Collective Nuclear Stabilization by Optically Excited Hole in Quantum Dot

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by  Wen Yang

Collective Nuclear Stabilization by Optically Excited Hole in Quantum Dot
Wen Yang and L. J. Sham
Center for Advanced Nanoscience, Department of Physics,
University of California San Diego, La Jolla, California 92093-0319, USA
We propose that an optically excited heavy hole in a quantum dot can drive the surrounding
nuclear spins into a quiescent collective state, leading to significantly prolonged coherence time for
theelectron spinqubit. Thisprovidesageneralparadigm tocombatdecoherencebyenvironmental
1 control without involving the active qubit in quantum information processing. It also serves as a
1 unified solution to some open problems brought about by two recent experiments [X. Xu et al.,
0 Nature459, 1105 (2009) and C. Latta et al.,NaturePhys. 5, 758 (2009)].
2
PACSnumbers: 78.67.Hc,72.25.-b,71.70.Jp,03.67.Lx,05.70.Ln
n
a
J
The hole, the removalofanelectronfromthe fully oc- remains “quiet” for a long time for electron spin manip-
0
cupiedvalenceband,isanelementaryexcitationinsemi- ulation. Intensive research efforts have led to successful
3
conductors. Because of the lower orbital symmetry, the nuclear stabilization in QD ensembles [15] and suppres-

] hole in the valence band has properties different from sion of the fluctuation of the nuclear field difference be-
l
l the electrons in the conduction band. In a quantum dot tween two neighboring QDs [16].
a
h (QD),thespinofalocalizedholeiscoupledtothespinsof Recently, three groups [7, 17, 18] reported significant
- thesurroundingatomicnucleiofthe hostlatticethrough nuclear stabilization in single QDs, most relevant for
s the long-range dipolar hyperfine interaction, as opposed quantum information processing. Vink et al. [18] de-
e
m tothestrongercontacthyperfineinteractionbetweenthe duced the stabilization from the observed electron spin
electron spin and the nuclear spins [1]. The hole-nuclear resonance locking, attributed to the electron-driven re-
.
at coupled system forms a different paradigm of the cen- verse Overhauser effect [14]. Xu et al. [7] and Latta
m tral spin model [2], which has been of interest for a long et al. [17] directly observed the stabilization by optical
- time in spin resonance spectroscopy, quantum dissipa- excitation of trion and blue exciton (both containing a
d tion, and classical-quantum crossover [3]. Currently, the hole spin) in the Voigt and Faraday geometries, respec-
n central hole spin in the QD has attracted increasing in- tively. Xu et al. also observed the resulting prolonged
o
terest due to its potential usage as carriers of long-lived electron spin coherence time. However, two key obser-
c
[ quantum information [4] and efforts are being directed vations, the nuclear stabilization observed by Xu et al.

towards understanding its interactions with the environ- and the bidirectional hysteretic locking of the blue exci-
2
ments [5], especially the nuclear spins [6]. Recently, a ton absorption peak observed by Latta et al. cannot be
v
0 symmetric hysteretic broadening of the absorption spec- explained by existing theories [7, 8, 13, 14, 17]. For a
6 trum was observed and attributed to a nuclear polar- complete understanding of the fundamental physics and
0 ization transient induced by an optically excited hole future optimization of the nuclear stabilization, a new
0
througha non-collinearterm ofthe dipolar hyperfine in- theory is needed.
.
2 teraction [7, 8]. In this Letter, we link the hole spin based quantum
1
Compared with the newly initiated hole spin based technology with the electron spin based quantum tech-
0
quantum technology, the electron spin based quantum nology by constructing a theory showing that an opti-
1
: technology has achieved a higher degree of maturity [9]. cally excited central hole spin can stabilize the nuclear
v
One critical obstacle is the short electron spin coher- spins and significantly prolong the coherence time of the
i
X ence time T∗ due to the randomly fluctuating nuclear electron spin qubit. This provides a general paradigm
2
r field produced by the “noisy” nuclear spins [10]. Pro- to combat decoherence by environmental control with-
a
longingthe electronspincoherencetime is ofparamount out involving the active qubit. The essential ingredients
importance in quantum informationprocessing. For this of this theory include a finite nuclear Zeeman splitting
purpose, two major approaches are under rapid devel- and the non-collinear dipolar hyperfine interaction [7],
opment. One aims at decoupling a general qubit from through which the fluctuating hole spin stabilizes the
theenvironmentsbyrepeatedpulsedcontrolofthequbit nuclear spins by driving them from the “noisy” thermal
[11]. The other aims at suppressing the fluctuation of equilibrium state into a “quiet” collective state, where
the nuclearfield, i.e., stabilizing the nuclearspins, byin- the strongthermalfluctuationis largelycancelledby the
ducing a steady-state nuclear polarization [12] through inter-spin correlation. This enables a flexible control of
the electron-driven Overhauser [13] and reverse Over- the nuclear fluctuation and hence the coherence time of
hauser [14] effects. The latter approach, being specifi- the electron spin qubit by engineering the hole spin fluc-
cally designed for the QD electron spin, has the merit tuationspectrum throughthe highly developedcoherent
that once the nuclear spin environment is stabilized, it opticaltechniques. Italsoprovidesa unifiedexplanation
2
(a) (b) (cid:39)>0 (c)(cid:39)<0 steady-state nuclear polarization
1 (cid:39) a(cid:4)h (cid:90)N 11,,(cid:112)(cid:110) (cid:39)(cid:39) 11,,(cid:112)(cid:110) ||(cid:39)(cid:39)|| 11,,(cid:110)(cid:112) hIˆzi0 ∝ WW++−+WW−− =−∆22∆+ωNγ22 +O(ε2) (1)
during a time scale characterized by the inverse of the
(cid:74)
(cid:58) 1 nuclear polarization buildup rate Γp ≡ W+ + W− =
R
0,(cid:110) 0,(cid:110) 0,(cid:110) O(a˜2hΩ2R), where ε ≡ ωN/γ2 ∼ 0.1 for a typical QD
0 (cid:90) under a magnetic field B = 1 T. The hole-driven in-
N
0,(cid:112) 0,(cid:112) 0,(cid:112) trinsic steady-state polarization hIˆzi shows a sign de-
0
pendence onthe detuning ∆, which is the key to explain
FIG. 1. (color online). (a) Energy levels for the optically the observation of Latta et al. on blue exciton excita-
excited hole coupled to a typical nuclear spin-1/2. (b) and tion [17]. By contrast, the intrinsic steady-state polar-
(c): Two competing nuclear spin-flip channels for (b) ∆> 0 izationhIˆzi =hSˆzi−hSˆzi (hIˆzi =hSˆzi )due to the
and (c) ∆<0. 0 e e eq 0 e eq
electron-drivenOverhauser[13](reverseOverhauser[14])
effect is equal to the nonequilibrium (equilibrium) part
of the electron spin polarization and is insensitive to the
to the two distinct experimental observations: the ap-
laser detuning, while the hole-driven nuclear polariza-
pearance of the “quiet” state explains the observationof tion transient hIˆzi ∝ ∆ [7] or random shift [8]
Xu et al. in the single pump experiment [19], while the transient
produces no intrinsic steady-state nuclear polarization.
switch between the “noisy” state and the “quiet” state
For the hole-drivenevolution of a typical nucleus with
explains the observationof Latta et al. on the blue exci-
spin I ≥1/2 under a general pumping intensity, we sin-
ton [17]. gle out the slow dynamics of the diagonal part Pˆ(t) of
The first step of the hole-drivencollective nuclear sta-
thereduceddensitymatrixρˆ (t)ofthisnucleusbyadia-
N
bilization is a steady-state nuclear polarization induced
batically eliminating the fast motion of the hole and the
by the optically excited hole. The essential physics of
off-diagonalpart and arrive at
this step is captured by the dynamics of a typical nu-
clearspin-1/2ˆIinaQDcoupledtoaheavyholestate|1i Pˆ˙ =−W+[Iˆ−,Iˆ+Pˆ]−W−[Iˆ+,Iˆ−Pˆ], (2)
through the non-collinear dipolar hyperfine interaction
σˆ11a˜h(Iˆ+ +Iˆ−) [7], where σˆji ≡ |jihi|, Iˆ± ≡ Iˆx ±iIˆy, valid up to O(a˜2), where the transition rates for the
h
and a˜h ≡ O(η2)ah, with η being the heavy-light hole down-to-up and up-to-down channels, W± = C(∓ωN),
mixing coefficient. An external magnetic field along the are determined by the steady-state hole fluctuation
z axisgivesrise to afinite nuclearZeemanfrequency ωN C(ω)= −∞∞dt eiωthσˆ11(t)σˆ11i.
and a continuous wave laser couples the ground state For IR= 1/2, the motion of the (degree of) nuclear
|0i to the excited hole state |1i with Rabi frequency polarization s(t)≡hIˆz(t)i/I follows from Eq. (2) as
Ω and detuning ∆. The hole state dephases with rate
R
γ and decays back to the ground state with rate γ s˙ =−Γ s−Γ (s−s ), (3)
2 1 1 p 0
[Fig. 1(a)]. The hole dephasing broadens |1,↑i and |1,↓i
to Lorentzian distribution L(γ2)(E)=(γ2/π)/(E2+γ22). where Γ1 is a phenomenological nuclear depolarization
In the weak pumping limit, two competing nuclear spin- rate for other nuclear spin relaxation mechanisms,
flip channels |0,↓i Ω→R |1,↓i →a˜h |1,↑i (down-to-up chan-
4a˜2 Wγ2
nel) and |0,↑i Ω→R |1,↑i →a˜h |1,↓i (up-to-down channel) Γp =W++W− = γh (cid:18)(γ +21W)3c1+O(ε2)(cid:19) (4)
1 1
are opened up to leading order [Figs. 1(b) and 1(c)].
For each channel, the transition rate is proportional to is the hole-driven nuclear polarization buildup rate, and
the square of the coupling strength times the final den-
sity of states of each step determined by the energy s = W+−W− =− ∆ωN γ1c0 +O(ε2) (5)
mismatch. For the down-to-up channel, the transition 0 W++W− ∆2+γ22γ2c1
rate W ∝ Ω2L(γ2)(∆) × a˜2L(γ2)(ω + ∆), where ∆
+ R h N is the intrinsic (i.e., when Γ = 0) steady-state nuclear
is the energy mismatch and L(γ2)(∆) is the final den- 1
polarization, in agreement with Eq. (1). Here c ≡
sity of states for the first step |0,↓i Ω→R |1,↓i, while 1/2+γ /γ +f+W/γ andc ≡1+[γ /(2γ )]f+W0/γ
2 1 1 1 1 2 1
(ωN +∆) and L(γ2)(ωN +∆) are corresponding quan- arenon-negativeconstants,f ≡(γ2−∆2)/(γ2+∆2),and
2 2
tities for the second step |1,↓i →a˜h |1,↑i [Fig. 1(b) or W ≡2π(Ω /2)2L(γ2)(∆)istheholeexcitationrate. The
R
1(c)]. For the up-to-down channel, the transition rate steady-state nuclear polarization is s(ss) = Γ s /(Γ +
p 0 p
W− ∝Ω2RL(γ2)(∆)×a˜2hL(γ2)(ωN −∆). The competition Γ1). These results agree well with the exact numerical
betweenthe twochannelsestablishes anintrinsic (i.e., in solutions to the coupled motion (see Fig. 2). For a gen-
theabsenceofothernuclearspinrelaxationmechanisms) eral nuclear spin-I, the nuclear polarization s still obeys
3

n ticassumption[13,14]inprevioustreatments,correlates
o
ati 0.05 Numerical the dynamics of different nuclear spins and drives the
z
ari Analytical nuclearspins fromthe “noisy”thermalequilibriumstate
e of nuclear pol -00..0050 ABN::< 0RR==01..20 nnss--11C BA -1 (s)p 000...048 A BC cicsnolaetlTiot)onhaiuasktc“pelqeehˆruipoipeI=tojt”hrt(e=ciNooenlalIxaee,pclIot/tωiso2vNi)tet,siˆjhosent(a=Nntseuiωmci[sl7Nep]at.,lehra,eep,jwonluea=mriczboaaenetrs,ioia˜donhfe,rjoQpi=Dedreanna˜ttouhir--,
gre C: R=2.0 ns-1 -10 -5 (n0s-1) 5 10 sˆ ≡ N Iˆz/(NI). First we consider the feedback of
e j=1 j
D -0.10-10 -5 0 5 10 themPacroscopicparth(t)bydroppingthefluctuationδhˆ
-1
Detuning (ns ) from Eq. (6) to obtain a mean-field description
FIG.2. (coloronline). s(ss) andΓp (inset)foranuclearspin-
1/2obtainedfrom Eqs. (3)-(5)(dashedlines) comparedwith h˙(t)=−Γp(∆(t))[h(t)−h(0I)(∆(t))], (7)
the exact numerical results (solid lines) for Γ = 0.2 s−1,
1
γ(c1or=resγp2on=di1ngntso−1a,tωyNpic=al −ho0l.e1mnisx−in1,ga˜ηh∼=0.42×[201]0).−5 ns−1 valid for I = 1/2 or |s(0I)| ≪ 1, where Γp(∆(t)) and
h(I)(∆(t))areobtainedfromΓ andh(I) ≡(Na I/2)s(I),
0 p 0 e 0
respectively,byreplacing∆with∆(t)=∆−h(t). Equa-
Eq. (3) by extending s to s(I) ≡ s h(Iˆx)2 +(Iˆy)2i/I, tion (7) shows that the dynamics of h(t) is highly non-
which reduces to 2(I+10)s /30for |s(I0)|≪1. linear,sinceΓp ands(0I) arehighlynonlinearfunctionsof
0 0 ∆(seeFig.2). Consequentlyforagivendetuning∆, the
The second step of the hole-driven collective nuclear
steady-statesolutionh(ss) maybe multi-valued,e.g.,two
stabilization is the feedback of the nuclear spins on the
stable solutions (black lines) and one unstable solution
hole excitation by shifting the energy of the excited hole
(gray line) in Fig. 3(a).
state |1i or the ground state |0i. For specificity, we con-
For a given detuning ∆, each stable solution h(ss) cor-
sider a negatively charged QD and identify |1i with the α
trionstate(stillreferredtoashole,despitetheadditional responds to a macroscopic (mixed) nuclear spin state
inert electronspin singlet)and |0i with the spin-up elec- with macroscopic polarization s(αss) ≡ 2h(αss)/(NaeI).
tronstate. Theholeiscoupledtothenuclearspinsinthe The noise on the electron spin qubit produced by this
QD through the non-collinear dipolar hyperfine interac- state is characterized by the fluctuation of sˆ around its
tionσˆ11 ja˜h,j(Iˆj++Iˆj−). Theelectroniscoupledtothe macroscopic value s(αss). To quantify this fluctuation,
nuclear sPpins through the diagonal part of the contact we define the probability distribution function p(s,t) ≡
hyperfine interaction σˆ a Iˆz/2 ≡ σˆ hˆ, with the TrPˆ(t)δ(s −sˆ) of sˆ. Then, instead of introducing the
00 j e,j j 00
off-diagonal terms involviPng the electron spin flip being fluctuationstochasticallybyassumingthatsˆexperiences
suppressed by the large electron Zeeman splitting. The a random walk described by a Fokker-Planck equation
feedback of the nuclear spins proceeds by shifting the (applied to nuclear spin-1/2’s only) [13, 14], we treat
energy of the electron state |0i by hˆ ≡ a Iˆz/2 (re- this feedback quantum mechanicallyby directly deriving
j e,j j
ferred to as Overhauser shift hereafter)P, which consists the equation of motion (which also assumes the Fokker-
of the macroscopic mean-field part h(t) ≡ TrPˆ(t)hˆ and Planck form) of p(s,t) from Eq. (6), valid for nuclei
the fluctuation δhˆ(t)≡hˆ−h(t), with Pˆ(t) being the di- with arbitrary spin I. The steady-state solution p(ss)(s)
agonal part of the reduced density matrix ρˆN(t) of the sharply peaks at each macroscopic value s(αss). The fluc-
nuclear spins. Then the laser detuning is changed from tuationofsˆinthe αthmacroscopicstateisquantifiedby
∆ to ∆ˆ ≡∆−ˆh≡∆(t)−δhˆ(t), where ∆(t)≡∆−h(t). the standard deviation of the Gaussian peak at s(ss):
α
This changes the nuclear spin dynamics from Eq. (2) to
1−(s(ss))2
Pˆ˙ =−Xj [Iˆj−,Iˆj+Wj,+(∆ˆ)Pˆ]−Xj [Iˆj+,Iˆj−Wj,−(∆ˆ)Pˆ], σα =σeqvuut1+dh(αs0s,)α/d∆, (8)
(6)
where Wj,±(∆ˆ) are obtained from W± by replacing where s(0s,sα) is obtained from s0 by replacing ∆ with
a˜ ,ω , and ∆ with a˜ , ω , and ∆ˆ, respectively. The ∆−h(ss) and σeq ≡ [(I +1)/(3NI)]1/2 is the thermal
h N h,j N,j α
feedbackis manifested inEq.(6) asthe Overhausershift equilibrium fluctuation of sˆ. Equation (8) is valid for
hˆ. Asshownbelow,thefeedbackofthemacroscopicpart I = 1/2 or |s(I)| ≪ 1 (generalization to arbitrary s(I)
0 0
h(t) leads to nonlinear, bistable nuclear spin dynamics, is straightforward). It unifies the different pictures of
experimentally manifested as hysteretic locking of the nuclear stabilization (e.g., a qualitative argument of nu-
electronicexcitation[17]. Thefeedbackofthefluctuation clear stabilization by the feedback of nuclear fluctua-
δhˆ(t), introduced by qualitative argument[7]or stochas- tion [7], nuclear stabilization by a strong polarization
4

-1 s) 80(a) N<0 (d) N>0 80 -1 s) shiftedhystereticallytofinitedetunings[Fig.3(f)]. Qual-
(ss) h (n 0 N Q N 0 (ss) h (n idtaattaivuelnydseirmrieladrebxechitaovnioerxscaitraetsioeenn[2in1]s.omeexperimental
-80 -80
eq 1.0(b) N<0 (e) N>0 1.0 eq
/ N N / This research was supported by NSF (PHY 0804114)
0.5 0.5 and U. S. Army Research Office MURI award
Q
Absorption0.0(c) N<0 (f) N>0 0.0 Absorption WHtho¨a9gn1ek1lesN,MFa0n.9dC1.0AZ4.0hI6am.nagmWaonegdlutYhf.aoWnrkafrnuRgi.tffouBrl.hdLeislipcuuf,uslsWidoi.nscsY.uasWsoi,o.nYAs...
-120 -60 0 60 120 -120 -60 0 60 120
-1 -1
Detuning (ns ) Detuning (ns )
FIG.3. (coloronline). Stable(blacklines)andunstable(grey
lines) h(ss) [(a), (d)]and σα/σeq in stable states [(b),(e)]. (c) [1] A.Abragam,ThePrinciples of Nuclear Magnetism (Ox-
and (f): Optical absorption spectra obtained by sweeping ∆
ford UniversityPress, New York,1961).
indifferentdirections(indicatedbythearrows). AtypicalQD
[2] N. V. Prokof’ev and P. C. E. Stamp, Rep. Prog. Phys.
withidenticalnuclearspin-9/2’sand(unit: ns−1)Nae =100, 63, 669 (2000).
γ1 =γ2 =ΩR =1, and ωN =−0.2 [(a)-(c)] or 0.2 [(d)-(f)] is [3] M. Schlosshauer, Decoherence and the quantum-to-
considered. The sharp Lorentzian peaks at ∆=0 in (c) and
classical transition (Springer-Verlag, Berlin, 2010).
(f) are absorption spectra in theabsence of thenuclei. [4] D. V. Bulaev and D. Loss, Phys. Rev. Lett. 98, 097202
(2007); A.J.Ramsayet al.,ibid.100, 197401 (2008); B.
s(ss) ≈ 1 [12], and nuclear stabilization by the compe- D. Gerardot et al., Nature 451, 441 (2008); D. Brunner
et al.,Science 325, 70 (2009).
tition between polarization and depolarization processes [5] S.Laurentet al.,Phys.Rev.Lett.94,147401 (2005); D.
[13, 14, 18]) by showing that the degree of stabilization V.BulaevandD.Loss,ibid.95,076805 (2005);D.Heiss
as quantified by the smallness of σα/σeq is the prod- et al.,Phys.Rev. B 76, 241306(R) (2007).
uct of the individual contribution [1 − (s(ss))2]1/2 [12] [6] J. Fischer et al., Phys. Rev. B 78, 155329 (2008); C.
0,α Testelin et al., ibid. 79, 195440 (2009); B. Eble et al.,
and the collective contribution(1+dh(αss)/d∆)−1/2. The Phys. Rev. Lett. 102, 146601 (2009); P. Fallahi, S. T.
individual contribution comes from the suppression of Yılmaz, and A. Imamoglu, ibid., 105, 257402 (2010); E.
the fluctuation of each individual nuclear spin by the A. Chekhovich et al.,ibid. 106, 027402 (2011).
steady-state nuclear polarization. The collective con- [7] X. Xu et al.,Nature 459, 1105 (2009).
tribution comes from the suppression of the collective [8] The sawtooth pattern in electron spin free-induction de-
cayobservedbyT.D.Laddet al. [Phys.Rev.Lett.105,
fluctuation of all the nuclear spins by the correlation
(hIˆzIˆzi−hIˆzihIˆzi)/I2 ≈ −(σeq)2 between different nu- 107401 (2010)] is attributed to a similar process.
i j i j [9] R. Hanson et al., Rev. Mod. Phys. 79, 1217 (2007); T.
clear spins, established by the correlated dynamics [Eq. D. Ladd, et al., Nature 464, 45 (2010); R. B. Liu, W.
(6)] driven by the feedback from the fluctuation δhˆ [7]. Yao, and L. J. Sham, Adv.Phys. 59, 703 (2010).
The collective contribution typically dominates the nu- [10] R.deSousa,andS.DasSarma,Phys.Rev.B,68,115322
clear spin stabilization, e.g., in the Q (i.e., “quiet”) (2003);R.B.Liu,W.Yao,andL.J.Sham,NewJ.Phys.
branch in Fig. 3(b), the nuclear spins are significantly 9, 226 (2007); J. Fischer et al., Solid State Commun.
149, 1443 (2009).
stabilizedcollectively,especiallyneartheresonancepoint
[11] For a review, see W. Yang, Z. Y. Wang, and R. B. Liu,
∆ = 0, where the slope dh(αss)/d∆ (and hence collective Front. Phys.6, 2 (2011).
stabilization) is maximal, but the magnitude |h(ss)| (and [12] W. A. Coish and D. Loss, Phys. Rev. B, 70, 195340
α
henceindividualstabilization)isnegligible. Thisexplains (2004).
the observation by Xu et al. in the single pump experi- [13] A. W. Overhauser,Phys. Rev.92, 411 (1953); J. Danon
andY.V.Nazarov,Phys.Rev.Lett.100,056603(2008).
ment [7]. For ω > 0 [Fig. 3(d)], the nuclear stabiliza-
N [14] E. A. Laird et al., Phys. Rev. Lett. 99, 246601 (2007);
tion is maximal at large detunings. M.S.RudnerandL.S.Levitov,ibid.99,246602(2007);
IntheQbranchofFig.3(a),h(ss) ∝∆alwaysshiftsthe J. Danon et al., ibid.103, 046601 (2009).
effective detuning ∆(ss) = ∆−h(ss) towards resonance, [15] A. Greilich et al.,Science 317, 1896 (2007).
while in the N (i.e., “noisy”) branch, h(ss) is very small. [16] D. J. Reilly et al., Science 321, 817 (2008).
Whensweepingthelaserfrequencyindifferentdirections, [17] C. Latta et al.,Nature Phys.5, 758 (2009).
[18] I. T. Vink et al., NaturePhys. 5, 764 (2009).
thenuclearspinstateswitchesbetweentheNbranchand
[19] Issler et al.[arXiv:cond-mat/1008.3507v1] propose that
the Q branch, leading to bidirectional hysteretic locking
electron-driven nuclear spin diffusion stabilizes the nu-
of the optical absorption peak [Fig. 3(c)]. This explains
clear spins in thetwo-pumpconfiguration of Ref. [7].
the observation by Latta et al. [17] upon blue exciton [20] B. Eble et al., Phys. Rev. Lett. 102, 146601 (2009).
excitationwhen we identify |0i as the vacuum and |1ias Dreiser et al. [Phys. Rev. B 77, 075317 (2008)] found
thespin-upexciton. InFig.3(d),h(ss) alwaystendstore- a small optically active mixing ηactive ≈0.02.
pel∆(ss) awayfromresonanceandtheabsorptionpeakis [21] A. H¨ogele, privatecommunication.