Lithuanian Journal of Physics, Vol.50, No. 4, pp. 377–394 (2010) doi:10.3952/lithjphys.50405

Review

QUASICLASSICAL THEORY OF QUANTUM DOTS

E. Anisimovasa and A. Matulisba Department of Theoretical Physics, Vilnius University, Sauletekio 9, LT-10222 Vilnius, Lithuania

E-mail: egidijus.anisimovas@ff.vu.ltb Semiconductor Physics Institute, Center for Physical Sciences and Technology, A. Goštauto 11, LT-01108 Vilnius, Lithuania

E-mail: amatulis@takas.lt

Received 30 November 2010; accepted 15 December 2010

We review the quasiclassical theory of quantum dots. The starting point of the developed approximate approaches isthe observation that in large (in comparison to the effective Bohr radius) quantum dots the energy of the classical Coulombinteractions dominates over the quantum-mechanical kinetic energy. This dominance is further enhanced by application of aperpendicular magnetic field. The classical regime is marked by the formation of structures (the Wigner crystal) and structuraltransitions. The nature of these phenomena is indeed classical, and they can be successfully tackled using classical approacheswhich are transparent and easy to understand. In this way heavy calculations typical of quantum-mechanical schemes areavoided and the quantum effects are included in an perturbative manner. We discus, in particular, the application of therenormalized perturbation series to the energy spectra, the structural transitions, the power law behaviour of the critical fields,the global (persistent) and local currents in quantum dots, and dissipation in mixed systems with both quantum and classicaldegerees of freedom.

Keywords: quantum dots, quasiclassical approximation, Wigner crystal

PACS: 73.21.La, 03.65.Sq

1. Introduction

Quantum dots [1, 2] are man-made nanostructureswhere the motion of charge carriers is confined in allthree spatial dimensions, and the confinement mani-fests itself in a discrete energy level spectrum. In thisrespect, quantum dots resemble atoms and, unsurpris-ingly, have been nicknamedartificial atoms[2]. Quan-tum dots are produced in a lab and thus – in stark con-trast to the natural atoms – have tunable parameters,which make them attractive objects for fundamentalstudies as well as applications [3].

As far as applications and mass-production is con-cerned, the most convenient are the so-calledself-assembledquantum dots [4, 5]. They can be producedin large quantities, and often come out nicely arrangedin regular arrays [4, 6] by simply growing layers of twolattice-mismatched materials (a common pair is GaAsand InAs) on top of each other. These quantum dotsare small objects with typical radii around 10 nm orless. Thus, the self-assembled quantum dots are mostclosely related to the natural atoms as the charge carrierconfinement length is comparable to the effective Bohrradius of the host material, which is also approximately10 nm for the gallium arsenide.

On the other hand, early attempts to fabricate con-fined pools of electrons typically employed applica-tion of electrostatic potentials to patterned gates placedabove a two-dimensional electron layer [7, 8]. Quan-tum dots defined in this way are commonly known aselectrostatic (gated), or parabolic. The latter term re-flects the fact that almost any potential distribution canbe reasonably well approximated by a parabola in thevicinity of its minimum point. Thus, we use the follow-ing form of the confinement potential:

Vconf(r) =m∗ω2

0r2

2, (1)

with m∗ representing the electron effective mass andω0 defining the confinement frequency. This frequencyis the oscillation frequency of either a single particleor the centre-of-mass mode of a many-particle sys-tem. The parabolic quantum dots are also significantlylarger. Here, the confinement length can exceed the ef-fective Bohr radius up to ten times. We define the con-finement length as the characteristic extension of theground-state wavefunction of a single particle

l0 =

√~

m∗ω0. (2)

c© Lithuanian Physical Society, 2010c© Lithuanian Academy of Sciences, 2010 ISSN 1648-8504

378 E. Anisimovas and A. Matulis / Lithuanian J. Phys.50, 377–394 (2010)

As a consequence of the size, the relative strength ofthe interparticle interactions is also much larger. Ele-mentary dimensional analysis shows that the kinetic en-ergy of a confined electron is of the order of~2/m∗l20.The typical energy of the interparticle Coulomb inter-action ise2/εl0, whereε denotes the dielectric constantof the medium. Thus, the relative importance of inter-actions can be estimated as the ratio of the characteris-tic energies

λ =e2m∗l0ε~2

=l0a∗B

with a∗B =ε~2

m∗e2, (3)

which is equal to the ratio of the confinement lengthand the effective Bohr radiusa∗B.

The length scales characterizing natural atoms arefixedand the coupling strengthλ defined in an analo-gous fashion turns out to be of the order of unity. Thisimplies that interactions are not particularly strong.The dimensions of quantum dots fabricated in a lab canbe tuned to some extent, thereby tuning also the ratioof the defining energy scales. Even greater possibilitiesof control are offered by the application of a perpen-dicular magnetic field, which influences the interactionparameter according to

λ→ λeff ∼ λ√γ . (4)

Here γ is the dimensionless magnetic field strength,defined as the ratio of the cyclotron frequencyωc =eB/m∗c to the confinement frequencyω0:

γ =ωc

ω0. (5)

In general, strong interactions are known to facili-tate the emergence of cooperative phenomena. Sincethese interactions are also tunable, the parabolic quan-tum dots have been recognized as an ideal laboratoryto study the collective effects brought about by thestrong Coulomb repulsion between the confined elec-trons. This has been a central topic of our studies ofquantum nanostructures. Therefore, the present reviewmostly focuses on the physics of parabolic – that is,large and strongly correlated – quantum dots and col-lective phenomena.

Early reports emphasizing the observation of dis-creet electronic states in artificial semiconductor struc-tures started appearing more than 20 years ago. Thediscreet nature of the electron energy spectrum wasconvincingly demonstrated using the resonant tun-nelling [9], capacitance [10] and optical (far-infrared)spectroscopies [11]. Already in these pioneering stud-ies physicists were quick to recognize the advantages

offered by the ability to localize a small and control-lable number of particles as well as the strong influ-ence of a perpendicularly applied magnetic field on themany-body states. At magnetic fields of the order of1 T, the cyclotron energy matches the typical energylevel spacing, and thus is sufficiently strong to controlthe behaviour of the quantum dot.

The influence of the electron interactions on the en-ergy spectrum, and in particular, on its magnetic-fielddependence was confirmed by exact diagonalizationstudies performed on simplest systems of three and fourelectrons [12]. A detailed exact-diagonalization studyof the interactions of two-particle system (the quantum-dot helium) was also performed [13] specifically con-centrating on peculiarities of the far-infrared response.The point is that in a perfect parabolic confinement thecentre-of-mass and the relative motion of electrons de-couple, and as a result, the effects of the electron inter-action are absent from the far-infrared absorption spec-trum. This effect is known as the generalized Kohntheorem [12, 14–16].

The aforementioned exact diagonalization approach[17–19] involves a numerical diagonalization of thefully interacting many-body Hamiltonian in the basisof configurations constructed by distributing electronsover single-particle states in all possible ways. In thiscontext, the wordexact is used to emphasize that noapproximations are used to describe the interparticleinteractions. However, in practical work the nominallyinfinite set of configurations has to be truncated at somepoint, which naturally introduces a certain error. Prob-lems involving a small number (let’s say,N 6 6)of particles, or restricted to extreme magnetic fields,can be successfully tackled with this technique sincepresent computational facilities allow to include a suf-ficient number of configurations in order to obtain con-vergent results. However, the computational demandsgrow exponentially with the size of the system, and pre-clude the use of the exact diagonalizations as a genericmethod.

Therefore, various single-particle approaches areroutinely applied to calculate the properties of few-electron quantum dots. In particular, standard methodsof atomic and molecular physics – such as the Hartree,Hartree–Fock, and density functional theory – havebeen extensively exploited [1, 20–23]. At high electrondensities the physics is dominated by quantum degener-acy, and the interparticle interaction acts a rather weakcorrection. Thus, in this limit the above-mentioned ap-proaches produce accurate and, one might say, unsur-prising results. In the opposite – dominated by strong

E. Anisimovas and A. Matulis / Lithuanian J. Phys.50, 377–394 (2010) 379

interactions – limit the spin- and space-unrestrictedHartree–Fock calculations were used to produce bro-ken symmetry states in quantum dots [24]. The ob-tained ground states showed the incipient Wigner crys-tallization, that is, the formation of charge densitylumps in spite of the fact that the confining potentialused in these calculations had a perfect circular sym-metry. Broken-symmetry solutions were also obtainedfrom calculations based on the density functional the-ory [25], which led to some controversy and a debate[23]. In contrast, the ground states obtained from theexact-diagonalization approach always have the sym-metry of the Hamiltonian. The internal structure of aquantum dot is somewhat hidden. It can be observedby considering the charge density correlation function,not the charge density itself [17, 26].

Leaving aside the subtleties of computational proce-dures, let us concentrate on the physical aspect: strongcorrelation between the particles leads to formationof structures. The ultimate limit of this process is,of course, the celebrated Wigner crystal [27–30] sug-gested as the ground state of a low-density electronicsystem back in 1934 [27]. However, even before theWigner crystal is formed in the extreme limit of strongcorrelations, its various precursors – such as chargedensity waves [1] and the angular momentum transi-tions [17, 26] – are detected.

The physics behind the angular momentum transi-tions is easy to explain without resorting to detailed cal-culations. It is not surprising that the electrons confinedwithin a quantum dot in the strong interaction regimetend to stay as far apart as possible. This makes thempopulate orbitals characterized by high values of theangular momentum. As a result, the total angular mo-mentum of the dot increases in abrupt jumps as the ef-fective coupling strength (set by the external magneticfield) increases. The ensuing charge redistribution in aquantum dot is significant and was successfully mea-sured in an experiment [31].

Let us emphasize that crystallization demonstratesthe dominance of the (classical) potential energy overthe quantum-mechanical kinetic energy. The ratio ofthese two characteristic energies is given by the param-eterλ. The reciprocal of the same parameter is the mea-sure of the importance of quantum effects. Thus, crys-tallization and quantum mechanics are mutually exclu-sive. The quasiclassical regime is the one where in-teractions and their consequences are important. Thequantum mechanical effects are weak, and can be ac-counted for in a perturbative approach. We were suc-cessful to build a coherent theory based on these ideas

[32, 33], and the main goal of the present review is todiscuss the quasiclassical regime.

In this review, we shall cover in detail the followingissues: (i) the enhancement of the relative strength ofthe electron interactions by the magnetic field, (ii) theWigner crystallization, (iii) the effect of the quantummechanics on the structural transitions, (iv) classicalpower-law behaviour in the energy spectra, (v) globaland local currents in a Wigner crystal, and (vi) the roleof dissipation in quantum phenomena.

2. Renormalized perturbation series

In the natural atoms the interaction parameterλ is ofthe order of unity, and the importance of correlationsis not overwhelming. Therefore, the atomic theory issuccessfully developed on the basis of effective singleparticle approaches. The correlation energy – which,by definition, includes everything that is overlooked bythe Hartree–Fock approximation – is a small correc-tion, and can be taken into account in a perturbativeapproach.

An analogous strategy is clearly unsuitable forparabolic quantum dots. The interaction constantλ is,in general, greater or even much greater than unity. Theelectron system in a quantum dot is indeed strongly cor-related, and a straightforward expansion in powers ofλis not a very promising way to proceed. A direct di-agonalization of the many-body problem, on the otherhand, is too demanding in terms of computational re-sources. Thus, development of reliable approximateschemes is of great interest.

We proposed to calculate the energy spectra ofstrongly correlated quantum dots using the renormal-ized perturbation series [34, 35]. The basic idea ofthe series renormalization is to supplement the ordi-nary few-term energy expansion in powers ofλ withthe information available from the consideration of theextreme opposite limit (λ→∞). The latter limit is par-ticularly advantageous due to its classical nature. Here,the kinetic energy is comparatively small, and a few-electron system may be represented as a small crystal-lite performing harmonic vibrations in the vicinity ofits lowest-energy configuration.

In both limits of small and large values of the inter-action constantλ, we were able to construct two-termexpansions. In the limitλ → 0 the behaviour of theenergy levels is described by the usual first-order per-turbation series

E(λ) = E0 + E1λ . (6)

380 E. Anisimovas and A. Matulis / Lithuanian J. Phys.50, 377–394 (2010)

The energyE0 characterizes a noninteracting system,and is straightforwardly calculated as the sum of single-electron energies. The second expansion coefficientE1

is obtained by diagonalizing a matrix built from two-particle Coulomb matrix elements.

In the limit of very strong interactions, the two-termexpansion becomes

E(λ) = c0λ2/3 + c1 . (7)

The form of the first term follows directly from the di-mensional considerations, which indicate that the ra-dius of the electron system scales asR ∼ λ1/3. Theinteraction energy is thus proportional toλ2/3. Thesecond term in the expansion (7) is obtained by consid-ering the vibrations of a ring ofN 6 5 particles. Thevibrational energy levels turn out to beλ-independent.

The interpolation between the two limits is facili-tated by introduction of a generalized Hamiltonian

H =12

N∑i=1

(−ξ2∇2i + r2

i ) + λ∑i>j

1|ri − rj |

. (8)

Here, the parameterξ plays the role of the inverseelectron mass and the original problem is recoveredby settingξ = 1. The generalized Hamiltonian, aswell as its eigenvalues, satisfies the scaling relationH(ξ, λ) = ξH(1, λξ−3/2). This relation helps mapthe strongly interacting regime of the original problem(λ→∞, ξ = 1) onto a generalized problem character-ized by finite parameter values,λ = 1 andξ → 0. Aconvergent four-term series is then constructed.

The technique of renormalized perturbation serieswas first applied to the simplest two-electron quantumdot [34]. The calculated energy values coincided withthe results of numerically exact calculation to within1% for all values of the coupling constantλ. Thissuccess motivated us to adapt the technique for largerparabolic quantum dots [35]. Here, the theory hasto be expanded to include a somewhat more involvedsymmetry classification of the energy levels in the twolimits. The obtained accuracy is also very satisfac-tory. We attribute the reasons of success to the soft-ness of the parabolic confinement since the applicationof the renormalized series to hard-wall quantum dotswas not as successful [36, 37]. This demonstrates theimportance of analytic properties of the potential forthe renormalization procedure. On the other hand, thetechnique turned to be insensitive to the particle statis-tics, and we successfully applied the renormalized per-turbation series to parabolically confined bosonic sys-tems [38], which are important in the Bose-Einsteincondensation studies.

3. Influence of quantization on the phase transition

Seeking to understand a complicated many-bodyphenomenon, such as crystallization, it is worthwhileto start from the simplest system of just two interact-ing electrons. Very often consideration of a simplifiedmodel problem produces analytical results and givesan approximate qualitative picture reflecting the phe-nomena that take place in the real many-electron sys-tem. Sometimes, one is even able to extract reliableestimates of the parameters that are responsible for thefeatures of the considered many-particle system.

In order to reveal the role of quantization in theelectron crystallization as an order-disorder phase tran-sition, we calculated the quantum-mechanical spec-trum of a two-electron artificial molecule composed oftwo vertically coupled two-dimensional single-electronquantum dots [39]. To be fully precise, in quantum dotsand other finite systems one should speak not of phasetransitions but rather of bifurcations that play the roleof phase transitions.

A classical study of the above artificial molecule wascarried out in Ref. [40]. The centre-of-mass motionof the two-electron system contributes only an energyshift, which is not important for our purposes. The rel-ative motion is then described by the potential

U(r) =14m∗ω2

0r2 +

e2

ε

√r2 + d2 , (9)

dependent on the relative coordinater = r1−r2. Herewe use the standard notation wherebym∗ is the effec-tive electron mass,ε is the dielectric constant of themedium, andω0 is the frequency of the radial electronconfinement.

In this context, it is convenient to introduce nonstan-dard dimensionless variables expressing the distancesin the units of(e2/εα)1/3, and energies in the unitsε0 = (e2/εα2)2/3. Hereα = mω2

0/2, and the sym-bol d stands for the vertical distance between the twodots. In this way, the potential reduces to

U(r) =12r2 +

1√r2 + d2

, (10)

and after minimization leads to the equilibrium radiusfor the ground-state configuration:r0(d) =

√1− d2

for d < 1 andr0(d) = 0 for d > 1. We see that thecalculated equilibrium radius plays the role of an or-der parameter. As long as the separation between thedots is larger than the critical valued = 1, the twoelectrons are located on the axis of the molecule. Thebifurcation, which occurs atd = 1, breaks the symme-try and moves the electrons into their new equilibrium

E. Anisimovas and A. Matulis / Lithuanian J. Phys.50, 377–394 (2010) 381

positions off the symmetry axis. Inserting the valuesof the radiusr0(d) into the potential (10) we also cal-culate the equilibrium energy and (aiming to determinethe nature of transition) the derivatives of the energywith respect to the distanced. The results read:

E = (3− d2)/2, E′ = −d, E′′ = −1, d < 1 ,E = 1/d, E′ = −1/d2, E′′ = 2/d3, d > 1 .

(11)

Fig. 1. The first and second energy derivatives as a function of thedistanced. The classical result: the first (second) derivative is plot-ted by the blue online dash-dotted (red online dotted) curve. Thequantum-mechanical result: the first (second) derivative is shown

by the blue online dashed (red online solid) line.

The behaviour of the energy derivatives (11) isshown in Fig. 1. The (blue online) dash-dotted curvecorresponds to the first derivative, and the (red online)dotted curve corresponds to the second derivative. Wesee that at the pointd = 1 the first derivative is contin-uous, but has a kink, while the second derivative is dis-continuous. This behaviour resembles the phase transi-tion of the second kind in the Ehrenfest classification.

Proceeding to the quantum-mechanical description,we construct the Hamiltonian and solve the eigenvalueproblem. In the dimensionless variables, the Hamilto-nian equals the sum of the classical potential (10) andthe operator

T = −κ2∇2 , (12)

which represents the kinetic energy of the relative mo-tion. The dimensionless parameterκ = ~ω0/ε0 isthe ratio of the characteristic quantum-mechanical andclassical energies at the transition point, and serves asthe measure of the importance of quantization. Thederivatives of the ground-state energy are plotted nextto the classical results on the same Fig. 1. We see thatquantization ruins the sharp phase transition. The kink(jump) in the first (second) energy derivative are elim-inated. Nevertheless, in the case of relatively small

κ . 0.1 values, the imprint of phase transition is stilldistinct.

Fig. 2. The quantum-mechanical (solid red online curves) and clas-sical frequencies (dashed blue online curves).

The excitation frequencies are also calculated, andthe result is shown in Fig. 2. The classical vibrationfrequencies are plotted in (blue online) dashed curvesand the corresponding quantum mechanical transitionsbetween the excited states are shown by the (red online)solid curves. It is generally known that the presence ofa soft mode is an inherent property of the second orderphase transition. Indeed, in the classical description ofthe structural transition in our molecule one of the vi-bration frequencies goes to zero when the distancedvalue approaches the critical point. The correspond-ing quantum-mechanical frequency, however, does notreach the zero value. This is yet another indication thatquantization destroys phase transitions.

4. Wigner crystallization

In this section, our goal is to define the criteria thatcould be used to judge about the formation of internalstructure in a quantum dot, and to relate this structure tothe involved energy scales and to the interparticle cou-pling strength. With this in mind, we solve a simplifiedmodel discussing the spectrum and the charge densityof a two-electron quantum dot [41].

As already mentioned in the previous section, in thecase of parabolic confinement the centre-of-mass andthe relative motion can be separated. The separation isachieved by introducing the radius-vector of the centreof massR = (r1 +r2)/2 and the relative radius vectorr = r1 − r2. As a result, the original problem splitsinto two independent single-particle problems. In addi-tion, the circular symmetry enables one to write the to-tal wavefunction as a radial part times the angular part

382 E. Anisimovas and A. Matulis / Lithuanian J. Phys.50, 377–394 (2010)

expi(mϕ + MΦ). Here, the uppercase symbolsMandΦ denote, respectively, the angular momentum andthe rotation angle of the centre-of-mass motion. Like-wise, the corresponding lowercase symbols have thesame meaning for the internal motion.

The radial wavefunction is, in its own turn, a productof two components describing the centre-of-mass andthe relative motion. These components satisfy the re-spective one-dimensional radial Schrödinger equationswith the following dimensionless Hamiltonians:

HR =− 14R

ddR

Rd

dR+

14

(M

R− γR

)2

+R2 , (13a)

Hr =−1r

r

drrr

dr+(m

r− γr

4

)2

+14r2 +

λ

r. (13b)

Here, all energies are measured in the units~ω0, andwe use the standard definitions for the confinement fre-quencyω0, the coupling strengthλ, and the dimension-less magnetic fieldγ.

Fig. 3. The potential for the relative motion whenγ = 0 (redonline solid curve), the approximate potential (14) (dashed blueonline curve), and the vibration and rotation terms indicated by the

horizontal lines.

Let us separately consider two variants of the formu-lated problem: with and without the magnetic field.

It is evident that in the absence of the magnetic field(γ = 0), the ground state of the centre-of-mass motioncorresponds to zero angular momentum (M = 0) andgives some constant energy shift which we eliminateby adjusting the origin of the energy axis. Meanwhile,the relative motion can be interpreted as the motion ofsome fictitious particle in a one-dimensional potentialwell

V (r, λ) =14r2 +

λ

r+m2

r2, (14)

shown in Fig. 3 by the (red online) solid curve.Restricting the consideration to the interesting limit

of strong interactions (λ → ∞) the potential (14) canbe simplified further. Disregarding the last rotational

term in Eq. (14), we find that the potential has its min-imum at the pointr0 = (2λ)1/3. The Taylor expansionof the potential in vicinity of the minimum point reads

V (r, λ) ≈ 34(2λ)2/3 +

34(r − r0)2 +

m2

(2λ)2/3. (15)

We see that there are three energy scales involved. Thelargest one appears in the first term (which is of the or-der ofλ2/3), and includes the interaction plus the con-finement. This classical potential is responsible for theelectron structure in the quantum dot. (The minimumof the potential can serve as a means for the Wignercrystal definition.) The second part defines the interme-diate energy scale. This potential term, together withthe kinetic energy operator in the Hamiltonian (13b),leads to the quantum-mechanical problem of the crys-tal vibrations. As this contribution does not depend onλ, the separation of the vibrational levels is of the or-der of unity. This result justifies the proposed schemeof calculation based on the potential expansion in pow-ers ofλ−2/3. It consists of the classical calculation ofthe electron system structure followed by the quantum-mechanical treatment of its vibrations in the harmonicapproximation. The remaining term in Eq. (15) con-tributes the fine rotational structure to the obtained vi-bration terms. All these energy scales are shown inFig. 3 by corresponding horizontal lines.

Fig. 4. Schematic view of (a) the electron density and (b) the cor-relation function (b) in the quasi-classical limit whenλ→∞.

In the simple case of two electrons we obtain anexplicit expression for the wavefunction in the groundstate

Ψ(r1, r2) ∼ exp(−R2) exp−a(r − r0)2 , (16)

wherea =√

3/4. Integrating the squared absolutevalue of the wavefunction (16) over the coordinates ofone electron, we obtain the electron density in the dot,which is shown schematically in Fig. 4(a). Due to thecircular symmetry of the dot, the crystal structure is notseen, however, the concentration of the electron densityon a ring indicates that such structure might be present.The Wigner crystal can be directly seen in the correla-tion function that is equal to the squared absolute value

E. Anisimovas and A. Matulis / Lithuanian J. Phys.50, 377–394 (2010) 383

of the same wavefunction (16) considered as a functionof the coordinates of one of the electron. The positionof the other electron is fixed at a certain pointr′ chosen,e. g., on the circle of radiusr0/2, as shown in Fig. 4(b).We see that in this case the density of the free electronis concentrated around the antipodal point on the samecircle.

Fig. 5. Energy spectrum in the case of strong magnetic field. Theenergy is given in~ωc units. Electron density in the inset.

In the case with a strong magnetic field, a similaranalysis can also be carried out, although it is more la-borious. Its main result is summarized in Fig. 5. Theapproach is based on minimization of two expressions:the term that appears in the parentheses in Eq. (13b),and the sum of the last two terms in the same equa-tion. This procedure leads to the equilibrium radiusr0 = (2λ)2/3 and the ground-state angular momentumm0 = γr20/4. The expansion of the potential in thevicinity of these values leads to(

m

r− γr

4

)2

≈ γ2

4(r − r0)2 +

(m−m0)2

(2λ)2/3. (17)

The first (largest) term together with the kinetic en-ergy operator in the Hamiltonian (13b) gives the largestcontribution to the energy. Actually it produces theLandau levels. So, in contrast to the situation with-out the magnetic field, now the main contribution tothe energy spectrum is quantum-mechanical. It leadsto the following dimensions of the radial wave func-tion ∆r ∼ B−1/2. Fortunately, in the limit of a strongmagnetic field and low temperature, the electrons arefrozen in the lowest Landau level, and the Wigner crys-tallization follows from the same minimum of the con-finement and the interaction potential. Comparing (15)and (17) we see that the Landau levels play the role ofthe Wigner crystal vibrations and – as we explainedabove – are frozen. The crystal dynamics contains onlythe rotational modes whose energies are of the same

order of magnitude as in the previous case without themagnetic field.

The electron density is shown in the inset of Fig. 5.It looks like the previous ring of the same radius, butthe thickness of this ring is significantly smaller. Thatis why we may to conclude that the strong magneticfield favours crystallization.

5. Angular momentum transitions

Formation of structures and structural transitions inquantum dots – as well as in any other many-particlesystem – are among the most complex and demandingphysical problems. The key factor involved in thesephenomena is the strong interaction between the parti-cles which makes the problem inseparable. We werefortunate to formulate several simple models, treatedin Sections 3 and 4, that are solvable analytically anddemonstrate structural transitions in few-electron quan-tum dots. When confronted with more complicatedmodels, however, we had to resort to numerical ap-proaches which define the subject of the present sec-tion.

As already mentioned in the Introduction, a ratherspecial role is played by the method of exact diagonal-izations [17–19]. In this approach, one first solves thesimple problem of a single electron moving in a quan-tum dot, and obtains a complete set of single-particleeigenfunctions and the corresponding energies. Thestarting basis for problems involvingN electrons isthen constructed in a combinatorial fashion, i. e., bydistributing the electrons over the available orbitals inall legitimate ways. This task is greatly simplified bythe presence of a circular symmetry, as many particlestates of different total angular momentaL are not cou-pled by the Hamiltonian and can be treated separately.

In this manner, the Hamiltonian operator is repre-sented as a set of matrices – one for each angular mo-mentum of interest – which can be diagonalized numer-ically. The matrix is, in principle, infinite dimensionaland has to be truncated by restricting the basis of many-particle states according to some criteria. In our work,we typically choose to include all many-body states forwhich the sum ofN single-particle energies does notexceed a given threshold energy. The accuracy of theresults depends essentially on our ability to use a suffi-ciently large threshold. To give a specific example, incalculations performed on four-electron quantum dots,we needed (sparse) matrices of around20 000 rows andcolumns, and were able to obtain energies accurate to

384 E. Anisimovas and A. Matulis / Lithuanian J. Phys.50, 377–394 (2010)

four significant digits [17]. We extended our calcu-lations to quantum dots containing five and six parti-cles [26]. The computational effort grows exponen-tially with the system size, and calculations for evenlarger systems turned out to be beyond reach.

We should note that due to its ability to provide accu-rate results, the exact-diagonalization approach provedto be very useful as a reference against which the reli-ability and accuracy of various approximation schemescould be tested and gauged. In particular, this methodplayed the role of a benchmark numerical experimentwhen developing the multicentre-basis Hartree–Fockapproach [20] as well as the quasiclassical theory fur-ther discussed in Section 8.

Fig. 6. Ground-state angular momentum transitions in a six-electron quantum dot.

Figure 6 shows a typical result obtained by means ofan exact-diagonalization study: the ground-state phasediagram of a six-electron parabolic quantum dot placedinto a perpendicular magnetic field. Phase diagramsof this type were obtained also for quantum dots con-taining a smaller number of electrons [17, 26]. Theysuccinctly summarize the transformation of the groundstate as a function of the effective coupling strengthλand the dimensionless magnetic field strengthγ.

The solid lines in Fig. 6 signify the phase bound-aries. They partition the two-parameter phase spaceinto areas corresponding to the different ground statesof the quantum dot. The ground states are labeled bythe pairs of numbers in parentheses(L, S). The quan-tum numberL is the total angular momentum of thestate andS = 0, 1, 2, 3 is the total spin. While the in-teraction strengthλ is difficult to modify, the magneticfield γ can be easily varied in a broad range of values.We see that with increasing magnetic field, states of

ever higher angular momenta are competing to becomethe ground state. This results in a sequence of angularmomentum (and spin) transitions and an ensuing redis-tribution of electron charge in a quantum dot [31].

The ground-state phase diagram in Fig. 6 shows that,in general, a large number of ground states are realizedat different values of the coupling strength and the mag-netic field. However, one notes that some of groundstates are significantly more stable than the others: theycontrol relatively large portions of the phase space. Themost conspicuous are the so-calledmaximum densitydroplet (MDD) states characterized by a peculiar dis-tribution of electrons over the single-particle orbitals[17, 21]. In these states, correlation arranges electronsin such a way that their single-particle angular mo-menta form an arithmetic progression with differenceequal to unity.

The first MDD state is the spin-unpolarized state(6, 0) seen in Fig. 6. Here, the electrons occupy orbitalswith the angular momental = 0, 1, and2. Each orbitalaccommodates two electrons of opposite spin. It is notdifficult to see that the single-particle angular momentaadd to six, and the total spin is zero. The second MDDstate is the spin-polarized state(15, 3). Here, the mag-netic field is sufficiently strong to force all spins to bealigned, and the electrons occupy six consecutive or-bitals with the single-particle angular momental = 0,1, 2, 3, 4, and5 so that the total angular momentumadds up toL = 15.

The exact-diagonalization method also provides thecomplete information about the many-particle wave-function. This wavefunction depends on the coordi-nates of all electrons and is hard to analyse. There-fore, reduced quantities must be constructed. An ob-vious first choice is the single-particle charge density.However, the charge density fails to fully demonstratethe crystallization of electrons in a quantum dot as thisquantity retains the circular symmetry of the Hamilto-nian. As a result, the charge density reveals only theformation of the ring structure and not the azimuthalcorrelations.

In order to fully uncover the internal structure of afew-electron system trapped by a circular confinement,one has to go one step further and consider the density-density correlation function. The physical meaning ofthe correlation function can be (somewhat loosely) in-terpreted as theconditional density. It shows the condi-tional probability to find an electron at a given pointrprovided that another electron is pinned at a fixed posi-tion r′. This result can be plotted on a two-dimensionalgraph and visualizes the formation of a Wigner crystal.

E. Anisimovas and A. Matulis / Lithuanian J. Phys.50, 377–394 (2010) 385

Fig. 7. Wigner crystallization in a six-electron quantum dot.

An illustrative example is given in Fig. 7. Here weshow how the crystal structure in a six-electron quan-tum dot gradually sets in as the magnetic field is in-creased. The four panels of Fig. 7 are labeled by thevalues of the ground-state angular momentum whichchanges fromL = 25 to 40. One of the electrons ispinned in the position marked by the black dot at the in-tersection of the maximum-density ring and thex axis.The remaining five electrons tend to localize close totheir classical lowest-energy positions, and the local-ization conspicuously improves with increasing mag-netic field.

6. Power law

In this section we will draw closer attention to thequantitative characteristics of the structural transitionsin strongly correlated quantum dots. As we learnedin the preceding section, the ground-state angular mo-mentum transitions play the role of a distinctive phe-nomenon which reflects the formation of structures in-side a quantum dot. In a nutshell, these transitions oc-cur when the external magnetic field strength reachescertain characteristic values, and the angular momen-tum quantum number in the ground state suddenlyjumps to a higher value.

We performed a numerical investigation in order todetermine how the critical values of the magnetic fielddepend on the interaction strengthλ. It turns out thatthe obtained dependences closely follow a power law[42]

γL = aLλb . (18)

Here,γL is the critical value of the dimensionless mag-netic field. As the magnetic field crosses this value, theground state of a quantum dot switches from the onecharacterized by the angular momentumL to the fol-lowing in order, typically with the angular momentumL+ 1. The quantityaL is the proportionality factor de-pendent on both the angular momentumL and the num-ber of electrons in the dot. Figure 8 presents the mostclear-cut illustration of the mentioned power-law be-

Fig. 8. Critical magnetic fields plotted as a function of the couplingstrength in a two-electron quantum dot.

haviour of the critical magnetic fields. This figure per-tains to the simplest two-electron quantum dot. Similarfigures have been obtained for three- and four-electronquantum dots as well [42].

The dependence (18) is very simple indeed; its va-lidity made us realize that the considered phenomenonof structure formation must be essentially classical. In-cidentally, let us mention that power laws also describethe internal structure of the many-particle correlationfunctions [43].

Naturally, having obtained such a simple relation(18) from a numerical calculation, we felt obliged toseek a more thorough understanding. Thus we at-tempted, and succeeded, to derive the power law an-alytically [42].

The analytical treatment starts from the basic factthat in the limit of strong interactions (in fact, strongmagnetic fields) a parabolic quantum dot containingnot more than five electrons is transformed to a polyg-onal electron ring. That is, allN electrons are arrangedequidistantly on a ring surrounding the centre of theconfinement. As the first step of our analytical solu-tion we are able to exclude the magnetic field. Thusthe energies of the original problemE(λ, γ, L) can beexpressed in terms of solutions of the modified model

386 E. Anisimovas and A. Matulis / Lithuanian J. Phys.50, 377–394 (2010)

E0(λ0, L) with no magnetic field but with a scaledvalue of the coupling constant

λ0 = λ

[1 +

14γ2]−1/4

. (19)

The relation between the two energies reads

E(λ, γ, L) = E0(λ0, L)[1 +

14γ2]1/2

− 12γL , (20)

and the dimensionless state energies obtained from thezero-field model are

E0(λ0, L) = L+ λ0

√N

LfN . (21)

HerefN is a geometric structure factor whose valuesaref2 = 1

2 , f3 =√

3, andf4 = 1 + 2√

2. The criti-calγ values can now be easily found from the definingequationE(λ, γL, L) = E(λ, γL, L + 1). The resultsshow that the critical magnetic fields indeed obey thepower law

γL =2

(Nf2N )1/3

(L+

12

)λ−2/3 . (22)

We would like to make two interesting observations.First, note that the power law exponent has a universalvalue b = −2

3 which is the same for any number ofelectrons in the quantum dot and any state. Second, itis generally not so surprising that the obtained powerlaw is valid in the classical limit of large values of theangular momentum quantum numbersL. However, itturns out to be working perfectly also in the quantum-mechanical regime for the angular momentaL 6 10.

This result convinced us that the classical approachesare very robust even in the traditionally quantum-mechanical realm where discreetness of quantum num-bers is significant. Therefore, it is worthwhile to fur-ther develop approximate calculation schemes based onclassical notions and valid in more general situationsthat were considered now. The main virtue of the clas-sical (or more precisely, quasiclassical) approaches liesin their relative simplicity and ability to provide resultsthat are accurate and easy to comprehend without re-sorting to heavy computations.

7. Local currents

Further development of the quasiclassical theory ofquantum dots leads naturally to the question of cur-rents that are induced in a quantum dot when a strongperpendicular magnetic field is applied. This problemarises from the representation of the Wigner crystal as

a rotating electron molecule [44]. One of the main is-sues here is the near compensation between a strongmagnetic field on the one hand, and the high angularmomentum value of the ground state on the other hand.The magnetic field strength can have an arbitrary valuewhereas the angular momentum is necessarily quan-tized. This conflict leads to the appearance of the so-calledpersistent currents[45] recently observed in ex-periments on nanoscopic ring-shaped conductors [46].Persistent currents are dissipationless electric currentsinduced by a static magnetic field. They do not re-quire an external power source, however, the survivalof the electron phase coherence along the path is cru-cial. These issues led us to consider the rotation of afew-electron quantum dot – to be more precise, a singleelectron ring – in a strong magnetic field [33, 47, 48].

When dealing with the rotation of the system asa whole, it is helpful to switch to the rotating frameof reference. The straightforward formulation of theHamiltonian of the problem is, however, not easy sincethe definitions of the canonical coordinates and mo-menta in a rotating frame may be tricky. Thus, follow-ing the suggestion of earlier authors [49] we: (i) startwith the Lagrangian, (ii) perform the change of coordi-nates in order to switch to the rotating frame, (iii) definethe canonical variables, and (iv) arrive at the Hamilto-nian using the standard procedure.

The system ofN two-dimensional electrons pos-sesses2N degrees of freedom. One of the modes isspecial – it corresponds, as expected, to the rotationof the system as a whole. The remaining2N − 1modes correspond to various vibrations. We were ableto present the total ground-state wavefunction in a re-markably simple form [47]

Ψ = eiMχe−γK/4 . (23)

The first exponent describes the rotation withχ beingthe collective rotation angle. The second factor is amultidimensional Gaussian which represents the vibra-tional ground state. Here,

K =N∑

i=1

(x2

n + y2n

)+

1N

(N∑

i=1

xn

)2

− 1N

(N∑

i=1

yn

)2

,

(24)andxn (yn) represent the local deviations of thenthelectron from the equidistant positions on a ring in theangular (radial) directions.

The knowledge of the complete ground-state wave-function enables us to calculate the charge and currentdistribution in the quantum dot which are shown inFig. 9. Due to the circular symmetry of the confinement

E. Anisimovas and A. Matulis / Lithuanian J. Phys.50, 377–394 (2010) 387

Fig. 9. Distribution of the charge and current density in a quantumdot.

Fig. 10. Persistent currents in a four-electron quantum dot.

these distributions are also circularly symmetric, andthe current density has only the azimuthal componentjϕ. Thus it suffices to plot only the radial dependencesof the charge densityρ andjϕ. Figure 9 correspondsto a three-electron quantum dot with the coupling con-stant set toλ = 4, and the dimensionless magnetic fieldis such (γ = 6.8) that the ground state has the angularmomentumL = 18. The radial dependence of the cur-rent density is nearly antisymmetric with respect to theclassical dot radius. Thus the current has two compo-nents flowing in the opposite directions in the interiorand the exterior of the electron ring. The two compo-nents nearly cancel each other, however, the cancella-tion is not complete, and there is a finite net azimuthalcurrent given by

I =∫ ∞

0dr jϕ(r) . (25)

The current (25) is precisely the quantum-dot ver-sion of the persistent currents in metal rings. The mag-netic field dependence of the persistent current is plot-ted in Fig. 10. The shown dependence clearly dis-

Fig. 11. Vorticity of the local current in a four-electron quantumdot.

plays a sawtooth-like behaviour, and the sudden dropsof the net current correspond to the abrupt changes ofthe ground state. The angular momenta of the involvedground states are indicated by numbers.

The persistent currents reflect the global rotation ofthe electron system in a quantum dot. Besides thisglobal motion, the local currents corresponding to theLarmor circulation of individual electrons in the vicin-ity of their classical positions can also be calculated.Due to the anisotropy of the local confinement felt byan electron these currents are not circular but rather el-liptic – elongated in the azimuthal direction. Figure 11shows the vorticity of the local currents (the curl of thecurrent density field). In order to rid of the global ro-tation and be able to concentrate on the local circula-tion we pin one of the electrons at the position markedby the black dot on the classical ring. The remainingthree electrons are localized close to their classical po-sitions. Dark (light) areas correspond to the areas ofpositive (negative) current vorticity. These areas are de-limited by the line of zero vorticity marked by a solidblack line. It is interesting to note that local currentswere also observed in density-functional calculationsthat produce broken internal symmetry states [25]. Inthis approach the currents are available directly fromthe Kohn-Sham orbitals and one does not need to con-sider correlation functions.

8. Quasiclassical theory

Our early work devoted to the quasiclassical treat-ment of quantum dots was restricted to the simplestcase – when the lowest energy configuration is a sin-gle polygonal ring accommodating all electrons. Thissimple picture is valid only as long as the number of

388 E. Anisimovas and A. Matulis / Lithuanian J. Phys.50, 377–394 (2010)

electrons in a parabolic confinement does not exceedfive. When the number of electrons is six or more, onetypically observes a system of concentric rings. Thus,for example, the ground state of a19-electron quantumdot features a single electron located at the centre ofthe quantum dot which is surrounded by an inner ringof six electrons plus an outer ring of twelve electrons.In order to be applicable to quantum dots with an arbi-trary number of electrons, the quasiclassical approachof the previous sections has to be generalized.

The essential idea of the quasiclassical theory [32,33] is to employ a perturbative approach based on theexpansion in the inverse powers of the dimensionlessmagnetic field. As mentioned before, the physics ofquantum dots is determined by the strength of the inter-particle interactions which are set by the magnetic field.These observations suggest the choice of the magneticfield strength as the control parameter.

The zeroth-order term in the expansion correspondsto the limit of an infinitely strong magnetic field. Inthis limit, the electron spectrum is transformed into aladder of Landau levels separated by very large (pro-portional to the magnetic field) gaps. The electronsbecome confined to the lowest Landau level and theirmotion is thus frozen out. The absence of dynamics en-ables us to significantly simplify the problem at handby disregarding the kinetic energy term. Note that thisin turn implies disregarding the quantum mechanics, asthe Planck’s constant is present only in the kinetic en-ergy term. Therefore, in this limit we are left with apurely classical problem.

The task of the classical limit is the determinationof the stable configurations defined as the minima ofthe total potential energy ofN interacting particles in aparabolic confinement

V =12

N∑i=1

r2i +N∑

i=1

N∑j>i

1|ri − rj |

. (26)

Note that since the kinetic energy is absent from theproblem, we define the scaled variables by balancingthe confinement and coupling terms. Thus, the cou-pling constantλ is included in the definition of thelength unitr0 = (λ/m∗ω2)1/3, which is used in thissection, and does not appear in the potential (26).

The problem posed by Eq. (26) is not a trivial one.The energy to be minimized is a complicated func-tion of 2N arguments, and typically, besides the globalenergy minimum, one finds a number of local energyminima – the so-called metastable configurations. Thenumber of metastable configurations is known to growas the exponent of the number of particles, and reaches

hundreds already for systems containing just a fewdozen particles [50].

The minimization of the potential energy in the mul-tidimensional configuration space can be successfullytackled by means of: (i) direct minimization tech-niques, such as the steepest descent, conjugate gradientor the Newton optimization [51], (ii) various thermo-dynamic approaches, most importantly the Simulatedannealing algorithm and Molecular dynamics [51, 52],and (iii) evolutionary approaches such as the genetic al-gorithm [53]. The first systematic study of the ground-state configurations of classical artificial atoms waspresented in Ref. [54]. Using the simulated annealingand direct minimizations, we are able to reliably deter-mine the ground state as well as all metastable statesof systems containing up to40 particles. Some of theground-state configurations are depicted in Fig. 12.

Fig. 12. Some ground-state configurations of classical systems in aparabolic confinement.

In a strong (but not infinitely strong) magnetic fieldone has to pay attention to the dynamics of elec-trons in the quantum dot. The dynamics primarily in-volves small collective oscillations in the vicinity of theground state configuration. However, due to the circu-lar symmetry of the Hamiltonian there is, in addition,a rotational mode. In spite of its relation to the sym-metry, in practical matters this mode is a complication,and one has to switch to the rotating frame of referencein order to decouple the rotational and vibrational mo-tion [32, 47–49].

Moreover, it turns out that rotation is not the onlyspecial mode. One of the vibrational modes is alsorather peculiar, and is known as the breathing mode[55]. In this mode, all electrons perform strictly radialoscillations to and from the centre, and the amplitudedeviations and velocities of all the electrons are pro-portional to their respective radial distances from thecentre of the quantum dot. In this sense, the breath-ing mode is indeed very similar to the rotational mode.Note that when the quantum dot rotates as a whole, thevelocities of individual electrons are also proportional

E. Anisimovas and A. Matulis / Lithuanian J. Phys.50, 377–394 (2010) 389

to their radial coordinates, however, their directionspoint perpendicularly to their radius-vectors.

Having successfully singled out the two specialmodes we are left with an ordinary vibrational problemin a magnetic field. The vibrational Hamiltonian can bepresented in a particularly elegant manner [32]. To thisend we collect all the generalized electron coordinatesin a column vector

w = (w1, w2, . . . , w2N−2)T , (27)

with the superscriptT denoting the transpose. The totalnumber of degrees of freedom is(2N−2) as two modeshave been separated, and thus the coordinateswi areorthogonal to the breathing and rotational modes. ThefirstN−1 of these are linear combinations of the radialdisplacements of the individual electrons, while the re-maining half are formed by identical combinations ofthe angular displacements. The canonical momenta areintroduced in a similar fashion

W = (∂/∂w1, ∂/∂w2, . . . , ∂/∂w2N−2)T . (28)

The vibrational Hamiltonian is then cast into the formof a multidimensional harmonic oscillator:

Hvib =12

(W − iγ

2Gw)2

+12wTV (2)w . (29)

The last term of (29) is a quadratic form written in amatrix notation, which represents the second order ex-pansion of the potential around its minimum point, andthe symbolG represents an auxiliary rotation matrix.

The ground state of (29) is a Gaussian function in2N − 2 dimensions

Ψvib = exp(−γ

4wTXw

), (30)

where the matrix of coefficientsX must be obtained bysolving the quadratic matrix equation

X 2 + GX − XG = I +4γ2V (2) , (31)

known as the matrix Riccati equation. We solve thequadratic matrix equation by mapping it onto the ordi-nary spectral problem of an extended(4N−4)×(4N−4) matrix [47]. The vibrational energy is proportionalto the trace of the matrixX

E =γ

4TrX . (32)

The knowledge of the many-body wavefunctionpaves the way for the investigation of the internal struc-ture of the correlated system of electrons. As explained

before, this structure is not directly manifest in the dis-tribution of the (charge) density – which retains the cir-cular symmetry of the Hamiltonian – and thus only thedensity-density correlation function is suitable for thatmatter.

Besides clearly demonstrating the crystal formation,our calculation leads to a classification of the emergingstructures. Thus when the structural stability is consid-ered, quantum dots may be categorized as eithersolidof liquid-like. A structure is liquid-like when easily ex-citable low-frequency modes are present.

Fig. 13. Solid and liquid-like quantum dots.

Often the presence of a low-frequency mode is con-nected to the commensurability of the numbers of elec-trons belonging to neighbouring shells [33]. An excel-lent illustration of this phenomenon is given by com-parison of eleven- and twelve-electron quantum dots inFig. 13. The system of eleven electrons in a parabolictrap serves as an example of a liquid-like structure.The lowest-energy configuration of the eleven-electronquantum dot consists of two rings: the inner one holdsthree electrons, and the outer one holds eight. Thenumbers3 and8 are incommensurate, that is, they haveno common divisors. As a consequence, the two elec-tronic shells have a nearly circular shape, and the inter-shell rotation is easily excitable thus introducing a low-frequency mode. In Fig. 13 the smooth rotation of thetwo rings is reflected by the elongated ellipses of theinner shell.

In contrast, the twelve-electron quantum dot is solidand has a rigid structure. The ground-state configura-tion is again composed of two rings, containing threeand nine particles, respectively. This distribution ofelectrons leads to the emergence of a triangular distor-tion of the shells, and as a consequence, the inter-shellrotation costs too much energy.

Low-frequency modes of a different nature are alsopossible [56]. In general, they relate to the presenceof two competing minima, that is, minima of compara-ble depth connected by a transition trajectory that runs

390 E. Anisimovas and A. Matulis / Lithuanian J. Phys.50, 377–394 (2010)

through a saddle point of a rather low energy. Typi-cal examples are: (i) the migration of an electron fromone ring into another and (ii) the inter-shell electron ex-change.

9. Dissipation in Schrödinger equation formalism

Dissipation is inherently present in any physical sys-tem, and dissipative processes play an important role inmost experimental measurements since the influx of theprobing energy into a system must be counterbalancedby its eventual loss. In many circumstances, one maydisregard the details of energy removal processes or ac-count for them in some simple way. However, this isusually not the case when one has to deal with ultrafastprocesses on the nanometric scale [57].

Fig. 14. The layout of the model system.

Traditionally, open quantum systems are treated interms of the density-matrix formalism [58]. However,the density-matrix formalism has a double set of vari-ables which results in increased computational com-plexity. We proposed [59–61] to embrace dissipationin a formalism based on the Schrödinger equation, andpresented the simplest illustrations, namely, the decayof an excited state of a 1D harmonic oscillator and thenonlinear power absorption.

The considered compound system – consisting of asmall quantum system plus a classical heat bath – isshown in Fig. 14. The quantum system is modelled asa harmonic oscillator of massm and spring constantk,depicted by the large red ball and described in terms ofthe quantum-mechanical position and momentum op-erators,x andp. The role of the heat bath is assigned toa semi-infinite chain of identical balls (shown as smallblue circles) of massM interconnected by springs ofequilibrium lengtha and spring constantK. The twosubsystems are linked by a weak spring (spring con-stantκ) connecting the quantum oscillator to the left-most – the ‘zeroth’ – ball of the classical chain. Thecorresponding Hamiltonian reads

H =1

2mp2 +

k

2x2 +

κ

2(x− x0)2

+∞∑

n=0

[1

2Mp2

n +K

2(xn − xn+1)2

],

(33)

with xn denoting the coordinate of thenth ball mea-sured with respect to its equilibrium position, andpn –the corresponding momentum.

We assume that the quantum oscillator is describedby its own wavefunctionΨ(x, t) that solves the Schrö-dinger equation

i~∂

∂tΨ(x, t) = HΨ(x, t) , (34)

with the above Hamiltonian (33), while the dynami-cal variables of the balls in the chain obey the classicalNewton equations

xn =∂

∂pn〈H〉, pn = − ∂

∂xn〈H〉 , (35)

with the Hamiltonian operator replaced by its quantum-mechanical average over the state of the quantum oscil-lator

〈H〉 =∫ ∞

−∞dxΨ∗(x, t)HΨ(x, t) . (36)

In this way, quantum variables and operators do not en-ter the equations for the classical degrees of freedom,and one obtains a consistent description. This averag-ing constitutes the main assumption of the quasiclassi-cal approximation, and was used earlier to treat the cou-pling of classical and quantum degrees of freedom [62].

The linear equations of the chain motion can be eas-ily solved, and the coordinatesxn of all balls are thusexpressed in terms of the zeroth ball coordinatex0.This enables us to obtain the set of dimensionless equa-tions

i∂

∂tΨ(x, t) = HΨ ≡

[12

(p2+x2

)−λxx0

]Ψ , (37a)

ddtx0 + λ(l0/l)x0 = λ〈x〉 , (37b)

where the classical coordinate is measured in unitsl0(l20 = ~/MΩ, Ω2 = K/M ) and the quantum coordi-nate in unitsl (l2 = ~/mω, ω2 = k/m). The cou-pling constantλ = (κ/k)

√ω0m/ΩM is small in the

adiabatic case (m M ), which justifies the proposedquasiclassical approach.

When considering the problem of the nonlinear reso-nance, we extend the Hamiltonian by adding a periodicexternal force and a nonlinear term:

∆H = −2√

2xF cos(ωt) + αx4 . (38)

The oscillator wavefunction is expanded into the seriesof the harmonic oscillator eigenfunctions

Ψ(x, t) =∞∑

n=0

an(t)ψn(x) , (39)

E. Anisimovas and A. Matulis / Lithuanian J. Phys.50, 377–394 (2010) 391

whereEn = n + 1/2, and the functionsψn can beexpressed in the standard way in terms of the Hermitepolynomials. Inserting (39) into Eq. (37) and assum-ing slow variation in time of the coefficientsan (therotating wave approximation), we arrive at the set ofequations

ian = −ηEn − α〈n|x2|n〉

an

−√n(F+iγa∗)an−1 −

√n+1(F−iγa)an+1 , (40a)

a =∞∑

n=0

ana∗n+1

√n+ 1, γ = λ2/2 , (40b)

whereη = ω − 1 is the detuning.

Fig. 15. Decay of a quasistationary state: the probability of theexcited state|a1|2 is shown by the thick solid curve, the probabilityof the ground state|a0|2 by the dashed curve, polarizationa =a0a

∗1 by the dotted curve, and the classical chain excitationx0 by

the thin solid curve.

The proposed method is illustrated by consideringthe decay of a quasistationary state in the two-level ap-proximation. The result is presented in Fig. 15. Wenote the characteristic time scale of the order of1/γduring which the system relaxes from the excited to theground state. It is remarkable that during this transitionan excitation pulse is generated in the classical chain.The pulse carries energy towards infinity which can beregarded as dissipation (energy loss) from the point ofview of the quantum subsystem. The energy of thispulse coincides with the difference of the energy levelsE1 − E0 which enables us to conclude that the quan-tum subsystem causes some quantization in the classi-cal one.

The second example is the resonant nonlinear powerabsorption which we calculate in the three-level ap-proximation. The typical resonance curves for variousdegrees of nonlinearity are shown in Fig. 16. We seethat in the linear-oscillator case (α = 0) the absorp-

Fig. 16. Power absorption in the three level approximation.

tion demonstrates the expected Lorentzian behaviour.When nonlinearity increases, the resonant peak movesto the right and becomes asymmetric. Saturation man-ifests itself as a widening absorption gap close to theshifted resonance frequency. This is in contrast to theclassical nonlinear resonance where the power absorp-tion peak is tilted and a hysteresis takes place. Thekinks of the resonance curve in the absorption gap arerelated to various synchronization types between thedriving force and the system response. This type ofbehaviour of the quantum system behaviour is due tothe infinite number of degrees of freedom, and remindsthe scenarios of appearance of the dynamical chaos inclassical systems.

We shall end the presentation of the quasiclassicalmethods briefly mentioning one more approach – thehydrodynamic method that was applied by Zarembaand co-workers [63] to the quantum dots with a largenumber of electrons. We applied the hydrodynamicmethod to the investigation of magnetoplasma exci-tations in two vertically aligned quantum dots [64].Two nontrivial results were found in the case whenthe parabolic confinement of the two dots is not thesame. First, the violation of the Kohn’s theorem leadsto the appearance of modes other than the centre-of-mass ones in the power absorption. Also, the so-callededge modes with small but nonzero oscillator strengthsare excited.

The hydrodynamic approach was extended to verti-cally coupled electron and hole quantum dots [65]. Themost conspicuous feature observed in the far-infraredabsorption spectra is the anticrossing of two centre-of-mass modes which takes places when the external mag-netic field aligns their frequencies. A number of weakedge modes are also present.

392 E. Anisimovas and A. Matulis / Lithuanian J. Phys.50, 377–394 (2010)

10. Conclusions

The following conclusions can be drawn from thediscussed issues:

1. In large quantum dots the coupling between elec-trons is the most important term, and accordingly,the kinetic energy and quantum mechanics play asecondary role.

2. The strong perpendicular magnetic field freezesout the kinetic energy of the electrons, which alsocontributes to the enhancement of the relative im-portance of the interparticle interactions.

3. Strongly interacting electron systems in quan-tum dots and quantum-dot molecules demonstrateformation of structures and structural transitions.Quantum-mechanical effects tend to smear out theotherwise sharp features of these transitions.

4. There exists a well defined quasiclassical regime.In this regime, a rather simple but still potent de-scription based on the classical concepts can bedeveloped. The quantum-mechanical features aresuccessfully included in an perturbative manner.

5. Quantum-mechanical systems are described byfields – the solutions of the Schrödinger equation.In this sense, quantum-mechanical systems havemore freedom, and consequently, their behaviouris less stable (more chaotic) than that of their clas-sical counterparts.

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KVAZIKLASIKIN E KVANTINIU TAŠKU TEORIJA

E. Anisimovasa, A. Matulisb

a Vilniaus universitetas, Vilnius, Lietuvab Fiziniu ir technologijos mokslu centro Puslaidininkiu fizikos institutas, Vilnius, Lietuva

Santrauka

Kvantiniai taškai yra dariniai, kuriuose elektrono judejimasyra apribotas visomis trimis erdves kryptimis, o energijos spek-tras diskretus. Taigi, kvantiniai taškai yra dirbtiniai atomai: val-domu parametru nat uraliu atomu analogai. Dideliu matmenu (lygi-nant su efektiniu Boro spinduliu) kvantiniuose taškuose klasikinekulonines st umos energija žymiai viršija kvantine kinetine ener-gija. Elektrostatines saveikos santykine svarba dar labiau išryškinastiprus taško plokštumai statmenas magnetinis laukas. Todel,viena vertus, šiuose kvantiniuose taškuose yra stebimi kolektyvi-niai reiškiniai (Vignerio kristalizacija), antra vertus, ju fizika yra

iš esmes klasikine. Straipsnyje apžvelgiama kvaziklasikine kvan-tiniu tašku teorija, išpletota pasinaudojus šiais pastebejimais. Tai-komi klasikiniai metodai leidžia išvengti sudetingu ir nevaizdžiukvantiniu mechaniniu skaiciavimu. Klasikiniais metodais gauti re-zultatai yra lengvai suprantami ir interpretuojami, o i kvantinespataisas atsižvelgiama kaip i mažus trikdžius. Aptariamas renor-malizuotos trikdžiu eilutes taikymas energiju spektrams skaiciuoti,strukt uru susidarymas ir virsmai kvantiniuose taškuose, laipsniniaidesniai, nusakantys kritiniu parametru elgesi, ir disipacija mišriosesistemose, pasižyminciose klasikiniu ir kvantiniu laisves laipsniusaveika.