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S hrödinger-equation formalism for a dissipative quantum system ∗ E. Anisimovas Department of Theoreti al Physi s, Vilnius University, Saule(cid:10)tekio al. 9, LT-10222 Vilnius, Lithuania † A. Matulis Semi ondu tor Physi s Institute, Go²tauto 11, LT-01108 Vilnius, Lithuania 7 (Dated: February 6, 2008) 0 0 We onsider a model dissipative quantum-me hani al system realized by oupling a quantum 2 os illator to a semi-in(cid:28)nite lassi al string whi h serves as a means of energy transfer from the os illator to thein(cid:28)nityand thus playstherole of a dissipative element. The oupling betweenthe n two(cid:22)quantumand lassi al (cid:22)partsofthe ompoundsystemistreatedinthespiritofthemean- a (cid:28)eld approximation and justi(cid:28) ation of the validity of su h an approa h is given. The equations J of motion of the lassi al subsystem are solved expli itly and an e(cid:27)e tive dissipative S hrödinger 9 equation for the quantum subsystem is obtained. The proposed formalism is illustrated by its appli ationtotwobasi problems: thede ayofthequasi-stationarystateandthe al ulationofthe ] r nonlinear resonan e line shape. e h PACSnumbers: 03.65.Sq,03.65.Yz,73.21.La t o . t a I. INTRODUCTION dissipative nonlinear S hrödinger equation [10℄, and Al- m bre ht[11℄ workedoutthegeneralprin ipleslimitingthe a eptable form of the nonlinear terms. - The rapid expansion of resear h frontiers into the d nanometer and femtose ond regime demands a areful The problem of dissipation in quantum me hani s is n o onsideration of dissipation as a way to ounterbalan e losely onne ted to the use of mixed quantum- lassi al c theenergyin(cid:29)uxfromtheexternal(cid:28)eldsinquantumme- treatments[12,13℄wherebyonlyafewdegreesoffreedom [ hani al systems. of a omplex system are treated quantum-me hani ally while the remaining ones are des ribed lassi ally. Al- 1 The subje t takes roots in the seminal work by Feyn- though Egorov and oworkers [14℄ point out that not v man and Vernon [1℄ who treated a quantum obje t ou- 3 pled to an in(cid:28)nite olle tion of os illators as a model always an a urate des ription is attained and so this 7 of a linear dissipative environment. By employing the approa h must be used with are, it is re ognized as a 1 method of hoi e when a signi(cid:28) ant redu tion of om- path-integral te hniques they were able to eliminate the 1 plexity is desired. environment variables and arrived at a dissipative mod- 0 i(cid:28) ation of the Green's fun tion of the quantum obje t. Inthispaperwepresentaderivationofarathersimple 7 0 This approa hwasfurther used by a number of authors, nonlinear S hrödinger equation that in ludes a dissipa- / most notably by Caldeira and Leggett [2℄, who applied tiveterm. Thisequationisobtainedfroma onsideration t a it to a spe i(cid:28) problem ofdissipative e(cid:27)e ts in tunneling of a physi al model: a quantum os illator onne ted to m on a ma ros opi s ale. A detailed a ount on the prob- an in(cid:28)nite lassi al string that a ts as a means to trans- - lem of dissipation in quantum-me hani al systems an ferenergyawayfromthequantumsystem(seealso[15℄). d befoundinamonographbyWeiss[3℄whilesemi lassi al The obtained S hrödinger-like equation is illustrated by n approa hesare also reviewed in [4℄. onsidering two spe i(cid:28) examples. The (cid:28)rst one, the de- o ay of a quasi-stationary state, is an auxiliary one and c Generally, the main mode of atta k uses the density- : matrixformalism[5℄andthepopularLindbladte hnique shows that the results obtained by using our analysis do v not ontradi t the ones derived by the usual and more [6℄. Thisapproa hisrelevantforopen quantumsystems. i X In ontrast, the onsiderably less arduous S hrödinger- omplexmethodsusingthedensity-matrixformalismand a double set of variables. The se ond example, the non- r equationformalismisregardedtobeappropriateforiso- a linear resonan e, is interesting on its own and demon- lated systems[7℄thataredes ribedbyahermitianHamil- stratesthatthequantum-me hani alnonlinearresonan e tonian. is physi ally ri her than its lassi al analog. Despite this fa t, thereweresu essfulattempts to in- lude dissipation dire tly in the S hrödinger equation. Our paper is organized in the following way. In Most often they based on the de Broglie-Bohm pi ture Se t. II, the model system is introdu ed and the dissi- [8℄ and employed modi(cid:28) ations to the quantum poten- pativeS hrödinger-likeequationforthequantumsystem tial [9℄. These ideas were used by Kostin to derive a is derived. In Se t. III, the obtained equation is simpli- (cid:28)ed employing the rotating-wave approximation. In the next two Se tions, some illustrations of the developed te hnique are presented: in Se t. IV we onsider the de- ∗ †Ele troni address: egidijus.anisimovas(cid:27).vu.lt ay of the quasi-stationary state, and Se t. V is devoted Ele troni address: amatulistakas.lt to the non-linear resonan e. Finally, our on lusions are 2 formulated in Se t. VI. obtains a onsistent des ription. This averaging onsti- tutes the main assumption of the quasi lassi alapproxi- mationandwasusedearliertotreatthe ouplingof las- II. INTERACTION OF THE QUANTUM si alandquantumdegreesoffreedom[16℄. Wegivesome OSCILLATOR WITH A CLASSICAL CHAIN further justi(cid:28) ation of this ru ial step below. Eqs.(3)leadtothestandard lassi alequationsforthe Let us begin by introdu ing our model of weakly ou- hain pled quantum and lassi al subsystems. The quantum Mx¨ +K(x x )+κx = κ x , m 0 0− 1 0 h i (5a) systemismodelekdasaharmoni os illatorofmass and Mx¨n K(xn−1 2xn+xn+1) = 0, n>1. spring onstant , des ribedby the quantum-me hani al − − (5b) xˆ pˆ position and momentum operators and . The role of The se ond of these equations (5b) is readily solved by the lassi al system is played by a semi-in(cid:28)nite lassi al the Fourier transform string whi h, for the sake of onvenien e, is dis retizMed 1 ∞ and represented as a hain of identi al balls of maass xn = 2π Z dqeß(qan−ωt)f(q), (6) inter onne ted by springs of equilibrium length and −∞ K spring onstant . The two subsystems are linked by a κ produ ing the standard dispersion law di(cid:27)erentspring(spring onstant ) onne tingthequan- 4K tum os illator to the ball sitting at the end of the hain ω2 = sin2(qa/2). M (7) as depi ted in Fig. 1. The orresponding Hamiltonian reads 1 k κ From here on we restri t ouqra ons1ideration to the long- H = pˆ2+ xˆ2+ (xˆ x0)2 wavelength approximation ( ≪ ) thus negle ting the 2m 2 2 − ∞ dis retization and e(cid:27)e tively returning to a ontinuous- + 1 p2 + K(x x )2 , (1) string model for the lassi al subsystem. In this limit, nX=0(cid:20)2M n 2 n− n+1 (cid:21) we have a linear dispersion law ω =vq x n (8) n with denoting the oordinate of thep -th ball mea- v = Ωa Ω = suredfromitsequilibriumposition,and n therespe tive witKh/Mthe wave-propagation velo ity and momentum. . p Inserting the Fourier transform (6) into the boundary ondition(5a)andusingthelong-wavelengthapproxima- tion we are able to simplify its left-hand side to 1 ∞ dqe−ßωt K 1 eßqa Mv2q2+κ f(q) 2π Z − − −∞ (cid:8) (cid:0) (cid:1) (cid:9) FIG. 1: The layout of themodel system. 1 ∞ dqeßvqt( iKqa+κ)f(q) ≈ 2π Z − (9) −∞ Ka Following [16℄, we assume that theΨq(uxa,ntt)um os illator = v x˙ +κx0. isdes ribedbyitsownwavefun tion whi hsolves the S hrödinger equation The resulting equation of motion for the (cid:28)rst ball oor- x ∂ 0 ß~ Ψ(x,t)=HΨ(x,t) dinate ∂t (2) Ka x˙ +κx =κ x 0 0 v h i (10) withtheaboveHamiltonian(1)whilethedynami alvari- ablesoftheballsinthe hainobeythe lassi alHamilton may be interpreted as the equation of motion for a equations dissipative lassi al mode. Eq. (10) together with the ∂ ∂ S hrödingerequation (2) for the quantum variables on- x˙ = H , p˙ = H , n ∂p h i n −∂x h i (3) stitutethe ompleteequationsetdes ribingthebehavior n n of the dissipative quantum system. Now we turn to the examination of the validity of the with the Hamiltonian operatorrepla edby its quantum- employed quasi lassi al approximation. A smooth way me hani al average over the state of the quantum os il- to pro eedis to introdu ethe dimensionlessvariables by lator ∞ means of s aling H = dxΨ∗(x,t)HΨ(x,t). ~ h i Z−∞ (4) xˆ→lxˆ, pˆ→ lpˆ, x0 →l0x0, t→ω0−1t, k ~ ~ (11) Inthisway,quantumvariablesandoperatorsdonotenter ω = , l= , l = . 0 rm rmω 0 rMΩ theequationsforthe lassi aldegreesoffreedomandone 0 3 l l 0 Here and are, respe tively, the hara teristi length for the des ription of the intera tion between the quan- ω 0 s ales of the quantum and lassi al subsystems, and tum and lassi al subsystems. A similar argument was stands for the frequen y of the quantum os illator. The alsopresentedinthedevelopmentofthemixedquantum- s alingenablesustorewritetheobtainedsetofequations lassi al dynami s [12℄. as Alternatively,onemayarguethatduethesmallnessof ∂ the adiabati parameter the hara teristi length of the ß Ψ(x,t)=HΨ(x,t), l0 ∂t (12a) wavefun tions ofheavlyballs ismu hsmallerthanthe 1 1 hara teristi length of the wave fun tion of the os il- H = pˆ2+ xˆ2 λxˆx , 2 2 − 0 (12b) lator. Consequently, the balls may be onsidered to be x˙ +λξx =λ x , following their well-de(cid:28)ned lassi al traje tories whereas 0 0 h i (12 ) theos illatorhasto bedes ribedquantum-me hani ally. with κ ω m l III. ROTATING-WAVE APPROXIMATION 0 0 λ= , ξ = . k rΩM l (13) 0 Let us now pro eed to some illustrations of the developed formalism based on a weakly nonlinear os illator. Note that we ex luded the terms whi h do not depend xˆ pˆ Namely,weassumethattheHamiltonianofthequantum on the operators and from the quantum Hamilto- subsystem onsists of a sum of Eq. (12b) and Eq. (14). nian(12b)asthey analwaysbeabsorbedintothewave- Besides, we restri t the treatment to the ase of a weak fun tion phase. The obtained equations des ribe the in- nonlinearity, that is, we assume that the anharmoni ity tera tion of a quantum-me hani al system with a single α f λ x 0 , the for e amplitude , and the oupling onstants dissipative lassi almode . Infa t,thederivationdoes ξ and aresmallquantities omparedtotheunity. Wealso not depend on the assumption that the quantum system restri t the frequen y of the driving (cid:28)eld to the immedi- is a harmoni os illator. Thus, the equationsare general ate vi inity of the resonant frequen y of the harmoni and anbe applied to anyquantum system withdissipa- os illatorwhi hisunityinthedimensionlessunits. Thus tion. For instan e, onsidering an anharmoni os illator we are able to treat the frequen y deviation from the driven by an external time-periodi for e (we treat this η =ω 1 resonan e as yet another small parameter − . ase as an illustration) the aboveHamiltonian is supple- We expand the sought wavefun tion into the seriesof mented with the following additional terms harmoni -os illatoreigenfun tions ∆H = xˆfcos(ωt)+αxˆ4, − (14) ∞ Ψ(x,t)= a (t)e−ßωEntψ (x). n n f α (15) with expressing the for e amplitude and being the nX=0 anharmoni ity oe(cid:30) ient. In order to have an appre iable (that is, not negligi- Here, the dimensionless energies and the orresponding blysmall)intera tionbetweenthequantumand lassi al eigenfun tions read subsωys0temsΩonehastoassumethe hara teristi frequen- E =n+ 1, ψ (x)= 1 e−x2/2H (x), ies and to beofthesameorderofmagnitude. Un- n n n 2 2nn!√π (16) der this assumption the quantum os illator will be able p to emit (cid:16)phonons(cid:17) into the string. Furthermore,the ou- κ H (x) n n pling onstant representingtheintera tionofquantum and the symbol standsforthe -th Hermite poly- kos illator with the hain should not ex eed the onstant noem−ißaEl.ntNote that instead of the usual time dependen e des ribing the os illator potential itself. Otherwise, ∼ of a harmoni -os illatoreigenfun tion Ee−q.ßω(E15n)t the os illator annot be onsidered as a separate entity. features a slightly modi(cid:28)ed exponential fa tor In view of the above onstraints, the dimensionless ou- thatabsorbsapartofthetimedependen eoftheexpan- λ ξ a (t) n pling onstants and depend essentially only on the sion oe(cid:30) ient . m/M adiabati parameter . Using the above expansion of the wave fun tion (15) Thus, in the ase of a light quantum os illator inter- the oordinate average an be presented as a ting with a hain of heavy balls the above adiabati ∞ parameter is small and so are the oupling onstants. hxi= eßω(En−Em)ta∗namhn|x|mi The smallness of the oupling onstant ensures a weak n,Xm=0 (17) perturbation of the hain. The hain under onsider- =√2Re eßωta ation may be treated as a olle tion of nonintera ting (cid:0) (cid:1) harmoni modes. Meanwhile, the quantum-me hani al des ription of a harmoni mode gives the same result as where ∞ the lassi al one. Therefore, one may on lude that the m/M a= a a∗ √n+1 smallness of the adiabati parameter guarantees n n+1 (18) the validity of the applied quasi lassi al approximation nX=0 4 is the usual polarization of the quantum os illator. Inserting the oordinate average (17) into Eq. (12 ), and solving it to the leading term in the small oupling onstants we obtain the following expression for the o- ordinate of the (cid:28)rst ball x =√2λ dtRe eßωta =√2λIm eßωta . 0 Z (19) (cid:0) (cid:1) (cid:0) (cid:1) Next, inserting the obtained expression (19) and the wave fun tion (15) into Eq. (12a), proje ting this equa- n tion onto the -th state, and negle ting all os illating terms (this amounts to the usual rotating-waveapproxi- mation)wearriveatthe(cid:28)nalsetofequationsforthetime evolution of the wave fun tion expansion oe(cid:30) ients ßa˙n = ηEn α nx4 n an |a1|2 √n(F−+(cid:8)ßγa∗−)an−h1| √| ni(cid:9)+1(F ßγa)an+1. FuIrGv.e,2:|a0D|2e (cid:21)aydaosfhqedua sui-rsvtea,tiaon(cid:21)ardyotstteadte :urve, an(cid:21)dthxi0 k(cid:21)stohliidn − − − (20) solid urve. Here, for the sake of simpler notation we denoted F =f/2√2, γ =λ2/2. (21) indu edduringthequantumtransition,andthethinsolid The obtained set of equations (20) was solved in the xlin(et)showsthe lassi al oordinateof(cid:28)rstballinthe hain 0 two- and three-level approximations, and the results are whi his al ulatedbymeansofexpression(19). As presented in the following se tions. a matter of fa t, this urve represents the oordinate of any ball in the hain be ause a ording to Eq. (6) the n oordinate of the -th ball an be expressed as IV. DECAY OF THE QUASI-STATIONARY x (t)=x (t an/v). n 0 STATE − (26) Therefore,duringthede ayofthequantumstateastring In order to illustrate the validity of the proposed ex itation of the shape shown by thin solid line is emit- method we onsider a well-known problem of the de ay ted and travelsawayfrom the quantum os illatorwith a of a quasi-stationary state. v onstantvelo ity . It anbeinterpreted asthe lassi al Forthis purpose, weassumethat the quantum system analogofanemittedphonon. Naturally,thetotalenergy has only two states and there is no anharmoni ity and F = η = α = 0 arriedbythestringex itationisequaltotheenergydif- no external for e ( ). At the beginning feren e between the two quantum levels. Consequently, the system is prepared in the upper level, that is, the we may on lude that the quantum subsystem provides boundary ondition reads the energy quantization in the lassi al one. t= , a =0, a =1. 0 1 −∞ (22) It is worth mentioanin2g+thaat2ta=kin1g into a ount the 0 1 norm onservation (| | | | ) Eqs. (20) an be A ordingto Eq. (20), the behaviorof this simplesys- transformed into another set of equations govIer=ninag t2he tem is des ribed by the following two equations baeh2aviorofthreerealvariables: theinvaersion | 1| − a˙0 =γaa1, a˙1 =−γa∗a0, with a=a0a∗1. (23) | 0| and the ( omplex) polarization . The linearized versionoftheseequationsinthevi inityofthestationary n=0 I = 1 These equations are readily solved analyti ally, and the state ( − ) reads solution reads I˙= 2γI, a˙ = γa. 1 1 − − (27) a 2 = , a 2 = . | 0| e−2γt+1 | 1| e2γt+1 (24) Itisremarkablethat theseequations oin idewith those onsidered in the density matrix formalism [17℄, more- These(cid:28)llingfa torsareshowninFig.2bythethi ksolid over, from our Eq. (27) we are able to extra t the relax- (ex ited level) and dashed (ground level) lines. We see T =1/2γ 1 ationtimes: thelongitudinalrelaxationtimeis that the quantum transition takγe−s1pla e during a (cid:28)nite T2 =1/γ andtheperpendi ularoneis . Weseethatthese period of time proportional to , the re ipro al fri - T = 2T 2 1 relaxation times obey the relation inherent to tion oe(cid:30) ient. The dotted urve shows the os illator the pure state when no ensemble average is arried out polarization andthisfa t on(cid:28)rmstheadequa yofourattempttode- 1 a= s ribe a dissipative quantum system in the S hrödinger- 2cosh(γt) (25) equation formalism. 5 Two more omments are in order. First, a ording to We observe the following typi al behavior: For a weak t = F < 0.5γ 0 Eqs. (22) we used the boundary ondition at −∞. for e ( ) the shape of the resonan e resembles Thisisrelatedtothe lassi aldes riptionofthe hainex- a Lorentzian, and its peak value is in reasing with in- itations when there is no spontaneous emission. When reasing for e. When the for e ex eeds the riti al value F =0.5γ 0 solving the equations numeri ally one has to add some the peak stops growingand is (cid:29)attened at the initial vibration (cid:28)eld (cid:29)u tuation in order to for e the topduetothesaturationatthe riti alpowerabsorption P =0.5γ 0 quantum transition. value indi ated by the thin dotted line. Another point is related to the Fourier transform of the oordinate of the (cid:28)rst ball 0.5 ∞ π/2γ x (ω)= dωeßωtx (t) F/g =5 0 0 Z ∼ cosh(ηπ/2γ) (28) g −∞ P/ 2 whi h gives us the shape of the emission line. We see η 1 0.25 that lose to the resonant frequen y, when | | ≪ , the line shape is Lorentzian oin iding with the result fol- 0.3 1 lowing from a simple al ulation based on the Fermi's 0.5 golden rule. However, further away from the resonan e, 0.2 the line shape deviates from the Lorentzianand features exponential tails. 0 0.1 -3 -2 -1 0 1 2 3 V. RESONANT POWER ABSORPTION h/g The following problem is that of the resonant power FIG. 3: Power absorption in the two-level approximation. absorbtion whi h we al ulated solving the equation set The numbers on urves indi ate the values of the external F/γ (20) using two-level for e amplitude . η a˙ = ß a +ß(F ßγa)a , 0 0 1 2 − (29a) 3η In the next two Figures the results for power ab- a˙ = ß a +ß(F +ßγa∗)a , 1 2 1 0 (29b) sorption al ulated in the three-level approximation are a = a a∗. shown. InFig.4weplotthepowerabsorptioninalinear 0 1 (29 ) α=0 os illator ( ). and three-level a˙ = ß(η/2 α)a +ß(F ßγa)a , 0 0 1 − − (30a) a˙ = ß(3η/2 5α)a 1 1 − +ß(F +ßγa∗)a +√2ß(F ßγa)a , 0 2 − (30b) a˙ = ß(5η/2 13α)a +√2ß(F +ßγa∗)a , 2 1 1 − (30 ) a = a a∗+√2a a∗. 0 1 1 2 (30d) approximations. Notethatweex ludedanonlinearterm from the equations of the two-level approximation be- ause in this ase it leads only to an inessential shift of the resonant frequen y. The power absorption was al- ulated by averaging the instantaneous power over the period of the external for e, that is, P = 2FIma. − (31) FIG.4: Powerabsorptioninthethree-levelapproximationin The power absorption in the two-level approximation the ase of the linear os illator. The numbers on the urves η F/γ as a fun tion of the deviation is shown in Fig. 3 for indi ate thevalues of theexternal for e amplitude . various for e amplitudes. Note that Eqs. (29) and (30) depend linearly on their parameters, therefore, by using an appropriate s aling of the time we may redu e The general shape of the absorption urves losely the number of parameters by one. Therefore, we hoose resembles those obtained in the two-level approxima- F P η α γ to express the quantities , , and in units. tion, ex ept that now the saturation sets in at a larger 6 F = γ 0 for e value , and the saturated power absorption of the quantum anharmoni os illator in the absorption P =1.5γ 0 is larger as well. gap more arefully. In Fig. 6 the right-hand side of this F = 0.5γ A more essential di(cid:27)eren e is seen in the uppermost absorption gap for the parameter values and F = α= 0.25γ urve orresponding to the for e whose amplitude ( is shown in more detail. The simplest way to 5γ ) strongly ex eeds the riti al value. Here we see a distinguish various types of dynami behavior is to plot narrowgap in the absorptionwhi h appears lose to the thephaseportraitofthesystem. Astherearethree om- resonant frequen y. It resembles an analogous gap that plexvariables onstrainedby asinglenormalization on- appears in the power absorption al ulated by means of dition the phase spa eis(cid:28)ve-dimensional. Therefore, we the density matrix equation te hnique [18℄ and is aused 0.5 by the intera tion of oherent light with an inhomoge- 1 neously broadened resonan e line. 2 0.4 3 0.5 a =0 0.3 0.4 P/g 0.2 P/g 0.3 0.1 0.2 0.075 4 0.1 0.05 0.0 0.525 0.550 0.575 0.600 0.025 h/g 0.0 0.5 0.0 0.5 1.0 1.5 F = h/g F0.I5Gγ. 6: Pαow=er0.a2b5sγorption gap for the parameter values and . FIG.5: Powerabsorptioninthethree-levelapproximationin the Fase=o0f.5aγn anharmoni os illator obtained for theparam- restri tourselvesto a pro(jRe etiao0n,Rofetah1e)phase spa e onto eter . The urvesarenumberedbythevaluesof the a two-dimensional plane shown in Fig. 7. α/γ anharmoni ity oe(cid:30) ient . The panel numbers orrespond to the frequen ies indi- atedbydotsinFig.6. Weseethatontheregular(cid:29)ankof In Fig. 5 the results for the nonlinear os illator are presented for a set of values of the anharmoni ity oe(cid:30)- ient and the for e amplitude orresponding to the riti- F = 0.5γ al value of the two-level approximation, i. e. . α = 0 We see that for the linear-os illator ase ( ) the absorptiondemonstratestheexpe tedLorentzianbehav- ior. When the anharmoni ity is in remented the resonant peak moves to the right, be omes asymmetri and the saturation manifests itself as a widening absorption gap lose to the shifted resonan e frequen y. The absorptiongapresemblesthe gapin nonlinear y- lotronresonan e[19℄whi hwasdes ribedusingthebal- an e equations. In that work the gap was related to the dynami al haos appearing in the lassi al nonlinear equations of motion. Quantum haos is a subje t of great interest as well (see [20℄ for referen es), and is largely on erned with the quantum signatures of lassi- ally haoti systems. Su h signatures are often sought by omparing the wave-fun tion patterns orresponding to various eigenstates with the lassi al traje tories [21℄. It follows from our quasi lassi al onsideration, FIG. 7: Two-dimensional proje tion of the phase spa e. The Eqs. (20), that the behavior of the quantum system an numbersonthepanels orrespondtothefrequen iesindi ated alsobevisualizedandinterpretedintermsofphase-spa e bydots in Fig. 6. traje tories obtained by treating the wave fun tion ex- a n pansion oe(cid:30) ients as generalized oordinates. In or- dertoillustratethispossibility,welookedatthebehavior theresonan e urve(point1)theos illatordemonstrates 7 a rather simple behavior (cid:22) the limit y le. As a mat- system. This equation is derived on the basis of a qua- a n ter of fa t, in this ase all three oe(cid:30) ients os illate si lassi al treatment of oupled quantum and lassi al a (t) = g exp(ßζt) n n with the same frequen y, namely, . subsystems and justi(cid:28)ed by the di(cid:27)eren e of masses of Therefore,inthis asethepowerabsorptionproblem an the two subsystems whi h leads to the smallness of adi- be simpli(cid:28)ed essentially and transformed into a nonlin- abati parameters. The onsidered de ay of a quasi- ζ eareigenvalueproblemde(cid:28)ningthe frequen y , and the stationarystatedemonstratesthattheproposeddes rip- g n stationary oe(cid:30) ients aresimply relatedto the (cid:28)lling tion gives identi al results to those obtained from the fa tors of the harmoni os illator levels. density-matrixapproa hwhen only homogeneousbroad- The behavior of the anharmoni os illator in the ab- eningistakenintoa ount. Themodel al ulationofthe sorption gap (points 2, 3, 4) is quite di(cid:27)erent. Here, the nonlinear absorption indi ates that many lassi al phe- a n oe(cid:30) ients os illate with multiple frequen ies, more- nomena(cid:22)su hastheasymmetryoftheresonan eshape over, the os illator an demonstrate both periodi be- and absorption dips (cid:22) brought about by the loss of o- havior(point 2) and non-periodi behavior(points 3, 4). heren e between the driving for e and the response may This non-periodi ity is in fa t responsible for the gap in be featured by a quantum system as well. We believe the power absorption as it auses the loss of oheren e that the proposed S hrödinger-equation formalism with betweenthe drivingfor e and the system response. This dissipation may (cid:28)nd appli ation to quantum nanostru - is the so- alled quasiperiodi behavior but the appear- tures and quantum haos. an e of a strange attra tor is also expe table at larger α values of the anharmoni ity oe(cid:30) ient . A knowledgments VI. CONCLUSIONS We thank Prof. K. Pyragas and J. 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