Tampere University of Technology

Controllable quantum scars in semiconductor quantum dots

CitationKeski-Rahkonen, J., Luukko, P. J. J., Kaplan, L., Heller, E. J., & Räsänen, E. (2017). Controllable quantum scarsin semiconductor quantum dots. Physical Review B, 96(9), [094204].https://doi.org/10.1103/PhysRevB.96.094204Year2017

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Published inPhysical Review B

DOI10.1103/PhysRevB.96.094204

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Controllable quantum scars in semiconductor quantum dots

J. Keski-Rahkonen,1 P. J. J. Luukko,2 L. Kaplan,3 E. J. Heller,4 and E. Rasanen1, 4

1Laboratory of Physics, Tampere University of Technology, Finland2Max Planck Institute for the Physics of Complex Systems, Dresden, Germany

3Department of Physics and Engineering Physics, Tulane University, New Orleans, USA4Department of Physics, Harvard University, Cambridge, USA

(Dated: October 3, 2017)

Quantum scars are enhancements of quantum probability density along classical periodic orbits.We study the recently discovered phenomenon of strong, perturbation-induced quantum scarring inthe two-dimensional harmonic oscillator exposed to a homogeneous magnetic field. We demonstratethat both the geometry and the orientation of the scars are fully controllable with a magnetic fieldand a focused perturbative potential, respectively. These properties may open a path into an exper-imental scheme to manipulate electric currents in nanostructures fabricated in a two-dimensionalelectron gas.

I. INTRODUCTION

Quantum scars1,2 are tracks of enhanced probabilitydensity in the eigenstates of a quantum system. Theyoccur along short unstable periodic orbits (POs) of thecorresponding chaotic classical system. While being aninteresting example of the correspondence between clas-sical and quantum mechanics, quantum scars also showhow quantum mechanics may suppress classical chaosand make certain systems more accessible for applica-tions. In particular, the enhanced probability density ofthe scars provides a path for quantum transport acrossan otherwise chaotic system.

A new type of quantum scarring was recently dis-covered by some of the present authors.3 It was foundthat local perturbations (potential “bumps”) on a two-dimensional (2D), radially symmetric quantum well pro-duce high-energy eigenstates that contain scars of shortPOs of the corresponding unperturbed (without bumps)system. Even though similar in appearance to ordinaryquantum scars, these scars have a fundamentally differ-ent origin. They result from resonances in the unper-turbed classical system, which in turn create semiclassi-cal near-degeneracies in the unperturbed quantum sys-tem. Localized perturbations then form scarred eigen-states from these near-degenerate “resonant sets” be-cause the scarred states effectively extremize the pertur-bation. These perturbation-induced (PI) scars are unusu-ally strong compared to ordinary scars, to the extent thatwave packets can be transported through the perturbedsystem, along the scars, with higher fidelity than throughthe unperturbed system, even with randomly placed per-turbations.3

In this work we study these perturbation-induced scarsin a 2D quantum harmonic oscillator exposed to a per-pendicular, homogeneous magnetic field. This systemhas direct experimental relevance as it is a prototypemodel for semiconductor quantum dots in the 2D elec-tron gas.4 Previous studies combining experiments andtheory have confirmed the validity of the harmonic ap-proximation to model the external confining potential of

electrons in the quantum dot, up to the quantum Hallregime reached with a strong magnetic field.5,6 Further-more, the role of external impurities in 2D quantum dotshas been quantitatively identified through the measureddifferential magnetoconductance that displays the quan-tum eigenstates.7 We show that once perturbed by Gaus-sian impurities, the high-energy eigenstates of the systemare strongly scarred by POs of the unperturbed system,and the shape of these scars can be conveniently tunedvia the magnetic field. Furthermore, in the last part ofthe work we replace the impurities with a single pertur-bation, representing a controllable nanotip,8 and use the“pinning” property of the scars to control the scars withthe nanotip. Together, these methods of controlling thestrong, perturbation-induced scars could be used to co-herently modulate quantum transport in nanoscale quan-tum dots.

II. MODEL SYSTEM

All values and equations below are given in atomicunits (a.u.). The Hamiltonian for a perturbed 2D quan-tum harmonic oscillator is

H =1

2

(− i∇+ A

)2+

1

2ω20r

2 + Vimp, (1)

where A is the vector potential of the magnetic field. Weset the confinement strength ω0 to unity for convenienceand assume the magnetic field B is oriented perpendicu-lar to the 2D plane. We model the perturbation Vimp asa sum of Gaussian bumps with amplitude M , that is,

Vimp(r) =∑i

M exp

[− (r− ri)

2

2σ2

].

We focus on the case where the the bumps are posi-tioned randomly with a uniform mean density of twobumps per unit square. In the energy range consideredhere, E = 50 . . . 100, hundreds of bumps exist in theclassically allowed region. The full width at half maxi-mum (FWHM) of the Gaussian bumps 2

√2 ln 2σ is 0.235,

arX

iv:1

710.

0058

5v1

[qu

ant-

ph]

2 O

ct 2

017

2

which is comparable to the local wavelength of the eigen-states considered. The amplitude of the bumps is set toM = 4, which causes strong PI scarring in this energyregime.

We solve the eigenstates of the Hamiltonian of Eq. (1)using the itp2d code.9 This code utilizes the imaginarytime propagation method, which is particularly suited for2D problems with strong perpendicular magnetic fieldsbecause of the existence of an exact factorization of theexponential kinetic energy operator in a magnetic field.10

The eigenstates and energies can be compared to thewell-known solutions of a unperturbed system. The un-perturbed energies correspond to the Fock-Darwin (FD)spectrum,11

EFDk,l = (2k + |l|+ 1)

√1 +

1

4B2 − 1

2lB, (2)

where k ∈ N and l ∈ Z. In the limit B → ∞, the statescondensate into Landau levels. The FD spectrum is ob-served experimentally in semiconductor quantum dots(see, e.g., Ref. 12).

Before considering further the quantum solutions of theHamiltonian we briefly discuss the corresponding classi-cal system. The unperturbed 2D harmonic oscillator ina perpendicular magnetic field is analytically solvable asdescribed in the Appendix and Ref. 13. The POs areassociated with classical resonances where the oscillationfrequencies of the radial and the angular motion are com-mensurable. In the following, the notation (vθ, vr) refersto a resonance where the orbit circles the origin vθ timesin vr radial oscillations. The resonances occur only atcertain values of the magnetic field given by

B =vr/vθ − 2√vr/vθ − 1

. (3)

By classical simulations using the bill2d program14

we have confirmed that a perturbation with ampli-tude M = 4 is sufficient to destroy classical long-termstability in the system. Any remaining structures in theotherwise chaotic Poincare surface of a section are van-ishingly small compared to ~ = 1.

III. QUANTUM SCARS

Next we describe the quantum solutions of the system.To visualize the spectrum of a few thousand lowest en-ergies as a function of B we show in Fig. 1 the densityof states (DOS) computed as a sum of the states witha Gaussian energy window of 0.001 a.u. The upper andlower panel correspond to the perturbed (with bumps)and unperturbed system, respectively. TheB values indi-cating resonances (vθ, vr) are marked by dashed verticallines. The upper panel of Fig. 1 shows that the bumpsare sufficiently weak to not completely destroy the FDdegeneracies.

(2,13)

(1,6)

(2,11)

(1,5)

(2,9)

(1,4)

(2,7)

(1,3)

(3,8)

(2,5)

(3,7)

FIG. 1. Scaled (arb. units) density of states (DOS) as a func-tion of the magnetic field for an unperturbed (lower panel)and perturbed (upper panel) two-dimensional harmonic os-cillator. The dashed vertical lines indicate the resonances(vθ, vr) that correspond to a substantial abundance of scarredeigenstates in the perturbed case (see Fig. 2).

As expected from the theory of PI scarring, the eigen-states of the perturbed system show clear scars corre-sponding to the POs of the unperturbed system. Becauseof the classical resonance condition (3) these scars appearat specific values of the magnetic field where short clas-sical periodic orbits are possible. Examples of the scarsare shown in Fig. 2. The eigenstate number varies be-tween 400 . . . 3900. The proportion of strongly scarredstates among the eigenstates varies from 10% to 60% atbump amplitude M = 4. The proportion and strength ofscarred states depend on the degree of degeneracy in thespectrum: more and stronger scars appear when moreenergy levels are (nearly) crossing. The specific shapeof the scar for a chosen rsonance (vθ, vr) depends on theenergy. For example, the two examples shown for the tri-angular geometry (1,3) in Fig. 2 correspond to orbits withopposite directions, which are not equivalent because themagnetic field breaks time-reversal symmetry.

As pointed out in Ref. 3, because the scarred states

3

(1,2) (1,3) (1,5)(1,4)

(2,5) (2,9) (3,7) (3,8)

FIG. 2. Examples of scars in a perturbed two-dimensionalharmonic oscillator in a magnetic field. The geometries of thescars can be attributed to the classical resonances (vθ, vr) andthe corresponding periodic orbits in the unperturbed system.

singlebump

(a) (b)

(c)

manybumps

FIG. 3. (a)-(b) Examples of scarred eigenstates at B ≈ 0.7found in a system with several impurities and a single impu-rity with M = 4, respectively. The locations and FWHMsof the impurities are denoted with blue dots. (c) Over-lap between the scarred state and the impurity potential,〈ψ|Vimp(θ)|ψ〉, as a function of the rotation angle with re-spect to the original orientation Vimp. The upper (blue) andlower (red) curves correspond to the situations in (a) and (b),respectively. The lower curve has been multiplied by a factorof six for visibility.

S D

conductingnanotip

semiconductorquantumdot

FIG. 4. Schematic figure on the utilization of scars in nanos-tructures. The magnetic field (here B = 0) determines thegeometry of the scar (here linear), the energy range refines thegeometry further (here longitudinal), and an external voltagegate generates a local perturbation (here the pink dot) thatpins the scar, thus determining the orientation.

maximize the perturbation Vimp they are oriented to co-incide with as many bumps as possible. This “pinning”effect is demonstrated in Fig. 3(a). Even a single pertur-bation bump can produce a strong scar that is pinned toits location, as illustrated in Fig. 3(b).

As another view on the pinning effect, Fig. 3(c) showsthe overlap between the scarred state and the impuritypotential, 〈ψ|Vimp(θ)|ψ〉, as a function of the rotation an-gle with respect to the original orientation Vimp. This isshown for both the multiple-bump (upper curve) and thesingle-bump (lower curve) cases, corresponding to situa-tions in Figs. 3(a) and (b), respectively. The three dis-tinctive maxima in both cases confirm that the scar ispinned to a location that maximizes the perturbation.

By creating a single perturbation with, e.g., a con-ducting nanotip,8 the pinning effect can be used to forcethe scars to a specific orientation. Together, the mag-netic field and the pinning effect provide complete exter-nal control over the strength, shape, and orientation ofthe scars. By selecting which parts of the system are con-nected by the scar paths, this provides a method to mod-ulate quantum transport in nanostructures in a schemedepicted in Fig. 4. A successful experiment requires thatthe system is otherwise clean from external impuritiessuch as migrated ions from the substrate.7 In addition,a sufficient energy range in transport is required as thescars are more abundant at high energies (level number& 500). In the future, we will study this effect in moredetail using realistic quantum transport calculations.

4

IV. SUMMARY

To summarize, we have shown that perturbation-induced quantum scars are found in a two-dimensionalharmonic oscillator exposed to an external magnetic field.The scars are relatively strong, and their abundance andgeometry can be controlled by tuning the magnetic field.By controlling the orientation of the scars though theirtendency to “pin” to the local perturbations, we reachperfect control of the scarring using external parame-

ters. This scheme, using a conducting nanotip as thepinning impurity, may open up a path into “scartronics”,where scars are exploited to coherently control quantumtrasport in nanoscale quantum systems.

V. ACKNOWLEDGMENTS

We are grateful to Janne Solanpaa for useful discus-sions. This work was supported by the Academy of Fin-land. We also acknowledge CSC – Finnish IT Center forScience for computational resources.

1 E. J. Heller, Phys. Rev. Lett. 53, 1515-1518 (1984).2 L. Kaplan, Nonlinearity 12, R1 (1999).3 P. J. J. Luukko, B. Drury, A. Klales, L. Kaplan, E. J.

Heller, and E. Rasanen, Sci. Rep. 6, 37656 (2016).4 S. M. Reimann and M. Manninen, Rev. Mod. Phys. 74,

1283 (2002).5 E. Rasanen, H. Saarikoski, A. Harju, M. Ciorga, and A. S.

Sachrajda, Phys. Rev. B 77, 041302(R) (2008).6 M. C. Rogge, E. Rasanen, and R. J. Haug, Phys. Rev. Lett.

105, 046802 (2010).7 E. Rasanen, J. Konemann, R. J. Haug, M. J. Puska, and

R. M. Nieminen, Phys. Rev. B 70, 115308 (2004).8 A. C. Bleszynski, F. A. Zwanenburg, R. M. Westervelt,

A. L. Roest, E. P. A. M. Bakkers, and L. P. Kouwen-hoven, Nano Lett. 7, 2559 (2007); E. E. Boyd, K. Storm,L. Samuelson, and R. M. Westervelt, Nanotechnology 22,185201 (2011); T. Blasi, M. F. Borunda, E. Rasanen, andE. J. Heller, Phys. Rev. B 87, 241303 (2013).

9 P. J. J. Luukko and E. Rasanen, Comp. Phys. Comm. 184,769 (2013).

10 M. Aichinger, S. A. Chin, and E. Krotscheck, Com-put. Phys. Commun. 171, 197 (2005).

11 V. Fock, Z. Phys. 47, 446 (1928); C. G. Darwin, Proc.Cambr. Philos. Soc. 27, 86 (1930).

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Appendix A: Classical periodic orbits of atwo-dimensional harmonic oscillator in a

perpendicular and homogeneous magnetic field

Here we present the solution of the equations of motionfor a classical particle in a harmonic potential with a per-pendicular, homogeneous magnetic field B [see also G. L.Kotkin and V. G. Serbo, Collection of Problems in Clas-sical Mechanics (Pergamon Press Ltd., 1971))]. Afterchoosing the vector potential as Aφ = 1

2Br, the system

is described, in polar coordinates, by the Lagrangian

L =1

2(r2 + r2φ2)− 1

2ω20r

2 +1

2Br2φ.

Its constants of motion are the energy E and the angularmomentum pφ.

In a rotating frame with θ = φ+ 12Bt, the Lagrangian

has the form of a simple harmonic oscillator,

L =1

2(r2 + r2θ2)− 1

2ω2r2,

where the effective frequency is

ω =

√ω20 +

1

4B2.

Thus, the system is reduced to a harmonic oscillatorwithout a magnetic field, the solution of which can befound in standard texts on classical mechanics [e.g., H.Goldstein, Classical Mechanics, 2nd ed. (Addison- Wes-ley, 1980)]. When the initial conditions are chosen asφ(0) = 0 and r(0) = a, the solution in the original coor-dinates (r, φ) is

r2 = a2 cos2(ωt) + b2 sin2(ωt)

and

φ(t) = −1

2Bt+ arctan

[ab

tan(ωt)],

where the branches of the arctangent are chosen suchthat the angle φ is a continuous function of time t. Themaximum and minimum radii, a and b, are determinedby the angular momentum pφ = ωab and the energy

E =1

2ω2(a2 + b2)− 1

2pφB.

The solution shows that the period of radial oscillationsT = π/ω is independent of E and pφ. The angle withwhich the radius vector turns during a period is

∆φ = π[ξ(pφ)− B

2ω

], (A1)

5

where the step function ξ(x) is

ξ(x) =

−1 if x < 00 if x = 01 if x > 0.

Note that ∆φ is also independent of E.The orbit is periodic exactly when ∆φ is a rational

multiple of 2π. Other orbits are quasiperiodic. When∆φ = 2πvθ/vr, the particle returns to its initial stateafter rotating vθ times around the origin and oscillatingvr times between the radial turning points. Thus, the

classical oscillation frequencies are said to be in vθ : vrresonance for this particular PO.

By solving Eq. (A1) for the magnetic field, we candetermine which value of B is required for a vθ : vr res-onance. This gives, after elementary algebra, the reso-nance condition

B =vr/vθ − 2√vr/vθ − 1

.