The two-dimensional D− complex in intense AC and strong magnetic fields

Physica E 3 (1998) 198–204

The two-dimensionalD− complex in intense AC and strongmagnetic �elds

Pawel Hawrylak ∗, Luis Rego

Institute for Microstructural Sciences, National Research Council of Canada, Ottawa, Canada K1A 0R6

Received 13 January 1998; accepted 29 June 1998

Abstract

The combined e�ect of intense time dependent far infrared �eld (FIR) and a strong static magnetic �eld on in-teracting electrons in a two-dimensional D− complex is studied. The two-electron problem is cast in the languageof the Center of Mass (CM) and relative particles, which in turn is mapped into a problem of coupled nonlin-ear harmonic oscillators. The time evolution of the CM and its coupling to the internal degrees of freedom via thecoulomb interaction and the Pauli exclusion principle is investigated. c© 1998 Elsevier Science B.V. All rights re-served.

PACS: 73.20Dx; 72.20Ht; 78.66.−w; 71.50.+t

Keywords: Far infrared �eld; High magnetic �eld; D− complex

1. Introduction

The quasi-two-dimensional D− complex consistsof two electrons bound to a donor in a quantum well[1,2]. The internal transitions of the singlet and triplettwo electron complex in a magnetic �eld have beenstudied by low intensity far infrared transmission spec-troscopy [1–3]. These optical studies have been re-cently extended using high intensity Free ElectronLaser (FEL) photon beams [4]. The D− complex in aFree Electron Laser and a strong magnetic �eld o�ers

∗ Corresponding author. Fax: +1 613 957 8734; e-mail: [email protected].

a prototype system for studying interacting electronsin intense AC and strong magnetic �elds under wellcontrolled conditions. Much e�ort has been devoted tothe study of atoms [5] and superlattices [6] in intenseAC �elds, but neither the e�ect of the magnetic �eldnor electron–electron interactions are well understood[7]. We study here the coherent time evolution of thedensity matrix for a free electron, and for one- andtwo- bound electron complexes. The analytical solu-tion of the free electron problem is combined with nu-merical integration of the time dependent Schrodingerequation. The e�ect of the electron–electron interac-tion on the coherent time evolution of theD− complexis studied.

1386-9477/98/$ – see front matter c© 1998 Elsevier Science B.V. All rights reserved.PII: S 1386 -9477(98)00234 -3

P. Hawrylak, L. Rego / Physica E 3 (1998) 198–204 199

2. The model

The problem of the quasi-two-dimensional D−

complex in a strong magnetic �eld has been studiedin detail [8–10]. The starting point for the time de-pendent problem is the Hamiltonian

H0 =12m

(� +

ecAext(t)

)2(1)

of the free electron with e�ective mass m moving inthe x; y plane (r=(x; y)), in a static magnetic �eldB along the z-direction, and a strong time dependentvector potential Aext(t) of the AC external �eld. Thegeneralized momentum �= p+ (e=c)A(r) and thestatic vector potential A(r)= 1

2B× r are de�ned inthe symmetrical gauge, and we take ˜=1. The AC�eld is assumed to vary slowly on the length scaleof the system, and therefore to depend only on time.It is de�ned by the vector potential Aext(t), switchedon at time t=0. The vector potential is relatedto the electric �eld by Eext(t)=−(1=c)9Aext(t)=9t,where c is the velocity of light. We assume anoscillating electric �eld with external frequency! and amplitude of the form: Eext(t60)=0, andEext(t¿0)=�[ey cos(!t)− ex sin(!t)].The Hamiltonian, Eq. (1), can be diagonalized

through a transformation into a pair of inter- andintra-Landau level creation=annihilation operatorsa+; a; b+; b de�ned as

x − iy= l0(a+ b+)21=29x + i9y = 1l0(b− a+)=21=2;

x + iy= l0(a+ + b)21=29x − i9y = 1l0(a− b+)=21=2;

where l0 is the magnetic length. Operators a; b; : : :satisfy boson commutation relations [a; a+]= 1,[b; b+]= 1 and [a; b] = 0. The single particle energies�n=!c(n+ 1

2) and eigenstates |m; n〉 =√(1=m!n!)

× (a+)n(b+)m |0; 0〉 are those of two harmonic oscil-lators, with !c = eB=mc being the cyclotron energy.The complex vector potential �(t) is related to

Aext(t) by �(t)= (Ay(t) + iAx(t))e=(21=2mcl0), whichis related to the oscillating electric �eld Eext(t)by (Ay(t) + iAx(t))− (Ay(0) + iAx(0)) = −c ∫ t0 (Ey(t) + iEx(t)) = −c ∫ t0 (�e−i!t) = (c�)(e−i!t − 1)=i!.We choose Ay(0) + iAx(0)= c�=(−i!) to give�(t)=−ie−i!t�0 with �0 = e�=(21=2ml0!).

The �nal form of the D− Hamiltonian is that ofdriven coupled harmonic oscillators:

H =!c(a+1 a1 +12) + �(t)a

+1 + �

+(t)a1

+∑qVD(q)e−|Q|2eiQ

∗a+1 eiQa1eiQb+1 eiQ

∗b1

+!c(a+2 a2 +12) + �(t)a

+2 + �

+(t)a2



∗b2

+∑qVee(q)e−2|Q|

2eiQ

∗(a+1 −a+2 )eiQ(a1−a2)

× eiQ(b+1 −b+2 )eiQ∗(b1−b2); (2)

where Q= l0(qx + iqy)=21=2 and the electron–electroninteraction potential is Vee(q)= 2�e2=�q, with � be-ing the dielectric constant. The VD(q) is the Fouriertransform of the con�ning potential, for a donorVD(q)=−Vee. The term quadratic in the vector po-tential �(t)�+(t)=!c has been removed via a simplegauge transformation and will be omitted in furtherdiscussion.The D− Hamiltonian can be further simpli�ed by

introducing Center of Mass A1 = (a1 + a2)=√2 and

relative A2 = (−a1 + a2)=√2 operators:

H =!c(A+1 A1 +12) +

√2�(t)A+1 +

√2�+(t)A1

+!c(A+2 A2 +12) +

∑qVee(q)e−2|Q|

2e−iQ

∗A+2

× e−iQA+2 e−iQB+2 e−iQ∗B2

+∑qVD(q)e−|Q|2 [eiQ


∗b1

+ eiQ∗a+2 eiQa2eiQb

+2 eiQ

∗b2 ]: (3)

The CM and relative operators are implicit in the lastexpression of Eq. (3).In the �nal general form of the total Hamiltonian the

coupling V (t) to the external �eld is only through theCenter of Mass operator A1 = (1=

√N )

∑i ai driven by

the external �eld. This guarantees that in the absenceof a con�ning potential a weak external �eld �(t) in-duces electronic transitions with frequency equal tothe cyclotron frequency !c, irrespective of the num-ber of carriers and interactions among them (Kohn’stheorem [11]).

200 P. Hawrylak, L. Rego / Physica E 3 (1998) 198–204

3. Density matrix

The time dependent wavefunction of the system,(t)=

∑j aj(t) |j〉, is expanded in the basis of exact

many-electron eigenstates |j〉 and energies Ej of theD− Hamiltonian. The time evolution of D− is bestdescribed by the density matrix Pi; j(t)= ai(t)a∗j (t).The density matrix satis�es an equation of motion:

i9Pdt= [H0 + V (t); P]: (4)

The conservation of the number of particles and theundetermined overall phase of the wavefunction leadto two constants of motion: Tr[P] and Tr[PP+].If we de�ne the real uij =Re(Pij) and imaginary

vij = Im(Pij) components of the density matrix, theysatisfy the following equations of motion:

9uijdt

=+(Ei − Ej)vij + Im(�(t))[〈i|V−|k〉 ukj−uik 〈k|V−|j〉]+Re(�(t))[〈i|V+|k〉 vkj − vik 〈k|V+|j〉];

9vijdt

=−(Ei − Ej)uij + Im(�(t))[〈i|V−|k〉 vkj−vik 〈k|V−|j〉]−Re(�(t))[〈i|V+|k〉 ukj − uik 〈k|V+|j〉]; (5)

where V± 〈i|a+ ± a|k〉. From the hermicity of thedensity matrix the diagonal matrix elements are real(vi; i=0) and satisfy ui; j = uj; i and vi; j =−vj; i.The problem with solving the equation of motion

for the density matrix is the size: for N states we have≈ 2N 2 real equations. The number of states of manyelectrons is already formidable, the time dependentproblem is obviously even more intractable.

4. Results and discussion

4.1. Free electron

The free electron Hamiltonian is that of a drivenharmonic oscillator:

H =!c(a+a+ 12) + �(t)a

+ + �+(t)a: (6)

The time evolution of the state of the free electron ina magnetic �eld is described by the evolution opera-tor U (t), i.e., (t)=U (t)(0). The time dependent

transformation U (t) for an arbitrary driving �eld �(t)switched on at t=0 is given by [7,12]:

Ui(t) = e−i!cta+i aie−iF(t)a

+i e−iF

+(t)ai

× e+i!cta+i aie−∫ t

0d� �(u)ei!cuF+(u) (7)

with F(u)=∫ u0 d� �(�)e

i!c�. Substituting �(t)=(e−i!t=i!)(e�)=(21=2ml0) into the amplitude F(t)gives F(t)=F0(ei(!c−!)t − 1) with F0 = e�=(21=2 ×ml0)=[!(!− !c)]. With our driving �eld the am-plitude of the displacement �eld, F(t60)=0 and|F(t¿0)|2 = (F0)22(1− cos(t)), oscillates with thedetuning frequency=(!c − !). The transformationU shifts the time dependence into the displaced bo-son operators a+i → a+i + i�∗(t) and ai→ ai − i�(t),where displacement �(t)= e−i!ctF(t). One could for-mulate the time evolution of an interacting system interms of displaced operators [13], based on a recentextension of the Kramers–Henneberger frame [14] tomagnetic �elds [7].Using the de�nition F(t)= |F(t)|ei�(t) the matrix

elements of the operatorU (t) in the free electron basis|m; n〉 (without a constant factor) are given by

〈n′; m′|U (t)|m; n〉= �m;m′⟨n′|e−i!cta+ae−iF(t)a+

× e−iF∗(t)ae+i!cta

+a|n⟩;

〈n′; m′|U (t)|m; n〉= �m;m′e−i(!ct−�(t))(n′−n)−i�(n′+n)=2

×min[n; n′]∑k=0

√n!√n′!|F(t)|n+n′−2k

k!(n− k)!(n′ − k)! :

(8)

If the electron is initially in the lowest Lan-dau level n=0 the time dependent probabilityPn;n(t)= |an(t)|2 = |〈n |U (t)| 0〉|2 of �nding an elec-tron in the nth Landau level is easily calculated. It isgiven by the Poisson distribution:

|an(t)|2 = e−|F(t)|2 (|F(t)|2)nn!

: (9)

The Poisson distribution describes a coherent stategenerated by the external source. The source is givenexactly by the amplitude F(t). The electron coherentlyredistributes itself between the n=0 and all higherLandau levels as a function of time through the ampli-tude F(t). An example of the time evolution of the free


Fig. 1. (a) The energy spectrum of a free electron as a functionof angular momentum R for �0 = 0, B=16 Tesla, (b) the timeevolution of the probability of occupation of states due to driving�eld with frequency !=2 and amplitude �0 = 0:707.

electron in a magnetic �eld is shown in Fig. 1. Fig. 1ashows the energy levels of the free electron E(n) andFig. 1b shows the time evolution of their probabil-ity distribution function |an(t)|2 for a magnetic �eldB=16T and external �eld with amplitude �0 = 0:707and frequency !=2. Here we set m=0 and set totalangular momentum R=m− n=−n. All energies aremeasured in units of E0 and time is measured in unitsof E−1

0 where E0 =Ry√2�a0=l0 is the binding energy

of an electron to a donor in the lowest Landau level,and Ry and a0 are the e�ective Rydberg and the e�ec-tive Bohr radius. The system oscillates with frequency=!c − !=0:78, and the electron wavefunction isredistributed into many Landau levels by the pumping�eld.The behaviour of the o�-diagonal density matrix

elements is also of interest. The o�-diagonal matrixelements correspond to a coherent polarization of thesystem 〈a+〉 = ∑

n P∗n+1; n

√n+ 1. They are given by

Pm;n=Um;0U+n;0. Simple algebra gives 〈a+〉 = i�∗(t)

i.e. a classical displacement of the oscillator. Note thatboth the diagonal and o�-diagonal elements of thedensity matrix oscillate with detuning frequency independently of the intensity of the driving �eld. Thisshould be contrasted with a driven two-level systemwhich oscillates with a Rabi frequency which dependson the intensity of the driving �eld.

Fig. 2. (a) The energy spectrum of D0 in the basis of singleparticle states |m=0; n〉 as a function of angular momentum R for�0 = 0, B=16 Tesla, (b) the time evolution of the probability ofoccupation of states due to a driving �eld with driving frequency!=2 and amplitude �0 = 0:707.

4.2. D0-electron bound to a donor

The Hamiltonian of a single electron bound to adonor driven by an external �eld is that of a pairof highly nonlinear harmonic oscillators, with inter-Landau level oscillators driven by the external �eld:

H =!c(a+a+ 12) + �(t)a

+ + �+(t)a


∗a+eiQaeiQb+eiQ

∗b: (10)

We investigate numerically the time evolution of thedensity matrix for D0. To make a direct comparisonwith the free electron problem the D0 basis has beenrestricted to states |0; n〉. The energies of D0 statesare shifted from the energies of Landau levels. Forthe lowest (R=0) and �rst excited (R=−1) statesthe energy shifts are −E0 and −E0=2, respectively.The transition energy is therefore !c + E0=2, blue-shifted from the cyclotron energy by 1=2 of the bind-ing energy E0. The allowed transition evolves from the1s− 2p+ transition at B=0. It has been observed ex-perimentally [1,2]. The energy levels of the D0 com-plex are shown in Fig. 2a for the same parametersas in Fig. 1. The time evolution of the probability ofoccupation of these levels is shown in Fig. 2b. The


detuning frequency is =0:28 which corresponds toa period T =22:4. We see that despite the fact thatthe detuning frequency is very di�erent from the freeelectron case, the overall time dependence is very sim-ilar. The pumping �eld populates many states of D0.The coherent system returns to the initial state muchfaster (higher frequency) than the detuning would in-dicate. We shall discuss this in greater detail for thetwo electron problem.

4.3. D−-two electrons bound to a donor

The Hamiltonian of Eq. (3) describes a CM par-ticle (1), a relative particle (2), and the interactionbetween them. The CM particle is free while therelative particle moves in a repulsive Coulomb �eld.CM and relative particles interact through the donorpotential. Only the CM particle is explicitly drivenby the AC �eld. However, due to interaction viathe donor potential the CM particle excites the rela-tive particle. In the absence of the driving �eld theHamiltonian is diagonalized in the basis of singleparticle states of the free CM and relative particles|M1; N1〉 |M2; N2〉. The values of the angular momen-tum M2 − N2 of the relative particle are restricted toeven(odd) values for the singlet(triplet) states. Theeigenstates |R; �〉 are labeled by total angular mo-mentum R=M1 − N1 +M2 − N2 and an eigenstateindex � in the Hilbert space of given R. Because onlythe |N1〉 oscillator is driven we start the analysis bybuilding Hilbert spaces only with states |0; N1〉 |0; N2〉.For small R we can easily enumerate the singletstates: [R=0; |0; 0〉 |0; 0〉], [R=−1; |0; 1〉 |0; 0〉];[R=−2; |0; 2〉 |0; 0〉, |0; 0〉 |0; 2〉]; : : : . There is only asingle ground state and a single �rst excited state dueto exclusion principle. These states are similar to theD0 case. The energy di�erence is also identical to theD0 case. Therefore the one photon transitions cannotdistinguish between the D− and D0, an accidentaldegeneracy removed in real systems by Landau levelmixing and �nite quantum well width. The �rst dif-ference between the D− and D0 appears at R=−2,where a state of the relative particle with higher an-gular momentum (lower energy) becomes possible.This Hilbert space is only accessible via two-photonabsorption, and is equivalent to a bi-exciton complex.The bi-exciton complex has two states, an opticallyactive one, and a “dark” one. The R=0;−1;−2

Fig. 3. (a) The energy spectrum of D− in the basis of singleparticle states |m=0; n〉 as a function of angular momentum R for�0 = 0, B=16 Tesla, (b) the time evolution of the probability ofoccupation of states due to a driving �eld with driving frequency!=2 and amplitude �0 = 1.

spaces, and an equivalent three-level system, are theonly ones needed to describe the time evolution up tosecond order in intensity of the �eld. We �nd howeversuch description inadequate. For higher total angularmomenta, the Hilbert space increases due to increas-ing number of states of the relative particle. For asinglet these states correspond to even values of angu-lar momentum of the relative particle which increasesin steps of two. The spectrum of D− generated in thisbasis is shown in Fig. 3a, and is to be compared withthe spectrum of D0 in the similar basis. The spectrumshows that for each optically active CM state thereare lower energy states which correspond to reducedrepulsive energy of the relative motion of electrons.These states become optically active due to Coulombinteractions. The time evolution of the occupationprobability of each state of the spectrum is shownin Fig. 3b. The amplitude �0 is 1.0 and the drivingfrequency is !=2:0. These parameters were chosento treat the CM particle on the same footing as thesingle electron in D0, including the lowest transitionenergy of wc + E0=2. We see that in such a limitedbasis only excited states of the CM particle are popu-lated, with only a very weak population of the excitedstates of the relative particle. The frequency of oscil-lation of e.g. the occupation probability of the ground


Fig. 4. The time dependence of the power in the system.

state is very similar to that of D0. In Fig. 4 we showthe power P(t)=Re[〈a+〉 · E(t)] for di�erent ampli-tudes of �0 for the D− system shown in Fig. 3. Forlow �0 = 0:1, the power oscillates (system absorbsand returns energy from a classical external �eld)with period T =2� corresponding to the detuningfrequency =wc + E0=2− !. As the intensity in-creases the oscillations become faster. We analyzethe frequency �! of these oscillations as a functionof intensity of the driving �eld. The intensity depen-dence of the frequency �! of oscillation of P(t) isshown in the inset of Fig. 4. The numerically obtainedfrequency �! appears to blue shift with the intensityof the �eld in a way similar to the Rabi frequency ofa two-level system. One might therefore be temptedto associate the blue shift with changes in energies ofthe “dressed D−” [7].Finally, in Fig. 5 we show the time evolution of

the D− complex in which the basis now includesintra-Landau level oscillators. One �nds an increasednumber of states in each Hilbert space of a giventotal angular momentum R. There are N =124 states

Fig. 5. Same as Fig. 3 but in the basis of single particle states|m; n〉.

in the basis, which requires a propagation in timeof ≈104 linear equations. The initial time evolutionis similar to that of D0 and D− in a limited Hilbertspace. However, once the higher Landau levels of theCM particle are partially populated, the CM particleinteraction with the relative particle signi�cantly re-distributes the spectral weight over many states ofrelative motion (t¿6). These states are always atlower energies than the corresponding CM states.

5. Conclusions

We investigated the coherent time evolution of thedensity matrix for the free electron, and of one- andtwo-electron complexes bound to a donor in quasi-two-dimensional systems in a strong magnetic �eld.An exact analytical solution of the free electron

problem was given in terms of the evolution ofthe driven harmonic oscillator. Both the analyticalsolution and the numerical propagation of the timedependent Schrodinger equation showed that a largenumber of Landau levels can be populated by an in-tense AC �eld. The rate of exchange of energy (Rabifrequency) between the free electron and the AC �elddoes not depend on the �eld intensity. However, forthe D− complex the Rabi frequency is found to in-crease with intensity. The time evolution of the D−

electrons di�ers from the time evolution of noninter-


acting electrons due to the transfer of energy from theAC �eld to the Center of Mass, and from the Centerof Mass to internal excitations via electron–electronand electron–donor interactions.These e�ects could be studied through pump and

probe or tunneling spectroscopies. Future work shouldinvestigate also relaxation processes.

Acknowledgements

We thank P.A. Schulz, J.A. Brum, G.C. Aers andB.D. McCombe for discussions.

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The two-dimensional D− complex in intense AC and strong magnetic fields