Fast single-electron transport in a double quantum dot

M. Førre,1 J. P. Hansen,1 V. Popsueva,1 and A. Dubois2

1Department of Physics and Technology, University of Bergen, N-5007 Bergen, Norway2Laboratoire de Chimie Physique-Matière et Rayonnement, Université Pierre et Marie Curie, F-75231 Paris Cedex 05, France

�Received 9 November 2005; revised manuscript received 24 May 2006; published 4 October 2006�

We analyze two realistic experimental setups for quantum-controlled single-electron transport betweencoupled two-dimensional parabolic quantum dots. The dynamics is induced by two time-dependent electro-magnetic switches: �i� a linear electric switch and �ii� an intense alternating microwave field. From solving thetime-dependent Schrödinger equation it is shown that these switches can move an initially localized wavefunction from one of the dot centers to the other with unit probability. In the case of �i� the transition is direct,while in the case �ii� the quasiclassical transition dynamics involves a large number of eigenstates coupled bymultiphoton emissions and absorptions.

DOI: 10.1103/PhysRevB.74.165304 PACS number�s�: 73.21.La, 03.67.�a, 73.23.Hk, 73.63.Kv

I. INTRODUCTION

Single-electron time-dependent quantum control is neces-sary to operate solid-state quantum computers and in high-precision electronic metrology. Within quantum informationsingle-particle operations are needed as pre- or post-processors before or after multiparticle operations involvingentangled states, such as, for example, the controlled-NOT

gate.1 In metrology a future accurate standard for electriccurrent may be directly related to single-electron transport.2

In both these areas the role of decoherence is an obstaclewhich can reduce the effect or the accuracy. In certain situ-ations it may even have detrimental consequences for thesystem. To maximize coherence the degree of isolation andthe degree of dynamic control of the electronic quantum sys-tem is therefore of major importance.3

Two-dimensional quantum dots, conveniently modeled byconfining harmonic forces, are one of the most promisingsystems for controlled quantum dynamics.4 Coupled quan-tum dots, in particular double dots with two quantum wellsseparated by nanometer distances, have been the subject ofintense experimental research.5–9 Single-electron dynamicsbased on multiphoton absorption in such systems was firstdemonstrated by Oosterkamp et al.,10 and very recently Gor-man et al.11 achieved a coherence time of about 200 ns, sig-nificantly longer than previously reported.12 This was ob-tained by performing quantum manipulation of the double-dot system with time-dependent gates operating a distancefrom the dots. Progress in this direction gives realistic hopeof obtaining coherence in the microsecond region or beyondwhich then will open for manipulation and controlled state-to-state dynamics with an accuracy similar to that obtained inisolated Rydberg atoms.13

In the experiment,11 an electron initially placed in onequantum well is partially transferred to the other well byexposing the system to a train of short external “top hat”pulses of electric field changes. In this work we suggest al-ternative time-dependent gates which can transfer the elec-tron with unit probability from one center to the other. Thetransfer is caused by linear switches of an electric field orwith an intense oscillating monochromatic microwave field.The latter is related to experiments.10–12 Here we will discuss

which intensities, frequencies, and pulse lengths are neededto obtain unit transfer probability.

The paper is organized as follows: In the next section wepresent the theoretical model, the numerical methods, andthe physical ideas behind the switches. In the subsequentsection we present the simulation results and discuss the im-pact of anharmonicity—i.e., deviations from the harmonicoscillator. We apply atomic units �a.u.� �=m*=e=1, with m*

the effective mass of the electron in the qubit. In real systemsthe confinement strength ��0 is typically about 1 meV andthe effective mass of an electron in a GaAs material is m*

=0.067me, where me is the electron mass. With these num-bers the atomic unit of time becomes about 0.66 ps and theunit of length about 34 nm.

II. THEORETICAL MODEL

Our starting point is the two-dimensional �2D� double-dotmodel14,15 where a single electron is confined by the poten-tial

V0�x,y ;d� =1

2m*�0

2min��x −d

2�2

+ y2,�x +d

2�2

+ y2� .

�1�

Here, the interdot distance is d, �0 is the confining trap fre-quency, and m* is the effective mass of the electron. Thesingle-electron Hamiltonian then reads

H�x,y,t;d� = −1

2��x

2 + �y2� + V0�x,y ;d� + Vext�x,y,t� , �2�

where Vext�x ,y , t� is the external time-dependent potential. Inthe case of a near-linear switch it is given by Vext�x ,y , t�=E0�t�x, where E0�t� describes a switching function whichchanges sign from −E0 to E0 during the time interval underconsideration. For a monochromatic microwave �laser� fieldthe external potential is given by, Vext�x ,y , t�=E0x sin �t andthe dynamics of an electron based on the latter perturbationis the main focus of the paper. The dynamics is governed bythe two-dimensional time-dependent Schrödinger equation

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i�t��x,y,t� = H�x,y,t���x,y,t� , �3�

which is solved in the Cartesian split-operator form16

��x,y,t + �t� � e−i�V0+Vext��t/2e−i��x2+�y

2��t/2e−i�V0+Vext��t/2

���x,y,t� + O��t3� . �4�

A numerical solution can alternatively be obtained by ex-panding the wave function in a basis set containing a largenumber of eigenstates of the d=0 a.u. harmonic oscillator inCartesian coordinates,

��x,y,t� = �n,m

anm�t��n,m�x,y� , �5�

which by standard projection techniques leads to a set ofcoupled differential equations for the amplitude vector a�t�= �a00,a01,a10, . . . �. By precalculating the matrix elementsthis method is used to diagonalize the Hamiltonian for vari-ous values of the interdot distance d.

In Fig. 1 the 20 lowermost energy levels for zero externalfield are shown. As expected, the energy curves behave simi-larly to two-center molecular energy curves as a function ofinternuclear distance: At large distances d7 a .u., the low-est energy levels are the degenerate levels of two isolatedharmonic oscillators. Similarly, in the limit d→0 a .u., thestate energies of a single harmonic oscillator are obtained.Between these two regions level splitting occurs, with a se-ries of �avoided� crossings of energy levels increasing ordecreasing with d depending on the reflection symmetry ofthe states.

The dynamical implication of this feature is the following:By placing an electron in the local lowermost state of one ofthe dot centers—for example, in the left well—and keeping asuitable initial separation d one obtains a quasistationarystate,17 which we denote by L. Correspondingly, we denotethe local lowermost state in the right well by R. If theelectric field is then turned on, the initial state would be in alinear combination of a gerade and an ungerade molecularstate with energies g and e, respectively. The charge cloudwould accordingly oscillate between the dot centers with fre-

quency proportional to the energy separation.18 In real sys-tems the interdot distance cannot be manipulated on a mo-lecular time scale. However, external fields can effectivelylower the interdot distance by lowering the barrier or bymultiple excitation: In the latter case we observe from Fig. 1that the ground state is degenerate at d=7, while for thestates corresponding to the n=4 manifold energy splittingsoccur at the same distance.

III. RESULTS

We start out by considering the linear electric switch,where the L→ R transition is guided by an electric fieldwhich through the switching time lowers the barrier and in-duces transitions. With an initial polarization of the doubledot by an external constant electric field E0 along the interdotdistance—i.e., Vext=E0x—the polarization may keep theelectron stationary in the state L for d as low as 3 a .u. Thecoupling strength between the two wells can then be con-trolled by tuning the electric field strength, and the electroncan be transported to the right quantum well by simply swap-ping the field direction. If the switching is done sufficientlyslow, the state is transferred adiabatically with 100% effi-ciency. The state then follows the lowermost adiabatic en-ergy curve marked with a thick line in Fig. 2. The actualshape of the pulse is not crucial for the transition. Here, wesimply assume the field is switched linearly with time. To

FIG. 2. �Color online�. Upper panel: energy levels of the double-dot system with d=3 a.u. placed in an electric field along the in-terdot direction �solid curves�. The dashed curves are the adiabaticenergy levels of the Landau-Zener approximation. Lower panel:snapshot of the initial �left� and final �right� electronic probabilitydensity for a linear switching of the electric field. The axis rangesare from x=−4 to x=4 a.u. and from y=−2 to y=2 a.u., and thetotal switching time was T=300 a .u.

FIG. 1. The lowest-energy curves as function of interdot dis-tance �no external fields� with �0=1.

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avoid nonadiabatic transitions to excited states and in par-ticular to the first excited state, the total switching time Tmust be longer than some critical time Tc. The transitionprobability to the excited state can be approximated by thetwo-state Landau-Zener model,19

P � exp�−�T

4

�E12

�E2� , �6�

where �E1 is the minimum splitting energy of the dynamicaladiabatic states and �E2 is the maximum splitting energyinduced by the electric field, both shown in the upper panelof Fig. 2. Hence, a sufficient condition for a completely adia-batic switch becomes

T � Tc �4�E2

��E12 . �7�

With �E1�0.2 a .u. and �E2�1.5 a .u., the critical timeTc�50 a .u.—i.e., about 30 ps. The lower panels of Fig. 2show the initial and final probability densities, and a com-

plete transition L→ R is seen to be achieved.We now turn to oscillating fields, and as recent experi-

ments have shown, transitions can be induced by oscillatingmicrowave pulses9,10 or trains of pulses.11 The characteristicintensity and frequencies of the pulse�s� naturally determinethe degree of fidelity of the transition. We will here point outthat intense field strengths can lead to a complete transferwhich can be understood from a sequence of classical overthe barrier20 transitions: We assume a single-harmonic,constant-intensity microwave field polarized in the x̂direction—i.e., Vext�t�=E0x sin �t. By choosing a frequency� which is near the “natural” frequency of the system—i.e.,���0—the system will, for sufficiently high intensities, ef-fectively couple to a large number of states by multiphotontransitions.

The transition probability versus number of field cycles isexplicitly seen in Fig. 3 for the case E0=0.1 a .u. �30 V/cm�and �=1 a.u. �f =240 GHz�. For each cycle, when the wavepacket approaches the barrier, a fraction of it is stripped off.This mechanism leads to the plateaus in the transition prob-ability seen in the figure. Remarkably, this dynamics contin-ues until a complete transfer is obtained after about 30 fieldperiods. After switching the field off at that point the electronis stable in the state R. The snapshots in Fig. 3 illustrate theover the barrier crossing dynamics. Again, the upper left andlower right panels show the initial and final probability dis-tributions, respectively. The remaining six snapshots aretaken within the 14th and 15th field cycles, in the regionwhere the average transition rate is at its highest. From thesecond snapshot in the left column to the second snapshot inthe right column the field completes one cycle. One can seethat the left charge cloud oscillates in phase with the fieldwith maximum displacement �4 a .u. along the x̂ axis,whereas the right charge cloud oscillates in the oppositephase. The transferred charge cloud develops from beingpartly delocalized in the right well to become maximallylocalized after about 30 oscillations of the field. At that timethe state is practically the Gaussian R state. We note thatthis type of dynamics is obtained with increasingly fastertransition times for increasing intensity, once the intensity isstrong enough to induce multiphoton absorption and emis-sion in the nonlinear regime. In the present case the transi-tion time is about 120 ps.

The large number of states involved allows us to discussthe dynamics in terms of classical mechanics: Let us assumea classical electron initially confined in the left well. Then, ifthe frequency of the field matches the natural oscillation fre-quency in the well, the electron is resonantly kicked by thefield; i.e., it oscillates in phase with the field with a steadilyincreasing displacement. At one stage the electron has gainedsufficient momentum to pass over the barrier. On the otherside of the barrier the electric field works against the motionof the electron since the electric force oscillates out of phasewith the electron motion in the right quantum well. Thus, theelectron is initially slowed down in the right well and willremain trapped in this region if the field is turned off beforeit has regained enough energy for recurrence. Quantum me-chanically the initial electronic state is determined by a wavepacket with a given momentum distribution. In the simpleclassical picture the highest components are peeled off first

FIG. 3. �Color online�. Upper panel: probability of finding theelectron in the state R as a function of field cycles for E0

=0.1 a .u. and �=�0=1 a .u. Lower panels: snapshot of the elec-tronic probability density at 8 instants, after 0 �upper left�, 14.25�second left�, 14.5 �third left�, 14.75 �lower left�, 15 �upper right�,15.25 �second right�, 15.5 �third right�, and 30 oscillations �lowerright� of the field. The axis ranges are from x=−10 to x=10 a.u.and from y=−2 to y=2 a.u., and the interdot separation d=10 a .u. �340 nm�.

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as the electron is dragged towards the barrier by the field. Inthis way the wave packet goes through a partial over thebarrier transition for each field cycle, accompanied by quan-tum mechanical tunneling. Once on the other side, the fieldturns against the motion and stabilizes the transferred part ofthe charge cloud rather than leading to a recurrent.

To explicitly demonstrate the classical nature of the quan-tum system we have solved Newton’s equations of motionfor a classical electron confined in a one-dimensional har-monic oscillator potential. The electron, which we assume tobe initially at rest in the bottom of the well at t=0, is drivenby the external harmonic field. For �=�0 the displacementx�t� from the equilibrium point at a later time is given by

x�t� =E0

2�02 ��0t cos �0t − sin �0t� . �8�

A phase-space plot of this dynamics is shown in Fig. 4, il-lustrating the position x�t� and velocity v�t� of the classicalelectron during a complete transfer cycle. The plot is similarto a phase-space diagram of a driven harmonic oscillator.

The path is a trajectory spiraling outwards and then inwards,with local radii proportional to the square root of the totalenergy. From Eq. �8� we can also get the classical transitiontime Tcl—i.e., the time it takes for an electron that is initiallyat rest in the left well to traverse the barrier and end up at restin the right well. It is easily derived from the constraintx�Tcl /2��d /2,

Tcl �2d�0

E0. �9�

Figure 5 shows two plots of the number of field oscilla-tions needed to achieve 100% transition versus the reciprocalof E0. The solid line is for the quantum mechanical cases,and the dashed line is for the classical cases. The differencesbetween the two may be attributed to quantum mechanicaltunneling processes, which cannot be accounted for using theclassical picture. We note that in the strong field limit, whererelatively short pulses are needed, the classical and quantum

FIG. 4. Phase-space plot of motion of a classical electron in thedouble-well potential �d=10 a.u . � for an electron that is initially atrest in the left well—i.e., x�t=0�=−5 a.u. The parameters � and E0

of the external driving field are identical to the ones used in Fig. 3.

FIG. 5. The number of field oscillations needed to achieve 100%transition from the left to the right well versus the reciprocal of E0,for the quantum mechanical �solid line� and classical �dashed line�cases, respectively. The differences between the two may be attrib-uted to quantum mechanical tunneling processes.

FIG. 6. As Fig. 3, but for three different choices of asymmetricpotentials: i.e., �=1.01�0 �solid line�, �=1.02�0 �dashed line�, and�=1.03�0 �dotted line�. The external field frequencies are �=1.005�0, �=1.01�0, and �=1.015�0, for the three cases,respectively.

FIG. 7. As Fig. 3, but for three different choices of anharmonicpotentials: i.e., A=−0.001 a .u. �solid line�, A=−0.002 a .u. �dashedline�, and A=−0.003 a .u. �dotted line�. The external field frequen-cies are �=0.983�0, �=0.970�0, and �=0.9545�0, for the threecases, respectively.

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mechanical lines coincide, showing that the role of tunnelingis negligible. On the other hand, as the field gets weaker, thepulse length grows and so does the contribution from thetunneling processes. As the field strength becomes verystrong—i.e., E0�0.1 a .u.—the dynamics ultimately be-comes strongly nonadiabatic and 100% transition is unlikelyto occur.

In real physical systems we expect to find small devia-tions from the harmonic potential well. We now discuss thestability of the switch with respect to small asymmetries andanharmonicities of the potential. First, consider the case withtwo slightly different harmonic quantum wells:

V�x,y ;d� =1

2m*min �0

2��x + d/2�2 + y2�,�2��x − d/2�2 + y2�� ,

�10�

with different confining frequencies �0 �left well� and ��right well�.

The transition probabilities as functions of the number ofcycles for three different cases are shown in Fig. 6, where theexternal field frequencies are tuned to achieve the optimaltransfer. The three curves demonstrate the switch efficiencyfor �=1.01�0 �solid line�, �=1.02�0 �dashed line�, and �=1.03�0 �dotted line�. We conclude that, although the switchis rather sensitive to the asymmetry, it is robust in a narrowrange of values of �. The complete transfer is possible whenthe difference in confining frequencies does not exceed 2%or so. A difference of 3% reduces the efficiency of the switchto about 80%.

We also investigate the effect of a slight anharmonicity A,

V�x,y ;d� = V0�x,y ;d� + A min �x − d/2�4 + y4,

�x + d/2�4 + y4� . �11�

The plots shown in Fig. 7 show a strong sensitivity of thesystem as in the case of asymmetric wells. The completetransfer is possible to achieve with A �0.002 a .u. Outsidethis range the efficiency is less than 90%. Experimental re-alizations of the present system thus require a high degree ofprecision in the manufacturing process: On the other hand,the sensitivity to anharmonicity effects as shown here maybe used in testing purposes in the process of designing suchsystems.

IV. CONCLUSION

In conclusion, we have shown that carefully chosen time-dependent electric fields can transfer an electron with 100%probability from one side of a double-quantum-dot system tothe other. This is true for direct Landau-Zener-type transi-tions induced by linear switches and for complex multipho-ton transitions in intense oscillating microwave fields. Thetransitions are realized in the picosecond range with realiz-able electromagnetic field strengths and corresponding elec-tronics. The switch is sensible to deviations from harmonicconfinement, but stable within a limited range of anharmo-nicity parameters. This, in principle, opens up possibilitiesfor precise metrology based on single-electron transport.With the experimental progress on limiting decoherence11,12

it appears that coupled quantum dots, controlled by time-dependent electromagnetic fields, are a very realistic routetowards real quantum computing.

ACKNOWLEDGMENT

The present research has been partially sponsored by NFRthrough the NANOMAT program.

1 M. A. Nielsen and I. L. Chuang, Quantum Computation andQuantum Information �Cambridge University Press, Cambridge,England, 2000�.

2 B. Sheridan, P. Cumpson, and M. Bailey, Phys. World 18, 37�2005�.

3 C. J. Myatt, B. E. King, Q. A. Turchette, C. A. Sackett, D.Kielpinski, W. M. Itano, C. Monroe, and D. J. Wineland, Nature�London� 403, 269 �2000�; L. B. Ioffe, M. V. Feigel’man, A.Ioselevich, D. Ivanov, M. Troyer, and G. Blatter, ibid. 415, 503�2002�.

4 S. M. Reimann and M. Manninen, Rev. Mod. Phys. 74, 1283�2002�.

5 F. H. L. Koppens, J. A. Folk, J. M. Elzerman, R. Hanson, L. H.Willems van Beveren, I. T. Vink, H. P. Tranitz, W. Wegscheider,L. P. Kouwenhoven, and L. M. K. Vandershypen, Science 309,1346 �2005�.

6 M. T. Woodside and P. L. McEuen, Science 296, 1098 �2002�; G.Schedelbeck, W. Wegscheider, M. Bichler, and G. Abstreiter,ibid. 278, 1792 �1997�.

7 R. Aguado and L. P. Kouwenhoven, Phys. Rev. Lett. 84, 1986�2000�.

8 J. C. Chen, A. M. Chang, and M. R. Melloch, Phys. Rev. Lett. 92,176801 �2004�.

9 J. R. Petta, A. C. Johnson, C. M. Marcus, M. P. Hanson, and A. C.Gossard, Phys. Rev. Lett. 93, 186802 �2004�.

10 T. H. Oosterkamp, T. Fujisawa, W. G. vander Wiel, K. Ishibashi,R. V. Hijman, S. Tarucha, and L. P. Kouwenhoven, Nature �Lon-don� 395, 873 �1998�.

11 J. Gorman, D. G. Hasko, and D. A. Williams, Phys. Rev. Lett. 95,090502 �2005�.

12 T. Hayashi, T. Fujisawa, H. D. Cheong, Y. H. Jeong, and Y.Hirayama, Phys. Rev. Lett. 91, 226804 �2003�.

13 M. Førre, D. Fregenal, J. C. Day, T. Ehrenreich, J.-P. Hansen, B.Henningsen, E. Horsdal-Pedersen, L. Nyvang, O. E. Povlsen, K.Taulbjerg, and I. Vogelius, J. Phys. B 35, 401 �2002�.

14 A. Wensauer, O. Steffens, M. Suhrke, and U. Rössler, Phys. Rev.B 62, 2605 �2000�.

15 A. Harju, S. Siljamäki, and R. M. Nieminen, Phys. Rev. Lett. 88,226804 �2002�.

16 M. D. Feit, J. A. Fleck, and A. Steiger, J. Comput. Phys. 47, 412�1982�.

FAST SINGLE-ELECTRON TRANSPORT IN A DOUBLE… PHYSICAL REVIEW B 74, 165304 �2006�

165304-5

17 The state is initially stationary on a time scale typically severalorders of magnitude longer than the manipulation time.

18 J. Caillat, A. Dubois, I. Sundvor, and J. P. Hansen, Phys. Rev. A70, 032715 �2004�.

19 C. Zener, Proc. R. Soc. London, Ser. A 137, 696 �1932�.20 H. Cederquist, L. H. Anderson, A. Barany, P. Hvelplund, H.

Knudsen, E. H. Nielsen, J. O. K. Pedersen, and J. Sorensen, J.Phys. B 18, 3951 �1985�.

FØRRE et al. PHYSICAL REVIEW B 74, 165304 �2006�

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