Pumping and electron-electron interactions
P. Devillard1,2 , V. Gasparian3, and T. Martin1,4

arXiv:0712.2797v2 [cond-mat.mes-hall] 13 Jun 2008

1

Centre de Physique Th´orique de Marseille CPT, case 907, 13288 Marseille Cedex 9 France
e
2
Universit´ de Provence, 3, Place Victor Hugo, 13331 Marseille cedex 03, France
e
3
Department of Physics, California State University, Bakersfield, CA, USA and
4
Universit´ de la M´diterran´e, 13288 Marseille Cedex 9, France
e
e
e

We consider the adiabatic pumping of charge through a mesoscopic one dimensional wire in the
presence of electron-electron interactions. A model of static potential in-between two-delta potentials
is used to obtain exactly the scaterring matrix elements, which are renormalized by the interactions.
Two periodic drives, shifted one from another, are applied at two locations of the wire in order
to drive a current through it in the absence of bias. Analytical expressions are obtained for the
pumped charge, current noise, and Fano factor in different regimes. This allows to explore pumping
for the whole parameter range of pumping strengths. We show that, working close to a resonance
is necessary to have a comfortable window of pumping amplitudes where charge quantization is
close to the optimum value: a single electron charge is transferred in one cycle. Interactions can
improve the situation, the charge Q is closer to one electron charge and noise is reduced, following
a Q (e − Q) behavior, reminiscent of the reduction of noise in quantum wires by T (1 − T ), where T
is the electron transmission coefficient. For large pumping amplitudes, this charge vanishes, noise
also decreases but slower than the charge.
PACS numbers: 73.23.-b, 72.70.+m, 71.10.Pm, 05.60.Gg

I.

INTRODUCTION

The suggestion that electrons can be supplied one by one by a mesoscopic circuit has been proposed over two
decades ago1 . Instead of applying a constant bias voltage to the system, it is possible to supply a.c. gate voltages
which perturb the system periodically. Under certain conditions, the charge transferred from one lead to the other,
during one period, can be almost quantized. Adiabatic pumping of electrons could in principle be used in future
nanoelectronics schemes based on single electron transfer, and it also has applications to quantum information physics.
Over the years, theoretical approaches to this adiabatic pumping based on scattering theory have become available2,3,4 .
These situations typically describe mesoscopic systems which are large enough, or sufficiently well connected to leads
that electronic interactions (charging effects for instance) can be discarded. Scattering theory has been applied4 to
calculate both the charge and noise in systems in the absence of electron-electron interactions.
On the experimental scene, Coulomb blockade effects have been successfully exploited to achieve pumping with
isolated quantum dots5 . To our knowledge, pumping experiments which are not entirely based on Coulomb blockade,
where the shape of the electron wave functions is modified in an adiabatic drive are rather scarce. A recent study6
has dealt with the transport through an open quantum dot where such interactions are minimized.
Besides Coulomb blockade physics, the effect of electron-electron interactions in conductors with reduced dimensionality have been discussed by several authors. The case of strong interactions in a one dimensional quantum wire was
presented in Ref. 7, using Luttinger liquid theory. Alternatively, Ref. 8 discussed the opposite limit, where the effect
of weak interactions can be included in a scattering formulation of pumping using renormalized transmission/reflection
amplitudes9 . However, the results for the pumped charge remain mostly numerical in this work.
The conditions under which pumping amplitude and interactions must be tuned to achieve quantized pumping are
not obvious. Many physical parameters enter this problem, such as the amplitude of the pumping potentials, the
phase difference between these, the possibility of a constant offset on these potentials, the overall conductance of the
unperturbed structure, and to what extent the strength of electron-electron interactions play a role. Analytical results
on this issues are highly desirable, as well as information about the noise.
With regard to the experiment of Ref. 6, there is clearly a need for further understanding the role of weak interactions
in such mesoscopic systems in the presence of pumping. The purpose of the present work goes in this direction, in
the sense that we provide analytical expressions for the pumped charge and the noise for a one dimensional wire
in the presence of interactions. This allows us to explore all pumping regimes10 (weak to strong pumping) and to
determine in which manner and to what extent the pumped charge can help to achieve single electron transfer. Besides
addressing the question of the ideal conditions for good charge quantization, we shall establish relationships between
charge and noise in different regimes. For concreteness, a two-delta potentials model will be used and interactions
will be added on top of it.

2
II.
A.

PUMPED CHARGE AND NOISE

Adiabatic pumping in non interacting systems

Here, we recall the formula which was established for the charge transferred during the single period of an adiabatic
pumping cycle through a quasi one-dimensional system. The system is in general described by a potential V (x) containing two internal parameters which are modulated periodically. The time dependence is assumed to be sufficiently
slow so that, although the scattering matrix depends on time, its variations are minute when an electron is scattered
in the mesoscopic wire.
At finite temperature, the pumped current reads11
2π
ω

dt

Q = e
0

f (E) T r S † (E, t)σz S(E, t) − I

dE
.
2π

(1)

where S(E, t) is the Wigner transform of the scattering matrix S(t, t′ ), and to, a good approximation the scattering
matrix for the problem with the potential frozen. σz is the ususal Pauli matrix, I the identity matrix and f the
Fermi-Dirac function.
∞

′

e−iE(t−t ) S(t, t′ )dt′ .

S(E, t) =

(2)

−∞

The pumping potential will generate sidebands at E ± ω and we assume that temperature is much smaller than ω,
i.e. kB T ≪ ω, so that we can approximate the Fermi function by a step function. In fact temperature dependence
occurs in two places in this problem. First in the Fermi function and second, the scattering matrices elements depend
on the temperature because of the renormalization due to the interactions (see next chapter). Formulas for averaged
current and zero frequency noise can be carried out using results of the literature in the “zero temperature” formalism,
except that the scattering matrix elements are in fact temperature dependent.
The pumped charge reduces to a time integral over a pumping cycle2,8
Q =

e
2π

2π

Im
0

∂s12 ∗ dX
∂s11 ∗
s11 +
s
+ (X ↔ Y ) dt,
∂X
∂X 12 dt

(3)

where s1i (i = 1, 2) are the elements of the scattering matrix s(E):

s(E) = e

iφ

√
√
−i √Reiθ
√T
T
−i Re−iθ

,

(4)

where φ is the phase accumulated in a transmission event and θ is the phase characterizing the asymetry between the
reflection from the left-hand-side and from the right-hand-side of the potential. Conservation of probabilities imposes
√ √
R + T = 1. We assume the quantities R, T , θ and φ to be functions of the Fermi energy EF and of the external
time-varying parameters X(t) and Y (t).
B.

Inclusion of weak interactions

In the case of weak interactions, the transmission and the reflection amplitudes s12 and s11 can be calculated
in the presence of Coulomb interaction via a renormalization procedure9 . High energy scales above a given cutoff
are eliminated. The high energy cutoff is lowered progressively. The renormalization has to be stopped when the
(0)
(0)
temperature becomes comparable to this cutoff. Finally, if s12 and s11 denote respectively, the transmission and
reflexion coefficient without interactions, s12 and s11 can be expressed in the form9 :
(0)

s12 =

s12 lα
,
1 + T0 (l2α − 1)

(5)

s11 =

s11
,
1 + T0 (l2α − 1)

(6)

(0)

3
where l = kB Θ/W ; Θ is the temperature, kB the Boltzmann constant and W the original bandwidth. α is a negative
exponent related to the strength of the screened Coulomb interaction potential9 . Specifically:
α=

Vc (2kF ) − Vc (0)
,
2πvF

(7)

with Vc (q) the Fourier transform of the screened Coulomb potential at q = 2kF and q = 0, respectively, Vc (0) is
(0)
finite due to screening. α = 0 corresponds to the absence of electron-electron interactions. T0 = |s12 |2 represents
the conductance of the wire in units of e2 /h. ¿From now on, Q will denote the pumped charge with interactions and
Q0 without interactions. The integrand of Eq. (3) is therefore modified by the presence of the interactions. Thus,
temperature dependence occurs through the renormalization of the S-matrix. We shall be interested in the regime
where temperature is much lower than pumping frequency, kB Θ ≪ ω, as said before, but renormalization of S-matrix
should not be too severe so that the renormalization of S-matrix still makes sense. At very low temperature, all
barriers become almost opaque and bosonization is required7 , so typically lα > 10−1 , that is kB Θ ≫ W 101/α . For
example, for nanotubes having α around −0.3, this gives ω ≫ kB Θ ≫ 10−3 W .
C.

Two-delta potentials model and pumped charge

Consider the pumping charge Q transferred during a single period through a 1D chain of an arbitrary potential
shape. Let two parameters of the system be modulated periodically. The single particle Hamiltonian reads
2 2

H =

k
+ V (x) + Vp (x, t),
2m

(8)

where Vp (x, t) is the time-dependent perturbation part of the arbitrary potential and has δ−like potentials form
Vp (x, t) = 2kF X(t)δ(x − xi ) + 2kF Y (t)δ(x − xf ),

(9)

with the amplitudes X(t) = Vi (t)/2kF , Y (t) = Vi (t)/2kF , where Vi (t) and Vf (t) have periodic time evolution with
the same period t0 = 2π/ω and kF denotes the Fermi wave-vector,. V (x) is the static potential in between two δ
potentials. Below, superscript indexes (0) indicate non-interacting systems.
Using the known relation12 ,
(0)

sαβ = −δαβ + 2ikF G0 (xα , xβ ),

(10)

where G0 (xα , xβ ) are the usual real space retarded Green’s functions and the fact that the functional derivative of
the Green’s function δG0 /δV (x) can be written as the product of two Green’s functions13
δG0 (xα , xβ )
= G0 (xα , xj )G0 (xj , xβ ).
δV (xj )

(11)

For the first bracket of Eq. (3), we get
(0)

(0)

∂s11 (0) ∗ ∂s12 (0) ∗
s
+
s
=
∂X 11
∂X 12
(0) (0) ∗

(0) (0) ∗

(0)

(0) ∗

1 + s11 1 + s11
2ikF

(0)

,

(12)

(0)

where we used the condition s11 s11 + s12 s12 = 1, s12 and s11 are the bare transmission and reflection amplitudes
from the disordered system, without taking account the electron-electron interactions. For the second bracket of Eq.
(3), we get
(0)

(0)

(0)

∂s11 (0) ∗ ∂s12 (0) ∗
s
s
+
s
= 12
∂Y 11
∂Y 12
2ikF

(0) (0) ∗

s12 s11

(0) (0) ∗

Here, we used the current conservation requirement s21 s11
right of the scaterrer and can be presented as
(0)

s22 = 2ikF

(0) ∗

(0)

+ s12 (1 + s22 )
(0) ∗ (0)

=

|T0 |2
.
2ikF

(13)

(0)

+ s12 s22 = 0. s22 is the reflection amplitude from the
(0)

∂lns12
− 1.
∂Y

(14)

4
Substituting Eq. (12) and (13) into Eq. (3), using the identity (14), we finally arrive at
e
Q0 = −
2π

2π
ω

0

(0) ∗

(0)

∂lns12 ∂lns12
∂X
∂X

dX
dY
+ T0
dt
dt

dt.

(15)

Similar expression for Q0 , can be found in Ref. 14.
Using the same method for Q, we obtain, in the presence of interactions
e(l2α − 1)
2π
(0)
∂ ln s12
Im
∂X

Q = Q0 −
2π
ω

×

0

−

(0) 2

∂ ln s12
∂X

dX
+
dt

(0)

Im

∂ ln s12
∂Y

− T0

dY
dt

T0
dt.
1 + T0 (l2α − 1)

(16)

For numerical simulations, we specialize to the case where the static potential V (x) = 0 in Eq. (8) and the time(0)
(0)
dependent perturbations are two δ-like potentials separated by a distance 2a. The expressions of s12 and s11 are
needed, when the single particle Hamiltonian reads
H = − 2 k 2 /2m + Vp (x, t).

(17)

The elements of the S-matrix in the absence of electron-electron interactions, are given by
(0)

s11 =
(0)

s12

=

(Y − X)sin(2kF a) − i 2X Y sin(2kF a) + (X + Y )cos(2kF a)
D
1
,
D

,

(18)
(19)

with
D =
(0)

1 − 2X Y sin2 (2kF a) + i X + Y + X Y sin(4kF a)
(0)

(0)

,

(20)

(0)

and X = X/2, same for Y . s21 = s12 and s22 is obtained by replacing Y by X in s11 .
D.

Noise

Ref. 14 studied adiabatic quantum pumping in the context of scattering theory. Their goal was to derive under
what conditions pumping could be achieved optimally, in a noiseless manner, with the assumption that the pumping
frequency is small compared to the temperature. This enabled the authors to derive expressions not only for the
pumped charge per cycle, but also for the pumped noise, the current-current correlation function, averaged over a
time which is long compared to the period of the adiabatic drive, at zero frequency. Specifically, the noise is defined
from the current-current time correlator:
S(t, t′ ) =

1
δI(t)δI(t′ ) + δI(t′ )δI(t) ,
2

(21)

with δI = I − I . This correlator is then averaged over n0 periods of the pumping drive with n0 large, and it is
taken at zero frequency by performing an integral over the remaining time argument. Setting τ0 = n0 2π/ω,
S(Ω = 0) =

ω
2π n0

τ0

∞

dt′ S(t, t′ ).

dt
0

(22)

−∞

Ref. 4 extended the results of Ref. 14 to the case where this limiting assumption is relaxed, yielding a complete
description of the quantum statistical properties of an adiabatic quantum pump, albeit restricted to small pumping
amplitudes. Results made use of the generalized emissivity matrix. These results were generalized to arbitrary
pumping amplitudes by Ref. 11. Here, our goal is to address the question whether electron-electron interactions
affect the pumping noise and how. Following formula (14) of Ref. 11 and applying their Eq. (15) without assuming

5
that the pumping amplitudes X(t) are small, it is possible to put the zero frequency noise S for arbitrary pumping
amplitudes into the form
S(Ω = 0) =

1 e2
(2π )2 τ0

τ0

∞

dt
0

−∞

∞

d(t′ − t)

∞

f (−ǫ1 )
−∞

f (ǫ2 )
−∞

× T r s(ǫ1 , t)† σz s(ǫ2 , t)s(ǫ2 , t′ )† σz s(ǫ1 , t′ ) − I ei

(t−t′ )(ǫ1 −ǫ2 )

dǫ1 dǫ2 ,

(23)

where s(ǫ1 , t) is the 2 × 2 S-matrix, for an incoming wave at energy ǫ1 + ǫF and value of the pumping parameter
2
X(t). τ0 is a time which is much larger than the period τ . ǫ1 = 2 k1 /2m − ǫF , where k1 is a wave vector and ǫF is
2 2
the Fermi energy ǫF = kF /2m. f is the Fermi-Dirac function. Here, s(ǫ1 , t) is taken for fixed t11 . Since we work at
temperature much smaller than ω/kB , as explained before, we can set f (−ǫ1 ) = 1 for ǫ1 > 0 and 0 otherwise. We
set ǫ′ = −ǫ2 so that both ǫ1 and ǫ′ will be positive. M is defined as
2
2
M (ǫ1 , ǫ′ , t) = s(ǫ1 , t)† σz s(−ǫ′ , t).
2
2

(24)

S now reads
S=

e2
limτ0 →∞
(2π )2

τ0
0

dt
τ0

∞

∞

−∞

∞

dǫ1

dt′

dǫ′ T r M (ǫ1 , ǫ′ , t)M (ǫ1 , ǫ′ , t′ )† − I e−i
2
2
2

0

0

(t−t′ )(ǫ1 +ǫ′ )
2

.

(25)

Now, for large pumping amplitudes, the above formula needs to be rearranged, using the fact that X(t) is a periodic
function of period 2π/ω. Note that the dependence of M on ǫ1 and ǫ′ prevents the direct use of fast Fourier
2
transform. Nevertheless, we can use the fact that, for given values of ǫ1 and ǫ′ , M (t) and M (t′ ) are periodic functions
2
of t. Switching to Fourier transform
ˆ
Mn (ǫ1 , ǫ′ ) =
2

ω
2π

2π
ω

0

M (ǫ1 , ǫ′ , t) e−inωt dt,
2

(26)

+∞

ˆ
Mn (ǫ1 , ǫ′ ) einωt .
2

M (ǫ1 , ǫ′ , t) =
2

(27)

n=−∞

Performing the trace, we arrive at
S=

e2
2π 2

∞
0

∞

∞

dǫ1
0

dǫ′
2
n=−∞

ˆ
ˆ
ˆ
ˆ
|M1,1 n |2 + |M1,2 n |2 + |M2,1 n |2 + |M2,2 n |2 − 2δn,0 δ

ǫ1 + ǫ′
2

− nω ,

(28)

ˆ
ˆ
where δn,0 is 1 if n = 0 and zero otherwise and Mi,j n is the (i, j) element of matrix of Mn (ǫ1 , ǫ′ ), where energy
2
dependences have been omitted to ease the notations.
When ω is much smaller than ǫF , formula (28) can be simplified. In this case, M (ǫ1 , ǫ′ , t) will be different from
2
M (0, 0, t) only when ǫ1 or ǫ′ are a non negligible fraction of ǫF . This occurs because M (0, 0, t) corresponds to
2
matrix M for incident wave and outgoing wave at energy ǫF . We denote by ǫ1F typical energies of the order ǫF .
ˆ
ǫ1F will correspond to n of the order ǫ1F / ω , which is very large. The Fourier transform Mn (ǫ1 , ǫ′ ) will decrease
2
exponentially with n for large n. Thus, we can neglect the dependence on ǫ1 and ǫ′ and replace them by zero, which
2
amounts to replacing the energies by ǫF , except in the argument of the δ function. Under these conditions, we have
S ≃ e2

ω
2π

n≥1

ˆ
ˆ
ˆ
ˆ
n |M1,1 n |2 + |M1,2 n |2 + |M2,1 n |2 + |M2,2 n |2

.

(29)

With our form of the S-matrix, this formula is equivalent to Eqs. (24a), (24b) and (24c) of Ref. 15, apart from am
overall factor 2. See Appendix C for details. For numerical simulations however, we did not make this simplification
and kept the dependence on ǫ1 and ǫ′ of Eq. (28).
2

III.

DISCUSSION OF PHYSICAL RESULTS

We now illustrate these formulas by computing the charge and noise, in the case of two-delta potentials model. The
two parameters of the drive X and Y (Eq.(9)) are chosen to vary periodically according to:
X = X0 + η cos(ωt),
Y = Y0 + η cos(ωt − ϕ),

(30)
(31)

6
where X0 is a constant offset potential and ϕ a phase difference. Note that X, X0 and η are all dimensionless, see
Eq. (9). To ensure maximal pumping, we shall specialize10 to ϕ = π/2.
A.

Zero offset

1.4
1.2

S

0.8

(2 π/ω)

Q

1.

0.6
0.4
0.2
0.

0.

5.

10.

η

15.

20.

FIG. 1: Q, charge with interactions, (solid line), Q0 , charge without interactions, (dashed line), both in units of e. S, noise with
interactions, (dotted line) and S0 without, (dashed dotted line), multiplied by 2π/ω, in units of e2 , vs. η. Essential parameters
are X0 = 0, no offset, l2α = 0.3, kF a = 0.5 and ω = 10−2 ǫF .

0.09
−2α

Q l
0

0.08

Q

0.06

0.04

0.02
0.2

0.4.

2α

0.6.

0.8.

1.

l

FIG. 2: Q, charge with interactions, (pluses), and Q0 l−2α , (upper dashed line), both in units of e vs. l2α , for η = 0.3, X0 = 0,
kF a = 0.5 and ω = 10−2 ǫF .

First, the case without offset X0 = Y0 = 0 is studied. To look at the influence of interactions, we plot in Fig. 1 the
pumped charge in units of e, with interactions and without, versus the amplitude of the drive η, for an interaction
parameter l2α = 0.3 (moderate electron-electron interactions). There are three regimes: weak pumping, η ≪ 1,
intermediate pumping, η of order 1 and large pumping amplitudes, η ≫ 1. The current noise times 2π/ω, in the limit
of small ω, is plotted together on the same figure in units of e2 . Analytically, for η ≪ 1, Q reads
Q =

e
sin(4kF a) l−2α η 2 .
4

(32)

As noted in Ref. 8, in the weak pumping regime, charge Q is larger with interactions by a factor l−2α , see Fig.
2. Results of Ref. 4 for the noise, valid for weak pumping and no interactions can be adapted in a straightforward
fashion to the case with interactions. We find the following formula for the noise for weak pumping.
S = e2 l−2α η 2

ω
.
2π

(33)

7
−3

2.2 10

−3

Q

2 10

−3

1.5 10

Q 0 l 2α

−3

10

4 10

−4

0.2

0.4.

l

2α

0.6.

0.8.

1.

FIG. 3: Q, charge with interactions, (pluses), and Q0 l2α , (lower dashed line), both in units of e, vs. l2α , for η = 15, X0 = 0,
kF a = 0.5 and ω = 10−2 ǫF . In this regime, Q is approximately larger than Q0 by a factor l−2α .

(2 π/ω) η 2 S

30

20

10

5

0.2

0.4.

l 2α

0.6.

0.8.

1.

FIG. 4: (2π/ω) η 2 S, noise with interactions scaled by η 2 and by the period, (+) and (2π/ω) η 2 S0 l2α , lower solid line, for
moderate η = 15. Also illustrated by the top two curves, is the very large pumping amplitude regime, (2π/ω) η 2 S (×) and
(2π/ω) η 2 S0 (∗), for very large η; η = 100. The top two curves are close to each other. An attempt to fit η 2 S by l2α η 2 S0
(dashed line) for η = 100, clearly fails for this regime of pumping amplitudes. kF a = 0.5 and ω = 10−2 ǫF for all cases.

The noise is thus increased by the same factor as the current. The Fano factor, defined as the ratio S/e I , is
4/sin(4kF a) and remains independent of the interactions, as long as we remain in the weak pumping regime. This
corresponds to the very left part of Fig. 1, for η smaller than 0.25, typically.
At intermediate pumping amplitudes, Q reaches a maximum value Qmax which is again larger than its noninteracting analog Q0 max . This maximum is of the order of the single electron charge, but less than it. Meanwhile,
the noise decreases. This is a reminder of the reduction of the noise by a factor T (1 − T ), where T is the electrons
transmission coefficient in quantum wires. This explains why the noise exhibits a first maximum around η close to
1, since Q gets closer to one electron charge, noise will decrease. Then, for moderate amplitudes, η around 6, charge
decreases and passes through the value 0.5e, this corresponds then to the second maximum of the noise.
For large, but not very large pumping amplitudes, typically η = 10, Q remains smaller than Q0 but behaves in the
same way, namely as η −3 , as noted in Ref. 10. As a function of the interaction parameter, Q behaves as Q0 l2α . See
Fig. 3. For very large pumping amplitudes, (typically order 100 or more), Q becomes practically equal to Q0 , Q − Q0
behaves as η −4 , see appendix B for details.
For the noise, we found numerically that S and its analog without interactions, S0 , both decrease as η −2 , much
slower than the charge. See below for analytical derivations. As concerns now the interaction dependence of the noise,
S is always smaller than S0 , but for very large η, S tends towards S0 . More precisely, for large but still reasonable η,
of the order 10 typically, S is almost equal to S0 l2α , whereas for very large η, of the order 100 or more, S and S0 are
practically the same. This is not surprising since Q and Q0 are then also practically equal in the end.
This dependence on the interaction parameter is shown in fig. 4. We have to plot (2π/ω) Sη 2 , vs. l2α , but the
overall factor 2π/ω is unimportant; the product Sη 2 can be compared to both l2α S0 η 2 and to S0 η 2 for η = 15 and

8
η = 200. For η = 15, we see that η 2 S is fairly well approximated by η 2 S0 l2α . On the contrary, for η = 200, such a fit
fails and instead, η 2 S is almost equal to η 2 S0 .
The results at large η can be derived from analytical formula for the charge and noise. An expansion for large η is
performed, X and Y behave as η, except at particular points where X or Y are zero. See Appendix A for details.
B.

Non-zero offset

We now turn to the case where X0 is non-zero, which enables to have regions where Q is almost quantized. There
are basically three cases, according to the value of kF a.
The first case corresponds to kF a = nπ/2 (rigorously), where n is an integer. In this case, it is impossible to pump
anything. The reason is given below. The second case corresponds to the case where kF a is small but non-zero, 0.1
typically. We first describe the behavior, then give numerical illustrations and last provide analytical justifications.
√
In this case, the charge is almost zero up to η = X0 . It rises quickly around η = 2X0 and reaches a value close to
quantized e for a wide range of values of η. This is the quantized region of η. The width of this region can be shown
to scale approximately as (kF a)−1 . After the end of this region, Q and Q0 first decrease abruptly and for even larger
values of η, decrease slower, as η −3 . The noise in the quantized region and around it seems to be well approximated
by Q(e − Q), reminiscent of the noise for fermions in narrow quantum wires. However, this does not last when η
becomes noticeably out of the region of almost quantized charge, since Q and Q0 behave as η −3 , whereas S and S0
decay only as η −2 .
Fig. 5 shows noise and charge with and without interactions, versus η for kF a = 0.1. Fig. 6 shows a comparison
between S and a least square fit of the form C Q(e−Q), in the quantized charge regime, where C is the only adjustable
parameter. See captions for details. We now turn to analytical justifications of the previous assertions.
1.4
1.2

(2 π/ω) S

1.

Q

0.8
0.6
0.4
0.2

0.

20.

40.

η

60.

80.

FIG. 5: Q, (solid line), curve with a plateau for η between 28 and 43, (the quantized region), Q0 , (dashed line). The two curves
with two spikes located at η around 28 and at η around 43 represent (2π/ω) S, (dotted line) and (2π/ω) S0 , (dashed dotted
line), vs. η. Essential parameters are X0 = 20, l2α = 0.3, kF a = 0.1 and ω = 10−2 ǫF . Electron charge e is set to 1. The solid
√
horizontal line of ordinate 1 and the thick vertical line at η = X0 2 ≃ 28.28 are guides to the eye.

We now explain why pumping is impossible for kF a = nπ/2. Since sin(2kF a) = 0, the scattering matrix now
(0) ′
depends only on a single parameter, the combination (X + Y ), see eqs. (18-20), so we denote by sij its derivative
with respect to X + Y . Thus,
e
Q0 =
π

2

2π
ω

(0) ′

0

(0) ∗

s1j (X + Y ) s1j (X + Y )

Im
j=1

(0)

(0)

d
(X + Y ) dt = 0,
dt

(34)

because the bracket is just (1/2) d/d(X + Y ) |s11 |2 + |s12 |2 . Then, we look at the case when kF a is close to nπ/2,
but different from it. Clearly, when sin(2kF a) is small, for η|sin(kF a)| < 1, we will be back to the former case, so
Q0 can start to level noticeably from 0 only for η values larger than a critical value ηc1 , which is proportionnal to
1/|sin(2kF a)|, independently of X0 . There is at least another scale, namely X0 . For X0 large, (typically larger than
10) and η smaller than X0 , the pumping contour is a circle which does not enclose the origin and both transmission

9
1.4

(2 π/ω) S

1.

0.5

0.

20.

30.

45.

η

60.

80

FIG. 6: S (solid line), in units of e2 , and best fit of the form C Q(e − Q) (dashed line) in the interval 20 ≤ η ≤ 45, for the same
situation as Fig. 5. The fit no longer works in the large pumping regime, for η between 45 and 80 here.
(0)

(0)

s12 and derivatives of the transmission coefficients, ∂s12 /∂X are small. Q0 will remain very small. Thus, to have
a significant Q0 , one needs η > max X0 , 1/|sin(2kF a)| , where max(x, y) is the larger of x and y. For even larger
η, when terms like sin(2kF a)X Y dominate over terms linear in X or Y , i.e. |sin(2kF a)| η ≫ 1, it is possible to
expand in η −1 and we are back to the large pumping regime where Q decays as η −3 . So, for |sin(2kF a)| smaller than
X0 , there will be a region between max(X0 , C1 /sin(2kF a)) and C2 /sin(2kF a), where C1 and C2 are constants, where
Q0 is appreciable. These very qualitative arguments however do not prove that the charge is almost quantized in this
interval, whose width is of the order |sin(2kF a)|−1 .
We now turn to the analytical explanation of the Q(e − Q) behaviour in the quantized charge regime. It is useful
to disentangle the effects of the fluctuations of T and those of the phase θ. We start from (29) and use the fact that
ˆ
for any periodic function f (x), if fn denotes the Fourier component at frequency nω, denoting x = ωt,

n≥1

2π

ˆ
n|fn |2 =

0

2π
0

|f (x) − f (x′ )| dx dx′
.
(x − x′ )2
4π 2

(35)

ˆ
ˆ
ˆ
ˆ
Applying the former equality to M1,1 n , M1,2 n , M2,1 n and M2,2 n , and denoting by M11 (x) the inverse Fourier
ˆ
transform of M1,1 n , we get
S = e2

ω
2π

2π
0

2π

|M11 (x) − M11 (x′ )|2 + |M12 (x) − M12 (x′ )|2 + |M21 (x) − M12 (x′ )|2 + |M22 (x) − M22 (x′ )|2

0

dx dx′
.
4π 2
In terms of θ and T , using (4) and (24), this reads
(x − x′ )−2

S = 8e2

(36)

ω
2π

I1 + I2 ,

(37)

with
2π

2π

I1 =
0

0
2π

2π

I2 =
0

with g(x) =

0

|T (x) − T (x′ )|2 dx dx′
,
(x − x′ )2
4π 2

(38)

|g(x) − g(x′ )|2 dx dx′
,
(x − x′ )2
4π 2

(39)

T (1 − T ) eiθ . We expand |g(x) − g(x′ )|2 and rewrite I2 in the form
I2 = J1 + J2 ,

(40)

with
2π

2π

J1 =
0

(x −

0
2π

2π

J2 =
0

T (x) 1 − T (x) −

0

T (x′ ) 1 − T (x′ )

x′ )

2

dx dx′
,
4π 2

(41)
′

T (x) 1 − T (x)

T (x′ )

1−

T (x′ )

|e−iθ(x) − e−iθ(x ) |2 dx dx′
.
(x − x′ )2
4π 2

(42)

10
Thus, noise breaks into three parts, one related to the fluctuations of T , another to the fluctuations of T (1 − T ),
and the last to the fluctuations of θ, in fact, to the variations of the slope dθ/dx, since θ has to vary by 2π in one
cycle to get appreciable pumped charge. The physical message is that if the fluctuations of T are much smaller than
the fluctuations of the phase, then noise shows a Q(e − Q) behaviour. Otherwise, fluctuations of T bring an extra
noise that does not contribute to the pumped charge and overall noise is thus larger than Q(e − Q). We now try to
establish this more firmly.
In this paragraph, we now show that, for Q0 ≤ e/2,
S0 ≥ 8

ω
Q0 (e − Q0 ).
2π

(43)

The first integral J1 involves solely the fluctuations of T and can be rewritten as
∞

J1 =
n=1

n| T (1 − T )n |2 .

(44)

We have a lower bound for J1 by replacing n by 1 in all terms of the sum except the term for n = 0,
∞

J1 ≥

n=0

| T (1 − T )n |2 − | T (1 − T )0 |2 .

Using then the Parseval identity for the first term and using the notation f ≡
T (1 − T )

J1 ≥

2

−

T (1 − T )

2

= T − T2 −

(45)
2π
0

f (x) dx/2π for the second,

T (1 − T )

2

.

(46)

Now for J2 , applying twice the H¨lder inequality,
o
2π

J2 =
0
2π

≥

0

≥

2π

dx′
2π

T (x′ ) 1 − T (x′ )

′

0

2π

′

dx
2π

T (x′ ) 1 − T (x′ )

T (1 − T )

2π

2
0

dx
2π

2π
0

|e−iθ(x) − e−iθ(x ) |2 dx
(x − x′ )2
2π

T (x) 1 − T (x)
T (1 − T )

0

′

|e−iθ(x) − e−iθ(x ) |2 dx
(x − x′ )2
2π
′

dx′ |e−iθ(x) − e−iθ(x ) |2
.
2π
(x − x′ )2

(47)

For the last double integral, we proceed as before, going again to Fourier transform, isolating the component of order
0 and using the Parseval identity, it is larger than |eiθ |2 − | eiθ |2 . Always, |eiθ | = 1 and, for reasonable θ(x), we
can assume symmetry x into −x which implies sinθ = 0. We can also assume symmetry when x is changed into
π − x, which implies cosθ = 0. We have
J2 ≥

T (1 − T )

2

.

(48)

Now, for I1 , using the same technique, (going to Fourier transform and isolating the n = 0 component)
I1 ≥ T 2 − T 2 .

(49)

Putting everything together
I1 + I2 ≥ T 2 − T
Now, assuming that

2π dφ dx
0 dx 2π

2

+ T − T2 −

T (1 − T )

2

+

T (1 − T )

2

= T − T 2.

(50)

= 0, (the circulation of φ is zero in one cycle),
Q = e 1− T

dθ
dx

,

(51)

which implies
Q(e − Q) = e2

T

dθ
dθ
− T
dx
dx

2

.

(52)

11
Moreover, by the H¨lder inequality
o
T

dθ
dx

≥

T

dθ
dx

= T .

(53)

dθ
Now, for any y ≥ 1/2, y(1 − y) is a decreasing function of y. Thus, if T dx

T − T
Thus S ≥ 8

ω
2π

2

≥ T

dθ
dθ
− T
dx
dx

2

.

≥ 1/2 , i.e. Q ≤ e/2, then
(54)

Q0 (e − Q0 ), which is (43).

Now we turn to the case of interest, when Q0 ≥ e/2, for example in the quantized region. We were not able to
provide a general analytical proof of Q0 (e − Q0 ) behaviour. We first look at simple limiting cases. Then, we examine
the particular case of the two delta-potentials model.
First, in a situtation where T (x) is a constant T , (with T small to have almost charge quantization), then Q0 =
ω
e(1 − T ) and S0 = 8C0 2π Q0 (e − Q0 ) with C0 depending on the shape of θ(x), but always C0 is greater than 1. In
the case of constant slope dθ/dx = 1, C0 = 1. Second, in the situation where dθ/dx is constant but T (x) arbitrary,
we have T (dθ/dx) = T and thus again (43) holds for any Q0 , not just for Q0 smaller than e/2.
However, in practise, θ varies abruptly by π in the vicinity of resonances. This is different from optimal pumping
strategies which have been studied before14,18 . We first give qualitative arguments and then give precise calculations
for the model studied here. Let us look at the contribution of J2 , Eq. (48), to the noise. When x and x′ are both close
′
to resonances |e−iθ(x) − e−iθ(x ) |2 /(x − x′ )2 behaves as (dθ/dx)2 for (x − x′ )(dθ/dx) < 1. If meanwhile, T (x) does not
vary too much and assumes the value Ti , the contribution of this region in the plane (x, x′ ) to J2 will be Ti (1 − Ti )/4.
The 1/4 comes from the fact that θ varies suddenly only by π and not by 2π at each resonance. Another contribution
will come from the regions where x is within the resonance but x′ just outside or vice versa. Let us take x′ outside
′
to be precise. Then |e−iθ(x) − e−iθ(x ) | = 2 and integration on x and x′ will give a contribution mainly from x′ just
outside, due to the rapidly decreasing factor (x − x′ )−2 . This will eventually give another factor Ti (1 − Ti )/4. Despite
the non-local character of the integrand in J2 , regions where x and x′ are both far from a resonance make very little
contribution to J2 . For Q, one gets Q = e(1 − i Ti /2). In the simple case where there are only two resonances and
the Ti ’s are equal, then Q = e(1 − T1 ) and J2 = T1 (1 − T1 ). If I1 and J1 , which are related to the fluctuations of T (x)
are much smaller than J2 , then, this leads to S0 = 8 (ω/2π) Q0(e − Q0 ).
We now test the former very qualitative ideas by analytical calculations on our particular model. A first thing to
be noted is that the phase φ does not intervene in the noise, which is normal since noise is related to the two-particle
scattering matrix. On the contrary, it does formally enter the equation for the pumped charge, see Eqs. (4),(15) and
(16). Nevertheless, the variation of φ when ωt varies in one period, is always zero so that φ does not play any role.
This is due to the fact that φ is the phase of D, see Eqs. (19) and (20). If the real part of D becomes negative, then,
its imaginary part cannot be zero and thus, φ can never be equal to −π or π. Moerover, φ varies by strictly less than
2π during one period and thus, its circulation is zero, giving no contribution to Q and Q0 .
√
We now turn to the variations of θ. Charge quantization necessitates η larger than 2 X0 and kF a small. We thus
approximate cos(2kF a) by 1 and set u = sin(2kF a) ≪ 1. θ can be written as
θ = arg(n),

(55)

n = X Y u + X + Y + iu(Y − X).

(56)

with

There exist two values of η, η1 and η2 , which play a particular role. For η1 ≤ η ≤ η2 , when √ varies by 2π, Re(n)
ωt
√
√
changes twice its sign and θ varies by 2π. In the model studied here, η1 = X0 2 and η2 = X0 2 + 2 2u−1 . Outside
this interval of η, the increase of θ when ωt varies by 2π is zero, not 2π. The reason is the following. For η ≤ η1 , Re(n)
is always positive and the phase θ remains confined in an interval contained in [− π , π ]. For η > η2 , Re(n) changes
2
2
four times its sign but the contour described by n, in the complex plane, as ωt is varies by 2π, does not enclose the
origin. It can be seen directly, for it is impossible to have Im(n) = 0 and Re(n) ≤ 0 at the same time. Thus, it is not
surprising that the quantized region corresponds approximately to the interval [η1 , , η2 ].
We now examine more precisely the variations of θ and T . Apart from a small variation around x ≡ ωt = −3π/4,
θ, as a function of x, is essentially flat, except around two points x1 and x2 . In practise, x1 is close to − π and x2
4

12
close to 3π/4. Around those two points, θ increases fastly by almost π each time. We can thus model the function θ
by
θ = −π, for
x ≤ x1 − l,
π x − x1
− 1 for x1 − l ≤ x ≤ x1 + l,
θ =
2
l
θ = 0,
for x1 + l ≤ x ≤ x2 − l,
x − x2
π
1+
for ≤ x2 − l ≤ x ≤ x2 + l,
θ =
2
l
θ = π,
for x ≥ x2 + l,

(57)

l being a small distance. As for T , T (x) shows a large peak around x = xs = −3π/4, and two smaller peaks,
practically identical, centered around x1 and x2 . Away from these values of x, T (x) is almost zero.
Now, we look at the implications for the charge and noise. When calculating the pumped charge without interaction
2π dθ
1
via the formula 2π 0 dx 1−T (x) dx, the region around x = xs does not bring much contribution because variations
of θ are small here. For the calculation of Q0 and S0 , we can ignore the large peak in T and thus model T (x) by
(x − xi )2
(x − xi )2 − l2
T (x) = 0 otherwise.
T (x) = T1 exp

for |x − xi | ≤ l,
(58)

T has to be derivable in order to avoid logarithmic divergencies due to the factor 1/(x − x′ )2 in Eqs. (38) and (39).
Then, it is possible to perform analytical calculations which give
Q0 = e 1 − C1 T1 ,
ω
S0 = C2
(e − Q0 ) e(1 − C3 ) + C3 Q0 ) .
2π

(59)
(60)

Detailed calculations, involving the integrals I1 and I2 and the constants C1 , C2 and C3 are given in the appendix.
C3 is smaller than 1, (approximately 0.58). Even if it is not exactly of the form Q0 (e − Q0), when there is good charge
quantization, i.e. when Q0 is not far from 1, S0 goes as (e − Q0 ). The essential thing is that T and θ vary rapidly
around certain values of ωt. We do not get exactly Q0 (e − Q0 ) because T varies substantially when θ jumps. One
might wonder if the results here are particular to our model. In fact, brisk variations of the phase are widely shared
by many types of models16,17 .
The third case corresponds to kF a not close to nπ/2. In this case, for large X0 , Q is almost zero except in the
√
vicinity of a value ηc , which is very near 2X0 ; numerically it seems that ηc is always a little less than this value. The
maximum pumped charge is of order e but no longer close to one electron charge. Noise has a double peak structure
around ηc . A rough qualitative picture of this can be seen in Eq. (19), because, as soon as the integration contour
does not get close to the point X(t) = Y (t) = 0, for any t, the integrands in Eqs. (15) and (16) are very small. An
example is shown in Fig 7.
1.2

(2 π/ω) S

Q

1.

0.5

0.

20.

30.

η

40.

45.

FIG. 7: Same as Fig. 5, except that kF a = 0.4. The solid horizontal line of ordinate 1 and the thick vertical line at abscissa
√
X0 2 are just guides to the eye.

13
IV.

CONCLUSION

We have studied the influence of weak electron-electron interactions on pumped charge and noise in the adiabatic
regime, in a mesoscopic one-dimensional disordered wire. Within the two-delta potentials barrier model, analytical
results were obtained for the charge and noise. Results were analyzed numerically for local pumping fields with a
harmonic dependence. Without any voltage offset, at weak pumping amplitudes, interactions tend to enhance the
pumped charge, as l−2α , where l is the interaction parameter. For fairly large pumping amplitudes, it is exactly the
reverse, Q and Q0 both decrease as η −3 , but Q remains smaller than Q0 by a factor l2α . At very large pumping
amplitudes, Q and Q0 are practically the same. As to the pumping noise, at weak amplitudes, it increases with
interactions, but in the same way as the charge, so that, the Fano factor remains constant, independent of the
interactions. For moderate pumping amplitudes, noise has a double peak structure around the maximum of pumped
charge. For large amplitudes, the noise decreases slower than the charge, as η −2 , and for very large amplitudes, noise
with and without interactions become approximately the same.
As emphasised in Ref. 8, interactions tend to make resonances sharper, which is conducive to obtaining an almost
quantized pumped charge. However, it is not sufficient to enclose a resonance, it is also necessary that the pumping
contour does not go too far from the resonance. Otherwise, the noise is appreciable and the signal Q can even be very
small.
In the case of constant offset X0 , the behavior depends if we are close to a resonance, kF a = nπ/2 in the two-delta
potentials model. Close to a resonance, there is a region of almost quantized pumped charge where the noise seems to
follow a Q(e − Q) behavior, reminiscent of the noise reduction in quantum wires for good transmission by a T (1 − T )
factor, where T is the modulus of the energy transmision coefficient. Quite generally, noise breaks up into pieces due
to fluctuations of T and those due to fluctuations of θ. We believe that in the quantized region, the fluctuations of θ
are predominant and give rise to the Q(e − Q) behaviour.
Interactions do help in having a charge closer to e and to reduce the noise. However, it does not change the range
of pumping amplitudes, where quantized charge is observed, i.e. the width of the quantized region practically does
not depend on the interactions. Qualitative arguments seem to indicate that this width scales as sin(2kF a)−1 close
to kF a = nπ/2. Further from the resonance, the maximum charge which can be pumped becomes of order, but less
than e. Moreover, the region of quantized charge shrinks to a narrow window of pumping amplitudes around a value
√
close to X0 2.
In summary, our study of noise shows that interactions tend to increase the quality of pumping. However, two
conditions need to be met; first, to operate at certain wavevectors favouring sharp resonances and second to have a
pumping contour which encircles the resonant point, passing not very far from it. Otherwise, only noise is produced
and the quantization of the charge is not achieved. These restrictions were not pointed out in previous works. In
addition, in the quantized charge region, noise vanishes as e − Q.
Acknowledgments

One of us (T.M) acknowledges the support from ANR grant “molspintronics” of the French ministry of research.
V.G. acknowledges the kind hospitality of the Centre de Physique Th´orique de Marseille, where inital parts of this
e
work were done.
APPENDIX A

In this appendix, we consider the limit of large pumping. In order to explain why, for offset X0 = 0, Q0 behaves
like η −3 at large η, we use Eqs. (15), (18), (19) and (20). Let us look first at the terms involving dX/dt.
X, Y , as well as their time derivatives will be of order η, except at isolated particular points. The term
(0)

∂ ln s12 /∂X

2

can be expanded in powers of 1/X and 1/Y :

(0) 2

∂ ln s12
∂X

=

1
X

2

−

cotan(2kF a)
X

3

+

3 cotan2 (2kF a) − 1
4X

4

−

1 + cotan2 (2kF a)
2

4X Y

2

+

cotan2 (2kF a) − 1
3

2X Y

+ o(η −5 ). (A1)
4

All terms multiplied by dX/dt and then, integrated over one period give zero. Note that a term like 1/(X Y ), which
is in η −5 , would not give 0. Thus, for large η, the term proportionnal to dX/dt in Q, behaves at least as η −3 .
For the term involving dY /dt, the situation is simpler. dY /dt goes as η, but since D goes as η 2 , T0 goes as η −4 ,
thus this term is at least in η −3 .

14
Now the remainder of contributions to Q−Q0 behave at least like η −4 for large η; it can be seen from (16). For large
(0)
η, since T0 goes as η −4 , the quantity T0 /(1 + T0 (A2α − 1)) is practically equivalent to T0 ∼ η −4 . Im{∂ ln s12 /∂X}
(0)
(0)
goes as η −1 for large η. The same holds for Im{∂ ln s12 /∂Y }. In the integral in the r.h.s. of (16), Im{∂ ln s12 /∂X}
(0)
(0)
is larger, by η, than |(∂ ln s12 /∂X)|2 . Then, Im{∂ ln s12 /∂X} is of order η −1 whereas T0 is of order η −4 . Finally, the
(0)
(0)
integrand is at least of order Im{∂ ln s12 /∂X}(dX/dt)T0 , or Im{∂ ln s12 /∂Y }(dY /dt)T0 , which are both at least of
−1
−4
−4
order η × η × η ∼ η .
We now evaluate the behavior of the noise for large η. When ω ≪ ǫF , only the low order Fourier components of
(0)
η(t) are important. ǫ1 and ǫ′ will be much smaller than ǫF . At ǫ1 = ǫ′ = 0, s11 ≃ −e−2ikF a 1 + i/X + o(η −2 ) and
2
2
(0)
(0)
s22 is the same but X is replaced by Y . s12 = −(1/2X Y ) 1 + i cotan(2kF a) + o(η −3 ). As a result, in Eq. (23),
s(ǫ1 , t)† σz s(ǫ2 , t) = σz + o(η −2 ), the same holds for s(ǫ2 , t′ )† σz s(ǫ1 , t′ ), so that the trace is o(η −2 ), which yields S of
order η −2 , at least.
APPENDIX B

In this appendix, we show the result of calculations for the integrals I1 and I2 , using Eqs. (57) and (58) for θ(x)
and T (x). The integrand of I1 and I2 are non-zero only if at least one of x or x′ is within distance l from an xi ,
i = 1, 2. We shall need the integrals
1

M0 =

exp
0
1

K1 =

exp
−1
1

1

x2
dx ≃ 0.603,
−1

x2
(1 − x)−1 dx ≃ 1.207,
−1

−1 −1
1

1

exp

L1 =
−1 −1

y2
x2
− exp 2
x2 − 1
y −1

1

2

exp

L2 =
−1 −1

cos

2

(x − y)−2 dx dy ≃ 2.088,

x2
y2
1
x2
y2
+ exp 2
− 2 exp
+ 2
x2 − 1
y −1
2 x2 − 1 y − 1

(x − y)−2 dx dy ≃ 7.997,
1

(B2)

x2

exp

K2 =

(B1)

x2

cos

π
(x − y)
2
(B4)

2

2

2

2y
1
x
y
2x
+ exp 2
− exp [ 2
+ 2
]
−1
y −1
2 x −1 y −1

x2

π
(x − y)
2

(B3)

(x − y)−2 dx dy ≃ 6.591.

2

exp(

2

x
y
) + exp( 2
)
−1
y −1

x2

(B5)

Then, plugging these values into (15), (38) and (39),
Q0 = e(1 − T1 C1 ),
ω
S0 = C2
(e − Q0 ) e − C3 (e − Q0 ) ,
2π

(B6)

C1 = M0 ≃ 0.603,
2
C2 = 2 (8K1 + 2L1 )/M0 ≃ 8.613,
π
L2 − K2
C3 =
≃ 0.58.
(L1 + 4K1 )M0

(B8)

(B7)

with

(B9)
(B10)

15
APPENDIX C

In this appendix, we show the equivalence of Eq. (29) with Eqs. (24a), (24b) and (24c) of Ref. 15. There, the noise
power Pαβ between leads α and β was given by
Pαβ = 2

e2
h

∞
sym
q ω Cαβ,q (µ),

(C1)

q=1

Cαβ,q (E) + Cαβ,−q (E)
,
2
s∗ (E)sαδ (E) q s∗ (E)sβγ (E)
Cαβ,q (E) =
αγ
βδ

sym
Cαβ,q (E) =

γ

(C2)
−q

,

(C3)

δ

where [A]q denotes the Fourier transform at frequency qω of a time dependent quantity A. In our case, there are only
two leads so that indices γ and δ are either 1 or 2. We are interested in P11 , so we make α = β = 1. Then, inserting
the value of the scattering matrix elements according to Eq. (4), leads to
P11 = 4e2

ω
2π

q≥1

√
1 √
|( RT eiθ )q |2 + |( RT e−iθ )q |2
2

.

(C4)

√
1 √
|( RT eiθ )n |2 + |( RT e−iθ )n |2
2

,

(C5)

q |Tq |2 +

On the other hand, using Eq.(29), we obtain
S = 8e2

ω
2π

n≥1

n |Tn |2 +

which is, apart from an overall factor 2, the same as Eq. (C4), with q changed to n.

1
2
3
4
5

6
7
8
9

10
11
12
13
14

15
16
17
18

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