Physica E 70 (2015) 217–224

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Physica E

http://d1386-94

n CorrE-m

tomasz.

journal homepage: www.elsevier.com/locate/physe

Charge and current beats in T-shaped qubit–detector systems

R. Taranko n, T. KwapińskiInstitute of Physics, M. Curie-Skłodowska University, 20-031 Lublin, Poland

H I G H L I G H T S

� We consider different T-shaped quantum dot detectors of the qubit charge.

� Very clear beat patterns are observed in the oscillations of the qubit occupancy.� The detector current reveals also the beating structure.� The period of beats decreases with the qubit–detector interaction.� The beating structures are sensitive on the initial detector occupancy (memory effect).

a r t i c l e i n f o

Article history:Received 23 December 2014Received in revised form9 March 2015Accepted 17 March 2015Available online 18 March 2015

Keywords:Charge qubitDouble quantum dotQubit dynamicsQuantum beatsEquation of motion method

x.doi.org/10.1016/j.physe.2015.03.01377/& 2015 Elsevier B.V. All rights reserved.

esponding author.ail addresses: ryszard.taranko@umcs.pl (R. Tarkwapinski@umcs.pl (T. Kwapiński).

a b s t r a c t

The time evolution of a charge qubit coupled electrostatically with different detectors in the forms ofsingle, double and triple quantum dot linear systems in the T-shaped configuration between two re-servoirs is theoretically considered. The correspondence between the qubit quantum dot oscillations andthe detector current is studied for different values of the inter-dot tunneling amplitudes and the qubit–detector interaction strength. We have found that even for a qubit coupled with a single QD detector, thecoherent beat patterns appear in the oscillations of the qubit charge. This effect is more evident for aqubit coupled with double or triple-QD detectors. The beats can be also observed in both the detectorcurrent and the detector quantum dot occupations. Moreover, in the presence of beats the qubit oscil-lations hold longer in time in comparison with the beats-free systems with monotonously decayingoscillations. The dependence of the qubit dynamics on different initial occupations of the detector sites(memory effect) is also analyzed.

& 2015 Elsevier B.V. All rights reserved.

1. Introduction

Recently coupled quantum dots (QDs) systems have been at-tracting much attention mainly because of their possible applica-tions in nanoscale devices. The transient and steady-state elec-tronic transport through various geometries of QDs coupled withtwo or more electron reservoirs were investigated, e.g. [1–15]. Alsothe charge or spin pumping in the linear QD systems was theo-retically predicted e.g. in the case of the train-like inter-dot cou-pling disturbance or due to the spin-flip interactions [14]. Gen-erally, for the QD systems coupled with electrodes, the transientcharge and current oscillations decrease monotonically with time.However, for QDs isolated from the leads, the QD occupationprobabilities vary in time with no-decreasing amplitudes. It isinteresting that for two coupled subsystems composed of double

anko),

QDs (DQD), the charge beat pattern can be observed which man-ifests by regular oscillations with alternate small and high ampli-tudes. The analysis of such beat patterns as well as the transientcurrent oscillations observed in the QDs systems can give usefulinformation about some system parameters defining the con-sidered nanoscopic devices, e.g. [1,16,17].

The simplest nontrivial quantum mechanical system whichplays a key role in quantum computation is the DQD, the so-calledqubit. In the case of finite tunneling amplitudes between QDs thesystem can be described by the coherent superposition of twocharge states. In order to analyze the qubit quantum state one hasto perform the measurement using the external nanoscopic de-vice. However, each measurement process introduces decoherenceof the qubit state and maintaining the coherence of this state is animportant task on the way of the construction of a quantum in-formation processor. The qubit is usually placed in close proximityto the charge sensitive detector and in such configuration thedetector current depends on the qubit QD occupancy. Detectorsare usually realized in the form of a single QD coupled with two

R. Taranko, T. Kwapiński / Physica E 70 (2015) 217–224218

electrodes (the so-called SET – single electron transistor), e.g. [18–22], in the form of DQD or triple-QD (TQD) in a linear configura-tion embedded between the source and drain electrodes, e.g.[19,23–26], or in the form of a quantum point contact, e.g. [27–30].Also other DQD systems in the horizontal geometry were con-sidered to investigate the qubit decoherence dynamics, e.g. theDQD gated by electrodes [31] or the system where qubit wasplaced between the detector QDs, [32]. Note that in one of ourprevious papers [26] short QD wires in the horizontal geometrywere proposed as the qubit charge detectors. We have found thatfor certain qubit–detector configurations one observes relativelylong time-dependent oscillations of the qubit charge.

In all papers mentioned above the qubit–detector interactionresults in damped oscillations of the qubit dots occupation prob-abilities. Depending on the system parameters and the detectorgeometry, the qubit decoherence can proceed relatively slow intime which is very important for the qubit measurements andpotential applications of such systems. In the literature chargequbits are usually coupled electrostatically with one (or more) ofthe detector sites which conducts electric current directly betweenelectrodes. Such connections determine relatively fast mono-tonously qubit decoherence. Note that for a single qubit or for aqubit coupled with a single QD (DQD–QD system) isolated fromthe leads the charge oscillation amplitude is constant in time anddoes not reveal any beat structure. However, for two interactingqubits (DQD–DQD system) isolated from the leads the occupationprobability can oscillate with regular beat patterns related withthe coupling parameters [20]. It is interesting if these beats canappear for a single qubit in the presence of different qubit chargedetectors. Thus in this work we propose a new charge qubit de-tector composed of two or three QDs in the T-shaped configurationbetween two electrodes. In our model the qubit interacts with theQD far-removed from the electrodes (see Fig. 1B, C) and the de-tector current does not disturb directly the qubit dynamics. In thatcase long-time qubit charge oscillations should be observed be-cause the detector current influences the qubit state to a relativelysmall extent. We predict that for the DQD detector coupled withthe leads, the qubit should reveal a beat structure. However, it isnot obvious for a single QD detector and the triple-QD systemconsidered here. Note that the beat patterns of the qubit occu-pancies were not reported before in the literature for a single qubitcoupled with a charge detector. The qubit beats can influence thecoherence in two opposite ways: they can destroy the qubit chargeoscillations very fast (undesirable phenomenon) or can sustainthese oscillations in time which extends the measurement processsignificantly. We discuss this problem in this paper. Additionally,we consider here the so-called ‘memory effect’ i.e. the role of the

Fig. 1. The sketch of the qubit–detector systems considered in the paper. The qubit(two coupled quantum dots: x and y) is coupled electrostatically via U parameterwith one of the detector QDs. Panels A, B and C correspond to the single-QD,double-QD and triple-QD detectors, respectively.

initial conditions (preparation of the detector state at the momentwhen the qubit begins to interact with the detector) which cansubstantially change the qubit dynamics and beating structure ofthe charge oscillations.

The paper is organized as follows. In Section 2 we present themodel and the calculation method. The main results of the paperare discussed in Section 3 for different qubit–detector setups.Section 4 is a summary.

2. Model and formalism

We consider the qubit (double quantum dot) interacting withdifferent detectors in the form of one, two or three QDs coupledwith two metallic leads in the T-shaped geometry, which isschematically shown in Fig. 1. Each QD is described by a singlelevel εi and can be coupled with other QDs by the inter-dot tun-neling amplitudes Vij (for detector QDs) or Vxy (for qubit QDs). Theelectrons residing on the neighboring qubit and the detector QDsinteract electrostatically with the strength U. The total Hamilto-nian of the system can be written as a sum of three parts:H H H Hqub qub det det= + +– , where Hqub corresponds to the DQDplaying the role of the qubit, Hdet describes the detector (QD, DQDor TQD coupled with leads) and Hqub det– stands for the qubit–de-tector interaction. All parts of the Hamiltonian can be written interms of the creation, annihilation and number operators for theelectrons localized on i-th QD and in the α lead (with momentumk), ci

+, ci, ckα+ ckα , ni, nkα, respectively. As an example, we write here

the Hamiltonian for the setup sketched in Fig. 1B (very similarrelations one can write for the other considered setups):

H n n V c c

Un n n n V c c

V c c n

h. c.

h. c.

h. c. .(1)

x x y y xy x y

x

k L Rk k

k L Rk k

2 1 1 2 2 12 1 2

, ,1

, ,

∑ ∑

ε ε

ε ε

ε

= + + +

+ + + + +

+ + +α

α αα

α α

+

+

=

+

=

Here, all terms in the first line correspond to Hqub, the term with Udescribes the qubit–detector interaction, Hqub det– , and the otherparts correspond to Hdet. The electron energies in the leads and thetunneling amplitudes between the α-lead and the detector QD aredenoted by kε α and Vkα, respectively. All system parameters, such asεi, kε α, Vxy, Vij, Vkα can vary in time.

We calculate the time-dependent QDs occupanciesn t c t c t( ) ( ) ( )i i i= ⟨ ⟩+ and the currents flowing in the system using theequation of motion (EM) method for appropriate correlationfunctions. The current flowing from the α-th lead, j t( )α , can beobtained from the evolution of the α-lead occupation numberoperator and reads (e.g. [33])

j t V t c t c t( ) 2Im ( ) ( ) ( ) ,(2)k

k k1∑= ⟨ ⟩α α α+

where the operators are given in the Heisenberg representation,the bracket ⟨⋯⟩ denotes the quantum-statistical average and theunits e 1= = are applied. Using the explicit expression for c t( )kα ,the current can be given in the form

⎛⎝⎜⎜

⎞⎠⎟⎟j t V t c t c t i dt K t t c t c t( ) 2Im ( ) ( ) ( ) ( , ) ( ) ( ) ,

(3)kk k

t

t

1 0 1 1 1 1 10

∫∑= ˜ ⟨ ⟩ − ⟨ ⟩α α α α+ +

where ( )V t V t i dt t( ) ( )exp ( )k k t

tk1 1

0∫ ε˜ = −α α α , V t V u t( ) ( )k =α α α and

⎛⎝⎜⎜

⎞⎠⎟⎟K t t V u t u t i dt t( , ) ( ) ( )exp ( ) .

(4)k t

t

k12

11

∫∑ ε= | | − ′ ′α α α α α

The function u t( )α is responsible for the initial switching on thecoupling parameters, i.e. u t( ) 0=α for t t0< and u t( ) 1=α for t t0≥ .

R. Taranko, T. Kwapiński / Physica E 70 (2015) 217–224 219

In the following we put t 00 = . Note that the expression for thecurrent is non-local in time and the integral kernel K t t( , )1α con-tains the information about the lead electron density of states,D ( )ωα , and can be written as follows:

K t t V u t u t e D t t( , ) ( ) ( ) ( ), (5)i dt t

12

1( )

1t

t

1∫= | | −α α α α

Δα

− ′ ′α

where D t t D e d( ) ( ) i t t1

( )1∫ ω ω− =α αω

−∞

∞ − − and we assume

t t( ) ( )k k0ε ε Δ= +α α α . Here t( )L R/Δ represents the time-dependence of

the single particle energies in the left or right lead. The timeFourier transform, D t t( )1−α , e.g. for the square or elliptical densityof states takes the form of the damping oscillatory function

w t t w t tsin( ( )/2)/ ( )/2− ′ − ′ or J w t t w t t( ( )/2)/ ( )/21 − ′ − ′ , respec-tively, where J1 is the Bessel function and w is the lead bandwidth.For sufficiently large w the so-called wide-band limit (WBL) can beused with the spectral density function

t u t V u t( ) 2 ( ) ( ) ( )k k2 2 0 2Γ π δ ε ε Γ= ∑ | | − ≡α α α α α α . Using the EM method one

obtains the following integro-differential equations for the QDoccupations:

⎪

⎪

⎪

⎪

⎧⎨⎩

⎛⎝⎜⎜

⎞⎠⎟⎟

⎫⎬⎭

dn tdt

V t c t c

i dt K t t K t t c t c t

iV c t c t

( )2Im ( ) ( ) (0)

( , ) ( , ) ( ) ( )

( ) ( ) ,(6)

k L RkL k

t

L R

1

, ,1

01 1 1 1 1 1

12 1 2

∫

∑= ˜ ⟨ ⟩

− + ⟨ ⟩

− ⟨ ⟩

αα

=

+

+

+

⎧⎨⎩⎫⎬⎭

dn tdt

V c t c t( )

2Im ( ) ( ) ,(7)

212 1 2= − ⟨ ⟩+

⎧⎨⎩⎫⎬⎭

dn tdt

V c t c t( )

2Im ( ) ( ) .(8)

xxy x y= ⟨ ⟩+

In the following we consider the case for which the lead energybandwidth is the largest energy in the system so the WBL is asufficiently good approximation. Now, Eq. (3) for the current andEq. (6) for n t( )1 become local in time, i.e. the second terms in theseequations (with time integrals) are reduced to i t n t( ) ( )/21Γ− α and

i t t n t( ( ) ( )) ( )/2L R 1Γ Γ− + , respectively. However, the set of Eqs. (6)–(8)is not closed and the corresponding equations for c t c t( ) ( )1 2⟨ ⟩+ ,c t c t( ) ( )x y⟨ ⟩+ or c t c( ) (0)k1⟨ ⟩α

+ generate the higher-order correlationfunctions. It appears that only two kinds of these functions arepresent in the subsequent equations of motion. The first kindcorresponds to the averages of a number of the QD electronannihilation and creation operators taken at a given time t, e.g.c t c t n t( ) ( ) ( )1 2 3⟨ ⟩+ , c t c t c t c t( ) ( ) ( ) ( )1 2 3 4⟨ ⟩+ + , …, and they form a closed set ofdifferential equations. For the system shown in Fig. 1 consisting ofN¼3, 4 or 5 QDs, the corresponding set of the coupled equationcontains 5, 15 or 49 different correlation functions of this type,respectively. The functions of the second kind correspond to theaverages of a number of QD electron operators taken at a giventime t and leads electron creation and annihilation operators takenat the initial time t¼0, e.g. c t c( ) (0)kL1⟨ ⟩+ , which appears in theformula for the current in Eqs. (3) and (6). This function satisfiesthe following integro-differential equation:

⎛⎝⎜

⎞⎠⎟

d c t cdt

i c t c iV c t c

iU c t n t c iV t n

dt K t t K t t c t c

( ) (0)( ) (0) ( ) (0)

( ) ( ) (0) ( ) (0)

( , ) ( , ) ( ) (0) ,(9)

kLkL kL

kL L kL

t

L R kL

11 1 12 2

1 2

01 1 1 1 1∫

ε⟨ ⟩

= ⟨ ⟩ + ⟨ ⟩

+ ⟨ ⟩ + ˜

− + ⟨ ⟩

++ +

+ ⁎

+

where n (0)kL is the initial Fermi distribution function in the leftlead at t¼0. Assuming the WBL approximation the last term in Eq.

(9) transforms to t t c t c( ( ) ( )) ( ) (0) /2L R kL1Γ Γ− + ⟨ ⟩+ . However, as usuallyin the EM method, new functions of the second type appear,c t n t c( ) ( ) (0)kL1 2⟨ ⟩+ and c t c( ) (0)kL2⟨ ⟩+ . Unfortunately, these functions donot form a closed set of equations and some approximations areneeded. We apply the decoupling procedure as follows [34,35]:

f c t c t c c f c t c t n( ( ), ( )) (0) (0) ( ( ), ( )) (0), (10)i j k k i j k k k,1 2 1 2 1δ δ⟨ ⟩ ≃ ⟨ ⟩α β αβ α

+ + +

f c t c t c c f c t c t c c( ( ), ( )) (0) (0) ( ( ), ( )) (0) (0) 0, (11)i j k k i j k k1 2 1 2⟨ ⟩ ≃ ⟨ ⟩⟨ ⟩ =α β α β

+ +

f c t c t c c f c t c t c c( ( ), ( )) (0) (0) ( ( ), ( )) (0) (0) 0, (12)i j k k i j k k1 2 1 2⟨ ⟩ ≃ ⟨ ⟩⟨ ⟩ =α β α β

+ + + + + +

where f c c( , )i j+ are the appropriate functions appearing in the

subsequent steps of the EM method. This scheme corresponds tothe Hubbard I approximation but is applied to the time-dependentprocesses, cf. [36]. Assuming the above decoupling procedure, thecorresponding set of differential equations for the second typefunctions consists of 4, 16 and 61 equations for the system with 3,4 or 5 QDs depicted in Fig. 1 (for the sake of brevity we do notshow them here). We solve these equations numerically and thenumber of k-vectors taken in the calculations of the correspondingsummations usually extends from 500 to 3000 dependingon the considered system. Similar equations can be also obtainedfor the qubit coupled with other detectors (Fig. 1A and C). We alsocalculate the inter-dot detector currentsj t V c t c t( ) 2Im ( ) ( )ij k ij i j= − ∑ ⟨ ⟩+ which is expressed by the appropriatecorrelation functions.

In order to check the correctness of the decoupling procedurewe have used in this paper to terminate the infinite hierarchy ofhigher-order functions we performed additional calculations ofthe QD occupation probability and the detector current for thequbit coupled with single and double-QD detectors considered inRefs. [18,19,22] where the method of the rate equations for thedensity matrix was used. The results we have obtained for thesame sets of the system parameters entirely reproduce the lit-erature results so in the following we shall use the EOM methodfor more complex systems of the double and triple-QD detectors inthe T-shaped configurations.

3. Results and discussion

In this section we present the results and discussion on the QDsoccupation probabilities and the electron current flowing betweenthe left and right leads. The central system (qubit charge detector)stands for single, double or triple QDs coupled with the leads inthe T-shaped configuration as is shown schematically in Fig. 1. Thequbit does not interchange electrons with any part of the detectorbut is coupled with it electrostatically. In our numerical calcula-tions we set e k 1B= = = and 10Γ = is the energy unit ( L RΓ Γ Γ= =is assumed). The current and time are expressed in the units ofeh/ 0Γ and / 0Γ , respectively. We consider mainly the qubit QD oc-cupation, nx(t), calculated for different parameters characterizingthe system i.e. the inter-dot tunneling couplings, the qubit–de-tector interaction strength, U, and the spectral density function, Γ.In our investigations we concentrate on the effect which was notreported in the literature before i.e. the beat patterns (of the cur-rent and QD occupations) in the qubit–detector systems. In someaspects appearing of such beats is very desirable as in this case thequbit oscillations are observed longer in time and also because thestructure of beats allows us to deduce about the couplingparameters.

In Fig. 2 we present the qubit occupation probability, nx(t), as afunction of time for the detectors shown in Fig. 1. The curves goingfrom the top to the bottom in both panels correspond to the qubitcoupled with a single dot, double-dot and triple-dot being in the

-1

0

1

n x(t)

ΓL/R=1A

B

C-1

0

1

n x(t)

ΓL/R=1A

B

C-1

0

1

n x(t)

ΓL/R=1A

B

C

-2

-1

0

1

15 20 25 30 35 40

nx(

t)

time

ΓL/R=0.2A

B

C

-2

-1

0

1

15 20 25 30 35 40

nx(

t)

time

ΓL/R=0.2A

B

C

-2

-1

0

1

15 20 25 30 35 40

nx(

t)

time

ΓL/R=0.2A

B

C

Fig. 2. Nearby qubit QD occupation, nx(t), as a function of time for different formsof the detector depicted in Fig. 1. The upper (bottom) panel corresponds to the

1L RΓ Γ Γ= = = ( 0.2Γ = ). The tunneling coupling between QDs is V¼1 for the de-tector and V 4xy = for the qubit, energy levels of all QDs are equal to 0iε = ,

10L Rμ μ= − = and U¼4. The qubit was ‘frozen’ in the configuration nx¼0, ny¼1 fort 15< , i.e. until the detector QD occupancies and currents jL and j12 achieved theirstationary values. The curves B and C are shifted down by 1 and 2 for clarity.

0

0.25

0.5

0.75

1

0.4

0.5

0.5

0.5

n x(t)

n 1(t)

, n2(

t) nx

n2

n1

0

0.25

0.5

0.75

1

0.4

0.5

0.5

0.5

n x(t)

n 12(

t) nx

n2

n1

0

0.25

0.5

0.75

1

0.48

0.5

0.52

0.54

n x(t)

n 12(

t) nx

n2

n1

-0.1

0

0.1

20 30 40 50

0.4

0.4

0.4

j 12

j L

time

j12

jL

-0.1

0

0.1

20 30 40 50

0.47

0.48

0.49

j 12

j L

time

j12

jL

Fig. 3. Nearby qubit QD occupation, nx(t), of the detector QDs, n t( )1 and n t( )2 (upperpanel), and the currents flowing from the left lead, jL(t), and between the detectorQDs, j t( )12 (bottom panel), as functions of time for the T-shaped DQD detectorshown in Fig. 1B. The bias voltage is 20L Rμ μ= − = , 1Γ = . The other parametersand initial conditions as in Fig. 2.

R. Taranko, T. Kwapiński / Physica E 70 (2015) 217–224220

T-shaped configuration between leads. Here we consider the fol-lowing initial conditions of the system (the results obtained fordifferent conditions will be discussed later). It is assumed that fort 0< all detector elements are isolated and at t¼0 the couplingsbetween them are switched on. After some time (we assumet¼15) the transient oscillations of the current and QD occupancies(due to the abrupt change of the coupling parameters) are washedout and the detector relaxes to its steady state. It is also assumedthat up to t¼15 the qubit electron does not oscillate and is loca-lized on the far-removed qubit dot (V 0xy = , ny¼1, nx¼0). Next, att¼15 we set V 0xy ≠ and the qubit electron begins to oscillate be-tween the dots x and y (we call this effect simply: the qubit os-cillations) and interacts with the detector. Such a procedure can beachieved experimentally applying the appropriate voltage pulse tothe gate defining the qubit QDs. The upper (bottom) panelcorresponds to a large (small) value of the spectral density func-tion 1Γ = ( 0.2Γ = ). As one can see for 1Γ = and for the single QDdetector the qubit oscillations disappear smoothly with time(curve A, upper panel). However, for the other detectors, DQD andTQD, clear beat patterns are visible (curves B and C). These beatsare characterized by two frequencies: the high one related to theisolated qubit oscillations and the low one which corresponds tothe common qubit–detector ones. Moreover, as one can see, hardlyvanishing beats are observed for the TQD detector (C curve) as inthis case the detector current flows directly through the first de-tector QD whereas the qubit interacts with the third detector dot(mostly removed from the leads). Thus the current does not dis-turb the qubit dynamics directly as there are two ‘buffer’ QDsbetween the qubit and the QD coupled with the leads (note thatthere is only one ‘buffer’ dot for the case B). To explain the natureof beats we discuss this effect in the next subsections in moredetail. As one can see beat patterns are not observed for the singleQD detector (upper panel for 1Γ = ). It is in accordance with a

qubit coupled with a single QD isolated from the leads i.e. for0Γ = . In such a case non-vanishing beats-free oscillations are

observed for both the occupied, n t( ) 11 = , and empty QD state,n t( ) 01 = . However, for the detector QD weakly coupled with theleads (bottom panel for 0.2Γ = ), the structure of beat patterns isvisible even for a single QD detector (curve A). Here an electronwhich moves from the left lead to the right one spends enoughtime at the QD (small Γ parameter) to interact effectively with thequbit electron (which oscillates between both qubit QDs). Thisleads to the beating structure in the qubit oscillations. For the DQDand TQD detectors (curves B and C) these oscillations are moreevident, with relatively large hardly vanishing oscillation ampli-tudes. In these cases an electron localized at the detector QDcoupled electrostatically with the qubit cannot leakage fast to theleads due to a small inter-dot coupling (V Vxy< ) and interacts re-latively long time with the qubit electron. This is the reason thatfor the DQD or TQD detectors the beating structure is observed fora small and large spectral density functions. However, the problemwill be further studied because the appearance of beats dependsalso on other system parameters. Looking at both panels we cansee that the qubit oscillations persist longer in time for morecomplex detectors (e.g. for the TQD detector) in comparison with asingle-dot detector (A curve). Thus, as a main feature of our studieswe have found very clear beat patterns of the qubit oscillations forall considered detectors – this effect has not yet been reported inthe literature.

It is worth noting that for the case of small bias voltage (notshown here), the qubit charge oscillations disappear more slowlyin time in comparison with large voltage bias. This effect is aconsequence of small electron current flowing through the de-tector in the case of small voltage which slightly disturbs the qubitdynamics. In this case, for all considered detectors, the charge doesnot show such clear beat patterns (cf. also the results shown inFig. 6).

-2

-1

0

1n x

(t)

Γ=0

Γ=0.05

Γ=0.5Γ=1-2

-1

0

1n x

(t)

Γ=0

Γ=0.05

Γ=0.5Γ=1-2

-1

0

1n x

(t)

Γ=0

Γ=0.05

Γ=0.5Γ=1-2

-1

0

1n x

(t)

Γ=0

Γ=0.05

Γ=0.5Γ=1

-3

-2

-1

0

1

0 15 30 45 60 75 90

nx(

t)

time

Γ=0

Γ=0.25

Γ=0.5

Γ=1-3

-2

-1

0

1

0 15 30 45 60 75 90

nx(

t)

time

Γ=0

Γ=0.25

Γ=0.5

Γ=1-3

-2

-1

0

1

0 15 30 45 60 75 90

nx(

t)

time

Γ=0

Γ=0.25

Γ=0.5

Γ=1-3

-2

-1

0

1

0 15 30 45 60 75 90

nx(

t)

time

Γ=0

Γ=0.25

Γ=0.5

Γ=1

Fig. 4. Nearby qubit QD occupation, nx(t), as a function of time for the detectorgeometry shown in Fig. 1B. Upper curves in both panels correspond to the DQD–DQD system isolated from the leads system i.e. 0Γ = and n n 1y1 = = , n n 0x2 = =for t 0< . Three lower curves in the upper (lower) panel correspond to the condi-tions 0Γ = up to t¼15 and 0.05, 0.5Γ = and 1 ( 0.25, 0.5Γ = and 1) for t 15≥ , re-spectively. The other parameters are 20L Rμ μ= − = , 0iε = and V V 1xy12 = = , U¼0.5(V V1, 4xy12 = = , U¼4) – upper (bottom) panel.

R. Taranko, T. Kwapiński / Physica E 70 (2015) 217–224 221

3.1. Double-QD detector

In this section we consider the qubit coupled with the detectorwhich stands for a DQD system in the T-shaped configuration. Forthis geometry the beat patterns of the qubit QDs occupationprobability are much more evident than for a single QD detector.We are interested if these beat patterns can be recorded by thedetector current which is very important from the experimentalpoint of view. Thus in Fig. 3 we show the time-oscillations of thequbit and detector QD occupation probabilities, the left currentand the detector inter-dot current. As one can see all presentedcharacteristics manifest very clear beat patterns. The oscillations ofthe detector QDs occupations are clearly correlated with the qubitQD occupations and the currents jL(t) and j t( )12 . We observe thatminima of beats in the qubit QD occupation, nx(t), coincidencewith the maxima of n t( )1 and n t( )2 and also with those of jL(t) andj t( )12 . Thus we have found that there exists a clear correspondencebetween the beat pattern in the qubit and the detector currentoscillations which can be used in possible qubit–detectorexperiments.

In order to examine the phenomenon of the qubit occupationbeats more carefully we analyze the role of the detector spectraldensity functions, i.e. the lead-QD couplings, Γ. In Fig. 4 we shownx(t) for the weak (strong) Coulomb interaction U¼0.5 andV Vxy12 = – upper panel (U¼4, V Vxy12 < – bottom panel) and fordifferent Γ couplings. The upper curves in both panels correspondto the qubit interacting with the DQD detector decoupled from theleads, 0Γ = . Here we assume that for t 0< the electrons occupytwo most outlying dots of the system i.e. n n 1y1 = = , n n 0x2 = =for t 0< and for t 0≥ electrons in both DQDs start to oscillate. Weobserve a very distinct beat pattern for the equal tunneling

amplitudes, V Vxy12 = , and a small interaction U¼0.5 (upper panel).For different couplings in both DQD systems, V Vxy 12> , and forlarger U, U¼4 (bottom panel), the oscillation period of nx(t) is verysmall and the beats of nx(t) are still visible although they are not asspectacular as in the former case. In the next subsection we dis-cuss this case analytically and show the origin of such behavior ofnx(t). Now we are interested how these beat patterns change if thecouplings with both leads are switched on. We expect that espe-cially for small coupling strengths, the beating structure of nx(t)should be preserved for some time before it is washed out due tothe coupling with leads. All other curves depicted in Fig. 4 presentnx(t) for the increasing Γ parameter as indicated in both panels.Here we assume the same initial conditions as for the upper curvesand at t¼15 the couplings with the leads are switched on ( 0Γ >for t 15≥ ). One can observe different pictures in the qubit electronoscillations depicted in both panels. For smaller values of the in-terdot tunneling amplitudes, V V 1xy12 = = (upper panel), thebeating structure of nx(t) survives only for sufficiently small Γ. Forlarger values of Γ the beats disappear and the amplitude of nx(t)oscillations slowly decreases. For larger values of the qubit inter-dot coupling, Vxy, and larger qubit–detector interaction, U (bottompanel), the beats of nx(t)are more visible even for larger values ofΓ, but the number of beats and oscillation amplitude decreasevery fast with Γ. Note the faster decreasing of the qubit electronoscillations for larger Vxy (bottom panel) in comparison with thecase of smaller Vxy (upper panel).

3.2. Occupation beats in the two-qubit system

In order to analyze the origin of the qubit beats structure inmore detail here we study the qubit–detector system shownschematically in Fig. 1B but without coupling with the leads. Sucha system corresponds to two DQDs in the linear configurationwhere two neighboring QDs (the second and the third) are cou-pled electrostatically via the Coulomb interaction U. The wavefunction describing this system can be written in the form

t a t j( ) ( )j j14Ψ| ⟩ = ∑ | ⟩= where the basis functions j| ⟩, j 1, 2, 3, 4= ,

correspond to the possible electron configurations i.e. the first andthe third dots occupied by electrons (1,3), and similar for the firstand the fourth dots (1,4), the second and the third dots (2,3) andthe second and the fourth dots (2,4), respectively. In the followingwe show analytically that for some system parameters, the time-dependent qubit occupancy exhibits very clear beat patterns. Tosimplify the calculations we assume the same values of the QDsenergy levels, 0jε = , the same coupling parameters, V V V12 34= = ,and the Coulomb interactionU V /2= . For this set of parameters wecalculate the QD occupancy, e.g. n t( )1 , which can be determinedfrom the knowledge of a t( )1 and a t( )2 coefficients:n t a t a t( ) ( ) ( )1 1

22

2= | | + | | . The considered model can be exactly diag-onalized and it is straightforward to show that the system energyeigenvalues are E V Vsin( /3) /61

73 0α π= − + + , E 02 = ,

E V Vsin /6373 0α= + , and ( )E V Vsin /3 /64

73 0π α= − + , where

arcsin013

549

α = , and the corresponding eigenstates j|˜ ⟩ take the form

j( 1, 3, 4)˜ = :

( ) ( )( )j f E E E( ) 1 2 ( ) 3 4 , 2 2 4 , (13)j j j12

η ξ˜ = + + + ˜ = −

where

( ) ( )f E E E E V E E V

E E E V V E V

E E V E V

( ) 2 (1 ( /2)) 1 /( /2) ,

( ) ( /2)/( (2 /2)),

( ) /( (2 /2)). (14)

2 2 2 2 2 1/2

2

ξ

η

= + + − + −

= − −

= −

− −

Assuming t( 0) 1Ψ| = ⟩ = | ⟩ the function t( )Ψ| ⟩ can be written as

-0.5

0

0.5

1

0.48

0.5

0.5

n x(t)

n 1, n

2, n

3

nx

n2

n1

n3

0.52

0.54

n x(t)

n

nx

n2

n1

n3

nx

n2

n1

n3

nx

n2

n1

n3

-0.1

0

0.1

15 20 25 30 35 40

-0.01

0

0.01

j 23

j 12,

j Ltime

j23

j12

jL (-0.46)

12

j23

j12

jL (-0.46)

j23

j12

jL (-0.46)

Fig. 5. The same as in Fig. 3 but for the triple-QD T-shaped detector, Fig. 1C. The leftcurrent jL is shifted down by �0.46 for clarity. The other parameters are

10L Rμ μ= − = , 0iε = and V V 112 23= = , V 4xy = , U¼4 and the initial conditions as inFig. 3.

-1

0

1

n x(t)

μ L/R=(1,-1)Vxy=1

2

4-1

0

1

n x(t)

μ L/R=(1,-1)Vxy=1

2

4-1

0

1

n x(t)

μ L/R=(1,-1)Vxy=1

2

4

-2

-1

0

1

15 20 25 30 35 40

nx(

t)

time

μ L/R=(10,-10) Vxy=1

2

4

-2

-1

0

1

15 20 25 30 35 40

nx(

t)

time

μ L/R=(10,-10) Vxy=1

2

4

-2

-1

0

1

15 20 25 30 35 40

nx(

t)

time

μ L/R=(10,-10) Vxy=1

2

4

R. Taranko, T. Kwapiński / Physica E 70 (2015) 217–224222

follows: t j e j( ) 1jiE t

14 jΨ| ⟩ = ∑ ⟨˜ | ⟩ |˜ ⟩=

− , and using Eqs. (13) one obtains

a t j e( ) 1jiE t

1 14 2 j= ∑ |⟨ |˜ ⟩|=

− and a t j j e( ) 2 1jiE t

2 14 j= ∑ ⟨ |˜ ⟩⟨˜ | ⟩=

− . Finally n t( )1

reads

n t f E E f E E E t

f E f E E E E E t

( )14

( )(1 ( )) ( )cos( )

2 ( ) ( )(1 ( ) ( ))cos(( ) ).(15)

jj j

jj j

i j j ii j i j i j

11,3,4

4 2

1,3,4

22

1,3 3,4( )

2 2

∑ ∑

∑ ∑

ξ

ξ ξ

= + + + −

+ + −

= =

= = >

For example for V¼1 this expression can be approximately writtenin the form

n t t t( ) 0.498 0.262cos(1.892 ) 0.230cos(2.146 ), (16)1 ≃ + +

and because there are two cosine functions with similar (but notequal) frequencies thus the system exhibits the beat patterns withthe period T 25beat ≃ . It is important to emphasize that the systemwe consider here is fully symmetrical with the same initial con-ditions for both DQD subsystems and observation of beats seemsrather surprising. However, for this simple system there still existtwo frequencies which are related to (i) isolated single DQD qubitand (ii) two coupled qubits via U parameter (the whole system).Superposition of these frequencies leads to beats in the qubitcharge oscillations. In the case when the system symmetry isbroken, e.g. for V V12 34≠ occupation oscillations have much morecomplicated form due to different frequencies of the isolated qu-bits (for U¼0). From the inspection of the numerical results ob-tained for any ratio of V U/ 1> the period of the beating structurecan be roughly estimated as

TVU

T V4 ( ), (17)beat qubit≃

where Tqubit(V) is the period of the occupation oscillations in theisolated qubit. Note that this formula works better for a larger ratioof V U/ .

Fig. 6. The nearby qubit QD occupation, nx(t), as a function of time for the triple-QDs detector shown in Fig. 1C for different values of the qubit tunneling amplitudeV 1, 2xy = and 4, respectively. The upper (bottom) panel corresponds to

1L Rμ μ= − = ( 10L Rμ μ= − = ). The other parameters are 0iε = , V V 112 23= = ,V 4xy = , U¼4 and the initial conditions as in Fig. 2.

3.3. Triple-QD detector

Here we consider the system composed of the qubit coupled

with three QDs detector (TQD). We expect that for this detectorthe qubit can stay longer in its coherent state especially in thepresence of the charge beating phenomenon. In Fig. 5 we presentthe QD occupancies (upper panel) for the TQD detector placed inthe T-shaped configuration between two leads (see Fig. 1C) andthe currents flowing between the leads and detector QDs (bottompanel). The beating structure of nx(t) is very clear and dumping ofoscillations is relatively small in this case. Note that the oscillationsof the detector QDs occupations, n t( )1/2/3 , show also the beatingstructure with the same beats period and maximum amplitudecorresponding to the nodes of nx(t) oscillations, cf. also Fig. 3. It isinteresting that the oscillation amplitudes of n t( )2 and n t( )3 arecomparable to each other but the oscillations of the detector QDcoupled with leads, n t( )1 , is substantially smaller (so the beatpattern is hardly visible). The current oscillations show also thebeating structure with maximum amplitudes corresponding tonodes of nx(t) beats. Note large oscillations of j t( )23 and relativelysmall oscillations of j t( )12 and jL(t). As it was expected for the TQDdetector, the oscillations of nx(t) (with the beats) are more ex-pressive in comparison with the case of the double-dot detectorand persists much longer in time. From the comparison of theresults shown in Figs. 3 and 5 one can say that the detector QDsplay a role of a ‘buffer’ which considerably suppresses the influ-ence of the qubit oscillations on the current or detector QDs oc-cupations. It is believed that for a longer chain of QDs being in theT-shaped configuration, the qubit electron oscillations persistlonger in time but unfortunately, the corresponding detector cur-rent oscillations should be smaller.

Now we are going to reveal the role of the qubit inter-dotcoupling, Vxy, on the beat patterns. This parameter determines thefrequency of qubit charge oscillations and can significantly changethe beat period. Thus in Fig. 6 we show the QD charge, nx(t), for thequbit coupled with the TQD detector but calculated for differentvalues of the qubit tunneling amplitudes, Vxy. We find that Vxy

influences the behavior of nx(t) independently of the bias voltage

-4

-3

-2

-1

0

1

20 30 40 50 60 70

nx(

t)

time

U=0

U=2

U=3

U=4

U=6-4

-3

-2

-1

0

1

20 30 40 50 60 70

nx(

t)

time

U=0

U=2

U=3

U=4

U=6-4

-3

-2

-1

0

1

20 30 40 50 60 70

nx(

t)

time

U=0

U=2

U=3

U=4

U=6-4

-3

-2

-1

0

1

20 30 40 50 60 70

nx(

t)

time

U=0

U=2

U=3

U=4

U=6-4

-3

-2

-1

0

1

20 30 40 50 60 70

nx(

t)

time

U=0

U=2

U=3

U=4

U=6

Fig. 7. Nearby qubit QD occupation, nx(t), as a function of time for the triple-QDdetector (see Fig. 1C) for different values of U parameter: U 0, 2, 3, 4= and 6, re-spectively. The bias voltage 10L Rμ μ= − = , other parameters and initial conditionsas in Fig. 6.

-4

-3

-2

-1

0

1

40 60 80 100

nx(

t)

time

a

b

c

d

e

-4

-3

-2

-1

0

1

40 60 80 100

nx(

t)

time

a

b

c

d

e

-4

-3

-2

-1

0

1

40 60 80 100

nx(

t)

time

a

b

c

d

e

-4

-3

-2

-1

0

1

40 60 80 100

nx(

t)

time

a

b

c

d

e

-4

-3

-2

-1

0

1

40 60 80 100

nx(

t)

time

a

b

c

d

e

Fig. 8. Qubit QD occupations, nx(t), versus time for the DQD (TQD) detector –

curves a–c (d, e) and for different initial conditions. Curves a and d: qubit is ‘frozen’in the state n n0, 1x y= = until t¼40 when the occupancies of all detector QDsachieve their steady state values. Curves b and e: qubit is ‘frozen’ in the staten n0, 1x y= = and also n n 02 3= = until t¼40 when the occupancy of the firstdetector QD, n1, achieves its steady state value. Curve c: all couplings in the qubit–detector system are switched on at t¼40 (i.e. n n0, 1x y= = , n n 01 2= = for t 40< ).The other parameters: V 4xy = , U¼4, V 0.5ij = , 1Γ = , 0iε = and 20L Rμ μ= − = .

R. Taranko, T. Kwapiński / Physica E 70 (2015) 217–224 223

(upper and bottom panels). Firstly, the period of nx(t) oscillations isroughly the same as for the isolated qubit and it decreases withthe increasing coupling Vxy. Secondly, the amplitudes of the qubitoscillations increase with Vxy which allows us to measure the qubitdynamics longer in time. Note that for the case of small voltages,the qubit charge oscillations do not reveal such evident beatstructure (due to small electron current flowing through the de-tector which hardly influences the qubit dynamics). However, inthe case of larger voltages (bottom panel), hardly vanishing beatsof nx(t) are observed (see e.g. the bottom curve in the lower panel).It confirms that for the system parameters for which the beats ofnx(t) are observed, the qubit can stay longer in its coherent statewhich is a very desirable effect for the measurement process.

As was shown in Section 3.2 the period of beats of nx(t) in thecase of two-interacting qubits (isolated from the leads) depends,among others, on the electrostatic coupling U between both qu-bits. In Fig. 7 we analyze the role of this parameter on the qubitdynamics. We depict nx(t) for different values of U for the samesetup as in Fig. 6, i.e. for the qubit-TQD detector system. For thecase of U¼0, the qubit is isolated from the detector and is char-acterized by non-vanishing regular oscillations of nx(t) with nobeats (the upper curve). These oscillations survive also for smallvalues of U (e.g. the curve for U¼2) but the amplitude of the qubitoscillations slowly decreases with time (similar to the single-QDdetector, upper curve in Fig. 2). For larger U the oscillation am-plitudes of nx(t) also decrease but they change non-monotonicallywith time. For these cases (U 2> ) very clear beating structures arevisible on the qubit occupation curves (three bottom curves). Theperiod of the high-frequency oscillations roughly corresponds tothe isolated qubit and the low-frequency period of beats is relatedto the qubit–detector system. Note that the period of the beatsdecreases with U but the damping of the oscillation amplitudes isvery slow and slightly depends on U. Moreover, as one can see, forrelatively large t, the amplitude oscillations of nx(t) in the presenceof very clear beat patterns (e.g. the curve for U¼6) are larger incomparison with the case where the beats do not appear (thecurve for U¼2). Thus the above results show that the qubit os-cillates longer in time in the presence of beats.

3.4. Initial occupations of the detector QDs

In this subsection we discuss the role of the initial detector QDsoccupancies on the qubit oscillations and beat patterns. Differentinitial occupations of QDs coupled with electrodes usually lead todifferences of the transient current oscillations, especially for smalltime, t (i.e. just after switching on the coupling parameters). Thesedifferences are related with the so-called system memory effect(memory about the system initial state). However, in many QDsystems for larger t, the results are independent of the initial oc-cupancies and we say that the system is memory-free (one can saynothing about the initial state of the system from the currentcharacteristics for larger t). As we can see below the memory effectappears in the qubit–detector system and can significantly changethe qubit dynamics. Up to now we have considered only the casewhen the qubit at some time t t0= was coupled with the detectorbeing already in its steady state. Such coupling can be realized e.g.taking V t( ) 0xy = , n t( ) 0x = and n t( ) 1y = for t t0< and V t( ) 0xy ≠ fort t0≥ – these initial conditions are called here IC1. The steady statemeans that all detector QDs occupations, ni(t), as well as the leadcurrents, j t( )L R/ , have achieved constant time-independent values.Now we consider also a different situation when the qubit starts tointeract with the empty (unoccupied) detector QD. We assumethat, as before, the qubit is ‘frozen’ up to t t0= (V 0xy = , nx¼0, ny¼1for t t0< ) and simultaneously the detector QDs are decoupledfrom each other (i.e. V 012 = for Fig. 1B configuration andV V 012 23= = for the TQD detector, Fig. 1C). It means that only thefirst detector QD, n t( )1 , which is coupled with the leads achievesthe steady value at t t0= and the other detector QD are un-occupied until t t0= (n t t( ) 02,3 0< = ) – such initial conditions arecalled IC2. After some time the occupation of the dot interactingwith the qubit achieves almost the same value as for the case IC1and for larger t one expects similar oscillations of the qubit charge,nx(t), for both conditions IC1 and IC2. Unfortunately, as we can seebelow, this prediction is not true. In Fig. 8 we study the influenceof the initial conditions on the qubit charge oscillations for thecase of the DQD detector (curves a, b and c) and for the TQD de-tector (curves d and e). Curves a and d are obtained for the initialconditions IC1 and very clear beat patterns of nx(t) are visible inthese cases. For new initial conditions, IC2, the correspondingcurves (b and e) are quite different and do not exhibit the beating

R. Taranko, T. Kwapiński / Physica E 70 (2015) 217–224224

structure. Moreover, the oscillations of nx(t) are dumped nearlymonotonously (curve b possesses some amplitude modulationconsistent with the beating structure but it is slightly visible). Thisdifference in the time-dependence of the qubit oscillations can beexplained by different initial occupation of the detector QD di-rectly interacting with the qubit QD – so the system consideredhere reveals the memory effect. Note that for the IC1 case (de-tector QDs achieve their steady state values and then the qubit is‘switched on’) the detector QD directly interacting with the qubitis half-filled. On the other hand, this detector QD (n2 for the DQDor n3 for the TQD detector) is empty for the initial conditions IC2and for larger t its occupancy tends slowly up to 0.5 (as in the caseIC1). Note also hardly vanishing oscillation amplitudes of nx(t) forthe TQD detector.

To study the problem of the initial conditions in more detail weconsider one more case (so-called IC3) i.e. all couplings in thequbit–detector system are switched on at t t 400= =(n n0, 1x y= = , n 01/2/3 = , V 0ij Γ= = for t 40< and V 0ij ≠ , 0Γ ≠ fort 40≥ ). Now the occupations of all detector QDs just after theswitching procedure increase with time (n1 is filled firstly, then theother detector QDs). Thus the detector QD which interacts directlywith the qubit holds almost empty for relatively long time so theresulting qubit QD occupation (Fig. 8, curve c for the DQD detector)is nearly the same as for the case IC2 (curve b), with no beatpatterns. These studies confirm that the initial occupancies of thedetector QDs can substantially change the qubit dynamics. Therole of the initial conditions (preparation of the detector state atthe moment when the qubit begins to interact with the detector)is therefore essential in determining the qubit electronoscillations.

4. Conclusions

We have considered the coherent oscillations of the qubitelectron in different qubit–detector systems. The detector consistsof a single QD, DQD or TQD placed in the T-shaped configurationbetween two leads. The QD occupancies and the currents flowingin the system were calculated using the equation of motionmethod for the appropriate correlation functions with the specialHubbard I-like decoupling procedure for higher-order functions.We have focused our attention on the qubit dynamics dependenceon the detector setup and its initial electron configuration.

As a main result of our studies we have found that for somequbit–detector configurations very clear beat patterns can appearin the oscillations of the qubit QD occupancy. Such beats are alsovisible on the occupancies of the detector QDs as well as in thedetector currents. In particular, for a qubit coupled with a singleQD between two leads (the simplest considered detector) thebeating structure has been observed (for the first time in the lit-erature) for the weak spectral density function. We have alsofound that for the qubit coupled electrostatically with the DQD orTDQ detectors, the beating structure is more distinct (larger am-plitude variation) especially for the smaller spectral densityfunction, Γ, and the smaller detector inter-dot tunneling cou-plings. Moreover, it was shown both analytically and numericallythat the period of beats decreases with the qubit–detector inter-action U. We have also found that in the presence of beats, the

qubit electron occupation oscillations decay much more slowly incomparison with the case for which oscillations decaymonotonously.

Additionally, the memory effect in the qubit–detector systemswas considered (i.e. the dependence of the results on differenttypes of the detector initial conditions). We have compared thecase when the qubit starts to interact electrostatically with thedetector QD (i) which is occupied (half-filled) in the equilibriumstate and (ii) which is unoccupied and starts to be filled just afterthe connection with the qubit. In the former case the quantumcoherent oscillations with clear beat patterns are observed. How-ever, in the second case the quantum coherent oscillations of thequbit occupancy does not show any beats, although some ampli-tude modulation appears.

Acknowledgement

This work has been partially supported by the National ScienceCenter, Poland, Grant no. 2014/13/B/ST5/04442.

References

[1] Z.Z. Sun, R.Q. Zhang, W. Fan, X.R. Wang, J. Appl. Phys. 105 (2009) 043706.[2] M.L. Ladron de Guevara, P.A. Orellana, Phys. Rev. B 73 (2006) 205303.[3] Z.Y. Zeng, F. Claro, A. Perez, Phys. Rev. B 65 (2002) 085308.[4] P. Trocha, J. Barnaś, Phys. Rev. B 78 (2008) 075424.[5] R. Zitko, Phys. Rev. B 81 (2010) 115316.[6] H. Pan, S.Q. Duan, W.D. Chu, W. Zhang, Phys. Lett. A 372 (2008) 3292.[7] S. Kohler, J. Lehmann, P. Hänggi, Phys. Rep. 406 (2005) 379.[8] Z.T. Jiang, J. Yang, Y. Wang, X.F. Wei, Q.Z. Han, J. Phys.: Condens. Matter 20

(2008) 445216.[9] T. Kwapiński, R. Taranko, Physica E 63 (2014);

T. Kwapiński, R. Taranko, Physica E 18 (2003) 402.[10] T. Kwapiński, J. Phys.: Condens. Matter 19 (2007) 176218.[11] Y. Tomita, T. Nakayama, H. Ishii, J. Surf. Sci. Nanotechnol. 7 (2009) 606.[12] V. Moldovenau, A. Mandescu, V. Gudmundsson, New J. Phys. 11 (2009)

073019.[13] E. Taranko, M. Wiertel, R. Taranko, J. Appl. Phys. 111 (2012) 023711.[14] T. Kwapiński, R. Taranko, J. Phys.: Condens. Matter 23 (2011) 405301.[15] H. Pan, Y. Zhao, J. Phys. Condens. Matter 21 (2009).[16] F.M. Souza, Phys. Rev. B 76 (2007) 205315.[17] G. Stefanucci, E. Prefetto, M. Cini, Phys. Rev. B 78 (2008) 155301.[18] T. Gilad, S.A. Gurvitz, Phys. Rev. Lett. 97 (2006) 116806.[19] S.A. Gurvitz, G.P. Berman, Phys. Rev. B 72 (2005) 073305.[20] R. Taranko, P. Parafiniuk, Physica E 40 (2008) 2765.[21] H.S. Goan, G.J. Milburn, H.M. Wiseman, H.B. Sun, Phys. Rev. B 63 (2001)

125326.[22] S.A. Gurvitz, D. Mozyrsky, Phys. Rev. B 77 (2008) 075325.[23] C. Kreisbeck, F.J. Kaiser, S. Kohler, Phys. Rev. B 81 (2010) 125404.[24] H.J. Jiao, X.Q. Lin, Y.J. Luo, Phys. Rev. B 75 (2007) 155333.[25] R. Taranko, T. Kwapiński, Physica E 48 (2013) 157.[26] T. Kwapiński, R. Taranko, Phys. Rev. A 86 (2012) 052338.[27] S.A. Gurvitz, Phys. Rev. B 56 (1997) 15215.[28] J.P. Zhang, S.H. Ouyang, C.H. Lam, J.Q. You, J. Phys.: Condens. Matter 20 (2008)

395206.[29] S.H. Ouyang, C.H. Lam, J.Q. You, J. Phys.: Condens. Matter 18 (2006) 115511.[30] Xin-Qi Li, Wen-Kai Zhang, Ping Cui, Jiushu Shao, Zhongshui Ma, YiJing Yan,

Phys. Rev. B 69 (2004) 085315.[31] M.W.Y. Tu, W.M. Zhang, Phys. Rev. B 78 (2008) 235311.[32] Z.T. Jiang, J. Yang, O.Z. Han, J. Phys.: Condens. Matter 20 (2008) 075210.[33] A.-P. Jauho, N.S. Wingreen, Y. Meir, Phys. Rev. B 50 (1994) 5528.[34] R. Taranko, P. Parafiniuk, Physica E 43 (2010) 302.[35] P. Parafiniuk, R. Taranko, Physica E 40 (2008) 3078.[36] A. Croy, U. Saalmann, A.R. Hernandez, C.H. Lewenkopf, Phys. Rev. B 85 (2012).