Transient probing of the symmetry and the asymmetry of electron interference

1.1 MB·

by  Matisse Wei-Yuan Tu

Transient probing of the symmetry and the asymmetry of electron interference
Matisse Wei-Yuan Tu,1 Amnon Aharony,2,3,∗ Ora Entin-Wohlman,2,3 Avraham Schiller †,4 and Wei-Min Zhang5,‡
1Department of Physics and Center of Theoretical and Computational Physics, University of Hong Kong, Hong Kong
2Physics Department, Ben Gurion University, Beer Sheva 84105, Israel
3Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
4Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel
5Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan
6 Thetransientprocessesofelectrontransportinnano-scaledevicesexhibitspecialphenomenathat
1 existonlyinthetransientregime. Besideshowfastthesteadystatesareapproached,oneinteresting
0 aspect of transient transport arises from its strong dependenceon the initial state of thesystem.
2 Hereweaddresstheissueofhowthesymmetriesembeddedintheinitialstateinterplaywiththose
of the system structure in the course of transient transports. We explicitly explore the transient
n
currentsarisingfrom variousinitialoccupationsinadouble-quantum-dotAharonov-Bohminterfer-
a
ometer. Wefindsymmetryrelationsbetweenthetransientin-tunnelingandout-tunnelingdynamics
J
for initially empty or full quantumdots when the energy levels in the electrodes are symmetrically
6
distributed with respect to the energy levels in the QDs. This is true for whatever applied fluxes.

Wealso findtheflux-evencomponentsof thecurrentsandtheflux-oddcomponentsof thecurrents
]
l exhibit distinct cross-lead symmetric relations.
l
a
h PACSnumbers: 73.23.-b,73.63.-b
-
s
e
I. INTRODUCTION channels are on the focus in some energy19 and material
m
applications.20
.
at Coherence of electron propagations in mesoscopic sys- Particularly for nanoelectronic systems, steering the
m tems is the foundation of developing quantum electronic transient currents can potentially offer versatile re-
devices. Thestudyofinterferencebetweendifferentelec- sources for timely operating quantum electronic circuits
-
d tron propagation paths is an important approach to in- at nanoscale. In general, the transient dynamics is sen-
n vestigate electron coherence. The manifestation of the sitive to the initial state of the system. In some cir-
o underlyingcoherenceinphysicalobservablesiscloselyre- cumstances,the startingpoint ofthe transientdynamics
c
latedtothesymmetryoftheinterferingstates. Bytuning is appropriately given by the steady state of the whole
[
the symmetry of the system, the interference results can transport setup before the turning on of the pulses.11–13

1 beutilizedtomodulateelectrontransportproperties,es- Inabroadercontextofquantumtechnology,besideshow
v sentialfordevicefunctionalities. Inadditiontotheman- fast one can switch on and off a current, utilization of
1
ifestation of electron coherence via steady-state current- the whole trajectory of a system from a prepared ini-
8
0 voltage characteristics, time-dependent coherent trans- tial state to a desired later state is also highly relevant.
1 port has also attracted much attention. In particular, Experimentally, preparation of desired charge states in
0 the availabilityoftime-resolvedmeasurementtechniques QDshavebeenrealized.4,21–23 Thereforeoneisnaturally
. in nanoelectronics makes it feasible to reveal the special motivated to investigatehow the transient transportdy-
1
0 roles played by the transient effects.1–5 In this article, namicsdependsontheinitialpreparations.24Inaddition
6 we explore the special transport properties,uniquely ex- tothis,tuninggeometricsymmetryhasbeenfounduseful
1 hibited in the transient regime, that reveal the underly- in modulating the stationary transport properties. For
: ingsymmetriesinducedbythedevicegeometries,viathe the purpose ofexploitingthe transienttransportproper-
v
i useofanAharonov-Bohm(AB)6 interferometerwithtwo ties as potentialresources,itis thus importantto under-
X quantum dots (QDs). stand how the symmetry embedded in the initial state
r interplays with the geometric symmetry of the interfer-
a On one hand, frequency-resolved responses to time-
ence device.
dependent periodic driving have been intensively
studied.7–10 On the other hand, the transient trans- Geometric symmetry of nanostructures can affect in-
port currents in response to pulsed voltagesprovide use- terference effects to changethe stationarytransportcur-
ful information for the switching behaviors of electronic rents. For example, working principles for molecular
devices.11–13 The transient currents induced by optical transistors have been proposed based on the interfer-
excitationsonmoleculartransportdevicesareinteresting ence effects governed by molecular geometries.25–27 Ex-
fortheirrelevancetoopticalswitches.14–18 Real-timemi- perimental observations of such effects rely on properly
grations of electrons between molecules and conduction arranging molecule-lead coupling configurations on the
molecular scale.28–30 Another widely applied approach
to manipulate stationary transport by tuning interfer-
ence is to thread a magnetic flux through a ring-shaped
†Deceased, June22nd,2013 nanostructure, rendering the AB effect. This was re-
2
alized earlier with metallic ring interferometry.31 With even in the flux (insensitive to the flux direction), the
a QD sitting on one arm, making the other arm of transient currents can exhibit components that are odd
the ring as a reference, studies on the resonant tun- intheflux. Theybehavedistinctlyfromtheirevencoun-
neling through the QD have been experimentally car- terparts in terms of the manifestation of the geometric
ried out.32–39 Ring-shaped natural molecules and arti- symmetry of the device. (ii) Initially empty QDs and
ficial molecules are useful nanostructures that can host initially occupied QDs can induce completely different
the AB effect and at the same time provide manipulable ways of electron motions, giving rise to distinguishable
geometricsymmetry. For ring-shapednaturalmolecules, in-tunnelingandout-tunnelingtransientcurrentsrespec-
effective use ofmagnetic fluxes either requiresunrealisti- tively. Their relationship is associated with the sym-
callyhighmagneticfields40 orotherspecialconditions.41 metry of the energy level distribution. (iii) Asymmetry
Evoking the AB effect with artificial molecules has been amongdifferentcomponentsofthetransientcurrentscan
realised by putting two QDs on the two arms of a meso- be either enhanced or suppressedby properly tuning the
scopic interferometer.42–44 This is known as the double- device geometry.
quantum-dot(DQD) AB interferometer. Suchsetups of- Thepaperisorganizedasfollows: InSec. II,webriefly
fer the advantages of incorporating molecular symmetry review the general model for mesoscopic transport sys-
into the flux-modulated interference effects. How to use tems and the nonequilibrium Green function formalism
theABoscillationstodiscernmolecularparitieshasbeen (NEGF) for calculating the currents. Methodologically,
theoretically45 and experimentally studied.44 By prop- there exist many sophisticated theoretical approaches
erly combining it with the spin-orbit interaction, mixed to study time-dependent quantum transport .12,13,59–69
actions of charge and spin interference have been shown Most of them are based on NEGF. Since many papers
to give rise to full spin-polarized transport.46,47 The AB onquantum transportuse the scatteringformalism,70–72
interference is also useful to affect the occupation differ- for pedagogical purposes, we also present a connection
ence between degenerate QDs48 and modulate the cur- to the scattering-state approach for deriving the tran-
rent noises.49,50 sient time-dependent transport properties. We pay spe-
cial attention to how the currents depend on the initial
Various time-dependent aspects of DQD AB inter-
occupationsinthecentralnanostructures. InSec.III,we
ferometers have been tackled. Real-time detection of
introduce our target system, an AB interferometer with
interference pattern formation has been experimentally
attempted.51 The flux-dependent decoherent and coher- twoQDs. Weidentifythepurelytransientcomponentsof
the tunneling currents and discuss generally their phys-
ent dynamics of the uncoupled DQD has been tackled
ical meanings. In Sec. IV, we show how the symmetry
by the exact fermion master equation for noninteracting
electrons.52,53 With inter-dot Coulomb repulsion consid- andasymmetry of the device parameterscan giverise to
distinguishable transient dynamics. Section V contains
ered, photoresponse of the coupled DQD in the AB in-
our conclusions.
terferometerhasbeeninvestigatedintheweak-tunneling
limit.54 The reduced density matrix dynamics under the
influence of the flux, including also inter-dot Coulomb
II. FORMALISM FOR TRANSIENT QUANTUM
interaction, has been studied by the use of the exact-
TRANSPORT
numerical-path-integral.55 The latter approach has also
been applied to examine the transientmagnetotransport
in similar systems.56 The effects of Fano resonances on A class of quantum transport systems is described by
the dynamics of wave scatterings with a coupled DQD the following general Hamiltonian,
molecule in the AB interferometer has been inspected in
the time domain.57 H=HS+HE+HT, (1a)
The choice of a simple uncoupled DQD placed in an where
ABinterferometerprovidesasuitableplatformtoclearly
definethegeometricsymmetryoftheDQDmoleculeand H = ε d†d , (1b)
S ij i j
the symmetry associated with initial occupations in the
i,j∈S
X
QDs. In a previous paper, we have comprehended the
flux responsesof the QD occupations,the transportcur- is the Hamiltonian for the central scattering region, and
rents as well as the circulating currents of the DQD AB
interferometers in the transient as well as in the steady- HE = Hα = εαkc†αkcαk, (1c)
stateregimes.58However,wehaveonlyworkedunderthe α α k∈α
X XX
conditionsthattunnelingsthroughthetwoQDsaresym-
isthesumofHamiltoniansoftheelectrodes,eachlabeled
metric and initially the two QDs are not occupied. Here
by α. The tunneling of electrons between the central
we remove all these restrictions and we show that DQD
region and the electrodes is described by
AB interferometers indeed provide several benefits that
diversify the uses of the transient currents. Our main
findings are summarized below: (i) Unlike the steady- HT = tαkic†αkdi+tiαkd†icαk . (1d)
state currents, which contain only components that are iX,αkn o
3
Here the subscripts i,j ∈ S = {1,2,··· ,D} enumerates where η →0+. The solution is found to be78
the D single-particle levels within the central regionand
k ∈ α stands for the continuum levels k within the elec- ψ† =
αk
trode α. The field operator d (d†) or c (c† ) annihi-
j j αk αk t c†
lates (creates) an electron on level j ∈ S or level k ∈ α. c† + t d†+ βqj βq G (ε +iη),
The hopping amplitude between a level i in the central αk iαkj ε −ε +iη ji αk
αk βq
i,j βq
part and a level k in lead α is given by t = t∗ . X X
αki iαk   (5)
Since we are targeting at the single-particle interference
effect, interactions leading to dephasing of transport are
wheretheGreenfunctionintheenergydomain,G (z)=
ignored.73–75 [G(z)] , ij
The essential quantity probed by transport measure- ij
ments is the tunneling current. The current tunneling
−1
out of lead α at time t is defined by G(z)= z1D−ε−Σ(z) , (6)
h i
d has been introduced with the selfe-energy,
I (t)=− hN (t)i=−ih[H,N (t)]i
α dt α α
=−i tiαkhd†i(t)cαk(t)i−tαkihc†αk(t)di(t)i , Σ(z)= dω Γ(ω), (7)
2π z−ω
Xi kX∈α(cid:16) (cid:17) Z
(2)
and the total levele-broadening function is,
whereN (t)= c† c is the totalparticle number
α k∈α αk αk Γ(ω)= Γα(ω), with
operator in lead α in the Heisenberg representation. We
have also set chParge unit e = 1 and ~ = 1. The bracket Xα
h·i=tr(·ρˆ(t0)) denotes the averageoverthe initial state, Γαij(ω)=2π δ(ω−εαk)tiαktαkj (8)
ρˆ(t0), at time t = t0, of the total system including the kX∈α
central scattering region plus the electrodes.
where [Γα(ω)] =Γα(ω).
The initial state of the total system ρˆ(t0) is ij ij
assumed12,76 to be a product state of the central area The states formed from ψ† |0i, where |0i denotes the
αk
and the electrodes, namely, vacuum of the total system, constitute a complete set in
the single-particle space and the relation ψ ,ψ† =
βq αk
ρˆ(t )=ρˆ (t ) ρˆ (t ), (3a)
0 S 0 α 0 δαk,βq is satisfied. One can therefore expnress dj foroall
α
Y j and c for all αk in terms of linear combinations of
αk
where theψαk’s. Thetime-dependenciesofdj(t)andcαk(t)are
then found through ψ (t) by ψ (t) = e−iεαk(t−t0)ψ
exp[−(H −µ N )/k T ] αk αk αk
ρˆα(t0)= trexp[−(Hαα−µααNαα)/kBBTαα], (3b) (a4s),ψwαkithsatthiesfieeisgetnheeneLrgipypεmαakn.-SSpchecwifiincgaellry,equation, Eq.
represents the thermal equilibrium of the electrode α,
d (t)= G (ε +iη)t e−iεαk(t−t0)ψ ,
eachwiththechemicalpotentialµ andthetemperature j ji αk iαk αk
α
αk i
T . Herek istheBoltzmannconstant. Theinitialstate XX
α B
of the central scattering area is denoted by ρˆS(t0) and is tαkjGji(εβq+iη)tiβq
notrestricted. Wewilldiscusshowthetransientcurrents cαk(t)= (δαk,βq+ εβq−εαk+iη 
depend on ρˆS(t0). Xβq Xi,j
 
×e−iεβq(t−t0)ψ . (9)
βq
A. Scattering-state method )
Replacing ψ on the right-hand side of Eq. (9) by
ThetimedevelopmentsofthetunnelingcurrentsI (t) αk
α Eq. (5), we obtain
can be calculated via different approaches.12,68 Here
we introduce the scattering-state method, which tack-
les the time developments of the Heisenberg field oper-
d (t)=i Gr (t,t )d + Gr (t,t )c ,
ators d (t)’s and c (t)’s through the scattering states. i  ij 0 j i,αk 0 αk
Itstartsi with solvinαgkthe Lippman-Schwingerequation77 Xj Xαk 
for the scattering states, as the eigenstate of the total
 
Hamiltonian, c (t)=i Gr (t,t )d + Gr (t,t )c ,
αk  αk,j 0 j αk,βq 0 βq
Xj Xβq 
ψ† ,H =−ε ψ† +iη c† −ψ† , (4) (10a)
αk αk αk αk αk  
h i (cid:16) (cid:17)
4
where The lesser Green function, defined by
Grij(t,t0)= ∞ d2ωπe−iω(t−t0)Gij(ω+iη), [G<(t1,t2)]ij =ihd†j(t2)di(t1)i, (13a)
Z−∞
∞ dω G (ω+iη)t is found via Eq. (10) to be
Gr (t,t )= e−iω(t−t0) ij jαk,
i,αk 0 2π ω−ε +iη
Z−∞ Xj αk G<(t,t′)=Gr(t,t0)G<(t0,t0)Ga(t0,t′)
Gr (t,t )= ∞ dωe−iω(t−t0) tαklGlj(ω+iη), t t′
αk,j 0 2π ω−ε +iη + dτ dτ′Gr(t,τ)Σ<(τ,τ′)Ga(τ′,t′),
Z−∞ l αk
X Zt0 Zt0
Gr (t,t )= −iδ e−iεαk(t−t0)+ (13b)
αk,βq 0 βq,αk
∞dωe−iω(t−tn0) tαkjGjl(ω+iη)tlβq . where
2π (ω−ε +iη)(ω−ε +iη)
Z−∞ Xjl (cid:20) αk βq (cid:21)o Σ<(τ,τ′)= Σ<(τ,τ′), (13c)
(10b) α
α
X
The result of Eq. (10) is identical to that obtained via andthelesserGreenfunctionatthe initialtime t shows
0
directly solving the Heisenberg equations. an explicit dependence on ρˆ (t ) as
S 0
UsingEq.(10)inEq.(2),withtheinitialstategivenby
Eq. (3), we obtain an expression for the time-dependent G<(t ,t ) =itr d†d ρˆ (t ) . (13d)
tunneling current, 0 0 ij j i S 0
(cid:2) (cid:3) h i
I (t)= Here tr means tracing over all the degrees of freedom
α
of the whole system (central scattering region plus the
t
−2ReTr dτ Σ<(t,τ)Ga(τ,t)+Σr(t,τ)G<(τ,t) , electrodes).
α α
Zt0
(cid:0) (1(cid:1)1a)
B. Transient dependence of the tunneling currents
where the retardedcomponentof the self-energyis given on the initial state of the central area
by,
The above result, Eqs. (11,12,13), shows that the de-
dω
[Σrα(t1,t2)]ij =−iθ(t1−t2) 2πΓαij(ω)e−iω(t1−t2), pendence of the tunneling currents on the initial state
Z of the central scattering region only comes from the sec-
(11b)
ond term in Eq. (11a) through the lesser Green func-
tion Eq. (13). By setting ρˆ (t ) = |0 ih0 | (such that
and the lesser component of that self-energy is S 0 S S
G<(t ,t )=0),where|0 irepresentstheemptystateof
0 0 S
dω thecentralscatteringarea,orlettingt →−∞,Eq.(11a)
[Σ<(t ,t )] =i f (ω)Γα(ω)e−iω(t1−t2), (11c) 0
α 1 2 ij 2π α ij becomes the current obtained in Ref. [12]. The expres-
Z sion Eq. (13b) can be obtained via the NEGF technique
wherefα(ω)=1/[e(ω−µα)/kBTα+1]isthe fermifunction by setting the coupling between the QDs and leads to
forelectrodeα. InEq.(11), the notationTrmeanstrace zero for t < t0.79 One can also obtain Eq. (13b) via the
over the indices of the central scattering area. The ad- influence functional theory.68 Substituting Eq. (13) into
vanced Green function is related to the retarded Green Eq. (11), the time-dependent tunneling current can thus
function by, be separated into two terms,
Gaij(t1,t2)=[Grji(t2,t1)]∗, (12a) Iα(t)=Iαem.(t)+Iαocc.(t), (14a)
where Ga(t ,t )=[Ga(t ,t )] and the retarded Green where
ij 1 2 1 2 ij
function, defined for arbitrary two times t ≥ t and
1 0 t
t2 ≥t0, Iem.(t)=−2ReTr dτ Σ<(t,τ)Ga(τ,t)
α α
Zt0
Grij(t1,t2)=−iθ(t1−t2)h di(t1),d†j(t2) i, (12b) + Σrα(t,τ)[G<(τ,t)]|0 (cid:0). (14b)
n o
(cid:1)
can be computed from Eq. (10), resulting in in which,
∞dω τ t
Gr (t ,t )=θ(t −t ) e−iω(t1−t2)G (ω+iη). [G<(τ,t)]| = ds ds′Gr(τ,s)Σ<(s,s′)Ga(s′,t),
ij 1 2 1 2 2π ij 0
Z−∞ Zt0 Zt0
(12c) (14c)
5
and
Iocc.(t)
α
t
=−2ReTr dτΣr(t,τ)Gr(τ,t )G<(t ,t )Ga(t ,t).
α 0 0 0 0
Zt0
(14d)
The first term Iem.(t) describes the tunneling current
α
that is independent of the initial occupation in the cen- FIG.1: (color online) A schematicsketchof theDQDdevice
tral scattering region. The superscript ”em” stands for underconsideration. Two single-level QDswith on-site ener-
initially”empty”states. ThesecondtermIocc.(t),where gies ε1 and ε2 are coupled tothetwoseparate leads with the
α
tunnelingamplitudest . Thedeviceis threadedbythethe
the superscript ”occ.” abbreviates ”initial occupation”, iαk
Aharonov-Bohm fluxφ.
explicitly includes this effect. The result Eq. (14d) fur-
ther indicates that Iocc.(t) is independent of the ini-
α
tial chemical potentials and temperatures of the elec-
To discuss the transient transport we first extract the
trodes. Aslongastherearenoboundstates,80theGreen
function, Ga(t ,t→∞) = 0, vanishes in the long-time components of the tunneling currents that are purely
0 transient. We identify the regime where only these com-
limit. The current purely induced by initial occupation,
ponents exist so that the steady-state currents vanish.
Iocc.(t), then only survives transiently.
α We elucidatethe physicalmeaningsofdifferenttransient
tunneling currents. This then clarifies the directions for
subsequent analysis of how their characters depend on
III. INITIAL-OCCUPATION-DEPENDENT
the interferometry parameters.
TRANSIENT INTERFEROMETRY WITH
PARALLEL QUANTUM DOTS Here we show the followings. (i) Apart from Iαocc.(t),
purelytransienteffectsinthetunnelingcurrentscanalso
be found from Iem.(t). These purely transient compo-
The target system in the present work consists of two α
nents can be extracted from the directly observable cur-
QDs arranged in parallel between two electrodes. It is
rents. (ii) Since the steady-state currents vanish in the
schematically shown in Fig. 1. For simplicity, we as-
zero-bias regime, it is convenient to study the transient
sumethateachQDcontainsasinglespinlessorbital. The
currents at zero bias. (iii) The currents purely induced
HamiltonianofthecentralnanostructureEq.(1b)isthen
by initial occupations can be analyzed by resolving con-
specified by
tributions from each QD.
H =ε d†d +ε d†d . (15)
S 1 1 1 2 2 2
The two QDs are uncoupled but tunneling through one A. transient zero-bias currents
QD can interfere with tunneling through the other. To
revealandcontrolsuchinterferenceforthestudyoftran- The presentsetup is a two-terminalsystem. The label
sient transport, we utilize a magnetic flux penetrating for reservoirα in Eq. (14) then takes the value α=L or
through the loop formed by the two QDs and the two α = R for a terminal on the left or on the right. In the
leads. ThisimplementstheABeffectviaproperlyattach- steady states, charge conservation asserts I (t → ∞) =
α
ing phases to the hopping amplitudes tiαk =t¯iαkeiφiα in −Iα¯(t → ∞), where α¯ refers to the opposite of α, i.e.,
Eq. (1d) such that the constraint L¯ = R and R¯ = L. Before the steady state is reached,
due to the time dependence of charge occupation in the
φ −φ =φ=2πΦ/Φ , (16)
L R 0 centralscatteringarea,onegenerallyhasI (t)6=−I (t).
α α¯
Therefore, it is necessary to characterize individually
where φ −φ = φ , is respected. Here Φ is the ap-
1α 2α α the currents on each side. Generically, we have
plied magnetic flux through the ring and Φ is the flux
0
quantum. ThisenterstheGreenfunctionsabovethrough Iem.(t)=Iem.(t)+∆Iem.(t), (18a)
the level-broadening functions, α α α
Γα(ω)=Γ¯α(ω)ei(φiα−φjα), (17a) where
ij ij
with Iem.(t)= ∞dωf¯(ω)T(+)(t,ω), (18b)
α 2π α
Z−∞
Γ¯α(ω)=2π δ(ω−ε )t¯ t¯ . (17b)
ij αk iαk αkj
and
k∈α
X
Without loss of generality, we assume Γ¯α(ω) to be real ∆Iem.(t)= ∞dω∆f (ω)T(−)(t,ω), (18c)
ij α 2π α α
for all α=L,R and i,j ∈{1,2}. Z−∞
6
are the contributions fromthe averageof the fermi func- I (t) obtained at reversedbiases, namely,
α
tions,
em. 1
I (t) = ×
f¯(ω)= fL(ω)+fR(ω), (18d) α (cid:12)(µL,µR)=(µA,µB) 2
2 Iαem.(t(cid:12)(cid:12))|(µ ,µ )=(µ ,µ )+ Iαem.(t)|(µ ,µ )=(µ ,µ )
L R A B L R B A
and from the deviation of the fermi distribution in each h (i21)
reservoir from the average,
∆f (ω)=f (ω)−f¯(ω), (18e)
α α B. Resolving level contributions to initial charge
induced currents
respectively. The transmission-like, energy- and time-
dependent functions in Eqs. (18b,18c) are
To study the effects of initial occupations on the tun-
nelingcurrentsviaIocc.(t),oneneedstoexaminevarious
t α
T(±)(t,ω)=2ImTr dτ Γα(ω)e−iω(t−τ)Ga(τ,t) attainable occupations. For the interferometer with two
α
Zt0 n QDs, particularly interesting initial occupations are the
+Σr(t,τ)× states with one electron either occupying QD1 or QD2,
α
τ t and the fully occupied configuration. We denote ni(t0)
dτ dτ Gr(t,τ )Γα(ω)±Γα¯(ω)e−iω(τ1−τ2)Ga(τ ,t) , as the initial occupation on QD i. These diagonal ini-
1 2 1 2
Zt0 Zt0 (cid:2) (cid:3) (18fo) tial occupations are then described by [G<(t0,t0)]ij =
iδ n (t ), in Eq. (14d) leading to
ij i 0
obtained by substituting Eq. (11c) into Eq. (14b) and
Iocc.(t)= n (t )I[i](t), (22a)
Eq. (14c). In Eq. (18b), T(+)(t,ω) is the transmission α i 0 α
α
i
ofelectrons withenergy ω fromreservoirα into the QDs X
at time t. The second part of the current, ∆Iem.(t), is with
α
nonzeroonlywhenf (ω)−f (ω)is nonzero. Itis there-
L R
t
foreattributedexclusivelytotheeffectinducedbyabias, I[i](t)=2Im dτ[Ga(t ,t)Σr(t,τ)Gr(τ,t )] ,
representedbythedifferencebetweenthetwofermifunc- α 0 α 0 ii
em. Zt0
tions. The other part, I (t), survives even without a (22b)
α
bias.
In the steady states, the current, I = I (t → ∞) = resolving the current specifically induced by an initial
L
−I (t → ∞), is reduced to the well-known expression, occupationonleveli. Consequently,the currentinduced
R
namely,81 by initially occupying both QDs, denoted as I[1+2](t), is
α
then
∞dω
I = (f (ω)−f (ω))T (ω)=ζ ∆Iem.(t→∞),
2π L R α α 2
Z−∞ I[1+2](t)≡ Iocc.(t)| = I[i](t), (23)
(19) α α (n1(t0),n2(t0))=(1,1) α
i=1
X
where
accordingto Eq. (22). The results Eq. (14) and Eq. (22)
1 imply that one only needs to separately track the dy-
T(ω)=2Tα(−)(t→∞,ω)=Tr Γα(ω)Gr(ω)Γα¯(ω)Ga(ω) , namics ofIα[i](t) for eachlevel i to see how variousinitial
(cid:8) (20(cid:9)) occupations influence the transient dynamics.
for both α = L,R. In Eq. (19), ζ = 1 and ζ = −1.
L R
Henceforth, from Eqs. (19,18) one sees unambiguously C. Physical meanings of transient currents
em.
thatI (t→∞)=0andthatthe steady-statecurrents
α
vanish at zero bias. Having identified distinct transient components of the
em.
WethusidentifythatI (t)isapurelytransientcom- tunneling currents, we now analyze their correspond-
α
ponentofthecurrentIem.(t)forwhateverbiases. Theef- ing underlying physical processes. Immediately after
α
fectsthatareexclusivelytransientsarecontainedonlyin t = t , the current Iem.(t) is contributed solely by the
Iem.(t)andIocc.(t). Giventhe abilities ofpreparingvar- in-tun0nelingprocessesα(tunneling ofelectronsfromreser-
α α
ious initial occupations of the QDs (including the initial voirsintothecentralscatteringarea). Sinceinthebegin-
empty state),4,21–23 the currents purely induced by ini- ning there were no electrons in the central part, it is not
tial occupations Iocc.(t) can be obtained by subtracting possible to have electrons tunnel out of the central area
α
Iαem.(t) from the total current Iα(t) starting fromemv.ari- into the reservoirs. On the contrary, Iαocc.(t) describes
ousoccupations. Theothertransientcomponent,I (t), currents exclusively induced by out-tunneling processes.
α
can be found from the difference between the currents Thecurrentsaredefinedasthe negativechangingrateof
7
theelectronnumbersintheelectrodes. In-tunnelingfrom Thegeometricsymmetryoftheinterferometerisspecified
reservoirsintotheQDsdecreasesthenumberofelectrons by the distribution ofthe fourbonds andthe two on-site
in the reservoirsand hence the changing rate of electron energiesoftheQDs. Itisalsoaffectedbytheappliedflux.
number in the reservoirs is negative. Consequently, we The geometry of the bonds can be classified according
anticipate the in-tunneling currents to be positive. On to the geometric symmetry they present: (i) left-right
the contrary,we expect the out-tunneling currents to be symmetry, namely, Γ¯L = Γ¯R = Γ¯ , and (ii) up-down
ii ii i
negative. symmetry, Γ¯α = Γ¯α = Γα. We denote the difference
11 22
Microscopically, the elementary processes underlying between the on-site energies of the two QDs by
the in-tunneling currents are induced by tunneling of
electronsfromoccupied levelsbelow the chemicalpoten- δε=ε1−ε2. (27)
tials in the reservoirs to the QDs. The positions of the
The geometry of the two on-site energies is categorized
chemical potentials relative to the energy levels of the
by being degenerate δε=0 or not δε6=0, exhibiting up-
QDs determine how much can tunnel into the QDs and
downsymmetryornot. Theinfluencesofthefluxcanbe
how much cantunnel out as well. Therefore the symme-
inspected by generally decomposing any flux-dependent
try of energy levels on the QDs relative to the chemical
quantity, Q(φ), into
potentials is important for the relationship between the
in-tunneling and the out-tunneling currents. In addition Q(φ)=Q+(φ)+Q−(φ), (28a)
tothat,theinterferometerisalsocharacterizedbythege-
ometricsymmetryofthedevice. Theindividualtransient with
components I[i](t,φ), Iem.(t,φ) are explicitly associated
α α Q±(φ)=±Q±(−φ) (28b)
with the geometric characterization of the interferome-
ter, namely, the up (i = 1), down (i = 2), left (α = L)
representingthe even”+”and the odd”−”responsesto
and right (α=R). Below we show how to modulate the
the flux.
propertiesofthesetransientcurrentsbyinvestigatingdif-
In Sec. IVA, we illustrate the special transient prop-
ferentcomponentsofthetransientcurrentsinassociation
erties of the odd components of I[i](t,φ), and Iem.(t,φ).
with these two symmetries of the system. α α
Subsequentdiscussionsrevealthat they arenot sensitive
to left-right asymmetry, unlike their even counterparts.
InSec.IVB,wediscusstheeffectsofup-downsymmetry
IV. SYMMETRY IN THE DYNAMICS FOR
THE TRANSIENT CURRENTS and we demonstrate how to affect the left-right asym-
metry in the time scales as well as the magnitudes of
the currents by the geometry of the system. The case
Inordertohaveanunambiguouspicture,weexplicitly
of left-right symmetry is presented in Sec. IVC together
specifytherelevantparametersofthesystem. Weassume
withgeometricfactorsthatinfluence the up-downasym-
the widely applied wide-band approximation. Within
this assumption, t¯iαk = t¯iα, becomes independent of k metry between Iα[1](t,φ) and Iα[2](t,φ). In Sec IVD we
in Eq. (17b) and consequently focus on the symmetry of the energy level distribution.
This symmetry is shown to play a key role in the rela-
Γ¯αij(ω)=Γ¯αij =2π̺αt¯iαt¯jα, (24) tionship between the main two parts of the purely tran-
where ̺α is the density-of-state of lead α. Fol- sientcomponentsofthecurrents,Iα[i](t,φ)andIeαm.(t,φ).
lowing Eq. (24), we have Γ¯α = 2π̺ t¯ t¯ = Note that the I[i](t,φ)’s do not depend on the chemi-
12 α 1α 2α α
(2π̺ t¯ t¯ )(2π̺ t¯ t¯ ) = Γ¯α Γ¯α . The solution cal potentials. As shown in Eq. (18b), the geometric
α 1α 1α α 2α 2α 11 22
to the Green functions and the consequent currents factors only enter Iem.(t,φ) through T(+)(t,ω,φ), hav-
p p α α
Eqs.(18),(22)areexplicitly obtainedandsummarizedin ingnothing to dowith µ andµ . Forsimplicity, weset
L R
em.
Appendix A. The parameters of the interferometry are µ =µ =µ =ε forinspectingI (t,φ)’sinSec.IVB
L R 0 0 α
then specified by the four bonds, Γ¯L , Γ¯L, Γ¯R, and Γ¯R, and Sec. IVC. Deviations from µ = µ = µ = ε are
11 22 11 22 L R 0 0
the two on-site energies of the QDs, ε and ε , the two considered in Sec IVD.
1 2
chemical potentials µ and µ of the reservoirs and the
L R
applied flux φ. The currents are functions of both time
and flux, namely, I (t)→I (t,φ). A. Transient odd components of the tunneling
α α
The symmetry of the energy level distributions con- currents
cerns the relative configuration of the four energy ref-
erences ε1 and ε2, µL and µR. We denote the energy Below we show that the odd components of the cur-
references by rents are insensitive to the left-right asymmetry in the
bonds. However, the up-down symmetry in terms of the
ε =(ε +ε )/2, (25)
0 1 2 on-siteenergiesofthe QDsiscrucialtomanifestthe odd
and components.
Substituting explicitly Eq. (A1) into Eq. (A9) and
µ =(µ +µ )/2. (26)
0 L R Eq. (A10) and extracting the parts that are odd in the
8
flux give The system under consideration is a two-terminal
setup. Therefore, the steady-state currents must
Iα[1],−(t,φ)=2ζαΓ¯α12Γ¯α1¯2sinφ obey phase rigidity making only the even components
2 remain.82,83 Taking the steady-state limit, t → ∞, in
b (t) b (t)
× Im − b∗ (t) −δε − , (29) Eqs. (29),(30), and (31), we see they all vanish (see Ap-
Γ (φ) + Γ (φ)
" (cid:18) g (cid:19) (cid:12) g (cid:12) # pendixA1). Henceforth,theeffectswhicharisefromthe
(cid:12) (cid:12)
(cid:12) (cid:12) odd components are exclusively transient. The above
(cid:12) (cid:12)
analysis then leads us to consider the effect of geometric
I[2],−(t,φ)=2ζ Γ¯α Γ¯α¯ sinφ
α α 12 12 symmetry for the even and the odd components sepa-
b (t) b (t) 2 rately according to whether δε=0 or δε6=0.
× −Im − b∗ (t) −δε − , (30)
Γ (φ) + Γ (φ)
" (cid:18) g (cid:19) (cid:12) g (cid:12) #
(cid:12) (cid:12)
(cid:12) (cid:12) B. Up-down symmetry of the bonds, Γ¯α =Γ¯α
and (cid:12) (cid:12) 11 22
Tα(±),−(t,ω,φ)=∓4ζαδεΓ¯α12Γ¯α1¯2sinφ Here we discuss the effects of up-down symmetry re-
2 alized by Eq. (32). Given the equally strong upper and
× Γ ˜b−(t,ω) +Re ˜b−(t,ω)˜b∗ (t,ω) . (31) lower bonds, the currents purely induced by occupying
 (cid:12) Γg(φ) (cid:12) Γg(φ) + ! one QD shall also exhibit up-down symmetry,84 namely,
 (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)  Iα[1](t,φ) = Iα[2](t,φ) (see Appendix A2 for a detailed
In Eqs. ((cid:12)29),(30),a(cid:12)nd (31), the ζ ’s are defined after proof using Eq. (32) only without Eq. (33)). Intuitively,
α
Eq. (19), Γ (φ) is given by Eq. (A7), b (t,φ) and a large asymmetry between the magnitudes of the cur-
g ±
˜b (t,ω) are defined in Eq. (A11) and Eq. (A12) respec- rents on the left and that on the right can be realized
±
by ΓL ≪ ΓR. More interestingly, we show that the left-
tively. Thefactor,Im b−(t)b∗ (t) ,inEqs.(29,30),van-
Γg(φ) + right asymmetry between the times of approaching the
ishes when the geome(cid:16)try of the sy(cid:17)stem satisfies either steady-state limit can be enhanced or suppressed by the
choices of δε in comparison to |Γ (φ)|.
Γ¯α =Γ¯α =Γα, (32) 12
11 22
or
1. degenerate QDs
δε=0. (33)
Figure2demonstratesthecurrentswithup-downsym-
(See Appendix A1 for a detailed derivation.) The re- metry in the bonds andwith degenerateQD levels, δε=
sults Eqs.(29),(30)and(31)showthatthe followingtwo 0. Inthis case,onlythe evencomponents ofthe currents
conditions must simultaneously hold for these odd com- arenonzero. Thecurrentsontheleftandontherightare
ponents to be nonzero: shown as a function of time at various fluxes on the first
two rows of Fig. 2 (plots (a1) for I[1](t,φ) = I[2](t,φ),
sin(φ)6=0. (34) (a2) for Iem.(t,φ), (b1) for I[1](t,φ)L= I[2](t,φ),L(b2) for
L R R
em.
and I (t,φ)).
R
At degeneracy, the rates of approaching steady states
δε6=0. (35) aregovernedonlyby the bonds. Therefore,the left-right
asymmetry in the bonds would be revealed also in the
The first condition Eq. (34) explicitly reveals that these
difference between the times needed for the currents on
odd components arise from the applied flux. The second
the left andonthe rightto reachthe steadystates. This
condition Eq.(35) canbe used to discern qualitatively if
is illustratedonthe firsttworowsofFig.2(comparethe
theQDsaredegenerateornotbyinspectingwhetherthe
timescalesin(a1),(a2)withthosein(b1),(b2)inFig.2).
odd components of the currents are always zero.
When the up-down symmetry is realized simultaneously
Furthermore, Eqs. (29),(30), and (31) also show that
byEq.(32)andEq.(33),Eqs.(A1),(A11),(A20),(A22),
regardless of the symmetry of the bonds, the odd com-
(A25) and (A18) yield
ponents always obey the following symmetry:
1 (Γα+Γα¯cosφ)
IL[i],−(t,φ)=IR[i],−(t,2π−φ), Iα[1](t,φ)=−Γα 2 1− δΓ(φ) e−(Γ−δΓ(φ))t
(cid:26) (cid:18) (cid:19)
T(±),−(t,ω,φ)=T(±),−(t,ω,2π−φ). (36) 1 (Γα+Γα¯cosφ)
L R + 1+ e−(Γ+δΓ(φ))t . (37)
2 δΓ(φ)
Note that Eq. (36) applies not only to the parts of the (cid:18) (cid:19) (cid:27)
currents that are purely transient, namely, Iα[i](t,φ) and The solution Eq. (37) shows that the initial-occupation-
em.
I (t,φ). Itis alsotrue forthe partofthe currentgiven inducedcurrentsonbothsidesα=L, Rcommonlypos-
α
by ∆Iem.(t) (see Eq. (18) with Eq. (36)). sess two exponential decay terms e−(Γ±δΓ(φ))t, each of
α
9
which is associated with an amplitude. The correspond-
ing decay rates are plotted in Fig. 2(c). Equation (37)
saysthatitisthedifferencebetweentheamplitudeswith I(cid:62)(cid:20)(cid:64)(cid:47)(t, )=I(cid:62)(cid:21)(cid:64)(cid:47)(t, ) (cid:68)(cid:20) I(cid:62)(cid:20)(cid:64)(cid:53)(t, )=I(cid:62)(cid:21)(cid:64)(cid:53)(t, ) (cid:69)(cid:20)
α=L and with α=R that is responsible for the asym- (cid:19) (cid:19)
metry between I[1](t,φ) and I[1](t,φ). The overall fac- (cid:16)(cid:19)(cid:17)(cid:19)(cid:21) (cid:16)(cid:19)(cid:17)(cid:21)
L R
tor Γα in front of the RHS of Eq. (37) is reflected on (cid:16)(cid:19)(cid:17)(cid:19)(cid:23)
(cid:16)(cid:19)(cid:17)(cid:23)
the apparent difference in the magnitudes of the left
(cid:16)(cid:19)(cid:17)(cid:19)(cid:25)
and the right currents (see magnitudes in Fig. 2(a1) (cid:16)(cid:19)(cid:17)(cid:25)
(cid:16)(cid:19)(cid:17)(cid:19)(cid:27)
and Fig. 2(b1)). The difference in the time scales arise
(cid:16)(cid:19)(cid:17)(cid:27)
from the difference in the relative importance of the am- (cid:16)(cid:19)(cid:17)(cid:20)
plitudes in front of the faster decay term e−(Γ+δΓ(φ))t (cid:19) (cid:20)(cid:19) (cid:21)(cid:19) (cid:22)(cid:19) (cid:23)(cid:19) (cid:19) (cid:21) (cid:23) (cid:25) (cid:27)
and the slower decay term e−(Γ−δΓ(φ))t. Furthermore, I(cid:72)(cid:80)(cid:17)(t, ) (cid:68)(cid:21) I(cid:72)(cid:80)(cid:17) (t, ) (cid:69)(cid:21)
(cid:47) (cid:53)
the currents on the left shown by Fig. 2(a1),(a2) are (cid:19)(cid:17)(cid:20) (cid:19)(cid:17)(cid:27)
clearly different for different fluxes. The currents on the (cid:19)(cid:17)(cid:19)(cid:27)
(cid:19)(cid:17)(cid:25)
right show less visible changes for different fluxes (see
(cid:19)(cid:17)(cid:19)(cid:25)
Fig. 2(b1),(b2)). These asymmetric features shown by (cid:19)(cid:17)(cid:23)
(cid:19)(cid:17)(cid:19)(cid:23)
Fig. 2(a1) and Fig. 2(b1) can be derived by imposing
ΓR ≫ΓL in Eq. (37) (see Appendix B1a). (cid:19)(cid:17)(cid:19)(cid:21) (cid:19)(cid:17)(cid:21)
In Fig. 2, one also observes a symmetry between (cid:19) (cid:19) t
I[1](t,φ) and Iem.(t,φ) showing that Iem.(t,φ) = (cid:19) (cid:20)(cid:19) (cid:21)(cid:19) (cid:22)(cid:19) (cid:23)(cid:19) (cid:19) (cid:21) (cid:23) (cid:25) (cid:27)
α α α
−I[1](t,φ). The interpretation of I[i](t,φ) and Iem.(t,φ)
α α α (cid:70)
as out-tunneling and in-tunneling currents in Sec. IIIC (cid:21)
is verified by the signs of these quantities (see the neg-
(cid:20)(cid:17)(cid:24)
ative signs in Fig. 2(a1),(b1) and the positive signs in
Fig.2(a2),(b2)). Asweshowlaterthataslongasµ =ε (cid:20)
0 0
with either Eq. (32) or Eq. (33) being satisfied, one can
(cid:19)(cid:17)(cid:24)
infer Iem.(t,φ) fromI[i](t,φ)’s (see detaileddiscussionin
α α
Sec.IVD).ThereforeinSec.IVB(wherethediscussions (cid:19)
are all for the case Eq. (32)) only the I[i](t,φ)’s are dis- (cid:16)(cid:21) (cid:16)(cid:20) (cid:19) (cid:20) (cid:21)
α
cussed.
FIG. 2: (color online) Various transient components of the
currents with up-down symmetry, Γ¯L = Γ¯L = 0.1Γ and
11 22
Γ¯R = Γ¯R = 0.9Γ at degeneracy δε = 0. In plots
2. non-degenerate QDs 11 22
(a1),(a2),(b1) and (b2) different line styles are for different
fluxesasindicatedontheright. Inplot(c)weshowthedecay
When the on-site energies of the QDs are not degen- rates as a function of flux under the bonds and on-site ener-
erate, δε 6= 0, then the odd components of the currents gies specified above. We have set µL =µR =µ0 =ε0 in this
appear when sinφ 6= 0, according to Eqs. (29),(30), and figureandaswellasinFig.6andFig.7. Thetemperatureis
(31). The time dependence of the currents relies on the takentobekBT =0.1Γ forall figures, suchthatthethermal
time dependence of the Green function Gr(t,t ). The effects dueto finitetemperature become unimportant.
0
latter, given by Eq. (A1), has its time dependence gov-
erned by the parameter Γ (φ) defined in Eq. (A7). The
g
condition of Eq. (32) makes Γ (φ) = |Γ (φ)|2−δε2
g 12
become either real, if |Γ (φ)|2 > δε2, or purely imag-
12
p
inary if |Γ (φ)|2 < δε2. Since the odd components
12
satisfy Eq. (36), to focus on the left-right asymmetry
we first discuss these two cases of |Γ (φ)|2 > δε2 and
12
|Γ (φ)|2 <δε2 separatelyfortheevencomponents. The Opening the energy splitting between the QDs intro-
12
effects of non-degeneracy on the odd components of the duces another energy scale. This suppresses the sole
currents are discussed later. In all the discussions below dominance of the bonds in the rates of approaching
for δε6=0,associatedwith Figs.3, 4, and5, the equality steady states. Consequently the difference between the
I[1],±(t,φ) = I[2],±(t,φ) under Eq. (32) is also numeri- times required by I[1](t,φ) and I[1](t,φ) to reach the
α α L R
cally verified. steady-state limits displayed in Fig. 3(a) and (b) is
not as obvious as those shown by Fig. 2(a1) and (b1).
a. Even components for δε6=0 with |Γ (φ)|2 >δε2 The same set of left-right asymmetric bonds is used in
12
both Figs. 2 and 3. The even component of the initial-
We first demonstrate the even components of the occupation-induced current with |Γ (φ)|2 > δε2 6= 0
12
transient currents in Fig. 3 with |Γ (φ)|2 > δε2 6= 0. now becomes (using the same derivation of Eq. (37)
12
10
without applying Eq. (33)):
(cid:62)(cid:20)(cid:64)(cid:15)(cid:14) (cid:62)(cid:21)(cid:64)(cid:15)(cid:14) (cid:68) (cid:62)(cid:20)(cid:64)(cid:15)(cid:14) (cid:62)(cid:21)(cid:64)(cid:15)(cid:14) (cid:69)
I (t, )=I (t, ) I (t, )=I (t, )
I[1],+(t,φ)=−Γα× (cid:19) (cid:47) (cid:47) (cid:19) (cid:53) (cid:53) (cid:32)(cid:19)
α
1 (Γα+Γα¯cosφ) δε 2 (cid:16)(cid:19)(cid:17)(cid:19)(cid:21) (cid:16)(cid:19)(cid:17)(cid:21)
1− + e−(Γ−δΓ(φ))t (cid:32) (cid:18)(cid:21)
2 δΓ(φ) δΓ(φ) (cid:16)(cid:19)(cid:17)(cid:19)(cid:23)
( (cid:18) (cid:19) ! (cid:16)(cid:19)(cid:17)(cid:23)
(cid:16)(cid:19)(cid:17)(cid:19)(cid:25)
+1 1+ (Γα+Γα¯cosφ) + δε 2 e−(Γ+δΓ(φ))t (cid:16)(cid:19)(cid:17)(cid:25) (cid:32)
(cid:16)(cid:19)(cid:17)(cid:19)(cid:27)
2 δΓ(φ) δΓ(φ)
(cid:18) (cid:19) ! (cid:16)(cid:19)(cid:17)(cid:27)
(cid:16)(cid:19)(cid:17)(cid:20) t (cid:32)(cid:22) (cid:18)(cid:21)
2
δε (cid:19) (cid:23) (cid:27) (cid:20)(cid:21) (cid:20)(cid:25) (cid:21)(cid:19) (cid:19) (cid:23) (cid:27) (cid:20)(cid:21) (cid:20)(cid:25) (cid:21)(cid:19)
− e−Γt , (38)
δΓ(φ)
)
(cid:18) (cid:19)
(cid:70)
(cid:21)
where δΓ(φ)= |Γ12(φ)|2−δε2 >0 (see Eqs.(A6) and (cid:20)(cid:17)(cid:25)
(A7)). In Eq. (38) in addition to the two decay terms,
e−(Γ±δΓ(φ))t, anpother exponential decay term e−Γt ap- (cid:20)(cid:17)(cid:21)
pears. This latter term has its amplitude determined by (cid:19)(cid:17)(cid:27)
the non-degeneracyof the QDs’ on-site energies. Setting
(cid:19)(cid:17)(cid:23)
δε = 0 in Eq. (38) reduces it to Eq. (37). In the degen-
(cid:19)
erate case we have learned that the choice of Γ ≫ Γ
R L
makesthe amplitude in I[1](t,φ) for the slowdecayterm (cid:16)(cid:21) (cid:16)(cid:20) (cid:19) (cid:20) (cid:21)
R
e−(Γ−δΓ(φ))t approach zero. However, when δε is finite,
FIG. 3: (color online) The even components of the initial-
all the amplitudes in Eq. (38) are not negligible. This
occupation-induced currents with up-down symmetry, Γ¯L =
explains why the left-right asymmetry in time scales for 11
Γ¯L =0.1Γ and Γ¯R =Γ¯R =0.9Γ are shown in plots (a) and
reachingthe steady-state limits shownin Fig.3 is not as 22 11 22
(b). Here we set δε = −0.5Γ such that |Γ (φ)|2 > δε2 and
12
drasticasthatshowninFig.2. Theleft-rightasymmetry
thecorresponding decay rates are plotted in (c).
inthemagnitudesofthecurrentsforδε6=0isstillclearly
seen (compare the magnitudes of Fig. 3(a) with those in
Fig. 3(b) respectively). This, again, reflects the overall condition |Γ (φ)|2 <δε2)
12
factor Γα in Eq. (38). The more visible flux dependence
in Fig. 3(a), in contrast to the less visible flux depen- I[1],+(t,φ)
α
dence in Fig. 3(b), can be explained by the same reason
2
δε
applied to Fig. 2 (see Appendix B1a). Letting δε 6= 0 =−Γαe−Γt 1− cos(ε (φ)t)
g
alsoraisesthe rateoftheslowdecaytermbyvaryingthe (" (cid:18)εg(φ)(cid:19) #
flux (compare the line for Γ−δΓ(φ) in Fig. 3(c) to that
Γα+Γα¯cosφ δε 2
inFig.2(c)). Thetimerequiredtoreachthesteady-state − sin(ε (φ)t)+ , (39)
g
limitisthusreduced(comparethetimescalesinFig.3(a) εg(φ) (cid:18)εg(φ)(cid:19) )
to that in Fig. 2(a1)).
b. Even components for δε6=0 with |Γ (φ)|2 <δε2 where εg(φ) = δε2−|Γ12(φ)|2 > 0. The result of
12
Eq. (39) is very different from Eq. (38). In addition
p
to the exponential decay factor e−Γt, time-oscillatory
The results of further increasing δε such that
|Γ (φ)|2 < δε2 are demonstrated in Fig. 4, still terms, cos(εg(φ)t) and sin(εg(φ)t), also appear. The
12
withΓR ≫ΓL. Similarto Fig.3,increasingδεdecreases step-like feature in the time evolution is a result of
the combined effect of the exponential decay and the
the asymmetry between the time scales while the asym-
oscillatory terms. The asymmetry between the left and
metryinthemagnitudesoftheleftandtherightcurrents
are unaffected. For |Γ (φ)|2 > δε2, Figs. 2 and 3 show the right currents in showing up the step-like feature in
12 different fluxes is explained by applying ΓR ≫ ΓL to
only smooth time evolutions of I[1],+(t,φ), described
α Eq. (39) (see Appendix B1b).
by exponential decays. However, for |Γ (φ)|2 < δε2,
12
a step-like feature in the time evolutions of IL[1],+(t,φ) c. The odd components of the initial-occupation-
is observed in Fig. 4(a) for φ = 0 (see the red solid induced currents
line) and for φ = π (see the green dashed-dotted
line). Asymmetrically for I[1],+(t,φ) in Fig. 4(b), this The odd components of the transient currents un-
R
feature exists for all the tested fluxes. Indeed, when der the same set of parameters used in Fig. 3 are
|Γ (φ)|2 < δε2 is obeyed, the even components of the displayed in Fig. 5(a) for |Γ (φ)|2 > δε2. The pa-
12 12
initial-occupation-induced current under Eq. (32) reads rameters discussed in Fig. 4 are shown in Fig. 5(b)
(using the same approach of deriving Eq. (38) with the for |Γ (φ)|2 < δε2. The numerical calculations in
12