Research ArticleSemiquantum Chaos in Two GaAs Quantum DotsCoupled Linearly and Quadratically by Two HarmonicPotentials in Two Dimensions

Emile Godwe ,1 Justin Mibaile,2 Betchewe Gambo,1,2 and Serge Y. Doka3

1Department of Physics, Faculty of Science, The University of Maroua, P.O. Box 814, Maroua, Cameroon2Higher Teachers’ Training College of Maroua, The University of Maroua, P.O. Box 46, Maroua, Cameroon3Department of Physics, Faculty of Science, The University of Ngaoundere, P.O. Box 454, Ngaoundere, Cameroon

Correspondence should be addressed to Emile Godwe; willgod7@yahoo.com

Received 24 November 2017; Revised 12 March 2018; Accepted 1 April 2018; Published 15 May 2018

Academic Editor: Xavier Leoncini

Copyright © 2018 Emile Godwe et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We analyze the phenomenon of semiquantum chaos in twoGaAs quantumdots coupled linearly and quadratically by two harmonicpotentials. We show how semiquantum dynamics should be derived via the Ehrenfest theorem.The extended Ehrenfest theorem intwo dimensions is used to study the system. The numerical simulations reveal that, by varying the interdot distance and couplingparameters, the system can exhibit either periodic or quasi-periodic behavior and chaotic behavior.

1. Introduction

Classical mechanics deals with the fluctuations of macro-scopic objects in phase space. It helps to drive and understandthe matter laws fluctuations of an object. The historicalevolution of dynamic systems evolves with the advent ofnonlinear dynamics. Scientists like Newton, Euler, Lagrange,Laplace, andmany others advocated the predictable behaviorof integrable systems [1].Those systems provide us later basicsof studying systems that are nonintegrable. Nevertheless, weneed to consider themanifestations of nonintegrable systems.That is the cause of apparition of chaos because an integrablesystem motion is either periodic or quasiperiodic. Whatproperties does chaotic system satisfy? For a dynamic systemto behave chaotically, it should fulfill three fundamentalproperties: boundedness, infinite recurrence, and sensitivedependence on initial conditions. For example, the authorsin [2] studied chaos in semiconductor band-trap impactionization, just to mention that.

However, many experimental observations did notexplain the fluctuation of very small objects like atomsand molecules, that is, objects of size 10−8m or smaller as

described by the wave function. Hence, it needs the help ofquantum mechanics, where the scale is set by the Planckconstant. The wave function is a complex value; it describesthe behavior and gives the fluctuation laws of microscopicconstituents of matter. In classical mechanics, time evolutioncan be described in terms of Hamilton’s equation as well asin quantum mechanics. The latter describes how atoms areput together and why they are stable. The problem of chaosreflected in quantum physics can be traced back to the earlydays of quantum mechanics, when Poincare worked on thethree-body problem [3]. The concept of quantum chaos isvery useful [4, 5]. It can be defined in numerous manners butchaotic quantum systems are characterized by their extremesensitivity both to initial conditions and to parameterschanges [6–8]. It is too earlier to make the concept of chaosfor quantum systems and its experimental manifestationsclear [9–12]. Nevertheless, there are three types of quantumchaos [13]: quantum chaology that refers to the study of thequantum signatures of classical chaos in the semiclassicalor high quantum number regimes; semiquantum chaos,which refers rigorously to chaotic dynamics in semiquantumsystem; and genuine quantum chaos, which rigorously refers

HindawiAdvances in Mathematical PhysicsVolume 2018, Article ID 6450687, 9 pageshttps://doi.org/10.1155/2018/6450687

2 Advances in Mathematical Physics

to chaotic behavior in full quantum systems.The applicationsof quantum chaos are too many such as the best securecommunication, the quantum cryptography, and the medicaldiagnosis.

Energy is optimizedwhen objects areminiaturized. Parti-clemoving in one-dimensional potentials has been studied bymany authors. In 1998, Loss and DiVincenzo have proposedtwo laterally coupled quantum dots with one valence electronper dot; particular attention has been suggested towardquantum computing [14]. Then, in 2004, Fortuna and Porto[15] were interested in coupled quantum dot cells. They havestudied the dynamic behavior by suitable variation of cou-pling parameter and initial conditions. Blum and Elze studiedthe semiquantum chaos in a double-well potential with thetime dependence of variational principle [16]. Lekeufack etal. studied quantum chaology in a multiple-well adjustablepotential heterostructure [17].They investigated the signatureof chaos in the semiquantum behavior of a classically regulartriple-well heterostructure [18]. Pattanayak and Schieve alsostudied the semiquantal dynamics of fluctuations: ostensiblequantum chaos [19, 20]. Besides, problems like independentSchrodinger’s equation have been studied for different 1Dpotentials in phase space representation. For instance, thereare free particle [21], linear potential [22], harmonic potential[23, 24], Morse potential [25], and so forth.

On the other hand, the quantum coupled dot potentialis a bilayer system [26]. It is a useful game footing forunderstanding numerous quantum mechanical structures[27]. For example, a quadratic 2D potential is modeled incoupled semiconductor quantum dots as quantum gates bythe author of [28] as 𝑉(𝑥, 𝑦) = (𝑚𝜔20/2)[(1/4𝑎2)(𝑥2 −𝑎2)2 + 𝑦2], where 𝜔0 is the confining frequency, one foreach dot, 𝑥 and 𝑦 are coordinates, 𝑚 is the effective massof electron, and 𝑎 is the half distance between the center ofthe dots and the effective Bohr radius of a single isolatedharmonic potential. An exchange coupling in semiconductornanostructures is modeled by 2D quadratic potential [29] as𝑉(𝑥, 𝑦) = (𝑚𝜔20/2)([𝑥2 − (𝑅/2)2]2/𝑅2 + 𝑦2). Here, the focusis on a linear coupled quantum dot to harmonic potential𝑉(𝑥, 𝑦) = 𝑥2(𝛼𝑥2−𝛽)+𝜅+𝑦(𝛾𝑦−𝜎𝑥); the coupling parameter(𝜎) represents the shift of dots from the 𝑥-axis [30, 31]; with𝑥 and 𝑦 being coordinates, 𝜅 is an interdot barrier; note that𝛼 is completely defined by 𝜔0 and the interdot distance; since𝛽 is set in such a way that the potential minimum could bezero, that is, 𝛽 = 2(𝛼𝜅)1/2 parameters on which the numerousvariety of configurations are dependent, 𝛾 is a constant.The interdot distance is physically equivalent to the interdotbarrier. Quantum dot has one degree of freedom as harmonicpotential. We note that the two dots are exactly identical.We also study a quadratic coupled quantum dot to harmonicpotential𝑉(𝑥, 𝑦) = 𝑥2(𝛼𝑥2−𝛽)+𝜅+𝑦(𝛾𝑦−𝜎𝑥2). Applicationsof these types of potential are found in study of manychemical reactions such as isomerization ofmalondialdehyde[32].They induce several configurations according to the timeevolution of quantum confinement of some rigid structuresand so forth. The theoretical investigation of a linear andquadratic coupled quantum dot to harmonic potential is

important, since it allows both optimizing information andinvestigating fundamental physics.

To be more specific, in this paper, we aim at extendingthe Ehrenfest theorem in one dimension to two dimensions.It is possible to extend dynamic systems theory to partialdifferential equations. We may apply this approach to thedynamics of a particle moving in 2D potentials. The studyof the semiquantal dynamics of a particle moving in a linearand quadratic coupled quantum dot to harmonic potentialis of great interest. Classical effective Hamiltonian is used todescribe the complete time evolution of the coupled classicaland quantum vibrations, which is the expectation value of thequantum Hamiltonian.

The rest of the paper is organized as follows: Section 2presents the methodology of the Ehrenfest theorem in twodimensions; in Section 3, we apply this methodology toa linear and quadratic coupled quantum dot to harmonicpotential and present the numerical study of the semi-quantum equations’ motion. Finally, Section 4 gives theconclusion.

2. Methodology

We propose an extended method to construct extendedeffective Hamiltonian equation for two dimensions. Thetime-dependent variational principle (TDVP) formulationis very necessary to understand many equivalent methodssuch as the Gaussian trial wave function [16, 17, 19, 20].Because its results are sometimes in good agreement withthe experimental physics, the TDVP may be performedwith the help of Dirac’s variational principle. Variationalparameters vanish at infinity and require the action Γ[Ψ] =∫ 𝑑𝑡⟨Ψ(𝑡)|𝑖ℏ(𝜕/𝜕𝑡) − 𝐻|Ψ(𝑡)⟩. The wave function providesinteresting results in both quantum chemistry and generalphysics. In ordinary quantum mechanics, a wave packet willbe of the form Ψ(𝑥, 𝑦) = 𝑁(𝜋ℏ)𝑛/4exp{[(−1/2ℏ)((𝑥 − 𝑞𝑥)2 +(𝑦 − 𝑞𝑦)2) + (𝑖/ℏ)(𝑝𝑥 + 𝑝𝑦)((𝑥 − 𝑞𝑥) + (𝑦 − 𝑞𝑦)) + 𝑖𝜆]}, if werestrict |Ψ(𝑡)⟩ to the family of coherent states [33, 34]. Here𝑁, 𝜆 are the normalization and phase constants, respectively;the parameters 𝑝𝑥, 𝑝𝑦, 𝑞𝑥, 𝑞𝑦 specify a point in the classicalphase space of dimension 2𝑛. In this system, the Hamilton’sequations for 𝑝𝑥, 𝑝𝑦, 𝑞𝑥, 𝑞𝑦, conserve the norm 𝑁 and yield�� = (𝑝𝑥 + 𝑝𝑦)(��𝑥 + ��𝑦) − 𝐻(𝑞𝑥, 𝑞𝑦, 𝑝𝑥, 𝑝𝑦), where 𝐻 is theclassical Hamiltonian. Therefore, 𝜆(𝑡) is equal to the classicalaction Γ = ∫ 𝑑𝑡[(1/2)((𝜕𝑥/𝜕𝑡)2 + (𝜕𝑦/𝜕𝑡)2) − 𝑉(𝑥, 𝑦)].

However, we will provide an alternative derivation ofsemiquantal dynamics via the Ehrenfest approach. Theauthors of [35, 36] said that, in classical mechanics or inquantum mechanics, the position and momentum are inde-pendent dynamical variables or operator. The configurationof the system in the 2𝑁-dimensional 𝑝 − 𝑞 is shown in phasespace. The time evolution of a system can be described interms of Hamilton equations 𝑑𝑞𝑗/𝑑𝑡 = 𝜕𝐻/𝜕𝑝𝑗 and 𝑑𝑝𝑗/𝑑𝑡 =−𝜕𝐻/𝜕𝑞𝑗, where 𝑗 = 1, 2; 𝑞𝑗 and 𝑝𝑗 are the position andmomentum, respectively. In quantum mechanics, there arealso several equivalentways of describing the dynamics.Here,quantities like position and momentum play the roles ofoperators as well as variables [35, 36]. Let us focus on a

Advances in Mathematical Physics 3

particle of unit mass moving in a two-dimensional time-independent bounded potential. The Hamiltonian on whicha system is configurable is

�� = 𝑝𝑥2 + 𝑝𝑦22 + 𝑉 (𝑞𝑥, 𝑞𝑦) , (1)

where 𝑞𝑥, 𝑞𝑦, 𝑝𝑥, and 𝑝𝑦 are operators; they depend on time𝑡 through these coordinates. In two dimensions, (1) can bewritten as 𝑑𝑑𝑡 ⟨𝑞𝑥⟩ = ⟨𝑝𝑥⟩ ,

𝑑𝑑𝑡 ⟨𝑝𝑥⟩ = −⟨𝜕𝑉(𝑞𝑥, 𝑞𝑦)𝜕𝑞𝑥 ⟩,𝑑𝑑𝑡 ⟨𝑞𝑦⟩ = ⟨𝑝𝑦⟩ ,𝑑𝑑𝑡 ⟨𝑝𝑦⟩ = −⟨𝜕𝑉(𝑞𝑥, 𝑞𝑦)𝜕𝑞𝑦 ⟩,

(2)

where ⟨ ⟩ indicates expectation values. Generally, the centroiddoes not go along the classical trajectory. We need to extendthe above equations around the centroid using the 2Didentity.

⟨𝐹𝑞𝑢 (𝑞𝑢, 𝑞V)⟩ = 1𝑛! ⟨𝑄𝑢𝑛⟩𝐹(𝑛)𝑞𝑢

, 𝑛 ≥ 0,⟨𝐹𝑞V (𝑞𝑢, 𝑞V)⟩ = 1𝑛! ⟨𝑄V

𝑛⟩𝐹(𝑛)𝑞V

, 𝑛 ≥ 0,(3)

where 𝐹(𝑛)𝑞𝑢

= 𝜕𝑛𝐹𝑞𝑢/𝜕𝑞𝑛𝑢|⟨𝑞𝑢⟩ and 𝑄𝑢 = 𝑞𝑢 − ⟨𝑞𝑢⟩; 𝐹(𝑛)𝑞V =𝜕𝑛𝐹𝑞V/𝜕𝑞𝑛V |⟨𝑞V⟩, and𝑄V = 𝑞V−⟨𝑞V⟩. Countably infinite numberof equations could be generated by using these operator rules.The space is rendered finite by the belief that the wave packetis a squeezed coherent; hence, these relations are provided.

⟨𝑄𝑢2𝑚⟩ = (2𝑚)!𝜇𝑚𝑢𝑚!2𝑚 (no summation) ,⟨𝑄𝑢2𝑚+1⟩ = 0,4𝜇𝑢 ⟨𝑃𝑢2⟩ = ℏ2 + 𝛼2𝑢,

⟨𝑄𝑢𝑃𝑢 + 𝑄𝑢𝑃𝑢⟩ = 𝛼𝑢;⟨𝑄V2𝑚⟩ = (2𝑚)!𝜇𝑚V𝑚!2𝑚 (no summation) ,

⟨𝑄V2𝑚+1⟩ = 0,

4𝜇V ⟨𝑃V2⟩ = ℏ2 + 𝛼2V ,⟨𝑄V𝑃V + 𝑄V𝑃V⟩ = 𝛼V.

(4)

The above equations are recognized as generalized Gaussianwave functions [33, 37–39]. After reducing the system to thedynamics of 𝑞𝑥, 𝑞𝑦, 𝑝𝑥, 𝑝𝑦, 𝜇𝑥, 𝜇𝑦, 𝛼𝑥, and 𝛼𝑦 (where 𝑞𝑥, 𝑞𝑦,𝑝𝑥, and 𝑝𝑦 are written for ⟨𝑞𝑥⟩, ⟨𝑞𝑦⟩, ⟨𝑝𝑥⟩, and ⟨𝑝𝑦⟩), weintroduce the change of variables 𝜇𝑥 = 𝜌2𝑥 and 𝛼𝑥 = 2𝜌𝑥𝜉𝑥and 𝜇𝑦 = 𝜌2𝑦 and 𝛼𝑦 = 2𝜌𝑦𝜉𝑦. This gives the following newequations:

𝑑𝑞𝑥𝑑𝑡 = 𝑝𝑥, (5a)

𝑑𝑞𝑦𝑑𝑡 = 𝑝𝑦, (5b)

𝑑𝑝𝑥𝑑𝑡 = − 𝜌2𝑚𝑥𝑚!2𝑚𝑉(2𝑚+1)𝑞𝑥(𝑞𝑥, 𝑞𝑦) , 𝑚 = 0, 1, . . . (5c)

𝑑𝑝𝑦𝑑𝑡 = − 𝜌2𝑚𝑦𝑚!2𝑚𝑉(2𝑚+1)𝑞𝑦(𝑞𝑥, 𝑞𝑦) , 𝑚 = 0, 1, . . . (5d)

𝑑𝜌𝑥𝑑𝑡 = 𝜉𝑥, (5e)

𝑑𝜌𝑦𝑑𝑡 = 𝜉𝑦, (5f)

𝑑𝜉𝑥𝑑𝑡 = ℏ24𝜌3𝑥 −𝜌2𝑚−1𝑥(𝑚 − 1)!2𝑚−1𝑉(2𝑚)𝑞𝑥 (𝑞𝑥, 𝑞𝑦) ,

𝑚 = 1, 2, . . .(5g)

𝑑𝜉𝑦𝑑𝑡 = ℏ24𝜌3𝑦 −𝜌2𝑚−1𝑦(𝑚 − 1)!2𝑚−1𝑉(2𝑚)𝑞𝑦 (𝑞𝑥, 𝑞𝑦) ,

𝑚 = 1, 2, . . . .(5h)

We observe that these new variables form an explicitcanonically conjugate set. We note that the classical degreesof freedom are the “average” variables 𝑞𝑥, 𝑞𝑦, 𝑝𝑥, and 𝑝𝑦 andthe “fluctuation” variables 𝜌𝑥, 𝜌𝑦, 𝜉𝑥, and 𝜉𝑦, respectively. Theextended Hamiltonian is

𝐻ext = 𝑝2𝑥 + 𝑝2𝑦2 + 𝜉2𝑥 + 𝜉2𝑦2+ 𝑉ext (𝑞𝑥, 𝑞𝑦, 𝜌𝑥, 𝜌𝑦) ,

𝑉ext (𝑞𝑥, 𝑞𝑦, 𝜌𝑥, 𝜌𝑦) = 𝑉 (𝑞𝑥, 𝑞𝑦) + ℏ28𝜌2𝑥 +ℏ28𝜌2𝑦

+ 𝜌2𝑚𝑥𝑚!2𝑚𝑉(2𝑚)𝑞𝑥 (𝑞𝑥, 𝑞𝑦)+ 𝜌2𝑚𝑦𝑚!2𝑚𝑉(2𝑚)𝑞𝑦 (𝑞𝑥, 𝑞𝑦) ,

𝑚 = 1, 2, . . . ,

(6)

where the subscript ”ext” indicates the “extended” poten-tial and Hamiltonian. This formulation is very useful and

4 Advances in Mathematical Physics

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Figure 1:The contour plots ((a) and (b)) and the mesh plots ((c) and (d)) of the potential with linear and quadratic coupling.The parametersare fixed: 𝛼 = 0.2, 𝜅 = 3, 𝛾 = 0.5, and 𝜎 = 0.3.

powerful because the gradient system is well explained byit, and the extended potential provides simple visualizationof the configuration of the semiquantal space. We couldget a qualitative feel for the semiquantal dynamics beforestarting the numerical analysis. First of all, the average andfluctuation variables are considered at the same footing andthe phase space is dimensionally consistent: 𝜌𝑥 and 𝜌𝑦 havethe dimension of lengths and 𝜉𝑥 and 𝜉𝑦 have the dimensionof momentum.

3. Double Quantum Dot PotentialCoupled Linearly and Quadratically toHarmonic Potential

The purpose of this section is to draw the basic set of equa-tions of a particle unit mass moving in linear and quadratic

two GaAs quantum dots coupled to harmonic potential.Physical phenomena related to two GaAs double quantumdots are of great interest. Hence, quantum mechanismsconcepts are defined in condensed matter physics [40]. Thetwo-dimensional quantum-mechanical toy model in which aparticle is moving in the upside down GaAs quantum dotsfor linear and quadratic coupling potential may be studied.First of all, the contour plots and the mesh plots of the2D potential in the cases of linear and quadratic couplingare shown in Figure 1. From the latter, we observe that thecoupling parameter 𝜎 is responsible for shift of dots fromthe 𝑥-axis. It stands that many physics phenomena can beobserved in the two potentials dimensions.Thepotential withlinear coupling is 𝑉(𝑞𝑥, 𝑞𝑦) = 𝑞2𝑥(𝛼𝑞2𝑥 − 2(𝛼𝜅)1/2) + 𝜅 +𝑞𝑦(𝛾𝑞𝑦 − 𝜎𝑞𝑥) and the potential with quadratic coupling is𝑉(𝑞𝑥, 𝑞𝑦) = 𝑞2𝑥(𝛼𝑞2𝑥 − 2(𝛼𝜅)1/2) + 𝜅 + 𝑞𝑦(𝛾𝑞𝑦 − 𝜎𝑞2𝑥).

Advances in Mathematical Physics 5

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Figure 2: Phase portrait (a) and Poincare sections (b) in the plane 𝜉𝑥, 𝜌𝑥 with the initial conditions 𝑞𝑥𝑜 = 1, 𝑞𝑦𝑜 = 1, 𝑝𝑥𝑜 = 3, 𝑝𝑦𝑜 = 1, 𝜌𝑥𝑜 = 1,𝜌𝑦𝑜 = 1, 𝜉𝑥𝑜 = 1, and 𝜉𝑦𝑜 = 3. We set the zero coupling parameters as 𝜅 = 3 and 𝛼 = 0.1.

3.1. Zero Coupling. We describe the zero coupling parameterGaAs quantum dots dynamics system with the Hamiltonian

�� = ��2𝑥 + ��2𝑦2 + 𝛼𝑞𝑥4 − 2 (𝛼𝜅)12 𝑞𝑥2 + 𝜅 + 𝛾𝑞𝑦2; (7)

we need to extend the Hamiltonian to

𝐻 = 𝑝2𝑥 + 𝑝2𝑦2 + 𝜉2𝑥 + 𝜉2𝑦2 + 𝛼 (𝑞4𝑥 + 6𝜌2𝑥𝑞2𝑥 + 3𝜌4𝑥)

− 2 (𝛼𝜅)12 (𝜌2𝑥 + 𝑞2𝑥) + 𝛾 (𝑞2𝑦 + 𝜌2𝑦)

+ 18 ( 1𝜌2𝑥 +1𝜌2𝑦) + 𝜅.

(8)

When the coupling parameter is equal to zero, theplots for either lower or higher interdot distance in thoseconfigurations display chaotic behavior. Figure 2 shows thephase portrait and Poincare plots in the plane 𝜉𝑥, 𝜌𝑥. In orderto determine the shape of trajectories of electrons, we findthe periodicity of the system for various interdot barriers. Itis almost the case for lower interdot barrier 𝜅 = 3, whereirregular chaotic motion was easily observed and shown inFigure 2. Almost similar behavior is also seen for upperinterdot distance (not shown).

Wehave to know the quantitativemeasure of the degree ofchaos in a given region; then we plot the maximal Lyapunovexponent of (5a)–(5h) versus the parameter 𝜅 as shown inFigure 3. In Figure 3, the maximal Lyapunov exponent has

a positive value for any value of parameter 𝜅. This means thatthe system described by (5a)–(5h) is chaotic.

3.2. Linear Coupling. The extended Hamiltonian for thedynamics of the linear coupled GaAs quantum dots systemis

𝐻 = 𝑝2𝑥 + 𝑝2𝑦2 + 𝜉2𝑥 + 𝜉2𝑦2 + 𝛼 (𝑥4 + 6𝜌2𝑥𝑞2𝑥 + 3𝜌4𝑥)

− 2 (𝛼𝜅)12 (𝜌2𝑥 + 𝑞2𝑥) + 𝛾 (𝑞2𝑦 + 𝜌2𝑦) − 𝜎𝑞𝑥𝑞𝑦

+ 18 ( 1𝜌2𝑥 +1𝜌2𝑦) + 𝜅.

(9)

The plots for lower interdot barrier and lower couplingparameter (not shown) display chaotic behavior for differentchosen initial conditions closer to the minimum of thepotential. Indeed, for the values of the control parameters𝜅 and 𝜎 are increased to medium or upper values, we findchaotic trajectories again. So, at 𝜅 = 200, the systemfinds itself to be chaotic as shown in Figure 4. Althoughlower interdot barrier parameter induces chaos, raisinginterdot barrier 𝜅 or coupling parameter 𝜎 to higher valuemakes a bend appear in the plane (𝑝𝑥, 𝑞𝑥) as shown inFigure 4.

The maximal Lyapunov exponent versus the couplingparameter 𝜎 is shown in Figure 5. We realize in Figure 5that the maximal Lyapunov exponent 𝜆max has a positivevalue. It means that the system described by (5a)–(5h) ischaotic.

6 Advances in Mathematical Physics

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q x

(b)

Figure 4: Phase portrait (a) and Poincare sections (b) in the plane 𝑝𝑥, 𝑞𝑥 with the initial conditions 𝑞𝑥𝑜 = 1, 𝑞𝑦𝑜 = 1, 𝑝𝑥𝑜 = 3, 𝑝𝑦𝑜 = 7, 𝜌𝑥𝑜 = 1,𝜌𝑦𝑜 = 1, 𝜉𝑥𝑜 = 4, and 𝜉𝑦𝑜 = 3. We set the linear coupling parameters as 𝜎 = 0.2, 𝜅 = 200, and 𝛼 = 0.1.

3.3. Quadratic Coupling. The extended Hamiltonian is

𝐻 = 𝑝2𝑥 + 𝑝2𝑦2 + 𝜉2𝑥 + 𝜉2𝑦2 + 𝛼 (𝑞4𝑥 + 6𝜌2𝑥𝑞2𝑥 + 3𝜌4𝑥)− 2 (𝛼𝜅)1/2 (𝜌2𝑥 + 𝑞2𝑥) + 𝛾 (𝑞2𝑦 + 𝜌2𝑦)− 𝜎 (𝑞2𝑥𝑞𝑦 + 𝑞𝑦𝜌2𝑥) + 18 ( 1𝜌2𝑥 +

1𝜌2𝑦) + 𝜅.(10)

For some specific value of the quadratic coupling param-eter, the trajectories of the system exhibit chaotic behavior

as shown in Figure 6. It is interesting to note that ourequations are coupled; classical and quantum interactionsare linked. At ℏ → 0 classical limit, only the first fourequations remain, confirming that the fluctuation variablesare responsible for quantum effects. As far as the couplingquadratic potential is concerned, it is the most useful andcomplicated.Wemay conclude by saying that more bends areobserved by projecting on the planes (𝑝𝑥, 𝑞𝑥) and (𝑝𝑦, 𝑞𝑥).Thus, it fascinates more and more spreading in each dot.Here again, the chaotic trajectories are very pronounced forhigher interdot barrier (not shown). Further, chaotic motionis found from medium interdot distance 𝜅 = 5 as shown inFigure 6.

Advances in Mathematical Physics 7

0.028

0.029

0.03

0.031

0.032

0.033

0.034

0.035

0.036

0.037

0.038

Ｇ；Ｒ

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

Figure 5: The maximal Lyapunov exponent in linear coupling case in (5a)–(5h). The parameter is fixed as 𝛼 = 0.1.

2 2.5 31.5px

−1.5

−1

−0.5

0

0.5

1

1.5

q x

(a)

1.8 2 2.2 2.4 2.6 2.8 31.6px

−1.5

−1

−0.5

0

0.5

1

1.5

q x

(b)

Figure 6: Phase portrait (a) and Poincare sections (b) in the plane 𝑝𝑥, 𝑞𝑥 with the initial conditions 𝑞𝑥𝑜 = 2, 𝑞𝑦𝑜 = 1, 𝑝𝑥𝑜 = 1, 𝑝𝑦𝑜 = 1, 𝜌𝑥𝑜 = 1,𝜌𝑦𝑜 = 2, 𝜉𝑥𝑜 = 1, and 𝜉𝑦𝑜 = 1. We set the quadratic coupling parameters as 𝜎 = 0.1, 𝜅 = 5, and 𝛼 = 0.1.

The maximal Lyapunov exponent versus the couplingparameter 𝜎 is shown in Figure 7. From Figure 7, we note thatthe maximal Lyapunov exponent 𝜆max has a positive value. Itmeans that the system described by (5a)–(5h) is chaotic.

4. Conclusion

In this paper, semiquantum chaos in two GaAs quantumdots coupled linearly and quadratically by two harmonicpotentials was investigated using the Ehrenfest theorem.Thiswork connected two distant concepts in physics: the quitepopular double-dot potential in semiconductor structures

proposed by Loss and DiVincenzo for qubits in a quantumcomputer and quantum chaos. The existence of irregularchaotic motion in nonintegrable systems was pointed out.To address this situation, we used a variational methodto extend the classical phase space in order to take intoaccount the quantum fluctuations and derive a set of effectiveHamiltonian equations in this extended space. By using thisextendedmethod, the existence of chaotic motion was shownin zero coupling, linear coupling, and quadratic coupling.Then, interdot distance parameter was used for controllingthe system. It was discovered that the effects of interdotbarrier parameter and coupling parameter induce irregular

8 Advances in Mathematical Physics

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.02

0.022

0.024

0.026

0.028

0.03

0.032

0.034

0.036

0.038

0.04

Ｇ；Ｒ

Figure 7: The maximal Lyapunov exponent in quadratic coupling case in (5a)–(5h). The parameter is fixed as 𝛼 = 0.1.

behavior such as chaos. This result enables us to estimate thedensity of particles that have chaotic trajectories in the nonin-tegrable systems, which might be useful for best performingquantum computing efficiently estimating potential errorsintroduced by Loss and DiVincenzo in quantum computer.The effect of high temperature was neglected in this workand will be studied in future works. The study of the fullquantum dynamics without classical counterpart will also beinteresting for better monitoring of quantum chaos.

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper.

Acknowledgments

The writers thank the members of the Fundamental Sci-ences Doctorate Training Unit of the University of Maroua,Cameroon, who provided pieces of advice and helpful discus-sions. Emile Godwe would like to thank Dr. Sifeu TakougangKingni (University of Maroua, Cameroon) for stimulatingdiscussions and Dr. Francois BAIMADA GIGLA (Universityof Maroua, Cameroon) for carefully editing the manuscript.

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