

where we used that the zero-mode overlap is unity forN N� � 0 1 and vanishes otherwise. The overlapintegrals between plasmon modes are, althoughstraightforward, quite tedious to calculate, and werefer to Appendix A for the details. The resultingoverlap of the plasmon states can be written as afunction of the mode occupations nm ,



| { }| � ( )| { } | | { }| � ( )| { } |† � ! � � ! �n X n n X nD D� �� 2 2

��

��

�

��

1L L

n nD D

D�1

�+

({ }, { }) (15)

with

1({ }, { })!

!

| |( )

( )n n

gm

n

nL

m

m

mn

nm nm

� ��

��

�

��

�

� �

�2� �

1

1 m

m mn n

gm( )| | ,

�

� �

��

�

��

"

#$

%

&'

12

(16)

where n n nm m m( ) ( , )� � �min and n n nm m m

( ) ( , )� � �max ,* D Dg� �

( )1 1 for the QD with one transport sector,and L xa

b ( ) are Laguerre polynomials. Additional tran-sport sectors would appear as multiplicative factors ofthe same form as 1({ }, { })n n� and result in a reductionof the exponent * D. Notice that in the low energyscale only the first few occupations nm and n m� in theproduct differ from zero, participating to the transi-tion rate (9) with nontrivial contributions.

2.3. Plasmon relaxation process in the quantum dot

In general, plasmons on the dot are excited by tun-neling events and have a highly nonequilibrium distri-bution. The coupling of the system to the environmentsuch as external circuit or background charge in thesubstrate leads to relaxation towards the equilibrium.While the precise form of the relaxation rate, �p , de-pends on the details of the relaxation mechanism, thephysical properties of our concern do not depend onthe details. Here we take a phenomenological modelwhere the plasmons are coupled to a bath of harmonicoscillators by

H g b b amn m nm n

plasmon bath h. c. � �� ( )†,

��

�

. (17)

In Eq. (17) a� and a�† are bosonic operators describ-

ing the oscillator bath and gmn� is the coupling con-

stants. We will assume an Ohmic form of the bathspectral density function

J gmn mn p( ) | | ( ) ,3 / 3 3 � 3�

�� 2 0� (18)

where 3� is the frequency of the oscillator cor-responding to a� , � p is a dimensionless constantcharacterizing the bath spectral density, and �0

1 �� 2 2 2v L t tF D L R(| | | | ) is the natural time scale ofthe system. Within the rotating-wave approximation,the plasmon transition rate due to the harmonic oscil-lator bath is given by

� �p pp p

Wn n

W /

p({ } { })� � �

��

��0

1e(19)

with

W n np p mm

m� � �� ( ) ,where � ��p � is the plasmon energy. Note that thesephenomenological rates obey detailed balance and,therefore, at low temperatures only processes thatreduce the total plasmon energy occur with apprecia-ble rates.

2.4. Matrix formulation

For later convenience, we introduce a matrix nota-tion for the transition rates �, with the matrix ele-ments defined by

[ � ( )] ( , { } , { }) ,{ },{ }� � ��

� � 0 � �N N n N nn n � 1 (20)

i.e, the element ({ }, { })�n n of the matrix block � ( )� �� N

is the transition rate ��( , { } , { })N n N n0 � �1 . Simi-larly,

[ � ] [ � � ]{ },{ } { },{ } { },{ }{ }

� � �� 0

n n n n n nn

� ��

��

� �/ , (21)and

[ � ( )] ({ }, { }){ },{ }�p n n pN n n� � � � �

��

�/{ },{ }{ }

({ }, { })n n pn

n n� . (22)

Master equation (1) can now be conveniently ex-pressed as

ddt

P t P t| ( ) � | ( )! � � !� (23)

with

� � ( � � � ),

� � � � ��

��pL R

� � �

�

0 ,

where | ( )P t ! is the column vector (not to be confusedwith the «ket» in quantum mechanics) with elementsgiven by ! �N n P t P N n t, { }| ( ) ( , { }, ). Therefore, thetime evolution of the probability vector satisfies

| ( ) exp( � )| ( )P t t P! � � !� 0 . (24)

In the long-time limit, the system reaches a steadystate | ( )P 4 !.

The ensemble averages of the matrices �� can then

be defined by

! � !� �� ( ) , { }| � | ( ),{ }

� �� t N n P tN n

. (25)

We will construct other statistical quantities such as av-erage current and noise power density based on Eq. (25).



3. Steady-state probability distributionof nonequilibrium plasmons

By solving the master equation (23) numerically(without plasmon relaxation), in Ref. 23, we obtainedthe occupation probabilities of the plasmonic many-body excitations as a function of the bias voltage andthe interaction strength. We found that in the weak tononinteracting regime, * � 0 or g /� �1 1 1( )* forthe wire with one transport sector, the nonequilibriumprobability of plasmon excitations is a complicatedfunction of the detailed configuration of state occupa-tions { } ( , , ..., , ...)n n n nm� 1 2 .

In contrast, the nonequilibrium occupation proba-bility in the strong interaction regime with (nearly)symmetric tunneling barriers depends only on the to-tal energy of the states, and follows a universal formirrespective of electron charge N in the QD. In theleading order approximation, it is given by

PZ eV p

( )( ) exp( )

log01 3 1

2�

* � ��

� ��

��

�

�� , (26)

where Z is a normalization constant. Notice that � isthe total energy, including zero-mode and plasmoncontributions. The distribution has a universal formwhich depends on the bias voltage and the interactionstrength. The detailed derivation is in Appendix B.This analytic form is valid for the not too low ener-gies ( , )� eV � 3�p and in the strongly interacting re-gime * � 1. More accurate approximation formula(Eq. (B.13)) is derived in Appendix B.

For symmetric junctions, the occupation probabili-ties fall on a single curve, well approximated by the an-alytic formulas Eqs. (26) and (B.13), as seen in the in-sets in Fig. 2, where P( )� is depicted as a function ofthe state energies for g � 0 2. (a) and g � 0 5. (b), withparameters R � 1 for the inset and R � 100 for the mainfigures (eV p� 6� , N /G � 1 2). For the asymmetricjunctions, the line splits into several branches, one foreach electric charge N, see the figure. However, as seenin Fig. 2,a, if the interaction is strong enough (g � 0.3for R � 100 and eV p� 6� ), each branch is, independ-ently, well described by Eq. (26) or (B.13). Forweaker interactions, g � 0.3 for R � 100 and eV p� 6� ,the analytic approximation is considerably less accurateas shown in Fig. 2,b. Even in the case of weaker inter-actions, however, the logarithms of the plasmon occu-pation probabilities continue to be nearly linear in � butwith a slope that deviates from that seen for symmetricjunctions.

4. Average current

In terms of the tunneling current matrices acrossthe junction L/R

� ( � � )IL/R L/R L/Re� ��

� � � , (27)

the average current I t tL/R L/R( ) � ( )� !I throughL/R-junction is

I t N n P tL/RN n

L/R( ) , { }| � | ( ),{ }

� !� I . (28)The total external current I t t( ) �( )� !I , which in-cludes the displacement currents associated withcharging and discharging the capacitors at the leftand right tunnel junctions, is then conveniently writ-ten as



a100

10–5

10–10

10–15

10–20

10–25

P( )� 010

–1010

–2010

0 2 4 6 8 10

0 5 10

� �/ p

N = 0N = 1N = 2N = 3

P(0)

P (1)

100

10–5

10–10

10–15

P( )�

0 2 4 6 8 10 � �/ p

N = 0N = 1N = 2N = 3

P(0)

P (1)

100

10–5

10–10

0 5 10

Fig. 2. The occupation occupation probability P( )� as afunction of the mode energy � �/ p. The energy � is abbrevia-tion for E N nD( ,{ }), the bias eV p� 6� , the asymmetry pa-rameter R � 100, and N /G � 1 2 (T � 0). Two analytic ap-proximations, Eq. (26) (blue curve) and Eq. (B.13) (cyancurve), are fitted to the probability distribution of thecharge mode N � 1 (blue circle). The interaction parameteris g � 02. (a) and 05. (b). In the inset the case of symmetricjunctions (R � 1) is plotted with the same conditions.

I t C/C I tL R

( ) ( ) ( ),,

��

�

� � (29)

where C C CL R

� 1 1 1. As the system reaches

steady-state in the long-time limit, the charge current isconserved throughout the system, I I� 4 �( ) IL( )4 �� 4IR( ).

One consequence of nonequilibrium plasmons is theincrease in current as shown in Fig. 3, where the aver-age current is shown as a function of the bias for dif-ferent interaction strengths. The currents are normal-ized by I I eV Ec C� �( )2 with no plasmon relaxation(� p � 0) for each interaction strength g, and we seethat the current enhancement is substantial in thestrong interaction regime (g � 0.5), while there is ef-fectively no enhancement in noninteracting limit g � 1(the two black lines are indistinguishable in the fig-ure). In the weak interaction limit the current in-creases in discrete steps as new transport channelsbecome energetically allowed, while at stronger inter-actions the steps are smeared to power laws with expo-nents that depend on the number of the plasmon statesinvolved in the transport processes.

Including the spin sector results in additional peaksin the average current voltage characteristic that canbe controlled by the transverse magnetic field [47,48].

The current-voltage characteristics show that, inthe noninteraction limit, the nonequilibrium approachpredicts similar behavior for the average current as thedetailed balance approach which assumes thermalequilibrium in the QD. In contrast, in the stronginteraction regime, nonequilibrium effects give rise toan enhancement of the particle current.

Experimentally, however, the current enhancementmay be difficult to attribute to plasmon distributionas the current levels depend on barrier transparenciesand plasmon relaxation rates, and neither of them canbe easily tuned. We now turn to another experimentalprobe, the shot noise, which is more sensitive tononequilibrium effects.

5. Current noise

Noise, defined by

S d t t tt

i( ) lim [ �( )�( ) �( ) ]3 �� ! � !��

�

�

(2 2e I I I , (30)describes the fluctuations in the current through aconductor [49,50]. Thermal fluctuations and thediscreteness of the electron charge are two fundamen-tal sources of the noise among others which are sys-tem specific [30,31]. Thermal (equilibrium) noise isnot very informative since it does not provide moreinformation than the equilibrium conductance of thesystem. In contrast, shot noise is a consequence of thediscreteness of charge and the stochastic nature oftransport. It thus can provide further insight beyondaverage current since it is a sensitive function of thecorrelation mechanism [29], internal excitations[51–53], and the statistics of the charge carriers[54–57]. Here we will particularly focus on the rolesof the nonequilibrium plasmon excitations in theSETs of 1D QWs [21–24].

Within the master equation approach, the correlationfunctions K t t

t�� !( ) lim � ( ) � ( ) I I in Eq. (30)

can be deduced from the master equation (23). In thematrix notation they can be written as [27,28]

K e N nN n

�� ( ) , { }| ( )� exp( � �,{ }

[ �2 . I I�

� 4 !� �� .( )� exp( � )� ( ) ( � � ) | ( ) ,] / /I I� � �� P

(31)

where .( )x is the unit step function. In principle,there is no well-known justification of the masterequation approach, which ignores the quantum coher-ence effects, for the shot noise. Actually, influence ofquantum coherence on shot noise is an intriguing is-sue [31]. However, we note that both quantum me-chanical approaches [58,59] and semiclassical deriva-tions based on a master equation approach predictidentical shot noise results [26,60,61], implying thatthe shot noise is not sensitive to the quantum coher-ence in double-barrier structures. Master equation ap-proach was used by many authors for shot noise inSET devices as well [26–28,60,62,63], and some pre-



0 0.5 1.0eV/2E c

4

3

2

1

cI/I

g = 0.3g = 0.3, �pg = 0.5g = 0.5, �pg = 0.7g = 0.7, �pg = 1.0g = 1.0, �p

Fig. 3. Average current I I/ c as a function of the bias volt-age eV/ EC2 and LL interaction parameter g for R � 100(highly asymmetric junctions) with no plasmon relaxation(�p � 0, solid lines) or with fast plasmon relaxation(�0

410� , dashed lines). The bias voltage is normalized bythe charging energy 2EC and current is normalized by thecurrent at eV EC� 2 with no plasmon relaxation for eachg. Other parameters are NG � 1/2, T � 0.

dictions of them [27] have been experimentally con-firmed [64]. On this ground, here we adopt the mas-ter equation approach and leave the precise test of thejustification open to either experimental or furthertheoretical test.

To investigate the correlation effects, the noisepower customarily compared to the Poisson valueS eIPoisson � 2 . The Fano factor is defined as the ratioof the actual noise power and the Poisson value,

FS

eI�

( )02

. (32)

Since thermal noise ( ( ))S k TG VB� �4 0 is not par-ticularly interesting, we focus on the zero frequencyshot noise, in the low bias voltage regime where theCoulomb blockade governs the electric transport;T � 0 and eV � 2EC.

5.1. Qualitative discussions

As we discuss below in detail, we have found in theabsence of substantial plasmon relaxation a giant en-hancement of the shot noise beyond Poissonian limit(F � 1) over wide ranges of bias and gate voltages;i.e., the statistics of the charge transport through thedevice is highly super-Poissonian. In the oppositelimit of fast plasmon relaxation rate, the shot noise isreduced below the Poissonian limit (still exhibitingfeatures specific to LL correlations) in accordancewith the previous work [22]. Before going directlyinto details, it will be useful to provide a possiblephysical interpretation of the result.

We ascribe the giant super-Poissonian noise to theopening of additional conduction channels via theplasmon modes. In the parameter ranges where the gi-ant super-Poissonian noise is observed, there are a con-siderable amount of plasmon excitations [23]. Itmeans that the charges can have more than one possi-ble paths (or, equivalently, more than two local statesin the central island involved in the transport) fromleft to right leads, making use of different plasmonmodes. Similar effects have been reported in conven-tional SET devices [29] and single-electron shuttles[65]. It is a rather general feature as long as the multi-ple transport channels are incoherent and have differ-ent tunneling rates. An interesting difference betweenour results and the previous results [29,65] is that thesuper-Poissonian noise is observed even in the sequen-tial tunneling regime whereas in Ref. 29 it was ob-served in the incoherent cotunneling regime and in[65] due to the mechanical instability of the elec-tron-mediating shuttle.

Notice that the fast plasmon relaxation preventsthe additional conduction channel opening, and in thepresence of fast plasmon relaxation, the device is

qualitatively the same as the conventional SET. Thenoise is thus reduced to the Poissonian or weaklysub-Poissonian noise.

To justify our interpretation, in the following twosubsections we compare two parameter regimes withonly a few states involved in the transport, which areanalytically tractable. When only two charge states(with no plasmon excitations) are involved (Sec. 5.2),the transport mechanism is qualitatively the same as theusual sequential tunneling in a conventional SET.Therefore, one cannot expect an enhancement of noisebeyond the Poissonian limit. As the bias voltage in-creases, there can be one plasmon excitation associatedwith the lowest charging level (Sec. 5.3). In this case,through the three-state approximation, we will expli-citly demonstrate that the super-Poissonian noise arisesdue to fluctuations induced by additional conductionchannels. Detailed analysis of the full model system isprovided in subsequent subsections.

5.2. Two-state model (eV p� � )

The electron transport involving only two lowestenergy states in the quantum dot are well studied bymany authors (see, for instance, Ref. 31). Neverthe-less, for later reference we begin the discussion of shotnoise with two-state process, which provides a reason-able approximation for eV p� � . At biases such thateV p� � and sufficiently low temperatures, the twolowest states | , | ,N n1 0 0! � ! and | ,10! dominate thetransport process and the rate matrix is given by

� ,� ��

�

"

#$$

%

&''

�

�

� �

� �(33)

where the matrix elements are � � � ��L( , , )10 0 0 and� � ��R( , , )0 0 10 .

With the current matrices defined by Eq. (27)

� ,IL �"

#$

%

&'�

0 0

0�� ,IR �

"

#$

%

&'

0

0 0

�

the noise power is obtained straightforwardly byEqs. (30) and (31) using the steady state probability

| ( )PP

P4 ! �

"

#$

%

&' �

"

#$$

%

&''�

�00

10

1

� �

�

�. (34)

The Fano factor (32) takes a simple form

F P P/

/2 00

2102

2

2

1

1� �

�

�

( )

( )

� �

� �, (35)

where



�

�

//

�

��

�

�

��

�

��

1 2

20

0R

eV/ E

eV/ E

and /E0 is the shift of the bottom of the zero modeenergy induced by the gate voltage,

/ / � /E N /g N N /G p G G0 1 2� � �( ) , . (36)

Note that Eq. (36) is valid for | |/E eV/0 25 , other-wise I � 0 and S � 0 due to Coulomb blockade. Wesee from Eq. (35) that the Fano factor is minimizedfor � � � �/ 1 and maximized for ( )� � � � �/ 1 0, withthe bounds 1 2 12/ F5 � . At the gate chargeN /G � 1 2, it is determined only by the junctionasymmetry parameter R: F R / R2

2 21 1� ( ) ( ) .Note that when only two states are involved in thecurrent carrying process (ground state to ground statetransitions), the Fano factor cannot exceed the Pois-son value F � 1.

As a consequence of the power law dependence ofthe transition rates on the transfer energy (13), theFano factor is a function of the bias voltage, and theinteraction strengts, it varies between the minimumand maximum values

FE

R

R

eV

EeV

/

/2

0

1

1

0

12

1

1 2

12

��

�

� 0

6

788

988

at

at

/

/

�

�(37)

(cf. Eq. (5) in Ref. 22). Figure 4 depicts the Fanofactor as

N / g eV/R

RG p

/

/dip �

�

1 2 2

1

1

1

1( )�

�

�. (38)

The Fano factor independently of the interactionstrength crosses F R / R2

2 21 1� ( ) ( ) at N /G � 1 2,and it approaches maximum F2 1� at N /G � 01 20 g eV/ p( )2� .

5.3. Three-state model (eV � 2�p)

The two-state model is applicable for bias voltagesbelow eV Epth � �2 0( | | )� / . Above this thresholdvoltage, three or more states are involved in the trans-port. For electron transport involving three lowest en-ergy states in the quantum dot | , | , , | ,N n1 0 0 10! � ! !, and| ,11! with nm � 0 for m : 2, the noise power can be cal-culated exactly if the the contribution from the (back-ward) transitions against the bias is negligible, as istypically the case at zero temperature. In practice,however, the backward transitions are not completelyblocked for the bias above the threshold voltage of theplasmon excitations, even at zero temperature: oncethe bias voltage reaches the threshold to initiateplasmon excitations, the high-energy plasmons in theQD above the Fermi energies of the leads are also par-tially populated, opening the possibility of backwardtransitions.

A qualitatively new feature that can be studied inthe three-state model as compared to the two-statemodel is plasmon relaxation: the system with a con-stant total charge may undergo transitions betweendifferent plasmon configurations.

We will show in this subsection that the analyticsolution of the Fano factor of the three-state processyields an excellent agreement with the low bias nu-merical results in the strong interaction regime, whileit shows small discrepancy in the weak interaction re-gime (due to non-negligible contribution from thehigh-energy plasmons). We will also show that withinthe three state model the Fano factor may exceed thePoisson value.

Analytic results

By allowing plasmon relaxation, the rate matrix in-volving three lowest energy states | , | , ,| , ,N n1 0 0 10! � ! !and | ,11! is given by

��

� � � �

� � �

� � �

0 1 0 1

0 0

1 10

� �

�

�

� �

� �

�

"

#

$$$$

%

&

'''

p

p '

, (39)

with the matrix elements � i L i� � �� ( , , ),1 0 0 � i

�� R i i( , , ), ,0 0 1 01 and � p introduced in Eq. (19).Current matrices defined by Eq. (27) are

[� ] , [� ]{ , } { , }I IL L21 0 31 1� �� ,

[� ] , [� ] ,{ , } { , }I IR R12 0 13 1� �

� �

with [� ]{ , }I� i j � 0 for other set of i j, , ,� 1 2 3, where� � L R, . The noise power is obtained straightfor-



0 0.2 0.4 0.6 0.8NG

1.5

1.0

0.5

F

g = 0.2g = 0.3g = 0.5g = 0.7g = 1.0

1 –2 –3 –

5 –4 –

1234

5

Fig. 4. Fano factor F S / eI� ( )0 2 as a function of the gatecharge NG and LL interaction parameter g for R = 100(highly asymmetric junctions) at eV p� � (T � 0).

wardly by Eqs. (30) and (31) using the steady stateprobability

| ( )

( )

(P

P

P

PZ

p

p4 ! �"

#

$$$

%

&

'''

�

� 00

10

11

0 1

0 11

� � �

� � � )

"

#

$$$$

%

&

''''

�

�

� �

� �

1

1 0

p , (40)

with normalization constant Z � � � � � � � � �0 1 1 0 0 1 � �( )� � � �0 0 1 p . Using the average current I �� e P P( )10 0 11 1� � , the Fano factor F S / eI� ( )0 2 isgiven by

F P P P PZ3 00

2102

112

110

0

21

� ;�

�

�

; � "

#$$

%

&''

� �

�

( ) ( ) .� � � �

� �

�� 0

21

20 1

0 1

01 p (41)

Compared to the Fano factor (35) in the two-stateprocess, complication arises already in the three-stateprocess due to the last term in Eq. (41) which resultsfrom the coupling of P11 and the rates which cannot beexpressed by the components of the probability vector.

In order to have | , | , , | , ,N n! � ! !0 0 10 and | ,11! as therelevant states, we assume � �0 0

� � or more explicitlyN NG G:

dip which is introduced in Eq. (38). In theopposite situation ( )� �0 0

� � , the relevant states are| , | , , | , ,N n1 0 0 01! � ! ! and | ,10!, and the above descriptionis still valid with the exchange of electron numberN �

over the other rates, ( , ) ( , )� � � �0 1 0 1� �� , which results

in 1 � P P P10 0 11 0�� ( , ) . The Fano factor (42) nowis reduced to

F P P30

10 110

21

2

0 1

1

0

1 2 1 2( )( ) ( )

.�

�

�

�

� �

� �

�

�(44)

Notice that while Eq. (42) is an exact solution for thethree-state process with no plasmon relaxation, Eq.(44) is a good approximation only sufficiently farfrom N /G � 1 2. In this range, Eq. (44) explicitlyshows that the opening of new charge transport chan-nels accompanied by the plasmon excitations causesthe enhancement of the shot noise (over the Poissonlimit).

Recently, super-Poissonian shot noises, i.e., the Fanofactor F � 1, were found in several different situations inquantum systems. Sukhorukov et al. [29] studied thenoise of the co-tunneling current through one or severalquantum dots coupled by tunneling junctions, in theCoulomb blockade regime, and showed that strong in-elastic co-tunneling could induce super-Poissonian shotnoise due to switching between quantum states carryingcurrents of different strengths. Thielmann et al. [66]showed similar super-Poissonian effect in a single-levelquantum dot due to spin-flip co-tunneling processes,with a sensitive dependence on the coupling strength.The electron spin in a quantum dot in the Coulombblockade regime can generate super-Poissonian shotnoise also at high frequencies [67]. The shot noise en-hancement over the Poisson limit can be observed bystudying the resonant tunneling through localized statesin a tunnel-barrier, resulting from Coulomb interactionbetween the localized states [68]. In nano-electro-me-chanical systems, in the semiclassical limit, the Fanofactor exceeds the Poisson limit at the shuttle threshold[65,69]. Commonly, the super-Poissonian shot noise isaccompanied by internal instability or a multi-channelprocess in the course of electrical transport.

In the limit of fast plasmon relaxation, on the otherhand, � �p i��

� and effectively no plasmon is excited,

| ( )P

P

P

P

4 ! �"

#

$$$

%

&

'''

�

� �

� �00

10

110 1 0

0

0 11

0� � �

�

� �

"

#

$$$$

%

&

''''

. (45)

Consequently, the Fano factor (41) is given by

F P3 00

212

212

12

1( ) ,� � ��

��

"#$

%&'

. (46)

The maximum Fano factor F � 1 is reached if one ofthe rates �0

� or �0

dominates, while the minimum

value F /� 1 2 requires that � � �0 0 1

� �� , i.e., that

the total tunneling-in and tunneling-out rates areequal; more explicitly,

11

2

21

12

0

0 0R

E

eV/ E g eV/ Ep�

�

��

�

��

�

��

�

��

//

�

/

� �

1.

(47)

5.4. Numerical results

The limiting cases of no plasmon relaxation(� p � 0) and a fast plasmon relaxation (� p � 10

4) aresummarized in Fig. 6,a and b, respectively. In the fig-ure the Fano factor is plotted as a function of gatevoltage and interaction parameter g in the strong in-teraction regime 0 3 0 6. .5 5g at eV p� 2� for R � 100(strongly asymmetric junctions).

As shown in Fig. 6,a, the Fano factor is enhancedabove the Poisson limit (F � 1) for a range of gatecharges not very close to N /G � 1 2, especially in thestrong interaction regime, as expected from thethree-state model. The shot noise enhancement is lostin the presence of a fast plasmon relaxation process, inagreement with analytic arguments, as seen in Fig. 6,bwhen F is bounded by 1 2 1/ F5 5 . Hence, slowly re-



F

gNG

2.5

2.0

1.5

1.0

0.5

0.7 0.6 0.5 0.4 0.3 0.2

0.30.4

0.50.6

F1.0

0.9

0.8

0.7

0.6

0.5

NG

g0.7 0.6 0.5 0.4 0.3 0.2 0.6

0.5

0.4

0.3

b

Fig. 6. Fano factor F S / eI� ( )0 2 as a function of the gatecharge NG and LL interaction parameter g for R � 100(highly asymmetric junctions) at eV p� 2� (T � 0), withno plasmon relaxation (�p � 0) (a) and with fast plasmonrelaxation (�p � 10

4) (b).

laxing plasmon excitations enhance shot noise, andthis enhancement is most pronounced in the stronglyinteracting regime.

In the limit of fast plasmon relaxation F exhibits aminimum value F /� 1 2 at positions consistent withpredictions of the three-state model: the voltage polar-ity and ratio of tunneling matrix elements at the twojunctions is such that total tunneling-in and tunnel-ing-out rates are roughly equal for small values of NG .If plasmon relaxation is slow, F still has minima at ap-proximately same values of NG but the minimal valueof the Fano factor is considerably larger due to thepresence of several transport channels.

5.5. Interplay between several charge states andplasmon excitations near eV EC� 2

So far, we have investigated the role of nonequi-librium plasmons as the cause of the shot noise en-hancement and focused on a voltage range when onlytwo charge states are significantly involved in trans-port. The question naturally follows what is the conse-quence of the involvement of several charge states. Dothey enhance shot noise, too?

To answer this question, we first consider a toymodel in which the plasmon excitations are absent dur-ing the single-charge transport. In the three-N-state re-gime where the relevant states are | | , |N! � � ! !1 0 and |1!with no plasmon excitations at all. At zero tempera-ture, the rate matrix in this regime is given by

��

�

� �

�

"

#

$$$$

%

&

''''

�

� �

�

� �

� � � �

� �

1 0

1 0 0 1

0 1

0

0

, (48)

where the matrix elements are

�i� � �L i i( , { } , { }), �1 0 0

� �i R i i

� � �( , { } , ),1 0 0 i � �1 0 1, , .

Repeating the procedure in Sec. 5.3, we arrive at aFano factor that has a similar form as Eq. (42),

F P P P PN3 1 1 1 11 2 1 1� � � � [( ) ( ) ]

�

��

�

��

�

�

2 1 11

1

1

1

P P�

�

�

�, (49)

with the steady-state probability vector

| ( )P

P

P

PZ

4 ! �"

#

$$$

%

&

'''

�

"

#

$$

�

� �

1

0

0 1

1 1

1 0

1� �

� �

� �$$

%

&

''''

, (50)

where Z � �

� �� 0 1 1 1 1 0 .

Despite the formal similarity of Eq. (49) withEq. (42), its implication is quite different. In terms ofthe transition rates, F N3 reads

FZ

N3 2 1 0 1 0 1 0 11

2� � �

� �

� � [ ( ( ) )� � � � � � �

�� 1 0 0 1 1 0 1( ( ) )] . (51)

Since � �� 1 0 and � �1 0

� in the three-N-state re-gime, the Fano factor F N3 is sub-Poissonian, i.e.,F N3 1� , consistent with the conventional equilibriumdescriptions [22,27].

We conclude that while plasmon excitations mayenhance the shot noise over the Poisson limit, theinvolvement of several charge states and the ensuingcorrelations, in contrast, do not alter the sub-Pois-sonian nature of the Fano factor in the low-energy re-gime. This qualitative difference is due to the fact thatcertain transition rates between different charge statesvanish identically (in the absence of co-tunneling): itis impossible for the system to move directly from astate with N � �1 to N � 1 or vice versa.

Therefore, we expect that for bias voltages neareV EC p� �2 2� , when both plasmon excitations andseveral charge excitations are relevant, the Fano fac-tor will exhibit complicated nonmonotonic behavior.Exact solution is not tractable in this regime since itinvolves too many states. Instead, we calculate theshot noise numerically, with results depicted inFig. 7, where the zero temperature Fano factor isshown as a function of the bias eV and LL interactionparameter g for R � 100 at N /G � 1 2, (a) with noplasmon relaxation ( )� p � 0 and (b) with fastplasmon relaxation (� p � 10

4).In the bias regime up to the charging energy

eV EC5 2 , the Fano factor increases monotonicallydue to nonequilibrium plasmons. On the other hand,additional charge states contribute at eV EC: 2 whichtends to suppress the Fano factor. As a consequence ofthis competition, the Fano factor reaches its peak ateV EC� 2 and is followed by a steep decrease asshown in Fig. 7,a. Note the significant enhancementof the Fano factor in the strong interaction regime,which is due in part to the power law dependence ofthe transition rates with exponent * � �( )1 1/g as dis-cussed earlier, and in part to more plasmon states be-ing involved for smaller g since E /gC p� � . The latterreason also accounts for the fact that the Fano factorbegins to rise at a lower apparent bias for smaller g:the bias voltage is normalized by EC so that plasmonexcitation is possible for lower values of eV/EC forstronger interactions.

In the case of fast plasmon relaxation, the richstructure of the Fano factor due to nonequilibrium



plasmons is absent as shown in Fig. 7,b, in agreementwith the discussions in previous subsections. The onlyremaining structure is a sharp dip around eV EC� 2that can be attributed to the involvement of addi-tional charge states at eV EC: 2 . Not only the mini-mum value of the Fano factor is a function of the in-teraction strength but also the bias voltage at which itoccurs depends on g. The minimum Fano factor occursat higher bias voltage, and the dip tends to be deeperwith increasing interaction strength. Note for g � 0.5,the Fano factor did not reach its minimum still at larg-est voltages plotted (eV/ EC2 12� . ).

The Fano factor at very low voltages eV p� 2� forN /G � 1 2 is F R / R

( ) ( ) ( )0 2 21 1� (see Eq. (35)),regardless of the plasmon relaxation mechanism. Asshown in Fig. 7,b, the Fano factor is bounded above bythis value in the case of fast plasmon relaxation. Noticethat in the case of no plasmon relaxation (Fig. 7,a), thedips in the Fano factor at eV EC� 2 reach below F

( )0 .See Fig. 1,a in Ref. 24 for more detail.

Since both the minimum value of F and the voltageat which it occurs are determined by a competition be-tween charge excitations and plasmonic excitations,

they cannot be accurately predicted by any of the sim-ple analytic models discussed above.

6. Full counting statistics

Since shot noise, that is a current-current correla-tion, is more informative than the average current, weexpect even more information with higher-order cur-rents or charge correlations. The method of countingstatistics, which was introduced to mesoscopic physicsby Levitov and Lesovik [70] followed by Muzy-kantskii and Khmelnitskii [71] and Lee et al. [72],shows that all orders of charge correlation functionscan be obtained as a function related to the probabil-ity distribution of transported electrons for a giventime interval. This powerful approach is known as fullcounting statistics (FCS). The first experimentalstudy of the third cumulant of the voltage fluctuationsin a tunnel junction was carried out by Reulet et al.[73]. The experiment indicates that the highercumulants are more sensitive to the coupling of thesystem to the electromagnetic environment. See alsoRefs. 74, 75 for the theoretical discussions on the thirdcumulant in a tunnel-barrier.

We will now carry out a FCS analysis of transportthrough a double barrier quantum wire system. Theanalysis will provide a more complete characterizationof the transport properties of the system than eitheraverage current or shot noise, and shed further lighton the role of nonequilibrium versus equilibriumplasmon distribution in this structure.

Let P M( , ) be the probability that M electronshave tunneled across the right junction to the rightlead during the time . We note that

P M( , ) �

� � ��

��lim ( , , { }, ; , , {, ,{ },{ }

tP M M N n t M N n

M N nN n

0 0 0 0

0 0 0

� }, ),t

where P M M N n t M N n t( , , { }, ; , , { }, )0 0 0 0 is cal-led joint probability since it is the probability that,up to time t, M0 electrons have passed across theright junction and N0 electrons are confined in theQD with { }n0 plasmon excitations, and that M M0 electrons have passed R-junction with ( , { })N n excita-tions in the QD up to time t . The master equationfor the joint probability can easily be constructedfrom Eq. (23) by noting that M M- 0 1 asN N- � 1 via only R-junction hopping

To obtain P M( , ) , it is convenient to define thecharacteristic function conjugate to the joint proba-bility as

g N n( , , { }, )= �



F7

5

3

10

12eV/E C 0.4

0.6 0.8 1.0

g

a

F

2.02.2

2.4eV/E C g0.4 0.6 0.8

1.0

1.0

0.9

0.8

0.7

0.6

0.5

b

Fig. 7. Fano factor F S / eI� ( )0 2 as a function of the biaseV and LL interaction parameter g for R � 100 (highlyasymmetric junctions) at N /G � 1 2 (T � 0): with no plas-mon relaxation (�p � 0) (a) and with fast plasmon relax-ation (�p � 10

4) (b).

� � ��

� �lim ( , , { }, ; , , {, ,{ }

t

i M

M M N n

P M M N n t M N ne � �0 0 0

0 0 0

0 }, ).t

The characteristic function satisfies the masterequation

��

! � � !

= = = | ( , ) �( )| ( , )g g�

with the initial condition | ( , ) | ( )g P= � ! � 4 !0 . The=-dependent �� in Eq. (52) is related to the previouslydefined transition rate matrices through

�( ) � ( � � � )� � � �= � � � � �p L L L0

� �� ( � � � ) .� � �R Ri

Ri0 e e� � (53)

The characteristic function G( , )= conjugate toP M( , ) is now given by G N n g

N n

( , ) , { }| ( , ),{ }

= = � !� ,or

G N n PN n

( , ) , { }| exp [ �( ) ]| ( ) .,{ }

= = � � 4 !� � (54)Finally, the probability P M( , ) is obtained by

P Md

Gdz

iG z

z

i MM

( , ) ( , )( , )

=�

=

�

�

�� ( ( �2 20

2

2 1e � (55)

with z i /� e � 2, where the contour runs counterclock-wise along the unit circle and we have used the symme-try property G z G z( , ) ( , ) � � for the second equality.

Taylor expansion of the logarithm of the character-istic function in i= defines the cumulants or irreduciblecorrelators > k( ):

ln ( , )( )

!( )G

ik

k

kk=

=> �

�

�

�1

. (56)

The cumulants have a direct polynomial relation withthe moments n n P nk k

n

( ) ( , ) � � . The first twocumulants are the mean and the variance, and thethird cumulant characterizes the asymmetry (or skew-ness) of the P M( , ) distribution and is given by

> / 33 3( ) ( ) ( ( ) ( ))� � �n n n . (57)

In this section, we investigate FCS mainly in thecontext of the probability P M( , ) that M electronshave passed through the right junction during thetime . Since the average current and the shot noiseare proportional to the average number of the tunnel-ing electrons !M and the width of the distribution ofP M( , ) , respectively, we focus on the new aspectsthat are not covered by the study of the average cur-rent or shot noise.

In order to get FCS in general cases we integratethe master equation (52) numerically (see Sec. 6.3).In the low-bias regime, however, some analytic calcu-lations can be made. We will show through the fol-lowing subsections that for symmetric junctions in thelow-bias regime (2 2�p CeV E� � ), and irrespective ofthe junction symmetry in the very low bias regime(eV p� 2� ), P M( , ) is given by the residue at z � 0alone,

P MM

d

dzG z

M

Mz

( , )( )!

( , ) . �,

,,

�

12

2

20

(58)

Through this section we assume that the gate chargeis N /G � 1 2, unless stated explicitly not so.

We will now follow the outline of the previous sec-tion and start by considering two analytically tracta-ble cases before proceeding with the full numerical re-sults.

6.1. Two-state process; eV p� �

For the very low bias eV p� 2� at zero temperature,no plasmons are excited and electrons are carried bytransitions between two states ( , ) ( , ) ( , )N n �

metry (skewness) by the ratio A /e� > >32

1 ( ) - 4 ,noticing the Fano factor F /e�

��lim ( ( )�

> � > 2 1 . Thefactor A F / /2 2

23 1 2 1 4� � ( ) is positive definite(positive skewness) and bounded by 1 4 12/ A5 5 . Itis interesting to notice that A2 is a monotonic func-tion of F2 and has its minimum A /2 1 4� for the mini-mum F /2 1 2� and maximum A2 1� for the maximumF2 1� . Notice A � 1 for a Poissonian and A � 0 for aGaussian distribution. Therefore, the dependence ofA2 on the gate charge NG is similar to that of theFano factor F2 (see Fig. 4) with dips at NG

dip in Eq.(38). Together with Eq. (37), it implies that the ef-fective shot noise and the asymmetry of the probabil-ity distribution per unit charge transfer have their re-spective minimum values at the gate charge NG

dip

which depends on the tunnel-junction asymmetry andthe interaction strength of the leads.

The integral in Eq. (55) is along the contour de-picted in Fig. 8. Notice that the contributions from thepart along the branch cuts are zero and we are left withthe multiple poles at z � 0. By residue theorem, thetwo-state probability P M2( , ) is given by Eq. (58).

The exact expression of P M2( , ) is cumbersome.For symmetric tunneling barriers with N /G � 1 2( , )/ � �� 0 0� , however, Eq. (60) is reduced to

G zz

z zs z z22 20 0 0

41 1( )( , ) [( ) ( ) ]

��

e e e

�� .

(62)

Accordingly, P Ms2( )( , ) is concisely given by

P MM M

sM M

20

2 10

20 1

2 2 1 2( )( , )

( )

( )!

( )

( )!

��

"

#$

e ��

$

%

&''

:�1

2 2 110

2 1( )

( )!for

� M

MM , (63)

with P /s2 00 1 20( )( , ) ( ) �� e � � , in agreement with

Eq. (24) of Ref. 76. While this distribution resemblesa sum of three Poisson distributions, it is not exactlyPoissonian.

For a highly asymmetric junctions R �� 1( )� �� , the first term in Eq. (60) dominates thedynamics of G z2( , ) and its derivatives, and the char-acteristic function is approximated by

G z za2 02 2( )( , ) exp ( ( ) ) � � /� � � � . (64)

Now, the solution of P Ma2( )( , ) is calculated by this

equation and Eq. (58). The leading order approxima-tion in � � �/ leads to the Poisson distribution,

P MM

aM

2( )( , )

( )!

.

� � ��

�e (65)

For a single tunneling-barrier, the charges are trans-ported by the Poisson process (F � 1) [31]. Therefore,we recover the Poisson distribution in the limit ofstrongly asymmetric junctions and in the regime ofthe two-state process, in which electrons see effec-tively single tunnel-barrier.

For the intermediate barrier asymmetry the proba-bility P M2( , ) of a two-state process is given by a dis-tribution between Eq. (63) (for symmetric junctions)and the Poissonian (65) (for the most asymmetricjunctions).

6.2. Four-state process; 2�p � eV � 2EC

Since we focus the FCS analysis on the caseN /G � 1 2, the next simplest case to study is a four-state-model rather than the three-state-model dis-cussed in the connection of the shot noise in the previ-ous section.

In the bias regime where the transport is governedby Coulomb blockade (eV EC� 2 ) and yet the plas-mons play an important role (eV p: 2� ), it is a fairlygood approximation to include only the four stateswith N � 0 1, and n1 0 1� , (nm � 0 for m : 2). Forgeneral asymmetric cases, the rate matrix �( )� = inEq. (52) is determined by ten participating transitionrates (four rates from each junction, and two relax-ation rates). The resulting eigenvalues of | ( , )g = ! arethe solutions of a quartic equation, which is in generalvery laborious to present analytically.



C2 C1

– /2 + /2

+ /2 – /2

C0

C 3 C4

Fig. 8. Contour of Eq. (55). The arguments of f z2( ) �� ( )� � ��/ z0

2 2 are / / / /2 2 2 2, , , along thebranch cuts C C C1 2 3, , , and C4, respectively.

For symmetric junctions (R � 1) with no plasmonrelaxation, however, the rate matrix is simplified to

�( )� =

� � � �

� � � ��

�

� �

� ��

00 10 002

012

01 11 102

112

00

0

0

z z

z z

� � �

"

#

$$$$$

%

&

'''''

� � ��

01 00 10

10 11 01 11

0

0

(66)

with the matrix elements � ij L i j� � �� ( , , )1 0� ��R i j( , , )0 1 . The steady-state probability is thengiven by

| ( ) | ( , )( )

( )P g zs401 10

01

10

10

01

24 ! � � ! �

"

#

$$$

� �

��

$

%

&

''''

. (67)

Solving Eq. (54) with this probability and the ratematrix (66) is laborious but straightforward and onefinds

G z /( , ) ( ) � � � � �� ; � � �e 00 10 01 11 2

;6

78

98

?@8

A8 - �

"

#$$

%

&''

G zzz

z zI ( , )( )

{ }

1

8

2, (68)

with

G zA zf zI

z z f z /( , )( )( )

( ( )) � � ��

��

�

��

67

� �e 00 11 4 2

41

9;

; ��

�

��

�

��?@A

- �12

4

01 104 4

A z f z

zf z f z

( ) ( )

( ){ ( ) ( )}

� �, (69)

where A z( ) and f z4( ) are given by

A z z( ) ( )� � � � � �� 00 10 01 11 00 11 012 ,

f z z a b4 00 112

10 012 24( ) ( ) ( )� � � � � � � � (70)

with dimensionless parameters a b, given by

a ��

� �

( )( )

( ),

� � � � � �

� � � �00 11 00 10 01 11

00 112

10 014

b � � �

� �

4

4

10 01 00 10 01 11

00 112

10 01

� � � � � �

� � � �

( )

( ).

Integral of G zI ( , ) along the contour | |z � 1 containstwo branch points at z a ibc � 0 , however, the inte-gral along the branch cuts cancel out due to the sym-metry under [ ( ) ( )]f z f z4 4- � . Therefore, the contri-bution from the branch cuts due to G zI ( , ) andG zI ( , )� is zero to the probability P M

s4( )( , ) , and it

is given by the residues only at z � 0, i.e., by

Eq. (58). The explicit expression of P Ms4( )( , ) is

cumbersome.The probability distribution P M2( , ) for the two-

state process deviates from Eq. (63) as a function ofthe asymmetry parameter R and reaches Poissonian inthe case of strongly asymmetric junctions. In a similarmanner, P Ms4

( )( , ) deviates from P Ms2( )( , ) as a

function of ratio of the transition rates � ij .

6.3. Numerical results

It is worth mentioning that for strongly asymmetricjunctions P M( , ) is Poissonian in the very low biasregime (eV p� 2� ), as seen from Eq. (65). It exhibitsa crossover at eV p� 2� : P M( , ) deviates from Pois-son distribution for 2 2�p CeV E� � while it isPoissonian for eV p� 2� (at T � 0), as shown by theshot noise calculation.

Voltage dependence

The analytic results presented above are useful ininterpreting the numerical results in Fig. 9, whereprobability P M( , ) for symmetric junctions (R � 1)and R � 100, in the case of LL parameter g � 0 5. withno plasmon relaxation (� p � 0), is shown as a functionof eV and M, that is the number of transported elec-trons to the right lead during such that during thistime ! � �M Ic 10 electrons have passed to the rightlead at eV EC� 2 .

The peak position of the distribution of P M( , ) isroughly linearly proportional to the average particleflow !M , and the width is proportional to the shotnoise but in a nonlinear manner. In a rough estimate,therefore, the ratio of the peak width to the peak posi-tion is proportional to the Fano factor. Two featuresare shown in the Fig. 9. First, the average particle flow(the peak position) runs with different slope when thebias voltage crosses new energy levels, i.e., at eV p� 2�and eV EC� 2 , that is consistent with the I V� study(compare Fig. 9,b with Fig. 3). Notice E /gC p� �� 2�p for g � 0 5. . Second, the width of the distributionincreases with increasing voltage, with different char-acteristics categorized by eV p� 2� and eV EC� 2 . Es-pecially in the bias regime eV EC� 2 in which severalcharge states participate to the charge transport, for thehighly asymmetric case, the peak runs very fast whileits width does not show noticeable increase. It causesthe dramatic peak structure in the Fano factor aroundeV EC� 2 as discussed in Sec. 5.

The deviation of the distribution of probabilityP M( , ) due to the nonequilibrium plasmons from itslow voltage (equilibrium) counterpart is shown in theinsets. Notice in the low-bias regime eV p� 2� , it fol-lows Eq. (63) for the symmetric case (Fig. 9,a, insetI), and the Poissonian distribution (65) for the highly



asymmetric case (Fig. 9,b, inset I). The deviation isalready noticeable at eV Ep C� �3 15� . for R � 1 (in-set II in Fig. 9,a), while it deviates strongly aroundeV EC� 2 for R � 100 (inset III in Fig. 9,a).

Interaction strength dependence

We have concluded in Sec. 5 that shot noise showsmost dramatic behavior around eV EC� 2 due to inter-play between the nonequilibrium plasmons and thecharge excitations. To see its consequence in FCS, weplot in Fig. 10 the probability P M( , ) as a function ofthe particle number M and the interaction parameter gfor �� 10 2 0e/I eV EC p( , ) with no plasmon re-laxation and with fast plasmon relaxation.

The main message of Fig. 10,a is that the shot noiseenhancement, i.e., the broadening of the distribution

curve, is significant in the strong interaction regimewith gradual increase with decreasing g. Fast plasmonrelaxation consequently suppresses the average currentand the shot noise dramatically as shown in Fig. 10,bimplying the Fano factor enhancement is lost. Effec-tively, the probability distribution of P M( , ) for dif-ferent interaction parameters maps on each other al-most identically if the time duration is chosen such that !M equals for all g.

7. Conclusions

We have studied different transport properties of aLuttinger-liquid single-electron transistor includingaverage current, shot noise, and full counting statis-tics, within the conventional sequential tunnelingapproach.

At finite bias voltages, the occupation probabilitiesof the many-body states on the central segment isfound to follow a highly nonequilibrium distribution.The energy is transferred between the leads and thequantum dot by the tunneling electrons, and the elec-tronic energy is dispersed into the plasmonic collective



40

30

20

10

0

M

0.5 1.0 1.5 2.0eV/E C

0.8

0.6

0.4

0.2

0

a1.0

0 0.5 1.0 1.5 2.0eV/E C

40

30

20

10

M

0.8

0.6

0.4

0.2

0

b1.0

Fig. 9. Probability P M( , )� with no plasmon relaxation��p � 0) during the time (� � 10/Ic, where Ic is the particlecurrent at eV EC� 2 with no plasmon relaxation (�p � 0):for symmetric junctions (R � 1) (a) and for a highly asym-metric junctions (R � 100) (b). Here g � 0 5. , N /G � 1 2,and T � 0. Inset shows cross-sectional image of P M( , )�(blue solid line) as a function of M and the reference dis-tribution function (a) Eq. (63) (magenta dashed line) and(b) the Poisson distribution Eq. (65) (red dashed), ateV EC� 05. (I), eV EC� 15. (II), and eV EC� 2 (III).

05 10 15 20

M

P(M, )�

0.20

0.15

0.10

0.05

0.4

0.6

0.8

1.0

g

a

05 10 15 20

M

0.4

0.6

0.8

1.0

g

P(M, )�

0.20

0.15

0.10

0.05

b

Fig. 10. The probability P M( , )� that M electrons havepassed through the right junction during the time� � 10 0/I , where I0 is the particle current with noplasmon relaxation (�p � 0); with no plasmon relaxation��p � 0) (a) and with fast plasmon relaxation (�p � 10

4)(b). Here eV E RC� �2 100, , N /G � 1 2 and T � 0.

excitations after the tunneling event. In the case ofnearly symmetric barriers, the distribution of the oc-cupation probabilities of the nonequilibrium plasmonsshows impressive contrast depending on the interac-tion strength: In the weakly interacting regime, it is acomplicated function of the many-body occupationconfiguration, while in the strongly interacting re-gime, the occupation probabilities are determined al-most entirely by the state energies and the bias volt-age, and follow a universal distribution resemblingGibbs (equilibrium) distribution. This feature in thestrong interaction regime fades out with the increasingasymmetry of the tunnel-barriers.

We have studied the consequences of these non-equilibrium plasmons on the average current, shotnoise, and counting statistics. Most importantly, wefind that the average current is increased, shot noise isenhanced beyond the Poisson limit, and full countingstatistics deviates strongly from the Poisson distribu-tion. These nonequilibrium effects are pronounced es-pecially in the strong interaction regime, i.e. g � 0.5.The overall transport properties are determined by abalance between phenomena associated with nonequi-librium plasmon distribution that tend to increasenoise, and involvement of several charge states andthe ensuing correlations that tend to decrease noise.The result of this competition is, for instance, anonmonotonic voltage dependence of the Fano factor.

At the lowest voltages when charge can be trans-ported through the system, the plasmon excitations aresuppressed, and the Fano factor is determined bycharge oscillations between the two lowest zero-modes.Charge correlations are maximized when the tunnel-ing-in and tunneling-out rates are equal, which forsymmetric junctions occurs at gate charge N /G � 1 2.At these gate charges the Fano factor acquires its low-est value which at low voltages is given by a half of thePoisson value, known as 1/2 suppression, as only twostates are involved in the transport, at somewhat largervoltages increases beyond the Poisson limit as plasmonexcitations are allowed, and at even higher voltages ex-hibits a local minimum when additional charge statesare important. If the nonequilibrium plasmon effectsare suppressed, e.g., by fast plasmon relaxation, onlythe charge excitations and the ensuing correlation ef-fects survive, and the Fano factor is reduced below itslow-voltage value. The nonequilibrium plasmon effectsare also suppressed in the noninteracting limit.

Acknowledgments

This work has been supported by the Swedish Founda-tion for Strategic Research through the CARAMEL con-sortium, STINT, the SKORE-A program, the eSSC atPostech, and the SK-Fund. J.U. Kim acknowledges par-

tial financial support from Stiftelsen Fru Mary vonSydows, f�dd Wijk, donationsfond. I.V. Krive gratefullyacknowledges the hospitality of the Department of Ap-plied Physies at Chalmers University of Technology.

Appendix A: The transition amplitudesin the quantum dot

In this appendix, we derive the transition ampli-tudes, Eqs. (15) and (16). As shown in Eq. (14), thezero-mode overlap of the QD transition amplitude iseither unity or zero. Therefore, we focus on the over-lap of the plasmon states. It is enough to consider

| { }| ( )| { } | , ,† � ! �n X n X X XD L R

� � �2

due to the symmetry between matrix elements of tun-neling-in and tunneling-out transitions, as in Eq. (15),

| { }| ( )| { } | [| { }| ( )| { } |† † � ! � � ! ��n X n n X nrr

� � �2 2B

� ! � ! �{ }| ( )| { } { }| ( )| { } ]†n X n n X nr rB B� �

� � !2 2| { }| ( )| { } |†n X nrB � , (A.1)

where r � �( ) denotes the right (left)-moving com-ponent, and the cross terms of oppositely movingcomponents cancel out due to fermionic anti-commu-tation relations.

The transition amplitudes at XL is identical to thatat XR. For simplicity, we consider the case at XL � 0only. The overlap elements of the many-bodyoccupations � { }| , , ..., ,... |n n n nm1 2 and | { }n� ! �� !| , ,..., ,...n n nm1 2 are

| { }| ( )| { } | | | | |† � � ! � � !�

�

2n x n n nrm

m m mB �C0

12

2

1

2

+,

(A.2)

where C Dm m m mb b� exp[ ( )]† with Dm i/ gmM� �

is the bosonized field operator at an edge of the wirewith open boundary conditions (see for instance Ref.44). Here g is the interaction parameter, m is the modeindex (and the integer momentum of it), and M is thenumber of transport sectors; if M � 1, the contribu-tions of the different sectors must be multiplied. Theoperators bm and bm

† denote plasmon annihilation andcreation and + is a high energy cut-off.

Using the Baker–Haussdorf formula

C D�

� �m m m m

a aa a m m m m� �

exp[ ( )]††

e e e

2

2 , (A.3)

and the harmonic oscillator states

|( )

!| , | |

!,

†n

a

nn

a

n

n n! � ! � 0 0 (A.4)



one can show that, if n n5 �,

| | | || |

! !!

( )!| |

( ) � ! �

� � ��

�� ;

�n n

n nn

n n

n nC

D�22 22

e

; � � �E( , ; | | )n n n1 1 2 2D , (A.5)

where E( , ; )x x z� is a degenerate hypergeometric func-tion defined by [77]

E( , ; )!( )!

( )!( )!

( )!x x z

z xx

x

x� �

� �

� ��

"

�

�

��

��

�

�0

11

11#

$$

%

&''

. (A.6)

If n n�5 , the indices n and n� are exchanged in Eq.(A.5). The function E( , ; )x x z� is a solution of theequation

z x z xz z� � � � � �2 0E E E( ) .

By solving this differential equation with the propernormalization constant, one obtains

E( , ; | | )!( )!

!(| || |n n n

n n nn

Lnn n � � � �

� � � �

�1 1 2 2

2D D�e ),

where L yab ( ) is the Laguerre polynomials [77]. In

terms of the Laguerre polynomials, therefore, thetransition amplitude (A.5) is written by

| | | |( )

!

!

/

| |

( )

( ) � ! � ;

�

�

�n n

gmM

n

nm m m

gmM

n n

m

mm mC 2

1e

;�

��

�

��

"

#$

%

&'�

� LgmMn

n n

m

m m( )

| | 12

, (A.7)

where n n n( ) ( , )� � �min and n n n( ) ( , )� � �max .We introduce a high frequency cut-off m k L /c F D~ �

to cure the vanishing contribution due to e 1/gmM ,

12

12

1 4

1�

��

++

e

�2 - �

��

�

��

/ mg

m

m

D

c

D

c

L L, (A.8)

where the exponent is * � �( )g /M1 1 . We arrive atthe desired form of the on-dot transition matrixelements,

| { }| ( { } |† )|� ! ��

��

�

�� ;n X n L La D D

�+

�

2 1 ��

;�

��

�

��

�

�

� �

�

�22 �11

g mM

n

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Nonequilibrium plasmons and transport properties of a ... · Keywords: quantum wire, Luttinger liquid, shot noise, full counting statistics. 1. Introduction The recent discovery of