Noise calculations within the second-order von Neumann approach

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by  Philipp Zedler

Noise calculations within the second-order von Neumann approach
Philipp Zedler,1 Clive Emary,1 Tobias Brandes,1 and Toma´ˇs Novotny´2,∗
1Institut fu¨r Theoretische Physik, Technische Universita¨t Berlin, D-10623 Berlin, Germany
2Department of Condensed Matter Physics, Faculty of Mathematics and Physics,
Charles University in Prague, Ke Karlovu 5, CZ-12116 Praha 2, Czech Republic
(Dated: January 25, 2012)
We extend the second-order von Neumann approach within the generalized master equation for-
2 malismforquantumelectronictransporttoincludethecountingfield. Theresultingnon-Markovian
1 evolution equation for the reduced density matrix of the system resolved with respect to the num-
0 ber of transported charges enables the evaluation of the noise and higher-order cumulants of the
2 full counting statistics. We apply this formalism to an analytically solvable model of a single-level
quantumdotcoupledtohighlybiasedleadswithLorentzian energy-dependenttunnelcouplingand
n
demonstrate that, although reproducing exactly the mean current, the resonant tunneling approx-
a
imation is not exact for the noise and higher order cumulants. Even if it may fail in the regime of
J
stronglynon-Markoviandynamics,thisapproachgenerically improvesresultsoflower-orderand/or
4
Markovian approaches.
2

PACSnumbers: 72.70.+m,73.63.-b,05.60.Gg,73.23.Ad
]
l
l
a
h Introduction. — Although electronic transport case.
- through interacting nanostructures has been a subject Altogether, 2vN/RTA is a powerful and successful ap-
s
e of intense study in the past decade, its theoretical de- proximation and it is, therefore, natural to ask whether
m scriptioningeneralsituationsremainsachallengingtask. it can be extended to the calculation of other transport
. Standard complementary theoretical approaches are the quantitiessuchasnoiseandfullcountingstatistics. This
at Landauer-Bu¨ttiker scattering formalism in the limit of is exactly the subject of this Brief Report — we extend
m negligibleCoulombcorrelationsandthe masterequation the second-order von Neumann approach to the evalua-
- for weak coupling to the leads, respectively.1,2 If both tionofthefullcountingstatistics,andnoiseinparticular,
d the single-particle coherence effects due to coupling to byincorporatingthecountingfieldinthetheoryandthen
n leads and many-body Coulomb correlations are impor- test the performance of this generalization for the sim-
o tant,highlycomplexphenomenacanoccur,suchas,e.g., plest possible model of a single-level quantum dot. We
[c the Kondo effect at low temperatures.2 Even well above findthat contraryto the stationarymeancurrent,which
the Kondo temperature the interplay between coherence is captured exactly by the 2vN, the noise and higher-
2 and correlations may lead to nontrivial consequences,3 order cumulants constituting the full counting statistics
v which are hard to describe theoretically. are not exact. Nevertheless, it does yield significant im-
3
provement over simpler, more standard approximations.
5 Thesecond-ordervonNeumannapproach(2vN)based
2 on decoupling of equations of motion for the density Generalized second-order von Neumann approach. —
3 matrix,4,5likeitsrelativetheresonanttunnelingapproxi- In our generalization of the second-order von Neumann
.
0 mation(RTA)basedonasuitableinfiniteresummationof scheme we use the equations of motion approach and
1 the perturbation series for the generalized master equa- closely follow the corresponding derivation without the
1 tion (GME),6,7 is capable of capturing such effects, as counting field in Ref. 4. We only summarize the main
1
has been confirmed also by the successful description of steps and associated approximations and show the final
:
v pertinent experiments.8,9 InRef. 8 the line-widths ofthe results. A fully detailed derivation can be followed in
i nonlinearconductanceforaquantumdotasymmetrically Ref. 12. We consider a generic transport Hamiltonian4
X
coupled to leads in the Coulomb blockade regime were consisting of three parts, H ≡ H + H +H ,
system leads T
r
a measured for both voltage-bias polarities and fully ex- with the central system (“dot”) Hamiltonian Hsystem ≡
plained by the RTA, while in Refs. 9 and 10 a canyonof E |aiha|expressedintermsofthesystemmany-body
a a
suppressedlinearconductancethroughanInSbnanowire eigenstates |ai and eigenenergies E , standard nonin-
a
P
quantum dot was correctly reproduced by the 2vN as a teracting free-electron Hamiltonian for the two mutu-
function ofthe gate voltageand magnetic field. Further- ally biased (µ = eV/2, µ = −eV/2) leads H ≡
L R leads
more,inaseriesofworks,3,5,11 the 2vNhasbeenapplied E c† c , and the tunneling Hamiltonian
k,α=L,R kα kα kα
to the single resonant level as well as double quantum H ≡ T∗(kα)c† |aihb|+T (kα)|biha|c .
dot models, finding that corrections beyond the lowest- PT kα,ab ba kα ba kα
order sequential tunneling (cotunneling) were described UsinPg the st(cid:2)andard prescription,13 we can write(cid:3)down
very well and conjecturing that the method yields exact the extended Liouville-von-Neumann equation for the
solutionsfornon-interactingmodels.5 Thisisanontrivial generalized (non-Hermitian) density matrix ρ(t,χ) =
feature as the method stems from the atomic limit (de- {Nα}ρ(t,{Nα})eiPαNαχα, ρ†(t,χ) = ρ(t,−χ), χ ≡
scribedbyGMEs)complementarytothenon-interacting (χ ,χ ) involving the counting fields χ dual to the
PL R α
2
number N ofchargespassedthroughjunctionα (leftor diagonal matrix elements of the total density matrix up
α
right) during time t. This equation reads (we set ~ ≡ 1 to the difference of one electron-hole pair in the leads”
and e≡1 throughout the rest of the paper) — i.e., for example, hag|c† c c† ρ(t,χ)|bgi ≈
g kα kα k′α′
f hag|c† ρ(t,χ)|bgi for kα 6= k′α′,
d kα g k′α′ P
idtρ(t,χ)=H+(χ)ρ(t,χ)−ρ(t,χ)H−(χ), (1) hag|Pck1α1c†k2α2ck3α3ρ(t,χ)|bgi ≈ 0, for k1α1, k2α2, and
k α all different. We obtain a closed (though infinite)
3 3
where the χ-dependent Hamiltonian is gener-
set of linear equations of motion for the reduced density
ated by modifying its tunnel part as follows:
matrix w (t,χ) and coherences φ (t,χ;kα), which
ab ab
H±(χ)≡Hsystem+Hleads+ kα,ab T¯b±a(kα,χ)c†kα|aihb|+ are more conveniently expressed in the Laplace picture
Tb±a(kα,χ)|biha|ckα ,with13PT¯b±a(kα(cid:2),χ)≡e±iχα/2Tb∗a(kα) (wab(z,χ) = 0∞dte−ztwab(t,χ) and analogously for
anFdorTbs±ata(kteαs,χof)t≡hee(cid:3)∓wiχhαo/l2eTsbyas(tkeαm).plus leads, we choose tφhaeb(azb,χbr;ekvαia))t.ioARnsssTubm±ain≡gTφb±aab((ktα=,χ0),,χφ;akbα≡)=φa0b(azn,dχ;uksiαn)g,
a basis of tensor products |agi ≡ |ai ⊗ |gi, with |gi φ¯ ≡ φ∗ (z¯,−χ;kα), T′± ≡ T±(k′α′,χ) and anal-
ab ab ab ab
a many particle state of the leads. We first eval- ogously for φ′ab, φ¯′ab, T¯ab, and T¯a′b, our equations of
uate the time-evolution of system matrix elements motion read
w (t,χ) ≡ hag|ρ(t,χ)|bgi and generalized system-
ab g
lead coherencPes φab(t,χ;kα) ≡ ghag|c†kαρ(t,χ)|bgi to (iz−Ea+Eb)wab(z,χ)−iwab(t=0,χ)
Awmthaitcrthhixiwsealblee(mvt,eeχl,n)tdswieroefcatpcly†kpαlcycoku′fαpa′lcρet(.otrW,iPχzea)ttihaonenndcacon†kndαtcint†kru′αue′nρfco(atr,ttiχoh)ne. =aX′,kα(cid:2)Ta+a′φ¯ba′ +T¯a+′aφa′b(cid:3)−bX′,kα(cid:2)Tb−′bφ¯b′a+T¯b−b′φ(a2ba′)(cid:3),
conditions analogousto Ref. 4 and,correspondingto the
resonant tunneling approximation of Ref. 7, keep “non-
(iz−Ea+Eb+Ek)φab = Ta+a′fkαwa′b(z,χ)− Tb−′b(1−fkα)wab′(z,χ)
a′ b′
X X
+ T′+ a′′ T¯a+′′a′(1−fk′α′)φa′′b−Ta+′a′′fkαφ¯′ba′′ + b′ T¯′−bb′fk′α′φa′b′ −Tb−′b(1−fkα)φ¯′b′a′
a′,k′α′6=kα aa′P (cid:2) iz−Ea′ +Eb(cid:3)+EPkα−h Ek′α′ i
X
+ T¯+ a′′ T′+a′a′′fk′α′φa′′b−Ta+′a′′fkαφa′′b + b′ T′−b′b(1−fk′α′)φa′b′ −Tb−′b(1−fkα)φ′a′b′
a′,k′α′6=kα a′aP h iz−Ea′i+EPb+hEkα+Ek′α′ i (2b)
X
a′ T¯′+a′a(1−fk′α′)φa′b′ −Ta+a′fkαφ¯′b′a′ + b′′ T¯′−b′b′′fk′α′φab′′ −Tb−′′b′(1−fkα)φ¯′b′′a
+ T−
b′,k′α′6=kα b′bP h iz−Ea+Eib′ +PEkαh−Ek′α′ i
X
+ T¯− a′ T′+aa′fk′α′φa′b′ −Ta+a′fkαφ′a′b′ + b′′ T′−b′′b′(1−fk′α′)φab′′ −Tb−′′b′(1−fkα)φ′ab′′ .
b′,k′α′6=kα bb′P h iz−Eia+PEb′h+Ekα+Ek′α′ i
X
Eqs. (2a) and (2b) constitute the generalization of For ease of notation, we will only explicitly demonstrate
Eqs. (10) and (11) in Ref. 4 to case with the counting this general procedure on the following example.
field; they are the main formal result of our paper and Single-resonant-level model. — We consider an
the starting point for the following studies. Analogously archetypicalmodelofspin-lesselectronsandasinglelevel
tothemeancurrent,4,7chargeconservationcanbeproven forming the system H ≡ E |1ih1|. The only non-
system d
for all stationary cumulants using the method described vanishing T (kα) is T (kα) = t and consequently
ba 10 kα
in Ref. 14. Eq. (2b) can be (at least formally) solved for T1±0 = e∓iχα/2tkα, T¯1±0 = e±iχα/2t∗kα. Defining analo-
φab intermsofwab(z,χ)andsubstitutedintoEq.(2a)to gously to Ref. 4 (apart from the factor of 2π) the quan-
giveaclosednon-Markoviangeneralizedmasterequation tity B (E) ≡ 2π t∗ φ (z,χ;kα)δ(E − E ) (with
α k kα 10 kα
for the reduced density matrix w (z,χ) only. Its evolu-
ab B¯ (E)≡B∗(E) )andintroducingtheconventional
tion kernel Wˆ(z,χ) (see below) can then be used in the α α χ→P−χ
machineryofRefs.15and16toproducethecurrentnoise tunneling rates (cid:12)(cid:12)Γαz→(Ez¯) ≡ 2π k|tkα|2δ(E − Ekα) and
andhigher-ordercumulantsofthefullcountingstatistics. Γ(E)≡Γ (E)+(cid:12)Γ (E), we get from Eqs. (2a) and (2b)
L R
P
3
dE
izw (z,χ)−iw (t=0,χ)= eiχα/2 B (E)−B¯ (E) ,
00 00 α α
2π
α Z
X (cid:2) (cid:3)
dE
izw (z,χ)−iw (t=0,χ)=− e−iχα/2 B (E)−B¯ (E) ,
11 11 α α
2π
α Z
X (cid:2) (cid:3)
E−Ed+iz− d2Eπ′E−Γ(EE′′+) iz Bα(E)=e−iχα/2fα(E)Γα(E) w00(z,χ)+ eiχα′/2 d2Eπ′E′B¯−α′E(E−′)iz
(cid:18) Z (cid:19) α′ Z !
X
−eiχα/2[1−fα(E)]Γα(E) w11(z,χ)− e−iχα′/2 d2Eπ′E′B¯−α′E(E−′)iz .
α′ Z !
X
(3)
We have verified12 that, for χ = 0, these equa- b (ǫ+iδ )=lim [E−(ǫ+iδ )]B (E)stemming
R R E→ǫ+iδR R R
tions coincide with the RTA obtained by the real-time fromtheaboveansatzforthedeterminationofb (ǫ+iδ )
R R
diagramatics.7 from the second of Eq. (4) and inserting the resultant
From now on, we focus on a specific example of an form of Eq. (4) into the first two lines of Eq. (3), we
infinite bias, i.e., f (E) ≡ 1, f (E) ≡ 0, with con- finally arrive at the non-Markovian kernel in the 2vN
L R
stant left tunnel rate Γ (E) ≡ Γ , and with the right approximation,
L L
tunnel rate having the Lorentzian energy dependence
vaΓacRtn(tEsaog)leu≡toiofn2b|eΩaisn|2gw(Eae−lnlEaδRalRys)2t+itchδaR2ell.yRTsToAlhviasebqlmueaoftdoioerlnbsh,oatyshettthhieet aeedxx--- Wˆ2vN(z,χ)= ΓL−eΓ−LiχL e−iχǫR2+ǫγ2+(zγ)[(γz()z[γ2)|+(2Ωz|z)Ω|+2+|γ2||Ω(Ωγz(||)22z//)((δδRR++zz//22))]]!,
(5)
hibits nontrivial non-Markovian dynamics. This model
isequivalent17,18 toanoninteractingdoubledotwithtwo with γ(z)≡δR+ΓL/2+z.
electroniclevelsatE andE ,mutuallycoherentlycou- For zero counting fields, this kernel is identical to the
d R
pled by the transfer amplitude Ω, with the left level at exact one, Wˆ2vN(z,0) = Wˆexact(z,0), with clear conse-
E coupled by Γ to the left (filled) lead, and with the quencethatthetime-dependentoccupationsandstation-
d L
rightlevelatE coupledtotheright(empty)leadbyan ary current are also exact in accordance with previous
R
energy-independentrate2δ . The double dotmodel can findings.4,11,18 The non-equilibrium rates are captured
R
be described exactly by a Markovian generalized mas- correctly including the effects of broadening of the level
ter equation, which can be then projected onto the left Ed due to the coherent coupling to the leads. However,
dot with the level E only resulting in an exact non- already at first sight the counting field is accounted for
d
Markovian kernel Wˆ (z,χ). Its explicit form is too in a rather primitive manner analogous to the lowest-
exact
lengthy and cumbersome to be presentedhere so we just ordersequentialtunnelingmodel,i.e.,theexactratesare
use it for the exact evaluation of the noise and skewness just multiplied by the exponentials with counting fields
shown in Fig. 1. (compare with Eq. (19) of Ref. 18). For non-Markovian
FortheLorentzianmodel,wecananalyticallysolvethe kernels,thisprescriptionispotentiallyproblematic16and
2vNequations(3),whichisinitselfremarkableanddoes the exact kernel indeed contains the counting fields also
notworkatfinitebias. Thissolutionisobtainedfromthe inthedenominatorsoftheexpressionsfortheratesensur-
ansatz B (E) = b (E)/(E−ǫ−iδ ), B (E) = b (E), ingconsistency. Therefore,the 2vNkernelwithcounting
R R R L L
whereb (E)hasnosingularitiesintheuppercomplex- fields is not exact and the noise and higher-order cumu-
L,R
E half-plane. Whenthisansatzisinsertedintothelastof lants it yields are not correct.
Eq.(3)adaptedtoourexample,oneobtains,consistently This is explicitly demonstrated in Fig. 1, where we
with the assumed analytic structure in E, the following comparetheexactsolutionforthenoiseandthirdcumu-
solution: lantwiththe2vN/RTAanditsMarkovianlimitobtained
byusingz →0inthekernel(5).16 Wealsocomparewith
B (E)=e−iχL/2Γ w00(z,χ)+ieiχR/2E¯b+Ri(zǫ−−ǫi+δRiδ)R , the first-order von Neumann approach (1vN) equivalent
L L E+iz−Σ(E+iz) tothestandardlowest-orderGMEdescribedinSec.IIIA
ofRef.18,anditsMarkovianlimit. We fixthe righttun-
B (E)=−Γ (E)w11(z,χ)eiχR/2−iE¯b−Rǫ(+ǫ−iδiRδR+)iz, nelrateΓR(0)byadjustingΩwhilechangingthe param-
R R E+iz−Σ(E+iz) eter δR controlling non-Markovianbehavior and observe
(4) theperformanceofvariousapproximationsforincreasing
degree of memory with decreasing δ . The Markovian
R
withǫ≡E −E andΣ(E)=−iΓ /2+|Ω|2/(E−ǫ+iδ), limit of the 1vN is left constant by this procedure and
R d L
theself-energyduetocouplingtotheleadscorresponding servesasa reference,while allother solutionsrespondto
to the sum of the tunneling rates. Using the relation the changeofδ . Obviously,noneofthe approximations
R
4
0.8 reproduces the exact solution for strong-enough non-
Exact
Markovian dynamics; all approximations perform quite
2vN
0.75 badly for strong memory at δ ∼ Γ (0) for both cumu-
ii 2vN + Markov R R
I 1vN lantswithextremeerrorsinthenon-Markovianversions.
hh These errorsaretypicalfor non-Markovianmaster equa-
/ 0.7 1vN + Markov
ii tions which have no Lindblad form and hence do not
2 guaranteethe conservationof probability18. However,in
I
0.65
hh the intermediate memory regime δR & ΓR(0) the non-
Markovianversionof2vN/RTA is clearlyby far the best
0.6 approach.
0 5 10 15 20
δ /Γ (0)
R R Conclusion. We have extended the second-order von
Neumann approach for GME kernels to the case with
counting field which has enabled us to study its perfor-
0.5
mance in the evaluation of the electronic current noise
ii0.45 andhighercumulants. Wehaveshownonananalytically
I solvable example of a single-level junction with struc-
0.4
hh tured coupling to the leads that 2vN is not exact for the
/
ii0.35 noise and higher-order cumulants and that it may fail
3I 0.3 significantly for highly non-Markovian dynamics. How-
hh ever, for intermediate degrees of memory, 2vN seems to
0.25
perform the best out of considered standard approxima-
0.2 tions. Thus,whensupplementedwithefficientnumerical
implementation,itwouldbeamethodofchoiceforevalu-
0 5 10 15 20
ationofnoiseandfullcountingstatisticsforsystemswith
δ /Γ (0)
R R strong interplay between correlations and coherence.
FIG. 1. (Color online) Second (top) and third (bottom) nor-
malized cumulants of the current as functions of the width
δR of the Lorentzian right tunneling rate measuring the de- Acknowledgments. We thank O. Karlstro¨m,
gree of non-Markovianity. Compared are the exact solution P. Samuelsson, and A. Wacker for useful discus-
(solid black line) and four tested approximations: 2vN/RTA sions. This work was supported by DFG Grants BR
(long-dashedredline),itsMarkovianlimit(short-dashedblue 1528/7-1, 1528/8-1, SFB 910, GRK 1558, the Heraeus
line),thefirst-ordervonNeumann/standardGME(long-dot- foundation, and the DAAD, and by the Czech Science
dashed green line), and its Markovian limit serving as the FoundationviaGrantNo.204/11/J042andtheMinistry
reference (short-dot-dashed magenta line). Parameters: ofEducationofthe CzechRepublicviathe researchplan
ǫ=ΓR(0)=4ΓL. MSM 0021620834(T. N.).
∗
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