A response of a system of strongly interacting particles to an inhomogeneous external field S.A.Mikhailov Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany (July 28, 1997) Abstract We consider a system of N strongly interacting quantum particles in an inhomogeneous external field, and find the general conditions under which the response of the system is independent of the inter-particle interaction and can therefore be calculated exactly. For a system of particles with an arbitrary parabolic conduction band, in an arbitrary parabolic confining potential we predict a phenomenon of a parametric resonance in an oscillating inhomogeneousexternalfield.weshowthatthepumpingofthesystembya quadrupole field leads to a generation or an amplification of dipole oscillations. Excitation frequencies, growth rates, and other parameters of the instability are independent of the number of particles and the inter-particle interaction. Possible applications of the predicted effect are discussed. PACS numbers: 78.66.-w, 42.65.Yj, 42.65.Ky, 85.30.Vw 1 Typeset using REVTEX
I. INTRODUCTION A response of a system of strongly interacting quantum particles to an external force depends, in general, on both the properties of individual particles and their interactions. In order to calculate the response one should solve the many particle Schrödinger equation which includes interactions of particles between each other and with external fields. In a system of non-interacting or weakly interacting particles the problem reduces to a single particle Schrödinger equation and can often be solved exactly. If the inter-particle interaction is not weak and cannot be treated as a perturbation, the problem becomes very complicated andcannotbesolvedinageneralform. However, there are situations when the response of a system of strongly interacting particles does not depend on inter-particle interactions and can be calculated exactly. A well-known example is the Kohn theorem[1], which states that in a translationally invariant system of electrons, the frequency of the cyclotron resonance is not affected by the electronelectron interaction. The Kohn theorem has been generalized in Refs.[2 6] to electron systems in a confining parabolic potential. It has been shown that the system exposed to a homogeneous external electric field absorbs at frequencies which do not depend on the number of particles and the inter-particle interaction. This statement has been proved for a system of electrons in a wide parabolic quantum well[2], in a circular[3 5] and an elliptic[6] quantum dots. In this Letter we consider the response of an arbitrary system of N interacting particles to a variable inhomogeneous external field with the aim to establish the most general conditions, under which the response does not depend on inter-particle interactions. For a uniform external field we obtain a further generalization of the Kohn theorem, showing that it can also be proved for particles with an arbitrary parabolic conduction band. A novel and important result is obtained for an inhomogeneous, quadrupole, external field[7]. We predict that the action of an oscillating external quadrupole field results in an instability of the system, so that the center-of-mass coordinate(and the dipole moment) of the system oscillates with an 2
exponentially growing amplitude(parametric resonance[8]). Our results rigorously follow from the many particle Schrödinger equation which includes an arbitrary interaction between particles. The predicted phenomenon can have interesting and promising applications which wediscussattheendoftheletter. II. RESPONSE EQUATIONS LetasystemofNidenticalinteractingparticleswiththekineticenergyT(p)isplaced inaconfiningpotentialv(r)andexposedtoanexternalvariablepotentialfieldφ ext (r,t). The dynamics of the system is determined by the time-dependent many particle Schrödinger equation i h Ψ(r 1,...,r N,t) t =ĤΨ(r 1,...,r N,t) (1) with the Hamiltonian Ĥ= i {T(ˆp i )+V(r i )+Φ ext (r i,t)}+ 1 U( r 2 i r j ), (2) i j wheretheinter-particleinteractionu( r i r j )canbeanarbitraryfunctionofthedistance between particles. Assuming that the kinetic energy T(p) is an arbitrary quadratic function of momenta, T(p)= 1 2 W αβp α p β, (3) [W αβ isa(symmetric)tensoroftheinverseeffectivemass]andcalculatingthetimederivativesoftheoperatorsofthecenterofmassr = (1/N) ir i andthetotalmomentum ˆP= iˆp i wefind Ṙ α =W αβˆpβ /N, (4) ˆ P α = i ( V(ri ) x i α + Φ ) ext(r i,t). (5) x i α 3
TheconfiningpotentialV(r)andthetime-dependentexternalpotentialΦ ext (r,t)canbe expanded in power series. If these expansions are restricted to linear and quadratic terms, V(r)= 1 2 K αβx α x β, (6) Φ ext (r,t)= F α (t)x α + 1 2 L αβ(t)x α x β, (7) thesumintherighthandsideofeq. (5)isexpressedviaR α,andthesystem(4) (5) becomesclosed.theequationofmotionofthecenterofmassrthenassumesasimpleform independent of the inter-particle interaction U R α +W αβ [K βγ +L αβ (t)]r γ =W αβ F β (t). (8) IftheTaylorexpansions(6),(7)containedhigherpowersofr( r n ),thederivativeˆ P α ineq. (5)woulddependonhighermoments( ir n 2 i x i α ).Calculatingtimederivativesofhigher moments one can reduce the problem to an infinite set of coupled U-dependent equations forthemomentsofthetype i(r n α,iˆp m β,i+ˆp m β,ir n α,i).thissetofequationsisequivalenttothe initialproblem(1) (2)andcannotbesolvedinageneralform.Asimilarsituationholds whenthehigherp n correctionstothekineticenergyoperator(3)aretriedtobetakeninto account. The U-independent closed equations(8) are thus separated from the whole set of equations and describe the dipole response of the system to an external field(7) only under the conditions(3),(6). Intheform(8)theresultobtainedisvalidforparticlesofanarbitrarynatureinaspace ofanarbitrarydimensionalityd(includingthecased>3). For charged particles in the three dimensional space the result(8) can be easily extended tothecasewhenthesystemisplacedinanarbitrarilydirecteduniformmagneticfieldb. Replacingˆp i ineq. (2)byˆp i (e/c)a(r i ),whereeisthechargeofparticlesanda(r)= B r/2isthevectorpotential,andproceedingthesamestepswiththenewhamiltonian, we obtain U-independent equations R α + e c W αβ(b Ṙ) β+w αβ [K βγ +L αβ (t)]r γ =W αβ F β (t), (9) 4
whichdescribethedipoleresponseofthesystemtothefield(7)inauniformmagneticfield. Equation(9)andallconsequencesfromitdonotdependonthenumberofparticlesand the inter-particle interaction. The statements of the original[1] and the generalized[2 6] Kohntheorems,concerningtheresponseofthesystemtoahomogeneousfield(L αβ =0), followsfromhereatascalareffectivemassofparticles(w αβ =δ αβ /m). Ofourparticular interest is however the behavior of the system under the action of a variable inhomogeneous (quadrupole) external field. III. PARAMETRIC RESONANCE IN A QUADRUPOLE FIELD Letthedipoletermintheexpansion(7)isabsent,F(t)=0,andthesystemisexposedto aquadrupolevariableexternalfield,φ ext (r,t)=l αβ (t)x α x β /2. Thecenter-of-massmotion is then described by a coupled set of linear differential equations with variable coefficients R α + e c W αβ(b Ṙ) β+w αβ [K βγ +L βγ (t)]r γ =0. (10) Asknown,iftheexternalforceisperiodicintimewithaperiodT,L αβ (t)=l αβ (t+t),the so called parametric resonance[8] can be observed in such systems: under certain conditions the solutions of Eqs. (10) exponentially grow with time which means an instability of the system. In our case this corresponds to a parametric excitation of the dipole oscillations by an oscillating quadrupole external field. In order to demonstrate the instability of the system in somewhat more detail, we consider a special case of particles with a scalar effective mass, in an isotropic confining potentialv(r)=kr 2 /2,atB=0,undertheactionofanexternalisotropicquadrupolefield L αβ (t)=kλ(t)δ αβ,whereλ(t)=λ(t+t).eqs.(10)thenassumetheform R α +ω 2 0[1+Λ(t)]R α =0, ω 2 0=K/m. (11) Equations of this type are encountered in different physical problems, such as the problem of a simple pendulum with an oscillating point of support[8], the motion of a quantum particle 5
in a one-dimensional periodic potential[9](if the time t is replaced by the coordinate x), the propagation of light in a periodically stratified medium[10], and others. The general solutionofeq.(11)iswrittenintheform(thefloquet,orblochtheorem[9]) R(t)=C 1 e iµt Π 1 (t)+c 2 e iµt Π 2 (t), (12) wherethefunctionsπ 1 (t)andπ 2 (t)areperiodicwiththeperiodt,andthecharacteristic exponent µ depends on parameters of the problem. The instability evolves in the system, when the function µ has a nonzero imaginary part, and one of the solutions(12) exponentially increases with time. For a harmonic external force, Λ(t) = λ cos(ωt), this happens inside instability regions, shown in Fig. 1 and centered, at a small amplitude of the external force (λ 1),atω 2ω 0 /n,n=1,2,...[8,9,11]. Intheproblemofaquantumparticleina one-dimensional periodic potential(of light propagating in a periodically stratified medium) the instability regions correspond to forbidden gaps in the energy spectrum of the particle (in the frequency spectrum of light). Ifthesystemispumpedbyanoscillatingquadrupoleforcewiththefrequencyω=2ω 0 (the first instability region), the solution R(t) has the form R(t) exp ( )[ ( ω0 λ 4 t sin ω 0 t π )+ λ ( 4 16 sin 3ω 0 t π ) ] +..., (13) 4 i.e.thesystemgeneratesoscillationswiththefundamentalharmonicoscillatorfrequencyω 0 anditsoddhigherharmonicsmω 0,m=3,5,... Thegrowthrateω 0 λ/4andtherelative amplitude of the third harmonics are proportional to the amplitude of the driving force λ. Forthepumpingquadrupolefieldwiththefrequencyω=ω 0 (thesecondinstabilityregion), allhigherharmonics(m=2,3,...)arepresentinthesolution,butthegrowthrateandthe widthoftheinstabilityregionaresmaller( λ 2 ). Inthepresenceofasmalldampingthe effectexistsifthegrowthratesexceedthedampingrateγ. In the general case of a finite magnetic field, anysotropic confining potential, and/or anysotropic external quadrupole field, the parametric resonance in the system is described by Eqs. (10). The dipole excitation spectrum of the system in this case typically consists 6
in several modes with B-dependent frequencies[2 6], and main instability regions exist at twice the frequencies of the eigen dipole modes. It should be emphasized that, in addition to the parametric excitation of dipole oscillations, an oscillating external quadrupole field induces an oscillating quadrupole moment and causes a quadrupole absorption[7](and a quadrupole radiation in a system of charged particles). The equation of motion of the quadrupole moment does depend on the inter-particle interaction. However, the value of the induced quadrupole moment is finite(proportional to the driving force), while the value of the parametrically induced dipole moment exponentially increases in time. At a large time scale Eq. (10) thus describes the dominant effect, and the induced quadrupole radiation can be neglected as compared to the parametrically excited dipole one. Under the action of both dipole and quadrupole time-dependent external fields the dynamicsofthecenterofmassofthesystemisdeterminedbythegeneralequations(9). Under these conditions one can realize a parametric amplification of the dipole oscillations by means of the oscillating quadrupole field. For the most effective operation of this type of a parametric amplifier, the frequency of the quadrupole oscillations should again be twice the frequency of eigen dipole modes of the system. The results obtained can be summarized in the form of the following Theorem. The action of an oscillating quadrupole external force on a system of N interacting particles results in the parametric excitation or the parametric amplification of dipole oscillations with the eigen dipole excitation frequencies and their higher harmonics. For a system of particles with an arbitrary parabolic conduction band(3), in an arbitrary parabolic external potential(6), and in an arbitrarily directed uniform magnetic field B, the effect is described by Eq. (9). Under these conditions excitation frequencies, growth rates, boundaries of the instability regions are independent of the number of particles and the inter-particle interaction. 7
IV. DISCUSSION Artificial electron structures with properties necessary for observation of the predicted effects are successfully fabricated due to a progress of modern semiconductor technology. These man-made atoms, or quantum dots, are realized in high-electron-mobility modulation-doped semiconductor heterostructures and have been intensively studied in recentyears[12]. Theconfiningpotentialinsuchsystemscanbeconsidered,toafirstapproximation,asparabolic,withthetypicalharmonicoscillatorfrequencyω 0 oforderofa fewmev.thenumberofelectronsinquantumdotstructurescanbevariedbyonestarting with a single electron in each dot[13] up to quite macroscopic values. Far-infrared transmission(absorption) experiments on quantum dot arrays have confirmed the validity of the generalized Kohn theorem[12]. These systems can be one of the candidates for experimental observation of the parametric resonance in a quadrupole oscillating external field. Another possibility to observe the predicted effects can be related to two-dimensional electron systems on the surface of liquid helium. An inhomogeneous quadrupole external field can be easily realized in such systems using an appropriate configuration of external electrodes(see, e.g. Ref.[14]). Typical frequencies and damping rates in these structures are smaller than in solid-state quantum-dot systems, which may facilitate the observation of the effect. If a parabolic quantum dot is exposed to an external dipole radiation with the frequency ω,itabsorbs,atb=0,theenergyoftheexternalfieldonlyatthefrequencyω=ω 0 (the generalizedkohntheorem). Ifthefrequencyωofanexternaldipoleradiationequals2ω 0, noresponsewillbeobserved. Ifhoweverthedotisexposedtoaquadrupolefieldwiththe frequency2ω 0,itwillemit,asshownabove,thedipoleradiationwiththeharmonicoscillator frequencyω 0 anditshigherharmonics.ifthequantumdotispumpedbyaquadrupolefield withthefrequency2ω 0 and,inaddition,isexposedtoadipoleradiationwiththefrequency ω 0,thedipoleradiationwillbeamplified. Experimentalobservationoftheseeffectswill probably open the way to a creation of quantum parametric generators(amplifiers) based 8
on single-electron or few-electron quantum dot arrays. To conclude, we have found the general conditions under which the dipole response of a system of strongly interacting quantum particles to an inhomogeneous external field is independent of the inter-particle interaction, and predicted the phenomenon of the parametric generation and amplification of the dipole radiation in the system with the help of the oscillating quadrupole field. ACKNOWLEDGMENTS I thank K. Richter for reading the manuscript and useful comments. 9
REFERENCES Electronicaddress:sam@mpipks-dresden.mpg.de [1]W.Kohn,Phys.Rev.123,1242(1961). [2]L.Brey,N.F.Johnson,andB.I.Halperin,Phys.Rev.B40,10647(1989). [3]P.A.MaksymandT.Chakraborty,Phys.Rev.Lett.65,108(1990). [4]P.Bakshi,D.A.Broido,andK.Kempa,Phys.Rev.B42,7416(1990). [5]A.O.GovorovandA.V.Chaplik,Zh.Eksp.Teor.Fiz.99,1853(1991)[Sov.Phys. JETP, 72(6), 1037(1991)]. [6]F.M.Peeters,Phys.Rev.B42,1486(1990). [7]M.Wagner,A.V.Chaplik,andU.Merkt,Phys.Rev.B51,13817(1995),havecalculated the frequencies and strengths of quadrupole excitations of a parabolic quantum dot. The phenomenon of a parametric resonance in a quadrupole field has not been discussed in that paper. [8] L. D. Landau and E. M. Lifshitz, Mechanics(Pergamon Press, Oxford, 1994), 27. [9]G.Floquet,Ann.ÉcoleNorm.Sup.12,47(1883);F.Bloch,Z.Physik,52,555(1928). [10] M. Born and E. Wolf, Principles of Optics(Pergamon Press, Oxford, 1993). [11] N. W. McLachlan, Theory and application of Mathieu functions(clarendon Press, Oxford, England, 1947). [12] for a review of fabrication methods and physical properties of quantum dot arrays see D.HeitmannandJ.P.Kotthaus,PhysicsToday46,56(June1993). [13]B.Meurer,D.Heitmann,andK.Ploog,Phys.Rev.Lett.68,1371(1992). [14]P.J.M.Peters,M.J.Lea,A.M.L.Janssen,A.O.Stone,W.P.N.M.Jacobs,P. Fozooni,andR.W.vanderHeijden,Phys.Rev.Lett.67,2199(1991). 10
FIGURES 2.5 2 n=1 ω/ω 0 1.5 Instability regions n=2 1 n=3 0.5 0 0.2 0.4 0.6 0.8 1 λ FIG.1. InstabilityregionsofsolutionsofEq. (11)inthecaseofasimpleharmonicexternal forceλ(t)=λcos(ωt)forn=1,2,3. 11