Bilayer quantum Hall systems can form collective states in which electrons
exhibit spontaneous interlayer phase coherence. We discuss the possibility of
using bilayer quantum dot many-electron states with this property to create
two-level systems that have potential advantages as quantum bits.Comment: 4 pages, 4 figures included, version to appear in Phys. Rev. B (Rapid
  Communications

A. H. MacDonald

A. Stern

A.H. MacDonald

A.W. Holleitner

B. G. Levi

B.E. Kane

D. Loss

D.G. Austing

E. Demler

I.B. Spielman

J. Hu

J. Schliemann

J.P. Eisenstein

John Schliemann

K. Moon

K. Yang

L. Balents

L.P. Kouwenhoven

M. Bayer

M. Fogler

R.H. Blick

S.-R. Eric Yang

S.Q. Murphy

T. Brandes

T. Fujisawa

X.G. Wen

Y. Joglekar

Y. Makhlin

Y. Nakamura

Y. Tokura

Z.F. Ezawa

English

arXiv

Bilayer quantum-Hall systems can form collective states in which electrons exhibit spontaneous interlayer phase coherence. We discuss the possibility of using bilayer quantum dot many-electron states with this property to create two-level systems that have potential advantages as quantum bits

Yang, S.-R. Eric

Schliemann, John

MacDonald, A. H.

University of Regensburg Publication Server

PHYSICAL REVIEW B 66, 153302 ~2002!Quantum-Hall quantum bitsS.-R. Eric Yang,1,2 John Schliemann,1,3 and A. H. MacDonald11Department of Physics, University of Texas, Austin, TX 787122Department of Physics, Korea University, Seoul, Korea3Department of Physics and Astronomy, University of Basel, Switzerland~Received 27 June 2002; published 2 October 2002!Bilayer quantum-Hall systems can form collective states in which electrons exhibit spontaneous interlayerphase coherence. We discuss the possibility of using bilayer quantum dot many-electron states with thisproperty to create two-level systems that have potential advantages as quantum bits.DOI: 10.1103/PhysRevB.66.153302 PACS number~s!: 03.67.Lx, 73.21.La, 73.43.LpOver the past several years there has been a great deal ofinterest in solid-state two-level systems that could serve asquantum computing bits ~qubits!,1 and thereby enable large-scale quantum computation. In order for a two-level systemto be useful as a quantum bit, it must be possible to maintaincoherent quantum evolution over time scales long comparedto those required for its intentional manipulation. This is par-ticularly difficult to achieve in a solid-state environment be-cause of the presence of many degrees of freedom, elec-tronic, photonic, and nuclear, which interact relativelystrongly. The spin degree of freedom of individual electrons2or nuclei3 in a semiconductor, and the total charge degree offreedom of a small superconductor4 are among the possibili-ties that have been been advanced for crafting solid statequbits. In this paper we propose a different possibility, qubitsbased on the the total pseudospin of double-layer quantumdots in the quantum-Hall regime. We suggest an experimentthat could be used to demonstrate their quantum coherence,and discuss some of their potential advantages.The two-level system we discuss is related to thespontaneous-coherence states that occur in bulk bilayerquantum-Hall systems.5–8 These states have a gap forcharged excitations, which is entirely due to electron-electron interactions. They can be viewed either as pseu-dospin ferromagnets or as a Bose condensate9 of pairsformed between electrons in one layer and holes in a Landaulevel of an adjoining layer. Their low-energy physics can bedescribed10 by an effective model usually expressed in termsof a pseudospin quantum field. The pseudospin in these sys-tems is the quantum which-layer degree of freedom. An elec-tron that is in an eigenstate of the zˆ -component of the pseu-dospin operator ~with eigenvalue 61/2) is definitely in oneof the layers, while one with quantum uncertainty in its layerindex is in a state with a pseudospin component in the xˆ -yˆplane. The qubits we propose are, however, based on many-electron eigenstates in small double-layer quantum dot11 sys-tems, like those realized recently by Tarucha andco-workers12 and illustrated schematically in Fig. 1. Theyhave cylindrical symmetry about an axis along which astrong magnetic field Bzˆ can be applied. The dot area iscontrolled by a side gate, which provides an approximatelyharmonic lateral confining potential that stabilizes finite areaquasi-two-dimensional quantum dots.0163-1829/2002/66~15!/153302~4!/$20.00 66 1533If we assume that the quantum dots of interest are suffi-ciently small to suppress spatial variation of the pseudospin,the effective-field theory10 has a single pseudospin collectivecoordinate (Tx ,Ty ,Tz) and an effective Hamiltonian:H5 8pl2bN Tz22DvTz2aD tTx . ~1!In Eq. ~1!, Tz52N/2,2N/211, . . . ,N/2 is half the differ-ence between electron numbers in right and left dots, N is thetotal number of electrons in the double dot, l is the magneticlength defined by 2pl2B5hc/e5F0, where F0 is themagnetic-flux quantum, Dv is an adjustable interlayer biasvoltage, D t is the splitting of single-particle energy levels inthe double-layer system due to interlayer tunneling, a,1FIG. 1. Schematic illustration of a pillar bilayer double-dot sys-tem. We consider the situation in which a strong magnetic field isapplied along the axis of the pillar. The barriers that confine elec-trons to the two GaAs layers in the pillar are established by theepitaxial growth of ~Al,Ga!As layers. The maximum phonon wave-lengths in the pillar (lmax) are the geometrical metrics associatedwith the structure’s fundamental vibration modes, represented inthis figure by the pillar height. The minimum wavelengths lmin forphonons that couple effectively to collective electronic states of thedot are comparable to the dot radius or the interlayer distance. Sincelmax is comparable to lmin , only the fundamental and at most offew harmonics will be important in decohering these collective statequbits.©2002 The American Physical Society02-1BRIEF REPORTS PHYSICAL REVIEW B 66, 153302 ~2002!accounts for microscopic quantum fluctuations in the many-electron state, and b is discussed below. For small D t and theratio DvN/8pl2b equal to an appropriate integer, the spec-trum of this Hamiltonian has two low-lying levels separatedfrom higher-lying states by a large gap. This two-level sys-tem is our proposed qubit.The mean-field theory of bilayer quantum-Hall ferromag-nets, like Ref. 9, the Bardeen-Cooper-Schrieffer ~BCS!theory of superconductivity, provides both valuable insightinto the nature of these states and a practical tool for quan-titative estimates. For bilayer quantum dot systems,13,14 themean-field state consists of a Slater determinant of bilayersymmetric, for example, single-particle orbitals with angularmomenta m50, . . . ,N21 that establish coherence betweenthe layers even when D t50. The bilayer coherent state isstable when the occupied symmetric single-particle orbitalshave mean-field eigenenergies that are ~i! lower than those ofsymmetric states with larger angular momenta and ~ii! lowerthan those of any antisymmetric state.14 Note that these qua-siparticle energies are split by a gap, analogous to the BCSgap of a superconductor, that can be entirely due to interac-tions and does not vanish for D t→0. For weak confinementor strong fields,14 electron-electron interactions dominate andfavor a fractional average occupation of angular-momentumstates, violating the first condition. For strong confinement orweak fields,14 average occupation numbers larger than 1 arefavored, violating the second condition. As illustrated in Fig.2, the double-dot coherent state is stable only over a finiterange of confinement strengths, g5mV2l2, which narrowswith increasing particle number N (V is the frequency of theharmonic potential!. The value of b can be estimated, usingeither mean field or exact diagonalization calculations, fromthe dependence of the ground-state pseudospin polarizationon bias potential. From the inset of Fig. 2, we see that2pl2b50.06e2/el at N514, compared to the bulk value2pl2b50.09e2/el .10 The results of Fig. 2 were obtainedwith the interlayer separation equal to l and with D t and Dvset to zero.FIG. 2. Hartree-Fock stability range of the bilayer coherent stateas a function of electron number N. The inset displays values of2pl2b obtained by fitting the effective Hamiltonian to exact-diagonalization and Hartree-Fock theory results. The unit of 2pl2bis e2/el .15330In the inset of Fig. 3, we plot the exact spectrum of themany-particle Hamiltonian for 12 electrons in a double-layerquantum dot system as a function of total angular momentumM. The collective state discussed above has quasiparticleswith angular momentum m50, . . . ,N21, and so occurs attotal angular momentum M DLCS5N(N21)/2. Exact diago-nalization calculations show that the double-layer coherentstate is the ground state over a narrow range of fieldstrengths. The results plotted in Fig. 3 are for g50.1e2/el ,which is in the middle of the stability range for N512. Com-paring with Fig. 2, we see that Hartree-Fock theory onlyslightly overestimates the confinement strength at which thedouble-layer coherent state occurs. The main part of Fig. 3shows the dependence of the three lowest eigenvalues on Dvfor D t equal to a small positive value. The first anticrossingtakes place near Dv50.02e2/el , consistent with the effectivemodel prediction, Dv58pl2b/N . These results demonstratethat the effective Hamiltonian accurately describes the low-energy part of the dependence of the microscopic Hamilto-nian’s spectrum on D t and Dv , which will figure importantlyin our discussion. The lowest-energy states that appear atM.M GS in Fig. 3 are edge excitations of the dot, whichform a part of our proposed qubit’s environment as discussedbelow.For N even, assumed in the following, we choose themany-body states u0& and u1& (Tz eigenvalues are 0 and 1)as the orthogonal states of the proposed qubit. ~For N odd,the optimal orthogonal states would be u1/2& and u21/2&.! Atzero bias voltage, the ground state is nondegenerate and, forsmall D t , has nearly definite Tz . The truncated two-levelHamiltonian is given up to a constant byHP5S 4pl2bN 2 Dv2 Dsz2 D2 sx , ~2!FIG. 3. Dependence of the three lowest eigenvalues with totalangular momentum M5M GS on bias voltage Dv . The inset showsthe low-energy part of the many-electron energy spectrum @in unitsof e2/el] as a function of total angular momentum M for 12 elec-trons at Dv50 and at the strength of the confinement potential g50.1e2/el . M GS566 for N512. This figure demonstrates that thetwo-level system of our proposed qubit is cleanly separated fromother many-electron states.2-2BRIEF REPORTS PHYSICAL REVIEW B 66, 153302 ~2002!where D5aD tA(N12)N/2, and sz and sx are Pauli spinmatrices in the space of the qubit. The tunneling term in theHamiltonian, which is very small for the quantum-dot sys-tems we have in mind, is effective only near the resonancesituation, Dv58pl2b/N , where it splits the degeneracy thatwould otherwise occur. From Fig. 3 we find that D52.5731023e2/el , corresponding to a50.86.In this paragraph, we propose an experiment that we be-lieve is feasible, and which can be used to study the coherentquantum evolution of this qubit. If the bias voltage is held atzero, where the zˆ component of the qubit effective fielddominates, for a sufficiently long time the qubit will reach itsground state u0&. When the bias voltage is tuned to reso-nance, the qubit evolves under the influence of thexˆ -direction Zeeman field. If after a time T, the bias voltagereturns to zero, the qubit will have a finite probability ampli-tude for being in the u1& state, i.e., for having transferred anelectron between layers of quantum dot. We require thatequilibrium should be reestablished by charge flowingthrough the external circuit used to control the bias voltage, aproperty that requires the dot contact resistances to besmaller than the off-resonance intradot resistance. If this pro-cedure is repeated many times with a repetition periodTrepeat that is longer than the equilibration time, the currentflowing between layers through the external circuit measuresthe probability for transferring an electron,I5eS 12 2 122p2 cos~DT/2\! D /Trepeat . ~3!Here pÞ0 is small if the resonance condition is establishedrapidly compared to \/D as we discuss below. A measure-ment of oscillating current as a function of time-on-resonance T would establish quantum coherence for this po-tential qubit.There are, of course, no genuine two-level systems in na-ture, and the one under discussion here is not an exception.Even discounting coupling to phonons, nuclear spins, andother degrees of freedom present in the solid-state environ-ment, the electron Hamiltonian has a large number of eigen-states. A practical requirement15 for an effective two-levelsystem is that its operational time scale \/D should be longcompared to the scale which breaks adiabaticity between thetwo-level system and its electronic environment. If the pseu-dospin effective Hamiltonian fully describes the double layerquantum dot, this time scale would be \/(2pl2b). If rota-tional symmetry of the Hamiltonian is broken, on the otherhand, edge states are the lowest-lying electronic excitationsoutside of the two-level system. As illustrated in Fig. 3, thesize of these energies for realistic double-layer quantum dotsis ;1.0 meV for e2/el;10 meV. The tunneling amplitudebetween bilayer systems can be varied over a very widerange because of its exponential dependence on Al content inthe barrier separating the quantum wells and on the width ofthis barrier. For D t50.0003 meV and N540, the gap D is0.001 meV, safely smaller than edge excitation energies, andthe operational time of the quantum bit is ;1029 s, makingit possible to adjust the bias potential sufficiently rapidly toachieve p!1 in Eq. ~3!.15330Bilayer quantum-Hall systems are similar7 in many re-spects to the Josephson coupled superconductors ~but alsoshow differences,8! and there are both similarities and differ-ences between the experiment we propose above and thecoherent single-Cooper-pair box effect realized16 by Naka-mura et al. In the Nakamura experiment, the qubit is formedby states that differ by 1 in the number of Cooper pairs onone side of the junction, whereas in the excitonic insulatorlanguage our qubit is formed by states that differ by 1 in thenumber of electron-hole pairs in the system. Since a newelectron-hole pair is formed by moving a single electronfrom one layer to the other, the current expression in our caseis smaller by a factor of 2. The two qubits share the propertythat, because of the collective behavior of many electrons,coherence can be established even though the systems arecomposed of a relatively large number of electrons. Thephase coherence time of bulk bilayer quantum-Hall systemshas been estimated6 to be tf;2310210 s, most likely8 dueto coupling between collective degrees of freedom and elec-tronic excitations near the boundaries between the incom-pressible and compressible regions that occur in all bulksamples. This coherence time is shorter than the period wepropose above for a typical Rabi-oscillating current madefrom quantum-Hall quantum bits. However, as explainedabove, this decoherence mechanism is not operative forquantum dot systems. Instead, the dominant decoherencemechanism is likely to be piezoelectric coupling to long-wavelength phonons, nabbed17 as the likely culprit forsingle-electron double dots.17,18 For collective qubits, like theones we propose, only phonons with wavelengths compa-rable to the entire electronic system can effectively decohere.We illustrate this point in Fig. 4 by plotting the dependenceof the change in Dv on N in the presence of a potentialFIG. 4. Dependence of the change in the qubit parameter Dv onN in the presence of a fluctuation in the potential difference betweenthe two layers that is localized at the center of the quantum dot.This quantity has been evaluated by performing self-consistentHartree-Fock calculations at D t50 and evaluating the energy dif-ference between states that differ by 1 in the number of electrons inthe top layer. The potential fluctuation strength has been chosen sothat it alters Dv by 0.01 e2/el for a single electron. Electroniccorrelations in large-N collective states suppress the system’s re-sponse to the fluctuation potential, and reduces the change in Dv ,which results from the same potential fluctuation.2-3BRIEF REPORTS PHYSICAL REVIEW B 66, 153302 ~2002!fluctuation located at the center of the quantum dot. For largeN, correlations between electrons in the collective state sup-press response to the localized potential fluctuation, and limitits effectiveness in decohering the qubit. We expect thatphonons will be an especially weak decoherence mechanismfor these qubits because the maximum phonon wavelength,established by size quantization, is only slightly larger thanthe minimum effective phonon wavelength, established bythe size of the double-dot system. The fact that the decoher-ence time is already ;10210 s in the bulk, suggests that verylong coherence times will be achievable.Solid-state quantum computing clearly presents evenmore daunting obstacles than merely coherent time evolutionof solid-state qubits. The problems that would need to besolved to control the entanglement of these quantum-Hallqubits are similar to entanglement strategies for the case of15330of Cooper pair box qubits.4 We believe that experimentalstudy of the novel qubit proposed here, which occurs in aphysical system that has already been achieved,12 couldmake valuable contribution to the growing understanding ofdecoherence in solids.We are grateful for valuable conversations with G. Aust-ing, S. Tarucha, D. Loss, Y. Nakamura, and P. Zoller.S.R.E.Y was supported by the KOSEF Quantum-functionalSemiconductor Research Center at Dongguk University andby the Korea Research Foundation under Grant No. 1998-015-D00114. J.S. was supported by the Deutsche Fors-chungsgemeinschaft. Work at the University of Texas wassupported by the Welch Foundation and the by the NationalScience Foundation under Grant No. DMR0115947.1 For a review see, for example, M.A. Nielsen and I.L. Chuang,Quantum Computation and Quantum Information ~CambridgeUniversity Press, Cambridge, England, 2000!.2 D. Loss and D.P. DiVincenzo, Phys. Rev. A 57, 120 ~1998!.3 B.E. Kane, Nature ~London! 393, 133 ~1998!.4 Y. Makhlin et al., Nature ~London! 398, 305 ~1999!.5 S.Q. Murphy et al., Phys. Rev. Lett. 72, 728 ~1994!; I.B. Spiel-man et al., ibid. 84, 5808 ~2000!; 87, 36803 ~2001!; L. Balentsand L. Radzihovsky, ibid. 86, 1825 ~2001!; A. Stern et al., ibid.86, 1829 ~2001!; J. Schliemann et al., ibid. 86, 1849 ~2001!; E.Demler et al., ibid. 86, 1852 ~2001!; K. Yang, ibid. 87, 056802~2001!.6 For reviews see J.P. Eisenstein, Phys. World 14~6!, 30 ~2001!; B.G. Levi, Phys. Today 54~5!, 14 ~2001!; S.M. Girvin, e-printcond-mat/0108181.7 X.G. Wen and A. Zee, Phys. Rev. Lett. 69, 1811 ~1992!; Z.F.Ezawa and A. Iwazaki, Int. J. Mod. Phys. B 19, 3205 ~1992!;M. Fogler and F. Wilczek, Phys. Rev. Lett. 86, 1833 ~2001!.8 Y. Joglekar and A.H. MacDonald, Phys. Rev. Lett. 87, 196802~2001!.9 A.H. MacDonald, Physica B 298, 129 ~2001!.10 K. Moon et al., Phys. Rev. A A51, 5138 ~1995!.11 For recent reviews see L.P. Kouwenhoven et al., Rep. Prog. Phys.64, 701 ~2001!; L.P. Kouwenhoven and P.L. McEuen, in Nano-science and Technology, edited by G. Timp ~Springer-Verlag,New York, 1999!.12 D.G. Austing et al., Physica B 249-251, 206 ~2000!; Y. Tokuraet al., Physica E ~Amsterdam! 6, 676 ~2000!.13 S.-R. Eric Yang et al., Phys. Rev. Lett. 71, 3194 ~1993!; A.H.MacDonald et al., Aust. J. Phys. 46, 345 ~1993!.14 J. Hu et al., Phys. Rev. B 54, 8616 ~1996!.15 J. Schliemann et al., Phys. Rev. B 63, 085311 ~2001!.16 Y. Nakamura et al., Nature ~London! 398, 786 ~1999!.17 T. Fujisawa et al., Science 282, 932 ~1998!.18 M. Bayer et al., Science 291, 451 ~2001!; R.H. Blick et al., Phys.Rev. Lett. 80, 4032 ~1998!; T. Brandes et al., Phys. Rev. B 64,035319 ~2001!; A.W. Holleitner et al., Phys. Rev. Lett. 87,256802 ~2001!.2-4

Quantum-Hall quantum bits

Crossref

Quantum-Hall Quantum-Bits

Quantum-Hall Quantum-Bits

Abstract

Similar works

Full text

Available Versions

University of Regensburg Publication Server

Crossref