Nonrenewal statistics in transport through quantum dots

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by  Krzysztof Ptaszynski

Nonrenewal statistics in transport through quantum dots
Krzysztof Ptaszyn´ski∗
Institute of Molecular Physics, Polish Academy of Sciences,
ul. M. Smoluchowskiego 17, 60-179 Poznan´, Poland
(Dated: January 10, 2017)
The distribution of waiting times between successive tunneling events is an already established
method to characterize current fluctuations in mesoscopic systems. Here, I investigate mechanisms
generating correlations between subsequent waiting times in two model systems, a pair of capaci-
tively coupled quantum dots and a single-level dot attached to spin-polarized leads. Waiting time
correlationsareshowntogiveaninsightintotheinternaldynamicsofthesystem,forexamplethey
7 allow distinction between different mechanisms of the noise enhancement. Moreover, the presence
1 of correlations breaks the validity of the renewal theory. This increases the number of independent
0 cumulants of current fluctuation statistics, thus providing additional sources of information about
2 the transport mechanism. I also propose a method for inferring the presence of waiting time cor-
relations based on low-order current correlation functions. This method gives a way to extend the
n
a analysis of nonrenewal current fluctuations to the systems for which single-electron counting is not
J experimentally feasible. The experimental relevance of the findings is also discussed, for example
reanalysis of previous results concerning transport in quantum dots is suggested.
9

] PACSnumbers: 72.70.+m,73.63.Kv,73.23.Hk
l
l
a
h I. INTRODUCTION imental studies of the electronic transport through mul-
- tilevel quantum dots [16–18] and organic metallic tunnel
s
e Investigation of current fluctuations is a valuable tool junctions [19].
m
for characterization of the underlying transport mech- Another mechanism of noise enhancement, which
t. anism. At low temperatures the main contribution to should be carefully distinguished from the former, is the
a low-frequency fluctuations comes from the shot noise as- stochasticswitchingbetweentransportchannelswithdif-
m sociated with charge quantization [1, 2]. The magnitude ferentconductances,referredtoasthetelegraphicswitch-
- of the shot noise depends on the quasiparticle charge ing [20]. It can be caused either by some external factor
d
and the correlation of transport events, which is associ- (e.g. a charge [5, 6] or spin [7] dynamics of a coupled
n
o ated with quasiparticle statistics or the presence of in- external system) or by some intrinsic causes (i.e. it can
c teractions [1]. In quantum dots the noise was found be a result of switching between charge [21], spin [13],
[ to provide information about the asymmetry [3] and vibrational [22–24], motional [25] or energy [20] states

spin-dependence[4]oftunnelingratesinCoulombblock- of a system itself). While this phenomenon has been
1
v ade systems, interactions with the external environment studied mostly theoretically, the noise enhancement re-
3 (with charge [5, 6], spin [7] and vibrational [8] degrees sulting from the switching between the ground and the
0 of freedom), the presence of quantum coherence [9] or metastable excited state of a quantum dot [21] or from
1 many-body quantum correlations (e.g. in the Kondo ef- the dependence of tunneling rates on a vibrational state
2 fect [10, 11]), or the existence of Majorana fermions [12]. of a single-molecule transistor [24] has been experimen-
0 There are three characteristic regimes of noise distin- tallyobservedusingchargecountingtechniquesandcon-
.
1 guished by the value of the Fano factor F, which is a ventional current measurements, respectively.
0 ratio of the noise power and the mean current (see e.g.
7 Refs. [1, 13, 14]). If transport events are uncorrelated Amoregeneralanalysisofthecurrentfluctuationscan
1 the noise is Poissonian (F = 1). In fermionic systems, beprovidedbytwootheralreadyestablishedapproaches:
v: correlations typically reduce the noise to sub-Poissonian thefullcountingstatistics(FCS)[26,27]andthewaiting
i values (F < 1). However, there are also cases for which time distribution (WTD) [28]. FCS analyses the distri-
X
the noise may become super-Poissonian (F >1). bution of the number of particles transferred in a given
ar One of the mechanisms generating the super- time interval, providing information not only about the
Poissonian noise is the dynamical channel blockade [4, noise, but also about higher zero-frequency current cor-
15], which is associated with the coexistence of trans- relators. High-order cumulants were found to be use-
port channels with different conductances. The electron ful for reconstructing generators of the time evolution of
occupying one channel may block transport through the systems with multiple degrees of freedom [29]. In a com-
remaining channels, which would lead to the noise en- plementary way, WTD studies the distribution of time
hancement. Thismechanismhasbeenreportedinexper- delays between subsequent physical events. WTD was
found to reveal the short-time dynamics of the system,
for example by showing the oscillatory behavior of the
distribution associated with a spin precession [30] or co-
∗ [email protected] herent oscillations of electrons in a system (e.g. in a
2
quantum dot molecule [28] or a quantum dot attached tached to spin polarized leads. These models enable a
to normal and superconducting leads [31]). In quantum relatively simple qualitative understanding of physical
dots both FCS and WTD can be experimentally stud- mechanisms generating nonrenewal current fluctuations,
ied using charge detection techniques [32]. While such and thus may provide a working basis for the analysis of
experiments are mostly conducted in ultra-low temper- more complex mesoscopic systems. Transport statistics
atures (down to several millikelvins), recently the real- of the studied models are analyzed using the methods
timechargetrapdynamicsmonitoringatroomtempera- based on the master equation [27, 28]. The nonrenewal
ture has been shown to be feasible [33]. However, single behavior is made evident by calculating the joint distri-
electron counting experiments are currently confined to bution and the cross-correlation of two subsequent wait-
very low currents, with frequencies of tunneling events ing times. These quantities are shown to give an insight
on the order of kHz [3, 32, 34, 35]. Due to this fact, into the internal dynamics of the system, for example by
the method determining indirectly WTD with the eas- providing a way to distinguish between the telegraphic
ier measurable quantities, like the second-order current switching and the dynamical channel blockade. The in-
correlation function, has been proposed [35]. While this fluenceofthenonrenewaldynamicsonFCSandWTDis
paper focuses on electronic transport, both FCS and also analyzed. This reveals some nontrivial effects, like
WTD are widely applied also beyond this field, for ex- the nonequivalence of waiting time distributions for in-
ample in quantum optics [36–38] and statistical kinetics coming and outgoing electrons which arises due to the
of biomolecular systems [39–41]. correlation between electron jumps. Moreover, in order
toextendtheanalysistothesystemsforwhichthesingle-
Inmanycasestransportstatisticscanbewelldescribed
electron counting is not experimentally feasible, I study
by the renewal theory, which assumes that successive
the influence of the nonrenewal dynamics on the second-
waiting times between transport events are statistically
ordercurrentcorrelationfunctionanditsrelationtoFCS
independent equally distributed random variables [42].
and WTD.
Insuchacase,thejointprobabilitydensityoftwosucces-
Thepaperisorganizedasfollows. Sec.IIdescribesthe
sivewaitingtimesw(τ ,τ )canbefactorizedintoaprod-
1 2
methods used to characterize the current fluctuations.
uct of two single-time distributions w(τ )w(τ ) [43, 44].
1 2
In Sec. III a model of the double quantum dot system
Whenthisassumptionissatisfied,thereareexactidenti-
is specified and corresponding results are presented and
ties between cumulants of FCS and WTD, enabling the
discussed. Sec. IV in a similar way deals with transport
reconstruction of FCS on the basis of WTD [42, 45].
within the Anderson model. Finally, Sec. V brings con-
Moreover, for the renewal dynamics a relation holds be-
clusions following from my results.
tween WTD and the second-order correlation function
g(2)(τ), which enables the reconstruction of the former
on the basis of the latter, even when the detector effi-
II. METHODS
ciency is below 100% [36]. However, studies of coherent
transport in a quantum point contact have shown that
waiting times between successive transport events can The paper reports investigation of electron transport
becorrelated[43,46]. Correlationswerefoundtoappear throughquantumdotsystemsinthesequentialtunneling
due to the fermionic statistics of electrons. The pres- regime. I focus on the situations when the coupling to
ence of correlations indicates that the renewal assump- theleadsisweakandtheseparationbetweenthediscrete
tion is not satisfied. In such a case, the aforementioned energy levels of the dot and the electrochemical poten-
relations between cumulants of FCS and WTD are no tials of the leads is high in comparison with kBT (the
longer valid, as shown by Albert et al. [46]. Beyond the assumption referred to as the infinite-bias limit [14, 28]).
field of electronic transport, nonrenewal statistics have As shown in Refs. [27, 51, 52], in such a case the tun-
been previously investigated in some systems described neling is unidirectional and can be exactly described by
by the Markovian master equation, for example in opti- the Markovian master equation. The master equation
cal [44, 47–49] or biochemical [50] ones. In particular, written in the Liouville space takes the form [53, 54]
the analysis of waiting times correlations was shown to
ρ˙(t)=Lρ(t), (1)
give an insight into the internal dynamics of fluorescent
molecules [47–49]. This suggests, that the investigation
where: ρ(t) is the column vector containing in general
of physical quantities characterizing the deviation from
both diagonal and non-diagonal elements of the density
the renewal behavior may be a useful tool for character-
matrixofthesystem(stateprobabilitiesandcoherences),
ization of the experimentally relevant systems.
and L is the square matrix representing the Liouvillian.
The aim of this paper is to extend the investigation In all studied systems the coherences are neglected and
of nonrenewal statistics to electronic transport in the se- only the dynamics of diagonal elements of the density
quential tunneling regime. To this end, I analyze two matrix is considered. The study is also confined to the
conceptually simple and experimentally feasible trans- systems having a unique stationary state ρ , which is a
0
port models: the system of two capacitively interacting solution of the equation Lρ=0.
quantum dots and the single-impurity Anderson model Tocharacterizethecurrentfluctuationsinthestation-
in the infinite-bias limit realized in a quantum dot at- ary state I calculate the following quantities: cumulants
3
of the zero-frequency full counting statistics, the wait- transform. In general w (τ) (cid:54)= w (τ). The n-th or-
kl lk
ing time distribution, the joint distribution of two suc- der moments and cumulants of the distribution w (τ),
kl
cessive waiting times and the second-order current cor- denoted as (cid:104)τn(cid:105) and κkl respectively, are given by ex-
kl n
relation function. To calculate FCS a counting field χ, pressions [28]
which counts the number of transport events, is intro-
(cid:20)dnw (s)(cid:21)
duced [26, 27]. The master equation is then written in (cid:104)τn(cid:105)=(−1)n kl , (6)
the counting field dependent form kl dsn
s=0
(cid:26)dnlog[w (s)](cid:27)
ρ˙(t,χ)=(L +Jeχ)ρ(t,χ), (2) κkl =(−1)n kl . (7)
0 n dsn
s=0
where J is the operator describing the considered set of In a similar way one can consider a joint distribution
jump processes (e.g. tunneling of electrons between a of two subsequent waiting times τ and τ separating
kl lm
dot and a chosen lead) and the operator L0 = L−J three successive jumps of types k, l and m. This distri-
describes the remainder of the system’s dynamics. In bution is denoted as w (τ ,τ ). Distributions of this
klm kl lm
this paper I focus on time-independent scaled cumulants type for systems described by the master equation have
of FCS. The n-th order scaled cumulant cn is defined in been already considered in Refs. [44, 48]. The Laplace
the following way: cn = limt→∞Cn(t)/t, where Cn(t) is transform of the distribution wklm(τkl,τlm) is given by
the n-th order cumulant of the number of jump events the expression
occurring in the time t. The scaled cumulants can be
(cid:90) ∞ (cid:90) ∞
calculated as follows [29, 55]: wklm(s,z)= dτlm dτkle−sτkl−zτlmwklm(τkl,τlm)
(cid:20) dn (cid:21) 0 0
cn = dχnλ(χ) , (3) = Tr[Jm(z−Ll0m)−1Jl(s−Lk0l)−1Jkρ0], (8)
χ=0 Tr[J ρ ]
k 0
where λ(χ) is the scaled cumulant generating function. where Lkl and Llm are remainders of the Liouvillian
0 0
Whenthesystemhastheuniquestationarystateρ0,λ(χ) defined in the analogous way as in Eq. (5). If sub-
is a dominant eigenvalue of the counting-field dependent sequent waiting times are uncorrelated the joint WTD
operator L0 +Jeχ [56]. In general it is impossible to can be factorized in the following way: wklm(τkl,τlm) =
find analytical expressions for λ(χ) if the rank of the w (τ )w (τ ) and w (s,z) = w (s)w (z), other-
kl kl lm lm klm kl lm
matrix is higher than 4. However, it is not necessary, wise it is not possible. Such a factorization is often re-
because all scaled cumulants up to the N-th order can ferred to as the renewal property [43]. Using the joint
be calculated using Eq. (3) and solving the following set WTDonecancalculatethecross-correlation(covariance)
of linear equations [29, 55]: of two successive waiting times
(cid:26) dn (cid:27) (cid:26) ∂ ∂ (cid:27)
dχn det[λ(χ)−L0−Jeχ] =0, (4) (cid:104)∆τkl∆τlm(cid:105)= ∂s∂z log[wklm(s,z)] , (9)
χ=0 s=0,z=0
where∆τ =τ −(cid:104)τ (cid:105)isthedeviationfromthemean. It
with n ranging from 1 to N. The knowledge of λ(χ) is ij ij ij
isalsousefultoconsiderthenormalizedcross-correlation
not required since λ(0)=0.
Now, the waiting time distribution is considered fol- (cid:104)∆τ ∆τ (cid:105)
NCC = kl lm , (10)
lowing the approach developed by T. Brandes [28]. One (cid:112)
(cid:104)∆τ2(cid:105)(cid:104)∆τ2 (cid:105)
can calculate the distribution of waiting times either be- kl lm
tween two successive jump events of the same type k or definedasthecovariancedividedbytheproductofstan-
between jumps of two different types k and l. In this dard deviations of waiting times. If the renewal assump-
paper the term ”jump of type k” refers to the tunneling tion is fulfilled NCC =0.
between a dot and a chosen lead; in general it may re- If the jumps associated with the operator J are
k
fer to an arbitrarily chosen set of transitions within the renewal processes, one can find a direct relation be-
Markovian model. The Laplace transform of the distri- tween WTD and the cumulant generating function of
bution of waiting times between jumps of types k and l FCS [28, 42], as well as between the cumulants of FCS
is given by the expression [28] and WTD [42, 45]. For nonrenewal processes such sim-
ple relations do not hold [45, 46]. In Markovian systems,
w (s)=(cid:90) ∞e−sτw (τ)dτ = Tr[Jl(s−L0)−1Jkρ0], liketheonesconsideredinthispaper,allelementarypro-
kl kl Tr[J ρ ] cesses (i.e the transitions between single states in the Li-
0 k 0
(5) ouville space) are renewal ones and one can always find
where L = L−J for k = l and L = L−J −J for a relation between the full counting statistics and the
0 k 0 k l
k (cid:54)= l, with J , J being operators describing jumps of waiting time distributions of such processes [28]. This
k l
typesk andlrespectively,andρ isthevectorofthesta- contrasts with the systems studied in Refs. [43, 46], in
0
tionary state. The waiting time distribution w (τ) can which the nonrenewal behavior was a result of the non-
kl
be easily obtained from w (s) using the inverse Laplace Markovian system-reservoir dynamics. However, also in
kl
4
Markovian systems the nonrenewal statistics can arise valid [14]. However, one can still use Eq. (16) to define
when the jump operator is associated with at least two the pseudo-waiting time distribution u(τ) and calculate
elementary transitions [44, 47–49]. Such complex forms its cumulants q in the following way:
n
of the jump operator are often necessary to describe the
experimentally observed processes, because in many sit- q =(−1)n(cid:26)dnlog[u(s)](cid:27) . (17)
uationstheelementarytransitionsarenotdirectlyacces- n dsn
s=0
sible due to the presence of hidden degrees of freedom.
Asshownbelow,thisfictitiousdistributionmaybeuseful
Finally,Iconsiderthesecond-ordercurrentcorrelation
for revealing the nonrenewal dynamics of the system.
function S(τ) which, in contrast to the waiting time dis-
tribution, is defined for the continuous current measure-
ments. I focus on the auto-correlation function – the
III. TWO CAPACITIVELY COUPLED DOTS
correlationofvaluesofthecurrentflowingthroughasin-
gle junction measured at two different moments of time.
It is expressed in the following way [14, 28]: A. Model
S(τ)=(cid:104)δI(τ +t)δI(t)(cid:105) , (11) In this section I consider the system of two capaci-
t
tively coupled quantum dots [Fig. 1 (a)] to illustrate the
where δI(t) is the deviation of the current from its aver-
generation of nonrenewal current fluctuations due to the
age in the stationary state. Its Fourier transform is the
telegraphic switching. Such a double quantum dot sys-
finite-frequency noise power [14, 28]
tem, despite its simplicity, exhibits a rich physics, and
therefore has been already extensively studied both ex-
(cid:90) ∞
S(ω)= dτeiωτS(τ). (12) perimentally[57]andtheoretically[58]. Previousstudies
−∞ have revealed, for example, phenomena like a Coulomb
drag [59–61], heat-to-current conversion [62, 63] and
For the unidirectional transport S(τ) can be expressed
even Maxwell’s-demon-like behavior [64] (confirmed ex-
as [14]
perimentally by using the system of two metallic is-
lands [65]). Moreover, recently the influence of the tele-
S(τ)=(cid:104)I(cid:105)2[g(2)(τ)−1], (13)
graphicswitchingonthefullcountingstatisticsinasim-
ilar system of two capacitively coupled single electron
where (cid:104)I(cid:105) is a mean current and g(2)(τ) is the second-
boxes has been experimentally investigated [6].
order correlation function defined as [14, 36]
The studied system is a four terminal one – each dot
(cid:104)P(t,t+τ)(cid:105) is coupled to separate left (source) and right (drain)
g(2)(τ)= t, (14) leads. As mentioned in Sec. II, the infinite-bias limit
(cid:104)P(t)(cid:105)2
t is taken. Tunneling rates between the dot and the leads
where P(t,t+τ) is the joint probability density of two are assumed to be energy-dependent, which is experi-
electron tunnelings (not necessarily subsequent) occur- mentally feasible [63], and thus their values are condi-
ringattimestandt+τ respectively, andP(t)isaprob- tioned on the occupancy of the other dot. Additionally,
abilitydensityofthesingleelectrontunnelingatthetime a strong intra-dot Coulomb repulsion is assumed – each
t. For the stationary state (cid:104)P(t)(cid:105) = (cid:104)I(cid:105). The Laplace dot can be occupied by at most one electron. Therefore,
t
transform of the function g(2)(τ) is given by the expres- the dynamics of the system can be described by a four-
sion [14] state Markovian model with states defined in the basis
{(0,0),(1,0),(0,1),(1,1)},wherethefirstandthesecond
Tr[J(s−L)−1Jρ ] position in the brackets corresponds to the upper and
g(2)(s)= 0 , (15)
Tr[Jρ ]2 the lower dot, while 0 and 1 refers to the empty and the
0
occupied dot, respectively [Fig. 1 (b)]. The Liouvillian
where the operator J corresponds to all tunneling pro- written in the considered basis reads
cesses contributing to the measured current. In contrast  
−Γ −γ Γ γ 0
to Eq. (5), the above expression contains full Liouvillian L L R R
L FinosrtetahdeMofatrhkeovrieamnarienndeewraLl0d.ynamics,g(2)(τ)andS(τ) L= ΓγLL −ΓR0−γLU −ΓUL0−γR ΓγRUUR .
0 γU ΓU −ΓU −γU
are related to the waiting time distribution w(τ). After L L R R
some algebra, presented in Ref. [36], one obtains (18)
(cid:104)I(cid:105)g(2)(s) (cid:104)I(cid:105)2+sS(s) In principle, by using two separate detectors, it is possi-
w(s)= = . (16) ble to simultaneously detect charge states of both dots.
1+(cid:104)I(cid:105)g(2)(s) (cid:104)I(cid:105)2+s[(cid:104)I(cid:105)+S(s)]
In this way one can observe the whole dynamics of the
Using this equality one can reconstruct the WTD on the system. Since the dynamics is Markovian, each transi-
basis of the second-order current correlation function. If tion between individual charge states is a renewal pro-
the dynamics is nonrenewal, this relation is no longer cess. However, in this paper I focus on the statistics of
5
Γ Γ
Γ(1-a)
Γ (E) Γ (E)
R
Γ Γ(1-a) Γ
R R
U U γ γ γ γ γL γR γL γR
L R L R
(E) (E)
L R ΓU Γ(1+a) ΓL
Γ(1+a) ΓU
ΓU R
R (a) (b)
(a) (b)
FIG. 2. (Color online). Four state models for different tun-
FIG.1. (a)Systemoftwocapacitivelycoupledquantumdots; neling rate symmetries considered in the main text: (a) the
Γ (E) and γ (E) – energy dependent tunneling rates. (b) symmetric system with γ = γU = γ, Γ = (1−a)Γ and
i i i i i
Four-state Markovian model of the dynamics of the system. ΓU =(1+a)Γ; (b) the asymmetric system with γ =γU and
i i i
Each state corresponds to a different charge configuration, ΓU = Γ . Arrows with shaped tails correspond to counted
L L
with black dots denoting the electrons occupying quantum transitions,theredarrowswithbullet-tailscorrespondtothe
dots. Transition rates with superscripts U refer to the cases tunneling from the left lead to the dot, and blue arrows with
when the other dot is charged. square-tails in (b) correspond to the tunneling from the dot
to the right lead.
tunneling events in the upper quantum dot only, treat-
ing the lower dot as an external source of a random tele- In order to calculate the statistics of current fluctua-
graph noise (similarly to Refs. [5, 6]). The main reason tions, I also define two jump operators
forthisistoprovideasimplemodelwhichwellillustrates
 0 0 0 0
the influence of telegraphic switching on current fluctu-
Γ 0 0 0
ations (and, in particular, on waiting time correlations). J = L , (21)
L  0 0 0 0
Already in Refs. [44, 47, 48] it has been reported, that
0 0 ΓU 0
the telegraphic switching may generate a correlation be- L
tween subsequent waiting times. Moreover, nonrenewal 0 Γ 0 0 
R
photon statistics in a similar four-state model have been 0 0 0 0
J = , (22)
already studied in Refs. [48, 49]. However, this paper R 0 0 0 ΓU
R
extends these previous studies in two ways. Firstly, it 0 0 0 0
analyses additional statistical quantities, like the second
describing the tunneling from the left lead to the upper
andthethirdcumulantofcurrentfluctuations. Secondly,
dot and the tunneling from the upper dot to the right
it presents qualitatively new phenomena resulting from
lead, respectively. As one can note, each of these op-
the reduction of symmetry of the model studied.
erators corresponds to two transitions within the four
To explain, why the random switching is expected to
state model. As mentioned in Sec. II, the waiting time
occur in the system, the Liouvillian is decomposed as
correlationscanoccuronly,whenthejumpoperatorcor-
follows:
responds to at least two elementary transitions.
L=L +L , (19)
Γ γ
where LΓ/Lγ are operators describing tunneling in B. Results
the upper/lower dot (containing, respectively, tunneling
rates {Γ ,ΓU} and {γ ,γU}). The operator L can be
i i i i Γ I begin my analysis with the simplest case illustrating
written in the block-diagonal form
thetelegraphicswitching: thesystemwithdotssymmet-
(cid:18) (cid:19) rically coupled to left and rights leads [Fig. 2 (a)]. The
L 0
LΓ = 011 L , (20) tunneling rates in the lower channel are assumed to be
22
energy independent and equal to γ. The tunneling rates
where matrices L and L contain transitions rates Γ betweentheupperdotandtheleadsareequalto(1−a)Γ
11 22 i
and ΓU, respectively. These matrices describe tunnel- whenthelowerdotisemptyand(1+a)Γwhenthelower
i
ing in two distinct transport channels corresponding to dotisoccupied,whereaistheasymmetryparameterde-
different charge configurations of the lower dot. The op- scribing the energy dependence of tunneling rates.
erator L , meanwhile, describes the switching between At first, I show that waiting time correlations occur
γ
these channels. Thus, the system is expected to exhibit in the system. I denote the waiting time between subse-
the telegraphic switching behavior. quent electron jumps from the left lead to the upper dot
6
probabilitythatashortwaitingtimeisfollowedbyalong
one and conversely. In the limiting case of γ → ∞, the
correlation vanishes. When the absolute value of a is
increased, the cross-correlation at first rises because the
deviationofwaitingtimesfromthemeanvalueincreases
in either channel. However, for high values of |a| the
NCC starts to decrease (expect for the limiting case of
γ → 0). In such a case the current flowing through a
slow channel is blocked, and transport is predominated
by thetunnelingin thefast channel, whichdecreases the
number of waiting times deviating from the mean value,
and consequently also the value of cross-correlation.
In the next step I analyze the influence of telegraphic
switching on the full counting statistics and the waiting
time distribution, and show that their joint analysis also
provides information about the nonrenewal behavior of
the system. It has been already reported by Albert et
al. [46], who studied FCS and WTD in a quantum point
contact. However, the system analyzed herein enables
to study the qualitative behavior of FCS and WTD in a
simpler way, and reveals also the specific physics of tele-
graphicswitchingphenomenon. Ifocusontheanalysisof
twoquantitiesassociatedwiththeFCSandWTDrespec-
tively, the Fano factor F and the randomness parameter
R [39, 40]:
c (cid:104)∆n(t)2(cid:105)
F = 2 = lim , (23)
c1 t→∞ (cid:104)n(t)(cid:105)
κLL (cid:104)∆τ2 (cid:105)
FIG. 3. (Color online). Dependence of (a) the normalized R= 2 = LL , (24)
(κLL)2 (cid:104)τ (cid:105)2
cross-correlation NCC of successive waiting times τLL, (b) 1 LL
theFanofactorF (blackcurves)andtherandomnessparame-
terR(greencurves)onaforthesymmetricsystem[Fig.2(a)]
for different values of γ/Γ.
where n(t) is the number of particles tunneling through
the upper dot within the time t. If the renewal assump-
tion is satisfied, these parameters are the same [42, 45].
as τLL. Fig. 3 (a) shows the dependence of the normal- As Fig. 3 (b) clearly shows, here this equality does not
izedcross-correlationoftwosubsequenttimesτLL onthe hold. In the considered system the Fano factor is always
asymmetryparameterafordifferentratiosγ/Γ. Onecan higher than or equal to the randomness parameter. The
observe that for a (cid:54)= 0 the cross-correlation can be pos- differenceF−Rdependsonγ/Γandainasimilarwayas
itive, which clearly indicates that the dynamics is non- the cross-correlation NCC. When the magnitude of the
renewal. As a matter of fact, the joint distribution of asymmetry parameter |a| is sufficiently high, and at the
two subsequent waiting times cannot be factorized into sametimetheratioγ/Γislowenough,thenoisebecomes
a product of two single-waiting time distributions. The super-Poissonian. Suchalargenoiseenhancementresult-
presence of the correlation results from the telegraphic ing from the telegraphic switching is an already known
switching between transport channels with different val- phenomenon [7, 13, 20, 22, 23]. In the limiting case of
uesoftunnelingrates[either(1+a)Γor(1−a)Γ]. When γ →0,theFanofactorbecomesinfiniteforarbitrarynon-
thechannelwithhighertunnelingrates(laterreferredto zero values of a. The randomness parameter, however,
asthefastchannel)isopen,thesubsequentwaitingtimes remains finite also at this limit. Let’s now explain this.
tends to be shorter than the mean value. Accordingly, For γ →0 the switching between transport channels be-
when the second (slow) channel is open, the subsequent comesinfinitesimallyslow,andthusonecanassumethat
waitingtimestendstobelongerthanthemeanvalue. As during an arbitrarily long time interval either the fast or
a result, the subsequent waiting times become positively the slow channel is open. Therefore, one can consider
correlated. two separate distributions of charge transmitted in the
The cross-correlation is reduced as the ratio γ/Γ rises, fast and the slow channel, denoted as n (t) and n (t),
f s
becausethefastswitchingbetweenchannelsincreasesthe respectively. The variance of the number of transmitted
7
particles is given by the following expression:
(cid:104)∆n(t)2(cid:105)=P (cid:104)[n (t)−(cid:104)n(t)(cid:105)]2(cid:105)+P (cid:104)[n (t)−(cid:104)n(t)(cid:105)]2(cid:105)
f f s s
=P {(cid:104)[n (t)−(cid:104)n (t)(cid:105)]2(cid:105)+[(cid:104)n (t)(cid:105)−(cid:104)n(t)(cid:105)]2}
f f f f
+P {(cid:104)[n (t)−(cid:104)n (t)(cid:105)]2(cid:105)+[(cid:104)n (t)(cid:105)−(cid:104)n(t)(cid:105)]2},
s s s s
(25)
where P and P denote the probabilities that the fast
f s
or the slow channel is open, respectively. For long
times the variance is predominated by the sum of terms
[(cid:104)n (t)(cid:105)−(cid:104)n(t)(cid:105)]2 = (I −I)2t2, where j ∈ {f,s}, I is
j j
the mean current and I is the mean current in the j-th
j
channel. These terms are quadratic functions of time,
while the mean value (cid:104)n(t)(cid:105) is a linear function of t, and
thusF =lim (cid:104)∆n(t)2(cid:105)/(cid:104)n(t)(cid:105)=∞. Therandomness FIG. 4. (Color online). Dependence of the quantities defined
t→∞
parameter, however, remains finite, which can be shown in Eq. (27) on a for the symmetric system [Fig. 2 (a)] for
in a similar way. One can write the variance of waiting γ/Γ=0.1. σ –blackline,ψ –reddashedline,ξ –greenline.
times as
(cid:104)∆τ2(cid:105)=p (cid:104)(τ −(cid:104)τ(cid:105))2(cid:105)+p (cid:104)(τ −(cid:104)τ(cid:105))2(cid:105)
f f s s accessible using continuous current measurements [66],
=pf[(cid:104)(τf −(cid:104)τf(cid:105))2(cid:105)+((cid:104)τf(cid:105)−(cid:104)τ(cid:105))2] (26) and the second-order current correlation function S(τ)
+p [(cid:104)(τ −(cid:104)τ (cid:105))2(cid:105)+((cid:104)τ (cid:105)−(cid:104)τ(cid:105))2], enablestoinferthepresenceofwaitingtimecorrelations.
s s s s
There is also a negative consequence: the identity (16),
where τ /τ are the waiting times measured in the
f s which enables the reconstruction of WTD on the basis
fast/slow channel, while pf = Pf(cid:104)τs(cid:105)/(Ps(cid:104)τf(cid:105)+Pf(cid:104)τs(cid:105)) of S(τ) or g(2)(τ), does not hold any longer. One should
and p = P (cid:104)τ (cid:105)/(P (cid:104)τ (cid:105)+P (cid:104)τ (cid:105)) are the probabilities
s s f s f f s be aware that measurements of S(τ), though in princi-
that the charge is transmitted in the fast or the slow
ple possible [67, 68], are in general very demanding [34].
channel, respectively. Itisasumoffinitequantities,and
However, I expect that experiments can give meaningful
thus the randomness parameter remains finite.
results at least for very slowly modulated systems. As a
As already mentioned in Sec. I, single electron count-
matter of fact, in such a case the step-like shape of the
ing experiments are currently confined to measurements
current-time trace could be even directly observed, as in
of low currents corresponding to tunneling frequencies
Refs. [6, 24].
on the order of kHz. Moreover, they are not directly
For the symmetric system the qualitative interpreta-
applicable for the study of electron transport beyond
tion of NCC, WTD and FCS was relatively intuitive.
the sequential tunneling regime. However, it appears
Now I consider an asymmetric system [see Fig. 2 (b)],
thatthenonrenewalcharacterofcurrentfluctuationscan
with different couplings of the dots to the left and right
beinferredusinglow-ordercurrentcorrelationfunctions,
leads, to show that the joint dynamics of the upper and
which in principle can be investigated by conventional
the lower dot may result in complex and non-trivial be-
current measurements. To show this, let’s consider cu-
haviorofcurrentfluctuations. Ascanbenoted,nowonly
mulants q of the pseudo-waiting time distribution u(τ)
n one of tunneling rates is assumed to be energy depen-
calculatedonthebasisofthesecond-ordercurrentcorre-
dent, i.e. the one between the upper quantum dot and
lationfunctionS(τ)(seetheendofSec.II).Thepseudo-
the right lead (cf. a similar case in Ref. [7], in which a
randomness parameter Π=q /q2 appears to be equal to
2 1 singletunnelingratewasmodulatedbyaspinimpurity).
theFanofactor. However,itappearsthatthenonrenewal
I begin the analysis by considering the correlation be-
dynamics may be revealed by the behavior of higher or-
tween subsequent times τ (the time during which the
der cumulants. Let’s analyze the following quantities: RL
dot remains empty) and τ (the time during which the
c LR
σ = 3, dot remains occupied – the electron dwell time). Similar
c
1 correlations in a double dot system, in which both tun-
ψ =3κ22 − κ3, (27) neling rates in the upper dot were assumed to be mod-
κ4 κ3 ulated, have been studied previously in Ref. [69]. As
1 1
Fig.5shows, themodulationofasingletunnelingrateis
q2 q
ξ =3 2 − 3, sufficient for generation of such correlations. This result
q4 q3
1 1 may seem to be non-intuitive, since the rate of tunneling
where for simplicity κLnL = κn. These three quantities through the left lead, associated with the time τRL, is
are equal to each other for renewal systems [42], but in now independent of the charge state of the lower dot.
general they are unequal (see Fig. 4). Thus, it may be To unveil the mechanism generating the correlations,
concludedthatthejointmeasurementofthethirdcumu- let’s consider the qualitative behavior of NCC as a func-
lant of the zero-frequency FCS, which is experimentally tion of the system parameters. Firstly, as expected, the
8
magnitudeofthecross-correlationrisesastheamplitude
ofthetunnelingratemodulation|Γ −ΓU|increases[see
R R
Fig. 5 (a)]. Let’s now focus on the influence of the dy-
namics of the lower dot. For the sake of simplicity, I
assume that γ = γ = γ. As follows, in contrast
L R
to the symmetric system, the cross-correlation depends
on γ in a non-monotonic way and reaches its maximum
for some finite value of γ [see Fig. 5 (b)]. Moreover,
for γ → 0 the cross-correlation vanishes, which suggests
that the correlation of waiting times is a result of cor-
related tunneling processes in the upper and the lower
dot. Let’s consider this in detail and analyze the follow-
ing sequence of events: electron jump from the dot to
the right lead – waiting time τ – jump from the left
RL
lead to the dot – waiting time τ – the second jump
LR
from the dot to the right lead. Two cases are consid-
ered: (i) when the time τ is short and (ii) when it
RL
is long. For simplicity, I assume that ΓU > Γ . At
R R
first let’s note that due to ΓU > Γ , the state (0,1)
R R
ratherthan(0,0)wouldbegeneratedatthebeginningof
the process. Thus, for the short time τ [the case (i)]
RL
the probability that successive evolution of the systems
follows the trajectory θ = (0,1) → (1,1) → (0,1) is
1
increased in comparison with the probability of the tra-
jectory θ = (0,0) → (1,0) → (0,0). As a result, the
2
time τ tends to be shorter than the mean value since
LR
the trajectory θ is associated with the occupancy of the
1
lowerdotwhich,duetoΓU >Γ ,allowsfastertunneling
R R
fromtheupperdot. Inconsequence,theshorttimeτ is
RL
correlatedtotheshortsuccessivetimeτ . Ontheother
LR
hand, for long times τ [the case (ii)] the initial imbal-
RL
ance of probabilities of states (0,1) and (0,0) is reduced
due to the tunneling through the lower dot. Therefore, FIG. 5. (Color online). Normalized cross-correlation of
the probabilities of the state (0,0) and the trajectory θ2 successive times τRL and τLR for the asymmetric system
rise up, and the subsequent time τLR tends to be longer [pFeingd.e2nc(eb)o]nw(iΓthUγ−LΓ=)/(γΓRU=+γΓ)anfodrΓdLiffe=renΓtRv=aluΓes: o(fa)γ/dΓe-;
than the mean value. Considering both cases together, R R
(b)dependenceon(γ−Γ)/(γ+Γ)fordifferentvaluesofΓU/Γ.
it is apparent that subsequent times τ and τ are R
RL LR
positively correlated.
Onemayask,ifthesamewaitingtimes(τ andτ ),
RL LR
but measured in the reversed order, are also correlated.
It appears, that they are not. As can be noted, in the
considered case the tunnel coupling of the upper dot to
the left lead is the same for either the (0,0) or the (0,1)
state. Thus, the rate of this process is in no way cor-
related to the previous course of the system’s dynamics.
As a result, subsequent times τ and τ are not cor-
LR RL
related ((cid:104)∆τ ∆τ (cid:105) = 0). Therefore, in general the
LR RL
correlation of two arbitrary waiting times τ and τ de-
ij ji
pends on their order: (cid:104)∆τ ∆τ (cid:105)(cid:54)≡(cid:104)∆τ ∆τ (cid:105).
ij ji ji ij
It has a direct consequence, which may also be sur-
prising: the waiting time distributions measured at the
left and the right junction, w (τ) and w (τ), can
LL RR
be in general unequal. To illustrate this, let us ex- FIG. 6. (Color online). Dependence of the quantities defined
press the variances of waiting times measured at the left inEq.(27)on(ΓU −Γ)/(ΓU +Γ)fortheasymmetricsystem
R R
and the right junction (τ and τ respectively) as fol- [Fig.2(b)]withγ =γ =γandΓ =Γ =Γforγ/Γ=0.3.
LL RR L R L R
lows: (cid:104)∆τ2 (cid:105) = (cid:104)∆τ2 (cid:105)+(cid:104)∆τ2 (cid:105)+2(cid:104)∆τ ∆τ (cid:105) and σ – black line, ψ – red line, ξ – green line. Continuous and
LL LR RL LR RL
(cid:104)∆τ2 (cid:105)=(cid:104)∆τ2 (cid:105)+(cid:104)∆τ2 (cid:105)+2(cid:104)∆τ ∆τ (cid:105). Asonecan dashedlinescorrespondtothequantitiesmeasuredattheleft
RR LR RL RL LR
and the right lead respectively.
easily observe, these expressions differ only in the cross-
9
correlation terms, and since these terms can be unequal,
the variances can differ as well. A
It also appears, that the functions g(2)(τ), S(τ) and
S(ω) measured at different junctions are also nonequiv- ΓAL ΓAR ΓA ΓAR
alent. At the same time, the mean currents and zero- L
frequency FCSs at both junctions are always the same,
ΓB ΓB 0
which is a consequence of the Kirchoff’s current law.
L R
Thepseudo-randomnessparameterΠ=q /q2,measured
at any junction, is equal to the Fano fac2tor1. However, ΓBL ΓB
R
the quantities ψ and ξ defined in Eq. (27) are junction-
dependent, as shown in Fig. 6. It should be mentioned,
B
that a similar difference in the time-dependent factorial
cumulants for incoming and outgoing electrons has been (a) (b)
recently reported by Stegmann and K¨onig [71].
FIG. 7. (Color online). (a) Scheme of the dynamical channel
Statistics of current fluctuations can be applied for
blockade in a single quantum dot. (b) The three-state model
the reconstruction of generators of dynamics of the sys-
of the dynamical channel blockade. Red arrows with bullet-
tems with hidden degrees of freedom. For example,
tails correspond to the tunneling from the dot to the right
Bruderer et al. [29] have proposed a procedure which
lead.
enables determination of n unknown parameters of the
Liouvillian using zero-frequency FCS cumulants up to
the n-th order. Although measurement of FCS cumu- negative–nocorrelationtakesplace. Toexplainthisone
lants up to the 15th order by single-electron counting should note an important difference between this model
techniques have been reported [70], such experiments and the one considered previously. The double quantum
are in general very demanding [29]. However, for non- dotexaminedbeforehandisamultiple-resetsystem[28]–
renewal systems the joint use of FCS and WTD may thereexisttwosetsofstatescorrespondingtothespecific
reduce the order of cumulants required for the recon- charge configurations of the upper dot. Each tunneling
struction of the Liouvillian thanks to the independence event corresponds to the transition between these two
of FCS and waiting time distributions measured at dif- sets. Because of the finite values of transition rates be-
ferent junctions. For example, all parameters of the Li- tween the states within the single set, the departure tra-
ouvillian of the analyzed asymmetric system can be de- jectory from the set of states is correlated to the entry
termined using the set of five independent quantities: trajectory. The current system is a single-reset one [28]
{(cid:104)τLR(cid:105),(cid:104)τRL(cid:105),(cid:104)∆τL2R(cid:105),(cid:104)∆τR2R(cid:105),F}. All these quantities – after each tunneling from the dot to the right lead the
are related only to the first and the second order cumu- system returns to the same empty state and the succes-
lants,whilethereconstructiononthebasisofFCSwould sive trajectory is in no way correlated to the previous
require the use of cumulants up to the fifth order. one. Thus, the dynamics of the system is renewal. This
suggeststhattestingoftherenewalpropertycanbeused
as a tool to differentiate between different mechanisms
C. Comparison with the dynamical channel leading to the noise enhancement, which can exhibit ei-
blockade ther the renewal or the nonrenewal dynamics.
As implied by the results presented above, in the sys-
tem exhibiting the telegraphic switching the noise en- IV. ANDERSON MODEL
hancement can be associated with nonrenewal transport
statistics(cf. Fig.3; oneshouldbeawarethatfor|a|=1 So far, I have focused on the telegraphic switching
the correlations vanish; thus, for example, the mecha- phenomenon generating the positive cross-correlation of
nismofnoiseenhancementreportedbySafonovetal.[72] waitingtimes. However,thisisbynomeanstheonlyone
wouldnotproducecorrelations). Tohighlightthesignifi- mechanism producing nonrenewal current fluctuations.
canceofthisfact,let’smakeacomparisonwiththeother Asamatteroffact,Caycedo-Soleretal.[48]havealready
knownmechanismofnoiseenhancement,namelythedy- investigatedthethree-levelcascade-likeopticalsystemin
namical channel blockade, focusing on the most simple which subsequent waiting times between photon emis-
case – a two-level quantum dot in the strong Coulomb sions were shown to be negatively correlated. The case
blockade regime [Fig. 7 (a)]. The dot levels are assumed analyzed therein has a direct electronic analogue: the
to be unequally coupled to the right lead. The infinite- Andersonsingleimpuritymodelintheinfinitebiaslimit,
bias limit is taken and the dynamics of the system is which can be realized using a single quantum dot with
described by a three-state Markovian model, with tun- a finite value of the intra-dot Coulomb interaction. Cur-
neling through a single lead corresponding to two tran- rent fluctuations within this model have been previously
sitions between states [Fig. 7 (b)]. One can ask if this investigated in Refs. [14, 28], and Brandes [28] has even
system exhibits the nonrenewal behavior. The answer is already mentioned its nonrenewal (multi-reset) charac-
10
ter. In this section I go, however, beyond the previous
studiesbyintroducingspinpolarizationofleadsattached Γ(1-a) Γ(1-a)
to the dot, which generates the telegraphic switching.
This enables us to observe the interplay of two mecha- Γ(1+a) Γ(1+a) 2 (1-a) (1-a)
nismsleadingtoeitherthenegativeorthepositivecross-
correlation of waiting times and analyze its influence on Γ(1+a) Γ(1+a)
cumulants of FCS and WTD.
(1+a) 2 (1+a)
A. Model
(a) (b)
Theanalyzedsystem[Fig.8(a)]consistsofaquantum
dot with a single electronic level. I consider unidirec-
tionaltransportfromthelefttotherightlead. Thedou- Γ(1-a)(1-p) Γ(1-a)(1+p)
ble occupancy of the dot by electrons with anti-parallel
spins is now allowed. Thus, there are four states within
the transport window: the empty state (denoted |0(cid:105)),
two singly-occupied states with different spin orienta-
tions (|↑(cid:105) and |↓(cid:105)) and the doubly occupied state (|↑↓(cid:105)).
Tunneling rates between the dot and the leads are as- Γ(1+a)(1+p) Γ(1+a)(1-p)
sumed to be energy dependent, which makes them con-
ditioned on the occupancy of the dot. As in the pre-
(c)
viously considered case, the energy-dependence of the
tunneling rates is described by the asymmetry param-
FIG. 8. (Color online). (a) Scheme of the transport within
eter a. For the sake of simplicity, I assume that the
theAndersonmodelintheinfinitebiaslimit. (b,c)Three-and
dot is symmetrically coupled to the left and the right
four-stateMarkovianmodelsofsystemswithoutandwithspin
lead. Forspin-independenttunnelingratesthedynamics polarizationoftheleads. Redarrowswithbullet-tailsdenote
of the system can be described by a three-state Marko- transitions corresponding to the tunneling from the left lead
vian model [Fig. 8 (b)], with states {(0),(1),(2)} refer- to the dot.
ring to the empty, the singly-occupied and the doubly-
occupied state, regardless of the spin. This is equivalent
to the models investigated in Refs. [14, 28, 48]. Incomparisonwiththemodelconsideredintheprevious
Moreover, the leads are assumed to be equally section [Fig. 1 (b)], the tunneling through a single junc-
spin polarized with the polarization defined as p = tion corresponds herein to four, instead of two, different
(ρ↑−ρ↓)/(ρ↑+ρ↓), where ρ↑, ρ↓ are the densities of transitions between states. As shown below, the way in
states for ↑ and ↓ electrons. The tunneling rates for whichelectrontunnelingsarecounted(andthustheform
different spin polarizations are then expressed as fol- ofthejumpoperator)hasasignificantimpactonresults.
lows: Γ = (1 + p)Γ and Γ = (1 − p)Γ , where
↑± ± ↓± ± In contrast to the optical systems (like the one con-
Γ = (1 ± a)Γ. The dynamics of the system is then
± sidered by Caycedo-Soler et al. [48]), in which emissions
described by a four-state Markovian model [Fig. 8 (c)].
of photons are detected, in the electronic systems one
The Liouvillian written in the basis {|0(cid:105),|↑(cid:105),|↓(cid:105),|↑↓(cid:105)} is
can experimentally detect charge states, and thus ana-
expressed as follows:
lyze not only jumps through a single lead, but also el-
−2(a+1) A(+)P(+) A(+)P(−) 0  ementary transitions (0) ↔ (1) and (1) ↔ (2). These
elementary processes are renewal ones even in the case
 A(+)P(+) −2ap−2 0 A(−)P(−) 
L=Γ , when the leads are spin-polarized. In view of this, the
 A(+)P(−) 0 2ap−2 A(−)P(+) 
analysis of nonrenewal statistics may seem superfluous.
0 A(−)P(−) A(−)P(+) −2(1−a)
However, Prance et al. [73] have shown that the detec-
(28) tion of electron jumps rather than charge states may be
preferred due to the lesser vulnerability to noise. There-
whereA(±) =1±a,P(±) =1±p. Afterwards,Ifocuson
fore,thetheoreticaldescriptionappliedinthispaperhas
thetunnelingfromtheleftleadtothedot. Theoperator
experimental relevance.
counting all transitions in the junction, irrespectively of
The analyzed model assumes that only the single level
the electron spin and the occupancy of the dot, is intro-
is available for the transport, and at the same time the
duced as
zero, single and double occupancy of the dot is allowed.
 
0 0 0 0 In real multi-level quantum dots this requires the intra-
 A(+)P(+) 0 0 0 dot Coulomb interaction energy U to be smaller than
JL =ΓA(+)P(−) 0 0 0.(29) the separation between electronic levels of the dot. One
0 A(−)P(−) A(−)P(+) 0 should be aware that in relatively large lateral quantum