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Coherent population transfer in a chain of tunnel coupled quantum dots∗

D. Petrosyan and P. LambropoulosInstitute of Electronic Structure & Laser, FORTH, 71110 Heraklion, Crete, Greece

We consider the dynamics of a single electron in a chain of tunnel coupled quantum dots, exploringthe formal analogies of this system with some of the laser-driven multilevel atomic or molecularsystems studied by Bruce W. Shore and collaborators over the last 30 years. In particular, wedescribe two regimes for achieving complete coherent transfer of population in such a multistatesystem. In the first regime, by carefully arranging the coupling strengths, the flow of populationbetween the states of the system can be made periodic in time. In the second regime, by employing a“counterintuitive” sequence of couplings, the coherent population trapping eigenstate of the systemcan be rotated from the initial to the final desired state, which is an equivalent of the STIRAPtechnique for atoms or molecules. Our results may be useful in future quantum computation schemes.

PACS numbers: 03.67.-a, 73.63.Kv, 73.23.Hk

I. INTRODUCTION

Population transfer in multistate quantum systems hasbeen an active topic of research over the last half a cen-tury. In the context of atomic and molecular physics, co-herent population transfer in optically-driven multilevelsystems has been studied since the invention of lasers[1]. Usually, the objective is to transfer the populationfrom the initial to a well defined final state of the atomor molecule, via one or more intermediate states, whileminimizing the loss of population through or its accu-mulation on the intermediate states. In early theoreticalwork, Shore and collaborators have studied populationtransfer in multilevel systems driven by resonant laserfields [2, 3, 4]. In particular, they have found that it ispossible to arrange the coupling strengths between theadjacent states in such a way that the system becomesanalogous to a spin-J in a magnetic field, whose dynamicevolution is known to be periodic for any J [3]. This cou-pling scheme was therefore named spin-coupling.Later, Hioe, Eberly, Bergmann and collaborators dis-

covered the technique of stimulated Raman adiabaticpassage (STIRAP) for three-level atomic/molecular sys-tems [5]. They have identified a specific eigenstate ofthe system, the so-called coherent population trapping(CPT) state, which contains a superposition of the ini-tial and final states, and dates back to Alzetta et al.and Arimondo and Orriols [6]. The STIRAP techniqueis then based on first preparing the system in its initialbare state, which coincides with the CPT state, and thenadiabatically rotating the CPT state towards the desiredfinal bare state of the system. This techniques has beensubsequently polished [7] and extended to multilevel sys-tems [8, 9, 10] with the active participation of Bruce W.Shore.While the above studies were conducted in the con-

text of multilevel atoms or molecules, here we show

∗This paper is dedicated to Bruce W. Shore on the occasion of his

70th birthday.

that similar effects can be found in the context of quan-tum transport in arrays of tunnel-coupled quantum dots[11, 12, 13, 14]. Often referred to as artificial atoms,semiconductor quantum dots offer an unprecedented pos-sibility of constructing at will and exploring situationsranging from practically single atom to a fully solid statemany-body systems [15]. The nanofabrication possibili-ties of tailoring structures to desired geometries and spec-ifications, and controlling the number and mobility ofelectrons confined within a region of space, are some ofthe features that make these structures unique tools forthe study of a variety of preselected set of phenomena,including the coherent population transfer in multistatesystems.

Given the controllable quantum properties of the elec-trons in such structures, the possibility of their applica-tion to schemes of quantum computers (QCs) [16] hasnot escaped attention [17, 18, 19]. The qubits of theQD-array based QC would be represented by the spin-states of single electrons confined in individual QDs,with the two-qubit nearest-neighbor coupling mediatedby the controlled spin-exchange interaction [17, 18]. Oneof the main difficulties with the existing proposals forintegrated solid-state based QCs is that there is no effi-cient way of transferring the information between distantqubits. We consider here a single-electron tunneling ina one-dimensional array of QDs and establish the con-ditions under which the complete transfer of the elec-tron wavepacket between two distant locations can beachieved. Our findings could therefore be relevant to thereliable information exchange between distant parts of anintegrated quantum computer [20].

In Section II we outline the mathematical formalismdescribing a chain of QDs, in terms of which, in SectionIII, we present the theory of coherent propagation andperiodic oscillations of the electron wavepacket betweenthe two ends of the chain. The single-electron transfervia an equivalent of multistate STIRAP is discussed inSection IV. In Section V we describe an envisioned im-plementation of a scalable quantum computer, followedby the concluding remarks.

2

∆ε

1 N2

∆ε

1t

. . .3

t t32 tN−1jt

FIG. 1: Schematic drawing of the chain of tunnel-coupledQDs.

II. MATHEMATICAL FORMALISM

We consider electron transport in a linear array ofN nearly identical QDs which are electrostatically de-fined in a two-dimensional electron gas by means ofmetallic gates on top of a semiconductor heterostructure(GaAs/AlGaAs) [11, 15]. This system is described bythe extended Mott-Hubbard Hamiltonian [12, 13], whichin its most general form is given by

H =∑

j,α

εjαa†jαajα +

1

2

∑

j

Unj(nj − 1)

+∑

i<j,α

tij,α(a†iαajα + aiαa

†jα) +

∑

i<j

Vijninj,(1)

where a†jα and ajα are the creation and annihilationoperators for an electron in state α with the single-particle energy εjα, U is the on-site Coulomb repulsion,

nj =∑

α a†jαajα the total electron number operator of

the jth dot, tij,α are the coherent tunnel matrix elementsbetween dots i and j, and Vij is the interdot electrostaticinteraction. In general, the index α refers to both or-bital and spin states of an electron. In the tight-bindingregime, when the on-site Coulomb repulsion and single-particle level-spacing ∆ε are much larger than the tun-neling rates, U > ∆ε ≫ tij,α, only the equivalent statesof the neighboring dots are tunnel-coupled to each other[21]. In the absence of a magnetic field, we can thus limitour consideration only to a single doubly- (spin-) degen-erate level per dot (α ∈ {↑, ↓}), assuming further thatthe tunneling rates do not depend on the electron spin.In this paper we are concerned with single-electron dy-

namics, considering a situation in which a preselected QDis initially doped with one mobile electron, while all of theother dots of the chain are empty, as indicated in Fig. 1.Our aim is to determine the conditions under which thecomplete coherent transfer of the electron between thetwo ends of the chain can be achieved. The populationtransfer in this system is mediated by the tunneling be-tween the neighboring QDs. The individual tunnelingrates tj ≡ tjj+1 are determined by the voltages appliedto the gates defining the corresponding interdot tunnelingbarriers. A chain of N tunnel-coupled QDs doped with asingle electron is described by the following Hamiltonian,

H1e =∑

j,α

εja†jαajα+

∑

j,α

tj(a†jαaj+1,α+ajαa

†j+1,α), (2)

which obviously does not contain terms responsible forelectrostatic interactions. Since this Hamiltonian pre-

serves the electron number and its spin, the total state-vector of the system reads

|ψ(τ)〉 =N∑

j,α

Aαj (τ) |jα〉, (3)

where |jα〉 ≡ a†jα |01, ..., 0N 〉 denotes the state with oneelectron having spin α at the jth dot. The time-evolutionof the system is governed by the Schrodinger equationi |ψ〉 = H1e |ψ〉 (~ = 1), which yields

idAα

j

dτ= εjA

αj + tj−1A

αj−1 + tjA

αj+1, (4)

where t0 = tN = 0. Obviously, the two sets of these am-plitude equations with α =↑ and α =↓ are equivalent anddecoupled from each other. As a result, if the electronis prepared in an arbitrary superposition of spin up and

spin down states, |ψ〉 = A↑j |j↑〉 + A↓

j |j↓〉, the two partsof the wavefunction evolve symmetrically and indepen-dently of each other. This assertion is valid as long asall the uncontrollable spin-flip processes are vanishinglysmall on the time scale of t−1. In semiconductor QDs, thespin decoherence originates mainly from the spin-phononcoupling, as well as the coupling of the electron spin withthe nuclear spins of the surrounding crystal (hyperfineinteraction) or stray magnetic fields. The first decoher-ence mechanism is suppressed at low temperatures [21],at which the density of crystal phonons is negligible [22].As for the uncontrollable hyperfine interactions, exper-imental measurements indicate spin-relaxation times inexcess of 100 µs, which can be further improved by ap-plying moderate magnetic fields or polarizing the nuclearspins [23]. Another mechanism for decoherence in theprocess of electron (charge) transfer in our system origi-nates from the structure imperfections and gate voltagefluctuations, which cause uncertainty in the intradot en-ergy levels and interdot couplings. These fluctuations,however, are typically slow on the time scale of t−1, andthe resulting disorder in the system may be consideredfrozen during its dynamic evolution, as we have discussedin a previous publication [20].Let us write the Hamiltonian for the electron with spin

α in the matrix form

Hα1e =

ε1 t1 0 · · ·t1 ε2 t20 t2 ε3...

. . ....

εN−1 tN−1

· · · tN−1 εN

, (5)

which is obviously tridiagonal. Inspection of the ampli-tude equations (4) or the Hamiltonian (5) indeed verifiesthat our system is formally analogous to the laser-drivenmultilevel atomic or molecular systems studied by Shoreand coworkers [2, 3, 4] and Bergmann, Shore and others[5, 7, 8, 9, 10]. Here, the tunneling rates tj between states

3

|j〉 and |j + 1〉 play the same role as the Rabi frequen-cies of the laser fields acting on the atomic transitions|j〉 ↔ |j + 1〉, while the energies εj of states |j〉 corre-spond to the cumulative detunings of the atomic levels.In the following Sections, we describe two methods forachieving complete population transfer from the initial|1〉 to the final |N〉 state of the system, which turn outto be the counterpart of those in Refs. [3] and [10].

III. PERIODIC OSCILLATIONS OF

POPULATION BETWEEN THE TWO END

STATES

In this Section we consider the electron wavepacketdynamics in the chain with static couplings between thedots. Assume that at time τ = 0 the electron is local-ized on the first dot, |ψα(0)〉 = |1α〉, and the tunnelcouplings are switched on. This switching should be fastenough on the time scale of t−1, so that no appreciablechange in the initial state of the system occurs duringthe switching time τsw, but slow on the time scale of ε−1,so that no nonresonant coupling between the dots is in-duced: ε−1 < τsw < t−1. The aim is to determine the setof couplings between the states of the systems which willachieve a complete transfer of the electron populationfrom the initial to the final dot.To determine the time-evolution of the state vector

(3) we need to solve the eigenvalue problem Hα1e |ψα〉 =

λ |ψα〉 which will yield the eigenvalues λk and corre-sponding eigenvectors |ψα

k 〉 of the Hamiltonian (5). Thestate vector |ψα(τ)〉 at any time τ ≥ 0 is given by

|ψα(τ)〉 =N∑

k

e−iλkτ |ψαk 〉〈ψα

k |ψα(0)〉 =N∑

j

Aαj (τ) |jα〉.

(6)Note that the matrix in Eq. (5) has the form of thetridiagonal Jacobi matrix. It is natural to first con-sider the case of equal tunneling rates between the dots:tj = t. Assuming equal energies εj = ε and makingthe transformation Aα

j → Aαj e

iετ , which is equivalentto the interaction picture, we find that the determinantDN (λ) ≡ det(Hα

1e − λI) is identical to the Chebyshevpolynomial of the second kind, which can be expressed asDN (λ) = ΠN

k=1(λ−λk). The eigenenergies of the systemare then given by the roots of this polynomial, namely

λk = 2t cos

(

kπ

N + 1

)

,

while the corresponding eigenvectors are

|ψαk 〉 =

√

2

N + 1

N∑

j

sin

(

jkπ

N + 1

)

|jα〉.

Using Eq. (6) and the initial conditions A1 = 1 andAj = 0 for j = 2, 3, . . .N , we obtain the solutions for the

amplitudes as,

Aαj =

2

N + 1

N∑

k=1

exp

[

−i2tτ cos

(

kπ

N + 1

)]

× sin

(

jkπ

N + 1

)

sin

(

kπ

N + 1

)

. (7)

It is thus evident that the eigenstates of the coupledsystem oscillate with incommensurate frequencies corre-sponding to the roots λk of DN , which in fact become in-creasingly densely spaced with increasing N . As a conse-quence, the system never revives fully to its initial state,as is illustrated in Fig. 2(a).Clearly, it is highly desirable to tailor the parameters

of the system so as to achieve a non-dispersive transferof the single-electron wavepacket between the two endsof the chain. Recall from the theory of angular momen-tum that a spin-J particle subject to a constant mag-netic field exhibits Larmor precession about the field di-rection. In particular, if one chooses the quantizationdirection along an axis perpendicular to the magneticfield direction and prepares the particle in its lowest spineigenstate |J,M = −J〉, it will oscillate between thisinitial and the final state |J,M = J〉 in a perfectlyperiodic way. The matrix elements for the transitions|J,M〉 ↔ |J,M + 1〉 between the neighboring states are

proportional to√

(J −M)(J +M + 1). It is thereforeclear that with the appropriate choice of the interdottunneling matrix elements, the dynamics of the single-electron in a chain of QDs can mimic that of a spin-J ina magnetic field. Indeed, if we formally set N = 2J + 1and j = J +M + 1, the tunneling rates tj should be ar-

ranged according to tj = t√

(N − j)j for j = 1, ..., N−1.Then again, by exploring the properties of the Jacobipolynomials, we find equally spaced eigenenergies of thesystem,

λk = t(2k −N − 1),

while the corresponding eigenvectors can be expressedthrough the rotation matrices commonly used in the rep-resentation theory of angular momentum. With the ini-tial conditions A1 = 1 and Aj = 0 for j = 2, 3, . . .N , forthe amplitudes of the state-vector (3), we then obtainsimple analytic expressions given by the binomial form

Aαj =

(

N − 1j − 1

)1/2

[−i sin (tτ)](j−1) cos (tτ)(N−j)

. (8)

Since the eigenstates of the system have commensurateenergies λk, the electron wavepacket oscillates in a per-fectly periodic way between the first and the last dots,whose occupation probabilities are given, respectively,

by |Aα1 |2 = cos (tτ)

2(N−1)and |Aα

N |2 = sin (tτ)2(N−1)

,which is illustrated in Fig. 2(b). In particular, if at timeτ = π/(2t) the tunneling rates are suddenly switched off,we obtain |Aα

1 |2 = 0 and |AαN |2 = 1, i.e. complete pop-

ulation transfer from the initial to the final state of the

4

13

5

7

9

0

10

20

30

0.5

1

13

5

7

9

0.5

1

13

5

7

9

02

46

8

10

0.5

1

13

5

7

9

0.5

1

Dot index

jPo

pula

tion

Time τDot in

dexj

Popu

latio

n

Time τ

(b)(a)

1 2 3 4 5 6 7 80

1

2

1 2 3 4 5 6 7 82

3

4

5

Tun

nel r

ates

j j

Tun

nel r

ates

FIG. 2: Time-evolution of a single-electron wavepacket in a chain of N = 9 QDs with static tunneling rates. (a) Population flowin the chain with equal interdot tunneling rates tj = t (shown in the inset). (b) Population flow in the chain with spin-model

tunneling rates tj = tp

(N − j)j (shown in the inset). The time τ is in units of t−1.

system. In a somewhat abstract sense, the behavior ofthe system is thus similar to that of a two-level systemsubject to a π pulse. Let us note at this point that thepopulation transfer between the two ends of the chaincan be achieved most straightforwardly by sequentiallypulsing the tunneling rates between the first and seconddots for time τ1 = π/(2t1), then the second and thirddots for time τ2 = π/(2t2), etc till reaching the Nth dot,which is equivalent to applying a sequence of π pulsesin a multistate atomic system. In the scheme describedabove, however, all the interdot tunnelings are switchedon and then off simultaneously, realizing thereby a fastand efficient transfer of the electron from the first to thelast QD.

IV. ADIABATIC POPULATION TRANSFER

BETWEEN THE TWO END STATES

While the above tunneling schemes, involving a se-quence of π pulses or an effective collective π pulse, re-quire both, careful control of the individual tunnelingrates and their timing, in this Section we describe a ro-bust adiabatic method for population transfer which isnot very sensitive to small uncertainties in the interdottunneling rates. Recall that a three-level atom inter-acting with two laser fields, under the condition of two-photon (Raman) resonance, possesses a coherent popula-tion trapping (CPT) state, which is decoupled from bothlaser fields [7]. Equivalently, for a chain of three tunnel-coupled quantum dots, assuming equal energies εj = ε,the eigenstate of Hamiltonian (5) with zero eigenvalue,λ0 = 0, is given by

|ψα0 〉 =

1√N0

[t2 |1α〉 − t1 |3α〉], N0 = t21 + t22. (9)

This is a CPT state that does not contain a contribu-tion from the intermediate state |2α〉. The other two

eigenstates

|ψα±〉 =

1√

N±

[t1 |1α〉 − λ± |2α〉+ t2 |3α〉],

N± = t21 + λ2± + t22 = 2N0,

with corresponding eigenvalues λ± = ±√

t21 + t22, containall three states |jα〉. If for a given coupling strengths t1and t2 the system is prepared in the CPT state (9), itwill remain in this state as long as the couplings are con-stant in time. But even for time-dependent couplings,the system initially prepared in the CPT state can adi-abatically follow this state, provided the tunneling rateschange slowly enough. More quantitatively, the nonadi-abatic coupling between the eigenstates of Hamiltonian(5) is small, if during the evolution the transition am-

plitude 〈ψα±|ψα

0 〉 remains much smaller than the energyseparation between the corresponding eigenstates [7],

|〈ψα±|ψα

0 〉| ≪ |λ± − λ0|. (10)

Our objective is to transfer the electron from the first tothe last QD using the time-dependent (pulsed) tunnel-couplings. From Eq. (9) one can see that if at an earlytime the tunnel coupling t2 is switched on while t1 ≪ t2,the CPT state coincides with the initial state |1α〉. Onethen slowly (adiabatically) decreases t2 while increasingt1, so that at a later time t1 ≫ t2 and the CPT statecoincides with the final state |3α〉. Assuming that t2and t1 are represented by partially overlapping pulses,each having temporal width τw, the adiabaticity condi-tion (10) requires tmax

1,2 τw ≫ 1.In the field of atomic/molecular physics, this tech-

nique, involving the so-called counterintuitive sequence ofpulses, is known as the stimulated Raman adiabatic pas-sage (STIRAP) that is commonly used for coherent pop-ulation transfer in three-state systems [7]. We note thatthe solid-state implementations of the CPT and STIRAPin a pair of coupled quantum dots driven by two electro-magnetic fields has been proposed in [24]. The single elec-tron transfer in a chain of three QDs via counterintuitive

5

13

57

9

010

2030

40

50

0

0.5

1

13

57

9

13

57

9

020

4060

80

100

0

0.5

1

13

57

9

Population

Dot index j

Time τDot in

dex jTime τ

Population

(a) (b)

0 10 20 30 40 500

0.5

1

0

0.5

1

0 20 40 60 80 1000

0.5

1

0

0.5

1teven todd teven todd

Tun

nel r

ates

τ

Popu

latio

n

Tun

nel r

ates

τ

Popu

latio

n

Dot 1 Dot 1 Dot 9Dot 9

FIG. 3: Time-evolution of a single-electron wavepacket in a chain of N = 9 QDs with time-dependent, counterintuitive tunnelingrates. (a) Population transfer is incomplete, |AN |2 ≃ 0.7, when the adiabatic condition is not very well satisfied. (b) Almostcomplete population transfer is achieved, |AN |2 ≃ 0.97, when the adiabatic condition is better satisfied by doubling the temporalwidths of the pulses and the total interaction time (note the different scales of the time axis in (a) and (b)). The insets showthe time-dependence of even and odd tunneling rates and the populations of the first and last QDs.

pulsing of tunnel-couplings as discussed above has beenstudied by Greentree et al. in [14], where it was termedcoherent tunneling by adiabatic passage (CTAP). Theseauthors also considered the extension of CTAP to mul-tidot systems employing the so-called straddling schemeof [9]. Other schemes for adiabatic electron transport intunnel-coupled QDs have been discussed in [25].

Another extension of the STIRAP technique to sys-tems containing more than just three states has beengiven in [10]. This scheme can easily be adapted to oursystem, as described below. We thus consider a chain ofN sequentially coupled QDs and assume that the individ-ual tunnel couplings can selectively and independently bemanipulated. When N is odd, i.e. N = 3, 5, 7, . . ., theHamiltonian (5) has a CPT eigenstate

|ψα0 〉 =

1√N0

[t2t4 . . . tN−1 |1α〉+ (−1)t1t4 . . . tN−1 |3α〉

+ . . .+ (−1)J t1t3 . . . tN−2 |Nα〉], (11)

J ≡ 1

2(N − 1),

with eigenvalue λ0 = 0. Thus the amplitude of the ini-tial state |1α〉 is proportional to the product of all theeven-numbered tunnel-couplings, while the amplitude ofstate |Nα〉 is given by the product of all odd-numberedtunnel-couplings, divided by the normalization parame-ter N0 = (t2t4 . . . tN−1)

2 + . . .+ (t1t3 . . . tN−2)2. There-

fore, if all the even-numbered tunnel-couplings are pulsedtogether first, the CPT state (11) would coincide with theinitial state |1α〉. This is then followed by switching-onall the odd-numbered tunnel-couplings, while the even-numbered ones decrease, which will result in a completetransfer of electron wavepacket to the state |Nα〉. If weassume that these two families of pulses are describedby common shape functions, t2, t4, . . . , tN−1 = teven and

t1, t3, . . . , tN−2 = todd, Eq. (11) takes a compact form

|ψα0 〉 =

1√N0

J∑

n=0

(−todd)n tJ−neven |(2n+ 1)α〉, (12)

N0 =

J∑

n=0

t2nodd t2(J−n)even ,

which makes the above discussion more transparent. Inparticular, complete population transfer from the initialstate |1α〉 to the final state |Nα〉 can be achieved byapplying first the teven pulses and then the todd pulses,the two sets of pulses partially overlapping in time, asshown in Fig. 3.In order to minimize the nonadiabatic coupling of the

CPT state to other eigenstates of the system, the rate ofchange of teven and todd, given approximately by the in-verse pulse-width τ−1

w , should be small compared to cor-responding eigenenergies |λ| ∼ |teven+ todd|, which yieldsthe same condition as above, tmax

even,oddτw ≫ 1. One can

see from the results in Fig. 3(a), which were obtained pre-cisely for this reason, that when this condition is not verywell satisfied, the population transfer is incomplete. Asexpected, when the tunneling rates are pulsed for longertimes, or, equivalently, have larger amplitudes, the adia-baticity condition is satisfied better, resulting in the com-plete population transfer from the initial to the final dotof the chain, as seen in Fig. 3(b). The remarkable advan-tage of this method over the one described in the previousSection is that as long as the two sets of partially overlap-ping pulses are strong enough, the adiabatic transfer ofpopulation is expected to be robust with respect to smalluncertainties and fluctuations of tunneling rates, just likeits atomic/molecular counterpart in Refs. [5, 7, 10]. Onthe other hand, the electron transfer via effective col-lective π pulse can be achieved with smaller tunnelingrates and/or reduced interaction times, provided a pre-cise control of the tunneling amplitudes and timings is

6

possible. Depending on the characteristics of the partic-ular system, one or the other method may prove to bemore practical.

V. CONCLUSIONS

In the above Sections, we have studied the dynamicsof a single-electron transport in a linear array of tunnelcoupled quantum dots. We have identified two regimesunder which a complete coherent transfer of electronwavepacket between the two ends of the array can beachieved. Our results could be used for reliable infor-mation exchange between distant parts of an integratedquantum computer. As already noted in the Introduc-tion, one of the difficulties with the existing proposalsfor integrated QD based QCs [17, 18] is that the qubits(electron spins) interact with the nearest neighbors only,and there is no efficient way of transferring the infor-mation between distant qubits. As a way around suchdifficulties, one can envision an integrated quantum reg-ister composed of a large number of sub-registers, eachcontaining two or more adjacent qubits, represented byspins of single electrons in individual QDs. The sub-registers are embedded in a two-dimensional array ofempty QDs. As we have shown in an earlier publication[20], through the mechanism of transient Heisenberg cou-pling, combined with the control of tunnel-coupling be-tween the dots studied in this paper, this two-dimensionalgrid could realize a flexible quantum channel, capable of

connecting any pair of qubits within the register. Thus,to transfer the information, one connects distant sub-registers by a chain of empty QDs and applies one of theprotocols described in the previous Sections to achieve anon-dispersive transfer of the qubit, followed by its con-trolled entanglement with a target qubit [17]. Note thatthis scheme is analogous to a proposal for an integratedion trap based QC [26], where, in order to circumvent thedifficulties associated with a single large ion trap quan-tum register, it has been proposed to use many smallsub-registers, each containing only a few ions, and con-nect these sub-registers to each other via controlled qubit(ion) transfer to the interaction region (entangler) repre-sented by yet another ion trap.

We should note that the coherent electron dynamics inarrays of tunnel-coupled QDs bears many analogies withspin-wave dynamics in spin chains [27] or electromag-netic field dynamics in periodic photonic crystals [28, 29],where some of the effects described above should be ob-servable. With an unprecedented control over systemparameters, arrays of QDs doped with more than oneelectron allow for studies of numerous coherence and cor-relation effects in many-body physics.

Acknowledgments

This work is an outgrowth of earlier collaborative workwith Dr. G.M. Nikolopoulos which we gratefully ac-knowledge.

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(Wiley, New York, 1990).[2] J. H. Eberly, B.W. Shore, Z. Bialynicka-Birula and I.

Bialynicki-Birula, Phys. Rev. A 16 (1977) 2038; Z.Bialynicka-Birula, I. Bialynicki-Birula, J.H. Eberly andB.W. Shore, Phys. Rev. A 16 (1977) 2048.

[3] R. Cook and B.W. Shore, Phys. Rev. A 20 (1979) 539.[4] B.W. Shore and R. Cook, Phys. Rev. A 20 (1979) 1958;

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(1984) 690; J.R. Kuklinski, U. Gaubatz, F.T. Hioe andK. Bergmann, Phys. Rev. A 40 (1989) 6741; U. Gaubatz,P. Rudecki, S. Schiemann and K. Bergmann, J. Chem.Phys. 92 (1990) 5363.

[6] G. Alzetta, A. Gozzini, L. Moi and G. Orriols, Nuovo Ci-mento 36B (1976) 5; E. Arimondo and G. Orriols, Lett.Nuovo Cimento 17 (1976) 333.

[7] K. Bergmann, H. Theuer and B.W. Shore, Coherent pop-ulation transfer among quantum states of atoms andmolecules, Rev. Mod. Phys. 70 (1998) 1003.

[8] Y.B. Band and P.S. Julienne, J. Chem. Phys. 95 (1991)5681; J. Oreg, K. Bergmann, B.W. Shore and S. Rosen-waks, Phys. Rev. A 45 (1992) 4888.

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