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Quantum dynamics, dissipation, and asymmetry effects in quantum dot arrays

F. Rojas and E. CotaCentro de Ciencias de la Materia Condensada – UNAM, Ensenada, Baja California, Mexico 22800

S. E. UlloaDepartment of Physics and Astronomy, and Nanoscale and Quantum

Phenomena Institute, Ohio University, Athens, OH 45701-2979

(Dated: 7/2/02)

We study the role of dissipation and structural defects on the time evolution of quantum dotarrays with mobile charges under external driving fields. These structures, proposed as quantumdot cellular automata, exhibit interesting quantum dynamics which we describe in terms of equationsof motion for the density matrix. Using an open system approach, we study the role of asymmetriesand the microscopic electron-phonon interaction on the general dynamical behavior of the chargedistribution (polarization) of such systems. We find that the system response to the driving field isimproved at low temperatures (and/or weak phonon coupling), before deteriorating as temperatureand asymmetry increase. In addition to the study of the time evolution of polarization, we explorethe linear entropy of the system in order to gain further insights into the competition betweencoherent evolution and dissipative processes.

PACS numbers: 73.21.La, 73.63.Kv, 85.35.Gv

I. INTRODUCTION

The size reduction of devices in microelectronics, andthe fundamental limitations encountered or anticipateddue to quantum mechanical effects, have turned attentionto understanding nanometer scale structures. Systemswhere intrinsic quantum mechanical effects can play animportant role and could even be used to advantage areunder intense scrutiny. One recent prominent example ofsuch systems are quantum dot cellular automata.

The original concept of “cellular automata” was in-troduced as n-dimensional arrays containing finite am-plitude (information) per cell (or site) and connected toone another according to certain rules. This simple defi-nition, however, governs the evolution of the entire arrayin rather complex ways.1 Quantum cellular automata area natural generalization of this concept, where each sitecontains a quantum mechanical probability amplitude.2

In 1993, Lent and coworkers,3 proposed the use of cellularautomata architectures, composed of nanometer-scaledquantum devices (quantum dots) coupled through care-fully chosen hopping and Coulomb interactions, to en-code classical binary information in the different chargeconfigurations of the system. The typical basic elementin these quantum-dot cellular automata (QCA) is a cellconsisting of four quantum dots (QD) located at the ver-tices of a square and connected via tunneling barriers totheir neighbors (see Fig. 1a). For two mobile electrons ineach four-dot cell, Coulomb repulsion between the elec-trons causes the charge in the cell, in the ground state,to align along either of the two diagonals. The symmet-rical arrangement of the system means that these two“polarization” states are degenerate and can be used torepresent logic 0 and 1 as the bits in this system. The de-generacy is split by an appropriately designed driver fieldwhich allows control of the polarization of the ground

state (and typically implemented as a second identicalcell with externally controlled polarization). Tunnelingbarriers between dots are designed in such a way thatintra-cell tunneling is possible, but inter-cell tunnelingis not, and only Coulomb interaction from cell to cell ispossible. Elemental cell designs have been experimen-tally realized using metal islands, and logic operationsAND, OR and NOT have been implemented using ar-rays of such cells suitably arranged.4,5,6 Also, a clever im-plementation of QCA using magnetic elements has beenrecently reported.7

Although originally proposed as ‘classical’ and neardissipationless bit operators, QCA could in principle alsobe used to implement a quantum computer, as discussedrecently,8 if the entire array system is coherent enough(and for sufficiently long times). In either the classicalor quantum operation mode, QCA computers (as well asany other) should keep errors to a manageable limit. Ofspecial significance in these systems, decoherence and dis-sipation are external sources of ‘error’ originating in thecoupling between the system and its surroundings. Otheraspects that have to be considered as possible sources ofloss of control or desirable operation are imperfectionsin the dot fabrication, stray charged impurities, and therole of switching fields or drivers. It is then of interestto study the dynamical evolution of the QCA and mon-itor the charge density distribution over the basic cellelements under the influence of a driver cell, with empha-sis on the effects of imperfections and dissipation. Thecharge configuration is conveniently monitored throughLent’s cell polarization,3 which is essentially a measure ofthe degree of alignment of the charges along the diagonalsof the square cell:

P =ρ1 + ρ3 − (ρ2 + ρ4)

ρ1 + ρ2 + ρ3 + ρ4, (1)

2

0.00 0.02 0.04 0.06 0.080.0

0.2

0.4

0.6

0.8

1.0

1.2

c

a Pdriver

=+1

t

4

3 2

1target driver

(a)

t

E

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

1.2

(b)t=0.03

δ

E

FIG. 1: (a) Energy level structure vs. t for a target cell withdriver at +1. Energy in units of V ≃ 1 meV, and δ = 0.Sketch at bottom: model QCA with four quantum dots percell. Two charges are allowed to tunnel between nearest neigh-bors, with probability t. Driver cell on right indicates a statewith polarization Pdriver = +1. (b) Level structure vs. dotsize asymmetry δ. Notice anticrossings for δ ≃ 0.1 and 0.2 inexcited states. Separation to driver, c = a in both panels.

where ρi is the charge density at site i (see Fig. 1a),and the numbering of the sites is in clockwise order.In previous works, the effect of dissipation on QCA hasbeen included in the quantum equations of motion in aphenomenological way,9 through a damping parameter.This model is shown to give qualitatively correct resultsonly in the low temperature limit, as it will be evidentin our calculations. Also, the temperature dependenceof the cell polarization has been studied for the case ofa cell composed of two Coulomb-coupled double dots,10

and the contribution of relaxation processes to the speedof the cell response was discussed using stochastic equa-tions for the population differences and dipole momentsof the double dots. The effects of imperfections have beenconsidered before, with11,12,13 and without14 dissipation,and found to have strong effects in the dynamical re-sponse, by delaying or slowing down the response of thetarget cell, and producing a less than optimal desired

polarization. In this paper, we include the important ef-fect of the environment through a microscopic model ofelectron-phonon interactions. The thermal bath is mod-elled with phonons in equilibrium at a given temperatureand structural asymmetries are introduced via size vari-ations of the quantum dots.

We should point out that in addition to their relevanceas a new computational paradigm, QCA arrays providea conceptually simple, yet intricate and subtle, quantummechanical system in which to study fundamental coher-ence and dissipative processes. The parametrical require-ments to make practical QCA devices yield a finite setof two-particle states which can be manipulated to yieldinsights into the loss of coherence in these and generalquantum systems. As such, QCA provide then a modelcondensed matter system where dynamics and decoher-ence can be explored in detail. To this effect, we usea quantum open systems approach12,15,16 to study thedynamical evolution of the reduced density matrix ρS(t)of the system, and evaluate the competition between co-herent (driver cell) and incoherent (thermal bath) fieldeffects in the response of the basic cell. We study the ef-fects of temperature, electron-phonon coupling intensity,and structural asymmetry on the behavior of the polar-ization as a function of time. In particular, we discussthe asymptotic polarization after switching events, andonce the driver is in a stationary configuration, for dif-ferent temperatures and structure parameters. An anal-ysis of asymptotic fixed points in the resulting dynamicsprovides information on the system, as we will explicitlydemonstrate and discuss later on.

A measure of the mixed character of a system describedby a density matrix ρ is provided by the linear entropy,s[ρ] = Tr{ρ− ρ2}.17 This quantity, which can be thoughtof as the lowest order approximation to the von Neumannentropy, is also called the idempotency defect or the pu-

rity of a state. Notice that the linear entropy is zero for apure state, since Tr{ρ} = Tr{ρ2} = 1 in that case. Non-zero values of s[ρ] provide then a quantitative measure ofnon-purity of the state of a system. When monitored intime, the linear entropy provides a convenient measureof how fast the loss of quantum purity occurs in a systemin contact with a bath, or the increase in coherence un-der appropriate ‘pumping’ conditions, as we will discussbelow. Other uses of s[ρ] include the study of quantummanifestations of chaotic behavior in a complex system.18

We explore the influence of temperature, electron-phononcoupling intensity and driver cell switching times on s[ρ],in order to understand the process of dissipation in theQCA system.

We find in all cases that high dissipation and asym-metry influence adversely the response of the basic cell,as one would intuitively anticipate. However, the behav-ior with temperature is quite subtle. For example, lowtemperature results in better switching and more faithfulfollowing of the driver by the target cell. Higher temper-atures eventually produce quenching of the polarizationand a dynamical evolution which basically ignores the

3

driver. On the other hand, size defects in the dots resultin lagging of the switching behavior with respect to thedriver, and a faster switching window. This is producedby the pinning of a given polarization state by the ener-getics of the size defect, so that the driver potential hasto apply a strong force before the target cell can follow.Similar lag and/or pinning, as well as sudden switch ofthe polarization is seen in the case of two target cells be-ing driven by a single cell. A given cell in the middleof a chain would clearly have to deal with the influenceof more than one cell, resulting in a more complex po-tential landscape and corresponding dynamical behavior.We also find for a single cell that in the case of low tem-perature or weak dissipation the cell recovers dynamicalcoherence, due to the ‘pumping’ of the driver, clearlyoverpowering the thermal bath. Other regimes will bediscussed in detail.

The behavior of the polarization of the basic cell asa function of time when the polarization of the driveris fixed, as well as when it changes linearly, has beenbriefly discussed recently.12 There, we studied the sym-metric (no defect) case, but taking account of dissipationin the system. In what follows, we present a full reportof the QCA dynamical behavior under different electron-phonon and temperature conditions, and analyze the lin-ear entropy of the system during these processes.

The paper is organized as follows: in section II we de-scribe the model for the entire system, composed of theQCA and the thermal bath. We use an extended Hub-bard Hamiltonian to describe the Coulomb interactionspresent in the system (target cell(s) + driver cell). Thethermal bath is represented by a set of harmonic oscilla-tors, and the electron-phonon interaction enters as emis-sion and absorption of phonons during electron tunnelingevents between quantum dots in the system. A Marko-vian approach to the dynamical evolution of the reduceddensity matrix is presented in II A, where the transitionrates for the electron-phonon interaction in this systemare explicitly obtained. In section III we present and dis-cuss the behavior of the polarization and linear entropy,as function of relevant parameters, with special emphasison effects of dissipation and asymmetry. Here, we alsodiscuss the dynamics of multiple cells, as the QCA ul-timate devices involve multiple cell geometries. Finally,we summarize and discuss our results in section IV.

II. MODEL

Consider a problem described by the Hamiltonian H =HS + HR + VSR, where the subscripts S and R stand forsystem and reservoir, respectively. In our case, HS de-scribes the target cell in the presence of a driver cell whichprovides external driving fields on the system. The inter-actions are introduced via an extended Hubbard Hamil-tonian with both intra- and inter-cell Coulomb repulsion

terms, as well as intra-cell tunneling,

HS =∑

j

ǫj nj −∑

<ij>

(tij c†i cj + c.c.)

+∑

i>j

Vij ninj +∑

i,j

Wijndi (t)nj . (2)

The first three terms describe the target or active cell andthe last term describes the Coulomb interaction betweenthe electrons in the target cell and a driver cell whichis externally controlled. Here, ǫj is the confinement en-ergy level in the jth quantum dot (QD) in the target cell.In this term, possible dot asymmetries or local poten-tial variations are introduced by changing the energy ina given kth QD, ǫk → ǫ−δ, where δ is the measure of theimperfection due to different dot size or a change in thelocal environment, and ǫ is the energy of the other (iden-tical) dots. The amplitude tij is the tunneling matrixelement between nearest neighbors < ij > on the same

cell, c†j and cj are creation and annihilation operators for

site j, nj = c†j cj is the electron number operator at sitej, Vij is the Coulomb interaction between sites i and jin the target cell, and Wij is that between site i in thedriver cell (with charge density nd

i (t), which in generalchanges in time) and site j in the target cell. A gen-eralization to multiple cells is straightforwardly given byinter-cell interactions.14 Following the notation of Lent et

al.,3 the distance between nearest-neighbor dots (lengthof the square side) is a, while the separation between thetarget and driver cells, located side by side, is c (see insetFig. 1a). Notice we consider spinless electrons, althoughconsidering two-particle states with given total spin isalso possible in our formalism.

The Hamiltonian for the reservoir is given by a set of

harmonic oscillators of frequency ωk, HR =∑

k ~ωkb†k bk,

where b†k(bk) are creation (annihilation) operators ofphonons. Finally, the electron-phonon interaction isgiven by a general expression,

VSR =∑

k

∑

<ij>

αkij c†i cj(b

†k + bk) , (3)

corresponding to emission/absorption of a phonon forelectron tunneling events and with matrix elements givenby the coefficients αkij . Notice that the physics of thephonon interactions in the system is well described bythe model chosen here. Although phonons are in prin-ciple present everywhere, and not only during tunnelingevents, they only provide an ‘on-site’ energy renormal-ization/shift and broadening, which will be ignored here,as it yields a uniform correction to the entire cell. Incontrast, the phonon-assisted tunneling events describedby VSR affect significantly the dynamics of the cell, asthey provide an effective coupling mechanism among thedifferent levels in the spectrum and allow for energy re-laxation processes.

We assume, as in [19], that the quantum dots have acharacteristic size d ≃ 50 nm defined on a GaAs/AlGaAs

4

heterostructure, with effective mass m∗ = 0.067. Thetypical distance between nearest-neighbor dots is a &100 nm, and the dielectric constant of the medium is ≈12. These values give an estimate of the Coulomb repul-sion between nearest neighbor dots of V ≈ 1 meV, whichis taken as the unit of energy in this paper. One shouldnotice that the first excited single-particle state in eachdot lies at quite a high energy, ∆ ≈ (~2/2m∗)(2π/d)2 ≈9 meV ≫ V , so that one can safely consider a single or-bital per site, although additional single-particle levels ineach dot could in principle be included. Notice also thatthe on-site Coulomb energy is quite high and preventsthe double occupation of the dot, so that our basis setof states ignores that possibility. We should commentthat successful implementations of QCA elements usingmetallic islands are in a multiple-orbital regime, since inthat case ∆ ≪ V for typical 1µm-size islands. The con-dition of overall electrical neutrality of the target (anddriver) cell is included in the calculation by distributinga uniform positive background charge +2e throughoutthe cell (e/2 in each dot), which results in multipolarfields from/at each cell.

Notice that we study the behavior of the target cellinteracting with the driver via Coulomb interactiononly (no inter-cell tunneling allowed). This interactionis described by the last term in the Hamiltonian HS ,Eq. (2), and assumed to be time dependent via nd

i (t).For a fixed driver and no dissipation, the solution canbe found by direct numerical diagonalization of HS .In that case, the effect of various parameters on thepolarization have been studied to show that the energyspectrum is strongly influenced by even small defects,which affect the bistability of the cell.20 In the case ofa time-dependent driver, we have to take into accountthat the driver promotes transitions to intermediateexcited states. In fact, the dynamical competitionbetween the driver (coherent terms in the equationof motion) and the bath (dissipative terms) causes anon-trivial time evolution of the system. We have stud-ied different ‘driving schemes’ but will mostly considerhere the case of a driver with a linear switching of itspolarization, Pdriver(t) = 1 − 2t/τ , for times 0 ≤ t ≤ τ ,where τ is the switching time. This corresponds to theconfiguration of charge densities in the driver cell ofnd

1(t) = nd3(t) = 1 − t/τ and nd

2(t) = nd4(t) = t/τ . For

the four-dot cell arranged on a square, we take Vij = V

for nearest-neighbor dots, while all others scale withtheir (inverse) separation. We also set all tij equal toa constant t between nearest neighbors only (not to beconfused with time).

A. Markovian master equation

In the interaction picture (I), the evolution of the totaldensity matrix is given by

∂ρ(I)(t)

∂t= −

i

~

[

V(I)SR (t), ρ(I)(0)

]

−1

~2

∫ t

0

dt′[

V(I)SR (t), [V

(I)SR (t′), ρ(I)(t′)]

]

, (4)

where V(I)SR is the cell-bath interaction operator, and the

expansion is valid up to second order in V(I)SR .16

Our approach uses a closed and time-local equa-

tion for the reduced density matrix (RDM): ρ(I)S (t) =

TrR{ρ(I)(t)} where the trace is carried out over the de-

grees of freedom of the reservoir, R. The following fun-damental assumptions are made in this approach: (1)the system S and reservoir R are initially uncorrelated,which implies ρ(0) = ρS(0)ρR(0) = ρ(I)(0); (2) a station-ary reservoir at temperature T with equilibrium densitymatrix ρR(0) = exp(−HR/kBT )/Tr{exp(−HR/kBT )},exists at all times; and (3) the correlation time τc inthe reservoir is much shorter than characteristic timesfor the RDM system to change appreciably, meaning that

ρ(I)S (t′) ≈ ρ

(I)S (t), if t′−t . τc, while the state of the reser-

voir at time t′ has already no correlation with the stateat time t (Markov approximation). Thus one arrives atthe time evolution equation for the RDM elements (de-tails of the derivation can be found in the literature12,16),which in the Schrodinger picture is given by

˙ρS(t)ss′ = −iωss′ ρS(t)ss′ +∑

mn

Rss′mn ρS(t)mn . (5)

The first term on the right represents reversible dynamics(“coherent” effects), in terms of the transition frequen-cies ωss′ = (Es−Es′)/~, while the second term describesrelaxation (irreversible dynamics). In this part, the re-

laxation tensor Rss′mn can be written explicitly as

Rss′mn =

{

δnm(1 − δms)Wsm − δmsδns

∑

k 6=s Wks, (s = s′)−γss′ δmsδns′ , (s 6= s′)

(6)

Here, Wnm are transition rates from state |m〉 to |n〉,which in terms of the properties of bath and target cell,

can be expressed as

Wmn =2π

~2D2|g(ωmn)|2D(ωmn)

{

|Snm|2n(ωmn)

+ |Smn|2[1+n(ωmn)]

}

(7)

5

where Smk are matrix elements of the electronic partof the target cell interacting with the phonon reservoir,

Eq. (3); n(ω) = 〈b†(ω)b(ω)〉 = (eβ~ω − 1)−1 is the av-erage phonon number in the reservoir at temperatureT = 1/kBβ. The first term (∝ n) describes the absorp-tion of phonons, while the second (∝ n + 1) describesphonon emission, both at a frequency given by the sys-tem’s transition energy ωmn. Notice that the matrix ele-ments Smn embody the appropriate selection rules for thedifferent states involved in the transitions. For simplicity,we consider a model where the amplitude of interactiondoes not depend on the electronic states of the targetcell, and has the form αkij = Dgk, where D is the con-stant prefactor of the electron-phonon interaction and isdetermined from the deformation potential model.21 Weshould emphasize that this model introduces dissipationeffects only at finite temperature, and cannot accountfor elastic decoherence effects.22 Here, gk = g(ωk) ∼

ω1/2k describes the frequency dependent amplitude of the

electron-phonon interaction in the deformation potentialmodel; D(ω) ∼ ω2 is the density of states in a Debyemodel and ωmn is the transition frequency between states|m〉 and |n〉. It is clear that other phonon channels withdifferent dispersion relations and/or matrix elements ex-ist, but we believe that other modes would not be aseffective in introducing dissipation in this system (e.g.,optical phonons occur at energies too high to couple ef-fectively), nor qualitatively change the results discussedhere. In Eq. (6), γss′ is a so-called non-adiabatic parame-ter whose real part gives a contribution to the time decayof the off-diagonal density matrix elements, and is thendirectly responsible for the loss of coherence. These pa-rameters can be written in terms of the transition rates byRe γss′ = (

∑

k 6=s Wks +∑

k 6=s′ Wks′ )/2. The imaginarypart of γss′ is an intrinsic relaxation rate for each transi-tion, which we assume negligible, Im γss′ = 0, as it couldbe incorporated in the transition frequencies, ωss′ .16

It is interesting to point out that Eq. (5), written forthe diagonal elements, takes the form:

˙ρS(t)ss =∑

m

Wsm ρS(t)mm − ρS(t)ss

∑

m

Wms , (8)

which has a simple interpretation: the probability of find-ing the level |s〉 occupied at time t, ρS(t)ss, increases dueto transitions from all other levels |m〉 to |s〉 (first term)and decreases due to transitions from |s〉 to all other lev-els |m〉 (second term).

Note also that the condition ˙ρS(t)ss′ = 0 in Eq. (5) cor-responds to ‘fixed points’ of the dynamics of the system,as both the terms related to coherence, (ρS)ss′ for s 6= s′,as well as the diagonal terms which determine the popu-lation in each state, do not change in time. These fixedpoint values depend only on the properties of the transi-tion rates and the energy spectrum of the system (as theexternal drive reaches a stationary configuration), andprovide a direct determination of the dissipation timesproduced by the thermal bath. It can be seen from Eq.

(7) that the transition rates satisfy

Wnm

Wmn

= exp

(

−~ωnm

kBT

)

, (9)

which guarantees the detailed balance condition. It isimportant to emphasize that we have derived an explicitexpression for the transition rates Wmn, which would be-have at low temperatures, T → 0, and/or low dissipation(small D2) as the form used by Zozoulenko et al., in aqualitative way.11 Notice however that the normalizationcondition in [11] is an added requirement in their phe-nomenological approach, while it is included naturally inour treatment. We should also mention that the detailedbalance ratio in (9) will determine the fixed point valuesof ρS and all the corresponding physical properties, in-cluding the different state populations. Since these willin general be asymmetric, the polarization value of theQCA is reduced, and as we will see later, this resultsalso in a limiting value for the linear entropy. A detailedanalysis of the asymptotic fixed points in terms of thestructure parameters will also shed light into the dynam-ical evolution of the system, as we will discuss.

The solution of the master equation (5) for the timeevolution of the RDM is carried out numerically usinga Runge-Kutta fourth order algorithm. We ensure, bya proper choice of the integration step, that the generalnormalization property Tr{ρS} = 1 for the RDM is satis-fied at all times. Notice that an instantaneous basis is in-corporated in the calculation to take into account the evo-lution of the ground state induced by the time-dependentdriver configuration. The basis functions are then definedby the eigenvalue/eigenvector problem of the coherent

Hamiltonian of the target cell, HS(t)|m〉 = ESm(t)|m〉, for

a given time t while the driver is in the process of switch-ing, and the instantaneous energy in the target cell isES

m(t). Notice also that the characteristic time variationof the driver τ is long compared with the target-cell’snatural frequencies ωll′ , so that τ ≫ ~/ωll′ . This is effec-tively the requirement for an adiabatic switching of thedriver. As is clear in Fig. 1, the characteristic frequenciesof the system are ωll′ . 1 meV, so that ~/ωll′ & 1 ps.This would require τ to be a few nanoseconds for theadiabatic switching regime to be valid.

The numerical solution of the RDM is used to calcu-late the dynamics of the polarization in the target cell, asthe physical observable of interest here. The dynamicalbehavior of P (t), calculated from the charge density ineach site as ρi(t) = Tr{ρS(t)ni}, and of the linear entropys[ρS ], are then determined for a given set of parameters.In order to get better insights in the problem, we alsocarry out a calculation of the fully coherent dynamics,ignoring the phonon thermal bath (setting D = 0 in Eq.7). In the coherent limit, the relevant parameters forthe dynamical evolution are those defining the structure,including the tunneling amplitude t, the separation oftarget and driver cell c, the switching time of the lineardriver τ , and the size of the dot asymmetry δ, if any. Forthe dissipative problem, on the other hand, it is clear that

6

in addition to the structure features, the important pa-rameters are the temperature of the thermal bath T , andthe amplitude of the electron-phonon interaction D. Ofcourse, the phonon density of states, as well as the inter-action details, will also play a role in the evolution of thesystem. Our model parameters provide an adequate andrealistic description of these systems in semiconductingQCA.21

III. RESULTS AND DISCUSSION

Figure 1a shows the level spectrum in the target cell fora fixed driver polarization, Pdriver = 1, as shown in theinset. We see that the four-fold symmetry of the struc-ture and consequent degeneracy is broken by the driver,even for t = 0.20 Moreover, increasing the intra-cell tun-neling t produces additional level splittings, as one wouldexpect. We should point out that the polarization of the

ground state follows that of the driver, P grndcell ≃ 1, but

increasing hopping t reduces that value somewhat. Thefirst excited state has P exc

cell ≃ −1, and is separated fromthe ground state by a gap of ≃ 0.2 in the range shown(our energy units are V ≃ 1 meV). The four higher stateshave vanishing polarization, as the charges are located onneighboring sites along the sides of the square. The basicoperation of the QCA requires adiabatic evolution froma polarized ground state to the other with opposite po-larization. Transitions to higher excited states allow fornon-adiabatic terms which yield to depolarization of thetarget cell. Figure 1b shows the influence of a “defect” indot size. Here, the dot at site 1 has an energy lower thanthe others by an amount δ (corresponding to a largerdot). Increasing this asymmetry produces a strong shiftof the spectrum, and perhaps most important for QCAoperation, a change of the polarization of the first excitedstate. This is apparent in the anticrossing seen in 1b forδ ≃ 0.1, where an opposite polarization state takes overas the first excited state.20 The defect then will producea pinning of the polarization when operating the QCAif the δ is small, and even prevent the switching of thetarget cell altogether if δ is large.

When the polarization of the driver cell changes lin-early with time, the polarization of the target cell evolvesas well, as presented in Fig. 2a for the non-dissipativecase, and for different temperatures. One can see thatthe polarization of the target cell follows the driver verywell, and switches smoothly over a short time window(≈ τ/5) centered about the time when Pdriver = 0. Thisgeneral behavior is not affected for different values of theswitching time τ we tested (over a wide range within theadiabatic regime; results not shown).23 We see that theresponse of the target cell deteriorates for increasing tem-perature. At low temperature, T = T1 ≃ 1 K, the targetcell has a slightly decreased polarization after the driveris fully switched, and the switching time of the cell is notgreatly affected. However, for T = 5T1, the target cell po-larization is quenched after 0.5 ns and ignores even a fully

0 1 2 3 4-1.0

-0.5

0.0

0.5

1.0

no dissipation

T = T1

T = 5T1

driver(a)

δ=0

Pol

ariz

atio

n

0 1 2 3 4-1.0

-0.5

0.0

0.5

1.0

driver

no dissipation

T = T1

T = 5T1

(b)

δ=0.1

time (ns)

Pol

ariz

atio

n

FIG. 2: Time evolution of target cell polarization as drivercell switches linearly from Pdriver = +1 to −1 in a time τ =2.5 ns, for different temperatures. (a) Symmetric cell withfour identical dots (δ = 0). (b) Asymmetric cell with onelower energy dot, δ = 0.1, at site 1. In all plots, t = 0.03 andD2 = 0.05.

polarized driver. [Notice that the natural unit of temper-ature in this problem is the difference in energy betweenthe first and second excited states, when the driver isfixed at Pdriver = 1 (Fig. 1b), as this transition has oneof the smallest natural frequencies ωll′ , and it producesdepolarization of the cell, as discussed above. In partic-ular, we take as characteristic its value for t = 0.03 inthe symmetric case, T1 = 0.0862 ≈ 1 K. Temperaturesare compared to this natural scale in the problem.] Thedeterioration of the response of the cell with temperatureis consistent with Eq. (9) where the different rates at lowtemperature would favor transitions to the ground state,while as temperature increases, transitions up and downbecome equally probable. This results in more symmetriclevel populations, more equal electron probabilities in alldots, and consequently the depolarization of the targetcell. It is somewhat surprising that even this low tem-perature would yield such strong quenching of the QCApolarizability (a value which of course depends also onhow strong D is in Eq. 7). Figure 2b shows an exam-

7

0 1 2 3 4-1.0

-0.5

0.0

0.5

1.0

(b)

time (ns)

Pol

ariz

atio

n

0 1 2 3 4 5 6-1.0

-0.8

-0.6

-0.4

-0.2

0.0

(c)

T / T1

Pa

δ=0.0 δ=.10 δ=.15

-1.0

-0.5

0.0

0.5

1.0

no dissipation(a)

T = T1

Pol

ariz

atio

n

δ=0 δ=.05 δ=.1

FIG. 3: Time evolution of target cell polarization with a linearswitch drive, τ = 2.5 ns, for different asymmetry δ. (a) Nodissipation (D = 0); (b) at temperature T = T1. Noticetarget cell lagging the switch of the driver increases with δ.(c) Long-time asymptotic polarization, Pa vs. temperature Tfor different δ values. In all panels, t = 0.03 and D2 = 0.05.

ple of the effects of asymmetry (quantum dot at site 1has lower energy than the rest), for the same parametervalues and temperatures as in Fig. 2a. We observe thatat low temperatures, the response of the target cell lagsbehind the driver, although the switching time is notaffected substantially. However, the target cell reachesasymptotic values of polarization which are smaller (inabsolute value) than in the ideal symmetric case. In thiscase too, changing τ over a wide range does not affectthis lagging behavior. It is evident that for T = 5T1, thetarget cell is almost completely depolarized, despite the“help” of the larger dot which pins the polarization ofthe ground and excited states to ≃ +1 (and induces thelag seen at low temperatures).

Figure 3 shows more of the effects of asymmetry andtemperature on the response of the target cell. Figure3a shows the results for no dissipation, and panel (b)those for T = T1. From these figures, we see again thatthe important effect of asymmetry is to produce a lag ordelay in the response of the target cell as asymmetry in-

0 1 2 3 4-1.0

-0.5

0.0

0.5

1.0

τ

time (ns)

Pol

ariz

atio

n

t=0.003 t=0.03 t=0.10

FIG. 4: Time evolution of polarization in a linear switchdriver, τ = 2.5 ns (see arrow). Increasing tunneling betweendots first improves (t = 0.03) and then deteriorates (t = 0.1)the final polarization. D2 = 0.05, T = T1, and δ = 0 in allplots.

creases. On the other hand, the effect of temperature isto decrease the asymptotic polarization in absolute value,associated with the delocalization of electrons within thecell. Notice also in Fig. 3b that for finite T , the polar-ization of the target cell continues evolving in time (de-teriorating), even after the driver is fully polarized. Inall cases, the polarization reaches asymptotic values Pa

(long after the driver has stopped changing) which area manifestation of ‘fixed points’ of the system dynam-ics for the quantum dot populations. These long-timevalues of the polarization depend on temperature (andindirectly on structure parameters and energy levels) asshown in Fig. 3c, for different asymmetries. The nearlyfull depolarization of the target cell at high temperatureis evident and anticipated. What is somewhat counterin-tuitive is the fact that the depolarization increases withasymmetry, as the pinning that yields the lag assists nowin the depolarization by favoring the population of thestates with the larger dot.

Figure 4 shows the effects of tunneling coupling in-tensity t on the time evolution of the polarization fora symmetric cell system (δ = 0). We see that increas-ing hopping between dots first introduces a smoothertransition following the linear driver (τ = 2.5 ns) anda better polarization. Increasing t however, delocalizesthe charges in the quantum dots and reduces the polar-ization of the target cell, as one can see from the longtime limits in the figure. In order to look more closelyat the non-monotonic behavior of Pa, we analyze thisquantity as a function of the tunneling coupling t in Fig.5a. We find that, for T = T1, Pa attains a minimumvalue at t ≈ 0.02. From this figure we can see that the

8

0.00 0.02 0.04 0.06 0.08

-1.0

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

δ = 0.1

(b)

T = T1 / 2

T = T1

Pa

t

-1.0

-0.9

-0.8

-0.7

-0.6

(a)

T = T1

T = T1 / 2

no dissipation

Pa

FIG. 5: Long-time asymptotic polarization Pa vs. hopping tat different temperatures, as in Fig. 4. The non-monotonicbehavior is seen only for T ≈ T1. (a) Symmetric cell (δ = 0);(b) asymmetric case, δ = 0.1. In both panels, D2 = 0.05.

drop in polarization at small values of t is due to dis-sipation processes: We expect that at low t values theelectrons would tend to be well localized at the cornersof the cell (ground state), giving P = ±1, dependingon the driver. This is indeed what is observed at lowtemperatures. However, as temperatures increases, de-polarization transitions from the ground state to excitedstates become more likely. As t increases substantially,the polarization decreases (in absolute value) due to thedelocalizing effects of both tunneling and dissipation, asexpected. However, for t . 0.02, the possibility of tran-sitions which in fact populate the ground state increases,and therefore improve slightly the asymptotic polariza-tion of the system. Qualitatively similar results are ob-tained in the asymmetric case (Fig. 5b), although withlower absolute values of the asymptotic polarization inthis case, as discussed above (Fig. 2 and 3), and a shal-lower oscillation.

We have also studied the dynamics of two target cellsand driver in line (Fig. 6). There are a number of in-teresting features in this case. We notice first that theswitching for both target cells lags behind the driver con-

siderably, as the cells switch only after Pdriver & 75%.Moreover, the switch occurs over a much shorter timescale than for the single target cell (≃ τ/25). This changeis understandable if one thinks that the target cell clos-est to the driver (cell 1) is in essence between two cellswith different polarization and has its wrong polariza-tion somewhat pinned (this is behavior also seen in otherwork11). Increasing temperature to T = T1 makes theswitching a bit smoother although still with a suddenchange, and with not as good final polarization. Noticein particular that the asymptotic polarization of targetcell 2 is poorer than that of cell 1. As the tempera-ture increases to T = 2T1, the target cells barely fol-low the drive, and the polarization of cell 2 is basicallyzero at long times. It is clear that control of the tun-neling process as the “signal” is transmitted down theline would be essential in proper operation of QCA ar-rays. Such “clocked switching” has been discussed byLent and coworkers as a solution to steer cells into bettercompliance.24 Coherent simulations of such steering,14 aswell as experiments in metallic QCA25 verify that this isa good operating scheme.

Knowing the time evolution of the density matrix wecan also calculate the linear entropy of the QCA sys-tem s[ρS ], to monitor the degree of non-purity introducedduring the switching process by the thermal bath. In Fig.7 we show the results for different temperatures, for thecase of weak and strong electron-phonon interactions, asmeasured by the value of D2 in Eq. (7). As in previouscases, the polarization of the driver cell is switched lin-early from +1 to −1 in a time τ = 2.5 ns. In Fig. 7a,the case of ‘weak’ electron-phonon coupling, D2 = 0.05,we see that at low temperatures s[ρS ] increases mono-tonically until it reaches a final stable value. It is in-teresting to see that as Pdriver = 0 or Pdriver = −1 (at1.25 and 2.5 ns, respectively), one can see kinks in theentropy curve, indicating the switch off of driving forces.The long time asymptotic value increases (coherence de-creases) with increasing temperature, as one would ex-pect. However, at still higher temperatures and/or forstrong electron-phonon coupling (Fig. 7b), s[ρS ] first in-creases and then decreases. In these cases, we see a max-imum in the linear entropy curve at ≈ τ/2, when thepolarization of the driver cell goes through zero. Noticein fact that whenever Pdriver ≃ 0, the level spectrum inthe target cell shows a great deal of degeneracies. Asthe driver turns negative (for time after τ/2) the split-tings are reinstated and the effect of the driver seems tobe one of a coherent ‘pump’ which produces a drop inthe purity, even as dissipative transitions are present. Inother words, the thermal dissipation seems to decreasewith respect to the coherent transitions induced by thedriver, producing a drop in the linear entropy of the sys-tem. After the driver is stationary, after time τ , thethermal transitions quickly produce the steady state inthe density matrix (or level populations) given by thedetailed balance equation (7). For stronger phonon cou-pling, D2 = 0.5, and high temperatures (Fig. 7b), the

9

-1.0

-0.5

0.0

0.5

1.0

cell 2 cell 1 driver

(b)

T = T1

Pol

ariz

atio

n

0 1 2 3 4-1.0

-0.5

0.0

0.5

1.0

(c)

T = 2 T1

time (ns)

Pol

ariz

atio

n

-1.0

-0.5

0.0

0.5

1.0

(a)

no dissipation

Cell 1 Cell 2

Pol

ariz

atio

n

FIG. 6: Polarization of two active cells and linear driver, asshown in inset in panel (b). (a) Coherent dynamics; (b) atT = T1; (c) at T = 2T1. Notice cells lag the driver, exceptat higher temperatures. More distant cell (2) switches moresuddenly than cell 1, and has smaller Pa. In all plots, t = 0.03and D2 = 0.05.

purity of the system deteriorates further and the driveris not able to ‘push’ the system back into a more coherentstate. Notice that the state of ‘maximal delocalization’would be achieved as ρSii ≃ 1/n, where n is the size ofthe basis, so that Tr{ρ2} ≃ 1/n. In that case, one wouldobtain s ≃ 1 − 1/n, giving 5/6 in our case. This value isstill higher than the asymptotic value of s for the highesttemperatures in Fig. 7b, so that even then the system isnot ‘fully’ decoherent.

Let us explore this competition between dissipationand coherent evolution in more detail. Since Tr{ρS} = 1is a general property of the density matrix in any rep-resentation, one can instead investigate the behavior ofTr{ρ2

S} = 1 − s[ρS ] directly as a function of the rele-vant parameters.26 Notice in fact that this trace has twocontributions

Trρ2S =

∑

i

(ρ2S)ii

=∑

i

(ρSii)2 +

∑

i6=j

|ρSij |2 , (10)

0.0

0.2

0.4

0.6

0.8

1.0

T = 2 T1

T = T1

T = 0.8 T1

(a) D2=0.05

s ( ρ

)

0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

1.0

T = 5 T1

T = 2 T1

T = T1

(b) D2=0.5

time (ns)

s ( ρ

)

FIG. 7: Time evolution of linear entropy or purity of the sys-tem in a linear switch, τ = 2.5 ns at different temperatures.(a) Weak coupling, D2 = 0.05, t = 0.03; (b) strong coupling,D2 = 0.5, t = 0.03. Notice entropy drop after τ/2 at interme-diate dissipation.

namely, the diagonal and non-diagonal elements of thedensity matrix. Comparing the two terms as a functionof time, for example, would give one further insights ofthe overall complex processes, since the behavior of thenon-diagonal elements gives information on the evolutionof coherence in the system. Figure 8 shows the trace asfunction of time for two values of the temperature, explic-itly showing the contributions from the diagonal and non-diagonal elements of ρS . We see in Fig. 8a an increase inthe contribution from the non-diagonal (“coherent”) ele-ments, with maximum at τ/2, exactly compensated by adecrease in the contribution from the diagonal elements,as the driver and the bath are promoting transitions. No-tice also that the values of the diagonal components be-fore and after τ/2 are approximately the same (with asimilar behavior for the non-diagonal terms). Thus forlow temperatures T = T1/10 (panel a), the sum remainsapproximately constant, equal to 1, and the system isclose to the pure limit with s ≈ 0. However, for T = T1,we see that this sum decreases with time, due essentiallyto the corresponding decrease in the diagonal elements.

10

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

(b) T=T1

Tr

{ ρ2 }

time (ns)

total diag non-diag

0.0

0.2

0.4

0.6

0.8

1.0

(a) T=0.1T1

Tr

{ ρ2 } total

diag non-diag

FIG. 8: Trace of ρ2 vs. time during a linear switch of the drivercell, τ = 2.5 ns, and different temperatures. (a) T = T1/10,(b) T = T1. Both diagonal and non-diagonal sums are shown,as well as the total trace, as in Eq. 10. In both panels, t = 0.03and D2 = 0.05.

Notice that as the driver switches and ‘pumps’ the targetcell system, the off-diagonal coherent terms contributeless to the sum as the temperature increases (its peakvalue is smaller in (b) than in (a); so that the system isless coherent indeed), and their contribution for longertimes is lower than at the start of the switch (time zero).This gradual decay in the coherent contribution showseven more substantially (and less intuitively) in the di-agonal terms, which one would expect to be less affectedperhaps. Since Tr{ρ} = 1, the drop in the diagonal termsof Tr{ρ2} means that the population of each level (ρSii)has been re-distributed and made somewhat more equal(although still much higher than the ‘maximal delocal-ization’ state discussed above with Tr{ρ2} ≃ 1/6).

It is also interesting to investigate the influence ofswitching time and of the function controlling the switch-ing. Figure 9 shows results for Tr{ρ2

S} for a smootherdriver transition. In this case, the driver has no rapidoff/onsets and the evolution of Pdriver is more adiabaticnear the ends of the sweep than before (see panel c).Panel (b) shows the trace at low temperatures, whichshows a similar rise (drop) in non-diagonal (diagonal)density matrix elements as in Fig. 8a. However, as thetime during which Pdriver ≃ 0 is shorter now (effectivelylarger slope in panel c), the window of variation is also re-duced. Higher temperature in panel 9a shows also a nar-rower time window variation, but an identical amplitudeas in Fig. 8b. Other runs with different driver switchingslopes follow this behavior too: narrower windows forlarger slopes near Pdriver ≃ 0 (not shown). This wouldsuggest that the amplitude variation of the diagonal/off-diagonal components is only a function of temperature,while the time window over which they vary is a function

0 1 2 3 4-1.0

-0.5

0.0

0.5

1.0

(c)

time (ns)

Pol

ariz

atio

n driver T = 0.1T

1

T = T1

0.0

0.2

0.4

0.6

0.8

1.0

diagonal

non-diagonal

(b) T = 0.1T1

Tr

{ ρ

2 }

0.0

0.2

0.4

0.6

0.8

1.0total

non-diagonal

diagonal

(a) T = T1

Tr

{ ρ

2 }

FIG. 9: Time evolution for a smoother driver switch. (a)Trace of ρ2

S at intermediate temperature, T = T1, showingdiagonal and non-diagonal contributions. (b) Trace contri-butions for T = T1/10. (c) Driver (solid curve) and targetcell polarizations for different temperatures. In all panels,t = 0.03 and D2 = 0.05.

of how long the driver is close to zero polarization.

IV. CONCLUSIONS

We have studied the behavior of the polarization of asquare array of quantum dots with two mobile electrons.The quantum dynamical evolution in response to a drivercell whose polarization is changing with time in a pre-scribed manner, has been shown to depend strongly onswitching or driver characteristics, symmetry of the celland on temperature. Our results show that the asymp-totic values of the polarization in a stationary driver arefixed points of the system that change with tempera-ture, tunneling probabilities and imperfections, but noton driving schemes. The optimal response of the QCAarrays is affected adversely by temperature and imper-fections, with a strong tendency toward depolarizationas temperature or asymmetry increase. Our calculationsindicate that good control of the QCA is only achieved for

11

temperatures of at most a few kelvins. From our two tar-get cell results, it is likely that “clocked switching” wouldbe required for faithful operation of a chain, so that thepolarization information is not degraded down the line.We have also analyzed dissipation effects on the responseof the basic cell through calculation of the linear entropyor purity of the system. We find that the study of thisquantity is a good tool in furthering the understandingof a quantum system as relatively simple, but still quitesubtle, as the QCA architecture. Apart from providinginsights as a function of time and driver characteristics,the entropy also shows clearly that at high temperatures,the basic cell is no longer in the required/desired final

state but in one where the polarization tends to zero andmaximizes the entropy.

Acknowledgments

This work was supported in part by CONACYTProject 27702–E. SEU is supported in part by US DOEgrant no. DE–FG02–91ER45334. We acknowledge sup-port from the Condensed Matter and Surface SciencesProgram at Ohio University.

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(1994).20 F. Ramırez, E. Cota and S. E. Ulloa, Phys. Rev. B 62,

1912 (2000)21 The value of D in Eq. (7) depends on the deformation po-

tential of the semiconductor, as we consider coupling toacoustic phonons. Typical values of 1 to 3 eV yield a valueof D2

≈ 7 to 20 in our units. As the potential confine-ment of the electrons in the dot yields poorer coupling toacoustic phonons in the solid with characteristic long wave-lengths, the effective value of D is likely much smaller. Weanticipate that the value of D is smaller by a factor of ≈ 5,and then use a value of D2 = 0.5 as our ‘strong coupling’regime.

22 A. O. Caldeira and A. J. Leggett, Phys. Rev. Lett. 46, 211(1981); 48, 1571 (1982).

23 In all the time evolution plots shown, the initial conditionof the target cell is chosen as the ground state of the sys-tem at the given hopping t, for Pdriver = +1. This yieldsa smooth non-oscillatory oscillations at the start of thedriver switchings. Other initial configurations with polar-ization similar to the ground state yield time evolutionswith similar characteristics in P and in linear entropy.

24 C. S. Lent and P. D. Tougaw, Proc. IEEE 85, 541 (1997);G. Toth and C. S. Lent, J. Appl. Phys. 85, 2977 (1999).

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26 K. Blum, Density Matrix Theory and Applications

(Plenum Press, NY, 1981).