# Excitation of collective modes in a quantum flute

**ABSTRACT** We use a generalized master equation (GME) formalism to describe the

non-equilibrium time-dependent transport of Coulomb interacting electrons

through a short quantum wire connected to semi-infinite biased leads. The

contact strength between the leads and the wire is modulated by out-of-phase

time-dependent potentials which simulate a turnstile device. We explore this

setup by keeping the contact with one lead at a fixed location at one end of

the wire whereas the contact with the other lead is placed on various sites

along the length of the wire. We study the propagation of sinusoidal and

rectangular pulses. We find that the current profiles in both leads depend not

only on the shape of the pulses, but also on the position of the second

contact. The current reflects standing waves created by the contact potentials,

like in a wind musical instrument (for example a flute), but occurring on the

background of the equilibrium charge distribution. The number of electrons in

our quantum "flute" device varies between two and three. We find that for

rectangular pulses the currents in the leads may flow against the bias for

short time intervals, due to the higher harmonics of the charge response. The

GME is solved numerically in small time steps without resorting to the

traditional Markov and rotating wave approximations. The Coulomb interaction

between the electrons in the sample is included via the exact diagonalization

method. The system (leads plus sample wire) is described by a lattice model.

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**ABSTRACT:**A Generalized Master Equation (GME) is used to study the thermoelectric currents through a quantum dot in both the transient and steady-state regime. The two semi-infinite leads are kept at the same chemical potential but at different temperatures to produce a thermoelectric current which has a varying sign depending on the chemical potential. The Coulomb interaction between the electrons in the sample is included via the exact diagonalization method. We observe a saw-teeth like profile of the current alternating with plateaus of almost zero current. Our calculations go beyond the linear response with respect to the temperature gradient, but are compatible with known results for the thermopower in the linear response regime.Physica E Low-dimensional Systems and Nanostructures 03/2013; 53. · 1.86 Impact Factor

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arXiv:1202.0566v1 [cond-mat.mes-hall] 2 Feb 2012

Excitation of collective modes in a quantum flute

Kristinn Torfason,1,2Andrei Manolescu,1Valeriu Molodoveanu,3and Vidar Gudmundsson2

1School of Science and Engineering, Reykjavik University, Menntavegur 1, IS-101 Reykjavik, Iceland

2Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland

3National Institute of Materials Physics, P. O. Box MG-7, Bucharest-Magurele, Romania

(Dated: February 6, 2012)

We use a generalized master equation (GME) formalism to describe the non-equilibrium time-

dependent transport of Coulomb interacting electrons through a short quantum wire connected to

semi-infinite biased leads. The contact strength between the leads and the wire is modulated by

out-of-phase time-dependent potentials which simulate a turnstile device. We explore this setup by

keeping the contact with one lead at a fixed location at one end of the wire whereas the contact with

the other lead is placed on various sites along the length of the wire. We study the propagation of

sinusoidal and rectangular pulses. We find that the current profiles in both leads depend not only on

the shape of the pulses, but also on the position of the second contact. The current reflects standing

waves created by the contact potentials, like in a wind musical instrument (for example a flute),

but occurring on the background of the equilibrium charge distribution. The number of electrons in

our quantum ”flute” device varies between two and three. We find that for rectangular pulses the

currents in the leads may flow against the bias for short time intervals, due to the higher harmonics

of the charge response. The GME is solved numerically in small time steps without resorting to

the traditional Markov and rotating wave approximations. The Coulomb interaction between the

electrons in the sample is included via the exact diagonalization method. The system (leads plus

sample wire) is described by a lattice model.

PACS numbers: 72.15.Nj, 73.23.Hk, 78.47.da, 85.35.Be

I. INTRODUCTION

The control of transient transport properties of open

nanodevices subjected to time-dependent signals is nowa-

days considered as the main tool for charge and spin ma-

nipulation. Pump-and-probe techniques allow the indi-

rect measurement of tunneling rates and relaxation times

of quantum dots in the Coulomb blockade1. Quantum

point contacts and quantum dots submitted to pulses

applied only to the input lead generate specific output

currents2–4. Single electrons pumping through a double

quantum dot defined in an InAs nanowire by periodic

modulation of the wire potential has been observed5, as

well as non-adiabatic monoparametric pumping in Al-

GaAs/GaAs gated nanowires6.

Modeling such short-time processes is a serious task

because even if the charge dynamics is imposed by the

time-dependent driving fields, the geometry of the sam-

ple itself and the Coulomb interaction play also an impor-

tant role. A well-established approach to time-dependent

transport relies on the non-equilibrium Greens’ function

(NEGF) formalism, the Coulomb effects being treated

either via density-functional methods7or within many-

body perturbation theory8. Alternatively, equation of

motion methods were used for studying pumping in fi-

nite and infinite-U Anderson single-level models9. The

numerical implementation of these formal methods in the

interacting case requires extensive and costly computa-

tional work if the sample accommodates more than few

electrons; as a consequence, accurate simulations for sys-

tems having a more complex geometry and/or complex

spectral structure are not easily obtained.

Recently we reported transport calculations for a two-

dimensional parabolic quantum wire in the turnstile

setup10, neglecting the Coulomb interaction between

electrons in the wire. The latter is connected to semi-

infinite leads seen as particle reservoirs. Let us remind

here that the turnstile setup was experimentally real-

ized by Kouwenhoven et al.11.

a time-dependent modulation (pumping) of the tunnel-

ing barriers between the finite sample and drain and

source leads, respectively. During the first half of the

pumping cycle the system opens only to the source lead

whereas during the second half of the cycle the drain

contact opens. At certain values of the relevant parame-

ters an integer number of electrons is transferred across

the sample in a complete cycle. More complex turnstile

pumps have been studied by numerical simulations, like

one-dimensional arrays of junctions12or two-dimensional

multidot systems13.

It essentially involves

In this work we perform a similar study for an inter-

acting one-dimensional quantum wire coupled to an in-

put (source) lead at one end, while the output (drain)

lead can be plugged at any point along its length. Both

contacts are modulated by periodic pulses (sinusoidal or

square shaped). Our study is motivated by the possibil-

ity to control the transient currents through the variation

of the drain contact. We shall see in fact that the flexi-

bility of the drain contact allows us to capture different

responses of the sample to local time-dependent pertur-

bations which can lead to transient currents with specific

shapes. In some sense, our system works like a ‘quan-

tum flute’, this fact being revealed when analyzing the

distribution of charge within the wire. In particular, we

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calculate and discuss the deviation of the charge density

from the mean value for each site of the quantum wire

and observe the onset of standing waves.

The effect of the electron-electron interaction is in-

cluded in the sample via the exact diagonalization

method while the time-dependent transport is performed

within the generalized master equation (GME) formal-

ism as it is described in Ref. 14. The implemented GME

formalism can be used to describe both the initial tran-

sient regime immediately after the coupling of the leads

to the sample and the evolution towards a steady state

achieved in the long time limit. The GME formalism

captures the transient charging of many-body states and

Coulomb blockade effects14.

To the best of our knowledge these are the first numer-

ical simulations of electronic transport through an inter-

acting quantum turnstile which is not a quantum dot.

We emphasize that most of the studies on pumping in

interacting systems are focused on single-level quantum

dots15,16in the Kondo regime. Here we consider a sys-

tem with spatial extension where the charge distribution

plays an important role in the transport processes.

We discuss for the first time the effect of contacts’ lo-

cation on the transient currents. More precisely, we show

that if the drain lead is attached to different regions of

the quantum wire the currents in both leads are consid-

erably affected.

The paper is organized as follows: The model and the

methodology are described in Section II, the numerical

results are presented in Section IV, and the conclusions

in Section V.

II. THE QUANTUM FLUTE MODEL

The physical system consists in a sample connected to

two leads acting as particle reservoirs. We shall adopt

a tight-binding description of the system: the sample is

a short quantum wire and the leads are 1D and semi-

infinite. In this work we consider a sample of 10 sites.

This number optimizes the computational time and the

physical phenomenology which we intend to describe. A

sketch is given in Fig. 1. The left lead (or the source,

marked as L) is contacted at one end of the sample and

the right lead (or the drain, marked as R) may be con-

tacted on any other site. The Hamiltonian of the coupled

and electrically biased system reads as

H(t) =

?

ℓ

Hℓ+ HS+ HT(t) = H0+ HT(t), (1)

where HSis the Hamiltonian of the isolated sample, in-

cluding the electron-electron interaction,

HS=

?

n

End†

ndn+1

2

?

mn

m′n′

Vmn,m′n′d†

md†

ndm′dn′ . (2)

The (non-interacting) single-particle basis states have

wave functions {φn} and discrete energies En. Hℓ, with

χL

χR

11234567789 10

µL

µR

FIG. 1. (Color online) A sketch of the system under study.

A 1D lattice with 10 sites (“the sample”) is connected to two

semi-infinite leads via tunneling. The left lead is connected to

the left end of lattice, while the position of the right lead can

be changed. The contacts (χL, χR) are modulated in time.

{ℓ} = (L,R), is the Hamiltonian corresponding to the left

and the right leads. The last term in Eq. (1), HT de-

scribes the time-dependent coupling between the single-

particle basis states of the isolated sample and the states

{ψqℓ} of the leads:

HT(t) =

?

n

?

ℓ

?

dq χℓ(t)(Tℓ

qnc†

qℓdn+ h.c.). (3)

The function χℓ(t) describes the time-dependent switch-

ing of the sample-lead contacts, while d†

ate/annihilate electrons in the corresponding single-

particle states of the sample or leads, respectively. The

coupling coefficient

nand cqℓ cre-

Tℓ

qn= V0ψ∗

qℓ(0)φn(iℓ), (4)

involves the two eigenfunctions evaluated at the contact

sites (0,iℓ), 0 being the site of the lead ℓ and iℓthe site

in the sample14. In our present calculations we keep the

left lead connected to the site iL= 1, while the position

of the right lead iR is varied. The parameter V0 plays

the role of a coupling constant between the sample and

the leads.

We will ignore the Coulomb effects in the leads, where

we assume a high concentration of electrons and thus

strong screening and fast particle rearrangements. The

Coulomb electron-electron interaction is considered in de-

tail only in the sample, where Coulomb blocking effects

may occur. The matrix elements of the Coulomb poten-

tial in Eq. (2) are given by,

Vmn,m′n′ =

?

d? xd? x′φ∗

m(? x)φ∗

n(? x′)

uC

|? x − ? x′|φm′(? x)φn′(? x′).

(5)

We calculate the many-electron states (MES) in the

sample by incorporating the Coulomb electron-electron

interaction following the exact diagonalization method,

i. e. without any mean field approximation. The MES

are calculated in the Fock space built on non-interacting

single-particle states14. Since the sample is open the

number of electrons is not fixed, but the Coulomb in-

teraction conserves the number of electrons. With 10

lattice sites we obtain 10 single-particle eigenstates and

Page 3

3

thus 210= 1024 elements in the Fock space spanned by

the occupation numbers. The Coulomb effects are mea-

sured by the ratio of a characteristic Coulomb energy

UC= e2/(κa) and the hopping energy ts= ?2/(2meffa2).

Here a denotes the inter-site distance (the lattice con-

stant of the discretized system), while κ and meff are

material parameters, the dielectric constant and the elec-

tron effective mass, respectively. In our calculations we

use the relative strength of the Coulomb interaction,

uC= UC/ts, which is treated as a free parameter.

III. THE TRANSPORT FORMALISM

The equation of motion for the our system is the quan-

tum Liouville equation,

i?˙W(t) = [H(t),W(t)] ,W(t < t0) = ρLρRρS. (6)

W(t), the statistical operator is the solution of the equa-

tion and completely determines the evolution of the sys-

tem. At times before t0the systems are assumed to be

isolated and W(t) is simply the product of the density

operator of the sample and the equilibrium distributions

of the leads.

Following the Nakajima-Zwanzig technique17we define

the reduced density operator (RDO), ρ(t), by tracing out

the degrees of freedom of the environment, the leads in

our case, over the statistical operatorof the entire system,

W(t)

ρ(t) = TrLTrRW(t),ρ(0) = ρS. (7)

The initial condition corresponds to a decoupled sample

and leads when the RDO is just the statistical opera-

tor of the isolated sample ρS. For a sufficiently weak

coupling strength (V0) one obtains the non-Markovian

integro-differential master equation for the RDO

˙ ρ(t) = −i

?[HS,ρ(t)]

?

−1

?2

?

ℓ

dq χℓ(t)

?

[Tqℓ,Ωqℓ(t)] + h.c.

?

,

(8)

where the operators Ωqℓand Πqℓare defined as

Ωqℓ(t) = e−itHS

?t

0

T†

ds χℓ(s)Πqℓ(s)ei(s−t)εqℓeitHS,

Πqℓ(s) = eisHS?

qℓρ(s)(1 − fℓ) − ρ(s)T†

qℓfℓ

?

e−isHS,

and fℓis the Fermi function of the lead ℓ describing the

state of the lead before being coupled to the sample. The

operators Tqℓand T†

qℓdescribe the ’transitions’ between

two many-electron states (MES) |α? and |β? when one

electron enters the sample or leaves it:

(Tqℓ)αβ=

?

n

Tℓ

qn?α|d†

n|β?.(9)

The GME is solved numerically by calculating the ma-

trix elements of the RDO in the basis of the interacting

MES, in small time steps, following a Crank-Nicolson al-

gorithm. More details of the derivation of the GME can

be found in Ref. 18. The calculation of the interacting

MES is described in Ref. 14

Mean values of observables can by obtained by tak-

ing the trace of product of the corresponding opera-

tor and the RDO. The time dependent charge den-

sity is obtained from the particle-density operator,

n(x) =?

particle wave functions,

l,mφ∗

l(x)φm(x)d†

ldm, where φl,m(x) are single-

?Q(t,x)? =

?

αβ

ραβ(t)

?

lm

φ∗

l(x)φm(x)?β|d†

ldm|α?. (10)

The total time dependent charge in the sample is found

by integrating over x or by using the number operator

N =?

?Q(t)? = eTr{ρN} = e

?

N

md†

mdm:

N

?

αN

?αN|ρ(t)|αN?, (11)

where αNdenotes the (Coulomb interacting) MESs with

fixed number of electrons N. Remark that one can also

calculate the partial charge accumulated on N-particle

MESs.

The currents in the system are then found by taking

the derivative of Eq. (11) with respect to time,

?J(t)? = JL(t)−JR(t) = e

?

N

N

?

αN

?αN| ˙ ρ(t)|αN?. (12)

The time derivative of the RDO can be substituted by

the right-hand side of the GME [Eq. (8)] and so it is

possible identify the currents in each lead,

?Jℓ(t)? = −1

?2

?

N

N

?

αN

?

dq χℓ(t)?αN|[Tqℓ,Ωqℓ(t)]|αN?

+ h.c.

(13)

We also introduce a p-indexed period average for the cur-

rents (the pth period covers the interval [tp−1,tp]):

jp=1

T

?tp

tp−1

dt Jℓ(t),(14)

which in the periodic phase, i. e. sufficiently long after

the initial transient stage, does not depend on k and on

the lead. T is the period of the pulses, and Qp = Tjp

is the total charge transferred through the sample within

the period p.

The switching-functions in Eq. (3) act on the contact

regions shaded blue in Fig. 1 and are used to mimic

potential barriers with time dependent height. In the

present study we use two kinds of switching-functions.

The first switching-function used in the study is a sine

function,

χℓ(t) = A

?

1 + sin(ω(t − s) + φℓ)

?

,(15)

Page 4

4

where A = 0.5 controls the amplitude and ω = 0.105 the

frequency. The phase shift between the leads is π, φL= 0

and φR= π. The last parameter s = 15 is used to shift

the functions as needed.

The second switching function corresponds to quasi-

rectangular pulses, and it is made by combining two quasi

Fermi functions that are shifted relatively to each other,

χℓ(t) = 1 −

1

et−γℓ

s−δ+ 1−

1

e−(t−γℓ

s)+(Tℓ

p+δ)+ 1,

t ∈ [0, 2Tℓ

p],

(16)

where γℓ

leads (ℓ = L,R) and Tℓ

in the two leads. The pulses are not built with perfect

rectangles for reasons related to numerical stability. The

parameter δ controls the shape of the pulse and is fixed

at the value δ = 10. The time unit used is ?/ts.

The time dependent contact functions are graphed at

the bottom of Figs. 7 and 3. The frequency of the func-

tions in Eq. (15) and (16) were chosen to be similar. The

initial values are χL,R(0) = 0, i. e. the leads and the

sample are initially disconnected.

s= {0, TL

p} defines the phase shift between the

p= 30 is the pulse length, the same

IV.RESULTS

We will use the relative Coulomb energy uC= 1.0. For

a material like GaAs this value would correspond to a

sample length of 9a ≈ 45 nm. Although quite short, this

is an experimentally attainable length. We believe our

results are also valid for longer samples, but the set of our

parameters is also restricted by the computational time

spent in solving the GME which grows very fast with the

number of MES. (A typical calculation took several days

of CPU.) The time unit is ?/ts≈ 0.029 picoseconds. The

lead-sample coupling parameter is also constant, V0= 1.0

(units of ts).

A. Energy spectrum

The MESs of the sample are characterized by the chem-

ical potentials µ(i)

ergy of the interacting MBS number i containing N par-

ticles, i = 0 indicating to the ground state and i > 0 the

excited states. In Fig. 2 we show the chemical potential

diagram for our system. The strength of the Coulomb

interaction is uC = 1. For the single-particle states

(N = 1) the chemical potentials are in fact the single-

particle energies. The effect of the Coulomb interaction

is clearly visible for N > 2.

chemical potential for N = 2, µ(0)

absence of Coulomb interaction it is equal to µ(1)

We select the bias window ∆µ = µL− µR such that

it includes the ground state with N = 3 electrons,

µL = 3.20 and µR = 2.98. The bias window includes

N:= E(i)

N− E(0)

N−1, where E(i)

Nis the en-

For example the lowest

≈ 2.58, whereas in the

2

1

≈ 2.32.

12

N

3

2.0

2.5

3.0

3.5

4.0

µ(i)

N[ts]

N = 1

N = 2

N = 3

µL = 3.37

µR = 3.15

FIG. 2. (Color online) The energy diagram. The blue circles

(

) correspond to single-particle states, the red squares

(

) to two-particle states and the brown diamonds (

to three-particle states. The green solid line (

window ∆µ = µL−µR = 3.20−2.98 = 0.22. The bias window

includes the three particle ground state, but also excited one

and two-particle states. So the expected number of electrons

in the steady state is slightly below three.

)

) is the bias

also excited single- and two-particle states. The single-

particle state is well below the top of the bias and so

the population of this state will be very small, and the

number of electrons in the sample is expected to be some-

where between 2 and 3. Therefore we also expect only

two- and three-particle states to be involved in the trans-

port of electrons through the sample.14

B. Sinusoidal pulses

We begin the time-dependent calculations with N = 3

electrons in the sample, initially assumed in the ground

state. This is done by initializing the diagonal density-

matrix element of the sample corresponding to this state

to one and all the other matrix elements to zero. The left

lead (L) is permanently in contact with the left end of the

sample, i. e. site 1. The right lead (R) is placed on various

other sites, as indicated in Fig. 1. The time evolution is

then followed in short time steps, by using the contact

functions χL,R(t). The charge accumulated in the sample

and the currents in the two leads are calculated at each

time step. In Fig. 3 we show the results with sinusoidal

pulses, corresponding to Eq. (15), and with two different

placements of the R lead: on sites 10 and 3.

We first observe the charge in the sample and its time

evolution shown in the upper panels of Fig. 3. After the

contacts begin to operate the initial charge of N = 3

electrons changes in time. Part of the charge flows into

Page 5

5

the leads, depending on which one is accessible, and the

average charge drops, until a periodic regime is estab-

lished. The lower panels of the figure show the contact

functions. In Fig. 3(a) the right contact is placed at

site 10. The charge in the sample has maxima and min-

ima at the time points when the contact functions are

equal. In between these time points, the charge increases

when the left contact opens further and the right one

closes, and decreases otherwise. The population of the

three-particle and two-particle states are also shown (the

population of the single-particle state being negligible)

and they oscillate in antiphase, i. e. the gain of one is

partly compensated by the loss of the other one.

The currents in the leads are shown in 3(b), and they

have similar shape as the contact functions, except in

the initial transient phase, before the periodic regime is

stabilized. In the first cycle the current in the left lead

is initially negative. The sign rule is that positive cur-

rents correspond to charge flow from the left to the right

lead, and negative currents correspond to the opposite

direction. The initial negative current in the left lead in-

dicates initial charge flow from the sample into that lead

during the first cycle as long as the contact to the right

lead is closed. The main impression of these results is

that the periodic regime qualitatively corresponds to a

linear response of the charge and currents to the contact

strength.

The situation may change when the right contact is

placed on another site, for example on site 3, as shown

in 3(c-d). In this case the oscillations of the charge and

currents are weaker. The current in the left lead is no

longer sinus-like. This shows now a non-linear behav-

ior of the charge response to the same pulses as before.

Some sort of standing waves are created in the sample

and the right contact creates a local perturbation of the

charge fluctuations at that point. Negative currents in

the left lead may occur now during more pulse cycles as

before. This is somewhat surprising, since such currents,

although very small, are actually driven against the bias.

Let’s mention that in the absence of a bias (∆µ = 0)

the currents in both leads oscillate between positive and

negative values, but with zero average, such that no real

pumping effect is obtained in this setup, irrespectively of

the placement of the leads (not shown).

The currents in the leads reflect the charging or dis-

charging of the sample, but these are actually complex

processes, because different states may be occupied with

different time constants, related to the tunneling matrix

elements, and thus the charging and the currents may

have short-time fluctuations. The fine structure of the

currents is thus a complicated issue, which will be dis-

cussed further.

C.Charge distribution in the sample

The charge distribution inside the sample is shown in

the Fig. 4 and it is far from homogeneous. The charge is

0

1

2

3

Q [e]

Left 1, Right 10

(a)

Left 1, Right 3

(c)

−0.005

0

0.005

0.01

I [ets/¯ h]

(b)

(d)

0 100 200 300 400

0

1

t [¯ h/ts]

χℓ(t)

40100 200 300 400

Left leadRight lead

Charge

N2

N3

FIG. 3. (Color online) Charge and current for µL = 3.37,

µR = 3.15, uC = 1.0 and χℓ(t) ∝ sin(ωt).

brown solid line ( ), charge for two particle states black

dashed (

), for three particle states violet dotted (

Current for the left lead blue dashed (

red solid ( ). We consider two locations of the right lead.

(a) Charge, left lead 1, right lead 10. (b) Current, left lead

1, right lead 10. (c) Charge, left lead 1, right lead 3. (d)

Current, left lead 1, right lead 3.

Total charge

).

), for the right lead

averagedin time over an entire period of the contact func-

tions when the system is in a periodic regime. In the case

shown the right contact is placed on site 10. The charge

distribution does not qualitatively change for other place-

ments of the right contact (not shown). The distribution

is symmetric along the sample, in spite of the presence

of the bias window, which shows that the contacts be-

tween the sample and the leads are actually weak in our

case. We can say that the charge distribution follows

the geometrical extend of those single-particle states that

contribute to the active two- and three-particle MBS.

Next, in Figs. 5 and 6, we show the deviation of the

charge density from the mean value, on each lattice site,

for selected time moments during half a cycle. For the

other half-cycle the reverse motion occurs.

placements of the right lead ate again selected at sites

10 and 3. Standing waves are clearly seen.

contact configuration L1R10 (Fig. 5) the standing-wave

pattern shows something between two and three wave-

lengths. Nodes and antinodes can be distinguished, and

also a global up and down motion mode seems to occur.

But it is clear that the amplitude of the charge oscilla-

tions at the contact sites is quite large.

When the right contact is on site 3 (Fig. 6) only about

two wavelengths may be seen, at least for t < 3T/8, but

also the charge seems to oscillate with different ampli-

The two

For the