Period Doubling and Strange Attractors in Quantum Wells

VOLUME 76, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 29 APRIL 1996

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Period Doubling and Strange Attractors in Quantum Wells

Bryan Galdrikian1 and Björn Birnir1,2

1Mathematics Department, University of California at Santa Barbara, Santa Barbara, California 932Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland

(Received 16 October 1995)

A realistic model of a doped quantum well heterostructure exhibits a subharmonic signal inpresence of intense far-infrared radiation. Nonlinearity enters the model through the effective potdue to the density of electrons in the well. Dephasing and energy relaxation are introduced throsimple density matrix approach. The resultant dynamics show the period-doubling routes to chathe form of strange attractors. All material parameters used are well within the present capabilitpresent quantum well technology, and the driving field amplitudes and frequencies are easily obtawith a free electron laser. [S0031-9007(96)00081-6]

PACS numbers: 05.45.+b, 73.20.–r, 78.66.Fd

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The term “quantum chaos” has come to describe the bhavior of a quantum system corresponding to a classicsystem which exhibits chaos. In this Letter, we describa many-electron quantum well system which is far fromhaving a classical correspondence. Electron-electron intactions and relaxation mechanisms may be approximaby an effective nonlinear density matrix equation of motion, with solutions that we will show to undergo a perioddoubling cascade [1] to a strange attractor, as the strenof the driving field is varied. Hamiltonian (nondissipativechaos has been studied in mean-field models of quantsystems [2], but this is the first description, as far as wknow, of chaotic behavior in a model of a driven quantumwell, with relaxation mechanisms included.

We have modeled a doped quantum well with design parameters within the capabilities of present matrials technology, and simulate such a well subjectedintense far-infrared radiation. The ubiquitous presenof period doubling in our simulations is a strong indicator that a subharmonic signal should be observable froa quantum well designed to match our theoretical prameters. Period-doubling cascades to strange attractwhich would indicate broadband spectra emitted frosuch a device, are also present in our simulations. Bcause of the fast relaxation rates present in quantum wesubharmonicsbeyondperiod-two and chaotic signals exiswithin narrow parameter regions and are less likely to bmeasured [3]. At the end of this Letter we describe somdesign strategies for improving the likelihood of observing period doubling in quantum wells.

The existence of bifurcation cascades towards stranattractors in classical nonlinear infinite-dimensional models (partial differential equations) is well known; see, foexample, [4] and the numerical references in [5]. In [6a nonlinear Schrödinger equation is driven. It is knowthat in the presence of damping the time evolution of thdriven soliton solution of this nonlinear Schrödinger equation period doubles and cascades to a strange attrac[5]. In the absence of damping, the soliton solution i[6] showed either regular or chaotic time evolution, de

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pending on the driving amplitude, and its presence didinhibit the quantum barrier to classical diffusion. Our ivestigation is, however, closest to that of [2] as it doespresuppose that a classical coherent state such a sois present, but rather investigates the bifurcations ofquantum mechanical density-dependent coherent moBecause of our inclusion of the relaxation mechanismthe quantum well, we are able to go farther than [2] adescribe the bifurcation cascade and the resulting chastate of the two-level model.

We choosez to be the growth direction of the semconductor heterostructure forming the quantum well. Twave function of the system may be considered to be plwaves inx andy, multiplying an envelope function inz.

We will consider only the conduction band, whicgives the initial “well shape”Wszd. In the undrivensystem, interactions between electrons in the wellaccounted for by a self-consistent potentialy0szd [7,8].Typically, y0szd consists of adirect part, which is theCoulomb potential due to the electronic charge distribtion, and anindirect part, or “exchange-correlation” potential. We will omit the indirect part from our modeSince it is usually a small correction to the direct potetial, our results do not suffer qualitatively.

We will work in the basis of eigenstates of the seconsistent potential, found by solving the Schrödingequation for thez component of the system∑

2h2

2mp

d2

dz21 W szd 1 y0szd

∏jnszd Enjnszd .

In GaAs, the effective electronic massmp is about 1y15the mass of a free electron. The dispersion relatis thus quantized into parabolic subbands, indexedthe z-quantum numbern and the wave vectorskx , kyd:Ekx ,ky ,n hsk2

x 1 k2y dy2mp 1 En.

The Coulomb potentialy0szd may be calculated fromintegrating Poisson’s equation:

y0szd 24pe2

k

Z z

2`

dz0Z z0

2`

dz00n0sz00d 12pe2NS

kz ,

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wherek is the static dielectric constant in GaAs, whicis approximately 13. The above potential has tboundary condition that whenz ! 6`, the electricfields 2s1yeddy0ydz are exactly opposite. The electronic densityn0szd is given byn0szd NSr0szd, whereNS is the sheet density in thex-y plane, andr0szd P`

n0 wnjjnszdj2. Here, wn is the fractional occupationof the nth subband. It is usually possible to dopa quantum well such that only the ground subbandn 0is occupied at zero temperature.

To model an external laser field, we must not onintroduce a time-dependent potential due to the labut also the change in the effective potential due tomotion of the electrons. Thus, in the electric dipoapproximation for the laser field, the time-dependeHamiltonian isHstd H0 1 V std, whereH0 is the self-consistent equilibrium Hamiltonian and

V std dysz, td 1 eE z sinsvtd.

The magnitude of the electric field due to the laser isE ,and the frequency isv. The time-dependent potentiadysz, td is obtained again by Poisson’s equation andchange in electron densitydnsz, td from n0:

dysz, td 2s4pe2ykdZ z

2`dz0

Z z0

2`dz00dnsz00, td.

The quantum well structure we will examine here is310 Å wide square well containing an off-center “step50 Å wide. The sides of the well have 30% Al cocentration, enclosing the pure GaAs well region. The spositioned 105 Å from one side of the well, has a coduction band energy 65 meV above that of pure GaDoping with a sheet densityNS 1.5 3 1011e2ycm2, theeffective potential becomes as shown in Fig. 1. Also sin this figure are the six bound envelope wave functioof the self-consistent well, plotted so that the asymptovalue of each wave function is set to its energy eigenva(The vertical scale of the wave functions are arbitrarilyfor greater clarity.) At this sheet density, only the grousubband is occupied at zero temperature. Notice thatlowest two subband energies are relatively close togetcompared to the energy differences between those two

FIG. 1. The stationary self-consistent potential, and its fisix eigenstates plotted with asymptotes set to their energies

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bands and the higher ones. This means that at low teperatures, when driven at frequencies at or below,sE1 2

E0dyh, the system is well approximated as having only tbottom two subbands, and therefore lacks a classical crespondence. The two-subband model has been showbe in excellent agreement with experiment for a wellsimilar design [9,10]. Confident that the dynamics mbe accurately captured with only two subbands, we restourselves to this approximation from here on. The stamodel parameters were calculated atT 30 K in orderto account for heating by the intense laser field duringexperiment, which would otherwise be kept below 10 K

To account for dissipation, we use the density matformalism in the basis of the self-consistent statesjnszd.The density matrix evolves in time according to

≠ry≠t 2siyhd fHstd, rstdg 2 Rfrstd 2 r0g , (1)

where we include a relaxation operatorR which sendsthe system to a diagonal equilibrium density matrixr0,the densityr0szd above being the trace of the matriP

nm jnszdjpmszdr0

nm. The form we use isRfdrgnm Rnmdrnm, whereR00 R11 G1 (the energy relaxationrate) andR10 R01 G2 (the decorrelation rate) [11].

We define the population difference between the twstates to beD r00 2 r11. Since the density matrix isHermitian with unit trace, the equation of motion (1) isset of coupled nonlinear ordinary differential equationsD andr10. With some work [12], we arrive at

ÙD s4yhdV10std Imr10 2 G1sD 2 D0d ,

Ùr10 2ifv10 1 szyhdV10g 2 G2r10 2 siyhdV10stdD .(2)

Here, vnm ; sEn 2 Emdyh, z ; sz11 2 z00dyz10 (mea-suring the system’s asymmetry),znm ; kjnjzjjml, andD0 ; fr0g00 2 fr0g11.

The matrix elementV10std is

V10std eE z10 sinsvtd

1 ahv10fRer10std 2z

4 sD 2 D0dg .

The parametera is the depolarization shift[13], definedasa s8pe2NSykv10dS1010, where

S1010 2Z `

2`

dzjp1szdj0szd

3Z z

2`

dz0Z z0

2`

dz00jp1 sz00dj0sz00d .

The single parametera describes the mean-field Coulominteractions in the two-subband system. Whena 0,our equations become those of the single-particle systcompletely linear inD andr10. In deriving (2), we haveassumed a real basis setjnszd, so thatV10 V01, and wehave used the two-state identitiesV11 2 V00 z V10 andS1110 2 S0010 z S1010 (see [12]).

Finally, nonzero temperatures are accountedby specifying the equilibrium populationsfr0g11 andfr0g00 1 2 fr0g11. The population of each subbanin thermal equilibrium at temperatureT is obtained by

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integrating the Fermi-Dirac distribution in theskx , kydplane. This gives the result (see [12])

fr0g11 t

sln

µq14 ss 1 1d2 1 ssr 2 1d 2

12 ss 2 1d

∂.

Here, r esyt and s e21yt, with t kBTyhv10 ands p hNSympv10 being the reduced temperature ansheet density, respectively. The latter is defined such tit is unity when the Fermi energy equalsE1.

Now we may evolve the equations of motion for aninitial condition rs0d, and calculate the expectation valuof any observableA usingkAl Tr srAd. We will scaleall frequencies henceforth byv10, and energies byhv10.This gives the unitless drive amplitudel eE z10yhv10,drive frequencyV vyv10, and dissipation ratesg1,2 G1,2yv10. In our model,v10 50.4 wave numbers, orabout 6.2 meV. Dissipation rates are typically very hig[10], with g2 . 0.8 in our units, andg1 ranging from0.01to g2, depending on temperature, impurities in the sampetc. We letg1 0.07. The sheet density we model gives 0.78, the depolarization shift is computed to bea 1.13, and the asymmetry parameter isz 1.82, sincejz11 2 z00j 110.6 Å andjz10j 61.0 Å. The drive fre-quency used in our simulation isV 0.33, correspondingto about 16.6 wave numbers, readily accessible with a frelectron laser (FEL). Computations were performed uing the fourth-order Runge-Kutta method to integrate thequations of motion, with 2048 steps per cycle of drive.

Since the time dependence of (2) is periodic, onfinds that sufficiently small initial conditionsrs0d evolvetoward a single periodic trajectoryrstd, when the driveis small. For larger values of the drive, there are twstable states, each with a basin of attraction of positi(Lebesque) measure. This bistability leads to hystereloops when some parameter is swept through. This wstudied for the two-state system in the rotating wav(complex drive) approximation by Załuzny [14].

For some values, however, the periodic solution bcomes unstable to a period-two orbit, and this bifurction may continue in a period-doubling cascade [1].our well, the scaled dissipation rates are so high that sucascades are hard to find. They do exist, however, asFig. 2. Here, the drive fieldl is swept through, from 1 to3. The valuel 3 corresponds to an ac electric field omagnitude 30.6 kVycm in the well, or roughly 110 kVycmexternally. This field strength is accessible with an FEbut is not likely to damage the quantum well [15].

Figure 2 shows the result of samplingkzl every drivecycle, sweeping throughl. At each newl, the system isallowed to evolve for 500 cycles before plottingkzlyz10.Plotting is done for 100 cycles, so that large-periomotion may be seen. As one can see, there are mvalues where period-two or higher motion exists (sewhere the single line splits into two or more branchesAlso seen are cascades to strange attractors, the leftm

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FIG. 2. Bifurcation diagram of the damped system, samplinkzlyz10 vs drive fieldl. Large dots indicate hysteresis.

of which is expanded in the inset. Regions of bistabiliare revealed by sweeping backwards,l 3 ! 1. Ifthe resultant trajectory is different from that obtained bsweeping forward, values plotted with thick dots.

Light emitted at a given frequency will have power proportional to the modulus squared of the Fourier transforof kzlstd. For four different drive strengths, we let thesystem run for hundreds of cycles to eliminate transienthen sampledkzlstd 16 times per cycle for 1024 cyclesResultant spectra are shown in Fig. 3. In part (a), tsteady state is period one, so we see a strong signal atdrive frequency and its harmonics. In part (b), the dnamics have bifurcated into period two, so that half hamonics are clearly observable. In part (c), after anothbifurcation, we have quarter harmonics. In part (d), whave cascaded to a strange attractor, and the output strum is broadband “noise.” In (d) a Hann window waused, since the dynamics were not periodic.

The nonlinearity in our model is an approximation ta linear many-body Schrödinger equation. The manbody equation, however, has a vast number of degreof freedom, and a solution which would exhibit a vasnumber of frequencies, most incommensurate with oanother. Thus, the linearity of the underlying dynamicdoes not prevent the result from being indistinguishabfrom the predictions of a nonlinear model, given anreasonable measurement accuracy and time scale.fact, similar nonlinear two-state models have had grepredictive accuracy for experiments [9,16]. The nonlinedensity matrix formulation [17,18] captures the collectivbehavior of the linear modes, and is therefore the relevdescription of the system.

The use of the two-level model is obviously restrictivand with high field intensities the higher levels will playrole, at least a quantitative one. A model including all thlevels in the quantum well is still being developed. Wdo not expect it to give qualitatively different results.

Since parameters such as dissipation rates, charge dsities, temperature, and drive strength are not measurato a great accuracy in FEL experiments on quantum wewe do not claim to predict exact behavior in a particula


FIG. 3. Power spectra of the damped system, at drive values (see Fig. 2) (a)l 1.60, (b) l 1.64, (c) l 1.65, (d) l 1.67.

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experimental arrangement. We do think, however, tthe model we present may be used to predict paramregions where period doubling is very likely to occuThis assertion comes from our experience, as we hfound parameters that make it very easy to see pertwo dynamics. Likely parameters include the followin(1) A large a, at least of order unity. (2) A drivestrengthl comparable toa. (If l ¿ a, then the dynamicsare dominated by the drive, so that the electron-elecinteractions do not play a great role, and ifa ¿ l, thedynamics are dominated by the internal fields, so thatsystem acts nearly undriven.) (3) A significantz . (4)Similar damping ratesg1 , g2, which are as small aspossible. (Arguably, a physical restriction isg1 # g2,since depopulation cannot occur without dephasing.)Low temperatures,t , 1, and low sheet densities,s , 1.

As mentioned before, in a real experiment the samheats up from the intense laser field. We have tried torealistic with the amount of this heating in our model [19A problem with heating is that it tends to equalize tstate populations, extinguishing the output signal and a“melting” the attractor through a period-halving cascaback to period one. We address this issue in [12].

In designing this well, we have sought a largea, z , andz10, and a smallv10, to increase the scaled drive strengl. This is necessary to observe a strange attractor, gthe large damping rates. If one is only interestedobserving period-two behavior,l may be decreased, anso increasingv10 is likely to be fruitful in the well design.

The two-level model is simple enough to have ratively few design parameters, but still accountsimportant physical effects such as electron-electron inactions and dissipation. It is our hope that the results psented here will lead to the observation of a subharmosignal in a quantum well experiment. Practically, susystems may act as a source of low-frequency monocmatic light. But it is appealing to think that if period doubling is observed, then the quantum well holds an obas exotic as a strange attractor in its depths, and its brband spectrum may also be experimentally observed.

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We thank Keith Craig, Andrea Markelz, and Mark SSherwin for their input and suggestions. Work supporteby NSF Grant No. DMS91-04532 (B. B. and B. G.) andINCOR Grant No. CNLSy91-0970 (B. G.).

[1] P. Collet and J.-P. Eckmann,Iterated Maps on the Intervalas Dynamical Systems(Birkhäuser, Boston, 1980).

[2] G. Jona-Lasinio et al., Phys. Rev. Lett. 68, 2269(1992).

[3] However, a period-doubling cascade to a strange attractis a robust chaotic phenomenon, in distinction to topological chaos [20] such as a Smale horseshoe. A stranattractor has a basin with a positive (Lebesque) measuwhile the basin of attraction of a Smale horseshoe has zemeasure.

[4] H. T. Moon et al., Phys. Rev. Lett.49, 458 (1982).[5] B. Birnir and R. Grauer, Commun. Math. Phys.162, 539

(1994).[6] F. Benvenutoet al., Phys. Rev. A44, R3423 (1991).[7] W. Kohn and L. J. Sham, Phys. Rev.140, 1133 (1965).[8] P. Hohenberg and W. Kohn, Phys. Rev.136, B864 (1964).[9] J. N. Heymanet al., Phys. Rev. Lett.72, 2183 (1994).

[10] J. N. Heymanet al., Phys. Rev. Lett.74, 2682 (1995).[11] K. Blum, Density Matrix Theory and Applications

(Plenum Press, New York, 1981).[12] B. Galdrikian and B. Birnir, “Chaos in Quantum Wells”

(to be published).[13] T. Ando, Rev. Mod. Phys.54, 437 (1982).[14] M. Załuzny, J. App. Phys.74, 4716 (1993).[15] A. Markelz, Physics Department, UCSB (private commu

nication).[16] K. Craig et al., Semicond. Sci. Technol.9, 627 (1994);

Phys. Rev. Lett.76, 2382 (1996).[17] B. Birnir and B. Galdrikian, “Time-Periodic and Quasi-

Periodic Density Functional Theory” (to be published).[18] B. Galdrikian, thesis, UCSB, 1994.[19] M. S. Sherwin, Physics Department, UCSB (private com

munication).[20] M. Levi, “A New Randomness-Generating Mechanism in

Forced Relaxation Oscillations” (to be published).

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Period Doubling and Strange Attractors in Quantum Wells