Dynamically Broken Symmetry in Periodically Gated Quantum Dots: Charge Accumulation and DC-current
Time-dependent electron transport through a quantum dot and double quantum dot systems in the presence of polychromatic external periodic quantum dot energy-level modulations is studied within the time evolution operator method for a tight-binding Hamiltonian. Analytical relations for the dc-current...
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irk-123456789-132892010-11-05T12:02:19Z Dynamically Broken Symmetry in Periodically Gated Quantum Dots: Charge Accumulation and DC-current Kwapiński, T. Kohler, S. Hänggi, P. Наносистеми Time-dependent electron transport through a quantum dot and double quantum dot systems in the presence of polychromatic external periodic quantum dot energy-level modulations is studied within the time evolution operator method for a tight-binding Hamiltonian. Analytical relations for the dc-current flowing through the system and the charge accumulated on a quantum dot are obtained for the zero-temperature limit. It is shown that, in the presence of periodic perturbations, the sideband peaks of the transmission are related to the combination of frequencies of the applied modulations. For a double quantum dot system under the influence of polychromatic perturbations, the quantum pump effect is studied in the absence of a source (drain) and static bias voltages. In the presence of a spatial symmetry, the charge is pumped through the system due to a broken generalized parity symmetry. Дослiджено залежний вiд часу транспорт електронiв через квантову точку i систему двох квантових точок при зовнiшнiй полiхромнiй перiодичнiй модуляцiї рiвнiв енергiї в межах методу оператора часової еволюцiї з гамiльтонiаном сильного зв’язку. Одержано аналiтичнi формули для постiйного струму через систему та для заряду, який накопичено на квантовiй точцi у границi нульової температури. Показано, що в присутностi перiодичних збурень боковi максимуми передачi залежать вiд спiввiдношення зовнiшнiх модуляцiй. Вивчено ефект квантової накачки за вiдсутностi джерела (стока) i статичних напружень змiщення. У випадку просторової симетрiї заряд накачується через систему внаслiдок порушення симетрiї узагальненої парностi. 2010 Article Dynamically Broken Symmetry in Periodically Gated Quantum Dots: Charge Accumulation and DC-current / T. Kwapiński, S. Kohler, P. Hänggi // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 85-94. — Бібліогр.: 34 назв. — англ. 2071-0194 PACS 05.69.Gg, 73.23.-b, 73.63.Nm http://dspace.nbuv.gov.ua/handle/123456789/13289 en Відділення фізики і астрономії НАН України |
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Наносистеми Наносистеми |
| spellingShingle |
Наносистеми Наносистеми Kwapiński, T. Kohler, S. Hänggi, P. Dynamically Broken Symmetry in Periodically Gated Quantum Dots: Charge Accumulation and DC-current |
| description |
Time-dependent electron transport through a quantum dot and double quantum dot systems in the presence of polychromatic external periodic quantum dot energy-level modulations is studied within the time evolution operator method for a tight-binding Hamiltonian. Analytical relations for the dc-current flowing through the system and the charge accumulated on a quantum dot are obtained for the zero-temperature limit. It is shown that, in the presence of periodic perturbations, the sideband peaks of the transmission are related to the combination of frequencies of the applied modulations. For a double quantum dot system under the influence of polychromatic perturbations, the quantum pump effect is studied in the absence of a source (drain) and static bias voltages. In the presence of a spatial symmetry, the charge is pumped through the system due to a broken generalized parity symmetry. |
| format |
Article |
| author |
Kwapiński, T. Kohler, S. Hänggi, P. |
| author_facet |
Kwapiński, T. Kohler, S. Hänggi, P. |
| author_sort |
Kwapiński, T. |
| title |
Dynamically Broken Symmetry in Periodically Gated Quantum Dots: Charge Accumulation and DC-current |
| title_short |
Dynamically Broken Symmetry in Periodically Gated Quantum Dots: Charge Accumulation and DC-current |
| title_full |
Dynamically Broken Symmetry in Periodically Gated Quantum Dots: Charge Accumulation and DC-current |
| title_fullStr |
Dynamically Broken Symmetry in Periodically Gated Quantum Dots: Charge Accumulation and DC-current |
| title_full_unstemmed |
Dynamically Broken Symmetry in Periodically Gated Quantum Dots: Charge Accumulation and DC-current |
| title_sort |
dynamically broken symmetry in periodically gated quantum dots: charge accumulation and dc-current |
| publisher |
Відділення фізики і астрономії НАН України |
| publishDate |
2010 |
| topic_facet |
Наносистеми |
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http://dspace.nbuv.gov.ua/handle/123456789/13289 |
| citation_txt |
Dynamically Broken Symmetry in Periodically Gated Quantum Dots: Charge Accumulation and DC-current / T. Kwapiński, S. Kohler, P. Hänggi // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 85-94. — Бібліогр.: 34 назв. — англ. |
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AT kwapinskit dynamicallybrokensymmetryinperiodicallygatedquantumdotschargeaccumulationanddccurrent AT kohlers dynamicallybrokensymmetryinperiodicallygatedquantumdotschargeaccumulationanddccurrent AT hanggip dynamicallybrokensymmetryinperiodicallygatedquantumdotschargeaccumulationanddccurrent |
| first_indexed |
2025-07-02T15:13:39Z |
| last_indexed |
2025-07-02T15:13:39Z |
| _version_ |
1836548589242810368 |
| fulltext |
NANOSYSTEMS
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 85
DYNAMICALLY BROKEN SYMMETRY IN PERIODICALLY
GATED QUANTUM DOTS: CHARGE ACCUMULATION
AND DC-CURRENT
T. KWAPIŃSKI,1, 2 S. KOHLER,1, 3 P. HÄNGGI1
1Institute of Physics, University of Augsburg
(Univeristatsstr. 1, D-86135 Augsburg, Germany)
2Institute of Physics, Maria Curie-Sk lodowska University
(Pl. M. Curie-Sk lodowskiej 1, 20-031 Lublin, Poland)
3Instituto de Ciencia de Materiales de Madrid, CSIC
(C/Sor Inés Juana de la Cruz 3, Cantoblanco, 28049 Madrid, Spain)
PACS 05.69.Gg, 73.23.-b,
73.63.Nm
c©2010
Time-dependent electron transport through a quantum dot and
double quantum dot systems in the presence of polychromatic ex-
ternal periodic quantum dot energy-level modulations is studied
within the time evolution operator method for a tight-binding
Hamiltonian.Analytical relations for the dc-current flowing through
the system and the charge accumulated on a quantum dot are
obtained for the zero-temperature limit. It is shown that, in the
presence of periodic perturbations, the sideband peaks of the trans-
mission are related to the combination of frequencies of the applied
modulations. For a double quantum dot system under the influ-
ence of polychromatic perturbations, the quantum pump effect is
studied in the absence of a source (drain) and static bias volt-
ages. In the presence of a spatial symmetry, the charge is pumped
through the system due to a broken generalized parity symmetry.
1. Introduction
Recently, considerable progress has been achieved in fab-
ricating low dimensional systems, and many experimen-
tal and theoretical works have been put forward. Es-
pecially interesting are quantum systems under the in-
fluence of external radio or microwave electromagnetic
radiation perturbations, where many interesting effects
are observed like photon-assisted tunneling (PAT) [1, 2],
turnstile effects and photon-electron quantum pumps [3–
5], conductance oscillations [6], and alike [7].
The symmetry of quantum dot (QD) systems (with
no source-drain voltage) plays the crucial role, as con-
cerns electron pumping effects. Generally, one can con-
sider symmetries like the time-reversal symmetry, time-
reversal parity, and generalized parity [7].
A single electron pump based on asymmetric couplings
between a QD and the left and right electrodes was con-
sidered in [4]. The couplings were switched on and off
alternatively from zero to maximal values (by means of
additional electrodes), and this led to the electron pump-
ing. A similar effect can be achieved for dipole driv-
ing forces applied to a double QD system (in the large
gate voltage regime) or to quantum wires. In this case,
one QD site is driven by the external dipole interaction
which is out of phase in comparison with the perturba-
tion applied to the second QD site (the QD sites are
not driven in homogeneous way), e.g. [7–12]. However,
in the presence of a spatial symmetry [and in absence
of a source (drain) and static gate voltages], it is also
possible to pump electrons, but the symmetry must be
broken in a dynamical way. The easiest way to break
the time-reversal symmetry is to add a second harmonic
to the driving system; i.e. the so-called “harmonic mix-
ing” drive [7, 13, 14], or, in general, the second external
perturbation with an arbitrary frequency [15]. In such
a case, depending on the parameters of these two time-
dependent perturbations, the generalized parity can be
broken and a nonvanishing current can flow through the
system [7, 13, 16].
There are few studies which address the electron trans-
port through low-dimensional systems in the presence of
several polychromatic external perturbations with arbi-
T. KWAPIŃSKI, S. KOHLER, P. HÄNGGI
trary frequencies. Due to numerical problems, most of
them concentrate on the case of commensurate frequen-
cies or only bi-harmonic perturbations. It was shown
that the external bi-harmonic time-dependent perturba-
tions can be used to control the noise level in quantum
systems [9, 17] or, as well, for routing optically induced
currents [18, 19]. The shot noise for a single-level quan-
tum dot under the influence of two ac external perturba-
tions (coherent or incoherent) was analyzed in [15]. The
coherent destruction of tunneling [20] and the associated
dynamical localization in quantum dots under the influ-
ence of a time-dependent perturbation with many har-
monics were investigated in [21]. Moreover, the dissipa-
tive quantum transport in one- or two-dimensional peri-
odic systems that are subjected to electric harmonic mix-
ing perturbations (bi-harmonic) were studied in [22–24].
The nonlinear signal consisted of, e.g., two rectangular-
like driving forces which allow one, in turn, to control
the overdamped transport in Brownian motor devices
[16, 25–27].
In this paper, we will investigate the influence of poly-
chromatic time-dependent energetic perturbations with
arbitrary (commensurate and incommensurate) frequen-
cies applied to a QD or a double QD system attached to
leads for charge accumulated on the QD and the time-
averaged dc-current flowing through the device. For a
double QD system, we propose a quantum pump which
is based on a scheme which mimics closely a dipole-
like perturbation. Thus, our work can be treated as
a generalization of the studies of the electron trans-
port through a QD or double QD systems affected by
one external perturbation or bi-harmonic electric time-
dependent ac-perturbations with arbitrary frequencies.
A tight-binding Newns–Anderson Hamiltonian and the
evolution operator method are used in our calculations.
The paper is organized as follows. In Sec. 2, the model
Hamiltonian and the theoretical description of a single-
level quantum dot are presented. The analytic relations
for a time-averaged current and a time-averaged charge
on the QD are obtained, and the numerical results are
depicted and interpreted. In Sec. 3, the current through
a double QD system is obtained, and the pumping effect
is discussed. The last section, Sec. 4, presents conclu-
sions.
2. Single-Level Quantum Dot
2.1. Theoretical description
In this section, by starting from the second quantization
Hamiltonian and using the evolution operator method,
we obtain the charge accumulated on a QD and the cur-
rent flowing through the system under the influence of
many external time-dependent perturbations. Our sys-
tem consists of a single-level quantum dot and two con-
necting electron electrodes, left (L) and right (R). The
total Hamiltonian is then given by H = H0 + V , where
H0 =
∑
kα=L,R
εkαc
+
kαckα + εd(t)c+d cd , (1)
V =
∑
kL
VkLc
+
kLcd +
∑
kR
VkRc
+
kRcd + h.c. (2)
The operators ckα(c+kα) and cd(c+d ) are the annihila-
tion (creation) operators of the electron in the lead α
(α = L,R) and at the QD, respectively. The QD is cou-
pled symmetrically to the leads through the tunneling
barriers with the transfer-matrix elements VkL and VkR
(hopping integrals). For the role of the asymmetric lead-
“molecule” coupling, see in [28,29]. The electron-electron
Coulomb interaction is neglected in our calculation, cf.
[7, 15, 30].
External perturbations are applied to the QD (the
QD energy level is driven in time by time-dependent ac-
voltages). We consider a harmonic modulation of the
external energy level perturbations applied to the QD,
i.e.
εd(t) = εd +
n∑
i=1
Δi cos(ωit+ φi), (3)
where ωi, Δi, and φi are the frequency, driving ampli-
tude, and phase of the i-th perturbation.
The current flowing through the system and the charge
localized at the QD can be described in terms of the
time evolution operator U(t, t0) given by the equation of
motion (in the interaction representation, ~ = 1), i.e.,
i
∂
∂t
U(t, t0) = Ṽ (t)U(t, t0) , (4)
where Ṽ (t) = U0(t, t0)V (t)U+
0 (t, t0) and U0(t, t0) =
T exp
(
i
∫
t
t0dt
′H0(t′)
)
. The knowledge of the appropri-
ate matrix elements of the evolution operator U(t, t0)
allows us to find the charge accumulated on the QD,
nd(t) (cf. [6, 31, 32]), which is given by
nd(t) =
∑
β
nβ(t0)|Ud,β(t, t0)|2, (5)
where nβ(t0) represents the initial filling of the corre-
sponding single-particle states (β = d,kL,kR). The
86 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
DYNAMICALLY BROKEN SYMMETRY
current flowing, e.g., from the left lead can be obtained
from the time derivative of the total number of electrons
in the left lead, cf. [33]:
jL(t) = −ednL(t)/dt , (6)
where nL(t) can be expressed as follows:
nL(t) =
∑
kL
nkL(t) =
∑
kL
∑
β
nβ(t0)|UkL,β(t, t0)|2. (7)
Using Eq. 4, the following differential equations for
Ud,β(t, t0) and UkL,β(t, t0) matrix elements which are
needed to obtain the current and the QD charge can
be written in the form
i
∂
∂t
Ud,β(t, t0) =
∑
kα=L,R
Ṽd,α(t)Uα,d(t, t0), (8)
i
∂
∂t
UkL,β(t, t0) = ṼkL,d(t)Ud,β(t, t0), (9)
where nonzero elements of the function Ṽ are
Ṽd,kα = Ṽ ∗kα,d = V ∗kα exp
i t∫
t0
(εd(t′)− εkα)dt′
. (10)
Next, using the wide band limit approximation
Γα(ε) = 2π
∑
kα VkαV
∗
kαδ(ε − εkα) = Γα and assuming
ΓL = ΓR = Γ, we can find the following relation for the
Ud,d(t) matrix element (t0 = 0): Ud,d(t) = exp (−Γt).
Similarly for Ud,kα(t) and UkL,β(t), we have
Ud,kα(t) = −i exp (−Γt)
t∫
0
dt′Ṽd,kα(t′) exp (Γt′), (11)
UkL,β(t) = −i
t∫
0
dt′ṼkL,d(t′)Ud,β(t′). (12)
Note that, as t → ∞, the element Ud,d(t) tends to
zero, and thus the charge accumulated on the QD does
not depend on the initial QD occupation nβ(t0) for
large t. Moreover, the current flowing through the
system is independent of nd(t0), because the element
d
dt
∑
kL nd(t0)|UkL,d(t)|2 tends to zero as t → ∞. Fi-
nally, the QD charge can be written in the form
nd(t) =
∑
kα=L,R
nkα(t0)|Ud,kα(t)|2, (13)
and the current through the system reads (e = 1)
jL(t) = Γnd(t) + Im
(∑
kL
nkL(t0)ṼkL,d(t)Ud,kL(t)
)
.
(14)
Equations 13 and 14 are very general relations which
should be analyzed using Eq. 10 and Eq. 11. The relation
for the current, Eq. 14, has the structure which can be
written by means of the transmission TLR and TRL, i.e.
jL(t) =
∑
kL nkL(0)TLR(t)−
∑
kR nkR(0)TRL(t), but we
note that, in general, TLR(t) 6= TRL(t). In order to
obtain the current, one should know the exact form of
the Ṽd,kα function. Using the time-dependence relation
for the QD-energy level, Eq. 3, and assuming φi = 0, the
elements Ṽd,kα, Eq. 10, can be expressed as
Ṽd,kα = V ∗kαe
i(εd−εkα)t
n∏
i=1
∑
mi
Jmi
(
Δi
ωi
)
eimiωit, (15)
and the solution of Eq. 11 for the evolution operator
elements can be written in the form
Ud,kα(t) = −V ∗kαei(εd−εkα)t×
×
∑
m1
. . .
∑
mn
Jm1
(
Δ1
ω1
)
. . . Jmn
(
Δn
ωn
)
eiΩt
εd − εkα + Ω− iΓ
, (16)
where mi ∈ (−∞,∞) and is an integer number, Ω =
m1ω1 + . . . + mnωn, and Ji is the Bessel function.
Next, we obtain the dc current through the system,
j0 = 〈j(t)〉 = limT→∞
1
T
∫ T/2
−T/2 j(t
′)dt′ which can be
symmetrized, j0 = 〈jL(t)〉 = (〈jL(t)〉 − 〈jR(t)〉) /2. Fi-
nally, we find
j0 =
Γ
2π
∞∫
−∞
(fR(ε)− fL(ε))T (ε), (17)
where fL/R(ε) is the Fermi function of the L/R elec-
trode, and the transmission reads
T (ε) = Γ2
∑
m1
. . .
∑
mn
∑
m
′
1
. . .
∑
m′
n
δΩ−Ω′×
×
Jm1
(
Δ1
ω1
)
. . . Jmn
(
Δn
ωn
)
Jm′
1
(
Δ1
ω1
)
. . . Jm′
n
(
Δn
ωn
)
(εd − ε+ Ω)2 + Γ2
,
(18)
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 87
T. KWAPIŃSKI, S. KOHLER, P. HÄNGGI
where Ω
′
= m
′
1ω1+...+m
′
nωn. Note that, for incommen-
surate frequencies, the Kronecker delta function δΩ−Ω′
is nonzero only for m
′
i = mi. Thus, the transmission can
be written in the following short form:
T (ε) = Γ2
∑
m1
. . .
∑
mn
J2
m1
(
Δ1
ω1
)
. . . J2
mn
(
Δn
ωn
)
(εd − ε+ Ω)2 + Γ2
. (19)
This equation is valid also in the case where there is
a large difference between the frequencies ωi. Note that
the transmission, Eqs. (18) and (19), corroborates with
the results obtained by means of the Green’s function
method for a quantum wire driven by homogeneous ex-
ternal perturbations [30].
The relation for the current, Eq. 17, has the struc-
ture of the Landauer formula. We note that the struc-
ture in Eq. (17) involves, in the clear contrast to a elec-
tric field dipole perturbation [7], no inelastic photon-
assisted tunneling events. This is so because, with the
time-dependent energy level perturbation used here, the
long time-average of TLR(t) and TRL(t) equals TLR(ε) =
TRL(ε). For the zero temperature and for incommensu-
rate frequencies of external perturbations, the dc-current
can be obtained analytically from the relation
j0 = Γ
∑
m1
...
∑
mn
J2
m1
(
Δ1
ω1
)
...J2
mn
(
Δn
ωn
)
×
×
(
arctan
εd − µL − Ω
Γ
− arctan
εd − µR − Ω
Γ
)
. (20)
Note that, for µL = µR, the dc current is zero.
The corresponding analytic relation for the accumu-
lated quantum dot dc-charge, i.e. n0 = 〈nd(t)〉 =
limT→∞
1
T
∫ T/2
−T/2 nd(t
′)dt′, reads
n0 =
1
2π
∑
m1
. . .
∑
mn
J2
m1
(
Δ1
ω1
)
. . . J2
mn
(
Δn
ωn
)
×
×
(
π − arctan
εd − µL − Ω
Γ
− arctan
εd − µR − Ω
Γ
)
.
(21)
It is worth noting that, for εd � µ (εd � µ), the charge
accumulated on the QD is maximal (minimal). The rela-
tions for the dc current, Eq. (20), and for the QD charge,
Eq. (21), constitute the main analytic relations of this
section.
2.2. QD accumulated charge and dc current
In this section, we analyze the QD charge and the dc
current flowing through a quantum dot driven by poly-
chromatic perturbations. All energies are expressed in
the units of Γ0. In order to obtain rather narrow side-
bands peaks, we assume Γ = 0.2Γ0 (taking the unit of
energy Γ0 = 0.05eV , this corresponds to Γ = 0.01 eV).
For larger Γ, all sideband peaks are wider, and it is dif-
ficult to observe many-perturbation effects. The cur-
rent and the conductance are given in units of 2eΓ0/~
and 2e2/~, respectively. Moreover, we show numerical
calculations for two external perturbations case but the
generalization for more perturbations is obvious.
In Fig. 1, the QD charge (upper panel) and the dc cur-
rent flowing through the system (lower panel) are shown
for two external perturbations applied to the system
(ω1 = 3, ω2 = 8, Δ1 = 4, Δ2 = 8) – thick lines. The fre-
quencies are commensurate but there is the large differ-
ence between them, and the transmissions obtained from
Eq. (18) and Eq. (19) are almost the same. Physically,
this means that the eight-photon adsorption/emission
process based on the third sideband peak should occur to
play the role in the transmission, (8ω1 = 3ω2), but this
is unlikely process. The broken lines correspond to the
single external perturbation case, i.e. Δ2 = 8 (Δ1 = 0)
– thin broken lines and Δ1 = 4 (Δ2 = 0) – thick broken
lines, respectively. The chemical potentials are rather
small, µL = −µR = 0.1, and the current peak for εd = 0
(lower panel) is observed for mono- and polychromatic
cases (it appears also for the time-independent case).
For only one external perturbation applied to the QD,
ω = 3 (or ω = 8), the sidebands peaks are visible for
εd = ±kω, where k is an integer number. However,
in the case where two external perturbations are applied
simultaneously to the QD, additional sideband peaks ap-
pear. Of course, the single peaks from the first and the
second fields are still visible, i.e. for εd = ω1, 2ω1 or ω2.
In Fig. 1, the additional dc current peaks are indicated
by points A1, A2 (first-order sidebands), and B1 (second-
order sideband). TheA1 (A2, B1) sideband peak appears
for εd = ω2 − ω1 (εd = ω2 + ω1, εd = ω2 − 2ω1) and is
related to the peak for εd = ω2 = 8 (but not to the main
peak for εd = 0). It is worth noting that, although the
frequencies ω1 and ω2 are not equal to 5 or 11, we observe
the sideband peaks for these values of εd. In general, we
observe sideband peaks for εd = ±k1ω1±k2ω2, where k1
and k2 are integer numbers. The structure of the dc cur-
rent curves are reflected also in the charge accumulated
on the QD (upper panel). The charge decreases with
εd, but there are many steps which are related to the
88 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
DYNAMICALLY BROKEN SYMMETRY
0
0.2
0.4
0.6
0.8
n 0
0
0.2
0.4
0.6
0.8
n 0
0
0.2
0.4
0.6
0.8
n 0
-0.2
-0.1
0
0.1
0.2
-2 0 2 4 6 8 10 12
j 0
εd
ω1
ω2
2ω1
A1 A2
B1
-0.2
-0.1
0
0.1
0.2
-2 0 2 4 6 8 10 12
j 0
εd
ω1
ω2
2ω1
A1 A2
B1
-0.2
-0.1
0
0.1
0.2
-2 0 2 4 6 8 10 12
j 0
εd
ω1
ω2
2ω1
A1 A2
B1
Fig. 1. QD charge (upper panel) and the dc current (lower panel)
as a function of εd for Δ1 = 4 and Δ2 = 8 (thick lines), Δ1 = 4,
Δ2 = 0 (thick broken lines) and Δ1 = 0 and Δ2 = 8 (thin broken
lines), respectively. The other parameters are ω1 = 3, ω2 = 8,
µL = −µR = 0.1, Γ = 0.2. The thick (thin) broken line on the
lower panel is shifted by −0.1 (−0.2) for better visualization
current sideband peaks. The additional sidebands, i.e.
points A1, A2, and B1 from the lower panel, are visible
on the charge curve (Fig. 1, upper panel, thick line).
Next, in Fig. 2, we present the dc current as a func-
tion of the driving strengths (amplitudes) of two external
perturbations applied to the system. For the second sig-
nal, the driving strength decreases with the amplitude
of the first perturbation, i.e. Δ2 = 10 −Δ1. The thick
(thin) solid line corresponds to εd = 5 (εd = 10). As
one can see for εd = 10, the dc current is very small and
almost independent of the amplitudes Δ1 and Δ2. This
conclusion is valid for every εd which is not any combi-
nation of ω1 and ω2. For εd = 5 (εd = ω2 − ω1), the
current is minimal for Δ1 = 0 and Δ1 = 10, but, for
the mixed regime of Δ1 and Δ2, it is characterized by a
local maximum, cf. the thick solid line. This is a very
interesting effect, because one can control the current
flowing through the system by applying an additional
time-dependent perturbation. Note that the maximal
value of the current is several times larger than that in
the case of one external perturbation, i.e. for Δ1 = 0 or
0
0.02
0.04
0 2 4 6 8 10
j 0
∆1, ∆2=10-∆1
εd=5, ∆2=0
εd=5, ∆1=0
εd=5
εd=10
0
0.02
0.04
0 2 4 6 8 10
j 0
∆1, ∆2=10-∆1
εd=5, ∆2=0
εd=5, ∆1=0
εd=5
εd=10
0
0.02
0.04
0 2 4 6 8 10
j 0
∆1, ∆2=10-∆1
εd=5, ∆2=0
εd=5, ∆1=0
εd=5
εd=10
0
0.02
0.04
0 2 4 6 8 10
j 0
∆1, ∆2=10-∆1
εd=5, ∆2=0
εd=5, ∆1=0
εd=5
εd=10
Fig. 2. Dc-current as a function of the amplitude Δ1 (Δ2 = 10−
Δ1) for εd = 5 (thick line) and εd = 10 (thin line), ω1 = 3,
ω2 = 8. The thin (thick) broken line corresponds to one external
perturbation case Δ1, ω = 3 with Δ2 = 0 (Δ2, ω = 8 with Δ1 = 0)
and εd = 5; µL = −µR = 0.1, Γ = 0.2
Δ1 = 10. For only one external perturbation applied to
the system, the dc current is shown by the broken lines –
the thin line for Δ2 = 0 as a function of Δ1 (ω = 3) and
the thick one for Δ1 = 0 as a function of Δ2 (ω = 8),
εd = 5. One can conclude that one external pertur-
bation slightly changes the current in this case. This
confirms that the current maximum, which appears for
εd = 5 (thick solid curve), is due to a combination of
two external perturbations applied to the system. The
similar conclusions are valid for other εd = ±k1ω1±k2ω2
(k1,2 6= 0).
To analyze the structure of sidebands peaks for poly-
chromatic perturbations, we show the dc current in
Fig. 3 as a function of the QD energy level, εd, and
the external perturbation frequency ω2 for Δ1 = 0 (only
one perturbation applied to the system with Δ2 = 8)
– upper panel, and for two external perturbations with
Δ1 = 3, ω1 = 3, Δ2 = 8 – lower panel. For one, as
well as for two time-dependent perturbations applied
to the system, we can distinguish between the adia-
batic regime for ω2 ≤ 1 and the non-adiabatic one for
ω2 > 1. For ω2 > 1, one can observe the main cur-
rent peak for εd = 0 and sidebands which are visible for
εd = ±kω2. The situation is more complex for two ex-
ternal perturbations applied simultaneously to the QD
(Δ1 = 3, ω1 = 3, Δ2 = 8). One observes that each
light line from the upper panel possesses satellite lines
which are localized at the distance ±kω1 from the orig-
inal lines. Thus, we observe a very reach structure of
the dc current in the presence of polychromatic pertur-
bations.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 89
T. KWAPIŃSKI, S. KOHLER, P. HÄNGGI
0
0.05
0.1
0.15
-12
-6
0
6
12
ε d
0
0.05
0.1
0.15
0 2 4 6 8 10
ω2
-12
-6
0
6
12
ε d
Fig. 3. Dc-current as a function of εd and ω2 for Δ1 = 0, Δ2 = 8
(one external perturbation – upper panel) and Δ1 = 3, Δ2 = 8
(two external perturbations – lower panel). The other parameters
are: ω1 = 3, µL = −µR = 0.1, Γ = 0.2
2.3. Results: Transmission asymmetry and
phase dependence
For one external perturbation applied to a QD, as well as
for two external perturbations with the same frequencies,
one observes fully symmetric current curves around the
value εd = 0. However, for two external time-dependent
perturbations, the dc current (or the transmission) can
be asymmetric, which appears in the case of commen-
surate frequencies. In Fig. 4, we show the dc current
obtained for the frequency ratios ω2/ω1 = 2, 1, and 0.5
using the transmission relations, Eq. (18), (solid lines)
and in the case where there is an infinitesimal shift of
the frequency ratio from an integer number, Eq. (19)
(broken lines); e.g. for ω2/ω1 = 2, we set ω1 = 3 and
ω2 = 5.999. Note that all broken curves in Fig. 4 are
symmetric. The solid lines, however, are asymmetric
around εd = 0 except for the case ω1 = ω2. Notably, for
ω1 = ω2, the system is equivalent to a single, effective
external perturbation case with the effective driving am-
plitude, Δ = Δ1 + Δ2, and thus the current curves turn
out symmetric again. The current obtained for incom-
0
0.05
0.1
0.15
-9 -6 -3 0 3 6 9
j 0
εd
ω2/ω1=2
ω2/ω1=1
ω2/ω1=0.5
Eq. 18
0
0.05
0.1
0.15
-9 -6 -3 0 3 6 9
j 0
εd
ω2/ω1=2
ω2/ω1=1
ω2/ω1=0.5
Eq. 18
Eq. 19
0
0.05
0.1
0.15
-9 -6 -3 0 3 6 9
j 0
εd
ω2/ω1=2
ω2/ω1=1
ω2/ω1=0.5
Eq. 18
Eq. 19
0
0.05
0.1
0.15
-9 -6 -3 0 3 6 9
j 0
εd
ω2/ω1=2
ω2/ω1=1
ω2/ω1=0.5
Eq. 18
Eq. 19
0
0.05
0.1
0.15
-9 -6 -3 0 3 6 9
j 0
εd
ω2/ω1=2
ω2/ω1=1
ω2/ω1=0.5
Eq. 18
Eq. 19
0
0.05
0.1
0.15
-9 -6 -3 0 3 6 9
j 0
εd
ω2/ω1=2
ω2/ω1=1
ω2/ω1=0.5
Eq. 18
Eq. 19
Fig. 4. Dc-current as a function of εd obtained for commensurate
frequency ratios ω2/ω1 = 2, 1, and 0.5 according to Eq. 17 with
the transmission given by Eq. 18 (solid lines) and for a slightly
off-commensurate frequency; i.e., ω1 = 3, ω2 = 5.999, see Eq. 19
(broken lines). All parameters are the same as in Fig. 3
mensurate frequencies differs from the ones in the case
of commensurate frequencies. It is worth noting that
the structure of the dc current curves for fixed ωi de-
pends on the amplitudes of external perturbations, but
the positions of peaks remain unchanged.
Next, it is necessary to explain the transmission and
current asymmetries for the case of commensurate fre-
quencies. In this case, to obtain the transmission one,
should use Eq. (18) which is expressed in terms of four
Bessel functions, an energy dependent factor, and the
Kronecker delta function. For commensurate frequen-
cies, this delta function produces off-diagonal elements of
the Bessel functions which are multiplied by the energy
factor. The energy factor depends on the order of the
Bessel functions and introduces an asymmetry for posi-
tive and negative values of εd (this means that the energy
factor possesses different values for positive and negative
integer orders of the Bessel functions). The off-diagonal
elements of the Bessel functions are also responsible for
different values of the dc current obtained for ω2/ω1 = 1,
90 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
DYNAMICALLY BROKEN SYMMETRY
according to Eq. (18) and Eq. (19). It is worth noting
that, in our calculation, we assume that the QD energy
level is driven according to the formula with the cosine
function (i.e. the cosine function is an odd function),
and the external time-dependent perturbation satisfies
the time-reversal symmetry, a(t) = a(−t).
In Fig. 5, we show the transmission for nonzero phase
factors of the external perturbations, i.e. φ1 and φ2,
cf. Eq. (3). The upper panel shows the results in the
case where both phases are changed in the same way, i.e.
φ1 = φ2. For φ1 = 0 (thin solid line), the asymmetry
in the transmission is observed, cf. also Fig. 4. If both
phases are equal to π/2 (broken line), the transmission is
symmetric (as a function of the energy) - in that case, the
external signals are sines. For φ1 = π, the transmission
is asymmetric again. The period of the external pertur-
bation for the case of the same phases, φ1 = φ2, is equal
to 2π. The situation is somewhat different, when the
phase of one external perturbation is constant (φ2 = 0 -
lower panel). Here, the transmission is symmetric only
for φ1 = π/4, but this function is asymmetric for φ1 = 0
and π/2. Note that the above conclusions are valid for
the case of commensurate frequencies. For incommensu-
rate frequencies, the transmission curves are symmetric,
cf. Fig. 4. Moreover, as one can see, the transmission
curves in the upper and lower panels are the same, only
the phases of external perturbations are different. This
means that the same effect can be obtained by changing
only one phase parameter instead of driving both phases
simultaneously.
3. Double Quantum Dot System and Pumping
Effect
The asymmetry effect in the transmission obtained in
the previous section can be used to construct a single
electron pump [12] based on a two-level fully symmet-
ric system with no source-drain and static bias voltages
applied to the system. In this section, we analyze the
electron transport through a double quantum dot in the
presence of external perturbations. The Hamiltonian of
our system can be written, in close analogy to Eq. (1)
and Eq. (2), as follows:
H0 =
∑
kα=L,R
εkαc
+
kαckα + ε1(t)c+1 c1 + ε2(t)c+2 c2 , (22)
V =
∑
kL
VkLc
+
kLc1 +
∑
kR
VkRc
+
kRc2 +V12c
+
1 c2 +h.c., (23)
where V12 is the tunnel coupling (hopping term) between
two QD sites. As before, we assume that there are only
0
0.1
0.2
0.3
T
a) φ1=φ2
φ1=0=2π
φ1=π/2
φ1=π
0
0.1
0.2
0.3
T
a) φ1=φ2
φ1=0=2π
φ1=π/2
φ1=π
0
0.1
0.2
0.3
T
a) φ1=φ2
φ1=0=2π
φ1=π/2
φ1=π
0
0.1
0.2
-6 -3 0 3 6
T
ε- εd
b) φ2=0
φ1=0=π
φ1=π/4
φ1=π/2
0
0.1
0.2
-6 -3 0 3 6
T
ε- εd
b) φ2=0
φ1=0=π
φ1=π/4
φ1=π/2
0
0.1
0.2
-6 -3 0 3 6
T
ε- εd
b) φ2=0
φ1=0=π
φ1=π/4
φ1=π/2
Fig. 5. Transmission versus the energy for two external perturba-
tions applied to the QD (ω1 = 3, ω2 = 6, Δ1 = 3, Δ2 = 8) with
phases: φ1 = φ2 = 0, π/2, π (upper panel) and φ1 = 0, π/4, π/2
and φ2 = 0 (lower panel)
two external perturbations applied to the system, and
one can write the following time dependence of the quan-
tum dot energy levels:
ε1(t) = ε1 + Δ1 cos(ω1t) + Δ2 cos(ω2t), (24)
ε2(t) = ε2 + Δ1 cos(ω1t+ φ) + Δ2 cos(ω2t+ φ), (25)
where φ is the phase difference between the external per-
turbations applied to the first and second QD sites which
play a similar role as dipole forces in the single external
harmonic field case, cf. [7, 9, 10]. We stress, however,
that here the driving is chosen uniform in the sense that
the amplitudes strengths for the driving of the two dots
are identical. Note that the role of the phase difference
between the first and second external perturbations was
considered in [13] for bi-harmonic perturbations. Here,
we concentrate on the phase difference between both QD
sites. The main reason we omit the phase difference be-
tween both external perturbations is that, in the pres-
ence of a spatial symmetry (ε1 = ε2, µL = µR; i.e., in
the absence of a static gate voltage and a source-drain
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 91
T. KWAPIŃSKI, S. KOHLER, P. HÄNGGI
voltage), it is impossible to pump electrons through the
system. By introducing the phase difference according
to Eq. (24) and Eq. (25), we change the symmetry at
the first and second QD sites, and thus electrons can be
pumped in the presence of external perturbations and
a spatial symmetry. The general formula for the time-
dependent current flowing from the left electrode can be
written as
jL(t)=
∑
kL
nkL(t0)
(
Γ|U1,kL(t)|2 + ImṼkL,1(t)U1,kL(t)
)
+
+
∑
kR
nkRL(t0)Γ|U1,kR(t)|2, (26)
where VkL,1(t) is defined by Eq. (15), and the evolution
operator matrix elements satisfy the following system of
differential equations (and similar for U1(2),kR):
∂
∂t
U1,kL(t) = −iV12e
i(ε1−ε2)tei(f1−f2)U2,kL(t)−
−iVkLe
i(ε1−εkL)teif1 − Γ
2
U1,kL(t), (27)
∂
∂t
U2,kL(t)=−iV12e
i(ε2−ε1)tei(f2−f1)U1,kL(t)−Γ
2
U2,kL(t),
(28)
where f1 = Δ1
ω1
sin(ω1t) + Δ2
ω2
sin(ω2t), f2 = Δ1
ω1
sin(ω1t+
φ) + Δ2
ω2
sin(ω2t + φ). Note that, for φ = 2πk and the
same on-site energies ε1 = ε2 = εd, one can obtain the dc
current and the transmission analytically. In this case,
the current satisfies the Landauer formula, and the time-
averaged left-right and right-left transmissions are equal.
This result is due to the above-mentioned chosen uni-
form driving strengths. For incommensurate frequencies
of the external perturbations, the formula for the trans-
mission simplifies and can be written as
T (ε) = Γ2
∑
m1
∑
m2
V 2
12×
×
J2
m1
(
Δ1
ω1
)
J2
m2
(
Δ2
ω2
)
(
(εd − ε+ Ω)2 − Γ
2
2 − V 2
12
)2
+ Γ2(εd − ε+ Ω)2
, (29)
where Ω = ω1m1 + ω2m2. In the case of commensurate
frequencies, the formula for the transmission assumes
no simple transparent from, but is similar to Eq. (18).
Note that, for φ = 0 and ε1 6= ε2, it is also possible to
analyze the system of differential equations analytically,
Eq. (27) and Eq. (28), however the solution is rather
complex. Generally, i.e. for φ 6= 0 and ε1 6= ε2, we solve
the system of differential equations for U1(2),kL(R) nu-
merically, then put the solution into the relation for the
current, Eq. (26), and average it over the common pe-
riod of external perturbations. Thus, this procedure can
be applied only in the case of commensurate frequencies.
In our calculations, we concentrate mainly on the phase
difference effect, as it can lead to the electron pumping
in the system with no source-drain and static bias volt-
ages. We take the phase difference, φ, between QD sites
into account, which leads, for φ = π/2, to a dipole-like
parametrization (the oscillations are out of phase). The
similar two-level system under the influence of one ex-
ternal perturbation in the case of a high bias voltage,
ε1 � ε2, was investigated in [34] and, in the presence of
a phase difference between both external perturbations
and dipole driving forces, in [13].
If only one external perturbation is applied to the dou-
ble dot system, the phenomenon electron pumping does
not occur in the case of the spatial symmetry ε1 = ε2, [7].
For a nonzero gate voltage and φ = π/2 (i.e. in the pres-
ence of the phase difference and the spatial asymmetry),
the current can flow through the system [34], but if there
is no phase difference between both QD sites, φ = 0, the
current vanishes again. In order to analyze the role of
polychromatic perturbations on the symmetric system,
i.e. ε1 = ε2 = 0 (no applied static gate voltages), we de-
pict the dc current in Fig. 6 as a function of the driving
amplitude, Δ2, for two external signals and for the phase
difference between QD sites φ = 0 = 2π (thick broken
line), φ = π (thin broken line) and φ = π/2 (solid line).
For Δ2 = 0, the current is zero and independent of the
phase φ; in this case, only one external perturbation is
acting, and no symmetry breaking takes place. For two
external perturbations and without phase difference be-
tween the first and second QD sites, the current also
does not flow, being equal to zero for all Δ2 (thick bro-
ken line). However, for φ ∈ (0, 2π), the current flows,
although there is no voltages applied to the system. In
that case, we realize a sort of a quantum pump which
works only in the presence of two time-dependent pertur-
bations, implying that the generalized parity is broken
in a dynamical way [7, 13]. Note that, depending on the
phase difference between both QD sites, the dc current
can be positive or negative, cf. the broken and solid
lines.
A remaining question is how the pump current de-
pends on the phase difference between both QD sites. In
92 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
DYNAMICALLY BROKEN SYMMETRY
-0.02
0
0.02
0.04
0 1 2 3 4 5 6 7
j 0
∆2
φ=π/2
-0.02
0
0.02
0.04
0 1 2 3 4 5 6 7
j 0
∆2
φ=π/2
φ=π
-0.02
0
0.02
0.04
0 1 2 3 4 5 6 7
j 0
∆2
φ=π/2
φ=π
φ=2π
Fig. 6. Dc-current flowing through a double QD symmetric sys-
tem under the influence of two external perturbations versus the
amplitude Δ2 for the phase difference φ = 0 = 2π (thick broken
line), φ = π (thin broken line) and φ = π/2 (solid line). The
other parameters are ω1 = 3, Δ1 = 3, ω2 = 6, V12 = 1, Γ = 0.2,
ε1 = ε2 = 0, µL = µR = 0
Fig. 7, we depict the dc current flowing through a dou-
ble QD symmetric system (ε1 = ε2 = 0) as a function
of the (relative) phase φ for three values of the driving
strength of the second perturbation, i.e. Δ2 = 3 (solid
line), Δ2 = 6 (thick solid line), and Δ2 = 9 (broken line).
In case of zero phase difference, φ = 0, 2π, the current
is zero, although two external perturbations are applied
to our system, cf. also Fig. 6. For values of φ different
from φ = 0, 2π, a finite pump current flows through the
double dot system: depending on the relative phase, it
can be either positive or negative.
It is also possible to stop the pumping current all to-
gether for specific values of φ. Note that, for very large
amplitudes of the external perturbation, Δ2, the cur-
rent takes on positive values for almost for all φ, cf. the
broken line.
4. Conclusions
The time-dependent electron transport through a quan-
tum dot and a double quantum dot system in the pres-
ence of polychromatic perturbations has been studied
within the evolution operator method. The QD has
been coupled with two electrodes, and the external time-
dependent energy perturbations have been applied to the
central region.
The analytic relations for the dc current flowing
through the system, Eq. (20), and the charge accumu-
lated on a QD, Eq. (21), have been obtained for incom-
-0.02
0
0.02
0.04
0 0.5 1 1.5 2
j 0
φ [π]
∆2=9
∆2=6
∆2=3
-0.02
0
0.02
0.04
0 0.5 1 1.5 2
j 0
φ [π]
∆2=9
∆2=6
∆2=3
-0.02
0
0.02
0.04
0 0.5 1 1.5 2
j 0
φ [π]
∆2=9
∆2=6
∆2=3
Fig. 7. Dc-current as a function of the phase difference φ for two
external perturbations ω1 = 3, Δ1 = 3, ω2 = 6 and for Δ2 = 3
(solid line), 6 (thick solid line), and 9 (broken line). The other
parameters are: V12 = 1, Γ = 0.2, ε1 = ε2 = 0, µL = µR = 0
mensurate external perturbations. In addition, the ana-
lytic relations for the transmission have been derived for
commensurate frequencies Eq. (18) and for incommen-
surate ones Eq. (19). It has been found that sideband
peaks appear for εd =
∑n
i=1±kiωi, where ki is an integer
number, and n stands for the total number of external
perturbations. In the case of two external perturbations,
this condition can be written as εd = ±k1ω1±k2ω2; and;
e.g. for ω1 = 3 and ω2 = 8, additional peaks appear for
εd = 2, 5, . . ., cf. Fig. 1. In the presence of external
perturbations, one can control the dc current by chang-
ing the amplitude strengths of acting perturbations, cf.
Fig. 2. In the presence of multiple external perturba-
tions, the current is characterized by satellite peaks, cf.
Fig. 3 for two acting energy perturbations.
Moreover, the dc current obtained for commensurate
frequencies, e.g. for bi-harmonic perturbations, strongly
differs from that obtained for frequencies slightly differ-
ent from commensurate ones, cf. Fig. 4. The asymmetry
in the transmission and the dc current versus the quan-
tum dot energy level εd for commensurate frequencies
case is detected (see Fig. 5). This asymmetry appears
only for commensurate external perturbations applied to
the system and is related to a phase difference between
time-dependent perturbations. For a double QD sys-
tem, the analytic formula for the transmission has been
obtained for incommensurate frequencies, Eq. (29). In
addition, an electron quantum pump based on a fully
symmetric double QD system has been proposed in the
absence of source-drain voltages and static bias voltages,
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 93
T. KWAPIŃSKI, S. KOHLER, P. HÄNGGI
for which the pump current varies as a function of the
relative phase shift φ, cf. Fig. 6 and Fig. 7.
This work has been partially supported by Grant
No. NN202 1468 33 of the Polish Ministry of Sci-
ence and Higher Education, the Alexander von Hum-
boldt Foundation (T.K.), the German-Israel-Foundation
(GIF) (P.H.), and the DFG priority program DFG-1243
“quantum transport at the molecular scale” (P.H., S.K.).
S.K. is supported by the Ramón y Cajal program of the
Spanish MICINN.
1. T.H. Oosterkamp, L.P. Kouwenhoven, A.E.A. Koolen,
N.C. van der Vaart, and C.J.P.M. Harmans, Phys. Rev.
Lett. 78, 1536 (1997).
2. W.G. van der Wiel, S. De Franceschi and J.M. Elzerman,
T. Fujisawa, S. Tarucha, and L.P. Kouwenhoven, Rev.
Mod. Phys. 75, 1 (2003).
3. C.A. Stafford, and N.S. Wingreen, Phys. Rev. Lett. 76,
1916 (1996).
4. L.P. Kouwenhoven, A.P. Johnson, N.C. van der Vaart,
A. van der Enden, C.J.P.M. Hermans, and C.T. Foxon,
Z. Phys. B: Condens. Matter 85, 381, (1991).
5. L.P. Kouwenhowen and P.L. McEuen, Nanotechnology,
eds. G. Timp, (Springer, New York, 1998).
6. T. Kwapiński, Phys. Rev. B 69, 153303 (2004).
7. S. Kohler, J. Lehmann, and P. Hänggi, Phys. Rep. 406,
379 (2005).
8. G. Platero and R. Aguado, Phys. Rep. 395, 1 (2004).
9. S. Camalet, J. Lehmann, S. Kohler, and P. Hänggi, Phys.
Rev. Lett. 90, 210602 (2003).
10. S. Kohler, S. Camalet, M. Strass, J. Lehmann, G.L. In-
gold, and P. Hänggi, Chem. Phys. 296, 243 (2004).
11. L. Arrachea, A.L. Yeati, and A. Martin-Rodero, Phys.
Rev. B 77, 165326 (2008).
12. M. Strass, P. Hänggi, and S. Kohler Phys. Rev. Lett. 95,
130601 (2005).
13. J. Lehmann, S. Kohler, P. Hänggi, and A. Nitzan, J.
Chem. Phys. 118, 3283 (2003).
14. M.M. Mahmoodian and M.V. Entin, EPL 77, 67002
(2007).
15. H.K. Zhao and J. Wang, Eur. Phys. J. B 59, 329 (2007).
16. P.Hänggi and F. Marchesoni, Rev. Mod. Phys. 81, 387
(2009).
17. S. Camalet, S. Kohler, and P. Hänggi, Phys. Rev. B 70,
155326 (2004).
18. J. Lehmann, S. Camalet, S. Kohler, and P. Hänggi,
Chem. Phys. Lett. 368, 282 (2003).
19. S. Kohler, J. Lehmann, and P. Hänggi, Superlatt. Mi-
crostruct. 34, 419 (2003).
20. F. Grossmann, T. Dittrich, P. Jung, and P. Hänggi, Phys.
Rev. Lett. 67, 516 (1991).
21. D.M. Basko, M.A. Skvortsov, and V.E. Kravtsov, Phys.
Rev. Lett. 90, 096801 (2003).
22. I. Goychuk and P. Hänggi, Europhys. Lett. 43, 503
(1998).
23. S. Sengupta, R. Guantes, S. Miret Artes, and P. Hänggi,
Physica A 338, 406 (2004).
24. M. Borromeo, P. Hänggi, and F. Marchesoni, J. Phys.:
Condens. Matter 17, S3707 (2005).
25. S. Savelev, F. Marchesoni, P. Hänggi, and F. Nori, Eur.
J. Phys. B 40, 403 (2004).
26. S. Savelev, F. Marchesoni, P. Hänggi, and F. Nori, Euro-
phys. Lett. 67, 179 (2004).
27. S. Savelev, F. Marchesoni, P. Hänggi, and F. Nori, Phys.
Rev. E 70, 066109 (2004).
28. E.G. Petrov, Y.R. Zelinskyy, V. May, and P. Hänggi, J.
Chem. Phys. 127, 084709 (2007).
29. E.G. Petrov, V. May, and P. Hänggi, Phys. Rev. B 73,
045408 (2006).
30. T. Kwapiński, S. Kohler, and P. Hänggi, Phys. Rev. B
79, 155315 (2009).
31. T.B. Grimley, V.C.J. Bhasu, and K.L. Sebastian, Surf.
Sci. 121, 305 (1983).
32. R. Taranko, T. Kwapiński, and E. Taranko, Phys. Rev.
B 69, 165306 (2004).
33. A.-P. Jauho, N.S. Wingreen, and Y. Meir, Phys. Rev. B
50, 5528 (1994)
34. F.J. Kaiser, M. Strass, S. Kohler, and P. Hänggi, Chem.
Phys. 322, 193 (2006).
Received 07.10.09
ДИНАМIЧНО ПОРУШЕНА СИМЕТРIЯ В ПЕРIОДИЧНО
КЕРОВАНИХ КВАНТОВИХ ТОЧКАХ: НАКОПИЧЕННЯ
ЗАРЯДУ I ПОСТIЙНИЙ СТРУМ
Т. Квапiнскi, С. Колер, П. Хангi
Р е з ю м е
Дослiджено залежний вiд часу транспорт електронiв через
квантову точку i систему двох квантових точок при зовнiшнiй
полiхромнiй перiодичнiй модуляцiї рiвнiв енергiї в межах мето-
ду оператора часової еволюцiї з гамiльтонiаном сильного зв’яз-
ку. Одержано аналiтичнi формули для постiйного струму через
систему та для заряду, який накопичено на квантовiй точцi
у границi нульової температури. Показано, що в присутностi
перiодичних збурень боковi максимуми передачi залежать вiд
спiввiдношення зовнiшнiх модуляцiй. Вивчено ефект квантової
накачки за вiдсутностi джерела (стока) i статичних напружень
змiщення. У випадку просторової симетрiї заряд накачується
через систему внаслiдок порушення симетрiї узагальненої пар-
ностi.
94 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
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