We investigate the dynamics of two interacting electrons confined to a pair
of coupled quantum dots driven by an external AC field. By numerically
integrating the two-electron Schroedinger equation in time, we find that for
certain values of the strength and frequency of the AC field we can cause the
electrons to be localised within the same dot, in spite of the Coulomb
repulsion between them. Reducing the system to an effective two-site model of
Hubbard type and applying Floquet theory leads to a detailed understanding of
this effect. This demonstrates the possibility of using appropriate AC fields
to manipulate entangled states in mesoscopic devices on extremely short
timescales, which is an essential component of practical schemes for quantum
information processing.Comment: 4 pages, 3 figures; the section dealing with the perturbative
  treatment of the Floquet states has been substantially expanded to make it
  easier to follo

C. E. Creffield

C.E. Creffield

F. Grossmann

G. Platero

H. Sambe

H. Wang

J.H. Shirley

M. Grifoni

M. Holthaus

M.A. Kastner

P. Zhang

P.I. Tamborenea

R.C. Ashoori

R.H. Blick

T.H. Oosterkamp

English

arXiv

Crossref

ac-driven localization in a two-electron quantum dot molecule

©2002 The American Physical Society.
C.E.C. thanks Sigmund Kohler for numerous stimulating discussions. This research was supported by the EU via Contract No. FMRX-CT98-0180, and by the DGES (Spain) through Grant No. PB96-0875.We investigate the dynamics of two interacting electrons confined to a pair of coupled quantum dots driven by an external ac field. By numerically integrating the two-electron Schrödinger equation in time, we find that for certain values of the strength and frequency of the ac field the electrons can become localized within just one of the dots, in spite of the Coulomb repulsion. Reducing the system to an effective two-site model of Hubbard type, and applying Floquet theory, leads to a detailed understanding of this effect. This demonstrates the possibility of using appropriate ac fields to manipulate entangled states in mesoscopic  devices on extremely short time scales, which is an essential component of practical schemes for quantum information processing.EUDGES (Spain)Depto. de Física de MaterialesFac. de Ciencias FísicasTRUEpu

Creffield, Charles

Platero, G.

Docta Complutense

PHYSICAL REVIEW B, VOLUME 65, 113304ac-driven localization in a two-electron quantum dot moleculeC. E. Creffield and G. PlateroInstituto de Ciencia de Materiales (CSIC), Cantoblanco, E-28049 Madrid, Spain~Received 21 November 2001; published 19 February 2002!We investigate the dynamics of two interacting electrons confined to a pair of coupled quantum dots drivenby an external ac field. By numerically integrating the two-electron Schro¨dinger equation in time, we find thatfor certain values of the strength and frequency of the ac field the electrons can become localized within justone of the dots, in spite of the Coulomb repulsion. Reducing the system to an effective two-site model ofHubbard type, and applying Floquet theory, leads to a detailed understanding of this effect. This demonstratesthe possibility of using appropriate ac fields to manipulate entangled states in mesoscopic devices on extremelyshort time scales, which is an essential component of practical schemes for quantum information processing.DOI: 10.1103/PhysRevB.65.113304 PACS number~s!: 73.23.2b, 03.67.Lx, 33.80.Betoim-rmoicvterhethfienrkurootrotnbleemdldilogpusmero-loueonudthinputhtedoficlefulclelyforthewithforepa-ofal-lic-enc-eofulyferls ofec-on--A set of electrons held in a semiconductor quantum doconceptually similar to a set of atomic electrons bound tnucleus, and for this reason these structures are somettermed ‘‘artificial atoms.’’1 To extend the atomic analogy further, we can consider joining quantum dots together to fo‘‘artificial molecules.’’ The simplest example consists of twcoupled dots containing a single electron. The dynamproperties of this system when driven by an ac field habeen extensively studied,2 and the technique of Floqueanalysis3 has proved to be particularly effective. The pioneing work in Ref. 4 first noted the dramatic effects on ttunneling rate arising from crossings and anticrossings ofFloquet quasienergies, and the termdynamical localizationwas coined to describe the phenomenon in which the acappears to trap the electron within just one dot over lotime scales. Subsequent analytical and numerical wo5,6showed that for high frequencies this localization occwhen the ratio of the field strength to the frequency is a rof the Bessel functionJ0.Adding a second electron to the coupled-dot system induces considerable complications, as for realistic devicesCoulomb interaction between the electrons cannot beglected, and a single-particle analysis is not applicaAchieving an understanding of the dynamics of this systis, however, extremely desirable, as the ability to rapicontrol the localization of electrons using ac fields immeately suggests possible applications to quantum metroand quantum information processing. In particular, manilating a pair of entangled electrons on short time scales igreat importance in the rapidly developing field of quantucomputation. Although numerical investigations have bemade of the rich behavior displayed by interacting electsystems driven by an ac field,7 there is, at present, little understanding of the observed effects beyond phenomenoand numerical experimentation.We address this problem here by applying the Floqformalism to a system ofinteracting particles. We initiallyconsider a realistic model of two interacting electrons cfined to a pair of coherently coupled quantum dots, and stits response to ac fields by a numerical method in whichCoulomb interaction is treated exactly. We find the surprisresult that, even in the presence of strong interelectron resion, suitable ac fields can localize both electrons within0163-1829/2002/65~11!/113304~4!/$20.00 65 1133isaesale-eldgst-hee-.y-y-ofnngyt-yegl-esame dot. We further show that this effect is also exhibiby a simple two-site model, obtained as a reduced formthe quantum dot system. As in the case of the single-partmodel, we find Floquet theory to be an extremely powertool, and we succeed in generalizing the single-partisolution2,5 to include interactions, from which we can easiand accurately identify the parameters of the ac fieldwhich localization occurs.We study a quasi-one-dimensional structure, in whichlinear dimensions of the quantum dots in the (x,y) plane aremuch smaller than their length. The energies associatedtransverse excitations are thus much higher than thoselongitudinal motion, and can be neglected. The dots are srated by a potential barrier which regulates the degreetunneling between the dots. Strongly coupling the dotslows the electrons to form extended states~‘‘molecular orbit-als’’! over the entire double-dot system, which can be expitly seen in transport experiments.8,9 Within the effectivemass approximation, the Hamiltonian for the system is givbyH52\22m*S ]2]z121]2]z22D 1V~z1!1V~z2!1e24peVC~ uz12z2u!2eE~ t !~z11z2!, ~1!wherez1 andz2 are the spatial coordinates of the two eletrons,m* is the effective mass, ande is the effective permi-tivity. GaAs material parameters are used, withm*50.067me and e510.9e0 . E(t) is the external ac field,E(t)5E cos(vt). To simplify the numerical treatment, thsystem was placed within an infinite square well of lengthLto prevent the electronic wave function from leaking outthe dot structure. So that these infinite barriers did not undinfluence the properties of the quantum dots, thick bufzones were included between these barriers and the walthe quantum dots, so that the electronic wave function efftively decays to zero before reaching the barriers. The cfining potentialV(z) is plotted in Fig. 1. The Coulomb potential used,VC(r ), had the formVC~r !51Ar 21l2, ~2!©2002 The American Physical Society04-1urthaousscesenefinldplvrfotoeld,lder-caneenale-neisnitiallueteeldd tocane ofrml-ofe.ed.butinBRIEF REPORTS PHYSICAL REVIEW B 65 113304wherel is a measure of the transverse width of the struct~taken to be 1 nm in this investigation!. This form for VCreproduces the normal 1/r fall-off of the Coulomb potential,but is nonsingular in the limitr→0.To study the time evolution of the system, we usedeigenstates of the static HamiltonianH0 as a basis, sinceexpanding in these states is a well-controlled procedure,also ensures that correlation effects arising from the Clomb interaction are automatically encoded within each bafunction. We note that as Eq. 1 contains no spin-flip termthere is no mixing between the singlet and triplet subspaIn particular, if the initial state has a definite parity this symmetry is retained throughout its time evolution, and conquently only basis functions of the same symmetry needbe included in the expansion. To obtain the eigensystemeither the singlet or triplet subspaces, we employed the Lazos technique described in Ref. 10, which is particularlyficient for systems in which the interaction is diagonalreal-space.The initial state used was the ground state ofH0 ~a sin-glet!. In the absence of the external electric field this wouhave a trivial time evolution, simply acquiring a phase. Aplying the field, however, causes the initial state to evointo a superposition of eigenstates:uc(t)&5(cn(t)uEn&.Substituting this expansion into the time-dependent Sch¨-dinger equation yields a first-order differential equationthe expansion coefficientscn ,i\dcndt5cn~ t !En2eE~ t ! (m51MFnmcm~ t !, ~3!where En is the nth eigenvalue ofH0, and Fmn are theoverlap integrals of the dipole operator,Fmn5^Emu(z1FIG. 1. Geometry of the double-dot system. The dotted lplots the charge density of the ground-state wave function.11330eend-is,s.--toofc---eor1z2)uEn&. A fourth-order Runge-Kutta method was usedtime evolve the expansion coefficients using Eq.~3!. Thenumber of basis states used in the expansion,M, required forconvergence depended on the strength of the electric fiand values ofM550 were required for the strongest fieconsidered.A similar model was studied recently in Ref. 7 for a largsystem size ofL5100 nm. An important advantage of considering a smaller system, however, is that finer structurebe resolved in the results due to the larger spacing betwthe energy levels. Following Ref. 7 we define a conditionprobability function, which gives an extremely useful dscription of the state of the system. The probability that oparticle is in the right dot while the other is in the leftgiven byPRL(t),PRL~ t !52ERdz1ELdz2uc~z1 ,z2!u2, ~4!where the notationR / L signifies that the integration is takeover the right/left quantum dot.PRL takes a value of 1 formaximally delocalized states~when one electron is in theright dot and the other is in the left!, and is zero for localizedstates when both electrons occupy the same dot. The instate plotted in Fig. 1 is highly delocalized, and has the vaPRL50.849. Our investigation consists of evolving this stathrough a time period of 18 ps while measuringPRL(t). Weterm the minimum value ofPRL attained during this periodPmin , and use this to quantify the degree to which the ac fibrings about localization.In Fig. 2~a! we present a contour plot ofPmin as a functionof the parameters of the ac field. Dark areas corresponlow values ofPmin , indicating a high probability that bothelectrons are occupying the same dot. Surprisingly thisoccur even at weak-field strengths, despite the presencthe Coulomb interaction. We note that the dark areas fohorizontal bands, indicating that, for various ‘‘resonant’’ vaues ofv, localization can be produced over a wide rangefields. The spacing of the bands decreases withv, and forvalues of\v,2.8 meV the structure is too fine to resolvBetween these bands strong localization is not producThese results are qualitatively similar to those of Ref. 7,eFIG. 2. Pmin as a function ofthe strengthE and energy\v ofthe ac field~a! for an interactingquantum dot system~\v in unitsof meV! and ~b! for a Hubbardmodel with U58 ~both axes inunits of t̃ ).4-2ateateasikeniourugpgntetlyerycoabletatedofol-tateeeglyhethetesses.stemtun-ner-ofthezinggls atesiedtatethg,theex-mthekesithinonsonale-orBRIEF REPORTS PHYSICAL REVIEW B 65 113304show finer detail, allowing us to observe additionally ththese bands are punctuated by narrow zones in whichfield does not create localization. Their form can be semore clearly in the cross section ofPmin given in Fig. 3~a!,which reveals them to be narrow peaks. These peaksapproximately equally spaced along each resonance,spacing increasing withv.We emphasize that these results are radically differfrom those obtained for noninteracting particles. In this can analogous plot of delocalization shows a fanlstructure,6,11 in which localization occurs along lines giveby v5E/xj , wherexj is the j th root of the Bessel functionJ0(x). As a test of our method, we repeated our investigatfor a quantum-dot system in which the interelectron Colomb repulsion was set to zero, and found that the fan stture was indeed reproduced.To account for these results it is thus necessary tobeyond the noninteracting case. We consider a highly simfied model in which each quantum dot is replaced by a sinsite. Electrons can tunnel between the sites, and importawe include interactions by means of a HubbardU term:H52 t̃(s~c1s† c2s1H.c.!1(i 512@Uni↑ni↓1Ei~ t !ni #.~5!Here t̃ is the hopping parameter. In this analysis we set\51,and measure all energies in units oft̃ . Ei(t) is the externalelectric potential applied to sitei. Clearly only the potentialdifferenceE12E2 is of importance, so we may choose thconvenient parametrizationE1~ t !5E2cosvt, E2~ t !52E2cosvt. ~6!A numerical investigation of such a model was recenmade in Ref. 12 for the case of very weak fields. The Hilbspace of Hamiltonian~5! is six dimensional, comprisingthree singlet states and a triplet. As with the double-dot stem, the singlet and triplet subspaces are completely depled, allowing us to study just the singlet space. In theFIG. 3. ~a! Quasienergy spectrum for the two-site model forU58 andv52: circles exact results; lines, perturbation theory.~b!Magnified view of exact results for a single anticrossing. The cresponding plots ofPmin appear beneath.11330thenrehenten-c-oli-lely,ts-u--sence of the ac field the eigenvalues of the singHamiltonian can be found analytically, and for largeU theyconsist of two almost degenerate excited states, separfrom the ground state by the Hubbard gapU. This mimicsthe eigenvalue structure of the lowest multiplet of statesthe full double-dot system. We time evolve the system flowing the same procedure as before, using the ground sas the initial state. As the singlet Hamiltonian is only thrdimensional, however, the computations can correspondinbe done more rapidly. In Fig. 2~b! we show the results forPmin obtained by settingU58, which clearly shows that thissimplified model strikingly reproduces the behavior of tfull system of Eq.~1!.As the driving field @Eq. ~6!# is a periodic function oftime, we can make use of Floquet analysis to describetime evolution of the system in terms of its Floquet staand quasienergies.3 Hamiltonian ~5! is invariant under thecombined parity operationx→2x, t→t1T/2, and so theFloquet states can also be classified into these parity clasClose approaches of the Floquet quasienergies as the syparameters are varied produce large modifications of theneling rate, and hence of the system’s dynamics. Quasiegies of different parity classes may cross, but if they arethe same class they form an anticrossing. Numericallyquasienergies can be conveniently obtained by diagonalithe time evolution operator for one period of the drivinfield, U(t1T,t). This is particularly suited to the numericaapproach we have used, asU(t1T,t) is simply the operatorobtained by time evolving the identity matrixI 3 over oneperiod of the driving field.In Fig. 3~a! we present the Floquet quasienergies afunction of the field strength forv52, one of the resonanfrequencies visible in Fig. 2~b!. Below the spectrum we alsoplot the behavior ofPmin . We see that the system possesstwo distinct regimes of behavior. For weak fields, as studin Ref. 12, the Floquet spectrum consists of one isolated s~which evolves from the ground state! and two states whichmake a set of exact crossings. In this regimePmin decaysslowly to zero, showing little structure. As the field strengexceedsU, however, this abruptly changes to an interestinpreviously unseen, behavior in whichPmin remains close tozero except at a series of narrow peaks, corresponding toclose approaches of two of the quasienergies. A detailedamination of these approaches@see Fig. 3~b!# reveals them tobe anticrossingsbetween Floquet states which evolve frothe ground state and the higher excited state, and havesame parity. The remaining state, of opposite parity, masmall oscillations around zero, but its exact crossings wthe other two states do not correlate with any structurePmin .To interpret this behavior we seek analytical expressifor the quasienergies. We choose to use a perturbatiapproach,5 starting from the Floquet equationS H2 i ]]t D uf j~ t !&5e j uf j~ t !&. ~7!HereH is the full Hamiltonian@Eq. ~5!#, e j are the Floquetquasienergies, anduf j (t)& are the Floquet states. Our procdure is first to find the eigenstates of the operator@HI2 i (]/]t)#, whereHI is the second term in Eq.~5! contain--4-3inoesrt-enamreattheinrsr-ichatrac-er-ak-yan-eso-rbe-aneeldtionforinarengngondngon-BRIEF REPORTS PHYSICAL REVIEW B 65 113304ing all the interaction terms, and then treat the tunnelcomponentHt as a perturbation. An important advantagethis approach is that the Floquet states are stationary statEq. ~7!. Consequently, by working in an extended Hilbespace ofT-periodic functions,13 the corrections can be evaluated easily by standard Rayleigh-Schro¨dinger perturbationtheory, without requiring more complicated time-dependmethods.In a real-space representation the interaction termsdiagonal, and so it can be readily shown that an orthonorset of eigenvectors of@HI2 i (]/]t)# is given byue0~ t !&5~exp@ i e0t#,0,0!,ue1~ t !&5S 0,expF2 i ~U2e1!t1 i Evsinvt G ,0D , ~8!ue2~ t !&5S 0,0,expF2 i ~U2e2!t2 i Evsinvt G D .ImposingT-periodic boundary conditions reveals the corsponding eigenvalues~modulo v! to be e050 ande65U.These eigenvalues represent the zeroth-order approximto the Floquet quasienergies, and for frequencies suchU5nv all three eigenvalues are degenerate. This degeracy is lifted by the perturbationHt , and to first order thequasienergies are obtained by diagonalizing the perturboperatorPi j 5^^e i uHtue j&&, where ^^•••&& denotes the inneproduct in the extended Hilbert space.5 By using the identityexp@2 ib sinvt#5 (m52``Jm~b!exp@2 imvt# ~9!to rewrite the form ofue6(t)&, the matrix elements ofP canbe obtained straightforwardly, and its eigenvalues subthvi11330gfoftreal-ionatn-ge-quently found to bee050 ande6562Jn(E/v). This solu-tion clearly reduces to the well-known solution for noninteacting particles whenU50. Figure 3~a! demonstrates theexcellent agreement between this result~with n54) and theexact quasienergies for strong and moderate fields, whallows the position of the peaks inPmin to be found by lo-cating the roots ofJn . Similar excellent agreement occursthe other resonances. For weak fields, however, the intetion terms do not dominate the tunneling terms and the pturbation theory breaks down, corresponding to the wefield regime in whichPmin decays smoothly to zero. Awafrom the resonances the first-order correction identically vishes, resulting in the lack of structure seen between the rnant bands in Fig. 2~b!, and it is necessary to go to higheorders in perturbation theory to obtain the quasienergyhavior.In summary, we have investigated the dynamics ofinteracting two-electron system driven by an ac field. Wfind that despite the Coulomb interaction a suitable ac fican nonetheless produce localized states. This localizaoccurs over a range of field strengths at frequencieswhich an integer number of quanta,n, is equal to the inter-ction energy. At high fields we find a regime of behaviorwhich this localization vanishes at the roots ofJn(E/v), andwe explain this using perturbation theory. These resultsof general applicability to ac-driven systems of interactielectrons, and hold out the exciting prospect of controlliand manipulating correlated quantum states on picosectime scales by means of applying ac fields.C.E.C. thanks Sigmund Kohler for numerous stimulatidiscussions. This research was supported by the EU via Ctract No. FMRX-CT98-0180, and by the DGES~Spain!through Grant No. PB96-0875.ys.1M.A. Kastner, Phys. Today46~1!, 24 ~1993!; R.C. Ashoori, Na-ture ~London! 379, 413 ~1996!.2J.H. Shirley, Phys. Rev.138, B979 ~1965!.3Space does not permit us to give a detailed explanation ofFloquet approach. Instead we refer the reader to a recent rearticle by M. Grifoni and P. Ha¨nggi, Phys. Rep.304, 219~1998!,and references within, for an in-depth exposition.4F. Grossmann, T. Dittrich, P. Jung, and P. Ha¨nggi, Phys. Rev. Lett.67, 516 ~1991!.5M. Holthaus, Z. Phys. B: Condens. Matter59, 251 ~1992!.eew6F. Grossmann and P. Ha¨nggi, Europhys. Lett.18, 571 ~1992!.7P.I. Tamborenea and H. Metiu, Europhys. Lett.53, 776 ~2001!.8R.H. Blick et al., Phys. Rev. Lett.80, 4032~1998!.9T.H. Oosterkampet al., Nature~London! 395, 873 ~1998!.10C.E. Creffield, W. Ha¨usler, J.H. Jefferson, and S. Sarkar, PhRev. B59, 10719~1999!.11H. Wang and X.-G. Zhao, J. Phys.: Condens. Matter7, L89~1995!.12P. Zhang and X.-G. Zhao, Phys. Lett. A271, 419 ~2000!.13H. Sambe, Phys. Rev. A7, 2203~1973!.4-4

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Coherent transport in a two-electron quantum dot molecule

Coherent transport in a two-electron quantum dot molecule

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